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2,877,628,090,754 | arxiv | \section{Introduction} \label{sec:intro}
Full-duplex (FD) radio has recently developed from long-studied theoretical concept to potential candidate solution to increase the performance of future wireless networks. By transmitting and receiving simultaneously on the same frequency band, FD radios can theoretically double the throughput with respect to their half-duplex (HD) counterparts \cite{Sab14}. Despite the potential of the FD approach, a critical issue hinders the effective achievement of the promised performance gains: the transmit and receive antennas of the FD radio need to be perfectly isolated, although physical limitations do not allow to attain this condition. As a result, strong self-interference (SI) appears at the receive chain, with consequent reduction of the signal-to-interference-plus-noise ratio (SINR) of the received signal. This issue implicitly sets an upper bound on the transmit power of the device. In this respect, small-cell base stations (BSs) prove especially suitable for the deployment of FD technology thanks to their low transmit power and the low mobility of the user terminals (UTs) \cite{Atz15a}.
Several efforts have been devoted in recent years to design effective SI cancellation (SIC) techniques as a means to approach the theoretical throughput of FD communications. In general, perfect SIC is assumed to be achieved whenever the SI is reduced to the same level as the noise floor. In this context, the best SIC results so far, for both single- and multiple-antenna FD settings, are accomplished by means of hybrid solutions based on both analog and digital signal processing~\cite{Bha13,Bha14}:
\begin{itemize}
\item[i)] Performing part of the cancellation via analog signal processing is beneficial to reduce problems such as saturation of the amplifiers and low dynamic range at the analog-to-digital converter of the receive chain~\cite{Knox12, Maso15FD};
\item[ii)] By resorting to digital signal processing at both transmit and receive side, one can exploit additional degrees of freedom and support higher transmit power of the FD device while preserving the effectiveness of SIC \cite{Ahm15, Kor14}.
\end{itemize}
In this regard, it is worth noting that the potential of digital signal processing for SIC is larger in case of multiple-input multiple-output (MIMO) systems, for which the advantages in terms of additional degrees of freedom can be exploited. The solutions based on this approach are typically referred to as \textit{spatial SIC} strategies.
As a matter of fact, a fundamental tradeoff exists between the effectiveness of spatial SIC and the achievable throughput. In practice, the more transmit (resp. receive) antennas are devoted to suppressing the SI, the less power is conveyed to (resp. from) the desired link. In addition, it is generally more convenient to perform such spatial SIC at the transmitter than at the receiver for a two-fold reason:
\begin{itemize}
\item[i)] The transmit power of the FD device is consistently higher than the power of the desired incoming signal and it is thus meaningful to fully exploit the receive antennas to maximize the signal-to-noise ratio (SNR) of the latter;
\item[ii)] An overabundance of SI may saturate the receiver circuitry, preventing any spatial SIC at the receive chain to be applied in the first place.
\end{itemize}
A possible approach is to design the transmit beamformer so as to apply full zero-forcing (ZF) to the SI channel, which allows to null the SI entirely in case of perfect channel state information (CSI) \cite{Rii11}. However, this solution does not exploit the aforementioned analog/digital SI capabilities at the receive chain: in fact, since the FD device can tolerate the SI power to be up to a certain threshold, nulling the SI completely via spatial SIC proves excessively aggressive and results in lower throughput. In this regard, an optimal transmit beamforming design in which the throughput of the downlink transmission is maximized under SI constraints was proposed in \cite{Zha12}. Therein, an iterative search algorithm based on well-known convex optimization techniques is proposed to identify the optimal transmit beamformer. Nevertheless, such scheme may not be adequate to compute the optimal solution within the coherence time of the wireless channel in realistic settings, especially in rich scattering environments populated by UTs whose channels can experience rapid variations in their fast-fading components.
Starting from these observations, in this paper we consider an FD MIMO radio with partial SIC capabilities at the receive chain: assuming perfect CSI of both the downlink and the SI channel, we propose a transmit beamforming design that leverages:
\begin{itemize}
\item[a)] The inherent SIC capability of the device;
\item[b)] The potential of multiple antennas in terms of spatial SIC.
\end{itemize}
Remarkably, the proposed solution is not only optimal in terms of throughput, but also enjoys a very low complexity that allows it to outperform state-of-the-art methods in terms of processing time and power requirements. In fact, the resulting transmit beamformer is characterized by a simple closed-form expression that follows from a practically relevant optimization problem formulation, whereas existing approaches for multiple-antenna FD radios achieve the desired SIC only at the expense of higher complexity, i.e., through more complex iterative algorithms (e.g. \cite{Zha12}). Evidently, this one-shot solution results in a complexity gain that is increasingly appealing as the number of antennas at the BS grows: in fact, the higher the number of transmit antennas at the BS, the larger the search set for iterative algorithms such as in \cite{Zha12}. Hence, the spatial SIC cancellation presented in this paper outperforms state-of-the-art counterparts in terms of both performance (cf. \cite{Rii11}) and complexity (cf. \cite{Zha12}). More specifically, numerical results show that the proposed approach achieves significant gains when the number of transmit antennas is small to moderate, which makes it particularly suitable for implementation in FD small-cell BSs.
\section{System Model} \label{sec:SM}
Consider a hybrid FD/HD scenario where, at each timeslot, a multiple-antenna FD BS serves one HD node in the uplink and one HD node in the downlink, both single-antenna. Figure~\ref{fig:conf} depicts three possible instances of such hybrid FD/HD scenario in the context of small-cell networks, namely:
\begin{itemize}
\item[i)] UTs in both uplink and downlink;
\item[ii)] Backhaul (BH) BS in the uplink and UT in the downlink;
\item[iii)] UT in the uplink and BH BS in the downlink.
\end{itemize}
\begin{figure}[t!]
\centering
\includegraphics[scale=1]{./img/conf}
\caption{Possible scenarios for the operation of FD small-cell (SC) BSs; solid and dashed lines indicate short- and long-range links, respectively.} \label{fig:conf}
\end{figure}
Two observations are in order at this stage. The hybrid FD/HD network configuration, i.e., a scenario where FD and HD devices operate side by side, is arguably the most suitable setting for our study, since it avoids strong inter-node interference while exploiting the full throughput gain provided by the FD paradigm at the BSs~\cite{Goyal15}. Additionally, the single-user communication assumed to occur in the uplink/downlink does not diminish the generality of our approach at the physical layer: in fact, this could be seen as the result of scheduling decisions, typically performed at upper layers in current networks. For clarity of presentation, in the rest of the paper we assume scenario (i).
Now, let $N_{R}$ and $N_{T}$ be the number of receive and transmit antennas, respectively. The task of the FD BS in this context is to exploit (all or a subset of) its receive antennas to maximize the uplink throughput and, simultaneously, to exploit (all or a subset of) its transmit antennas to maximize the downlink throughput. Moreover, as previously discussed, it is assumed that the FD BS has preexisting hybrid SIC capabilities, such as the ones described in Section~\ref{sec:intro}. In this context, the FD BS needs to guarantee that the power of the SI experienced during the reception of the incoming signal does not exceed a certain threshold, which guarantees the full effectiveness of the preexisting hybrid SIC capabilities at the receive chain. It is worth noting that the latter are typically characterized in the literature in terms of maximum SI attenuation/cancellation they can provide, expressed in dB~\cite{Bha13,Bha14}. In practice, each of these strategies sets an implicit upper bound on the maximum transmit power that can be adopted by the FD radio in order to preserve the full effectiveness of the preexisting SIC algorithms (and the throughput of the incoming transmission). In particular, this can be straightforwardly obtained by summing the maximum SIC capability to the noise floor and by subsequently subtracting inter-antenna distance-dependent pathloss attenuation experienced by the transmitted signal during its propagation from the transmit to the receive antennas.
Concerning the notation adopted throughout the paper, it is convenient to begin by clearly differentiating between uplink
(i.e., from the UT served in the uplink to the FD BS) and downlink communication (i.e., from the FD BS to the UT served in the downlink). In this context, we let $\mathbf{h}_{\mathrm{u}} \in \mbox{$\mathbb{C}$}^{N_{R}}$ and $\mathbf{h}_{\mathrm{d}} \in \mbox{$\mathbb{C}$}^{N_{T}}$ be the uplink and the downlink channels, respectively. Likewise, we define $p_{\mathrm{u}}$ and $p_{\mathrm{d}}$ as the uplink and downlink transmit powers, respectively. We use $s_{\mathrm{u}}$ and $s_{\mathrm{d}}$ to denote the uplink and downlink data symbols, respectively, with $\mathbb{E} [|s_{\mathrm{u}}|^{2}] = 1$ and $\mathbb{E} [|s_{\mathrm{d}}|^{2}] = 1$. Furthermore, we let $\mathbf{n}_{\mathrm{u}} \sim \mathcal{C} \mathcal{N}(0,\sigma^{2} \mathbf{I}_{N_{R}})$ and $n_{\mathrm{d}} \sim \mathcal{C} \mathcal{N}(0,\sigma^{2})$ be the additive noise in the uplink and in the downlink, respectively. Finally, we let $\H \in \mbox{$\mathbb{C}$}^{N_{R} \times N_{T}}$ be the SI channel at the FD BS and denote the receive combiner and the transmit beamformer used by the FD BS with the vectors $\v \in \mbox{$\mathbb{C}$}^{N_{R}}$ and $\mathbf{w} \in \mbox{$\mathbb{C}$}^{N_{T}}$, respectively.
We assume that $\mathbf{h}_{\mathrm{d}}$ is subject to Rayleigh fading with elements distributed independently as $\mathcal{C} \mathcal{N} (0, 1)$, whereas $\H$ is subject to Ricean fading \cite{Dua12} and, therefore, its elements are distributed independently as $\mathcal{C} \mathcal{N} (\mu, \nu^{2})$. In this regard, one can measure the Ricean $K$-factor and the pathloss $\Omega$ between transmit and receive antennas and determine the mean and standard deviation of $\H$ as (cf. \cite{Atz15a})
\begin{align*}
\mu \triangleq \sqrt{\frac{K \Omega}{K+1}}, \qquad \nu \triangleq \sqrt{\frac{\Omega}{K+1}}.
\end{align*}
We assume that the FD BS has perfect CSI related to both $\mathbf{h}_{\mathrm{d}}$ and $\H$. The uplink and downlink signals can be expressed as
\begin{align*}
y_{\mathrm{u}} & \triangleq \sqrt{p_{\mathrm{u}}} \v^{\mathrm{H}} \mathbf{h}_{\mathrm{u}} s_{\mathrm{u}} + \sqrt{p_{\mathrm{d}}} \v^{\mathrm{H}} \H \mathbf{w} s_{\mathrm{d}} + \v^{\mathrm{H}} \mathbf{n}_{\mathrm{u}} \\
y_{\mathrm{d}} & \triangleq \sqrt{p_{\mathrm{d}}} \mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w} s_{\mathrm{d}} + n_{\mathrm{d}},
\end{align*}
respectively. Lastly, we let $\varepsilon > 0$ be the SI threshold, i.e., the maximum tolerable power of the SI experienced at the receive antennas to preserve the full effectiveness of the pre-existing SIC algorithms. In a practical case, this is given by
\begin{align} \label{eq:epsilon}
\varepsilon = r_{\mathrm{n}} - c
\end{align}
where $r_{\mathrm{n}}$ represents the noise floor and $c$ is the SIC capability at the receive chain.
\section{Transmit Beamforming Design} \label{sec:TBD}
Similarly to \cite{Zha12}, we aim at maximizing the downlink spectral efficiency, i.e., from the FD BS to the served UT, while keeping the SI below a certain threshold. The resulting optimization problem can be written as follows:
\begin{align} \label{eq:opt1} \tag{$\mathrm{P}$}
\begin{array}{ccll} \vspace{2mm}
\displaystyle \max_{\mathbf{w}} & & \log_{2} \big( 1 + \rho |\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w}|^{2} \big) & \\
\displaystyle \mathrm{s.t.} & & |\v^{\mathrm{H}} \H \mathbf{w}|^{2} \leq \varepsilon \\
\displaystyle & & \| \mathbf{w} \|^{2} \leq 1
\end{array}
\end{align}
where $\rho \triangleq p_{\mathrm{d}}/\sigma^{2}$ is the SNR at the downlink UT. Furthermore, switching our focus to the constraints in \eqref{eq:opt1}, we note that the first constraint enforces that the SI experienced at the receive antennas does not exceed the threshold that guarantees the full effectiveness of the pre-existing SIC algorithms,\footnote{Herein, we assume that the receive combining vector $\v$ is independent from the SI channel as in, e.g., maximum ratio combining.} whereas the second constraint ensures that the solution does not induce any undesired amplification of the transmit signal.
Now, let $(\cdot)^{\sharp}$ denote the Moore-Penrose pseudoinverse operator and let $\mathbf{I}$ be the $N_{T}$-dimensional identity matrix. In the next theorem, we provide the closed-form expression of the optimal transmit beamformer for the considered problem.
\begin{theorem} \rm{
The transmit beamformer that solves \eqref{eq:opt1} is given by
\begin{align} \label{eq:w_star}
\mathbf{w}^{\star} & \triangleq \frac{( \mathbf{I} - \alpha^{\star} \H^{\mathrm{H}} \v (\H^{\mathrm{H}} \v)^{\sharp}) \mathbf{h}_{\mathrm{d}}}{\| ( \mathbf{I} - \alpha^{\star} \H^{\mathrm{H}} \v (\H^{\mathrm{H}} \v)^{\sharp}) \mathbf{h}_{\mathrm{d}} \|}
\end{align}
with
\begin{align}
\label{eq:alpha_star} \alpha^{\star} & \triangleq 1 - \min \bigg( 1, \sqrt{\frac{\max(0, \zeta - \eta)}{\zeta}} \bigg)
\end{align}
where we have defined
\begin{align}
\label{eq:zeta} \zeta \triangleq \ & \bigg( 1 - \frac{\varepsilon}{\v^{\mathrm{H}} \H \H^{\mathrm{H}} \v} \bigg) \mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \H^{\mathrm{H}} \v \v^{\mathrm{H}} \H \mathbf{h}_{\mathrm{d}} \\
\label{eq:eta} \eta \triangleq \ & \mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \H^{\mathrm{H}} \v \v^{\mathrm{H}} \H \mathbf{h}_{\mathrm{d}} - \varepsilon \mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{h}_{\mathrm{d}}.
\end{align}}
\end{theorem}
\begin{figure}[!t]
\centering
{\def\columnwidth{\columnwidth}
\import{img/}{model_conversion.pdf_tex}}
\caption{Virtual model conversion from a cognitive interference channel with interference temperature constraint (a) to an SI FD channel (b).}
\label{fig:conversion}
\end{figure}
\begin{IEEEproof}
The key enabler to derive a closed-form expression of the optimal transmit beamformer is the intuition that the considered FD SI channel is equivalent to a cognitive interference channel with an interference temperature constraint. In this respect, Figure~\ref{fig:conversion}(a) depicts a licensed system consisting of a primary transmitter/receiver pair (PT/PR) that coexists with an opportunistic system consisting of a secondary transmitter/receiver pair (ST/SR), both operating in the same bandwidth. Let us assume an interference temperature constraint imposed on the ST, i.e., the maximum interference that the opportunistic transmission can generate towards the PR is upper bounded. Now, the equivalence between this setting and the FD SI channel considered in this paper becomes evident from Figure~\ref{fig:conversion}(b), where the same labels as in Figure~\ref{fig:conversion}(a) are used for convenience. In particular, we note that:
\begin{itemize}
\item[-] The transmit antennas of the FD BS serve a UT in the downlink, and such transmission causes SI at the receive antenna array, which is simultaneously communicating with another device in the uplink.
\item[-] In order to guarantee the full effectiveness of the SIC algorithms at the receive chain, an upper bound on the maximum tolerable power of the SI is imposed. Such upper bound has the same role as the interference temperature constraint in the aforementioned cognitive interference channel.
\end{itemize}
Thus, it is not difficult to see that the receive and transmit antennas of the FD BS are equivalent to a virtual PR and ST, respectively. Similarly, the UT served in the downlink operates as virtual SR, whereas the device communicating in the uplink acts as virtual PT. Therefore, we can cast \eqref{eq:opt1} into a rate maximization problem for a cognitive interference channel with interference temperature constraint, whose solution is known to have the form \cite{Lv12}
\begin{align} \label{eq:w_alpha}
\mathbf{w}(\alpha) & \triangleq \frac{\alpha \mathbf{w}_{\mathrm{ZF}} + (1 - \alpha) \mathbf{w}_{\mathrm{MRT}}}{\| \alpha \mathbf{w}_{\mathrm{ZF}} + (1 - \alpha) \mathbf{w}_{\mathrm{MRT}} \|}
\end{align}
where we have defined (see Remark~\ref{rem:pzf})
\begin{align}
\label{eq:w_pzf} \mathbf{w}_{\mathrm{ZF}} & \triangleq (\mathbf{I} - \H^{\mathrm{H}} \v (\H^{\mathrm{H}} \v)^{\sharp}) \mathbf{h}_{\mathrm{d}} \\
\mathbf{w}_{\mathrm{MRT}} & \triangleq \mathbf{h}_{\mathrm{d}}. \nonumber
\end{align}
In other words, the optimal solution of \eqref{eq:opt1} can be expressed as the (normalized) linear combination of a maximum ratio transmission (MRT) beamformer, obtained as a function of the downlink channel, and a ZF beamformer, obtained as a function of both the SI and the downlink channel. In particular, we note that the normalization constraint is satisfied by construction of $\mathbf{w}(\alpha)$.
Now, it is easy to observe that both the objective and the SI constraint in \eqref{eq:opt1} are monotonically decreasing functions of $\alpha$. Then, it follows that the optimal solution of \eqref{eq:opt1} satisfies the SI constraint with equality. In this regard, let $\widetilde{\alpha}$ be the (unbounded) solution of $|\v^{\mathrm{H}} \H \mathbf{w}(\alpha)|^{2} = \varepsilon$ given by
\begin{align} \label{eq:alpha_tilde}
\widetilde{\alpha} \triangleq 1 - \sqrt{\frac{\zeta - \eta}{\zeta}}
\end{align}
with $\zeta$ and $\eta$ defined in \eqref{eq:zeta} and \eqref{eq:eta}, respectively; then, we readily obtain $\alpha^{\star}$ in \eqref{eq:alpha_star} from \eqref{eq:alpha_tilde} by imposing $\widetilde{\alpha} \in [0,1]$. Finally, $\mathbf{w}^{\star}$ is computed as in \eqref{eq:w_star}.
\end{IEEEproof}
\begin{remark} \label{rem:pzf} \rm{
The transmit beamformer $\mathbf{w}_{\mathrm{ZF}}$ in \eqref{eq:w_pzf} is chosen as the (non-normalized) projection of $\mathbf{h}_{\mathrm{d}}$ onto the null space of $\H^{\mathrm{H}} \v$, i.e., one antenna nulls the SI entirely whereas the remaining $N_{T} - 1$ antennas are used to maximize the power signal to the desired link. On the other hand, with $\mathbf{w}_{\mathrm{MRT}}$, all $N_{T}$ antennas are used to maximize the power signal to the desired link. Therefore, increasing $\alpha$ in \eqref{eq:w_star} (resp. \eqref{eq:w_alpha}) corresponds to applying more SIC and results in less power conveyed to the desired link.}
\end{remark}
\begin{remark} \label{rem:low_compl} \rm{
The model conversion from \eqref{eq:opt1} to its cognitive radio-based interpretation casts a $N_{T}$-dimensional problem into a one-dimensional problem, for which we are able to derive the optimal solution in closed-form. On the one hand, such closed-form expression does not require any iterative algorithm; instead, it consists in a single operation with no matrix inversion whatsoever,\footnote{Observe that $\v^{\mathrm{H}} \H \H^{\mathrm{H}} \v$ is a scalar value and, therefore, there is no matrix inversion involved either in \eqref{eq:w_star}, i.e., in the computation of $(\H^{\mathrm{H}} \v)^{\sharp}$, or in \eqref{eq:zeta}.} which enables the computation of the optimal transmit beamformer within the limited coherence time allowed by the fluctuations of the wireless channel. On the other hand, the model conversion brings full scalability with the number of transmit antennas $N_{T}$ in terms of computational complexity, since it only requires the closed-form computation of the one-dimensional optimal variable $\alpha^{\star}$. In that sense, the advantages brought over state-of-the-art solutions for optimal transmit beamforming design proposed in \cite{Zha12} are evident.}
\end{remark}
\begin{figure}[t!]
\centering
\includegraphics[scale=1]{./img/figure1} \\ \vspace{2mm}
\includegraphics[scale=1]{./img/figure2}
\caption{Average number of iterations and average CPU time using CVX for the computation of the optimal transmit beamformer, with $c=-110$~dB, $\rho=0$~dB, and for different number of transmit antennas.} \label{fig:complexity} \vspace{-1mm}
\end{figure}
\section{Numerical Results} \label{sec:Num}
In this section, we analyze the benefits of the proposed transmit beamforming design with respect to: i) \cite{Zha12} in terms of complexity, and ii) \cite{Rii11} in terms of performance. The reported numerical results are obtained by means of Monte Carlo simulations over $10^{4}$ channel realizations. In particular, all simulations are run on a single core of a 2.50~GHz CPU with 8~GB of memory. The transmit power and the noise floor of the FD BS are set in compliance with the Long-Term Evolution (LTE) radio frequency planning for outdoor small cells \cite{Holma10}, i.e., $p_{\mathrm{d}} = 30$~dBm and $r_{\mathrm{n}} = -116.4$~dBm. Furthermore, we assume $\Omega=-30$~dB, and $c=-110$~dB as in \cite{Bha13,Bha14}, and that maximum ratio combining (MRC) is adopted at the receive chain of the FD BS, i.e., $\v = \mathbf{h}_{\mathrm{u}}$.
\vspace{2mm}
\noindent{\textbf{Complexity gains.}} We first illustrate the reduced complexity of our approach with respect to \cite{Zha12}. Therein, \eqref{eq:opt1} is solved by applying the following SDP relaxation:
\begin{align} \label{eq:opt3}
\begin{array}{ccl} \vspace{2mm}
\displaystyle \max_{\mathbf{W}} & & \log_{2} \big( 1 + \mathrm{tr}(\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{W} \mathbf{h}_{\mathrm{d}}) \big) \\
\displaystyle \mathrm{s.t.} & & \mathbf{W} \succeq 0 \\
& & \mathrm{tr}(\v^{\mathrm{H}} \H \mathbf{W} \H^{\mathrm{H}} \v) \leq \varepsilon \\
& & \mathrm{tr}(\mathbf{W}) \leq 1
\end{array}
\end{align}
with $\mathbf{W} \triangleq \mathbf{w} \mathbf{w}^{\mathrm{H}}$. We begin our comparison by solving~\eqref{eq:opt3} numerically with the Matlab-based convex optimization solver CVX \cite{cvx14} and study the complexity of this approach in terms of number of iterations and CPU time. From Figure~\ref{fig:complexity}, we observe that, as could have been intuitively expected, the computation requires an increasing number of iterations and CPU time as the number of antennas increases. In particular, even for low number of antennas, i.e., $N_{T} \in [2,6]$, the CPU time ranges around 200~ms; this is not a suitable value for accommodating the requirements of real-world cellular system implementations, typically characterized by channel coherence times whose largest value is in the order of a few hundreds of ms, even if very low mobility settings are considered \cite{Ghosh10}. In other words, by the time the solution of \eqref{eq:opt1} is found by the algorithm in \cite{Zha12}, the CSI may be outdated and may require a substantial update, de facto rendering the optimal solution useless. Remarkably, this is not the case for our low-complexity optimal transmit beamforming design based on the closed-form expression \eqref{eq:w_star}. In fact, the latter does not require any iterative algorithm to be computed and allows to obtain $\mathbf{w}^{\star}$ with one-shot computation (see Remark~\ref{rem:low_compl}).
\begin{figure}[t!]
\centering
\includegraphics[scale=1]{./img/figure3} \\ \vspace{2mm}
\includegraphics[scale=1]{./img/figure4}
\caption{Average throughput gain and average power saving with respect to applying ZF to the SI, with $c=-110$~dB and for different numbers of transmit antennas $N_{T}$.} \label{fig:gain1}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[scale=1]{./img/figure5} \\ \vspace{2mm}
\includegraphics[scale=1]{./img/figure6}
\caption{Average throughput gain and average power saving with respect to applying ZF to the SI, with $\rho=0$~dB and for different values of the SIC capability at the receive chain $c$.} \label{fig:gain2}
\end{figure}
\vspace{2mm}
\noindent{\textbf{Performance gains.}} We compare now our optimal transmit beamforming design and the ZF approach presented in \cite{Rii11}, whose complexities can be considered the same as a first approximation. The performance gain of the proposed method in this case can be seen from two perspectives:
\begin{enumerate}
\item \textit{Average throughput gain}, i.e., the increase in terms of throughput that is obtained for identical transmit power, defined as
\begin{align*}
\overline{\mathrm{TG}} & \triangleq \mathbb{E} \bigg[ \frac{\log_{2} \big( 1 + \rho |\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w}^{\star}|^{2} \big)}{\log_{2} \big( 1 + \rho |\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w}_{\mathrm{ZF}}|^{2} \big)} \bigg] - 1;
\end{align*}
\item \textit{Average power saving}, i.e., the power reduction that can be supported by the FD BS assuming a target throughput as the one achieved by the ZF approach in \cite{Rii11}, defined as
%
%
\begin{align*}
\overline{\mathrm{PS}} & \triangleq 1 - \mathbb{E} \bigg[ \frac{|\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w}_{\mathrm{ZF}}|^{2}}{|\mathbf{h}_{\mathrm{d}}^{\mathrm{H}} \mathbf{w}^{\star}|^{2}} \bigg].
\end{align*}
In this regard, we note that $\overline{\mathrm{PS}}$ is independent of the SNR, as intuitively should be, confirming its consistency.
\end{enumerate}
These metrics are depicted in Figure~\ref{fig:gain1} for $N_T \in [2,10]$ and SNR $\rho \in [-10, 20]$~dB. We start by focusing on $\overline{\mathrm{TG}}$ and observe that, for $c=-110$~dB and the simplest BS setup with $N_{T}=2$ transmit antennas, we obtain an average throughput gain of $29.66\%$ and $11.1\%$ for $\rho=-10$~dB and $\rho=20$~dB, respectively. Two observations are in order at this stage. First, $\overline{\mathrm{TG}}$ decreases as $N_T$ increases: this effect is expected and is due to the impact of the loss of one degree of freedom over the increasing number of available ones (i.e., one transmit antenna is sacrificed for nulling the SI) that characterizes the ZF approach. In this sense, the proposed transmit beamforming design proves particularly suitable for FD radios equipped with small to moderate number of antennas, e.g. FD small-cell BSs. Second, $\overline{\mathrm{TG}}$ decreases as the SNR increases: this is intuitively due to the different impact that the same power gain at the receiver has on the spectral efficiency of the link for different SNR values. In other words, the power gain induced by the proposed solution over \cite{Rii11} results in lower spectral efficiency gains as the SNR increases. Switching our focus on $\overline{\mathrm{PS}}$, we observe that the smaller $N_T$, the larger the average power saving, e.g. $\overline{\mathrm{PS}}=17.87\%$ for $N_T=2$. As for $\overline{\mathrm{TG}}$, the performance gains brought by our optimal transmit beamforming design with respect to applying ZF to the SI decrease as the number of transmit antennas at the FD BS increases, for the same aforementioned reasons.
Lastly, we study the impact of the SI threshold $\varepsilon$ on both $\overline{\mathrm{TG}}$ and $\overline{\mathrm{PS}}$, by computing these metrics in Figure~\ref{fig:gain2} for different pre-existing SIC capabilities of the FD device, i.e., $c \in [-120, -90]$~dB (see \eqref{eq:epsilon}), and SNR $\rho=0$~dB. We first observe that the proposed technique is more beneficial in terms of $\overline{\mathrm{TG}}$ as the pre-existing SIC capabilities increase, and allows $\overline{\mathrm{TG}}$ to range up to $110.21\%$. This result is rather intuitive to explain, given that the larger the pre-existing SIC capabilities, the more the degrees of freedom loss (due to using the legacy ZF-based method) affects the achievable downlink throughput. The same holds for $\overline{\mathrm{PS}}$, as previously discussed. In practice, and as it could have been expected, more sophisticated SIC strategies allow for larger power saving in terms of transmit power of the FD device. Finally, we note that larger values of $\overline{\mathrm{PS}}$, i.e., up to a remarkable $36.12\%$, are achievable for small values of $N_T$, confirming the findings in Figure~\ref{fig:gain1}.
\vspace{2mm}
\section{Conclusions} \label{sec:Concl}
In this paper, we consider full-duplex (FD) radios with multiple antennas and analyze the problem of identifying the optimal transmit beamforming that maximizes the downlink throughput, while fulfilling the self-interference (SI) cancellation requirements imposed by the receive chain. In this context, the FD radio is subject to strong limitations in terms of transmit power, which cannot exceed a certain threshold in order to protect the incoming signal from the SI. Given the current state of the art solutions, this problem has particular relevance for outdoor small cells populated by mobile users with rapid variations of their fast fading component and for vehicular small cells. In this regard, we derive a closed-form expression for the optimal solution to the considered problem. Remarkably, our numerical findings confirm that the proposed method improves the state of the art in terms of downlink throughput (with respect to applying zero-forcing to the SI) or complexity (with respect to existing solutions based on iterative algorithms). Quantitatively, the magnitude of the achievable gains depends on the pre-existing SI cancellation capabilities and the number of transmit antennas at the FD radio. In particular, both the performance enhancement and power saving grow as the number of antennas decreases and as the SI cancellation at the receive chain increases.
\vspace{2mm}
\balance
\addcontentsline{toc}{chapter}{References}
\bibliographystyle{IEEEtran}
|
2,877,628,090,755 | arxiv | \section{Introduction}
The structure, chemistry, and kinematics of the stellar halo of the Galaxy, with its
predominantly old and metal-poor populations, collectively preserve a detailed record
of our Galaxy's formation in the early universe \citep[e.g.,][]{eggen:62,searle:78}.
Thanks to large-area surveys such as the Sloan Digital Sky Survey
\citep[SDSS;][]{york:00,edr,dr1,dr2,dr3,dr4,dr5,dr6,dr7}, recent studies have revealed
that the halo is marked by numerous stellar substructures. The presence of these lumpy
and complex substructures (both in real and phase space) are in qualitative
agreement with models for the formation of the stellar halo through the hierarchical
merging and accretion of low-mass subhalos \citep[e.g.,][]{bullock:05}.
Among the various substructures discovered to date, the Virgo Overdensity (VOD) is one
of the most striking. It was discovered as a stellar overdensity of main-sequence
stars in SDSS; starcounts in the region toward Virgo are enhanced by a factor of two
above the background stellar distribution \citep{juric:08}. The overdensity covers
almost $1000\ \deg^2$, and it appears to span a wide range of heliocentric distances
of $\sim5$~kpc--$20$~kpc. The overdensity seems to be associated with clumps of RR Lyrae
stars \citep{vivas:01,duffau:06} and turn-off stars \citep{newberg:02}, but it is less
likely to be connected with the leading tidal tail of the Sagittarius dwarf galaxy
\citep{newberg:07}.
At present, metallicity estimation from broadband photometry is the only practical
means of obtaining metal abundances for a large number of faint objects such as those
in the VOD. Such methods are based on the relative sensitivity of stellar colors to
photospheric abundances over a wide wavelength baseline. The clear advantage of using
a photometric metallicity technique is the efficiency of estimating metallicities for
individual main-sequence stars, which are the most plentiful and representative sample
of stellar populations.
\citet{ivezic:08a} constructed photometric metallicity relations in the $u - g$
vs.\ $g - r$ plane using SDSS filter passbands, and studied the abundance structures
of the Galaxy with an accuracy of $\sim0.2$~dex at $g < 17$. This approach is similar
to the traditional $UBV$ method \citep[e.g.,][]{carney:79}, which relies on the strong
dependence of $U$-band
magnitudes on metal abundance. However, the $u$-band photometry in SDSS is limited
to $u \approx 22$ (99\% detection limit). This, and the greatly deteriorating errors
in SDSS $u$-band magnitudes near the faint limit, restricts photometric metallicity
estimates to stars at $r \la 20.8$, an insufficient depth to fully explore the VOD
\citep[see Fig.~37 in][]{juric:08}.
In this letter we overcome the limitations of the $u$-band photometry in SDSS by
exploring less metallicity-sensitive, but better-determined, color indices in the $gri$
passbands.\footnote{The $z$-band data were not used due to the bright
survey limit in SDSS for this filter.} Turn-off stars in globular clusters have
$M_r \sim 4$, so the SDSS survey limit in $gri$ ($95\%$ completeness limit at $r = 22.1$)
allows us to probe the halo out to $\sim20$~kpc using stars that are $\sim2$~mag below
the main-sequence turn-off ($\sim0.6\ M_\odot$).
\section{Method}
\subsection{Photometry}
We employed SDSS photometry from DR7 \citep{dr7}. SDSS measures the
brightnesses of stars with accurate astrometric positions \citep{pier:03},
using a dedicated 2.5-m telescope \citep{gunn:06} in five broadband filters
\citep[$ugriz$;][]{edr}, on 6 columns of CCDs
\citep{gunn:98}, under photometric conditions \citep{hogg:01}. Photometric
calibration is carried out using observations of stars in the secondary patch
transfer fields \citep{tucker:06}, based on the \citet{smith:02} sample of
standard stars.
The rms photometric precision is $0.02$~mag for sources not limited by photon
statistics, and the photometric calibration is accurate to $\sim2\%$ in the $gri$
bands, and $\sim3\%$ in $u$ and $z$ \citep{ivezic:03,ivezic:04,an:08}.
We used photometry of stellar objects \citep[identified as {\tt STAR} in the
standard SDSS photometric pipelines;][]{lupton:02} that were detected (at the
$5\sigma$ level) in all of the $gri$ passbands. The observed magnitudes are
corrected for extinction adopting reddening values in the \citet{schlegel:98}
dust maps and the extinction coefficients given by \citet{an:09}.
\subsection{Photometric Metallicity}
The isochrones in \citet{an:09} were used to determine photometric
metallicity (${\rm [Fe/H]_{phot}}$) estimates based on color-color relations for
main-sequence stars. In the following analysis, we adopted the same $\alpha$-element
enhancement scheme as in \citet{an:09}, motivated by the observed behavior of
these elements among field dwarfs and cluster stars from spectroscopic studies
\citep[e.g.,][]{venn:04,kirby:08a}:
[$\alpha$/Fe]$ = +0.4$ at [Fe/H]$ = -3.0$,
[$\alpha$/Fe]$ = +0.3$ at [Fe/H]$ = -2.0, -1.5, -1.0$,
[$\alpha$/Fe]$ = +0.2$ at [Fe/H]$ = -0.5$, and
[$\alpha$/Fe]$ = +0.0$ at [Fe/H]$ = -0.3, -0.2, -0.1, +0.0, +0.1, +0.2, +0.4$.
A linear interpolation was used in this metallicity grid to obtain isochrones
at intermediate [Fe/H] values. We adopted an age of 12.6~Gyr for ${\rm [Fe/H]}
\leq -1$, and 4.6~Gyr at ${\rm [Fe/H]} \geq 0$, and a linear interpolation
between these two values. To expedite the computational process, we derived
$5^{th}$-order polynomials to describe the color-magnitude
relations, with intervals in abundance of ${\rm \Delta [Fe/H]} = 0.01$~dex.
To reliably estimate a photometric metal abundance, it is
necessary to use stellar models that match cluster main-sequences and give
internally consistent distances from multiple color-magnitude
diagrams (CMDs). However, despite improvements in theoretical models, there
still exist small but significant mismatches between calculated isochrone colors
and the best photometry in well-studied open clusters \citep{pinsono:04,an:07a,an:07b}.
Therefore, we applied empirical corrections on theoretical color-$T_{\rm eff}$
relations for ${\rm [Fe/H]} \geq -0.8$ to match the observed main sequence of
the solar-metallicity cluster M67 \citep{an:09}. Constant correction factors
were adopted at ${\rm [Fe/H]} \geq 0$, and a linear ramp was used between
${\rm [Fe/H]} = -0.8$ and ${\rm [Fe/H]} = 0$, so that the correction becomes
zero at ${\rm [Fe/H]}=-0.8$ and below, as the models are in good agreement with
the data for globular clusters \citep{an:09}.
\begin{figure}
\epsscale{1.2}
\plotone{f1.eps}
\caption{Illustration of the photometric metallicity estimation technique.
{\it Top}: Distance moduli from two CMDs as a function of a metallicity. For
this example, ${\rm [Fe/H]} =-1.6$ was assumed for the true metallicity of
a star. {\it Bottom}: Difference in distance modulus from the two CMDs. A
photometric metallicity is defined as the [Fe/H] that results in the same
distance modulus from the two CMDs. The gray region shows the error bound
for a $\sim2\%$ photometric color error. Note that only the upper [Fe/H] bound
is defined in this example.
\label{fig:fehdist}}
\end{figure}
A photometric metallicity \citep{an:07b} was computed for each star by requiring
distances from main-sequence fitting to be the same from two different CMDs,
with $g - r$ and $g - i$ as color indices and $r$ as a luminosity index
(Fig.~\ref{fig:fehdist}). We searched the entire [Fe/H] grid from $-3.0$ to
$+0.4$ to find [Fe/H] with a minimum $\chi^2$ value, defined as
\begin{equation}
\chi^2 = \frac{(\mu_{g-r} - \bar{\mu})^2}{\sigma_{\mu_{g-r}}^2}
+ \frac{(\mu_{g-i} - \bar{\mu})^2}{\sigma_{\mu_{g-i}}^2},
\end{equation}
for each star. Here, $\mu$ and $\sigma_{\mu}$ are the distance modulus and its
error for each CMD, respectively. The quantity $\bar{\mu}$ is a weighted average
distance modulus from the $(g - r, r)$ and $(g - i, r)$ CMDs. Since only three passbands
are considered here, the problem is reduced to the traditional manner of
determining metal abundances from a color-color diagram. In some cases, a star
becomes bluer than the main-sequence turnoff as the metallicity increases,
either due to a large photometric error or a younger age than our assumed values
in the models. If a minimum $\chi^2$ was not found, ${\rm [Fe/H]_{phot}}$ was
estimated based on a mean relation between [Fe/H] and $\Delta (m - M)_0$.
\subsection{Accuracy of Photometric Metallicity Determinations}
Photometric metallicity estimates become insensitive for very metal-poor stars,
as illustrated in Figure~\ref{fig:fehdist}. Therefore, ${\rm [Fe/H]_{phot}}$ could
be biased due to systematic effects such as
photometric errors, unresolved binary stars, and dust extinction, which alter
the differential color estimates at a small level. We tested our photometric
metallicity estimates using field stars with well-measured spectroscopic estimates,
as discussed below, and adjusted ${\rm [Fe/H]_{phot}}$ to correct for the bias.
\begin{figure}
\epsscale{1.2}
\plotone{f2.eps}
\caption{Comparison between spectroscopic and photometric estimates of [Fe/H].
Open circles are median ${\rm [Fe/H]_{phot}}$ for the SSPP field star sample with
available spectroscopic metallicities; the gray region represents interquartile
ranges. Solid lines are piecewise linear fits to the open circles.
\label{fig:cluster}}
\end{figure}
Figure~\ref{fig:cluster} shows the comparison with low-resolution ($R\sim 2000$)
spectroscopic measurements for field dwarfs \citep{carollo:07,yanny:09} from
the most recent version (Data Release planned in December 2010)
of SSPP \citep[SEGUE Stellar Parameter Pipeline;][]{lee:08a,lee:08b}.
This version of the SSPP partially solves the under- and over-estimation of metallicity
at higher and lower ends of [Fe/H] that the earlier version (DR7) showed.
Comparisons are shown for 46,983 stars, after applying cuts at ${\rm S/N} > 20$ and
$\log{g} \geq 4.2$ to select dwarfs with good spectroscopic abundance measurements.
We further restricted the sample to those with $| 1.4(g - r)_0 - (g - i)_0 | \leq 0.1$.
Open points show a median metallicity in ${\rm [Fe/H]}=0.2$~dex bins; the gray
region represents interquartile ranges. Because the metallicity sensitivity
essentially disappears below the lower limit of our metallicity grid, a significant
number of stars are found at ${\rm [Fe/H]_{phot}} = -3.0$.
The large uncertainty in ${\rm [Fe/H]_{phot}}$ is mainly due to photometric errors.
We performed artificial star tests by generating stars from
the isochrones with Gaussian errors of $0.02$~mag in $r$, $g - r$, and $g - i$. The
interquartile range from the test showed a reasonable agreement with those for the
SSPP sample, indicating that the observed dispersion is at least in part due to the
$\sim2\%$ photometric error in SDSS. Despite the large uncertainty in
${\rm [Fe/H]_{phot}}$ for individual stars, a meaningful constraint on a median
${\rm [Fe/H]_{phot}}$ can be made by applying the technique to a large number of
stars ($N\sim500$, see below).
For the SSPP sample in Figure~\ref{fig:cluster}, we estimated the error in the median
of $\la0.1$~dex at ${\rm [Fe/H]_{spec}} > -2$ in each [Fe/H] bin; this was done by
computing the median absolute deviation (MAD)\footnote{${\rm MAD} \equiv 1.483\ {\rm median}
(|x_i - {\rm median}(x_i)|)$.} from stars with ${\rm [Fe/H]_{phot}}$ above the median,
divided by $\sqrt{N}$ (although ${\rm [Fe/H]_{phot}}$ does not strictly follow a normal
distribution). We explored the effects of differing stellar ages ($\sigma = 20\%$),
[$\alpha$/Fe] ratios ($\sigma \approx 0.1$~dex at ${\rm [Fe/H]} \la -1$), and dust
extinctions ($\sigma = 20\%$), but they change
${\rm [Fe/H]_{phot}}$ by $\la0.1$~dex.
As shown in Figure~\ref{fig:cluster}, our photometric solution underestimates [Fe/H] by
as much as $\Delta {\rm [Fe/H]} \sim 0.5$~dex for ${\rm [Fe/H]} \la -1$ with respect to
the SSPP results. One reason for this offset is likely due to the presence of
unresolved binary populations in the sample, which will have a portion of their light
from a cooler secondary \citep{an:07b}. To evaluate the effect of binaries, we performed
artificial star tests with binaries generated from a flat mass function for the secondaries,
and found that the $\Delta {\rm [Fe/H]} \sim 0.5$~dex offset can be explained with
a $\sim40\%$ binary fraction. On the other hand, the difference could also be induced
by a systematic color offset in the isochrones. Although we found a good agreement of
the models with globular cluster data within the errors \citep{an:09}, a small
color offset ($\la0.01$~mag) in the models can still change the photometric metallicity
estimate by $\Delta {\rm [Fe/H]} \sim 0.5$~dex at the low metallicity end. Although
possible, the low-resolution spectroscopic values are less likely the source of the
problem, given the extensive tests that have been applied in their validation
\citep[see][]{lee:08a,lee:08b,allende:08}. Nevertheless, the relative
metallicity comparison would be robust for stellar populations with the same binary
fraction, even if there is an offset with the SSPP.
In the following analysis, we adjusted our ${\rm [Fe/H]_{phot}}$ to be on the same
scale as the SSPP results by deriving a piecewise linear fit to the median
${\rm [Fe/H]_{phot}}$ as a function of metallicity (solid lines in Fig.~\ref{fig:cluster}).
This adjustment implicitly assumes the same fraction of unresolved binaries applies
to both the SSPP and the field halo samples, if binaries are solely
responsible for the difference between the ${\rm [Fe/H]_{phot}}$ and the SSPP results.
No statistically significant trend was found in the systematic difference between
${\rm [Fe/H]_{phot}}$ and ${\rm [Fe/H]_{spec}}$ for various sets of the binned data with
different magnitude and color ranges.
\section{Results}
\begin{figure}
\epsscale{2.4}
\plottwo{f3a.ps}{f3b.ps}
\caption{{\it Top}: Number density of stars at distances from $\sim10$~kpc to
$\sim20$~kpc from the Sun in the Lambert projection of the Galactic coordinates.
The North Galactic pole is at the center, and the Galactic Center is to the
bottom. Concentric circles represent $b = 0\arcdeg$, $30\arcdeg$, and $60\arcdeg$,
respectively. The VOD is the feature seen at $(l,b) \sim (300\arcdeg,70\arcdeg)$.
{\it Bottom}: Median metallicity of the same stars as in the {\it top} panel. The
median occupancy of each pixel is $544$ for both maps.
\label{fig:halo}}
\end{figure}
Figure~\ref{fig:halo} shows the distribution of stellar number density ({\it top})
and photometric metallicity ({\it bottom}) for 740,658 stars detected in SDSS.
The maps are Lambert equal-area projections of the Northern Galactic hemisphere,
and the distributions are projected as seen from the Sun (i.e., a view from the
inside of the Galaxy). The North Galactic Pole is located at the center, and the
direction of the Galactic Center is toward the bottom in each panel. Each pixel
has an area of $12.96\ \deg^2$. To avoid any possible bias, we restricted our
sample to stars with $0.3 \leq (g - r)_0 \leq 0.4$ and the same color-color cut
as those for the SSPP comparison sample. We used stars with
$15.0 \leq (m - M)_0 \leq 16.5$ ($10 \leq d \leq 20$~kpc).
Although a distance modulus was derived from the $\chi^2$ minimization for each
star, we used a 12.6~Gyr old model with ${\rm [Fe/H]} = -1.6$ for all of the stars
to bracket the distance range in the sample. This was because the uncertainty in
${\rm [Fe/H]_{phot}}$ for individual stars is large enough that the strong
correlation between ${\rm [Fe/H]_{phot}}$ and distance could lead to a biased result.
The feature at $(l,b) \sim (300\arcdeg,70\arcdeg)$ is the VOD \citep{juric:08},
and the long strip that crosses the sky from $(l,b) \sim (210\arcdeg,30\arcdeg)$
to the VOD is the leading tidal tail of Sgr. The feature at $(l,b) \sim (40\arcdeg,
40\arcdeg)$ is the Hercules-Aquila Cloud, another large-area overdensity of halo
stars discovered from SDSS \citep{belokurov:07}.
The median metallicity of stars in the VOD area at $270\arcdeg \leq l \leq 330\arcdeg$
and $60\arcdeg \leq b \leq 70\arcdeg$ is ${\rm [Fe/H]} = -2.0\pm0.1$ from the
metallicity map in Figure~\ref{fig:halo}, where the error is from a pixel-to-pixel
dispersion. The photometric zero points in SDSS vary at the $\sim2\%$ level over
a $\sim10\arcmin$ scale along the scan line \citep{an:08}. They are also known to vary
at the same $\sim2\%$ level over a larger angular scale along the stripe
(a $2.5\arcdeg$ wide SDSS stripe typically runs from the $1^{st}$ to the $3^{rd}$
Galactic quadrant), and from one strip to the other \citep{ivezic:03,ivezic:04}.
Our hope is that these components are properly averaged out over a large area in
the sky, such as the solid angle covered by the VOD.
Field halo stars located at the mirrored position ($30\arcdeg \leq l \leq 90\arcdeg$,
$60\arcdeg \leq b \leq 70\arcdeg$) exhibit ${\rm [Fe/H]} = -1.9\pm0.1$.
Although half of the stars in the direction toward Virgo are likely associated
with a progenitor dwarf galaxy or a tidal stream, the median [Fe/H] value
essentially remains unchanged; it is the same as that for the field halo stars
within the precision of the technique.
It is perhaps of interest that the field-star
metallicities are as low as they appear to be, as \citet{carollo:07} have
argued that the peak metallicity of the outer-halo population is ${\rm [Fe/H]} = -2.2$,
and that this component is expected to dominate over the more metal-rich inner-halo
population (with a peak metallicity at ${\rm [Fe/H]} = -1.6$) at Galactocentric
distances greater than $15$~kpc -- $20$~kpc.
Note that metallicity estimates at the second Galactic quadrant
are even lower than ${\rm [Fe/H]} \approx -2$.
\begin{deluxetable}{lcc}
\tablewidth{0pt}
\tablecaption{Error budget in ${\rm [Fe/H]_{phot}}$ for the VOD\label{tab:sys}}
\tablehead{
\colhead{Source of Error} &
\colhead{$\Delta$ Quantity} &
\colhead{$\Delta {\rm [Fe/H]_{phot}}$}
}
\startdata
Internal & \nodata & $\pm0.1$ \nl
Age & $\pm20\%$ & $\pm0.3$ \nl
Dust extinction & $\pm20\%$ & $\pm0.1$ \nl
$[\alpha{\rm /Fe}]$ & $\pm0.1$~dex & $\pm0.3$ \nl
Contamination by giants & \nodata & $-0.1$ \nl
Alternative approach & \nodata & $-0.1$ \nl
Total (systematic) & \nodata & $\pm0.5$
\enddata
\tablecomments{These estimates are slightly different from those for the SSPP
sample (\S~2.3) due to different color ranges used.}
\end{deluxetable}
In Table~\ref{tab:sys} we list the sources of several systematic errors and their
contributions to errors in ${\rm [Fe/H]_{phot}}$ for the VOD. The effects of
the age, extinction, and $[\alpha/{\rm Fe}]$ were tested by constructing similar
${\rm [Fe/H]_{phot}}$ maps to Figure~\ref{fig:halo} with different parameters in
the models. Our photometric technique is valid only for main-sequence stars,
but giants constitute approximately $10\%$ of the stars in the sample \citep[see][]{an:08}. Since
photometry alone cannot be used to adequately distinguish giants from dwarfs,
we estimated a bias due to the presence of giants using photometry for a sample of
nearby globular clusters \citep{an:08}. In Table~\ref{tab:sys} we also list the
error from an alternative approach where we
used a median difference in distance modulus in each pixel of Figure~\ref{fig:halo}
to determine ${\rm [Fe/H]_{phot}}$. The total systematic error is the quadrature sum
of all of the error contributions. The relative metallicity comparison is more robust,
if stellar populations in the field halo and the VOD have the same age and [$\alpha$/Fe]
distributions: the difference in ${\rm [Fe/H]_{phot}}$ remains within
$\Delta {\rm [Fe/H]_{phot}} \la 0.2$.
\section{Discussion}
This initial application of the photometric metallicity technique
demonstrated that there is no metallicity difference between the field
halo stars and those in the VOD within the precision of the method. Our
estimate can be compared with previous measurements for
a handful of RR Lyrae variables that are likely associated with the VOD
\citep{duffau:06,vivas:08,prior:09}. These estimates range from ${\rm [Fe/H]} = -1.55$
to $-1.95$, based on the pseudo-equivalent width of the \ion{Ca}{2}~K line.
From principal axes on the $u - g$ vs.\ $g - r$ diagram, \citet{juric:08} argued
that the VOD metallicity is lower than that of thick-disk stars and similar to that
of halo stars.
It is tempting to place the VOD in the observed trend of the luminosity-metallicity
relation among surviving dwarf galaxies in the Local Group \citep[e.g.,][]{grebel:03,
munoz:06,kirby:08b}. If we take our median ${\rm [Fe/H]}$ from Figure~\ref{fig:halo}
at $M_V \sim 10$ \citep{juric:08,prior:09}, the VOD follows this trend, supporting
the idea that [Fe/H] can serve as a luminosity indicator for an accreting dwarf
galaxy, in the process of building up the stellar halos of large spiral galaxies
like the Milky Way \citep[e.g.,][]{johnston:08}. However, this should be taken with
caution, as previous studies often report average abundances for dwarf galaxies
rather than median values.
Future imaging surveys, such as the the Large Synoptic Survey Telescope
\citep[LSST;][]{ivezic:08b} will use similar photometric bandpasses as those in SDSS,
providing even deeper (and {\it far} more accurate) photometric data than SDSS over
a larger fraction of the sky. Our photometric metallicity method will be useful to
exploit these databases for understanding the chemical evolution of progenitor dwarf
galaxies that are identified, as well as for the bulk populations of field stars.
\acknowledgements
We thank James Bullock, \v{Z}eljko Ivezi\'{c}, Heather Morrison, and Katie Schlesinger
for useful discussions.
T.C.B. and Y.S.L. acknowledge partial funding of this work
from grant PHY 08-22648: Physics Frontiers Center/Joint Institute for Nuclear
Astrophysics (JINA), awarded by the U.S. National Science Foundation.
Funding for the SDSS and SDSS-II has been provided by the Alfred P.\ Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S.\ Department of Energy, the National Aeronautics and Space Administration,
the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education
Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions. The Participating Institutions are the American Museum of Natural
History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge,
Case Western Reserve University, University of Chicago, Drexel University, Fermilab,
the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University,
the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory, and
the University of Washington.
|
2,877,628,090,756 | arxiv | \section{Introduction}
The magnetorotational instability (MRI), first derived by \citet{VelikhovMRI} and \citet{ChandrasekharMRI}, has been studied extensively as a possible mechanism of enhanced angular momentum transport in accretion disks \citep{BalbusHawley91,BalbusReview}. In an idealized form with no dissipative mechanisms, this instability occurs when conducting fluid in a spinning disk is threaded by a weak magnetic field. In order to excite the instability the fluid must have an angular velocity profile which is decreasing with radius, $\partial\Omega^{2}/\partial\ln{r}<0$, and the magnetic field must be sufficiently weak as to not stabilize perturbations, $({\bf k}\cdot{\bf V}_{A})^{2}<-\partial\Omega^{2}/\partial\ln{r}$. Dissipation enters through both the fluid viscosity, $\nu$, and the magnetic diffusivity, $\eta$. These dissipative mechanisms are commonly parameterized by the fluid and magnetic Reynolds numbers: $Re=VL/\nu$ and $Rm=VL/\eta$, respectively. When $Re$ is small, fluid viscosity acts to dampen any instabilities, thus there is a minimum $Re$ threshold for observing the MRI. Likewise, when the magnetic field advection is small compared to diffusion, i.e. $Rm$ is small, magnetic field lines are not affected by flows and there is no feedback mechanism for the MRI, thus a minimum $Rm$ value must be met as well.
The ratio between fluid viscosity and magnetic diffusivity is also key to understanding the MRI. This ratio is described by the magnetic Prandtl number, $Pm=Rm/Re=\mu_{0}\nu/\eta$ and can vary by many orders of magnitude in different astrophysical systems: $Pm\lesssim10^{-5}$ in disks around young stellar objects while $Pm\gtrsim10^5$ around active galactic nuclei. Shearing box simulations have shown that varying $Pm = 0.06-4$ can cause the magnitude of the MRI driven turbulent radial momentum transport to vary by two orders of magnitude \citep{Longaretti_2010}, suggesting that the MRI may not be efficient at small Prandtl numbers. Unfortunately for liquid metal experiments, $Pm$ is fixed at $\sim10^{-5}$, thus making them susceptible to Ekman circulation driven fluid turbulence (caused by large $Re$) and very small saturated amplitudes (caused by small $Rm$) \citep{Gissinger2012}. However, in plasmas $Pm\propto n^{-1}T_{e}^{3/2}T_{i}^{-5/2}$ and can be adjusted by varying the density and temperature.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure1a.pdf}\includegraphics[scale=1]{Figure1b.pdf}}
\caption{Left: A cutaway view of PCX showing the major subsystems. Right: The magnetic field geometry of PCX showing electrodes at both the inner and outer boundaries.}
\label{fig:Figure1}
\end{figure}
Motivated by this control of $Pm$ in plasmas, the Plasma Couette Experiment (PCX) explores the magnetohydrodynamic stability of hot, fast-flowing plasmas in a Taylor-Couette flow (TCF) geometry. In PCX, steady-state plasmas are created and confined in a magnetic field-free volume and spun via electrostatic stirring at inner and outer boundaries of a cylindrical plasma volume \citep{CamiPRL,CamiPoP}. Differential flows with peak velocities up to 12 km s$^{-1}$ in helium, densities of 10$^{11}$ cm$^{-3}$, and $T_{e}\sim8$ eV are routinely created, corresponding to $Rm\sim65$. TCF involves boundary driven flows, where momentum injected at inner and outer boundaries (via rotating walls in liquid metal experiments or electromagnetic drive in PCX) is viscously coupled to the bulk fluid. This is distinctly different from body forces like gravity or electromagnetic forces. To observe the MRI, TCF flow profiles must be adjusted to mimic Keplerian-like (body force) flows of astrophysical accretion disks, where angular velocity is decreasing with radius while still maintaining Rayleigh hydrodynamic stability with angular momentum increasing with radius (the Rayleigh criterion). In PCX, the range of achievable densities and temperatures set $Pm\approx10^{-1}-10^{2}$.
PCX operates at densities of $10^{10}-10^{11}$ cm$^{-3}$, which correspond to an ion inertial length on the order of the vessel size or larger. Under these conditions the Hall effect, where ions become decoupled from the magnetic field, has a strong influence on the MRI. One of the most dramatic consequences of Hall-MRI is the requirement that the magnetic field threading the flow be antiparallel to the axis of rotation \citep{wardle99,balbus01_aj}. Additionally, PCX works at rather low ionization fractions, $f_{\%}\lesssim1\%$ in helium. The high neutral density acts as a retarding body force on the bulk ion flow via charge-exchange collisions and can have dramatic effects on the boundary driven profiles, thus the stability of PCX flows. Recent work by \citet{Kunz2013,Lesur2014} suggests that two-fluid and neutral effects are necessary for qualitatively correct descriptions of the MRI in protoplanetary discs. Because of the importance of two-fluid and neutral collision effects in real astrophysical systems and the adjustable $Pm$, plasma experiments, such as PCX, are ideal for laboratory study of the MRI.
This article focuses on the effects of dissipation, the Hall term, and the neutral drag body force on the onset of the MRI in PCX. First, a description of PCX and its current operation is provided. Extensive work has been spent understanding the effect of the neutral drag body force on boundary driven flow profiles. In order to mitigate the effects of neutrals on PCX flows, a confinement and input power upgrade, PCX-U, is described. Using the estimated parameters of PCX-U a global stability analysis of the MRI including the neutral drag force and Hall effect is presented. Results show that even in the improved PCX-U, the neutral drag body force can drive hydrodynamic instability of flows, thus the final portion of this article focuses on the equilibrium and stability of a body driven flow scheme which we simply call electrically driven flow (EDF).
\section{Description of Experiment}\label{sec:descrip}
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure2.pdf}
\caption{An example discharge from PCX. In (a) plasma is created with a neutral gas puff and ECH heating. The plasma is spun with four outer cathodes (each 350 V, $\sim1.1$ A) and five inner cathodes (each 575 V, $\sim0.75$ A). In (b) electron temperature and density as measured by the triple Langmuir probe. (c) The angular velocity profile is shown at two different times in the shot (shown as dashed vertical lines in (a) and (b)) along with the azimuthal velocity (d) and angular momentum (e) profiles. The error bars are standard deviation of fluctuation in time.}
\label{fig:Figure2}
\end{figure}
PCX consists of a roughly 1 m tall by 1 m diameter cylindrical vacuum chamber. The inner walls of the chamber are lined with toroidal rings of permanent magnets. This magnet geometry is shown in Fig. \ref{fig:Figure1}. The rings alternate polarity creating a strong multi-cusp magnetic field isolated to the edge of the vessel. This field confines the plasma losses to a small cusp on the face of the magnets, while leaving the bulk volume unmagnetized. If desired, an external Helmholtz coil can produce nearly uniform axial magnetic fields in the chamber up to $B_{0}\sim50$ G.
A plasma discharge is created by biasing emissive thoriated tungsten cathodes to cold molybdenum anodes at up to 600 V. The plasma is then heated with up to 6 kW of electron cyclotron heating (ECH). Table \ref{tab:params} shows typical PCX plasma parameters along with the relevant dimensionless quantities for this paper, while Fig. \ref{fig:Figure2} shows the time evolution of plasma parameters in a typical PCX discharge.
The hot cathodes also act to stir the plasma by drawing a current across the multi-cusp magnetic field at the edge. This ${\bf J}\times{\bf B}$ torque viscously couples inward to the unmagnetized bulk plasma. Due to the axisymmetric configuration of the magnets, particle drifts in the magnetized edge act to symmetrize the system, such that the toroidal location of the cathodes and anodes does not affect the flow drive \citep{Katz2012RSI}. Flow can be driven at the inner boundary by a center core assembly consisting of 22 stacked magnet rings and up to 8 smaller thoriated tungsten cathodes biased with respect to cold anodes. This assembly can be entirely removed via a gate valve for maintenance of the cathodes.
\begin{table}
\begin{center}
\def~{\hphantom{0}}
\begin{tabular}{l || c | c }
{}&PCX&PCX-U\\ \hline
$n_{e}$ (cm$^{-3}$)&$10^{10}-10^{11}$&$10^{11}-10^{12}$\\
$T_{e}$ (eV)&$5-10$&$10-20$\\
$T_{i}$ (eV)&$0.1-0.5$&$0.5-1$\\
$f_{\%}$&$0.1-10\%$&$1-99\%$\\
$M\equiv V_{1}/C_{s}$&$0.5-0.7$&$0.3-0.5$\\
$M_{A}\equiv V_{1}/V_{A}$&$0-4$&$0-12$\\
$Rm\equiv (V_{1} R_{1})/\eta$&$0-500$&$0-1600$\\
$Pm\equiv Rm/Re$&$0.2-200$&$0.5-350$\\
$Ha\equiv \sqrt{ReRm}/M_{A}$&$40-80$&$30-50$\\
$\mu\equiv R_{1}/L_{\nu}$&$0.1-10$&$0.02-2$\\
$\delta_{i}\equiv c/(\omega_{pi}R_{1})$&$5-50$&$2-15$
\label{tab:params}
\end{tabular}
\caption{Plasma parameter and dimensionless variable ranges for both PCX and PCX-U. For the Hartmann number ($Ha$) and Alfv\'{e}n Mach number ($M_{A}$), a magnetic field of $B_{0}=-2$ Gauss is used. The Mach numbers used are based on velocity at the inner radius. The unit of length used in this paper is the inner stirring radius $R_{1}=0.1$ m.}
\label{tab:params}
\end{center}
\end{table}
Plasmas in PCX are routinely diagnosed using a swept or triple tip Langmuir probe for electron temperature and density, a cold cathode gauge for neutral pressure, and Mach probes for flow velocity. In addition to these routine diagnostics, PCX has implemented Optical Emission Spectroscopy (OES) and a Fabry-Perot interferometer to measure electron and ion temperature, respectively. Both of these optical diagnostics are non-invasive, operating by simply sampling a single chord of emitted plasma light. The OES system takes a broadband spectrum of the plasma and uses line ratios to determine the electron temperature. OES temperature measurements in PCX agree within 15\% of the routine Langmuir probe. The Fabry-Perot interferometer finely samples a small ($<1$ nm) range of the emission spectrum centered at 468.6 nm in helium and 488 nm in argon. Ion emission lines at these wavelengths reflect the entire ion distribution function, such that the ion temperature and velocity can be deduced. In initial measurements of non-flowing argon plasmas, the ion temperature was determined to be $T_{i}=0.2-1$ eV.
\subsection{Taylor-Couette Flow (TCF) Profiles with Ion-Neutral Drag}\label{sec:TCF}
In weakly-ionized plasmas charge exchange collisions between neutrals and ions impose a body drag force that affects the equilibrium (steady-state) velocity profiles. This neural drag effect has been directly observed in PCX flows \citep{CamiPRL}. An analytical expression for neutral drag modified profiles can be found by treating the neutral drag as a momentum sink in the toroidal momentum balance equation. These profiles are a departure from ideal TCF yet still have a simple Bessel function form:
\begin{equation}\label{eq:TCF}
V_{\phi}(r)=A\mathcal{I}_{1}(r/L_{\nu})+B\mathcal{K}_{1}(r/L_{\nu})
\end{equation}
\\
where the constants $A$ and $B$ depend on the outer and inner boundary locations and the velocities applied, $L_{\nu}^{2}\equiv\tau_{i0}\nu$ is the momentum diffusion length, $\tau_{i0}=(n_{0}<\sigma_{cx}v>)^{-1}$ is the ion-neutral collision time, $\sigma_{cx}$ is the ion-neutral charge exchange cross section, and $\nu$ is the kinematic viscosity due to ion-ion collisions. The momentum diffusion length represents the combined effects of viscous momentum diffusion and the neutral drag momentum sink. When $\mu\equiv R_{1}/L_{\nu}\gg1$, the neutral drag dominates the viscous diffusion, rotation is confined to inner and outer edges of the plasma. These heavily affected profiles have a much lower average velocity across the profile and increased shear near the boundaries that can drive hydrodynamic instability. In the opposite limit $\mu\ll 1$, the velocity profile becomes the ideal Taylor-Couette profile: $V_\phi(r)=ar+b/r$.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure3.pdf}}
\caption{Synthetic profiles of the (a) toroidal velocity, (b) angular frequency, and (c) angular momentum of modified Taylor-Couette flow with different $\mu\equiv R_{1}/L_{\nu}$ values. The effect of the neutral drag is increased at greater values of $\mu$.}
\label{fig:Figure3}
\end{figure}
Finite neutral drag has a large qualitative effect on the velocity profile of boundary driven flows and greatly affects the stability of this modified TCF. Figure \ref{fig:Figure3} highlights the effect of ion-neutral collisions on TCF profiles of toroidal velocity, angular frequency and angular momentum. The boundary velocities for these profiles are chosen to be $V_{1}=10$ km s$^{-1}$ at $R_{1}=0.1$ m and $V_{1}=2.5$ km s$^{-1}$ at $R_{1}=0.4$ m. With no neutrals this gives a toroidal velocity profile $V_\phi(r)\propto 1/r$, which is marginally stable to the Rayleigh criterion and meets the ideal MRI condition. For heavily affected profiles ($\mu\gg1$), the Rayleigh criterion is not met across the whole profile and hydrodynamic instability is expected. Such a choice of boundary velocities also allows comparison of TCF with electrically driven flow, where $V_\phi(r)\propto 1/r$ is driven via a body force (Sec. \ref{sec:EDF}).
The profiles shown in Fig. \ref{fig:Figure3} have not been created on PCX due to the current limitations of the centerstack stirring assembly, but they are expected to be achieved in the upgraded experiment. TCF profiles of up to 12.3 km s$^{-1}$ on the outer boundary and 3.4 km $^{-1}$ on the inner have been driven in helium PCX plasmas \citep{CamiPoP}. These flows, shown in Fig. \ref{fig:Figure2}, have areas of decreasing angular frequency as required by the ideal-MRI criteria. They also have increasing angular momentum, thus meet the Rayleigh criterion for hydrodynamic stability. At PCX plasma parameters, these flows correspond to a Reynolds number of $Re\approx26$ (using the outer boundary velocity) and a magnetic Prandtl number of $Pm\approx2.5$. However, these flows have thus far been obtained without applied axial magnetic field so no instabilities have yet been observed.
\subsection{Proposed System Upgrades}\label{sec:PCX_U}
An upgrade to the confinement and heating systems for PCX is underway. The goals are to improve confinement by increasing the volume to loss area ratio, $\mathbb{V}/A_{\ell}$, to input more power, and to improve the temperature tolerances of PCX in order to accommodate longer and higher power discharges. These upgrades, referred to as PCX-U from here on, constitute a major change for PCX that collectively aims to push the experiment into MRI relevant parameter regimes. Expected parameters are presented in Table~\ref{tab:params}.
The largest portion of the PCX upgrade consists of an entirely new magnet assembly. Approximately 2000 samarium cobalt (SmCo) magnets, like those used in the Madison Plasma Dynamo Experiment (MPDX) \citep{MPDXPOP}, will replace the current array of ceramic magnets. These relatively inexpensive magnets have a much higher field strength and temperature tolerance. For the sizes and grade proposed, the field will be approximately 4 kG at the face of the magnets (the ceramic magnets are 1 kG) and the maximum operating temperature will be nearly 300$^{\circ}$C.
The new magnets themselves will be a significant improvement over the current system. Currently, the length of PCX discharges is mostly limited by the temperature that the magnets reach, which is ideally kept below about 60$^{\circ}$C. With a higher temperature tolerance, the SmCo magnets will allow for longer discharges. In turn, good vacuum pumping in conjunction with longer discharges will remove more neutrals, leading to higher ionization fractions. Additionally, the four-fold increase in magnetic field strength will reduce the cusp loss width by a factor of four. The cusp width can be estimated as $w=4\sqrt{\rho_{i}\rho_{e}}\propto B^{-1}$, where $\rho_{i}$ and $\rho_{e}$ are the ion and electron gyroradii, respectively \citep{Cusp}.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure4a.pdf}\includegraphics[scale=1]{Figure4b.pdf}}
\caption{Left: A design image of the proposed new magnet assembly on PCX-U. Right: A comparison of the current PCX magnet geometry (left) and the proposed geometry (right). These plots show the magnitude of the magnetic field on a scale from Earth's field to 1 T. The pink contour is the 875 Gauss ECH resonance.}
\label{fig:Figure4}
\end{figure}
The left image of Fig. \ref{fig:Figure4} shows a design image of the new magnet assembly. Approximately 1000 magnets are bolted to 14 individually water cooled aluminum rings equally spaced in the axial direction. At the top and bottom of the chamber, 8 concentric rings attached by spokes carrying water cooling support an additional 350 magnets each. Finally, two rings pitched at 45$^{\circ}$ are supported in the corners between the top and bottom rings and the side rings. These angled rings serve to strengthen the field at the corners where the current assembly suffers from the most magnet heating. The whole assembly is supported by 6 rods attached to a top flange which doubles as the water cooling vacuum feedthrough. The inner three rings on the top and bottom assemblies may be removed in order to place future centerstack assemblies of up to 16in in diameter into PCX-U.
When describing confinement a good figure of merit is the ratio between plasma volume and loss area, representing the particle balance between volumetric ionization and particle flux to loss areas. For this figure of merit, PCX-U represents a significant improvement over PCX. On the right in Fig. \ref{fig:Figure4}, a side by side comparison of the current and new magnet assemblies highlights the roughly 150\% increase in plasma volume (i.e. PCX-U is 1.5 times the volume of PCX). Due to the extra rings of magnets, the cusp length of the new assembly is roughly 130\% longer, but the stronger magnets make the cusp width smaller by a factor of four, so the $\mathbb{V}/A_{\ell}$ ratio is about 4.5 times larger on PCX-U. This favorable improvement will increase ionization fractions in PCX-U significantly.
In addition to the new magnet assembly, PCX-U includes a second 6 kW magnetron for increased plasma heating, doubling the available microwave power to 12kW. At densities above, $n=7.4\times10^{10}$ cm$^{-3}$, PCX plasmas are above the O-mode cutoff for 2.45 GHz, yet it has been observed that microwave power still heats the plasma possibly via surface wave discharges \citep{CamiPoP}. PCX-U is estimated to operate in this overdense mode. Additionally, the stronger field on the magnets will place the 875 Gauss ECH resonance further into the plasma volume, thus the surface wave discharge will have a larger volume to fill. This extra power will be managed by the high temperature tolerance of the SmCo magnets.
\section{Model for Global Stability Analysis}\label{sec:Stab}
Here we present a model for studying the global stability of flows in PCX-U as a method for determining the threshold for MRI. Previous global stability analyses for PCX have included the Hall term, but not the neutral drag body force \citep{Ebrahimi2011}. In order to capture both of these effects, an incompressible dissipative Hall MHD model with the neutral body drag force is used. The only two-fluid effect included is the Hall term. Ambipolar diffusion is negligible in PCX since recently charge-exchanged neutrals have a mean free path much longer than the machine size and thus are not coupled to the ions. The neutral drag body force is included as a momentum sink term dependent on the momentum diffusion length. While the incompressibility assumption may not be completely valid near the inner cylinder, we leave detailed study of this effect for future work and assume for this analysis a uniform density. The governing equations used for this analysis are:
\begin{eqnarray}
\frac{\partial {\bf V}}{\partial t}=-({\bf V}\cdot\nabla){\bf V}-\nabla \frac{P}{\rho} +\frac{1}{\mu_{0} \rho}(\nabla \times {\bf B})\times{\bf B}+\nu\nabla^{2}{\bf V}-\frac{1}{\tau_{i0}}{\bf V}\label{eq:MHD}\\
\nabla\cdot{\bf V}=0\\
\frac{\partial {\bf B}}{\partial t}=\nabla\times\left[{\bf V}\times{\bf B}-\frac{1}{\mu_{0} n e}(\nabla\times{\bf B})\times{\bf B}\right]+\eta\nabla^{2}{\bf B}\\
\nabla\cdot{\bf B}=0\label{eq:MHD_end}
\end{eqnarray}
\\
where $P$ is the scalar pressure, $\rho$ is the mass density and $\eta$ is the magnetic diffusivity (in m$^{2}$ s$^{-1}$). In these equations, plasma parameters $n_{e}$, $\rho$, $\tau_{i0}$, $\nu$ and $\eta$ are assumed to be constant and uniform throughout the volume. Profiles of $T_{e}$ and $n_{e}$ from Langmuir probes and the OES system support this assumption \citep{CamiPoP}. The equilibrium modified TCF profile (\ref{eq:TCF}) results from the steady-state balance between the last two terms in (\ref{eq:MHD}). In order to study linear stability these equations are cast in terms of non-dimensional parameters and linearized near the equilibrium state: ${\bf V}_{eq}=V_{\phi}(r)\,{\bf e}_{\phi}$ and ${\bf B}_{eq}=B_{0}\,{\bf e}_{z}$. The unit of length is the radius of the inner cylindrical boundary $R_{1}$ and the unit of velocity is the plasma velocity at this boundary, so ${\bf V}=V_{1}{\bf v}$ and $P=\rho V_{1}^{2}p$. The magnetic field is normalized by the applied axial field: ${\bf B}=B_{0}{\bf b}$. The dimensionless equations are
\begin{eqnarray}
\label{lin_v}
\gamma {\bf v}=-\left({\bf v}_{eq}\cdot \nabla\right){\bf v}-({\bf v}\cdot\nabla){\bf v}_{eq}-\nabla p+\frac{1}{M_{A}^{2}}(\nabla\times{\bf b})\times{\bf b}_{eq}+\frac{1}{Re}\left(\nabla^{2}-\mu^{2}\right){\bf v}\\
\nabla\cdot{\bf v}=0\\
\label{lin_b}
\gamma {\bf b}=\nabla\times\left[{\bf v}_{eq}\times{\bf b}+{\bf v}\times{\bf b}_{eq}-\frac{\delta_{i}}{M_{A}}(\nabla\times{\bf b})\times{\bf b}_{eq}\right]+\frac{1}{Rm}\nabla^{2}{\bf b}\\
\label{end_lin}
\nabla\cdot{\bf b}=0
\end{eqnarray}
where $\gamma$ is the growth rate in units of angular frequency $\Omega_{1}\equiv V_{1}/R_{1}$. The dimensionless parameters that enter these equations are: the fluid Reynolds number, $Re\equiv V_{1}R_{1}/\nu$; the magnetic Reynolds number, $Rm\equiv V_{1}R_{1}/\eta$; the Alfv\'{e}n Mach number, $M_{A}\equiv V_{1}/V_{A} \equiv V_{1}\sqrt{\mu_{0}\rho}/B_{0}$; the normalized ion inertial length for the Hall effect, $\delta_{i}\equiv d_{i}/R_{1}\equiv c/(\omega_{pi}R_{1})$; and the normalized momentum diffusion length for the neutral collision effect, $\mu\equiv R_{1}/L_{\nu}\equiv R_{1}/\sqrt{\tau_{i0}\nu}$.
Note that in this stability analysis the effect of the neutral drag enters consistently both through the modification of the equilibrium rotation profile and as a drag force in the linearized momentum equation. Equations (\ref{lin_v})-(\ref{end_lin}) are solved for axisymmetric modes using a standard finite difference method assuming no-slip, non-conducting side walls and periodicity in the axial direction.
We do not take into account the multicusp magnetic field, details of the plasma confinement and driving near the walls. For TCF study we assume that the boundary toroidal velocities are given. By assuming axial periodicity we also ignore the presence of top and bottom endcaps, thus neglecting the possibility of the Ekman circulation and Hartmann layers. These assumptions greatly simplify present analysis, allowing us to focus on the global MRI physics and not on the boundary effects.
The neglected boundary effects are clearly of great importance in the MRI experiments (e.~g., \cite{Gissinger2012}), but they deserve a separate detailed study. In particular, correct boundary conditions near the multicusp edge must be determined by values of viscosity and resistivity in that magnetized region. Empirical measurements and PIC simulations will be needed in order to understand this rather complicated non-MHD boundary condition. In our current model viscosity and resistivity are spatially uniform and calculated for unmagnetized plasma core. Nonetheless, we expect that such approximation still gives us reliable estimates for the parameters required for observing the MRI in PCX.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure5.pdf}}
\caption{Growth rate, $\gamma$ in units of $\Omega_{1}$, plotted as a function of applied magnetic field. The different curves present the single fluid MHD case, the inclusion of the Hall term and the inclusion of neutral charge exchange collisions. For this plot $n_{e}=10^{12} cm^{-3}$ and $P_{0}=10^{-5}$ torr corresponding to $\delta=4.55$ and $\mu=0.95$. (a) The full range of magnetic field that gives positive MRI growth rates. (b) A close-up view near $B_0=0$ shows the small (less than earth field) positive $B_{0}$ branch for the Hall and Hall+Neutral cases.}
\label{fig:Figure5}
\end{figure}
\section{Magnetorotational Instability in Taylor-Couette Flow}\label{sec:Stab_results}
The results of the global stability code from the previous section are shown below. Unless otherwise noted, all the analysis in this section is done assuming a singly ionized ($Z=1$ and $n_{e}=n_{i}\equiv n$) helium plasma with $T_{e}=12$ eV and $T_{i}=0.4$ eV. Fixing the temperatures allows viscosity to vary only with density and resistivity, $\eta$, to be mostly fixed (there is a weak density dependence). The dimensions of this system are $R_{1}=0.1$ m, $R_{2}=0.4$ m, and $H=0.8$ m, where the height determines the axial wave-number $k_{z}=2\pi/H$. A neutral-modified TCF profile, as defined in (\ref{eq:TCF}), is used as an initial equilibrium profile for this analysis. The boundary flow velocities are chosen to give a $v_{\phi}\propto1/r$ profile when no neutrals are present. A $v_{\phi}\propto1/r$ profile marginally meets the Rayleigh criterion (but is fully stable when viscosity is included) and meets the ideal-MRI condition. For this analysis an inner velocity of $V_{1}=10$ km s$^{-1}$ was chosen, which sets $V_{2}=2.5$ km s$^{-1}$ when a $v_{\phi}\propto1/r$ profile is desired. All of these fixed values fall into the range of expected parameters and flows for PCX-U.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure6.pdf}}
\caption{Toroidal velocity profiles of the most unstable eigen-modes in the Taylor-Couette flow. Corresponding normalized growth rates are $\gamma=0.087$ for single fluid MHD, $\gamma=0.043$ for Hall MHD, $\gamma=0.063$ for Hall MHD with neutrals. Plasma parameters are the same as in Fig. \ref{fig:Figure5}. The magnetic field is $B_0=-2$ Gauss and the mode numbers are $m=0$ and $k_{z}=2\pi/H$.}
\label{fig:Figure6}
\end{figure}
The Hall and neutral drag momentum sink terms both produce large qualitative effects on the stability of flows as shown in Fig. \ref{fig:Figure5} (corresponding eigen-modes are shown in Fig. \ref{fig:Figure6}). For the case when neither of these terms are included (single fluid MHD) positive MRI growth rates occur for very small magnitude seed fields (on the order of Earth's field), but the field orientation with respect to the axis of rotation does not matter. When the Hall term is included, positive MRI growth rates are found for stronger seed fields and only when the field is antiparallel to the axis of rotation (negative values of $B_{0}$ in this analysis). If the neutral drag term is added as well, the growth rate is slightly reduced and the magnetic field at the peak growth rate is smaller in magnitude. In the case of all terms being included, increased shear in the modified velocity profile drives a hydrodynamic instability (positive growth rate at $B_{0}=0$) at the particular plasma density and pressure shown in Fig. \ref{fig:Figure5}.
When the dimensionless parameter, $\mu\equiv R_{1}/L_{\nu}$, gets large the neutral-drag modified TCF profile becomes hydrodynamically unstable. Under this condition, the momentum injected at the inner and outer boundaries does not couple across the whole profile. This leads to increased shear at the boundaries of the flow. If the shear is great enough, the Rayleigh criterion is violated for a portion of the profile near the inner boundary, causing the hydrodynamic instability (instability at $B_{0}=0$). This analysis attempts to find a region in parameter space where this hydrodynamic instability is not present ($\mu$ is sufficiently small) and the MRI can still be excited with a weak $B_{0}$.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure7.pdf}}
\caption{(a) Stability curves as functions of $n$ and $P_{0}$. For each $n$ and $P_{0}$ the maximum growth rate in the range $B=-300$ to $100$ Gauss is used to determine stability. Region I is stable. Region II is hydrodynamically stable, but unstable to the MRI. Region III is hydrodynamically and MRI unstable. Contours of $\mu$ are also plotted. On the right, the density dependence of (b) $Re$ for both $V_{1}$ (solid) and $V_{2}$ (dashed), (c) $Pm\equiv Rm/Re$, and (d) $\delta_{i}$ are plotted to highlight how relevant dimensionless parameters scale with density.}
\label{fig:Figure7}
\end{figure}
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure8.pdf}}
\caption{Contours of normalized growth rate $\gamma$ as a function of density and applied magnetic field for the case with (a) $P_{0}=10^{-5}$ torr, (b) $P_{0}=10^{-6}$ torr and (c) no neutrals. The dashed vertical line represents the density at these neutral pressures above which the flow becomes hydrodynamically unstable due to a decrease in viscosity and increased shear caused by neutral drag.}
\label{fig:Figure8}
\end{figure}
\subsection{Parametric Study of Stability}
By fixing, $T_{e}$, $T_{i}$, $V_{1}$, $V_{2}$ and the dimensions of the system, the remaining variable plasma parameters are the plasma density, $n$; the neutral pressure, $P_{0}$; and the magnetic field $B_{0}$. Such a scan provides a range over which plasma physics phenomena can affect the onset of the MRI. Given these three variables, the dimensionless parameters of interest to this analysis have the following dependencies:
\begin{eqnarray}
&&\delta_{i}\propto n^{-1/2}\\
&&\mu\propto (nP_{0}\ln{\Lambda_{ii}})^{1/2}\\
&&M_{A}\propto n^{1/2}B_{0}^{-1}\\
&&Re\propto n\log{\Lambda_{ii}}\\
&&Pm\propto (n\log{\Lambda_{ii}}\log{\Lambda_{ei}})^{-1}
\end{eqnarray}
where $\log{\Lambda_{ii}}$ and $\log{\Lambda_{ei}}$ are the weakly density dependent Coulomb logarithms for ion-ion and electron-ion collisions, respectively.
In neutral pressure-density phase space, regions of stability can be mapped out with respect to the MRI and hydrodynamic instabilities as shown in Fig. \ref{fig:Figure7}. Region II represents the region where a MRI experiment would need to operate. Here flowing plasmas are hydrodynamically stable, but an applied axial magnetic field sets off the MRI. It is clear that to ensure that an experiment is in region II with these fixed temperatures and flow velocities, the neutral pressure must be as low as possible and the density must be neither too low nor too high. At higher neutral pressures and higher densities, $\mu$ can become of order unity, at which point the shear in the velocity profile caused by neutral drag is large enough to trigger hydrodynamic instabilities (region III). The MRI threshold between region I and region II appears to be set mainly by the density, i.e. viscosity. For low enough densities, the viscosity is large enough to damp out any instabilities. PCX currently operates in region I, but by boosting confinement (increasing the volume to loss-area ratio), PCX-U is expected to fall inside region II.
The onset of the hydrodynamic instability can be seen when scanning density at a fixed neutral pressure as in Fig. \ref{fig:Figure8}. For a given neutral pressure there is a density (i.e. viscosity) at which the shear caused by neutral drag becomes great enough to drive a hydrodynamic instability. As the amount of neutrals are decreased this threshold density becomes larger, because less shear is caused by neutrals. In the case when there are no neutrals present (plot (c) in Fig. \ref{fig:Figure8}), there is no hydrodynamic stability and larger MRI growth rates can be reached by increasing the density (lowering the viscosity). With the improved confinement in PCX-U the effect of neutrals can be reduced leading to conditions where the viscosity can be low enough to allow the MRI while still maintaining hydrodynamic stability.
\subsection{Nonlinear Saturation and Detectability}
In order to show the saturated state and determine the detectability of the MRI a nonlinear analysis following the model presented above was carried out. Fixing the magnetic field at $B=-2$ Gauss, density at $n_{e}=10^{12}$ cm$^{-3}$ and assuming no neutrals, the full nonlinear system (\ref{eq:MHD})-(\ref{eq:MHD_end}) is evolved in time for single-fluid and Hall MHD cases. Figure \ref{fig:Figure9} shows the kinetic and magnetic energy time-dynamics for these two respective cases and Fig. \ref{fig:Figure10} shows the structure of velocity and magnetic field in the saturated phase of MRI for single-fluid MHD and Hall MHD cases.
\begin{figure}
\begin{center}
{\includegraphics[scale=1]{Figure9a.pdf}\includegraphics[scale=1]{Figure9b.pdf}} \\
{\includegraphics[scale=1]{Figure9c.pdf}\includegraphics[scale=1]{Figure9d.pdf}}
\end{center}
\caption{Time evolution of the magnetic and kinetic energies for both the single fluid and Hall-MHD cases.}
\label{fig:Figure9}
\end{figure}
\begin{figure}
\begin{center}
{\includegraphics[scale=1]{Figure10a.pdf}\includegraphics[scale=1]{Figure10b.pdf}} \\
{\includegraphics[scale=1]{Figure10c.pdf}\includegraphics[scale=1]{Figure10d.pdf}}
\end{center}
\caption{Saturated structure of the velocity and magnetic fields for $n_{e}=10^{12}$ cm$^{-3}$ with no neutrals in both the single fluid and Hall-MHD cases. The applied field is $B_{0}=-2$ Gauss. }
\label{fig:Figure10}
\end{figure}
The interesting point here is that the saturation level of magnetic energy for non-zero $k_{z}$ is much lower in the Hall MHD case than in the MHD case. This difference in the energy levels can be explained by noticing that in the Hall MHD case for $\delta_{i}\gg1$ the saturation of the magnetic field occurs when the Hall term is balanced by the induction term in Eq.~(3.3), so $\tilde{b}\sim\tilde{v}/\delta_{i}$, where tilde denotes the parts of magnetic and velocity fields with non-zero $k_{z}$. In the MHD case, the saturation is due to a balance in Eq.~(3.1), so $\tilde{b}\sim\tilde{v}$ and the fields are in equipartition. Also the odd axial harmonics ($k_{z}=2\pi/H,~6\pi/H,\ldots$) turn out to be stable in the saturated Hall MHD state; they are decreasing exponentially in time.
The saturated magnetic field in Hall MHD is not much different from the originally applied vertical field, as should be expected from consideration of energy evolution plots. However, in the Hall MHD saturated state strong (up to 2 km s$^{-1}$) equatorial jets are produced. Because of lower $Re$ than similar liquid metal MRI experiments, we expect parasitic flows to very minimal, thus allowing this radial jet to be very pronounced. Using either mach probes or the Fabry-Perot system (which is theoretically capable of $\sim10$ m s$^{-1}$ velocity resolution), these strong radial jets will be easily diagnosed, providing evidence of the MRI in PCX-U. Using this signature, onset parameters such as the magnitude of the applied field could be swept to compare to predictions made by this stability analysis.
\section{Electrically Driven Flow}\label{sec:EDF}
As an alternative to boundary driven TCF, which is greatly affected by the presence of neutrals via the neutral drag body force, we present a scheme for driving flows in PCX-U using a body ${\bf J}\times{\bf B}$ force. This flow scheme, so called electrically driven flow (EDF), is driven by drawing a radial current across the small axial magnetic field provided by external Helmholtz coils.
EDF in a cylindrical volume produces a $v_{\phi}\propto1/r$ profile in the bulk of the plasma. As opposed to boundary driven flows, the EDF profile shape is unchanged for different driving parameters (total current and ${\bf B}_{0}$). The peak velocity of this profiles is set by the total current that can be drawn radially across the magnetic field. These EDF $1/r$ profiles (as discussed in Sec.~\ref{sec:Stab_results}) are hydrodynamically stable under the Rayleigh criterion when viscosity is included and meet the ideal-MRI conditions, thus constitute a good flow for MRI studies.
In PCX-U, the Helmholtz coil supplies an axial magnetic field while a single cathode or anode can be placed in the center of the vessel and biased to electrodes on the outer wall to draw the cross field current. The simplicity of this system is experimentally very attractive because an inner boundary electrode assembly is not required as in TCF.
In order to drive EDF, the current output of PCX cathodes will be increased by replacing the emitting material with lanthanum hexaboride (LaB6). This material has an extremely low work function and when heated is a superb electron emitter, thus making it an ideal candidate for emissive plasma cathodes. LaB6 cathodes used in MPDX routinely draw the maximum current of power supplies ($\sim100$ A), which represents a two order of magnitude increase over the tungsten PCX cathodes \citep{MPDXPOP}.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure11.pdf}}
\caption{Equilibrium electrically driven flow in single fluid MHD: (a) angular momentum profile, (b) current lines in $r-z$ plane. Narrow Hartmann layers are visible near the top and bottom endcaps. Calculations are done for a helium plasma with density $n=10^{12}$ cm$^{-3}$, electron temperature $T_e=12$ eV, ion temperature $T_i=0.4$ eV, total radial current $I_0=100$ A and axial magnetic field $B_0=10$ Gauss (corresponding to $Re=18$, $Rm=10$, $\emph{Ha}=385$).}
\label{fig:Figure11}
\end{figure}
\subsection{Equilibrium}
Analysis of EDF in relation to the MRI has been presented in various parameter regimes using dissipative single-fluid MHD \citep{Noguchi_2003, Khalzov_2006, IvanDeanStab}. In this analysis Hall and neutral collision terms are included to reflect the regime of weakly-ionized sparse plasmas. As an initial step, the axisymmetric equilibrium state ($\partial/\partial t\rightarrow0$) described by the system of equations (\ref{eq:MHD}-\ref{eq:MHD_end}) is found assuming perfectly conducting inner and outer cylinders, insulating top and bottom ends, and no-slip boundary conditions for velocity at the walls. The system is discretized in the two-dimensional $r$-$z$ domain and solved by an iterative method outlined in \citep{IvanDeanStab}.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure12.pdf}}
\caption{The radial profiles of (a) azimuthal velocity, (b) angular frequency, and (c) angular momentum of electrically driven flow showing the effects of neutral drag and the Hall term. Parameters are the same as in Fig.~\ref{fig:Figure11}.}
\label{fig:Figure12}
\end{figure}
Several key features are present in the equilibrium EDF profiles as shown in Fig. \ref{fig:Figure11}. First, there are thin Hartmann layers near the top and bottom ends, which scale like $1/\emph{Ha}$ (where $Ha\equiv\sqrt{RmRe}/M_{A}$) and parallel layers near side walls, which scale like $1/\emph{Ha}^{1/2}$ \citep{IvanDeanStab}. The poloidal current and associated toroidal field are localized in these boundary layers, which are narrow if $\emph{Ha}\gg1$ and, thus, their presence can be neglected in the following stability analysis. The rest of the plasma rotates with azimuthal velocity $v_\phi\propto1/r$ and is practically current-free.
Second, since the driving ${\bf J}\times{\bf B}$ force for EDF is a body force applied across the whole profile, the neutral drag body force acts only to uniformly lower the magnitude of the profile. Increased neutral density no longer results in a qualitative change to the shape of the profile like it does for TCF. Because of the fixed profile shape, the bulk EDF is guaranteed to meet the Rayleigh criterion for any finite viscosity. This effectively removes the hydrodynamic instability limit encountered in stability analysis for TCF. Additionally, the Hall term acts to uniformly increase the magnitude of the profile (at fixed total radial current) and does not affect the radial profile shape. This is illustrated in Fig. \ref{fig:Figure12}.
Third, as shown in \citep{IvanDeanStab}, the magnitude of the secondary flows (poloidal circulation) scales as $Re/\emph{Ha}^2$. At large Hartmann numbers these flows become small, therefore they can be neglected in the stability analysis along with the small boundary layers. Additionally it is assumed that the plasma rotates in the uniform axial field $B_0$ with azimuthal velocity $V_\phi(r)=b/r$, where amplitude $b$ is determined from the equilibrium solver (it depends on total current $I_0$, neutral drag and Hall effect).
Note that the neglected boundary layers may lead to some additional parasitic instabilities, especially the layer near the outer wall where the angular momentum is decreasing (and therefore violating the Rayleigh criterion). For large $\emph{Ha}$, these layers will be small and localized to a region of lower viscosity in the magnetized cusp region. It is expected that any instability here would not couple into the bulk flow. For completeness, these boundary layers can be removed experimentally by adjusting the positions of the near-wall electrodes so that in addition to bulk forcing, they produce a local drive in the cusp region (like in the TCF) and equalize the boundary and bulk angular momenta.
\subsection{Stability}
The stability of the equilibrium flow with $V_\phi(r)=b/r$ in an uniform axial field $B_0$ is studied by solving the eigenvalue problem resulting from the linearized system (3.5-3.6). The following results are obtained for singly ionized helium plasma with density $n=10^{12}$ cm$^{-3}$, electron temperature $T_e=12$ eV, ion temperature $T_i=0.4$ eV, and neutral pressure $P_0=10^{-5}$ torr in a cylindrical volume with dimensions $R_1=0.1$ m, $R_2=0.4$ m, $H=0.8$ m. This gives dimensionless parameters, $\mu=0.95$ and $\delta=4.55$.
The MRI boundary as a function of external field $B_0$ and total current $I_0$ is given in Fig. \ref{fig:Figure13}. The familiar Hall effect is also present in the electrically driven flow where the axis of the flow must be antiparallel to the axial magnetic field to obtain positive MRI growth rates. This means that current must flow from the inner to outer boundary of the plasma. Experimentally, this means that cathodes at the outer edge must be negatively biased with respect to an anode close to ground placed in the center.
Figure \ref{fig:Figure13} also demonstrates that several modes with different $k_z$ will be excited almost simultaneously when the MRI threshold is reached at total radial currents of $\sim100$ A. Such a multi-mode instability may lead to a fast development of turbulence, which is another intriguing objective for the proposed plasma MRI experiment since in astrophysical systems it is MRI turbulence which leads to enhanced angular momentum transport ultimately.
\begin{figure}
\centerline{\includegraphics[scale=1]{Figure13.pdf}}
\caption{(a) MRI regions in electrically driven flow for modes with different axial wave-numbers $k_z$. Dashed lines of the same color denote stability boundaries for modes with the same $k_z$ but different radial mode numbers. (b) Azimuthal velocity at the inner wall. Plasma parameters are listed in the text. Negative sign of $I_0$ corresponds to current flowing from the inner to outer wall. }
\label{fig:Figure13}
\end{figure}
\section{Summary and Future Work}
The prospects for observing the MRI in PCX are good. In this paper, the effects of plasma dynamics pertinent to the PCX regime on the MRI have been studied via a global stability analysis. From this study, an experimentally achievable regime has been found where the MRI should be excited in a laboratory plasma. Experimental considerations have led to an upgrade to put PCX in this regime that is currently underway. Finally, so-called electrically driven flow (EDF) is considered as an alternative to boundary driven Taylor-Couette flow and is shown to lead to the MRI with PCX-U parameters.
A global stability analysis for weakly-ionized sparse plasmas has highlighted the combined effects of ion-neutral collisions and the Hall term on the MRI. As previously noted, the Hall term leads to a requirement that the seed magnetic field be antiparallel to the axis of rotation for positive MRI growth rates. Ion-neutral collisions effectively add a body drag force to the momentum equation. This drag quantitatively changes the profile shape of boundary driven flows and leads to a small region in parameter space where flows are unstable to the MRI while maintaining hydrodynamic stability.
Finally, an experimental configuration for EDF provides an alternative to boundary driven Taylor-Couette flow that is experimentally attractive due to its simplicity and has a fixed profile shape that is unaffected by the presence of neutrals. EDF can be easily realized in PCX-U with high power LaB6 cathodes capable of driving high $Rm$ flows. Preliminary global stability analysis of EDF profiles shows that the MRI and possibly MRI turbulence can be excited at experimentally achievable parameters. It should be noted that Keplerian flows are body force driven (as is EDF), such that many of the difficulties associated with exciting the MRI in boundary driven flows (such as the neutral drag) are not present in the true astrophysical systems under study. In this way using EDF to study the MRI removes details that are not present in real accretion disks.
We emphasize again that the presented results of the MRI analysis are based on numerical modeling with a number of simplifying assumptions. In particular, we avoid complications related to the presence of top and bottom end-caps by assuming axial periodicity of the system, we simplify the boundary conditions of the magnetized cusp, and we neglect the small boundary layers in EDF and electrical currents flowing in them. We recognize that these are important issues and plan to address them in more detail in future work, especially with respect to determining the proper boundary conditions for the magnetized cusp region. Despite these simplifying assumptions, we believe all the analysis in this paper provides good estimates for MRI onset in PCX that can help to inform the design of PCX-U and other plasma MRI experiments. \\
This work was funded in part by the National Science Foundation (NSF) and the Center for Magnetic Self Organization in Laboratory and Astrophysical Plasmas (CMSO).
\bibliographystyle{jpp}
|
2,877,628,090,757 | arxiv | \section{Introduction}
Correctly capturing key patterns and relationships in economic history often requires the measurement of entire distributions of variables. For example, consider human demographics. Stunting, or very low heights owing to impaired growth in childhood, is associated with lower cognitive ability in later life, poorer health during adulthood, and reduced labour market earnings \citep{JayachandranPande2017,fogel2004,floudetal2011}. And while stature is known to be broadly associated with health and other measures of well-being in many populations and time-periods \citep{Costa2004,VOGL201484,Deaton13232}, these relationships are not constant over the entire range of heights, nor even montonic in all populations. Thus, if one wishes to evaluate some historical event as an input to height or other demographic measures, it is necessary to determine how the event has impacted heights across the entire distribution. Similar phenomena are encountered in many applications in economic history, such as economic growth and its impacts on inequality in income distributions, determinants of historical salary distributions, and the study of mortality declines in higher versus lower mortality settings. In each case, the distributional effects of policies or events can be as important as their mean effects in the population under study.
In this article we seek to motivate the importance of considering the distributional impacts of historical events, and specifically survey a broad series of methods which are ideally designed for such analyses. In particular, we lay out a range of methods related to quantile analyses, such as quantile regression, and the estimation of quantile treatment effects. These methods have emerged in a long line of theoretical and computational advances in the econometric literature, first fully described in \citet{KoenkerBassett1978}, and comprehensively discussed in a range of papers or books since \citep{Koenker2017,KoenkerHallock2001,Lamarche2019,koenker_2005}. While four decades have passed since the publication of \citet{KoenkerBassett1978}'s seminal paper on quantile regression, this is still an area of active research, in particular with recent advances considering quantile treatment effects, and identification in broader circumstances.
Despite being well-suited to applications in economic history where dependent variables of interest are often continuously distributed, these are arguably under-utilised in empirical applications. As well as providing an overview of these methods including recent extensions which are likely to be of particular interest to practitioners in economic history, we provide a survey of their usage in economic history, an analysis of their use, and \emph{potential} for their use in papers published over the past 30 years in the field of economic history. In a survey across all principal economic history journals we find around 50 papers that have used quantile regression in some way. However, in a deeper analysis of all the papers published in the journal \emph{Explorations in Economic History}, we find that while around 60\% of papers are based on quantitative analyses of continuous dependent variables and thus potentially suited to analysis using quantile regression or related methods, only around 1\% of papers actually apply these tools to consider effects beyond the mean.
This article thus seeks to motivate the ``use'' of quantile analyses in economic history in two ways. The first is to describe how they can be productively used by practitioners in empirical studies to capture relationships of interest which may not be gleaned from simple mean or other average estimators. And the second is to document how they have been used (and arguably under-used) in the literature on economic history up to this point.
In what remains of this paper, we first provide a summary of key methods for distributional analyses in empirical methods in section \ref{scn:theory}. In section \ref{scn:hist} we provide both an overview of papers based on quantile analysis methods in economic history, as well as a review of all papers in a highly cited economic history journal, and their suitability for methods of this type. Finally, in section \ref{scn:example} we document a specific example based on microdata on human height covering around 20,000 individuals exposed to divergent rates of economic growth throughout their life. In closing, we make a number of points related to the computational implementation of these methods.
\section{Quantile Regression}
\label{scn:theory}
Generally, in empirical applications one wishes to consider the impact of some group of independent variables $x_k$ for $k\in\{1,\ldots,K\}$ on a specific dependent variable, denoted $y$. Where observations $i$ refer to units (such as individuals), this is often parameterised using a linear regression model:
\begin{equation}
\label{eqn:linMod}
y_i = \beta_1 + \beta_2x_1 + \ldots + \beta_Kx_K + u_i,
\end{equation}
where $u_i$ is an unobserved error term. Frequently, and indeed, in a the majority of papers in economic history (see Section \ref{scn:hist}) estimation is implemented using the ordinary least squares (OLS) method. Mathematically, this procedure simply consists of finding the $K$ parameter estimates of $\beta$ which minimize the following problem:
\[
\widehat\beta=\arg\min_{\beta\in\mathbb{R}^K} \sum_{i=1}^N(y_i-\bm{x_i}^\prime\beta)^2,
\]
where $\bm{x_i}$ refers to the vector of values of each the independent variables $x_k$ for unit $i$. This optimization returns a vector of parameters capturing the mean impacts of each $x_k$ on the outcome of interest $y$. While mean impacts across the distribution of $y_i$ may be a logical summary parameter in many cases, it is not the only point with meaningful empirical content. Often, other points of the distribution of $y$ may be as, or more important, than the mean, particularly in historical processes where extreme outcomes may be of particular interest. This suggests the importance of modelling options allowing for additional heterogeneity.\footnote{Heterogeneity can be observed for a number of reasons in empirical studies. In this paper we are interested in heterogeneity in the impacts of some independent variable(s) over the full distribution of a dependent variable of interest. A useful alternative example of the interest of heterogeneity in economic history is provided by \citet{BisinMoro2020} who discuss heterogeneity in the context of differential take-up of some treatment, and resulting sample-based heterogeneity due to the estimation of Local Average Treatment Efects (LATEs).} Below we describe a range of quantile estimation techniques which allow for focus across any points of the distribution of outcomes of interest.
\subsection{The Linear Quantile Regression Model}
Quantile regression considers the full distribution of some dependent variable $y$. Thus, at a minimum, $y$ requires a distribution with considerable variation, and is inappropriate in cases where $y$ is a binary or categorical measure. We will denote as $\tau$ the quantiles of the distribution, such that (for example) $\tau=0.5$ indicates the median of the distribution, and $\tau=0.1$ indicates the 10\textsuperscript{th} percentile, or the point of the distribution below which 10\% of the observations of $y$ are observed. As originally documented in \citet{KoenkerBassett1978}, we can estimate the parameter vector $\beta(\tau)$ capturing the impact of each $x_k$ on the $\tau$\textsuperscript{th} quantile of $y$ based on the following minimization problem:
\begin{equation}
\label{eqn:qreg}
\widehat\beta(\tau)=\arg\min_{\beta\in\mathbb{R}^k}\left[\sum_{i:y_i\geq \bm{x_i}^\prime\beta} \tau\left|y_i-\bm{x_i}^\prime\beta\right|+\sum_{i:y_i<\bm{x_i}^\prime\beta} (1-\tau)\left|y_i-\bm{x_i}^\prime\beta\right| \right].
\end{equation}
Note that here, estimation is based on absolute deviations of $\bm{x_i}^\prime\beta$ from $y_i$ rather than quadratic distances in OLS, and indeed, in the case that $\tau=0.5$ (the median) this formula collapses to the Least Absolute Deviations estimator. In all cases except for the median, this minimization problem uses the quantity $\tau$ to `tilt' the estimates towards data which is lower or higher in the distribution of $y$, as can be observed in the two terms within the parentheses in equation \ref{eqn:qreg}: when $\tau$ is between 0 and 0.5, more weight is given to the right-hand summation for units $i$ whose $y_i$ is less than the conditional mean $\bm{x_i}^\prime\beta$, whereas when $\tau$ is between 0.5 and 1, more weight is assigned to the left-hand summation considering units whose $y_i$ is greater than the conditional mean.
Equation \ref{eqn:qreg} is the well-known linear quantile regression model which is implemented as standard in many computational languages. Frequently, rather than focusing on a particular quantile of interest, estimates $\widehat\beta(\tau)$ are documented over a range of quantiles in a graphical manner (refer for example to the illustration in section \ref{scn:example} of this paper. What's more, with modern computational tools, standard errors can be calculated quite simply along the range of the distribution of $y$, which permits formal hypothesis tests and other inferential procedures at each quantile considered. Formally, following notation from \citet{Lamarche2019}, the approximate distribution of the vector $\widehat\beta(\tau)$ can be written as: \[
\widehat\beta(\tau)\sim \mathcal{N}\left(\beta,\frac{1}{N}\tau(1-\tau)H^{-1}JH^{-1}\right),
\]
where $\mathcal{N}(\cdot)$ refers to a Gaussian distribution and the matrices $J=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^Nx_ix_i^\prime$, and $H = \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^Nf_i(x_i^\prime\beta(\tau))x_ix_i^\prime$, with $f_i(\cdot)$ being the conditional density function of $y$. Note here two key implications: firstly, that this inference holds asymptotically, ie as the sample size grows, and secondly that a key `ingredient' is the estimate of the density $f_i(x_i^\prime\beta(\tau))$ at quintile $\tau$. If one is willing to assume that error terms are identical and independently distributed (iid), the $f_i$ term is identical among all units $i$, and estimation is simplified, for example using the fitted value of the density at the quantile of interest.\footnote{There are also other reasonable approches generally included as standard in statistical software, such as using a kernel density estimator or residual quantile function to estimate the density at quantile $\tau$. These are all available in programs such as Stata with the \texttt{qreg} program, or R with the \texttt{quantreg} library.} However, if heteroscedasticity robust estimates are desired, more complex `sandwich' estimates are necessary. While these are also implemented as standard in computational routines in languages such as Stata or R, further background on these procedures can be found in \citet[chapter 3]{koenker_2005}. Alternative options which avoid the estimation of these matrices consist of using bootstrap resampling methods for inference. A review of such resampling methods in quantile regression is provided by \citet{He2018}. Finally, note that solutions have also been proposed to resolve cases where clustered inference is desired, allowing for correlated shocks within groups, as well as heteroscedasticity \citep{Hagemann2017}, once again based on bootstrap resampling procedures.
\subsection{Quantile Treatment Effects}
\label{sscn:QTE}
While this quantile regression can be used for $x_k$ of arbitrary forms and dimensions, a particular case of interest is that of treatment effect models, with some binary `treatment' of interest, which we denote $D_i$. Note that while quantile regression requires continuous distributions for the outcome variable of interest, there is no limit on the nature of independent variables $x_k$, including binary and categorical measures. The standard average treatment effect aims to identify the mean impact of receiving treatment (versus not receiving treatment) on dependent variable $y$:
$\Delta=E(y_i|D=1)-E(y_i|D=0)$. This assumes independence of $D$ from unobservables. Extending this average treatment effect to its quantile treatment effect (QTE) analogue gives.
\begin{equation}
\label{eqn:QTW}
\Delta(\tau)=Q_y(\tau|D=1)-Q_y(\tau|D=0),
\end{equation}
where $Q_y(\tau)$ refers to the value of $y$ at a particular quantile $\tau$ of the distribution of $y$.\footnote{Formally, this is $Q_y(\tau)=\inf\{y:F(y)\geq\tau\}$, where $F(\cdot)$ refers to the cumulative density function of $y$.} Such an estimate, beyond the mean is likely to be relevant for a range of historical policies, for example examining whether exposure to a specific type of economic or political system, exposure to a historical environmental shock, or exposure to a historical policy has divergent impacts across the distribution of individual outcomes capturing well-being. All such examples refer to binary `treatment' statuses, and hence are potentially appropriate for QTE methods and their extensions discussed in the following sub-section, provided that outcomes of interest $y$ are continuous measures. In practice, these estimates can be generated using regression following the procedure laid out in the previous section. We return to discuss computational implementations in the final section of this paper.
This setting can be extended considerably, for example to include covariates, and/or to explicitly consider selection into treatment. This can be done in a number of ways, such as by attempting to correct for selection non-parametrically \citep{Bitleretal2006,Firpo2007}, or extending to panel data settings and using recent methods such as difference-in-differences \citep{CallawayLi2019}.
A particular case where QTE methods are extended to deal with endogeneity is laid out in the following sub-section when discussing instrumental variables and Local QTEs.
While the inclusion of covariates can complicate the QTE setup \citep{Koenker2017}, suggestions on how to deal with this have been proposed in \citet{Firpo2007,FrolichMelly2010,Callaway2018} based on propensity score methods and re-weighting techniques. For applied work, a particularly useful reference discussing a range of practical estimation methods for quantile treatment effects is the paper of \citet{FrolichMelly2010}.
\subsection{Quantile Regression Extensions}
While standard quantile regression and QTE implementations can provide illustrative results which allow for an understanding of effects beyond the mean, recent theoretical and computation advances in these methods offer considerable additional benefits, including the possibility to address a number of issues in standard quantile regression models, and more explicitly address questions of causality. We discuss the broad scope of these recent advances here, providing key references. Full modeling considerations are available in the papers discussed below, however particularly useful overviews of these methods at more length can also be found in the survey articles or handbook chapters of \citet{Koenker2017} (discussing all below points), or \citet{Wei2017,Lamarche2019,MellyWuthrich2018,Chernozhukovetal2018} with focuses on particular points.
\paragraph{Models to Correct for Measurement Error}
In the case of variables collected over a considerable time-frame from diverse data sources as is often the case in studies in economic history, measurement error or issues of sample selection are valid concerns. Fortunately, a number of methods suggest ways that measurement error can be accounted for within the framework of quantile regression (under assumptions about the nature of these errors). For example, \citet{WeiCarroll2009} document potential estimation methods when in the presence of mesurement error in independent variables, and alternative methods are proposed by \citet{Wangetal2012}. The crux of these models is that if rather than observing a true variable $x$ we observe some noisy proxy, we must consider the conditional \emph{expectation} of $x$ based on the noisy proxy, rather than simply using the proxy in the regression. Additionally, even if there is limited error in the measurement of variables, samples may be selected in certain ways, for example if only certain types of records survive following collection of historical data, or if data are based on suriving individuals. \citet{ArrellanoBonhomme2017} propose methods for quantile regression under selection based on an exclusion restriction for estimation, while \citet{Blancoetal2013} (see also \citet{Lamarche2019}) propose partial identification methods as way ahead in such circumstances. Computational implementations of such selection models are available, for example via the programs of \citet{BE2020,Siravegna2020}. Interested readers are pointed to the handbook chapter of \citet{Wei2017} which provides a deeper discussion of these issues.
\paragraph{Endogeneity, Instrumental Variables and Causality} In many settings of interest in economic history, independent variables of interest will be endogenous---corrleated with unobserved factors which are themselves correlated with outcomes of interest. As is the case in standard econometric models, quantile regressions or QTE models do not allow for causal inference when in the presence of endogeneity. However, there is a rich stream of work which seeks to extend standard linear instrumental variable (IV) models into a quantile framework. Here in particular we discuss the Local Quantile Treatment Effect (LQTE) approach, which, under a number of key assumptions provides consistent (causal) estimates for certain sub-groups of the population across the entire distribution of the outcome variable.
The LQTE framework extends the well-known Local Average Treatment Effect (LATE) framework laid out in \citet{ImbensAngrist1994}. Here, causal estimation using an instrumental variable is based on a monotonocity assumption that requires that the instrumental variable induces exogenous variation in the dependent variable of interest which shifts all individuals weakly in the same direction, for example acting as an as-good-as-randomly assigned incentive or disincentive to uptake an endogenous variable of interest. Here we summarise this and the other identifying assumptions, as well as how an LQTE is estimated, with a binary instrumental variable denoted $Z$. A chapter length discussion of these methods, as well as extensions to other circumstances is provided by \citet{MellyWuthrich2018}. Following the notation of section \ref{sscn:QTE}, consider an outcome variable of interest $y$, a binary treatment variable of interest $D$, as well as the instrumental variable $Z$. Under 5 assumptions; instrumental independence (or validity), the exclusion restriction, instrumental relevance, monotonicity and the stable unit treatment value assumption (SUTVA) limiting interaction between treatment statuses, \citet{ImbensAngrist1994} prove that the following Wald estimator gives the LATE:
\[
\frac{E(y_i|Z_i=1)-E(y_i|Z_i=0)}{E(y_i|D_i=1)-E(y_i|D_i=0)}
\]
Note that this LATE is a consistent estimate of the impact of treatment ($D_i$) on outcomes ($y_i$) for those individuals whose treatment status would be changed by the instrument (the `compliers').
In order to extend this framework to a quantile framework, distributional results are provided by \citet{ImbensRubin1997,Abadie2002}. These give cumulative density functions specifically for the compliers, in this case which we denote $F^c_{Y_1}$ and $F^c_{Y_0}$. Here superscript $c$ refers to the population of compliers, and $F$ refers to the cumulative density of outcomes for cases where treatment is received ($F^c_{Y_1}$) or not received ($F^c_{Y_0}$).\footnote{In extensive form, these can be represented \citep{Abadie2002} as:
\[
F^c_{Y_1}(y)=\frac{E[\mathbbm{1}(Y\leq y)D|Z=1]-E[\mathbbm{1}(Y\leq y)D|Z=0]}{E(D|Z=1)-E(D|Z=0)},
\]
and
\[
F^c_{Y_0}(y)=\frac{E[\mathbbm{1}(Y\leq y)(1-D)|Z=1]-E[\mathbbm{1}(Y\leq y)(1-D)|Z=0]}{E(1-D|Z=1)-E(1-D|Z=0)}.
\]
}
Given these definitions based on an IV and compliers, it is very easy to return to the notation from section \ref{sscn:QTE}, which gives the LQTE estimator. As a clear parallel to equation \ref{eqn:QTW}, this is defined as
\begin{equation}
\label{eqn:LQTE}
\Delta^c(\tau)=Q^c_{Y_1}(\tau)-Q^c_{Y_0}(\tau),
\end{equation}
where as previously, $\tau$ refers to quantiles of the relevant conditional density function above.\footnote{In the interests of completeness, this is:
\begin{eqnarray}
Q^c_{Y_1}(\tau)&=&\inf\{y:F^c_{Y_1}(y)\geq\tau\} \nonumber \\
Q^c_{Y_0}(\tau)&=&\inf\{y:F^c_{Y_0}(y)\geq\tau\} \nonumber
\end{eqnarray}
where all notation follows that of section \ref{sscn:QTE}.}
It is worth noting that these methods, while more demanding than standard QTE models given the required conditions of instrumental validity, are potentially well suited to applications in economic history where concerns exist relating to endogeneity. For example, \citet{Aaronsonetal2020} estimate the impacts of the number of children born to women -- a clearly endogenous variable -- on maternal labour supply, using data over more than two centuries, by leveraging the birth of twins as an IV. In this case monotonicity assumptions are very likely met given that twin births should have weakly positive impacts on completed fertility.\footnote{Note however that in general, the assumptions necessary for identification with instruments are not trivial. \citet{BhalotraClarke2019} discuss a number of considerations related to these instruments in a standard linear model. An entirely different take on IV style models which precluded the LQTE approach described here and allows IVs to generate variation locally in endogenous variables is the work of \citet{Chesher2005,MaKoenker2006}. This suggests productive ways forward in a quantile framework even if instruments generate shifts at only specific points of endogenous variables of interest.} Very recent work by \citet{ValenciaCaicedo2021} provides considerable other examples of the use of IV in economic history (though not discussing quantile methods), describing among other examples IVs based on map borders or other geographical features such as rivers, or slopes of terrain, geographic suitability indexes, or Bartik-type instruments. All such instruments in economic history could be productively introduced into a quantile framework if distributional outcomes are considered.
Finally, prior to turning to applied examples of the use of quantile regression, we note that this particular implementation of IV and endogeneity corrections via LQTE methods is not the only way forward. Regression discontinuity designs can be similarly cast in this ``Local'' framework, while the instrumental variable quantile regression models (IVQR) of \citet{ChernozhukovHansen2005} provides estimation methods which return average Quantile Treatment Effects (rather than LQTEs), but requiring alternative assumptions.\footnote{In particular, these methods require a rank preservation assumption, restricting the ranks which individuals can take in terms of the ordering of the outcome variable to be the same across differing potential IV assignments.} These methods are further discussed in \citet{MellyWuthrich2018} and \citet{Chernozhukovetal2018} respectively, and are likely to more productive ways forward when non-binary endogenous variables are considered--in which case LQTE models are less appropriate--or in cases when broader \emph{non}-local QTEs are desired, at the cost of alternative assumptions. A valuable review of these methods is provided in \citet{ChernozhukovHansen2013}.
\paragraph{Other Extensions and Methods} A range of other contexts which are potentially of use in quantitative studies of economic history can be productively studied in quantile settings. This includes settings such as longitudinal or panel data and difference-in-difference models, regression discontinuity designs, and non-parametric analyses. We briefly discuss these settings in turn below, pointing interested readers to relevant references.
Extensions of quantile regressions to panel data following individuals over time have been proposed in \citet{KOENKER2004} where challenges arise given desires to estimate movements across distributions \emph{within} individuals (or panel units) with potentially few repeated individual level data-points. \citet{KOENKER2004} proposes using a penalized estimator to control for relevant individual-level effects, and much additional work has been conducted to take forward these techniques (a complete discussion is provided in \citet[pp. 11-14]{Lamarche2019}). Recent work of \citet{GU201968} has suggested using a grouped fixed effect approach, also based on penalized estimators to group similar individuals in a panel setting. Alternative lines of work, such as \citet{ArellanoBonhomme2016}, suggest viewing (potential individual-level) variation as a problem of unobserved heterogeneity and estimating using an iterative process, super-imposing simulation based estimation procedures on top of standard quantile regression procedures. A particular setting of interest in cases of longitudinal data consists of the estimation of difference-in-differences models where exposure to some treatment varies within a panel over time such that baseline differences between exposed and unexposed individuals can be captured using pre-treatment periods. These models have been extended to a quantile setting, see for example \citet{CALLAWAY2018395} for a setting with two time periods and \citet{CallawayTong2019} for a broader panel setting. Computational software is also available to implement these methods in \citet{Callaway2019}
The regression discontinuity design provides credible identification in cases where some dependent variable of interest is moved discontinuously by some arbitrary cut-off. The use of these methods in economic history has been surveyed by \citet{ValenciaCaicedo2021}, where examples are often based on distance to geographical features or map boundaries. The regression discontinuity design has been extended to a quantile framework in a very flexible way by \citet{FRANDSEN2012382}, which allows for the estimation of quantile treatment effects `local' to a particular discontinuity, and accompanying computational routines are available to implement this method.
Standard non-parametric regression implementations can capture fully flexible relationships between the mean of some variable $y$ and some independent variable $x$. Rather than parametric assumptions and linear functional forms (as imposed in equation \ref{eqn:linMod}, the relationship is allowed to vary freely along the support of $x$, allowing for $x$ to have a non-linear impact on the mean of $y$. This logic can be extended to a quantile regression, if rather than considering a non-parametric relationship between the mean of $y$ and $x$, a non-parametric relationship between specific \emph{quantiles} $\tau$ of $y$ and $x$ are estimated. This is a particularly flexible way to model heterogeneity which may be well-suited to historical outcomes over which there are few prior assumptions related to the nature of the relationship under study. A review of the methods is available in \citet[section 3]{Koenker2017}, while computational resources for the implementation of such routines are available in, among others, \citet{Koenker2021,Lipsitzetal2017}.
\section{Applications in Economic History}
\label{scn:hist}
\subsection{Quantile Regression in Economic History Research}
To have some idea about the extensiveness of the use of quantile regression in economic history, we begin by running a search within the main economic history journals, covering the period from 2000 until the present.\footnote{As we see in the following sub-section, there is even less use of quantile regression prior to 2000. Two notable exceptions are the studies of \citet{ConleyGalenson1998,ConleyGalenson1994} which provided early illustrations of the power of quantile regression in historical analyses.} Namely, we searched within the Journal of Economic History, Explorations in Economic History, Economic History Review, Cliometrica, and the European Review of Economic History. But we also complemented these searches with searches of other economic history journals as discussed below, estimating that around 50-55 journal articles have made use of quantile regression as an analytical tool during the last two decades, although only a handful of these are dated pre-2005.
The journal which has most frequently published articles employing this technique is the Journal of Economic History, which published 19 articles using quantile regressions for a wide range of topics from 2000, including: Max Weber's hypothesis on the role of Protestantism for economic development \citep{Kersting2020}; the market for paintings in Florence and Italy between 1285 and 1550 \citep{Etro2018}; the interaction between inequality and financial development in the US during the late nineteenth century \citep{Jaremski2018}; fluctuations in technology during the Great Depression in the US \citep{Watanabe2016}; the credibility of fixed exchange rates during the classical gold standard era \citep{Mitchener2015}; and inequality of wealth in the Ottoman Empire \citep{Cosgel2012}; just to mention those published after 2010. These few examples make clear the broad applicability of quantile methods to questions of interest in economic history, covering issues in micro, macro and financial economics.
Explorations in Economic History has also published articles using quantile regression based methods with some frequency. A scoping review identified six such article (see also the following sub-section). Namely, \citet{Walker2000}, which explores the degree of economic opportunities in San Francisco compared to other regions around mid-nineteenth century; \citet{Dupont2007}, which tests for contagion in bank runs in Kansas during the panic of 1893; \citet{Canaday2008}, which deals with the relationship between wealth and wealth accumulation by both blacks and whites in South Carolina between 1910 and 1919, and its determinants; \citet{Drelichman2014}, which reconstructs housing costs for various social groups and traces the effect of exogenous shocks on the rental market for Toledo, Spain, between 1489 and 1600; \citet{Alvarez2018}, which deals with the relationship between human capital and male labour earnings in eighteenth-century Spain; and \citet{Callaway2018}, which measures the union wage premium for several US-cities circa 1950, using unconditional quantile methods.
The Economic History Review, in turn, has published five articles where quantile regressions were used: \citet{Temin2008} analyses the cost and availability of private bank credit between 1702 and 1724; \citet{Gazeley2011}, in turn, estimates urban poverty among working families in the British Isles circa 1904; \citet{Brown2018} deals with the causes of fluctuations in infant mortality rates in Bavaria during the 1820s-1910s; \citet{Artunc2019} examines the composition of firm ownership and entrepreneurship in Egypt between 1910 and 1949; while \citet{Karagedikli2021} estimates real hedonic house prices and urban wealth inequality for the housing market between 1720 and 1814 in the Ottoman Empire. Note that as above, these articles show both a broad scope of themes, as well as a broad scope of geographic and temporal settings which have been productively analysed with these models.
The European Review of Economic History has also published five articles making use of quantile regression: \citet{Koepke2005}, which provides the first anthropometric estimates of the biological standard of living in Europe during the first millennium AD; \citet{Dincecco2009}, which performs a statistical analysis of political regimes and sovereign credit risk in Europe from 1750 to 1913; \citet{Dribe2009}, who analyses the importance of demand and supply factors in the Swedish fertility transition between 1880 and 1930; \citet{Kholodilin2016}, that analyses the housing rental dynamics of Berlin during World War I; and more recently, \citet{Jorge-Sotelo2020} focused on the impact of currency depreciation on international capital flows in Spain between 1928 and 1931 crisis.
Cliometrica has published six articles using quantile regressions, the first of these less than a decade ago: \citet{Carson2012}, compared body mass index values of late 19th- and early 20th-century amongst African-Americans groups (i.e. blacks versus mulatto); \citet{Ogasawara2015}, which deals with the impact of social workers on reducing infant mortality rates in inter-war Tokyo; \citet{DuPlessis2015}, who for the period 1700-1725, estimated hedonic slave price indices and the value of their marginal productivity; \citet{GonzalezVal2017}, who analysed the impact of market potential on the structure and growth of some Spanish cities during 1860–1960; \citet{Ogasawara2019}, which deals with the treatment effects of piped on diseases in industrializing Japan (1920s-1930s); and finally, \citet{Keywood2020}, who tests the relationship between elite numeracy and elite violence in Europe from 500 to 1900.
We also run searches in other economic history journals, but where quantile regression was observed to be very infrequently used. For example, in the main Spanish economic history journal, Revista de Historia Economica -- Journal of Iberian and Latin American Economic History, quantile regressions were used in only four papers; in the Australian Economic History Review in only one article; in the Economic History of Developing Regions, similarly in only one article; while in the Scandinavian Economic History Review it has appeared twice. Finally, in the three main business history journals (Business History, Business History Review and Enterprise \& Society), it was used in three articles, twice at Business History and once at Enterprise \& Society, although this is rather unsurprising since this sub-discipline cultivates a less quantitative approach to history.
\subsection{The Potential for Quantile Regression Methods in Economic History}
Reviewing cases where quantile regression \emph{is} used suggests the broad applicability of these methods across themes and settings in economic history, but does not indicate the additional scope for use in papers which focus principally on mean effects. To see the \emph{potential} for Quantile Regression, we carried out a review based on all papers published over the last 30 years in a specific economic history journal (Explorations in Economic History, hereafter EEH). This allows us to gain a more complete picture of trends in published research in a specific highly-cited economic history journal, consider the methods used, and the potential for distributional analyses in economic history. In this review, we read each paper published over the last 3 decades (the first volume of 1990 to the last volume of 2020), and classified each paper according to whether it contained empirical methods, or was theoretical. In the case of empirical papers, we then classified these by method (OLS, Logit, Quantile Regression, and so forth) and finally whether principal dependent and independent variables are continuous, discrete, ordinal, etc.
\begin{figure}[htpb!]
\caption{Trends in the dependent variable in papers published in \emph{Explorations in Economic History}}
\label{fig:area}
\subfloat[Large groups\label{fig:aggregated}]{%
\includegraphics[scale=0.57]{./results/grouped_dep_var.eps}%
}
\subfloat[Small groups\label{fig:disaggregated}]{%
\includegraphics[scale=0.57]{./results/group_dep_var.eps}%
}
\floatfoot{\textsc{Notes:} Based on our reading of each paper published in EEH we classify each paper as belonging to a particular (single) area, as described in the legend in panel (b). These classes are further aggregated in panel (a), where `Macroeconomic Measures' refers to Growth/Policy, National Accounting and Production; `Human Capital/Well Being' refers to Health/Welfare, Household/Education and Labour Market; `Social Change' refers to Population/Migration, War/Crime and Votes; and Finance and Firms refers to Finance and Firms/Prices. A file summarising all papers in terms of a number of characteristics discussed in this review is provided as an online appenix to this paper.}
\end{figure}
Prior to considering trends in measures and methods, it is useful to provide descriptive evidence of the key themes studied over the past 30 years in this particular journal. In Figure \ref{fig:area}, we plot the broad areas which papers study, as classified based on our reading of the literature. These classifications refer to the dependent variable under study in all empirical papers published in EEH. The right-hand plot classifies papers into one of four aggregate areas: macroeconomic measures, human capital and well-being, social change (which includes papers studying topics such as conflict, migration, and changing political landscapes), and finance and firms. In general we note the largest of these groups is that related to micro-economic areas such as education, health and labour markets (the shaded light gray area), followed by themes related to finance and firms, with relatively less work focused principally on macroeconomic indicators. In the left-hand panel we provide more disaggregated classes, seeing more noisy patterns, and evidence of broad coverage of topics within economic history.
While it is illustrative to observe trends in the published record in economic history, here we are most interested in considering how the types of questions and measures used in these papers relate to their potential for using quantile methods. At minimum, this implies that dependent variables need to be continuous, given that quantile methods examine impacts of some independent variable across the distribution of the dependent variable.
Figure \ref{fig:trends} plots the number and share of papers published in \emph{EEH}, classifying them by whether they are empirical or not, and in the case of empirical papers, whether the dependent variable is continuous, and hence suitable for quantile analyses. Finally, we plot the number and share of papers that effectively do undertake quantile analyses. In Figure \ref{fig:trends} panel (a) we can see how the amount of total articles published in \emph{EEH} was around 20 annual publications, with an increase to around 30 from 2005 onwards. We can see how the number of empirical papers follows a similar trend to that of total articles, and corresponds to 93.1\% on average of the number of total articles. In recent years, this value is even higher, with 100\%, or close to 100\% of articles containing at least some considerable empirical portion, pointing to the growing frequency of empirical evidence in studies in economic history.
\begin{figure}[htpb!]
\caption{Trends in Publications in an Economic History Journal}
\label{fig:trends}
\subfloat[All Papers\label{fig:allPapers}]{%
\includegraphics[scale=0.57]{./results/papers.eps}%
}
\subfloat[Proportion of Papers\label{fig:propPapers}]{%
\includegraphics[scale=0.57]{./results/papers_prop.eps}%
}
\floatfoot{\textsc{Notes:} The total number of papers (panel A) and the proportion of all papers (panel B) refer to all papers in the journal EEH, based on the isssue in which they are published. This data was collected based on a reading of each paper and classification of the methods and `principal' dependent variables.}
\end{figure}
While this is illustrative of the importance of quantitative analyses in understanding historical processes in economics, it does not necessarily imply a suitability for analysis with quantile methods. However, if we observe the articles that are based on continuous dependent variables and hence amenable to quantile-based methods, we observe that they correspond to about 56.5\% of the quantity of total articles (over the past 30 years). Despite these two stylised facts: (i) that there is considerable quantitative evidence brought to bear in economic history papers, and (ii) that many of these analyses are based on variables which have continuous distributions and hence potentially interesting distributional features, we observe little use of quantile methods. Among the 738 papers examined, 692 of which were empirical, only 6 studies used quantile regression (as described in the previous sub-section). Unless there is a particular interest in the mean, these facts taken together point to a considerable margin for the adoption of other estimation methods such as quantile regression or QTE methods within economic history.
\begin{table}[htpb!]
\centering
\caption{Classification of Dependent and Independent variables}
\label{tab:codvar}
\scalebox{0.84}{
\begin{tabular}{llcccc}\toprule
&&\multicolumn{2}{c}{Dependent}&\multicolumn{2}{c}{Independent} \\
\cmidrule(r){3-4}\cmidrule(r){5-6}
& Type &Number&Proportion&Number&Proportion\\ \midrule
& Continuous & 418 & 76.70 & 207 & 37.91\\
& Binary & 58 & 10.64 & 47 & 8.61\\
& Discrete & 23 & 4.22 & 12 & 2.20\\
& Ordinal & 3 & 0.55 & 0 & 0\\
& Multiple & 10 & 1.83 & 115 & 21.06\\
& Continuous-Binary & 19 & 3.49& 77 & 14.10\\
& Continuous-Discrete & 9 & 1.65& 21 & 3.85 \\
& Continuous-Dummy & 0 & 0 & 25 & 4.58 \\
& Continuous-Ordinal & 1 & 0.18& 0 & 0 \\
& Dummy-Dummies & 0 & 0 & 30 & 5.49 \\
& Discrete-Binary & 4 & 0.73& 12 & 2.20 \\
\bottomrule
\multicolumn{6}{p{11.8cm}}{{\footnotesize Notes: Multiple classification is article with more than two types of variables. The classification that presents 2 types of variables, correspond to papers that present more than one regression in which its dependent variables are continuous plus one or more variables of another type, and both regressions are relevant in the research.}}
\end{tabular}}
\end{table}
Finally, before providing a simple illustrative example of patterns observable in quantile regressions in a particular study of historical economic processes, we note how these continuous dependent variables are situated within analyses. In Table \ref{tab:codvar} we provide a full break down of all the classes of dependent variables observed across all studies from EEH. 418 papers had a single continuous outcome variable of interest, while another 29 had a continuous outcome variable as well as other non-continuous measures. Of these 447 studies, 201 had independent variables which were also continuous, and so more suited to quantile regression methods, while 33 had a binary independent variable of interest, and so are potentially well suited to QTE methods. Many other studies have various types of independent variables, and as such either QTE or quantile regression methods may be appropriate.
\section{An Illustrative Example Based on 19\textsuperscript{th} Century Demographics and Economic Growth}
\label{scn:example}
As a brief illustration of the use of quantile regression, and how it can shed light on patterns across the distribution of outcomes which are hidden by standard analyses, we consider the empirical setting described in \citet{llorcajanaetal2019,llorcajanaetal2021}. As laid out there, we gathered information on all the 36,371 records of Military Personnel born in the 20\textsuperscript{th} century and 3,283 record of Military Personnel born in the 19\textsuperscript{th} century in Chile.\footnote{There are many historical precedents to the study of height in economic history, although less work extending to quantile analyses. Among many other references \citet{WachterTrussell1982} discuss historical measurement of height, though work goes back much further, for example Quetelet's discussion of classifying populations.} These records relate mainly to soldiers or low-ranking officers (the Army's Historical Archive). Full information on this process and these data are available in \citet{llorcajanaetal2019,llorcajanaetal2021}. Of those individuals, we generate a final database of the Chilean-born individuals aged between 17-55, which here we cross with rates of economic growth in the province in which the individual was born. These rates of economic growth are calculated from historical evidence collected by \citet{badiamiro2008}, which is the best sub-national evidence of economic conditions in Chile, available covering the periods of 1890-1950. From these data sources, we are able to combine a large micro-level sample of human height as well as measures of economic growth by decade and province of birth. Finally, we have a database of 17.293 individuals. The reduction of the database is explained given the data availability of growth rates (1890-1940).\footnote{We note that the sample reduction from the initial 36,371 digitized records owes to the availability of historical measures of growth. We calculate growth records in each decade as: \[\frac{GDPpc_{p,t+1}-GDPpc_{p,t}}{GDPpc_{p,t}},\] where $p$ indexes provinces and $t$ indicates decades, and as such this will not be defined in the decade of 1950, given that comparable records are not available for 1960. We thus limit the final estimation sample to all individuals born in the 60 years between the 1890s and 1940s.}
Descriptive plots of these data on height and exposure to economic growth are provided in Figure \ref{fig:hist}, suggesting considerable variation in observed heights, and also in changes in economic conditions within provinces over time.
\begin{figure}[htpb!]
\caption{Distributions of individual height and sub-national rates of historical growth}
\label{fig:hist}
\subfloat[Adult height in cm\label{fig:height}]{%
\includegraphics[scale=0.55]{./results/hist_height.eps}%
}
\subfloat[Growth rate over all decades\label{fig:growth}]{%
\includegraphics[scale=0.55]{./results/hist_growth.eps}%
}
\floatfoot{\textsc{Notes:} Histogram: Panel (a) distribution of adult male height in Chile, 1890s–1940s (in centimetres, 17,293 observations); Panel (b) distribution of growth rate in Chile, 1890s–1940s.\\ Source of data: Panel (a) \citet{llorcajanaetal2019,llorcajanaetal2021}, Panel (b) \citet{badiamiro2008}.}
\end{figure}
To document these methods, we consider a particular model seeking to determine the effect of economic growth during an individual's formative years on their adult height. Relationships between height and economic development have been discussed in the past, including over long periods, such as \citet{Peracchi2008}'s study of Italian height from the 1730s-1980s. Here, we are interested in a model of the following type:
\begin{equation}
\label{eqn:heightReg}
\text{Height}_{ipt} = \beta_0 + \beta_1 \text{Growth}^0_{pt} + \beta_2 \text{Growth}^6_{pt} + \beta_3 \text{Growth}^{12}_{pt} + \beta_4 \text{Growth}^{18}_{pt} + \mu_p + \lambda_t + \varepsilon_{pt},
\end{equation}
where the height of an individual $i$ born in province $p$ and year $t$ is regressed on the growth rate in that province when the individual is born (Growth$^0$), when they are aged 6 years (Growth$^6$), when they are aged 12 years (Growth$^{12}$) and when they are aged 18 years (Growth$^{18}$). It is important to note that given challenges in collecting data on economic patterns around 200 years in the past, these measures of growth are the best available estimates, but should be considered as noisy measure of the economic conditions during an individual's growing years.\footnote{We also note that we consistently use measures related to an individual's province of birth, given that from Military data we know where they are born. However, in the case of individuals moving between provinces in the country, these measures are noisy proxies of exposure to local economic conditions.} This model includes province and decade fixed effects ($\mu_p$ and $\lambda_t$ respectively), capturing idiosyncratic regional or temporal factors, such that measures of growth are not simply proxying regional or time-specific factors that correlated with height.
\begin{figure}[htpb!]
\caption{Quantile Regression}
\label{fig:qreg_hat}
\includegraphics[scale=1.1]{./results/qreg_hat.eps}%
\floatfoot{\textsc{Notes:} Each sub-plot presents the OLS and quantile regression estimates of the effect of local rates of economic growth at particular points of an individual's life on their adult height. Solid black lines and dashed lines refer to OLS point estimates and 95\% CIs, while gray solid lines and shaded areas refer quantile regression point estimates and 95\% CIs.}
\end{figure}
To consider what can be gained from estimating quantile regression in this particular setting of interest in economic history, we present estimates from equation \ref{eqn:heightReg} by both standard OLS, and using quantile regression following equation \ref{eqn:qreg} at quantiles $5, 10, \ldots, 95$. These results are displayed in Figure \ref{fig:qreg_hat}, where we plot estimates on each of the parameters $\beta_1, \beta_2, \beta_3$ and $\beta_4$ on average (OLS, indicated by the solid horizontal black line), and across the distribution (quantile regression, indicated by the solid gray line). 95\% confidence intervals are indicated by dashed lines (OLS) or shaded intervals (quantile regression).
The top-left panel documents a null effect of exposure to growth in the decade of birth on adult height when considering average effects, although marginally negative effects lower in the distribution of adult height, and marginally positive effects higher in the distribution. Where results are most striking is in the top right-hand panel. While exposure to growth in the decade in which a child is aged 6 years is observed to significantly increase adult height (OLS estimates suggest a 1\% higher rate of growth is associated with nearly 1cm in additional height), this effect is very heterogeneous, with large impacts at the lower end of the distribution of height, and null impacts among taller individuals. Here the value of considering quantile regression is clear. These results point to a particular value of economic growth in human condition, which -- at least in this early life period where children are growing significantly -- is most relevant among those who are shorter. This is significant, as adult height is a well known marker of health, suggesting that economic growth during this period of childhood in this context has done the most to pick up individuals who have the worst health stocks. While some similar patterns may even persist when children are older (bottom panels), these effects are observed to be most notable in the early years of life when children are particularly sensitive to their conditions. While these results are descriptive, and presented in part to illustrate the utility of quantile regression in research in economic history, they are nonetheless able to point to key distributional factors which are of relevance in understanding human demographics and sensitive periods of human capital accumulation, as well as the value of historical periods of growth -- beyond just population averages.
\section{Conclusions and Ways Ahead}
This paper seeks to provide a preview of a range of empirical methods which are relevant to consider impacts of some independent variable(s) of interest, across the entire distribution of a continuous dependent variable of interest. We specifically seek to describe these methods, and motivate their adoption more widely in literature in economic history where considerable work is often spent to collect rich (continuous) outcome measures of interest. These methods can thus contribute to fully taking advantage of such data collection processes or existing data respositories which have been collected based on considerable efforts in collating, systematising, or digitising historical records.
We discuss both standard quantile regression methods originally laid out by \citet{KoenkerBassett1978} and also document how these have been fruitfully applied in a more recent ``treatment effects'' literature, with the application of Quantile Treatment Effects. We discuss a number of other extensions which may be of particular interest for researchers in economic history such as quantile regression with measurement error, additional ways to loosen parametric assumptions, and potential solutions to endogeneity in these models. This is a very large and ever-growing literature in econometrics, and so here, while aiming to provide a broad overview of the field, we do not claim to comprehensively survey the entire field nor the full depth of all models. Fortunately, there are a number of full textbook or handbook references such as work of \citet{KOENKER2004,Koenkeretal2017}, to which we point interested readers in cases where a more comprehensive econometric base of these models is desired.
While we argue that these models are well suited to research in economic history, we suggest that there is scope for considerably more work in this line. Fortunately, these methods are accompanied by a range of computational tools which mean that their application can be viewed as part of a quite standard toolbox for interested practitioners. In closing, it is worth pointing to the functionality of these packages, which are significant and generally open source contributions of the methods discussed in this paper, allowing for these methods to be adopted at relatively low cost.
Computational languages widely used in economics and economic history such as R, Stata, Julia, Python, MATLAB and so forth generally all have a standard implementation of quantile regression allowing for simple implementation of these models. For example, \citep{Koenker2021} provides the R \texttt{quantreg} package which contains (among \emph{many} other things) a standard quantile regression interpretation as \texttt{rq}, while Stata's \texttt{qreg} provides a stable option for both estimation and various inference procedures. However, many extensions to these commands' `standard' procedures are available, including packages to extend analyses to quantile treatment effects such as \texttt{ivqte} in Stata \citep{FrolichMelly2010} or the \texttt{QTE} package in R \citep{Callaway2019}. Each of these QTE libraries extends in numerous ways to cases where treatment assignment is endogenous, and, in the case of \texttt{QTE} to difference-in-differences models \citep{Callaway2019}. Within the `universe of \texttt{quantreg} in R there are many other extensions, including a wide range of inference procedures, non-linear models, LASSO models, and plotting functions. Given the highly applied nature of these methods, and the considerable recent extensions in the field, many new papers are also accompanied by computational code -- most frequently in R or Stata -- including the examples discussed in the paper such as recent advances in selection models \citep{Siravegna2020,BE2020}, and quantile analyses in regression discontinuity designs \citep{FRANDSEN2012382}. All in all, these models present an extremely flexible, accessible and extendable series of analytical routes to researchers in economic history, and should be viewed as a key component of analyses, allowing for a much richer consideration of the distributional effects of historical phenomena in economic processes.
\clearpage
\end{spacing}
|
2,877,628,090,758 | arxiv | \section{Introduction}
\label{sec:introduction}
In this paper, we study the online algorithms.
Online algorithms get their input one-by-one during their computation unlike their offline counterparts which are given the whole input before their computation.
\textit{Competitive analysis} is concerned with how an online algorithm performs with respect to its offline counterpart. \textit{Competitive ratio} is the ratio between the
performance of an online algorithm and its offline counterpart. An online algorithm is called \textit{competitive} if its competitive ratio is bounded by some constant. More formally,
we call an online algorithm $c \textit{-competitive}$ if there exists an arbitrary constant $a$ such that for all finite input sequences $I$, we have $$ C(ALG(I)) \leq c \cdot C(OPT(I)) + a, $$ where $C(OPT(\cdot))$ is the cost
of the optimal offline algorithm and $C(ALG(\cdot))$ is the cost of the online algorithm. We note that an optimal offline algorithm knows the
entire sequence in advance and can process it with minimum possible cost.
An important concept about online algorithms is \textit{recourse}, i.e. being able to change past decisions.
Indeed, not being able to see future inputs can be problematic for an online algorithm. A single new input might be inconsistent with the solution that the algorithm has built until that point and can increase
the competitive ratio greatly. The main idea is, by allowing the algorithm to change some of its past decision, to make the solution more compatible with arriving inputs and thus, to lower the competitive ratio.
Certainly, for an online algorithm there is a tradeoff between the number of allowed recourse operations and its competitive ratio. If an arbitrary number of recourse operations are allowed, then the online algorithm can
simply simulate the offline optimal algorithm and achieve $1$-competitiveness. All it has to do is to make some arbitrary decisions as the inputs arrive and to run the offline optimal algorithm
once it has all the inputs. Due to that, we want to keep the number of recourse operations bounded from above to maintain the online nature of online algorithms.
The rest of the paper is organized as follows: in Section 2, we present the online edge orientation and two different algorithms, the shortest path algorithm
and the all-flip algorithm, that achieve constant competitive ratio under certain assumptions. In Section 3, we establish some results regarding the shortest path algorithm through analyzing
it against an adversary. In Section 4, we present the online bipartite $b$-matching and an algorithm that achieves a constant competitive ratio for it.
In the last section, we give some concluding remarks.
\newpage
\section{Online Edge Orientation}
In the online edge orientation, we are given a set of nodes and a set of undirected edges. The edges are not initially present but rather, they are added one-by-one.
When an edge is added, we must orient it towards one of its endpoints. The objective is to orient the edges in a way such that the maximum in-degree of the graph is minimized after $n$ edge additions.
More precisely, the cost of the problem is defined as the maximum in-degree in the graph after $n$ arrivals.
Note that $n$ is the number of edges in the graph after $n$ edge additions and there is no upper bound on the number of nodes.
For this problem, a recourse operation is simply reorienting an edge (which we also call as flip sometimes) after it was given an initial orientation at its addition.
The first algorithm we present for this problem assumes the set of arriving edges are acyclic. Before presenting the algorithm, we present a lemma that gives a tight bound on the competitive
ratio of any algorithm under that assumption when we do not allow recourse.
\begin{lemma}\label{lwbnd}
If the given edge set is acyclic and if the reorientation of the edges are not allowed, the best competitive ratio that can be achieved for the online edge orientation is $\Theta(\log n)$.
\end{lemma}
\begin{proof}
We first show the lower bound by constructing a sequence of the edges. First assume that $n \neq 1$ and divisions below yield an integer value.
We keep track of the maximum in-degree round by round. In the first round, we put n/2 edges (that arrive one-by-one) such that none of them are incident.
This will create n/2 components that consist of an edge and two nodes in which one has in-degree 0 and the other has in-degree 1.
In the second round, we put n/4 edges between the nodes that have in-degree $1$. After this round we have n/4 components that
consist of two edges and four nodes such that two has in-degree 0, one has in-degree 1 and one has in-degree 2.
If we continue to keep putting edges in this fashion, we see that maximum in-degree of
the graph increases by 1 after each round. The number of rounds is simply the number of times we can halve $n$ which is $\log_2 n$.
If dividing $n$ by powers of $2$ does not yield an integer, we take ceil of the divisions if $n = 2^k - 1$ for some
integer $k \geq 2$ and we take floor of them otherwise. Then, the number of rounds and consequently
the maximum in-degree is $\lceil \log_2 n \rceil$ if $n = 2^k - 1$ or $\lfloor \log_2 n \rfloor$ otherwise. Finally for $n = 1$, the
maximum in-degree is simply $1$.
For the upper bound, consider the simple algorithm that orients edges towards the node with smaller in-degree.
Now observe that forcing the maximum in-degree to increase by 1 requires connecting two nodes that has maximum in-degree.
For any other case, the algorithm orients the edge towards the node with smaller in-degree and avoids increasing the maximum in-degree.
Then, it is easy to prove any sequence that achieves $k$ in-degree requires at least $2^k - 1$ edges by induction.
Base case $k = 1$ trivially holds. Assume any sequence that achieves $k-1$ in-degree requires at least $2^{k-1} -1$ edges. Then we can construct a sequence
to achieve $k$ in-degree with two $k-1$ achieving sequences and a single edge. Note that $k-1$ achieving sequences must be defined on distinct set of
nodes in order to ensure to have two distinct nodes with $k-1$ in-degree.
Thus, we need at least $2\cdot(2^{k-1} -1) + 1 = 2^k -1$ edges.
Of this we conclude that any sequence that achieves $\log_2 n$ in-degree requires at least $2^{\log_2 n} - 1 = n-1$ edges. Since the last remaining
edge can not possibly increase the maximum in-degree by itself, this is the upper bound.
Finally, remember that our edge sequence is acyclic. This means they form a forest.
Further, it implies there exists an orientation of the edges such that the maximum in-degree is 1. One can find such orientation by arbitrarily picking a node as the root and orienting all the edges
away from it in every tree in the forest. Hence, the competitive ratio is $\Theta (\log n)$.
\end{proof}
\begin{figure}[ht] \label{fig:lowerBound}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=1.5cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] (1) {};
\node[main node] (2) [right of=1] {};
\node[main node] (3) [right of=2] {};
\node[main node] (4) [right of=3] {};
\node[main node] (5) [right of=4] {};
\node[main node] (6) [right of=5] {};
\node[main node] (7) [right of=6] {};
\node[main node] (8) [right of=7] {};
\path[every node/.style={font=\sffamily\small}]
(1) edge node [right][above] {1} (2)
(3) edge node [right][above] {1} (4)
(5) edge node [right][above] {1} (6)
(7) edge node [right][above] {1} (8)
(2) edge[bend right] node [left][below] {2} (4)
(6) edge[bend right] node [left][below] {2} (8)
(4) edge[bend left] node [left][above] {3} (8);
\end{tikzpicture}
\caption{Construction for the lower bound with $n = 8$ where the last edge is redundant. Numbers on the edges indicate their addition round. As can be seen, maximum in-degree is $\log_2 8 = 3$.}
\end{figure}
We have showed the competitive ratio for this problem is $\Theta(\log n)$ when there are no reorientations.
We now present an algorithm that achieves $O(1)$ competitive ratio by doing $O(n)$ reorientations over $n$ arrivals.
\subsection{Shortest Path Algorithm}
We remind that we assume the set of arriving edges are acyclic. The algorithm given in \cite{Gupta:2014:MAO:2634074.2634109} works in a greedy
fashion to achieve $O(1)$ competitiveness under that assumption. Basically, the algorithm maintains a set of nodes such that no node has more than $c$ in-degree
over $n$ edge additions for some constant $c \geq 2$ which is referred as the in-degree constraint of the graph. Hence, it is $c$-competitive.
We note that achieving $1$-competitiveness is possible under the given edge set however,
it is known that there exists instances which might require any algorithm to do $\Omega(n \log n)$ flips in that case (\cite{Grove:1995:OPM:645930.758390}).
So by relaxing the constraint, authors try to approximate the optimal solution within a constant degree by doing fewer number of flips.
For the rest of the section we let the in-degree constraint be $c=2$ and call a node \textit{saturated} if it has 2 in-degree.
We now present a simple lemma which is rather an observation about our graph.
\begin{lemma}
Let $u$ be a node in the forest. We denote the tree that $u$ belongs to by $T_u$ and the number of nodes it has by $|T_u|$. Then, there exists a node $u'$ in $T_u$ such that it is unsaturated and there is a path
from $u'$ to $u$, i.e. $P_{u' \rightarrow u}$. We denote the length of that path by $|P_{u' \rightarrow u}|$.
\end{lemma}
\begin{proof}
Let $|T_u| = k$. Since $T_u$ is a tree, it has $k-1$ edges. To saturate all the nodes in $T_u$ we need $2k$ edges. Hence, there must be some unsaturated nodes in it.
If $u$ is not saturated, we simply have $u = u'$. If $u$ is saturated then it has $2$ edges coming to it.
By doing a tree traversal that starts at $u$, we will eventually arrive to an unsatured node as we know all the nodes
in $T_u$ can not be saturated. The node we arrive is $u'$ and we can extract the path $P_{u' \rightarrow u}$ from the traversal.
\end{proof}
As its name indicates, the algorithm handles constraint violations by flipping the edges towards the shortest of the available paths. Now
suppose some edge $(u, v)$ arrives. If both endpoints are unsaturated, the algorithm picks one randomly.
If one of them is not saturated, then it orients the edge to it. Now assume both $u$ and $v$ are saturated. Since we assumed that the set of arriving edges are acyclic, $T_u \neq T_v .$
By the lemma above, there are unsaturated nodes in $T_u$ that have a path to $u$. Let $u'$ be any of the closest among such nodes by $|P_{u' \rightarrow u}|$. Similarly let $v'$ be such node in $T_v$. It is clear that flipping
process must end in an unsaturated node. This implies the minimum number of flips we can have is the minimum of $|P_{u' u}|$ and $|P_{v'v}|$.
So the algorithm finds $u'$ and $v'$, computes $|P_{u' \rightarrow u}|$ and $|P_{v' \rightarrow v}|$, picks the path with distance $\min(|P_{u' \rightarrow u}|, |P_{v' \rightarrow v}|)$
and flips all the edges on that path until it reaches an unsaturated node which is either $u'$ or $v'$.
Then, the number of flips caused by the addition of $(u, v)$ is simply given by $\min(|P_{u' \rightarrow u}|, |P_{v' \rightarrow v}|).$
We now give an upper bound to the number of flips that $(u, v)$ can cause.
Suppose the algorithm oriented the towards $u$ and consequently, flipped all the edges on $P_{u' \rightarrow u}$.
We observe that every node on that path must be saturated except $u'$. This is simply from the definition of $u'$, i.e. $u'$ is the closest unsaturated node to $u$.
So $u$ has two edges pointing to it, the nodes that are at the other endpoints of those two edges must have two edges pointing to them too and so on
until $u'$. Essentially, the tree $T_u$ contains a complete binary tree at least up to depth $|P_{u' \rightarrow u}| - 1$. So $|P_{u' \rightarrow u}| \leq \log_2 |T_u| $
and therefore there can be at most $\log_2 |T_u|$ flips. Consequently, we can upper bound the number of flips that any edge addition can cause by $\log_2 n$.
As last, we bound the total number of flips over $n$ edge additions. From the result above, a straightforward upper bound is $O(n \log n)$. However, by using the fact that the algorithm always
flips the shortest of the available paths, it is possible to obtain a tighter bound. The worst case is simply the case where the edges
are stacked in just two trees over $n$ additions. We write the recurrence accordingly and present its solution.
\begin{lemma}
Let $T(k)$ be an upper bound on the number of flips after $k$ edge additions. Then,
$$ T(n) \leq \max_{1 \leq n_1 < n}\{T(n_1) + T(n-n_1) + \log_2 \text{min}(n_1, n - n_1)\} $$ with base case $T(1) = 0$ solves to $T(n) \leq n - \log_2 n - 1 .$
\end{lemma}
\begin{proof}
Base case trivially holds. Now assume $T(k) \leq k - \log_2 k - 1$ for $k < n$. We will show
$T(n) \leq n - \log_2 n - 1$.\\
1) If $n_1 > n - n_1$:
\begin{align*}
T(n) & \leq n_1 - \log_2 n_1 -1 + (n-n_1) - \log_2 (n-n_1) - 1 + \log_2 (n-n_1) \\
& = n - (\log_2 n_1 + 1 ) - 1 \\
& = n - \log_2{2n_1} - 1 \\
& < n - \log_2 n - 1 \text{ as } 2n_1 > n.
\end{align*}
2) If $n_1 \leq n - n_1$:
\begin{align*}
T(n) & \leq n_1 - \log_2 n_1 -1 + (n-n_1) - \log_2 (n-n_1) - 1 + \log_2 (n_1) \\
& = n - (\log_2(n- n_1) + 1) - 1 \\
& = n - \log_2{2(n- n_1)} - 1 \\
& \leq n - \log_2 n - 1 \text{ as } n \geq 2n_1.
\end{align*}
\end{proof}
Note that we improved the analysis of the original authors. We restate the main theorem presented by them accordingly.
\begin{theorem}\label{shortest_mainTheo}
Given the edge set is acyclic, the shortest path algorithm maintains at most $2$ in-degree in all nodes by doing at most
$n - \log_2 n - 1$ edge reorientations over $n$ edge additions. Consequently, the shortest path algorithm achieves $O(1)$ competitiveness with $O(n)$ edge reorientations over $n$ arrivals.
\end{theorem}
We finally note that generalizing results for an arbitrary in-degree constraint $c \geq 2$ is possible by changing the base of the logarithm.
As we have showed, we have binary trees up to unsaturated nodes when $c=2$. For an arbitrary $c \geq 2$, we would have $c$-ary trees.
We can upper bound the maximum number of flips per step as $\log_c n$ and upper bound the total number of flips over $n$ edge additions as $\cfrac{n - log_2 n - 1}{log_2 c}$.
\subsection{All Flip Algorithm}
We now present the algorithm given in \cite{Brodal99dynamicrepresentations} which achieves $O(1)$ competitive ratio with $O(n)$ reorientations when the arboricity of the graph is bounded by $c$ during the entire edge addition sequence.
Like the previous algorithm, algorithm keeps the in-degree of all nodes below some constant $c$ over $n$ edge additions.
The arboricity $c$ of a graph is defined as $$ c = \max_{J} \frac{|E(J)|}{|V(J)| - 1}, $$ where $J$ is any subgraph in $G = (V,E)$ with $|V(J)| \geq 2$ nodes and $|E(J)|$ edges.
The importance of the bound on arboricity is due to the following theorem.
\begin{theorem} (Nash-Williams~\cite{Nash-Williams01011964})
A graph $G = (V, E)$ has arboricity $c$ if and only if $c$ is the smallest number of sets $E_1, \dots, E_c$ that $E$ can be partitioned into, such that each subgrah $(V, E_i)$ is a forest.
\end{theorem}
In other words, if the arboricity of $G$ is bounded by $c$ then we can partition $G$ into $c$ forests. In each of these forests, we can pick an arbitrary node as the root and orient all the edges away from it.
So, in-degree of each node can be at most one in each forest and at most $c$ in the entire graph as a node can be in all of the forests.
Thus for any graph $G$ whose arboricity is bounded by $c$, there exists an orientation of the edges such that no node has more than $c$ in-degree.
The set of edge orientations such that no node has more than $\delta$ in-degree is called $\delta$-orientations.
Assuming the arboricity of the graph is bounded by $c$ during the entire edge addition sequence, the algorithm given by them finds a $\Delta$-orientation if the given edge set allows a $\delta$-orientation for $\delta \geq c$ and if $\Delta \geq 2\delta$.
It runs as follows: when the algorithm gets a new edge $(u, v)$, it orients it arbitrarily to one of its endpoints.
Assume it is oriented towards $v$. Then, if the in-degree of $v$ is still bounded by $\Delta$, algorithm proceeds to the next input.
Otherwise ($v$ has $\Delta + 1$ in-degree), the algorithm flips all the incoming edges at $v$. If this causes
some other nodes to have $\Delta + 1$ in-degree, algorithm flips their incoming edges too. This flipping process continues until all the nodes have at most $\Delta$ in-degree.
Before presenting the proof that shows the algorithm indeed works, i.e. the flipping process eventually terminates in a state where all nodes have at most $\Delta$ in-degree,
we give an example case to build some intuition.
Assume the set of arriving edges form a tree. So, our input allows a $\delta$-orientation for $\delta = 1$ and say we want a $\Delta$-orientation for
$\Delta = 2$.
Define potential $\Psi(t)$ as the number of edges that have a different orientation with respect to a fixed $\delta$-orientation in our graph after $t$ edge additions and note that this potential
can increase at most by 1 with a new addition.
Assume after $t$'th edge addition a node $v$ gets 3 incoming edges.
We know at most 1 of those edges can have the same orientation with respect to our fixed $\delta$-orientation.
Hence, at least 2 of these incoming edges must have a different orientation.
We see that that if we flip all the $3$ incoming edges at $v$, $\Psi(t)$ must decrease at least by 1. As noted before, flipping those edges
might cause problems at some other nodes. However, by the same argument we can keep reducing $\Psi(t)$ by repeating the process at them. The following
lemma gives an upper bound on the number of flips over $n$ edge addition and proves that the algorithm eventually achieves the desired state.
\begin{lemma}
For an arboricity c preserving edge addition sequence $\sigma$ on an empty graph, let $G_t = (V, E_t)$ denote the graph after $t$'th edge addition and let length of the sequence be $n$.
If the given sequence $\sigma$ allows a $\delta$-orientation, then the algorithm does at most $$ n\frac{\Delta+1}{\Delta + 1 - 2\delta} $$ edge reorientations on $\sigma$ given that $\Delta \geq 2\delta$ and $\delta \geq c$.
\end{lemma}
\begin{proof}
Consider a fixed $\delta$-orientation built by $\sigma$ on the node set $V$. Let $\bar{E_t}$ denote the edge set of the fixed $\delta$-orientitation after its $t$'th edge addition.
An edge in $E_t$ is denoted \textit{good} if it has the same orientation in $\bar{E_t}$. Otherwise it is denoted \textit{bad}. Define a potential function as the following
$$ \Psi(t) = \text{the number of bad edges in } E_t .$$
Note that this potential is non-negative and $\Psi(0) = 0$.
Assume after $t$'th addition, a node $v$ gets $\Delta + 1$ incoming edges.
We note that at most $\delta$ of them can be good and consequently, at least $\Delta + 1 - \delta$ of them are bad.
Thus, if all the incoming edges are flipped at $v$, at most $\delta$ edges may become bad and at least $\Delta + 1 - \delta$ edges become good.
So by flipping all the incoming edges at $v$, we reduce the bad edges by at least $(\Delta + 1 - \delta) -\delta = \Delta + 1 - 2\delta$.
After $t$'th addition, the number of bad edges can be at most $\Psi(t-1) + 1$ and we can make at most $\Psi(t)$ of them good (simply from the definition, i.e.
$\Psi(t)$ is the number of bad edges after $t$'th addition) in that step. Since we know an all-flip (flipping all the incoming edges at some node) decreases bad edges by
at least $\Delta + 1 - 2\delta$, we can have at most $(\Psi(t-1) + 1 - \Psi(t))/\Delta + 1 - 2\delta$ all-flips after $t$'th addition.
Summing this over $n$ edge additions gives us an upper bound on the total number of all-flips.
We have
$$\sum_{t=1}^n \frac{\Psi(t-1) + 1 -\Psi(t)}{\Delta + 1 - 2\delta} = \frac{\Psi(0) + n -\Psi(n)}{\Delta + 1 - 2\delta} \leq \frac{n}{\Delta + 1 - 2\delta},$$
and since an all-flip is simply flipping $\Delta + 1$ edges, the total number of flips are bounded from above by
$$ n\frac{\Delta+1}{\Delta + 1 - 2\delta} .$$
\end{proof}
For our example case, we had $\delta = 1$ and $\Delta = 2$. Hence, the algorithm would do at most $3n$ flips over $n$ additions.
\section{Adversary Against the Shortest Path Algorithm}
We now prove some lower bounds for the shortest path algorithm. We do this
by analyzing the algorithm against an adversary who supplies edges that forces the algorithm to make
expensive choices. We again assume the in-degree constraint is $2$ and the set of arriving edges are acyclic.
We remind that a node is called saturated if it has $2$ in-degree and start by giving a few definitions.
\begin{definition} \label{path-def}
{\textit{Variant of the shortest path algorithm.}}
An algorithm is called a variant of the shortest path algorithm if it satisfies the following conditions.
\begin{enumerate}
\item It reorients edges only when it is necessary (i.e. when there is a constraint violation)
and always towards the shortest of the available paths. It can arbitrarily chose among such paths when there is a tie.
\item It can orient an edge between two unsaturated nodes in any manner.
\end{enumerate}
\end{definition}
The shortest path algorithm we presented in the previous section of course fits this description.
\begin{definition}
{\textit{Adversary}.} An adversary is defined as a player who has complete knowledge about the algorithm and can pass any valid input to it.
\end{definition}
\begin{definition}{\textit{$T^m.$}}
$T^m$ is the set of trees with saturated root and shortest path of length $m$ where $m \geq 1$.
Further for a $t_m \in T_m$, we define its size as the number of edges it has and denote it by $|t_m|$.
\end{definition}
We note that we can express the state of our graph at each step as a combination of $t_m$ trees.
We now show how an adversary can force the construction of such trees. Basically by using them, an adversary would be
able to force more complex constructions.
\newpage
\begin{lemma} \label{sizeLemma}
An adversary can force any variant of the shortest path algorithm to construct a $t_m \in T_m$ by using at most $5 \cdot 2^{m-1} - 2$ edges.
\end{lemma}
\begin{proof}
We give a proof by induction. Consider the base case $m = 1$. Adversary needs at most $4$ single nodes. Let them be $a, b, c$ and $d$. Adversary first gives the edge $(a, b)$. By point 2 of the
Definition \ref{path-def}, this edge can be oriented to any direction. As everything is symmetric, we can assume it is oriented towards $b$ without loss of generality. Adversary then gives the edge $(c, d)$. By the same argument,
assume it is oriented towards $d$ without loss of generality. Then by giving the edge $(b, d)$, adversary will have a $t_1$ as either orientation will cause a saturated
node with shortest path of length $1$. Following figure illustrates this process.
\begin{figure}[ht] \label{figt1}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=3cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] (1) {a};
\node[main node] (2) [right of=1] {b};
\node[main node] (3) [right of=2] {c};
\node[main node] (4) [right of=3] {d};
\path[every node/.style={font=\sffamily\small}]
(1) edge node [right][above] {1} (2)
(3) edge node [right][above] {2} (4)
(2) edge[bend right] node [left][below] {3} (4);
\end{tikzpicture}
\caption{Construction of a $t_1$ with root $d$. Numbers on the edges indicate their addition order.}
\end{figure}
Assume the construction of $t_{m-1}$ can be forced by using at most $5 \cdot 2^{m-2} - 2$ edges.
To force the construction of a $t_m$, adversary first forces the construction of two $t_{m-1}$ using disjoint sets of nodes. Let $a$ be the root of the first $t_{m-1}$, let $b$ be the root
of the second and let $c$ be a single node separate from the constructed trees. Adversary first gives the edge $(a, c)$. By point 1 of the Definition \ref{path-def}, the edge will orient towards $c$.
Adversary then gives the edge $(b, c)$ and by the same argument, it will be oriented towards $c$. After these two edges, $c$ is a saturated
node with shortest path of length $m$ giving us a $t_m$ with root $c$.
Hence, the total number of edges used to force the construction of a $t_{m}$ is at most $2|t_{m-1}| + 2 = 5 \cdot 2^{m-1} - 2$.
\end{proof}
\begin{corollary}
An adversary can force any variant of the shortest path algorithm to flip $\log_2 m$ edges in a single step using at most $5m-3$ edges in total.
\end{corollary}
\begin{proof}
Adversary first forces the construction of two $t_{\log_2 m}$ using Lemma \ref{sizeLemma}. Then by giving the edge that connects the roots of these trees,
adversary will cause $\log_2 m$ flips. In total, $2|t_{\log_2 m}| + 1 = 5m-3$ edges are used.
\end{proof}
Notice that if we set the total number of edges as $n = 5m-3$ then we have $\Omega(\log n)$ flips.
Recall that we have the upper bound $O(\log n)$ for the number of edge flips in a single step
so, we conclude that this bound is tight.
\begin{lemma}
Any variant of the shortest path algorithm can be forced to flip $\Omega(n)$ edges in total after $n$ edge additions.
\end{lemma}
\begin{proof}
The adversary first constructs $k+1$ disjoint $t_m$ trees using Lemma \ref{sizeLemma}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[{root/.style={circle,draw,fill,inner sep=2 pt}}]
\node[root] (rootA) {};
\node[root,xshift=3cm] (rootB) {};
\node[root,yshift =-5cm,xshift=3cm] (rootC) {};
\draw (rootA) -- +(-1,-2) -- node[yshift=0.75cm]{$t_m$} +(1,-2) -- (rootA);
\draw (rootB) -- +(-1,-2) -- node[yshift=0.75cm]{$t_m$} +(1,-2) -- (rootB);
\draw [dotted] (3, -2.5) -- (3, -4.5);
\draw (rootC) -- +(-1,-2) -- node[yshift=0.75cm]{$t_m$} +(1,-2) -- (rootC);
\end{tikzpicture}
\caption{A single $t_m$ on the left, k $t_m$ on the right.}
\end{figure}
Now when we look at the figure below, we see that for every $t_m$ on the right,
an edge can be added between its root and the root of $t_m$ on the left. For every such edge, there will be $m$ flips due to
point 1 of Definition \ref{path-def}. Thus, forcing the algorithm to make $km$ flips requires $k$ edges after reaching the construction in the figure.
Total number of edges required to do $km$ flips is then given by $$f(k, m) = k + (k+1)|t_m| \leq n .$$ Clearly, $f(k, m)$ is maximized when $m=1$ as $|t_m|$ grows exponentially with $m$.
Hence, an upper bound on $k$ when $m=1$ will be an upper bound on the total number of flips for this construction. Solving the inequality for $k$ with $m = 1$ gives $$k \leq \frac{n-|T_1|}{|T_1| + 1} = \frac{n-3}{4}.$$
So, we have $f(k, m) \leq \frac{n-3}{4}$ with equality iff $m = 1$ and $\frac{n-3}{4}$ is an integer.
\end{proof}
By combining this result with Theorem \ref{shortest_mainTheo}, we see that the total number of flips is $\Theta(n)$. So, we can not asymptotically do better by handling edges between unsaturated nodes cleverly.
\begin{lemma}
A single edge can be forced to flip $\Omega(\log n)$ after $n$ edge additions.
\end{lemma}
\begin{proof}
Consider the initial setting where the adversary puts an edge between single nodes $a$ and $b$. Without loss of generality, assume the algorithm directed it towards $b$.
\begin{figure}[ht] \label{mrkd}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=3cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] (1) {a};
\node[main node] (2) [right of=1] {b};
\path[every node/.style={font=\sffamily\small}]
(1) edge[red] node [right][above] {} (2);
\end{tikzpicture}
\caption{Initial setting. The edge we count flips for is in red.}
\end{figure}
To flip the red edge back to $a$, adversary connects 2 $t_1$ to $b$ due to point 1 of Definition \ref{path-def}.
The first one will saturate $b$ and the second one will force the flip in the
direction of $a$ and consequently will flip the red edge. Note that we now have a $t_2$ rooted at node $b$.
To flip the red edge again, adversary can connect 2 $t_3$ to $a$. By the same reasoning, this will flip the edge towards $b$. Adversary can continue to flip the red edge in this fashion.
The total number of edges required to do $k$ flips on the red edge is then given by $$f(k) = 1 + \sum_{m=1}^{k} 2|t_{2m -1}| + 2.$$
By using Lemma \ref{sizeLemma}, it can be rewritten as $$f(k) = -ax + b\exp\{cx\} + d$$ for some positive constants $a, b, c, d$ and for Euler constant $\exp$.
Finally, solving $f(k) \leq n$ for $k$ will give $k = \Omega(\log n).$
\end{proof}
As last, we consider a slightly different lower bound. Namely, we wonder whether a shortest path algorithm can maintain in-degree constraint
over $n$ additions by doing constant number of flips at each step. As we have showed, an adversary can force any number of flips up to $O(\log n)$ at a step
if the shortest path algorithm satisfies Definition \ref{path-def}. So, we define a \textit{fixing} shortest path algorithm. It is, the algorithm
is now allowed to do unforced flips in order to fix the graph but is still flipping the edges on the shortest path at a constraint violation.
We show maintaining constraint with a single flip at a step is not possible.
\begin{lemma} Given sufficient number of edges, there exists a sequence of edge additions such that a fixing shortest path algorithm
is forced to flip $2$ edges at a step. Consequently given sufficient number of edges, no fixing shortest path algorithm can maintain in-degree constraint $c = 2$ by doing
at most $1$ flip at a step.
\end{lemma}
\begin{proof}
Assume the adversary has constructed eight trees with shortest path of length $1$, i.e. we have $t_{1, i} \in T^1$ for $i \in \{ 1, 2, \dots, 8\}$. Let $t_{1, i} \rightarrow t_{1, j}$ denote
the edge addition that connects the roots of $t_{1, i}$ and $t_{1, j}$.
Without loss of generality, we can assume such edges are oriented towards the trees with higher index.
Then to force $2$ flips, adversary first gives the following sequence to the algorithm:
$$ (t_{1, 1} \rightarrow t_{1, 2}), (t_{1, 3} \rightarrow t_{1, 4}), (t_{1, 5} \rightarrow t_{1, 6}), (t_{1, 7} \rightarrow t_{1, 8}) .$$
Then by giving the edges $(t_{1, 2} \rightarrow t_{1, 4})$ and $(t_{1, 6} \rightarrow t_{1, 8})$, adversary would have two trees with shortest path of length $2$ rooted at
the roots of $t_{1, 4}$ and $t_{1, 8}$. Note that each addition causes a flip and renders the algorithm null.
Finally by giving the edge $t_{1, 4} \rightarrow t_{1, 8}$, adversary causes 2 flips.
So after constructing 8 $t_1$ trees, adversary can force $2$ flips by using $7$ more edges.
What remains is to show how to construct $t_1$ trees.
Clearly, constructing them like in Figure 2 would not work as the algorithm could break the tree with a single flip. For
the given figure, the algorithm would just flip $(c, d)$ after adversary adds $(b, d)$.
The idea is to construct long chains and connect their heads (the node with 0 out-degree). Observe that there is nothing algorithm
can do to prevent adversary from constructing such chain without creating a $t_1$.
The algorithm can pick an arbitrary node at somewhere on the chain and orient both sides to opposite directions but then the adversary can
pick the longest side of the chain and proceed. Clearly, flipping an edge in-between would create a $t_1$.
\begin{figure}[ht] \label{fig:chain}
\centering
\begin{tikzpicture}[->,>=stealth',auto,node distance=1.5cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] (1) {};
\node[main node] (2) [right of=1] {};
\node[main node] (3) [right of=2] {};
\node[main node] (4) [right of=3] {};
\node[main node] (5) [right of=4] {};
\path[every node/.style={font=\sffamily\small}]
(1) edge node [right][above] {} (2)
(2) edge node [right][above] {} (3)
(3) edge node [right][above] {} (4)
(4) edge node [right][above] {} (5);
\end{tikzpicture}
\caption{A chain with length 4.}
\end{figure}
We see that if we connect the heads of two chains, we would have a $t_1$ rooted at one of the heads.
To break this $t_1$, the algorithm has to flip one of the edges that point to the heads of chains. But then, that would just create a new $t_1$. We see that the algorithm has to flip
all the edges that lie on one of the two chains.
Let $L$ be the length of the chains. The adversary first gives the input sequence that constructs $16$ distinct chains of length $L$ and algorithm
orients them in some way. By the argument above, adversary has some combination of $t_1$ trees and
chains of length at least $L/2$. If there are $8$ $t_1$ trees or more, adversary can force $2$ flips as described before. For all other cases,
adversary can group the chains by $2$ (some chains might not be used), connect their heads and can have $8$ $t_1$ trees.
We know explain what the sufficient number of edges mean. Assume that after inputting the algorithm, adversary gets $16$ chains of length $L/2$.
After that point, adversary needs $8$ more edge additions to have $8$ $t_1$ trees. This simply says the chains must be at least of length 9, i.e. $L/2 \geq 9.$
Thus, we see that we need at least $16 \cdot 18 + 8 + 7 = 303$ edges in total to construct $8$ $t_1$ trees and to force $2$ flips by connecting them in this pattern.
\end{proof}
\section{Online Bipartite $b$-Matching}
We now analyze the online bipartite $b$-matching.
First, we will give the definition of the problem.
Then, we will present the algorithm given in \cite{Gupta:2014:MAO:2634074.2634109} which achieves $O(1)$ competitive ratio by doing $O(n)$ recourse operations over $n$ arrivals.
As last, we will supply its analysis.
We are given a bipartite graph $G=(L, R, E)$ where nodes in $R$ are initially present and nodes in $L$ arrive one-by-one.
When a $v_L \in L$ arrives, it specifies a set of nodes in $R$ whose size is at least $1$. An edge is put between $v_L$ and all of the nodes in the set that $v_L$ specified.
Those nodes are the potential matches of $v_L$.
Among those potential matches, we have to select a single $v_R \in R$ and match $v_L$ to it by putting the edge $(v_L, v_R)$ to the matching set $M \subseteq E$.
While doing this over $n$ arrivals, we want to keep every $v_L$ matched and the degree of nodes in $R$ below some constant $b$.
More formally, $deg_G(v)$ is the degree of $v$ in $G$ and $deg_M(v)$ is the degree of $v$ in $M$. The latter one is also called $load(v)$ if $v \in R$.
Then for $n$ arrivals, we would like to maintain a subset of edges $M$ such that for every $v_L$, we have $deg_M(v_L) = 1$. That is, every $v_L$ is matched with only one node in $R$.
While satisfying this constraint, we also want to have $\max_{v_R \in R}$ $load(v_R) \leq b$ which is defined as the cost of the problem.
We note that the bound on the competitive ratio given for the online edge orientation holds for this problem too.
If all the arriving nodes specify 2 possible matches, the upper bound and the lower bound on the competitive can be proven exactly as in the Lemma \ref{lwbnd}.
Again, we need recourse if we want to improve the competitive ratio.
For this problem, a recourse operation is defined as changing the match of a $v_L \in L$ from some $v_R \in R$ to a different $v_R' \in R$
where $v_R$ and $v_R'$ are among the potential matches of $v_L$. This is called as \textit{swap}.
The operation is defined so because every node in $L$ has to stay matched with some node in $R$ after its arrival.
In what follows, we present an algorithm that finds a solution with cost $CK$ for some $C \geq 2$ even though the given input allows a solution
with cost $K$. For the analysis, we set $C=2$ without loss of generality. We now give some preliminaries before presenting the authors algorithm.
\subsection{Preliminaries}
The algorithm runs on the residual graph $G^{res} = (L, R, A)$ with $A$ being a directed set of edges (arcs) that is constructed from the given bipartite graph $G$ and
with $L$ and $R$ being the same node sets in $G$. The set $A$ is constructed as follows: whenever we add the edge $(v_L, v_R)$ to $M$, we add
the arc $(v_R, v_L)$ (points to $v_L$) to $G^{res}$. For every other edge $(v_L', v_R') \in E \setminus M$, we add the arc $(v_L', v_R')$ to $G^{res}$.
Let $N(S)$ denote the set of nodes that have an incident edge whose other endpoint is in some node set $S$.
Similarly, let $N_M(S)$ denote the set of nodes that have an incident edge in $M$ whose other endpoint is in some node set $S$.
Respectively, those sets are called \textit{neighborhood of node set S} and \textit{M-neighborhood of node set S}.
A node $v_R \in R$ is called saturated if $2K$ $v_L \in L$ are matched to it.
This is equivalent to $deg_M(v_R) = 2K$ and it means the out-degree of $v_R$ in $G^{res}$ is $2K$ as each of those matches will give rise to
an arc $(v_R, v_L)$ in $G^{res}$.
For any node $v$ in the given bipartite graph, $height(v)$ denotes the length of a shortest path in $G^{res}$ from $v$ to some unsaturated node.
\begin{lemma} \label{halls}
For any $S\subseteq L$, $N(S) \geq |S|/K $.
\end{lemma}
\begin{proof}
By Hall's theorem (\cite{Hall01011935}) and using the fact that the input allows a $K$ matching, we can write $N(S) \geq |S|$ for all $S \subseteq L$ when $K = 1$.
For an arbitrary $K \geq 2$, we just observe that if we duplicate every node in $N(S)$ for $K$ times we can treat the problem as if $K = 1$. So, if we apply the theorem
after the duplication we see that $K \cdot N(S) \geq |S|$.
\end{proof}
\begin{lemma} \label{sizeofsat}
For any $T\subseteq R$ whose all nodes are saturated, $N_M(T) = |T|\cdot 2K$
\end{lemma}
\begin{proof}
Each node in $T$ is matched to $2K$ nodes in $L$. Hence, the total number of matches of $T$ is simply $|T|\cdot 2K$.
\end{proof}
\subsection{Shortest Augmenting Path Algorithm}
Consider the arrival of a node $v_L$. When it arrives, it specifies its neighborhood set $N(v_L)\subseteq R$.
Algorithm then puts edges $(v_L, v_R)$ for all $v_R \in N(v_L)$ in $G^{res}$. If some nodes in $N(v_L)$ are unsaturated, the algorithm
picks one arbitrarily and puts the corresponding edge to $M$ (which also orients the edge accordingly). Now assume all the nodes in $N(v_L)$ are saturated.
Let $P$ be a shortest path to some unsaturated $y_k \in R$ from $v_L$.
From the construction of $G^{res}$, we know we can write the sequence of nodes on that path as,
$$ v_L=x_0, y_1, x_1, y_2, \dots, x_{k-1}, y_{k} $$ where each $x_i \in L$ and each $y_i \in R$.
We note that all the $y_i$ must be saturated except for $y_k$ by our assumption.
By looking at the sequence, we see that $y_1$ is a possible match for $x_0$ (as we have the arc $(x_0, y_1)$) but it is currently saturated.
Moreover, $x_1$ is currently matched to $y_1$ but it could be matched to $y_2$ if $y_2$ becomes unsaturated.
Looking at the end of the path we see that $x_{k-1}$ is currently matched to $y_{k-1}$ but it could be matched to $y_k$.
Since $y_k$ is unsaturated, we can match $x_{k-1}$ to $y_k$ and unsaturate $y_{k-1}$. Then, we can match $x_{k-2}$ to $y_{k-1}$ and unsaturate $y_{k-2}$ and so on.
Eventually, we can unsaturate $y_1$ and match $x_0$ to $y_1$.
Note that each $(x_i, y_i)$ pair in $M$ is replaced with $(x_{i-1}, y_i)$ for $i \in \{ 1, 2, \dots, k-1 \}$ and we added the new pair $(x_{k-1}, y_{k})$.
This is the main idea behind the algorithm. When a new node $v_L$ arrives, algorithm finds a shortest path from it to some unsaturated node and flips all the edges on that path.
This corresponds to making $(height(v_L) - 1)/2$ swaps.
\subsection{Analysis}
We will present the proof of the following theorem which gives an upper bound on the number of swaps over $n$ arrivals.
\begin{theorem} \label{btheo}
The total number of swaps done by the shortest augmenting path algorithm over $n$ arrivals is at most $2n$.
\end{theorem}
To prove the theorem, authors make use of a series of lemmas. We first present those. For the part below, $d_G(u, v)$ denotes the
length of a shortest directed path from $u$ to $v$ in a directed graph $G$, $|P|$ denotes the length of some path $P$ and
$P_{a \rightarrow b}$ denotes the subpath $a \rightarrow b$ that lies on $P$. The symbol $\circ$ denotes the concatenation of two directed paths.
\begin{lemma} \label{dhref}
Upon arrival of some $a \in L$, assume the algorithm flips the shortest path $P$ that ends at some unsaturated node $b$.
If $u$ and $v$ are two nodes on $P$ such that $u$ appears before $v$, then we have
$$ d_{H^{res}}(u, v) \geq d_{G^{res}}(u, v) $$
where $G^{res}$ is the residual graph before the flip of $P$ and $H^{res}$ is residual the graph after the flip of $P$.
\end{lemma}
\begin{proof}
First note that $H^{res}$ is just $G^{res}$ with the edges on $P$ flipped. This reversed path is denoted by $P^{rev}$ and
we write $P$ as $$ P = P_{a \rightarrow u} \circ P_{u \rightarrow v} \circ P_{v \rightarrow b} .$$
Let $Q$ be a shortest $u \rightarrow v$ path in $H^{res}$. Then there are two cases. Either $Q$ do not share any edge with $P^{rev}$ or it shares at least an edge with $P^{rev}$.
If $Q$ do not share any edge with $P^{rev}$, then we can write a path $P'$ from $a$ to $b$ in $G^{res}$ as
$$ P' = P_{a \rightarrow u} \circ Q \circ P_{v \rightarrow b} .$$
Since $P$ is a shortest $a \rightarrow b$ path in $G^{res}$ by our assumption, we have $|P'| \geq |P|$.
Moreover, $|P_{u \rightarrow v}| = d_{G^{res}}(u, v)$ as $P$ is a shortest path. \\
Thus, $ |Q| = d_{H^{res}}(u, v) \geq d_{G^{res}}(u, v) = |P_{u \rightarrow v}| .$
For the second case, we observe that if we concatenate the edges in $Q \setminus P^{rev}$, we get a path that starts at $u$ and ends at $v$.
Since this path is also present in $G^{res}$, its length is at least $d_{G^{res}}(u, v)$.
This implies $|Q| = d_{H^{res}}(u, v)$ is strictly greater than $d_{G^{res}}(u, v)$ in this case.
Hence, in general we have $d_{H^{res}}(u, v) \geq d_{G^{res}}(u, v).$
\end{proof}
\begin{lemma} \label{hnondec}
For any node x, height(x) is non-decreasing over arrivals.
\end{lemma}
\begin{proof}
We use the definitions from the previous lemma. Again, assume some $a \in L$ arrives and the algorithm flips a shortest $a \rightarrow b$ path $P$.
If $height(x) = h$ in $G^{res}$, then the lemma says that for any unsaturated node $y$ in $H^{res}$, the length of a shortest $x \rightarrow y$ path in $H^{res}$ is at least of length $h$.
Let $T$ be a shortest $x \rightarrow y$ path in $H^{res}$. We give a proof by showing $|T| \geq h$ in all possible cases.
First assume that $T$ does not share any edge with $P^{rev}$. If $y \neq b$, then this means flipping $P$ does not affect $T$. So, $T$ is also a shortest $x \rightarrow y$ path in $G^{res}$
and consequently, its length must be $h$. If $y = b$, then it is possible for $y$ to become saturated after the flip. If that happens, the shortest path from $x$ to some unsaturated node can not
possibly decrease as this would imply there exists a node $b'$ such that $x$ is closer to $b'$ than $b$ in $G^{res}$.
In short, $|T| \geq h$ if $T$ does not share any edge with $P^{rev}$.
Now assume $T$ shares at least an edge with $P^{rev}$. Let $u$ and $v$ be the first and the last node that appears in $T \cap P$ as we traverse the path $x \rightarrow y$.
First assume that $u$ appears before $v$ in $P$.
So if $Q$ denotes a shortest $u \rightarrow v$ path then, $$ T = T_{x \rightarrow u} \circ Q \circ T_{v \rightarrow y} .$$
Now we observe that $$ T' = T_{x \rightarrow u} \circ P_{u \rightarrow v} \circ T_{v \rightarrow y}$$ is a $x \rightarrow y$
path in $G^{res}$ and hence, $|T'| \geq h$. \\
The length of $T$ is given by
$$ |T| = |T_{x \rightarrow u}| + d_{H^{res}}(u, v) + |T_{v \rightarrow y}|$$
and the length of $T'$ is given by
$$ |T'| = |T_{x \rightarrow u}| + d_{G^{res}}(u, v) + |T_{v \rightarrow y}| .$$
Due to Lemma \ref{dhref}, we must have $|T| \geq |T'| \geq h$.
Now let $v$ appear before $u$ in $P$. First, we observe that
$ P_{a \rightarrow v} \circ T_{v \rightarrow y} $ is a path from $a$ to an unsaturated node. Hence, we must have
$$ |P_{a \rightarrow v}| + |T_{v \rightarrow y}| \geq |P| .$$
If we write $P$ as $ P = P_{a \rightarrow v} \circ P_{v \rightarrow u} \circ P_{u \rightarrow b} $, we can see that this inequality implies
$$ |T_{v \rightarrow y}| \geq |P_{v \rightarrow u}| + |P_{u \rightarrow b}| .$$
We further observe that $T_{x \rightarrow u} \circ P_{u \rightarrow b}$ is a path
from $x$ to an unsaturated node. Hence,
$$ |T_{x \rightarrow u}| + |P_{u \rightarrow b}| \geq h .$$
Finally, by summing the last two inequality we see that
$$ |T| \geq |T_{x \rightarrow u}| + |T_{v \rightarrow y}| \geq h .$$
\end{proof}
\begin{lemma} \label{largeheight}
The number of nodes in $L$ with height at least $2h + 1$ is at most $|L|/2^h.$
\end{lemma}
\begin{proof}
The claim holds for $h = 0$. We analyze the case $h \geq 1$. Let $S_0 \subseteq L$ denote
the nodes with height at least $2h + 1 \geq 3$. Let $S_1$ be $N(S_0)$. Then, by Lemma \ref{halls} we have $|S_1| \geq |S_0|/K$.
Now observe that all the nodes in $S_1$ must be saturated. If they were not, some of the nodes in $S_0$ would have height 1
which would contradict with our assumption. Let $S_2$ be the set of nodes that are matched with nodes in $S_1$. Note that $S_2$ is
the set of nodes with height at least $2h-1$.
By Lemma \ref{sizeofsat}, we have $|S_2| = 2K \cdot |S_1| \geq 2|S_0|.$ By defining the sets $S_3, S_4, \dots, S_{2h}$ similary and repeating the argument up to
$S_{2h}$, we see that $2^h \cdot |S_0| \leq |S_{2h}|$. Since $|S_{2h}| \leq |L|$ as $S_{2h} \subseteq L$, we conclude that $2^h \cdot |S_0| \leq |L|$.
\end{proof}
\begin{lemma} \label{sumheight}
Assume the algorithm flips the path $P$ upon arrival of some node $a \in L$.
Let $height'(\cdot)$ denote the height function after the flip and let $height(\cdot)$ denote the height function before the flip.
If the match of $y$ is changed from $x$ to $x'$ after the flip, then $height'(x') \geq height(x) + 2.$
\end{lemma}
\begin{proof}
We again write the sequence of nodes in $P$ from $a$ to some unsaturated node $b$ as,
$$ a = x_0, y_1, x_1, y_2, \dots, x_{k-1}, y_{k} = b $$
where every $x_i \in L$ and every $y_i \in L$.
We observe that $height(y_k) = 0$ and $height(x_i) = 2(k - i) -1$ as heights decrease by $2$ for each $x_i$ when traversing from $a$ to $b$.
After flipping $P$, $y_i$ gets matched with $x_i$ and it was previously matched with $x_{i+1}$.
Using the fact that $height(\cdot)$ is non-decreasing (Lemma \ref{hnondec}), we see
\begin{align*}
height'(x_i) &\geq height(x_i) \\
& = height(x_{i+1}) + 2
\end{align*}
\end{proof}
The last lemma shows that each swap increases the sum of heights at least by $2$. Further, we have a bound on the number nodes with height $2h + 1$ for any $h \geq 0.$
Authors make use of these two observations to prove Theorem \ref{btheo}. We restate it below.
\begin{proof}[Proof of Theorem \ref{btheo}]
Define the potential as
$$ \Phi = \sum_{x \in L}\frac{height(x) -1}{2} .$$
Clearly, it starts at $0$ as the first arriving nodes height is $1$.
A key observation about this potential is that each swap causes it to increase by at least 1 due to Lemma \ref{sumheight}. To see this more clearly, notice that
instead of summing over nodes in $L$, we could have summed over nodes in $R$ and then sum over their matches in $L$. We know from the Lemma \ref{sumheight} that
each swap causes a node in $R$ to match with a node in $L$ whose height is at least $2$ greater. Due to division by $2$, this will increase
potential at least by $1$. Since the potential starts at $0$ and since each swap causes it to increase at least by $1$, the value of this potential after $n$
arrivals is an upper bound on the number of swaps after $n$ arrivals. We compute the potential after $n$ arrivals and see that
\begin{align*}
\Phi &= \sum_{x \in L}\frac{height(x) -1}{2} \\
& = \sum_{h=0}^{\infty} h \cdot (\text{number of nodes in L with height 2h + 1}) \\
& \leq |L| \cdot \sum_{h=0}^{\infty} \frac{h}{2^h} \\
& = 2|L| \\
& = 2n,
\end{align*}
where the third line follows from Lemma \ref{largeheight}.
\end{proof}
As before, we note the tradeoff between the number of recourse operations and the constraint. For a $CK$ matching such that $C \geq 2$,
the result of Lemma \ref{largeheight} can be generalized as $|L|/C^h$ as the result of Lemma \ref{sizeofsat} would have changed to $|L| \cdot CK$.
Then, we would have $$ |L| \cdot \sum_{h=0}^{\infty} \frac{h}{C^h} = n \cdot \frac{C}{(C-1)^2} $$ as an upper bound on the number of swaps
after $n$ arrivals.
\section{Conclusion}
Throughout the paper, the common theme was improving the competitive ratio by recourse. We presented
two problems in which no algorithm can do better than logarithmic competitive ratio when there is no recourse.
Moreover, we have seen even if we allow recourse to improve the competitive ratio, achieving the optimal offline solution can be costly.
In contrast to that, we observed approximating the optimal solution within a constant factor costs much less, e.g. the
algorithms we presented just needed a linear number of recourse operations in terms of arrivals.
We also did a brief analysis in an adversary against the algorithm setting for the shortest path algorithm and showed that it can be helpful to establish bounds.
In general, we have observed recourse can be a good approximation tool.
\newpage
|
2,877,628,090,759 | arxiv | \section{Introduction}
\label{intro}
Among the best understood class of features in Saturn's rings are density waves. These structures are usually tightly wound multi-armed spiral patterns that are generated at mean-motion resonances with either Saturn's moons or structures in the planet's interior. The theory behind these patterns is very well developed \citep{GT82, Shu84}, enabling ring parameters like the local surface mass density to be derived from observable wave properties \citep{Cuzzi81, Esposito83, Tiscareno07, Esposito10}. At the same time, the predictable properties of these waves allow wavelet-based filtering techniques to identify wave-like structures that are not apparent in individual observations \citep{HN16}. These tools are also starting to reveal unexpected waves with unusual properties, including several features that appear to be {\em axisymmetric} density waves (i.e.\ waves with azimuthal wavenumber $m=0$).
The clearest example of an axisymmetric density wave lies between the Bessel and Barnard Gaps in the Cassini Division. This region had been designated the 1.994 $R_s$ ringlet prior to the Cassini mission \citep{NCP90, French93} when the two gaps were thought to be a single gap inhabited by a ringlet nearly as wide as the gap. However, Cassini observations later revealed that the Bessel and Barnard gaps likely had different dynamical origins \citep{Colwell09, French10, Hedman10, French16} and so it seemed more appropriate to consider them as separate gaps. Hence the region in between them was not given a formal name.
\pagebreak
As shown in Figure~\ref{baedge}, the region between the Bessel and Barnard gaps contains quasi-periodic optical depth variations and the locations of the peaks and troughs vary from one observation to another. This is reminiscent of typical density waves, but this particular structure is anomalous in that there is no resonance with any of Saturn's moons that would naturally generate this wave. While the Prometheus 5:4 inner Lindblad resonance lies at 120,304 km (near the inner edge of the Barnard gap) the wave generated by this resonance should propagate outwards, away from the region of interest. \citep[In fact, this resonance instead excites a five-lobed radial variation in the edge's position, see ][]{French16}.
A wavelet-based analysis of this structure provides strong evidence that it is an axisymmetric density wave consisting of concentric zones of varying density. Such a wave is a valid solution to the relevant equations of motion for ring material, and the standard theory of density waves can naturally be extended to this case, yielding sensible estimates of the local surface mass density. However, such a structure is not easily excited by any mean-motion resonance with a satellite because the gravitational perturbations from a moon must always vary with longitude. An axisymmetric wave instead requires some process that induces all the particles to have finite orbital eccentricities and to pass through pericenter at the same time regardless of longitude. We propose that interference among the observed normal-mode oscillations in the position of the Barnard gap's inner edge can give rise to perturbations with the correct form to generate an axisymmetric density wave.
\begin{figure}
\centerline{\resizebox{3.5in}{!}{\includegraphics{profplot_4panel_pr54_011419.pdf}}}
\caption{A few representative optical depth profiles of the region between the Bessel and Barnard Gaps derived from stellar occultations obtained by the VIMS instrument onboard the Cassini spacecraft.}
\label{baedge}
\end{figure}
\begin{figure}
\centerline{\resizebox{3.5in}{!}{\includegraphics{profplot_4panel_dawes_011419.pdf}}}
\caption{A few representative optical depth profiles of the Dawes ringlet derived from stellar occultations obtained by the VIMS instrument onboard the Cassini spacecraft.}
\label{dawes}
\end{figure}
\begin{figure}
\centerline{\resizebox{3.5in}{!}{\includegraphics{profplot_4panel_bringedge_011419.pdf}}}
\caption{A few representative optical depth profiles of the B ring outer edge derived from stellar occultations obtained by the VIMS instrument onboard the Cassini spacecraft.}
\label{bedge}
\end{figure}
This wave would not be the first axisymmetric structure to be found in planetary rings. Prior analyses of occultation data for Uranus' rings revealed that the $\gamma$ ring exhibits axisymmetric variations in its radial position \citep{French91}. The $\gamma$ ring is also found close to a mean motion resonance with one of Uranus' moons (specifically, the 6:5 inner Lindblad resonance with Ophelia), so the dynamical environment of the $\gamma$ ring is similar to that of the material around the Barnard Gap. The axisymmetric motion of Uranus' $\gamma$ ring and the axisymmetric wave in the Cassini Division could therefore represent a previously unrecognized class of ring features that are excited in the vicinity of resonantly-confined ring material.
Indeed, other resonantly-confined edges of Saturn's rings, like the outer edges of the B ring and the Dawes ringlet (which are perturbed by the Mimas 2:1 and 3:1 inner Lindblad resonances, respectively) are associated with periodic optical depth variations that may be additional examples of axisymmetric density waves. Unfortunately, the wavelet-based techniques that could characterize the patterns near the Barnard gap do not appear to work on these structures, making their interpretation less certain.
Figure~\ref{dawes} shows that the Dawes ringlet contains periodic opacity variations of similar scale to those seen interior to the Barnard Gap. These variations also extend interior to the Mimas 3:1 and Pandora 2:1 resonances in this region, and so cannot just be density waves driven by those satellites (which would propagate outwards). This feature might be an axisymmetric density wave, but other structures in the region complicate its interpretation.
The outer part of the B ring is far more complex (see Figure~\ref{bedge}), and is perturbed by a number of strong resonances. However, some quasi-periodic signals can be seen around 117,300 km. These are not in the right place to be generated by the known satellite resonances. Furthermore, imaging data support the idea that these structures are another example of an axisymmetric density wave (see Section~\ref{bim} below).
The rest of this paper provides a detailed analysis of these structures. Section~\ref{background} provides the relevant theoretical background for density waves in order to clarify the expected properties of axisymmetric waves and how they can be distinguished from other ring features. Section~\ref{barnard} describes our analysis of the region between the Bessel and Barnard gaps and the evidence that there is an axisymmetric density wave in this region. Section~\ref{model} then discusses how this sort of axisymmetric wave could be excited by interference among the various normal modes on resonantly-confined gap edges. Section~\ref{others} examines the material in the vicinity of other resonantly confined gap edges and whether these might also contain axisymmetric density waves. Finally, Section~\ref{summary} summarizes the results and implications of these analyses.
\section{Theoretical Background}
\label{background}
The theory behind spiral density waves is described in several classic papers \citep[e.g.][]{Shu84}. However, for the sake of convenience we will review the relevant equations here in order to clarify how axisymmetric waves should behave.
Generically, density waves arise when a dense ring of material is subjected to a periodic perturbing force that generates a term $\mathcal{U}$ in the potential that varies periodically with time $t$ and/or inertial longitude $\lambda$:
\begin{equation}
\mathcal{U} \propto \exp[i(|m|\lambda-\omega t)],
\label{perturb}
\end{equation}
where $\omega$ is the perturbation frequency and $m$ is a integer that can be positive, negative or 0, with $m=0$ corresponding to an axisymmetric perturbation\footnote{Note that the sign convention used here for the phase parameter in the exponential is the opposite of that used in \citet{Shu84}}. This sort of perturbation has its strongest effect on the orbital motion of the ring particles at locations where the ring-particles' radial epicyclic frequency $\kappa$ and orbital mean motion $n$ satisfy this relationship (for either choice of sign):
\begin{equation}
\omega=|m|n\mp \kappa,
\label{resrel}
\end{equation}
At these locations, we can re-write the frequency $\omega$ in Equation~\ref{perturb} in terms of the mean motion and radial epicyclic frequency. Furthermore, for each ring particle $\lambda-nt$ is a constant, and so the perturbation has the following form:
\begin{equation}\mathcal{U} \propto
\exp[i(|m|\lambda-|m|nt\pm\kappa t)]=\exp[\pm i\kappa t].\end{equation} The perturbations on each particle therefore have the same frequency as the particle's radial epicyclic motion, which is the appropriate condition for a Lindblad resonance that will excite and organize the ring particles' epicyclic motions. Furthermore, if the ring has a finite surface mass density $\sigma$ where this resonance occurs, then these localized disturbances will propagate radially across the rings, forming the pattern of surface mass density variations known as a density wave.
In general, the density variations associated with any density wave can be quantified using the following expression.
\begin{equation}
\sigma =\sigma_0 + \Re\left[A_\sigma e^{i(\phi_r+\phi_{\lambda t})}\right],
\end{equation}
where $\sigma_0$ is the background surface mass density and $A_\sigma$ is a (radius-dependent) amplitude of the density variations, while $\phi_{\lambda t}$ and $\phi_r$ are phase parameters that depend on longitude/time and radius, respectively. The functional forms of these two phases are determined by the characteristics of the perturbation and the ring's background surface mass density.
First, consider the longitude/time dependent phase $\phi_{\lambda t}$. Steady state solutions for the density variations only exist if the pattern as a whole tracks the potential, so $\phi_{\lambda t}$ must have the same form as the phase in the perturbing potential shown in Equation~\ref{perturb}:
\begin{equation}
\phi_{\lambda t}=|m|\lambda -\omega t
\label{phidef}
\end{equation}
The entire pattern therefore has azimuthal wavenumber $|m|$. Also, if $m \ne 0$, then this pattern rotates at a speed $\Omega_p=\omega/|m|$, which can be written in terms of the mean motion and radial epicyclic frequency at the resonant location $n_L$ and $\kappa_L$:
\begin{equation}
\Omega_p(m\ne 0)=n_L\mp\frac{1}{|m|}\kappa_L=n_L-\frac{1}{m}\kappa_L
\label{eqpat}
\end{equation}
where for the second equality we have chosen the sign of $m$ such that positive $m$ correspond to situations where the pattern speed is slower than the orbital mean motion (i.e. an Inner Lindblad Resonance) while negative $m$ correspond to cases where the pattern speed is faster than the local orbital mean motion (i.e. an Outer Lindblad Resonance). Of course, for the axisymmetric case ($m=0$), there is no sensible definition of a real azimuthal pattern speed. However, since in this case $\phi_{\lambda t}=-\omega t$, a reasonable analog of the pattern speed is
\begin{equation}
\Omega_p(m =0)=\omega=\kappa_L
\label{eqpat0}
\end{equation}
where the second equality uses the general definition of $\omega$, and we have chosen the positive sign in order to ensure that $\Omega_p$ and $\omega$ are sensibly positive.
While $\phi_{\lambda t}$ depends only on the properties of the perturbation, the radius-dependent part of the wave phase $\phi_r$ is determined by a dispersion relation that relates the wave frequency $\omega$ to the radial wavenumber of the pattern $k$, which is simply the radial derivative of the wave phase at a fixed longitude and time:
\begin{equation}
k(r)=\frac{\partial \phi_r}{\partial r}.
\end{equation}
In the limit where the velocity dispersion of the ring particles is negligible (which is appropriate for the ring regions considered here), the relevant dispersion relation is \citep{Shu84}:
\begin{equation}
(\omega-|m|n)^2=\kappa^2-2\pi G\sigma_0|k|.
\label{dispersion}
\end{equation}
So long as the wave is observed at radii $r$ close to the radius of the resonance $r_L$ (i.e $|r-r_L|<<r_L$) and the apsidal precession rate is much less than the mean motion (i.e. $\kappa \simeq n$), this expression yields the standard expression for the wavenumber of a wave (valid for all $m \ne 1$):
\begin{equation}
|k(r)|=\frac{3(m-1) M_P(r-r_L)}{2\pi\sigma_0r_L^4},
\label{keq}
\end{equation}
where $M_P$ is the mass of the planet. Note that if $m>1$ then the right hand side of this equation is only positive if $r>r_L$, which is consistent with the fact that such density waves only exist outside Inner Lindblad Resonances. By contrast, if $m<1$ (including $m=0$), then the wave only exists interior to the resonance. Furthermore, in order for the waves to propagate away from the resonant location in the appropriate direction, their group velocity $v_g=\partial \omega /\partial k$ also needs to be positive if $m>1$ and negative if $m<1$. Taking the appropriate derivatives of Equation~\ref{dispersion} reveals that these conditions will be satisfied provided $k$ is positive (again, opposite the Shu 1984 convention). Note that since $m=0$ waves behave like those generated by standard Outer Lindblad Resonances in this regard, it is sensible to regard $m=0$ resonances as a member of that group.\footnote{Note also that the dispersion relation in Equation~\ref{dispersion} implies that $m=0$ waves can only propagate where
$\kappa > |\omega|$, i.e., interior to the Lindblad resonance} Also, for all $m$, the magnitude of the group velocity is given by the standard expression:
\begin{equation}
|v_g|=\frac{\pi G\sigma_0}{\kappa_L}.
\label{group}
\end{equation}
Since $k=\partial \phi_r/\partial r$ is positive for both types of waves, the phase always increases with increasing radius, which is consistent with how phases are defined for the wavelet transformations (see below). Furthermore, at a fixed longitude, the location of maxima or minima will drift outwards over time for all $m$. In other words, the phase velocity $v_p=\omega/k$ is positive definite for both types of waves, consistent with all the waves with $m\ne 0$ being trailing spiral patterns. Finally, we can integrate Equation~\ref{keq} for $|k|=k$ to obtain the following expression for the radial phase parameter in the vicinity of the resonance:
\begin{equation}
\phi_r(r)=\frac{3(m-1) M_P(r-r_L)^2}{4\pi\sigma_0 r_L^4} +\phi_0
\label{phir}
\end{equation}
where $\phi_0$ is a constant phase offset.
In summary, standard density wave theory predicts that axisymmetric $m=0$ density waves should have the following properties:
\begin{itemize}
\item They should exist interior to the resonant location where $\omega=\kappa$.
\item They should have negative group velocities, which means they should propagate and carry energy and angular momentum inwards.
\item They should have positive phase velocities, which means the positions of individual peaks and troughs should move outwards over time.
\item They should have radial wavenumbers given by the expression $k(r)=\frac{3 M_P|r-r_L|}{2\pi\sigma_0r_L^4}$
\end{itemize}
\section{Analysis of the region between the Barnard and Bessel Gaps}
\label{barnard}
The region between the Barnard and Bessel Gaps shown in Figure~\ref{baedge} contains the feature that can be most convincingly identified as an axisymmetric density wave based on wavelet analysis of multiple occultation profiles. The relevant occultation data are described in Section~\ref{obs}, while Section~\ref{wave} provides the wavelet-based methods employed to characterize this particular pattern. Finally, Section~\ref{results} presents the resulting evidence that the region interior to the Barnard Gap does indeed contain an axisymmetric wave.
\begin{table*}
\caption{Summary of occultations used in this study}
\label{occs}
\centerline{\resizebox{6.5in}{!}{
\begin{tabular}{|c|c|c|c||c|c|c|} \hline
Star & Rev$^a$ & i/e$^b$ & B$^c$ & ET$^d$ & Longitude$^e$ & m=0 Phase$^f$ \\ & & & (degrees) & (seconds) & (degrees) & (degrees) \\ \hline
RHya & 036 & i & -29.4 & 220943101. & 195.4 & 349.6 \\
alpSco & 013 & i & -32.2 & 177810304. & 288.6 & 324.7 \\
alpSco & 013 & e & -32.2 & 177817667. & 358.0 & 262.6 \\
alpSco & 029 & i & -32.2 & 212528274. & 194.6 & 359.9 \\
alpAur & 041 & i & 50.9 & 227942859. & 8.9 & 27.2 \\
RCas & 065 & i & 56.0 & 262010525. & 29.6 & 144.0 \\
gamCru & 071 & i & -62.3 & 266187755. & 186.7 & 213.3 \\
gamCru & 072 & i & -62.3 & 266804308. & 186.3 & 56.2 \\
gamCru & 073 & i & -62.3 & 267420458. & 186.0 & 262.6 \\
gamCru & 077 & i & -62.3 & 269852643. & 185.1 & 281.2 \\
gamCru & 078 & i & -62.3 & 270461132. & 184.9 & 192.1 \\
betGru & 078 & i & -43.4 & 270509701. & 263.3 & 142.7 \\
RSCnc & 080 & i & 30.0 & 271864841. & 56.1 & 240.0 \\
RSCnc & 080 & e & 30.0 & 271884854. & 155.6 & 71.3 \\
gamCru & 081 & i & -62.3 & 272314456. & 183.2 & 50.1 \\
gamCru & 082 & i & -62.3 & 272950229. & 182.9 & 91.0 \\
RSCnc & 085 & i & 30.0 & 275048824. & 57.9 & 41.5 \\
RSCnc & 085 & e & 30.0 & 275068331. & 153.4 & 237.1 \\
gamCru & 086 & i & -62.3 & 275497768. & 182.2 & 217.3 \\
RSCnc & 087 & i & 30.0 & 276322450. & 58.8 & 105.9 \\
RSCnc & 087 & e & 30.0 & 276341685. & 152.5 & 303.7 \\
gamCru & 089 & i & -62.3 & 277402848. & 181.9 & 359.0 \\
gamCru & 093 & i & -62.3 & 280038476. & 203.3 & 102.7 \\
gamCru & 094 & i & -62.3 & 280675202. & 192.0 & 135.7 \\
epsMus & 094 & i & -72.8 & 280700017. & 257.0 & 286.5 \\
epsMus & 094 & e & -72.8 & 280717828. & 316.7 & 136.4 \\
gamCru & 096 & i & -62.3 & 282008576. & 187.6 & 56.4 \\
gamCru & 097 & i & -62.3 & 282697254. & 187.5 & 11.4 \\
gamCru & 100 & i & -62.3 & 285027485. & 213.1 & 169.4 \\
gamCru & 101 & i & -62.3 & 285854639. & 213.1 & 37.1 \\
gamCru & 102 & i & -62.3 & 286679833. & 212.9 & 281.4 \\
betPeg & 104 & i & 31.7 & 288910932. & 353.7 & 195.0 \\
RCas & 106 & i & 56.0 & 291029840. & 50.6 & 334.3 \\
betPeg & 108 & i & 31.7 & 292213793. & 4.3 & 74.5 \\
alpAur & 110 & i & 50.9 & 295141170. & 317.1 & 239.1 \\
alpAur & 110 & e & 50.9 & 295160215. & 241.7 & 78.6 \\
alpSco & 115 & i & -32.2 & 302010739. & 170.6 & 294.1 \\
betPeg & 170 & e & 31.7 & 397982124. & 87.5 & 251.9 \\
betPeg & 172 & i & 31.7 & 401611701. & 302.8 & 257.5 \\
lamVel & 173 & i & -43.8 & 403829302. & 131.2 & 284.9 \\
RLyr & 176 & i & 40.8 & 407915130. & 219.9 & 44.6 \\
RLyr & 176 & e & 40.8 & 407945440. & 164.1 & 149.1 \\
WHya & 179 & i & -34.6 & 411898308. & 130.1 & 309.6 \\
RLyr & 180 & i & 40.8 & 412507880. & 217.2 & 211.4 \\
RLyr & 180 & e & 40.8 & 412535706. & 166.4 & 336.9 \\
WHya & 180 & i & -34.6 & 413047699. & 130.4 & 341.2 \\
WHya & 181 & i & -34.6 & 414196938. & 130.4 & 14.0 \\
RHya & 185 & i & -29.4 & 418337590. & 45.1 & 31.6 \\
RHya & 185 & e & -29.4 & 418351569. & 329.7 & 273.8 \\
RCas & 185 & i & 56.0 & 418059013. & 313.3 & 219.8 \\
muCep & 185 & e & 59.9 & 418027873. & 67.6 & 122.3 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|c||c|c|c|} \hline
Star & Rev$^a$ & i/e$^b$ & B$^c$ & ET$^d$ & Longitude$^e$ & m=0 Phase$^f$ \\
& & & (degrees) & (seconds) & (degrees) & (degrees) \\ \hline
WHya & 186 & e & -34.6 & 419161471. & 292.1 & 286.9 \\
RDor & 186 & i & -56.3 & 419057192. & 161.2 & 85.9 \\
gamCru & 187 & i & -62.4 & 419913820. & 130.8 & 65.2 \\
gamCru & 187 & e & -62.4 & 419936428. & 246.4 & 234.7 \\
RDor & 188 & i & -56.3 & 420711012. & 160.7 & 185.6 \\
RDor & 188 & e & -56.3 & 420716266. & 195.7 & 141.3 \\
WHya & 189 & e & -34.6 & 421642104. & 291.2 & 257.2 \\
RCar & 191 & i & -63.5 & 423385180. & 126.8 & 324.5 \\
RCas & 191 & i & 56.0 & 423126448. & 291.5 & 345.4 \\
muCep & 191 & i & 59.9 & 423048531. & 286.9 & 282.1 \\
muCep & 193 & i & 59.9 & 425115241. & 287.0 & 141.4 \\
RCas & 194 & e & 56.0 & 426266707. & 89.6 & 155.5 \\
2Cen & 194 & e & -40.7 & 426591441. & 249.1 & 298.2 \\
muCep & 195 & i & 59.9 & 427182862. & 335.0 & 353.0 \\
WHya & 196 & i & -34.6 & 430037069. & 157.8 & 54.3 \\
WHya & 196 & e & -34.6 & 430051310. & 228.2 & 294.3 \\
WHya & 197 & i & -34.6 & 432104346. & 158.7 & 268.8 \\
WHya & 197 & e & -34.6 & 432118271. & 227.3 & 151.5 \\
RLyr & 198 & i & 40.8 & 435185387. & 267.4 & 218.1 \\
L2Pup & 199 & e & -41.9 & 438488221. & 318.4 & 97.9 \\
RLyr & 199 & i & 40.8 & 439304272. & 248.7 & 59.2 \\
RLyr & 199 & e & 40.8 & 439331815. & 132.9 & 187.0 \\
RLyr & 200 & i & 40.8 & 442043470. & 264.0 & 9.9 \\
L2Pup & 201 & i & -41.9 & 446129569. & 96.3 & 127.4 \\
RLyr & 202 & i & 40.8 & 447535799. & 303.7 & 154.0 \\
RLyr & 202 & e & 40.8 & 447559049. & 70.3 & 318.0 \\
lamVel & 203 & i & -43.8 & 449043024. & 40.0 & 49.3 \\
lamVel & 203 & e & -43.8 & 449096026. & 336.7 & 322.5 \\
L2Pup & 205 & i & -41.9 & 456837673. & 130.3 & 226.6 \\
RLyr & 206 & i & 40.8 & 458836881. & 294.6 & 294.9 \\
RLyr & 208 & e & 40.8 & 464399829. & 64.3 & 203.7 \\
WHya & 236 & i & -34.6 & 517599124. & 116.5 & 336.1 \\
2Cen & 237 & i & -40.7 & 519786996. & 96.4 & 254.1 \\
2Cen & 237 & e & -40.7 & 519835092. & 28.8 & 208.7 \\
alpSco & 237 & i & -32.2 & 520137664. & 228.1 & 178.2 \\
alpSco & 237 & e & -32.2 & 520153854. & 270.7 & 41.8 \\
betPeg & 237 & i & 31.7 & 520424269. & 265.5 & 282.4 \\
betPeg & 237 & e & 31.7 & 520430571. & 231.7 & 229.3 \\
alpSco & 238 & i & -32.2 & 522206348. & 229.6 & 20.9 \\
alpSco & 238 & e & -32.2 & 522221476. & 269.2 & 253.4 \\
alpSco & 239 & i & -32.2 & 523688965. & 135.7 & 123.6 \\
RCas & 239 & i & 56.0 & 523881209. & 29.9 & 303.1 \\
RCas & 239 & e & 56.0 & 523893640. & 91.9 & 198.3 \\
rhoPer & 239 & e & 45.3 & 523938719. & 148.2 & 178.4 \\
alpSco & 241 & i & -32.2 & 525831418. & 112.4 & 64.4 \\
alpSco & 241 & e & -32.2 & 525849007. & 2.4 & 276.2 \\
alpSco & 243 & e & -32.2 & 527916676. & 1.8 & 127.4 \\
RCas & 243 & i & 56.0 & 528004792. & 351.2 & 104.6 \\
alpSco & 245 & i & -32.2 & 529597290. & 108.3 & 1.1 \\
alpSco & 245 & e & -32.2 & 529610004. & 359.1 & 254.0 \\
gamCru & 245 & e & -62.4 & 529546263. & 296.8 & 71.3 \\
\hline
\end{tabular}}}
\bigskip
$^a$ Cassini Orbit Around Saturn
$^b$ i=ingress portion of occultation, e=egress portion of occultation
$^c$ Ring opening angle to the star (positive numbers correspond to stars in Saturn's northern hemisphere)
$^d$ Ephemeris Time in seconds past J2000 (TDB) when the spacecraft observed the inner edge of the Barnard Gap.
$^e$ Inertial longitude measured from the ascending node of Saturn's ringplane on the J2000 reference plane
$^f$ Expected phase of an $m=0$ wave launched from the Prometheus 5:4 resonance (i.e. with $\omega= 728.28^\circ$/day), measured relative to the epoch time 2008-001T12:00:00 (ET 252460865.184)
\vspace{2in}
\end{table*}
\subsection{Occultation Observations}
\label{obs}
This examination of the region around the Barnard Gap uses stellar occultation data obtained by the Visual and Infrared Mapping Spectrometer (VIMS) onboard the Cassini Spacecraft \citep{Brown04}. While in its standard operating mode VIMS obtains spatially resolved spectra of various objects in the Saturn system, this instrument can also operate in a mode where it repeatedly measures the spectrum of a star as the rings pass between it and the spacecraft. In this mode, a precise timestamp is appended to each spectrum to facilitate reconstruction of the observation geometry \citep{Brown04}.
We compute both the radius and inertial longitude in the rings that the starlight passed through using a combination of the timing information accompanying each brightness measurement and the appropriate SPICE kernels \citep{Acton96}. Note that the information encoded in these kernels has been determined to be accurate to within one kilometer, and fine-scale adjustments based on the positions of circular ring features enable these estimates to be refined to an accuracy of $\sim$ 150 m \citep{French17}.
Table~\ref{occs} lists the 101 occultation observations that we used in this analysis. This includes essentially all the occultations obtained prior to 2017 that covered the relevant part of the Cassini Division with adequate resolution and signal-to-noise to discern the quasi-periodic structures shown in Figure~\ref{baedge}. This large set of occultations spans an entire decade and includes observations obtained over a broad range of inertial longitudes, which ensures that we can uniquely identify the $m$-number and pattern speed of wave-like structures in the ring. However, for some aspects of this analysis we focus our attention on a sub-set of these observations: the occultations of the star $\gamma$ Crucis obtained between Cassini ``Revs" (that is, orbits) 71 and 102 in late 2008 and early 2009. All of these occultations use the same star and occurred at about the same inertial longitude, and have very good signal-to-noise, all of which facilitates comparisons among these profiles.
For this particular study, we only consider data obtained at wavelengths between 2.87 and 3.00 microns, where the rings are especially dark and so provide a minimal background to the stellar signal. Together with the highly linear response of the instrument \citep{Brown04}, this low ring background means that the raw signal is directly proportional to the transmission $T$ through the rings. In practice, the transmission is estimated by normalizing the signal to unity during a time when the star was not obscured by the rings. Whenever possible, the selected time period corresponds to when the star was visible through the Huygens Gap in the Cassini Division (i.e. 117,700-117,750 km from Saturn center), and if this region was not available, a time period when the star was outside the main rings (i.e. more than 145,000 km from Saturn's center) was used. Also, any instrumental background level was removed by subtracting a constant offset from the data equal to the mean signal level when the star was behind an opaque part of the B ring (105,700-106,100 km). The resulting transmission values $T$ can then be transformed into the ring's normal optical depth $\tau_n$ using the standard expression $\tau_n=-\ln(T)\sin|B|$, where $B$ is the elevation angle of the star above the ring (see Table~\ref{occs}). Note that for low optical depth regions like the Cassini Division, $\tau_n$ should be largely independent of the observation geometry and directly proportional to the surface mass density $\sigma$.
\subsection{Wavelet analysis methods}
\label{wave}
We analyze these occultation data using wavelet-based tools developed in \citet{HN16} for isolating wave signatures in Saturn's B ring. These tools are designed to take multiple occultation profiles and combine the data in a manner that isolates signals with pattern speeds and $m$-values consistent with specified density waves. Details of this approach are provided in \citet{HN16}, but for the sake of completeness we will summarize the basic method here.
We begin by taking each occultation profile, interpolating the transmission estimates onto a regular grid of radii with a spacing of 100 meters, converting the transmission values to normal optical depth,\footnote{Note that \citet{HN16} applied the wavelet transformation directly to the transmission profiles instead of the optical depth profiles. This was a sensible choice because that work only considered occultations obtained with similar geometries and ring regions with high optical depth. However, in this case we are considering a broader range of occultation geometries and a ring region with low optical depth. Converting the profiles to optical depth helps make the signals observed at different times more comparable and simplifies the interpretation of the reconstructed wave profiles.} and transforming the profile into a wavelet using the IDL {\tt wavelet} routine \citep{TC98} with a Morlet mother wavelet and parameter $\omega_0=6$.This yields a complex two-dimensional wavelet for each profile $\mathcal{W}_i=\mathcal{A}_ie^{i\Phi_i}$ where $\mathcal{W}_i, \mathcal{A}_i$ and $\Phi_i$ are all functions of both radius $r$ and radial wavenumber $k$. For the signal from a density wave the wavelet phase $\Phi_i$ is equivalent to the local wave phase $\phi_r+\phi_{\lambda t}$. Given the observed longitude $\lambda_i$ and observation time $t_i$ for each occultation, we can compute the following phase parameter
\begin{equation}
\phi_i=|m|\lambda_i-\omega (t_i-t_0)
\end{equation}
where $|m|$ and $\omega$ are the assumed $m$-number and frequency of the perturbation, and $t_0$ is an arbitrary epoch time. Here $t_0$ corresponds to 2008-001T12:00:00 UTC, in order to be consistent with the epoch time used by \citet{French16}. For a wave with the selected $m$ and $\omega$ values, the phase difference $\Phi_i-\phi_i$ will be the same function of radius $r$ ($\phi_r$) for every occultation, and so we can define a phase-corrected wavelet:
\begin{equation}
\mathcal{W}_{\phi,i}(r,k)=\mathcal{W}_i(r,k)e^{-i\phi_i}=\mathcal{A}_i(r,k)e^{i(\Phi_i(r,k)-\phi_i)}
\end{equation}
For any signal with the selected values of $m$ and $\omega$, the phase of this corrected wavelet should be the same for all the occultation profiles. The average phase corrected wavelet of $N$ profiles:
\begin{equation}
\langle \mathcal{W}_\phi(r,k) \rangle=\frac{1}{N}\sum_{i=1}^N\mathcal{W}_{\phi,i}(r,k)
\end{equation}
will therefore be nonzero for such patterns while any other structure will average to zero. Only patterns with the desired $m$ and $\omega$ should therefore remain in the power of the average phase corrected wavelet
\begin{equation}
\mathcal{P}_\phi(r,k)=|\langle \mathcal{W}_\phi(r,k) \rangle|^2=\left|\frac{1}{N}\sum_{i=1}^N\mathcal{W}_{\phi,i}(r,k)\right|^2
\end{equation}
while all other signals are only seen in the average wavelet power:
\begin{equation}
\bar{\mathcal{P}}(r,k)=\langle |\mathcal{W}_\phi(r,k)|^2 \rangle=\frac{1}{N}\sum_{i=1}^N\left|\mathcal{W}_{\phi,i}(r,k)\right|^2
\end{equation}
We also use the ratio of these two powers $\mathcal{R}(r,k)=\mathcal{P}_\phi/\bar{\mathcal{P}}$, which ranges between 0 and 1 \citep{HN16}, as a measure of how much of the signal at a given $r$ and $k$ is consistent with the assumed $m$ and $\omega$.
\subsection{Results}
\label{results}
We used the above tools to search for patterns in the region between the Bessel and Barnard Gaps with values of $m$ between $-10$ and 10 and, for each $m$, a range of pattern speeds corresponding to the expected values of $\omega$ within 100 km of the Barnard gap inner edge (cf. Equation~\ref{eqpat}). The only strong signal found with this approach was obtained with $m=0$, corresponding to an axisymmetric wave with a pattern speed $\Omega_p=\omega$ equivalent to the local epicyclic rate. Figure~\ref{wavegam} shows the signal in the $\gamma$ Crucis occultations obtained in 2008 and 2009. Both the power of the average phase-corrected wavelet and the power ratio show a clear diagonal band running from 120,260 km to 120,290 km, consistent with an inward-propagating density wave launched from somewhere close to the inner edge of the Barnard gap. Also, the signal in the power ratio is strongest when we assume the appropriate pattern speed for an $m=0$ wave launched from that location, providing further evidence that this region does indeed contain an $m=0$ wave (but see the Appendix for a potential ambiguity between $m=0$ density waves and $m=2$ bending waves).
\begin{figure}
\centerline{\resizebox{3.5in}{!}{\includegraphics{waveletcombprocm_barnardset_gamcru_011419.pdf}}}
\caption{Wavelet analysis of the structures interior to the Barnard gap using data from $\gamma$-Crucis occultations obtained in 2008-2009. The top panel shows a representative occultation profile for reference. The second panel shows the average wavelet power, which contains strong signals near the various sharp edges, as well as a suggestive diagonal band between 120,260 and 120,290 km. The third panel shows the power of the average phase-corrected wavelet assuming $m=0$ and a pattern speed of 728.28$^\circ$/day, which corresponds to the radial epicyclic frequency at the location of the Prometheus 5:4 resonance (i.e. 120304 km from Saturn center, marked by a dotted vertical line). The fourth panel shows the power ratio, which only contains the signal from the region interior to the Barnard gap. Note that the second and third panels use a common set of logarithmically-spaced greyscale levels, while the ratio uses linearly-spaced levels between 0 and 1. The bottom panel shows the peak power ratio as a function of radius and assumed epicyclic frequency (expressed in terms of a shift in the resonant radius), which shows that the signal is strong only when the pattern speed is close to the assumed value (marked with a horizontal dotted line). }
\label{wavegam}
\end{figure}
\begin{figure}
\centerline{\resizebox{3.5in}{!}{\includegraphics{waveletcombprocm_barnardset_all_011419.pdf}}}
\caption{Wavelet analysis of the structures interior to the Barnard gap using the full set of occultations. See Figure~\ref{wavegam} for detailed descriptions of the panels. Note that the vertical range of the bottom plot is expanded in this case to better show the signal. There is no peak interior to 120,275 km outside the illustrated range.}
\label{wavefull}
\end{figure}
Figure~\ref{wavefull} shows the same analysis for the full set of 101 occultations. For this full data set, the same wave signal can be seen in the average phase-corrected wavelet between 120,260 and 120,290 km, although the signal in the power ratio plot is quite a bit weaker than it is for the 2008-2009 $\gamma$ Crucis occultations (especially interior to 120,275 km, see below). Even so, it is still the case that no other $m$ value yields a sensible signal for this structure. Moreover, this full data set removes the ambiguity between $m=0$ density waves and $m=2$ bending waves that exists when only the $\gamma$ Crucis occultations are considered (see Appendix). Furthermore, the extended data set yields a much tighter constraint on the wave's pattern speed, which for radii between 120,275 and 120,290 km is 728.28$\pm$0.02$^\circ$/day. This narrow range of pattern speeds implies that the resonance responsible for generating this wave lies at 120304.0 +/- 2 km, which includes the nominal location of the Prometheus 5:4 resonance at 120304.0 km and
the mean position of the Barnard gap inner edge at 120303.7 km (French {\it et al.} 2016a). This location is also consistent with the observed trends in the pattern's wavenumber (that is, $k$ must approach zero at the exact resonance).
In order to verify that this wave-like signal is indeed an axisymmetric density wave, we can take the average phase-corrected wavelet $\langle \mathcal{W}_\phi \rangle$ and apply the inverse wavelet transform to obtain a reconstructed profile of the $m=0$ signal. Note that this profile is itself complex, but the real and imaginary parts are simply the wave profiles at times when $\phi_{\lambda t}$ = 0 and $\pi/2$, respectively. Figure~\ref{profmod} shows the reconstructed profile derived from the full set of occultation profiles.\footnote{Reconstructed profiles generated with subsets of the data like that shown in Figure~\ref{wavegam} have a generally similar structure. However, the wavelength trends are noisier because residual background noise from other structures is not as cleanly removed in the average phase-corrected wavelet when fewer occultations are considered.} This profile was computed using only wavenumbers between $2\pi/1$ and $2\pi/10$ km$^{-1}$ in order to filter out slow variations and high-frequency noise. The resulting profile indeed looks like an inward-propagating density wave, with a wavelength that steadily decreases with distance from the putative resonance.
\begin{figure}
\resizebox{3.5in}{!}{\includegraphics{waveletcombproc3nnp_barnardset_all_011419.pdf}}
\caption{Reconstructed wave profile based on the full set of all occultations used to make Figure~\ref{wavefull}. The top panel shows the mean and range of normal optical depths among the profiles. The second panel shows the reconstructed profile of the fractional optical depth variations for the $m=0$ wave extracted from the average phase-corrected wavelet assuming a pattern speed of 728.28$^\circ$/day, corresponding to a resonant radius of 120304 km (marked by the dotted line). The third profile shows the radial wavenumber of this wave, while the fourth and fifth panels show the estimated surface mass density and opacity derived from this wave (again, assuming a fixed resonant radius of 120304 km). The latter two values are consistent with other estimates of these parameters for comparable regions of the Cassini Division.}
\label{profmod}
\end{figure}
To better quantify the trends in this feature's wavelength, we compute the phase $\phi_r$ from the real and imaginary parts of the profile and then take the radial derivative of this quantity to determine the radial wavenumber of the signal $k$. This parameter shows a linear trend, consistent with Equation~\ref{keq} for a density wave in a region of near-uniform mass density. We can then use this equation, assuming $m=0$ and $r_L=120,304$ km, to estimate both the surface mass density $\sigma$ and the opacity parameter $\tau_n/\sigma$. We find that over the region covered by the wave the surface mass density varies between 1.0 and 1.5 g/cm$^2$. This surface mass density is comparable to estimates derived from the Prometheus 9:7, Pan 6:5 and Atlas 5:4 waves in the inner Cassini Division, which all fall between 1.1 and 1.4 g/cm$^2$ and occur in regions of comparable optical depth \citep{Colwell09}. These consistent numbers provide further evidence that this structure is indeed an $m=0$ density wave.
Intriguingly, the surface mass density remains nearly constant despite the optical depth having a clear peak around 120,270 km, which means the opacity $\tau_n/\sigma$ varies substantially across this region. This behavior is actually consistent with previous analyses of the A, B and C rings, which show that the surface mass density is far less variable than the optical depth \citep{Tiscareno13, HN14, HN16}.
While the wavelength trends associated with this structure are perfectly consistent with those expected for a $m=0$ density wave, closer inspection of the wavelets and the reconstructed wave profile in Figures~\ref{wavegam}-\ref{profmod} reveal some surprising variations in the pattern's amplitude and pattern speed. The average wavelet power $\bar{\mathcal{P}}$ for both the $\gamma$ Crucis occultations and the full data set shows the strongest wave signal interior to 120,275 km. By contrast, for both figures the power of the average phase-corrected wavelet ${\mathcal{P}}_\phi$ has a more uniform strength between 120,260 and 120,290 km. In fact, for the full data set (Figure~\ref{wavefull}) the signal in ${\mathcal{P}}_\phi$ is higher around 120,280 km than it is around 120,265 km, which is consistent with the reconstructed profile (Figure~\ref{profmod}) having a higher amplitude around 120,280 km than around 120,265 km. Finally, the signal in the power ratio $\mathcal{R}$ is strongest between 120,275 and 120,290 km, and is much weaker interior to 120,275 km. This trend in $\mathcal{R}$ is consistent with the above trends in $\bar{\mathcal{P}}$ and $\mathcal{P}_\phi$, and implies that the signals exterior to 120,275 km are more consistent with those expected for an axisymmetric density wave with the given pattern speed.
The plots of the power ratio versus radius and pattern speed reveal another important difference between the inner and outer parts of this wave. In both Figures~\ref{wavegam} and~\ref{wavefull}, the peak signal exterior to 120,275 km occurs at a pattern speed consistent with that expected for an $m=0$ wave launched from the inner edge of the Barnard gap. By contrast, interior to 120,275 km the peak signal seems to shift to smaller resonant radii (that is, higher pattern speeds). For the $\gamma$ Crucis data shown in Figure~\ref{wavegam}, the best-fit pattern speed for this part of the wave would correspond to a resonant location 20 km interior to the gap edge. Other sub-sets of the data show a similar general trend, but the best-fit resonant radius varies by several kilometers. For the full data set the peak power ratio is considerably weaker than that found outside 120,275 km, and the best fit resonant radius is only 8 km interior to the gap edge.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nc_barnardset_gamcru_091118.pdf}}
\caption{Selected $\gamma$ Crucis occultation profiles sorted by phase of the $m=0$ pattern (phase increases upwards). Note that the position of many peaks shift from left to right as one moves from top to bottom of this plot, consistent with an $m=0$ wave.}
\label{prof1}
\end{figure}
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nc_barnardset2_gamcru_091118.pdf}}
\caption{Selected $\gamma$ Crucis occultation profiles sorted by phase of the $m=0$ pattern (phase increases upwards). The overlaid green curves show the predicted variations expected to arise from the $m=0$ wave based on the average phase-corrected wavelet and the appropriate phase factors $\phi_i$ (these are scaled by a factor of 2 for clarity). Note that interior to 120,275 km there are noticeable mismatches between the expected and observed profiles. Such misalignments are less common at larger radii.}
\label{prof1p}
\end{figure}
Together, these trends imply that while the periodic optical depth variations are more prominent interior to 120,275 km, the signals exterior to 120,275 km are more coherently organized. Examinations of the wave profiles from individual occultations confirm these findings. Figure~\ref{prof1} shows occultation profiles derived from the 2008-2009 $\gamma$ Crucis occultations sorted by the phase $\phi_{\lambda t}=-\omega (t-t_0)$ for the $m=0$ wave with $\omega=728.28^\circ$/day. The comparable viewing geometries of these occultations facilitates comparisons among them. For all these profiles, the most obvious wave-like signals lie between 120,260 and 120,275 km. By contrast, between 120,275 and 120,290 km the situation is more complicated because the peaks are not strictly periodic. These aspects of the individual profiles primarily reflect the trends in the average wavelet power $\bar{\mathcal{P}}$, which also indicate a stronger periodic signal interior to 120,275 km.
Assessing how consistent these structures are with an $m=0$ wave is not as straightforward. Since the profiles are sorted by phase, the positions of optical depth maxima and minima should shift from left to right as we proceed from the top to the bottom of the plot. The peaks interior to 120,275 km do generally follow this trend, while the situation further out is less obvious. In order to clarify this situation, Figure~\ref{prof1p} shows the same occultation profiles together with profiles of what the predicted $m=0$ wave pattern for each observation should look like based on the average phase-corrected wavelet of these occultations (these predicted wave signals are superimposed on the average background ring profile to facilitate comparisons). While the variations associated with the wave are easier to see interior to 120,275 km, the locations of the individual peaks and troughs in this region deviate from those expected for the $m=0$ wave in several of the profiles. By contrast, exterior to 120,275 km the peaks are less obvious, but their positions are generally much better aligned with those expected for the $m=0$ wave. These comparisons therefore support the notion that while the wave amplitude is higher in the inner part of the wave, the outer part of the wave has a more coherent $m=0$ pattern.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nc_barnardset_12-14_091118.pdf}}
\caption{Occultation profiles from 2012-2014 sorted by phase of the $m=0$ pattern (phase increases upwards). Note that the locations of the prominent density variations around 120,265 km are shifted slightly inwards relative to those shown in Figure~\ref{prof1}.}
\label{prof2}
\end{figure}
Other subsets of the occultation data show similar differences between the inner and outer parts of this wave, and provide evidence for longer-term changes in the wave's structure. Figure~\ref{prof2} shows profiles obtained between 2012 and 2014. Note that these occultations have more heterogeneous viewing geometries, which makes the signal-to-noise of the density variations more variable. Despite this, the radial extent of the strong wave pattern around 120,265 km in these profiles appears to differ systematically from that found in Figure~\ref{prof1}. In the earlier $\gamma$ Crucis occultations, the strong peaks are found in a region between 120,262 km and 120,272 km, with peaks rarely being seen interior to 120,260 km. However, for the data obtained between 2012 and 2014 the strong peaks are often found around 120,260 km and now are rarely found exterior to 120,270 km. This packet of waves therefore appears to have shifted roughly 2 km inwards during the 4 years between these observations.
The observed inward shift of this wave packet is consistent with the expected inward group velocity for this density wave. Assuming a surface mass density of around 1.2 g/cm$^2$, Equation~\ref{group} yields a group velocity of roughly 0.5 km/year, which would be consistent with a shift of around 2 km in the 4 years between the two groups of observations. We may therefore posit that the amplitude of the perturbation driving the wave changed over time, and a packet of high-amplitude waves created sometime in the past (ending around 70 years ago) has propagated inwards over the course of the Cassini mission.
Time variations in how this wave is excited could also help explain why the wave becomes less well organized further away from the nominal resonance. If not only the perturbation's amplitude, but also its frequency changed over time, then this wave may have some similarities with the waves excited by resonances with the satellite Janus, whose orbit period changes periodically due to interactions with its coorbital companion Epimetheus \citep{Yoder83}. The waves generated by Janus exhibit unusual variations in their pattern speeds that likely arise because parts of the wave have partially decoupled from the satellite perturbations \citep{Porco05, Tiscareno06, HN16}, and a similar phenomenon could be operating here. In principle, these variations in the wave's pattern speed could impact the estimates of the ring's surface mass density and opacity shown in Figure~\ref{profmod}. However, since there is not yet a complete theory for how waves with variable pattern speeds propagate in shearing disks, there are not yet any quantitative estimates for how big these effects may be. Developing such a theory is well beyond the scope of this article, but we can explore this idea further by considering how this axisymmetric wave could be excited in the first place.
\section{Excitation of axisymmetric density waves from resonantly-confined edges}
\label{model}
The above observations provide reasonably firm evidence that an axisymmetric $m=0$ wave is being launched from the inner edge of the Barnard gap. Hence there must be a perturbation that is independent of longitude but oscillates in time at a frequency equal to the radial epicyclic frequency of the particles within two kilometers of the gap edge. In principle, such a perturbation could arise from collisions or the gravitational potential, but for the sake of simplicity we will assume here that it is due to a term in the gravitational potential that can be written as:
\begin{equation}
\mathcal{U}_0=C_0e^{\pm i\kappa_L t}
\end{equation}
where $\kappa_L$ is the radial epicyclic frequency at the gap edge and $C_0$ is a constant that determines the strength of this wave.
The fact that the wave is launched from an edge that is perturbed by the 5:4 resonance with the satellite Prometheus strongly suggests that the perturbations from that satellite play a role in generating this wave. We may designate the term in the potential associated with this resonance as:
\begin{equation}
\mathcal{U}_
=C_Pe^{i5(\lambda-n_Pt)} =C_Pe^{i(5\lambda-(5 n_L-\kappa_L) t)}
\end{equation}
where $n_P$ is the mean motion of Prometheus, and the last equality uses the definition of the resonant mean motion $n_L$ and radial epicyclic frequency $\kappa_L$ (see Equation~\ref{eqpat}). The constant $C_P$ quantifies the strength of the perturbation from this moon.
Of course, the perturbation from Prometheus does not have the same form as $\mathcal{U}_0$,
but \citet{French16} determined that the inner edge of the Barnard gap also exhibits a diverse array of normal modes (see Table~\ref{modes}). Each of these normal modes has an azimuthal wavenumber $m$ and a pattern speed $\Omega_p=n_L-\kappa_L/m$. The only detectable normal modes have pattern speeds slower than the local orbital rate (i.e. positive values of $m$). While we do not yet have a complete understanding of what determines the relative amplitudes of these normal modes, it is likely that the observable variations in the edge position correspond to cavity modes trapped between the edge and a nearby location within the ring where the resonant condition is exactly satisfied. For material close to the gap edge, the variations in the edge position should generate terms in the local gravitational potential that have $m-$fold symmetry and rotate at the corresponding pattern speed:
\begin{equation}
\mathcal{U}_m =C_m e^{i(m\lambda-(mn_L-\kappa_L)t)}.
\end{equation}
Each of these modes could potentially produce azimuthal variations in the local gravitational field or perturb the motions of particles via interparticle collisions. Since the latter is a dissipative process, it is not strictly appropriate to represent this force in terms of a potential. Future studies should examine this more carefully, but for the sake of simplicity we will assume here that the perturbation associated with each edge mode can be written in the above form.
\begin{table}
\caption{Modes observed on the inner edge of the Barnard Gap \citep{French16}}
\label{modes}
{\begin{tabular}{|c c c|}\hline
Mode & Amplitude (km) & Phase et Epoch$^a$\\ \hline
1 & 0.44$\pm$0.06 & 200.07$\pm$8.92 \\
2$^b$ & 0.61$\pm$0.07 & 44.12$\pm$3.25 \\
3 & 1.31$\pm$0.06 & 108.47$\pm$ 1.02 \\
4 & 1.64$\pm$0.07 & 46.00$\pm$ 0.57 \\
5 & 1.36$\pm$0.06 & 27.61$\pm$ 0.56$^c$ \\
6 & 0.59$\pm$0.07 & 24.56$\pm$ 0.57 \\
7 & 0.55$\pm$0.06 & 46.93$\pm$ 0.95 \\
8 & 0.30$\pm$0.07 & 10.41$\pm$ 1.59 \\
9 & 0.71$\pm$0.07 & 8.38$\pm$ 0.55 \\
10 & 0.42$\pm$0.06 & 1.75$\pm$ 0.98 \\
13 & 0.36$\pm$0.06 & 26.84$\pm$ 0.86 \\ \hline
\end{tabular}}
\bigskip
$^a$ Epoch is UTC 2008-001T12:00:00
$^b$ Free $m=2$ mode, note that the $m=2$ mode driven by the Mimas 2:1 resonance has an amplitude that is about three times smaller.
$^c$ Phase determined from the data, the expected phase of the $m=5$ resonance with Prometheus is 23$^\circ$.
\end{table}
It is important to note that \citet{French16} found no evidence for an $m=0$ mode in the position of the Barnard Gap's inner edge. This is probably because such a mode, like Outer Lindblad Resonances, produces disturbances that propagate inwards from the edge, and so cannot become a self-excited cavity mode trapped close to the edge. Instead, we posit that the $m=0$ wave is generated through a non-linear mixing of the above terms in the local gravitational potential. A detailed model of how this mixing could occur is beyond the scope of this paper, and so we instead simply demonstrate that appropriate mixtures of $\mathcal{U}_P$ and two different $\mathcal{U}_m$ can produce the desired source term for the $m=0$ wave.
Since each normal mode on the edge has a different prescribed dependence on longitude, there is no way to mix $\mathcal{U}_P$ with a single $\mathcal{U}_m$ to produce a term that is independent of $\lambda$ like $\mathcal{U}_0$. However, if we consider a process that mixes $\mathcal{U}_P$ with two $\mathcal{U}_m$ terms, we can get terms that look like:
\begin{equation}
\mathcal{F}\mathcal{U}_P\mathcal{U}^*_m\mathcal{U}^*_{m'} = \mathcal{F}C_PC^*_mC^*_{m'}e^{i(5-m-m')(\lambda-n_Lt)-i\kappa_Lt},
\end{equation}
where asterisks denote complex conjugates, and $\mathcal{F}$ is a factor that describes the efficiency of the interference. This expression will have the desired form so long as we chose $m+m'=5$. Similar terms can be obtained from combinations like $\mathcal{U}_P\mathcal{U}^*_m\mathcal{U}_{m'}, \mathcal{U}_P\mathcal{U}_m\mathcal{U}^*_{m'}$, etc.,\footnote{All these terms are possible because the physical potential (i.e. the real part of $\mathcal{U}$) contains products of terms that go like $\sin[m\lambda-(mn_L-\kappa_L)t]$, and the expansion of these products into sum and difference frequencies yields terms equivalent to the real parts of all these terms.} which would yield suitable terms if $m=5+m'$ or $m'=5+m$. However, since the strongest normal modes observed on the edge have $m<5$ (see Table~\ref{modes}), we will focus on the first option here. Note that there are two different mode combinations that can produce the desired mixture, one with $m=1$ and $m'=4$ and another with $m=2$ and $m'=3$. Both of these pairs include one of the highest amplitude modes ( $>1$ km in Table~\ref{modes}) and another mode with amplitude around 0.5 km.
Of course, without an actual theory for how these perturbations actually interact with each other, we cannot make quantitative predictions for the amplitude or the phase of this perturbation to see if they are sufficient to produce the observed wave. However, we can at least determine what the amplitude of these perturbations would need to be in order to be consistent with the observed wave signature.
The fractional optical depth variations associated with the axisymmetric wave are around 50\% (see Figure~\ref{baedge} and Figure~\ref{profmod}), which is comparable to the variations seen in nearby density waves generated by Pan. This implies that the ratio $\mathcal{U}_0/\mathcal{U}_{P}$ should be of order the mass ratio between Pan and Prometheus, which is about 3\% \citep{Porco07, Jacobson08}. Hence we can conclude that in order for the edge modes to produce the observed waves $|\mathcal{F}\mathcal{U}^*_m\mathcal{U}^*_{m'}| \sim 0.03$. This is not many orders of magnitude less than one and so implies that the mixing between the modes needs to be fairly strong.
Even if the mixing between the relevant modes is reasonably strong, the axisymmetric terms arising from different combinations of terms could potentially interfere with each other. Fortunately, we can estimate the relative phases of the axisymmetric terms by re-writing them in the following form:
\begin{equation}
\mathcal{F}\mathcal{U}_P\mathcal{U}^*_m\mathcal{U}^*_{m'} = \mathcal{F}C_PC^*_mC^*_{m'}e^{-i(5\lambda_P-m\lambda_m-m'\lambda_{m'})}
\end{equation}
where $\lambda_P$ is the longitude of Prometheus, while $\lambda_m$ and $\lambda_{m'}$ are longitudes that track the two normal modes. Since $\lambda_P$ corresponds to a minimum in the radial position of the Barnard Gap edge, $\lambda_m$ and $\lambda_{m'}$ should also correspond to minima of their corresponding edge modes. These numbers correspond to the phase parameters provided in Table~\ref{modes} from \citet{French16}. Hence $5\lambda_P-m\lambda_m-m'\lambda_{m'}$ is the analog of $m\lambda_{sat}$ for standard Lindblad resonances. Using the phases given in Table~\ref{modes}, we find that for the $m=2/m'=3$ mode combination that this phase is 61$^\circ$-83$^\circ$, while for the $m=1/m'=4$ mode combination it is 91$^\circ$-116$^\circ$ (the lower numbers use the actual numbers for Prometheus' longitude, while the higher ones use the observed location of the $m=5$ edge mode at epoch). These phase parameters are very close to each other, indicating that these two perturbations are nearly in phase with each other. Thus these two different mode combinations could reinforce each other, supporting the formation of an $m=0$ wave.
Finally, we can note that this basic scenario can potentially accommodate the unusual trends in the wave's amplitude and coherence. Note that the $m=5$ structure on the Barnard gap edge is not perfectly aligned with Prometheus (the phase of the pattern at epoch is about 5$^\circ$ away from its expected orientation relative to the moon). Similar offsets have been observed in the $m=2$ structure in the B-ring's outer edge, which occur because there are actually two separate $m=2$ patterns on this edge, a ``forced'' pattern that tracks Mimas and a ``free'' pattern moving at a slightly different pattern speed \citep{SP09, Nicholson14b}. Together, these two patterns produce a combined $m=2$ edge structure whose amplitude and orientation relative to Mimas varies slowly over time. A similar phenomenon could potentially occur on the Barnard gap inner edge, causing the amplitude and phase of the $m=5$ term to vary and thus producing variations in the wave amplitude that propagate inwards. Looking at the reconstructed wave profile in Figure~\ref{profmod}, we can posit that the peaks in amplitude around 120,280 km and 120,265 km could represent parts of the waves generated during times when the $m=5$ edge mode was particularly high. Assuming a group velocity of order 0.5 km/year, this would imply the $m=5$ pattern's amplitude varies with a period of order 30 years. This period is significantly longer than the full span of the Cassini mission and so the variations in the edge shape might have eluded detection thus far. Further study will therefore be needed to determine if the wave's detailed structure is consistent with the recent history of the gap edge.
\section{Candidates for additional axisymmetric density waves}
\label{others}
If axisymmetric density waves can be generated by combinations of normal modes on resonantly confined ring edges, then other edges confined by resonances could potentially produce axisymmetric waves. In Section~\ref{saturn} below we discuss additional candidate axisymmetric waves in Saturn's rings, while in Section~\ref{uranus} we briefly discuss possible analogs in the Uranian rings.
\subsection{Other axisymmetric waves in Saturn's rings}
\label{saturn}
A wavelet-based survey of the entire ring system failed to reveal any additional signals consistent with $m=0$ density waves that were as strong as the signal from the region interior to the Barnard Gap. This is not necessarily so surprising if such waves are indeed generated by mode mixing near edges, since most edges do not show the rich spectrum of modes seen on the inner edge of the Barnard Gap \citep{Nicholson14b, Nicholson14c, French16}. Instead, most inner edges of gaps are dominated by a single mode (usually $m=1$), which would probably not be ideal for generating the axisymmetric perturbations needed to produce a detectable wave. Even so, one might hope to see such waves generated at other resonantly-confined outer ring or inner gap edges, which are the closest analogs to the Barnard Gap. The other four edges in Saturn's rings that are clearly associated with satellite resonances are:
\begin{itemize}
\item The outer edge of the A ring, which is close to the 7:6 resonance with the coorbital moons Janus and Epimetheus \citep{SP09, EM16}
\item The inner edge of the Keeler Gap in the outer A ring, which is influenced by the 32:31 resonance with Prometheus \citep{Tajeddine17}.
\item The outer edge of the B ring, which is held in place by the 2:1 resonance with Mimas \citep{SP10, Nicholson14b}
\item The outer edge of the Dawes ringlet in the C ring, which is held in place by the 3:1 resonance with Mimas \citep{Nicholson14c}
\end{itemize}
Note that while other features are located close to strong satellite resonances --like the Laplace Ringlet in the Cassini Division, or the Bond and Colombo Ringlets in the C ring-- for these features the relevant resonances are located close to the center of the ringlet and so perturb the internal structure or global shape of the ringlet, rather than the position of one of its edges. We will not consider those features further here.
Section \ref{aring} below briefly considers the edges of the A ring and the Keeler gap, neither of which seem to produce an axisymmetric density wave. However, as mentioned in the Introduction, both the Dawes ringlet and the outer B ring possess structures that could represent axisymmetric density waves. Section~\ref{bcedge} examines the occultation data in more detail, which provide evidence that these patterns are distorted by long-wavelength $m=1$ perturbations that complicate their interpretation. Section~\ref{bim} then discusses evidence from selected imaging sequences that the structures in the outer B ring do have some properties consistent with axisymmetric density waves.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{profplot_4panel_aringedge_011419.pdf}}
\caption{A few representative optical depth profiles of the A ring outer edge derived from stellar occultations obtained by the VIMS instrument onboard the Cassini spacecraft.}
\label{aedge}
\end{figure}
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{profplot_4panel_kgapedge_011419.pdf}}
\caption{A few representative optical depth profiles of the Keeler Gap inner edge derived from stellar occultations obtained by the VIMS instrument onboard the Cassini spacecraft.}
\label{kedge}
\end{figure}
\subsubsection{No evidence for axisymmetric waves associated with edges in the A ring}
\label{aring}
Figures~\ref{aedge} and ~\ref{kedge} show representative profiles of the Keeler Gap and A-ring edges. While there are a small number of peaks near both of these edges, these can reasonably be attributed to outward-propagating density waves generated by the Pandora 19:18/Prometheus 35:34 or Pandora 18:17 resonances, respectively. Hence there is no evidence for axisymmetric density waves associated with either of these edges.
Axisymmetric waves may not exist on these particular edges because their dynamical environments are somewhat different from that of the Barnard gap edge. While the Keeler gap is clearly perturbed by a resonance with Prometheus \citep{Tajeddine17}, this edge also falls close to the orbit of the much smaller moon Daphnis, and that moon may prevent the edge from developing a complex spectrum of normal modes. On the other hand, the outer edge of the A ring is perturbed by the co-orbital moons Janus and Epimetheus, whose periodic orbital changes clearly influence the shape of the edge over the course of a few years \citep{SP09, EM16}. A tesseral resonance with the planet may further complicate this situation \citep{EM16}. This edge's structure is therefore more time-dependent than the other edges considered here, which could inhibit the formation of additional edge modes and axisymmetric waves.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nt_b_dawes_m1_uni_gamcru_090417.pdf}}
\caption{Plot showing a collection of profiles of the Dawes ringlet derived from occultations of the star $\gamma$ Crucis obtained in 2008 and 2009. The profiles are sorted by an $m=1$ phase computed using the indicated pattern speed. Note that both the shape and spacing of the periodic structure vary systematically from profile to profile.}
\label{dawes1}
\end{figure}
\subsubsection{Occultation profiles of the candidate axisymmetric waves in the Dawes ringlet and the outer B ring}
\label{bcedge}
As mentioned in the introduction to this paper, the Dawes ringlet does contain a series of peaks and dips that extend on either side of the Pandora 2:1 resonance, which is inconsistent with a wave launched from that resonance, and so could be due to a wave propagating inwards from the gap edge (see Figure~\ref{dawes}). Also, the complex structure of the B ring does include some periodic optical depth variations around 117,300 km that could represent parts of a similar wave (see Figure~\ref{bedge}). Furthermore, if axisymmetric waves are generated by interference among resonant perturbations and edge normal modes, then the Dawes ringlet and outer B ring are the most promising places to find additional examples of these waves. Both edges exhibit normal modes with amplitudes that are not much smaller than the $m=2$ patterns generated by the relevant satellite resonances \citep{Nicholson14b, Nicholson14c}.
Applying the above wavelet-based analyses to the Dawes ringlet and outer B-ring failed to reveal any clear $m=0$ patterns in these regions. Closer inspection of the relevant patterns, however, reveals that the structures of both of these regions are probably much more heavily distorted by $m=1$ disturbances than is the region near the Barnard gap. Figures~\ref{dawes1}-\ref{bwave2} show multiple profiles of the Dawes ringlet and the outer B ring sorted by the phase of an $m=1$ pattern generated near the relevant edge (i.e. $\lambda-\dot{\varpi}(t-t_0)$). In Figure~\ref{dawes1} the profiles of the Dawes ringlet clearly indicate that both the shape and spacing of the periodic opacity peaks vary systematically from profile to profile. For the outer B ring, the situation is a bit more complicated. Figure~\ref{bwave1} shows profiles derived from occultations of the star $\gamma$ Crucis obtained in 2008-2009. Here there are systematic trends in the locations where small-scale periodic optical depth variations occur (e.g. the cluster of peaks around 117,200 km), but variations in wavelength are harder to discern. However, in later observations (shown in Figure~\ref{bwave2}) the periodic patterns are more prominent, and show variations in shape and spacing similar to those seen in the Dawes ringlet.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nct_b_bout_gamcru_uni_122617.pdf}}
\caption{Plot showing a collection of profiles of the outer B ring oderived from occultations of the star $\gamma$ Crucis obtained in 2008 and 2009. The profiles are sorted in order of an $m=1$ phase computed using the indicated pattern speed. Note that the locations and extent of regions containing fine-scale structures vary systematically among these profiles.}
\label{bwave1}
\end{figure}
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{waveprofplot2nct_b_bout_aslate_uni_122617.pdf}}
\caption{Plot showing a collection of profiles of the outer B ring derived from occultations of the star $\alpha$ Scorpii in 2016. The profiles are sorted in order of an $m=1$ phase computed using the indicated pattern speed. Note that the shapes and spacing of the peaks vary systematically among these profiles.}
\label{bwave2}
\end{figure}
It is important to note that while the shape and spacing of the periodic optical depth variations are modulated by something with $m=1$, this does not mean that periodic structures are themselves $m=1$ spiral patterns. Recall that $m=1$ waves propagate outwards, so such waves would need to be generated interior to the relevant edges, and there are no known $m=1$ resonances that could excite waves in these regions. Instead, these periodic structures probably have another $m$-number and are distorted by the $m=1$ normal modes associated with the nearby edge. Indeed, $m=1$ edge modes are expected to penetrate much further into the ring than other normal modes,\footnote{This is for the same reason that the radial wavelengths of $m=1$ density waves are much larger than those with other values of $m$.} so it is not unreasonable that the observed distortions are mostly $m=1$ \citep{SP10, Nicholson14b}. Also, the $m=1$ normal modes on the outer edges of the Dawes ringlet and the B-ring are over an order of magnitude larger than that found on the inner edge of the Barnard Gap, being $6.10\pm0.12$ km and $20.44\pm0.99$ km,. respectively \citep{Nicholson14b, Nicholson14c}, as opposed to $0.44\pm0.06$ km \citep{French16}. This would naturally explain why these distortions did not interfere with our wavelet analysis of the latter region.
In principle, the distortions in the pattern's wavelength induced by the $m=1$ edge modes could be corrected for and removed using techniques such as normalizing the radius scale \citep{Graps95, French16c}. However, thus far we have not had any success in using these techniques to obtain a signal that can be clearly identified with wavelet tools like those discussed above. This is probably because the $m=1$ distortion is not simply a uniform telescoping of the entire region, but is instead a more complex distortion involving gradients in both the eccentricity and pericenter position. For example, if we examine the Dawes ringlet profiles in Figure~\ref{dawes1}, we can see that the wavelength of the periodic structures around 90,150 km reaches its maximum value when the pattern's wavelength around 90,170 km is near its minimum value. Thus we cannot use the same sorts of wavelet analyses done above to ascertain if these patterns are truly due to a $m=0$ waves launched from the relevant edges.
\begin{figure}
\resizebox{3.4in}{!}{\includegraphics{bmoviedisp_206_ra004-014_091818.pdf}}
\caption{Image showing the brightness of the outer part of Saturn's B-ring as a function of radius and time at one inertial longitude, derived from a series of Cassini images obtained on day 198 of 2014 (i.e. near a maximum in the amplitude of the $m=2$ amplitude). Note the bright and dark bands between 117,200 km and 117,400 km. The phase speed of this pattern is consistent with an $m=0$ wave launched from the outer edge of the B ring.}
\label{bmovie}
\end{figure}
Rather than try to correct for these distortions, we could instead select observations made at nearly the same phase of the $m=1$ distortion. This strategy greatly reduces the number of possible comparisons among occultations, and unfortunately has not yet been productive for the occultation data. However, circumstantial evidence for an axisymmetric density wave in the outer part of the B ring can be obtained from imaging data.
\subsubsection{Evidence for an axisymmetric wave in the outer B-ring from images}
\label{bim}
For the occultations shown in Figure~\ref{bwave2}, the wavelength of the pattern around 117,250 km is 10-30 km. which is large enough to be resolved in images obtained by the Narrow Angle Camera onboard the Cassini spacecraft \citep{Porco04}. This instrument made several observations where it stared at a fixed longitude in the outer part of the B ring over the course of an orbital period (roughly 12 hours) and watched material rotate through the field of view. These particular observations are especially useful for this investigation because all the images were taken at the same inertial longitude. Since the apsidal precession rates in this part of the ring are only about 5$^\circ$/day, this means they were all taken at basically the same phase of any $m=1$ structure. By contrast, the pattern speed for any other $m$ value is comparable to the orbital mean motion, and so changes associated with the changing phases of other patterns should be visible in these data.
A thorough investigation of all the relevant imaging data is beyond the scope of this work, but we conducted a preliminary study of one observation sequence called BMOVIE from Rev 206, which consisted of 100 images (filenames N1784298322-N1784343322) of the unlit side of the outer B ring obtained on day 198 of 2014. We navigated these images with the appropriate SPICE kernels and adjusted estimates of the pointing so that the outer edge of the Jeffreys gap in the Cassini Division was at the expected position in all the images. We then averaged the brightness over longitude to generate a radial brightness profile from each image. Figure~\ref{bmovie} shows a mosaic of these brightness profiles that gives the brightness of the ring as a function of radius and time at the observed inertial longitude. Note that the total timespan of the observation corresponds to approximately one local orbital period.
The most dramatic feature in this image is the variation in the location of the outer edge between 117,470 km and 117,630 km, which is dominated by perturbations driven by the Mimas 2:1 resonance. This is an $m=2$ pattern moving around the planet at around half the local orbital speed, so we only see a single maximum and minimum in the radial position over the course of the observation.
For the purposes of this analysis, however, the more interesting structures are the alternating bright and dark bands between 117,220 km and 117,400 km. These bands are in the same place as the periodic opacity variations seen in the occultation data in Figure~\ref{bwave2}, and so are almost certainly the same structure. With the imaging data, we can clearly trace brightness maxima and minima over time, and while the bands seem to drift inwards and outwards at different times, the overall trend is for the bands to move outwards over time. This generally outwards motion of brightness maxima and minima is consistent with the expected behavior of crests and troughs of a density wave (recall from Section~\ref{background} that density waves always have a positive (outwards) phase velocity). The variations around this main trend, by contrast, are likely due to distortions in the ring associated with the various edge modes. Indeed, the wiggles in the positions of these dark bands seem to track the $m=2$ variations in the position of the edge.
If these periodic bands are in fact a density wave, we can use these data to constrain the value of $m$ for this pattern. Recall that the frequency $\omega$ of such a wave can be written as a function of the pattern speed or of the resonant mean motion and epicyclic frequencies, so that the phase of the pattern $\phi_r+\phi_{\lambda t}$ can be written as (cf. Equations~\ref{phidef}-\ref{eqpat0}):
\begin{equation}
\phi \simeq \phi_r(r)+|m|\lambda-|m-1|n_L t
\end{equation}
This means that at a given radius and longitude, the phase should go through $|m-1|$ cycles in one orbital period, or equivalently, $|m-1|$ bright or dark bands should cross each radius in an orbit period. For the pattern in Figure~\ref{bmovie}, if we neglect the distortions associated with the edge modes, then we find that only a single bright or dark band crosses each radius in an orbit period. This means that $|m-1|$=1, which means that $m$ should be either 0 or 2. At first, $m=2$ seems like a more logical choice, given the strong $m=2$ Inner Lindblad Resonance near the edge, but an $m=2$ structure would propagate outwards, leading to smaller wavelengths closer to the edge, which is not the case. Also, an $m=2$ resonant cavity mode should not extend this far inwards from the edge \citep{SP10, Nicholson14b}. Instead, the wavelength seems to decrease inwards, which is more consistent with an $m=0$ outer-Lindblad-resonance-like wave launched from the edge itself.\footnote{Recall that for density waves with $m<1$ the phase and group velocities are in opposite directions.}
Again, further work will be needed to completely understand this structure in the outer B ring, but imaging data does provide evidence that an axisymmetric density wave is likely to be part of the structure in the outer B ring. Unfortunately, there are no movies with sufficient resolution of the Dawes ringlet to provide evidence that the periodic structures here are also an inward-propagating axisymmetric wave, but analogies with both the Barnard gap and the B ring make this likely. Also note that both the Dawes ringlet and B-ring edges show strong $m=1, 2$ and 3 normal modes in addition to the resonantly excited $m=2$ modes \citep{Nicholson14b, Nicholson14c}, which could generate the $m=0$ wave through the same basic mechanism as described above for the Barnard Gap (in this case having the $m=2$ pattern generated by Mimas mix with normal modes with $m=1$ and 3).
\bigskip
\subsection{Axisymmetric structures in the Uranian rings}
\label{uranus}
Additional axisymmetric structures might also be found in Uranus' narrow dense rings. Of course, the available data on these rings are much more limited, but there are at least four potential examples of resonantly-confined edges in Uranus' rings \citep{PG87, French91}:
\begin{itemize}
\item The outer edge of the $\epsilon$ ring is close to the 14:13 inner Lindblad resonance with Ophelia
\item The inner edge of the $\epsilon$ ring is close to the 24:25 outer Lindblad resonance with Cordelia
\item The outer edge of the $\delta$ ring is close to the 23:22 inner Lindblad resonance with Cordelia
\item The inner edge of the $\gamma$ ring is close to the 6:5 inner Lindblad resonance with Ophelia
\end{itemize}
The $\eta$ ring is also perturbed by the 3:2 inner Lindblad resonance with Cressida \citep{Chancia17}, but this resonance is sufficiently distant from the ring that it affects the overall shape of the ring, rather than a specific edge.
Since axisymmetric density waves propagate inwards, the most likely places these waves would arise is from the outer edges of the $\epsilon$ and $\delta$ rings. Interestingly, inward-propagating density waves have been identified in both of these rings \citep{Horn88, YF92}. These waves were previously suggested to be generated by resonances with interior satellites, but the possibility that they are due to an axisymmetric wave generated by edge modes is worth exploring. One potential argument against such an interpretation is that the short wavelengths of these patterns suggest that each must represent a wave with a large $|m|$ \citep{Horn88, YF92}.
Of course, the most famous axisymmetric structure in the Uranian rings is the $m=0$ ``breathing mode'' in the mean position of the $\gamma$ ring. Strangely, the nearest satellite resonance appears to be closer to the inner edge of this ring than its outer edge, so it may not be a perfect analog to the waves seen in Saturn's rings. However, there is evidence that the width of this ring has $m=4$ and $m=6$ patterns \citep{Showalter11}, and it could be that these represent suitable analogs of edge modes for this narrow ring. It may even be that this ring is so narrow that some modes primarily affect the width of the ring while others like the $m=0$ and $m=1$ primarily affect its mean radius \citep{Longaretti89}. Understanding how these modes could mix and interfere in this context could be a useful test for any model developed to explain the axisymmetric density waves in Saturn's rings.
\section{Summary}
\label{summary}
The basic results of the above investigation are:
\begin{itemize}
\item An inward-propagating axisymmetric density wave is being launched from the inner edge of the Barnard Gap in the Cassini Division.
\item This wave could be generated by interference between perturbations from the 5:4 resonance with Prometheus and normal modes on the gap edge.
\item Another example of an axisymmetric density wave probably exists in the outer B ring, being launched from that ring's outer edge near the Mimas 2:1 Lindblad Resonance.
\item A third example may be present in the Dawes ringlet in the C ring.
\item Additional examples may be present in the $\gamma$, $\delta$ and/or $\epsilon$ Uranian rings.
\end{itemize}
Future studies will be needed to develop a proper physical model of the mode mixing that could generate axisymmetric waves, to confirm or deny the existence of the other possible waves associated with resonantly-confined edges, and to determine whether these waves play any significant role in angular momentum and energy transport near these edges \citep{Tajeddine17b}
\section*{Acknowledgements}
This work was partially supported by a NASA Cassini Data Analysis Program Grant NNX14AD50G. We thank the Cassini project, as well as the VIMS and Imaging Teams for generating the data used in this analysis. We also wish to acknowledge three undergraduate students, S. Graven, R. Miller, and R. Buckingham, whose examinations of the B ring helped put the axisymmetric structures in that region in context.
\section*{Appendix: A potential ambiguity in wave identifications}
While the 2008-2009 $\gamma$ Crucis occultation data shown in Figure~\ref{wavegam} are consistent with an $m=0$ density wave, for these particular occultations we cannot rule out the possibility that this is an $m=2$ bending wave.
Recall that the expected phase shifts for an $m=0$ density wave are $\phi_i(m=0) = -\kappa_L(t_i-t_0)$. The analogous expression for an $m=2$ bending wave is $\phi_{i,v}(m=2) = 2\lambda-(2n_L-\nu_L)(t_i-t_0)$, where $\nu_L$ is the vertical epicyclic frequency. Expressed in terms of the apsidal precession rate $\dot{\varpi}_L$ and the nodal regression rate $\dot{\Omega}_L$ these expressions become
\begin{equation}
\phi_i(m=0) = -(n_L-\dot{\varpi}_L)(t_i-t_0)
\end{equation}and
\begin{equation}
\phi_{i,v}(m=2) = 2\lambda-(n_L+\dot{\Omega}_L)(t_i-t_0).
\end{equation}
For all parts of Saturn's rings, $\dot{\Omega}_L \simeq -\dot{\varpi}_L$, and so the time-dependent parts of these two phases are nearly identical. This means that a set of occultations obtained at a single longitude $\lambda$ cannot distinguish between these two different types of waves because the predicted phase shifts differ by an additive constant $2\lambda$ that does not contribute to the wavelet power levels (In general, similar ambiguities can arise between spiral density waves with $|m|$ arms and bending waves with $|m+2|$ arms).
Fortunately, we can resolve this ambiguity and confirm our identification of this wave by considering the full set of occultations, which cut this region at different longitudes as well as different times (they also had a range of opening angles, which also influence the inferred phase of any potential bending wave). This extended data set yields no signal for the $m=2$ vertical wave option but is consistent with an $m=0$ density wave (see Figure~\ref{wavefull}).
|
2,877,628,090,760 | arxiv | \section{Introduction}
Blazars are active galactic nuclei (AGNs) with a radio-loud behaviour and a relativistic jet pointing towards the observer \citep{Abdo01, fra}. These sources are divided into two main classes: BL Lacertae objects (BL Lacs) and Flat Spectrum Radio Quasars (FSRQs), which show very different optical spectra. FSRQs have strong, broad emission lines, while BL Lacs show mostly weak or no emission lines. Compact radio cores, flat radio spectra, high brightness temperatures, superluminal motion, high polarization, and strong and rapid variability are also commonly found in BL Lacs and FSRQs. Blazars emit variable, non-thermal radiation across the whole electromagnetic spectrum, featuring components forming two broad humps in a $\nu f{_\nu}$ representation, where $\nu$ is the observing frequency and $f{_\nu}$ the spectral energy density. The low-energy hump is attributed to synchrotron radiation, and the high-energy one is usually thought to be due to inverse Compton radiation \citep{Ghisellini}.
The \textit{Fermi} Large Area Telescope (LAT) has been continuously observing the $\gamma$-ray sky since 2008 August in the 100 MeV--300 GeV energy range. The latest \textit{Fermi}-LAT catalog is the LAT 8-year Source Catalog \textit{4FGL} \citep{4fgl}, which lists 5066 $\gamma$-ray sources, about 2000 more than the previous 3FGL catalog \citep{3fgl}, which was based on four years of data. Out of the 5066 4FGL sources, 3131 are blazars: 1116 BL Lacs, 686 FSRQs, and 1329 blazar candidates of uncertain type (BCUs). If we compare the 4FGL with previous LAT catalogs we can see the significant increase of the number of unclassified sources. The percentage of BCUs increased from 14$\%$ in 1FGL \citep{1fgl} to 42$\%$ in 4FGL. In Table~\ref{<fgl>} we show the growth of the number of blazar sources detected by \textit{Fermi}-LAT. The increased difficulty to have sufficiently extensive optical observation campaigns for rigorous classification of BCUs emphasizes the importance of finding alternative ways to classify blazars.
\begin{table}
\label{<fgl>}
\begin{footnotesize}
\begin{tabular}{lccccr}
\hline
\hline
\bf{Class} &\bf{1FGL} & \bf{2FGL} & \bf{3FGL}& \bf{4FGL} \\
\hline
BL Lac & 295 (44\%) & 436 (41\%) & 660 (38\%) & 1116 (36\%)\\
FSRQ & 278 (42\%) & 370 (35\%) & 484 (28\%) & 686 (22\%) \\
BCU & 92 (14\%) & 257 (24\%) & 573 (34\%) & 1329 (42\%)\\
\hline
Total & 665 & 1063 & 1717 & 3131\\
\hline
\end{tabular}
\end{footnotesize}
\caption{Blazar class distribution in \textit{Fermi}-LAT catalogs.}
\end{table}
Since more than 1300 $\gamma$-ray sources in the 4FGL remain unassociated with any plausible source class, the full nature of almost half the sources in the 4FGL catalog remains undetermined. Classifying BCUs remains a strategic goal not only to enlarge the number of detected BL Lacs and FSRQs but also to confirm the extragalactic background light absorption of high energy photons that will be strategic in the next Cherenkov Telescope Array (CTA) extragalactic survey, which will investigate the physics of high-energy emission from relativistic AGN jets. For this reason, studies and methods for hunting and characterizing BCUs are very useful for the scientific community. When optical spectra or multiwavelength information needed for a rigorous classification are not available, a statistical approach to the problem, including machine learning, can be very useful for classification of BCUs.
Machine learning is a method of recognizing patterns within data in order to achieve goals such as classification. In a type of machine learning called \textit{supervised} machine learning, an algorithm classifies unknown objects by comparing their characteristics with characteristics of known objects.
Machine learning has been applied by \citet{ack2012, lee2012, hassan, doert2014, bflap, einecke, mirabal2016, pablo, yi, lefau, zoo, kang, kovacevic3fgl, kaur} and other studies in order to classify unassociated sources and/or BCUs from the LAT catalogs. Some of the most commonly used machine learning techniques in the above cited works, and astrophysics in general, include: Random Forest \citep{rf}, Artificial Neural Network (ANN) \citep{ann}, Support Vector Machines \citep{cor, vap}, and Boosted Decision Trees \citep{fri}.
Following \citet{bflap, zoo, kovacevic3fgl} (hereinafter \textit{C16}, \textit{S17}, \textit{K19}) in which ANN was used to classify BCUs and BCU candidates from 3FGL catalog, here we used ANN in order to classify BCUs from the 4FGL catalog. For input parameters to the network we used $\gamma$-ray parameters present in the the 4FGL catalog\footnote{\url{https://fermi.gsfc.nasa.gov/ssc/data/access/lat/8yr_catalog/}.} which is publicly available. For ANN we used \textit{TensorFlow}\footnote{\url{https://www.tensorflow.org}. TensorFlow is an open source library for machine learning. It is relatively fast, easy to use, and transparent.} \citep{tf} which was implemented in \textit{Python}\footnote{\url{https://www.python.org/}}.
The paper is organized as follows: in Section ~\ref{<2>} we present the ANN method used. In Section ~\ref{<3>} we discuss the network outputs and caveats. In Section ~\ref{<4>} and Section ~\ref{<5>} we present and validate the results.
\section{The ANN method}
\label{<2>}
\begin{figure}
\begin{center}
\includegraphics[width=.35\textwidth]{figures/ann.png}
\caption{
Schematic view of a simple feedforward ANN with one hidden layer. Circles represent neurons where information is processed and arrows represent travel direction of information through the network.
}
\label{ann}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=.9\textwidth, trim={0 0 5cm 0}]{figures/FluxHistory.pdf}
\caption{
The annual fluxes of 4FGL blazars sorted from lowest to highest values. Each curve represents a single source. Vertical axes present annual flux values for the energy range 0.1--100 GeV. The lower and upper plots correspond to flux ranges of 0--1 $\times$ 10$^{- 8}$ ph cm$^{-2}$ s$^{-1}$ and 0--10 $\times$ 10$^{- 8}$ ph cm$^{-2}$ s$^{-1}$. Horizontal axes present 8 annual time bins. For each source the curve is made by sorting annual flux values from lowest (1st time bin) to highest (8th time bin). Therefore, lower time bins correspond to years of lower activity while higher to years of higher activity for each source. BL Lacs are in the first plot column (left-hand panel), FSRQs in the second, both are in the third and BCUs are in the fourth (right-hand panel). For clarity only one third of sources for each class are plotted.
}
\label{FluxHistory}
\end{center}
\end{figure*}
The ANN technique is modeled by the way biological neural systems in the brain work. The schematic view of a simple ANN is presented in Fig.~\ref{ann}. The information enters the input layer and is sent to neurons in hidden layer(s) where it is processed. Finally it exits the output layer producing a desired outcome (classification of objects, for example).
Basically, ANN is a mathematical function over an $N$-dimensional space, where $N$ is the number of input parameters to the network. Input parameters are values which describe an object (blazars in our case). ANN produces a likelihood for the object to belong to a certain class (when ANN is used for classification). The network is trained on already classified objects (known BL Lacs and FSRQs in our case). Training the network involves adjusting the very large number of ANN parameters in order to find a function which best separates objects belonging to different classes. The network is then tested on classified objects which were not used in training in order to evaluate the trained network. After that the trained network can be used to classify unknown objects (BCUs in our case).
More detailed information on general characteristics of ANN, and particularly ANN for classifying BCUs, is present in \textit{C16}, \textit{S17}, \textit{K19}. The following method mostly follows the ones from the 3 cited works (particularly \textit{K19}).
Spectra and variability (obtained from the light curve) are two main features by which BL Lacs and FSRQs are distinguished in $\gamma$-ray band \citep{4fgl, 4lac}. Therefore, for input parameters we used $\gamma$-ray light curves and spectra present in the 4FGL catalog. More precisely we used 8 energy-integrated fluxes corresponding to 1-year observation periods sorted by increasing value, and time-integrated flux values in 7 different energy bands. This produced a set of $N=15$ input parameters to the network for each source.
\subsection{Gamma-ray light curves}
\label{<variability>}
We use the $\gamma$-ray light curves with sorted flux values from lowest to highest for each source, which is in line with an Empirical cumulative distribution function. In the 3FGL catalog, time bins had a duration of one month. This created a set of (12 months $\times$ 4 yr) 48 sorted monthly flux values for each source, which were used in previous studies. The 4FGL catalog contains light curves with a bin duration of 1 yr. This created a set of (1 yr $\times$ 8 yr) 8 sorted annual flux values for each source. While the light curves in the 4FGL catalog have smaller time resolution, each flux value is obtained from a 12 times longer observational period; therefore they are more precisely determined. Consequently, there are no undetermined fluxes with only upper limits in the 4FGL light curves as was the case with the 3FGL light curves. Also, the twice as long observational period allows us to better capture true characteristics of blazar light curves. Although the 4FGL also has two-month-long light curves, we choose to focus on the longer duration time bins for the reasons described above.
Sorting the flux values from lowest to highest is one way of making blazar activities comparable. The 8 annual time bins corresponding to 8 years of \textit{Fermi}-LAT observations are random time intervals in the life of each blazar. Fluxes in the same observational time bin go into the same input node of the network, but there is no physical meaning for this. By sorting the flux values, we are directly comparing fluxes of dimmest, average, and brightest periods for each blazar and relationships between them.
The corresponding curves are presented in Fig.~\ref{FluxHistory}. Most of the sources occupy the range of flux values in the 0--10 $\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$ interval (upper plots). In order to capture characteristics at lower flux values, the range 0--1 $\times$ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$ has been plotted separately below.
The curves contain information on average brightness, maximum annual-averaged activity, variability of sources, flaring patterns, etc.
BL Lacs are on average dimmer than FSRQs in the \textit{Fermi}-LAT energy range. Their activity tends to be more continuous over time than that of FSRQs. Quick comparison between BL Lacs and FSRQs shows several features. In the lower right part of the plots there is an area where mostly BL Lacs are found. Sources passing through this area are ones which have lower flux ($\lesssim$ 1 $\times$ 10$^{- 8}$ ph cm$^{-2}$ s$^{-1}$) during their brightest years. Both dimmer and brighter BL Lacs, on average, have more \textit{horizontal} curves with respect to FSRQs (of similar average flux), which reflects their more continuous emission over time and lower variability.
Similar behaviour was present with 3FGL blazars with a few differences. In general the resolution is higher (time bins smaller) for 3FGL blazars, so the differences between BL Lacs and FSRQs are more obvious. For example, the area of lower flux values during brightest periods where mostly BL Lacs can be found is more clear for 3FGL BL Lacs ($\lesssim$ 2 $\times$ 10$^{- 8}$ ph cm$^{-2}$ s$^{-1}$) than for 4FGL BL Lacs. 3FGL BL Lacs and especially BCUs have large numbers of time bins, during dimmer periods, with only upper limits while 4FGL BL Lacs and BCUs have relatively small but defined flux values thanks to the larger time bins of 4FGL blazars.
\begin{figure*}
\begin{center}
\includegraphics[width=.9\textwidth]{figures/EnergyBands.pdf}
\caption{
Time-integrated fluxes in 7 energy bands: Band 1: 0.05--0.1 GeV; Band 2: 0.1--0.3 GeV; Band 3: 0.3--1 GeV; Band 4: 1--3 GeV; Band 5: 3--10 GeV; Band 6: 10--30 GeV; Band 7: 30--300 GeV. Each curve represents a single source. BL Lacs (blue) are in the top-left, FSRQs (red) in the top-right, both are in the lower left and BCUs (green) are in the lower right. For clarity only one third of sources for each class are plotted.
}
\label{energybands}
\end{center}
\end{figure*}
\subsection{Gamma-ray spectra}
\label{<spectrum>}
We used spectral information in addition to light curves with sorted flux values. The 4FGL catalog contains time-integrated fluxes in 7 energy bands: 0.05--0.1, 0.1--0.3, 0.3--1, 1--3, 3--10, 10--30, 30--300 GeV (Fig.~\ref{energybands}). This is a wider energy range (0.05--300 GeV) than the one from the 3FGL catalog (0.1--100 GeV), which contained 5 energy bands. Energy bins 2, 3, 4, 5 (0.1--0.3--1--3--10 GeV) for 4FGL blazars are the same as energy bins 1, 2, 3, 4 for the 3FGL ones. Energy bin 1 (0.05--0.1 GeV) covers a new energy range in 4FGL while bins 6 and 7 (10--30--300 GeV) correspond partly to bin 5 in 3FGL (10--100 GeV). The improvement is due to longer observation period, i.e. better statistics and improvements in analysis techniques \citep{4fgl}. This set of parameters contains information of average spectral index, spectral curvature, spectral breaks, hardness ratios and other spectral information.
In the previous case fluxes were sorted in ascending order so that, among other reasons, there would be physical meaning for comparing fluxes (and relationships between them) that go into the same network input node. Here the fluxes of blazars in the same energy band go into the same network input node so the physical meaning is already there.
Quick comparison between BL Lacs and FSRQs shows several features: there is a difference in slope, i.e. average power-law index, with BL Lacs having a lower one; BL Lacs on average have higher flux values than FSRQs in the highest energy band and vice versa for lowest; some blazars show sharp breaks in slopes at lower and/or higher energy bands, and this behavior is mostly different for BL Lacs and FSRQs.
Comparing the spectral relationship of 4FGL BL Lacs to FSRQs with their relationship in 3FGL, it is mostly similar with several differences mainly related to spectral breaks thanks to the widening of the energy range. For example bin 1 in 4FGL (0.05--0.1 GeV) covers a new energy range and shows that some blazars peak in the energy range 0.1--0.3 GeV, which was not clear before. These blazars seem to be BL Lacs and FSRQs in similar proportion as the rest of the two classes. It also shows that some blazars (mainly BL Lacs) have a sharp decrease in flux from bin 1 to bin 2, and then sharp increase in bin 3, with the second feature also being present in 3FGL blazars.
\subsection{The Network}
\label{<network>}
Here we briefly describe the network architecture and the training strategy. They mostly follow the architecture and training strategy in \textit{K19} and are explained in more detail there, particularly how overfitting was handled.
We used 8 annual fluxes sorted in ascending order and 7 flux values in different energy bands as input parameters. This produces a $N=15$ dimensional parameter space in which each blazar occupies a certain position. We noted some obvious differences between BL Lacs and FSRQs when comparing their annual fluxes (Section~\ref{<variability>}) and spectra (Section~\ref{<spectrum>}). The purpose of the ANN algorithm is to fully determine the differences and to quantify them. It does so not just for sorted light curves and spectra separately but also taking into account relationships between them by examining the whole 15D parameter space.
The number of input neurons was 15 (8 for 8 annual sorted fluxes plus 7 for 7 fluxes in energy bands). The hidden layer had 40 neurons. The output layer had 2 neurons. The two output neurons produce likelihood that a source is BL Lac $L_B$ or an FSRQ $L_F$ such that $L_B + L_F = 1$ for each source. The larger the $L_B$, more likely that the source is a BL Lac and vice-versa. The Loss/Cost function used was the mean squared error. The number of ANN parameters, which are adjusted during network training, for this architecture is on the order of $\sim$700.
The training set consisted of 70\% and the test set of 30\% of the 4FGL classified blazars. The process of training the network and results from testing the network may depend on which sources were selected for the training sample and which for the testing sample. For this reason, we performed training and testing the network on 300 different combinations of training and testing samples and compared the results.
\section{Network outputs}
\label{<3>}
\subsection {Test sample sources vs BCUs}
\label{<3.1>}
In order to better present the results of the full analysis, we show here results from a single train and test sample which is representative of the full analysis. The histogram of $L_B$ for BL Lacs and FSRQs from the test sample is shown in the upper plot in Fig.~\ref{<hist>}. It is obtained by inputting parameters of sources from test sample (which the network never "saw") into the trained network. As expected, BL Lacs concentrate towards $L_B \rightarrow 1$ while FSRQs $L_B \rightarrow 0$. The number of BL Lacs and FSRQs is 30\% of the total sample and the ratio of BL Lacs to FSRQs is the same as the ratio in the total sample.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{figures/HistTestBCU.pdf}
\caption{
\textbf{Top:} Histogram of $L_B$ for BL Lac (blue) and FSRQ (red) sources from the test sample obtained from inputting test sample source parameters into the trained network.
\textbf{Middle:} Histogram of $L_B$ for BCUs obtained from inputting BCU parameters into the trained network.
\textbf{Bottom:} Histogram of BCUs (green) and sum of BL Lacs and FSRQs from the test sample (purple). Both histograms are normalized such that surface of each equals 1 (the number of sources in both is the same).
}
\label{<hist>}
\end{center}
\end{figure}
In Fig.~\ref{<TestPrec>} the cumulative precision versus $L_B$ is shown. Sources from the test sample are sorted by their $L_B$.
The two curves practically meet at 90\% precision value, meaning that almost all sources from the test sample can be separated with 90\% precision.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{figures/TestPrec.pdf}
\caption{
\textbf{Lower bar:} 335 BL Lacs (blue; vertical lines in upper half of the bar) and 206 FSRQs (red; vertical lines in lower half of the bar) from the test sample sorted by increasing $L_B$ and at equal horizontal distance from each other. The $L_B$ does not increase linearly in the plot. The lowest (left) and highest (right) obtained $L_B$ are shown in the upper plot.
\textbf{Upper plot:} change of cumulative precision with the $L_B$ for BL Lacs (blue) and FSRQs (red).
}
\label{<TestPrec>}
\end{center}
\end{figure}
Inputting BCUs into the trained network produces a histogram (middle plot in Fig.~\ref{<hist>}) with peaks towards $L_B \rightarrow 1$ and $L_B \rightarrow 0$, imitating the distribution of BL Lacs and FSRQs from the test sample (upper plot in Fig.~\ref{<hist>}). This is expected since the large majority of BCUs are either BL Lacs or FSRQs. We can expect that BCUs with large $L_B$ are mostly BL Lacs and vice versa.
In order to construct the same precision vs. $L_B$ relation (Fig.~\ref{<TestPrec>}) to BCUs, the BCU distribution with respect to $L_B$ should be as similar as possible to the combined distribution of BL Lacs and FSRQs from the test sample with respect to $L_B$. This is not entirely the case. In the bottom plot in Fig.~\ref{<hist>}, the histograms of BCUs and test sample sources are compared. Both histograms are normalized to the number of sources. While the peak at $L_B \rightarrow 1$ on histogram from the test sample sources and BCUs is very similar, the peak at $L_B \rightarrow 0$ is less pronounced for BCUs. In the middle range $0.2 \gtrsim L_B \gtrsim 0.8$ there are more sources with respect to both peaks for BCUs than for the test sample sources. In order to quantify these differences we use differential precision.
In Fig.~\ref{<TestPrecDiff>} the differential precision, obtained from the test sample, is compared to $L_B$ of BCUs. The lower bar is the same as in Fig.~\ref{<TestPrec>} and presents test sample BL Lacs and FSRQs sorted by increasing $L_B$. The middle plot shows the differential precision $P_B$. It is obtained by binning sources from the lower bar in equal bins of 20 (the last bin has 21 sources). Then, $P_B$ is calculated for each bin as the ratio of BL Lacs to the number of sources and vice versa for FSRQs. This produces a set of $P_B$ (a step function) for BL Lacs (blue line) and FSRQs (red line) with resolution of 0.05 (1/20) such that their sum is 1 for each bin. Then the BCUs are sorted in each bin based on their $L_B$ (upper plot) and a value of $P_B$ (middle plot) is assigned to each BCU (upper plot). In this way $P_B$ of BCUs can be considered a as a probability that the given BCU is BL Lac or FSRQ.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{figures/TestPrecDiff.pdf}
\caption{
\textbf{Lower bar:} 335 BL Lacs (blue; vertical lines in upper half of the bar) and 206 FSRQs (red; vertical lines in lower half of the bar) from the test sample sorted by increasing $L_B$ and at equal horizontal distance from each other. The $L_B$ does not increase linearly.
\textbf{Middle:} differential precision for each bin which contains 20 test sample sources (last bin has 21). Differential precision is ratio of BL Lacs to all sources in each bin and vice-versa for FSRQs.
\textbf{Top:} number of BCUs in each bin. Each BCU is assigned to a bin such that its $L_B$ is in between $L_B$ of the bin edges.
}
\label{<TestPrecDiff>}
\end{center}
\end{figure}
Examining Fig.~\ref{<TestPrecDiff>}, the BCU distribution is not uniform across bins, meaning that the BCU distribution with respect to $L_B$ is not the same as that of the combined BL Lacs and FSRQs from the test sample, which number 20 in each bin. A larger than average number of BCUs are in the range where $P_B$ for either class is less than 90\%.
To overcome peculiarities of a single train and test sample and resolution lost to binning, the same process of training and testing the network was repeated for 300 different train-test samples. The final differential precision for each BCU $\bar{P_B}$ is then calculated as the average of 300 $P_B$ values. The value $\bar{P_B}$ can then be considered a probability of a given BCU to be BL Lac (or $\bar{P_F} = 1 - \bar{P_B}$ to be FSRQ) taking into account fluctuations due to train-test sample selections.
Lower and upper values of the error interval are 2 values corresponding to $\approx$16th and $\approx$84th percentile of 300 $P_B$ sorted from lowest to highest\footnote{Since all 300 values have resolution of 0.05, the same values were linearly extrapolated in [-0.025, +0.025] range.}. The interval in between these values can then be considered 1$\sigma$ errors due to differences in train-test sample selections.
We used differential precision to obtain probabilities for BCUs and classify them instead of thresholds obtained from test sample cumulative precision. Therefore we did not apply any cut to the test sample BL Lacs and FSRQs which occupy same parts of the parameter space (which makes them hardly distinguishable) in which BCUs are hardly present.
\subsection {Differential precision versus Likelihood}
\label{<3.2>}
When using mean squared error (MSE) \citep{gis90, ric91} or cross-entropy \citep{ric91} for Loss/Cost function, the network output $L_B$ can be considered an approximation to class probabilities. The accuracy of approximation depends on characteristics of the network architecture and training data \citep{ric91}. Since we used MSE, we also calculate $\bar{L_B}$ as an average of 300 $L_B$ and lower and upper limits as $\approx$16th and $\approx$84th percentile of 300 sorted $L_B$. The $\bar{L_F}$ value is just $\bar{L_F} = 1 - \bar{L_B}$. The comparison of $\bar{P_B}$ and $\bar{L_B}$ is shown in Fig.~\ref{PvsL}. Both quantities for all 1329 BCUs are very close in value. There are obviously some small systematic differences, but they do not change the overall results by much. The differences are probably due to resolution lost to binning for $\bar{P_B}$ and the above-mentioned approximation accuracy for $\bar{L_B}$. When experimenting with Loss/Cost functions which are not MSE or cross-entropy, the separation of test sample sources (Fig~\ref{<TestPrec>}) and $\bar{P_B}$ of BCUs remain similar while $\bar{L_B}$ of test sample sources and BCUs may change significantly. In these cases $\bar{L_B}$ cannot be considered in absolute terms as direct probability; instead it can be used in relative terms to compare sources to each other.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{figures/PvsL.pdf}
\caption{
Average differential precision of 1329 BCUs to be BL Lac $\bar{P_B}$ with respect to average network output to be BL Lac $\bar{L_B}$.
}
\label{PvsL}
\end{center}
\end{figure}
Showing equivalence between $\bar{P_B}$ and $\bar{L_B}$ in this case, from here on out we will use $\bar{L_B}$, since its interpretation as direct network output is more obvious and it gives more precisely defined errors (which are additionally affected by binning for $\bar{P_B}$).
\subsection {Caveats}
\label{<3.3>}
Here we note some caveats in the supervised learning approach. The issues have to do with how the parameters of known sources (known BL Lacs and FSRQs) compare to parameters of unknown (BL Lacs and FSRQs among BCUs), and this is related to astronomical observations.
As a simple example, known LAT BL Lacs are 44\% more present in the northern Galactic hemisphere than in the southern one because larger and better optical spectroscopic data, required to identify BL Lacs so that LAT blazars can be associated to them, are more available for the Northern hemisphere \citep{4lac}. The \textit{Fermi}-LAT sweeps the whole $\gamma$-ray sky continually, and there is no reason to think that the fraction of LAT BL Lacs is larger for the Northern hemisphere. If the Galactic latitude was used as a parameter, the machine learning algorithms would wrongly assume that BCUs in the Northern hemisphere are more likely to be BL Lacs.
Regarding the parameters used in this work, one of the obvious differences is that BCUs have lower flux values compared to known BL Lacs and FSRQs. This means that BCU population density in the parameter space is different than that of combined known BL Lacs and FSRQs. However this is not an issue since the ANN function is defined for each part of the parameter space. It just means that $\bar{L_B}$ of BCUs will be differently distributed than those of combined known BL Lacs and FSRQs, but they will still be accurate. What is important is that the fraction of unknown BL Lacs and FSRQs among BCUs is similar to the fraction of known BL Lacs and FSRQs in each part of the parameter space, and that is a potential caveat.
Another important factor is the redshift/distance. The parameters in the 4FGL catalog are observational parameters. A different redshift for the same source would change its flux values, observational time bin intervals, and energy bin intervals. It would, of course, be more accurate to take into account these effects in the analysis, but the majority of BCUs do not have measured redshift. In any case the difference in observational parameters does exist for BL Lacs and FSRQs. What is important is that unknown BL Lacs and FSRQs among BCUs have a similar redshift distribution as known BL Lacs and FSRQs. This is part of the previous requirement that the fraction of unknown BL Lacs and FSRQs among BCUs is similar to that of known BL Lacs and FSRQs throughout parameter space.
\section {Results}
\label{<4>}
In Table~\ref{<tab2>} an example of 7 classified BCU sources is shown. The complete list of 1329 BCUs is available in electronic format. The table contains Galactic coordinates, $\bar{L_B}$ and upper and lower values of the error interval.\\
\begin{table*}
\begin{center}
\caption{
Example of 7 classified BCU sources. The full list is available in electronic format. Columns: 4FGL name, Galactic latitude, Galactic longitude, $\bar{L_B}$, lower value of error interval $\bar{L_B}^{low}$, upper value of error interval $\bar{L_B}^{up}$.
}
\label{<tab2>}
\begin{tabular}{l|cc|ccc}
\hline
Name & $b$ (deg) & $l$ (deg) & $\bar{L_B}$ & $\bar{L_B}^{low}$ & $\bar{L_B}^{up}$ \\
\hline
4FGL J1224.7$-$8313 & -20.397 & 302.096 & 0.039 & 0.031 & 0.048 \\
4FGL J0804.5+0414 & 18.180 & 217.568 & 0.105 & 0.089 & 0.124 \\
4FGL J0914.1$-$0202 & 30.177 & 233.058 & 0.431 & 0.327 & 0.540 \\
4FGL J0709.0+4304 & 21.177 & 174.289 & 0.830 & 0.769 & 0.893 \\
4FGL J1514.6$-$2044 & 30.895 & 342.539 & 0.920 & 0.898 & 0.942 \\
4FGL J0538.2$-$3910 & -30.297 & 244.438 & 0.977 & 0.969 & 0.986 \\
4FGL J2251.7$-$43208 & -63.607 & 14.738 & 0.997 & 0.995 & 0.999 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\subsection {Classification}
In Fig.~\ref{BCUPrecDiff} $\bar{L_B}$ of 1329 BCUs is shown along with the error. The quantities $\bar{L_B}$ and $\bar{L_F}$ (1 - $\bar{L_B}$) are probabilities for a BCU to be a BL Lac or a FSRQ. In order to present results as number of sources classified by precision metric, cumulative $\bar{L_B}$ ($\bar{L_B}_c$) and $\bar{L_F}$ ($\bar{L_F}_c$) are calculated as average $\bar{L_B}$ of all BCUs which have the same or higher $\bar{L_B}$ and vice-versa for cumulative $\bar{L_F}$. Cumulative values are shown on the right-hand vertical axis in Fig.~\ref{BCUPrecDiff}.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{figures/BCUPrecDiff.pdf}
\caption{
BL Lac probability $\bar{L_B}$ (1 - $\bar{L_F}$) of 1329 BCUs. Each BCU is presented by a green dot. The right-hand vertical axis shows the corresponding cumulative $\bar{L_B}$ (blue) and $\bar{L_F}$ (red). Values where cumulative $\bar{L_B}$ and $\bar{L_F}$ reaches 0.9 are marked by two horizontal blue and red dashed lines. Cumulative $\bar{L_B}$ and $\bar{L_F}$ values where $\bar{L_B} \geq 0.9$ and $\bar{L_F} \geq 0.9$ (high probable candidates) are also shown. The light green area corresponds to 1$\sigma$ error due to differences in train-test sample selections.
}
\label{BCUPrecDiff}
\end{center}
\end{figure}
Selecting BL Lac and FSRQ candidates with a 90\% precision metric ($\bar{L_B}_c \geq 0.9$ and $\bar{L_F}_c \geq 0.9$; $\bar{L_B} \geq 0.528$ and $\bar{L_F} \geq 0.701$), 801 BCUs are classified as BL Lacs and 406 as FSRQs, leaving 122 unclassified. If only highly probable candidates are selected ($\bar{L_B} \geq 0.9$ and $\bar{L_F} \geq 0.9$; $\bar{L_B}_c \geq 0.979$ and $\bar{L_F}_c \geq 0.962$), then 534 BCUs are classified as BL Lacs and 245 as FSRQs. The second classification corresponds to 98\% precision for BL Lacs and 96\% for FSRQ.
The ratio of BL Lac to FSRQ candidates is about 2. For 90\% precision candidates it is 1.7, and for high probability candidates 2.2. Looking at Table~\ref{<fgl>}, it is clear that the ratio of known BL Lacs to FSRQs has steadily increased (1FGL: 1.1; 2FGL: 1.2; 3FGL: 1.4; 4FGL: 1.6). Since BL Lacs are on average dimmer in $\gamma$-rays than FSRQs, at first they were hard to detect but as \textit{Fermi}-LAT sensitivity increased due to its longer observational period, more BL Lacs started to be discovered with respect to FSRQs. For this reason, it is reasonable to assume that the true ratio among BCUs is larger than the current ratio of known BL Lacs to FSRQs.
Looking at Fig.~\ref{BCUPrecDiff}, the network can classify many more BCUs as almost certain BL Lacs ($\bar{L_B} \rightarrow 1$) than FSRQs ($\bar{L_F} \rightarrow 1$). This is because some BL Lacs occupy parts of parameter space where there are no FSRQs, i.e. certain group of BL Lacs are easily distinguishable from FSRQs.
The error is naturally small for sources with high $\bar{L_B}$ or $\bar{L_F}$ and the network classified them as probable BL Lac/FSRQ irrespective of train-test sample selection. For sources with intermediate $\bar{L_B}$, errors are larger. This is expected because classification of BCUs with properties (input parameters to the network) not clearly corresponding to either class will be more affected by fluctuation due to train-test sample selection.
\subsection {Classification vs galactic latitude}
The number of known BL Lacs and FSRQs within the Galactic plane region $|b| < 10^{\circ}$ is about 5\%. The number of BCUs within the $|b| < 10^{\circ}$ region is 18\%. The optical spectroscopy which is required to fully classify blazars is harder to do for sources near the Galactic plane.
In Fig.~\ref{<bcumap>}, the sky distribution in Galactic coordinates of 1329 BCUs is shown together with their classification $\bar{L_B}$. Galactic diffuse $\gamma$-ray emission from the Galactic disk and many point and extended sources inside it make it more difficult to detect $\gamma$-ray blazars and measure their flux. Here we look at differences in classification between sources inside the Galactic diffuse emission area ($|b| < 10^{\circ}$) and those outside ($|b| > 10^{\circ}$).
\begin{figure}
\begin{center}
\includegraphics[width=.5\textwidth]{figures/BCUmap.pdf}
\caption{
\textbf{Upper plot:} sky distribution in Galactic coordinates of 1329 BCUs from the 4FGL catalog. Colors correspond to $\bar{L_B}$. \textbf{Bottom plots:} $\bar{L_B}$ vs. Galactic longitude (left-hand panel) and latitude (right-hand panel). \textbf{Bottom-left plot:} the two black dashed vertical lines around $b = 0^{\circ}$ correspond to $|b| = 10^{\circ}$.
}
\label{<bcumap>}
\end{center}
\end{figure}
The threshold of $\bar{L_B} \approx 0.42$ corresponds to a precision of about 87\% at which all BCUs can be classified (865 BL Lacs and 464 FSRQs). Then MSE is defined as $\sum (1 - \bar{L_B})^2 / N$ for BL Lac candidates ($\bar{L_B} > 0.42$) and $\sum (0 - \bar{L_B})^2 / N$ for FSRQ candidates ($\bar{L_B} < 0.42$). This quantity is an average measure of uncertainty of BCUs classification, i.e. how far away $\bar{L_B}$ of BCUs is from the peaks at $\bar{L_B}$ = \{0,1\}. We found that this value is not bigger for BCUs at $|b| < 10^{\circ}$ than the ones at $|b| > 10^{\circ}$, meaning that BCUs near the Galactic plane are not classified with less certainty by the network.
The average integrated (in time and energy) flux value of BL Lac and FSRQ candidates near the plane region is about two times larger than for candidates outside it. The same is true for known BL Lacs and FSRQs. This is expected since the $\gamma$-ray emission from the disk makes it harder to detect sources with lower flux. Known FSRQs have on average larger flux than known BL Lacs and the same is true for FSRQ and BL Lac candidates. For this reason, the fraction of FSRQ candidates inside the region ($112:464 \approx 0.24$) is larger than the fraction of BL Lacs ($125:865 \approx 0.14$). Considering only highly probable candidates ($\bar{L_B} > 0.9$ and $\bar{L_F} > 0.9$) the difference increases to 0.28:0.14. Therefore BCUs near the Galactic plane are made of a larger fraction of FSRQs when compared to BCUs outside of it. While the ratio of BL Lac to FSRQ candidates is about 2 for the whole sky, it is about 1 for the Galactic plane.
\section {Validation}
\label{<5>}
\begin{figure}
\includegraphics[width=0.45\textwidth]{figures/LvsPLI.pdf}
\caption{
Comparison of BCU classifications with power-law (PL) indexes. The higher the probability of a BCU to be a BL Lac $\bar{L_B}$, the lower the PL index and vice-versa. The distribution of PL indexes of BL Lac and FSRQ candidates is in agreement with distribution of PL indexes of known BL Lacs and FSRQs.
\textbf{Upper plot:} $\bar{L_B}$ of 1329 BCUs with respect to their PL indexes. The blue and red vertical lines are mean values of PL indexes of known BL Lacs (2.02) and FSRQs (2.47) and their 1$\sigma$ distribution widths which in both cases is 0.21. Horizontal blue and red dashed lines correspond to BL Lac and FSRQ candidates with 90\% precision metric.
\textbf{Bottom plots:} Histograms of PL indexes for BL Lacs (left-hand panel) and FSRQs (right-hand panel). BL Lac and FSRQ candidates are selected such that precision value is 90\%. Candidates number 2 are highly probable candidates ($\bar{L_B} > 0.9$ for BL Lacs and $\bar{L_F} > 0.9$ for FSRQs). Candidates number 3 are highly probable candidates which are confined to Galactic plane region $|b| < 10^\circ$.
}
\label{LvsPLI}
\end{figure}
It was discovered that BL Lacs and FSRQs are characterized by different $\gamma$-ray spectral properties. Usually BL Lacs show harder spectra than FSRQs \citep{3lac, 4lac}. Fitting 4FGL blazars, assuming a power-law (PL) spectral model, it was observed that the best-fit photon spectral index distribution is rather dissimilar for the two subclasses, making this observable an important $\gamma$-ray parameter to distinguish the two blazar classes. Since we did not include this parameter in our algorithm\footnote{We did use fluxes in different energy bands which contain information on average power-law index, but the power-law index per se was never used as an input parameter.}, in order to validate the performance of our algorithm (as a sanity check), we compared the PL index distribution of BCUs vs their $\bar{L_B}$ together with the PL distribution of known BL Lacs and FSRQs.
A clear correlation between $\bar{L_B}$ and PL index of BCUs exists (upper plot in Fig.~\ref{LvsPLI}) such that higher $\bar{L_B}$ corresponds to lower PL index, i.e. harder spectrum, which is expected. Mean values and 1$\sigma$ spread for known BL Lacs and FSRQs is also shown. In the bottom plots in Fig.~\ref{LvsPLI}, the same correlation is shown in the form of histograms. BL Lac and FSRQ candidates follow the PL index distribution of known BL Lacs and FSRQs. High probability BL Lac candidates have even lower PL index and vice-versa for FSRQs, which is expected. Finally, high probability candidates within the Galactic plane region $|b| < 10^\circ$ follow the same distribution as high probability candidates in total, showing that correctness of classification is no different for the Galactic plane even though there are differences when it comes to integrated flux values of blazars.
The good agreement of the PL index distribution for our candidates with the PL index distribution for known blazars confirms the correctness of the algorithm. We also plot the Variability Index distributions for known BL Lacs and FSRQs (Fig.~\ref{piv}). This parameter, unlike the PL index, is not as efficient at distinguishing blazar subclasses, so we did not use it for validation.
\begin{figure}
\includegraphics[width=0.45\textwidth]{figures//4FGLvarTotal.pdf}
\caption{
Variability index distribution for the known 4FGL blazars: BL Lacs (blue histogram) and FSRQs (red histogram). The evident overlap of the histograms show it to be inefficient at distinguishing blazar subclasses.
}
\label{piv}
\end{figure}
\section {Conclusion}
\label{<6>}
In this study we used a neural network method for the classification of uncertain blazars. We studied effects of selecting different training and testing samples, differences in test sample and BCU sample and discussed the meaning of network outputs. In the end, classification probabilities for each of 1329 BCUs are obtained along with error due to train-test sample selection. In terms of number of classified sources, 1207 BCU are classified compared to 1329 original BCUs, classifying 91\% of the sample with 90\% precision. Ratio of BL Lac candidates to FSRQ candidates is about 2:1 for the whole sky, and 1:1 for the Galactic plane. This result confirms that machine learning techniques are powerful methods to classify uncertain astrophysical objects and particularly blazars.
In this work we used sets of $\gamma$-ray parameters that are spectra and light curves since these two features are known to be different for BL Lacs and FSRQs. It is, of course, possible to use other $\gamma$-ray parameters from the 4FGL catalog (including the PL index\footnote{Information on PL index is already contained in 7 fluxes in energy bands so it is not expected to bring new information. However the PL index is obtained from the likelihood fit over whole energy interval and it might be different for some blazars than PL index that would be obtained from 7 fluxes in energy bands.}) as well as multiwavelength data, such as X-ray and radio flux\footnote{It is possible to use radio and X-ray flux even if they are present for only a subset of LAT blazars (K19).} present in the upcoming Fourth Catalog of Active Galactic Nuclei \textit{4LAC} \citep{4lac} or other catalogs. This can be addressed in a future appendix to this paper.
Due to the increasing number of uncertain blazars during the \textit{Fermi}-LAT mission, the ANN technique could be a very worthwhile opportunity for the scientific community to quickly select promising targets for multiwavelength rigorous classification and related studies at different energy ranges, mainly at very high energies by the present generation of Cherenkov telescopes and the forthcoming Cherenkov Telescope Array\footnote{\url{www.cta-observatory.org.}} \citep{cta}.
\section {Acknowledgments}
The authors would like to thank David J. Thompson (NASA Goddard Space Flight Center, Greenbelt, MD, USA) for review of the paper. For science analysis during the operation phase we acknowledge the {\it Fermi}-LAT collaboration for making the results available in such a useful form. The authors also acknowledge their institutions for providing opportunity to carry out research. We thank the anonymous reviewer for suggestions leading to improvement of this work.
|
2,877,628,090,761 | arxiv | \section{Introduction}
We have acquired large sets of both written and spoken data during the implementation of campaigns aimed at assessing the proficiency, at school, of Italian pupils learning both German and English. Part of the acquired data has been included in a corpus, named "Trentino Language Testing" in schools (TLT-school), that will be described in the following.
All the collected sentences have been annotated by human experts in terms
of some predefined ``indicators'' which, in turn, were used to assign the proficiency level to each student undertaking the assigned test. This level is expressed
according to the well-known Common European Framework of Reference for Languages (Council of Europe, 2001) scale. The CEFR defines $6$ levels of proficiency: A1 (beginner), A2, B1, B2, C1 and C2. The levels considered in the evaluation campaigns where the data have been collected are: A1, A2 and B1.
The indicators measure the linguistic competence of test takers both in relation to the content (e.g.\ grammatical correctness, lexical richness, semantic coherence, etc.) and to the speaking capabilities (e.g.\ pronunciation, fluency, etc.). Refer to Section~\ref{sec:acq} for a description of the adopted indicators.
The learners are Italian students, between 9 and 16 years old. They took proficiency tests by answering question prompts provided in written form. The ``TLT-school'' corpus, that we are going to make publicly available, contains part of the spoken answers (together with the respective manual transcriptions) recorded during some of the above mentioned evaluation campaigns. We will release the written answers in future. Details and critical issues found during the acquisition of the answers of the test takers will be discussed in Section~\ref{sec:acq}.
The tasks that can be addressed by using the corpus are very challenging and pose many problems, which have only partially been solved by the interested scientific community.
From the ASR perspective, major difficulties are represented by: {\it a)}
recognition of both child and non-native speech, i.e.\ Italian pupils speaking both English and German, {\it b)} presence of a large number of spontaneous speech phenomena (hesitations, false starts, fragments of words, etc.), {\it c)} presence of multiple languages (English, Italian and German words are frequently uttered in response to a single question), {\it d)} presence of a significant level of background noise due to the fact that the microphone remains open for a fixed time interval (e.g.\ 20 seconds - depending on the question), and {\it e)} presence of non-collaborative speakers (students often joke, laugh, speak softly, etc.). Refer to Section~\ref{sec:spoken}
for a detailed description of the collected spoken data
set.
Furthermore, since the sets of data from which ``TLT-school'' was derived were primarily acquired for measuring proficiency of second language (L2) learners, it is quite obvious to exploit the corpus for automatic speech rating. To this purpose, one can try to develop automatic approaches to reliably estimate the above-mentioned indicators used by the human experts who scored the answers of the pupils (such an approach is described in \cite{icassp2019}). However, it has to be noticed that scientific literature proposes to use several features and indicators for automatic speech scoring, partly different from those adopted in ``TLT-school'' corpus (see below for a brief review of the literature). Hence, we believe that adding new annotations to the corpus, related to particular aspects of language proficiency, can stimulate research and experimentation in this area.
\COMMENT{
that can be addressed using the ``TLT-school'' corpus, which was also the main purpose for developing the corpus itself, is automatic rating of the proficiency levels of students who undertook the tests.
First of all it is not easy to define which are the most suitable ``indicators'' to be used for estimating the CEFR levels and, then, it has to be defined how to weigh them to compose the final judgements. Note that the indicators adopted in the evaluation campaigns mentioned above are equally weighted to give the CEFR level and do not allow to consider important aspects of the proficiency of L2 learners \SB{Stefano si puo` accennare qualcosa in piu` circa i limiti relativi agli indicatori che usa IPRASE? Da specificare meglio nella section conclusions and future works}. Finally, it has to be addressed the problem of achieving reliable estimates of the indicators given the collected data.
}
Finally, it is worth mentioning that also written responses of ``TLT-school'' corpus are characterised by a high level of noise due to: spelling errors, insertion of word fragments, presence of words belonging to multiple languages, presence of off-topic answers (e.g. containing jokes, comments not related to the questions, etc.). This set of text data will allow scientists to investigate both language and behaviour of pupils learning second languages at school.
Written data are described in detail in Section~\ref{sec:written}
{\bf Relation to prior work.} Scientific literature is rich in approaches for automated assessment of spoken language proficiency. Performance is directly dependent on ASR accuracy which, in turn, depends on the type of input, read or spontaneous, and on the speakers' age, adults or children (see \cite{eskenazi2009} for an overview of spoken language technology for education). A recent publication reporting an overview of state-of-the-art automated speech scoring technology as it is currently used at Educational Testing Service (ETS) can be found in \cite{zechner2019}.
In order to address automatic assessment of complex spoken tasks requiring more general communication capabilities from L2 learners, the AZELLA data set \cite{cheng2014}, developed by Pearson, has been collected and used as benchmark for some researches~\cite{angeliki2014,cheng2014}. The corpus contains $1,500$ spoken tests, each double graded by human professionals, from a variety of tasks.
A public set of spoken data has been recently distributed in a spoken CALL (Computer Assisted Language Learning) shared task\footnote{https://regulus.unige.ch/spokencallsharedtask\_3rdedition/ for details.}
where Swiss students learning English had to answer to both written and spoken prompts. The goal of this challenge is to label students' spoken responses as ``accept'' or ``reject''. Refer to \cite{Baur2018} for details of the challenge and of the associated data sets.
Many non-native speech corpora (mostly in English as target language) have been collected during the years. A list, though not recent, as well as a brief description of most of them can be found in \cite{noeth2007}. The same paper also gives information on how the data sets are distributed and can be accessed (many of them are available through both LDC\footnote{https://www.ldc.upenn.edu/} and ELDA\footnote{http://www.elra.info/en/about/elda/} agencies). Some of the corpora also provide proficiency ratings to be used in CALL
applications. Among them, we mention the ISLE corpus \cite{menzel2000isle}, which also contains transcriptions at the phonetic level and was used in the experiments reported in~\cite{icassp2019}.
Note that all corpora mentioned in \cite{noeth2007} come from adult speech while, to our knowledge, the access to publicly available non-native children's speech corpora, as well as of children's speech corpora in general, is still scarce.
Specifically concerning non-native children's speech, we believe worth mentioning the following corpora. The PF-STAR corpus (see \cite{batliner2005}) contains English utterances read by both Italian and German children, between 6 and 13 years old. The same corpus also contains utterances read by English children. The {\em ChildIt} corpus \cite{russell2007} contains English utterances (both read and imitated) by Italian children.
By distributing ``TLT-school'' corpus, we hope to help researchers to investigate novel approaches and models in the areas of both non-native and children's speech and to build related benchmarks.
\section{Data Acquisition}
\label{sec:acq}
In Trentino, an autonomous region in northern Italy, there is a series of evaluation campaigns underway for testing L2 linguistic competence of Italian students taking proficiency tests in both English and German. A set of three evaluation campaigns is underway, two having been completed in 2016 and 2018, and a final one scheduled in 2020.
Note that the ``TLT-school'' corpus refers to only the 2018 campaign, that was split in two parts: 2017 try-out data set (involving about 500 pupils) and the actual 2018 data (about 2500 pupils).
Each of the three campaigns (i.e. 2016, 2018 and 2020) involves about 3000 students ranging from 9 to 16 years, belonging to four different school grade levels and three proficiency levels (A1, A2, B1). The schools involved in the evaluations are located in most part of the Trentino region, not only in its main towns;
Table~\ref{tab:plan} highlights some information about the pupils that took part to the campaigns. Several tests, aimed at assessing the language learning skills of the students, were carried out by means of multiple-choice questions, which can be evaluated automatically. However, a detailed linguistic evaluation cannot be performed without allowing the students to express themselves in both written sentences and spoken utterances, which typically require the intervention of human experts to be scored.
\input{tablesData.tex}
Tables~\ref{tab:datawritten} and \ref{tab:dataspeech} report some statistics extracted from both the written and spoken data collected so far in all the campaigns. Each written or spoken item received a total score by human experts, computed by summing up the scores related to $6$ indicators in 2017/2018 (from $3$ to $6$ in the 2016 campaign, according to the proficiency levels and the type of test).
Each indicator can assume a value 0, 1, 2, corresponding to
bad, medium, good, respectively.
The list of the indicators used by the experts to score written sentences and spoken utterances in the evaluations, grouped by similarity, is reported in Table~\ref{tab:indicators}.
Since every utterance was scored by only one expert, it was not possible to evaluate any kind of agreement among experts. For future evaluations, more experts are expected to provide independent scoring on same data sets, so this kind of evaluation will be possible.
\subsection{Prompts}
The speaking part of the proficiency tests in 2017/2018 consists of 47 question prompts provided in written form: 24 in English and 23 in German, divided according to CEFR levels. Apart from A1 level, which differs in the number of questions (11 for English; 10 for German), both English and German A2 and B1 levels have respectively 6 and 7 questions each.
As for A1 level, the first four introductory questions are the same ({\it How old are you?}, {\it Where do you live?}, {\it What are your hobbies?}, {\it Wie alt bist du?}, {\it Wo wohnst du?}, {\it Was sind deine Hobbys?}) or slightly different ({\it What's your favourite pet?}, {\it Welche Tiere magst du?}) in both languages, whereas the second part of the test puts the test-takers in the role of a customer in a pizzeria (English) or in a bar (German).
\COMMENT{
Highlighting the difference between the two environments might seem nitpicking, but this aspect together with the degree of openness of the questions has a strong and direct influence on the test-takers’ answers. For example the expected answers to ‘Would you like to order a pizza?’ might be either ‘Yes, please.’ or ‘No, thanks.’, while its German counterpart ‘Was m\"ochtest du essen?’ provides a much wider range of answers. It is no coincidence that ‘pizza’ is the fourth most frequent word in the English A1 test, but such word is not really worthy of interest from a phonetic and lexical point of view for obvious reasons.
}
A2 level test is composed of small talk questions which relate to everyday life situations. In this case, questions are more open-ended than the aforementioned ones and allow the test-takers to interact by means of a broader range of answers.
Finally, as for B1 level, questions are similar to A2 ones, but they include a role-play activity in the final part, which allows a good amount of freedom and creativity in answering the question.
\COMMENT{
There is another issue that emerges from the answers of both A2 and B1 level tests: the question prompts do not require test takers to shift tense. This lack is quite remarkable not just for the impact that has on the data collection itself, but also on the performance of the ASR system.
The written part of the proficiency tests is made up of 5 questions for each language: 1 for A1, 2 for A2 and 2 for B1. The question prompt of the A1 level test is in Italian and asks the test-takers to write a letter to a friend, that should include a short descriptive text.
A similar task is also required – again in Italian – in one of the two questions of A2 level, whereas the second question is prompted in the target language and is a short text message from a friend who invites the test-taker to go out and play sport.
Likewise, the B1 test consists of two questions. Once again, one asks the test-takers to write a letter, while the other requires them to write a brief argumentative text. Similarly to the A2 test, the question prompts of the German B1 test are in Italian and German, whereas the respective ones of the English B1 test are only in Italian. The presence of both source and target languages in the question prompts is quite confusing and problematic.
}
\subsection{Written Data}
\label{sec:written}
Table~\ref{tab:datawritten} reports some statistics extracted from the written data collected so far.
In this table, the number of pupils taking part in the English and German evaluation is reported, along with the number of sentences and tokens, identified as character sequences bounded by spaces.
It is worth mentioning that the collected texts contain a large quantity of errors of several types: orthographic, syntactic, code-switched words (i.e. words not in the required language), jokes, etc.
Hence, the original written sentences have been processed in order to produce ``cleaner'' versions, in order to make the data usable for some research purposes (e.g.\ to train language models, to extract features for proficiency assessment, \ldots).
To do this, we have applied some text processing, that in sequence:
$\bullet$ removes strange characters;
$\bullet$ performs some text normalisation (lowercase, umlaut, numbers, \ldots) and tokenisation (punctuation, etc.)
$\bullet$ removes / corrects non words (e.g.\ {\it hallooooooooooo} becomes {\it hallo}; {\it aaaaaaaaeeeeeeeeiiiiiiii} is removed)
$\bullet$ identifies the language of each word, choosing among Italian, English, German;
$\bullet$ corrects common typing errors (e.g. {\it ai em} becomes {\it i am})
$\bullet$ replaces unknown words, with respect to a large lexicon, with the label {\it $<$unk$>$}.
Table~\ref{tab:writtensamples} reports some samples of written answers.
\subsection{Spoken Data}
\label{sec:spoken}
Table~\ref{tab:dataspeech} reports some statistics extracted from the acquired spoken data.
Speech was recorded in classrooms, whose equipment depended on each school. In general, around 20 students took the test together, at the same time and in the same classrooms, so it is quite common that speech of mates or teachers often overlaps with the speech of the student speaking in her/his microphone. Also, the type of microphone depends on the equipment of the school. On average, the audio signal quality is nearly good, while the main problem is caused by a high percentage of extraneous speech.
This is due to the fact that organisers decided to use a fixed duration - which depends on the question - for recording spoken utterances, so that all the recordings for a given question have the same length. However, while it is rare that a speaker has not enough time to answer, it is quite common that, especially after the end of the utterance, some other speech (e.g. comments, jokes with mates, indications from the teachers, etc.) is captured.
In addition, background noise is often present due to several sources (doors, steps, keyboard typing, background voices, street noises if the windows are open, etc). Finally, it has to be pointed out that many answers are whispered and difficult to understand.
\section{Manual Transcriptions}
\label{sec:annotation}
In order to create both an adaptation and an evaluation set for ASR,
we manually transcribed part of the 2017 data sets. We defined an initial set of guidelines for the annotation, which were used by 5 researchers to manually transcribe about 20 minutes of audio data. This experience led to a discussion, from which a second set of guidelines originated, aiming at reaching a reasonable trade-off between transcription accuracy and speed. As a consequence, we decided to apply the following transcription rules:
\begin{itemize
\item
only the main speaker has to be transcribed; presence of other voices (schoolmates, teacher) should be reported only with the label ``@voices'',
\item
presence of whispered speech was found to be significant, so it should be explicitly marked with the label ``()'',
\item
badly pronounced words have to be marked by a ``\#'' sign, without trying to phonetically transcribe the pronounced sounds; ``\#*'' marks incomprehensible speech;
\item speech in a different language from the target language has to be reported by means of an explicit marker {\it ``I am 10 years old @it(io ho gi\`a risposto)''}.
\end{itemize}
\input{tablesManual.tex}
Next, we concatenated utterances to be transcribed into blocks of about 5 minutes each. We noticed that knowing the question and hearing several answers could be of great help for transcribing some poorly pronounced words or phrases. Therefore, each block contains only answers to the same question, explicitly reported at the beginning of the block.
We engaged about 30 students from two Italian linguistic high schools (namely ``C'' and ``S'') to perform manual transcriptions.
After a joint training session,
we paired students together. Each pair first transcribed, individually, the same block of $5$ minutes. Then, they went through a comparison phase, where each pair of students discussed their choices and agreed on a single transcription for the assigned data. Transcriptions made before the comparison phase were retained to evaluate inter-annotator agreement.
Apart from this first 5 minute block, each utterance was transcribed by only one transcriber.
Inter-annotator agreement for the 5-minute blocks is shown in Table~\ref{tab:agreement} in terms of words (after removing hesitations and other labels related to background voices and noises, etc.). The low level of agreement reflects the difficulty of the task.
In order to assure quality of the manual transcriptions, every sentence transcribed by the high school students was automatically processed to find out possible formal errors, and manually validated by researchers in our lab.
Speakers were assigned either to training or evaluation sets, with proportions of $\frac{2}{3}$ and $\frac{1}{3}$, respectively; then training and evaluation lists were built, accordingly.
Table~\ref{tab:TestAsrV6} reports statistics from the spoken data set. The id {\em All} identifies the whole data set, while {\em Clean} defines the subset in which sentences containing background voices, incomprehensible speech and word fragments were excluded.
\section{Usage of the Data}
\label{sec:expe}
From the above description it appears that the corpus can be effectively used in many research directions.
\subsection{ASR-related Challenges}
\label{sec:asr}
The spoken corpus features non-native speech recordings in real classrooms and, consequently, peculiar phenomena appear and can be investigated. Phonological and cross-language interference requires specific approaches for accurate acoustic modelling. Moreover, for coping with cross-language interference it is important to consider alternative ways to represent specific words (e.g.\ words of two languages with the same graphemic representation).
Table~\ref{table:icassp_wer_results}, extracted from \cite{icassp2019}, reports WERs obtained on evaluation data sets with a strongly adapted ASR, demonstrating the difficulty of the related speech recognition task for both languages. Refer to \cite{icassp2018} for comparisons with a different non-native children speech data set and to scientific literature \cite{WilJac96,DasNixPic98,LiRus01,GiuGer03,PotNar03,GerGiuBru07,GerGiuBru09,liao2015,serizel2016} for detailed descriptions of children speech recognition and related issues. Important, although not exhaustive of the topic, references on non-native speech recognition can be found in \cite{Wang2003a,Wang2003b,Oh2006,strik2009,Steidl2004,bouselmi2006,duan2017,li2016,lee2015,das2015}.
As for language models, accurate transcriptions of spoken responses demand for models able to cope with not well-formed expressions (due to students' grammatical errors). Also the presence of code-switched words, words fragments and spontaneous speech phenomena requires specific investigations to reduce their impact on the final performance.
We believe that the particular domain and set of data pave the way to investigate into various ASR topics, such as: non-native speech, children speech, spontaneous speech, code-switching, multiple pronunciation, etc.
\subsection{Data Annotation}
The corpus has been (partly) annotated using the guidelines presented in Section~\ref{sec:annotation} on the basis of a preliminary analysis of the most common acoustic phenomena appearing in the data sets.
Additional annotations could be included to address topics related to other spurious segments, as for example: understandable words pronounced in other languages or by other students, detection of phonological interference, detection of spontaneous speech phenomena, detection of overlapped speech, etc.
In order to measure specific proficiency indicators, e.g. related to pronunciation and fluency, suprasegmental annotations can be also inserted in the corpus.
\subsection{Proficiency Assessment of L2 Learners}
The corpus is a valuable resource for training and evaluating a scoring classifier based on different approaches.
Preliminary results \cite{icassp2019} show that the usage of suitable linguistic features mainly based on statistical language models allow to predict the scores assigned by the human experts.
The evaluation campaign has been conceived to verify the expected proficiency level according to class grade; as a result, although the proposed test cannot be used to assign a precise score to a given student, it allows to study typical error patterns according to age and level of the students.
Furthermore, the fine-grained annotation, at sentence level, of the indicators described above is particularly suitable for creating a test bed for approaches based on ``word embeddings'' \cite{chen2018,oh2017,russell2019} to automatically estimate the language learner proficiency. Actually, the experiments reported in \cite{chen2018} demonstrate superior performance of word-embeddings for speech scoring with respect to the well known (feature-based) SpeechRater system \cite{zechner2009,zechner2019}. In this regard, we believe that additional, specific annotations can be developed and included in the ``TLT-school'' corpus.
\input{tablesUsage.tex}
\subsection{Modelling Pronunciation}
By looking at the manual transcriptions, it is straightforward to detect the most problematic words, i.e.\ frequently occurring words, which were often marked as mispronounced (preceded by label ``\#''). This allows to prepare a set of data composed by good pronounced vs. bad pronounced words.
A list of words, partly mispronounced, is shown in Table~\ref{tab:mispronwords}, from which one can try to model typical pronunciation errors (note that other occurrences of the selected words could be easily extracted from the non-annotated data).
\COMMENT{
Manually transcribed sentences were analysed to extract occurrences of
the same word (e.g.\ favourite or lieblingsessen) that, when marked with a "#", were
badly pronounced. Data are divided into train and test.
From this analysis, some table was obtained, see Table, that could be used:
Note that other occurrences of the selected words could be easily extracted from the non-annotated data.
Some further manual checking and annotation could be carried out to model typical pronunciation errors.
}
Finally, as mentioned above, further manual checking and annotation could be introduced to improve modelling of pronunciation errors.
\section{Distribution of the Corpus}
The corpus to be released is still under preparation, given the huge amount of spoken and written data; in particular, some checks are in progress in order to:
\begin{itemize}
\item remove from the data responses with personal or inadequate content (e.g.\ bad language);
\item normalise the written responses (e.g.\ upper/lower case, punctuation, evident typos);
\item normalise and verify the consistency of the transcription of spoken responses;
\item check the available human scores and - if possible - merge or map the scores according to more general performance categories (e.g.\ delivery, language use, topic development) and an acknowledged scale (e.g.\ from 0 to 4)\footnote{https://www.ets.org/s/toefl/pdf/toefl\_speaking\_rubrics.pdf}.
\end{itemize}
In particular, the proposal for an international challenge focused on non-native children speech recognition is being submitted where an English subset will be released and the perspective participants are invited to propose and evaluate state-of-art techniques for dealing with the multiple issues related to this challenging ASR scenario (acoustic and language models, non-native lexicon, noisy recordings, etc.).
\COMMENT{
\section{Citing References in the Text}
\subsection{Bibliographical References}
All bibliographical references within the text should be put in between
parentheses with the author's surname followed by a comma before the date
of publication,\cite{Martin-90}. If the sentence already includes the author's
name, then it is only necessary to put the date in parentheses:
\newcite{Martin-90}. When several authors are cited, those references should be
separated with a semicolon: \cite{Martin-90,CastorPollux-92}. When the reference
has more than three authors, only cite the name of the first author followed by
``et al.'' (e.g. \cite{Superman-Batman-Catwoman-Spiderman-00}).
\section{Figures \& Tables}
\subsection{Figures}
All figures should be centred and clearly distinguishable. They should never be
drawn by hand, and the lines must be very dark in order to ensure a high-quality
printed version. Figures should be numbered in the text, and have a caption in
Times New Roman 10 pt underneath. A space must be left between each figure and
its respective caption.
Example of a figure enclosed in a box:
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.5]{lrec2020W-image1.eps}
\caption{The caption of the figure.}
\label{fig.1}
\end{center}
\end{figure}
Figure and caption should always appear together on the same page. Large figures
can be centred, using a full page.
\subsection{Tables}
The instructions for tables are the same as for figures.
\begin{table}[!h]
\begin{center}
\begin{tabularx}{\columnwidth}{|l|X|}
\hline
Level&Tools\\
\hline
Morphology & Pitrat Analyser\\
\hline
Syntax & LFG Analyser (C-Structure)\\
\hline
Semantics & LFG F-Structures + Sowa's\\
& Conceptual Graphs\\
\hline
\end{tabularx}
\caption{The caption of the table}
\end{center}
\end{table}
}
\section{Conclusions and Future Works}
We have described ``TLT-school'', a corpus of both spoken and written answers collected during language evaluation campaigns carried out in schools of northern Italy. The procedure used for data acquisition and for their annotation in terms of proficiency indicators has been also reported. Part of the data has been manually transcribed according to some guidelines: this set of data is going to be made publicly available.
\COMMENT{A new campaign is going to be conducted in 2020, with which we hope to increase the size of the corpus. At the same time, we aim at augmenting the annotations included in the corpus in order to cope with tasks more related to the L2 learning research area. To this purpose we also hope
to arouse sufficient interest from the interested scientific community.}
With regard to data acquisition, some limitations of the corpus have been observed that might be easily overcome during next campaigns. Special attention should be paid to enhancing the elicitation techniques, starting from adjusting the questions presented to test-takers. Some of the question prompts show some lacks that can be filled in without major difficulty: on the one hand, in the spoken part, questions do not require test-takers to shift tense and some are too suggestive and close-ended; on the other hand, in the written part, some question prompts are presented both in source and target language, thus causing or encouraging code-mixing and negative transfer phenomena. The elicitation techniques in a broader sense will be object of revision (see \cite{cooke1994} and specifically on children speech \cite{beckman2017}) in order to maximise the quality of the corpus.
As for proficiency indicators, one first step that could be taken in order to increase accuracy in the evaluation phase both for human and automatic scoring would be to divide the second indicator (pronunciation and fluency) into two different indicators, since fluent students might not necessarily have good pronunciation skills and vice versa, drawing for example on the IELTS~\footnote{https://www.ielts.org} Speaking band descriptors.
Also, next campaigns might consider an additional indicator specifically addressed to score prosody (in particular intonation and rhythm), especially for A2 and B1 level test-takers.
Considering the scope of the evaluation campaign, it is important to be aware of the limitations of the associated data sets: proficiency levels limited to A1, B1 and B2 (CEFR); custom indicators conceived for expert evaluation (not particularly suitable for automated evaluation); limited amount of responses per speaker. Nevertheless, as already discussed, the fact that the TLT campaign was carried out in 2016 and 2018 in the whole Trentino region
makes the corpus a valuable linguistic resource for a number of studies associated to second language acquisition and evaluation. In particular, besides the already introduced proposal for an ASR challenge in 2020, other initiatives for the international community can be envisaged: a study of a fully-automated evaluation procedure without the need of experts' supervision; the investigation of end-to-end classifiers that directly use the spoken response as input and produce proficiency scores according to suitable rubrics.
\section{Acknowledgements}
This work has been partially funded by IPRASE (http://www.iprase.tn.it) under the project ``TLT - Trentino Language Testing 2018''. We thank ISIT (http://www.isit.tn.it) for having provided the data and the reference scores.
\COMMENT{
\subsection{Bibliographical References}
Bibliographical references should be listed in alphabetical order at the
end of the article. The title of the section, ``Bibliographical References'',
should be a level 1 heading. The first line of each bibliographical reference
should be justified to the left of the column, and the rest of the entry should
be indented by 0.35 cm.
The examples provided in Section \secref{main:ref} (some of which are fictitious
references) illustrate the basic format required for articles in conference
proceedings, books, journal articles, PhD theses, and chapters of books.
\subsection{Language Resource References}
Language resource references should be listed in alphabetical order at the end
of the article.
\section*{Appendix: How to Produce the \texttt{.pdf} Version}
In order to generate a PDF file out of the LaTeX file herein, when citing
language resources, the following steps need to be performed:
\begin{itemize}
\item{Compile the \texttt{.tex} file once}
\item{Invoke \texttt{bibtex} on the eponymous \texttt{.aux} file}
\item{Compile the \texttt{.tex} file twice}
\end{itemize}
}
\section{Bibliographical References}
\label{main:ref}
\bibliographystyle{lrec}
|
2,877,628,090,762 | arxiv | \section{Introduction}
Through Hamilton's program \cite{Ham93b}, the study of ancient solutions emerged into the field. Perelman's seminal work \cite{Per02} manifests not only that the understanding of ancient solutions is crucial to the analysis of the singularity formation, but also that ancient solutions are interesting in their own right. The ancient solution can be approached from different directions: one is the classification (c.f. \cite{DHS12}, \cite{Bre20}, \cite{Li20}, etc.), which, in the higher-dimensional non-K\"ahler case, is impossible without any strong curvature assumption; another is the construction of examples (\cite{BKN12}, \cite{Lau13}, \cite{LW16}, etc.), whose immense quantity with their diverse behaviors discloses the difficulty of the whole field; a third approach is the least specific---to study the geometric and analytic properties of ancient solutions, among which Perelman's monotonicity formulas are very attracting ones. Works in this category include \cite{Y09}, \cite{N10}, \cite{CZ11}, \cite{Zhang21}, to list but a few.
Bamler's recent ground-breaking works \cite{Bam20a}---\cite{Bam20c} have shed more light on the field of Ricci flow. The estimates therein within the smooth category can be regarded as some good improvements of Perelman's entropy techniques, whereas the weak version of the Ricci flow (called the \emph{metric flow}) provides an intriguing object for the future study. While the latter is fascinating and promising, it is yet beyond the scope of the present article, and we shall focus on the former, just as the last two authors have done in \cite{MZ21}.
As is well-known now, for ancient solutions under certain curvature conditions, Perelman's monotonicity formulas \cite{Per02} imply the existence of an asymptotic shrinker (c.f. \cite[Proposition 11.2]{Per02} and \cite{N10}). On the other hand, Bamler \cite{Bam20c} has proved that an ancient solution has a tangent flow at infinity, and that, if we further assume that the Nash entropy is uniformly bounded and that the underlying manifold is closed, then the tangent flow is a metric soliton with a singular set of co-dimension 4 (see \cite[Section 1.8]{Bam20c}; c.f. \cite[Theorem 7.8]{Bam20c}). A natural question arises that how Perelman's asymptotic shrinker and Bamler's tangent flow at infinity are related. Bamler himself remarks that, should the ancient solution be a $\kappa$-solution, they two would then coincide. But what about other cases? What is the weakest possible condition for this to be true? These are the questions that motivated the current study. In fact, one of our results is that the asymptotic shrinker and the tangent flow at infinity coincide if the former exists. We do not assume any other curvature condition for the ancient solution except for its time-wise boundedness.
Let $(M^n,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow. Throughout this article, we shall make a technical assumption that $g(t)$ has bounded curvature within each compact time interval, namely,
\begin{eqnarray}\label{curvaturebound}
\sup_{M\times[t_1,t_2]}\big|{\Rm}_{g_t}\big|<\infty \quad \text{ for all }\quad -\infty<t_1\leq t_2\leq 0.
\end{eqnarray}
Note that the curvature bound may depend on the interval $[t_1,t_2]$, and is not uniform on $M\times(-\infty,0]$. This assumption is only for the validity of the classical formulas (especially the integration by parts at infinity) concerning conjugate heat kernels and entropies. In particular, we do not assume the ancient solution in question to be a $\kappa$-solution or to be Type I.
Let $p_0\in M$ be a fixed point. Then, for a sequence $\{\tau_i\}_{i=1}^\infty$ with $\tau_i\nearrow\infty$, we may find $\{p_i\}_{i=1}^\infty$ such that $(p_i,-\tau_i)$ are $\ell$-centers of $(p_0,0)$, namely, $\ell_{p_0,0}(p_i,\tau_i)\leq\frac{n}{2}$; see section 2.3 for more details of the definitions.
For most of the time in this article, we make the following assumption for the ancient solutions which we consider.
\\
\noindent\textbf{Assumption B:} For the fixed point $(p_0,0)$ and the sequences $\{\tau_i\}_{i=1}^\infty$ and $\{p_i\}_{i=1}^\infty$ as described above, there exists a smooth and complete Ricci flow $\big(M_\infty,g_\infty(t),p_\infty\big)_{t\in[-2,-1]}$, such that
\begin{eqnarray}\label{smoothconvergence}
\big(M,g_i(t),p_i\big)_{t\in[-2,-1]}\xrightarrow{\makebox[1cm]{}} \big(M_\infty,g_\infty(t),p_\infty\big)_{t\in[-2,-1]}
\end{eqnarray}
in the smooth Cheeger-Gromov-Hamilton sense, where the Ricci flow $g_i(t)$ is obtained by the Type I scaling
\begin{eqnarray}\label{TypeIscale}
g_i(t):=\tau_i^{-1}g(\tau_it).
\end{eqnarray}
\\
The statement of Assumption B is involved with a base point $(p_0,0)$, a sequence of positive scales $\{\tau_i\}_{i=1}^\infty$, and the choices of $\ell$-centers $(p_i,-\tau_i)$. Hence, if necessary, we shall refer to an ancient solution as ``satisfying Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$''. For the limit Ricci flow $(M_\infty,g_\infty(t))$ in (\ref{smoothconvergence}), we do not assume any shrinker structure, neither do we make any geometric assumption except for its completeness. However, it will soon be clear that, because of \cite{CZ20}, this limit is Perelman's asymptotic shrinker; see the statement of Theorem \ref{Theorem_main}(1) below.
We are now ready to state the first of our main theorems.
\begin{Theorem}\label{Theorem_main}
Under Assumption B, the following hold (see section 2 for all the definitions involved).
\begin{enumerate}[(1)]
\item $\big(M_\infty,g_\infty(t),p_\infty\big)_{t\in[-2,-1]}$ admits a shrinker structure, which makes it an asymptotic shrinker in the sense of Perelman (\cite[Proposition 11.2]{Per02}).
\item We have
\begin{eqnarray*}
\lim_{\tau\rightarrow\infty}\mathcal{N}_{p_0,0}(\tau)=\mu_\infty,
\end{eqnarray*}
where $\mathcal{N}_{p_0,0}$ is the Nash entropy based at $(p_0,0)$ and $\mu_\infty$ is the entropy of the asymptotic shrinker. In particular, $\mu_\infty$ is the infimum of $\mathcal{N}_{p_0,0}(\tau)$, $\tau>0$.
\item
Any $\mathbb{F}$-limit of the sequence $\left\{\left((M,g_i(t))_{t\in[-2,-1]},(\nu_t^i)_{t\in [-2,-1]}\right)\right\}_{i=1}^\infty$, where $\nu^i_t:=\nu_{p_0,0\,|\,\tau_it}$, given by Bamler's compactness theorem \cite[Theorem 7.6]{Bam20b} is of the form
\[
\left(
(M_\infty,g_\infty(t))_{t\in[-2,-1]},
(\nu^\infty_t)_{t\in [-2,-1)}
\right),
\]
up to isometry, where $(\nu_t^\infty)_{t\in [-2,-1)}$ is a conjugate heat flow made of a shrinker potential function.
\end{enumerate}
\end{Theorem}
\bigskip
The proof of the above theorem consists of several elements. First of all, the smooth convergence (\ref{smoothconvergence}) in Assumption B implies locally uniform geometry bounds for $g_i(t)$ around points $p_i$, which, after scaling back, implies that $(M,g(t))$ is locally uniformly Type I (see Definition \ref{def}) along $\{(p_i,-\tau_i)\}_{i=1}^\infty$. Hence, by \cite{CZ20}, the limit $(M_\infty,g_\infty(t))$ admits a shrinker structure, and it is an asymptotic shrinker in the sense of Perelman. Then, applying Theorem \ref{Coro_nash_2} to the the $\ell$-centers $(p_i,-\tau_i)$, we obtain a uniform lower bound for the Nash entropy. Thirdly, the Nash entropy bound provides a Gaussian upper bound for the conjugate heat kernel \cite[Theorem 7.2]{Bam20a}, which is applied to show Theorem \ref{Theorem_main}(2). Finally, the Gaussian upper bound of the conjugate heat kernel also implies that an $H_n$-center is always not far from an $\ell$-center; this is the main idea in the proof of Theorem \ref{Theorem_main}(3).
The last two authors \cite{MZ21} have observed that, on an ancient solution with bounded curvature within each compact time interval, the limit of the Nash entropy $\lim_{\tau\rightarrow\infty}\mathcal{N}_{x,t}(\tau)$ and the limit of Perelman's entropy $\lim_{\tau\rightarrow\infty}\mathcal{W}_{x,t}(\tau)$ are independent of the base point $(x,t)$. What is new in the corollary below is that the limit of the reduced volume $\lim_{\tau\rightarrow\infty}\mathcal{V}_{x,t}(\tau)$ is also independent of the base point $(x,t)$. This fact follows not from \cite{MZ21}, but from an observation that in the statement of Assumption B, the base point $(p_0,0)$ is not so important as the sequence of positive scales $\{\tau_i\}_{i=1}^\infty$.
\begin{Corollary}\label{entropynoloss}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (\ref{curvaturebound}) and Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. Then the following holds.
\begin{eqnarray*}
\lim_{\tau\rightarrow\infty}\mathcal{N}_{p'_0,t'_0}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{W}_{p'_0,t'_0}(\tau)=\lim_{\tau\rightarrow\infty}\log\mathcal{V}_{p'_0,t'_0}(\tau)=\mu_\infty,
\end{eqnarray*}
where $\mathcal{W}$ and $\mathcal{V}$ stand for Perelman's entropy and reduced volume, respectively, $(p'_0,t'_0)$ is an arbitrary point in $M$, (in particular, $(p'_0,t'_0)$ is not necessarily $(p_0,0)$), and $\mu_\infty$ is the entropy of the asymptotic shrinker $(M_\infty,g_\infty(t))$ (see section 2.2 and section 2.3 for the definitions).
\end{Corollary}
As a complement of Theorem \ref{Theorem_main}, we also consider its reciprocal problem: is a tangent flow of an ancient solution also Perelman's asymptotic shrinker? This, of course, is not true if the tangent flow is not smooth. However, if the tangent flow is smooth, then this problem can be easily answered with Bamler's results in \cite{Bam20c}, which are currently only proved to be true for a sequence of Ricci flows on closed manifolds and with bounded entropies. Hereby we would like to clarify that the following theorem is true \textbf{only if} the results in \cite{Bam20c} (especially Theorem 1.6 therein) are valid for noncompact Ricci flows with bounded geometry within each compact time interval (as it is conventional, by \textbf{bounded geometry} we always mean that the sectional curvature is bounded from above and from below, and that the volumes of unit balls are uniformly bounded away from $0$). This generalization is not among the goals of the present article. However, its validity, with relatively strong assurance, is not only our speculation, but expectation.
\begin{Theorem}[The reciprocal of Theorem \ref{Theorem_main}]\label{Thm_main_reciprocal}
Let $(M^n,g(t))_{t\in(-\infty,0]}$ be an ancient solution with bounded geometry within each compact time interval and with bounded Nash entropy, that is, there is a point $p_0\in M$ such that $\mathcal{N}_{p_0,0}(\tau)\geq -Y$ for all $\tau>0$, where $Y$ is a constant. If a tangent flow of $(M,g(t))_{t\in(-\infty,0]}$ at infinity (c.f. Definition \ref{tangentflowatinfinity}) is smooth, then it is also an asymptotic shrinker in the sense of Perelman.
\end{Theorem}
\textbf{Remark:} Of course, if we assume the underlying manifold $M^n$ (not necessarily the underlying manifold of the tangent flow) to be closed, then the proof of the above theorem can easily go through, since the techniques of \cite{Bam20c} do apply in this case.
\\
Under Assumption B, we also consider Perelman's $\nu$-functional (see section 2.2 for the definition) on the ancient solution. We prove that on such an ancient solution, the $\nu$-functional is uniformly bounded everywhere, and a lower bound (indeed, the infimum) is the entropy of the asymptotic shrinker.
\begin{Theorem}\label{nu-functional}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow satisfying (\ref{curvaturebound}) and Assumption B. Then we have
\begin{eqnarray}\label{nu_nonsense_00}
\inf_{t\leq 0}\nu(g(t))=\mu_\infty>-\infty,
\end{eqnarray}
where $\mu_\infty$ is the entropy of the asymptotic shrinker in Theorem \ref{Theorem_main}(1). In particular, this result is true for all noncollapsed Type I ancient solutions and all $\kappa$-solutions.
\end{Theorem}
Once the $\nu$-functional is known to be bounded, the following logarithmic Sobolev inequalities and Sobolev inequalities are simply consequences of straightforward computations (c.f. \cite{Zhq07, LW20}).
\begin{Corollary}[The logarithmic Sobolev and Sobolev inequalities]
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow satisfying (\ref{curvaturebound}) and Assumption B. Then, for any $t\in(-\infty,0]$, the following are true.
\begin{enumerate}[(1)]
\item Logarithmic Sobolev inequality: for any compactly supported locally Lipschitz function $u$ and positive scale $\tau>0$, we have
\begin{align*}
\int_M u^2\log u^2dg_t-\left(\int_Mu^2dg_t\right)\int_M u^2dg_t+\left(\mu_\infty+n+\frac{n}{2}\log(4\pi\tau)\right)\int_Mu^2dg_t&
\\
\leq \tau\int_M(4|\nabla u|^2+Ru)dg_t&.
\end{align*}
\item Sobolev inequality: for any compactly supported locally Lipschitz function $u$, we have
\begin{eqnarray*}
\left(\int_M |u|^{\frac{2n}{n-2}}dg_t\right)\leq C(n)e^{-\frac{2\mu_\infty}{n}}\int_M(4|\nabla u|^2+Ru)dg_t.
\end{eqnarray*}
\end{enumerate}
Here $\mu_\infty$ is the entropy of the asymptotic shrinker in Theorem \ref{Theorem_main}(1). In particular, this result is true for all noncollapsed Type I ancient solutions and all $\kappa$-solutions.
\end{Corollary}
Obviously, we may also use Theorem \ref{nu-functional} to obtain a finer version of the main results in \cite{Zhang20} and \cite{MZ21}. We include the following corollary, and the details of the proof are merely combinations of Theorem \ref{nu-functional} with \cite{Zhang20} and \cite{MZ21}.
\begin{Corollary}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (\ref{curvaturebound}). Furthermore, assume \emph{either one} of the following conditions is true.
\begin{enumerate}[(1)]
\item $g(t)$ satisfies a Type I curvature bound, that is, there is a constant $C$ such that $$|\Rm_{g_t}|\leq\frac{C}{|t|}\quad\text{ for all }\quad t\in(-\infty,0).$$
\item $g(t)$ satisfies Hamilton's trace Harnack $$\frac{\partial R}{\partial t}+2\langle X,\nabla R\rangle+2\Ric(X,X)\geq 0\quad\text{ for all vector field } X,$$
and there is a constant $C$ such that
$$|{\Rm}|\leq CR\quad\text{ everywhere on }\quad M\times(-\infty,0].$$
\end{enumerate}
Then, $(M,g(t))_{t\in(-\infty,0]}$ is strongly $\kappa$-noncollapsed on all scales if and only if $$\inf_{t\leq 0}\nu(g(t))\geq -\beta,$$ where $\kappa$ and $\beta$ are mutually dependent on.
\end{Corollary}
The next two results imply that the Nash entropy depends only on the local geometry around the $\ell$-centers or the $H_n$-centers, and does not depend on the geometry near the base point. These theorems are particularly useful in the long-time analysis of the Nash entropy.
\begin{Theorem}\label{Coro_nash_2}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each time interval compact in $I$. Let $s$, $t\in I$, $s<t$, and assume that $s-(t-s)\in I$. Let $x\in M$ and let $(z,s)$ be an $\ell$-center of $(x,t)$. Furthermore, assume that
\begin{gather*}
|{\Ric}|\leq\frac{C_0}{t-s}\quad \text{ on }\quad B_s(z,\sqrt{t-s})\times[s-(t-s),s],
\\
\operatorname{Vol}_{g_s}\big(B_s(z,\sqrt{t-s})\big)\geq\alpha (t-s)^{\frac{n}{2}}.
\end{gather*}
Then we have
\begin{eqnarray*}
\mathcal{N}_{x,t}(t-s)\geq-\beta,
\end{eqnarray*}
where $\beta$ is a positive constant depending only on $n$, $\alpha$, and $C_0$. Consequently, under the same assumption of the theorem, we have
\begin{eqnarray*}
K(x,t\,|\,y,s)\leq\frac{C}{(t-s)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_s^2(z,y)}{C(t-s)}\right)\quad\text{ for all }\quad y\in M,
\end{eqnarray*}
where $C$ depends only on $n$, $\alpha$, and $C_0$, and $K$ is the conjugate heat kernel (see section 2.1 for the definition.)
\end{Theorem}
\begin{Theorem}\label{Thm_nash}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each time interval compact in $I$. Let $s$, $t\in I$, $s<t$, and assume that $s-(t-s)\in I$. Let $x\in M$ and let $(z,s)$ be an $H_n$-center of $(x,t)$. Furthermore, assume that
\begin{gather*}
\operatorname{Vol}_{g_s}\big(B_s(z,\sqrt{t-s})\big)\geq\alpha (t-s)^{\frac{n}{2}}.
\end{gather*}
Then all the conclusions in Theorem \ref{Coro_nash_2} are true, with constants $\beta$ and $C$ depending on $n$ and $\alpha$.
\end{Theorem}
\textbf{Remarks:} \begin{enumerate}
\item In Theorem \ref{Coro_nash_2}, the assumptions can be replaced by\begin{gather*}
|{\Ric}|\leq\frac{C_0}{\varepsilon^2(t-s)}\quad \text{ on }\quad B_s(z,\varepsilon\sqrt{t-s})\times[s-\varepsilon^2(t-s),s],
\\
\operatorname{Vol}_{g_s}\big(B_s(z,\varepsilon\sqrt{t-s})\big)\geq\alpha \varepsilon^n(t-s)^{\frac{n}{2}},
\\
s-\varepsilon^2(t-s)\in I;
\end{gather*}
in Theorem \ref{Thm_nash}, the assumptions can be replaced by\begin{gather*}
\operatorname{Vol}_{g_s}\big(B_s(z,\varepsilon\sqrt{t-s})\big)\geq\alpha \varepsilon^n(t-s)^{\frac{n}{2}},
\\
s-\varepsilon^2(t-s)\in I,
\end{gather*}
where $\varepsilon$ is any positive constant. If so, then the constants in the conclusions also depend on $\varepsilon$.
\item Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash} do not imply each other. The reason is because unless the Nash entropy is known to be bounded, one does not know whether an $\ell$-center is close to an $H_n$-center. In fact, the statement of Theorem \ref{Thm_nash} is much stronger than Theorem \ref{Coro_nash_2}. This is because of Bamler's good gradient estimates for heat kernels.
\item Comparing Theorem \ref{Thm_nash} with \cite[Theorem 6.2]{Bam20a}, we have that, if $(z,t-r^2)$ is an $H_n$-center of $(x,t)$, then $r^{-n}\Vol_{g_{t-r^2}}\left(B_{t-r^2}(z,\sqrt{2H_n}r)\right)$ and $\mathcal{N}_{x,t}(r^2)$ mutually bound each other.
\end{enumerate}
Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash} show an interesting behavior of the Nash entropy: it is the geometry around the $H_n$-centers or the $\ell$-centers that determines (a lower bound of) the Nash entropy. This is somewhat in contrast to our first impression, since the Nash entropy is defined as a global integral. But the $H_n$-concentration property provides a likely explanation: as we know, the $H_n$-centers are points around which the conjugate heat flow (regarded as a probability measure) accumulates its measure (\cite[Proposition 3.13]{Bam20a}), and hence the region far away from an $H_n$-center should make little contribution to the Nash entropy. Furthermore, this ``locality'' behavior is also in time. In other words, $\mathcal{N}_{x,t}(t-s)$ is bounded so long as the geometry near $(z,s)$, an $H_n$-center of $(x,t)$, is bounded, regardless of what is happening to the Ricci flow on $M\times[s,t]$. We hope these results to be useful in future studies.
This paper is organized as follows. In section 2 we review some basic definitions and results. In section 3 we prove Bamler's conjugate heat kernel estimates for noncompact Ricci flows. In section 4 we prove Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash}. In section 5 we prove some basic properties for ancient solutions satisfying Assumption B, including the boundedness of the Nash entropy; Theorem \ref{Theorem_main}(1) is proved in this section. In section 6 we explore the relation between smooth convergence and $\mathbb{F}$-convergence. In section 7 we prove Theorem \ref{Theorem_main}(2)(3). In section 8 we prove Corollary \ref{entropynoloss}. In section 9 we prove Theorem \ref{nu-functional}. In section 10 we consider two classical cases in which the asymptotic shrinker is known to exist, and show that they are special cases of Theorem \ref{Theorem_main}. In section 11 we prove Theorem \ref{Thm_main_reciprocal}.
\\
\emph{Acknowledgement:} The authors are much indebted to Professor Bennett Chow for many insightful discussions. The first author would also like to thank Professor Richard Bamler for many good suggestions.
\section{Preliminaries}
\subsection{Conjugate heat kernel}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow.
We denote by
$K(x,t\,|\, y,s)$
the unique minimal fundamental solution to the heat equation coupled with $g(t)$, that is,
\begin{align*}
\Box_{x,t} K(x,t\,|\,y,s)=0,
& \quad \lim_{t\rightarrow s+} K(x,t\,|\,y,s)
= \delta_y(x),\\
\Box^*_{y,s} K(x,t\,|\,y,s)=0,
& \quad \lim_{s\rightarrow t-} K(x,t\,|\,y,s)
= \delta_x(y),
\end{align*}
where
\[
\Box_{x,t}:= \partial_t - \Delta_{g_t,x},
\quad
\Box^*_{y,s} :=-\partial_s - \Delta_{g_s,y} + R(y,s).
\]
Throughout this paper, the laplacians and the covariant derivatives are all time-dependent, computed using the evolving metric of the Ricci flow. We shall suppress the subindices in the notations such as $\Delta_{g_t,x}$ when the metric and the variables are understood. It is well known that, whenever the integration by parts at infinity is valid, the conjugate heat equation preserves the integral. Hence, at least for Ricci flows with bounded curvature within each compact time interval, the measure $\nu_{x,t\,|\,s}$ defined as
\begin{eqnarray}\label{CHK_measure}
\nu_{x,t\,|\,s}(A):=\int_AK(x,t\,|\, \cdot, s) dg_s,\quad A\subset M
\end{eqnarray}
is always a probability measure, where $x\in M$, $s,t\in I$, and $s\le t.$ In this article, both the fundamental solution $K(x,t\,|\,\cdot,\cdot)$ and the evolving probability measure $(\nu_{x,t\,|\,s})_{s\in I\cap(-\infty,t]}$ will be referred to as the \emph{conjugate heat kernel}, whenever there is no ambiguity. One can also use other positive solutions to the conjugate heat equation instead of a fundamental solution to construct time-dependent probability measures as (\ref{CHK_measure}), and such evolving probability measures are usually referred to as \emph{conjugate heat flows}. For the conjugate heat kernel, there are the following logarithmic Sobolev and Poincar\'e inequalities proved in \cite{HN14}.
\begin{Proposition}[Hein-Naber's logrithmic Sobolev and Poincar\'e inequalities \cite{HN14}]\label{Hein-Naber-log-Sobolev}
Suppose that $(M^n,g(t))_{t\in [-T,0]}$ is a complete Ricci flow with bounded curvature. Let $x_0\in M$ be a fixed point and $\nu_s:=\nu_{x_0,0\,|\,s}$ for $s\in [-T,0).$ Then, for any function $u\ge 0, \sqrt{u}\in C_0^{0,1}(M)$, we have
\begin{equation}
\label{ineq: log-Sobolev}
\int_M u\log u \, d\nu_{s}
-\left( \int_M u\, d\nu_s\right)
\log \left( \int_M u\, d\nu_s\right)
\le |s| \int_M \frac{|\nabla u|^2}{u} d\nu_s.
\end{equation}
For any $u\in C_0^{0,1}(M),$ we have
\begin{equation}
\int_M u^2d\nu_s-\left(\int_M ud\nu_s\right)^2\leq 2|s|\int_M |\nabla u|^2d\nu_s.\label{ineq: poincare}
\end{equation}
\end{Proposition}
In the original statement of the above results, Hein-Naber \cite{HN14} assumed bounded geometry for the Ricci flow. However, given all the estimates for the conjugate heat kernel (c.f. \cite[Corollary 26.26, Theorem 26.31]{RFV3}, \cite[Theorem 10]{EKNT08}, and \cite{Zhq06,BCP10}), it is not difficult to weaken the assumption to bounded curvature alone. We shall include the proof in Appendix A.
Hein-Naber also proved the following Gaussian concentration theorem using Davies' technique and their logarithmic Sobolev inequalities.
\begin{Proposition}[Hein-Naber's Gaussian concentration \cite{HN14}]\label{prop: gaussian concentration}
Under the same assumption as the above proposition, we have
\begin{equation}
\label{ineq: gaussian concentration}
\nu_s(A)\nu_s(B)
\le \exp\left\{
- \frac{{\rm dist}^2_s(A,B)}{8|s|}
\right\},
\end{equation}
for any measurable subsets $A$, $B\subset M$. Here ${\rm dist}_s$ means the classical distance between two sets computed using the metric $g(s)$, not the Hausdorff distance.
\end{Proposition}
\subsection{Perelman's entropy functionals}
Let us recall Perelman's definition of the $\mathcal{W}$-functional:
\begin{eqnarray}\label{Perelmansentropy}
\mathcal{W}(g,f,\tau):=\int_M\Big(\tau\big(|\nabla f|^2+R\big)+f-n\Big)(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg,
\end{eqnarray}
where $(M^n,g)$ is a Riemannian manifold, $f$ is a smooth function, and $\tau>0$ is a positive scale. If we let $u:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$, then we may rewrite
\begin{eqnarray}\label{anotherPerelmansentropy}
\overline{\mathcal{W}}(g,u,\tau)&:=&\int_M\left(\tau\left(\frac{|\nabla u|^2}{u^2}+R\right)-\log u-\frac{n}{2}\log(4\pi\tau)-n\right)u\,dg.
\end{eqnarray}
In fact, this form of the $\mathcal{W}$ functional is what we will apply in the discussion of section 9. Note that for any $c>0$, we have
\begin{eqnarray*}
\mathcal{W}(cg,f,c\tau)=\mathcal{W}(g,f,\tau),\quad \overline{\mathcal{W}}(cg,c^{-\frac{n}{2}}u,c\tau)=\overline{\mathcal{W}}(g,u,\tau).
\end{eqnarray*}
Perelman's $\mu$-functional and $\nu$-functional are defined as follows.
\begin{eqnarray*}
\mu(g,\tau)&:=&\inf\left\{\overline{\mathcal{W}}(g,u,\tau)\ \bigg|\ \sqrt{u}\in C^\infty_0(M),\ u\geq 0,\ \text{ and } \int_Mudg=1\right\},
\\
\nu(g)&:=&\inf_{\tau>0}\mu(g,\tau).
\end{eqnarray*}
Here for the $\mu$ functional we are adopting the definition in \cite{RFV1}; see the formula above Lemma 6.28 therein. It is well understood that $\mu(g,\tau)$ is the logarithmic Sobolev constant at scale $\tau$, and that $\nu$ is the Sobolev constant.
Let us fix a point $(p_0,t_0)$ in the space-time of a Ricci flow $(M^n,g(t))_{t\in I}$, and denote the conjugate heat kernel based at $(p_0,t_0)$ as
\begin{eqnarray*}
K(p_0,t_0\,|\,x,t):=(4\pi(t_0-t))^{-\frac{n}{2}}e^{-f(x,t)}, \quad t\in I\cap(-\infty,t_0).
\end{eqnarray*}
Then, \emph{Perelman's entropy} and the \emph{Nash entropy} are respectively defined as
\begin{eqnarray}
\mathcal{W}_{p_0,t_0}(\tau)&:=& \mathcal{W}\big(g(t_0-\tau),f(\cdot,t_0-\tau),\tau\big)
\\\nonumber
&=&\int_M\Big(\tau\left(|\nabla f|^2+R\right)+f-n\Big)(\cdot,t_0-\tau)\,d\nu_{p_0,t_0\,|\, t_0-\tau},
\\
\mathcal{N}_{p_0,t_0}(\tau)&:=&\int_M f(\cdot,t_0-\tau)\,d\nu_{p_0,t_0\,|\, t_0-\tau}-\frac{n}{2},
\end{eqnarray}
for all $\tau>0$ and $t_0-\tau\in I$. The point $(p_0,t_0)$ is called the \emph{base point}. It is well known from Perelman \cite{Per02} that both Perelman's entropy and the Nash entropy are increasing in time (and hence decreasing in $\tau$). Although the monotonocity of $\mathcal{W}_{p_0,t_0}$ or $\mathcal{N}_{p_0,t_0}$ requires the integration by parts at infinity, yet this is valid at least for Ricci flows with bounded curvature.
\subsection{Perelman's reduced distance}
We briefly review Perelman's reduced distance and reduced volume. Let $(M,g(t))_{t\in[-T,0]}$ be a Ricci flow with bounded curvature. Let $(p_0,t_0)\in M\times(-T,0]$ be a fixed point in space-time. Then, Perelman's \emph{reduced distance} is defined as
\begin{eqnarray}\label{definitionofl}
\ell_{p_0,t_0}(x,\tau):=\frac{1}{2\sqrt{\tau}}\inf_{\gamma}\int_0^\tau\sqrt{s}\left(|\dot{\gamma}(s)|_{g(t_0-s)}^2+R(\gamma(s),t_0-s)\right)ds,
\end{eqnarray}
where $x\in M$, $\tau\in(0,T-|t_0|]$, and the infimum is taken over all piecewise smooth curves $\gamma:[0,\tau]\rightarrow M$ satisfying $\gamma(0)=p_0$ and $\gamma(\tau)=x$. The minimizer of (\ref{definitionofl}) is usually called a minimal $\mathcal{L}$-geodesic from $(p_0,t_0)$ to $(x,t_0-\tau)$. $(p_0,t_0)$ is called the base point of $\ell$, and whenever the base point is understood, we shall suppress the subindex in the notaion $\ell_{p_0,t_0}(\cdot,\cdot)$. The \emph{reduced volume based at $(p_0,t_0)$} is defined as
\begin{eqnarray}
\mathcal{V}_{p_0,t_0}(\tau):=\int_M(4\pi\tau)^{-\frac{n}{2}}e^{-\ell_{p_0,t_0}(\cdot,\tau)}dg_{t_0-\tau}.
\end{eqnarray}
Perelman's reduced distance and reduced volume satisfy many nice equations and inequalities. Among them the most important one is the monotonicity of the reduced volume.
\begin{Proposition}
Perelman's reduced volume $\mathcal{V}(\tau)$ is an increasing function in time (and hence a decreasing function in $\tau$).
\end{Proposition}
The underlying reason for the monotonicity of the reduced volume is the fact that its integrand is a ``sub''-conjugate heat kernel.
\begin{Proposition}\label{basic_l}
Let $\ell_{p,0}(x,\tau)$ be the reduced distance based at $(p,0)$. Then, $\displaystyle u(x,t):=(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{p,0}(x,|t|)}$ is a subsolution to the conjugate heat equation $-\partial_t u-\Delta u+Ru=0$ which also converges to the Dirac delta measure based at $p$ as $\tau\rightarrow 0+$. Precisely, this means
\begin{gather*}
\frac{\partial\ell}{\partial\tau}-\Delta_{g_{-\tau}}\ell+\left|\,\nabla_{g_{-\tau}}\ell\,\right|_{g_{-\tau}}^2-R_{g_{-\tau}}+\frac{n}{2\tau}\geq 0,\\
\lim_{\tau\rightarrow 0+}(4\pi\tau)^{-\frac{n}{2}}e^{-\ell_{p,0}(\cdot,\tau)}=\delta_{p}.
\end{gather*}
Both of the formulas above are understood in the sense of distribution. In consequence, we have
\begin{eqnarray}\label{subsolution}
(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{p,0}(x,|t|)}\leq K(p,0\,|\,x,t)\quad \text{ for all } (x,t)\in M\times [-T,0),
\end{eqnarray}
where $K(p,0\,|\,\cdot,\cdot)$ is the conjugate heat kernel based at $(p,0)$.
\end{Proposition}
By an elementary application of the maximum principle, Perelman \cite{Per02} proved that $\ell(\cdot,\tau)$ always attains its minimum. This minimum point should be viewed as the ``center'' of the reduced distance.
\begin{Proposition}
Let $\ell_{p,0}$ be the reduced distance based at $(p,0)$. Then we have
\begin{eqnarray}\label{lcenter}
\min_{M}\ell(\cdot,\tau)\leq\frac{n}{2} \quad \text{ for all } \quad \tau\in(0,T].
\end{eqnarray}
\end{Proposition}
The point(s) where the minimum in formula (\ref{lcenter}) is attained plays an important role in our arguments. In most of the cases, it turns out that such a minimum point is not far from Bamler's $H_n$-centers. Hence, we would like to assign a special term to these points.
\begin{Definition}
Let $(M^n,g(t))_{t\in I}$ be a Ricci flow, and let $(x,t)\in M\times I$ be a point in space time. Let $s\in I\cap(-\infty,t)$. Then, $(z,s)$ is called an \emph{$\ell$-center} of $(x,t)$ if
\begin{eqnarray*}
\ell_{x,t}(z,t-s)\leq\frac{n}{2}.
\end{eqnarray*}
\end{Definition}
\textbf{Remark:} Similar to the case of the $H_n$-center, the $\ell$-center is not necessarily unique at a fixed time $s$ for a fixed base point $(x,t)$. Furthermore, in practice (especially when considering the base points for the blow-down sequence from which we obtain an asymptotic shrinker), a sequence of space-time points along which $\ell$ is uniformly bounded serves equally well as a sequence of $\ell$-centers; see Definition \ref{def}.
\subsection{Shrinking gradient Ricci soliton and its entropy}
A shrinking gradient Ricci soliton (or Ricci shrinker for short) is a tuple $(M^n,g,f)$, where $(M^n,g)$ is a smooth Riemannian manifold, and $f$ is a smooth function on $M$ called a \emph{shrinker potential}, satisfying
\[
\Ric + \nabla^2 f = \tfrac{1}{2}g.
\]
Apparently, the shrinker potential is not unique, for one may always add a constant to it. If we normalize $f$ such that
\[
(4\pi)^{-n/2}\int_M e^{-f}dg = 1,
\]
then
\[
\mu:= f-(|\nabla f|^2 + R)
\]
is a constant called the \emph{shrinker entropy}.
Let $\Phi_t$ be the 1-parameter group of self-diffeomorphisms generated by $\nabla f$ with $\Phi_0:M\rightarrow M$ being the identity map.
Define $\phi_t:= \Phi_{-\log|t|}$,
$g(t) := |t| \phi^*_t g$, and $f_t=f\circ\phi_t$. Then, $(M,g(t))_{t\in(-\infty,0)}$ is an ancient solution to the Ricci flow, and $(4\pi|t|)^{-\frac{n}{2}}e^{-f_t}$ is a solution to the conjugate heat equation. The evolving tuple $(M,g(t),f_t)_{t\in(-\infty,0)}$ is called the \emph{canonical form} of the Ricci shrinker. The canonical form satisfies
\begin{align}\label{canonicalformnormalization}
\Ric_{g_t}+\nabla^2 f_t=\frac{1}{2|t|}g(t),&
\\\nonumber
-|t|\big(|\nabla f_t|^2_{g_t}+R_{g_t}\big)+f_t\equiv \mu,&
\end{align}
where $\mu$ is the shrinker entropy. It turns out that the entropy of a fixed shrinker is unique (even if the normalized potential is not necessarily unique). In fact, Li-Wang \cite{LW20} proved the following strong statement.
\begin{Proposition}[\cite{LW20}]\label{shrinkernufunctional}
Let $(M^n,g,f)$ be a Ricci shrinker, where $f$ is normalized so that $\int_M (4\pi)^{-\frac{n}{2}}dg=1$. Let $\mu$ be the shrinker entropy. Then we have
\begin{eqnarray*}
\mu=\mathcal{W}(g,f,1)=\mu(g,1)=\nu(g).
\end{eqnarray*}
Furthermore, if we let $(M,g(t),f_t)_{t\in(-\infty,0)}$ be the canonical form, then we also have
\begin{eqnarray*}
\mu=\mathcal{W}(g(t),f_t,|t|)=\mu(g(t),|t|)=\nu(g(t))\quad\text{ for all }\quad t\in(-\infty,0).
\end{eqnarray*}
\end{Proposition}
\begin{comment}
Set
\[
d\nu := (4\pi)^{-n/2}e^{-f}dg,\quad
d\nu_t := (4\pi|t|)^{-n/2}e^{-f_t}dg(t),
\]
where $f_t=\phi_t^*f.$
Then
\[
\phi_t:(M,g(t),\nu_t)\to (M,|t|g,\nu)
\]
is an isometry.
Let $K(x,t|\,y,s)$ be the heat kernel coupled with $(M,g_t)_{t<0}$ and write
$\nu_{x,t;s}=K(x,t|\,\cdot,s)dg_s$. Let $H(x,y,t)$ be the heat kernel of the operator $\partial_t - \Delta_{f}$ coupled with the static metric $g$ and $d\nu'_{x;t} := H(x,\cdot, t)\, dg$.
Then for any $s<t<0,x\in M,$
\[
\nu_{x,t;s} = \phi_s^*\, \nu'_{\phi_t(x);\log(s/t)}.
\]
\end{comment}
\subsection{Locally uniformly Type I ancient solution}
Cheng-Zhang \cite{CZ20} studied a geometric condition for ancient Ricci flows, called the locally uniformly Type I condition. Precisely, it is defined as follows.
\begin{Definition}[Locally uniformly Type I ancient solutions]\label{def}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow. Fix $p_0\in M$ and let $\{(p_i,-\tau_i)\}_{i=1}^\infty\subset M\times(-\infty,0)$ be a sequence of space-time points with $\tau_i\nearrow\infty$. Then, $(M,g(t))_{t\in(-\infty,0]}$ is called \emph{locally uniformly Type I} along the sequence $\{(p_i,-\tau_i)\}_{i=1}^\infty$, if the following hold.
\begin{enumerate}[(1)]
\item $\ell_{p_0,0}$ is bounded along $\{(p_i,-\tau_i)\}_{i=1}^\infty$, that is, \begin{eqnarray*}\label{def1}
\limsup_{i\rightarrow\infty}\ell(p_i,\tau_i)<\infty.
\end{eqnarray*}
\item Around the space-time points $(p_i,-\tau_i)$, the curvature has a locally uniformly Type I bound. More precisely, there exists a positive function $C:(0,\infty)\rightarrow(0,\infty)$ with the following property: for all $A>0$, there exists $i_0\in\mathbb{N}$, depending on $A$, such that
\begin{eqnarray*}\label{def2}
\sup_{B_{-\tau_i}(p_i,r\sqrt{\tau_i})\times[-2\tau_i,-\tau_i]}|{\Rm}|\leq \frac{C(r)}{\tau_i} \quad\text{for all}\quad i\geq i_0\quad \text{and for all}\quad r\leq A.
\end{eqnarray*}
\item There is a time-wise Ricci curvature lower bound for $g(t)$. In other words, there exists a continuous positive function $K:(-\infty,0]\rightarrow(0,\infty)$, such that
\begin{eqnarray*}\label{def3}
\Ric_{g_t}\geq-K(t)g(t)\quad\text{ for all }\quad t\in(-\infty,0].
\end{eqnarray*}
\item $g(t)$ is noncollapsed along $(p_i,-\tau_i)$. In other words,
\begin{eqnarray*}\label{def4}
\liminf_{i\rightarrow\infty} \Big((\tau_i)^{-\frac{1}{2}}\text{inj}\big(g(-\tau_i),p_i\big)\Big)>0,
\end{eqnarray*}
where $\text{inj}(g, x)$ stands for the injectivity radius of the Riemannian metric $g$ at $x$.
\end{enumerate}
\end{Definition}
With the locally uniformly Type I condition, Cheng-Zhang \cite{CZ20} obtained locally uniformly $C^0$ and $C^1$ estimates for the reduced distance $\ell$, and consequently proved the existence of an asymptotic shrinker. We shall include these results below.
\begin{Proposition}[Proposition 5.1 in \cite{CZ20}]\label{C0-l-estimate-LUTypeI}
Let $(M,g(t))_{t\in(-\infty,0]}$ be a locally uniformly Type I ancient solution as described in Definition \ref{def}. Let $\ell_i(\cdot,\tau):=\ell_{p_0,0}(\cdot,\tau_i\tau)$ and $g_i(t):=\tau_i^{-1}g(\tau_it)$. Then, for any $\varepsilon\in(0,\frac{1}{4})$, there is a positive function $C(\cdot,\varepsilon):(0,\infty)\rightarrow(0,\infty)$ with the following property: for any $r>0$, it holds that $0\leq\ell_i(x,|t|)\leq C(r,\varepsilon)$ for all $(x,t)\in B_{g_{i,-1}}(p_i,r)\times[-2,-1-\varepsilon]$ whenever $i$ is large enough.
\end{Proposition}
\begin{Proposition}[Asymptotic Shrinker \cite{CZ20}]\label{TIshrinker}
Let $g_i(t)$ and $\ell_i$ be as described in the above proposition, then, after possibly passing to a subsequence, the sequence of tuples $\displaystyle\Big\{\big(M,g_i(t),\ell_i(\cdot,|t|)\big)_{t\in[-2,-1]}\Big\}_{i=1}^\infty$ converges to (the canonical form of) a shrinking gradient Ricci soliton $\big(M_\infty,g_\infty(t),\ell_\infty(\cdot,|t|)\big)_{t\in(-2,-1)}$, satisfying
\begin{eqnarray*}
\Ric_{g_\infty}(t)+\nabla^2\ell_\infty(\cdot,|t|)=\frac{1}{2|t|}g_\infty(t).
\end{eqnarray*}
The convergence of the Ricci flows is in the pointed smooth Cheeger-Gromov-Hamilton sense, and $\ell_i\rightarrow\ell_\infty$ in the $C^{0,\alpha}_{\operatorname{loc}}$ sense or in the weak $*W^{1,2}_{\operatorname{loc}}$ sense on $M_\infty\times(1,2)$, where $\ell_i$ should be understood to be pulled back by the defining diffeomorphisms of the Cheeger-Gromov-Hamilton convergence, and $\alpha$ is any number in $(0,1)$.
\end{Proposition}
\begin{Corollary}[Noncollapsing \cite{CZ20}]\label{noncollapsing}
A locally uniformly Type I ancient solution is (weakly) $\kappa$-noncollapsed on all scales, where $\kappa>0$ depends only on the entropy of the asymptotic shrinker.
\end{Corollary}
\subsection{Metric flow and
$H_n$-center}
In this subsection we briefly review the notion of the metric flow introduced by Bamler \cite{Bam20b}. Since here (and in the next subsection) we are not attempting to be comprehensive about all the nuances, the reader should always resort to \cite{Bam20b} for more details. We shall use the notation $\mathcal{P}(X)$ to represent the space of probability measures on a metric space $X.$ The metric flow is introduced as a natural generalization of the Ricci flow space-time. A metric flow over $I\subset \mathbb{R}$ is a tuple
\[\big(\mathcal{X},\mathfrak{t}, ({\rm dist}_t)_{t\in I}, (\nu_{x\,|\,s})_{x\in \mathcal{X}, s\in I, s\le \mathfrak{t}(x)}\big),\]
where $\mathfrak{t}:\mathcal{X}\to I$ is the time function and $\mathcal{X}_t:= \mathfrak{t}^{-1}(t)$ is called the time slice at $t$, ${\rm dist}_t$ is a metric on $\mathcal{X}_t$, $\nu_{x\,|\,s}\in \mathcal{P}(\mathcal{X}_s)$ is a family of probability measures called the \emph{conjugate heat kernel} based at $x\in\mathcal{X}$, and it satisfies the usual reproduction formula: for any $t_1\le t_2\le t_3$ in $I$ and for any $x\in \mathcal{X}_{t_3}$, we have
\[
\nu_{x\,|\,t_1}
= \int_{\mathcal{X}_{t_2}} \nu_{\cdot\,|\,t_1}\, d\nu_{x\,|\,t_2}.
\]
The sharp gradient estimate of Bamler \cite[Theorem 4.1]{Bam20b} is also axiomized into the definition of the metric flow. More details could be found in \cite[Definition 3.2]{Bam20b}. The last two authors generalized \cite[Theorem 4.1]{Bam20b} to noncompact Ricci flows with bounded curvature within each compact time interval (c.f. \cite{MZ21}). Hence, the following observation is straightforward.
\begin{Theorem}[Bamler]
\label{thm: smooth flows are metric flows}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each time interval compact in $I$.
Then it induces a canonical metric flow in the sense of \cite[Definition 3.2]{Bam20b}.
\end{Theorem}
\begin{proof}
By \cite[Lemma 26.16]{RFV3}, the reproduction formula holds for the minimal heat kernel $K(x,t\,|\,y,s)$ coupled with the flow $(M,g(t))$.
So \cite[Definition 3.2 (7)]{Bam20b} is satisfied if we choose $d\nu_{x,t\,|\,s} := K(x,t\,|\,\cdot,s)\, dg_s.$
As mentioned in the above paragraph, \cite[Definition 3.2 (1)-(6)]{Bam20b} are satisfied.
\end{proof}
We shall then introduce some definitions and results for the metric flow. In particular, they can be applied to smooth Ricci flows satisfying the condition of the above theorem. Before we proceed to define the notion of $H$-concentration, let us recall the \emph{variance} of two probability measures. Let $\mu,\nu\in \mathcal{P}(X)$, where $X$ is a metric space, then their variance is defined as
\[
{\rm Var}(\mu,\nu)
:= \int_{X\times X} {\rm dist}^2(y_1,y_2)
\, d\mu(y_1)d\nu(y_2),
\]
where ${\rm dist}$ is the metric on $X$. If $\mu$ is the same as $\nu$, then we also denote ${\rm Var}(\mu):={\rm Var}(\mu,\mu)$. $H$-concentration is defined as follows.
\begin{Definition}[Definition 3.30 in \cite{Bam20b}] A metric flow $\mathcal{X}$ over $I\subset\mathbb{R}$ is said to be $H$-concentrated, where $H$ is a positive number, if for any $s, t\in I$, $s\leq t$, and $x_1, x_2$ $\in \mathcal{X}_t$, we have,
\begin{equation}\label{thedefinitionofHconcentrarion}
\text{Var}(\nu_{x_1\,|\,
s}, \nu_{x_2\,|\, s})\leq \text{dist}_{t}^2(x_1, x_2)+H(t-s).
\end{equation}
\end{Definition}
Combining (\ref{thedefinitionofHconcentrarion}) with the reproduction formula, we have that, on an $H$-concentrated metric flow $\mathcal{X}$, for any $x_1,x_2\in \mathcal{X}_t,$
\[
{\rm Var}(\nu_{x_1\,|\,s},\nu_{x_2\,|\,s}) + H s
\]
is non-decreasing in $s\in I\cap(-\infty, t]$, and is bounded from above by ${\rm dist}_t^2(x_1,x_2)+Ht$ (c.f. \cite[Proposition 3.34]{Bam20b}). This fact guarantees the existence of $H$-centers (c.f. \cite[Proposition 3.36]{Bam20b}), which are defined as follows.
\begin{Definition}[Definition 3.35 in \cite{Bam20b}]\label{def_H_n_center}
Let $s,t\in I$ and $s\leq t$. A point $z$ $\in \mathcal{X}_s$ is called an $H$-center of $x$ $\in \mathcal{X}_t$ if
\begin{equation*}
\text{Var}(\delta_z, \nu_{x\,|\,s})\leq H(t-s).
\end{equation*}
\end{Definition}
The $H$-center adopted its name partially because the conjugate heat kernel accumulates its measure around it. Precisely, we have (\cite[Proposition 3.13]{Bam20a} and \cite[Lemma 3.37]{Bam20b}):
\begin{Proposition}[Bamler]\label{measureaccumulationofHcenter}
Let $\mathcal{X}$ be an $H$-concentrated metric flow over $I\subset\mathbb{R}$. Let $x\in \mathcal{X}_t$ be a fixed point, and let $z\in\mathcal{X}_s$ be an $H$-center of $x$, where $s<t$ and $s,t\in I$. Then, for any $A>1$, we have
\begin{eqnarray*}
\nu_{x\,|\, s}\big(B_s(z,\sqrt{AH(t-s)})\big)\geq 1-\frac{1}{A}.
\end{eqnarray*}
\end{Proposition}
\textbf{Remark:} Bamler \cite[Proposition 3.2]{Bam20a} proved that if $\mathcal{X}=M^n\times I$ is a Ricci flow space-time on a closed manifold $M^n$ over an interval $I$, then $\mathcal{X}$ must be $H_n$-concentrated, where $$H_n:=\frac{(n-1)\pi^2}{2}+4.$$ The same argument also works when the Ricci flow is complete and has bounded curvature within compact time intervals. \emph{This is a fact which shall be used throughout this article.} In general, given $x$ $\in \mathcal{X}_t$, and $s\leq t$, $H$-centers of $x$ may not be unique in $\mathcal{X}_s$.
\subsection{Metric flow pair and $\mathbb{F}$-convergence}
Suppose $\mathcal{X}$ is a metric flow over $I\subset\mathbb{R}$.
A family of probability measures $\mu_s\in \mathcal{P}(\mathcal{X}_s)$, where $s\in I'\subset I$, is called a \emph{conjugate heat flow} if it satisfies the reproduction formula: for any $s,t\in I',s\le t,$ we have
\[
\mu_s
= \int_{\mathcal{X}_t} \nu_{x\,|\,s}\, d\mu_t(x).
\]
A metric flow pair is then defined to be a metric flow coupled with a conjugate heat flow.
\begin{Definition}[Definition 5.1 in \cite{Bam20b}] A pair $\left(\mathcal{X}, (\mu_t)_{t\in I'}\right)$ is called a \emph{metric flow pair} over $I\subset \mathbb{R}$ if the following conditions are satisfied:
\begin{itemize}
\item $I'\subset I$, and $|I\setminus I'|=0$, where $|\,\cdot\,|$ is the Lebesgue measure;
\item $\mathcal{X}$ is a metric flow over $I'$;
\item $(\mu_t)_{t\in I'}$ is a conjugate heat flow on $\mathcal{X}$ with $\spt\mu_t=\mathcal{X}_t$ for all $t\in I'$.
\end{itemize}
\end{Definition}
The definition of $\mathbb{F}$-convergence requires the notions of coupling and $1$-Wasserstein distance between probability measures. Let $X, Y$ be metric spaces.
For any $\mu\in \mathcal{P}(X)$ and $\nu\in \mathcal{P}(Y),$ we denote by $\Pi(\mu,\nu)$ the space of \emph{couplings} between $\mu$ and $\nu$, namely, the set of all the probability measures $q\in \mathcal{P}(X\times Y)$ satisfying
\[
q(A\times Y)=\mu(A),\quad
q(X\times B)=\nu(B),
\]
for any measurable subsets $A\subset X$ and $ B\subset Y$. The $1$-Wasserstein distance between $\mu,\nu\in \mathcal{P}(X)$ is defined to be
\[
{\rm dist}_{W_1}(\mu,\nu)
:= \inf_{q\in \Pi(\mu,\nu)} \int_{X\times X} {\rm dist}(x,y) \, dq(x,y).
\]
By the Kantorovich-Rubinstein Theorem, this definition is equivalent to
\[
{\rm dist}_{W_1}(\mu,\nu)
= \sup_{f}\left( \int f\, d\mu
- \int f\, d\nu\right),
\]
where the supremum is taken over all bounded $1$-Lipschitz functions $f$ on $X.$
It is to be noted that for any metric flow $\mathcal{X}$, the $1$-Wassernstein distance between two conjugate heat flows satisfies a monotonicity property (\cite[Proposition 3.24(2)]{Bam20b}), namely, for any conjugate heat flows $(\mu^1_s)_{s\in I'}$ and $(\mu^2_s)_{s\in I''}$, we have
\begin{eqnarray}\label{monotoneofdW1}
{\rm dist}_{W_1}^{\mathcal{X}_s}
(\mu^1_2,\mu^2_s)\quad\text{ is non-decreasing in }\quad s\in I'\cap I''.
\end{eqnarray}
Consequently, for any $x_1$ and $x_2\in\mathcal{X}_t$, we have
\begin{eqnarray}
{\rm dist}_{W_1}^{\mathcal{X}_s}(\nu_{x_1\,|\,s},\nu_{x_2\,|\,s})\leq {\rm dist}_t(x_1,x_2)\quad\text{ for all }\quad s< t.
\end{eqnarray}
In fact, this monotonicity property is
a consequence of the prescribed gradient estimate in the definition of the metric flow (\cite[Definition 3.2(6)]{Bam20b}).
We now introduce the definition of $\mathbb{F}$-convergence within a correspondence, because this is the only version we shall use, and because it is essentially equivalent to the definition of $\mathbb{F}$-convergence itself. See more details in \cite[Section 6]{Bam20b}. Let $\displaystyle\big\{(\mathcal{X}^i,(\mu_t^i)_{t\in I'^{, i}})\big\}_{i\in \mathbb{N}\cup\{\infty\}}$ be a sequence of metric flow pairs over a finite interval $I\subset\mathbb{R}$. A correspondence $\mathfrak{C}$ is a collection of complete and separable metric spaces $(Z_t,{\rm dist}^{Z_t})_{t\in I}$ together with isometric embeddings $\phi_t^i:(\mathcal{X}_t^i,{\rm dist}_t^i)\to (Z_t,{\rm dist}^{Z_t})$ for $t\in I'^{,i}.$
Then, $(\mathcal{X}^i,(\mu_t^i)_{t\in I'^{, i}})$ $\mathbb{F}$-converges to $(\mathcal{X}^\infty,(\mu_t^\infty)_{t\in I'^{, \infty}})$ within the correspondence $\mathfrak{C}$ uniformly on $J\subset I$, denoted as
\[
\left(\mathcal{X}^i,(\mu_t^i)_{t\in I'^{, i}}\right)
\xrightarrow{\makebox[1.5cm]{$\mathbb{F},\mathfrak{C},J$}}
\left(\mathcal{X}^\infty,(\mu_t^\infty)_{t\in I'^{,\infty}}\right),
\]
if for any $\varepsilon>0,$ there is an $\bar i\in \mathbb{N},$ such that if $i\ge \bar i,$
there is a measurable subset $E_i\subset I$ with $$J\subset I\setminus E_i\subset I'^{,i}\cap I'^{,\infty},$$ and there are couplings $q_t^i\in \Pi(\mu_t^i,\mu_t^\infty)$ for $t\in I\setminus E_i$ with the following properties:
\begin{itemize}
\item $|E_i| \le \varepsilon^2$.
\item For any $s,t\in I\setminus E_i,s\le t,$ it holds that
\[
\int_{\mathcal{X}^i_t\times \mathcal{X}^\infty_t}
{\rm dist}^{Z_s}_{W_1}
\left(
\phi^i_{s*}\nu^i_{x_1\,|\,s},
\phi^\infty_{s*}\nu^\infty_{x_2\,|\,s}
\right)\, dq_t^i(x_1,x_2)
\le \varepsilon.
\]
\end{itemize}
If $J$ above can be taken as any compact sub-interval of $I$, we say that the convergence is uniform over any compact sub-intervals.
The tangent flow at infinity of an ancient solution is then defined to be the $\mathbb{F}$-limit of a sequence of Ricci flows obtained by a Type I scaling process for the ancient solution.
\begin{Definition}[Tangent flow at infinity \cite{Bam20b}]\label{tangentflowatinfinity}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution. A metric flow pair $\big(\mathcal{X},(\mu_t)_{t\in (-\infty,0)}\big)$, where $\mathcal{X}$ is a metric flow over $(-\infty,0]$, is called a \emph{tangent flow at infinity} of $(M,g(t))_{t\in(-\infty,0]}$, if there exist a fixed point $(p_0,t_0)\in M\times(-\infty,0]$ and a sequence $\tau_i\nearrow\infty$, such that
\begin{eqnarray*}
\left((M,g_i(t))_{t\in(-\infty,0]},(\nu^i_t)_{t\in(-\infty,0]}\right)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\left(\mathcal{X},(\mu_t)_{t\in (-\infty,0)}\right),
\end{eqnarray*}
where
\begin{gather*}
g_i(t):=\tau_i^{-1}g(t_0+\tau_it),
\\
\nu^i_t:=\nu_{p_0,t_0\,|\,t_0+\tau_it},
\end{gather*}
for all $t\in(-\infty,0]$. Since the interval $(-\infty,0]$ is infinite, the $\mathbb{F}$-convergence here should be understood to be the $\mathbb{F}$-convergence on each finite subinterval of $(-\infty,0]$.
\end{Definition}
If we assume that the ancient solution in question has bounded curvature within each compact time intervals, then it must be $H_n$-concentrated. By \cite[Theorem 7.6]{Bam20b}, a tangent flow always exists for any fixed base point $(p_0,t_0)$ and for any sequence of scaling factors $\{\tau_i\}_{i=1}^\infty$. However, without a uniform lower bound on the Nash entropy, the tangent flow could be collapsed.
\section{Conjugate heat kernel estimates on noncompact manifolds}
In this section, we shall verify that \cite[Theorem 7.2]{Bam20a} and \cite[Proposition 5.13]{Bam20a} are still valid for Ricci flows with bounded curvature within each compact time interval. Bamler's original proof was for Ricci flows on closed manifolds. Given the assumptions which we make, these generalizations are but straightforward arguments, yet they are very important to the proofs of our main theorems.
\subsection{The Gaussian upper bound}
\begin{Theorem}[Theorem 7.1 in \cite{Bam20a}]\label{aCHKcoarseupperboundofbamler}
Let $(M^n,g(t))_{t\in I}$, where $I$ is a compact interval, be a complete Ricci flow with bounded curvature. Suppose $[s,t]\subset I$ and $R\geq R_{\operatorname{min}}$ on $M\times[s,t]$. Then for any $x,y\in M$, we have
\begin{eqnarray*}
K(x,t\,|\,y,s)\leq\frac{C}{(t-s)^{\frac{n}{2}}}\exp(-\mathcal{N}_{x,t}(t-s)),
\end{eqnarray*}
where $C:=C(R_{\operatorname{min}}(t-s))$ is a constant depending on $R_{\operatorname{min}}(t-s)$.
\end{Theorem}
\begin{Theorem}[Theorem 7.2 in \cite{Bam20a}]\label{gaussianupperbound}
Let $(M^n,g(t))_{t\in I}$, where $I$ is a compact interval, be a complete Ricci flow with bounded curvature. Suppose $[s,t]\subset I$ and $R\geq R_{\operatorname{min}}$ on $M\times[s,t]$. Let $(z,s)\in M\times I$ be an $H_n$-center of a point $(x,t)\in M\times I$. Then for any $\varepsilon>0$ and $y\in M$, we have
\begin{eqnarray*}
K(x,t\,|\,y,s)\leq\frac{C\exp(-\mathcal{N}_{x,t}(t-s))}{(t-s)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_s^2(z,y)}{(8+\varepsilon)(t-s)}\right),
\end{eqnarray*}
where $C:=C(R_{\operatorname{min}}(t-s),\varepsilon)$ is a constant depending on $R_{\operatorname{min}}(t-s)$ and $\varepsilon$.
\end{Theorem}
\begin{proof}[Proof of Theorem \ref{aCHKcoarseupperboundofbamler}]
The proof is not essentially different from \cite[Theorem 7.1]{Bam20a}, since the latter does not rely heavily on the fact that the background manifold is closed. The only potential issue is the application of the maximum principle. Hence, we shall verify \cite[(7.11)]{Bam20a} below. By parabolic rescaling, we may assume that $s=0$ and $t=1$. It suffices to show
\begin{equation} \label{ineq: v is subsolution}
v(x,t)
\le \int_{M} K(x,t\,|\,y,\tfrac{1}{2})
v(y,\tfrac{1}{2})\, dg_{\frac{1}{2}}(y),
\end{equation}
for all $(x,t)\in M\times [\frac{1}{2},1]$, where
\begin{equation} \label{ultranonsense}
v := (t-\tfrac{1}{2})|\nabla u|^2 + u^2,\quad
u(x,t):= K(x,t\,|\,y,0),
\end{equation}
and $y\in M$ is a fixed point.
In fact, by elementary computations, we have that $v$ is a subsolution to the heat equation, that is, $$\Box v=|\nabla u|^2-2(t-\tfrac{1}{2})|\nabla^2u|^2-2|\nabla u|^2\leq 0.$$Hence, to prove (\ref{ineq: v is subsolution}), we need only to verify that the maximum principle can be applied to $v$ on $M\times[\frac{1}{2},1]$.
By \cite[Lemma 26.17]{RFV3}, we can find a positive constant $J>0$, depending on the curvature bound on $M\times[0,1]$ and $\Vol_{g_0}\big(B_{g_0}(y,1)\big)>0$, such that $0<u(x,t)\leq J$ for all $(x,t)\in M\times[\frac{1}{4},1]$. We may then apply \cite[Theorem 3.2]{Zhq06} (for the proof in our noncompact setting, see \cite[Lemma 2.4]{Zhang21}) to obtain
\begin{eqnarray*}
\frac{|\nabla u|^2}{u^2}\leq\frac{1}{t-\tfrac{1}{4}}\log\frac{J}{u}\quad\text{ on }\quad M\times(\tfrac{1}{4},1].
\end{eqnarray*}
Hence, we have the following estimate on $M\times[\frac{1}{2},1]$.
\begin{eqnarray*}
|\nabla u|^2\leq 4\Big(u^2\log J-u^2\log u\Big)\leq C(J),
\end{eqnarray*}
where $C(J)$ is a constant depending only on $J$. Here we have applied the fact that $0<u\leq J$ on $M\times[\frac{1}{2},1]$. It then follows that the function $v$ as defined in (\ref{ultranonsense}) is bounded on $M\times[\frac{1}{2},1]$, and the standard maximum principle (c.f. \cite[Theorem 12.10]{RFV2}) implies that
\begin{equation*}
(t-\tfrac{1}{2})|\nabla u|^2(x,t)\leq v(x,t)
\leq \int_{M}K(x,t\,|\, y, \tfrac{1}{2})u^2(y,\tfrac{1}{2})d g_{\tfrac{1}{2}}.
\end{equation*}
This is exactly \cite[(7.11)]{Bam20a}.
For the rest of the proof, the reader may follow \cite{Bam20a} line by line, and we shall not include all the details therein. Besides the argument above, Bamler's original proof also applied \cite[Theorem 5.9]{Bam20a}, \cite[Corollary 5.11]{Bam20a}, and \cite[Theorem 6.2]{Bam20a}; the first two of them are already proved in \cite[Theorem 4.4, Corollary 4.5]{MZ21} for our case, and the third result can be easily generalized to our case using (\ref{Nashintegral_2}) below.
\end{proof}
The proof of Theorem \ref{gaussianupperbound} also follows closely after Bamler's argument, for it does not depend on anything holding exclusively on closed manifolds. In the course of the proof, all the auxiliary results have already been established in our case, as described in the last paragraph above. We shall not exhibit all these details here, since this could be a distraction from our main purpose, and since this type of generalizations are usually understood to be straightforward.
\subsection{An integral inequality involved in the Nash entropy}
Next, we prove \cite[Proposition 5.13]{Bam20a} on a complete Ricci flow with bounded curvature on compact time intervals. The reader will see later that formula (\ref{Nashintegral_2}) is particularly useful for the local analysis of the Nash entropy, that is, the estimation of the Nash entropy using local geometry (see section 4 below). Interestingly, we shall apply this formula in a somewhat reverse way as Bamler did in the proof of \cite[Theorem 6.2]{Bam20a}.
\begin{Proposition}[Proposition 5.13 in \cite{Bam20a}]\label{Nashintegral}
Let $(M^n,g(t))_{t\in I}$, where $I$ is a compact interval, be a Ricci flow with bounded curvature. Let $t_0-\tau$, $t_0\in I$, where $\tau>0$, and let $d\nu:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg$ be the conjugate heat kernel based at $(x_0,t_0)$. Furthermore, assume $R(\cdot,t_0-\tau)\geq R_{\operatorname{min}}$, then we have
\begin{eqnarray}
\int_M\tau(|\nabla f|^2+R)d\nu_{t_0-\tau}&\leq&\frac{n}{2},\label{Nashintegral_1}
\\
\int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}&\leq& n-2R_{\operatorname{min}}\tau.\label{Nashintegral_2}
\end{eqnarray}
\end{Proposition}
\begin{proof}
(\ref{Nashintegral_1}) follows from the fact that the Nash entropy is always no smaller than Perelman's entropy, which is always true when the Ricci flow has bounded curvature. (\ref{Nashintegral_2}) follows from (\ref{Nashintegral_1}) together with Hein-Naber's Poincar\'e inequality (\ref{ineq: poincare}). To apply the Poincar\'e inequality to the function $f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}$, let us first of all recall the coarse heat kernel Gaussian upper and lower bounds (c.f. Theorem 26.25 and Theorem 26.31 in \cite{RFV3}), that is,
\begin{eqnarray*}
C^{-1}{\rm dist}^2_{t_0-\tau}(x,x_0)-C\leq f(x,t_0-\tau)\leq C{\rm dist}^2_{t_0-\tau}(x_0,x)+C \quad \text{ for all }\quad x\in M,
\end{eqnarray*}
where $C$ is a constant depending only on the curvature bounds on $M\times[t_0-\tau,t_0]$, the value of $\tau$, and $\Vol_{g_{t_0}}\big(B_{g_{t_0}}(x_0,1)\big)$. It then follows from some straightforward computations that
\begin{align}\label{H-N-bound}
&\big|\,\mathcal{N}_{x_0,t_0}(\tau)\,\big|=\left|\,\int_Mfd\nu_{t_0-\tau}-\frac{n}{2}\,\right|<\infty,
\\\nonumber
& \int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}<\infty.
\end{align}
Let us then fix an arbitrary number $A>0$, and let $\phi^A:M\rightarrow[0,1]$ be a smooth cut-off function such that $\phi^A\equiv 1$ on $B_{t_0-\tau}(x_0,A)$, $\phi^A\equiv 0$ on $M\setminus B_{t_0-\tau}(x_0,2A)$, and $|\nabla\phi^A|_{g_{t_0-\tau}}\leq 2A^{-1}$ everywhere on $M$. We may then apply the PoincaR\'e inequality (\ref{ineq: poincare}) to $u=\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)$ and obtain
\begin{align}\label{H-N-f}
&\int_M\left(\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)\right)^2d\nu_{t_0-\tau}-\left(\int_M\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)d\nu_{t_0-\tau}\right)^2
\\\nonumber
&\leq2\tau\int_M\left|\nabla \left(\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)\right)\right|^2d\nu_{t_0-\tau}.
\end{align}
Taking $A\rightarrow\infty$ and using (\ref{H-N-bound}), we have the following computations for all terms in formula (\ref{H-N-f}).
\begin{align*}
&\int_M\left(\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)\right)^2d\nu_{t_0-\tau}\rightarrow \int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau},
\\
&\int_M\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)d\nu_{t_0-\tau}\rightarrow \int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)d\nu_{t_0-\tau}=0,
\\
&\int_M\left|\nabla \left(\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)\right)\right|^2d\nu_{t_0-\tau}=\int_M|\nabla\phi^A|^2\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}
\\
&\quad +\int_M(\phi^A)^2|\nabla f|^2d\nu_{t_0-\tau}+2\int_M\langle\nabla\phi^A,\nabla f\rangle\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)d\nu_{t_0-\tau},
\end{align*}
and for the last three terms, we have
\begin{align*}
&\int_M(\phi^A)^2|\nabla f|^2d\nu_{t_0-\tau}\rightarrow \int_M|\nabla f|^2d\nu_{t_0-\tau},
\\
&\int_M|\nabla\phi^A|^2\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}\leq\frac{4}{A^2}\int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}\rightarrow 0
\\
&\left|\int_M\langle\nabla\phi^A,\nabla f\rangle\phi^A\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)d\nu_{t_0-\tau}\right|
\\
&\quad\quad \leq\frac{2}{A}\left(\int_M|\nabla f|^2d\nu_{t_0-\tau}\right)^{\frac{1}{2}}\left(\int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}\right)^{\frac{1}{2}}\rightarrow 0.
\end{align*}
Hence, (\ref{Nashintegral_2}) follows from (\ref{H-N-f}).
\end{proof}
\begin{comment}
\begin{proof}[\textbf{Slightly different proof for Prop \ref{Nashintegral}:}]
(\ref{Nashintegral_1}) follows from the fact that the Nash entropy is always no smaller than Perelman's entropy, which is always true when the Ricci flow has bounded geometry. (\ref{Nashintegral_2}) follows from (\ref{Nashintegral_1}) together with Hein-Naber's Poincare inequality. Recall that if $u\in C_0^{\infty}(M)$ with $\int_M u d\nu_{t_0-\tau}=0$, then Hein-Naber (Theorem 1.10) proved that
\begin{eqnarray}\label{H-N}
\int_Mu^2d\nu_{t_0-\tau}\leq2\tau\int_M|\nabla u|^2d\nu_{t_0-\tau}.
\end{eqnarray}
To apply the above formula to the function $f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}$, let us first of all recall the coarse heat kernel Gaussian upper and lower bounds proved by [Chau-Tam-Yu, cite CCCG+], that is,
\begin{eqnarray*}
C^{-1}{\rm dist}^2_{t_0-\tau}(x,x_0)-C\leq f(x,t_0-\tau)\leq C{\rm dist}^2_{t_0-\tau}(x_0,x)+C \quad \text{ for all }\quad x\in M,
\end{eqnarray*}
where $C$ is a constant depending only on the curvature upper bound and the unit ball volume lower bound on $M\times[t_0-\tau,t_0]$. It then follows that
\begin{align}\label{H-N-bound}
&|\mathcal{N}_{x_0,t_0}(\tau)|=\left|\int_Mfd\nu_{t_0-\tau}-\frac{n}{2}\right|<\infty,
\\\nonumber
& \int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau}<\infty.
\end{align}
Let us then fix an arbitrary number $A>0$, and let $\phi:M\rightarrow[0,1]$ be a smooth cut-off function such that $\phi\equiv 1$ on $B_{t_0-\tau}(x_0,A)$, $\phi\equiv 0$ on $M\setminus B_{t_0-\tau}(x_0,2A)$, and $|\nabla\phi|_{g(t_0-\tau)}\leq 2A^{-1}$ everywhere on $M$. There exists positive constant $c(A)$ with $\lim_{A\to\infty} c(A)=0$ and
\begin{eqnarray*}
\int_M \phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right) d\nu_{t_0-\tau}=c(A)\int_M \phi d\nu_{t_0-\tau}.
\end{eqnarray*}
We may then apply (\ref{H-N}) to $u=\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)$ and obtain
\begin{align}\label{H-N-f}
&\int_M\left(\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)\right)^2d\nu_{t_0-\tau}
\\\nonumber
&\quad \leq2\tau\int_M\left|\nabla \left(\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)\right)\right|^2d\nu_{t_0-\tau}.
\end{align}
Taking $A\rightarrow\infty$ and using (\ref{Nashintegral_1}) and (\ref{H-N-bound}), we have the following analysis for all terms in formula (\ref{H-N-f}).
\begin{align*}
&\int_M\left(\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)\right)^2d\nu_{t_0-\tau}\rightarrow \int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}\right)^2d\nu_{t_0-\tau},
\\
&\int_M\left|\nabla \left(\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)\right)\right|^2d\nu_{t_0-\tau}=\int_M|\nabla\phi|^2\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)^2d\nu_{t_0-\tau}
\\
&\quad +\int_M\phi^2|\nabla f|^2d\nu_{t_0-\tau}+2\int_M\langle\nabla\phi,\nabla f\rangle\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)d\nu_{t_0-\tau},
\end{align*}
and for the last formula, we have
\begin{align*}
&\int_M\phi^2|\nabla f|^2d\nu_{t_0-\tau}\rightarrow \int_M|\nabla f|^2d\nu_{t_0-\tau},
\\
&\int_M|\nabla\phi|^2\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)^2d\nu_{t_0-\tau}\leq\frac{4}{A^2}\int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)^2d\nu_{t_0-\tau}\rightarrow 0
\\
&\left|\int_M\langle\nabla\phi,\nabla f\rangle\phi\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)d\nu_{t_0-\tau}\right|
\\
&\quad\quad \leq\frac{2}{A}\left(\int_M|\nabla f|^2d\nu_{t_0-\tau}\right)^{\frac{1}{2}}\left(\int_M\left(f-\mathcal{N}_{x_0,t_0}(\tau)-\frac{n}{2}-c(A)\right)^2d\nu_{t_0-\tau}\right)^{\frac{1}{2}}\rightarrow 0.
\end{align*}
Hence, (\ref{Nashintegral_2}) follows from (\ref{H-N-bound}).
\end{proof}
\end{comment}
\section{A pseudolocality for the Nash entropy}
In this section we prove Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash}. These results shall be important tools in the proof of Theorem \ref{Theorem_main}. We emphasize again that though the statements of Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash} are closely related, yet without a Nash entropy bound, there is no Gaussian upper bound for the conjugate heat kernel, and it is not clear whether the $H_n$-centers are close to the $\ell$-centers (compare with Proposition \ref{H_n_l_n}). Therefore, Theorem \ref{Coro_nash_2} and Theorem \ref{Thm_nash} do not imply each other.
\begin{proof}[Proof of Theorem \ref{Coro_nash_2}] In the proof, we shall use the lower case letter $c$ to represent a positive estimate constant which is intuitively small, and the capital letter $C$ to represent a positive estimate constant which is intuitively large. These constants may vary from line to line, we nevertheless use the same letters in order not to cause notational complexity. Throughout the proof, we define
\begin{eqnarray*}
r:=\sqrt{t-s},\quad s':=s-\tfrac{r^2}{9}.
\end{eqnarray*}
First of all, we state a consequence of the Bishop-Gromov comparison theorem, which will be used in the estimation of the local upper and lower bounds for the conjugate heat kernel.
\\
\noindent\textbf{Claim 1.} There are positive constants $c$ and $C$, depending only on $\alpha$ and $C_0$, such that the following hold for all $\displaystyle y\in B_s(z,\tfrac{r}{3})$.
\begin{gather}\label{claim1_1}
|{\Ric}|\leq C_0r^{-2} \quad \text{ on } \quad B_s(y,\tfrac{r}{3})\times [s',s],
\\
\Vol_{g_{s}}\left(B_s(y,\tfrac{r}{3})\right)\geq c\left(\tfrac{r}{3}\right)^n. \label{claim1_2}
\end{gather}
\begin{proof}[Proof of Claim 1]
(\ref{claim1_1}) is almost a restatement of an assumption of the theorem. We will prove (\ref{claim1_2}) by the Bishop-Gromov comparison theorem. Fixing any point $y\in B_s(z,\frac{r}{3})$, since $\Ric\geq -C_0r^{-2}$ on $B_s(z,r)$, we may argue as follows. Note that all the volumes are computed using $g(s)$.
\begin{eqnarray*}
\frac{\Vol\big(B_s(y,\tfrac{r}{3})\big)}{(\tfrac{r}{3})^n}&\geq& c\frac{\Vol\big(B_s(y,\tfrac{2r}{3} )\big)}{(\tfrac{2r}{3})^n}\geq c\frac{\Vol\big(B_s(z,\tfrac{r}{3} )\big)}{(\tfrac{2r}{3} )^n}
\\
&=&2^{-n}c\frac{\Vol\big(B_s(z,\tfrac{r}{3} )\big)}{(\tfrac{r}{3} )^n}\geq c\frac{\Vol\big(B_s(z,r)\big)}{ r^n}\geq c,
\end{eqnarray*}
where the constant $c$ depends only on $\alpha$ and $C_0$.
\end{proof}
Next, we shall obtain some estimates for the conjugate heat kernel $K(x,t\,|\,\cdot,\cdot)$. The local upper bound follows straightforwardly from the standard mean value inequality, and the local lower bound is a consequence of (\ref{subsolution}).
\\
\noindent\textbf{Claim 2.} There is a constant $C$, depending only on $\alpha$ and $C_0$, such that
\begin{eqnarray}
K\left(x,t\,\left|\,y,s'\right.\right)\leq Cr^{-n}\leq\frac{C}{(t-s')^{\tfrac{n}{2}}}\quad \text{ for all }\quad y\in B_s(z,\tfrac{r}{3}).
\end{eqnarray}
\begin{proof}[Proof of Claim 2]
Let us fix an arbitrary $y\in B_s(z,\frac{r}{3})$. Since $K(x,t\,|\,\cdot,\cdot)$ is a conjugate heat kernel, we have
\begin{eqnarray*}
\int_{s-\frac{1}{9}r^2}^s\int_{B_s(y,\frac{r}{3})}K(x,t\,|\,\cdot,\eta)dg_\eta d\eta\leq \int_{s-\frac{1}{9}r^2}^s\int_{M}K(x,t\,|\,\cdot,\eta)dg_\eta d\eta=\int_{s-\frac{1}{9}r^2}^s1 d\eta=\frac{1}{9}r^2.
\end{eqnarray*}
On the other hand, the bounds provided by (\ref{claim1_1}) and (\ref{claim1_2}) are sufficient for the implementation of the standard mean value inequality (c.f. \cite[Theorem 25.2]{RFV3}) on the parabolic disk $B_s(y,\frac{r}{3})\times[s-\frac{1}{9}r^2,s]$. We then obtain that
\begin{eqnarray*}
K\left(x,t\,\left|\,y,s'\right. \right)&\leq&\frac{C}{\frac{1}{9}r^2\Vol_{g_{s}}\big(B_s(y,\frac{r}{3})\big)}\int_{s-\frac{1}{9}r^2}^s\int_{B_s(y,\frac{r}{3})}K(x,t|\cdot,\eta)dg_\eta d\eta
\\
&\leq&\frac{C}{\frac{1}{9}r^2\cdot c(\frac{r}{3})^n}\cdot\tfrac{1}{9}r^2\leq C r^{-n}.
\end{eqnarray*}
\end{proof}
\noindent\textbf{Claim 3:} There is a constant $c$, depending only on $\alpha$ and $C_0$, such that
\begin{eqnarray*}
K(x,t\,|\,y,s')\geq cr^{-n}\geq\frac{c}{(t-s')^{\frac{n}{2}}}\quad\text{ for all }\quad y\in B_s(z,\tfrac{r}{3}).
\end{eqnarray*}
\begin{proof}[Proof of Claim 3]
Let us fix an arbitrary $y\in B_s(z,\frac{1}{3}r)$, and let $\gamma_2:[0,\frac{1}{9}r^2]\rightarrow B_s(z,\frac{1}{3}r)$ be the minimal constant-speed $g(s)$-geodesic, such that $\gamma_2(0)=z$ and $\gamma_2(\frac{1}{9}r^2)=y$. Then, obviously we have
\begin{eqnarray*}
|\gamma_2'(\tau)|^2_{g_{s}}\leq\left(\frac{r/3}{r^2/9}\right)^2= 9r^{-2}\quad \text{ for all }\quad \tau\in[0,\tfrac{1}{9}r^2].
\end{eqnarray*}
Therefore, by the Ricci curvature bound in (\ref{claim1_1}), we have
\begin{eqnarray}\label{maximusnonsense001}
|\gamma'_2(\tau)|^2_{g_{s-\tau}}\leq|\gamma'_2(\tau)|^2_{g_{s}}\exp\left(\int_0^{\tfrac{1}{9}r^2}\left(\sup_{B_s\left(z,\tfrac{1}{3}r\right)\times[s',s]}|\Ric|\right)d\eta\right)\leq Cr^{-2},
\end{eqnarray}
for all $\tau\in[0,\tfrac{1}{9}r^2]$. Let $\gamma_1:[0,r^2]\rightarrow M$ be a minimal $\mathcal{L}$-geodesic from $(x,t)$ to $(z,s)$. Then, we may use
\begin{eqnarray*}
\gamma(\tau):=\left\{\begin{array}{lcr}
\gamma_1(\tau) &\text{if} &\tau\in[0,r^2], \\
\gamma_2(\tau-r^2)&\text{if}&\tau\in[r^2,\tfrac{10}{9}r^2].
\end{array}\right.
\end{eqnarray*}
as a test curve in (\ref{definitionofl}) and estimate
\begin{eqnarray*}
\ell_{x,t}(y',t-s')&=&\ell_{x,t}\left(y',\tfrac{10}{9}r^2\right)
\\
&\leq&\frac{1}{2\sqrt{\tfrac{10}{9}r^2}}\bigg\{\int_0^{r^2}\sqrt{\tau}\left(|\gamma_1'(\tau)|^2_{g_{t-\tau}}+R_{g_{t-\tau}}(\gamma_1(\tau))\right)d\tau
\\
&&\quad\quad\quad +\int_{0}^{\tfrac{1}{9}r^2}\sqrt{\tau+r^2}\left(|\gamma_2'(\tau)|^2_{g_{s-\tau}}+R_{g_{s-\tau}}(\gamma_2(\tau))\right)d\tau\bigg\}
\\
&\leq&\frac{3}{2\sqrt{10}r}\left(2\sqrt{r^2}\ell_{x,t}(z,r^2)+\int_0^{\frac{1}{9}r^2}\sqrt{\tau+r^2}Cr^{-2}d\tau\right)
\\
&\leq&\frac{3}{2\sqrt{10}r}\left(nr+Cr\right)
\\
&\leq& C.
\end{eqnarray*}
Here we have used the facts that $\ell_{x,t}(z,t-s)\leq\frac{n}{2}$ and that $\displaystyle\sup_{\tau\in[0,\frac{1}{9}r^2]}\left|R_{g_{s-\tau}}(\gamma_2(\tau))\right|\leq C_0r^{-2}$, where the latter is because $\gamma_2\subset B_s(z,\frac{r}{3})$. Claim 3 then follows from (\ref{subsolution}).
\end{proof}
With all the preparations above, we are ready to estimate the Nash entropy. By (\ref{claim1_1}), we have
\begin{eqnarray*}
\Vol_{g_{s'}}\big(B_s(z,\tfrac{r}{3})\big)&\geq&\Vol_{g_{s}}\big(B_s(z,\tfrac{r}{3})\big)\exp\left(-\int_0^{\tfrac{1}{9}r^2}\left(\sup_{B_s(z,\tfrac{r}{3})\times[s',s]}|R|\right)d\eta\right)
\\\nonumber
&\geq&cr^n\geq c(t-s')^{\tfrac{n}{2}}.
\end{eqnarray*}
Combining this with Claim 3, we have
\begin{eqnarray}\label{P1}
\nu_{x,t\,|\,s'}\big(B_s(z,\tfrac{r}{3})\big)=\int_{B_s(z,\tfrac{r}{3})}K(x,t\,|\,\cdot,s')dg_{s'}\geq c.
\end{eqnarray}
Since $[s-r^2,s]\subset I$, we have, by the maximum principle,
\begin{eqnarray}\label{P2}
R(\cdot,s')\geq-\frac{n}{2(s'-(s-r^2))}\geq-Cr^{-2}\geq-\frac{C}{t-s'}.
\end{eqnarray}
Let us define function $f$ by
\begin{eqnarray*}
K(x,t\,|\,\cdot,s'):=\frac{1}{(4\pi(t-s'))^{\frac{n}{2}}}e^{-f(\cdot,s')}.
\end{eqnarray*}
Then, Claim 2 and Claim 3 imply
\begin{eqnarray}\label{P3}
-C\leq f(\cdot,s')\leq C\quad\text{ on }\quad B_s(z,\tfrac{r}{3}).
\end{eqnarray}
With the estimates (\ref{P1})---(\ref{P3}), we may compute by using (\ref{Nashintegral_2})
\begin{align*}
& \quad\quad n+2(t-s')\cdot\frac{C}{t-s'}
\\
&\geq\int_M\left(f(\cdot,s')-\mathcal{N}_{x,t}(t-s')-\tfrac{n}{2}\right)^2d\nu_{x,t\,|\,s'}
\\\nonumber
&\geq\int_{B_s(z,\tfrac{r}{3})}\left(f(\cdot,s')-\mathcal{N}_{x,t}(t-s')-\tfrac{n}{2}\right)^2d\nu_{x,t\,|\,s'}
\\\nonumber
&=\int_{B_s(z,\tfrac{r}{3})}\bigg(f^2(\cdot,s')-2f\left(\mathcal{N}_{x,t}(t-s')+\tfrac{n}{2}\right)+\left(\mathcal{N}_{x,t}(t-s')+\tfrac{n}{2}\right)^2\bigg)d\nu_{x,t\,|\,s'}
\\\nonumber
&\geq\int_{B_s(z,\tfrac{r}{3})}\left(-f^2(\cdot,s')+\tfrac{1}{2}\left(\mathcal{N}_{x,t}(t-s')+\tfrac{n}{2}\right)^2\right)d\nu_{x,t\,|\,s'}
\\\nonumber
&\geq\tfrac{1}{2}\left(\mathcal{N}_{x,t}(t-s')+\tfrac{n}{2}\right)^2\nu_{x,t\,|\,s'}\big(B_s(z,\tfrac{r}{3})\big)-\nu_{x,t\,|\,s'}\big(B_s(z,\tfrac{r}{3})\big)\sup_{B_s(z,\tfrac{r}{3})}f^2
\\\nonumber
&\geq c\left(\mathcal{N}_{x,t}(t-s')+\tfrac{n}{2}\right)^2-C.
\end{align*}
It then follows that
\begin{eqnarray*}
\mathcal{N}_{x,t}(t-s)\geq\mathcal{N}_{x,t}(t-s')\geq -\beta.
\end{eqnarray*}
This finishes the proof of the theorem.
\end{proof}
Before we proceed to prove Theorem \ref{Thm_nash}, a few preparatory results are needed. These results are all proved in \cite{Bam20a} for Ricci flows on closed manifolds, but they can be easily generalized to our case without much effort.
\begin{Proposition}[Corollary 5.11 in \cite{Bam20a}; c.f. Corollary 4.5 in \cite{MZ21}]\label{harnackofnashentropy}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each time interval compact in $I$. Assume $s$, $t^*$, $t_1$, and $t_2\in I$ with $s<t^*\leq t_1, t_2$. Moreover, assume $R(\cdot,t^*)\geq R_{\operatorname{min}}$. Then, for any $x_1$ and $x_2\in M$, we have
\begin{eqnarray}
\mathcal{N}^*_s(x_1,t_1)-\mathcal{N}_s^*(x_2,t_2)\leq\left(\frac{n}{2(t^*-s)}-R_{\operatorname{min}}\right)^{\frac{1}{2}}{\rm dist}_{W_1}^{g_{t^*}}(\nu_{x_1,t_1\,|\,t^*},\nu_{x_2,t_2\,|\,t^*})+\frac{n}{2}\log\left(\frac{t_2-s}{t^*-s}\right),
\end{eqnarray}
where we have defined $\mathcal{N}_s^*(x,t):=\mathcal{N}_{x,t}(t-s)$.
\end{Proposition}
\begin{Theorem}[Theorem 8.1 in \cite{Bam20a}]\label{localnoncollapsing}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each time interval compact in $I$. Assume $[t-r^2,t]\subset I$ and $R\geq R_{\operatorname{min}}$ on $M\times[t-r^2,t]$. Then for all $A\in[1,\infty)$, we have
$$\Vol_{g_t}\left(B_{g_t}(x,Ar)\right)\leq C(R_{\operatorname{min}}r^2)\exp(\mathcal{N}_{x,t}(r^2))\exp(C_0A^2)r^n,$$ where $C_0$ is a constant depending only on the dimension.
\end{Theorem}
\begin{proof}
In the original proof of \cite[Theorem 8.1]{Bam20a}, the results applied are \cite[Theorem 5.9]{Bam20a} and \cite[Theorem 7.5]{Bam20a}, and the former is already established in \cite[Theorem 4.4]{MZ21} under the assumption of bounded curvature within each compact time interval. For \cite[Theorem 7.5]{Bam20a}, the results applied in the original proof are \cite[Theorem 5.9]{Bam20a} (c.f. \cite[Theorem 4.4]{MZ21}), \cite[Corollary 5.11]{Bam20a} (c.f. \cite[Corollary 4.5]{MZ21}), the Gaussian concentration theorem (c.f. Proposition \ref{prop: gaussian concentration} above), \cite[Theorem 7.1]{Bam20a} (c.f. Theorem \ref{aCHKcoarseupperboundofbamler} above), and Proposition 4.2 (c.f. \cite[Proposition 3.4]{MZ21}). In conclusion, the proof in our case follows step by step after \cite[Theorem 8.1]{Bam20a}; all these results hold under our current assumption. Since the proofs of both \cite[Theorem 7.5]{Bam20a} and \cite[Theorem 8.1]{Bam20a} are short, we shall leave the verification to the readers.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Thm_nash}]
Under the assumption of the theorem, we denote
$$r:=\sqrt{t-s}.$$
Since $s-r^2\in I$, by the maximum principle, we have $R(\cdot,s)\geq -\frac{n}{2r^2}$. Then, applying Theorem \ref{localnoncollapsing} to the point $(z,s)$ and $A=1$, we have
$$\alpha r^n\leq \operatorname{Vol}_{g_s}\big(B_s(z,r)\big)\leq C(-\tfrac{n}{2})\exp(\mathcal{N}_{z,s}(r^2))\exp(C_0)r^n,$$
that is,
\begin{eqnarray}\label{thecomparisonofthenashentropy}
\mathcal{N}^*_{s-r^2}(z,s)=\mathcal{N}_{z,s}(r^2)\geq -C_1,
\end{eqnarray}
where $C_1$ depends only on $\alpha$ and $n$.
Next, applying Proposition \ref{harnackofnashentropy} with $(z,s)\to (x_1,t_1)$, $(x,t)\to (x_2,t_2)$, $ s\to t^*$, $s-r^2\to s$, $-\frac{n}{2r^2}\to R_{\operatorname{min}}$, we have
\begin{eqnarray*}
\mathcal{N}_{s-r^2}^*(z,s)-\mathcal{N}_{s-r^2}^*(x,t)&\leq& \left(\frac{n}{r^2}\right)^{\frac{1}{2}}{\rm dist}_{W_1}^{g_s}(\nu_{x,t\,|\, s},\delta_z)+\frac{n}{2}\log\left(\frac{t-s+r^2}{s-s+r^2}\right)
\\
&\leq& \left(\frac{n}{r^2}\right)^{\frac{1}{2}}\sqrt{H_n(t-s)}+\frac{n}{2}\log\frac{2r^2}{r^2}
\\
&\leq& C(n),
\end{eqnarray*}
where we have applied the definition of the $H_n$-center and the fact ${\rm dist}_{W_1}^{g_s}(\nu_{x,t\,|\, s},\,\delta_z)\leq \sqrt{\operatorname{Var}(\nu_{x,t\,|\, s},\,\delta_z)}\leq\sqrt{H_n(t-s)}$ (c.f. \cite[Lemma 3.2]{Bam20a}). In combination with (\ref{thecomparisonofthenashentropy}), we have
$$\mathcal{N}_{x,t}(2r^2)=\mathcal{N}_{s-r^2}^*(x,t)\geq \mathcal{N}_{s-r^2}^*(z,s)-C(n)\geq -C_1-C(n).$$
This finishes the proof of the theorem.
\end{proof}
\section{Properties of ancient solutions satisfying Assumption B}
In this section, we prove several properties for an ancient solution satisfying (\ref{curvaturebound}) and Assumption B. We first of all verify that such an ancient solution must be locally uniformly Type I, which implies a shrinker structure on the blow-down limit in (\ref{smoothconvergence}). This fact together with Theorem \ref{Coro_nash_2} implies that the ancient solution has bounded Nash entropy. Finally, by using Theorem 3.1, we show that an $H_n$-center is always not far from an $\ell$-center.
Throughout this section, we let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (\ref{curvaturebound}) and Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. Let $(M_\infty,g_\infty(t))_{t\in[-2,-1]}$ be the limit Ricci flow in (\ref{smoothconvergence}). From the smooth convergence (\ref{smoothconvergence}), we may easily check that $(M,g(t))_{t\in(-\infty,0]}$ satisfies the locally uniformly Type I condition.
\begin{Proposition}\label{LUTypeI}
$(M,g(t))_{t\in(-\infty,0]}$ is locally uniformly Type I along the sequence $\{(p_i,-\tau_i)\}_{i=1}^\infty$.
\end{Proposition}
\begin{proof}
By the following Cheeger-Gromov-Hamilton convergence in the statement of Assumption B
\begin{eqnarray*}
\big(M,g_i(t),p_i\big)_{t\in[-2,-1]}\xrightarrow{\makebox[1cm]{}} \big(M_\infty,g_\infty(t),p_\infty\big)_{t\in[-2,-1]},
\end{eqnarray*}
where $g_i(t)=\tau_i^{-1}g(\tau_it)$, we may find $\varepsilon_i\searrow 0$, open sets $U_i\subset M_\infty$, and diffeomorphisms
\begin{eqnarray}\label{CGH-COMVERGENCE-11111}
\Psi_i:M_\infty\supset U_i\rightarrow V_i\subset M,
\end{eqnarray}
such that the following hold
\begin{gather}
U_i\supset B_{g_{\infty, -1}}(p_\infty, \varepsilon_i^{-1}) \quad \text{ and }\quad V_i=\Psi_i(U_i)\supset B_{g_{i, -1}}(p_i, \varepsilon_i^{-1}),
\\
\Psi_i(p_\infty)=p_i,\label{base_B}
\\
\big\|\Psi_i^*g_i-g_\infty\big\|_{C^{[\varepsilon_i^{-1}]}(U_i\times[-2,-1])}\leq \varepsilon_i.\label{norm}
\end{gather}
As a consequence of (\ref{base_B}) and (\ref{norm}), we can find a positive function $C:(0,\infty)\rightarrow(0,\infty)$, such that
\begin{gather}\label{supranonsense008}
\lim_{i\rightarrow\infty}\operatorname{inj}_{g_{i, -1}}(p_i)=\operatorname{inj}_{g_{\infty, -1}}(p_\infty)>0,\\
\lim_{i\rightarrow\infty} \sup_{B_{g_{i, -1}}(p_i,r)\times[-2,-1]}|\Rm_{g_i}|=\sup_{B_{g_{\infty, -1}}(p_\infty,r)\times[-2,-1]}|\Rm_{g_{\infty, -1}}|\leq C(r)<\infty,\label{supranonsense009}
\end{gather}
where the latter holds for all $r>0$. Scaling (\ref{supranonsense008}) and (\ref{supranonsense009}) with factor $\tau_i$, we have verified Definition \ref{def}(2)(4) for $(M,g(t))$. Furthermore, by (\ref{curvaturebound}), we have a time-wise Ricci curvature lower bound for $g(t)$. Lastly, since $(p_i,-\tau_i)$'s are $\ell$-centers of $(p_0,0)$, we also have $\ell_{p_0,0}(p_i,\tau_i)\leq\frac{n}{2}$ for all $i$. Therefore, $(M,g(t))$ is locally uniformly Type I along the sequence $\{(p_i,-\tau_i)\}_{i=1}^\infty$.
\end{proof}
The following proposition and corollary are but restatements of Proposition \ref{TIshrinker} and Corollary \ref{noncollapsing}, respectively. Theorem \ref{Theorem_main}(1) follows from Proposition \ref{shrinkerstructure}.
\begin{Proposition}\label{shrinkerstructure}
The limit $(M_\infty,g_\infty(t))_{t\in[-2,-1]}$ has a shrinker structure. More precisely, $\ell_i\rightarrow\ell_\infty$ in the $C^\alpha_{\operatorname{loc}}$ sense or in the weak $*W^{1,2}_{\operatorname{loc}}$ sense on $M_\infty\times(1,2)$, where $\ell_i(\cdot,\tau)=\ell(\cdot,\tau_i\tau):M\rightarrow\mathbb{R}$ for $\tau\in[1,2]$, $\ell$ is the reduced distance based at $(p_0,0)$ with respect to the Ricci flow $(M,g(t))$, and $\ell_\infty$ is a smooth shrinker potential satisfying
\begin{eqnarray*}
\Ric_{g_\infty}(-\tau)+\nabla^2\ell_\infty(\cdot,\tau)=\frac{1}{2\tau}g_\infty(-\tau)\quad\text{ for all }\quad \tau\in(1,2).
\end{eqnarray*}
\end{Proposition}
\begin{Corollary}\label{LUTypeInoncollapsed}
$(M,g(t))_{t\in(-\infty,0)}$ is weakly $\kappa$-noncollapsed on all scales. As a consequence, $(M,g(t))_{t\in(-\infty,0]}$ has bounded geometry in each compact time-interval.
\end{Corollary}
\textbf{Remark:} From this point on, any ancient solution satisfying (\ref{curvaturebound}) and Assumption B is understood to have bounded geometry within each compact time interval.
\begin{Theorem}\label{entropybound}
The Nash entropy of $(M,g(t))_{t\in(-\infty, 0]}$ based at $(p_0,0)$ is bounded independent of time, that is,
\begin{eqnarray*}
\mathcal{N}_{p_0,0}(\tau)\geq -Y\quad\text{ for all }\quad\tau>0,
\end{eqnarray*}
where $Y\in(0,\infty)$ depends on the local geometry bounds around the $\ell$-centers $(p_i,-\tau_i)$.
\end{Theorem}
\begin{proof}
By the assumption (\ref{curvaturebound}), we have that each $(M,g_i(t))_{t\in[-2,0]}$ is a complete Ricci flow with bounded curvature, where $g_i(t):=\tau_i^{-1}g(\tau_it)$. Since $(p_i,-1)$ is an $\ell$-center of $(p_0,0)$ with respect to the Ricci flow $(M,g_i(t))$, and since (\ref{supranonsense008}) and (\ref{supranonsense009}) imply that
\begin{gather*}
\big|\Rm_{g_i}\big|\leq C\quad\text{ on }\quad B_{g_{i,-1}}(p_i,1)\times[-2,-1],
\\
\Vol_{g_{i,-1}}(p_i,1)\geq c,
\end{gather*}
where $c$ and $C$ are positive constants independent of $i$, we have, by Theorem \ref{Coro_nash_2},
\begin{eqnarray*}
\mathcal{N}^i_{p_0,0}(1)\geq -Y \quad\text{ for all }\quad i\in\mathbb{N},
\end{eqnarray*}
where $\mathcal{N}^i$ is the Nash entropy of the Ricci flow $(M,g_i(t))$ and $Y$ depends only on $c$ and $C$. Therefore, by the scaling property of the Nash entropy, we have $$\mathcal{N}_{p_0,0}(\tau_i)\geq -Y\quad\text{ for all }\quad i,$$and the theorem follows from the monotonicity of $\mathcal{N}$.
\end{proof}
Next, we show that the $H_n$-centers and the $\ell$-centers are not far from each other, and consequently the base points in (\ref{smoothconvergence}) can be replaced by $H_n$-centers. First of all, the following result is merely a consequence of Theorem \ref{gaussianupperbound} and Theorem \ref{entropybound}.
\begin{Proposition}\label{gauss0}
There exists a positive number $C$ depending only on the number $Y$ in the conclusion of Theorem \ref{entropybound}, such that
\begin{eqnarray}\label{gauss00}
K(p_0,0\,|\,x,t)\leq\frac{C}{|t|^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_{t}^2(x,z)}{C|t|}\right)\quad\text{ for all }\quad x\in M\quad\text{ and }\quad t\in (-\infty,0),
\end{eqnarray}
where $(z,t)$ is an $H_n$-center of $(p_0,0)$.
\end{Proposition}
Let us find another sequence of points $\{z_i\}_{i=0}^\infty\subset M$, such that each $(z_i,-\tau_i)$ is an $H_n$-centers of $(p_0,0)$. For $H_n$-concentrated Ricci flows (this category obviously contains all Ricci flows with bounded curvature
in compact time intervals; see the remark below Proposition \ref{measureaccumulationofHcenter}),
one may always find an $H_n$-center at each time for any base point. We then have the following estimate.
\begin{Proposition} \label{H_n_l_n}
There is a constant $C$ depending on the number Y in the conclusion of Theorem \ref{entropybound}, such that the following holds.
\begin{eqnarray}\label{boundeddist}
{\rm dist}_{-\tau_i}(p_i,z_i)\leq C\sqrt{\tau_i} \quad\text{ for all }\quad i.
\end{eqnarray}
\end{Proposition}
\textbf{Remark:} Indeed, the same argument of the proof below shows that, for any $t<0$, we have $${\rm dist}_t(p,z)\leq C\sqrt{|t|},$$where $(p,t)$ and $(z,t)$ are an $\ell$-center and an $H_n$-center of $(p_0,0)$, respectively.
\begin{proof}
Combining (\ref{subsolution}) and (\ref{gauss00}), we have
\begin{align*}
&\frac{C}{\tau_i^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_{-\tau_i}^2(p_i,z_i)}{C\tau_i}\right) \geq K(p_0,0\,|\,p_i,-\tau_i)\geq\frac{1}{(4\pi\tau_i)^{\frac{n}{2}}}e^{-\ell_{p_0,0}(p_i,\tau_i)}\geq \frac{1}{(4\pi\tau_i)^{\frac{n}{2}}}e^{-\frac{n}{2}},
\end{align*}
where we have used the fact that $\ell_{p_0,0}(p_i,\tau_i)\leq\frac{n}{2}$. It then follows that
\begin{eqnarray*}
{\rm dist}_{-\tau_i}^2(p_i,z_i)\leq C\tau_i\quad\text{ for all }\quad i.
\end{eqnarray*}
\end{proof}
\begin{Corollary}\label{change-l-center-to-Hn-center}
In the convergence in (\ref{smoothconvergence}), possibly after passing to a subsequence, one may replace $p_i$ by $z_i$. If so, then the limit Ricci flow is still the same, whose base point is correspondingly replace by some $z_\infty$ satisfying ${\rm dist}_{g_{\infty, -1}}(p_\infty,z_\infty)\leq C$, where $C$ is the constant in (\ref{boundeddist}).
\end{Corollary}
\begin{proof}
After the parabolic scaling $g_i(t)=\tau_i^{-1}g(\tau_it)$, (\ref{boundeddist}) becomes
\begin{eqnarray*}
{\rm dist}_{g_{i,-1}}(p_i,z_i)\leq C\quad\text{ for all }\quad i.
\end{eqnarray*}
The corollary then follows from the definition of Cheeger-Gromov-Hamilton convergence.
\end{proof}
\section{Smooth convergence and $\mathbb{F}$-convergence}
For the purpose of proving Theorem \ref{Theorem_main}(3), we shall prove a more general statement in this section, namely, that from a sequence of complete smooth Ricci flows, if, using a sequence of $H_n$-centers as base points, one obtains a smooth Cheeger-Gromov-Hamilton limit, then, this smooth limit is the same as the $\mathbb{F}$-limit.
Let $\{(M_i,g_i(t))_{t\in(-T_i,0]}\}_{i=1}^\infty$ be a sequence of complete Ricci flows, each one with bounded curvature within each compact time interval, where $\infty\geq T_i>c>0$ for some constant $c$. For each $i$, let $p_i\in M_i$ be a fixed point and $(\nu^i_{t})_{t\in(-T_i,0]}$ the conjugate heat kernel on $(M_i,g_i)$ based at $(p_i,0)$. According to \cite[Theorem 7.4, Corollary 7.5]{Bam20b},
$\displaystyle\left\{\big(M_i,g_i(t))_{t\in(-T_i,0]},(\nu^i_t)_{t\in(-T_i,0]}\big)\right\}_{i=1}^\infty$ has an $\mathbb{F}$-convergent subsequence.
\begin{Theorem}\label{convergence}
Assume that there exist a
compact
interval $I=[a,b]\subset (-T_\infty,0)$, where $T_\infty:=\limsup_{i\rightarrow\infty }T_i$, and a smooth Ricci flow $(M_\infty,g_\infty(t),z_\infty)_{t\in I}$, such that
\begin{eqnarray}\label{smoothconvergence1}
\left(M_i,g_i(t),z_i\right)_{t\in I}\xrightarrow{\makebox[1cm]{}}\left(M_\infty,g_\infty(t),z_\infty\right)_{t\in I}
\end{eqnarray}
in the smooth Cheeger-Gromov-Hamilton sense, where $(z_i,b)$ is an $H_n$-center of $(p_i,0)$ for each $i\in\mathbb{N}$. Then $(M_\infty,g_\infty(t))_{t\in [a,b]}$ induces an $H_n$-concentrated continuous metric flow $\mathcal{X}^\infty$ and
there is a conjugate heat flow
$(\nu_t^\infty)_{t\in [a,b)}$ on $\mathcal{X}^\infty$, such that, by passing to a subsequence, we have
\begin{eqnarray}\label{theequivalentFconvergence}
\left((M_i,g_i(t))_{t\in [a,b]},(\nu^i_t)_{t\in [a,b]}\right)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\left(\mathcal{X}^\infty,(\nu_t^\infty)_{t\in [a,b)}\right),
\end{eqnarray}
where the convergence is uniform over any compact sub-interval of $[a,b).$
\end{Theorem}
\textbf{Remarks:}\begin{enumerate}
\item This theorem implies that, for any subsequence of $\left\{\left((M_i,g_i(t))_{t\in [a,b]},(\nu^i_t)_{t\in [a,b]}\right)\right\}_{i=1}^\infty$, there is a further subsequence that $\mathbb{F}$-converges to $\left(\mathcal{X}^\infty,(\nu_t^\infty)_{t\in [a,b)}\right)$, where $\nu_t^\infty$ is a conjugate heat flow on $\mathcal{X}^\infty$.
So any continuous metric flow pair $(\mathcal{Y}^\infty, (\mu_t)_{t\in [a,b)})$ that arises as an $\mathbb{F}$-limit given by the compactness theorem \cite[Theorem 7.6]{Bam20b} should be of the form $\left(\mathcal{X}^\infty, (\nu_t^\infty)_{t\in [a,b)}\right)$. However, this does not imply that the $\mathbb{F}$-convergence in $(\ref{theequivalentFconvergence})$ holds without passing to a subsequence, since we are not able to show that $\nu_t^\infty$ is independent of the subsequence.
\item We may replace the $H_n$-centers $(z_i,b)$ in the assumptions with $\ell$-centers and the conclusions still hold. This is because $H_n$-centers are not far away from $\ell$-centers by the arguments in Proposition \ref{H_n_l_n}. This point will become clear from the proof.
\end{enumerate}
Throughout this section, we assume that the conditions of Theorem \ref{convergence} hold. Just as we have done in (\ref{CGH-COMVERGENCE-11111})---(\ref{norm}), by the definition of smooth convergence, we may find an increasing sequence of pre-compact open sets $U_i\subset M_\infty$ with $\cup_{i=1}^\infty U_i=M_{\infty}$ and a sequence of diffeomorphisms $\Psi_i:U_i\to V_i\subset M_i$,
such that $\Psi_i(z_\infty)=z_i$ and
\begin{eqnarray}\label{ultranonsense1}
\|\Psi_i^*g_i - g_\infty\|_{C^{[\varepsilon_i^{-1}]}(U_i\times [a,b])}
< \varepsilon_i,
\end{eqnarray}
for some $\varepsilon_i\searrow 0.$
The proof of Theorem \ref{convergence} is divided into several components. We shall first of all show that the smooth limit flow is indeed a metric flow. This is not as obvious as it appears to be, since we do not make any geometric assumption for $(M_\infty,g_\infty(t))_{t\in [a,b]}$ except for its smoothness and completeness. The idea is to construct a metric flow structure using the converging sequence.
\begin{Lemma}
\label{lem: limit is metric flow}
$(M_\infty,g_\infty(t))_{t\in [a,b]}$ induces an $H_n$-concentrated continuous metric flow $\mathcal{X}^\infty$.
\end{Lemma}
\begin{proof}
As mentioned in Theorem \ref{thm: smooth flows are metric flows},
each $(M_i,g_{i}(t))$ induces an $H_n$-concentrated metric flow (see also the remark below Proposition \ref{measureaccumulationofHcenter}). Let us then verify \cite[Definition 3.2]{Bam20b} and the $H_n$-concentrarion condition for $(M_\infty,g_{\infty}(t))_{t\in I}$.
The conjugate heat kernel $K_\infty(\cdot,\cdot\,|\,\cdot,\cdot)$ on $(M_\infty,g_{\infty}(t))_{t\in I}$ is given by Theorem \ref{thm: HK conv under CGH}. Furthermore, by Theorem \ref{thepropertiesoftheCHK}, if we define $d\nu^\infty_{x,t\,|\,s}=K(x,t\,|\,\cdot,s)dg_{\infty,s}$, then we have that $\nu^\infty_{x,t\,|\,s}\in\mathcal{P}(M_\infty)$ and \cite[Definition 3.2(1)---(5),(7)]{Bam20b} holds.
To verify that $\left(M_\infty,g_\infty(t),\nu^\infty_{x,t\,|\,s}\right)$ satisfies \cite[Definition 3.2(6)]{Bam20b}, we shall apply \cite[Lemma 3.8]{Bam20b}. Let us fix $s, t\in I$ with $s<t$, a positive number $T$, and a measurable function $u: M_\infty\rightarrow (0,1)$ such that $f:=\Phi^{-1}\circ u$ is $T^{-\frac{1}{2}}$-Lipschitz with respect to the metric $g_\infty(s)$, where $\Phi:\mathbb{R}\rightarrow(0,1)$ is defined as in \cite[(3.1)]{Bam20b}. Let $\eta:[0,\infty)\rightarrow[0,1]$ be the standard cut-off function such that $\eta\equiv 1$ on $[0,1]$, $\eta\equiv 0$ on $[2,\infty)$, and $0\geq\eta'\geq -2$ everywhere. Let $\zeta:\mathbb{R}\rightarrow[-1,1]$ be the function satisfying $\zeta(t)=t$ for $|t|<1$, $\zeta|_{[1,\infty)}\equiv1$, and $\zeta|_{(-\infty,-1]}\equiv-1.$ For any $A>0$, we define
\begin{gather*}
f^A:=\eta\left(\tfrac{1}{A^2}{\rm dist}_{g_{\infty,s}}(z_\infty,\cdot)\right)\cdot A\zeta\left(\tfrac{f}{A}\right),
\\
u^A:=\Phi\circ f^A.
\end{gather*}
Then we have
\begin{gather}\label{supernonsense_001}
u^A\in[\Phi(-A),\Phi(A)]\subset(0,1)\quad \text{ everywhere on }\quad M_\infty,
\\
\big|\,u-u^A\,\big|\leq 1-\Phi(A)=\Phi(-A)\quad\text{ on }\quad B_{g_{\infty,s}}(z_\infty,A^2),\label{supernonsense_002}
\\
u^A\equiv 1/2\quad\text{ on }\quad M_\infty\setminus B_{g_{\infty,s}}(z_\infty,2A^2),\label{supernonsense_003}
\\
\operatorname{Lip}f^A\leq \tfrac{2}{A^2}\sup\left|A\zeta\left(\tfrac{f}{A}\right)\right|
+\sup|\eta|\cdot A\operatorname{Lip}\zeta \cdot\operatorname{Lip} \tfrac{f}{A}
\leq T^{-\frac{1}{2}}+\tfrac{2}{A}.\label{supernonsense_004}
\end{gather}
Let us fix two points $x, y\in M_\infty$. By (\ref{supernonsense_001}), (\ref{supernonsense_002}), and the bounded convergence theorem, we have
\begin{eqnarray}\label{supernonsense006}
\lim_{A\rightarrow\infty}u^A_t(x),\ \lim_{A\rightarrow\infty}u^A_t(y)= u_t(x),\ u_t(y),
\end{eqnarray}
where$$u_t^A(\cdot):=\int_{M_\infty}u^A
\,d\nu^\infty_{\cdot,t\,|\, s},\quad u_t(\cdot):=\int_{M_\infty}u\,d\nu^\infty_{\cdot,t\,|\, s}.$$
In view of (\ref{supernonsense_003}), for all $i$ large enough we may define
\begin{eqnarray*}
f_i^A&=&f^A\circ\Psi_i^{-1}\quad\text{ on }\quad V_i=\Psi_i(U_i)
\\
f_i^A&\equiv&
0
\quad\text{ on }\quad M_i\setminus V_i.
\\
u_i^A&:=&\Phi\circ f_i^A.
\end{eqnarray*}
Then, by the smooth convergence (\ref{ultranonsense1}), we have
\begin{eqnarray*}
\operatorname{Lip}f_i^A\leq T^{-\frac{1}{2}}+\frac{2}{A}+\varepsilon_i,
\end{eqnarray*}
and consequently (c.f. \cite[Theorem 3.1]{MZ21}),
\begin{align}\label{supernonsense005}
&\Big|\,\Phi^{-1}\circ u^A_{i,t}(\Psi_i(x))-\Phi^{-1}\circ u^A_{i,t}(\Psi_i(y))\,\Big|
\\\nonumber
&\quad\quad\leq \left(\left(T^{-\frac{1}{2}}+\tfrac{2}{A}+\varepsilon_i\right)^{2}+t-s\right)^{-\frac{1}{2}}{\rm dist}_{g_{i,t}}\big(\Psi_i(x),\Psi_i(y)\big),
\end{align}
for all $i$ large enough, where $u_{i,t}^A:=\int_{M_i}u_i^Ad\nu^i_{\cdot,t\,|\,s}$. In view of (\ref{supernonsense_001}) and Proposition \ref{measureaccumulationofHcenter}, the integral of $\int_{M_i}u_i^Ad\nu^i_{\cdot,t\,|\,s}$ is uniformly negligible outside a large disk. Then, by the locally smoothly convergence of the conjugate heat kernels, we have (c.f. the proof of (\ref{someimportantconvergenceargument})) $$\lim_{i\rightarrow\infty}u^A_{i,t}(\Psi_i(x)),\ \lim_{i\rightarrow\infty}u^A_{i,t}(\Psi_i(y))=u_t^A(x),\ u_t^A(y).$$
Hence, (\ref{supernonsense005}) implies that
\begin{eqnarray*}
\big|\Phi^{-1}\circ u_t^A(x)-\Phi^{-1}\circ u_t^A(y)\big|\leq \left(\left(T^{-\frac{1}{2}}+\tfrac{2}{A}\right)^{2}+t-s\right)^{-\frac{1}{2}}{\rm dist}_{g_{\infty,t}}(x,y).
\end{eqnarray*}
Therefore, by (\ref{supernonsense006}) and taking $A\rightarrow\infty$, we have $$\big|\Phi^{-1}\circ u_t(x)-\Phi^{-1}\circ u_t(y)\big|\leq \left(T+t-s\right)^{-\frac{1}{2}}{\rm dist}_{g_{\infty,t}}(x,y).$$
Since $x,y\in M_\infty$ are arbitrary, we have verified \cite[Definition 3.2(6)]{Bam20b}.
Finally, it is also easy to verify the $H_n$-concentration property for $\mathcal{X}^\infty$, since it is but a consequence of Fatou's lemma.
\end{proof}
\begin{Lemma}
\label{lem: dnu^oo_s}
There is a positive solution to the conjugate heat equation $v:M_\infty\times [a,b)\rightarrow\mathbb{R}$ coupled with $(M_\infty,g_{\infty}(t))$, satisfying $d\nu^\infty_s := v_s \, dg_{\infty,s}\in \mathcal{P}(M_\infty)$ and
\[
\Psi_i^* K_i(p_i,0\,|\, \cdot,\cdot)
\to v
\]
locally smoothly on $M_\infty\times [a,b).$
\end{Lemma}
\begin{proof}
Arguing in the same way as the proof of \cite[Theorem 2.1]{Lu12}, we can find a nonnegative solution $v:M_\infty\times[a,b)\rightarrow\mathbb{R}$ to the conjugate heat equation, such that
\begin{eqnarray}\label{supernonsense009}
\Psi_i^*K_i(p_i,0\,|\, \cdot, \cdot)
\to v
\end{eqnarray}
locally smoothly on $M_\infty\times [a,b)$.
In this case, Theorem \ref{thm: HK conv under CGH} does not imply that $\int v_s \, dg_{\infty,s}=1$ as it did in the proof of formula (\ref{theintegralisequaltoone}). This is because the base point $(p_i,0)$ of the conjugate heat kernel $K_i(p_i,0\,|\,\cdot,\cdot)$ is not in the region of the Cheeger-Gromov-Hamilton convergence. We first of all observe from (\ref{ultranonsense1}) that, there is a positive function $C:(0,\infty)\rightarrow(0,\infty)$ with the following property: for any $r>0$, we have
\begin{eqnarray}\label{supranonsense001}
\left|\Rm_{g_{i}}\right|\leq C(r)\quad\text{ on }\quad B_{g_{i,b}}(z_i,r)\times[a,b]
\end{eqnarray}
whenever $i$ is large enough. Since we also have \begin{gather*}
\liminf_{i\rightarrow\infty}\Vol_{g_{i,b}}\big(B_{g_{i,b}}(z_i,1)\big)=\Vol_{g_{\infty,b}}\big(B_{g_{\infty,b}}(z_\infty,1)\big)>0,
\end{gather*}
we may apply Theorem \ref{Thm_nash} and the remark below it to the $H_n$-center $(z_i,b)$ of $(p_i,0)$, and consequently, we can find a positive number $Y$ independent of $i$, such that
\begin{eqnarray*}
\mathcal{N}^i_{p_i,0}(|b|)\geq -Y \quad\text{ for all }\quad i\in\mathbb{N},
\end{eqnarray*}
where $\mathcal{N}^i$ is the Nash entropy of the Ricci flow $(M_i,g_i(t))$. The argument in Proposition \ref{H_n_l_n} then implies that
\begin{eqnarray}\label{supranonsense002}
{\rm dist}_{g_{i,b}}(z_i,p'_i)\leq C\quad\text{ for all }\quad i\in\mathbb{N},
\end{eqnarray}
where $(p'_i,b)$ is an $\ell$-center of $(p_i,0)$, and $C$ is a constant depending only on $Y$.
Now we will use (\ref{subsolution}) to obtain a uniform lower bound for $K_i(p_i,0\,|\,p'_i,b-\varepsilon)$, where $\varepsilon$ is an arbitrarily fixed number in $(0,b-a]$. Let us fix such an $\varepsilon$. (\ref{supranonsense001}) and (\ref{supranonsense002}) imply
$$\sup_{t\in
[a,b]
}\left|R_{g_i}(p'_i,t)\right|\leq C\quad\text{ for all }i\in\mathbb{N},$$
where $C$ is a constant independent of $i$.
We may concatenate a minimal $\mathcal{L}$-geodesic from $(p_i,0)$ to $(p'_i,b)$ and
the static curve from $(p'_i,b)$
to $(p'_i,b-\varepsilon)$ as a test curve in (\ref{definitionofl}). This yields
\begin{eqnarray*}
\ell_{p_i,0}(p'_i,|b|+\varepsilon)&\leq&\frac{1}{2\sqrt{|b|+\varepsilon}}\left(2\sqrt{|b|}\ell_{p_i,0}(p'_i,|b|)
+ \int_{|b|}^{|b|+\varepsilon}
\sqrt{\tau} R_{g_i}(p'_i,-\tau)\,d\tau
\right)
\\
&\leq&\frac{1}{2\sqrt{|b|+\varepsilon}}\left(2\sqrt{|b|}\cdot\frac{n}{2}
+C(|b|+\varepsilon)^{3/2}
\right)
\\
&\leq& C\quad\text{ for all }\quad i\in\mathbb{N}.
\end{eqnarray*}
Hence, by (\ref{subsolution}), we have
$$K_i(p_i,0\,|\,p'_i,b-\varepsilon)\geq\frac{1}{4\pi(|b|+\varepsilon)^{\frac{n}{2}}}e^{-\ell_{p_i,0}(p'_i,|b|+\varepsilon)}\geq c(\varepsilon)>0\quad\text{ for all }\quad i\in\mathbb{N}.$$
By (\ref{supranonsense002}) again, we may find a point $p'_\infty\in M_\infty$, such that $\Psi_i^{-1}(p'_i)\rightarrow p'_\infty$ after possibly passing to a subsequence. Consequently
$$v(p'_\infty,b-\varepsilon)=\lim_{i\rightarrow\infty}K_i(p_i,0\,|\,p'_i,b-\varepsilon)\geq c(\varepsilon)>0.$$
Since $\varepsilon\in(0,b-a]$ is arbitrary, it then follows from the strong maximum principle that $v>0$ everywhere on $M_\infty\times[a,b)$.
Once we know that $v$ is positive, by exactly the same argument as in the proof of (\ref{theintegralisequaltoone}), we have
\[
\int_{M_\infty} v_{s}\, dg_{\infty,s}
=1,
\]
for any $s\in [a,b).$
Indeed, here one may use the fact that $$\lim_{i\rightarrow\infty} \int_{B_{g_{i,s}}(z_i,1)}K_i(p_i,0\,|\,\cdot,s)dg_{i,s}=\int_{B_{g_{\infty,s}}(z_\infty,1)}v(\cdot,s)dg_{\infty,s}>0$$ to show that $(z_i,s)$ is not far from an $H_n$-center of $(p_i,0)$, for any $s\in[a,b)$, and the rest of the argument is the same as the proof of (\ref{theintegralisequaltoone}).
\end{proof}
\begin{Lemma}
\label{lem: compact pyramid}
Let $(M^n,g(t))_{t\in I}$ be a complete Ricci flow over a compact interval $I$ and $o\in M$.
Then for any $A>0,\Omega:=\cup_{t\in I} \bar B_t(o,A)$ is compact.
\end{Lemma}
\begin{proof}
Let $x_j\in \bar B_{s_j}(o,A)\subset \Omega$ be an arbitrary sequence. Assume that $s_j\to \bar s\in I.$ We shall prove that $\{x_j\}$ has a convergent subsequence.
Suppose that
\[
\sup_{B_{\bar s}(o,10A)\times I} |{\Ric}| \le \Lambda.
\]
We claim that for any $\epsilon\in (0,1),$ there is $\bar j,$ such that if $j\ge \bar j,$ $x_j\in B_{\bar s}(o,(1+\epsilon)A)$.
Suppose not. Then by passing to a subsequence, we may assume that there are $g(s_j)$-minimal geodesics $\gamma_j:[0,\sigma_j]\to M$ with
$\gamma_j(0)=o,\gamma_j(\sigma_j)=x_j,\sigma_j<A$ but there is a first time $\lambda_j<\sigma_j$ such that $\gamma_j(\lambda_j)\in \partial B_{\bar s}(o,(1+\epsilon/2)A).$
Pick $\delta>0$ such that $e^{-\Lambda \delta} > 1- \epsilon/4.$
When $|s_j-\bar s|< \delta$, we have
\begin{align*}
A\ge L_{s_j}(\gamma_j|_{[0,\lambda_j]})
&\ge e^{-\Lambda |s_j-\bar s|}
L_{\bar s}(\gamma_j|_{[0,\lambda_j]})
\ge (1 - \epsilon/4) {\rm dist}_{\bar s}(o,\gamma_j(\lambda_j))\\
&= (1 - \epsilon/4)(1+\epsilon/2)A
> (1+ \epsilon/8)A,
\end{align*}
which is a contradiction. Hence, after passing to a subsequence, we have $x_j\to \bar x$ for some $\bar x\in \bar B_{\bar s}(o,A).$
\end{proof}
We are now ready to prove Theorem \ref{convergence}.
\begin{proof}[Proof of Theorem \ref{convergence}]
We divide the proof into several steps.\\
\noindent
\textbf{Step 1:} Construction of the correspondence. For $t\in I$, set $Z^i_t:=M_i\sqcup M_\infty$ and we shall extend the metrics on $(M_i,g_i(t))$ and $(M_\infty, g_\infty(t))$ to $Z^i_t.$ For any $y_i\in M_i,y\in M_\infty,$ define
\[
{\rm dist}^{Z^i_t}(y,y_i)
= {\rm dist}^{Z^i_t}(y_i,y)
:= \inf_{w\in U_i} {\rm dist}_{g_{\infty,t}}(y,w)
+ {\rm dist}_{g_{i,t}}(\Psi_i(w),y_i)
+ \varepsilon_i.
\]
It is routine to verify that this is indeed a metric, and $M_i, M_\infty\rightarrow M_i\sqcup M_\infty$ are isometric embeddings.
By \cite[Lemma 2.13]{Bam20b}, we may assume that $Z_t^i$ are isometrically embedded into a common metric space $Z_t$ that is complete and separable.
Let $\phi^i_t:(M_i,g_i(t))\to Z_t$ be the isometric embedding for $i\in \mathbb{N}\cup\{\infty\}.$
Note that for any $x\in U_i,$
\[
{\rm dist}^{Z_t}(\phi_t^\infty(x),
\phi_t^i(\Psi_i(x))) = \varepsilon_i.
\]
\bigskip
\noindent
\textbf{Step 2:} Construction of the couplings. Henceforth until the end of the proof of the theorem, we shall fix an arbitrarily small $\varepsilon >0$ and denote $E=(b-\varepsilon^2,b].$ By Lemma \ref{lem: dnu^oo_s}, there is a conjugate heat flow
\[
d\nu^\infty_s := v_s\, dg_{\infty,s},
\]
where $\nu^\infty_s\in \mathcal{P}(M_\infty)$ for $s\in [a,b)$, and $\Psi_i^*\nu_s^i\to \nu_s^\infty$ on $M_\infty\times[a,b)$ in the $C_c^\infty$-topology as smooth $n$-forms.
\\
\noindent
\textbf{Claim 1:}
For $i=\infty$ or for all $i$ sufficiently large, if $s\in I\setminus E$ and $r\ge 10\sqrt{|a|}$, then
\begin{equation}
\label{ineq: gaussian decay}
\nu^i_s(M_i\setminus B_{g_{i,s}}(z_i,r))
\le Ce^{-cr^2},
\end{equation}
where $c$ and $C$ are constants depending only on the geometry of $(M_\infty,g_{\infty}(t))_{t\in I\setminus E})$.
\begin{proof}[Proof of Claim 1.] By the smooth convergence of $\Psi_i^*K_i(p_i,0\,|\,\cdot,\cdot)$ and the fact that $v>0$, there is $c_0>0,$ such that for any $s\in I\setminus E$, we have
\begin{align}
\label{ineq: noncollapsing for nu_s^i}
\nu_{s}^i\left(B_{g_{i,s}}(z_{i},\sqrt{|s|})\right)
\ge \tfrac{1}{2} \nu_s^\infty\left(B_{g_{\infty,s}}(z_\infty, \sqrt{|s|})\right) \ge c_0,
\end{align}
if $i\ge \bar i$ is sufficiently large.
For $i\ge\bar i$ and $r\ge 10\sqrt{|a|},$ by the Gaussian concentration \eqref{ineq: gaussian concentration} and
\eqref{ineq: noncollapsing for nu_s^i}, we have
\begin{align*}
c_0\nu_{s}^i\left(M_i\setminus B_{g_{i,s}}(z_i,r)\right)\leq \nu_s^i\left(B_{g_{i,s}}(z_i,\sqrt{|s|})\right) \nu_{s}^i\left(M_i\setminus B_{g_{i,s}}(z_i,r)\right)
\le
\exp\left\{- \frac{(r-\sqrt{|s|})^2}{8|s|}
\right\}\leq e^{-cr^2},
\end{align*}
for some $c$ depending on $a$ and $b$.
Note that the Gaussian concentration is also true for $\nu^\infty_s$ by Fatou's lemma. Thus, \eqref{ineq: gaussian decay} also holds for $i=\infty.$
\end{proof}
For all $s\in I\setminus E$, set
$\Omega_s=\bar B_{g_{\infty,s}}(z_\infty, A)$, where $A$ is some large number to be determined. Let us also denote $\Omega:=\cup_{s\in I\setminus E} \Omega_s$, which is compact by Lemma \ref{lem: compact pyramid}. For all $s\in I\setminus E$ and for $i$ large enough or $i=\infty$, define
\[
\mu_s^\infty:= \nu_s^\infty|_{\Omega_s}
+ \eta_s \delta_{z_\infty},\quad
\mu_s^i := \Psi_{i*}(\mu_s^\infty),
\]
where the push-forward by $\Psi_i$ makes sense because $\spt\mu_s^\infty\subset U_i$ when $i$ is large enough.
\\
\noindent
\textbf{Claim 2:} We can fix $A$ large enough, such that for $i=\infty$ or for $i\in \mathbb{N}$ sufficiently large, we have
\[
\sup_{s\in I\setminus E} {\rm dist}_{W_1}^{(M_i,g_{i}(s))}(\nu_s^i, \mu_s^i)
<\varepsilon.
\]
\begin{proof}[Proof of Claim 2.]
For any $s\in I\setminus E$ and any $1$-Lipschitz function $\phi$ on $(M_\infty,g_{\infty}(s)),$
by \eqref{ineq: gaussian decay}, we have
\begin{align*}
\int \phi \, d(\nu_s^\infty-\mu_s^\infty)&=\int (\phi-\phi(z_\infty)) \, d(\nu_s^\infty-\mu_s^\infty)
\le \int_{M_\infty\setminus \Omega_s} {\rm dist}_{g_{\infty,s}}(z_\infty,x) \, d\nu_s^\infty(x)
\\
&\le AC e^{-cA^2}
+ C\int_A^\infty r e^{-cr^2} dr < \varepsilon,
\end{align*}
if $A$ is large enough. Here we have used a standard real analysis result (c.f. \cite[Lemma 3.3]{MZ21}).
Let $s\in I\setminus E$ and $\phi$ be any $1$-Lipschitz function on $(M_i,g_{i}(s)).$
By \eqref{ineq: gaussian decay}, if $A$ is fixed large enough, then whenever $i$ is sufficiently large (depending on $A$), we have
\begin{align*}
&\int \phi \, d(\nu_s^i-\mu_s^i)
= \int [\phi-\phi(z_i)] \, d(\nu_s^i-\mu_s^i)\\
\le& A \int_{\Omega_s}d(\Psi_i^*\nu_s^i-\nu_s^\infty)
+ \int_{M_i\setminus \Psi_i(\Omega_s)}
{\rm dist}_{g_{i,s}}(z_i, x) \, d\nu^i_s(x)
\\
\le & 2A|\Omega|_{g_{\infty,s}}
\Big\|K_i(p_i,0\,|\,\Psi_i(\cdot),s)-v_s^\infty\Big\|_{C^0(\Omega)}
+ AC e^{-cA^2} + C\int_A^\infty r e^{-cr^2} dr < \varepsilon.
\end{align*}
Here we have used the smooth convergence (\ref{supernonsense009}) and the fact $\Psi_i(z_\infty)=z_i$. This finishes the proof of the claim.
\end{proof}
Next, we define a sequence of coupling by
\[
\tilde q_s^i := ({\rm id}, \Psi_i)_*(\mu_s^\infty)\in \Pi(\mu_s^\infty,\mu_s^i).
\]
Then, for any $s,t\in I$ with $s<t,$ we have
\begin{align*}
& \int_{M_\infty\times M_i}
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{y,t\,|\,s}
\right)\, d\tilde q_t^i(x,y)\\
=& \int_{\Omega_t}
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{\Psi_i(x),t\,|\,s}
\right)\, d\nu^\infty_t(x)
+ \eta_t \cdot d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{z_\infty,t\,|\,s},
\phi^i_{s*} \nu^i_{z_i,t\,|\,s}\right).
\end{align*}
\noindent\textbf{Claim 3:}
There is a large $\bar i\in\mathbb{N}$, such that if $i\ge \bar i$, then, for any $s,t\in I\setminus E = [a, b-\varepsilon^2]$ with $s<t$ and for any $x\in \Omega_t$, we have
\[
{\rm dist}^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{\Psi_i(x),t\,|\,s}\right)
<\varepsilon.
\]
\begin{proof}[Proof of Claim 3]
Suppose not. By passing to a subsequence, we may assume that there are $s_i,t_i\in I\setminus E, s_i<t_i, x_i\in \Omega_{t_i},$ such that
\begin{equation}
\label{ineq: 1-W dist no conv}
{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{x_i,t_i\,|\,s_i},
\phi^i_{s_i*} \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right)
\ge \varepsilon.
\end{equation}
By passing to a further subsequence, we may assume that $t_i\to \bar t, s_i\to \bar s\le \bar t, x_i\to \bar x\in \Omega_{\bar t}.$
\noindent \textbf{Case A:} $\bar s=\bar t.$
Write $\bar x_i = \Psi_i(\bar x).$
\begin{align*}
&{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{x_i,t_i\,|\,s_i},
\phi^i_{s_i*} \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right)\\
\le&\ {\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{x_i,t_i\,|\,s_i},
\phi^\infty_{s_i*} \delta_{\bar x}\right)
+ {\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \delta_{\bar x},
\phi^i_{s_i*} \delta_{\Psi_i(\bar x)}
\right)
+{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^i_{s_i*} \delta_{\Psi_i(\bar x)}, \phi^i_{s_i*}\nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right)\\
\le&\ {\rm dist}^{g_{\infty,s_i}}_{W_1}\left(
\nu^\infty_{x_i,t_i\,|\,s_i},
\delta_{\bar x}\right)
+ \varepsilon_i
+{\rm dist}^{g_{i,s_i}}_{W_1}\left(
\delta_{\bar x_i},
\nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right) \\
\le&\ {\rm dist}^{g_{\infty,s_i}}_{W_1}\left(
\nu^\infty_{x_i,t_i\,|\,s_i},\nu^\infty_{\bar x,t_i\,|\,s_i}
\right)+{\rm dist}^{g_{\infty,s_i}}_{W_1}\left(
\nu^\infty_{\bar x,t_i\,|\,s_i},
\delta_{\bar x}\right)+\varepsilon_i
\\
&\quad\quad +{\rm dist}^{g_{i,s_i}}_{W_1}\left(
\nu^i_{\bar x_i,t_i\,|\,s_i},
\nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right)+{\rm dist}^{g_{i,s_i}}_{W_1}\left(
\nu^\infty_{\bar x_i,t_i\,|\,s_i},
\delta_{\bar x_i}\right)
\\
\le&\ 2{\rm dist}_{g_{\infty,t_i}}(x_i,\bar x)
+2\varepsilon_i
+ {\rm dist}^{g_{\infty,s_i}}_{W_1}\left(
\nu^\infty_{\bar x,t_i\,|\,s_i},
\delta_{\bar x}\right)
+ {\rm dist}^{g_{i,s_i}}_{W_1}\left(
\nu^\infty_{\bar x_i,t_i\,|\,s_i},
\delta_{\bar x_i}\right) \to 0,
\end{align*}
as $i\to \infty$, which is a contradiction to \eqref{ineq: 1-W dist no conv}. Here we have also applied (\ref{monotoneofdW1}).
The last convergence above is due to \cite[Proposition 9.5]{Bam20a} which is but a consequence of Proposition \ref{measureaccumulationofHcenter}.
\noindent \textbf{Case B:} $\bar s<\bar t.$
By Theorem \ref{thm: HK conv under CGH}, after possibly passing to a subsequence, we have
\[
\Psi_i^*K_i(\Psi_i(x_i),t_i\,|\,\cdot,\cdot)
\to K_\infty(\bar x,\bar t\,|\, \cdot,\cdot)
\]
in the $C_c^\infty$-topology and the convergence is uniform on compact subsets of $M_\infty\times [a,\bar t).$ In particular,
\[
\left\|\Psi_i^*\nu_{\Psi_i(x_i),t_i\,|\,s_i}- \nu^\infty_{\bar x,\bar t\,|\,s_i}\right\|_{C^0(\mathcal{K})} \to 0,
\]
as $n$-forms for any compact subset $\mathcal{K}\subset M_\infty.$
Let $(z,\bar s)$ be an $H_n$-center of $(\bar x,\bar t)$ and let $B:=B_{g_{\infty,\bar s}}(z,10D) $ for some large constant $D$ to be determined. First choose $D$ large enough so that $\nu^\infty_{\bar x,\bar t\,|\,\bar s}(M_\infty\setminus B) < \frac{\varepsilon}{10D}.$ This is possible because of Proposition \ref{measureaccumulationofHcenter}. Then we have
\begin{align*}
&{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{x_i,t_i|s_i},
\phi^i_{s*} \nu^i_{\Psi_i(x_i),t_i|s_i}\right)\\
\le &\ {\rm dist}^{g_{\infty,s_i}}_{W_1}\left(
\nu^\infty_{x_i,t_i|s_i},
\nu^\infty_{\bar x,\bar t|s_i} \right)
+{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{\bar x,\bar t|s_i},
\phi^i_{s*} \nu^i_{\Psi_i(x_i),t_i|s_i}\right).
\end{align*}
The first term above clearly converges to 0.
For the second term, we argue in the same way as Claim 2 above.
By the local distance distortion estimates, we may assume that $B_i=B_{g_{\infty,s_i}}(z,D)\subset B.$
Consider any bounded $1$-Lipschitz function $\phi$ defined on $Z_{s_i}.$ We may assume that $\phi(\phi_{s_i}^\infty(z))=0,$ for otherwise we may replace it with $\phi-\phi(\phi_{s_i}^\infty(z))$. Then, we compute
\begin{align*}
& \int_{M_\infty} \phi\circ \phi^\infty_{s_i}\, d\nu^\infty_{\bar x,\bar t\,|\,s_i}
- \int_{M_i} \phi\circ \phi^i_{s_i}\, d \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\\
\le & \ \int_{B_i} \phi\circ \phi^\infty_{s_i} \,d\nu^\infty_{\bar x,\bar t\,|\,s_i}
- \int_{\Psi_i(B_i)} \phi\circ \phi^i_{s_i}\, d \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\\
&\ + CD e^{-cD^2} + C \int_D^\infty s e^{-cs^2}\, ds,
\end{align*}
where we used the Gaussian concentration as in Claim 1. We can fix $D$ so that the last line above is less than $\varepsilon/4$ for all $i$ large. Then, we have
\begin{align*}
& \int_{B_i} \phi\circ \phi^\infty_{s_i} \,d\nu^\infty_{\bar x,\bar t\,|\,s_i}
- \int_{\Psi_i(B_i)} \phi\circ \phi^i_{s_i}\, d \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\\
\le &\ \int_{B_i} \left\{ \phi\circ \phi^\infty_{s_i}
- \phi\circ \phi^i_{s_i}\circ \Psi_i\right\}
\,d\nu^\infty_{\bar x,\bar t\,|\,s_i}
+ \int_{B_i} \phi\circ \phi^i_{s_i}\circ \Psi_i
\left\{ d\nu^\infty_{\bar x,\bar t\,|\,s_i}
- \Psi_i^*d \nu^i_{\Psi_i(x_i),t_i\,|\,s_i} \right\}\\
\le &\ \varepsilon_i
+ 2D \left\|\Psi_i^*\nu_{\Psi_i(x_i),t_i\,|\,s_i}- \nu^\infty_{\bar x,\bar t\,|\,s_i}\right\|_{C^0(B)}.
\end{align*}
Note that the last line does not depend on $\phi$ and converges to $0$ since $B$ is compact.
Hence, we have
\[
{\rm dist}^{Z_{s_i}}_{W_1}\left(
\phi^\infty_{s_i*} \nu^\infty_{x_i,t_i\,|\,s_i},
\phi^i_{s_i*} \nu^i_{\Psi_i(x_i),t_i\,|\,s_i}\right)
< \varepsilon/2,
\]
when $i$ is large enough; this is a contradiction agains \eqref{ineq: 1-W dist no conv}.
\end{proof}
By the definition of the $1$-Wassernstein distance, there are couplings $\theta^i_s\in \Pi(\mu_s^i,\nu_s^i)$ such that
\[
\int_{M_i\times M_i} {\rm dist}_{g_{i,s}}(x,y) \, d\theta^i_s(x,y)
< {\rm dist}_{W_1}^{(M_i,g_{i,s})}(\nu_s^i, \mu_s^i) + \varepsilon < 2 \varepsilon,
\]
if $i\ge \bar i.$
Applying \cite[Lemma 2.2]{Bam20b} for three times, there is $Q_s^i\in \mathcal{P}(M_i\times M_i \times M_\infty\times M_\infty)$ such that the marginal into the first and second factors equals $\theta^i_s,$ the marginal into the third and first factors equals $\tilde q^i_s,$ and the marginal into the third and fourth factors equals $\theta^\infty_s.$
Define $q_s^i$ to be the marginal of $Q_s^i$ into the second and fourth factors. Then $q_s^i\in \Pi(\nu_s^i,\nu_s^\infty).$
\bigskip
\noindent
\textbf{Step 4:} Final verification.
For any $s,t\in I\setminus E=[a,b-\varepsilon^2], s\le t,$ we have
\begin{align*}
&\int_{M_\infty\times M_i}
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{y,t\,|\,s}
\right)\, d q_t^i(x,y)\\
= &\ \int_{M_i\times M_i\times M_\infty\times M_\infty}
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{y,t\,|\,s}
\right)\, dQ_t^i(y,y_1,x,x_1)\\
\le & \ \int
\left\{{\rm dist}_{g_{\infty,t}}(x,x_1) +
{\rm dist}_{g_{i,t}}(y,y_1) +
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x_1,t\,|\,s},
\phi^i_{s*} \nu^i_{y_1,t\,|\,s}
\right)\right\}\, dQ_t^i(y,y_1,x,x_1)\\
=&\ \int_{M_\infty\times M_\infty} {\rm dist}_{g_{\infty,t}}\, d\theta^\infty_t
+ \int_{M_i\times M_i} {\rm dist}_{g_{i,t}}\, d\theta^i_t
+ \int_{M_\infty\times M_i}
d^{Z_s}_{W_1}\left(
\phi^\infty_{s*} \nu^\infty_{x,t\,|\,s},
\phi^i_{s*} \nu^i_{y,t\,|\,s}
\right)\, d\tilde q_t^i(x,y)\\
< &\ 10 \varepsilon,
\end{align*}
if $i\ge \bar i,$ where we have also used the monotonicity formula (\ref{monotoneofdW1}).
\end{proof}
As a special case of this Theorem, under the assumptions of Theorem \ref{Theorem_main}, we have, after passing to a subsequence,
\begin{eqnarray*}
\Big((M,g_i(t))_{t\in[-2,-1]},(\nu^i_s)_{s\in[-2,-1]}\Big)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\Big((M_\infty,g_\infty(t))_{t\in[-2,-1]},(\nu_s^\infty)_{s\in[-2,-1)}\Big).
\end{eqnarray*}
To finish the proof of Theorem \ref{Theorem_main}, we only need to show that any $\nu_s^\infty$ in the proof of Theorem \ref{convergence}
is induced by the shrinker potential function. We defer the proof of this fact to Proposition \ref{f-potential} in the next section.
\section{Conergence of the Nash entropy and Smooth tangent flow at infinity}
In this section, we prove Theorem \ref{Theorem_main}(2)(3). To this end, we shall show that the conjugate heat kernel based at $(p_0,0)$, after scaling, converges to the shrinker potential of the asymptotic shrinker. Throughout this section, we still consider an ancient solution $(M,g(t))_{t\in(-\infty,0]}$ satisfying Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. Let $(M_\infty,g_\infty(t))_{t\in[-2,-1]}$ be the limit in (\ref{smoothconvergence}), which is known to be Perelman's asymptotic shrinker by Prposition \ref{shrinkerstructure}. Recall that at this moment we only have a Gaussian upper bound for $K(p_0,0\,|\,\cdot,\cdot)$. The lack of a Gaussian lower bound is because of the fact that we currently do not have a good estimate for the reduced distance, which, however, is not needed now. For the sake of convenience, we define the following notations for the scaled conjugate heat kernel
\begin{gather}\label{somenonsenseofscaledheatkernels}
u_i(x,t):=\tau_i^{\frac{n}{2}}K(p_0,0\,|\,x,\tau_it)=\frac{1}{(4\pi|t|)^{\frac{n}{2}}}e^{-f_i(x,t)}
\\\nonumber
d\nu^i_t:=K(p_0,0\,|\,\cdot,\tau_it)dg_{\tau_it}=u_i(\cdot,t)dg_{i,t}.
\end{gather}
\begin{comment}
Indeed, $u_i(x,t)$ is the conjugate heat kernel on the scaled Ricci flow $g_i(t):=\tau_i^{-1}g(\tau_it)$ based at $(p_0,0)$. Because of the gaussian upper bound Theorem \ref{gaussianupperbound}, we have that all $u_i$'s are uniformly bounded on $M\times [-2,-1]$. On the other hand, the locally uniformly Type I condition together with Proposition 5.1(1) in \cite{CZ20} implies the existence of a sequence $A_i\nearrow\infty$, such that the following holds: for all $\varepsilon>0$, there are positive functions $c, C:(0,\infty)\rightarrow(0,\infty)$, satisfying the following property
\begin{eqnarray*}
& \left.\begin{array}{ll}
\ell_i(x,|t|)\leq C(r)\\ \nonumber
u_i(x,t)\geq (4\pi|t|)^{-\frac{n}{2}}e^{-\ell_i(x,|t|)}\geq c(r),
\end{array}\right\}\\ \nonumber
&\text{ for all }\quad (x,t)\in B_{g_i(1)}(z_i,r)\times[-2,-(1+\varepsilon)]\quad\text{ and whenever }\quad r\in(0,A_i).
\end{eqnarray*}
With the uniform upper bound and locally uniform lower bound for function $u_i$, while at the same time applying standard gradient estimates for the conjugate heat equation, we may find a smooth function $u_\infty: M\times[-2,-1)\rightarrow\mathbb{R}$, such that
$u_i\rightarrow u_\infty$ locally smoothly. Obviously, $u_\infty$ is positive everywhere because of the locally uniformly lower bound for $u_i$. Hence we may write
\begin{eqnarray*}
u_\infty(\cdot,t):=\frac{1}{(4\pi|t|)^{\frac{n}{2}}}e^{-f_\infty (\cdot,t)},
\end{eqnarray*}
and we have $f_i\rightarrow f_\infty$.
Here we remark that it is not known from the outset that $f_\infty$ serves as a shrinker potential for the limit Ricci flow (though the limit is already known to be a shrinker by the reduced volume monotonicity). This is because we do not know whether the entropy constructed by $f_\infty$ is a constant, since we do know whether the Nash entropy of the ancient solution exactly converge to $\int_{M_\infty}f_\infty u_\infty dg_\infty-\frac{n}{2}$; this is exactly the object of this subsection.
Following Proposition \ref{H_n_l_n}, we are still using notation $z_i\in M$ to denote the point such that $(z_i,-\tau_i)$ is an $H_n$-center of $(p_0,0)$ with respect to the Ricci flow $g(t)$.
\end{comment}
\begin{Lemma}\label{unitmeasure}
There is a smooth function $u_\infty: M_\infty\times[-2,-1)\rightarrow \mathbb{R}$, satisfying
\begin{enumerate}[(1)]
\item $u_i\rightarrow u_\infty$ locally smoothly on $M_\infty\times[-2,-1)$, where $u_i$ should be understood to be pulled back by the defining diffeomorphisms in the Cheeger-Gromov-Hamilton convergence.
\item $u_\infty$ is a solution to the conjugate heat equation coupled with $g_\infty(t)$.
\item $u_\infty>0$ everywhere on $M_\infty\times[-2,-1)$.
\item $\displaystyle\int_{M_\infty}u_\infty(\cdot,t)dg_{\infty,t}\equiv1 \quad\text{ for all }\quad t\in[-2,-1).$
\end{enumerate}
\end{Lemma}
\begin{proof}
In view of Corollary \ref{change-l-center-to-Hn-center}, this lemma is but a restatement of Lemma \ref{lem: dnu^oo_s}.
\end{proof}
Since $u_\infty>0$ everywhere, we may define the function $f_\infty$ by
\begin{eqnarray*}
u_\infty(x,t):=(4\pi|t|)^{-\frac{n}{2}}e^{-f_\infty(x,t)}\quad \text{ for all }\quad (x,t)\in M_\infty\times[-2,-1).
\end{eqnarray*}
Then, we obviously have
$$f_i\rightarrow f_\infty\quad\text{ locally smoothly on } \quad M_\infty\times[-2,-1).$$
\begin{Lemma}\label{lowerquadratic}
For any $\varepsilon\in(0,1)$, there exists a constant $C$ independent of $i$, such that the following holds for each $i\in\mathbb{N}\cup\{\infty\}$.
\begin{eqnarray*}
f_i(x,t)\geq\frac{1}{C}{\rm dist}_{g_{i,t}}^2(x,z_i)-C\quad\text{ for all }\quad (x,t)\in M\times[-2,-(1+\varepsilon)].
\end{eqnarray*}
\end{Lemma}
\begin{proof}
Fixing an $\varepsilon \in(0,1)$, we have that $\cup_{t\in[-2,-1-\varepsilon]}B_{g_{\infty,t}}(z_\infty,1)\times\{t\}$ is a precompact set in $M_\infty\times [-2,-1)$, and hence $u_i\rightarrow u_\infty$ uniformly on this set. Consequently we have
\begin{eqnarray*}
\liminf_{i\rightarrow\infty}\int_{B_{g_{i,t}}(z_i,1)}u_i(\cdot,t)dg_{i,t}&=&\int_{B_{g_{\infty,t}}(z_\infty,1)}u_\infty(\cdot,t)dg_{\infty,t}
\\
&\geq& \inf_{s\in[-2,-1-\varepsilon]}\int_{B_{g_{\infty,s}}(z_\infty,1)}u_\infty(\cdot,s)dg_{\infty,s}>0,
\end{eqnarray*}
for all $t\in[-2,-1-\varepsilon]$. Hence, we can find a positive number $c_0>0$, such that
\begin{eqnarray}\label{extranonsense001}
\int_{B_{g_{i,t}}(z_i,1)}u_i(\cdot,t)dg_{i,t}\geq c_0\quad\text{ for all }\quad t\in[-2,-1-\varepsilon]\quad\text{ and for all }\quad i\in\mathbb{N}.
\end{eqnarray}
Let $t\in[-2,-1-\varepsilon]$ and let $(z'_i,t)$ be an $H_n$-center of $(p_0,0)$ with respect to the scaled Ricci flow $(M,g_i(t))$. By Proposition \ref{measureaccumulationofHcenter} and (\ref{extranonsense001}), we have that
\begin{eqnarray}\label{extranonsense002}
{\rm dist}_{g_{i,t}}(z_i,z'_i)\leq\sqrt{\frac{2}{c_0}H_n|t|}+1\leq C\quad\text{ for all }\quad t\in[-2,-1-\varepsilon]\quad\text{ and for all }\quad i\in\mathbb{N}.
\end{eqnarray}
The case when $i\in\mathbb{N}$ follows from Proposition \ref{gauss0} and (\ref{extranonsense002}). If $i=\infty$, then this case follows from the fact that $f_\infty$ is the local smooth limit of $\{f_i\}_{i=1}^\infty$.
\end{proof}
\begin{Lemma}
There is a constant $C$ independent of $i$, such that the following hold for all $t\in[-2,-1)$ and $i\in\mathbb{N}\cup\{\infty\}$.
\begin{eqnarray}
\int_{M_i}f_i^2(\cdot,t)u_i(\cdot,t)dg_{i,t}\leq C.\label{lem5.6.2}
\end{eqnarray}
Here we have let $M_i\equiv M$ for all $i\in\mathbb{N}$.
\end{Lemma}
\begin{proof}
By (\ref{Nashintegral_2}) and the Nash entropy bound in Theorem \ref{entropybound}, we have that (\ref{lem5.6.2}) hold for $i\in\mathbb{N}$. The case when $i=\infty$ then follows from Fatou's lemma and the locally smooth convergences of $f_i$ and $u_i$.
\end{proof}
\begin{Proposition}\label{Nconst}
The following statements are true.
\begin{enumerate}[(1)]
\item $\displaystyle \lim_{i\rightarrow\infty}\mathcal{N}_i(|t|)=\int_{M_\infty}f_\infty(\cdot,t)u_\infty(\cdot,t)dg_{\infty,t}-\frac{n}{2}$ for all $t\in[-2,-1)$.
\item $N_\infty:=\displaystyle \int_{M_\infty}f_\infty(\cdot,t)u_\infty(\cdot,t)dg_{\infty,t}-\frac{n}{2}$ is a constant in $t\in[-2,-1)$.
\end{enumerate}
Here $\mathcal{N}_i(\tau):=\mathcal{N}_{p_0,0}(\tau_i\tau)$ stands for the Nash entropy of the scaled Ricci flow $(M,g_i(t))$ based at $(p_0,0)$.
\end{Proposition}
\begin{proof}
We shall only prove part (1), since part (2) obviously follows from part (1) and the monotonicity of the Nash entropy. Let us fix an arbitrary $t\in[-2,-1)$. Applying Lemma \ref{lowerquadratic} and (\ref{lem5.6.2}), we may find a positive number $C$, such that the following holds for all $A>C$.
\begin{eqnarray*}
C\geq\int_Mf_i^2u_idg_{i,t}\geq \int_{M\setminus B_{g_{i,t}}(z_i,A)}f_i^2u_idg_{i,t}\geq \frac{A^2}{2C}\int_{M\setminus B_{g_{i,t}}(z_i,A)}f_iu_idg_{i,t}.
\end{eqnarray*}
Hence
\begin{eqnarray}\label{nonsense5.5}
0<\int_{M\setminus B_{g_{i,t}}(z_i,A)}f_iu_i\,dg_{i,t}\leq\frac{C}{A^2}\quad\text{ for all }\quad A>C.
\end{eqnarray}
Since $f_i\rightarrow f_\infty$ locally smoothly, the following holds for all $A>0$.
\begin{eqnarray*}
\int_{B_{g_{\infty,t}}(z_i,A)}f_\infty u_\infty dg_{\infty,t}&=&\lim_{i\rightarrow \infty}\int_{B_{g_{i,t}}(z_i,A)}f_i u_i dg_{i,t}
\\
&=&\lim_{i\rightarrow \infty}\left(\int_{M}f_i u_i dg_{i,t}-\int_{M\setminus B_{g_{i,t}}(z_i,A)}f_iu_idg_{i,t}\right)
\\
&=&\lim_{i\rightarrow \infty}\left(\mathcal{N}_i(|t|)+\tfrac{n}{2}+O(A^{-2})\right).
\end{eqnarray*}
Here we have applied (\ref{nonsense5.5}). Note that the constant $C$ in formula (\ref{nonsense5.5}) is independent of both $i$ and $A$. Finally, taking $i\rightarrow\infty$ first and then $A\rightarrow\infty$, the conclusion follows.
\end{proof}
Therefore, if we use $u_\infty$ to construct a Nash entropy, then what we obtain is a constant---the critical point of Perelman's monotonicity formula. This should imply that $f_\infty$ is a shrinker potential. However, since we do not have any geometric condition on $(M_\infty,g_\infty(t))$ except for the fact that it is a shrinker, and since Perelman's monotonicity formula depends heavily on the integration by parts at infinity, we cannot so easily conclude that $f_\infty$ is a shrinker potential. One way to resolve this is to apply the cut-off function constructed by \cite[Lemma 3]{LW20}. We shall use an alternative method. Recall the following result of Bamler.
\begin{Proposition}[Proposition 6.1 in \cite{Bam20c}]\label{almostselfsimilar}
For any $Y<\infty$ and $\varepsilon>0$, there is a $\bar{\delta}(Y,\varepsilon)>0$, such that the following holds whenever $\delta\in (0,\bar{\delta})$. Let $(M,g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature within each compact time interval. Let $(x_0,t_0)\in M\times I$ and $r>0$ be such that $[t_0-\delta^{-1}r^2,t_0-\delta r^2]\subset I$. Suppose $\mathcal{N}_{x_0,t_0}(r^2)\geq -Y$. If $$\mathcal{N}_{x_0,t_0}(\delta^{-1}r^2)\geq \mathcal{N}_{x_0,t_0}(\delta r^2)-\delta,$$ then we have
$$\int_{t_0-\varepsilon^{-1} r^2}^{t_0-\varepsilon r^2}\int_M \tau\left|\,\Ric+\nabla^2 f-\frac{1}{2\tau}g\,\right|^2d\nu_{x_0,t_0\,|\,t}dg_tdt\leq \varepsilon,$$
where $\tau(t):=t_0-t$ and $d\nu_{x_0,t_0\,|\,t}:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg_t$ is the conjugate heat kernel based at $(x_0,t_0)$.
\end{Proposition}
\begin{proof}
In fact, in the proof of \cite[Proposition 6.1]{Bam20c}, the conclusion stated in the current proposition relies only on Perelman's monotonicity formulas for entropies, which are obviously true for Ricci flows with bounded curvature. Hence, Bamler's proof can be directly applied to our case.
\end{proof}
We are then ready to show that $f_\infty$ is a shrinker potential.
\begin{Proposition}\label{f-potential}
$f_\infty$ is a shrinker potential function, satisfying
\begin{eqnarray*}
\Ric_\infty+\nabla^2f_\infty=\frac{1}{2|t|}g_\infty\quad\text{ for all }\quad t\in [-2,-1).
\end{eqnarray*}
\end{Proposition}
\begin{proof}
By Theorem \ref{entropybound}, we have that, for any $\delta>0$, it holds that
\begin{eqnarray*}
\lim_{i\rightarrow\infty}\mathcal{N}_{p_0,0}(\delta^{-1}\tau_i)-\mathcal{N}_{p_0,0}(\delta\tau_i)=0.
\end{eqnarray*}
Hence, Proposition \ref{almostselfsimilar} implies the existence of a sequence $\varepsilon_i\searrow 0$, such that
$$\int_{-\varepsilon_i^{-1} \tau_i}^{-\varepsilon_i \tau_i}\int_M |t|\left|\,\Ric+\nabla^2 f-\frac{1}{2|t|}g\,\right|^2d\nu_{p_0,0\,|\,t}dg_tdt\leq \varepsilon_i,$$
where $d\nu_{p_0,0\,|\,t}=(4\pi|t|)^{-\frac{n}{2}}e^{-f}dg_t$ is the conjugate heat kernel based at $(p_0,0)$. Scaling by $\tau_i$, we immediately have
\begin{align}\label{extranonsense004}
&\quad\int_{-2 }^{-1}\int_M |t|\left|\,\Ric_{g_i}+\nabla^2 f_i-\frac{1}{2|t|}g_i\,\right|^2u_idg_{i,t}dt
\\\nonumber
&\leq\int_{-\varepsilon_i^{-1} }^{-\varepsilon_i }\int_M |t|\left|\,\Ric_{g_i}+\nabla^2 f_i-\frac{1}{2|t|}g_i\,\right|^2u_idg_{i,t}dt
\\\nonumber
&\leq \varepsilon_i \rightarrow 0,
\end{align}
where $g_i(t):= \tau_i^{-1}g(\tau_it)$, $f_i$ and $u_i$ are defined in (\ref{somenonsenseofscaledheatkernels}). Because of the local smooth convergence of $g_i$, $f_i$, and $u_i$, taking a limit for (\ref{extranonsense004}) and applying Fatou's lemma, we have
$$\int_{-2 }^{-1}\int_{M_\infty} |t|\left|\,\Ric_{g_\infty}+\nabla^2 f_\infty-\frac{1}{2|t|}g_\infty\,\right|^2u_\infty dg_{\infty,t}dt=0.$$
Since $u_\infty>0$ everywhere, the Proposition follows immediately.
\end{proof}
\begin{comment}
Once we have Proposition \ref{f-potential} proved, Theorem \ref{convergence} implies Theorem \ref{Theorem_main}(3). We shall break the proof of Proposition \ref{f-potential} into several lemmas. First of all, the derivation of Perelman's monotonicity formula intensely depends on integration by parts. Since we do not have any geometric assumption on $(M_\infty,g_\infty)$, except for the fact that it is a shrinker (with potential function $\ell_\infty$; see Theorem \ref{shrinkerstructure}), the validity of integration by parts is not clear from the outset. To resolve this problem, recall that Li-Wang \cite{LW20} constructed a cut-off function on all Ricci shrinkers.
\begin{Lemma}[Lemma 3 in \cite{LW20}]\label{cutoff}
For any $A$ large enough, there exists a cut-off function $\phi^A: M_\infty\times(-\infty,0)\rightarrow[0,1]$, such that the following hold for all $t\in(-\infty,0)$.
\begin{align*}
& \phi^A(\cdot,t)=1 \quad\text{ on }\quad B_{g_{\infty,t}}(x_\infty, A)\\
& \phi^A(\cdot,t)=0 \quad\text{ on }\quad M_\infty\setminus B_{g_{\infty,t}}(x_\infty, CA)\\
& |\nabla \phi^A|\leq\frac{C}{A},\quad |\Box \phi^A|\leq\frac{C}{A^2},
\end{align*}
where $C$ is a dimensional constant, and $x_\infty$ is a fixed point on $M_\infty$ (namely, the minimum point of $\ell_\infty$).
\end{Lemma}
Using the above cut-off function, we may easily verify the following formulas.
\begin{Lemma}\label{intbyparts}
\begin{enumerate}[(1)]
\item (Integration by parts.) \\$$ \int_{M_\infty}\Delta f_\infty(\cdot,t) u_\infty(\cdot,t) dg_{\infty,t}=\int_{M_\infty}|\nabla f_\infty(\cdot,t)|^2u_\infty(\cdot,t)dg_{\infty,t}\quad\text{ for all }\quad t\in[-2,-1).$$
\item (Integral bound for $\Delta f_\infty$.)
$$|t|\int_{M_\infty}|\Delta f_\infty(\cdot,t)|u_\infty(\cdot,t)dg_{\infty,t}\leq -N_\infty+\frac{n}{2}\quad\text{ for all }\quad t\in[-2,-1),$$
where $N_\infty$ is as defined in Proposition \ref{Nconst}(2).
\end{enumerate}
\end{Lemma}
\begin{proof}
Part (1) is merely a consequence of (\ref{lem5.6.1}) together with the cut-off function in Lemma \ref{cutoff}. Indeed, fixing an arbitrary $t\in[-2,-1)$ and an $A>0$ large enough, we may compute
\begin{eqnarray*}
\int_{M_\infty}\phi^A\Delta f_\infty u_\infty dg_{\infty,t}=\int_{M_\infty}\phi^A|\nabla f_\infty|^2u_\infty dg_{\infty,t}-\int_{M_\infty}\langle\nabla\phi^A,\nabla f_\infty\rangle u_\infty dg_{\infty,t},
\end{eqnarray*}
while by Lemma \ref{cutoff} and (\ref{lem5.6.1}) we also have
\begin{eqnarray}\label{nonsense003}
\left|\int_{M_\infty}\langle\nabla\phi^A,\nabla f_\infty\rangle u_\infty dg_{\infty,t}\right|&\leq&\big(\sup|\nabla \phi^A|\big)\int_{M_\infty}|\nabla f_\infty| u_\infty dg_{\infty,t}
\\\nonumber
&\leq&\big(\sup|\nabla \phi^A|\big)\left(\int_{M_\infty}|\nabla f_\infty|^2 u_\infty dg_{\infty,t}\right)^{\frac{1}{2}}\left(\int_{M_\infty} u_\infty dg_{\infty,t}\right)^{\frac{1}{2}}
\\\nonumber
&\leq&\frac{C}{A}.
\end{eqnarray}
Taking $A\rightarrow\infty$, (1) follows.
To prove part (2), recall that Perelman's Harnack estimate (section 9 of \cite{Per02}, see also \cite{CTY11} for the proof in the complete noncompact case) holds for the conjugate heat kernel $u_i=(4\pi|t|)^{-\frac{n}{2}}e^{-f_i}$:
\begin{eqnarray*}
|t|\big(2\Delta f_i-|\nabla f_i|^2+R_{g_i}\big)+f_i-n\leq 0 \quad\text{ on }\quad M\times(-\infty,0).
\end{eqnarray*}
Since $f_i\rightarrow f_\infty$ and $g_i\rightarrow g_\infty$ locally smoothly, the above inequality is carried to the limit. Hence we have
\begin{eqnarray}
|t|\big(2\Delta f_\infty-|\nabla f_\infty|^2+R_{g_\infty}\big)+f_\infty-n\leq 0 \quad\text{ on }\quad M_\infty\times[-2,-1),
\end{eqnarray}
and consequently
\begin{eqnarray*}
(2\Delta f_\infty)_+\leq |\nabla f_\infty|^2-\frac{1}{|t|}f_\infty+\frac{n}{|t|}\quad\text{ on }\quad M_\infty\times[-2,-1).
\end{eqnarray*}
Here we have used the fact that $R_{g_\infty}\geq 0$ (\cite{CBl09}). Then, fixing an arbitrary $A>0$ large enough and a $t\in[-2,-1)$, we have
\begin{eqnarray}\label{nonsense5}
\int_{M_\infty}\phi^A|\Delta f_\infty|u_\infty dg_{\infty,t}&=&\int_{M_\infty}\phi^A\big(2(\Delta f_\infty)_+-\Delta f_\infty\big)u_\infty dg_{\infty,t}
\\\nonumber
&\leq& \int_{M_\infty}\phi^A\left(|\nabla f_\infty|^2-\frac{1}{|t|}f_\infty+\frac{n}{|t|}\right)u_\infty dg_{\infty,t}
\\\nonumber
&&\quad -\int_{M_\infty}\phi^A|\nabla f_\infty|^2u_\infty dg_{\infty,t}+\int_{M_\infty}\langle\nabla\phi^A,\nabla f_\infty\rangle u_\infty dg_{\infty,t}
\\\nonumber
&\leq& -\frac{1}{|t|}\int_{M_\infty}\phi^A f_\infty u_\infty dg_{\infty,t}+\frac{n}{|t|}+\frac{C}{A},
\end{eqnarray}
where we have also applied (\ref{nonsense003}). Recall that by (\ref{lem5.6.3}), $f_\infty u_\infty$ is absolutely integrable. Hence, taking $A\rightarrow\infty$, (\ref{nonsense5}) becomes (2).
\end{proof}
\begin{Lemma}
Perelman's entropy constructed using $f_\infty$
\begin{eqnarray*}
W_\infty:=\int_M\Big(|t|\big(|\nabla f_\infty(\cdot,t)|^2+R_\infty(t)\big)+f_\infty(\cdot,t)-n\Big)u_\infty(\cdot,t)dg_{\infty,t}
\end{eqnarray*}
is also a constant in $t\in[-2,-1)$. Furthermore, $W_\infty=N_\infty$, where $N_\infty$ is defined as in Proposition \ref{Nconst}(2).
\end{Lemma}
\begin{proof}
Let us fix arbitrary $t_1$ and $t_2\in[-2,-1)$, such that $t_1<t_2$. Since, by (\ref{lem5.6.3}), $f_\infty u_\infty$ is absolutely integrable on $M_\infty\times[t_1,t_2]$, Proposition \ref{Nconst}(2) implies
\begin{eqnarray}\label{nonsense001}
0&=&\int_{M_\infty}f_\infty(\cdot,t)u_\infty(\cdot,t)dg_{\infty,t}\Bigg|_{t=t_1}^{t=t_2}
\\\nonumber
&=&\lim_{A\rightarrow\infty}\left(\int_{M_\infty}\phi^A f_\infty(\cdot,t)u_\infty(\cdot,t)dg_{\infty,t}\Bigg|_{t=t_1}^{t=t_2}\right)
\\\nonumber
&=&\lim_{A\rightarrow\infty}\int_{t_1}^{t_2}\left(\frac{d}{dt}\int_{M_\infty}\phi^A f_\infty u_\infty dg_{\infty,t}\right)dt
\\\nonumber
&=&\lim_{A\rightarrow\infty}\int_{t_1}^{t_2}\int_{M_\infty}\Box\big(\phi^A f_\infty\big) u_\infty dg_{\infty,t}dt
\\\nonumber
&=&\lim_{A\rightarrow\infty}\int_{t_1}^{t_2}\int_{M_\infty}\Big(\phi^A \Box f_\infty+f_\infty\Box\phi^A -2\langle\nabla\phi^A,\nabla f_\infty\rangle \Big) u_\infty dg_{\infty,t}dt.
\end{eqnarray}
Here we have used the fact that $u_\infty$ is a solution to the conjugate heat equation. On the other hand, by Lemma \ref{cutoff} and (\ref{lem5.6.3}), we have
\begin{eqnarray}\label{nonsense002}
\left|\int_{M_\infty}f_\infty\Box\phi^A u_\infty dg_{\infty,t}\right|&\leq&\big(\sup|\Box\phi^A|\big)\int_{M_\infty}|f_\infty| u_\infty dg_{\infty,t}
\leq\frac{C}{A^2}.
\end{eqnarray}
Combining (\ref{nonsense003}), (\ref{nonsense001}), (\ref{nonsense002}), taking $A\rightarrow\infty$, we have
\begin{eqnarray}\label{nonsense004}
0&=&\int_{t_1}^{t_2}\int_{M_\infty} (\Box f_\infty) u_\infty dg_{\infty,t}dt
\\\nonumber
&=&\int_{t_1}^{t_2}\int_{M_\infty} \left(-2\Delta f_\infty+|\nabla f_\infty|^2-R_\infty+\frac{n}{2|t|}\right) u_\infty dg_{\infty,t}dt
\\\nonumber
&=&\int_{t_1}^{t_2}\frac{1}{|t|}\left(\frac{n}{2}-|t|\int_{M_\infty}\big(|\nabla f_\infty|^2+R_\infty\big)dg_{\infty,t}\right)dt,
\end{eqnarray}
where we have applied Lemma \ref{intbyparts}(1) and the fact $u_\infty=(4\pi|t|)e^{-f_\infty}$ is a solution to the conjugate heat equation. Since, by (\ref{lem5.6.1}), $\displaystyle \frac{n}{2}-|t|\int_{M_\infty}\big(|\nabla f_\infty|^2+R_\infty\big)dg_{\infty,t}\geq0$ for all $t\in[-2,-1)$, and since $t_1$, $t_2\in[-2,-1)$ are arbitrarily chosen, we then have, from (\ref{nonsense004}), that
\begin{eqnarray*}
\frac{n}{2}-|t|\int_{M_\infty}\big(|\nabla f_\infty|^2+R_\infty\big)dg_{\infty,t}\equiv 0\quad\text{ for all }\quad t\in[-2,-1),
\end{eqnarray*}
and consequencely
\begin{eqnarray*}
W_\infty =N_\infty-\left(\frac{n}{2}-|t|\int_{M_\infty}\big(|\nabla f_\infty|^2+R_\infty\big)dg_{\infty,t}\right)\equiv N_\infty\quad\text{ for all }\quad t\in[-2,-1).
\end{eqnarray*}
We have finished the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Proposition \ref{f-potential}]
We consider the following quantity defined by Perelman
\begin{eqnarray*}
v_\infty:=|t|\big(2\Delta f_\infty-|\nabla f_\infty|^2+R_\infty\big)+f_\infty-n \quad\text{ on }\quad M_\infty\times[-2,-1).
\end{eqnarray*}
As computed by Perelman, we have
\begin{eqnarray}\label{nonsense0001}
\Box^*(v_\infty u_\infty)=-2|t|\left|\,\Ric_\infty+\nabla^2f_\infty-\frac{1}{2|t|}g_\infty\,\right|^2u_\infty.
\end{eqnarray}
By Lemma \ref{intbyparts}(1), we also have
\begin{eqnarray}\label{nonsense0002}
\int_{M_\infty}v_\infty u_\infty dg_{\infty,t}\equiv W_\infty\quad\text{ for all }\quad t\in[-2,-1).
\end{eqnarray}
Next, combining (\ref{lem5.6.1}), (\ref{lem5.6.3}), and Lemma \ref{intbyparts}(2), we can find a constant $C$ independent of $t\in[-2,-1)$, such that
\begin{eqnarray}\label{nonsense0003}
\int_{M_\infty}|v_\infty|u_\infty dg_{\infty,t}\leq C\quad \text{ for all }\quad t\in[-2,-1).
\end{eqnarray}
Finally, fixing arbitrary $t_1$ and $t_2\in[-2,-1)$ such that $t_1<t_2$, we have, by (\ref{nonsense0002}),
\begin{eqnarray}\label{nonsense0004}
0&=&\int_{M_\infty}v_\infty u_\infty dg_{\infty,t}\Bigg|_{t=t_1}^{t=t_2}=\lim_{A\rightarrow\infty}\left(\int_{M_\infty}\phi^A v_\infty u_\infty dg_{\infty,t}\Bigg|_{t=t_1}^{t=t_2}\right)
\\\nonumber
&=&\lim_{A\rightarrow\infty}\int_{t_1}^{t_2}\left(\frac{d}{dt}\int_{M_\infty}\phi^A v_\infty u_\infty dg_{\infty,t}\right)dt
\\\nonumber
&=&\lim_{A\rightarrow\infty}\int_{t_1}^{t_2}\int_{M_\infty}\Big(\big(v_\infty u_\infty\big)\Box\phi^A +\phi^A\Box^*(v_\infty u_\infty) \Big)dg_{\infty,t}dt
\end{eqnarray}
By Lemma \ref{cutoff} and (\ref{nonsense0003}), we have
\begin{eqnarray*}
\int_{M_\infty}\big|\big(v_\infty u_\infty\big)\Box\phi^A\big|dg_{\infty,t}&\leq&\left(\sup|\Box\phi^A|\right)\left(\int_{M_\infty}|v_\infty|u_\infty dg_{\infty,t}\right)\leq\frac{C}{A^2}
\end{eqnarray*}
for all $t\in[t_1,t_2]$. Hence, taking $A\rightarrow\infty$, (\ref{nonsense0004}) becomes
\begin{eqnarray*}
0&=&\int_{t_1}^{t_2}\int_{M_\infty}\Box^*(v_\infty u_\infty) dg_{\infty,t}dt
\\
&=&-2\int_{t_1}^{t_2}|t|\int_{M_\infty}\left|\,\Ric_\infty+\nabla^2f_\infty-\frac{1}{2|t|}g_\infty\,\right|^2u_\infty dg_{\infty,t}dt,
\end{eqnarray*}
where we have applied (\ref{nonsense0001}). This finishes the proof of the Proposition.
\end{proof}
\end{comment}
\begin{proof}[Proof of Theorem \ref{Theorem_main}(2)]
To prove Theorem \ref{Theorem_main}(2), we need only to show $N_\infty=\mu_\infty$, where $N_\infty$ is defined in Proposition \ref{Nconst}(2) and $\mu_\infty$ is the entropy of the asymptotic shrinker. Because of Lemma \ref{unitmeasure}(4), by (\ref{canonicalformnormalization}) we have
\begin{eqnarray*}
N_\infty&=&\int_{M_\infty} f_\infty u_\infty dg_{\infty,t}-\frac{n}{2}
\\
&=&\int_{M_\infty}|t|(|\nabla f_\infty|^2+R_\infty)u_\infty dg_{\infty,t}-\frac{n}{2}+\mu_\infty
\\
&=&\int_{M_\infty}|t|(\Delta f_\infty+R_\infty)u_\infty dg_{\infty,t}-\frac{n}{2}+\mu_\infty
\\
&=&\int_{M_\infty}|t|\frac{n}{2|t|}u_\infty dg_{\infty,t}-\frac{n}{2}+\mu_\infty
\\
&=&\mu_\infty.
\end{eqnarray*}
Here we have applied integration by parts at infinity. This is valid, since both $f_\infty$ and $|\nabla f_\infty|^2$ have quadratic growth bounds (\cite{CZ10}), and since a Ricci shrinker has at most Euclidean volume growth (\cite{CZ10, Mun09}); a standard cut-off argument easily verifies the integration by parts at infinity.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Theorem_main}(3)]
Since any Ricci flow with bounded curvature within each compact time interval is $H_n$-concentrated, we have
\begin{eqnarray*}
\big((M,g_i(t))_{t\in(-\infty,0]},(\nu^i_s)_{s\in(-\infty,0]}\big)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\mathcal{X},
\end{eqnarray*}
where $d\nu_s^i=u_i(\cdot,s)dg_{i,s}$ and $\mathcal{X}$ is a metric flow pair over $(-\infty,0]$ (c.f. \cite[Theorem 7.8]{Bam20b}). Here the $\mathbb{F}$-convergence should be understood to be the $\mathbb{F}$-convergence over each finite subinterval of $(-\infty,0]$.
By Corollary \ref{change-l-center-to-Hn-center}, we have
\begin{eqnarray*}
\big(M,g_i(t),z_i\big)_{t\in[-2,-1]}\xrightarrow{\makebox[1cm]{}}\big(M_\infty,g_\infty(t),z_\infty\big)_{t\in[-2,-1]}.
\end{eqnarray*}
Therefore, Theorem \ref{convergence} implies that
\begin{eqnarray*}
\mathcal{X}_{[-2,-1)}=\big((M_\infty,g_\infty(t))_{t\in[-2,-1)},(\nu^\infty_s)_{s\in[-2,-1)}\big),
\end{eqnarray*}
where $\nu_s^\infty:=u_\infty(\cdot,s)dg_s$ is a conjugate heat flow on $(M_\infty,g_\infty(t))$.
Finally, the proof of Theorem \ref{convergence} indicates that $u_i\rightarrow u_\infty$ locally smoothly, where $u_i$ is defined in (\ref{somenonsenseofscaledheatkernels}). Hence, by Proposition \ref{f-potential}, $u_\infty=(4\pi|t|)^{-\frac{n}{2}}e^{-f_\infty}$ is indeed a conjugate heat flow made of a shrinker potential function; this proves Theorem \ref{Theorem_main}(3).
\end{proof}
\section{Independence of the base point}
In this section, we prove Corollary \ref{entropynoloss}. Recall that in the statement of Assumption B, the choices of the base point $(p_0,0)$ and the sequence of $\ell$-centers $\{(p_i,-\tau_i)\}_{i=1}^\infty$ are involved. If an ancient solution satisfies Assumption B with respect to some certain base point $(p_0,0)$ and some certain sequence of scales $\{\tau_i\}_{i=1}^\infty$, it is not immediately clear whether Assumption B is still valid if we alter the base point and the sequence of scales. And even if it were, it is not yet clear whether the limit is the same as before. Although the choices of the $\ell$-centers $p_i$ are not unique either, yet, according to what we have developed by far, this fact does not affect the validity of Assumption B or the limit flow in (\ref{smoothconvergence}). What we shall prove next is that for an ancient solution satisfying Assumption B, we may freely alter the base point, while the asymptotic shrinker is still the same. (But we may not change the sequence of scales $\{\tau_i\}_{i=1}^\infty$ at will.) This leads to the proof of Corollary \ref{entropynoloss}.
\begin{Theorem}\label{basepointindep}
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (\ref{curvaturebound}) and Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. Then, for any $(p'_0,t'_0)\in M\times(-\infty,0]$, the following holds.
\begin{eqnarray*}
\big(M,g'_i(t),p'_i\big)_{t\in[-2,-1]}\xrightarrow{\makebox[1cm]{}} \big(M_\infty,g_\infty(t),p'_\infty\big)_{t\in[-2,-1]},
\end{eqnarray*}
where $(p'_i,-\tau_i)$ is an $\ell$-center of $(p'_0,t'_0)$ with respect to the Ricci flow $(M,g(t))$, $g'_i(t):=\tau'^{-1}_ig(t'_0+\tau'_it)$, $\tau'_i:=\tau_i+t'_0$ for all $i$ large, and $(M_\infty,g_\infty(t))$ is the same limit as in (\ref{smoothconvergence}).
\end{Theorem}
\textbf{Remark:} In fact, if the existing interval of $(M,g(t))$ extends beyond $t=0$, then $t_0'$ can be taken to be greater than $0$, so long as the curvature remains to be bounded up to $t'_0$. This can be easily observed from the proof of this theorem.
\begin{proof}
Since $\displaystyle g'_i(t)=\frac{\tau_i}{\tau'_i}g_i\left(\frac{\tau'_i}{\tau_i}t+\tau_i^{-1}t'_0\right)$, where $g_i(t):=\tau_i^{-1}g_i(\tau_it)$, $$\tau_i^{-1}t'_0\rightarrow 0\quad\text{ and } \quad \frac{\tau_i}{\tau'_i}\rightarrow 1$$ as $i\rightarrow\infty$. It is then clear that we need only to show the equivalence of the base points, that is
\begin{eqnarray}\label{nonsense6-2-0}
\limsup_{i\rightarrow\infty}{\rm dist}_{g'_{i,-1}}(p_i,p'_i)<\infty.
\end{eqnarray}
Obviously, our concern is only in the case when $i$ is large. Hence, throughout this proof, we assume
\begin{eqnarray*}
\tau_i\gg|t'_0|\quad\text{ and }\quad \left|\frac{\tau_i}{\tau'_i}-1\right|\ll 1.
\end{eqnarray*}
Let $(z'_i,-\tau_i)$ be an $H_n$-center of $(p'_0,t'_0)$, and $(z_i,-\tau_i)$ and $H_n$-center of $(p_0,0)$. By \cite[Proposition 4.6]{MZ21} and Theorem \ref{entropybound}, we have
\begin{eqnarray*}
\lim_{\tau\rightarrow\infty}\mathcal{N}_{p'_0,t'_0}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{N}_{p_0,0}(\tau)\geq -Y>-\infty.
\end{eqnarray*}
It then follows from Proposition \ref{H_n_l_n} that
\begin{eqnarray*}
{\rm dist}_{g_{i,-1}}(p_i,z_i),\ {\rm dist}_{g'_{i,-1}}(p'_i,z'_i)\leq C,
\end{eqnarray*}
where $C$ is independent of $i$. Hence, to prove (\ref{nonsense6-2-0}), it suffices to show that
\begin{eqnarray*}
{\rm dist}_{g_{i,-1}}(z_i,z'_i),\ {\rm dist}_{g'_{i,-1}}(z_i,z'_i)\leq C,
\end{eqnarray*}
or equivalently,
\begin{eqnarray}\label{nonsense6-2-1}
{\rm dist}_{g_{-\tau_i}}(z_i,z'_i)\leq C\sqrt{\tau_i},
\end{eqnarray}
for some constant $C$ independent of $i$.
Let us prove (\ref{nonsense6-2-1}) by contradiction. Assume that there are positive numbers $A_i\nearrow\infty$ such that
\begin{eqnarray*}
{\rm dist}_{g_{-\tau_i}}(z_i,z_i')\geq 10\sqrt{A_iH_n\tau_i}.
\end{eqnarray*}
Then, whenever $i$ is large enough, we may find a nonnegative $1$-Lipschitz function $\varphi_i$, satisfying the following properties:
\begin{enumerate}[(1)]
\item $\varphi_i$ is compactly supported on $B_{g_{-\tau_i}}(z_i,2\sqrt{A_iH_n\tau_i})$,
\item $\varphi_i\geq \sqrt{A_iH_i\tau_i}$ on $B_{g_{-\tau_i}}(z_i,\sqrt{A_iH_n\tau_i})$,
\item $0\leq \varphi_i\leq 2\sqrt{A_iH_n\tau_i}$ everywhere,
\item the support of $\varphi_i$ is disjoint from $B_{g_{-\tau_i}}(z'_i,\sqrt{A_iH_n\tau'_i})$.
\end{enumerate} Using $\varphi_i$ as a test function, we may compute
\begin{align*}
&\quad\quad{\rm dist}_{W_1}^{g_{-\tau_i}}(\nu_{p_0,0\,|\,-\tau_i},\nu_{p'_0,t'_0\,|\,-\tau_i})
\\
&\geq \int_M \varphi_i d\nu_{p_0,0\,|\,-\tau_i}-\int_M \varphi_i d\nu_{p'_0,t'_0\,|\,-\tau_i}
\\
&\geq\int_{B_{g_{-\tau_i}}(z_i,\sqrt{A_iH_n\tau_i})} \varphi_i d\nu_{p_0,0\,|\,-\tau_i} -\int_{M\setminus B_{g_{-\tau_i}}(z'_i,\sqrt{A_iH_n\tau'_i})}\varphi_i d\nu_{p'_0,t'_0\,|\,-\tau_i}
\\
&\geq \left(1-\frac{1}{A_i}\right)\sqrt{A_iH_n\tau_i}-\frac{1}{A_i}2\sqrt{A_iH_n\tau_i}
\\
&=\left(1-\frac{3}{A_i}\right)\sqrt{A_iH_n\tau_i}\rightarrow\infty,
\end{align*}
as $i\rightarrow\infty$; here we have applied Proposition \ref{measureaccumulationofHcenter}. On the other hand, by (\ref{monotoneofdW1}), we have
\begin{eqnarray*}
d_{W_1}^{g_{-\tau_i}}(\nu_{p_0,0\,|\,-\tau_i},\nu_{p'_0,t'_0\,|\,-\tau_i})\leq d_{W_1}^{g_{t'_0}}(\nu_{p_0,0\,|\,t'_0},\delta_{p'_0})<\infty.
\end{eqnarray*}
This is a contradiction, and (\ref{nonsense6-2-1}) follows; we have finished the proof of the theorem.
\end{proof}
We are now ready to prove Corollary \ref{entropynoloss}. As before, we still consider an ancient solution $(M,g(t))_{t\in(-\infty,0]}$ satisfying (\ref{curvaturebound}) and Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. We first of all need to deal with the case when the base point is fixed at $(p_0,0)$.
\begin{Lemma}\label{somethingmaybeuseful}
Let $\ell_\infty: M_\infty\times(1,2]\rightarrow\mathbb{R}$ be the shrinker potential of $(M_\infty,g_\infty(t))_{t\in[-2,-1)}$ as given by Proposition \ref{shrinkerstructure}. Then we have
\begin{eqnarray*}
V_\infty:=\int_{M_\infty}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_\infty(\cdot,|t|)}dg_{\infty,t}\equiv\lim_{\tau\rightarrow\infty}\mathcal{V}_{p_0,0}(\tau)
\end{eqnarray*}
for all $t\in [-2,-1)$.
\end{Lemma}
\begin{proof}
Let $t\in[-2,-1)$ be arbitrarily fixed. We need only to show
\begin{eqnarray*}
\int_{M_\infty}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_\infty(\cdot,|t|)}dg_{\infty,t}=\lim_{i\rightarrow\infty}\int_{M}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_i(\cdot,|t|)}dg_{i,t},
\end{eqnarray*}
where, as before, $g_i(t)=\tau_i^{-1}g(\tau_it)$ and $\ell_i(\cdot,\tau)=\ell_{p_0,0}(\cdot,\tau_i\tau)$.
By (\ref{subsolution}) and Proposition \ref{measureaccumulationofHcenter}, we have
\begin{eqnarray*}
\int_{M\setminus B_{g_{i,t}}(z_i,\sqrt{AH_n|t|})} (4\pi|t|)^{-\frac{n}{2}}e^{-\ell_i(\cdot,|t|)}dg_{i,t}\leq \int_{M\setminus B_{g_{i,t}}(z_i,\sqrt{AH_n|t|})} u_i(\cdot,t)dg_{i,t}\leq\frac{1}{A},
\end{eqnarray*}
for all $A>1$, where $u_i$ is the conjugate heat kernel based at $(p_0,0)$ with respect to Ricci flow $g_i$. The conclusion then follows from the same argument as in the proof of Proposition \ref{Nconst}.
\end{proof}
\begin{Lemma}\label{somethingmaybeuseful1}
We have
\begin{eqnarray*}
\lim_{\tau\rightarrow\infty}\log \mathcal{V}_{p_0,0}(\tau)=\log V_\infty=\mu_\infty,
\end{eqnarray*}
where $\mu_\infty$ is the entropy of the asymptotic shrinker $(M_\infty,g_\infty,\ell_\infty)$.
\end{Lemma}
\begin{proof}
Since $\ell_\infty$ is a shrinker potential normalized in the way that
\begin{eqnarray*}
\Delta \ell_\infty -|\nabla\ell_\infty|^2+R_\infty+\frac{\ell_\infty-n}{\tau}=0,
\end{eqnarray*}
we have $\log V_\infty=\mu_\infty$; the details of this argument can be found in the proof of \cite[Theorem 1.2]{Zhang20}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{entropynoloss}]
By \cite[Proposition 4.6]{MZ21} and Theorem \ref{Theorem_main}(2), we already have
$$\lim_{\tau\rightarrow\infty}\mathcal{N}_{p'_0,t'_0}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{W}_{p'_0,t'_0}(\tau)=\mu_\infty$$
for any $(p'_0,t'_0)\in M\times(-\infty,0]$. By Theorem \ref{basepointindep}, $(M,g(t))_{t\in(-\infty,0]}$ satisfies Assumption B with respect to $\big(p'_0,t'_0, \{\tau'_i\}_{i=1}^\infty,\{p'_i\}_{i=1}^\infty\big)$, and the smooth limit is the same as the one in (\ref{smoothconvergence}). Since the entropy of a Ricci shrinker is unique, the last conclusion follows from applying Lemma \ref{somethingmaybeuseful1} to $\mathcal{V}_{p'_0,t'_0}(\tau)$.
\end{proof}
\section{Perelman's $\nu$-functional on ancient solutions satisfying Assumption B}
In this section, we shall prove Theorem \ref{nu-functional}. We still consider an ancient solution $(M^n,g(t))_{t\in(-\infty,0]}$ satisfying (\ref{curvaturebound}) and Assumption B with respect to $\big(p_0,0,\{\tau_i\}_{i=1}^\infty,\{p_i\}_{i=1}^\infty\big)$. Let $(M_\infty, g_\infty, \ell_\infty)$ be the asymptotic shrinker from Proposition \ref{shrinkerstructure}. By the remark below Corollary \ref{LUTypeInoncollapsed}, the ancient solution in question has bounded geometry withing each compact time interval, and this will be a convenient condition to be applied in this section.
Let us fix an arbitrary $t_0\in(-\infty,0]$. Without loss of generality, we may assume $t_0<0$. The reason is because $(M,g(t))$ has bounded curvature at each time and can be extended pass $t=0$, and we may replace our base point $(p_0,0)$ by some $(p_0,\varepsilon)$, where $\varepsilon>0$, if ever $t_0=0$. By Theorem \ref{basepointindep}, this the change of base point does not affect anything, except for slight modifications of the scales $\{\tau_i\}_{i=1}^\infty$.
Let us fix a scale $\tau_0>0$ and consider an arbitrary function $u_0$ which is qualified to be a test function for $\mu(g(t_0),\tau_0)$, that is, $u_0\geq 0$, $\sqrt{u_0}\in C_0^\infty(M)$, and $\int_M udg_{t_0}=1$. We may then solve the conjugate heat equation with $u_0$ being its initial data and obtain the solution
\begin{eqnarray}\label{thedefinitionofu}
u(x,t):=\int_M u_0 K(\cdot,t_0\,|\,x,t)dg_{t_0}.
\end{eqnarray}
Since $u$ is positive on $M\times(-\infty,t_0)$, we write
\begin{eqnarray*}
u(x,t):=(4\pi(T-t))^{-\frac{n}{2}} e^{-f(x,t)}\quad \text{ for all }\quad (x,t)\in M\times(-\infty,t_0),
\end{eqnarray*}
where $$T=t_0+\tau_0.$$
For the notational convenience, we shall use $\spt$ to denote $\spt u_0=\spt \sqrt{u_0}$, and shall fix a point $x_0\in \spt$ and a positive number $D_0>0$, such that
\begin{eqnarray*}
\spt\subset B_{t_0}(x_0,D_0).
\end{eqnarray*}
We shall consider the time-dependent function $\overline{\mathcal{W}}(g(t),u(\cdot,t),T-t)=\mathcal{W}(g(t),f(\cdot,t),T-t)$ for all $t\in(-\infty,t_0)$, where $\mathcal{W}$ and $\overline{\mathcal{W}}$ are defined in (\ref{Perelmansentropy}) and (\ref{anotherPerelmansentropy}), respectively. Since $u$ is a positive solution to the conjugate heat equation, and since $(M,g(t))$ is a Ricci flow with bounded geometry on compact time-intervals, the classical monotonicity formula of Perelman is still valid. We include the following lemma without a tedious proof. The interested reader could easily verify it; note that the estimate (\ref{EKNT_gradient}) below is helpful.
\begin{Lemma}
We have
\begin{eqnarray*}
\frac{d}{dt}\overline{\mathcal{W}}(g(t),u(\cdot,t),T-t)=2(T-t)\int_M \left|\Ric+\nabla^2f-\frac{1}{2(T-t)}g\right|^2udg_t\geq 0
\end{eqnarray*}
for all $t\in(-\infty, t_0)$.
\end{Lemma}
The result of this section consists of the following two theorems, and we shall prove them one-by-one.
\begin{Theorem}\label{backwardlimit}
We have
$$\lim_{t\rightarrow-\infty}\overline{\mathcal{W}}(g(t),u(\cdot,t),T-t)\geq \mu_\infty,$$ where $\mu_\infty$ is the entropy of the asymptotic shrinker $(M_\infty,g_\infty,\ell_\infty)$.
\end{Theorem}
\begin{Theorem}\label{forwardlimit}
We have $$\lim_{t\rightarrow t_0-}\overline{\mathcal{W}}(g(t),u(\cdot,t),T-t)=\overline{\mathcal{W}}(g(t_0),u_0,\tau_0).$$
\end{Theorem}
Recall that the ancient solution $(M,g(t))$ is locally uniformly Type I along the space-time sequence $\{(p_i,-\tau_i)\}_{i=1}^\infty$. We shall use this condition to obtain a locally uniformly lower bound for $u$ around $(p_i,-\tau_i)$. From this point on until the completion of the proof of Theorem \ref{backwardlimit}, we shall assume that
\begin{eqnarray*}
\tau_i\gg |t_0|,\quad \tau_i\gg \tau_0.
\end{eqnarray*}
\begin{Proposition}
For any $\varepsilon\in(0,\frac{1}{4})$, there is a decreasing positive function $c(\cdot,\varepsilon):(0,\infty)\rightarrow(0,1)$ with the following property: for any $r>0$, there is an $i_0$, such that
whenever $i\geq i_0$ and for all $y\in B_{t_0}(x_0,D_0)$, we have
\begin{eqnarray}\label{nonsense8004001}
K(y,t_0\,|\,x,t)\geq c(r,\varepsilon)\tau_i^{-\frac{n}{2}}\quad\text{ for all }\quad(x,t)\in B_{-\tau_i}(p_i,r\sqrt{\tau_i})\times[-2\tau_i,-(1+\varepsilon)\tau_i],
\end{eqnarray}
and consequently
\begin{eqnarray}\label{nonsense8004002}
u(x,t)\geq c(r,\varepsilon)\tau_i^{-\frac{n}{2}}\quad\text{ for all }\quad(x,t)\in B_{-\tau_i}(p_i,r\sqrt{\tau_i})\times[-2\tau_i,-(1+\varepsilon)\tau_i].
\end{eqnarray}
\end{Proposition}
\begin{proof}
Let us fix an arbitrary $y\in B_{t_0}(x_0,D_0)$, then we have
\begin{eqnarray*}
{\rm dist}^{g_{-\tau_i}}_{W_1}(\nu_{y,t_0\,|\,-\tau_i},\nu_{p_0,0\,|\,-\tau_i})&\leq&{\rm dist}^{g_{t_0}}_{W_1}(\delta_y,\nu_{p_0,0\,|\,t_0})
\\
&\leq& {\rm dist}_{t_0}(x_0,y)+{\rm dist}^{g_{t_0}}_{W_1}(\delta_{x_0},\nu_{p_0,0\,|\,t_0})\leq C,
\end{eqnarray*}
where $C$ is independent of both $i$ and $y$.
Next, we let $(z_i,-\tau_i)$ and $(z'_i,-\tau_i)$ be $H_n$-centers of $(p_0, 0)$ and $(y, t_0)$, respectively. Arguing in the exactly same way as in the proof of formula (\ref{nonsense6-2-1}), we have
\begin{eqnarray}
{\rm dist}_{-\tau_i}(z_i,z'_i)\leq C\sqrt{\tau_i} \quad\text{ for all } i\in\mathbb{N},
\end{eqnarray}
where $C$ is a constant independent of both $i$ and $y$.
Let $(p'_i,-\tau_i)$ be an $\ell$-center of $(y, t_0)$. By Proposition \ref{H_n_l_n} (note that Proposition \ref{H_n_l_n} relies only on a Nash entropy lower bound, which is true by Corollary \ref{entropynoloss}), we have ${\rm dist}_{-\tau_i}(p_i,z_i)\leq C\sqrt{\tau_i}$ and ${\rm dist}_{-\tau_i}(z'_i,p'_i)\leq C\sqrt{\tau_i+t_0}\leq C\sqrt{\tau_i}$. Hence, we have
\begin{eqnarray}\label{closebycenters}
{\rm dist}_{-\tau_i}(p_i,p'_i)\leq C\sqrt{\tau_i}\quad\text{ for all } i\in\mathbb{N}.
\end{eqnarray}
Therefore, the locally uniformly Type I condition along $\{(p_i,-\tau_i)\}_{i=1}^\infty$ (c.f. Proposition \ref{LUTypeI}) also implies the existence of a positive function $C:(0,\infty)\rightarrow(0,\infty)$, such that, for any $r>0$, there is an $i_0\in\mathbb{N}$ with the property that
\begin{eqnarray}\label{anotherlutypei}
|\Rm|\leq\frac{C(r)}{\tau_i}\quad\text{ on }\quad B_{-\tau_i}(p'_i,r\sqrt{\tau_i})\times[-2\tau_i,-\tau_i]
\end{eqnarray}
for all $i\geq i_0$; in particular, $i_0$ depends on the constant in (\ref{closebycenters}) and the original locally uniformly Type I condition along $\{(p_i,-\tau_i)\}_{i=1}^\infty$. It is to be emphasized that this $i_0$ is independent of $y$, and this is due to the fact that the constant $C$ in (\ref{closebycenters}) is independent of $y$. Since $\ell_{y,t_0}(p'_i,-\tau_i)\leq\frac{n}{2}$, arguing in the same way as the proof of \cite[Proposition 5.1(1)]{CZ20} and using (\ref{anotherlutypei}), we have that for any $r>0$, if $i$ is large enough (independent of $y$), then
\begin{eqnarray*}
\ell_{y, t_0}(x,|t|)\leq C(r,\varepsilon)\quad \text{ for all }\quad (x,t)\in B_{g_{-\tau_i}}(p'_i,r\sqrt{\tau_i})\times[-2\tau_i,-(1+\varepsilon)\tau_i],
\end{eqnarray*}
where $C(\cdot,\varepsilon):(0,\infty)\rightarrow(0,\infty)$ is a function depending on $\varepsilon$. Therefore, (\ref{subsolution}) implies that
\begin{eqnarray*}
K(y,t_0\,|\,x,t)\geq \frac{1}{(4\pi(t_0-t))}e^{-\ell_{y, t_0}(x,|t|)}\geq c(r,\varepsilon)\tau_i^{-\frac{n}{2}},
\end{eqnarray*}
for all $t\in[-2\tau_i,-(1+\varepsilon)\tau_i]$ and for all $x\in B_{g_{-\tau_i}}(p'_i,r\sqrt{\tau_i})\subset B_{g_{-\tau_i}}\big(p_i,(r-C)\sqrt{\tau_i}\big)$; here we have used (\ref{closebycenters}) again. This finishes the proof of (\ref{nonsense8004001}), and (\ref{nonsense8004002}) follows from (\ref{thedefinitionofu}), (\ref{nonsense8004001}), and the fact that $\int_Mu_0dg_{t_0}=1$.
\end{proof}
Next, we prove a Gaussian upper bound for $u$.
\begin{Proposition}\label{upperbound}
There is a constant $C_0$, depending on the geometry bounds on $M\times[t_0,0]$, the value of $D_0$, ${\rm dist}_0(p_0,x_0)$, and the upper bound of $u_0$, such that
\begin{eqnarray*}
u(x,t)\leq C_0K(p_0,0\,|\,x,t) \quad\text{ for all }\quad (x,t)\in M\times(-\infty,t_0).
\end{eqnarray*}
\end{Proposition}
\begin{proof}
Since $K(p_0,0\,|\,\cdot,t_0)$ is a positive function on $M$, we may the take
\begin{eqnarray*}
C_0:=\frac{\sup u_0}{\inf_{\spt}K(p_0,0\,|\,\cdot,t_0)}\in(0,\infty).
\end{eqnarray*}
Then we have $u_0\leq C_0 K(p_0,0\,|\,\cdot,t_0)$ everywhere on $M$, and the conclusion follows from the maximum principle.
\end{proof}
According to the scaling property of the $\overline{\mathcal{W}}$ functional, we have $\displaystyle\overline{\mathcal{W}}(g(\tau_it),u(\cdot,\tau_it),T-\tau_it)=\overline{\mathcal{W}}\left(\tau_i^{-1}g(\tau_it),\tau_i^{\frac{n}{2}}u(\cdot,\tau_it),\tfrac{T}{\tau_i}-t\right)$. Hence, for $t\in[-2,-1]$, we may define
\begin{eqnarray*}
g_i(t)&:=&\tau_i^{-1}g(\tau_it),
\\
f_i(\cdot,t)&:=&f(\cdot,\tau_it)+\frac{n}{2}\log\left(\frac{T+\tau_i|t|}{\tau_i|t|}\right),
\\
\mathcal{W}_i(t)&:=&\overline{\mathcal{W}}\left(\tau_i^{-1}g(\tau_it),\tau_i^{\frac{n}{2}}u(\cdot,\tau_it),\tfrac{T}{\tau_i}-t\right)+\frac{n}{2}\log\left(\frac{T+\tau_i|t|}{\tau_i|t|}\right)
\\
&=&\int_M\left(\big(\tfrac{T}{\tau_i}+|t|\big)\big(|\nabla f_i|^2+R_i\big)+f_i-n\right)(4\pi|t|)^{-\frac{n}{2}}e^{-f_i}dg_{i,t}.
\end{eqnarray*}
Since $T/\tau_i\rightarrow 0$, to prove Theorem \ref{backwardlimit}, it suffices to show that
\begin{eqnarray*}
\liminf_{i\rightarrow\infty} \mathcal{W}_i(t)\geq \mu_\infty\quad\text{ for some } \quad t\in[-2,-1].
\end{eqnarray*}
By Proposon \ref{upperbound} (arguing as the proof in \cite[Theorem 2.1]{Lu12} again), we have that $u_i=(4\pi|t|)^{-\frac{n}{2}}e^{-f_i}\rightarrow u_\infty$ locally smoothly on $M_\infty\times [-2,-1)$, where $u_\infty$ is a nonnegative solution to the conjugate heat equation. (\ref{nonsense8004002}) then implies $u_\infty>0$ everwhere on $M_\infty\times[-2,-1)$ and we may define $f_\infty: M_\infty\times[-2,-1)\rightarrow\mathbb{R}$ by $u_\infty:=(4\pi|t|)^{-\frac{n}{2}}e^{-f_\infty}$. Then we have $f_i\rightarrow f_\infty$ locally smoothly on $M_\infty\times[-2,-1)$. By Proposition \ref{upperbound} and arguing as in the proof of Lemma \ref{lem: dnu^oo_s}, we have that
\begin{eqnarray}\label{anotherunitmeasurenonsense}
\int_{M_\infty}u_\infty(\cdot,t)dg_{\infty,t}\equiv 1\quad\text{ for all }\quad t\in[-2,1).
\end{eqnarray}
Since we have $\int_M u(\cdot,t)dg_t\equiv 1$ for all $t\in(-\infty,t_0)$ by the property of the conjugate heat equation, the proof of (\ref{anotherunitmeasurenonsense}) is in fact somewhat like the dominated convergence theorem,.
Proposition \ref{upperbound} also implies that $f_i$ has at least quadratic growth, that is, there is a constant $C$, such that
\begin{eqnarray}\label{quadratic_1}
f_i(x,t)&\geq& \frac{1}{C}{\rm dist}_{g_{i,t}}^2(p_i,x)-C\quad\text{ for all }\quad (x,t)\in M\times[-2,-1],
\\
f_\infty(x,t)&\geq& \frac{1}{C}{\rm dist}_{g_{\infty,t}}^2(p_\infty,x)-C\quad\text{ for all }\quad (x,t)\in M_\infty \times[-2,-1).\label{quadratic_2}
\end{eqnarray}
If we denote $$\mathcal{W}_\infty(t)=\int_{M_\infty}\big(|t|(|\nabla f_\infty|^2+R_\infty)+f_\infty-n\big)u_\infty dg_{\infty,t},$$ then (\ref{quadratic_1}) and (\ref{quadratic_2}) are sufficient to show
\begin{eqnarray}\label{wbconvergence}
\liminf_{i\rightarrow\infty} \mathcal{W}_i(t)\geq \mathcal{W}_\infty(t)\quad\text{ for all }\quad t\in[-2,-1).
\end{eqnarray}
To see this, let us fix an arbitrary large number $A$, such that the integrand of both $\mathcal{W}_i(t)$ and $\mathcal{W}_\infty(t)$ are positive outside a disk with radius $A$; this is possible because of (\ref{quadratic_1}) and (\ref{quadratic_2}). Then we may compute
\begin{align*}
&\quad\quad\int_{B_{g_{\infty,t}}(p_\infty,A)}\big(|t|(|\nabla f_\infty|^2+R_\infty)+f_\infty-n\big)u_\infty dg_{\infty,t}
\\
&=\lim_{i\rightarrow\infty}\left(\int_{B_{g_{i,t}}(p_i,A)}\left(\big(\frac{T}{\tau_i}+|t|\big)\big(|\nabla f_i|^2+R_i\big)+f_i-n\right)(4\pi|t|)^{-\frac{n}{2}}e^{-f_i}dg_{i,t}+\frac{n}{2}\log\left(\frac{T+\tau_i|t|}{\tau_i|t|}\right)\right)
\\
&\leq\liminf_{i\rightarrow\infty}\mathcal{W}_i(t),
\end{align*}
taking $A\rightarrow\infty$, (\ref{wbconvergence}) follows.
\begin{Lemma}\label{anotherl2nonsese}
For all $t\in[-2,-1)$, we have
\begin{eqnarray*}
\int_M |\nabla f_\infty|^2 u_\infty dg_{\infty,t}<\infty.
\end{eqnarray*}
\end{Lemma}
\begin{proof}
(\ref{wbconvergence}) implies that $\mathcal{W}_\infty(t)<\infty$. Furthermore, by (\ref{quadratic_2}), we have $f_\infty(\cdot,t)>-C$ for some constant $C$. Hence
\begin{eqnarray*}
|t|\int_M(|\nabla f_\infty|^2 +R_\infty)u_\infty dg_{\infty,t}&=&\mathcal{W}_\infty(t)-\int_M(f_\infty-n)u_\infty dg_{\infty,t}
\\
&\leq& \mathcal{W}_\infty(t)+C+n<\infty.
\end{eqnarray*}
This finishes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{backwardlimit}]
Let us fix an arbitrary $t\in[-2,-1)$. By (\ref{quadratic_2}), we have that $u_\infty$ also has a Gaussian upper bound. Hence, by the Euclidean volume growth bound for shrinkers (\cite{Mun09, CZ10}), we have
\begin{eqnarray*}
\int_{M_\infty}{\rm dist}_{g_{\infty,t}}^2(p_\infty,\cdot)u_\infty(\cdot,t)dg_{\infty,t}<\infty.
\end{eqnarray*}
Consequently, by (\ref{anotherunitmeasurenonsense}) and Lemma \ref{anotherl2nonsese}, $u_\infty$ can be used as a test function for the $\mu$-functional (see formulas (91) and (92) in \cite{LW20}). Then, by Proposition \ref{shrinkernufunctional}, we have
\begin{eqnarray*}
\mathcal{W}_\infty(t)=\mathcal{W}\big(g_\infty(t),f_\infty(\cdot,t),|t|\big)\geq\mu(g_\infty(t),|t|)=\mu(g_\infty(-1),1)=\mu_\infty.
\end{eqnarray*}
Taking (\ref{wbconvergence}) into account, this finishes the proof.
\end{proof}
From this point on, we will proceed to prove Theorem \ref{forwardlimit}. It turns our that the following coarse Gaussian estimates are very helpful.
\begin{Proposition}[Theorem 26.25 and Theorem 26.31 in \cite{RFV3}]\label{coarsegaussboundthatisuseful}
There is a constant $C$ depending only on the geometry bounds on $M\times[t_0-1,t_0]$, such that the following holds
\begin{eqnarray}
\frac{1}{C(t-s)^{\frac{n}{2}}}\exp\left(-\frac{C{\rm dist}_t^2(x,y)}{(t-s)}\right)\leq K(x,t\,|\,y,s)\leq \frac{C}{(t-s)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_t^2(x,y)}{C(t-s)}\right),
\end{eqnarray}
for all $x, y\in M$ and for all $t_0-1\leq s<t\leq t_0$. Here ${\rm dist}_t$ can be replaced by ${\rm dist}_{t'}$ for any $t'\in[t_0-1,t_0]$.
\end{Proposition}
\textbf{Remark:} Because of the boundedness of geometry, we have that for any $x\in M$ and $t_0-1\leq s<t\leq t_0$, it holds that
\begin{eqnarray*}
\Vol_{g_t}\big(B_t(x,\sqrt{t-s})\big)\geq c(t-s)^{\frac{n}{2}},
\end{eqnarray*}
where $c$ depends only on the geometric bounds on $M\times[t_0-1,t_0]$.
\begin{Lemma}\label{coarsebound}
There exists positive constant $C<\infty$, such that
\begin{eqnarray}\label{nonsense_grad}
|\nabla ^k u|\leq C \quad\text{ on }\quad M\times[t_0-1,t_0],
\end{eqnarray}
where $k\in\{0,1,2,3\}$.
\end{Lemma}
\begin{proof}
Since $u_0$ is smooth and compactly supported, we have $|\nabla^k u_0|\leq C_k$ everywhere on $M$. Then, one may use the same argument as in the proof of \cite[Theorem 10]{LT} to prove (\ref{nonsense_grad}). This is a standard Shi-type estimate and the proof is left to the readers.
\end{proof}
\begin{Lemma}
There exists a constant $C<\infty$, such that
\begin{align}\label{quadratic_nonsense}
u(x,t)\leq \frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{C(t_0-t)}\right),
\end{align}
for all $(x,t)\in M\times[t_0-1,t_0)$. Here $\spt=\spt u_0$ denotes the compact support set of $u_0$.
\end{Lemma}
\begin{proof}
To see this, we compute using Proposition \ref{coarsegaussboundthatisuseful}
\begin{eqnarray*}
u(x,t)&=&\int_{M}u_0(y)K(y,t_0\,|\,x,t)dg_{t_0}(y)
\\
&\leq&\int_{\spt}u_0(y)\cdot \frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,y)}{C(t_0-t)}\right)dg_{t_0}(y)
\\
&\leq&\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{C(t_0-t)}\right)\int_Mu_0dg_{t_0},
\end{eqnarray*}
the lemma then follows.
\end{proof}
\begin{Proposition}\label{ulogunonsense}
\begin{eqnarray*}
\lim_{t\rightarrow t_0-}\int_M u(\cdot,t)\log u(\cdot,t)\, dg_{t}=\int_M u_0\log u_0\,dg_{t_0}.
\end{eqnarray*}
\end{Proposition}
\begin{proof}
First of all, we show that
\begin{eqnarray}\label{uniform_nonsense}
u(\cdot,t)\log u(\cdot,t)\rightarrow u_0\log u_0 \quad \text{ uniformly on }\quad M.
\end{eqnarray}
Let us fix an arbitrary small $\delta>0$. Since $u$ is bounded from above, and $u\rightarrow u_0$ uniformly (note that $|\nabla^2 u|\leq C$ by Lemma \ref{coarsebound}, we have that $|\partial_t u|\leq |\nabla^ 2u|+Ru \leq C$ near $t=t_0$), we have that $u\log u\rightarrow u_0\log u_0$ uniformly on $\{u_0\geq \delta\}$. Hence, we may find a small $\varepsilon\in(0,\delta)$, such that
\begin{eqnarray*}
\sup_{\{u_0\geq\delta\}}\big|u(\cdot,t)\log u(\cdot,t)-u_0\log u_0\big|\leq \delta\quad\text{ for all }\quad t\in[t_0-\varepsilon,t_0].
\end{eqnarray*}
On the other hand, we may take $\delta$ and $\varepsilon$ small enough, such that $a\log a\leq a^{\frac{1}{2}}$ whenever $a\in(0,\delta+C\varepsilon)$, where $C$ is the constant in (\ref{nonsense_grad}). Therefore, if $x\in\{u_0<\delta\}$ and $t\in[t_0-\varepsilon,t_0]$, we have $0<u(x,t)\leq u_0(x)+C|t_0-t|< \delta+C\varepsilon$ and $u\log u\leq u^{\frac{1}{2}}\leq\sqrt{\delta+C\varepsilon}$. This implies
\begin{eqnarray*}
\sup_{\{u_0<\delta\}}\big|u(\cdot,t)\log u(\cdot,t)-u_0\log u_0\big|\leq \sqrt{\delta+C\varepsilon}+\delta\quad\text{ for all }\quad t\in[t_0-\varepsilon,t_0].
\end{eqnarray*}
This proves (\ref{uniform_nonsense}).
Next, we shall prove that for any $A>2D_0$ (recall that $D_0>0$ is such that $\spt\subset B_{t_0}(x_0,D_0)$), we have
\begin{eqnarray*}
\lim_{t\rightarrow t_0-}\int_{M\setminus B_{t_0}(\spt, A)}\left|u(\cdot,t)\log u(\cdot,t)\right|dg_t\rightarrow 0.
\end{eqnarray*}
Here $B_{t_0}(\spt,A):=\{x\ |\ {\rm dist}_{t_0}(\spt,x)< A\}$. Apparently, this is sufficient for the lemma. Fixing any $A>2D_0$, we may, by (\ref{quadratic_nonsense}), find a positive number $\varepsilon$, such that $u$ is small enough on $M\setminus B_{t_0}(\spt, A)\times[t_0-\varepsilon,t_0)$ and satisfies
\begin{eqnarray*}
0&<&\left|u(x,t)\log u(x,t)\right|<u^{\frac{1}{2}}(x,t)\leq (t_0-t)^{\frac{n}{4}}\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{2C(t_0-t)}\right)
\\
&\leq&(t_0-t)^{\frac{n}{4}}\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{({\rm dist}_{t_0}(x,x_0)-D_0)^2}{2C(t_0-t)}\right)
\\
&\leq&(t_0-t)^{\frac{n}{4}}\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_{t_0}^2(x,x_0)}{4C(t_0-t)}\right)
\end{eqnarray*}
for all $(x,t)\in M\setminus B_{t_0}(\spt, A)\times[t_0-\varepsilon,t_0]$. Therefore, we have
\begin{eqnarray*}
\int_{M\setminus B_{t_0}(\spt, A)}\left|u(\cdot,t)\log u(\cdot,t)\right|dg_t&\leq& (t_0-t)^{\frac{n}{4}}\int_{M\setminus B_{t_0}(x_0,A)}\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}_{t_0}^2(x,x_0)}{4C(t_0-t)}\right) dg_t
\\
&\leq& C(t_0-t)^{\frac{n}{4}}\rightarrow 0 \quad \text{ as }\quad t\rightarrow t_0-.
\end{eqnarray*}
The integral estimate in the last inequality above is a standard result using the Bishop-Gromov volume comparison theorem. Similar estimates will appear many times later.
\end{proof}
Next, we deal with the integral of the gradient term. First of all, we recall the following gradient estimate from \cite[Theorem 10]{EKNT08}. Although $u$ is not positive at $t=t_0$, yet one may apply this theorem to $u$ on $M\times [t,\frac{t+t_0}{2}]$, for any $t\in[t_0-1,t_0)$.
\begin{Lemma}[Theorem 10 in \cite{EKNT08}]
\label{lem: grad est on CHF}
There is a constant $C$ such that the following holds
\begin{eqnarray}\label{EKNT_gradient}
\frac{|\nabla u|^2}{u}\leq\frac{Cu}{t_0-t}\left(1+\log\frac{C}{u}\right)^2\leq\frac{C}{t_0-t}(u+u\log^2 u) \quad \text{ on }\quad M\times[t_0-1,t_0).
\end{eqnarray}
\end{Lemma}
\begin{Lemma}\label{someothergradientl2nonsense}
For all $A>2D_0+1$, we have
\begin{eqnarray}
\lim_{t\rightarrow t_0-}\int_{M\setminus B_{t_0}(\spt, A)}\frac{|\nabla u|^2}{u}(\cdot,t)dg_{t}= 0.
\end{eqnarray}
\end{Lemma}
\begin{proof}
Let us fix an arbitrary $A>2D_0+1$. By (\ref{quadratic_nonsense}), we may let $\varepsilon\in(0,1]$ be small enough, such that $u\log^2 u\leq u^\frac{1}{2}$ on $M\setminus B_{t_0}(\spt, A)\times [t_0-\varepsilon,t_0)$. Hence we have
\begin{eqnarray*}
\frac{|\nabla u|^2}{u}(x,t)&\leq& \frac{C}{t_0-t}\Big(u(x,t)+u(x,t)\log^2 u(x,t)\Big) \leq\frac{C}{t_0-t}\big(u(x,t)+u^{\frac{1}{2}}(x,t)\big)
\\
&\leq&\frac{C}{t_0-t}\left(\frac{1}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{C(t_0-t)}\right)+\frac{1}{(t_0-t)^{\frac{n}{4}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{2C(t_0-t)}\right)\right)
\\
&\leq&\frac{C}{(t_0-t)^{\frac{n}{2}+1}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{2C(t_0-t)}\right)
\end{eqnarray*}
for all $(x,t)\in M\setminus B_{t_0}(\spt, A)\times [t_0-\varepsilon,t_0)$. Integrating the above inequality, we have
\begin{eqnarray*}
\int_{M\setminus B_{t_0}(\spt, A)}\frac{|\nabla u|^2}{u}dg_t&\leq& \int_{M\setminus B_{t_0}(\spt, A)}\frac{C}{(t_0-t)^{\frac{n}{2}+1}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{2C(t_0-t)}\right) dg_t(x)
\\
&\leq&\sqrt{t_0-t}\int_{M\setminus B_{t_0}(\spt, A)}\frac{C{\rm dist}^3_{t_0}(x,\spt)}{(t_0-t)^{\frac{n}{2}+\frac{3}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,\spt)}{2C(t_0-t)}\right)dg_t(x)
\\
&\leq&\sqrt{t_0-t}\int_{M\setminus B_{t_0}(x_0, A)}\frac{{\rm dist}^3_{t_0}(x,x_0)}{(t_0-t)^{\frac{3}{2}}}\cdot\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,x_0)}{4C(t_0-t)}\right)dg_t(x)
\\
&\leq& C\sqrt{t_0-t}\quad\text{ for all }\quad t\in[t_0-\varepsilon,t_0).
\end{eqnarray*}
This finishes the proof.
\end{proof}
\begin{Proposition}\label{grad u l2}
We have
\begin{eqnarray*}
\lim_{t\rightarrow t_0-}\int_M\frac{|\nabla u|^2}{u}(\cdot,t)dg_{t}=\int_M\frac{|\nabla u_0|^2}{u_0}dg_{t_0}.
\end{eqnarray*}
\end{Proposition}
\begin{proof}
Since, by our assumption, $\sqrt{u_0}\in C_0^\infty(M)$, we have
\begin{eqnarray*}
|\nabla u_0|^2\leq C u_0.
\end{eqnarray*}
Then, arguing in the same way as \cite[Lemma 4.3]{Wang18}, we have that, fixing any $\varepsilon\in(0,1)$, there is a constant $C$, such that
\begin{eqnarray*}
|\nabla u|^2\leq C u \quad \text{ on }\quad M\times [t_0-\varepsilon,t_0).
\end{eqnarray*}
In view of Lemma \ref{someothergradientl2nonsense}, if we can prove
\begin{eqnarray}\label{grad u pw convergence}
\frac{|\nabla u|^2}{u}(\cdot,t)\rightarrow \frac{|\nabla u_0|^2}{u_0}\quad \text{ pointwise on $M$ as } t\rightarrow t_0-,
\end{eqnarray}
then the current proposition follows from the bounded convergence theorem. We shall then prove (\ref{grad u pw convergence}) below.
Obviously, (\ref{grad u pw convergence}) is true on $\{u_0>0\}$, since $u\rightarrow u_0$ and $|\nabla u|^2\rightarrow |\nabla u_0|^2$ uniformly, because of their evolution equations, and because of Lemma \ref{coarsebound}. We shall then consider a point $x\in M$ such that $u_0(x)=0$. Since $\sqrt{u_0}\in C_0^\infty(M)$ is a nonnegative smooth function, we have that $|\nabla\sqrt{u_0}|(x)=0$ and $|\nabla^2\sqrt{u_0}|\leq C$ everwhere on $M$. It then follows that
\begin{eqnarray*}
|\nabla \sqrt{u_0}|(y)\leq C{\rm dist}_{t_0}(x,y),\quad \sqrt{u_0}(y)\leq C{\rm dist}^2_{t_0}(x,y),\quad \text{ for all }\quad y\in M,
\end{eqnarray*}
in other words,
\begin{eqnarray}\label{u_0 growth}
u_0(y)\leq C {\rm dist}^4_{t_0}(x,y)\quad \text{ for all }\quad y\in M.
\end{eqnarray}
Hence, for any $t\in [t_0-1,t_0)$, we may compute using (\ref{u_0 growth})
\begin{eqnarray*}
u(x,t)&=&\int_M K(y,t_0\,|\,x,t)u_0(y)dg_{t_0}(y)
\\
&\leq& \int_M C{\rm dist}^4_{t_0}(x,y)\cdot\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,y)}{C(t_0-t)}\right)dg_{t_0}(y)
\\
&\leq&(t_0-t)^2\int_M\left(\frac{{\rm dist}_{t_0}(x,y)}{\sqrt{t_0-t}}\right)^4\cdot\frac{C}{(t_0-t)^{\frac{n}{2}}}\exp\left(-\frac{{\rm dist}^2_{t_0}(x,y)}{C(t_0-t)}\right)dg_{t_0}(y)
\\
&\leq& C(t_0-t)^2.
\end{eqnarray*}
Applying the above result to (\ref{EKNT_gradient}), we have that if $t_0-t$ is small is small enough, then
\begin{eqnarray*}
\frac{|\nabla u|^2}{u}(x,t)&\leq& \frac{C}{t_0-t}(u+u\log^2 u)\leq \frac{C}{t_0-t}(u+u^{\frac{3}{4}})
\\
&\leq&\frac{C}{t_0-t}\big((t_0-t)^2+(t_0-t)^{\frac{3}{2}}\big)\leq C(t_0-t)^{\frac{1}{2}}
\\
&\rightarrow &0=\frac{|\nabla u_0|^2}{u_0}(x).
\end{eqnarray*}
This finishes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{forwardlimit}]
This theorem is now but a consequence of Proposition \ref{ulogunonsense}, Proposition \ref{grad u l2}, and the definition of $\overline{\mathcal{W}}(g(t),u(\cdot,t),T-t)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{nu-functional}]
Combining Theorem \ref{backwardlimit} and Theorem \ref{forwardlimit}, we have that, for any $t_0\in(-\infty,0]$, for any $\tau_0>0$, and for any $u_0:M\rightarrow\mathbb{R}$ satisfying $u_0\geq 0$, $\sqrt{u_0}\in C_0^\infty(M)$, and $\int_M u_0dg_0=1$, it holds that $$\overline{\mathcal{W}}(g(t_0),u_0,\tau_0)\geq \mu_\infty.$$ Hence we have $$\inf_{t\leq 0}\nu(g(t))\geq \mu_\infty.$$
To see that the equality in (\ref{nu_nonsense_00}) holds, recall that $(M,g_i(-1))\rightarrow (M_\infty,g_\infty(-1))$ in the smooth Cheeger-Gromov sense, and that the $\mu$-functional is upper semi-continuous with respect to the Cheeger-Gromov convergence (see Lemma 6.28 in \cite{RFV1}; this fact can be easily observed by taking an arbitrary compactly supported function on the limit, and pull it back to the sequence using the defining diffeomorphisms of the Cheeger-Gromov convergence), and these two facts imply
\begin{eqnarray*}
\limsup_{i\rightarrow\infty}\mu(g(-\tau_i),\tau_i)=\limsup_{i\rightarrow\infty}\mu(g_i(-1),1)\leq \mu(g_\infty(-1),1)=\mu_\infty.
\end{eqnarray*}
This then finishes the proof.
\end{proof}
\section{Synthesis with the classical cases}
In this section, we consider some classical cases in which Perelman's asymptotic shrinker exists. We shall see that, in all these cases, Assumption B is satisfied. Hence, Perelman's asymptotic shrinker is identical to Bamler's tangent flow at infinity. We consider an ancient solution $(M,g(t))_{t\in(-\infty,0]}$ with bounded curvature within each compact time interval, satisfying \emph{either one} of the following conditions:
\begin{enumerate}[(1)]
\item $g(t)$ satisfies a Type I curvature bound, that is, there is a constant $C$ such that $$|\Rm_{g(t)}|\leq\frac{C}{|t|}\quad\text{ for all }\quad t\in(-\infty,0).$$
\item $g(t)$ satisfies Hamilton's trace Harnack, that is, $$\frac{\partial R}{\partial t}+2\langle X,\nabla R\rangle+2\Ric(X,X)\geq 0\quad\text{ for all vector field } X,$$
and there is a constant $C$ such that
$$|\Rm|\leq CR\quad\text{ everywhere on }\quad M\times(-\infty,0].$$
\end{enumerate}
Except for the assumption above, we would also like to impose a $\kappa$-noncollapsing assumption. This is actually equivalent to
\begin{eqnarray*}
\mathcal{N}_{p_0,t_0}(\tau)\geq-Y\quad\text{ for all }\quad \tau>0,
\end{eqnarray*}
where $(p_0,t_0)$ is an arbitrarily fixed point on $M\times(-\infty,0]$.
\begin{Theorem}
Under the assumptions of this section, Perelman's asymptotic shrinker is identical to Bamler's tangent flow at infinity.
\end{Theorem}
\begin{proof}
Let us fixed a point $p_0\in M$ and a sequence $\tau_i\nearrow\infty$. Let $\ell$ be the reduced distance based at $(p_0,0)$ and let $\{(p_i,-\tau_i)\}_{i=1}^\infty$ be a sequence of $\ell$-centers of $(p_0,0)$. We will still use $\ell_i(\cdot,\tau):=\ell_i(\cdot,\tau_i\tau)$ and $g_i(t):=\tau_i^{-1}g(\tau_it)$ to denote the scaled reduced distances and the scaled Ricci flows. Note that $\ell_i$ is the reduced distance based at $(p_0,0)$ with respect to the flow $g_i$. Recall that by \cite{Per02} and \cite{N10}, the following formulas hold everywhere on $M\times(-\infty,0)$ in the barrier sense or in the sense of distribution
\begin{align}\label{nonsense7_1_0}
\big|\nabla\ell_i(\cdot,|t|)\big|^2+R_i(\cdot,t)\leq\frac{C\ell_i(\cdot,|t|)}{|t|},&
\\\label{nonsense7_1_1}
-2\frac{\partial\ell_i(\cdot,|t|)}{\partial t}+\big|\nabla\ell_i(\cdot,|t|)\big|^2-R_i(\cdot,t)+\frac{\ell_i(\cdot,|t|)}{|t|}=0.&
\end{align}
Combining (\ref{nonsense7_1_0}) and (\ref{nonsense7_1_1}), we have
\begin{eqnarray}\label{nonsense7_1_2}
\left|\frac{\partial\ell_i(\cdot,|t|)}{\partial t}\right|\leq\frac{C\ell_i(\cdot,|t|)}{|t|}.
\end{eqnarray}
Since $\ell_i(p_i,1)\leq\frac{n}{2}$, we may integrate (\ref{nonsense7_1_2}) and obtain that, for any $\varepsilon\in(0,1)$, there is a constant $C=C(\varepsilon)$, such that
\begin{eqnarray*}
\ell_i(p_i,|t|)\leq C\quad\text{ for all }\text t\in[-\varepsilon^{-1},-\varepsilon].
\end{eqnarray*}
It then follows from (\ref{nonsense7_1_0}) that for $t\in[-\varepsilon^{-1},-\varepsilon]$, all the functions $\ell_i(\cdot,|t|)$ have uniformly quadratic growth bounds around $p_i$, and all curvatures $|\Rm_{g_{i,t}}|$ have uniform growth bounds around $p_i$ (indeed, uniform upper bound in the Type I case). Hence, $(M,g(t))_{t\in(-\infty,0]}$ satisfies a stronger version of Assumption B, with the $[-2,-1]$ time interval in formula (\ref{smoothconvergence}) replaced by $[-\varepsilon^{-1},-\varepsilon]$ for any $\varepsilon\in(0,1)$, and we will let $(M_\infty,g_\infty(t),\ell_\infty)_{t\in(-\infty,0)}$ denote the asymptotic shrinker as given by Proposition \ref{shrinkerstructure}.
Let $(\nu^i_s)_{s\in(-\infty,0]}$ be the conjugate heat kernel based at $(p_0,0)$ on the Ricci flow $g_i$, then we have
\begin{eqnarray*}
\big((M,g_i(t))_{t\in(-\infty,0]},(\nu^i_s)_{s\in(-\infty,0]}\big)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\mathcal{X},
\end{eqnarray*}
where $\mathcal{X}$ is a conjugate flow pair. By the arguments in sections 5---7, we have that, for any $\varepsilon\in(0,1)$, it holds that
\begin{eqnarray*}
\mathcal{X}_{[-\varepsilon^{-1},-\varepsilon)}=\big((M_\infty,g_\infty(t))_{t\in[-\varepsilon^{-1},-\varepsilon)},(\nu^\infty_s)_{s\in[-\varepsilon^{-1},-\varepsilon)}\big),
\end{eqnarray*}
where $(\nu^\infty_s)_{s\in(-\infty,0)}$ is a conjugate heat flow on $(M_\infty,g_\infty)$ made of a shrinker potential. This then finishes the proof.
\end{proof}
\section{Ancient Ricci flows with bounded entropy and smooth tangent flows}
In this section we prove Theorem \ref{Thm_main_reciprocal}. We would like to emphasize again that the arguments in this section are true only if the results in \cite{Bam20c} (especially Theorem 1.6) are true for noncomapct Ricci flows with bounded geometry within each compact time interval (or if the ancient solution in question is on a closed manifold).
Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution with bounded geometry within each compact time interval. Let $p_0\in M$ be a fixed point and let $Y>0$ be a constant such that
\begin{eqnarray}\label{bddentropychpt10}
\mathcal{N}_{p_0,0}(\tau)>-Y\quad\text{ for all }\quad \tau>0.
\end{eqnarray}
Let $\{\tau_i\}_{i=1}^\infty$ be a sequence of positive numbers such that $\tau_i\nearrow\infty$, and we shall denote
\begin{eqnarray*}
g_i(t):= \tau_i^{-1}g(\tau_it),\quad \nu^i_t:=\nu_{p_0,0\,|\,\tau_it},\quad\text{ for all }\quad t<0.
\end{eqnarray*}
Let $\big((M_\infty,g_\infty(t))_{t\in(-\infty,0)},(\nu^\infty_s)_{s\in(-\infty,0)}\big)$ be the smooth tangent flow mentioned in the statement of the theorem, that is,
\begin{eqnarray*}
\big((M,g_i(t))_{t\in(-\infty,0]},(\nu^i_s)_{s\in(-\infty,0]}\big)\xrightarrow{\makebox[1cm]{$\mathbb{F}$}}\big((M_\infty,g_\infty(t))_{t\in(-\infty,0)},(\nu^\infty_s)_{s\in(-\infty,0)}\big).
\end{eqnarray*}
Since $(M_\infty,g_\infty(t))$ is smooth, we have, by \cite[Theorem 1.6]{Bam20c}, that the above convergence is smooth. Precisely, this means that one can find an increasing open sets in space-time $U_1\subset U_2\subset\cdots\subset M_\infty\times(-\infty,0)$ with $\cup_{i=1}^\infty U_i=M_\infty\times(-\infty,0)$, open subsets $V_i\subset M\times(-\infty,0)$, time-preserving diffeomorphisms $\psi_i:U_i\rightarrow V_i$, and a sequence $\varepsilon_i\searrow 0$, such that \cite[Theorem 9.31]{Bam20b} holds.
Let us then fix a point $x_\infty\in M_\infty$ and a positive radius $D$ (which could be very large), such that
$$\nu^\infty_{-1}\big(B_{g_{\infty,-1}}(x_\infty,D)\big)\geq \frac{1}{2}.$$
Note that this is possible since $\nu^\infty_{-1}$ is a probability measure. Since $\cup_{i=1}^\infty U_i=M_\infty\times(-\infty,0)$, we have that $$B_{g_{\infty,-1}}(x_\infty, D)\times\{-1\}\subset U_i\quad\text{ and } \quad \nu^i_{-1}\Big(\psi_{i,-1}\big(B_{g_{\infty,-1}}(x_\infty,D)\big)\Big)\geq\frac{1}{3}$$
for all $i$ large enough. Here (and below) $\psi_{i,-1}$ denotes the $t=-1$ time slice of $\psi_i$. Since $\psi_i$ is almost isometry, we can find $D_0\geq D$, such that
\begin{eqnarray*}
\nu^i_{-1}\big(B_{g_{i,-1}}(\psi_{i,-1}(x_\infty),D_0)\big)\geq \frac{1}{3}\quad\text{ for all $i$ large enough.}
\end{eqnarray*}
Hence, by Proposition \ref{measureaccumulationofHcenter}, letting $(z_i,-1)$ be an $H_n$-center of $(p_0,0)$ with respect to the Ricci flow $g_i(t)$, we have
\begin{eqnarray}\label{10nonsense00001}
{\rm dist}_{g_{i,-1}}\big(z_i,\psi_{i,-1}(x_\infty)\big)\leq D_0+\sqrt{3AH_n}\quad \text{ for all $i$ large enough}.
\end{eqnarray}
By Proposition \ref{H_n_l_n} and (\ref{bddentropychpt10}), we can find a constant $C$ depending only on $Y$, such that
\begin{eqnarray}\label{10nonsense00002}
{\rm dist}_{g_{i,-1}}(z_i,p_i)\leq C\quad \text{ for all $i$ large enough},
\end{eqnarray}
where $(p_i,-1)$ is an $\ell$-center of $(p_0,0)$ with respect to the Ricci flow $g_i(t)$.
Combining (\ref{10nonsense00001}) and (\ref{10nonsense00002}), we have
\begin{eqnarray*}
{\rm dist}_{g_{i,-1}}\big(\psi_{i,-1}(x_\infty),p_i\big)\leq C\quad \text{ for all $i$ large enough}.
\end{eqnarray*}
Arguing in the same way as in the proof of Proposition \ref{LUTypeI}, we have that $(M,g(t))$ is locally uniformly Type I along $(p_i,-\tau_i)$, and the smooth Cheeger-Gromov-Hamilton limit of $\{(M,g_i(t),p_i)_{t\in[-2,-1]}\}_{i=1}^\infty$ is an asymptotic shrinker in the sense of Perelman. Obviously, this limit must be $(M_\infty,g_\infty(t))_{t\in[-2,-1]}$.
Next, we shall show that $\nu^\infty_t$ is a conjugate heat flow made of a shrinker potential. Because of (\ref{bddentropychpt10}), \cite[Proposition 6.1]{Bam20c} (see Proposition \ref{almostselfsimilar} above) implies that, there is a sequence $\delta_i\searrow 0$, such that
\begin{eqnarray}\label{selfsimilar}
\int_{-\delta_i^{-1}}^{-\delta_i}\int_M|t|\left|\,\Ric_i+\nabla^2f_i-\frac{1}{2|t|}g_i\,\right|^2d\nu^i_tdt<\delta_i,
\end{eqnarray}
where $$d\nu^i_t:=(4\pi|t|)^{-\frac{n}{2}}e^{-f_i}dg_{i,t}.$$
Since, by \cite[Theorem 9.31]{Bam20b}, the conjugate heat kernel and the Ricci flow converge smoothly along the sequence, we may take a limit for (\ref{selfsimilar}) and apply Fatou's lemma to obtain
\begin{eqnarray*}
\int_{-\delta^{-1}}^{-\delta}\int_M|t|\left|\Ric_\infty+\nabla^2f_\infty-\frac{1}{2|t|}g_\infty\right|^2d\nu^\infty_tdt=0, \quad\text{ for any }\quad \delta\in(0,1),
\end{eqnarray*}
where $$d\nu^\infty_t:=(4\pi|t|)^{-\frac{n}{2}}e^{-f_\infty}dg_{\infty,t}.$$
This finishes the proof Theorem \ref{Thm_main_reciprocal}.
\begin{comment}
\section{Concluding remarks}
\subsection{An alternative version of Assumption B}
In the statement of Assumption B, we have assumed $(p_i,-\tau_i)$ to be $\ell$-centers of $(p_0,0)$. However, given what we have established in this article, one can replace the $\ell$-centers by $H_n$-centers, in other words, we may assume the following in place of Assumption B.
\\
\noindent\textbf{Assumption B':} Let $\tau_i\nearrow\infty$ and let $(z_i,-\tau_i)$ be $H_n$-centers of $(p_0,0)$. Then there is a smooth Ricci flow $\big(M_\infty,g_\infty(t),z_\infty\big)_{t\in[-2,-1]}$, such that
\begin{eqnarray*}
\big(M,g_i(t),z_i\big)_{t\in[-2,-1]}\xrightarrow{\makebox[1cm]{}} \big(M_\infty,g_\infty(t),z_\infty\big)_{t\in[-2,-1]}
\end{eqnarray*}
in the smooth Cheeger-Gromov-Hamilton sense, where the Ricci flow $g_i(t)$ is obtained by the following Type I scaling
\begin{eqnarray*}
g_i(t):=\tau_i^{-1}g(\tau_it).
\end{eqnarray*}
\bigskip
Indeed, Assumption B and Assumption B' imply each other by the following argument:
\begin{enumerate}[(1)]
\item Assuming either one of Assumption B and Assumption B', one can prove that $$\mathcal{N}_{p_0,0}(\tau)\geq-Y\quad\text{ for all }\quad \tau>0$$ where $Y$ is a constant.
\item By part (1), one may prove that the $\ell$-centers and $H_n$-centers are not far from each other, i.e., Proposition \ref{H_n_l_n}.
\item Then, the other one of these two assumptions follows.
\end{enumerate}
\subsection{A property of the $H_n$-center}
Let $(M,g(t))_{t\in I}$ be a Ricci flow with bounded geometry within each compact time interval. Let us fix $x\in M$ and $s,t\in I$ with $s<t$. Let $(z,s)$ be an $H_n$-center of $(x,t)$ and $r\in(0,\sqrt{t-s})$. Then
\begin{enumerate}[(1)]
\item Let $(z',s-r^2)$ be another $H_n$-center of $(x,t)$. Then the distance between $z$ and $z'$ can be estimated using the geometry bounds on $B_s(z,r)\times[s-r^2,s]$.
\item Let $(z',s+r^2)$ be another $H_n$-center of $(x,t)$. Then the distance between $z$ and $z'$ cannot be estimated using the geometry bounds on $B_s(z,r)\times[s,s+r^2]$ alone.
\end{enumerate}
This phenomenon is somewhat similar to the triangle inequality of the reduced distance. If we fix $t_1<t_2<t_3$ and $x,y,z\in M$, then, the concatenation of a minimal $\mathcal{L}$-geodesic from $(z,t_3)$ to $(y,t_2)$ and
\end{comment}
\def \mathcal{M} {\mathcal{M}}
|
2,877,628,090,763 | arxiv | \section{Introduction}
The excellent spatial resolution of $Chandra$ X-ray Observatory has opened a new era to study the large scale jets in powerful extragalactic radio sources. At the time of this writing, more than 40 radio-loud AGNs are known to possess X-ray counterparts of radio jets on kpc to Mpc scales (Harris \& Krawczynski 2002, Stawarz 2004 and references therein; see also \texttt{http://hea-www.harvard.edu/XJET/}). Bright X-ray knots (hereafter ``jet-knots'') are most often detected, but the X-ray emissions from the hotspots and radio lobes are also reported in a number of FR II radio galaxies and quasars (e.g., Wilson, Young \& Shopbell 2000; 2001; Hardcastle et al. 2002b; 2004, Tashiro et al. 1998; Isobe 2002).
The broad-band spectra of jet-knots, hotspots, and lobes detected by $Chandra$ show great variety between radio and X-ray energy bands. In nearby FR I sources, typical X-ray-optical-radio spectrum of the jet-knots is consistent with a single smoothly broken power-law continuum, suggesting that this broad-band emission is entirely due to non-thermal synchrotron radiation from a single electron population (e.g., Marshall et al. 2002 and Wilson \& Yang 2002 for M~87). In most other sources, however, the X-ray knots' spectra are much harder than expected from a simple extrapolation of the radio-to-optical fluxes. These situations are believed that both the radio and optical emissions are due to synchrotron radiation, whereas X-ray photons are produced via the inverse-Compton scattering of either synchrotron photons (SSC) or cosmic microwave background photons (EC; Tavecchio et al. 2000, Celotti, Ghisellini \& Chiaberge 2001). Other (synchrotron) models have been also proposed to explain intense X-ray emission of the large-scale quasar jets (e.g., Dermer \& Atoyan 2002, Stawarz \& Ostrowski 2002). In the case of the hotspots in powerful sources one finds an analogous controversy regarding the X-ray emission: although in many objects this emission is consistent with the standard SSC model (see, e.g., Wilson et al. 2000 for Cygnus~A), in some other sources it cannot be simply explained in this way, suggesting most likely a synchrotron origin of the detected X-ray photons (see, e.g., Hardcastle et al. 2004). For the extended lobes of quasars and FR IIs the X-ray radiation is established to be produced by the EC process involving CMB target radiation. In some cases, however, infrared target photons from quasar cores may contribute to the inverse-Compton lobes' emission at keV photon energy range (Brunetti, Setti \& Comastri 1997).
In the standard picture of FR II radio galaxies and quasars, the
relativistic jet is decelerated in a hotspot converting part of its
energy into relativistic electrons and part in magnetic field. Then the
shocked plasma moves inside the head region just behind the hotspot, and
expands almost adiabatically to form diffuse, extended radio lobes. Even
though this picture appears to be simple, much of the fundamental
physics behind it remains unclear (see, e.g., recent monograph by De
Young 2002a). For example, the velocity and dynamics of the large-scale
jets is unknown. From the analogy to sub-pc jets in blazar-type AGNs, it
is plausible that some of the FR II and quasar jets are highly
relativistic even on kpc/Mpc scales. Recent studies on the optical
emission of the large-scale jets seem to justify this hypothesis (e.g.,
Sparks et al. 1995, Scarpa \& Urry 2002, Jester 2003), and the usually
discussed versions of the EC model for the X-ray jet-knots indeed
require the jet bulk Lorentz factors $\Gamma_{\rm BLK} \geq 10$ (e.g.,
Harris \& Krawczynski 2002). Yet, the exact velocity structure both
along and across large-scale jets in FR II radio galaxies and quasars
remains an open issue. The strong terminal shocks at the hotspots are
unlikely to be moving with high bulk Lorentz factors, but moderately
relativistic motions ($\Gamma_{\rm BLK}$ $\le$ a few) are permitted by
hydrodynamic simulations (e.g., Aloy et al. 1999). We note, that such
simulations repeatedly reveal a complex hotspots' morphology, especially
at the late stages of the jet evolution (e.g., Marti et al. 1997,
Mizuta, Yamada \& Takabe 2004). Finally, the main-axis expansion of
radio lobe is thought to be sub-relativistic;
$\Gamma_{\rm BLK}$ $\simeq$ 1. However, detailed transport and spatial
distribution of the radiating particles within the lobes of powerful
radio sources is still being debated (e.g., Blundell \& Rawlings 2000,
Kaiser 2000, Manolakou \& Kirk 2003).
As for the velocity of jet plasma, the strength of magnetic field in radio galaxies is an open matter. Assuming an equipartition field value in the lobes (1$-$10 $\mu$G), which seems to be supported by the X-ray lobes' observations, a simple flux conservation argument predicts the magnetic field in the jets as high as 0.01$-$1 G (De Young 2002b). Such a strong magnetic field is problematic, since numerical simulations of Poynting-flux dominated jets (e.g., Komissarov 1999) cannot correctly reproduce the observed large-scale morphologies of powerful radio sources. Thus, an amplification of the magnetic field to the equipartition value in strong jet terminal shock and in its turbulent downstream region is required, although only little theoretical investigations of this issue has been reported (see De Young 2002b). Let us mention in this context, that turbulent processes that may lead to amplification of the magnetic field can manifest in formation of the flat-spectrum synchrotron X-ray features, such like the ones discovered recently in the hotspots of Cygnus A radio galaxy (Ba\l uci\'nska-Church et al. 2004). On the other hand, the equipartition of energy between the magnetic field and the radiating electrons, established for some high-luminosity sources, may not be valid in general, especially in the case of low-luminosity hotspots (Hardcastle et al. 2004). Finally, we note that the configuration of the magnetic field within the lobes is also not well understood (see a discussion in Blundell \& Rawlings 2000).
Unfortunately, present radio-to-X-ray observations are not sufficient to discriminate conclusively between different models proposed in order to explain multiwavelength emission of the large-scale structures of powerful radio sources, and of their kpc/Mpc jets in particular. However, we believe that a systematic comparison between broad-band radiative properties of the jet-knots, hotspots, and lobes will provide important clues to dynamics and the physics of large scale jets, and to put some constraints on the models discussed in the literature. Keeping these motivations in mind, the purpose of this paper is to obtain a rough, but unified picture which may link the jet-knots, hotspots and radio lobes, rather than modeling individual sources in a sufficiently detailed manner. Obviously, detailed studies on individual cases are irreplaceable. In fact, many controversial issues briefly touched in this analysis will remain open until such detailed investigations, based on long multiwavelength observations, are performed. We emphasize, that our analysis confirms many results known from the literature (see, e.g., Stawarz 2004 for a review), although for a large number of sources modeled in addition in a uniform way. Basing on this homogeneous approach, we explore however some new, hardly discuss in the literature aspects of the physics behind the X-ray emission models for the considered objects. Let us also mention, that in this paper we do not consider hadronic models for the broad-band emission of the large-scale jets and their hotspots (see, e.g., Aharonian 2002, Atoyan \& Dermer 2004).
Our presented study is based on data analysis for a sample consisting of 26 radio galaxies, 14 quasars, and 4 blazars. We collected all existing data at well sampled radio (5 GHz) and X-ray (1 keV) frequencies and analyzed them in a systematic manner. In $\S$2, we defined sample selection and observables used in this paper. In $\S$3, we presented a simple formulation of calculating the ``expected'' X-ray flux densities for the SSC and EC models taking the relativistic beaming effect into account. We then compared the physical quantities (beaming factor and the magnetic field) of the jet-knots, hotspots, and lobes. In $\S$4 we discuss the results and the summary is presented in $\S$5.
\section{Data and analysis}
\subsection{Sample}
Table 1 compiles a list of ``X-ray jet sources'' in which jet-knots,
hotspots and/or radio lobes are detected by $Chandra$ and $ASCA$. The
first pioneer work have been reported by Harris \& Krawczynski (2002),
where the emission mechanisms of 18 X-ray jet sources (mainly jet-knots)
are discussed in the framework of relativistically moving jet
model. They continue to maintain current information at
\texttt{http://hea-www.harvard.edu\\
/XJET/}, which conveniently summarize
the name, coordinate, distance, and morphology of the X-ray jet
sources. Our sample contains all of the sources listed in this page,
with additional information on the X-ray observations of radio lobes
mainly organized by $ASCA$.
Before compiling the data, we have performed quick re-analysis of $Chandra$ data (if already archived) to check the published results, and found no discrepancy. We therefore refer to published results (fluxes and spectral indices) unless otherwise stated in this paper. This gives a large number of objects known to us as of 2004 June, which contains 44 X-ray jet sources (56 jet-knots, 24 hotspots, and 18 radio lobes: see Table 2). We are aware that our sample is still incomplete, as the known X-ray jet sources are increasing their number day by day. Nevertheless such a list provides a convenient overview of X-ray jet sources detected so far, and provide a useful hint to $predict$ fluxes of unobserved X-ray jet sources. We also note that Hardcastle et al. (2004) recently summarizes the X-ray emission properties of the hotspots in FR II radio galaxies.
The basic information about each source are listed in Table 1: (1) source name, (2) redshift $z$, (3) luminosity distance to the source $d_{\rm L}$ adopting $H_0$ = 75 km s$^{-1}$ Mpc$^{-1}$ and $q_0$ = 0.5, (4) classification, and (5) references. RG denotes radio galaxy of either Fanaroff \& Riley type I (FR I) or type II (FR II), QSO denotes either core dominated (CD) quasars or lobe dominated (LD) quasars, and BLZR denotes blazar-class.
More detailed information on each source are listed in Table 2. In the second column we denote ``knot (K)'' to indicate a distinct structure in the jet, ``hotspots (HS)'' as a terminal bright enhancement at an end of the FR II jet or as one of the multiple features associated with a termination of the jet, and ``lobe (L)'' as a diffuse extended structure associated with a radio lobe. A suffix after K, HS, and L means the identification of each structure. For example ``K-A'' denotes `knot-A'' and ``HS-SE'' means ``South-east hotspot''. In succeeding columns of Table 2, we have listed 6 observables: (1) $\alpha_{\rm R}$: radio spectral index measured at 5 GHz, (2) $f_{\rm R}$: radio flux density at 5 GHz in mJy, (3) $\alpha_{\rm X}$: X-ray spectral index at 1 keV, (4) $f_{\rm X}$: X-ray flux density at 1 keV in nJy, (5) $f_{\rm O}$: optical flux density at 5$\times$10$^{14}$ Hz in $\mu$Jy, and (6) $\theta$: radial size of the emitting region in arcsec. When observations have not been reported at 5 GHz or 1 keV, we calculate the flux by extrapolating the nearest measured frequency by assuming the spectral index listed in the table. A suffix $f$ means that we have assumed the fixed value for this calculation.
\subsection{Radio/X-ray comparison}
Figure 1 shows the distribution of the spectral indices in the radio band ($\alpha_{\rm R}$; $upper$) and in the X-ray band ($\alpha_{\rm X}$; $lower$), respectively. Note that radio spectral index shows a relatively narrow distribution centered at 0.8 and there is no clear difference between the jet-knots, hotstpots and radio lobes. As is widely believed, the radio emissions of these sources are most likely due to the synchrotron radiation from the low-energy population of relativistic electrons. In other words, energy index of accelerated electron is narrowly distributed around $s$ = ($\alpha_{\rm R}$ +1)/2 $\simeq$ 2.6, which is slightly steeper than the one expected from a diffusive acceleration at nonrelativistic shocks, $s$ = $2$. Let us note in this context, that analytical and numerical studies of particle acceleration at relativistic shocks (reviewed by, e.g., Kirk \& Duffy 1999 and Ostrowski 2002), indicate that in such a case one can expect variety of particle spectra, with the asymptotic power-law inclination $s$ = $2.2$ for the strong turbulence condition and ultrarelativistic shock velocity. We also note, that stochastic second-order Fermi processes do not favor any universal value of the power-law spectral index characterizing accelerated electrons.
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig1.eps}
\caption{Distribution of the energy index measured at 5 GHz and at 1 keV.}
\end{figure}
Meanwhile, the X-ray energy index, $\alpha_{\rm X}$, is widely distributed from 0.2 to 1.6. Part of the reason may be relatively large uncertainties in determining the spectral shape of faint X-ray sources compared to the radio spectral shape, but even if only bright (i.e., small error bars) X-ray sources are plotted, the same trend is obtained. Steep X-ray sources are most frequently found in nearby FR I radio galaxies. As discussed in the literature, the X-ray fluxes in these sources may smoothly connect with radio/optical fluxes and hence are considered to be the highest energy tail of the synchrotron radiation. For the X-ray emission from other jet-knots the situation is less clear. Flat X-ray spectral indices may indicate pile-up effects at the high-energy part of the electron energy distribution, advocating thus synchrotron origin of the keV photons (see Harris, Mossman \& Walker 2004), or, oppositely, spectral flattenings occurring at the low-energy part of the electron continuum thus being consistent with the EC interpretation of the X-ray knots' emission. Clearly, spectral information alone are not sufficient at the moment to distinguish either between synchrotron and inverse-Compton origin of the keV photons from the jet-knots in most of the cases, or to indicate the appropriate particle acceleration process.
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig2.eps}
\caption{Distribution of the ratio between $L_{\rm 1keV}$ and $L_{\rm 5GHz}$.}
\end{figure}
Figure 2 presents the distribution of luminosity ratio of $L_{\rm R}$
and $L_{\rm X}$, where $L_{\rm R}$ = 4$\pi$$d_{\rm L}^2$$f_{\rm
R}$$\nu_{\rm R}$ and $L_{\rm X}$ = 4$\pi$$d_{\rm L}^2$$f_{\rm
X}$$\nu_{\rm X}$, respectively. Note that a clear difference can be seen between the jet-knot and hotspot or radio lobes. The jet-knots tend to be much brighter in X-rays compared to the hotspots and radio lobes. This trend is seen more clearly in Figure 3, where the correlation between $L_{\rm R}$ and $L_{\rm X}$ is plotted in two dimensional space. One finds several important tendencies which cannot be accounted by the sampling bias effect. First, hotspots and radio lobes occupy only the high-luminosity part of the plot, namely $\ge$ 10$^{40}$ erg s$^{-1}$. Secondly, low-luminosity hotspots tend to be brighter in X-ray, as has been pointed out by Hardcastle et al. (2004). Thirdly, $L_{\rm R}$ $\ge$ $L_{\rm X}$ for most of the hotspots and radio lobes, but most of the jet-knots show an opposite trend.
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig3.eps}
\caption{Relation between the luminosities $L_{\rm 5GHz}$ and $L_{\rm 1keV}$.}
\end{figure}
We should note that due to limited sensitivity of $Chandra$ (typically 0.1 nJy at 1 keV for 10 ksec exposure), we would not expect to detect the X-ray emission from the ``X-ray faint'' jet-knots. Therefore the lack of the X-ray faint (i.e., $L_{\rm R}$ $\ge$ $L_{\rm X}$) jet-knots at the bottom left corner of Figure 3 would be biased by the sensitivity of $Chandra$ detector. In fact, we can find a few X-ray faint jet-knots at the top right corner, where the luminosity is the highest. However, even if only high-luminosity sources are selected, we can see a clear difference between the jet-knots, hotspots and radio lobes, namely ``X-ray bright'' sources are found only in jet-knots. Apparently, this is not due to the sampling effect since we certainly would have been able to detect ``X-ray bright'' hotspots if they existed.
\section{Model application to data}
In this section we present a simple formulation of computing an equipartition magnetic field strength $B_{\rm eq}$ from an observed radio flux $f_{\rm R}$ measured at a radio frequency $\nu_{\rm R}$. Next, we calculate the ``expected'' SSC and EC luminosities for $B_{\rm eq}$, to compare them with the observed X-ray luminosities. In the analysis, we include possible relativistic bulk velocity of the jet plasma. Taking the obtained results into account, and analyzing additionally the observed broad-band spectral properties of the compiled sources (including optical fluxes), we follow the ``conservative'' classification of the compiled X-ray sources into three groups, namely (i) synchrotron involving single/broken power-law electron energy distribution (SYN), (ii) synchrotron self-Compton (SSC) and (iii) external Compton of CMB photons (EC). Finally, we discuss the validity of the applied classification scheme, and compare it briefly with the classification introduced in the literature.
\vspace{5mm}
\subsection{Equipartition magnetic field}
In order to determine the X-ray emission properties of the large-scale jets, we first estimated the magnetic field strength under the minimum-power hypothesis using the observed radio luminosities measured at 5 GHz. As we have reviewed in $\S$1, it is generally believed that the magnetic field energy density $u_{\rm B}$ and the particle energy density $u_{\rm e}$ may be close to the equipartition in a number of radio sources. Therefore our approach is that we first assume an equipartition to calculate ``predicted'' inverse-Compton X-ray luminosities, and then compare them to the observed ones. If a large discrepancy occurs, this may suggest that equipartition is strongly violated, that the inverse-Compton origin of the observed keV photons is not the case, or that we have to consider another correction factor, such as Doppler beaming factor $\delta$, as we will discuss below.
Since the synchrotron luminosity, $L_{\nu}$, is proportional to $u_{\rm e}$$u_{\rm B}$$V$, where $V$ is the volume of the emitting regions, we can estimate the equipartition magnetic field $B_{\rm eq}$ for a given luminosity observed at a radio frequency $\nu$. Under the assumption of $no$ relativistic beaming ($\delta$ = 1), $B_{\rm eq}$ is expressed as
\begin{equation}
\begin{aligned}
B_{\rm eq, \, \delta=1} = \left[ \frac{3 \mu_0}{2} \frac{G(\alpha) \eta L_\nu}{V} \right]^{2/7}\\
\propto \left( \frac{\eta L_\nu}{V} \right)^{2/7} \nu^{1/7} \quad ,
\end{aligned}
\end{equation}
\noindent
where $\mu_0$ is permeability of free space, $G(\alpha)$ is a function given in Longair (1994), $\alpha$ is the spectral energy index and $L_{\nu}$ is the synchrotron luminosity measured at a frequency $\nu$. $\eta$ is the ratio of energy density carried by proton and electrons to the energy density of the electrons, i.e., $\eta$ = 1 for the leptonic ($e^{-}$$e^{+}$) jet and $\eta$ = 1836 for the hadronic ($e^{-}$$p^{+}$) jet for which the ratio of proton to electron energy densities equals the ratio of their rest masses. In the last approximation in the equation (1) we put minimum synchrotron frequency $\nu_{\rm min}$ = $\nu$ and $\alpha$ = 0.75. The latter choice is justified by a narrow distribution of the radio spectral indices in the compiled dataset (Figure 1, $upper$). The former choice gives the $minimum$ value of $B_{\rm eq}$ for the observed $L_{\nu}$ at some given frequency $\nu$. Below we consider $\nu$ = $\nu_{\rm R}$ = 5 GHz, although it is obvious that the minimum radio frequency has to be lower than this (especially in the case of the EC model, which requires presence of low-energy electrons with energies below GeV). However, the difference between equipartition magnetic field computed for $\nu_{\rm min}$ = $\nu$ and for $\nu_{\rm min}$ $\neq$ $\nu$, respectively, is rather small, $\propto$ $(\nu_{\rm min} / \nu)^{1/14}$. In addition, the expected spectral flattenings at the low-energy part of the synchrotron continuum are likely to make this difference even smaller.
In general, an emission volume, $V$, is quite uncertain for astrophysical sources due to the limited angular resolution of detectors and projection effect. We have assumed that the emitting region has a spherical volume of a certain angular radius $\theta$[''] for all the jet structures. This is obviously an over-simplified assumption, however it significantly reduces the complexity of the models. Most of the jet-knots and hotspots are point-like sources when observed with $Chandra$. We therefore set an upper limit of $\theta$ = 0.3'', unless there are additional radio/optical observations obtained with better angular resolution. Meanwhile a number of radio lobes show extended structures, but morphology is more complicated than a homogeneous sphere. We therefore calculated the volume by assuming a cylinder or a rotational ellipse, and then approximated it as a sphere which has an equal volume.
For a relativistically moving plasma, the equipartition magnetic field measured in the emitting plasma rest frame is related to the equipartition value computed for no beaming by the relation (Stawarz, Sikora \& Ostrowski 2003)
\begin{equation}
\begin{aligned}
B_{\rm eq} = B_{\rm eq, \, \delta =1} \delta^{-5/7}.
\end{aligned}
\end{equation}
\noindent
The above expression can be more conveniently written as
\begin{equation}
\begin{aligned}
B_{\rm eq} = 123 \hspace{3mm} \eta^{2/7} (1+z)^{11/7} \left( \frac{d_{\rm L}}{\rm 100 Mpc} \right)^{-2/7}\\
\times\left( \frac{\nu_{\rm R}}{\rm 5GHz} \right)^{1/7}
\left( \frac{f_{\rm R}}{100{\rm mJy}} \right)^{2/7}
\left( \frac{\theta}{0.3''} \right)^{-6/7}\\
\times\delta^{-5/7} {\rm [\mu G]},
\end{aligned}
\end{equation}
\noindent
where $d_{\rm L}$ is the luminosity distance to the source, and $f_{\rm R}$ is the observed radio luminosity measured at frequency $\nu_{\rm R}$. $B_{\rm eq, \, \delta=1}$ for various jet sources are calculated in Table 2 for $\eta = 1$, what gives again the $minimum$ value of $B_{\rm eq, \, \delta = 1}$. We note that this particular choice does not refer exclusively to the leptonic jet model. For example, it may refer to the case of energy equipartition between jet magnetic field and radiating electrons solely. We note, that the discussion on the jet composition is still open, and the situation may be quite complex as, for example, the jet can be composed predominantly from the $e^{-}$$e^{+}$-pairs, but still remain dynamically dominated by the (cold) hadrons (see Sikora \& Madejski 2000).
\vspace{5mm}
\subsection{Synchrotron (SYN) case}
We first consider the case where the X-ray emissions are due to the synchrotron radiation emitted by the electrons with the Lorentz factor $\gamma_{\rm X}$. We assume that the magnetic field in the jet moving plasma is close to equipartition $B_{\rm eq}$, and its relativistic beaming factor is $\delta$. Then the observed X-ray frequency is given by
\begin{equation}
\begin{aligned}
\nu_{\rm X} \simeq 1.2 \times 10^6 \gamma_{\rm X}^2 B_{\rm eq} (1+z)^{-1}
\delta\\
\simeq 1.2 \times 10^6 \gamma_{\rm X}^2 B_{\rm eq, \delta=1} (1+z)^{-1}
\delta^{2/7}.
\end{aligned}
\end{equation}
\noindent
The respective electron Lorentz factor, $\gamma_{\rm X}$, is hence given as
\begin{equation}
\begin{aligned}
\gamma_{\rm X} \simeq 4.5 \times 10^7 \left( \frac{\nu_{\rm X}}{\nu_{\rm 1keV}} \right)^{1/2} \left( \frac{B_{\rm eq, \delta=1}}{100 {\rm \mu G}} \right)^{-1/2}\\
\times(1+z)^{1/2} \delta^{-1/7},
\end{aligned}
\end{equation}
\noindent
where ${\nu_{\rm 1keV}}$ is 2.4$\times$10$^{17}$ Hz. Therefore, even though $\delta$ is quite uncertain, the estimated value of $\gamma_{\rm X}$ is not affected significantly since $\gamma$ is only weakly dependent on $\delta$, i.e., $\propto$ $\delta^{-1/7}$.
\vspace{5mm}
\subsection{Synchrotron self-Compton (SSC) emission}
The observed radio frequency is approximately
\begin{equation}
\begin{aligned}
\nu_{\rm R} \simeq 1.2 \times 10^6 B_{\rm eq} \gamma_{\rm R}^2 (1+z)^{-1} \delta ,\\
\end{aligned}
\end{equation}
\noindent
where $\gamma_{\rm R}$ is a Lorentz factor of electrons which emit synchrotron photons at $\nu_{\rm R}$. In the SSC case, electrons upscatter synchrotron photons to a frequency
\begin{equation}
\begin{aligned}
\nu_{\rm IC} \simeq \frac{4}{3} \gamma_{\rm R}^2 \nu_{\rm R} = 2.8 \times10^{17} \left( \frac{\nu_{\rm R}}{\rm {5 GHz}} \right)^2\left( \frac{B_{\rm eq}}{100{\rm \mu G}} \right)^{-1}\\
\times(1+z) \delta^{-1} \quad {\rm [Hz]} \\
= 2.3 \times 10^{17} \eta^{-2/7} (1+z)^{-4/7} \left( \frac{d_{\rm L}}{\rm 100 Mpc} \right)^{2/7}\\
\times\left(\frac{\nu_{\rm R}}{\rm 5GHz} \right)^{13/7}
\left( \frac{f_{\rm R}}{100 {\rm mJy}} \right)^{-2/7}\left( \frac{\theta}{0.3''} \right)^{6/7}\\
\times\delta^{-2/7} \quad {\rm [Hz]}.
\end{aligned}
\end{equation}
\noindent
Note that, $\nu_{\rm IC}$ depends both on the observed radio frequency
and magnetic field strength $B_{\rm eq}$. To calculate the X-ray flux at
an observed frequency $\nu_{\rm X}$, we have to extrapolate the
inverse-Compton flux calculated for $\nu_{\rm IC}$ by assuming the
observed X-ray spectral index $\alpha_{\rm X}$. In the SSC case, we
expect $\alpha_{\rm X}$ $\simeq$ $\alpha_{\rm R}$, if the synchrotron
continuum can be well approximated by a single power-law with $\alpha$
$\simeq$ $\alpha_{\rm R}$. The ratio of X-ray (SSC) luminosity to the
radio (synchrotron) luminosity is therefore
\begin{equation}
\begin{aligned}
\frac{\nu_{\rm IC} f_{\rm IC}}{\nu_{\rm R} f_{\rm R}} \simeq \frac{\nu_{\rm X} f_{\rm X}}{\nu_{\rm R} f_{\rm R}} \left( \frac{\nu_{\rm IC}}{\nu_{\rm X}} \right)^{1 - \alpha_{\rm R} } \simeq \frac{u'_{\rm sync}}{u'_B},
\end{aligned}
\end{equation}
\noindent
where $u'_{\rm sync}$ and $u'_B$ are the synchrotron photon energy density and the magnetic field density, respectively, both evaluated in the emitting region rest frame denoted by primes. $u'_{\rm sync}$ is given by
\begin{equation}
\begin{aligned}
u'_{\rm sync} = \frac{d_{\rm L}^2 \nu_{\rm R} f_{\rm R}}{R^2 c \delta^4} = 7.9 \times 10^{-13} (1+z)^4 \left( \frac{\nu_{\rm R}}{5 {\rm GHz}} \right)\\
\times \left( \frac{f_{\rm R}}{100 {\rm mJy}} \right) \left(\frac{\theta}{0.3''} \right)^{-2} \delta^{-4} \quad \rm{[erg/cm^3]},
\end{aligned}
\end{equation}
\noindent
if we assume that the emission regions (jet-knots) are $moving$ sources (see a discussion in Stawarz et al. 2004). From the equations (7)$-$(9), we predict the SSC flux density measured at $\nu_{\rm X}$ to be roughly
\begin{equation}
\begin{aligned}
f_{\rm X} = 2.8 \times 10^{-3} \eta^{-1/2} (1+z) \left( \frac{d_{\rm L}}{\rm 100 Mpc} \right)^{1/2}\\
\times\left( \frac{\nu_{\rm R}}{\rm 5GHz} \right)^{5/4} \left( \frac{\nu_{\rm X}}{\nu_{\rm 1keV}} \right)^{-3/4}\left( \frac{f_{\rm R}}{100{\rm mJy}} \right)^{3/2}\\
\times \left( \frac{\theta}{0.3''} \right)^{-1/2} \delta^{-5/2} \quad {\rm [nJy]}.
\end{aligned}
\end{equation}
\noindent
Here we have assumed $\alpha_{\rm R}$ = 0.75 taking the result of Figure 1 into account. Note that $f_{\rm X}$ goes as $\propto$ $\delta^{-5/2}$, meaning that the SSC flux significantly $decreases$ as the beaming factor increases. Note also that $f_{\rm X}$ $\propto$ $\theta^{-1/2}$, i.e. that for a given $f_{\rm R}$ and $B$ = $B_{\rm eq}$ clumping of the emission region leads to the $increase$ of the SSC X-ray flux.
\vspace{5mm}
\subsection{External Compton (EC) emission on CMB photon field}
Similarly to the SSC case, we can estimate the expected EC flux at a certain X-ray frequency $\nu_{\rm X}$. In the EC model, electrons upscatter CMB photons to frequencies peaked at $\nu_{\rm IC}$, which, in a Thomson regime, is simply
\begin{equation}
\begin{aligned}
\nu_{\rm IC} \simeq \frac{4}{3} \gamma_{\rm R}^2 \nu_{\rm CMB} (1+z)^{-1} \delta^2 \kappa = 7.3 \times 10^{18} \eta^{-2/7}\\
\times\kappa (1+z)^{-4/7} \left( \frac{d_{\rm L}}{\rm 100 Mpc} \right)^{2/7}\left(\frac{\nu_{\rm R}}{\rm 5GHz} \right)^{6/7}\\
\times\left( \frac{f_{\rm R}}{100 {\rm mJy}} \right)^{-2/7} \left( \frac{\theta}{0.3''} \right)^{6/7}\delta^{12/7} \quad {\rm [Hz]},
\end{aligned}
\end{equation}
\noindent
where $\kappa$ = (1 + $\mu$)(1 + $\beta)^{-1}$ and $\nu_{\rm CMB}$ =
1.6$\times$10$^{11}$(1+$z$) [Hz]. Here we may set $\kappa$ $\simeq$ 1
for simplicity, since the value of $\kappa$ is always an order of unity
for any choice of $\Gamma_{\rm BLK}$ and $\delta$. The ratio of X-ray
(EC) luminosity to the radio (synchrotron) luminosity is approximately
given by (Stawarz et al. 2003)
\begin{equation}
\begin{aligned}
\frac{\nu_{\rm IC} f_{\rm IC}}{\nu_{\rm R} f_{\rm R}} \simeq \frac{\nu_{\rm X} f_{\rm X}}{\nu_{\rm R} f_{\rm R}} \left( \frac{\nu_{\rm IC}}{\nu_{\rm X}} \right)^{1 - \alpha_{\rm R}} \simeq \frac{u_{\rm CMB}}{u'_B} \kappa^2 (1+z)^4 \delta^2,
\end{aligned}
\end{equation}
\noindent
where $u_{\rm CMB}$ = 4.1$\times$10$^{-13}$ erg/cm$^3$. From the equations (11) and (12), the EC flux density measured at $\nu_{\rm X}$ can be expressed as
\begin{multline}
\begin{aligned}
f_{\rm X} = 5.9 \times 10^{-4} \kappa^{7/4} \eta^{-1/2} (1+z)
\left( \frac{d_{\rm L}}{\rm 100 Mpc} \right)^{1/2}\\
\times\left( \frac{\nu_{\rm R}}{\rm 5GHz} \right)^{1/2} \left( \frac{\nu_{\rm
X}}{\nu_{\rm 1keV}} \right)^{-3/4}
\times\left( \frac{f_{\rm R}}{100{\rm mJy}} \right)^{1/2}\\
\times\left( \frac{\theta}{0.3''} \right)^{3/2} \delta^{3} \quad {\rm [nJy]} .
\end{aligned}
\end{multline}
\noindent
for $\alpha_{\rm R}$ = 0.75. It is interesting to note that $f_{\rm X}$ goes as $\propto$ $\delta^{3}$, meaning that the EC flux significantly $increases$ as the beaming factor increases, which is the exact opposite trend in the SSC case. Note also, that in the case of the EC emission $f_{\rm X}$ $\propto$ $\theta^{3/2}$, i.e. for smaller emission region with given $f_{\rm R}$ and $B$ = $B_{\rm eq}$ the EC X-ray emission decreases, again opposite to what is expected in the case of the SSC process.
\subsection{Source classification}
First we group the sources by the X-ray spectral index $\alpha_{\rm X}$ and the X-ray flux $f_{\rm X}$ observed at 1 keV. If the X-ray emission smoothly connects with the radio/optical spectra, we consider the X-rays to be produced via the synchrotron emission as for the radio to optical photons, and that only the highest energy tail of the electron population contributes to the X-ray emission. Good examples are the knots in M~87, where the X-ray spectral indices are $\alpha_{\rm X}$ $\simeq$ 1.3$-$1.6 and the X-ray fluxes are consistent with radio-optical-X-ray synchrotron continua of a broken power-law form. As listed in Table 3, we find 25 ``synchrotron'' jet-knots and 7 hotspots, but none was found for the radio lobes. Figure 4 plots the distribution of $\gamma_{\rm X}$$\delta^{1/7}$, calculated from the equation (5) derived in $\S$ 3.2. Note that for all the synchrotron sources, electrons must be accelerated very efficiently up to $\gamma_{\rm X}$ $\simeq$ 10$^{7-8}$ for $B$ = $B_{\rm eq}$ (and to even higher energies if only $B$ $<$ $B_{\rm eq}$), and that the highest population is occupied by the hotspots.
\begin{table}
\begin{center}
Table 3. Source classification of Jets, hotspots, and lobes.
\vspace{3mm}
\begin{tabular}{lccc}
\tableline\tableline
& Jet-knot & Hotspot & Lobe \\
\tableline
QSO(CD) & 19 & 2 & 0 \\
QSO(LD) & 7 & 9 & 6 \\
RG(FR I) & 22 &0 & 3 \\
RG(FR II) & 1 &13 & 9 \\
BLZR & 7 &0 & 0 \\
\tableline
SYN & 25 & 7 & 0 \\
SSC & 4 & 16 & 1 \\
EC & 27 &1 & 17 \\
\tableline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig4.eps}
\caption{Distribution of the electron Lorentz factor, $\gamma_{\rm X}$, for the SYN sources.}
\end{figure}
Meanwhile, remaining sources show flat X-ray spectra which cannot connect smoothly with the radio and optical spectra in terms of single (or broken) power-law continuum. Let us follow the ``conservative'' hypothesis that the X-rays in these sources are due to the inverse-Compton emission of either synchrotron itself (SSC), or the CMB photons (EC). We therefore compare the ratio between the observed flux density to that expected one from SSC and EC models (c.f., $\S$ 3.3; 3.4), $R_{\rm SSC}$ and $R_{\rm EC}$, to determine which process may dominate for the X-ray production. For example, the hotspot of 3C~123 is well explained by SSC, because $R_{\rm SSC}(1)$ = 1.5 and $R_{\rm EC}(1)$ = 140. This means that observed X-ray luminosity is 1.5 times larger than that expected from the SSC model under equipartition hypothesis, whereas 140 times of the expected EC flux if $\delta$ = 1. In contrast, a good example of the EC source are the lobes in 3C~15, where $R_{\rm SSC}(1)$ = 59 and $R_{\rm EC}(1)$ = 1.2. The results of classification are given in the last column of Table 2.
Resultant group of jet-knots, hotspots and radio lobes are summarized in Table 3. Note that most of the jet-knots are either the synchrotron or the EC sources, whereas majority of the hotspots are the SSC sources. Moreover, almost all of the radio lobes emit X-rays via the EC (CMB) process. However, in a number of jet-knots classified as SSC and EC, the observed X-ray luminosities cannot be reproduced satisfactorily. For example, modelling of the jet-knot in PKS~0637 results in $R_{\rm SSC}(1)$ = 600 and $R_{\rm EC}(1)$ = 1,600, meaning that the observed X-ray flux is about 1,000 times brighter than those expected from both the EC and SSC models. As we have derived in $\S$ 3.3 and 3.4, and is well known from the literature, such discrepancy could be reduced by taking the relativistic beaming effect into account, $\delta$ $\neq$ 1, by giving up the equipartition hypothesis, $B$ $<$ $B_{\rm eq}$, or by postulating a synchrotron origin of the X-ray photons due to an additional flat-spectrum electron population. None of these possibilities can be simply excluded. We will comment more about it in the next section.
Let us mention briefly, that due to differences in the model fitting procedure
adopted in this paper as compared to the previous studies reported in the literature,
some differences may occur in either specific values for the obtained model parameters
(e.g., cf. Sambruna et al. 2004 for the case of NGC~6251), or even in classification of
some particular sources (e.g., cf. Fabian et al. 2003 for 3C~9).
Yet another case is the knot A1 in quasar 3C 273. Marshall et al. (2001) claimed that
its X-ray emission is consistent with the extrapolation of the radio-to-optical single
power-law synchrotron continuum. However, Jester et al. (2002) have shown that this is
not the case, as the observed X-ray flux thereby $exceedes$ the one expected from such
an extrapolation. Here we classify the 3C~273 knot A1 as the SYN source,
in accordance with
Marshall et al. (2001), although it should be emphazise that --
in face of the detailed optical studies by Jester et al. (2002, 2004) --
this particular choice already involves non-standard energy distribution of the radiating
electrons.
\section{Discussion}
In the previous sections, we have followed ``conservative'' classification of the discussed
sources based on their radio and X-ray emission properties. SYN sources are mainly found as
jet-knots in nearby low-luminosity radio galaxies in agreement with previous studies (e.g.,
Hardcastle et al. 2001a, Pesce et al. 2001, Birkinshaw et al. 2002).
If the magnetic field strength is not far from the equipartition
value in these objects, the electrons must be accelerated very efficiently up to 10$-$100 TeV,
in accordance with the general expectation that radio galaxies may be one of the most efficient
particle accelerators in the Universe (see a discussion in Kataoka et al. 2003). If the
electrons are actually accelerated to such high energies, the electrons emitting via synchrotron
in the X-ray band have relatively short radiative life times. The synchrotron cooling time of
the electrons can be expressed as
\begin{equation}
t_{\rm sync} = 250 \left( \frac{B_{\rm eq}}{\rm 100 \mu G} \right)^{-2} \left( \frac{\gamma}{\rm 10^7} \right)^{-1} \quad {\rm [yr]}.
\end{equation}
\noindent
Since Comptonization of the synchrotron photons, CMB photons and of the galactic
photon fields also cool electrons (what can be especially significant if the
considered jets are relativistic on kpc scales), the above estimate would be an
upper limit for the electron cooling time scale. Indeed, Stawarz et al. (2003)
estimated the energy density of the starlight emission at 1 kpc from the center of
average elliptical galaxy -- where typically the X-ray bright knots of the low-powerful jets
are located -- to be $u_{\rm star}$ $\sim$ $10^{-9}$ erg cm$^{-3}$. Now, for the
25 FR I jet-knots classified as SYN sources in this paper the median equipartition magnetic field
computed for non-relativistic bulk velocities is $B_{\rm eq, \, \delta=1}$ = 130 $\mu$G (see Table 2),
what gives the comoving energy density of the magnetic field $u'_{B}$ = 6.7 $\times$ 10$^{-10}$
$\delta^{-10/7}$ erg cm$^{-3}$. Thus, the relative importance of the inverse-Compton to synchrotron
radiative losses for the electrons within the FR I jets is roughly
\begin{equation}
{u'_{\rm star} \over u'_{B}} \sim \Gamma_{\rm BLK}^2 \, \delta^{10/7} \, .
\end{equation}
\noindent
That is, radiating electrons within nearby FR I jets possessing X-ray (and optical) counterparts
(which are believed to be at least moderately beamed toward the observer, $\delta > 1$), cool mainly
due to inverse-Compton losses on the starlight photon fields of the host galaxies unless $B$ $\gg$ $B_{\rm eq}$.
Hence, for the highest energy electrons in the FR I jets one can safely put the radiative cooling
spatial scale $ct_{\rm cool}$ $<$ 100 pc. In general, this is consistent with the visual
sizes of the jet-knots, but significantly smaller than the typical knots' distances from the nucleus
($\gtrsim$ kpc in the case of FR I sources), and also than the typical inter-knot separation distances.
Therefore, the jet electrons have to be accelerated $in$ $situ$, most probably due to stochastic processes
connected with strong turbulence occurring within those jets as a result of the mass entrainment from the
surrounding medium (De Young 1986).
One can ask if in the case of the SYN jet-knots in the nearby FR I galaxies magnetic field can be much smaller than the equipartition value. This possibility could be verified by means of detecting the inverse-Compton radiation of the synchrotron-emitting electrons, which is expected to peak at high-energy $\gamma$-ray band. Interestingly, we can already put some meaningful limits on such high-energy component in the case of the M~87 jet. Nearby radio galaxy M~87 was detected at TeV photon energies by $HEGRA$ Cherenkov Telescope (Aharonian et al. 2003), and it was shown that the kpc-scale jet in this object can produce very high energy $\gamma$-ray photons at the required level via comptonization of the starlight photon field (Stawarz et al. 2003). However, the latest non-detection of M~87 by $Whipple$ Telescope (Le Bohec et al. 2004) suggests variability of the discussed TeV radiation, indicating that the kpc-scale jet in M~87 cannot account for the $HEGRA$ signal. The implied upper limits indicate in turn that the magnetic field within the kpc-scale jet of M~87 radio galaxy cannot be smaller than the equipartition value (Stawarz et al. 2004, in preparation). Thus, one can expect that also in the case of the other FR I jets $B$ $\gtrsim$ $B_{\rm eq}$.
For the SSC and EC sources, a number of jet-knots seem extremely bright in the X-rays, as we have seen in Figure 2 and 3. This inevitably causes a large discrepancy between the ``expected'' and ``observed'' X-ray fluxes as we see in Table 2 and $\S$ 3.5, and what is again well known from the previous studies. One formal possibility is that equipartition hypothesis may not be valid in the considered jet-knots. For a given synchrotron luminosity $L_{\rm sync}$ $\propto$ $u_{\rm e}$$u_B$ and for a given emitting region volume $V$, an expected SSC luminosity is $L_{\rm SSC}$ $\propto$ $u_{\rm e}$. We therefore expect ratio $R_{\rm SSC}$(1) $\propto$ $L_{\rm SSC}^{-1}$ $\propto$ $u_B$. Similarly, for the EC case, $R_{\rm EC}$(1) $\propto$ $L_{\rm EC}^{-1}$ $\propto$ $u_B$. Hence, in both models, the expected X-ray luminosity will be increased by decreasing the magnetic field strength.
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig5.eps}
\caption{Distribution of the evaluated magnetic field, $B$, for the case of $no$ relativistic beaming ($\delta$=1).}
\end{figure}
Figure 5 shows the distribution of the ``best-fit'' magnetic field $B$
if we allow for the deviation from the equipartition condition and
assume nonrelativistic velocities for the emitting regions (what, in the
case of the jet-knots, is rather only a formal hypothesis). One finds
that both the non-SYN jet-knots and radio lobes are distributed around
$B$$\simeq$ 1$-$10$\mu$G, whereas hotspots have a relatively narrow peak
at higher field strength, $B$ $\simeq$ 50$-$300$\mu$G, plus a ``tail''
extending down to $\sim$$\mu$G. Figure 6 shows the ratio of $B$ to the
equipartition value. Interestingly, $B$ in the lobe and most of the
hotspots are almost consistent with the equipartition ($B$/$B_{\rm
eq,\delta=1}$ $\sim$ 1), whereas that of the non-SYN jet-knots and of
some of the hotspots is much weaker from what is expected ($B$/$B_{\rm
eq, \delta=1}$ $\sim$ 0.01$-$0.1).
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig6.eps}
\caption{Distribution of the ratio between the evaluated magnetic field $B$ (for $\delta$ = 1) and the equipartition value $B_{\rm eq, \, \delta = 1}$.}
\end{figure}
As an alternative idea, we also consider a case when the difference between the ``expected'' and ``observed'' X-ray fluxes is due to the relativistic beaming effect, and the minimum-power condition is fulfilled, as suggested by Tavecchio et al. (2000) and Celotti et al. (2001). Relativistic beaming changes the observed X-ray luminosities significantly as $f_{\rm X}$ $\propto$ $\delta^{-5/2}$ for the SSC and $\propto$ $\delta^{3}$ for the EC (equation (10) and (13)). Deviation from equipartition may not be formally required so long as an appropriate beaming factor is assumed. The Doppler factors thus calculated are shown in Table 2 and Figure 7. One can see that the lobes and the hotspots exhibit relatively narrow distribution at $\delta$ $\sim$ 1, whereas for most of the jet-knots large beaming factors of $\sim$ 10 are required, as noted before by many authors. We note, that the obtained $\delta$ $\sim$ 0.1 for some of the hotspots is rather a formal possibility. Figure 8 shows the distribution of equipartition magnetic field in the framework of relativistically moving jet model. Similarly to Figure 5, we find again that the narrowly distributed strength of the magnetic field in the hotspots, $B$ $\sim$ 100$-$500$ \mu$G, is an order of magnitude larger than that of the jet-knots and radio lobes.
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig7.eps}
\caption{Distribution of the required beaming factor $\delta$ for $B$ = $B_{\rm eq}$.}
\end{figure}
\begin{figure}[htb]
\includegraphics[angle=90,scale=.40]{fig8.eps}
\caption{Equipartition magnetic field for relativistically moving jet model.}
\end{figure}
Apparently, the above considerations imply that the inverse-Compton X-ray emissions from the lobes and hotspots are relatively close to that expected from the magnetic field--radiating electrons energy equipartition, with at most mildly-relativistic bulk velocities of the radiating plasma. A number of jet-knots in powerful sources requires however significant bulk Lorentz factor of $\Gamma_{\rm BLK}$ $\ge$ $\delta$/2 $\sim$ 5 to agree the inverse-Compton origin of the X-ray photons with the minimum-power condition $B$ = $B_{\rm eq}$. We note, that our evaluation gives the $minimum$ value of $B_{\rm eq}$, as we set $\nu_{\rm min}$ = $\nu$ in the equation (1) and $\eta = 1$ in the subsequent analysis. Therefore, any more realistic derivation would result in an even larger deviation from the energy equipartition, and thus in an even larger value for the jet Doppler factor $\delta$ required to satisfy the minimum power condition. Let us mention, that the alternative two-population synchrotron models do not require violation of the energy equipartition (Stawarz \& Ostrowski 2002, Dermer \& Atoyan 2004).
Usually, in applying the EC model to the quasar jet-knots' X-ray emission, the idea of sub-equipartition magnetic field is rejected since it implies a very high kinetic power of the jets. For this reason, large values for the jet Doppler factors are invoked. However, as discussed by, e.g., Atoyan \& Dermer (2004), such an approach does not solve all the problems with the total energy requirements (see also a discussion in Stawarz 2004). Let us mention in this context another important issue. It is well known, that the $VLA$ studies of the large-scale jets in quasars and FR IIs indicate that bulk Lorentz factors of the radio-emitting plasma in these sources cannot be much greater than $\Gamma_{\rm BLK}$ $\sim$ 3 (Wardle \& Aaron 1997). The discrepancy between this result and the requirement of the minimum-power EC model for $\Gamma_{\rm BLK}$ $>$ 10 is typically ascribed to the jet radial velocity structure, namely that the radio emission originates within slower-moving jet boundary layer and the inverse-Compton X-ray radiation is produced within the fast jet spine (e.g., Ghisellini \& Celotti 2001). While it is true that jet radial stratification can indeed significantly influence jet-counterjet brightness asymmetry ratio, one should be aware that by postulating different sites for the origin of radio and X-ray photons, homogeneous one-zone models for the broad-band knots' emission can no longer be preserved. In particular, in such a case one has to specify exactly what fraction of the jet radio emission is produced within the spine and what fraction within the boundary layer, what is exactly the jet velocity radial profile, and what is the magnetic field strength in each jet components, etc. Without such a discussion one cannot simply use the observed radio flux of the jet to construct the broad-band spectral energy distribution of the knot region, i.e. simply estimate the expected inverse-Compton flux by means of equipartition magnetic field derived from the radio observations. If one insists on applying the homogeneous one-zone model (as a zero-order approximation), as presented in this paper, self-consistency requires a consideration of $\Gamma_{\rm BLK}$ $\leq$ 5. In such a case, a departure from the minimum power conditions within the non-SYN X-ray jets is $inevitable$.
Accordingly to the discussion above, if the X-ray emission of the powerful jets is due to the EC process, these jets are most likely $particle$ $dominated$ ($u_e$$\gg$$u_B$). The jet magnetic field must be then significantly amplified in the hotspot, where an approximate equipartition is expected to be reached. Then the shocked plasma moves slowly to the radio lobe, where the equipartition field becomes close to the intergalactic value ($B$$\sim$ a few $\mu$G). Let us comment in this context on the following issue. Pressure of radio-emitting electrons within the lobes of quasars and FR IIs computed from the equipartition condition is often found to be below the thermal pressure of the ambient medium (Hardcastle \& Worrall 2000), what challenges the standard model for the evolution of powerful radio sources. Such a discrepancy can be removed only by postulating deviations from the equipartition condition, or presence of non-radiating relativistic protons within the lobes. The presented analysis of the X-ray data confirms for a large number of sources the anticipated result that the magnetic field--radiating electrons energy equipartition within the lobes is generally fulfilled, and thus that the relativistic protons are very likely to constitute a significant fraction of the lobes' non-thermal pressure. Interestingly, this would mean that the total energy outputs of powerful jets are systematically $larger$ from what is implied by the analysis of the lobes' radio emission solely (Rawlings \& Saunders 1991). This, in turn, would be consistent with deviation from the minimum-power condition within the considered jets themselves. We note that viscous acceleration of cosmic rays taking place at the turbulent boundary layers of relativistic jets, discussed by, e.g., Ostrowski (2000) and Rieger \& Mannheim (2002), could provide energetically important flat-spectrum population of ultrarelativistic protons within the lobes of powerful radio sources.
We have discussed two different versions of the EC model to account for extremely bright X-ray jet-knots: (1) non-equipartition case and (2) significant relativistic beaming case. Both of these options are in many ways problematic. Our next concern is to attempt to prove in general the postulated inverse-Compton origin of the X-ray photons. Note in this context, that for the EC sources the radio-to-X-ray flux ratio is proportional to $u'_B$$^{-1}$$(1+z)^4$$\delta^2$ (equation 12). Therefore, for a large sample of EC sources one should expect to observe $L_{\rm R}$/$L_{\rm X}$ $\propto$ $(1+z)^4$ behavior, if only $B$ and $\Gamma_{\rm BLK}$ do not have large scatter from source-to-source (cf. Figures 5 and 7). We therefore expect the high-redshift EC sources to be brighter in X-rays than nearby EC sources (see Schwartz 2002, Cheung 2004).
\begin{figure}[htb]
\includegraphics[angle=90,scale=.33]{fig9.eps}
\caption{Luminosity ratio, $L_{\rm 1keV}$/$L_{\rm 5GHz}$, as a function of redshift for SYN, SSC and EC soures.}
\end{figure}
Figure 9 shows the distribution of the flux ratio ($L_{\rm 1keV}$/$L_{\rm 5GHz}$) as a function of $z$. The dotted line shows $\propto$ $(1+z)^4$ relation which fits the highest $z$ data point (GB~1508+5714; $z$ = 4.3) just to help guide the eyes. Although data sample is still poor, we may say that no clear trend can be seen in this plot. Furthermore, we notice that the discussed ratio is widely distributed even in the same objects. For example, in cases of knots in 4C~19.33 ($z$ = 0.72), ``conservatively'' classified as the EC sources, the X-ray-to-radio luminosity ratio changes of about an order of magnitude (Table 2). Such a difference is not easy to explain in the framework of model (1), since we have to assume an order of magnitude increase in the magnetic fields along the jet. In a framework of relativistic beaming hypothesis (2) one may possibly explain such variation by postulating the decrease of the bulk Lorentz factor along the flow and only moderate changes in magnetic field (Georganopoulos \& Kazanas 2004). In this case, however, one has to explain what causes significant deceleration of the jet, which preserves its excellent collimation, with no significant radiative energy losses. We need more data obtained in various energy bands, and a more sophisticated analysis to conclude this further. However, we note that recent observations of high-redshift quasars by Bassett et al. (2004) did not reveal any evidence for the enhanced X-ray emission of the distant large-scale jets due to the increased energy density of the cosmic microwave background. Since such an effect is expected in a framework of the EC model, one may conclude that the X-ray photons from the powerful quasar jets are not inverse-Compton in origin. Recent detailed re-analysis of the $Chandra$ data for 3C~120, again ``conservatively'' classified as an EC source, strongly support this idea (Harris et al. 2004).
\begin{figure}[htb]
\includegraphics[angle=90,scale=.33]{fig10.eps}
\caption{Expected beaming factor $\delta_{\rm EC}$ for $B$ = $B_{\rm eq}$,
as a function of redshift for EC jet-knot sources. $Open$ $circle$ shows
FR~I radio galaxy NGC~6251, but its X-ray emission seems to be
problematic in a framework of the EC model. Full details are given in
the text.}
\end{figure}
Let us finally discuss yet another issue regarding the EC scenario for
the quasar jets' X-ray emission. Figure 10 shows the Doppler beaming factor $\delta$ required
in this model to obtain $B$ $=$ $B_{\rm eq}$, versus the redshifts $z$ of the jet-knots classified
here as the EC ones. One can clearly see a significant anticorrelation between $\delta$ and $z$,
meaning that the high-$z$ sources require much smaller $\delta$ for the magnetic field-radiating electrons
energy equipartition than the sources located closer to the
observer.\footnote{Nearby FR I radio galaxy NGC~6251 ($open$ $circle$) constitutes the only exception from this trend. However, this peculiar source does not belong to
the quasar class, and, in general, its X-ray emission is particularly problematic (especially in
a framework of the EC model).} There are two possible explanations for the noted $\delta$--$z$
anticorrelation. If reflecting physical property, it would mean that the distant large-scale quasar jets
are less relativistic than their nearby analogues but similarly close to the equipartition, $or$ that
both low- and high-$z$ quasar jets are only mildly relativistic on large scales but closer to the
minimum-power condition when located at large redshifts. None of this options appear to be particularly
inartificial, especially as the high-$z$ quasar cores seem to be comparable to their low-$z$ counterparts (e.g.,
Bassett et al. 2004). On the other hand, differences in velocity and energy content of the large-scale jets may
not reflect differences in the central engines, but more likely differences in the surrounding galactic or
intergalactic medium. The second possibility for understanding $\delta$--$z$ anticorrelation is however that
it is simply an artifact of the applied but inappropriate EC model. This issue has to be discussed carefully
for a larger number of sources.
\section{Conclusion}
We have studied the statistical properties of the large-scale jet-knots,
hotspots and lobes in more than 40 radio galaxies recently observed with
$Chandra$ and $ASCA$. For the jet-knots in nearby low-luminosity radio
galaxies and for some of the hotspots, X-ray photons are most likely
synchrotron in origin, being then produced by ultrarelativistic
electrons with energies 10$-$100 TeV that must be accelerated in
situ. For the other objects X-ray photons are inverse-Compton in origin,
or, alternatively, are due to synchrotron emission of very high energy
electrons with a non-standard energy distribution. In this paper we
examine in more detail the former possibility. We
first calculated the ``expected'' SSC or EC fluxes by assuming
equipartition magnetic field and nonrelativistic velocity of the
emitting plasma, and then compared them to the observed fluxes.
We confirmed that the observed X-ray fluxes from the hotspots and
radio lobes are approximately consistent with the expected ones,
whereas a number of the jet-knots in powerful sources is too bright
at X-rays. We examined two possibilities to account for this discrepancy
in a framework of the inverse-Compton model. The first idea is that
equipartition hypothesis may not be valid for the considered sources.
In this case, the X-ray bright jets are particle dominated and therefore
far from the minimum-power condition. The jet magnetic field must be
then significantly amplified in the hotspots where an approximate energy
equipartition with the radiating particles is expected to be reached.
An alternative idea is that the jets are highly relativistic
($\Gamma_{\rm BLK}$ $\ge$ 5) even on kpc/Mpc scales, but significantly
decelerate in the hotspots. This however, in addition to other problems,
challenges the homogeneous one-zone emission region model adopted in this
paper, as discussed in the text. Unfortunately, the comparison of the
observed radio-to-X-ray flux ratios for various $z$ sources
\emph{from the compiled dataset} does not provide definite constraints
on the X-ray emission process dominating within the quasar and FR II jets.
\acknowledgments
We would like to thank F. Takahara and M. Ostrowski for fruitful
discussion and constructive comments. J.K. acknowledges a support by
JSPS.KAKENHI (14340061). \L .S. was supported by the grant PBZ-KBN-054/P03/2001
and partly by the Chandra grants G02-3148A and G0-09280.01-A.
|
2,877,628,090,764 | arxiv | \section{Introduction}
In this paper we consider local linear smoothing in an additive model
\begin{align}\label{model:add}
E[Y_i | X_i ] = m_0+m_1(X_{i1})+\cdots +m_d(X_{id}),
\end{align}
where $(Y_i,X_i)$ $(i=1,\dots,n)$ are $iid$ observations with values in $\mathbb R \times \mathcal X$ for a bounded connected open subset $\mathcal X\subseteq\mathbb R^d$.
Here, $m_j$ $(j=1,\dots,d)$ are some smooth functions which we aim to estimate and $m_0\in\mathbb{R}$. Below, we will add norming conditions on $m_0,\dots,m_d$ such that they are uniquely defined given the sum.
In \cite{mammen1999existence} a local linear smooth backfitting estimator based on smoothing kernels was proposed for the additive functions $m_j$. There, it was shown that their version of a
local linear estimator $\hat m_j$ of the function $m_j$ has the same pointwise asymptotic variance and bias as a classical local linear estimator in the oracle model, where one observes i.i.d. observations $(Y_i^\ast,X_{ij})$ with
\[
E[Y^\ast_i | X_{ij} ] = m_j(X_{ij}), \quad Y^\ast_i = Y_i-\sum_{k \neq j} m_k({X_{ik}}).
\]
In this respect the local linear estimator differs from other smoothing methods where the asymptotic bias of the estimator of the function $m_j$ depends on the shape of the functions $m_k$ for $k\not = j$. An example for an estimator with this disadvantageous bias property is the local constant smooth backfitting estimator which is based on a backfitting implementation of one-dimensional Nadaraya-Watson estimators. It is also the case for other smoothing estimators as regression splines, smoothing splines and orthogonal series estimators, where in addition also no closed form expression for the asymptotic bias is available.
Asymptotic properties of local linear smoothing simplify the choice of bandwidths as well as the statistical interpretation of the estimators $\hat m_j$. These aspects have made local linear smooth backfitting a preferred choice for estimation in additive models.
Deriving asymptotic theory for local linear smooth backfitting is typically complicated by an overloaded notation that is required for detailed proofs. In this note we will use that the local linear smooth backfitting estimator has a nice geometric interpretation. This simplifies mathematical arguments and allows for a more intuitive derivation of asymptotic properties. In particular, we will see that the estimator can be characterized as a solution of an empirical integral equation of the second kind as is the case for local constant smooth backfitting, see \cite{mammen2009nonparametric}.
Our main point is that the local linear estimator can be seen as an orthogonal projection of the response vector $Y= (Y)_{i=1,\dots,n}$ onto a subspace of a suitably chosen linear space. A similar point of view is taken in \cite{mammen2001general} for a related construction where it was also shown that regression splines, smoothing splines and orthogonal series estimators can be interpreted as projection of the data in an appropriately chosen Hilbert space. Whereas this interpretation is rather straight forward for these classes of estimators it is not immediately clear that it also applies for kernel smoothing and local polynomial smoothing, see \cite{mammen2001general}. In this paper we will introduce a new and simple view of local linear smoothing as data projection. In the next section we will define the required spaces together with a corresponding semi-norm. We will also introduce a new algorithm motivated by our interpretation of local linear smooth backfitting. The algorithm will be discussed in Section \ref{sec:exist}. In Section \ref{sec:asymp} we will see that our geometric point of view allows for simplified arguments for the asymptotic study of properties of the
local linear smooth backfitting estimator.
The additive model \eqref{model:add} was first introduced in \cite{friedman1981projection} and enjoys great popularity for two main reasons.
The first is estimation performance.
While not being as restrictive as a linear model, in contrast to a fully flexible model, it is not subject to the curse of dimensionality. Assuming that $E[Y_i | X_i =x]$ is twice continuously differentiable, the optimal rate of convergence of an estimator of $E[Y_i | X_i =x]$ is $n^{-2/(d+4)}$ if no further structural assumptions are made, see \cite{stone1982optimal}. This means the rate deteriorates exponentially in the dimension of the covariates $d$. Under the additive model assumption \eqref{model:add} and assuming that each function $m_j$, $j=1,\dots, d$ is twice continuously differentiable, the optimal rate of convergence is $n^{-2/5}$.
The second reason is interpretability. In many applications it is desirable to understand the relationship between predictors and the response.
Even if the goal is prediction only, understanding this relationship may help detect systematic biases in the estimator, so that out of sample performance can be improved or adjusted for.
While it is almost impossible to grasp the global structure of a multivariate function $m$ in general, the additive structure \eqref{model:add} allows for visualisation of each of the univariate functions, providing a comprehensible connection between predictors and the response.
Though the setting considered in this paper is fairly simple, it can be seen as a baseline for more complicated settings.One main drawback is the additive structure which cannot account for interactions between covariates.
It is assuring however that even if the true model is not additive, the smooth backfitting estimator is still defined as the closest additive approximation. This will be shown in the next section.
If the true regression function is far away from an additive structure, then a more complex structure may be preferable. This could be done by adding higher-dimensional covariates, products of univariate functions or considering a generalized additive model. For testing procedures that compare such specifications, see also \cite{Hardle2001,MamSper21}. Besides such structural assumptions, other directions the ideas in this paper can be extended to are the consideration of time-series data or high dimensional settings. Settings using more complicated responses like survival times, densities or other functional data may also be approached. Some of these cases have been considered, e.g., in \cite{mammen2003generalised}, \cite{yu2008smooth}, \cite{mammen2009nonparametric}, \cite{mammen2014additive}, \cite{han2018smooth}, \cite{MamSper21}, \cite{han2019additive}, \cite{jeon2020additive}, \cite{hiabu2020smooth} and \cite{Gregory2020optimal}.
We hope that a better understanding of local linear estimation in this simple setting will help advance theory and methodology for more complicated settings in the future.
\section{Local linear smoothing in additive models} \label{sec:local linear}
The local linear smooth backfitting estimator $\hat{m}=(\hat{m}_0,\hat{m}_1,\dots,\hat{m}_d,\hat{m}_1^{(1)},\dots,\hat{m}_d^{(1)})$ is defined as the minimizer of the criterion
\begin{eqnarray*}
&&S(f_0,\dots,f_d, f_1^{(1)},\dots,f_d^{(1)}) \\
&& \qquad = n^{-1}\sum_{i=1}^n \int_{\mathcal X} \left \{ Y_i - f_0 -\sum_{j=1}^d f_j(x_j) -\sum_{j=1}^d f_j^{(1)}(x_j) (X_{ij}-x_j)\right \} ^2 \\
&& \qquad \times K_h^{X_i}(X_i-x) \mathrm dx
\end{eqnarray*}
under the constraint
\begin{eqnarray} \label{eq:constr}
\sum_{i=1}^n \int_{\mathcal X} f_j(x_j) K_h^{X_i}(X_i-x) \mathrm dx= 0
\end{eqnarray}
for $j=1,\dots,d$. The minimization runs over all values $f_0 \in \mathbb R$ and all functions $f_j, f_j^{(1)}: \mathcal X_j \to \mathbb R$ with $\mathcal X_j = \{ u \in \mathbb R:$ there exists an $ x \in \mathcal X$ with $ x_j = u\}$. Under the constraint (\ref{eq:constr}) and some conditions introduced in Section \ref{sec:exist}, the minimizer is unique. For $j=1,\dots, d$ the local linear estimator of $m_j$ is defined by $\hat{m}_j$.
\noindent In the definition of $S$ the function $K_h^{u}(\cdot)$ is a boundary corrected product kernel, i.e.,
\[
K_h^{u}(u-x)= \frac{\prod_{j=1}^d \kappa \left(\frac{u_j-x_j}{h_j}\right)}{\int_{\mathcal X} \prod_{j=1}^d \kappa \left(\frac{u_j-v_j}{h_j}\right) \mathrm dv_j }.
\]
Here, $h=(h_1,\dots,h_d)$ is a bandwidth vector with $h_1,\dots,h_d > 0$ and $\kappa :\mathbb{R}\rightarrow\mathbb{R}$ is some given univariate density function, i.e., $\kappa (t)\geq 0$ and $\int \kappa (t) \mathrm dt=1$.
We use the variable $u$ twice in the notation because away from the boundary of $\mathcal X$, the kernel $K_h^{u}(u-x)$ only depends on $u-x$.\\
It is worth emphasizing that the empirical minimization criterion $S$ depends on a choice of a kernel $\kappa$ and a smoothing bandwidth $h$. While the choice of $\kappa$ is not of great importance, see similar to e.g. \cite[Section 3.3.2]{silverman2018density},
the quality of estimation heavily depends on an appropriate choice of the smoothing parameter $h$.
We will not discuss the choice of a (data-driven) bandwidth in this paper, but we note that
the asymptotic properties of the local linear smoothing estimator do simplify the choice of bandwidths compared to other estimators. The reason is that the asymptotic bias of one additive component does not depend on the shape of the other components and on the bandwidths used for the other components.
\noindent We now argue that the local linear smooth backfitting estimator can be interpreted as an empirical projection of the data onto a space of additive functions. We introduce the linear space
\[
\mathcal H = \left \{ (f^{i,j})_ {i=1,\dots,n;\ j=0,\dots,d} | \ f^{i,j}: \mathcal X \mapsto \mathbb R, \ {\norm f }_n< \infty \right \}
\]
with inner product
\begin{eqnarray*}
{\innerproduct fg}_n &=& n^{-1}\sum_{i=1}^n \int_{\mathcal X} \left \{f^{i,0}(x) +\sum_{j=1}^df^{i,j}(x) (X_{ij}-x_j)\right \} \\
&& \qquad \times \left \{g^{i,0}(x) +\sum_{k=1}^d g^{i,k}(x) (X_{ij}-x_j)\right\} K_h^{X_i}(X_i-x) \mathrm dx
\end{eqnarray*}
and norm $\| f\|_n = \sqrt {{\innerproduct f f}_n}$.\newline
We identify the response $Y= (Y_i)_{i=1,\dots,n}$ as an element of $\mathcal H$ via $Y^{i,0}\equiv Y_i$ and $Y^{i,j}\equiv 0$ for $j\geq1$.
We will later assume that the functions $m_j$ are differentiable. We identify the regression function
\[
m:\mathcal{X}\rightarrow\mathbb{R},\ m(x)=m_0+m_1(x_1)+\dots+m_d(x_d)
\]
as an element of
$\mathcal H$ via $m^{i,0}(x)=m_0+\sum_j m_j(x_j)$ and $m^{i,j}={\partial m_j(x_j)}/{\partial x_j}$ for $j\geq 1$. Note that the components of $m\in\mathcal{H}$ do not depend on $i$.
We define the following subspaces of $\mathcal H$:
\begin{align*}
\mathcal H_{full} &= \left \{ f \in \mathcal H | \text{ the components of }\ f \ \text{do not depend on} \ i \right \}, \\
\mathcal H_{add} &= \left \{ f \in \mathcal H_{full} | \ f^{i,0}(x)=f_0 + f_1(x_1)+\cdots + f_d(x_d), f^{i,j}(x) =f^{(1)}_j(x_j) \text{ for}\right. \\ & \qquad \text{some } f_0 \in \mathbb R \text { and some univariate functions}\ f_j, f_j^{(1)}: \mathcal X_j \to \mathbb R, j=1,\dots,\\
&\qquad \left. d\text{ with }\sum_{i=1}^n \int_{\mathcal X} f_{j}(x_j) K^{X_i}_h(X_i-x) \mathrm dx =0 \right \}.
\end{align*}
For a function $f \in \mathcal H_{add}$ we write $f_0 \in \mathbb R$ and $f_j, f_j^{(1)}: \mathcal X_j \to \mathbb R$ for $j=1,\dots,d$ for the constant and functions that define $f$. In the next section we will state conditions under which the constant $f_0$ and functions $f_j, f_j^{(1)}$ are unique given any $f \in \mathcal H_{add}$. By a slight abuse of notation we also write $f_j$ for the element of $\mathcal H$ given by $f^{i,0}(x)= f_j(x_j)$ and $f^{i,k}(x) \equiv 0$ for $k=1,\dots,d$. We also write $f_j^{(1)}$ for the element of $\mathcal H$ with $f^{i,k}(x) \equiv 0$ for $k \not = j$ and $f^{i,j}(x) = f_j^{(1)}(x_j)$. Furthermore, we define $f_{j+d}:=f_j^{(1)}$ for $j=1,\dots,d$ for both interpretations. Thus, for $f \in \mathcal H_{add}$ we have
\begin{align}\label{eq:add}
f= f_0+\dots+f_{2d}.
\end{align}
Recall that the linear smooth backfitting estimator $$\widehat m=(\hat m_0, \hat m_1, \dots,\hat m_d, \hat m_1^{(1)},\dots,\hat m_d^{(1)})$$ is defined as the minimizer of the criterion $S$ under the constraint
\eqref{eq:constr}.
By setting $\hat{m}^{i,0}(x)=\hat{m}_0+\sum_{j=1} ^d \hat{m}_j(x_j)$ and $m^{i,j}(x)=\hat{m}_j^{(1)}(x_j)$ for $j\geq 1$ it can easily be seen that
\begin{align}\label{minimiser}
\widehat m = \argmin_{f\in \mathcal H_{add}} {\norm{Y-f}}_n.
\end{align}
In the next section we will state conditions under which the minimization has a unique solution.
Equation \eqref{minimiser} provides a geometric interpretation of local linear smooth backfitting. The local linear smooth backfitting estimator is an orthogonal projection of the response vector $Y$ onto the linear subspace $\mathcal H_{add}\subseteq\mathcal H$. We will make repeated use of this fact in this paper.\\
We now introduce the following subspaces of $ \mathcal H$:
\begin{align*}
\mathcal H_0 &= \left\{ f \in \mathcal H | f^{i,0}(x)\equiv c \text{ for some } c \in \mathbb R, f^{i,j}(x)\equiv 0 \ \text{ for }\ j\neq0 \right \},\\
\mathcal H_k &= \left\{ f \in \mathcal H | \ f^{i,j}(x)\equiv 0 \ \text{ for }\ j\neq0, \text { and } f^{i,0}(x)=f_k(x_k) \text{ for some univariate}\right .\\
& \left . \qquad \text{function } f_k: \mathcal X_k \to \mathbb R \text{ with }
\sum_{i=1}^n \int_{\mathcal X} f_{k}(x_k) K^{X_i}_h(X_i-x) \mathrm dx =0 \right\},\\
\mathcal H_{k'} &= \left\{ f \in \mathcal H | \ f^{i,j}(x)\equiv 0 \ \text{ for }\ j \not =k, f^{i,k}(x)=f^{(1)}_k(x_k) \text{ for some univariate}\right .\\
& \left . \qquad \text{function } f^{(1)}_k: \mathcal X_k \to \mathbb R \right\}
\end{align*}
for $k=1,\dots,d$ and $k':=k+d$.
Using these definitions we have $\mathcal H_{add}=\sum_{j=0} ^{2d} \mathcal H_j$ with $\mathcal H_j\cap \mathcal H_k=\{0\}, j\neq k$. In particular, the functions $f_j$ in \eqref{eq:add} are unique elements in $\mathcal H_j, j=0,\dots,2d.$
For $k=0,\dots,2d$ we denote the orthogonal projection of $ \mathcal H$ onto the space $\mathcal H_k$
by $\mathcal P_k$.
Note that for $k=0,\dots,d$ the operators $\mathcal P_k$ set all components of an element $f = (f^{i,j})_ {i=1,\dots,n;\ j=0,\dots,d}\in\mathcal{H}$ to zero except the components with indices $(i,0), i=1,\dots,n$. Furthermore, for $k=d+1,\dots, 2d$, only components with index $(i,k-d)$ are not set to zero.
Because $\mathcal H_0 $ is orthogonal to $\mathcal H_k$ for $k=1,\dots,d$, the orthogonal projection onto the space $\mathcal H_k$ is given by $\mathcal P_k = P_k - \mathcal P_0$ where $P_k$ is the projection onto $\mathcal H_0+\mathcal H_k$. In Appendix \ref{sec: projection operators} we will state explicit formulas for the orthogonal projection operators.\newline
The operators $\mathcal P_k $ can be used to define an iterative algorithm for the approximation of $\hat m$. For an explanation observe that $\hat m$ is the projection of $Y$ onto $\mathcal H_{add}$ and $\mathcal H_k$ is a linear subspace of $\mathcal H_{add}$. Thus $\mathcal{P}_k(Y) = \mathcal{P}_k(\hat m) $ holds for $k=0,\dots, 2d$. This gives
\begin{equation} \label{eq:back} \mathcal P_k(Y) =\mathcal P_k(\hat m) = \mathcal P_k\left ( \sum_{j=0} ^{2d} \hat m_j\right) = \hat m_k + \mathcal P_k\left( \sum_{j\not = k} \hat m_j\right)\end{equation}
or, equivalently,
$$\hat m_k = \mathcal P_k (Y) - \sum_{j\not = k} \mathcal P_k( \hat m_j)= P_k (Y) - \bar Y - \sum_{j\not = k} \mathcal P_k( \hat m_j),$$
where $\bar Y=\mathcal P_0 (Y)=P_0 (Y)$ is the element of $\mathcal H$ with $(\bar Y)^{i,0}\equiv \frac 1 n \sum_{i=1}^n Y^i$, $(\bar Y)^{i,j}\equiv 0$ for $j\geq1$.
This equation inspires an iterative algorithm where in each step approximations $\hat m^{old}_k$ of $\hat m_k$ are updated by
$$\hat m_k^{new} = P_k (Y) -\bar Y - \sum_{j\not = k} \mathcal P_k( \hat m^{old}_j).$$
\begin{algorithm}
\caption{Smooth Backfitting algorithm}
\begin{algorithmic}[1]
\State \textbf{Start:} $\hat m_k(x_k)\equiv 0, \widetilde m_k =\mathcal P_k( Y), error=\infty$ \Comment{$k=0,\dots,2d$}
\While {$error>tolerance$}
\State $error \gets 0$
\For{$k=0,\dots,2d$}
\State $\hat m_k^{old} \gets \hat m_k$
\State $\hat m_k \gets \widetilde m_k - \sum_{j\not = k} \mathcal P_k( \hat m_j)$
\State $error \gets error + | \hat m_k - \hat m_k^{old}|$
\EndFor\label{euclidendwhile}
\EndWhile
\State \textbf{return} $\hat m=(\hat m_0, \hat m_1,\dots, \hat m_{2d})$
\end{algorithmic}
\end{algorithm}
Algorithm 1 provides a compact definition of our algorithm for the approximation of $\hat m$. In each iteration step, either $\hat m_j$ or $ \hat m_j^{(1)}$ is updated for some $j=1,\dots, d$. This is different from the algorithm proposed in \cite{mammen1999existence} where in each step a function tuple $(\hat m_j, \hat m_j^{(1)})$ is updated. For the orthogonal projections of functions $m \in \mathcal H_{add} $ one can use simplified formulas. They will be given in Appendix \ref{sec: projection operators}.
Note that $\tilde m_k, k=0,\dots,2d$ only needs to be calculated once at the beginning. Also the marginals
$p_k(x_k)$, $p^\ast_k(x_k)$, $p^{\ast \ast}_k(x_k)$, $p_{jk}(x_j,x_k)$, $p^{\ast}_{jk}(x_j, x_k)$ and $p^{\ast \ast}_{jk}(x_j,x_k)$ which are needed in the evaluation of $\mathcal P_k$ only need to be calculated once at the beginning. Precise definitions of these marginals can be found in the following sections. In each iteration of the for-loop in line 4 of Algorithm 1, $O(d \times n \times gs)$ calculations are performed. Hence for a full cycle, the algorithm needs $O(d^2 \times n \times gs\times \log (1/ \mathrm{tolerance}))$ calculations. Here $\mathrm{grid.size}$ is the number of evaluation points for each coordinate $x_k$. \newline
Existence and uniqueness of the local linear smooth backfitting estimator will be discussed in the next section. Additionally, convergence of the proposed iterative algorithm will be shown.
\section{Existence and uniqueness of the estimator, convergence of the algorithm} \label{sec:exist}
In this section we will establish conditions for existence and uniqueness of the local linear smooth backfitting estimator $\widehat m$. Afterwards we will discuss convergence of the iterative algorithm provided in Algorithm 1. Note that convergence is shown for arbitrary starting values, i.e., we can set $\hat{m}_k(x_k)$ to values other than zero in step 1 of Algorithm 1. For these statements we require the following weak condition on the kernel.
\begin{enumerate}
\item[(A1)]
The kernel $k$ has support $[-1,1]$. Furthermore, $k$ is strictly positive on $(-1,1)$ and continuous on $\mathbb R$.
\end{enumerate}
For $k=1,\dots,d$ and $x\in \mathbb{R}^d$ we write
\[
x_{-k}:=(x_1,\dots,x_{k-1},x_{k+1},\dots,x_d).
\]
In the following, we will show that our claims hold on the following event:
\begin{align*}
\mathcal E &= \bigg \{ \text{ For } k=1,\dots,d \text{ and } x_k\in \overline{\mathcal {X}}_k \text{ there exist two observations}\ i_1,i_2 \in \{1,\dots,n\} \\
& \text{such that}
\ X_{i_1,k} \not = X_{i_2,k}\ ,\ |X_{i,k} - x_k| < h_k, (x_k,X_{i,-k})\in\overline{\mathcal{X}}\ \text{for}\ i=i_1,i_2.\\
&\text{Furthermore, there exist no } b_0,\dots,b_d \in \mathbb R \text{ with } b_0 +
\sum_{j=1} ^d b_j X_{ij} = 0\ \forall i=1,\dots,n
\bigg \},
\end{align*}
where $\overline{\mathcal{X}}_k$ is the closure of $\mathcal{X}_k$ and by a slight abuse of notation $$(x_k,X_{i,-k}):=(X_{i,1},\dots, ,X_{i,k-1},x_k,X_{i,k+1},\dots,,X_{i,d}).$$
Throughout this paper, we require the following definitions.
\begin{eqnarray*}
\hat p_k(x_k) &=& \frac 1 n \sum_{i=1} ^n \int _{\mathcal X_{-k}(x_k) } K^{X_i}_h(X_i-x)\mathrm dx_{-k}, \\
\hat p^{*}_k(x_k) &=& \frac 1 n \sum_{i=1} ^n \int _{\mathcal X_{-k}(x_k) } (X_{ik}-x_k) K^{X_i}_h(X_i-x)\mathrm dx_{-k},\\
\hat p^{**}_k(x_k) &=& \frac 1 n \sum_{i=1} ^n \int _{\mathcal X_{-k}(x_k) } (X_{ik}-x_k) ^2 K^{X_i}_h(X_i-x)\mathrm dx_{-k},
\end{eqnarray*}
where ${\mathcal X_{-k}(x_k) }=\{ u_{-k}\ |\ (x_k,u_{-k}) \in \mathcal X \}$.
\begin{lemma} \label{lem:uni Hadd}
Make Assumption (A1). Then, on the event $\mathcal E$ it holds that $\|f\|_n =0$ implies $f_0 =0$ as well as $f_j\equiv 0$ almost everywhere for $j=1\dots 2d$ and all $f \in \mathcal H_{add}$. \end{lemma}
One can easily see that the lemma implies the following. On the event $\mathcal{E}$, if a minimizer $\hat m=\hat m_0+\cdots +\hat m_{2d}$ of ${\norm{Y-f}}_n$ over $f \in \mathcal H_{add}$ exists, the components $\hat m_0,\dots, \hat m_{2d}$ are uniquely determined: Suppose there exists another minimizer $\tilde m \in \mathcal H_{add}$. Then it holds that $\langle Y- \hat m, \hat m - \tilde m\rangle_n = 0$ and $\langle Y- \tilde m, \hat m - \tilde m\rangle_n = 0$ which gives $\|\hat m - \tilde m\|_n =0$. An application of the lemma yields uniqueness of the components $\hat m_0,\dots, \hat m_{2d}$.
\begin{figure}
\centering
\includegraphics[width=5cm]{Beispiel_fuer_E.png}
\caption{An example of a possible data set $\mathcal{X}\subseteq\mathbb{R}^2$ including data points where the conditions of the event $\mathcal E$ are not satisfied and where the components of functions $f\in \mathcal H_{add}$ are not identified. The data is visualized by blue dots. The size of the parameter $h=h_1=h_2$ is showcased on the right hand side. For explanatory reasons, the interval $a$ is included.} \label{fig:addsmooth}
\end{figure}
\begin{remark}
In Figure 1 we give an example where a set $\mathcal X$ and data points $X_i$ do not belong to the event $\mathcal E$ and where the components of the function $f\in \mathcal H_{add}$ are not identified.
Note that in this example for all $k=1,2$ and $x_k\in\overline{\mathcal{X}_k}$ there exist $i_1,i_2\in\{1,\dots,n\}$ such that$X_{i_1}\neq X_{i_2}$ and $|X_{i,k}-x_k|<h$, for $i=i_1,i_2$. However, for $x_1\in a$ the condition $(x_1,X_{i,2})\in\overline{\mathcal{X}}$ is not fulfilled for any $i=1,\dots,n$ with $|X_{i,1}-x_1|<h$. Therefore, $K_h^{X_i}(X_i-x)=0$ for all $x\in\overline{\mathcal{X}}$ with $x_1\in a$. Thus, any function satisfying $f\in\mathcal{H}_1$ with $f_1(x)=0$ for $x\in\mathcal{X}_1\backslash\{a\}$ has the property $\norm{f}_n=0$.
\end{remark}
\begin{proof} [of Lemma \ref{lem:uni Hadd}]
First, for each pair $i_1,i_2=1,\dots,n$ define the set
\[
M_{i_1,i_2}:=\{x_1\in\mathcal{X}_1\ |\ |X_{i,1} - x_1| < h, (x_1,X_{i,-1})\in\mathcal{X}\ \text{for}\ i=i_1,i_2\}
\]
if $X_{i_1,1}\neq X_{i_2,1}$ and $M_{i_1,i_2}=\emptyset$ otherwise.
It is easy to see that $M_{i_1,i_2}$ is open as an intersection of open sets. Note that on the event $\mathcal E$ we have
\begin{equation} \label{eq:Ueberdeckung}
\bigcup_{i_1,i_2}M_{i_1,i_2}=\mathcal{X}_1.
\end{equation}
Now, suppose that for some $f \in \mathcal H_{add}$ we have $\|f\|_n =0$. We want to show that $f_0 =0$ and that $f_j \equiv 0$ for $j=1,\dots,2d$. From $\|f\|_n =0$ we obtain
$$ \bigg \{ f_0 + \sum_{j=1} ^d f_j(x_j) + \sum_{j=1} ^d f_{j'}(x_j) (X_{ij}-x_j)\bigg \}^2 K^{X_i}_h(X_i -x) = 0$$
for $i=1,\dots, n$ and almost all $x \in \mathcal X$. Let $i_1,i_2\in\{1,\dots,n\}$. Then
\begin{equation} \label{eq:Haddhelp} f_0 + f_1(x_1) + \sum_{j=2} ^d f_j(X_{ij}) + f_{d+1}(x_{1}) (X_{i1}-x_1) =0\end{equation}
holds for all $i=i_1,i_2$ and $x_1\in M_{i_1,i_2}$ almost surely. By subtraction of Equation \eqref{eq:Haddhelp} for $i=i_1$ and $i=i_2$ we receive
$$f_{d+1}(x_{1}) = v_1$$
with constant $v_1 = - \sum_{j=2} ^d (f_j(X_{i_1,j}) - f_j(X_{i_2,j})) / (X_{i_1,1} - X_{i_2,1} )$ for $x_1\in M_{i_1,i_2}$.
Furthermore, by using \eqref{eq:Haddhelp} once again we obtain
$$f_{1}(x_{1}) = u_1 + v_1 x_1$$
with another constant $u_1 \in \mathbb R$. Following \eqref{eq:Ueberdeckung}, since $\mathcal{X}_1$ is connected and the sets $M_{i_1,i_2}$ are open we can conclude
$$f_{d+1}(x_{1}) = v_1 \text{ and } f_{1}(x_{1}) = u_1 + v_1 x_1$$
for almost all $x_1 \in \mathcal X_1$ since the sets must overlap. Similarly one shows $$f_{j'}(x_{j}) = v_j \text{ and } f_{j}(x_{j}) = u_j + v_j x_j$$
for $j =2,\dots, d$ and almost all $x_j \in \mathcal X_j$. We conclude that
\begin{eqnarray*} 0 &=& \|f\|_n ^2 \\
&=& \frac 1 n \int \sum_{i=1} ^n \bigg \{ f_0 + \sum_{j=1} ^d f_j(x_j) + \sum_{j=1} ^d f_{j'}(x_j) (X_{ij}-x_j)\bigg \}^2 K^{X_i}_h(X_i -x) \mathrm d x\\
&=& \frac 1 n \int \sum_{i=1} ^n \bigg \{ f_0 + \sum_{j=1} ^d u_j + \sum_{j=1} ^d v_jX_{ij}\bigg \}^2 K^{X_i}_h(X_i -x) \mathrm d x\\
&=& \bigg \{ f_0 + \sum_{j=1} ^d u_j + \sum_{j=1} ^d v_jX_{ij}\bigg \}^2 .
\end{eqnarray*}
On the event $\mathcal E $ the covariates $X_i $ do not lie in a linear
subspace of $ \mathbb R^d$. This shows $v_j=0$ for $1\leq j\leq d$. Thus $f_j \equiv 0$ for $d+1 \leq j \leq 2d$ and $f_j = u_j$ for $1 \leq j \leq d$.\newline
\noindent Now, $\int f_j(x_j) \hat p_j(x_j) \mathrm d x_j = 0$ implies that $f_j \equiv 0$ for $1 \leq j \leq d$ and $f_0=0$. This concludes the proof of the lemma.
\end{proof}
Existence and uniqueness of $\widehat m$ on the event $\mathcal E$ under Assumption (A1) follows immediately from the following lemma.
\begin{lemma} \label{lem:Hadd closed}
Make Assumption (A1). Then, on the event $\mathcal E$, for every $D\subseteq\{0,\dots,2d\}$ the linear space
$\sum_{k\in D}\mathcal{H}_k$ is a closed subset of $\mathcal H$. In particular, $\mathcal H_{add}$ is closed.
\end{lemma}
For the proof of this lemma we make use of some propositions introduced below. In the following, we consider sums $L=L_1 + L_2$ of closed subspaces $L_1$ and $ L_2$ of a Hilbert space with $L_1\cap L_2=\{0\}$. In this setup, an element $g\in L$ has a unique decomposition $g=g_1+g_2$ with $g_1\in L_1$ and $g_2\in L_2$. Thus, the projection operator from $L$ onto $L_1$ along $L_2$ given by
\begin{align*}
\Pi_1(L_2):L\rightarrow L_1,\ \Pi_1(L_2)(g)=g_1
\end{align*}
is well defined.
\begin{proposition}
For the sum $L=L_1 + L_2$ of two closed subspaces $L_1$ and $L_2$ of a Hilbert space with $L_1\cap L_2=\{0\}$,
the following conditions are equivalent
\begin{enumerate}
\item[(i)] $L$ is closed.
\item[(ii)] There exists a constant $c>0$ such that for every $g=g_1+g_2\in L$ with $g_1 \in L_1$ and $g_2 \in L_2$ we have
\begin{equation}\label{cond2:closed}
\|g\| \geq c \max \{\|g_1\|, \|g_2\|\}.
\end{equation}
\item[(iii)] The projection operator $\Pi_1(L_2)$ from $L$ onto $L_1$ along $L_2$ is bounded.
\item[(iv)] The gap from $L_1$ to $L_2$ is greater than zero, i.e.,
\[\gamma(L_1,L_2):=
\inf_{g_1 \in L_1} \frac{dist(g_1,L_2)}{\norm{g_1}} >0,
\]
where $\mathrm{dist}(f,V):= \inf_{h \in V} \norm{f-h}$ with the convention $0/0=1$.
\end{enumerate}
\label{lprop:Hadd}
\end{proposition}
\begin{remark}
A version of Proposition \ref{lprop:Hadd} is also true if $L_1\cap L_2\neq\{0\}$. In this case, the quantities involved need to be identified as objects in the quotient space $L/(L_1\cap L_2)$.
\end{remark}
\begin{proposition} \label{lprop:Hadd closed help1}
The sum $L=L_1 + L_2$ of two closed subspaces $L_1$ and $L_2$ of a Hilbert space with $L_1\cap L_2=\{0\}$ is closed if the orthogonal projection of $L_2$ on $L_1$ is compact.\end{proposition}
The proofs of Propositions \ref{lprop:Hadd} and \ref{lprop:Hadd closed help1} can be reconstructed from \cite[A.4 Proposition 2]{bickel1993efficient}, \cite[Chapter 4, Theorem 4.2]{kato2013perturbation} and \cite{kober1940theorem}. For completeness, we have added proofs of the propositions in Appendix \ref{app:B}.\newline
\noindent We now come to the proof of Lemma \ref{lem:Hadd closed}.
\begin{proof} [of Lemma \ref{lem:Hadd closed}] First note that the spaces $\mathcal H_k$ are closed for $k=0,\dots, 2d$.\newline
\noindent We show that $\mathcal H_k + \mathcal H_{k'}$ is closed for $1 \leq k \leq d$. Consider
$R= \min M$ where
\[
M:=\{ r\geq 0\ |\ (\hat p_k^*(x_k))^2 \leq r \hat p_k(x_k) \hat p_k^{**}(x_k) \text{ for all } x_k\in\overline{\mathcal{X}}_k \text{ and } 1 \leq k \leq d\}
\]
and
\[
I_{x_k}:=\bigg\{i\in\{1,\dots,n\}\ \bigg|\ \int_{u \in \mathcal X_{-k}(x_k) }K^{X_i}_h(X_i-u)\mathrm du_{-k} > 0\bigg\}
\]
for $x_k\in\overline{\mathcal X}_k$. By the Cauchy-Schwarz inequality we have $(\hat p_k^*(x_k))^2 \leq \hat p_k(x_k) \hat p_k^{**}(x_k)$ for $x_k\in\overline{\mathcal{X}}_k$ and $1\leq k\leq d$. This implies $R \leq 1$ . Now, equality in the inequality only holds if $X_{ik} - x_k$ does not depend on $i\in I_{x_k}$. On the event $\mathcal E$ for $x_k\in \overline{\mathcal{X}}_k$ there exist $1\leq i_1,i_2\leq n$ with $|x_k-X_{i,k}|<h$ for $i=i_1,i_2$ and $X_{i_1,k}\neq X_{i_2,k}$. Thus, $X_{ik}-x_k$ depends on $i$ for $i\in I_{x_k}$ and the strict inequality holds for all $x_k$. Furthermore,
because the kernel function $k$ is continuous, we have that $\hat p_k$, $\hat p_k^{*}$ and $\hat p_k^{**}$ are continuous. Thogether with the compactness of $\overline{\mathcal{X}}_k$ this implies that $R < 1$ on the event $\mathcal E$.\newline
Now let $f \in \mathcal H_k$ and $g \in \mathcal H_{k'}$ for some $1\leq k\leq d$. We will show
\begin{equation} \label{eq: 2 closed}
\|f+g\|_n^2 \geq (1-R) (\|f\|_n^2 + \|g\|_n^2).
\end{equation}
By application of Proposition \ref{lprop:Hadd} this immediately implies that $\mathcal H_k + \mathcal H_{k'}$ is closed. For a proof of \eqref{eq: 2 closed} note that
\begin{eqnarray*}
\|f+g\|_n^2 &=& n^{-1} \sum_{i=1} ^n (f_k(x_k) + g_0 + (X_{ik} -x_k) g_{k'}(x_k))^2 K_h^{X_i}(X_i -x) \mathrm d x\\
&=& \int ( f_k(x_k) +g_0)^2 \hat p_k(x_k) \mathrm d x_k +
2 \int ( f_k(x_k) + g_0) g_{k'}(x_k) \hat p^{*}_k(x_k) \mathrm d x_ k \\
&& \qquad+ \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k\\
&\geq& \int ( f_k(x_k) +g_0)^2 \hat p_k(x_k) \mathrm d x_k + \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k
\\
&& \qquad -
2 R \int | f_k(x_k) + g_0| |g_{k'}(x_k)| (\hat p_k(x_k)\hat p^{**}_k(x_k))^{1/2} \mathrm d x_ k\\
&\geq& \int ( f_k(x_k) +g_0)^2 \hat p_k(x_k) \mathrm d x_k + \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k
\\
&& -
2 R \left ( \int ( f_k(x_k) +g_0)^2 \hat p_k(x_k) \mathrm d x_k \right )^{1/2} \left ( \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k\right) ^{1/2}
\\
&\geq& (1-R) \int ( f_k(x_k) +g_0)^2 \hat p_k(x_k) \mathrm d x_k + (1-R) \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k
\\
&=& (1-R) \left ( \int f_k(x_k)^2 \hat p_k(x_k) \mathrm d x_k + g_0^2+ \int g_{k'}(x_k)^2 \hat p^{**}_k(x_k) \mathrm d x_ k \right )\\
&\geq& (1-R) (\|f\|_n^2 + \|g\|_n^2),
\end{eqnarray*}
where in the second to last row, we used that $f\in\mathcal{H}_k$.
This concludes the proof of \eqref{eq: 2 closed}.\newline
Note that the statement of the lemma is equivalent to the following statement: For $D_1, D_2\subseteq\{1,\dots,d\}$ and $\delta\in\{0,1\}$ the space $\mathcal \delta\mathcal{H}_0+\sum_{k\in D_1} \mathcal{H}_k +\sum_{k\in D_2}\mathcal{H}_{k'}$ is closed. We show this inductively over the number of elements $s=|D_2\cap D_1|$ of $D_1\cap D_2$.\newline
\noindent For the case $s=0$, note that
for $D_1\cap D_2=\emptyset$, the space $\mathcal \delta\mathcal{H}_0+\sum_{k\in D_1} \mathcal{H}_k+\sum_{k\in D_2}\mathcal{H}_{k'}$ is closed, which can be shown with similar but simpler arguments than the ones used below.
Now let $s\geq 1$, $\delta\in\{0,1\}$, $D_1,D_2\subseteq \{1,\dots,d\}$ with $|D_2\cap D_1|=s-1$ and assume $L_2 = \delta\mathcal{H}_0+\sum_{j\in D_1}\mathcal{H}_j+\sum_{j\in D_2}\mathcal{H}_{j'}$ is closed. Without loss of generality, let $k\in\{1,\dots,d\}\backslash (D_1\cup D_2)$. We will argue that on the event $\mathcal E$ the orthogonal projection of $L_2$ on $L_1 = \mathcal H_{k}+ \mathcal H_{k'}$ is Hilbert-Schmidt, noting that a Hilbert-Schmidt operator is compact.
Using Proposition \ref{lprop:Hadd closed help1} since $L_1$ and $L_2$ are closed, this implies that $L = \mathcal \delta \mathcal{H}_0+\sum_{j\in D_1\cup\{k\}}\mathcal{H}_j+\sum_{j\in D_2\cup\{k\}}\mathcal{H}_{j'}$ is closed which completes the inductive argument.\newline
For an element $f\in L_2$ with decomposition $f= f_0 + \sum_{j\in D_1}f_j+\sum_{j\in D_2}f_{j'}$ the projection onto $L_1+\mathcal{H}_0$ is given by univariate functions $g_k, g_{k'}$ and $g_0 \in \mathbb R$ which satisfy
\begin{eqnarray*}
0&=& \sum_{i=1} ^n \int \bigg (f_0 + \sum_{j\in D_1} f_j(x_j) + \sum_{j\in D_2} f_{j'}(x_j) (X_{ij}-x_j) {- g_0} - g_k(x_k) \\
&& \qquad - g_{k'}(x_k) (X_{ik}-x_k) \bigg )\left(\begin{array}{c}1 \\X_{ik}-x_k\end{array}\right)
K^{X_i}(X_i -x) \mathrm d x_{-k}.
\end{eqnarray*}
Note that $f_0=0$ if $\delta=0$. This implies
\begin{eqnarray*}
&&\left(\begin{array}{c}{g_0+}g_k(x_k) \\g_{k'}(x_k)\end{array}\right)
= \frac 1 {(\hat p_k \hat p_k^{**} - ( \hat p_k^{*})^2)(x_k)} \left(\begin{array}{cc}\hat p_k^{**} & -\hat p_k^{*} \\ -\hat p_k^{*} & \hat p_k\end{array}\right)(x_k)\\
&&\qquad \times \bigg \{f_0 \left(\begin{array}{c}\hat p_k\\\hat p_k^{*}\end{array}\right) (x_k) +
\sum_{j\in D_1}\int f_j(x_j) \left(\begin{array}{c}\hat p_{jk} (x_j,x_k) \\ \hat p_{kj} ^{*}(x_k,x_j)\end{array}\right) \mathrm d x_{j}
\\ && \qquad \qquad+\sum_{j\in D_2}\int f_{j'}(x_j) \left(\begin{array}{c}\hat p^{*}_{jk} (x_j,x_k) \\ \hat p_{jk} ^{**}(x_j,x_k)\end{array}\right) \mathrm d x_{j}\bigg \},
\end{eqnarray*}
where $g_k$ and $g_0$ are chosen such that $\int g_k(x_k) \hat p_k(x_k) \mathrm d x_k =0$.
We now use that the projection of $g_0$ onto $L_1 = \mathcal H_{k}+ \mathcal H_{k'}$ is equal to
\begin{eqnarray*}
&&\left(\begin{array}{c}r_k(x_k) \\r_{k'}(x_k)\end{array}\right)
= \left(\begin{array}{c} g_0 (1 -c_k s_k(x_k) \hat p_k^{**}(x_k)) \\g_0 c_k s_k(x_k) $ $\hat p_k^{*} (x_k)\end{array}\right),
\end{eqnarray*}
where $s_k(x_k) = \hat p_k(x_k)/ ( \hat p_k(x_k) \hat p_k^{**}(x_k)-( \hat p_k^{*})^2(x_k))$ and $c_k = (\int s_k(x_k) \hat p^{**}_k(x_k)$ $ \hat p_k(x_k) \mathrm d x_k) ^{-1} $.
Thus, the projection of $f$ onto $L_1$ is defined by $(g_{k}(x_k) + r_{k}(x_k),$ $ g_{k'}(x_k)+ r_{k'}(x_k)) ^\intercal$.
Under our settings on the event $\mathcal E$ this is a Hilbert-Schmidt operator. This concludes the proof.
\end{proof}
We now come to a short discussion of the convergence of Algorithm 1. The algorithm is used to approximate $\hat m$. In the lemma we denote by
$\hat m^{[r]}$ the outcome of the algorithm after $r$ iterations of the while loop (see Algorithm 1).\newline
\noindent We prove the algorithm for arbitrary starting values, i.e. we can set the $\hat{m}_k(x_k)$ to values other than zero in step 1 of Algorithm 1. The vector of starting values of the algorithm is denoted by $\hat m^{[0]}\in\mathcal{H}_{add}$.
\begin{lemma} \label{lem:conv alg}
Make Assumption (A1). Then, on the event $\mathcal E$, for Algorithm 1 and all choices of starting values $\hat m^{[0]}\in\mathcal{H}_{add}$ we have
$$\| \hat m^{[r]} - \hat m \|_n \leq V ^r \| \hat m^{[0]} - \hat m \|_n,$$
where $0 \leq V=1-\prod_{k=0}^{2d-1}\gamma^2(\mathcal{H}_k,\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})< 1$ is a random variable depending on the observations. \end{lemma}
\begin{remark}
On the event $\mathcal E$, the algorithm converges with a geometric rate where in every iteration step the distance to the limiting value, $\hat m$, is reduced by a factor smaller or equal to $V$. If the columns of the design matrix $X$ are orthogonal, $V$ will be close to zero and if they are highly correlated, $V$ will be close to 1.
The variable $V$ depends on $n$ and is random. Under additional assumptions, as stated in the next section, one can show that with probability tending to one, $V$ is bounded by a constant smaller than 1.
\end{remark}
\begin{proof} [of Lemma \ref{lem:conv alg}]
For a subspace $\mathcal{V}\subseteq\mathcal{H}_{add}$ we denote by $\mathcal{P}_\mathcal{V}$ the orthogonal projection onto $\mathcal{V}$.
For $k=0,\dots,2d$ let $\mathcal{Q}_k:=\mathcal{P}_{\mathcal{H}_k^\perp}=1-\mathcal{P}_k$ be the projection onto the orthogonal complement $\mathcal{H}_k^\perp$ of $\mathcal{H}_k$. The idea is to show the following statements.
\begin{enumerate}
\item[(i)] $Y-\hat{m}^{[r]}=(\mathcal{Q}_{2d}\dots \mathcal{Q}_0)^r(Y-\hat{m}^{[0]})$,
\item[(ii)] $(\mathcal{Q}_{2d}\dots\mathcal{Q}_0 )^r(Y-\hat{m})=Y-\hat{m}$,
\end{enumerate}
This then implies
\begin{align*}
\norm{\hat{m}^{[r]}-\hat{m}}_n &=\norm{\hat{m}^{[r]}-Y+Y-\hat{m}}_n= \norm{(\mathcal{Q}_{2d}\dots\mathcal{Q}_0)^r(\hat{m}^{[0]}-\hat{m})}_n.
\end{align*}
The proof is concluded by showing
\begin{equation}\label{eq: Sinus}
\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_0g}^2_n\leq \bigg(1-\prod_{k=0}^{2d-1}\gamma^2(\mathcal{H}_{k},\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})\bigg)\norm{g}^2_n
\end{equation}
for all $g\in \mathcal{H}_{add}$. Note that $0\leq V:=1-\prod_{k=0}^{2d-1}\gamma^2(\mathcal{H}_k,\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})<1$ by Lemma \ref{lem:Hadd closed} and Proposition \ref{lprop:Hadd}.\newline
\noindent For (i), observe that for all $r\geq 1$ and $k=0,\dots,2d$ we have
\begin{align*}
& Y-\hat{m}^{[r-1]}_0-\dots-\hat{m}_{k-1}^{[r-1]}-\hat{m}_{k}^{[r]}-\dots-\hat{m}^{[r]}_{2d}\\
& =(1-\mathcal{P}_k)(Y-\hat{m}^{[r-1]}_0-\dots-\hat{m}_{k-1}^{[r-1]}-\hat{m}_{k+1}^{[r]}-\dots-\hat{m}^{[r]}_{2d})\\
& =(1-\mathcal{P}_k)(Y-\hat{m}^{[r-1]}_0-\dots-\hat{m}_{k-1}^{[r-1]}-\hat{m}_{k}^{[r-1]}-\hat{m}_{k+1}^{[r]}-\dots-\hat{m}^{[r]}_{2d})\\
&=\mathcal{Q}_k(Y-\hat{m}^{[r-1]}_0-\dots-\hat{m}_{k}^{[r-1]}-\hat{m}_{k+1}^{[r]}-\dots-\hat{m}^{[r]}_{2d}).
\end{align*}
The statement follows inductively by beginning with the case $r=1, k=0$. Secondly, (ii) follows from
\begin{align*}
\mathcal{Q}_r\dots\mathcal{Q}_0(Y-\hat{m})& =\mathcal{Q}_r\dots\mathcal{Q}_0\mathcal{P}_{\mathcal{H}_0^\perp\cap\dots\cap \mathcal{H}_{2d}^\perp}(Y) = \mathcal{P}_{\mathcal{H}_0^\perp\cap\dots\cap \mathcal{H}_{2d}^\perp}(Y)=Y-\hat{m}.
\end{align*}
It remains to show the inequality in \eqref{eq: Sinus}.\newline
\noindent For $0\leq k\leq 2d$ define $\mathcal{N}_k:=\mathcal{H}_{k}+\dots+ \mathcal{H}_{2d}$. We prove $\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_jg}_n^2\leq (1-\prod_{k=j}^{2d-1}\gamma^2(\mathcal{H}_{k},\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d}))\norm{g}_n^2$ for all $g\in\mathcal{H}_{add}$ and $0\leq j\leq 2d$ using an inductive argument.\newline
The case $j=2d$ is trivial. For $0\leq j<2d$ and any $g\in \mathcal{H}_{add}$ let $g_{j}^\perp:=\mathcal{Q}_jg=g'+g''$ with $g':=\mathcal{P}_{\mathcal{N}_{j+1}^\perp}(g)$ and $g'':=\mathcal{P}_{\mathcal{N}_{j+1}}(g)$. Then, by orthogonality, we have
\[
\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_{j+1}g_j^\perp}_n^2=\norm{g'+\mathcal{Q}_{2d}\dots\mathcal{Q}_{j+1}g''}^2_n=\norm{g'}^2_n+\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_{j+1}g''}^2_n.
\]
Induction gives
\[
\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_{j+1}g''}^2_n\leq \bigg(1-\prod_{k=j+1}^{2d-1}\gamma^2(\mathcal{H}_k,\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})\bigg)(\norm{g^\perp_j}_n^2-\norm{g'}_n^2)
\]
which implies
\begin{align*}
\norm{\mathcal{Q}_{2d}\dots\mathcal{Q}_{j+1}g_j^\perp}_n^2\leq\bigg(& 1-\prod_{k=j+1}^{2d-1}\gamma^2(\mathcal{H}_k,\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})\bigg)\norm{g^\perp_j}_n^2\\
& +\prod_{k=j+1}^{2d-1}\gamma^2(\mathcal{H}_k,\mathcal{H}_{k+1}+\dots+\mathcal{H}_{2d})\norm{g'}_n^2.
\end{align*}
By Lemma \ref{lem:Hadd closed} and Lemma \ref{lemma gamma} we have
\[
\norm{g'}_n^2\leq \norm{\mathcal{P}_{\mathcal{N}_{j+1}^\perp}\mathcal{Q}_1}_n^2=\norm{\mathcal{P}_{N_{j+1}}\mathcal{P}_{\mathcal{H}_j}}_n^2=1-\gamma^2(\mathcal{H}_j,\mathcal{H}_{j+1}+\dots+\mathcal{H}_{2d}).
\]
This concludes the proof by noting that $\norm{g_j^\perp}_n\leq\norm{g}_n$.
\end{proof}
\section{Asymptotic properties of the estimator} \label{sec:asymp}
In this section we will discuss asymptotic properties of the local linear smooth backfitting estimator.
For simplicity we consider only the case that $\mathcal X$ is a product of intervals $\mathcal X_j= (a_j, b_j) \subset \mathbb R$.
We make the following additional assumptions:
\begin{enumerate}
\item[(A2)] The observations $(Y_i, X_i)$ are i.i.d. and the covariates $X_i$ have one-dimansional marginal densities $p_j$ which are strictly positive on $[a_k, b_k]$. The two-dimensional marginal densities $p_{jk}$ of $(X_{i,j}, X_{i,k})$ are continuous on their support $[a_j, b_j] \times [a_k, b_k] $.
\item[(A3)] It holds
\begin{equation} \label{eq:model}Y_i= m_0 + m_1(X_{i1}) + \dots + m_d(X_{id})+ \varepsilon_i, \end{equation}
for twice continuously differentiable functions $m_j : \mathcal X_j \to \mathbb R$ with
$\int m_j(x_j)$ $ p_j(x_j) \mathrm d x_j =0$. The error variables $\varepsilon_i$ satisfy
$\mathbb E[\varepsilon_i | X_i] =0$ and $$\sup_{x \in \mathcal X } \mathbb E[|\varepsilon_i| ^{5/2} | X_i=x] < \infty.$$
\item[(A4)]
There exist constants $c_1,\dots,c_d> 0$ with $n^{1/5} h_j \to c_j$ for $n \to \infty$. To simplify notation we assume that $h_1=\dots=h_d$. In abuse of notation we write $h$ for $h_j$ and $c_h $ for $c_j$.
\end{enumerate}
From now on we will write $\hat m^n =(\hat m^n_0, \hat m^n_1, \dots, \hat m^n_{2d})$ for the estimator $\hat m$ to indicate its dependence on the sample size $n$. The following theorem states an asymptotic expansion for the components $ \hat m^n_1, \dots, \hat m^n_{d}$.
Later in this section we will state some lemmas which will be used to prove the result.
\begin{theorem} \label{theo: asymp}
Make assumptions (A1) -- (A4). Then
\begin{align*} &\bigg| \hat m^n_j(x_j) - m_j(x_j)- \left (\beta_j(x_j) -\int \beta_j(u_j) p_j(u_j) \mathrm d u_j\right )
- v_j(x_j)\bigg| \\ &= o_P(h^2+\{nh\}^{-1/2}) = o_P(n^{-2/5}),
\end{align*}
holds uniformly over ${1 \leq j \leq d}$ and ${a_j \leq x_j \leq b_j }$,
where $v_j$ is a stochastic variance term
\[
v_j(x_j)= \frac { \frac 1 n \sum_{i=1} ^n h^{-1} k(h^{-1}(X_{ij} -x_j))\varepsilon_i} { \frac 1 n \sum_{i=1} ^n h^{-1} k(h^{-1}(X_{ij} -x_j))}=O_p(\{nh\}^{-1/2})
\]
and $\beta_j$ is a deterministic bias term
\[
\beta_j(x_j)=\frac 1 2 h^2 m_j^{''}(x_j)\ \frac {b_{j,2}(x_j) ^2- b_{j,1}(x_j) b_{j,3}(x_j)}{b_{j,0}(x_j) b_{j,2}(x_j)- b_{j,1}(x_j) ^2}=O(h^2),
\]
with
$b_{j,l}(x_j)= \int_{\mathcal X_j} k(h^{-1} (u_j-x_j))(u_j-x_j)^l h^{-l-1}b_j(u_j)^{-1}\mathrm du_j$ and $ b_{j}(x_j)= \int_{\mathcal X_j} k(h^{-1} (x_j-w_j)) h^{-1}\mathrm dw_j$ for $0 \leq l \leq 2$.
\end{theorem}
The expansion for $\hat m^n_j$ stated in the theorem neither depends on $d$ nor on functions $m_k$ ($k \not = j$). In particular, this shows that the same expansion holds for the local linear estimator $\widetilde m^n_j$ in the oracle model where the functions $m_k$ ($k \not = j$) are known. More precisely, in the oracle model one observes i.i.d. observations $(Y_i^\ast,X_{ij})$ with
\begin{align}\label{oracle}
Y^\ast_i = m_j(X_{ij}) + \varepsilon_i , \quad Y^\ast_i = Y_i-\sum_{k \neq j} m_k({X_{ik}}),
\end{align}
and the local linear estimator $\widetilde m^n_j$ is defined as the second component that minimises
the criterion
\begin{eqnarray*}
\widetilde S(f_0, f_j, f^{(1)}_j)&=& \sum_{i=1}^n \int_{\mathcal X} \left \{ Y_i^\ast - f_0 -f_j(x_j) - f _j^{(1)}(x_j) (X_{ij}-x_j)\right \} ^2 \\ && \qquad \times \kappa_{h}^{X_{ij}}(X_{ij}-x) \mathrm dx_j
\end{eqnarray*}
with boundary corrected kernel
\[
k_h^{u}(u-x)= \frac{ \kappa \left(\frac{u-x}{h}\right)}{\int_{\mathcal X_j} \kappa \left(\frac{u-v}{h}\right) \mathrm dv }.
\]
We conclude that the local linear smooth backfitting estimator $\hat m_j$ is asymptotically equivalent to the local linear estimator $\widetilde m^n_j $ in the oracle model. We formulate this asymptotic equivalence as a first corollary of Theorem \ref{theo: asymp}. In particular, it implies that the estimators have the same first order asymptotic properties.
\begin{corollary} \label{corr: compare tilde}
Make assumptions (A1) -- (A4). Then it holds uniformly over ${1 \leq k \leq d}$ and ${a_j \leq x_j \leq b_j }$ that
\begin{eqnarray*} &&\bigg| \hat m^n_j(x_j) - \widetilde m^n_j(x_j) \bigg| = o_P(h^2).
\end{eqnarray*}
\end{corollary}
For $x_j \in (a_j + 2 h, b_j - 2h)$ the bias term $\beta_j$ simplifies and we have that
\[
\beta_j(x_j) =h^2 \frac 1 2 m_j^{''}(x_j) \int k(v)v^2\mathrm dv.
\]
This implies the following corollary of Theorem \ref{theo: asymp}.
\begin{corollary} \label{corr: asymp}
Make assumptions (A1) -- (A4). Then it holds uniformly over ${1 \leq k \leq d}$ and ${a_j + 2 h \leq x_j \leq b_j - 2h}$ that
\begin{eqnarray*} &&\bigg| \hat m^n_j(x_j) - m_j(x_j) - \frac 1 2 \left (m_j^{''}(x_j) - \int m_j^{''}(u_j) p_j(u_j) \mathrm d u_j\right ) h^2 \ \int k(v)v^2\mathrm dv \\
&& \qquad - v_j(x_j)\bigg| = o_P(h^2).
\end{eqnarray*}
\end{corollary}
Corollary \ref{corr: asymp} can be used to derive the asymptotic distribution of $\hat m^n_j(x_j)$ for an $x_j \in (a_j, b_j)$. Under the additional assumption that $\sigma_j^2(u) = \mathbb E[\varepsilon_i^2| X_{ij}=u]$ is continuous in $u=x_j$ we get under (A1) -- (A4) that $n^{2/5} ( \hat m^n_j(x_j) - m_j(x_j) )$ has an asymptotic normal distibution with mean $c_h \frac 1 2 \left (m_j^{''}(x_j) - \int m_j^{''}(u_j) p_j(u_j) \mathrm d u_j\right ) \int k(v)v^2\mathrm dv$ and variance $c_h^{-1} \sigma_j^2(x_j) p_j^{-1}(x_j) \int k(v)^2\mathrm dv$. This is equal to the asymptotic limit distribution of the classical local linear estimator in the oracle model in accordance with Corollary \ref{corr: compare tilde}.\newline
Now, we come to the proof of Theorem \ref{theo: asymp}.\newline
First, we define the operator $\mathcal S_{n} = (\mathcal S_{n,0} , \mathcal S_{n,1} ,\dots, \mathcal S_{n,2d} ):\mathcal{G}^n\rightarrow\mathcal{G}^n$ with
\[
\mathcal G^n= \{(g_0,\dots,g_{2d})| g_0\in \mathbb R, g_l, g_{l'} \in \text{L}_2(p_l)\ \text{with } P_0(g_l) =0\ \text{for } l=1,\dots,d\},
\]
where $\mathcal S_{n,k}$ maps $g=(g_0,\dots,g_{2d})\in\mathcal{G}^n$ to $f_k$ with
\begin{align*} f_0 & = \mathcal P_0\left( \sum _{1 \leq l \leq 2d} g_l\right)=\mathcal P_0\left( \sum _{d+1 \leq l \leq 2d} g_l\right) \in \mathbb R,\\
f_k(x_k)&= \mathcal P_k\left( \sum _{0 \leq l \leq 2d, l\not = k} g_l\right) (x), \end{align*}
for $1 \leq k \leq 2d$.
With this notation we can rewrite
the backfitting equation \eqref{eq:back} as
\begin{equation} \label{eq:back2} \bar m^n(Y)= \hat m^n +\mathcal S_{n} \hat m^n,
\end{equation}
where for $z \in \mathbb R ^n$ we define $\bar m^n_0(z) = \bar z = \frac 1 n \sum_{i=1} ^n z_i$ and for $1 \leq j \leq d,$
\begin{eqnarray*}
\bar m^n_j(z) (x_j) &=& \hat p_j(x_j) ^{-1} \frac 1 n \sum_{i=1} ^n (z_i - \bar z) \int K_h^{X_i}(X_i -x) \mathrm d x_{-j}, \\\bar m^n_{j'}(z) (x_j) &=& \hat p^{**}_j(x_j) ^{-1} \frac 1 n \sum_{i=1} ^n (X_{ij} -x_j) z_i \int K_h^{X_i}(X_i -x) \mathrm d x_{-j}.
\end{eqnarray*}
The following lemma shows that $I+\mathcal S_{n}$ is invertible on the event $\mathcal E$. Here we denote the identity operator by $I$.
\begin{lemma} \label{lem:invert}
On the event $\mathcal E$ the operator $I+ \mathcal S_{n}: \mathcal G^n \to \mathcal G^n$ is invertible.
\end{lemma}
\begin{proof}
Suppose that for some $g \in \mathcal G^n $ it holds that $(I+\mathcal S_{n})(g)=0$. We have to show that this implies $g=0$.\newline
For the proof of this claim note that $g_k + \mathcal S_{n,k} (g) $ is the orthogonal projection of $\sum_{j=0} ^{2d} g_j$ onto $\mathcal H_k$. Furthermore, we have that $g_k$ is an element of $\mathcal H_k$. This gives that
$$\left\langle g_k , \sum_{j=0} ^{2d} g_j\right\rangle_n =0.$$ Summing over $k$ gives
$$\left\langle \sum_{j=0} ^{2d} g_j , \sum_{j=0} ^{2d} g_j\right\rangle_n =0.$$
According to Lemma \ref{lem:uni Hadd} on the event $\mathcal E$ we have $g_0 = 0$ and $g_j \equiv 0$ for $j=1,\dots,2d$. This concludes the proof of the lemma.
\end{proof}
One can show that under conditions (A1) -- (A4) the probability of the event $\mathcal E$ converges to one. Note that we have assumed that $\mathcal X=\prod_{j=1} ^d \mathcal X_j$. We conclude that under (A1) -- (A4) $I+\mathcal S_{n}$ is invertible with probability tending to one.\newline
Thus we have that with probability tending to one
\begin{eqnarray} \label{eq:back3} &&\hat m^n - m -\bar m^n(\varepsilon)- \beta_n + \Delta_n m + \Delta_n \beta_n \\ \nonumber
&& \qquad= ( I+ \mathcal S_{n})^{-1} ( I+ \mathcal S_{n}) (\hat m^n - m - \bar m^n(\varepsilon)-\beta_n + \Delta_n m + \Delta_n \beta_n),\end{eqnarray}
where
$m$ has components $m_0,\dots,m_{2d}$ with $m_0,\dots, m_d$ as in \eqref{eq:model} and with $m_{j'} = m_j^{\prime}$ for $1 \leq j \leq d$.
Furthermore, $\beta(x)$ has components $\beta_0=0$, $\beta_j(x_j)$ and
\begin{align*}
\beta_{j'}(x_j)&=\frac 1 2 m_j^{''}(x_j)\ \frac {b_{j,0}(x_j) b_{j,3}(x_j)- b_{j,1}(x_j) b_{j,2}(x_j)}{b_{j,0}(x_j) b_{j,2}(x_j)- b_{j,1}(x_j) ^2} h
\end{align*}
for $j=1,\dots,d$ with $ b_{j,l}(x_j)$ defined above.
Additionally, the norming constants are given by
\begin{align*}
&(\Delta_n \beta)_j = \int \beta_j (x_j) \hat p_j(x_j) \mathrm d x_j,\\
&(\Delta_n m)_j = \int m_j (x_j) \hat p_j(x_j) \mathrm d x_j, \\
&(\Delta_n \beta)_{j'} = (\Delta_n m)_{j'}=0
\quad \text{for}\ j=1,\dots,d,\\
&(\Delta_n \beta)_0 = \sum_{j=1} ^{d} \int \beta_{j'} (x_j) \hat p^*_j(x_j) \mathrm d x_j,\\
&(\Delta_n m)_0 = \sum_{j=1} ^{d} \int m_{j'} (x_j) \hat p^*_j(x_j) \mathrm d x_j.
\end{align*}
One can verify that for $a_j+ 2 h_j \leq x_j \leq b_j - 2 h_j$ one has $\beta_{j'}(x_j)=o_P(h)$. We have already seen that $\beta_j(x_j) = \frac 1 2 m_j^{''}(x_j) \int k(v)v^2\mathrm dv + o_P(h^2)$ holds for such $x_j$.\newline
For the statement of Theorem \ref{theo: asymp} we have to show that for $1 \leq j \leq d$ the $j$-th component on the left hand side of equation \eqref{eq:back3} is of order $o_P(h^2)$ uniformly for $a_j +2h \leq x_j \leq b_j - 2h$.\newline
For a proof of this claim we first analyze the term
\begin{align} \label{eq:back4} D_n&=( I+ \mathcal S_{n}) (\hat m^n - m -\bar m^n(\varepsilon) - \beta_n+ \Delta_n m + \Delta_n \beta_n) \\ \nonumber
& = \bar m^n(Y) - ( I+ \mathcal S_{n}) (m +\bar m^n(\varepsilon) + \beta_n - \Delta_n m - \Delta_n \beta_n).\end{align}
For this sake we split the term $ \bar m^n(Y)$ into the sum of a stochastic variance term and a deterministic expectation term:
\begin{align}\label{error_decomposition}
\bar m^n(Y) = \bar m^n(\varepsilon) + \sum_{j=0}^d\bar m^n(\mu_{n,j}),
\end{align}
where
\begin{align*}
\varepsilon &= Y-\sum_{j=0}^d \mu_{n,j},\\
\mu_{n,j} &=(m_j(X_{ij}))_{i=1,..,n} \quad \text{for}\ j=1,\dots,d,\\
\mu_{n,0} &=(m_0)_{i=1,..,n} .
\end{align*}
\newline
We write $D_n=D_n^{\beta}+D_n^\varepsilon$, with
\begin{align*}
D_n^{\beta}&= \sum_{j=0}^d\bar m^n(\mu_{n,j}) - ( I+ \mathcal S_{n}) (m + \beta_n - \Delta_n m - \Delta_n \beta_n) , \\
D_n^{\varepsilon}&= \mathcal S_{n} (\bar m^n(\varepsilon)).
\end{align*}
The following lemma treats the conditional expectation term $D^{\beta}$.
\begin{lemma}\label{lemma:bias} Assume (A1) -- (A4).
It holds
$D^\beta_{n,0} = o_p(h^2) $ and
\[
\sup_{x_k\in \mathcal X_k}\big|D^\beta_{n,k}(x_k)\big|=
\begin{cases} o_p(h^2) &\text{for } \ 1\leq k\leq d, \\
o_p(h) &\text{for } \ d+1\leq k\leq 2d .
\end{cases}
\]
\end{lemma}
\begin{proof}
The lemma follows by application of lengthy calculations using second order Taylor expansions
for $m_j(X_{ij})$ and by application of laws of large numbers.
\end{proof}
We now turn to the variance term.
\begin{lemma}\label{lemma:var} Assume (A1) -- (A4).
It holds $D^\varepsilon_{n,0} = o_p(h^2) $ and
\[
\sup_{x_k\in \mathcal X_k}\big|D^\varepsilon_{n,k}(x_k)\big|=
\begin{cases} o_p(h^2) &\text{for } \ 1\leq k\leq d, \\
o_p(h) &\text{for } \ d+1\leq k\leq 2d .
\end{cases}
\]
\end{lemma}
\begin{proof}
One can easily check that $D^\varepsilon_{n,k}(x_k)$ consists of weighted sums of $\varepsilon_i$ where the weights are of the same order for all $1 \leq i \leq n$. For fixed $x_k$ the sums are of order $O_P(n^{-1/2})$ for $1 \leq k \leq d$ and of order $O_P(h^{-1} n^{-1/2})$ for $d+1 \leq k \leq 2d$. Using the conditional moment conditions on $\varepsilon_i$ in Assumption (A3) we get the uniform rates stated in the lemma.
\end{proof}
It remains to study the behaviour of $( I+ \mathcal S_{n})^{-1} D_n^\varepsilon$ and $( I+ \mathcal S_{n})^{-1} D_n^\beta$. We will use a small transformation of $\mathcal S_{n}$ here which is better suitable for an inversion. Define the following $2 \times 2$ matrix $A_{n,k}(x)$ by
$$A_{n,k}(x) =\frac 1 {\hat p_k \hat p_k^{**} - (\hat p_k^{*})^2} \left(\begin{array}{cc}\hat p_k^{**}\hat p_k & \hat p_k^{**}\hat p_k^{*} \\\hat p_k^{*}\hat p_k & \hat p_k\hat p_k^{**}\end{array}\right)(x_k).$$
Furthermore, define the $2d \times 2d$ matrix $A_n(x)$ where the elements with indices $(k,k), (k,k'),$ $ (k', k),(k',k')$ are equal to the elements of $A_{n,k}(x)$ with indices $(1,1), (1,2), (2,1),(2,2)$. We now define $\tilde {\mathcal S}_{n}$ by the equation $I+ \tilde {\mathcal S}_{n} = A_n(I+ \mathcal S_{n})$. Below we will make use of the fact that $\tilde {\mathcal S}_{n}$ is of the form
\begin{eqnarray} \label{eq:AQ1} \tilde {\mathcal S}_{n,k}m(x) = \sum_{l \not \in \{k,k'\}} \int q_{k,l} (x_k,u) m_l(u) \mathrm d u+ \sum_{l \in \{k,k'\}} \int q_{l} (u) m_l(u) \mathrm d u,\\
\label{eq:AQ2} \tilde {\mathcal S}_{n,k'}m(x) = \sum_{l \not \in \{k,k'\}} \int q_{k',l} (x_k,u) m_l(u) \mathrm d u+ \sum_{l \in \{k,k'\}} \int q_{l} (u) m_l(u) \mathrm d u\end{eqnarray}
for $1 \leq k \leq d$ with some random functions $q_{k,l},q_{l}$ which fulfill that $\int q_{k,l}(x_k, u ) ^2 \mathrm d u$ and $\int q_k( u ) ^2 \mathrm d u$ are of order $O_P(1)$ uniformly over $1 \leq k,l\leq 2d$ and $x_k$.
Note that we need $\tilde {\mathcal S}_{n}$ because ${\mathcal S}_{n}$ can not be written in the form of \eqref{eq:AQ1} and \eqref{eq:AQ2}. The operator $\tilde {\mathcal S}_{n}$ differs from ${\mathcal S}_{n}$
in the $h$-neighbourhood of the boundary by terms of order $h^2$. Otherwise the difference is of order $o_p(h^2)$.
Outside of the $h$-neighbourhood of the boundary, for $n \to \infty$, the matrix $A_n(x)$ converges to the identity matrix.
Thus $\tilde {\mathcal S}_{n}$ is a second order modification of ${\mathcal S}_{n}$ with the advantage of having \eqref{eq:AQ1}-\eqref{eq:AQ2}.\newline
For our further discussion we now introduce the space $\mathcal G^0$ of tuples $f=(f_0, f_1,\dots,$ $f_{2d})$ with $f_0=0$ and $f_k,f_{k'} : \mathcal X_k \to \mathbb R$ with $\int f_k(x_k) p_k(x_k) \mathrm d x_k =0$ and endow it with the norm $\|f\|^2 = \sum _{k=1} ^{d} (f_k(x_k)^2 + f_{k'}(x_k)^2) p_k(x_k) \mathrm d x_k $.
The next lemma shows that the norm of $H_n( I+ \mathcal S_{n})^{-1} D_n^\varepsilon$ and $H_n( I+ \mathcal S_{n})^{-1} D_n^\beta$ is of order $o_P(h^2)$. Here $H_n$ is a diagonal matrix where the first $d+1$ diagonal elements equal 1. The remaining elements are equal to $h$.
\begin{lemma}\label{lemma:rate of norm} Assume (A1) -- (A4).
Then it holds that
$\|H_n( I+ \mathcal S_{n})^{-1} D_n^*\| = \|H_n( I+\tilde{ \mathcal Q}_{n})^{-1} A_nD_n^*\|=o_P(h^2)$ for $D_n^*=D_n^\varepsilon$ and $D_n^*= D_n^\beta$.\end{lemma}
\begin{proof}
Define $\bar D_n^\varepsilon$ and $\bar D_n^\beta$ by $\bar D_{n_k}^\varepsilon(x_k) = D_{n,k}^\varepsilon(x_k) - \int D_{n,k}^\varepsilon(u_k) p_k(u_k) \mathrm d u_k$ and $\bar D_{n,k}^\beta(x_k) = D_{n,k}^\beta(x_k) - \int D_{n,k}^\beta(u_k) p_k(u_k) \mathrm d u_k$ for $1 \leq k \leq d$ and $\bar D_{n,k}^\varepsilon= D_{n,k}^\varepsilon$ and $\bar D_{n,k}^\beta=D_{n,k}^\beta$, otherwise. It can be checked that it suffices to prove the lemma with $ D_n^\varepsilon$ and $D_n^\beta$ replaced by $\bar D_n^\varepsilon$ and $\bar D_n^\beta$. Note that $\bar D_n^\varepsilon$ and $\bar D_n^\beta$ are elements of $\mathcal G^0$. For the proof of this claim we compare the operator $\tilde{ \mathcal S}_{n}$ with the operator
$\mathcal S_{0}$ defined by $\mathcal S_{0,0}(g) =0$, $\mathcal S_{0,k'}(g)(x_k) = 0$ and $$\mathcal S_{0,k}(g)(x_k) = \sum_{j\not = k} \int_{\mathcal X_j} g_j(u_j) \frac {p_{j,k} (u_j,x_k)}{p_{k} (x_k)} \mathrm d u_j$$
for $1 \leq k \leq d$.
By standard kernel smoothing theory one can show that
$$ \sup _{g\in \mathcal G^0, \|g \| \leq 1} \| (\mathcal S_{0} - H_n \tilde{ \mathcal S}_{n} H_n^{-1}) g \| = o_P(1).$$
For the proof of this claim one makes use of the fact that non-vanishing differences in the $h$-neighbourhood of the boundary are asymptotically negligible in the calculation of the norm because the size of the neighbourhood converges to zero.\newline
In the next lemma we will show that $I + \mathcal S_{0}$ has a bounded inverse. This implies the statement of the lemma by applying the following expansion:
\begin{eqnarray*}
&&(I + H_n \tilde{ \mathcal S}_{n} H_n^{-1})^{-1}- (I + \mathcal S_{0})^{-1} \\
&& \qquad = (I + \mathcal S_{0})^{-1} ((I + \mathcal S_{0}) (I + H_n \tilde{ \mathcal S}_{n} H_n^{-1})^{-1} -I)\\
&& \qquad = (I + \mathcal S_{0})^{-1}(((I + H_n \tilde{ \mathcal S}_{n} H_n^{-1}) (I + \mathcal S_{0})^{-1} )^{-1} -I)\\
&& \qquad = (I + \mathcal S_{0})^{-1} ( (I + (H_n \tilde{ \mathcal S}_{n} H_n^{-1}- \mathcal S_{0}) (I + \mathcal S_{0})^{-1} )^{-1} -I)
\\
&& \qquad = (I + \mathcal S_{0})^{-1} \sum_{j=1} ^\infty (I + (-1)^j(H_n \tilde{ \mathcal S}_{n} H_n^{-1}- \mathcal S_{0}) (I + \mathcal S_{0})^{-1} )^j.
\end{eqnarray*}
This shows the lemma because of $H_n( I+ \tilde{ \mathcal S}_{n})^{-1}A_nD_n^* = H_n( I+ \tilde{ \mathcal Q}_{n})^{-1}H_n^{-1}H_n A_nD_n^* = (I+H_n\tilde{ \mathcal S}_{n} H_n^{-1} )^{-1}H_n A_nD_n^* $ for $D_n^*=D_n^\varepsilon$ and $D_n^*=D_n^\beta$. \end{proof}
\begin{lemma}\label{lemma:bd inverse} Assume (A1) -- (A4).
The operator $I+ \mathcal S_{0}: \mathcal G^0 \to \mathcal G^0$ is bijective and has a bounded inverse.
\end{lemma}
\begin{proof}
For a proof of this claim it suffices to show that the operator $I+ \mathcal S_{*}: \mathcal G^* \to \mathcal G^*$ is bijective and has a bounded inverse where $\mathcal G^*$ is the space of tuples $f=(f_1,\dots,f_{d})$ where $f_k : \mathcal X_j \to \mathbb R$ with $\int f_k(x_k) \mathrm d x_k =0$ with norm $\|f\|^2 = \sum _{k=1} ^{d} f_k(x_k)^2 p_k(x_k) \mathrm d x_k $ and $$\mathcal S_{*,k}(g)(x_k) = \sum_{j\not = k} \int_{\mathcal X_j} g_j(u_j) \frac {p_{j,k} (u_j,x_k)}{p_{k} (x_k)} \mathrm d u_j$$
for $1 \leq k \leq d$. We will apply the bounded inverse theorem. For an application of this theorem we have to show
that $I+ \mathcal S_{*}$ is bounded and bijective. It can easily be seen that the operator is bounded. It remains to show that it is
surjective. We will show that
(i) $(I+ \mathcal S_{*}) g^n \to 0$ for a sequence $g^n \in \mathcal G^*$ implies that $g^n \to 0$.
(ii) $ \int g_k (I+ \mathcal S_{*})_k r (x_k) p_k(x_k) \mathrm d x_k =0$ for all $ g \in \mathcal G^*$ implies that $r=0$.
\noindent Note that (i) implies that $ \mathcal G^{**} = \{ (I+ \mathcal S_{*}) g: g \in \mathcal G^{*}\}$ is a closed subset of $ \mathcal G^{*} $. To see this suppose that $(I+ \mathcal S_{*}) g^n \to g$ for $g,g^n \in \mathcal G^{*} $. Then (i) implies that $ g^n$ is a Cauchy sequence and thus $ g^n$ has a limit in $\mathcal G^{*} $ which implies that $(I+ \mathcal S_{*}) g^n$ has a limit in $ \mathcal G^{**}$. Thus $ \mathcal G^{**} $ is closed.\newline
From (ii) we conclude that the orthogonal complement of $ \mathcal G^{**}$ is equal to $\{0\}$. Thus the closure of $ \mathcal G^{**}$ is equal to $ \mathcal G^{*}$. This shows that $ \mathcal G^{*}= \mathcal G^{**}$ because $\mathcal G^{**}$ is closed. We conclude that $(I+ \mathcal S_{*})$ is surjective.\newline
It remains to show (i) and (ii). Fo a proof of (i) note that $(I+ \mathcal S_{*}) g^n \to 0$ implies that
$$\int g_k^n(x_k) (I+ \mathcal S_{*})_k g^n(x_k) p_k(x_k) \mathrm d x_k \to 0$$
which shows $$\sum_{k=1}^d \int g_k^n(x_k) ^2 p_k(x_k) \mathrm d x_k + \sum_{k\not = j} g_k^n(x_k) p_{kj}(x_k,x_j) g_j(x_j) \mathrm d x_k \mathrm d x_j \to 0.$$
Thus we have $$\mathbb E[ (\sum _{k=1} ^d g_k(X_{ik}))^2] \to 0.$$
By application of Proposition \ref{lprop:Hadd} (ii) we get that $\max_{ 1 \leq k \leq d} \mathbb E[ g_k(X_{ik})^2] \to 0$, which shows (i).\newline
Claim (ii) can be seen by a similar argument. Note that $ \int g_k (I+ \mathcal S_{*})_k r p_k(x_k) \mathrm d x_k =0$ for all $ g \in \mathcal G^*$ implies that $ \int r_k (I+ \mathcal S_{*})_k r (x_k)p_k(x_k) \mathrm d x_k =0$.
\end{proof}
We now apply the results stated in the lemma for the final proof of Theorem \ref{theo: asymp}.
\begin{proof}[of Theorem \ref{theo: asymp}]
From \eqref{eq:back3} and Lemma \ref{lemma:rate of norm} we know that the L$_2$ norm of $\hat m^n - m -\bar m^n(\varepsilon)- \beta_n + \Delta_n m + \Delta_n \beta_n=H_n( I+ \mathcal S_{n})^{-1} (D_n^\varepsilon + D_n^\beta) = H_n( I+\tilde{ \mathcal S}_{n})^{-1} A_n(D_n^\varepsilon + D_n^\beta)$ is of order $o_P(h^2)$. Note that $H_n( I+\tilde{ \mathcal S}_{n})^{-1} = H_n - H_n \tilde{ \mathcal S}_{n} ( I+\tilde{ \mathcal S}_{n})^{-1}. $ We already know that the sup norm of all components in $H_nA_n(D_n^\varepsilon + D_n^\beta)$ are of order $o_P(h^2)$. Thus, it remains
to check that the sup norm of the components of $H_n \tilde{ \mathcal S}_{n} ( I+\tilde{ \mathcal S}_{n})^{-1}A_n(D_n^\varepsilon + D_n^\beta)$ is of order $o_P(h^2)$. But this follows by application of the just mentioned bound on the L$_2$ norm of $H_n( I+ \mathcal S_{n})^{-1} (D_n^\varepsilon + D_n^\beta)$, by equations \eqref{eq:AQ1} --
\eqref{eq:AQ2}, and the bounds for the random functions $q_{k,l}$ and $q_{l}$ mentioned after the statement of the equations. One gets a bound for the sup norms by application of the Cauchy Schwarz inequality.
\end{proof}
|
2,877,628,090,765 | arxiv | \section*{Preliminary}Fluid dynamics equations written in terms of vorticity favourably differ from ones covering velocity evolution. So, for both 2D and 3D Navier-Stokes systems the vorticity dynamics involves a fewer number of equations rather than in the velocity-pressure formulation. For barotropic fluid the curl removes the most problematic term - pressure. If the system doesn't involve boundary condition (e.g. Cauchy problem), then the $\operatorname{curl}$ operator significantly simplifies corresponding {{\it initial-value problem}. But in the case of {\it initial-boundary-value problem} it isn't true since Dirichlet no-slip boundary turns out to be a sophisticated integral condition. And so, vorticity boundary conditions mostly are deduced in vorticity-stream formulation\cite{Wu}\cite{And}\cite{WJ}\cite{Suh}. In \cite{Zakh} was studied the parametric boundary condition covering vorticity and stream. Some new results on Newman and Dirichlet boundary conditions for vorticity on solid walls were given in \cite{OH}.
We will investigate the boundary vorticity which corresponds to the no-slip condition. At first sight there is no analogous to no-slip condition only in terms of vorticity without additional functions involved (e.g. stream function, velocity). Biot-Savar law restores the solenoidal velocity field $\mathbf{v}(\mathbf{x})$ induced by vorticity $w(\mathbf{x})$. It expresses vorticity via velocity field by some integral relation. If the flow interacts with solid by no-slip condition, then it turns to zero integral relation in Biot-Savar law which doesn't admit explicit integration. If we look at dolphins, then their skin can read vorticity distribution in order to prevent turbulence. In a playful way we can say that dolphins know something about boundary vorticity.
In this paper will be established vorticity boundary condition for both linear and nonlinear Helmholtz equations without any stream function involved in. For the linear Stokes system in the exterior of the disc such boundary condition along with explicit formula to Stokes problem was obtained by the author in \cite{AG}. In this article we show that for nonlinear vorticity equation in the exterior of the disc of radius $r_0$ the boundary condition can be defined in terms of vorticity Fourier coefficients $w_k(t,r)$ as Robin-type boundary problem:
\begin{align}\label{_3:bound:nonlin}
\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = u_k(t).
\end{align}
And the same boundary-value problem will be actual in more general domains with help of Riemann mapping.
For exterior flows it is typical to study solutions with infinite energy. These solutions with finite Dirichlet integral were studied in \cite{Abe}\cite{MS}. For Cauchy problem global existence and bounds on velocity as time $t\to \infty$ were established in \cite{GW1}\cite{Zel}\cite{Th}. In this paper we will research the vorticity equation for infinite energy solutions. The main difficulty is the fact that the space $L_2$ is not enough to describe vorticity dynamics. The solutions with finite vorticity energy but nonzero total circulation may have infinite kinetic energy. But in case of zero circularity it's not getting considerably better. In order to restore velocity via Biot-Savar law and correctly define the nonlinear term $(\mathbf{v}, \nabla w)$ we need to impose additional requirements on the phase space.
The Laplace operator with Robin boundary condition (\ref{_3:bound:nonlin}) possesses the non-trivial kernel and as result it causes the presence of the stationary solution for the corresponding evolution equation. But from the no-slip condition follows an orthogonality relation between the solution and the kernel (see \cite{AG} for more details). It is entirely consistent with the Stokes Paradox.
The paper is arranged as follows. In Section 1 we study Biot-Savar law and the integral relation for no-slip condition. Then in Section 2 we will derive the precise integral boundary condition for Stokes system and its local approximation for Helmholtz vorticity equation. In Section 3 we will prove the local existence of no-slip condition for vorticity. In Appendix the solvability of the vorticity equation with boundary (\ref{_3:bound:nonlin}) will be established.
{\bf Notations:}
In the paper we will exploit $\mathbb{R}^2$ as real as well as a complex plane depending on the context.
For points from $\mathbb{R}^2$ we will use different real and complex notations including polar coordinates $r,\varphi$ such as $\mathbf{x}=(x_1,x_2)$, $z=x_1+ix_2=r e^{i\varphi}$. For functions defined on $E\subset \mathbb{R}_+$ along with classic spaces $L_p(E)$, $L_p(E)$ we will use $L_p(E; r)$, $L_p(E; \lambda)$ of square-integrable functions with infinitesimal elements $r\operatorname{dr}$, $\lambda \dlambda$, supplied with norms
\begin{align*}
\|f\|^p_{L_p(E; r)}=\int\limits_E |f(r)|^p r\operatorname{dr}, \\
\|f\|^p_{L_p(E; \lambda)}=\int\limits_E |f(\lambda)|^p \lambda \dlambda.
\end{align*}
Velocity $\mathbf{v}$ will be used in both Cartesian $(v_1, v_2)$ and polar coordinates $(v_r, v_\phi)$. Fourier coefficients for function $f$ will be referred $f_k$ with prefix $k$. For Laplace operator its Fourier expansion will involve $\Delta_k$ defined as
\begin{align} \label{fourierlaplace}
\Delta_k w(t,r) = \frac 1r \frac {\partial}{\partial r}\left(r \frac {\partial}{\partial r}w(t,r)\right) - \frac{k^2}{r^2} w(t,r).
\end{align}
\section{Biot-Savar law in exterior domains and no-slip integral condition}
Now we study when the solenoidal velocity field $\mathbf{v}(\mathbf{x})$ can be uniquely restored from its vorticity $w(\mathbf{x})$. Consider the following elliptic problem in exterior domain $\Omega$:
\begin{eqnarray}
&&\rm{div}~ \mathbf{v}(\mathbf{x}) = 0, \label{freediv} \\
&&\rm{curl}~ \mathbf{v}(\mathbf{x}) = w(\mathbf{x}), \label{curleq} \\
&&\mathbf{v}(\mathbf{x})=0,~\mathbf{x} \in \partial \Omega, \label{bound}\\
&&\mathbf{v}(\mathbf{x})\to\mathbf{v}_\infty,~|\mathbf{x}|\to \infty, \label{boundinf}.
\end{eqnarray}
Exterior domains are not simply connected and the problem above could haven't unique solution. For example, equations (\ref{freediv}), (\ref{curleq}) supplied with slip condition on the boundary
\begin{equation}\label{slip}
\left(\mathbf{v}(\mathbf{x}), \mathbf{n} \right) = 0,~\mathbf{x} \in \partial \Omega,
\end{equation}
and fixed flow at infinity (\ref{boundinf}) have a unique solution only if we fix circularity at infinity ($\mathbf{n}$ is an outer normal to boundary). No-slip condition (\ref{bound}) is stronger than (\ref{slip}), and so some additional restrictions on $w(\mathbf{x})$ are required. These restrictions can be realized via moment relations for vorticity. In \cite{AG} the solvability of the system above was researched in detail for slip and no-slip conditions in the exterior of the disc. Here we extend these results on more general domains and obtain integral no-slip condition.
So, we need to fix circularity at infinity. From a physical point of view it is natural to suppose zero-circularity:
\begin{equation} \label{zerocirculation}
\lim_{R\to\infty}\oint_{|\mathbf{x}|=R} \mathbf{v} \cdot d\mathbf{l} = 0.
\end{equation}
Then the solution of the above problem if it exists is given by Biot-Savar formula
\begin{equation} \label{BSformula}
\mathbf{v}(\mathbf{x}) =\frac 1{2\pi} \int_\Omega \frac{(\mathbf{x}-\by)^\perp}{|\mathbf{x}-\by|^2} w(\by) \operatorname{d\by} + \mathbf{v}_\infty,
\end{equation}
which we rewrite in polar coordinates. Boundary condition (\ref{bound}) obliges the vorticity $w(\by)$ to satisfy some additional equities.
The relationship between Cartesian and polar coordinate systems for $\mathbf{v}_\infty=(v_{\infty,x},v_{\infty,y})$ is given by formulas:
\begin{align*}
&v_{\infty,r}=v_{\infty,x}\cos \varphi + v_{\infty,y}\sin \varphi, \\
&v_{\infty,\phi}=v_{\infty,y}\cos \varphi - v_{\infty,x}\sin \varphi.
\end{align*}
Then its Fourier coefficients are determined as
\begin{align}
&v_{\infty,r,k}=\frac{\delta_{|k|,1}}2 (v_{\infty,x} - i k v_{\infty,y}) \label{fouriercoeffr}, \\
&v_{\infty,\phi,k}=\frac{\delta_{|k|,1}}2 (v_{\infty,y} + i k v_{\infty,x}) \label{fouriercoeffphi},
\end{align}
and
\begin{align} \label{vrvphi}
v_{\infty,\phi,k} = \operatorname{sign}(k) iv_{\infty,r,k}.
\end{align}
All Fourier coefficients of the external flow equal to zero except $k=\pm 1$. For horizontal flow $\mathbf{v}_\infty=(v_\infty,0)$
\begin{align*}
&v_{\infty,r,k}=\frac{\delta_{|k|,1}}2 v_\infty, \\
&v_{\infty,\phi,k}=i k\frac{\delta_{|k|,1}}2 v_\infty.
\end{align*}
\subsection{Biot-Savar law in exterior of the disc}
In this subsection the domain under investigation will be the exterior of the disc $B_{r_0}=\{\mathbf{x} \in \mathbb{R}^2,~|\mathbf{x}| > r_0 \},~r_0>0$. We will derive Biot-Savar law and no-slip boundary condition for Stokes and Navier-Stokes systems in integral form.
In polar coordinates equations (\ref{freediv}),(\ref{curleq}) can be written in Fourier coefficients $v_{r,k}$, $v_{\varphi,k}$:
\begin{eqnarray*}
&&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{r,k}\right)+{\frac {ik}{r}} v_{\varphi,k} = 0,\\
&&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{\varphi,k}\right)-{\frac {ik}{r}} v_{r,k} = w_k.
\end{eqnarray*}
The basis for solutions of homogeneous system when $w_k \equiv 0$ consists of two vectors:
\begin{align*}
\begin{pmatrix}
v^1_{r,k} \\
v^1_{\varphi,k}
\end{pmatrix}
=
\begin{pmatrix}
ir^{-k-1} \\
r^{-k-1}
\end{pmatrix}
, \\
\begin{pmatrix}
v^2_{r,k} \\
v^2_{\varphi,k}
\end{pmatrix}
=
\begin{pmatrix}
ir^{k-1} \\
-r^{k-1}
\end{pmatrix}.
\end{align*}
The solution of this system with boundary relations (\ref{bound}), (\ref{boundinf}) and zero-circularity condition (\ref{zerocirculation}) was derived in \cite{AG} as Biot-Savar law in exterior of the disc in the following form for $k \in \mathbb{Z}$:
\begin{eqnarray} \label{BiotSavar1}
&&v_{r,k} = \operatorname{sign}(k) \frac{ir^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}w_k(s)\operatorname{ds} \\
&&~~~~~~~~~~~~~~~~~~+ \operatorname{sign}(k) \frac{ir^{|k|-1}}2 \int_r^\infty s^{-|k|+1}w_k(s)\operatorname{ds} + v_{\infty,r,k} \nonumber \\ \label{BiotSavar2}
&&v_{\varphi,k} = \frac{r^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}w_k(s)\operatorname{ds} \nonumber \\
&&~~~~~~~~~~~~~~~~~~- \frac{r^{|k|-1}}2 \int_r^\infty s^{-|k|+1}w_k(s)\operatorname{ds} + v_{\infty,\phi,k}.
\end{eqnarray}
No-slip condition (\ref{bound}) and (\ref{vrvphi}) lead to moment relations for vorticity ($k\in \mathbb{Z}$):
\begin{equation} \label{noslipcondintegral}
\int_{r_0}^\infty s^{-|k|+1}w_k(s)\operatorname{ds} = 2ik v_{\infty,r,k} = 2 v_{\infty,\phi,k}.
\end{equation}
From (\ref{fouriercoeffr}), (\ref{fouriercoeffphi}) these moments don't equal to zero only if $|k|=1$.
The above formulas (\ref{BiotSavar1}), (\ref{BiotSavar2}) represent Fourier coefficients of Biot-Savar formula (\ref{BSformula}).
Since for $p >1$ $\nabla \mathbf{v}$ is obtained from $\omega$ via a singular integral kernel of Calderon-Zygmund type\cite{CZ}\cite{St}, then
\begin{align}\label{bsest2}
\|\nabla \mathbf{v}(\cdot)\|_{L_p} \leq C \| w \|_{L_p}.
\end{align}
The case $p=2$ causes most difficulties in estimates of Biot-Savar law. The following lemma gives an estimate for $w\in H^1$.
\begin{lem}\label{bsest}
Let $w(\cdot) \in L_2(B_{r_0})$, the Fourier coefficients at $k=-1, 0, 1$ belong to $L_1(r_0,\infty)$, $\mathbf{v}(\cdot)$ - be the solution of (\ref{freediv}) - (\ref{boundinf}), (\ref{zerocirculation}). Then the following estimate holds
with some $C>0$:
\begin{align*}
&\operatorname{vraisup}_{r\in [r_0,\infty)} \| \mathbf{v}(r,\cdot) - \mathbf{v}_\infty \|^2_{H^{1/2}(S_{r})} \leq \\ &~~~~~~~~~~~C
\left (\|w(\cdot)\|^2_{L_2(B_{r_0})} + \sum_{k=-1,0,1}\|w_k(\cdot)\|^2_{L_1(r_0,\infty)} \right),
\end{align*}
where $S_r = \{\mathbf{x} \in \mathbb{R}^2,~|\mathbf{x}|=r\}$.
\end{lem}
\begin{proof}
We will estimate one of the terms in (\ref{BiotSavar1}) when others can be processed in a similar way.
For $|k| > 1:$
\begin{align}\label{bsest_proof}
\left | r^{|k|-1} \int_r^\infty s^{-|k|+1}w_k(s)\operatorname{ds} \right |^2 \leq \frac {\|w_k\|^2_{L_2(r_0,\infty; r)}}{2|k|-2}.
\end{align}
For $|k| = 1:$
\begin{align*}
\left | \int_r^\infty w_{\pm 1}(s)\operatorname{ds} \right |^2 \leq (1+1/r_0) \|w_{\pm 1}\|^2_{L_1(r_0,\infty, r)}.
\end{align*}
For $|k| =0:$
\begin{align*}
\left | r^{-1} \int_r^\infty s w_0(s)\operatorname{ds} \right |^2 \leq r_0 \|w_0\|^2_{L_1(r_0,\infty; r)}.
\end{align*}
Fractional differentiation of order $\frac 12$ corresponds to multiplier $\sqrt k$ for Fourier coefficients. Summarizing by $k$ we obtain the required estimate.
\end{proof}
We rewrite moment relationship in terms of vorticity $w(\mathbf{x})$ for $k\geq 0$:
\begin{align*}
\int_{r_0}^\infty s^{-|k|+1}w_k(s)\operatorname{ds} = \frac 1{2 \pi} \int_{r_0}^\infty \int_0^{2\pi} s^{-k+1}w(s,\varphi)e^{-ik\varphi}\operatorname{ds} d\varphi \\ =\frac 1{2 \pi} \int_{B_{r_0}} \frac {w(\mathbf{x})}{z^k} \operatorname{d\mathbf{x}} = 2ik v_{\infty,r,k},
\end{align*}
where $z=x_1+ix_2=s e^{i\varphi}$.
For $k<0$ we have absolutely the same moments equity:
\begin{align*}
\int_{r_0}^\infty s^{-|k|+1}w_k(s)\operatorname{ds} = \frac 1{2 \pi} \int_{r_0}^\infty \int_0^{2\pi} s^{k+1}w(s,\varphi)e^{-ik\varphi}\operatorname{ds} d\varphi \nonumber \\ =\frac 1{2 \pi} \int_{B_{r_0}} \overline{z^k} w(\mathbf{x}) \operatorname{d\mathbf{x}} = 2ik v_{\infty,r,k},~k<0.
\end{align*}
The last formula for $k<0$ is just complex conjugation of the analogous formula for $k \geq 0$ due to equities
$$
v_{\infty,r,k} = \overline{v_{\infty,r,-k}}, v_{\infty,\phi,k} = \overline{v_{\infty,\phi,-k}}.
$$
Then for no-slip condition we have affine subspace $M$ which must be invariant under vorticity flow:
\begin{align}\label{noslipcondintegral2}
M=\left \{ w(\mathbf{x})\in L_1(B_{r_0})~\Big |~
\int_{B_{r_0}} \frac {w(\mathbf{x})}{(x_1+ix_2)^k} \operatorname{d\mathbf{x}} =
4 \pi ik v_{\infty,r,k},~k \in \mathbb{Z}_+ \right \}.
\end{align}
Note that from (\ref{fouriercoeffr}), (\ref{fouriercoeffphi}) the integral relations in the definition of $M$ are non-zero ones only if $k=1$.
For the Fourier coefficients the invariance of $M$ means that for vorticity flow which is described by the coefficients $w_k(t,\cdot)$ holds
$$
\int_{r_0}^\infty s^{-|k|+1}w_k(t,s)\operatorname{ds} = const,~k \in \mathbb{Z}.
$$
In \cite{AG} was proved
\begin{thm}[Biot-Savart Law in polar coordinates]\label{Biotpolarnoslip}
If $w(\mathbf{x}) \in M$ then there exists the unique solution of (\ref{freediv}) - (\ref{boundinf}), (\ref{zerocirculation}) given by (\ref{BiotSavar1}), (\ref{BiotSavar2})
with Fourier coefficients $v_{r,k}$, $v_{\varphi,k} \in L_\infty(r_0,\infty)$.
\end{thm}
\subsection{Biot-Savar Law in the simply connected domains}
Here we derive Biot-Savar law in more general domains. We will establish that the Robin-type boundary (\ref{robin_bound}) stays actual for these domains.
Let $\Omega=\mathbb{R}^2 \setminus B$, where $B$ is bounded simple-connected domain with smooth boundary and $\Phi$ be a Riemann mapping from $\Omega$ into exterior of the disc $B_{r_0}$ such that
\begin{align*}
\Phi(z)=z+O\left (\frac 1z \right ), \\
\Phi'(z)=1+O\left (\frac 1{z^2} \right ).
\end{align*}
Then $\mathbf{v}=\mathbf{v}(\Phi^{-1}(z))=\mathbf{v}(\operatorname{Re} \Phi^{-1}(x_1+ix_2), \operatorname{Im} \Phi^{-1}(x_1+ix_2))$ defines vector field in $B_{r_0}$. We will not use the tensor form of divergence and consider the vector field $(v_r(\mathbf{x}),v_\phi(\mathbf{x}))$ as a set of two scalar functions. Then with help of Cauchy–Riemann relationship the equations (\ref{freediv}), (\ref{curleq}) turn to relation
\begin{equation}
\operatorname{curl} \mathbf{v} + i \operatorname{div} \mathbf{v} = \frac {\Phi'(z)}{|\Phi'(z)|^2}w.
\end{equation}
Then the system (\ref{freediv})-(\ref{boundinf}) after Riemann mapping in $B_{r_0}$ takes the form
\begin{eqnarray}
&&\rm{div}~ \mathbf{v}(\mathbf{x}) = \operatorname{Im} \overline {\Phi'^-1(z)} w(\mathbf{x}) \label{freediv3}\\
&&\rm{curl}~ \mathbf{v}(\mathbf{x}) = \operatorname{Re} \overline {\Phi'^{-1}(z)} w(\mathbf{x}) \label{curleq3}\\
&&\mathbf{v}(\mathbf{x})=0,~|\mathbf{x}|=r_0 \label{bound3}\\
&&\mathbf{v}(\mathbf{x})\to\mathbf{v}_\infty,~|\mathbf{x}|\to \infty. \label{boundinf3}
\end{eqnarray}
Let
\begin{align*}
r_k(r)=[\operatorname{Im} \overline {\Phi'^{-1}(z)} w(\mathbf{x})]_k,\\
q_k(r)=[\operatorname{Re} \overline {\Phi'^{-1}(z)} w(\mathbf{x})]_k,
\end{align*}
where subscript $k$ denotes $k$-th Fourier harmonic and $z=re^{i\varphi}=x_1+ix_2$.
Rewrite (\ref{freediv3}),(\ref{curleq3}) in polar coordinates in terms of Fourier coefficients $v_{r,k}$, $v_{\varphi,k}$:
\begin{eqnarray*}
&&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{r,k}\right)+{\frac {ik}{r}} v_{\varphi,k} = r_k(r),\\
&&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{\varphi,k}\right)-{\frac {ik}{r}} v_{r,k} = q_k(r).
\end{eqnarray*}
Under the assumption of zero-circularity (\ref{zerocirculation}) from Stoke's theorem we have
\begin{align*}
\lim_{R\to\infty}\oint_{|\mathbf{x}|=R} \mathbf{v}(\mathbf{x}) \cdot d\mathbf{l}= \frac 1{2 \pi} \int_{\Omega} w(\mathbf{x}) \operatorname{d\mathbf{x}} \\ = \frac 1{2 \pi} \int_{B_{r_0}} \frac {w(\mathbf{x})}{|\Phi'|^2} \operatorname{d\mathbf{x}} =
\int_{r_0}^\infty \left [\frac {w(\mathbf{x})}{|\Phi'|^2} \right ]_{k=0} s \operatorname{ds} = 0.
\end{align*}
It guarantees the uniqueness of the above system. Existence will be provided by the moment relations below. The solution of the above system for $k \in \mathbb{Z}$ is derived by the similar formulas (\ref{BiotSavar1}), (\ref{BiotSavar2}):
\begin{eqnarray} \label{BiotSavar1_2}
&&v_{r,k} = \operatorname{sign}(k) \frac{ir^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}(q_k-i\operatorname{sign}(k) r_k)\operatorname{ds} \\
&&~~~~~~~~~~~~~~~~~~+ \operatorname{sign}(k) \frac{ir^{|k|-1}}2 \int_r^\infty s^{-|k|+1}(q_k+i\operatorname{sign}(k)r_k)\operatorname{ds} + v_{\infty,r,k} \nonumber \\ \label{BiotSavar2_2}
&&v_{\varphi,k} = \frac{r^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}(q_k-i\operatorname{sign}(k)r_k)\operatorname{ds} \nonumber \\
&&~~~~~~~~~~~~~~~~~~- \frac{r^{|k|-1}}2 \int_r^\infty s^{-|k|+1}(q_k+i\operatorname{sign}(k)r_k)\operatorname{ds} + v_{\infty,\phi,k}.
\end{eqnarray}
Formulas (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}) combined with (\ref{bound}) lead to the relations on vorticity ($k\in \mathbb{Z}$):
\begin{equation} \label{noslipcondintegralrieman}
\int_{r_0}^\infty s^{-|k|+1}\left ( q_k(s)+i\operatorname{sign}(k)r_k(s) \right )\operatorname{ds} = 2ik v_{\infty,r,k} = 2 v_{\infty,\phi,k}.
\end{equation}
The above formulas (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}) are the Fourier coefficients of Biot-Savar formula
\begin{equation}\label{_1:BS}
\mathbf{v}(\mathbf{x})=\frac 1{2\pi} \int_{B_{r_0}} \left ( \frac{(\mathbf{x}-\by)^\perp}{|\mathbf{x}-\by|^2} \operatorname{Re} \overline {\Phi'^{-1}(z)} +
\frac{\mathbf{x}-\by}{|\mathbf{x}-\by|^2} \operatorname{Im} \overline {\Phi'^{-1}(z)} \right ) w(\by)d\by +\mathbf{v}_\infty.
\end{equation}
Following by the same way as in Theorem \ref{invthm} we define affine subspace $M$ via vorticity moments which must be invariant under vorticity flow:
\begin{align}\label{noslipcondintegral2_omega}
M=\Big \{ w(\mathbf{x})\in L_1(B_{r_0})~\Big |~
\int_{B_{r_0}}\overline{\Phi'^{-1}(x_1+ix_2)} \frac {w(\mathbf{x})}{(x_1+ix_2)^k} \operatorname{d\mathbf{x}} = \nonumber \\
4 \pi ik v_{\infty,r,k},~k \in \mathbb{Z}_+ \Big \}.
\end{align}
In view of (\ref{fouriercoeffr}), (\ref{fouriercoeffphi}) all moments in the definition of $M$ must be equal to zero except $k=1$.
\begin{prop}[Biot-Savart Law in exterior domain]\label{Biotpolarnoslip2}
If $w(\Phi^{-1}(z)) \in M$ then there exists the unique solution of (\ref{freediv}) - (\ref{boundinf}), (\ref{zerocirculation}) given by (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}) with Fourier coefficients $v_{r,k}$, $v_{\varphi,k} \in L_\infty(r_0,\infty)$.
\end{prop}
\begin{proof}All we need is to prove the validity of no-slip condition. Set $r=r_0$ in (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}). Since with $z=se^{i\varphi}=x_1+ix_2$
$$
q_k(s)+ir_k(s) = [\overline{\Phi'^{-1}(z)} w(x)]_k,
$$
equities (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}) can be written in terms of $w(\mathbf{x})$. Then
\begin{align} \label{_1:BiotSavar_int}
&\int_{r_0}^\infty s^{-|k|+1}\left ( q_k(s)+ir_k(s) \right )\operatorname{ds} \\&= \frac 1{2 \pi} \int_{r_0}^\infty \int_0^{2\pi} s^{-|k|+1}\overline{\Phi'^{-1}(z)}w(s,\varphi)e^{-ik\varphi}\operatorname{ds} d\varphi \nonumber \\ &=\frac 1{2 \pi} \int_{B_{r_0}} \overline{\Phi'^{-1}(z)} \frac {w(\mathbf{x})}{z^k} \operatorname{d\mathbf{x}} = 2ik v_{\infty,r,k}. \nonumber
\end{align}
\end{proof}
\begin{lem} \label{lembsest} In the exterior simple-connected domain $\Omega$ with smooth boundary let $\Phi$ be a Riemann mapping from $\Omega$ into exterior of the disc $B_{r_0}$, $w(\cdot) \in L_1(B_{r_0})\cap L_2(B_{r_0})$, $\mathbf{v}=\mathbf{v}(\Phi^{-1}(z))=\mathbf{v}(\operatorname{Re} \Phi^{-1}(x_1+ix_2), \operatorname{Im} \Phi^{-1}(x_1+ix_2))$ defines vector field in $B_{r_0}$ of the the solution of (\ref{freediv}) - (\ref{boundinf}), (\ref{zerocirculation}) with vorticity $w(\Phi^{-1}(z))$. Then the following estimate holds with some $C>0$:
\begin{align*}
\operatorname{vraisup}_{r\in [r_0,\infty)} \| \mathbf{v}(r,\cdot) - \mathbf{v}_\infty \|^2_{H^{1/2}(S_{r_0})} \leq C
(\|w(\cdot)\|_{L_2(B_{r_0})} + \|w(\cdot)\|_{L_1(B_{r_0})}).
\end{align*}
\end{lem}
Since $|\Phi'(z)|$ is bounded, then the proof of this lemma is the same as for Lemma \ref{bsest} in the previous subsection.
\subsection{Invariant affine manifolds for a no-slip condition}
Consider the flow of 2D velocity field $\mathbf{v}(t,\mathbf{x})$ and its vorticity $w(t,\mathbf{x})=\operatorname{curl} \mathbf{v}(t,\mathbf{x})$. The set of relations (\ref{noslipcondintegral2_omega}) is the integral form of the no-slip condition (\ref{bound}).
\begin{prop} \label{invthm}
Given initial datum $\mathbf{v}_0(\mathbf{x})$ satisfying no-slip condition (\ref{bound}), infinity condition (\ref{boundinf}), zero-circularity (\ref{zerocirculation}), such that \\$w_0=\operatorname{curl} \mathbf{v}_0(\Phi^{-1}(z))$ $\in L_1(\Omega)$, and $w(t,\cdot)=\operatorname{curl} \mathbf{v}(\Phi^{-1}(z))$ $\in L_1(\Omega)$ be the vorticity flow. Then $w_0 \in M$ and in order to conserve no-slip condition the affine subspace $M$ must be invariant under the flow, e.g. for any time $t>0$ $w(t,\cdot) \in M$.
\end{prop}
\begin{proof}
Since $\mathbf{v}_0(\mathbf{x})$ satisfies no-slip condition, then from (\ref{BiotSavar1_2}), (\ref{BiotSavar2_2}), (\ref{_1:BiotSavar_int}) follows $w(t,\cdot) \in M$.
\end{proof}
If $\mathbf{v}_\infty=0$ then this affine manifolds $M$ path through zero and becomes the invariant subspace. If we limit infinite set of relations in (\ref{noslipcondintegral2_omega}) only by $k=1,...,N$, we obtain affine manifold $M_N$ of finite codimension:
\begin{align}\label{noslipcondintegral2_omega_N}
M_N=\Big \{ w(\mathbf{x})\in L_1(B_{r_0})~\Big |~
\int_{B_{r_0}}\overline{\Phi'^{-1}(z)} \frac {w(\mathbf{x})}{z^k} \operatorname{d\mathbf{x}} =
4 \pi ik v_{\infty,r,k},\\ k =0, 1, \dots, N \Big \}. \nonumber
\end{align}
The following lemma says that since $M$ corresponds to a no-slip condition, then $M_N$ is the approximation of this boundary condition.
\begin{lem}Let for any fixed $t\in [0,T]$ $w_N(t,\cdot)$ are uniformly bounded in $H^1(\Omega)$ by $N$. Then the vector field $\mathbf{v}^N(t,\cdot)$ given by (\ref{_1:BS}) converges weakly in $H^{1/2}(\partial \Omega)$ to zero for $t\in [0,T]$ as $N\to \infty$.
\end{lem}
\begin{proof}
Without loss of generality assume that $\Omega=B_{r_0}$.
\begin{align*}
&v_{r,k}(t,r_0) = \operatorname{sign}(k) \frac{ir^{|k|-1}}2 \int_{r_0}^\infty s^{-|k|+1}w_k(t,s)\operatorname{ds} + v_{\infty,r,k} \\
&v_{\varphi,k}(t,r_0) = \frac{r^{|k|-1}}2 \int_{r_0}^\infty s^{-|k|+1}w_k(t,s)\operatorname{ds} + v_{\infty,\phi,k}.
\end{align*}
From $w(t,\cdot) \in M_N$ $v_{r,k}(t,r_0) = v_{\varphi,k}(t,r_0)=0$ for $k =-N, \dots, N$. For $k>N$
\begin{align*}
\left | \int_{r_0}^\infty s^{-|k|+1}w_k(t,s)\operatorname{ds} + v_{\infty,r,k} \right |=
\left | \int_{r_0}^\infty s^{-|k|+1}w_k(t,s)\operatorname{ds} \right |.
\end{align*}
Then from (\ref{bsest_proof})
\begin{align*}
\| \mathbf{v}^N(t,\cdot)\|_{H^{1/2} (\partial \Omega)} \leq C \sum_{k=N+1}^\infty \|w^N_k(t,\cdot)\|^2_{L_2(r_0,\infty, rdr)}.
\end{align*}
For $w_N(t,\cdot) \in M_N$ the Fourier coefficients of $\mathbf{v}_r$, $\mathbf{v}_\phi$ with index $k =-N, \dots, N$ equal to zero which implies
\begin{align}\label{weakconv}
\mathbf{v}(t,\mathbf{x}') \rightharpoonup 0,~\mathbf{x}' \in \partial \Omega,~(weakly)
\end{align}
in $H^{1/2}(\partial \Omega)$.
\end{proof}
\begin{remark} If $w_N(t,\cdot)$ are uniformly bounded in $H^2$ one can prove that $v(t,\mathbf{x}')$ converges weakly in $H^{3/2}(\partial \Omega)$ to zero. And from the Rellich–Kondrachov Theorem due to the compact embedding follows that for any $t\in [0,T]$
\begin{align}\label{strongconv}
\mathbf{v}(t,\mathbf{x}') \to 0,~\mathbf{x}' \in \partial \Omega~(strongly)
\end{align}
in $L_2(\partial \Omega)$.
\end{remark}
\section{No-slip integral condition for vorticity in exterior domains}
Consider the initial-boundary-value problem for the Navier-Stokes system defined in exterior domain $\Omega$ modelling flow around solid with given constant horizontal flow at infinity $\mathbf{v}_\infty = (\mathbf{v}_{1,\infty},\mathbf{v}_{2,\infty}) \in \mathbb{R}^2$:
\begin{eqnarray}
&&\partial_t \mathbf{v} - \Delta \mathbf{v} +(\mathbf{v},\nabla)\mathbf{v} = \nabla p \label{maineqns}\\
&&{\rm div}~\mathbf{v}(t,\mathbf{x})=0 \label{freedivns}\\
&&\mathbf{v}(0,\mathbf{x})=\mathbf{v}_0(\mathbf{x}) \label{initns}\\
&&\mathbf{v}(t,\mathbf{x})=0,~|\mathbf{x}|=r_0 \label{boundns}\\
&&\mathbf{v}(t,\mathbf{x}) \to \mathbf{v}_\infty,~|\mathbf{x}|\to \infty. \label{boundinfns}
\end{eqnarray}
Here $\mathbf{v}(t,\mathbf{x})=(v_1(t,\mathbf{x}),v_2(t,\mathbf{x}))$ is the velocity field and $p(t,\mathbf{x})$ is the pressure.
Applying the curl operator
$w(t,\mathbf{x})=$ ${\rm curl}~\mathbf{v}(t,\mathbf{x})$ $=\partial_{\mathbf{x}_1}v_2 - \partial_{\mathbf{x}_2}v_1$ we get boundary problem for vorticity equation
\begin{eqnarray}
\frac{\partial w(t,\mathbf{x})}{\partial t} - \Delta w + (\mathbf{v},\nabla)w = 0, \label{maineqw} \\
w(0,\mathbf{x})=w_0(\mathbf{x}) \label{initw}\\
\operatorname{curl}^{-1} w(t,\mathbf{x}) \Big|_{|\mathbf{x}|=r_0} = 0, \label{boundw}\\
w(t,\mathbf{x}) \to 0,~|\mathbf{x}|\to \infty \label{boundinfw}
\end{eqnarray}
with initial datum $w_0(\mathbf{x})={\rm curl}~\mathbf{v}_0(\mathbf{x})$.
Vector field $\mathbf{v}(t,\mathbf{x})$ can be derived from $w(t,\mathbf{x})$ using Green function $G(\mathbf{x},\by)$ for Laplace operator $\Delta$:
$$
\mathbf{v}(t,\mathbf{x}) = \int_\Omega \nabla_x^\perp G(\mathbf{x},\by) w(\by) \operatorname{d\mathbf{y}} + \mathbf{v}_\infty,
$$
where
\begin{equation}\label{_2:GreenProp}
\Delta_x G(\mathbf{x},\by)=0,~\mathbf{x}\neq \by.
\end{equation}
Indeed
$$
{\rm div}~\mathbf{v}(t,\mathbf{x}) = \int_\Omega (\nabla_x,\nabla_x^\perp) G(\mathbf{x},\by) w(\by) \operatorname{d\mathbf{y}} = 0,
$$
and
$$
{\rm curl}~\mathbf{v}(t,\mathbf{x}) = \int_\Omega \Delta_x G(\mathbf{x},\by) w(\by) \operatorname{d\mathbf{y}} = w.
$$
Then no-slip condition gives the following integral expression
$$
\mathbf{v}(t,\mathbf{x}') = \int_\Omega \nabla_x^\perp G(\mathbf{x}',\by) w(t,\by) \operatorname{d\mathbf{y}} = 0,~\mathbf{x}'\in \partial \Omega,~\forall t>0.
$$
Velocity field for Stokes system satisfies
\begin{align} \label{stokeseqvelocity}
\frac{\partial v(t,\mathbf{x})}{\partial t} - \Delta v(t,\mathbf{x}) = \nabla p
\end{align}
when vorticity evolution for Stokes system is described by the heat equation
\begin{align} \label{stokeseq}
\frac{\partial w(t,\mathbf{x})}{\partial t} - \Delta w(t,\mathbf{x}) = 0.
\end{align}
Multiplying it by $\nabla_x^\perp G(\mathbf{x}',\by)$ and integrating over exterior domain using Green formula with help of (\ref{_2:GreenProp}) we will have integral boundary condition
$$
\int_{\partial \Omega} \left (
\nabla_x^\perp G(\mathbf{x}',\by) \frac{\partial w(t,\by)}{\partial n} -w(t,\by) \frac{\partial }{\partial n}\nabla_x^\perp G(\mathbf{x}',\by) \right ) \operatorname{d\mathbf{y}}=0,~\forall \mathbf{x}' \in \partial \Omega.
$$
It is still an integral condition but only with the surface integral over $\partial \Omega$ involved.
For cylindrical domains this surface integral turns into boundary condition on Fourier coefficients. In this section for 2D Stokes and Navier-Stokes we derive boundary condition in terms of Fourier harmonics in exterior simply-connected domains.
\subsection{Linear vorticity equation in the exterior of the disc}
Consider Stokes flow for vorticity (\ref{stokeseq}) and supply it with Robin-type boundary condition:
\begin{equation}\label{robin_bound}
r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = 0,~k \in \mathbb{Z}
\end{equation}
and
\begin{equation}
w(t,\mathbf{x}) \to 0,~|\mathbf{x}|\to \infty. \label{boundinfvorticity}
\end{equation}
Then $M$ is invariant under the flow $w(t,\cdot)$. Indeed, fix $k>0$ and divide equation (\ref{stokeseq}) by $z^k$ and integrate over the exterior of the disc $B_{r_0}$. From moment relations (\ref{noslipcondintegral2}) follows
$$
\frac d{dt}\int_{B_{r_0}} \frac {w(t,\mathbf{x})}{z^k} \operatorname{d\mathbf{x}} = 0
$$
and thus
$$
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}}=0.
$$
In other hand
\begin{align}\label{integrationbypart_dissip}\nonumber
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}} = \int_{r_0}^\infty \int_0^{2\pi} \frac {\Delta w}{s^ke^{ik\phi}}sdsd\varphi = 2\pi \int_{r_0}^\infty s^{-k+1}\Delta_k w_k(t, s) ds \\ \nonumber=-2\pi \int_{r_0}^\infty s^{-|k|} \left ( \frac {\partial}{\partial s}\left(s \frac {\partial}{\partial s}w_k(t,s)\right) - \frac{k^2}{s} w_k(t,s) \right ) \operatorname{ds} \\ \nonumber=
- r_0^{-|k|+1}\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + \int_{r_0}^\infty s^{-|k|} \left ( |k| \frac {\partial}{\partial s}w_k(t,s) - \frac{k^2}{s} w_k(t,s) \right ) \operatorname{ds} \\ = -2\pi r_0^{-k}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) \right )=0,
\end{align}
where $\Delta_k w(t,r)$ is defined in (\ref{fourierlaplace}).
In a similar way using complex conjugation of moment relations (\ref{noslipcondintegral2}) we obtain boundary condition for $k<0$:
\begin{align*}
\int_{B_{r_0}} {\Delta w}\overline{z^{k}}\operatorname{d\mathbf{x}} = \int_{r_0}^\infty \int_0^{2\pi} \frac {\Delta w}{s^{-k}e^{ik\phi}}sdsd\varphi = 2\pi \int_{r_0}^\infty s^{k+1}\Delta_k w_k(s) ds \\= -2\pi r_0^{k}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} - k w_k(t,r_0) \right )=0.
\end{align*}
\subsection{Helmholtz equation in the exterior of the disc}
Now we are ready to derive the boundary condition to the vorticity equation. In fact it will be an integral condition. But its first approximation will be the same boundary condition (\ref{robin_bound}) as for Stokes flow.
Consider Helmholtz equation for vorticity
\begin{align} \label{helmholtzeq}
\frac{\partial w(t,\mathbf{x})}{\partial t} - \Delta w(t,\mathbf{x}) + (\mathbf{v},\nabla w) = 0,
\end{align}
where $\mathbf{v}$ is redtored from $w$ via Biot-Savar law (\ref{BSformula}).
\begin{thm}Given initial datum $\mathbf{v}_0(\mathbf{x})$, $\operatorname{curl} \mathbf{v}_0\in L_1(B_{r_0})$ satisfying no-slip condition (\ref{bound}), infinity condition (\ref{boundinf}), zero-circularity (\ref{zerocirculation}) and $\mathbf{v}(t,\mathbf{x})$ be the solution of (\ref{maineqns})-(\ref{boundinfns}) in $B_{r_0}$. Then $w(t,\mathbf{x})=\operatorname{curl} \mathbf{v}(t,\mathbf{x})$ satisfies
\begin{align}
&r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = \nonumber \\
&~~~~\left \{ { \begin{matrix}
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} (\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C}){z^{-|k|-1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k\geq 0, \\
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \overline{(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C})z^{-|k|-1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k<0.
\end{matrix} } \right . \label{robin_bound_ns}
\end{align}
\end{thm}
\begin{proof}
Fix $k>0$ and divide this equation by $z^k$ and integrate over the exterior of the disc $B_{r_0}$. From moment relations (\ref{noslipcondintegral2}) follows
$$
\frac d{dt}\int_{B_{r_0}} \frac {w(t,\mathbf{x})}{z^k} \operatorname{d\mathbf{x}} = 0
$$
Denote $$\mathbf{v}^\mathbb{C} = v_1+iv_2,~\mathbf{v}^\mathbb{C}_\infty = v_{1,\infty}+iv_{2,\infty}.$$
Then
\begin{align*}
\int_{B_{r_0}} \frac {(\mathbf{v},\nabla w)}{z^k} \operatorname{d\mathbf{x}} = -\int_{B_{r_0}} (\mathbf{v},\nabla z^{-k})w(t,\mathbf{x}) \operatorname{d\mathbf{x}} \\
=k \int_{B_{r_0}} \frac{v_1+iv_2}{z^{k+1}} w(t,\mathbf{x}) \operatorname{d\mathbf{x}}=k \int_{B_{r_0}} \frac{\mathbf{v}^\mathbb{C}}{z^{k+1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}}
\end{align*}
From (\ref{noslipcondintegral2})
$$
k \int_{B_{r_0}} \frac{\mathbf{v}^\mathbb{C}_\infty}{z^{k+1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}} = 0
$$
and
$$
\int_{B_{r_0}} \frac {(\mathbf{v},\nabla w)}{z^k} \operatorname{d\mathbf{x}} = k \int_{B_{r_0}} \frac{\mathbf{v}^\mathbb{C}-\mathbf{v}^\mathbb{C}_\infty}{z^{k+1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}}.
$$
Using (\ref{integrationbypart_dissip}) we get
$$
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}}=-2\pi r_0^{-k}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) \right ).
$$
Then for $k\geq 0$
\begin{align*}
\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) \right ) = \frac {k r_0^k}{2\pi}
\int_{B_{r_0}} \frac{\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C}}{z^{k+1}} w(t,\mathbf{x})\operatorname{d\mathbf{x}}.
\end{align*}
Since $w_{-k}(t,r) = \overline{w_k(t,r)}$ then (\ref{robin_bound_ns}) holds for $k<0$. Theorem is proved.
\end{proof}
\begin{remark}
If $\|\mathbf{v} - \mathbf{v}_\infty\|$ is small in some integral norm for a well-streamlined body , then the right side in (\ref{robin_bound_ns}) transfers to boundary condition (\ref{robin_bound}). So, (\ref{robin_bound}) becomes a rather accurate approximation for Navier-Stokes system. From this fact naturally occurs boundary control problem with unknown function $u_k(t)$:
\begin{equation}\label{robin_bound_control}
r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = u_k(t),~k \in \mathbb{Z},
\end{equation}
where feedback controls $u_k(t)$ are treated as a small correction of zero boundary condition.
\end{remark}
\subsection{Linear vorticity equation in a simply connected domains}Here we find out that Robin-type boundary (\ref{robin_bound}) works well as the no-slip boundary condition in exterior domains. It keeps moment relations (\ref{noslipcondintegralrieman}) under the Stokes flow, and thus $M$ is an invariant affine subspace.
We impose additional requirement on $\Omega$ that $\Phi^{-1}$ can be represented by absolutely convergent series
\begin{equation} \label{phireq}
\Phi^{-1}(z) = z+\sum\limits_{n=1}^\infty \frac {b_n}{z^n}.
\end{equation}
\begin{thm}\label{stokesomegathm}Let $\Omega=\mathbb{R}^2 \setminus B$, where $B$ is a bounded simple-connected domain with smooth boundary and $\Phi$ be a Riemann mapping from $\Omega$ into exterior of the disc satisfying (\ref{phireq}), $\mathbf{v}_0(\mathbf{x})$ - is the initial datum, satisfying no-slip condition (\ref{bound}), infinity condition (\ref{boundinf}), zero-circularity (\ref{zerocirculation}), $\operatorname{curl} \mathbf{v}_0\in L_1(B_{r_0})$, and $\mathbf{v}(t,\mathbf{x})$ be the solution of (\ref{maineqns})-(\ref{boundinfns}) in $\Omega$. Then $w(t,\mathbf{x})$=$\operatorname{curl} \mathbf{v}(t,\Phi^{-1}(\mathbf{x}))$ satisfies (\ref{robin_bound}).
\end{thm}
\begin{proof}
After Riemann mapping Stokes equations reduce to scalar equation on vorticity
$$
|(\Phi^{-1})'(z)|^2\partial_t w(t,x) - \Delta w=0.
$$
Fix $k>0$ and divide this equation by $z^k=(x_1+ix_2)^k$. Then integrate it over the exterior of the disc $B_{r_0}$. From moment relations (\ref{noslipcondintegral2_omega}) follows
$$
\frac d{dt}\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'(z)}}{z^k} w(t,\mathbf{x}) \operatorname{d\mathbf{x}} = 0.
$$
Using $|(\Phi^{-1})'(z)|^2=(\Phi^{-1})'(z)\overline{(\Phi^{-1})'(z)}$ and
$$
(\Phi^{-1})'(z) = \sum\limits_{n=0}^\infty \frac {c_n}{z^n}
$$
with some coefficients $c_n$ we have
\begin{align*}
\frac d{dt}\int_{B_{r_0}} \frac{|(\Phi^{-1})'(z)|^2 }{z^k} w(t,x) \operatorname{d\mathbf{x}} = \frac d{dt}\int_{B_{r_0}} \sum\limits_{n=0}^\infty \frac {c_n \overline{(\Phi^{-1})'(z)}}{z^{n+k}}w(t,\mathbf{x}) \operatorname{d\mathbf{x}}\\=\sum\limits_{n=0}^\infty c_n \frac d{dt}\int_{B_{r_0}} \frac { \overline{(\Phi^{-1})'(z)}}{z^{n+k}}w(t,\mathbf{x}) \operatorname{d\mathbf{x}}=0,
\end{align*}
and thus
$$
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}}=0.
$$
In other hand
\begin{align*}
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}} = \int_{r_0}^\infty \int_0^{2\pi} \frac {\Delta w}{s^ke^{ik\phi}}sdsd\varphi = 2\pi \int_{r_0}^\infty s^{-k+1}\Delta_k w_k(s) ds \\= -2\pi r_0^{-k}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) \right )=0.
\end{align*}
Using $w_{-k}(t,r) = \overline{w_k(t,r)}$ we will have condition (\ref{robin_bound}) for $k<0$.
\end{proof}
\subsection{Helmholtz equation in a simply connected domains}
For the linear vorticity equation we found the boundary condition. Here we find it for nonlinear equation. Vorticity equation (\ref{maineqw}) corresponding to Navier-Stokes system after Riemann mapping reduces to
\begin{align}\label{helmholtzeq2}
|(\Phi^{-1})'(z)|^2\partial_t w(t,\mathbf{x}) - \Delta w + B(v,w) =0,
\end{align}
where
$$
B(v,w) = \operatorname{Re} (\Phi^{-1})' (\mathbf{v},\nabla w) - \operatorname{Im} (\Phi^{-1})' (\mathbf{v}^\perp,\nabla w).
$$
with $\mathbf{x} \in B_{r_0}$.
Supply this equation with
\begin{align}
&r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = \nonumber \\
&~~~~\left \{ { \begin{matrix}
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} (\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C}){z^{-|k|-1}} \overline{(\Phi^{-1})'} w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k\geq 0, \\
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \overline{(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C})z^{-|k|-1}} (\Phi^{-1})' w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k<0.
\end{matrix} } \right . \label{robin_bound_ns_general}
\end{align}
\begin{thm}Let $\Omega$, $\Phi$ as in Theorem \ref{stokesomegathm}, given initial datum $\mathbf{v}_0(\mathbf{x})$, $\operatorname{curl} \mathbf{v}_0\in L_1(\Omega)$ satisfying no-slip condition (\ref{bound}), infinity condition (\ref{boundinf}), zero-\\circularity (\ref{zerocirculation}) and $\mathbf{v}(t,\mathbf{x})$ be the solution of (\ref{maineqns})-(\ref{boundinfns}) in $\Omega$. Then vorticity $w(t,\cdot)=$ $\operatorname{curl} \mathbf{v}(t,\Phi^{-1}(z))$ satisfies (\ref{robin_bound_ns_general}).
\end{thm}
\begin{proof}
Apply the same steps as in previous subsections: for fixed $k>0$ divide this equation by $z^k$ and integrate over $B_{r_0}$. In the previous subsection we obtained
$$
\frac d{dt}\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'(z)}}{z^k} w(t,\mathbf{x}) \operatorname{d\mathbf{x}} = 0,
$$
and
\begin{align*}
\int_{B_{r_0}} \frac {\Delta w}{z^{k}}\operatorname{d\mathbf{x}} = -2\pi r_0^{-k}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) \right ).
\end{align*}
From (\ref{freediv3}), (\ref{curleq3})
$$
\int_{B_{r_0}} \frac {\operatorname{Re} (\Phi^{-1})'}{z^{k}} w \operatorname{div} \mathbf{v} \operatorname{d\mathbf{x}} = -
\int_{B_{r_0}} \frac {\operatorname{Im} (\Phi^{-1})'}{z^{k}} w \operatorname{curl} \mathbf{v} \operatorname{d\mathbf{x}}.
$$
Then using Cauchy-Riemann equations
\begin{align*}
\int_{B_{r_0}} \frac {B(v,w)}{z^{k}}\operatorname{d\mathbf{x}} = \int_{B_{r_0}} \frac {\operatorname{Re} (\Phi^{-1})'}{z^{k}}(\mathbf{v},\nabla w) -
\frac {\operatorname{Im} (\Phi^{-1})'}{z^{k}} (\mathbf{v}^\perp,\nabla w) \operatorname{d\mathbf{x}}\\=
\int_{B_{r_0}} \left ( -\frac {\operatorname{Re} (\Phi^{-1})'}{z^{k}} w \operatorname{div} \mathbf{v} -
\frac {\operatorname{Im} (\Phi^{-1})'}{z^{k}} w \operatorname{curl} \mathbf{v} \right ) \operatorname{d\mathbf{x}} \\
-\int_{B_{r_0}} w \left ( \left ( v, \nabla \frac {\operatorname{Re} (\Phi^{-1})'}{z^{k}} \right ) - \left ( v^\perp, \nabla \frac {\operatorname{Im} (\Phi^{-1})'}{z^{k}} \right ) \right ) \operatorname{d\mathbf{x}} \\ =
k\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'}}{z^{k+1}} (v_1+iv_2) w(t,\mathbf{x}) \operatorname{d\mathbf{x}}
=k\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'}}{z^{k+1}} \mathbf{v}^\mathbb{C} w(t,\mathbf{x}) \operatorname{d\mathbf{x}}
\end{align*}
Since from (\ref{noslipcondintegral2_omega})
$$
k\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'}}{z^{k+1}} \mathbf{v}^\mathbb{C}_\infty w(t,\mathbf{x}) \operatorname{d\mathbf{x}} = 0,
$$
then
$$
\int_{B_{r_0}} \frac {B(v,w)}{z^{k}}\operatorname{d\mathbf{x}} = k\int_{B_{r_0}} \frac {\overline{(\Phi^{-1})'}}{z^{k+1}} (\mathbf{v}^\mathbb{C}-\mathbf{v}^\mathbb{C}_\infty ) w(t,\mathbf{x}) \operatorname{d\mathbf{x}},
$$
from which follows (\ref{robin_bound_ns_general}) for $k\geq 0$.
For $k<0$ (\ref{robin_bound_ns_general}) follows from the identity $w_{-k}(t,r) = \overline{w_k(t,r)}$. Theorem is proved.
\end{proof}
This theorem says, that for Navier-Stokes system if $\|\mathbf{v}^\mathbb{C}-\mathbf{v}^\mathbb{C}_\infty\|$ is small, then (\ref{robin_bound}) works well. Nevertheless this boundary condition requires some corrections like boundary control (\ref{robin_bound_control}) in order to stay on invariant affine subspace $M$.
\section{No-slip boundary condition for vorticity in exterior domains}
In this section we construct boundary condition for vorticity which approximates no-slip condition. With help of Riemann mapping we had reduced the Helmholtz equation (\ref{maineqw}) defined in $\Omega$ to (\ref{helmholtzeq2}) defined in $B_{r_0}$. Formulas (\ref{robin_bound_ns_general}) present integral condition which approximately equal to boundary condition (\ref{robin_bound}) for well streamlined obstacle when $\mathbf{v} \simeq \mathbf{v}_\infty$. In this section using fixed point theorem we will derive boundary condition for vorticity in the form like (\ref{_3:bound:nonlin}) which ensure the solution to satisfy $w_N(t,\cdot) \in M_N$ defined in (\ref{noslipcondintegral2_omega_N}).
Then the velocity $\mathbf{v}(t,\mathbf{x})$ restored from $w(t,\mathbf{x})$ via Biot-Savar law (\ref{BSformula}) will be the solution of Navier-Stokes system with approximate no-slip boundary condition satisfying (\ref{weakconv}).
Fix $N\in \mathbb{N}$. We supply vorticity equation (\ref{helmholtzeq2}) with
\begin{align}
&w(0,\mathbf{x})=w_0(\mathbf{x}), \label{_3:initw}\\
&w(t,\mathbf{x}) \to 0,~|\mathbf{x}|\to \infty, \label{_3:boundinfw}
\end{align}
and boundary condition
\begin{equation}\label{_3:bound1}
r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) = u_k(t),~k = 0,1,\dots, N,
\end{equation}
where $\vec u(t) = \{u_k(t)\}_{k=0}^N$ is the set of unknown boundary functions.
Then since $w_{-k}(t,r) = \overline{w_k}(t,r)$ for $k=-N, -N-1, \dots, -1$
\begin{equation}\label{_3:bound2}
r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} - k w_k(t,r_0) = \overline{u_k(t)}.
\end{equation}
For $|k|>N$ set
\begin{equation}\label{_3:bound3}
r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + k w_k(t,r_0) = 0.
\end{equation}
We construct approximate boundary condition for Helmholtz equation under the assumption of unique resolvability of (\ref{helmholtzeq2}), (\ref{_3:initw})-(\ref{_3:bound3}). For the exterior of the disc the solvability theorems are proved in Appendixes \ref{oseen_sect}, \ref{ns_sect}.
For the Helmholtz problem we should use Sobolev space $H^1(B_{r_0})$ as the phase space for $w(t,\cdot)$. Difficulty lies in the fact that the velocity field $\mathbf{v}$ isn't well defined in $H^1(B_{r_0})$.
In order to obtain $\mathbf{v}(t,\cdot) \in L_\infty(B_{r_0})$ from Lemma \ref{lembsest} we need involve $L_1$ into phase space for $w_{-1}, w_0, w_1$. So, we will use phase space
$$
W = \left \{w \in H^1(B_{r_0}),~w_{-1}, w_0, w_1 \in L_1(r_0,\infty;r) \right \}.
$$
\begin{thm} Suppose that for fixed $N>0$, $T>0$ there exists the solution $w_N(t,\mathbf{x}) \in C \left ( [0,T], W) \right )$ of Helmholtz equation (\ref{helmholtzeq2}), (\ref{_3:initw})-(\ref{_3:bound3}) which is locally Lipschitz continuous due to $u_k \in C[0,T]$, $k=0,\dots, N$, $w_0(\mathbf{x}) \in H^1(B_{r_0})$. Then for some $M>0$ and any initial datum with $\|w_0(\cdot)\|_{H^1}\leq M$ there exists $\vec u(t) = \{u_k(t)\}_{k=0}^N$, that $w_N(t,\cdot) \in M_N$ for all $t \in [0,T]$.
\end{thm}
\begin{proof}
The relation (\ref{robin_bound_ns_general}) can be rewritten as
\begin{align}\label{boundary_equation}
&\vec u(t) = F(\vec u)
\end{align}
with the mapping $F(\vec u) : \left (C[0,T]\right )^{2N+1} \to \left (C[0,T]\right )^{2N+1}$:
\begin{align*}
&F[\vec u(\cdot)] = \nonumber \\
&~~~~\left \{ { \begin{matrix}
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} (\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C}){z^{-|k|-1}} \overline{(\Phi^{-1})'} w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k = 0, \dots, N \\
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \overline{(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C})z^{-|k|-1}} (\Phi^{-1})' w(t,\mathbf{x})\operatorname{d\mathbf{x}},~k = -N, \dots, -1.
\end{matrix} } \right .
\end{align*}
where $w(t,\mathbf{x})$ is the solution of boundary-value problem (\ref{helmholtzeq2}), (\ref{_3:initw})-(\ref{_3:bound3}) with boundary condition $\vec u(t)$. $F$ is well-defined since $k/z^{|k|+1} \in L_2(B_{r_0})$ and in virtue of Lemma \ref{lembsest} from
\begin{align*}
\left \| k (\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}^\mathbb{C}(t,\cdot) ){z^{-|k|-1}} \right \|_{L_2(B_{r_0})} \leq C \|w(t,\mathbf{x})\|_{W}
\end{align*}
follows
\begin{align*}
\left |F[\vec u(t)] \right | \leq C \|w(t,\cdot)\|^2_{W}
\end{align*}
with some new constant $C>0$. From $w(t,\mathbf{x}) \in C\left([0,T];H^1(B_{r_0}) \right)$ follows $F[\vec u(t)] \in \left (C[0,T]\right )^{2N+1}$.
Fix $M>0$ and
take $\vec u_1(\cdot)$, $\vec u_2(\cdot)$ from $\left (C[0,T]\right )^{2N+1}$ with $\|\vec u_i\|_{L_\infty}<M$, $i=1,2$. Let $w_1$, $w_2$ - be the corresponding solutions of (\ref{helmholtzeq2}), (\ref{_3:initw})-(\ref{_3:bound3}) with $w_0(\mathbf{x})$ satisfying $\|w_0(\mathbf{x})\|\leq M$.
$F$ is the contraction mapping with respect to $\vec u$. Indeed
\begin{align*}
&\left | F[\vec u_1(\cdot)] - F[\vec u_2(\cdot)] \right | \leq \\&~~~~~~~~
\frac {|k| r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \Big | (\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}_1^\mathbb{C}){z^{-|k|-1}} \overline{(\Phi^{-1})'} w_1(t,\mathbf{x}) - \\&~~~~~~~~
(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}_2^\mathbb{C}){z^{-|k|-1}} \overline{(\Phi^{-1})'} w_2(t,\mathbf{x})\Big | \operatorname{d\mathbf{x}} \leq \\&~~~~~~~~ \frac {r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \left | k(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}_1^\mathbb{C}){z^{-|k|-1}} (w_1(t,\mathbf{x}) - w_2(t,\mathbf{x})) \right | \operatorname{d\mathbf{x}} + \\&~~~~~~~~ \frac {r_0^{|k|}}{2\pi}
\int_{B_{r_0}} \Big | k(\mathbf{v}_2^\mathbb{C} - \mathbf{v}_1^\mathbb{C}){z^{-|k|-1}} w_2(t,\mathbf{x}) \Big | \operatorname{d\mathbf{x}} .
\end{align*}
From Lemma \ref{lembsest} again we have with some $C>0$
\begin{align*}
\left \| k \left (\mathbf{v}_1^\mathbb{C}(t,\cdot) - \mathbf{v}_2^\mathbb{C}(t,\cdot)\right ){z^{-|k|-1}} \right \|_{L_2(B_{r_0})} \leq C \|w_1(t,\mathbf{x}) - w_2(t,\mathbf{x})\|_{W}
\end{align*}
and
$$
\| k(\mathbf{v}^\mathbb{C}_\infty - \mathbf{v}_1^\mathbb{C}(t,\cdot)){z^{-|k|-1}}\|_{L_2(B_{r_0})}
\leq C\| w_1(t,\mathbf{x}) \|_{W}.
$$
Note, that from Lipschitz continuity of $w$ due to $\vec u$ the norms $\| w_1(t,\mathbf{x}) \|_{W}$, $\| w_2(t,\mathbf{x}) \|_{W}$ can be arbitrary small for small $M>0$. Then the estimate
\begin{align*}
\left | F[\vec u_1(\cdot)] - F[\vec u_2(\cdot)] \right | \leq
C(M) \|w_1(t,\cdot) - w_2(t,\cdot)\|_{W}
\end{align*}
holds with $C(M) \to 0$ as $M \to 0$.
Since with some $L>0$
$$
\|w_1 - w_2\|_{C \left ( [0,T], W) \right )} \leq L \| u_1 - u_2 \|_{\left (C[0,T]\right )^{2N+1}}
$$
then for small $M>0$ we will finally have
$$
\left \| F[\vec u_1(\cdot)] - F[\vec u_2(\cdot)] \right \|_{\left (C[0,T]\right )^{2N+1}} \leq K \|\vec u_1(\cdot) - \vec u_2(\cdot)\|_{\left (C[0,T]\right )^{2N+1}},
$$
with $K<1$ and from the fixed-point theorem there exists the solution of (\ref{boundary_equation}). The theorem is proved.
\end{proof}
\begin{remark} If $w_0(\mathbf{x}) \in H^2(B_{r_0})$ then the solution will satisfy $w(t,\cdot)\in H^2(B_{r_0})$ and the velocity on the boundary will tend to zero in $L_2(\partial \Omega)$ as $N\to\infty$ according to (\ref{strongconv}).
If we further increase the smoothness of the initial datum then we can obtain the uniform convergence of velocity to no-slip condition on the boundary.
\end{remark}
\section{Appendix}
Here we study semigroup for Stokes flow and prove uniqueness solvability for Oseen and Helmholtz equations using fixed-point argument.
\subsection{Stokes semi-group estimates}
The boundary-value problem (\ref{stokeseq}), (\ref{robin_bound}), (\ref{boundinfvorticity}) can be solved using special Weber-Orr transform (see \cite{AG} for more details):
\begin{equation}\label{int:weberorr}
W_{k,l}[f](\lambda) = \int_{r_0}^\infty \frac{J_{k}(\lambda s)Y_{l}(\lambda r_0) - Y_{k}(\lambda s)J_{l}(\lambda r_0)}{\sqrt{J_{l}^2(\lambda r_0) + Y_{l}^2(\lambda r_0)}} f(s) s \operatorname{ds},~k\in \mathbb{Z},
\end{equation}
where
$J_{k}(r)$, $Y_{k}(r)$ - are the Bessel functions of the first and second type (see \cite{BE}).
\vskip 5pt
The inverse transform is defined by the formula
\begin{equation}\label{int:weberorrinv}
W^{-1}_{k,l}[\hat f](r) = \int_{0}^\infty \frac{J_{k}(\lambda r)Y_{l}(\lambda r_0) - Y_{k}(\lambda r)J_{l}(\lambda r_0)}{\sqrt{J_{l}^2(\lambda r_0) + Y_{l}^2(\lambda r_0)}} \hat f (\lambda) \lambda \dlambda.
\end{equation}
The case of $W_{|k|,|k|-1}$ is of special interest and this transform can be considered as hydrodynamical. In \cite{AG} was proved the following
\begin{thm} Let vector field $\mathbf{v}_0(\mathbf{x})$ satisfies (\ref{freediv}), (\ref{bound}), (\ref{boundinf}), (\ref{zerocirculation}), the vorticity $\operatorname{curl} \mathbf{v}_0(\mathbf{x})$ $ \in L_1(B_{r_0})$, and its Fourier series as well as Fourier series for its vorticity $w_0(\mathbf{x})$ with coefficients $w_k^0(r)$ converges, $\mathbf{v} (t, \mathbf{x})$ be the solution of (\ref{stokeseqvelocity}), (\ref{freedivns})-(\ref{boundinfns}). Then $w(t,\mathbf{x})=\operatorname{curl} \mathbf{v} (t, \mathbf{x})$ satisfies equation (\ref{stokeseq}), boundary conditions (\ref{robin_bound}), (\ref{boundinfvorticity}) and is given via Fourier coefficients:
\begin{align}\label{MyGreatFormula}
w_k(t,r) = W^{-1}_{|k|,|k|-1} \left [ e^{-\lambda^2 t} W_{|k|,|k|-1} [w^0_k(\cdot)](\lambda) \right ](t,r).
\end{align}
\end{thm}
This formula gives an explicit form of the solution for Stokes system in the exterior of the disc in terms of vorticity. From Biot-Savar law one can get the velocity field.
Generalised Weber-Orr transform has a non-trivial kernel. So, the transform (\ref{int:weberorrinv}) is the inverse one to (\ref{int:weberorr}) only for functions, which are orthogonal to the kernel. But from no-slip condition (\ref{noslipcondintegral}) follows orthogonality of Fourier coefficients $w_k(t,\cdot)$ to the kernel of $W_{|k|,|k|-1}$. The invertibility of the generalised Weber-Orr transform was studied in detail in \cite{AG}.
So, Weber-Orr transforms satisfy Bessel-type inequality instead of the Plancherel equity:
\begin{align}\label{besselineq}
\| W_{k,l}[f] \|_{L_2(0,\infty; \lambda) }^2 \leq \|f\|_{L_2(r_0,\infty; r)}^2.
\end{align}
From differentiation rules for Bessel functions follows
\begin{align}
\frac{\partial}{\partial r} W^{-1}_{k,k-1} [f] = \frac 1 2 \left ( W^{-1}_{k-1,k-1}[\lambda f] - W^{-1}_{k+1,k-1}[\lambda f] \right ), \label{diffweber1}\\
\frac kr W^{-1}_{k,k-1}[ f] = \frac 12 \left ( W^{-1}_{k+1,k-1}[\lambda f] + W^{-1}_{k-1,k-1}[\lambda f] \right ).\label{diffweber2}
\end{align}
In polar coordinates with unit vectors $\mathbf{e_r}$, $\mathbf{e_\varphi}$ the gradient is defined as $\nabla = \frac{\partial}{\partial r} \mathbf{e_r} + \frac 1r \frac{\partial}{\partial \varphi} \mathbf{e_\varphi}$ with polar coordinates $\nabla_r$, $\nabla_\varphi$. Multiplier $\frac kr$ corresponds to differentiation with respect to $\varphi$ of function $f(r)e^{ik\varphi}$.
Formula (\ref{diffweber2}) involves Weber transform $W^{-1}_{k+1,k-1}$ which also possesses non-trivial kernel and satisfy Bessel inequality
$$
\| W_{k+1,k-1}[f] \|_{L_2(0,\infty; \lambda) }^2 \leq \|f\|_{L_2(r_0,\infty; r)}^2.
$$
And vice versa, multiplication by $\lambda$ transfers to differentiation but in more general sense including not only $\frac{\partial}{\partial r}$ but also angle derivative $\frac{\partial}{\partial \varphi}$ expressed by multiplier $\frac kr$. Indeed, using
\begin{align*}
\lambda J_k(\lambda r) = \frac{k-1}r J_{k-1}(\lambda r) - \lambda J'_{k-1}(\lambda r), \\
\lambda Y_k(\lambda r) = \frac{k-1}r Y_{k-1}(\lambda r) - \lambda Y'_{k-1}(\lambda r)
\end{align*}
we will have
\begin{equation} \label{diffweber3}
\lambda W_{k,k-1} [f] = W_{k-1,k-1} [\frac {k f}r] + W_{k-1,k-1} [f'(\cdot )].
\end{equation}
Formula (\ref{MyGreatFormula}) in theorem defines the Stokes semigroup $S(t)$ which corresponds to problem (\ref{stokeseq}), (\ref{robin_bound}), (\ref{boundinfvorticity}). The estimates of this semi-group are given by the following
\begin{prop} \label{stokes_semigroup_thm}For $t > 0$ Stokes semigroup $S(t)$ for vorticity $\operatorname{curl}$ satisfies
\begin{align*}
\|S(t)w_0\|_{L_2(B_{r_0})} \leq \|w_0\|_{L_2(B_{r_0})}, \\
\|\nabla S(t)w_0\|_{L_2(B_{r_0})} \leq \frac 1{\sqrt {et} }\|w_0\|_{L_2(B_{r_0})},\\
\|S(t)w_0\|_{H^1(B_{r_0})} \leq \sqrt 3 \|w_0\|_{H^1(B_{r_0})}.
\end{align*}
\end{prop}
\begin{proof}
From Bessel inequality we have
\begin{align*}
\|w_k(t,r)\|^2_{L_2(r_0,\infty, r)} \leq \| e^{-\lambda^2 t} W_{|k|,|k|-1} [w^0_k(\cdot)](\lambda)\|^2_{L_2(0,\infty, \lambda)} \\
\leq \| W_{|k|,|k|-1} [w^0_k(\cdot)](\lambda)\|^2_{L_2(0,\infty, \lambda)} \leq
\| w^0_k(r)\|_{L_2(r_0,\infty, r )}.
\end{align*}
Summarizing by $k$ we obtain the first estimate.
Fix $k \geq 0$. Then from estimate
$$
|\lambda e^{-\lambda^2 t}| \leq \frac 1{\sqrt {2et} }
$$
and Bessel inequality (\ref{besselineq}) from (\ref{diffweber1}) we have
\begin{align*}
\left \|\frac{\partial}{\partial r} w_k(t,r) \right \|_{L_2(r_0,\infty, r)} \leq \| \lambda e^{-\lambda^2 t} W_{k,k-1} [w^0_k(\cdot)](\lambda)\|_{L_2(0,\infty, \lambda)} \\ \leq
\frac {\| w^0_k(r)\|_{L_2(r_0,\infty, r )}}{\sqrt {2et}}
\end{align*}
and
$$
\|\nabla_r S(t)w_0\|_{L_2(B_{r_0})} \leq \frac 1{\sqrt {2et} }\|w_0\|_{L_2(B_{r_0})}.
$$
From (\ref{diffweber2}) in a similar way follows
\begin{align*}
\left \|\frac kr w_k(t,r) \right \|_{L_2(r_0,\infty, r)} \leq \frac {\| w^0_k(r)\|_{L_2(r_0,\infty, r )}}{\sqrt {2et}}.
\end{align*}
Then we have
$$
\|\nabla_\varphi S(t)w_0\|_{L_2(B_{r_0})} \leq \frac 1{\sqrt {2et} }\|w_0\|_{L_2(B_{r_0})}
$$
and the second estimate of the proposition is proved.
Now we will prove the last inequality. Using (\ref{diffweber1}),(\ref{diffweber3})
\begin{align*}
\left \|\frac{\partial}{\partial r} w_k(t,r) \right \|_{L_2(r_0,\infty, r)} \leq \| \lambda e^{-\lambda^2 t} W_{k,k-1} [w^0_k(\cdot)](\lambda)\|_{L_2(0,\infty, \lambda)} \\ = \left \| e^{-\lambda^2 t} W_{k-1,k-1} [\frac {k w^0_k}r + \frac{\partial}{\partial r} w^0_k(\cdot )] \right \| \leq \left \|\frac {k w^0_k}r \right \| + \left \| \frac{\partial}{\partial r} w^0_k(\cdot ) \right \|
\end{align*}
and so
\begin{align} \label{nablarsemigroup}
\|\nabla_r S(t)w_0\|_{L_2(B_{r_0})} \leq \sqrt 2 \|\nabla w_0\|_{L_2(B_{r_0})}.
\end{align}
In a similar way from (\ref{diffweber2}), (\ref{diffweber3})
\begin{align*}
\left \|\frac kr w_k(t,r) \right \|_{L_2(r_0,\infty, r)} \leq \| \lambda e^{-\lambda^2 t} W_{k,k-1}
[w^0_k(\cdot)](\lambda)\|_{L_2(0,\infty, \lambda)}
\\ \leq
\left ( \left \|\frac {k w^0_k}r \right \| + \left \| \frac{\partial}{\partial r} w^0_k(\cdot ) \right \| \right )
\end{align*}
and finally
$$
\|\nabla_\varphi S(t)w_0\|_{L_2(B_{r_0})} \leq \sqrt 2 \|\nabla w_0\|_{L_2(B_{r_0})}
$$
combined with (\ref{nablarsemigroup}) and the first estimate of the proposition gives the last estimate.
\end{proof}
\subsection{Existence theorem for Oseen equation in exterior of the disc} \label{oseen_sect}
For a fixed irrotational velocity field $\tilde \mathbf{v}(t,\mathbf{x}) $ consider Oseen equation
\begin{equation}
\frac{\partial w(t,\mathbf{x})}{\partial t} - \Delta w + (\tilde \mathbf{v},\nabla)w = 0, \label{oseeneqw}
\end{equation}
with initial datum (\ref{_3:initw})
and condition at infinity (\ref{_3:boundinfw}).
We supply the problem (\ref{oseeneqw}), (\ref{_3:initw}), (\ref{_3:boundinfw}) with boundary conditions (\ref{_3:bound1})-(\ref{_3:bound3}).
Define for $k, l\in \mathbb{Z}$
$$
R_{k,l}(\lambda, r) = \frac{J_{k}(\lambda r)Y_{l}(\lambda r_0) - Y_{k}(\lambda r)J_{l}(\lambda r_0)}{\sqrt{J_{l}^2(\lambda r_0) + Y_{l}^2(\lambda r_0)}}.
$$
From properties of Bessel functions
\begin{align*}
J_{k - 1}(r)={\frac {k }{r}}J_{k }(r) + J'_{k }(r) \\
Y_{k - 1}(r)={\frac {k }{r}}Y_{k }(r) + Y'_{k }(r)
\end{align*}
follows
$$
k {R_{k,k-1}(\lambda, r_0)} + \lambda r_0 {R'_{k,k-1}(\lambda, r_0)} = r_0 {R_{k-1,k-1}(\lambda, r_0)} = 0.
$$
Also holds (\cite{BE}):
$$
R_{k,k-1}(\lambda, r_0) = \frac{J_{k}(\lambda r_0)Y_{k-1}(\lambda r_0) - Y_{k}(\lambda r_0)J_{k-1}(\lambda r_0)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} = \frac 2{\pi r_0 \lambda }.
$$
We apply Weber-Orr transform $W_{k,k-1}$ (\ref{int:weberorr}) to Helmholtz equation for Oseen flow (\ref{oseeneqw}). First we find how it acts on Laplace operator. Then using integration by parts we will have
\begin{align*}
&W_{k,k-1}[\Delta_k w_k(t, r)] = - r_0 \frac{\partial w_k(t, r)}{\partial r} \frac{R_{k,k-1}(\lambda, r)}{\sqrt{J_{k-1}^2(\lambda, r_0) + Y_{k-1}^2(\lambda, r_0)}} \Bigg |_{r_0}^\infty \\ & + r_0 w_k(t, r) \frac{\lambda R'_{k,k-1}(\lambda, r)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} \Bigg |_{r_0}^\infty - \lambda^2 W_{k,k-1}[w_k(t, r)] \\ &
=\left (u_k(t) - k w_k(t,r_0) \right) \frac{ R_{k,k-1}(\lambda, r_0)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} \\ & - r_0 w_k(t, r_0) \frac{\lambda R'_{k,k-1}(\lambda, r)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} - \lambda^2 \hat w_k(t, \lambda)\\ &
= - \frac{w_k(t,r_0)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} \left (k {R_{k,k-1}(\lambda, r_0)} + \lambda r_0 {R'_{k,k-1}(\lambda, r_0)} \right ) \\ &- \lambda^2 \hat w_k(t, \lambda) + u_k(t)\frac{ R_{k,k-1}(\lambda, r_0)}{\sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} \\ & =
\frac{2 u_k(t)}{\pi r_0 \lambda \sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}} - \lambda^2 \hat w_k(t, \lambda).
\end{align*}
Then the Helmholtz equation (\ref{oseeneqw}) in terms of Fourier coefficients $w_k$ after Weber-Orr transform can be written as
$$
\partial \hat w_k(t, \lambda) + \lambda^2 \hat w_k(t, \lambda) + r_k(\lambda) u_k(t) + W_{k,k-1}[(\mathbf{v},\nabla)w]_k = 0
$$
where
$$
r_k(\lambda) = \frac{2}{\pi r_0 \lambda \sqrt{J_{k-1}^2(\lambda r_0) + Y_{k-1}^2(\lambda r_0)}}.
$$
Finally with the help of integral transform we reduced the Helmholtz equation to the following integral relation
\begin{align*}
\hat w_k(t, \lambda) =
e^{-\lambda^2 t} \hat w_{0,k}(\lambda) + r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \nonumber \\ + \int_0^t e^{-\lambda^2(t-\tau)}W_{k,k-1}[(\tilde \mathbf{v},\nabla)w]_k(\tau, \lambda)d\tau,
\end{align*}
where $\hat w_k(t, \cdot) = W_{k,k-1}[w_k(t, \cdot)]$, $[(\tilde \mathbf{v},\nabla)w]_k$ - $k$-th Fourier coefficient of the term $(\tilde \mathbf{v},\nabla)w$. Take the inverse transform $W_{k,k-1}$ and rewrite it in terms of Stokes semigroup $S(t)$ corresponding to the problem (\ref{stokeseq}), (\ref{robin_bound}), (\ref{boundinfvorticity}):
\begin{align}\label{_3:int_rel}
w(t,\mathbf{x}) =
S(t)w_0(\mathbf{x}) + \sum_{k=-N}^N e^{ik\varphi} W_{k,k-1} \left [ r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \right ] \nonumber \\ + \int_0^t \left (\nabla S(t - \tau), \tilde \mathbf{v} \right) w(\tau, \mathbf{x}) d\tau.
\end{align}
\begin{thm}[resolvability of Oseen equation]
For given $\tilde \mathbf{v}(t,\mathbf{x}) \in C(R_+\times B_{r_0})\cap L_\infty^{loc}\left(R_+; L_\infty(B_{r_0})\right)$ satisfying no-slip condition (\ref{boundns}), $u_k(t) \in L_\infty^{loc}(\mathbb{R}_+)$, $k=0,\dots, N$,
$w_0(\mathbf{x}) \in L_2(B_{r_0})$, the problem
(\ref{oseeneqw}), (\ref{_3:initw})-(\ref{_3:bound3}) has a unique global solution $w(t,\mathbf{x}) \in C \left ( [0,\infty), L_2(B_{r_0}) \right )$ which is locally Lipschitz mapping due to $\{u_k\}_{k=1}^N$, $w_0(\mathbf{x})$.
\end{thm}
\begin{proof}First we need prove the local resolvability of the equation (\ref{_3:int_rel}). Fix $T>0$. For given $\tilde \mathbf{v} \in L_\infty(B_{r_0})$, $u_k\in C[0,T]$, $k=0,\dots, N$. Consider the map
\begin{align*}
F\left(w(\tau, \cdot)\right) = S(t)w_0(\mathbf{x}) + \sum_{k=-N}^N e^{ik\varphi} W_{k,k-1} \left [ r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \right ] \nonumber \\ - \int_0^t \left (\nabla S(t - \tau), \tilde \mathbf{v} \right) w(\tau, \mathbf{x}) d\tau.
\end{align*}
Consider the space $Q=C \left ( [0,T], L_2(B_{r_0}) \right )$. Since asymptotical behaviour of $r_k(\lambda)$ is $1/\sqrt \lambda$ then with some $C>0$
$$
\| r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \|_{L_2(0,\infty, \lambda d\lambda)} \leq
C\|u_k(\tau)\|_{C[0,T]}.
$$
Set
$$M=\| w_0(\cdot)\|_{L_2} + C\sum_{k=-N}^N \|u_k(\tau)\|_{C[0,T]}.$$
Using estimates on $S(t)$ from Proposition \ref{stokes_semigroup_thm} we have
\begin{align*}
&\left \|F\left(w(t, \cdot)\right) \right\|_{L_2} \leq \| w_0(\cdot)\|_{L_2} + C\sum_{k=-N}^N \|u_k(\tau)\|_{C[0,T]} \nonumber + \\ & \int_0^t \left \| (\nabla S(t - \tau), \tilde \mathbf{v} ) w(\tau, \mathbf{x}) \right\|_{L_2} d\tau \leq \\ & M +\|\tilde \mathbf{v}\|_{L_\infty(R_+\times B_{r_0})} \|w\|_{Q} \int_0^t \frac{d\tau}{\sqrt{e(t-\tau)}} = \\ &
M + 2 \sqrt \frac T e \|\tilde \mathbf{v}\|_{L_\infty([0,T]\times B_{r_0})} \|w\|_{Q}.
\end{align*}
So we deduced that $F:Q \to Q$ is well-defined and maps $Q$ into itself. Now we prove that $F$ is a strict contraction in $Q$.
\begin{align*}
F\left(w_1(\tau, \cdot)\right) - F\left(w_2(\tau, \cdot)\right) = \int_0^t \left (\nabla S(t - \tau), \tilde \mathbf{v} \right) \left (w_1(\tau, \mathbf{x}) - w_2(\tau, \mathbf{x}) \right )d\tau
\end{align*}
and
\begin{align*}
&\|F\left(w_1(\tau, \cdot)\right) - F\left(w_2(\tau, \cdot)\right)\|_Q \leq \\ & \operatorname{vraisup}_{t\in[0,T]} \int_0^t \left\| (\nabla S(t - \tau), \tilde \mathbf{v} ) \left (w_1(\tau, \mathbf{x}) - w_2(\tau, \mathbf{x}) \right )\right \|_{L_2}d\tau \leq \\ &
2\sqrt \frac T e \|\tilde \mathbf{v}\|_{L_\infty([0,T]\times B_{r_0})} \|w_1-w_2\|_Q.
\end{align*}
Then for small $T$ by the Banach fixed point theorem, the map $F$ has a unique fixed point $w(t,\mathbf{x})$.
Next, we prove, that $\|w(t,\cdot)\|_{L_2(B_{r_0})}$ cannot blow up in finite time and the solution is global. Since $F(w)=w$ then from estimates above
\begin{align*}
\|w(t,\cdot)\|_{L_2} \leq M + \|\tilde \mathbf{v}\|_{L_\infty([0,t]\times B_{r_0})} \int_0^t \left ( \frac{1}{\sqrt{2e(t-\tau)}} \right )\|w(\tau,\cdot)\|_{L_2}d\tau
\end{align*}
and from Gronwall’s Lemma
\begin{align*}
\|w(t,\cdot)\|_{L_2} \leq M e^{2 \sqrt \frac t e \operatorname{vraisup}_{\tau\in [0,t] }\|\tilde \mathbf{v}(\tau, \cdot)\|_{L_\infty(B_{r_0})}}
\end{align*}
and $\|w(t,\cdot)\|_{L_2}$ stays finite for all time $t>0$.
No, we prove uniqueness. If $w_1(t,\cdot)$, $w_2(t,\cdot)$ are two solutions of (\ref{oseeneqw}), (\ref{_3:initw})-(\ref{_3:bound3}), then its difference $w_1-w_2$ is the solution of the same problem with zero initial and boundary conditions. Then $M$ defined above equals zero and from the same Gronwall’s Lemma
$$
\|w_1(t,\cdot)-w_2(t,\cdot)\|_{L_2} \leq 0,
$$
and the theorem is completely proved.
\end{proof}
\subsection{Existence theorem for Helmholtz equation in exterior of the disc} \label{ns_sect}
We consider the map
\begin{align} \label{mapF}
F\left(w(t, \cdot)\right) = S(t)w_0(\mathbf{x}) + \sum_{k=-N}^N e^{ik\varphi} W_{k,k-1} \left [ r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \right ] \nonumber \\ + \int_0^t S(t - \tau) (\mathbf{v}, \nabla w) d\tau
\end{align}
associated with the boundary-value problem for the Helmholtz equation (\ref{helmholtzeq}). In virtue of Lemma \ref{lembsest} $F$ will be well-defined in $Q=C \left ( [0,T], W) \right )$, where $W = \{w \in H^1(B_{r_0})$, $w_{-1}, w_0, w_1 \in L_1(r_0,\infty;r) \}$. Denote cylinder $Z_{r_0,T} = [0,T]\times B_{r_0}$.
\begin{thm}[resolvability of Helmholtz equation]
For any $L>0$ there exists $T=T(L)>0$ such that for all $w_0 \in H^1(B_{r_0})$, $u_k\in C[0,T]$, $k=0,\dots, N$, $\|w_0\|_{H^1} \leq L$, $\| u_k(\cdot) \|_{C[0,T]}<L$ the problem (\ref{helmholtzeq}), (\ref{_3:initw})-(\ref{_3:bound3}) has a unique solution $w(t,\mathbf{x}) \in Q$.
\end{thm}
\begin{proof}
First we derive that $F\left(w(t, \cdot)\right) \in H^1(B_{r_0})$ for fixed $t\in [0,T]$. From the proposition \ref{stokes_semigroup_thm} $S(t)$ is continuous mapping in $H^1(B_{r_0})$.
From the estimate
$$
\| \lambda r_k(\lambda) \int_0^t e^{-\lambda^2(t-\tau)}u_k(\tau)d\tau \|_{L_2(0,\infty, \lambda d\lambda)} \leq
C\|u_k(\tau)\|_{C[0,T]}
$$
the second term in (\ref{mapF}) belongs to $H^1(B_{r_0})$.
With help of the proposition \ref{stokes_semigroup_thm} we have
\begin{align*}
\left \|F\left(w(t, \cdot)\right) \right\|_{H^1(B_{r_0})} \leq \sqrt 3 \| w_0(\cdot)\|_{H^1(B_{r_0})} + C\sum_{k=-N}^N \|u_k(\tau)\|_{ C[0,T]} \nonumber + \\ \int_0^t \left \| S(t - \tau) \left ( \mathbf{v}, \nabla w(\tau, \mathbf{x}) \right ) \right\|_{H^1(B_{r_0})} d\tau \leq \\
M + \int_0^t \left (1+\frac1{\sqrt {e(t-\tau)}} \right )\left \|\left ( \mathbf{v}, \nabla w(\tau, \mathbf{x}) \right ) \right\|_{H^1(B_{r_0})} d\tau
\end{align*}
with
$$M=\sqrt 3 \| w_0(\cdot)\|_{H^1} + C\sum_{k=-N}^N \|u_k(\tau)\|_{ C[0,T]}.$$
We invoke the following Sobolev inequality:
$$
\|\mathbf{v}(t,\cdot) - \mathbf{v}_\infty\|_{L_\infty(B_{r_0})} \leq C \|\mathbf{v} - \mathbf{v}_\infty(t,\cdot)\|^\frac 12 _{L_4(B_{r_0})} \|\nabla \mathbf{v}(t,\cdot)\|^\frac 12_{L_4(B_{r_0})}.
$$
From (\ref{bsest2}) follows
$$
\|\nabla \mathbf{v}(t,\cdot)\|_{L_4(B_{r_0})} \leq C \|w(t,\cdot) \|_{L_4(B_{r_0})}.
$$
Then in view of
$$
\|w(t,\cdot) \|_{L_4(B_{r_0})} \leq C \|w(t,\cdot)\|^\frac 12_{L_2(B_{r_0})} \|\nabla w(t,\cdot)\|^\frac 12_{L_2(B_{r_0})}
$$
we have
$$
\|\nabla \mathbf{v}(t,\cdot)\|_{L_4(B_{r_0})} \leq C \|w(t,\cdot)\|^\frac 12_{L_2(B_{r_0})} \|\nabla w(t,\cdot)\|^\frac 12_{L_2(B_{r_0})}.
$$
For any $r$ on circle $S_r = \{\mathbf{x} \in \mathbb{R}^2,~|\mathbf{x}|=r\}$, $r \geq r_0$ holds:
$$
\operatorname{vraisup}_{r\in [r_0,\infty)} \|\mathbf{v}(t,\cdot) - \mathbf{v}_\infty\|_{L_4(S_r)} \leq C \|\mathbf{v}(t,\cdot) - \mathbf{v}_\infty\|^\frac12_{L_2(S_r)} \|\nabla \mathbf{v}(t,\cdot)\|^\frac12_{L_2(S_r)}
$$
Then from Lemma \ref{bsest}
\begin{align*}
&\int_{B_{r_0}} |\mathbf{v}(t,\cdot) - \mathbf{v}_\infty|^4d\mathbf{x} \leq \\ &C \left (\|w(t,\cdot)\|^2_{L_2(B_{r_0})} + \sum_{k=-1,0,1}\|w_k(t,\cdot)\|^2_{L_1(r_0,\infty)} \right) \|\nabla \mathbf{v}(t,\cdot)\|^2_{L_2(B_{r_0})}.
\end{align*}
Finally in virtue of (\ref{bsest2})
\begin{align*}
&\|\mathbf{v}(t,\cdot) - \mathbf{v}_\infty\|_{L_\infty(B_{r_0})} \leq \\
&C \left (\|w(t,\cdot)\|^2_{L_2(B_{r_0})} + \sum_{k=-1,0,1}\|w_k(t,\cdot)\|^2_{L_1(r_0,\infty)} \right)^\frac 18 \times \\ &\|w(t,\cdot)\|^\frac 12 \|\nabla w(t,\cdot)\|^\frac 14 _{L_2(B_{r_0})}
\end{align*}
and
\begin{align}\label{vwest}
\|\mathbf{v}\|_{L_\infty(Z_{r_0,T})} \leq C \|w\|_Q.
\end{align}
So, with new constant $C>0$
\begin{align*}
&\left \| \int_0^t S(t - \tau) ( \mathbf{v} , \nabla w) d\tau \right \|_{H_1(B_{r_0})} \leq \\
&C \left(T+2\sqrt \frac Te \right) \|\mathbf{v}(t,\cdot)\|_{L_\infty(Z_{r_0,T})} \|w(\tau, \mathbf{x})\|_{C \left ( [0,T], H^1(B_{r_0}) \right )}
\end{align*}
and $F\left(w(t, \cdot)\right) \in H^1(B_{r_0})$.
Now we prove that the first Fourier coefficients for $k=-1, 0, 1$ of the function $F\left(w(\tau, \cdot)\right)$ belong to $L_1(r_0,\infty;r)$.
From (\ref{robin_bound}) zero Fourier coefficient of the Stokes semi-group $[S(t)]_0$ generates radially symmetrical solution $w(t,\mathbf{x})$ of the heat equation with Newman boundary condition
$$
\frac{\partial w(t,\mathbf{x}')}{\partial n} = 0,~|\mathbf{x}'|=r_0.
$$
Semigroup coefficients $[S(t)]_{\pm 1}$ correspond to solutions of the heat equation with Robin boundary
$$
\frac{\partial w}{\partial n} + w(t,\mathbf{x}') = 0,~|\mathbf{x}'|=r_0.
$$
From $L_p-L_q$ estimates for heat equation \cite{Dv} with some $C>0$
$$
\left \| [S(t)]_k f \right \|_{L_1(r_0,\infty, r)} \leq C\sqrt t \|f\|_{L_2(r_0,\infty, r)}
$$
for $k=-1,0,1$.
So we deduced that $F:Q \to Q$ is well-defined and maps $Q$ into itself. Now we prove that $F$ is a strict contraction in $Q$ in some ball $B=\{ w \in Q~|~||w||_Q<L\}$.
Take $w_1, w_2 \in B$ with corresponding velocity fields $\mathbf{v}_1$, $\mathbf{v}_2$:
\begin{align*}
&F\left(w_1(\tau, \cdot)\right) - F\left(w_2(\tau, \cdot)\right) =\\ &\int_0^t S(t - \tau) \left ( \mathbf{v}_1 , \nabla \left (w_1(\tau, \mathbf{x}) - w_2(\tau, \mathbf{x}) \right ) \right )d\tau +\\
&\int_0^t S(t - \tau) \left ( \mathbf{v}_1 - \mathbf{v}_2 , \nabla w_2(\tau, \mathbf{x}) \right ) d\tau
\end{align*}
Then from (\ref{vwest}) with some constants $C_1$, $C_2$
\begin{align*}
&\|F\left(w_1(\tau, \cdot)\right) - F\left(w_2(\tau, \cdot)\right)\|_{H^1} \leq \\ &\operatorname{vraisup}_{t\in[0,T]} \int_0^t \left\| S(t - \tau) \left ( \mathbf{v}_1 , \nabla \left (w_1(\tau, \mathbf{x}) - w_2(\tau, \mathbf{x}) \right ) \right ) \right \|_{H^1}d\tau + \\& \operatorname{vraisup}_{t\in[0,T]} \int_0^t \left\| S(t - \tau) \left ( \mathbf{v}_1 - \mathbf{v}_2 , \nabla w_2(\tau, \mathbf{x}) \right ) \right \|_{H^1}d\tau \leq \\&
\left (T+2\sqrt \frac T e \right )\left (\|
\mathbf{v}_1\|_{L_\infty(Z_{r_0,T})} \|w_1-w_2\|_Q + \|\mathbf{v}_1 - \mathbf{v}_2\|_{L_\infty(Z_{r_0,T})} \|w_2\|_Q\right )\leq \\&
C_1\left (T+2\sqrt \frac T e \right )\left (\|
\mathbf{v}_1\|_{L_\infty(Z_{r_0,T})} + \|w_2\|_Q\right ) \|w_1-w_2\|_Q
\leq \\& C_2 L \left (T+2\sqrt \frac T e \right ) \|w_1-w_2\|_Q.
\end{align*}
Estimates of Fourier coefficients for $k=-1,0,1$ can be held in a similar way using $L_p-L_q$ estimates. So, for any $L>0$ we can find such a small $T>0$ that $F$ becomes the strict contraction map in $Q$. Then for small $T$ by the Banach fixed point theorem, the map $F$ has a unique fixed point $w(t,\mathbf{x})$. The theorem is completely proved.
\end{proof}
|
2,877,628,090,766 | arxiv | \section{Introduction}
\label{intro}
This article is concerned with the following discrete-time non-auto\-no\-mous dynamical system on the plane $\mathbb{R}^2$:
\begin{align} \label{eq:napdintrotwo}
x_{n+1}&=\dfrac{n}{Nrx_n}-\dfrac{1}{r}-x_n-y_n \\
y_{n+1}&=x_n \nonumber,
\end{align}
where $r$ and $N$ are parameters. Equation \eqref{eq:napdintrotwo} defines
a sequence of birational maps, $\phi_n$ on the $(x,y)$-plane, for $n \geq 1$, which can be extended to $n \leq 0$. For $n \ne 0$ these maps are undefined along the $y$-axis; however, this does not affect our principal considerations, as we will explain.
In many areas of mathematics \eqref{eq:napdintrotwo} is well known as the discrete Painlev\'e I equation (or just dP1). There is a vast literature on the relevance of Painlev\'e equations, both continuous and discrete, to various subfields of mathematics and we refer the reader to \cite{bib:cm20}, \cite{bib:kny17}, and \cite{bib:va18} for recent reviews of many of these connections.
Painlev\'e equations in general were originally characterized by the special, restricted behavior of the singularities that their solutions may have. Indeed, in the autonomous limit of (\ref{eq:napdintrotwo}), which we may specify by setting $N = \alpha^{-1} \, n$ for some non-zero real constant $\alpha$, the system is analytically completely integrable, by which we mean that its solutions lie on closed curves and may be explicitly written in terms of meromorphic functions. For instance, in the case of autonomous dP1, solutions are expressed in terms of rational functions on an elliptic curve or one of its degenerations \cite{bib:be19}. For general continuous Painlev\'e equations, this translates to solutions being expressible in terms of functions whose global analytic continuations are restricted only to have poles as singularities. This criterion is known as \textit{the Painlev\'e property}. For general discrete Painlev\'e equations, this property can be re-expressed dynamically in terms of {\it singularity confinement}: solutions may become arbitrarily large in a finite number of steps before returning to a bounded region of the plane \cite{bib:grp91,bib:lg03}. That number of steps is fixed for almost all solutions and excursions to infinity can only take place along a fixed set of directions in the plane.
System \eqref{eq:napdintrotwo} is not known to be analytically completely integrable and the existence of a closed form analytic expression of its solutions remains an open problem. Nevertheless, dP1 continues to possess many if not all of the well-known properties typically associated with integrable systems, such as singularity confinement of its generic solutions \cite{bib:grp91,bib:otgr99,bib:gr04}, zero algebraic entropy \cite{bib:bv99,bib:otgr99}, as well as Lax pair \cite{bib:fik91,bib:pap92} and Hirota birational \cite{bib:kny17} formulations. We will refer to solutions that leave the first quadrant and exhibit singularity confinement as {\it polar solutions}. There are, however, certain initial conditions leading to solutions that increase without bound for all time. We will refer to these as {\it non-polar} solutions, similar to the \textit{pole-free} terminology originally used in \cite{bib:jos97}. We do not claim that this dichotomy is exhaustive although numerical explorations (a few examples of which are provided in this manuscript) suggest this is the case. The primary focus of this paper is on non-polar solutions.
An important example of a non-polar solution, which originally motivated our interest in this work, stems from the connection of dP1 to approximation theory, more explicitly to analyzing the structure of orthogonal polynomials with exponential weights.
Recall that a family of orthonormal polynomials $\{ p_{m}\}$ associated to an exponential weight
$w(\lambda) = \exp(-V(\lambda))$ is a complete basis of polynomials for the weighted $L^2$ space such that
\begin{eqnarray*}
\int_{\mathbb{R}} p_n(\lambda) p_m(\lambda) w(\lambda) d \lambda &=& \delta_{nm}.
\end{eqnarray*}
It is known that such polynomials are algebraically specified as solutions to a three-term recurrence relation
with real coefficients. When $V$ is even, this recurrence takes the form
\begin{eqnarray} \label{eq:threetermintro}
\lambda p_n(\lambda) = b_{n+1} p_{n+1}(\lambda) + b_n p_{n-1}(\lambda)
\end{eqnarray}
and the recurrence coefficients $\{b_n\}$ themselves are solutions to a
{\it nonlinear} difference equation (see for instance \cite{bib:va18}). In the particular case where $V(\lambda) = N \left( \frac12 \lambda^2 + \frac{r}4 \lambda^4\right)$ \cite{bib:fre76,bib:mag99,bib:va18}, this difference equation reads
\begin{equation}
\label{eq:Freudeq}
rb_n^2\left( b_{n+1}^2 + b_n^2 + b_{n-1}^2 \right) + b_n^2 = \frac{n}{N},
\end{equation}
which, setting $x_n = b_n^2$, is precisely equivalent to the dP1 system (\ref{eq:napdintrotwo}). Freud \cite{bib:fre76} went on to pin down that $x_n \propto n^{1/2}$, using properties of orthogonal polynomials. We call the corresponding orbit $\{x_n = b_n^2\}$, the \textit{Freud orbit}.
Subsequently, M\'at\'e, Nevai and Zaslavsky \cite{bib:mnz85} built on Freud's work to show the existence of a full asymptotic expansion of the orthogonal polynomial solution in powers of $n^{1/2}$, although they did not describe the coefficients in that expansion. Later Ercolani, McLaughlin, and Pierce \cite{bib:emp08} developed an alternative approach to the asymptotic expansion of the recursion coefficients using Riemann-Hilbert analysis. This work provided a geometric characterization of the recursion coefficients, in terms of a graphical enumeration problem related to diagrammatic expansions in mathematical physics. Since $x_n = b_n^2 >0$, Freud's special orbit remains in the first quadrant for all time. This feature motivated Lew and Quarles \cite{bib:lq83} to seek {\it all} solutions of
\begin{equation}
\label{eq:LQN}
x_n\, (x_{n+1} + x_n + y_n) = n, \quad y_{n+1} = x_n, \quad n > 0
\end{equation}
that possess this property. For
$x_n \ne 0$, \eqref{eq:LQN} is equivalent to \eqref{eq:napdintrotwo} without the term in $1/r$ on the right-hand side of the equation for $x_{n+1}$. In \cite{bib:lq83}, Lew \& Quarles established, by an elegant contraction mapping argument, that there is a one parameter family of such non-polar solutions. According to \cite{bib:ansva15}, addressing the existence and uniqueness of the positive solution of \eqref{eq:LQN} with $y_1 = 0$ was a problem posed by Nevai, and independently solved by him in \cite{bib:n83}. In Appendix \ref{app:LQ_construction}, we extend the Lew-Quarles construction to describe a broad class of non-polar orbits of dP1, which we refer to as the Lew-Quarles orbits.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{TwoD_System.png}
\caption{Phase portrait of the discrete system \eqref{eq:u=0_plane} defined in the invariant plane at infinity $u = 0$. Different colors correspond to different orbits (most of which repeatedly escape the field of view), each with $3000$ iterates.}
\label{fig:u=0}
\end{figure}
In this paper, we adopt a dynamical systems perspective for \eqref{eq:napdintrotwo}. Doing so enables us to simultaneously describe and compare polar and non-polar solutions, thereby illuminating novel features of the fuller class of non-polar solutions of dP1. To this end, we use the Painlev\'e property to complete the phase space at infinity with an asymptotic change of variables based on the Riemann-Hilbert scalings used in \cite{bib:emp08}. A further transformation allows us to define the
{\sl asymptotic change of coordinates} $(x,y,n) \to (s, f, u)$,
\begin{equation}
\label{eq:xyn_usf}
s = \frac{y}{x} + 1 + \frac{1}{r x}, \qquad f = \frac{n}{N r x^2} - \frac{y}{x}, \qquad u = - \frac{1}{r x},
\end{equation}
which reveals the existence of an {\it invariant plane at infinity} ($u=0$). The dynamics in this invariant plane, which organizes the various asymptotic behaviors of the full system, is shown in Figure \ref{fig:u=0} and corresponds to the reduced discrete system
\begin{equation}
\label{eq:u=0_plane}
s_{n+1} = Z_n f_n, \quad
f_{n+1} = Z_n^2 s_n, \quad Z_n = (f_n -1)^{-1}.
\end{equation}
The picture we will develop is that the Freud orbit, when extended to negative discrete times, is a singular heteroclinic connection from the origin (which we call $P_{- \infty}$) of the $(s,f,u)$ space to the fixed point $P_\infty$ with coordinates $s = f = 2$ and $u=0$. In addition, the Lew-Quarles orbits form a family of non-singular heteroclinic connections that converge to the Freud orbit, both as $n \to -\infty$ and $n \to \infty$. The asymptotic points $P_{-\infty}$ and $P_\infty$ lie in the plane $u=0$ and are marked as black dots in Figure \ref{fig:u=0}. Given the phase plane structure suggested by this figure, heteroclinic connections from $P_{-\infty}$ to $P_\infty$ are trajectories that ``escape'' into the third dimension before asymptotically returning to the invariant plane. Our goal is to characterize the structure of these connections.
The rest of this article is organized as follows. Section \ref{sec:FO} defines the Freud orbit, explains how it is initialized, and presents a high precision numerical simulation of forward and backward iterates from the selected initial condition, illustrating convergence to $P_{-\infty}$ as $n \to -\infty$, and $P_\infty$ as $n \to \infty$. Section \ref{sec:usf_coord} introduces the change of variables that transforms system \eqref{eq:napdintrotwo} into an autonomous system in $(s,f,u)$ coordinates, and reviews the basic properties of the resulting discrete dynamical system. Section \ref{sec:NP} presents a detailed study, both analytical and numerical, of the Lew-Quarles orbits, and characterizes these solutions as heteroclinic connections between $P_{-\infty}$ and $P_\infty$. In particular, we provide strong evidence that these trajectories live on the center and center-stable manifolds of these two points respectively, and converge exponentially to the Freud orbit as $n \to \infty$. We also build on these results to propose a unique characterization of the Freud orbit as a singular limit of the Lew-Quarles orbits. In Section \ref{sec:Freud}, we prove that orbits that converge to $P_{-\infty}$ and $P_\infty$ along specific invariant curves track sequences of points formed by the period-2 points (as $n \to -\infty)$ and fixed points (as $n \to \infty$) of the autonomous dP1 system, in which $n$ appears as a parameter. This system is further described in Appendix \ref{app:Aut}. In addition, we obtain expansions of these orbits in powers of $|n|^{-1/2}$, valid as $n \to -\infty$ and $n \to \infty$. Applying these results to the Freud orbit, as legitimized by our numerical observations, provides a simple means to obtain its asymptotic expansions to arbitrary order in powers of $|n|^{-1/2}$ as $n \to -\infty$ or $n \to \infty$. Finally, Section \ref{sec:conclusion} summarizes our findings and identifies future directions that build on the dynamical perspective introduced in this work. Numerical methods are presented in Appendix \ref{app:Num}, and asymptotic expansions for the invariant curves near $P_{-\infty}$ and $P_\infty$ are given in Appendix \ref{app:CM}.
\section{Definition of the Freud orbit}
\label{sec:FO}
\begin{figure}[hbtp]
\centering
\includegraphics[width=\textwidth]{Comb}
\caption{Forward and backward iterates of the Freud orbit with $r = N = 1$ in the $(x,y,n)$ coordinates (top left panel) and in the asymptotic coordinate system $(s,f,u)$ (top right panel). Values of $n$ range from $-224$ to $225$. The bottom panels are projections on the $(x,y)$ (left) and $(s,f)$ (right) planes (highlighted in gray in the top panels). Colors range from dark blue to dark red as $n$ increases (see color bar). For negative values of $n$, iterates alternate between the two branches (in blue) in the second and fourth quadrants of the $(x,y)$ plane. Points corresponding to integer values of $n$ near $0$ are in light green and are contoured in grey for added visibility. In the right column, the black dots represent $P_{-\infty}$ and $P_\infty$ (top), as well as their projections (bottom). A 10\textsuperscript{th} order expansion (Equations \eqref{eq:CMP2}) of an invariant curve transverse to the plane $u=0$ valid near $P_{-\infty}$, and a 6\textsuperscript{th} order expansion (Equations \eqref{eq:CMP1}) valid near $P_\infty$, are also plotted (solid black curves). The reader is referred to Appendix \ref{app:Num} for details on the numerical simulations.}
\label{fig:comb}
\end{figure}
As mentioned above, Freud's asymptotic analysis of the recurrence coefficients for orthogonal polynomials with quartic weight (\ref{eq:threetermintro}) determines a particular solution of (\ref{eq:napdintrotwo}) with the property that in forward time the orbit always lies in the first quadrant of the phase plane and, further, that $x_n \propto \sqrt{n}$. Using the orthonormality of the $p_n$ (see Appendix \ref{app:OP}), one finds that the initial conditions for this orbit are
\begin{equation}
y_1^F = b_0^2 = 0, \ \ \ x_1^F=b_1^2=\dfrac{\mu_2}{\mu_0}, \label{eq:introxnintialcondition}
\end{equation} where the $\mu_i$ are the moments:
\begin{equation}
\mu_i=\int_{\mathbb{R}} \lambda^i w(\lambda) d\lambda. \label{eq:intromomentsdef}
\end{equation}
We will refer to iterates of $(x_1^F, y_1^F)$ under \eqref{eq:napdintrotwo} as the {\it Freud orbit} and use the superscript $F$ to distinguish this specific set of points. Another form of initial conditions is given by $(x_2^F, x_1^F)$ (See Appendix \ref{app:OP}).
Figure \ref{fig:comb} shows a high-precision numerical simulation of the Freud orbit in the $(x,y,n)$ (top left) and $(s,f,u)$ (top right) coordinates, for $n \in [-224, 225]$ and parameter values $r = N = 1$. The color scale goes from dark blue (large, negative values of $n$) to dark red (large, positive values of $n$), with lighter colors corresponding to negative and positive values of $n$ that are smaller in magnitude. The bottom row shows projections of the orbit on the $(x,y)$ plane (left), and on the $(s,f)$ plane (right). In the right column, the points, $P_{-\infty}=(0,0,0)$ and $P_\infty = (2,2,0)$, as well as their projections on the $(s,f)$ plane, are marked with black dots. In the bottom left panel, the point $(x_1^F,y_1^F)$ is shown in bright green on the $x$-axis with $x_1^F \simeq 0.47$. The image of $(x_1^F,y_1^F,n=1)$ in the $(s,f,u)$ space (top right panel) is the point of approximate coordinates $(3.1,4.6,-2.1)$ in the bottom right corner of the plot. As detailed in Section \ref{sec:NP}, we will define the pre-image of $(x_1^F,y_1^F)$ as the point on the $y$-axis of coordinates $x_0^F = 0$, $y_0^F \simeq -1.5$. In the $(s,f,u)$ space, iterates blow up when $n=0$ but are well defined for $n \ne 0.$ The forward part ($n > 0$) of Freud's orbit is in green, yellow, and red, with $x \simeq y \simeq n^{1/2}$. Its backward iterates (negative values of $n$) alternate between the second and fourth quadrants (points shown in green as well as light and dark blue). In the asymptotic coordinate system (right column), the Freud orbit moves away from $P_{-\infty}$ while alternating between each side of the plane $u=0$, and converges to $P_\infty$ as $n \to \infty$. The thin solid curves correspond to Equations \eqref{eq:CMP1} and \eqref{eq:CMP2} (see Section \ref{sec:NP}), which capture the dynamics near $P_\infty$ and $P_{-\infty}$ respectively.
As is clearly illustrated in Figure \ref{fig:comb}, the $(s,f,u)$ coordinate system provides a natural framework to analyze the properties of the Freud orbit. We now explain the origins of this change of coordinates.
\section{Asymptotic change of coordinates}
\label{sec:usf_coord}
The non-autonomous dP1 mapping \eqref{eq:napdintrotwo} may be transformed into an autonomous 3-dimensional system by introducing the variable $\alpha_n = n / N$. The choice of denominator here is motivated by a fundamental re-scaling used in the Riemann-Hilbert analysis of \cite{bib:emp08}, which amounts to considering a limit in which $n$ and $N$ go to infinity together at a fixed rate. We first set
\[
\theta_1 = \psi^2 \alpha, \qquad \theta_2 = \psi \sqrt N y, \qquad \psi = \frac{1}{\sqrt N x}.
\]
The singling out of the coordinates $\theta_1 = n / (N x)^2$ and $\theta_2 = y / x$ stems from an alternative approach to extending the phase space at infinity developed in \cite{bib:kny17}, based on a resolution of singularities (see also \cite{bib:tip20}). The above change of variables leads to
\begin{align}
\theta_{1,n+1} &= Z_n^2 \left(\theta_{1,n} + \frac{1}{N} \psi_n^2\right) \nonumber \\
\theta_{2,n+1} &= Z_n = \gamma \left(\theta_{1,n} - \frac{1}{\sqrt N} \psi_n - \gamma - \gamma \theta_{2,n}\right)^{-1} \label{eq:infty} \\
\psi_{n+1} &= Z_n \psi_n,
\nonumber
\end{align}
where $\gamma = r/N.$ This system has two fixed points,
\[
P_\infty = (3 \gamma, 1, 0), \qquad P_{-\infty} = (-\gamma, -1, 0),
\]
corresponding to positive ($P_\infty; \ \theta_1 > 0$) and negative ($P_{-\infty}; \ \theta_1 < 0$) values of $n$. The associated values of $Z$ are $1$ and $-1$, respectively. The eigenvalues and eigenvectors of the linearization of \eqref{eq:infty} about these fixed points are
\begin{align*}
P_\infty:\quad & \lambda = 1, & e_1 &= \left(1, 0, \sqrt N\right)^T \\
& \lambda = -2 \pm \sqrt 3, & e_\pm &= \left(\gamma (3 \mp \sqrt 3), 1, 0\right)^T \\
P_{-\infty}:\quad & \lambda = -1, & \xi_1 &= \left(\gamma, 1, -\gamma \sqrt N\right)^T\\
& \lambda = \pm i, & \xi_\pm &= \left(\gamma (1 \mp i), 1, 0\right)^T.
\end{align*}
We now introduce a change of coordinates centered on the fixed point $P_{-\infty}$ and consistent with the basis of eigenvectors of the linearization of \eqref{eq:infty} near $P_{-\infty}.$ Specifically, setting
\begin{equation}
\theta_1 = - \gamma + \gamma (u + s + f), \quad \theta_2 = - 1 + u + s, \quad \psi = - \gamma \sqrt N u,
\label{eq:usf_coord}
\end{equation}
transforms \eqref{eq:infty} into the following discrete dynamical system
\begin{align}
s_{n+1} &= Z_n f_n, \qquad Z_n = (u_n + f_n -1)^{-1} \nonumber\\
f_{n+1} &= Z_n^2 \left(s_n + \gamma u_n^2 \right), \label{eq:usf_sys}\\
u_{n+1} &= Z_n u_n \nonumber
\end{align}
with which we will now work.
System \eqref{eq:usf_sys} has exactly two fixed points, which in the $(s,f,u)$ coordinates are given by
\[
P_{-\infty} = (0, 0, 0), \qquad P_\infty = (2, 2,0),
\]
and therefore lie in the invariant plane $u = 0$. Looking for periodic orbits of higher order, we find by direct calculation that there is only one genuine period-2 orbit, given by
\[
(2, 0, 0) \to (0, 2, 0) \to (2, 0, 0),
\]
and a line of period-3 orbits such that $u_n = 0$ and $s_n + f_n = 1$, with $s_n \ne 0$ and $f_n \ne 0$. They are of the form
\begin{align}
\label{eq:P3} \left(s_0, 1 - s_0, 0 \right) \to \left(1 - s_0^{-1}, s_0^{-1}, 0\right) & \to \left((1 - s_0)^{-1}, \left(1 - s_0^{-1}\right)^{-1}, 0\right) \\ & \to \left(s_0, 1 - s_0, 0 \right), \nonumber
\end{align}
where $s_0 \in \mathbb{R} \setminus \{0,1\}.$ Moreover, these are the only period-3 orbits in the invariant plane $u = 0$, other than the two fixed points. The points $s_n = 0$ and $f_n = 0$ are special, in the sense that they are part of a singular period-3 orbit of the form
\begin{equation}
\label{eq:so}
(0, 1, 0) \to (\infty, - \infty, 0) \to (1, 0, 0) \to (0, 1, 0).
\end{equation}
Our numerical explorations, in which we set $\gamma=1$, indicate that polar orbits (see examples in Appendix \ref{app:Num}) track the phase plane structure shown in Figure \ref{fig:u=0} as soon as $|x|$ is larger than a few units (thus corresponding to $|u| < 0.5$). We also observe almost periodic orbits associated with large values of $x$ and $y$ ($|x|,\ |y| > 1000$), which in $(s,f,u)$ coordinates correspond to solutions close to \eqref{eq:P3} on the line $s + f = 1$; in that case, $u$ remains small ($|u| < 10^{-3}$) and a slight drift is observed across the line $s+f = 1$. Polar orbits for which there exists a value of $n$ such that $x_n = {\mathcal O}(\epsilon)$ and $y_n = {\mathcal O}(1)$ display generic singularity confinement: iterates of $x_n$ become large before returning to values of order $y_n$. Specifically,
\begin{align*}
x_{n-1} = y_n &= {\mathcal O}(1), \quad x_n = \epsilon, \quad x_{n+1}=\frac{n}{N r \epsilon}+{\mathcal O}(1), \quad x_{n+2}=-\frac{n}{N r \epsilon} + {\mathcal O}(1), \\
&x_{n+3} = {\mathcal O}(\epsilon), \quad x_{n+4} = {\mathcal O}(y_n) = {\mathcal O}(1), \quad \text{as } \epsilon \to 0.
\end{align*}
In $(s,f,u)$ coordinates, this translates to
\begin{align*}
&s_{n+1} = 1 + \frac{N \epsilon}{n} + {\mathcal O}(\epsilon^2), \quad f_{n+1} = {\mathcal O}(\epsilon^2), \quad u_{n+1} = - \frac{N \epsilon}{n} + {\mathcal O}(\epsilon^2)\\
&s_{n+2}= {\mathcal O}(\epsilon^2), \quad f_{n+2} = 1 - \frac{N \epsilon}{n} + {\mathcal O}(\epsilon^2), \quad u_{n+2} = \frac{N \epsilon}{n} + {\mathcal O}(\epsilon^2),
\end{align*}
as $\epsilon \to 0$. In other words, during singularity confinement, iterates of dP1 visit neighborhoods of the points $(1,0,0)$ and $(0,1,0)$ of the singular period-3 orbit \eqref{eq:so}.
The linearization about the line of period-3 orbits \eqref{eq:P3} is given by
\begin{align*}
s_{n+3}-s_n &= - 2 \mu_n - 3 \nu_n -3 u_n + O(\epsilon) \\
f_{n+3} - f_n &= 3 \mu_n + 4 \nu_n + 4 u_n + O(\epsilon) \\
u_{n+3}-u_n &= O(\epsilon)
\end{align*}
for $s_n = s_0 + \mu_n$, $f_n = 1 - s_0 + \nu_n$, $u_n = O(\epsilon)$, $\mu_n = O(\epsilon)$, and $\nu_n = O(\epsilon).$
The matrix
\[
A = \left(\begin{array}{ccc} -2 & -3 & -3 \\ 3 & 4 & 4 \\ 0 & 0 & 0 \end{array}\right)
\]
has eigenvalues $\lambda_A = 0$ with eigenvector $\zeta_0 = (0, -1,1)^T$ and $\lambda_A = 1$ with eigenvector $\zeta_1 = (1, -1,0)^T$. In addition, since $\lambda_A = 1$ has algebraic multiplicity 2 but geometric multiplicity 1, we define the generalized eigenvector $\zeta_2$ such that
\[
\zeta_2 = (-1/3,0,0), \quad (A - I) \zeta_2 = \zeta_1.
\]
With these conventions, any initial condition of the form $\zeta = a\, \zeta_0 + b\, \zeta_1 + c\, \zeta_2$ evolves according to
\[
A^k \zeta = \left(\begin{array}{c} b + k c - \frac{c}{3} \\ - (b + k c) \\ 0 \end{array}\right),
\]
so that a perturbation transverse to the line of period-3 orbits, of the form $\zeta = a\, \zeta_0 + c\, \zeta_2$ will linearly drift along this line according to $s_{3 p} = s_0 + 3\, p\, c - c/3$, $f_{3 p} = 1 - s_0 - 3\, p\, c$, $u_{3p} = 0$, and eventually reach the vicinity of the singular orbit \eqref{eq:so}.
Finally, we note that the change of coordinates from $(s,f,u)$ to $(x,y,n)$ is given by
\begin{equation}
\label{eq:usf_xyn}
x=-\frac{1}{r u}, \qquad y = - \frac{s+u-1}{r u}, \qquad \alpha = \frac{s+f+u-1}{r u^2},
\end{equation}
with $\alpha = n/N$.
\section{Characterization of the Lew-Quarles orbits}
\label{sec:NP}
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{xyLQ}
\caption{Forward and backward iterates of the Lew-Quarles orbits in the $(x,y)$ plane. A subset of the set $\mathcal S$ of initial conditions is shown in brown (along the curve with $y>0$ and $0<x<x_1^F \simeq 0.47$). Iterates of $\mathcal S$ under \eqref{eq:napdintrotwo} (also in brown) remain in the first quadrant and quickly converge to the Freud orbit (in green, orange, and red). The pre-image of $\mathcal S$, or equivalently $\psi_0({\mathcal S})$, is the set of brown colored points in the fourth quadrant. Images of $\mathcal S$ under $\psi_n,\ n \le -1$ are in brown and alternate between the second and fourth quadrants. They are only visible in the second quadrant because they are are all ``aligned'' with, and thus hidden by, $\psi_0({\mathcal S})$ in the fourth quadrant. Backward iterates of $\mathcal S$ converge to the backward iterates of the Freud orbit, shown in green and blue.}
\label{fig:xyLQ}
\end{figure}
We now restrict our attention to the family of Lew-Quarles orbits, whose proof of existence is summarized in Appendix \ref{app:LQ_construction}. The goal of this section is to show that these orbits, which are initiated in the first quadrant, can be extended to heteroclinic connections between $P_{-\infty}$ and $P_\infty$, and that the Freud orbit is a singular limit of such solutions. For reference, Figure \ref{fig:xyLQ} shows the Lew-Quarles (in brown) and Freud (in color) orbits in the $(x,y)$ plane.
\subsection{Forward iterates}
In Appendix \ref{app:LQ_construction} we provide an adapted version of Lew and Quarles' construction that leads to the following existence and uniqueness theorem for a broad class of non-polar orbits for dP1. A different proof, for a more general class of systems that includes dP1, is provided in \cite{bib:ansva15}.
\begin{theorem}
For any $\xi_0 = y_1 \geq 0$ there is a unique solution of (\ref{eq:napdintrotwo}) that remains in the first quadrant. It is defined in terms of a sequence $\{\xi_n\}_{n \ge 0}$ which is a fixed point of a contraction mapping, has initial value $(x_1, y_1) = (\xi_1, \xi_0)$, and is such that
$\xi_n = x_n = y_{n+1} > 0$ for all $n \ge 0$.
\label{th:LQ}
\end{theorem}
\noindent We define $\mathcal{S}$ to be the set of initial conditions described in Theorem \ref{th:LQ}. As explained in Appendix \ref{app:Num}, any point on $\mathcal S$, including the initial condition for the Freud orbit $F$, may be numerically estimated with arbitrary precision using the Lew-Quarles contraction mapping. Then, we repeatedly apply the forward mapping $\phi_n, \ n\ge 1$ defined in \eqref{eq:napdintrotwo}, to find the associated orbit. Figure \ref{fig:xyLQ} shows that as $n$ increases, the Lew-Quarles orbits quickly collapse onto $F$. It is natural to conjecture that such a convergence is exponentially fast. This is confirmed by our simulations, which in fact provide strong numerical evidence that these orbits approach $F$ at a {\it uniform} exponential rate.
\begin{figure}[ht]
\centering
\includegraphics[width=.9\textwidth]{turnaround}
\caption{Log distance $\delta_n$ between the orbit of a point $Q$ on $\mathcal{S}$ (corresponding to $\xi_0 = 20$) and the Freud orbit, for varying number of contractions $N_c$ used to numerically compute the initial condition
$Q \in \mathcal{S}$.}
\label{fig:turnaround}
\end{figure}
Figure \ref{fig:turnaround} shows how the Lew-Quarles orbit originating from a numerically approximated initial condition $Q$ on $\mathcal{S}$, in this case prescribed by $\xi_{0} = 20$, converges exponentially to $F$ in forward time. The $y$-axis displays the quantity:
\begin{equation}
\delta_n = \log || \Phi^n (Q)- \Phi^n(x_1^F,y_1^F) ||,
\end{equation}
the log distance between the $n\textsuperscript{th}$ iterate of the orbit of $Q$ and the $n\textsuperscript{th}$ iterate of the Freud initial condition ($\xi_0=0$), as a function of $n$ (the iterate number). With our previously introduced notation, $\Phi^n = \phi_n \circ \cdots \circ \phi_1$. Different traces, with increasing numbers of Lew-Quarles contractions used to define the point $Q$, are provided to illustrate how $\delta_n$ behaves under improved initial condition calculation. In all of these instances, the initial condition for the Freud orbit is calculated with 800 Lew-Quarles contractions. Turnaround in the trend, at the vertices of each of the graphs shown in Figure \ref{fig:turnaround}, is indicative of accumulation of numerical error, as this turnaround time increases with the number of contractions used to estimate $Q$. These graphs are truncated just before the approximated orbit exits the first quadrant, indicating a critical accumulation of numerical error (the true orbits never exit the first quadrant, by virtue of $Q$ being on $\mathcal{S}$, the set of initial conditions that, by Theorem \ref{th:LQ}, lead to positive iterates).
\begin{figure}[ht]
\centering
\includegraphics[width=.9\textwidth]{logdisslopes}
\caption{Secant estimate of the slope of the log distance $\delta_n$ as a function of $n$, for Lew-Quarles orbits associated with different initial conditions $\xi_0$. Top panel: all curves are indistinguishable, suggesting that the convergence to the Freud orbit is uniform in $\xi_0$. The bottom row shows enlargements near $n = 1$ and $n = 224$, where differences between the curves are visible. In all panels, the black line corresponds to $s = \log(|\lambda_-|)$, where $\lambda_- = -2 + \sqrt 3$ is the stable eigenvalue of the linearization of dP1 near $P_\infty$.}
\label{fig:logdisslopes}
\end{figure}
We witness the same turnaround behavior when replacing $\xi_0$ with different values ranging from 10 to 6,400. In Figure \ref{fig:logdisslopes}, we record the local secant approximation of the slope $s$ of the log distance $\delta_n$, for different values of $\xi_0$. In this case, all initial conditions were computed using 800 contractions. This figure strongly suggests that the rate of convergence of the Lew-Quarles orbits to the Freud orbit is uniform as all of the sub-graphs corresponding to different values of $\xi_0$ are effectively the same, giving the appearance of a single graph. Moreover, the thin black lines indicate that this rate is close to $|\lambda_-|$, where $\lambda_- = -2 + \sqrt 3$ is the stable eigenvalue of the linearization of dP1 about $P_\infty$. The bottom row shows regions where the slopes visibly depend on $\xi_0$. This occurs at the beginning of each orbit (small values of $n$, bottom left panel) and near the turnaround point (bottom right panel), when the accumulation of numerical errors starts to become noticeable and $s$ changes sign (for $n$ near 224 in the case of initial conditions calculated with 800 iterations of the Lew-Quarles contraction mapping, as is the case for Figure \ref{fig:logdisslopes}).
We now turn to a description of the Lew-Quarles solutions in the $(s,f,u)$ space. The right panel of Figure \ref{fig:sfuLQ} shows these orbits (in brown) as well as the Freud orbit (in color) in that space. This plot was obtained by applying the change of coordinates \eqref{eq:xyn_usf} to the initial conditions on $\mathcal S$ and their iterates, which were all numerically evaluated in the $(x,y,n)$ space.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{sfuLQ}
\caption{Lew-Quarles orbits in $(s,f,u)$ coordinates, near $P_{-\infty}$ (left) and $P_\infty$ (right). Backwards iterates of $\mathcal S$ as well as the Freud orbit converge to $P_{-\infty}$ in a direction perpendicular to the plane $u=0$. Forward iterates quickly collapse onto the Freud orbit and converge to $P_\infty$ in a direction transverse, but not perpendicular, to the plane $u=0$. The solid curves represent approximations of the invariant curve given by equations \eqref{eq:CMP2} near $P_{-\infty}$ (left) and of the center manifold $F$ described by equations \eqref{eq:CMP1} near $P_\infty$ (right).}
\label{fig:sfuLQ}
\end{figure}
In Appendix \ref{app:LQ_construction}, we extend Lew's and Quarles' results to arrive at the following theorem.
\begin{theorem} \label{th:LQ2}
All of the Lew-Quarles orbits, each defined by its initial condition on the set $\mathcal S$ of Theorem \ref{th:LQ}, converge to $P_\infty$ as $n \to \infty$.
\end{theorem}
\noindent It is natural to ask, at this stage, what dynamical systems theory can tell us about invariant manifolds containing $P_\infty$. In the autonomous limit of dP1 one knows that the phase space is foliated by invariant curves due to integrability. In the non-autonomous version such a foliation does not exist, but there are still invariant manifolds associated to fixed points such as $P_\infty$. We formulate the relevant results for us in this regard in the following two theorems which are consequences of the general center manifold theorem.
\begin{theorem} \cite{bib:i79} \label{th:CM}
Let $E_{cs}, E_u$ denote, respectively, the center-stable and unstable subspaces of $P_\infty$, and let $\Phi$ denote the 3-D map (\ref{eq:usf_sys}).
\begin{enumerate}
\item Then there exists a smooth map $\chi: E_{cs} \to E_u$ whose graph $\mathcal{M}$ is tangent to $E_{cs}$ at $P_\infty$ and invariant under $\Phi$. Such an $\mathcal{M}$ is not necessarily unique, but we will refer to any such as a {\it center-stable} manifold for $\Phi$.
\item Though $\mathcal{M}$ may not be unique, the coefficients of the Taylor series of $\chi$ are unique.
\item If $Q$ is a point in the phase space such that all its iterates $\Phi^n(Q)$ for $n$ larger than some $n_0 \in \mathbb{N}$ are in a certain fixed bounded neighborhood of $P_\infty$, then for any particular choice of center manifold $\overline{\mathcal M}$, $\text{dist}(\Phi^n(Q), \overline{\mathcal M}) \to 0$ as $n \to \infty$.
\end{enumerate}
\end{theorem}
\begin{theorem} \cite{bib:i79} \label{th:CM2}
Let $E_{c}, E_s$ denote, respectively, the center and stable tangent subspaces of $P_\infty$, in a local chart for a center-stable submanifold $\overline{\mathcal M}$ of $P_\infty$ and let $\widehat{\Phi}$ denote the restriction of $\Phi$ to $\overline{\mathcal M}$ in this chart.
\begin{enumerate}
\item Then there exists a smooth map $\widehat{\chi}: E_{c} \to E_s$ whose graph $\mathcal{C}$ is a curve tangent to $E_{c}$ at $P_\infty$ and invariant under $\widehat{\Phi}$. Such a $\mathcal{C}$ is again not necessarily unique, but we will refer to any such as a {\it center} manifold (curve) for $\widehat{\Phi}$.
\item Though $\mathcal{C}$ may not be unique, the coefficients of the Taylor series of $\widehat \chi$ are unique.
\item If $Q$ is a point on $\overline{\mathcal M}$ such that all its iterates $\widehat{\Phi}^n(Q)$ for $n$ larger than some $n_0 \in \mathbb{N}$ are in a certain fixed bounded neighborhood of $P_\infty$ in $\overline{\mathcal M}$, then $\text{dist}(\widehat{\Phi}^n(Q), \overline{\mathcal C}) \to 0$ as $n \to \infty$, for any particular choice $\overline{\mathcal C}$ of $\mathcal C$.
\end{enumerate}
\end{theorem}
The orbits described in Theorem \ref{th:LQ} satisfy the conditions of statement 3 of Theorem \ref{th:CM} and so are necessarily attracted to $\overline{\mathcal M}$. It is tempting to believe that in our case they are all actually {\it contained in} the same $\overline{\mathcal M}$ and determine it uniquely. Indeed it is difficult to imagine that a multiplicity of manifolds, comprised of orbits all limiting to
$P_\infty$ could coexist with structures of generic polar orbits. Unfortunately, the limitations of the center manifold theorem for maps do not enable us to conclude that on general grounds. (However, this is just one indication of the value in studying concrete examples of non-autonomous dynamics such as dP1.)
Despite these vagaries, the theoretical background provided by Theorems \ref{th:LQ}, \ref{th:LQ2}, and \ref{th:CM} together with the detailed numerical studies presented in this section will enable us to build a coherent and self-consistent picture of the dynamic state of affairs for non-polar orbits. In this picture ${\mathcal M}$ is unique and so contains all of the Lew-Quarles orbits. The latter orbits are then naturally interpreted as distinct center curves for $P_\infty$ as described in Theorem \ref{th:CM2}. In addition, the Freud orbit is chosen to represent the center manifold $\mathcal C$.
We now look for an expansion of $\mathcal C$ in powers of $u$, by seeking a curve that is invariant under $\Phi$ and is tangent to the center direction of its linearization about $P_\infty$. We obtain
\begin{align}
s_\infty(u)&=2-u-\frac{\gamma}{6} u^2-\frac{\gamma}{36} u^3-\frac{\gamma (3 \gamma +1)}{216} u^4+{\mathcal O}(u^{5}), \nonumber \\
& \label{eq:exp_sf_P1} \\
f_\infty(u)&=2-u+\frac{\gamma}{6} u^2+\frac{\gamma}{36} u^3-\frac{\gamma (3 \gamma -1)}{216} u^4+{\mathcal O}(u^{5}) \nonumber
\end{align}
as $u \to 0.$ The iterative process leading to the above formulas can be continued to arbitrary order and expressions valid to 6\textsuperscript{th} order are provided in Equations \eqref{eq:CMP1} of Appendix \ref{app:CM}. These expressions are used to plot an approximation (black curve) of $F$ near $P_\infty$ in Figure \ref{fig:comb} and in the right panel of Figure \ref{fig:sfuLQ}. As shown in Figure \ref{fig:P1_expansions} of Appendix \ref{app:CM}, they capture the Freud orbit extremely well, even for values of $u$ of order one, thereby providing numerical support to our conjecture that the Freud orbit $F$ is well approximated by the center manifold $\mathcal C$ of $P_\infty$. We note that by the center manifold theorem the above expansions are independent of the particular choice of $\mathcal{C}$.
\subsection{Backward iterates}
\label{sec:backward_iterates}
The map sequence $\phi_n$ given by (\ref{eq:napdintrotwo}) has a sequence of inverse maps $\psi_n$ given by
\begin{align} \label{eq:INV}
x_n&= y_{n+1}\\
y_n&= \dfrac{n}{N r y_{n+1}}-\dfrac{1}{r}-x_{n+1}-y_{n+1}. \nonumber
\end{align}
The map $\psi_n$ is a birational mapping singular along the $x$-axis. We are interested in studying extensions of the above non-polar orbits in reverse time using $\psi_n$. In this case the singularities enter our consideration since the initial point of the Freud orbit, $(x_1, y_1 = 0)$ corresponding to $\xi_0 = 0$, lies on the $x$-axis. This is the only Lew-Quarles orbit that has this issue. Both the $\phi_n$ and $\psi_n$ mappings can be extended to be defined along their respective singular axes by a resolution of singularities process (see \cite{bib:kny17}). However we take a simpler approach that is more relevant to our asymptotic change of variables
We address the singularity of the map \eqref{eq:INV} on the $y=0$ axis by taking a limit from the backward iterates of the Lew-Quarles orbits for which $x_0 =\xi_0 > 0$. In that case, $y_1>0$ and the term $n / (N r y_1)$ in the inverse mapping $\psi_0$ is well defined and equal to $0$ since $n=0$. Therefore, for $y_{1}=0$ and $n=0$, we define $y_0$ in the mapping $\psi_0$ by
\[
y_0 = 0 -\dfrac{1}{r}-x_{1}-y_{1} = -\dfrac{1}{r}-x_{1}.
\]
Going back to Figure \ref{fig:xyLQ}, $\psi_0({\mathcal S})$, which is well defined since $y_1 > 0$, is the collection of points shown in brown in the fourth quadrant. As mentioned above, this set of points limits to a point on the $y$-axis, which we defined to be the image of $(x_1^F,y_1^F)$ under $\psi_0$. Successive applications of $\psi_n$, $n < 0$, to $\psi_0(x_1^F,y_1^F)$ define the backward iterates $\left\{\psi_n \circ \psi_{n+1} \circ \cdots \circ \psi_0(x_1^F,y_1^F)\right\}$ of the Freud orbit. The Freud orbit is shown in green and blue as $n$ becomes more negative, while backward iterates of $\mathcal S$ under $\psi_n, n \le -1$ are in brown. They alternate between the second and fourth quadrants. As was the case for its forward iterates, the backward iterates of $\mathcal S$ converge to the Freud orbit as $n$ becomes more negative. The numerically estimated rate of convergence appears to be sub-exponential, which is consistent with all of the eigenvalues of the linearization of dP1 at $P_{-\infty}$ being on the unit circle. Backward iterates of $\mathcal S$ in the fourth quadrant are not visible because they are superimposed with $\psi_0({\mathcal S})$ when projected onto the $(x,y)$ plane. Because the dP1 system \eqref{eq:napdintrotwo} is singular along the Freud orbit when $n=0$ (since $x_0 = 0$) but not along the other Lew-Quarles orbits, the above construction allows us to view the former as a singular limit of the latter.
As illustrated in Figure \ref{fig:sfuLQ}, our numerical simulations show that in the $(s,f,u)$ coordinates, backward iterates of $\mathcal S$ converge to the fixed point $P_{-\infty}$ in a direction perpendicular to the invariant plane $u=0$. Recall that the linearization of \eqref{eq:usf_sys} about $P_{-\infty}$ has eigenvalues $-1$ and $\pm i$, and the eigendirection associated to $-1$ is perpendicular to the plane $u=0$. To better understand the dynamics near $P_{-\infty}$, we look for an invariant curve parametrized by $u$, of the form
\[
s=s_{-\infty}(u) = \sum_{k=2}^\infty a_k u^k, \qquad f=f_{-\infty}(u) = \sum_{k=2}^\infty b_k u^k.
\]
Requiring that $s_{n+1} = s_{-\infty}(u_{n+1})$ and $f_{n+1} = f_{-\infty}(u_{n+1})$ be satisfied when ($s_n = s_{-\infty}(u_n)$, $f_n=_{-\infty}(u_n)$, $u_n$) and $(s_{n+1},f_{n+1},u_{n+1})$ are related by the mapping \eqref{eq:usf_sys}, leads to a consistency relation in powers of $u$, which in turn defines a set of equations for the coefficients $a_k$ and $b_k$. This procedure gives
the following formal expressions for $s_{-\infty}$ and $f_{-\infty}$ as $u \to 0$ near $P_{-\infty}$,
\begin{align}
s=& s_{-\infty}(u) = -\frac{\gamma}{2} u^2 - \frac{\gamma}{4} u^3 + \frac{\gamma}{8} (\gamma - 1) u^4 + {\mathcal O}(u^{5}), \nonumber \\
& \qquad \label{eq:exp_sf} \\
f=& f_{-\infty}(u) = +\frac{\gamma}{2} u^2 + \frac{\gamma}{4} u^3 + \frac{\gamma}{8} (\gamma + 1) u^4 + {\mathcal O}(u^{5}).\nonumber
\end{align}
The above relationships can be pushed to higher order with the help of computer algebra software. Equations \eqref{eq:CMP2} in the appendix, valid to order 10, are used to plot the thin black invariant curve through $P_{-\infty}$ displayed in Figures \ref{fig:comb} and \ref{fig:sfuLQ}. They capture the Freud orbit very well near $P_{-\infty}$, even for relatively large values of $u$, as further illustrated in Figure \ref{fig:P2_expansions} of Appendix \ref{app:CM}.
In summary, the combination of numerical and analytical investigations presented in this section suggests the following picture: in $(s,f,u)$ coordinates, the Lew-Quarles orbits form a family of heteroclinic connections between $P_{-\infty}$ and $P_\infty$. All iterates are defined for all values of $n$, but $u_0 \to -\infty$ (equivalently $x_0 \to 0$) as $\xi_0 \to 0$, where $\xi_0$ parametrizes the family of Lew-Quarles orbits. This limit defines the Freud orbit, which leaves $P_{-\infty}$ along an invariant curve perpendicular to the plane $u=0$ and converges to $P_\infty$ along its center direction.
\section{Characterization of the Freud orbit}
\label{sec:Freud}
This section provides the conceptual framework for understanding a striking observation: invariant curves near the fixed points of \eqref{eq:usf_sys} {\it track} periodic points of the associated autonomous limits of dP1. This conceptual framework is grounded in a novel process that realizes a simple and elegant mechanism for generating classical asymptotic expansions of the Freud orbit.
In Section \ref{sec:NP}, we made the conjecture that near $P_{-\infty}$ and $P_\infty$, the Lew-Quarles orbits lie on invariant curves whose expansions are given by \eqref{eq:CMP2} and \eqref{eq:CMP1} respectively. We now use this assumption to determine the behavior of $u_n$ as a function of $n$ along the Freud orbit near $P_{-\infty}$ and $P_\infty$.
Near $P_{-\infty}$, we look for a sequence $\{u_n\}$ that converges to $0$ as $n \to -\infty$ and is consistent with the last equation of \eqref{eq:usf_xyn}. We thus require that
\begin{equation}
\frac{n}{N} = \frac{s_{- \infty}(u_n) + f_{- \infty}(u_n) + u_n - 1}{r u_n^2} \Longleftrightarrow \gamma n u_n^2 - u_n + 1 = s_{- \infty}(u_n) + f_{- \infty}(u_n).
\label{eq:P2_un}
\end{equation}
Since per \eqref{eq:exp_sf} $s_{- \infty}(u_n) + f_{- \infty}(u_n) = {\mathcal O}(u_n^{4})$, we see that at dominant order, the iterates $\{u_n\}$ solve $\gamma\, n\, u_n^2 - u_n + 1 = 0$, which is the equation defining the period-2 solutions of the autonomous dP1 system in $(s,f,u)$ coordinates. As further described in Appendix \ref{app:Aut}, this system is obtained from dP1 by setting $\alpha = n/N$ and then assuming that $\alpha$ is a constant parameter. We call the resulting autonomous map $\alpha$-dP1. Figure \ref{fig:p2pts} of Appendix \ref{sec:period2points} illustrates the numerical convergence of the Freud orbit to the sequence of period-two points of $\alpha$-dP1 defined in Equation \eqref{eq:peralpha}, as $n \to -\infty$. Equation \eqref{eq:P2_un} also provides an expansion of $u_n$ as a function of $n$,
\begin{equation}
\label{eq:unP2}
u_{-\infty,n} = u_{-\infty,n}^\pm = \pm \frac{1}{\sqrt{- \gamma n}} + \frac{1}{2 \gamma n} \pm \frac{1}{8 (- \gamma n)^{3/2}} + {\mathcal O}\left((-n)^{-5/2}\right) \quad \text{as } n \to -\infty,
\end{equation}
which is obtained by writing $u_{-\infty,n}$ as a Laurent expansion in powers of $\sqrt{-n}$, substituting into the right-hand equation of \eqref{eq:P2_un}, and solving term by term. At this point, this expansion is formal because we do not have a proof of the existence of the smooth functions $s_{-\infty}$ and $f_{-\infty}$ that appear in expansions \eqref{eq:exp_sf}, and therefore do not have enough control to bound the remainder in \eqref{eq:unP2}. The approach however, is very general. The definition of $\alpha$ in terms of $s$, $f$, and $u$, together with Equation \eqref{eq:CMP2}, immediately leads to two results: that solutions of dP1 on the invariant curve associated with \eqref{eq:CMP2} track the sequence of period-two points of the autonomous dP1 system as $n \to -\infty$, and the expansion \eqref{eq:unP2}.
Near $P_\infty$, we proceed in a similar fashion, although in that case the existence of the center manifold $\mathcal C$ makes the resulting expansions asymptotic. Equation \eqref{eq:P2_un} is replaced by
\[
s_{\infty}(u_n) + f_{\infty}(u_n) = \gamma\, n\, u_n^2-u_n+1.
\]
Using \eqref{eq:exp_sf_P1}, we see that at leading order, the iterates $\{u_n\}$ solve $\gamma\, n\, u_n^2+u_n-3= 0$, which is the equation defining the fixed point solutions of the autonomous dP1 system in $(s,f,u)$ coordinates (see Appendix \ref{app:Aut}). Solving for $u_n$ as a function of $n$, we find
\begin{equation}
\label{eq:unP1}
u_{\infty,n} = u_{\infty,n}^\pm = \pm \sqrt{\frac{3}{\gamma n}}-\frac{1}{2 \gamma n} \pm \frac{1}{\left(8 \sqrt{3}\right) (\gamma n) ^{3/2}} + {\mathcal O}\left(n^{-5/2}\right), \quad \text{as } n \to \infty.
\end{equation}
We note that control of the big $\mathcal O$ term follows from the center manifold theorem that guarantees the existence of all Taylor coefficients of $s_\infty$ and $f_\infty$, and then a straightforward application of Taylor's remainder theorem. Plots of $u_n$ as a function of $n$ for the numerically estimated Freud orbit, and of the expansions $u_{-\infty,n}^\pm$ and $u_{\infty,n}^-$ given above are provided in Figure \ref{fig:Comb3}; they show excellent agreement, even for values of $n$ near 0. This figure dramatically reinforces the fact that the Freud orbit is singled out among the Lew-Quarles orbits by its singularity at $n=0$.
\begin{figure}[ht]
\centering
\includegraphics[width=.9\textwidth]{Comb3}
\caption{Values of $u_n$ for iterates of the Freud orbit (black, connected dots), together with the asymptotic expansions $u_{-\infty,n}^\pm$ defined in \eqref{eq:unP2} for $n < 0$ (solid red and yellow curves), and $u_{\infty,n}^-$ defined in \eqref{eq:unP1} for $n > 0$ (solid green curve).
\label{fig:Comb3}}
\end{figure}
Given their uniform exponential convergence to the Freud orbit exhibited above, it is natural to further conjecture that
all the Lew-Quarles orbits defined in Theorem \ref{th:LQ} have this same asymptotic expansion, and so differ only in terms that are beyond all orders with respect to the algebraic gauge $n^{-1/2}$.
\section{Conclusions}
\label{sec:conclusion}
By introducing the asymptotic change of coordinates \eqref{eq:xyn_usf}, we transformed the dP1 mapping \eqref{eq:napdintrotwo} into the 3-dimensional discrete autonomous dynamical system \eqref{eq:usf_sys}, whose two fixed points, $P_{-\infty}$ and $P_\infty$, correspond to solutions $\{x_n\}$ of dP1 that grow without bounds as $|n| \to \infty$. This transformation, together with a combination of analytical and numerical investigations, allowed us to characterize known non-polar orbits of dP1 as heteroclinic connections between these two fixed points. In particular, we described the Freud orbit as a singular limit of the Lew-Quarles orbits. By understanding how these solutions leave $P_{-\infty}$ to converge to $P_\infty$ as $n$ increases, we discovered that they track sequences of points constructed from period-2 (near $P_{-\infty}$) and period-1 (near $P_\infty$) points of the autonomous counterpart of dP1. Moreover, our results are consistent with and in many aspects support the conjecture that the Lew-Quarles orbits are on center-stable manifolds of $P_\infty$ and, as $n \to \infty$, exponentially converge to the Freud orbit, which itself lives on a center manifold of $P_\infty$. The presence of invariant curves that contain the Lew-Quarles (as $n \to \infty$) and Freud orbits provides a method to find explicit expansions of these solutions in powers of $|n|^{-1/2}$ as $|n| \to \infty$. These expansions are asymptotic near $P_\infty$ and formal near $P_{-\infty}$.
Compared to their continuous counterparts, many aspects of discrete dynamical systems remain relatively unexplored, especially in the non-autonomous case. In this environment, the study of concrete examples takes on an added value. Due to its analytically completely integrable discrete limit, dP1 is of particular interest. The resulting rich inherent structure therefore provides an auspicious environment for the exploration of explicit features of non-autonomous discrete dynamics that cannot be directly attacked in a general setting. This paper does that in the context of what we have termed non-polar orbits and relates its findings to previously known aspects of Painlev\'e systems. This approach also leads to interesting applications regarding the behavior of specific orbits as $|n| \to \infty$.
The autonomous dP1 system has orbits that are degenerations of its quasi-periodic elliptic solutions and correspond to trigonometric solutions along a separatrix. This separatrix, and the solutions along it, were examined in detail from a dynamical systems perspective in \cite{bib:be19} and used to better understand their relation to the combinatorial problem of {\it geodesic distance} in the enumeration of planar graphs. That paper also examined a related system, $x_n (x_{n+1} + x_{n-1} + 1/r) =n / Nr$, which has relevance to the enumeration of labelled trees and super-Brownian excursions. Both of these are instances of the application of dP1 type systems to other areas of mathematics. The present manuscript reveals a subtle connection between the separatrices of autonomous dP1 discussed in \cite{bib:be19} and Freud's orbit. Specifically, we discover a novel and deep connection between the mapping \eqref{eq:napdintrotwo} and its autonomous counterpart, as the sequence $\{x_n\}$ in Freud's recurrence follows the fixed points of a collection of autonomous mappings. In Appendix \ref{app:Aut}, we provide a geometric description of the important role played by these separatrices in making it possible for the Freud orbit to grow without bounds as a solution of dP1. In addition, the principal message of this article is that Freud's orbit coincides with a center manifold of a fixed point at infinity. This property of having, effectively, a limiting fixed point at infinity is highly atypical. We do not believe that such a connection between autonomous fixed points and an asymptotic non-autonomous fixed point has been noticed before in the literature.
Another novel discovery is the realization that recursive constructions of invariant curves provide an elegant mechanism for generating asymptotic expansions of non-polar solutions of dP1 as $|n| \to \infty$ (see \eqref{eq:unP2} and \eqref{eq:unP1}). For $n > 0$, these have relevance for another combinatorial problem related to random tilings of Riemann surfaces (or geometric foams and quantum gravity in the physics literature). Together with the results of \cite{bib:be19}, we therefore now have three examples of dynamical systems structures (specifically two instances of a stable manifold of a hyperbolic fixed point in \cite{bib:be19}, and a center manifold here) associated with solutions to combinatorial problems. Such connections between different areas of mathematics are highly intriguing. In a subsequent paper we will develop some of these connections using our recursive construction of invariant curves and their associated asymptotic expansions. Methods developed in \cite{bib:tip20} will enable us to demonstrate the key feature of uniform validity in $r$ and $N$, within appropriate ranges, for these expansions.
As mentioned in the introduction, M\'{a}t\'{e}, Nevai, and Zaslavsky in 1985 \cite{bib:mnz85} established the existence of an asymptotic expansion for the Freud orbit in powers of $n^{1/2}$, whose leading behavior can be shown to be \cite{bib:tip20}
\begin{align} \label{eq:introlikewn}
x_n& = \sqrt{\dfrac{n}{3rN}}-\dfrac{1}{6r}+\dfrac{\sqrt{12}}{144}\sqrt{\dfrac{N}{n}}\dfrac{1}{r^{3/2}} + O(1/n).
\end{align}
We also note that in \cite{bib:jos97} Joshi carried out a formal Painlev\'e dominant balance analysis, seeking asymptotic fixed points of dPI in terms of an asymptotic constant of motion. From the perspective of this paper one can see that this formal approach correctly captured the leading order behavior at $P_\infty$ and $P_{-\infty}$. It would be interesting to determine if and where this approach might match the ``level set" structure evident in Figure \ref{fig:u=0}, and possibly even approximate our exact trajectories in the vicinity of $P_{\pm\infty}$.
The mechanisms just referred to in the previous paragraph reproduce and extend this type of asymptotic analysis. Indeed, in these connections, our results may say more about the asymptotics of orthogonal polynomials.
Our subsequent work will also make central use of the realization that the leading order term in the large $n$ asymptotic expansion of \cite{bib:emp08} in fact coincides with our expression for the fixed points of autonomous dP1. This, together with the analysis developed in Section \ref{sec:Freud},
enables us to devise an efficient scheme for explicitly counting topologically
distinct quadrangulations, involving a fixed number of tiles, of compact Riemann surfaces. Other, higher dimensional, Painlev\'e systems have orbits of Freud type corresponding to degree $2 \nu$ potentials that in turn yield information about the enumeration of more general polygonal tilings. For potentials of odd dominant degree traditional methods of orthogonal polynomial theory break down and one must consider generalizations such as non-Hermitian orthogonal polynomials \cite{bib:ep12}, which presents obstacles to asymptotic analysis. The particular case of a cubic potential, which relates to topological triangulations of surfaces, is of special interest and involves a non-autonomous planar Painlev\'e dynamical system. The dynamical systems approach developed in this paper may help to overcome some of these obstacles.
Finally, we note that our work has focused on the dP1 regime with $r >0$. However, there is also interest in the singular regime where $r \to -1/12$, the so-called ``Boutroux regime'' that has been explored in \cite{bib:jl15} and \cite{bib:dk06}. This is the so-called double scaling limit of random matrix theory, which provides a bridge between the discrete and continuous versions of Painlev\'e I. A problem of general physical importance is to study and relate the behavior of a correlation function for a statistical mechanical system at small values of a scaling parameter to its behavior at large values. This is called a {\it connection problem}. For a class of explictly solvable statistical mechanical problems this is related to connection problems for continuous Painlev\'e equations. For Painlev\'e I and II a systematic study of this problem was carried out in \cite{bib:jk}.
A natural extension of our work is to explore the potentially interesting connection problem between the limits $r \to \infty$ and $r \to -1/12$ from a dynamical systems perspective.
As stated at the outset of these Conclusions, the work presented here combines theory and asymptotics with high-precision numerical simulations to arrive at a detailed and compelling picture for the structure of non-polar dynamics in the dP1 system. The main work needed to convert this picture into fully rigorous statements centers on establishing our stated conjectures about center-stable and center manifolds laid out in Section \ref{sec:NP}. This challenge is of independent interest. Its resolution will have significance for discrete dynamical systems theory broadly speaking. Within that scope particular open questions of importance include
\begin{enumerate}
\item The rigorous verification of the numerically evident fact that all of the Lew-Quarles orbits converge exponentially to the Freud orbit at a uniform rate as $n \to \infty$.
\item A more general conceptual understanding of the unique singularity formation in the
Freud orbit as compared to the other Lew-Quarles orbits. In other (continuum) examples of center manifolds (see e.g. \cite{bib:meiss07} Section 5.6) one observes that singularities form in finite time along those center manifolds whose asymptotic expansions do not have any corrections beyond all orders. Could this be the case for Freud's orbit?
\end{enumerate}
|
2,877,628,090,767 | arxiv | \section{Conclusion}
In this work, we propose \textit{Se}gment representation learning for long documents \textit{D}ense \textit{R}etrieval (SeDR) to tackle the long document issues in DR models. Concretely, we propose Segment-Interaction Transformer that encodes document into document-aware and segment-sensitive representations, which is verified to be superior to other segment-interaction patterns. To address the negative limitation of GPU memory, we devise Late-Cache Negative to provide additional negative instances that is shown to be necessary for long documents training. Finally, our model SeDR outperforms other DR models on MS MARCO and TREC-DL.
\section{Introduction}
Document retrieval is a crucial component in information retrieval (IR) and benefits for various IR-related tasks, e.g. web search~\cite{xiao2022progressively} and question answering~\cite{inbatch1}. Despite its significance, document retrieval is a challenging problem that requests to rapidly retrieve top documents from scale documents collection. Traditional bag-of-words (BoW) models such as BM25~\cite{bm25} conduct it by term exact matching and therefore suffer from the term mismatch problem~\cite{star}. Recently, with the progress of pre-trained language models like BERT/RoBERTa~\cite{devlin2018bert,liu2019roberta}, dense retrieval (DR) has shown promising results in document retrieval that significantly outperforms BoW models, benefiting from their powerful semantic representative ability. For a typical DR model, query and document are encoded into single low-dimension dense embedding. Then a dot-product or cosine function is applied to them for computing their relevance score. With the offline-constructed index of document embedding, dense retrieval can fulfill efficient semantic search in milliseconds online.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\columnwidth]{./figure/plt4.pdf}
\caption{\label{plt4} Document Length Distribution of MS MARCO}
\end{figure}
However, current DR models are unfit for long documents due to the following problems: \textbf{(1) Length Limitation.} As Transformer-based pre-trained models are used in DR models, the length limitation in Transformer is inevitable. Since the memory and computational requirements scale quadratically $O(n^2)$ with text length $n$, long documents can't be input directly. For instance, as shown in Figure~\ref{plt4}, only 29.5\% documents are less than 512 tokens, which is a maximum length of BERT-base. To tackle this issue, prior work adopts bandit solutions such as truncating~\cite{inbatch1,star} or splitting-and-pooling~\cite{ance}, both of which cause information loss. Truncating the long documents to meet input length loses the information from unused segments. Splitting the documents into text segments and max-pooling the segment scores can obviate unused segments, but violate the consistency and relevance of segments in the document by encoding each segment independently. \textbf{(2) Finite Representation Capacity.} According to the analysis in \citet{mebert} and \citet{Multi-View}, the finite capacity of a single low-dimension embedding is awkward to represent a document well, especially for the long document that covers more diverse information. To enhance the representation capacity, recent studies resort to multiple representations for document~\cite{Multi-View}. Nevertheless, the effectiveness of multiple representation comes with extra computation and memory demand in searching. Due to the dispersed length of realistic documents as shown in Figure~\ref{plt4}, these methods are not reasonable to use fixed-number vectors to represent both short and long documents. Moreover, our empirical experiment reveals that multiple representations tend to collapse into the same one, which constrains the capacity of multiple representations~\cite{Multi-View}. \textbf{(3) Memory Bottleneck.} Recent studies make progress on negative sampling strategies for DR training. ~\citet{morenegative} found that more contrastive negatives often lead to better performance. Hence, the prior works introduce in-batch negative sampling~\cite{inbatch1,inbatch2}, which utilizes other document representations in mini-batch as negatives without extra encoding cost. However, in-batch negative is limited to the batch size that GPU memory can sustain. Along with text length increasing, long document encoding consumes more GPU memories, thus leading to insufficient negatives for representation learning.
To address these problems, we propose \textit{Se}gment representation learning for long documents \textit{D}ense \textit{R}etrieval (SeDR), which split long document to segments and transfer them into document-aware segment embeddings for DR. Since segment embedding represents a fixed-length segment so that the number of embeddings depends on document length, segment representations can exert their capacity to represent non-redundant information. Meanwhile, the relevance matching between queries and documents usually takes place at text segments of document~\cite{QDS}, which illustrates that segments of document are distinct for searching. Therefore, segment representation learning can be a more effective and efficient paradigm for DR.
Despite these merits, segment representations suffer from the independence bias on splitting-and-pooling as stated before. To solve this issue, we propose Segment-Interaction Transformer. It upgrades the Transformer with a simple modification, enabling segments from the same document to interact with each other in self-attention mechanism for generating document-aware segment representations with less information loss from document splitting. Moreover, Segment-Interaction Transformer preserves the capacity of pre-trained model like BERT/RoBERTa with negligible extra parameters, and hold the complexity $O(n_s^2)$, where $n_s$ denotes the segment length, instead of original $O(n^2)$. Extensive experiments show that Segment-Interaction Transformer outperforms other segment-interaction patterns, e.g. Longformer\cite{beltagy2020longformer} and global-attention~\cite{globalattention1}, on DR with the smallest parameters and highest efficiency.
There is also a problem with memory limitation that result in insufficient negatives for training on long documents. To tackle it, we present Late-Cache Negative, which stores late encoding representations as cache negatives and introduces them to improve subsequent training in a free-cost way. Since the cache negatives don’t require gradient, the number of cache negatives is not limited to GPU memory so that it can provide extra negatives for training. Our experiments show that the extra cache negatives are necessary to improve the performance for long document training. The code is available at \url{https://github.com/jymChen/SeDR}.
Our main contributions can be summarized as follows:
\begin{itemize}
\item To the best of our knowledge, this is the first study to address the issues of long document on DR. And we propose a Segment-Interaction Transformer to encode long documents efficiently into document-aware segment representations that outperforms other segment-interaction patterns on DR.
\item We propose Late-Cache Negative to break down the constraint of GPU memory and provide additional negative instances, which is verified to be necessary for long documents DR in the ablation study. We also further investigate the effectiveness of different negative settings.
\item Our model SeDR achieves superior retrieval performances among DR models on the MSMARCO-Document dataset as well as TREC-DL 19' and 20'.
\end{itemize}
\section{Preliminaries}
\subsection{Task Definition}
Given a query $q$ and a massive document collection $D$, a document retrieval task aims to find the positive document $d^+ \in D$, or provide high-quality candidates for further reranking tasks where it served as a first-stage retrieval.
\subsection{Dense Retrieval Architecture}
By precomputing the index of document representations, DR has been proved effective and efficient in document searching. We start with introducing a common dense retrieval architecture for document retrieval. A typical DR model encodes query $q$ and document $d$ separately into dense representations using a query encoder $E_Q$ and a document encoder $E_D$ respectively. In this process, these encoders widely adopt \emph{[CLS]} representations from pre-trained language models such as BERT/RoBERTa. Then, a similarity function $sim(\cdot)$, usually inner product, is used to perform efficient retrieval by predicting the similarity score $f(q, d)$ of query $q$ and document $d$:
\begin{equation}
\label{sim}
f(q,d)=sim(E_{Q}(q),E_{D}(d))
\end{equation}
Supervised by the training set of the target retrieval task, query and document encoders are trained with a contrastive-learning loss. In our models, we use InfoNCE~\cite{infonce} that computes the loss for a given instance $q$ as:
\begin{equation}
\label{loss1}
\mathcal{L}(q,d^+)=-log{\frac{e^{f(q,d^+)}}{e^{f(q,d+)}+\sum\limits_{d\in D^-_q}{e^{f(q,d)}}}}
\end{equation}
where $D_q^-$ is the negative document set that contrasts with the positive one $d^+$ for representation learning. In practice, $|D_q^-|<<|D|$ since the cost for computation of all negative documents is prohibitive. To improve the performance, hard negative and in-batch negative are introduced to sample negatives. Hard negative strategy is used to sample top-K documents of $q$ as hard negative $d^-$:
\begin{equation}
\label{sample}
d^-=\mathrm{Sample}(\mathrm{Top}{-}K(q))
\end{equation}
Thus we call K as the hardness of hard negative to distinguish learning. In-batch negative sampling leverages the positive document and the negative document from other queries in the same mini-batch, served as random negatives in a cost-free way, where each training instance is used for the augmentation of other instances' negative samples. With these two negative sampling strategies, the negative sample set $D_q^-$ become:
\begin{equation}
\label{negatives}
D^-_q={d^-}\cup\{d^+,d^-\}_{q^\prime \in B\land q^\prime \neq q}
\end{equation}
where B denotes the query set in a mini-batch.
\section{SeDR}
SeDR is built on the above DR architecture and proposed to enhance it on long documents. It comprises of Segment-Interaction Transformer and Late-Cache Negative, both essential to improve retrieval performance for long documents.
\subsection{Segment-Interaction Transformer}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{./figure/SeDR-transformer.pdf}
\caption{\label{overview}Illustration of segment representation learning with Segment-Interaction Transformer.}
\end{figure*}
Given a n-tokens document $d=[t_1,t_2,...,t_n]$, it firstly splits the document to $m$-tokens segments batch $[s_1,...,s_k]$, where the segment number $k=\left \lceil \frac{n}{m} \right \rceil$ and the last segment $s_k$ will be padded to $m$ tokens. Then each segment will be concatenated with the special tokens according to the pre-trained model as:
\begin{equation}
\label{segment}
s_{i}=[CLS,t_{im+1},...,t_{im+m},SEP]
\end{equation}
where $s_i$ indicates the $i$-th segment in the segment batch. Unlike the splitting-and-pooling method, the segment batch will be input into the Segment-Interaction Transformer to obtain the document-aware segment representations.
\paragraph{Segment-individual Embedding}
After the input tokens are converted to token embedding, position embedding is added to them to signal the position so that the model can perceive the order of sequence. Similarly, we introduce the segment embedding to indicate the segment order in documents. Specifically, the input representation $H_{s_i}^0$ of segment $s_i$ is computed by summing token embedding, position embedding and the $i$-th segment embedding.
\paragraph{Segment-Interaction mechanism}
Following previous DR models, we use BERT/RoBERTa architecture that comprises of multiple stacked bidirectional transformer layers. The key point of the transformer layer is self-attention mechanism~\cite{vaswani2017attention} that calculates the $l$-th layer output $H^l$ taking the input $H^{l-1}$ from the previous layer:
\begin{equation}
\label{attention1}
\begin{gathered}
Q^T;K^T;V^T = (W^q;W^k;W^v)\cdot H^{l-1}_{s_i} \\
H^l_{s_i} = \mathrm{softmax}(\frac{Q K^T}{\sqrt{d_k}})\cdot V^T
\end{gathered}
\end{equation}
where $H^l_{s_i}$ denotes the $l$-th layer output of segment $s_i$ and $W^q;W^k;W^v$ are three linear projections of query, key and value. In the self-attention layer, each token can attend to other tokens in the same segment, but fail to attend to other segment tokens. To fulfill segments interaction, we propose a Segment-Interaction mechanism that is simple but effective. As shown in Figure~\ref{overview}, the hidden representation $H_{s_i}^{l-1}$ is integrated with the \emph{[CLS]} representations from other segments to produce a new $H_{s_i}^{l}$ in each layer as:
\begin{equation}
\label{attention2}
\begin{gathered}
H_{s_i}^{l-1}=[h_{CLS}^{s_i,l-1},h_{t_{im+1}}^{s_i,l-1},...,h_{t_{im+m}}^{s_i,l-1},h_{SEP}^{s_i,l-1}] \\
Q^T = W^q\cdot H^{l-1}_{s_i} \\
\dot{K}^T;\dot{V}^T = (W^k;W^v)\cdot [H^{l-1}_{s_i}\circ [h_{CLS}^{s_j,l-1}]_{j\neq i}] \\
H^l_{s_i} = \mathrm{softmax}(\frac{Q\dot K^T}{\sqrt{d_k}})\cdot \dot{V}^T
\end{gathered}
\end{equation}
where $\circ$ is the concatenation operation, $h_{CLS}^{s_i,l-1}$ denotes the \emph{[CLS]} representation of segment $s_i$ on ($l-1$)-th layer output and $[h_{CLS}^{s_j,l-1}]_{j\neq i}$ denotes the list of \emph{[CLS]} representations on other segments built automatically. In this way, each \emph{[CLS]} token can attend to its segment tokens as well as other segment \emph{[CLS]} tokens, while segment tokens can also attend to other segment \emph{[CLS]} tokens. Hence, different segments interact across segment batch using the bond of \emph{[CLS]} representations, where segment tokens can capture other segments information and focus on its own segment content. Meanwhile, \emph{[CLS]} are controlled to attend to its segment content and interact with other segment representation in
every Transformer layers to perceive global information of the document. More importantly, this upgrade is very cost effective in practice. Then we use a linear layer on the final output \emph{[CLS]} representations to gain the segment representations:
\begin{equation}
\label{segInteraction}
[\vec{s_1},...,\vec{s_k}]=[h_{CLS}^{s_1},...,h_{CLS}^{s_k}]\times W+b
\end{equation}
where $\vec{s_i}$ denotes the $i$-th segment representation and $h_{CLS}$ denotes the final output of \emph{[CLS]} representations.
\paragraph{Pooling}
To compute the score of query $q$ and document $d$, a max-pooling is applied to multiple segment representations of $d$ to calculate the final score as:
\begin{equation}
\label{score}
f(q,d)=\max\limits_{i}\left \{ \mathrm{sim}(E_Q(q), \vec{s_i} )\right \}
\end{equation}
Thus, the top-score documents can be retrieved with one fast approximate nearest neighbor (ANN) search operation during inference, using the index of segment embedding that is built offline. Therefore, segment representations can remain the advantage of fast search in dense retrieval.
\begin{figure}[t]
\centering
\includegraphics[width=0.83\columnwidth]{./figure/SeDR-cache.pdf}
\caption{\label{cacheneg}Schematic of Late-Cache Negative. }
\end{figure}
\subsection{Late-Cache Negative}
A longer document demands more memory when encoding, resulting in a smaller batch-size $B$, as well as fewer negatives $|D_q^-|$ for $\mathcal{L}(q,d^+)$ on DR training. Despite the efficiency of In-batch negative, it's not flexible that the number of negatives is limited to the batch-size setting. Thus, we propose Late-Cache Negative that provides more negatives in a cost-free way without the limitation of GPU memory or batch-size setting.
The point of the idea is to keep a late-cache queue $Q$, that stores the late-computed representations of documents and queries. For a training instance $\{q,d^+,d^-\}$, where $d^+$ and $d^-$ indicate the positive document and hard negative of $q$, the embeddings of $\{q,d^+,d^-\}$ will detach the training gradient and enqueue to the queue $Q$ after the training step. Because the cache embeddings don't require gradients, the $Q$ can store a large number of embeddings to provide enough cache negatives. And the cache-size $C$ is a hyperparameter that can flexibly control the number of cache negatives used in training. As illustrated in Figure~\ref{cacheneg}, the cache negatives in $Q$ are used in two ways. Firstly, the cache negatives provide the cache documents to $D^-_q$ that:
\begin{equation}
\label{cache1}
D^-_q={d^-}\cup\{d^+,d^-\}_{q^\prime \in B\land q^\prime \neq q}\cup\{\hat d^+,\hat d^-\}_{\hat q \in Q}
\end{equation}
Consequently, the extra negatives of $Q$ can supply extra random negatives to replenish the insufficient negatives set for long document representation learning. On the other hand, the cache query $\hat q$ in $Q$ is used to further constrict the document representations learning as:
\begin{equation}
\label{cache2}
\begin{gathered}
\mathcal{L}(\hat q, \hat{d}^+) = -log{\frac{e^{f(\hat q, \hat{d}^+)}}{e^{f(\hat q, \hat{d}^+)}+\sum\limits_{d\in D^-_{\hat{q}}}e^{f(\hat{q},d)}}} \\
D^-_{\hat{q}}={\hat{d}^-}\cup\{d^+,d^-\}_{q \in B}\cup\{\hat{d}^+,\hat{d}^-\}_{\hat{q} \prime \in Q \land \hat q^\prime \neq \hat q}
\end{gathered}
\end{equation}
where the document embeddings are trained to keep away from previous document embeddings and query embeddings and adapt the distribution in the dense space. In the end, the final training loss of a training batch is computed as:
\begin{equation}
\label{cache4}
\mathcal{L}_{\mathrm{batch}}= \sum_{q\in B}{\mathcal{L}(q, d^+)}+\sum_{\hat q\in Q}{\mathcal{L}(\hat q,\hat d^+)}
\end{equation}
\section{Related Work}
\paragraph{Dense Retrieval}
Dense retrieval has been significantly promoted by Negative sampling. Recent studies prove that sampling "hard" negatives for contrast learning benefit for DR. \citet{bm25neg,inbatch1} sample hard negatives from BM25 top documents. In \citet{qu2020rocketqa,ance}, they sampled harder negative from the top documents retrieved by the warm-up DR model. \citet{star} is proposed to optimize query encoder with dynamic hard negative sampled from ANN-searched documents. Moreover, the in-batch negative sampling is introduced to augment negatives~\cite{in-batch3,in-batch4}. Later on, \citet{qu2020rocketqa,xiao2021matching} adopt the cross-device negative sampling to promote in-batch negatives by using the negatives from other distributed devices, which is still confined to device memory. Besides, there are some studies that try to employ multiple representations to improve the representation ability of documents. \citet{mebert} utilizes the first k document token embeddings as the document representation. \citet{Multi-View} is proposed to use multiple viewer tokens replacing single \emph{[CLS]} to generate document vectors. \citet{colbert} stores each token embeddings for later interaction, which is unfit to document retrieval due to the tokens scale of long documents.
\paragraph{Long Document Solution} For long documents, most previous DR models take the first 512 tokens as input~\cite{star,qu2020rocketqa,inbatch1}. \citet{ance} extra use the MaxP operation~\cite{maxp} to split the document into segments and max-pooled the segment scores. Despite the convenience of these bandit solutions, they cause significant information loss of long documents~\cite{QDS}. There are several works proposed to promote the long document ranking on long documents~\cite{TKL,QDS,zhou2022socialformer,fu2022leveraging}. However, these reranking models are impractical to rank all documents in corpus to do retrieval that would incur severe calculation delay. Another idea to solve the long document problem is to use sparse attention patterns~\cite{beltagy2020longformer,zhou2022socialformer,kitaev2020reformer}, which can avoid quadratic complexity in self-attention mechanism. One of the most successful sparse attention is the sliding window attention~\cite{hofstatter2020local,qiu2019blockwise,beltagy2020longformer}, which allows each token to attend to its surrounding tokens. In \citet{globalattention1,bigbird,gupta2020gmat}, global attention is proposed to fit the specific tasks. Among long-document Transformers, Longformer~\cite{beltagy2020longformer} is a prevalent model that integrates the sliding window attention and the global attention.
\section{Experiments}
\subsection{Datasets}
We evaluate SeDR on the TREC Deep Learning Track benchmarks~\cite{DL2019,DL2020} for document retrieval. Specifically, we use its MSMARCO-Document corpus~\cite{msmarco} that contains 3,213,835 documents, 367,013 training queries and 5,193 Dev queries (MARCO Dev), where each query has one relevant-judged document. In addition, we utilize two differently-constructed test datas: 2019 Deep Learning track~\cite{DL2019} (TREC 19' DL) with 43 queries and 2020 Deep Learning track~\cite{DL2020} (TREC 20' DL) with 45 queries, where each query involves multiple relevant documents. Following \citet{ance}, we truncate the documents to a maximum of 2048 tokens and report the official metrics and Recall@100 from the full-corpus retrieval results.
\subsection{Baselines}
\paragraph{Retrieval Models}
BM25~\cite{bm25} and docT5query~\cite{nogueira2019doc2query} are used as two BoW baselines, where we use Anserini implementation~\cite{yang2018anserini}. For DR models, we adopt three DR baselines with different sampling strategies: Rand Neg~\cite{randneg}, BM25 Neg~\cite{bm25neg} and In-batch Neg~\cite{inbatch1,inbatch2}. Other competitive DR models are ANCE~\cite{ance}, which iteratively samples hard negatives of the model, and STAR~\cite{star} that improve ANCE by introducing in-batch negative. To fit long documents, we apply MaxP~\cite{maxp} operation to upgrade ANCE and STAR, where the document is split into 512-token segments and the scores are max-pooled. To evaluate the multiple representations, STAR-multi is introduced to train STAR to generate multiple vectors using viewer tokens in \citet{Multi-View}, where the vector number is set to 4. We also compare the DR models under ADORE~\cite{star} mechanism, which is used to further optimize the query encoder to better fit on the document index.
\paragraph{Segment-Interaction Baselines}
To evaluate our Segment-Interaction Transformer, we change the segment-interaction pattern of SeDR and obtain 4 comparable models. 1) SeDR-MaxP, where MaxP operation is adopted on SeDR meaning that all the segment representations are independent. 2) SeDR-Transformer-Head, where the segment representations can interact via a single extra Transformer layer on top of SeDR-MaxP. 3) SeDR-Global-Attention, where the global attention is utilized to enable the each segment \emph{[CLS]} token attend to whole document tokens. 4) SeDR-Longformer, where Longformer is used to encode the long document and generate the segment representation by inserting \emph{[CLS]} tokens in document.
\subsection{Implementation Details}
All DR models are based on RoBERTa-base model. For document splitting, we set the segment length as 512 and maximum number of segments as 4. For training settings, we use STAR as warm-up model to initialize SeDR as well as other upgrade models including STAR-multi and STAR(MaxP) to further train it on long document input for fair comparisons. Thus, we use STAR to generate top candidates as the static hard negatives. Following \citet{star}, we sample one hard negative from top-200 documents. When using Late-Cache Negative, we set the cache size as 50 and sample hard negative from top-100 documents. We use Lamb optimizer~\cite{lamb} with a learning rate 5e-5 for all experiment settings. All model training are conducted on Tesla V100 (32G) GPUs and every model is limited to access to one GPU. Thus, the training batch size is set to 17, a maximum can be took in the GPU memory. For inference, we adopt the Faiss library~\cite{faiss} to perform efficient similarity search.
\begin{table*}[t]
\centering
\begin{tabular}{lccccccccc}
\toprule
\multirow{2}{*}{} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Long \\input\end{tabular}} & \multicolumn{2}{c}{MARCO Dev} & \multicolumn{2}{c}{TREC 19' DL} & \multicolumn{2}{c}{TREC 20' DL} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}index \\size\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Latency\\~(ms)\end{tabular}} \\
\cmidrule(l){3-4} \cmidrule(l){5-6} \cmidrule(l){7-8}
& & MRR & Recall & NDCG & Recall & NDCG & Recall & & \\
\midrule
\multicolumn{10}{l}{\textbf{BoW model }} \\
BM25 & \scriptsize{\Checkmark} & 0.277 & 0.807 & 0.519 & 0.385 & 0.506 & 0.586 & - & 87.2 \\
docT5query & \scriptsize{\Checkmark} & 0.327 & 0.861 & 0.597 & 0.399 & 0.582 & 0.618 & - & 91.5 \\
\midrule
\multicolumn{10}{l}{\textbf{Dense Retrieval}} \\
In-Batch Neg & \scriptsize{\XSolidBrush} & 0.320 & 0.864 & 0.544 & 0.295 & 0.509 & 0.479 & 9.2G & 1.3 \\
Rand Neg & \scriptsize{\XSolidBrush} & 0.330 & 0.859 & 0.572 & 0.284 & 0.500 & 0.484 & 9.2G & 1.3 \\
BM25 Neg & \scriptsize{\XSolidBrush} & 0.360 & 0.877 & 0.597 & 0.268 & 0.569 & 0.464 & 9.2G & 1.3 \\
ANCE & \scriptsize{\XSolidBrush} & 0.377 & 0.893 & 0.615 & 0.277 & 0.580 & 0.497 & 9.2G & 1.3 \\
STAR & \scriptsize{\XSolidBrush} & 0.390 & 0.913 & 0.605 & 0.313 & 0.575 & 0.513 & 9.2G & 1.3 \\
STAR-Multi & \scriptsize{\XSolidBrush} & 0.404 & 0.913 & 0.616 & 0.309 & 0.596 & 0.508 & 36.8G & 4.8 \\
ANCE(MaxP) & \scriptsize{\Checkmark} & 0.384 & 0.906 & 0.628 & 0.323 & 0.612 & 0.540 & 21.5G & 2.7 \\
STAR(MaxP) & \scriptsize{\Checkmark} & 0.394 & 0.909 & 0.615 & 0.302 & 0.586 & 0.510 & 21.5G & 2.7 \\
\hdashline
SeDR(Ours) & \scriptsize{\Checkmark} & \textbf{0.409} & \textbf{0.921} & \textbf{0.632} & \textbf{0.343} & 0.607 & 0.527 & 21.5G & 2.7 \\
\hspace{1em} \small{w/o Segment-Interaction} & \scriptsize{\Checkmark} & 0.403 & 0.917 & 0.611 & 0.312 & 0.594 & 0.511 & 21.5G & 2.7 \\
\hspace{1em} \small{w/o Late-Cache Negative} & \scriptsize{\Checkmark} & 0.400 & 0.915 & 0.616 & 0.340 & \textbf{0.617} & \textbf{0.551} & 21.5G & 2.7 \\
\midrule
\multicolumn{10}{l}{\textbf{Dense Retrieval w/ ADORE}} \\
STAR + ADORE & \scriptsize{\XSolidBrush} & 0.405 & 0.919 & 0.628 & 0.317 & 0.604 & 0.519 & 9.2G & 1.3 \\
STAR-Multi + ADORE & \scriptsize{\XSolidBrush} & 0.417 & 0.919 & 0.634 & 0.326 & 0.594 & 0.519 & 36.8G & 4.8 \\
ANCE(MaxP) + ADORE & \scriptsize{\Checkmark} & 0.396 & 0.915 & 0.627 & 0.315 & 0.612 & 0.542 & 21.5G & 2.7 \\
STAR(MaxP) + ADORE & \scriptsize{\Checkmark} & 0.414 & 0.921 & 0.631 & 0.319 & 0.577 & 0.515 & 21.5G & 2.7 \\
SeDR + ADORE & \scriptsize{\Checkmark} & \textbf{0.421} & \textbf{0.933} & \textbf{0.645} & \textbf{0.353} & \textbf{0.626} & \textbf{0.549} & 21.5G & 2.7 \\
\bottomrule
\end{tabular}
\caption{Evaluation of retrieval models on TREC Deep Learning Track for document retrieval. MRR, NDCG and Recall are set to MRR@100, NDCG@10 and Recall@100. Latency indicates average time requirement for one query searching.}
\label{res1}
\end{table*}
\subsection{Comparison with Retrieval Models}
In this section, we analyse the effectiveness of different document retrieval models. We conduct the experiments on TREC Deep Learning Track for document retrieval and report the results in Table~\ref{res1}. In the results, DR models outperform BoW models by a large margin, except for the Recall@100 metric on TREC DL Doc, which may be caused by unlabeled relevant documents~\cite{ance}. Besides, there is much less retrieval latency on DR models due to their GPU acceleration.
Among DR models, SeDR achieves remarkable performance. Firstly, it significantly surpasses the previous strongest model STAR by a large margin on MARCO Dev and TREC DL, which is consistent with expectation to make full use of long document information. Secondly, for the models taking long document input without truncation, SeDR outperforms ANCE(MaxP) and STAR(MaxP) within the same index size and retrieval latency. Specifically, we further compare the retrieval performance in different document lengths and report the results in Figure~\ref{plt2}. In Figure~\ref{plt2}, SeDR obviously outperforms other models on longer documents, while performing worse than the ANCE(MaxP) and STAR on the documents that are less than 512 tokens. It demonstrates that SeDR can considerably improve the DR performance on the retrieval of longer documents, and better learn long document representations than conventional MaxP fashion. Thirdly, compared with multi-vector model STAR-Multi, SeDR captures better retrieval results with much less index size. Larger index introduces more retrieval latency as well as more memory occupation. To alleviate the redundancy of multiple vectors in short documents, SeDR uses segment vectors whose number depends on the document length, instead of generating fixed-number vectors for each document. Fourthly, with ADORE, SeDR achieves greater performance and overwhelms other models. Since the ADORE only optimizes the query encoder, the significant improvement on SeDR is due to its document index, i.e. segment representations. It indicates that SeDR can learn better representations for long documents dense retrieval.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\columnwidth]{./figure/plt2.pdf}
\caption{\label{plt2} Performance with query sets of different document length ranges on MARCO Dev.}
\end{figure}
Besides, we conduct an ablation study with SeDR to evaluate the contribution of its two components: Segment-Interaction Transformer and Late-Cache Negative. As shown in Table~\ref{res1}, we can see that without Segment-Interaction Transformer or Late-Cache Negative, SeDR performs worse than the original model but still outperforms STAR(MaxP). It verifies that Segment-Interaction Transformer and Late-Cache Negative both contribute to long document DR. Furthermore, it can be observed that SeDR without Late-Cache Negative performs best in TREC 20' DL, which may be caused by greater generalization without Late-Cache Negative that would introduce more negatives for training.
\begin{table*}[t]
\centering
\begin{tabular}{lcccccccccc}
\toprule
\multirow{2}{*}{} & \multicolumn{2}{c}{MARCO Dev~} & \multicolumn{2}{c}{TREC 19' DL} & \multicolumn{2}{c}{TREC 20' DL} & \multicolumn{2}{c}{Time(hour)} & \multirow{2}{*}{\#Param} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Latency\\~(ms)\end{tabular}} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
& MRR & Recall & NDCG & Recall & NDCG & Recall & \small{Training} & \small{Indexing} & \\
\midrule
SeDR-MaxP & 0.403 & 0.917 & 0.611 & 0.312 & 0.594 & 0.511 & 15.8 & 20.6 & 125M & 2.7 \\
SeDR-Transformer-Head & 0.405 & 0.915 & 0.622 & 0.334 & 0.605 & 0.519 & 15.8 & 20.7 & 132M & 2.7 \\
SeDR-Global-Attention & 0.406 & 0.920 & 0.600 & 0.310 & 0.584 & 0.499 & 21.2 & 26.9 & 149M & 3.4 \\
SeDR-Longformer & 0.408 & \textbf{0.922} & 0.625 & 0.315 & 0.578 & 0.518 & 88.2 & 65.2 & 149M & 21.1 \\
SeDR & \textbf{0.409} & 0.921 & \textbf{0.632} & \textbf{0.343} & \textbf{0.607} & \textbf{0.527} & 15.8 & 20.7 & 125M & 2.7 \\
\bottomrule
\end{tabular}
\caption{Comparing with different segment-interaction pattern. The compared models are derived from SeDR with different segment-interaction patterns to generate same-scale segment index (21G) for dense retrieval. Training time indicate time consumption per training epoch. MRR, NDCG and Recall are set to MRR@100, NDCG@10 and Recall@100.}
\label{res2}
\end{table*}
\subsection{Segment-Interaction Pattern}
To investigate the effectiveness of Segment-Interaction Transformer, we compare it with different segment-interaction patterns on SeDR. All these variant models use Late-Cache Negative and the same hyper-parameters setting. As shown in Table~\ref{res2}, all the segment-interaction patterns improve the performance of original SeDR-MaxP, in which segments encode independently. It demonstrates that segment-interaction can essentially alleviate information loss of document
splitting.
Here we analyze segment-interaction models. SeDR-Transformer-Head adopts an extra Transformer layer on the segment representation, where the segment-interaction is limited to the encoder output and requests extra parameters. In contrast, SeDR-GLobal-Attention and SeDR-Longformer use global attention mechanism that allows every segment \emph{[CLS]} token to attend to whole document tokens. However, global attention mechanism request more computations and parameters. Compared with SeDR-Global-Attention, the sparse attention architecture of Longformer requests much more training time and retrieval latency. Among segment-interaction models, SeDR outperforms other models with the smallest parameters and highest efficiency.
To further investigate the upgrade of Segment-Interaction Transformer, we show the segment embeddings distribution in dense space using t-SNE in Figure~\ref{plt1}. As shown in Figure~\ref{plt1}, the segment embeddings tend to collapse into one point for those using global attention. As \citet{Multi-View} claims, the collapse of segment embedding can reduce the performance of multiple representations. Instead, the segment embeddings of SeDR-MaxP diffuse arbitrarily due to independently encoding. By contrast, SeDR and SeDR-Transformer-head learn the global-aware and segment-sensitive segment representations that scatter to a document area in Figure~\ref{plt1}. The area distribution of SeDR and SeDR-Transformer-head benefit DR, where it's consistent with the greater performance on TREC DL 19 and 20.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./figure/plt1.pdf}
\caption{\label{plt1}The t-SNE plot of query and segment representations for models with different segment-interaction patterns. The query id used is 201376 from MARCO Dev.}
\end{figure}
\subsection{Negative Setting}
In this section, we discuss the negative setting for Late-Cache Negative and hard negative sampling. Different from the in-batch negative, the number of cache negatives is not limited to GPU memory or batch-size settings. It gives the way to flexibly set the negative number. Therefore, we explore how to set the cache size $C$ of Late-Cache Negative. As Late-Cache Negative can increase training on discriminating random negatives, we further explore how to adjust the hard negative setting by controlling the top-$K$ document sampling, which we call hardness.
Figure~\ref{plt3} illustrates the results of different settings of cache size $C$ and hardness $K$. As for the cache size, too small cache size ($C<50$) can significantly degrade the performance. If $C>50$, larger cache size yields near performance. It also demonstrates that it's necessary to replenish the insufficient negatives for long documents training.
In terms of hardness, we can see that less hardness (larger K) contributes to higher Recall scores, while less hardness (larger K) reduces MRR@100 scores, which are used to evaluate the top-ranking performance. Since too large hardness (small K) hurt performance, we set the $K$ as 100 and $C$ as 50 to keep a high MRR@100 score and Recall@100 score at the same time. It suggests setting high $K$ in training when applying DR to tasks that need more relevant documents recall.
\begin{figure}[t]
\centering
{ \label{fig:b}
\includegraphics[width=0.48\columnwidth]{./figure/plt3c.pdf}
}
{ \label{fig:a}
\includegraphics[width=0.48\columnwidth]{./figure/plt3t.pdf}
}
\caption{\label{plt3}Evaluation of cache size $C$ and hardness $K$ on MARCO Dev. Default $C$ is 50 and default $K$ is 100.}
\label{fig}
\end{figure}
|
2,877,628,090,768 | arxiv | \section{Introduction}
The numerical study of the dynamics of core-collapse supernovae allowed in the recent decades to identify specific hydrodynamics mechanisms which control the evolution of the shock wave. Among these dynamics, one that is expected to produce signatures both in the neutrino luminosity and the gravitational wave emission is the Standing Accretion Shock Instability (SASI) \cite{Blondin:2002sm,Foglizzo:2015dma}.
SASI is a hydrodynamical mode with a typical
frequency, phase and possibly varying amplitude that develops when a deformed stalled shock front precesses around the newly formed proto-neutron star (PNS). Such precession in turn induces an asymmetric accretion onto the PNS, resulting in fluctuations in the luminosity of the emitted neutrinos, and the emission of gravitational waves (GW)
(see, e.g., \cite{Mirizzi:2015eza,Kuroda:2017trn,Andresen:2016pdt,bernhard14} and references therein).
Indications of SASI were first identified in two-dimensional (2D) numerical simulations \cite{Blondin:2002sm,Blondin:2006fx,Marek:2008qi,Marek:2007gr,bernhard14,Nakamura:2014caa,Summa:2015nyk}, and then confirmed by three-dimensional (3D) simulations as well \cite{Blondin:2006yw,Iwakami:2008qj,Fernandez:2010db,Hanke:2013jat,OConnor:2018tuw,Vartanyan:2019ssu,Walk:2019miz}.
The precession frequency (and therefore the frequency of the neutrino\ modulations) was found to be between a few tens of Hz and 200 Hz \cite{Tamborra:2013laa,Kuroda:2017trn,Walk:2018gaw,Walk:2019miz} depending on the progenitor mass, nuclear equation of state (EOS), and progenitor rotation. A possible
correlation of the SASI-modulated neutrino and GW\ signals has been studied in \cite{Kuroda:2017trn},
which also demonstrated that a GW\ SASI signature could be contaminated by other effects (e.g., neutrino-driven convection and the associated turbulence). A multi-messenger analysis joint with neutrinos, which could clarify the presence of SASI in GW, is particularly motivated.
While the frequency of the SASI is expected to be mostly related to the mechanical properties of the PNS, the duration of SASI signatures in neutrinos\ and GW reflects the duration of the phase when the shock wave is stalled, before either being launched to drive an explosion, or dying out so the star implodes directly into a black hole (failed supernova).
Indeed, progenitors at the interface of the successful and failed explosions tend to exhibit longer periods with SASI \cite{Walk:2019miz}.
At this moment, SASI is a hypothesis -- supported by numerical simulations -- that awaits observational tests. Neutrinos and GW\ are the only messengers that can, for a future galactic supernova, directly probe this phenomenon and provide measurements the relevant parameters (such as the SASI frequency and amplitude).
Such measurements will clarify the properties of the PNS, the nuclear EOS, ultimately the yet-uncertain supernova explosion mechanism. The phase difference between GW and neutrino luminosity observed at Earth could also in principle (for an uncertainty-free signal at the source) probe propagation effects, like the time delay due to the neutrinos\ being massive \cite{Lund:2010kh,Beacom:1998yb}. It also carries the potential to estimate the different depths of the main production zone of neutrinos (the neutrinosphere) and of GW \cite{Kuroda:2017trn}.
The theme of this paper is the detectability of SASI signatures in the neutrino luminosity as recorded at neutrino detectors on Earth, and the potential of estimation of its main phenomenological parameters. The SASI-induced modulation of neutrino emission has been studied previously on the base of both two-dimensional \cite{Rampp:2002bq,Buras:2005rp} and three-dimensional \cite{Hanke:2013jat,Mueller:2012sv} SASI-dominated supernova simulations. The neutrino signal in terms of its Fourier power spectrum was analyzed \cite{Lund:2010kh,Lund:2012vm,Tamborra:2013laa,Walk:2019miz,Migenda:2016xnc} in order to assess the detectability of SASI activity. The minimum requirement for signal detection was established by stating that the power spectrum of signal has to exceed the one of the background \cite{Lund:2010kh,Lund:2012vm,Tamborra:2013laa,Tamborra:2014aua}.
In this work we advance the topic to a more quantitative level, by establishing a framework which is new in the context of neutrino\ data analyses. This methodology is an implementation of the maximum likelihood principle, and uses the probability distribution of the observed power at different frequencies. As part of the likelihood-based analysis
we also address the question of parameter estimation, and compare the results for the parameter variances to the optimally possible variance according to a Fisher matrix analysis of the problem.
The present paper is intended as a first step towards a joint description of the problem for neutrinos\ and GW, which is left for future work.
The paper is structured as follows. In Sec. \ref{sec:general}, generalities are given on SASI and on neutrino\ detection. In Sec. \ref{sec:maxlik}, our methodology to establish the presence of SASI in a neutrino\ signal is presented, and results are shown using a specific numerical simulation as a test-bed of the method. Parameter estimation is then addressed in Sec. \ref{sec:parameter}, and a discussion follows in Sec. \ref{sec:disc}. Three appendices offer proofs and technical details to the interested reader.
\section{Generalities}
\label{sec:general}
\subsection{Supernova neutrino detection}
We consider neutrino\ detections in two different experimental settings. The first is a water Cherenkov\ detector at the Megaton mass scale, like the planned Hyper-Kamiokande (Hyper-K\ from here on) \cite{Abe:2018uyc}.
For simplicity, only the main detection channel, inverse beta decay ($\mathrel{{\bar \nu}_e} + p \rightarrow n + e^+$), is included here. Individual positrons are detected via their Cherenkov\ photon signature with high efficiency and excellent time resolution (microseconds or less
\cite{Abe:2018uyc}). Therefore, here
an ``event" from a supernova\ burst indicates an individual neutrino\ interacting within the volume of the detector. Background events due to other neutrino\ sources, cosmic rays or detector impurities -- which in principle could mimic supernova\ neutrinos\ events -- are negligible for a galactic supernova\ \cite{Scholberg:2012id}.
The number of supernova\ neutrino\ events in the detector is directly proportional to the number of target particles in the detector (and therefore to its mass), and it scales like the neutrino\ number flux, i.e., proportionally to $D^{-2}$, with $D$ being the distance to the star.
As a reference, here the expected Hyper-K\ mass of 0.44 Mt and 100\% detector efficiency will be used;
results for different detector masses can thus be obtained by rescaling $D$.
Given the microsecond recording time scale, the number of events $n_i$ in each millisecond time bin $[t_i , t_i+
\Delta t]$ is subject to Poisson statistical fluctuations (standard deviation $\sigma_i=\sqrt{n_i}$), with negligible correlations between different time bins.
The second experimental setting refers to the kilometer-scale antarctic detector IceCube\ \cite{Kopke:2017req}.
There, the detection concept is designed for multi-TeV neutrinos, and is based on Digital Optical Modules (DOMs) positioned in geometrically sparse arrays in the antarctic ice. For a flux of $\sim 10$ MeV supernova\ neutrinos, individual neutrino\ interactions (mostly from inverse beta decay, like in water) can not be resolved, however a surge of total photon count rate in the optical modules can be observed as a signal. In this context, an event is intended to be the observation of a photon in a DOM.
In contrast with Hyper-K, in IceCube\ the background level is relatively high, at a rate of $\dot n \simeq 1340 ~{\rm ms^{-1}}$ \cite{Kowarik:2009qr,Lund:2010kh}. Therefore, the number events $n_i$ in each time bin is the number of photons recorded in the entire detector in that time bin, and is the sum of the contributions of the supernova\ signal (scaling like $D^{-2}$) and of background (fixed, and constant in time). Note that in this work we focus on the dominant emission signatures of anti-electron neutrinos (in both IceCube and Hyper-K), and we shall leave the consideration of multi-flavor interactions (albeit important, see an in-depth review by Mirizzi et al. \cite{Mirizzi:2015eza}) for future work
\subsection{SASI: physics and numerical predictions}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.45\textwidth]{KnurateFull10new.pdf}
\caption{Predicted neutrino event rate at Hyper-K\ from the KKHT\ model of a 15 $M_{\odot}$\ progenitor with the SFHx equation of state \cite{Kuroda:2017trn}, for a star at distance $D=10$ kpc. }
\label{fig:knuratefull}
\end{figure*}
We use the numerically calculated neutrino\ event rates for IceCube\ and Hyper-K\ from a 3D general relativistic (GR) simulation (model SFHx, where SFHx indicates the equation of state by \cite{Steiner:2012rk}) by Kuroda, Kotake, Hayama and Takiwaki (KKHT\ from here on) \cite{Kuroda:2017trn} as a test bed of a realistic scenario where SASI\ effects are present in the neutrino luminosity.
They are shown in Fig. \ref{fig:knuratefull}. For simplicity, the observer's direction is taken along the polar (e.g., the $z$) axis of the source
as a fiducial case, where the flux-projection effects and the detection efficiencies for estimating the event rates are taken into account
following \citet{Tamborra:2014aua}.
In the KKHT model, the 3D hydrodynamics evolution is self-consistently
followed from the onset of
core-collapse of a $15M_\odot$ star \cite{Woosley:1995ip}, through core bounce,
up to $\sim$ 350 ms after bounce. As consistent with the outcomes from recent 3D models (e.g., \cite{Hanke:2013jat,Andresen:2016pdt,Yakunin:2017tus}), the
hydrodynamic evolution is characterized by the prompt convection phase shortly after bounce
($T_{\rm pb}\lesssim 20$ ms with $T_{\rm pb}$ the postbounce time),
then the linear (or quiescent) phase ($20 \lesssim T_{\rm pb}\lesssim 140$ ms),
which is followed by the non-linear phase when the vigorous activity of SASI was
observed for the model. The dominance of the SASI over neutrino-driven convection persists over $140 \lesssim T_{\rm pb} \lesssim 300$ ms, after which neutrino-driven convection dominates over the SASI
(see \cite{Kuroda:2016bjd} for more details).
In \cite{Kuroda:2016bjd},
the SASI frequency was roughly estimated as $\dot{M}/{M} \sim 100\, {\rm Hz}$,
where $M \sim 10^{-3} M_{\odot}$ and $\dot{M} \sim 0.1 M_{\odot}/{\rm s}$ denote the typical mass and mass accretion rate in the gain region, respectively, which is consistent with the numerically obtained SASI-modulated neutrino frequency (e.g., Figure 7 of \cite{Kuroda:2017trn}).
In the simulation, the Baumgarte-Shibata-Shapiro-Nakamura formalism was employed to evolve the metric
\citep{Shibata:1995we,Baumgarte:1998te}, and the GR neutrino transport was solved
by an energy-integrated M1 scheme \citep{Kuroda:2012nc}. For simplicity,
effects of neutrino flavor oscillations (e.g., the Mikheyev-Smirnov-Wolfenstein (MSW) effect \cite{Mikheev:1986gs}, and collective neutrino oscillations, see \cite{Mirizzi:2015eza} and references therein for a review) are neglected in this study (see, e.g., \cite{Tamborra:2013laa} for a brief discussion of the validity of this approximation).
Here the supernova\ burst simulated by KKHT\ will be used as representative of a future SASI-carrying signal in the two detectors of interest. It will be compared with a similar signal that has no SASI features in it.
Such null model is constructed by smoothing out the SASI oscillations from the original KKHT model. The smoothing is done by taking the event rates averaged over eight time bins, each of 1 ms width, and performing a polynomial interpolation of these averaged rates.
A zoomed-in plot of the KKHT\ and smoothed out rates is given in Fig. \ref{fig:compare} (black solid lines in left panes).
\section{Testing for SASI: likelihood ratio method}
\label{sec:maxlik}
In this section, we
set up the formalism necessary to our statistical method.
Considering the oscillatory character of the SASI signatures, we choose to work in the frequency space, and establish the discretized power spectrum of the neutrino\ time profile as the observable of interest.
The statistical behavior of the power spectrum is then presented. Finally, the likelihood ratio as test-statistics is defined and used to assess the detectability of the SASI.
We use the likelihood ratio as deciding statistic for the hypothesis test because of its optimality properties, which are described by the Neymann-Pearson Lemma \cite{2009fundamentals}.
For clarity, in what follows the symbols with tilde (e.g., $\tilde N$) will indicate an actual outcome of a measurement, which is affected by statistical fluctuations. The same symbol without tilde (e.g., $N$) will be used for the mean, ``true" value of the same quantity.
\subsection{Neutrino time profile templates}
\label{sub:templates}
When data from a supernova\ burst are analyzed, it can be useful -- as it is often done in neutrino\ data analyses, see, e.g., \cite{Lund:2010kh} -- to fit the event rate time profile with simplified analytical templates that, while necessarily inaccurate, will allow to gain analytical understanding and to estimate the main phenomenological parameters. The latter can then be compared with predictions of detailed numerical simulations for greater insights into the microphysics at play. In this work, we use two parametric templates which characterize the main features of neutrino signals with and without the SASI activity respectively, to study the potential of a data analysis algorithm to identify the presence of SASI.
For the case with SASI activity
we choose a single frequency function:
\begin{equation}
R_2(t)=(A-n)(1+a \sin(2\pi f_S t))+n~,
\label{eq:mod2}
\end{equation}
where $A$ is the time-averaged event rate (the ``DC component") in the detector including instrumental noise (after possible experimental cuts), $a$ is the relative SASI amplitude, $n$ is the mean value of the background rate ($n=0$ for Hyper-K), and $f_S$ is the nominal frequency of the SASI.
The second template, for the case without SASI, is a constant:
\begin{equation}
R_{0}(t)=A~,
\label{eq:mod0}
\end{equation}
(with $A$ having the same meaning as in Eq. (\ref{eq:mod2})).
In our method, only $f_S$ and $a$ will be treated as free parameters with respect to which the likelihood will be maximized. We assume that other relevant quantities, such as the DC component, $A$, and the starting time ($t_0$) and duration ($\tau$) of the SASI\ activity, can be determined separately, by using theoretical priors, visual inspection, or a separate algorithm.
For $A$, it is immediate to see that it can be measured with high precision (i.e., negligible uncertainty), without the need of a fit. Its relative uncertainty is $\delta A/A=1/\sqrt{N_{ev}}\ll 1$ where $N_{ev}$ is the total number of events, and $N_{ev}\gtrsim 2500$ in all the cases examined here
(we also assume that systematic uncertainties on $n$ are negligible, because background rates can be measured precisely over years of data-taking).
With regard to $t_0$ and $\tau$, here they are fixed to be $t_0=155$ ms post-bounce, and $\tau= 55$ ms, consistently with the KKHT\ simulation results (fig. \ref{fig:knuratefull}).
Fixing these quantities is legitimate in the spirit of answering the question whether there is indication of single-frequency fluctuations in a signal between two chosen (generic) instants of time.
Realistically, in the context of a more specific search for SASI\ effects, $t_0$ and $\tau$ could be at first set using rough estimations from visual inspections of the data, in conjunction with expectations from the theory. Indeed, a delay in the onset of SASI\ (relative to the bounce time) is expected considering that SASI\ requires the shockwave to come to a stalling point. We checked that 3D numerical simulations roughly place $t_0$ in the interval $\sim 0.1-0.4$ s post-bounce \cite{Radice:2018usf,Powell:2018isq,Kuroda:2016bjd,OConnor:2018tuw,Muller:2011yi}, with $\tau$ being even more uncertain. It is possible that, by the time the next galactic supernova\ is observed, theoretical progress will be able to place stronger priors on $t_0$ and $\tau$.
The method proposed here will be applicable to data with externally-estimated (not optimized) $t_0$ and $\tau$; the lack of optimization of these parameters will result in certain loss of power of the method, which can be overcome by generalizing the method to include $\tau$ and $t_0$ as fit parameters.
\subsection{Time series and power spectrum}
\label{sub:discretized}
Let us consider the events that are recorded in a detector after an initial time $t_0$, in time bins of width $\Delta=1~{\rm ms}$. The $j$-th time bin then corresponds to the time $t_j=t_0+j\Delta$.
The observed number of events in the same bin will then be $\tilde{N}(t_j)$, which is a random variable fluctuating around its mean $N(t_j)\simeq R(t_j)\Delta$.
Following \cite{Lund:2010kh,Lund:2012vm}, we perform a discrete Fourier transform of the time series $\{\tilde N(t_j) \}$ over the time interval $[t_0,t_0+ \tau ]$, containing $N_{bins}=\tau/\Delta$ time bins. The discrete frequency resolution is then:
\begin{equation}
\delta =\frac{1}{\tau}~,
\label{eq:tau}
\end{equation}
which represents the minimum width of frequency bins for which statistical independence between adjacent bins can be realized (see the discussion in Appendix A).
For our fiducial value $\tau= 55~{\rm ms}$, the resolution is $\delta= 18~{\rm Hz}$~\footnote{Eq. (\ref{eq:tau}) implies that a that much longer SASI signature will result in more precise estimation of the frequency; this might be relevant for future work on long-stalling shockwaves.}.
The Nyquist frequency becomes \cite{2009fundamentals}
\begin{equation}
f_{\textit{Nyq}}=\frac{1}{2\Delta},
\label{eq:fnyq}
\end{equation}
which corresponds to the frequency index
\begin{equation}
k_{\textit{Nyq}}=\frac{f_{\textit{Nyq}}}{\delta }=\frac{\tau}{2 \Delta}=\frac{1}{2}N_{bins}~.
\label{eq:knyq}
\end{equation}
We define the discrete Fourier-transformed neutrino\ signal as:
\begin{equation}
\tilde h(k\delta )=\sum_{j=0}^{N_{bins}-1}\tilde{N}(t_j)e^{i2\pi j \Delta k\delta }~,
\label{eq:h}
\end{equation}
and the one-sided power spectrum, similarly to \cite{2009fundamentals} as:
\begin{equation}
\tilde P(k\delta )=\begin{cases}
2|\tilde h(k\delta )|^2/N_{bins}^2~~~ \text{for } 0<k\delta <f_{Nyq}~,\\
\\
|\tilde h(k\delta )|^2/N_{bins}^2 ~~~\text{for } k\delta =0 ~
\end{cases}
\label{eq:power1}
\end{equation}
(here the identity $(|\tilde h(k\delta )|^2+|\tilde h(-k\delta )|^2)=2|\tilde h(k\delta )|$ was used).
The factor of $1/N^2_{bins}$ is included in order to fix the normalization, so that at $k=0$ we have $\tilde P(0)=(\tilde N_{ev}/N_{bins})^2$ (here $\tilde N_{ev}= \sum^{N_{bins}-1}_{j=0} \tilde{N}(t_j)$).
Fig. \ref{fig:compare} shows an illustration of the discretized time profile, and the corresponding power spectra, for the KKHT\ model, with and without SASI (as well as for the two templates in Eqs. (\ref{eq:mod2}) and (\ref{eq:mod0})). For the latter, the parameters have been fit to maximize the likelihood (see eq. (\ref{eq:likeli}) in the following section) to best reproduce the general features of the neutrino\ event rates predicted by the KKHT\ model.
The figure shows that, qualitatively, the templates capture the main features of the realistic, numerically calculated time and frequency profiles. An exception is the peak at $f\sim 60$ Hz in the power spectrum of the no-SASI\ model, which is not reproduced by the template. We checked that this peak is due to the ``wavy'' structure at $t\sim 180-200$ ms in the numerical model.
\begin{figure*}[htp]
\centering
\includegraphics[width=0.45\textwidth]{KurodCompareTSASInew}
\includegraphics[width=0.45\textwidth]{KurodCompareFSASInew}
\includegraphics[width=0.45\textwidth]{KurodCompareTnoSASInew}
\includegraphics[width=0.45\textwidth]{KurodCompareFnoSASInew}
\caption{Neutrino event rate (left panels) and its power spectrum (right panels) at Hyper-K\, for distance $D=1$ kpc. Shown as solid black lines are a case where there is SASI\ (upper panes, from the KKHT\ model), and no SASI (lower panes, derived from the KKHT\ model with smoothing, see text). We also show (solid, purple curves) the predictions of the 2-parameter template (2P, Eq. (\ref{eq:mod2})) and of the 0-parameter template (0P, Eq. (\ref{eq:mod0})), for estimated best-fitting parameters
($f_S=119.72$ Hz, $a=0.049$ and $A=6141.54$, see Eq. (\ref{eq:mod2}) and Table \ref{tab:sasiPfa10}). The shaded (blue) bands characterize the probability density distributions with the width of one standard deviation. }
\label{fig:compare}
\end{figure*}
\subsection{The SASI-meter}
\label{sub:likelihoodR}
Let us now consider the series of power spectrum values at the discrete frequencies $k\delta$, $P(k\delta)$, and their statistical properties.
Considering that (i) the probability that a single neutrino interacts in the detector is very small, (ii) event counts in different time bins are statistically independent (see Sec. \ref{sec:general}), and (iii) $N(t_j) \gtrsim 10$ (large number approximation), we conclude that the binomial distribution for $N(t_j)$ approaches a Gaussian distribution with a variance proportional to the square root of the mean number (Poisson process): $s^2(t_j)=N(t_j)$.
This implies (see the proofs in Appendices A and B) that the real part and imaginary part of the discrete Fourier transform, $h(k\delta)$ (Eq. (\ref{eq:h})), are also Gaussian-distributed,
and the probability distribution of the power spectrum $\tilde{P}$ at a given frequency is given by
\begin{equation}
\begin{split}
Prob(\tilde{P})&=\frac{N_{bins}^2}{4\sigma^2} \exp{ \left[ -\frac{N_{bins}^2}{4\sigma^2} \left(\tilde{P} + P \right)\right]}\\
&\times I_0\left( \frac{N_{bins}^2}{2\sigma^2} \sqrt{ \tilde{P}P} \right)~,
\label{eq:prob}
\end{split}
\end{equation}
where $I_0$ is the modified Bessel function of the first kind, and
\begin{equation}
\begin{split}
\sigma^2=\frac{N_{ev}}{2}~.
\end{split}
\label{eq:sigma2}
\end{equation}
The object of this study is to perform a hypothesis test for the presence of SASI.
There is evidence from numerical simulations that the SASI only develops within a certain range of frequencies from a few tens of Hz to about 250 Hz \cite{Mueller:2003fs,Murphy:2009dx,Yakunin:2010fn,Mueller:2012sv,Andresen:2016pdt,Kuroda:2016bjd}. Therefore, we apply a frequency cut, and restrict the analysis to the interval from 54 Hz to 216 Hz. The corresponding range of wavenumbers is $k=3,4,5,...,12$.
In addition to being motivated by estimates of the SASI\ frequency, the cut is instrumental to exclude a large peak at low frequency due to the spectral leakage \cite{2009fundamentals} from $0$ Hz.
Let us now define the likelihood that a given observed power series vector, $\mathcal{\tilde P}= \{ \mathcal{\tilde P}_k \}$ (i.e., the series of powers for discrete wavenumbers $k$) is a realization of a certain hypothesis, which can be described by a parametric template.
It is defined as:
\begin{equation}
L(\mathcal{\tilde P},\Omega)=\prod_{k=3}^{12}Prob(\mathcal{\tilde P}_k, P_k(\Omega))~,
\label{eq:likeli}
\end{equation}
where $P_k(\Omega)$ is the power predicted by the template, and $\Omega$ indicates the set of parameters of the template.
Given two hypotheses (i.e., two templates) with parameters $\Omega$ and $\Omega_0$, and a fixed observed set $\mathcal {\tilde P}$, the likelihood ratio is:
\begin{equation}
\mathcal{L}(\mathcal{\tilde P})=\frac{Max_{\Omega}[L(\mathcal{\tilde P},\Omega)]}{Max_{\Omega_0}[L(\mathcal{\tilde P},\Omega_0)]}~.
\label{eq:likeR}
\end{equation}
In the numerator (denominator), the first (second) hypothesis is used and the likelihood is maximized with respect to the parameters $\Omega$ ($\Omega_0$).
In this work, the templates in Eqs. (\ref{eq:mod2}) and (\ref{eq:mod0}) will be used as representative of the SASI\ and no-SASI\ cases.
Their parameters are $\Omega=\{a, f_S \}$ and $\Omega_0=\{ Null\}$ respectively.
It is intuitive to see how the likelihood ratio in Eq. (\ref{eq:likeR}) is sensitive to SASI. Since our templates $R_2$ (Eq. (\ref{eq:mod2})) and $R_0$ (Eq. (\ref{eq:mod0})) capture well the main features of the neutrino event rates of the models with and without SASI respectively, as the SASI\ features in the data become more pronounced, the numerator Eq. (\ref{eq:likeR}) is likely to increase (generally better fit for the $R_2$ template), while at the same time the denominator is likely to decrease (poorer fit for the $R_0$ template), so $\mathcal{L}$ is likely to increase. Vice-versa, $\mathcal{L}$ will take lower values if the SASI\ signatures in the data become weaker.
Therefore, Eq. (\ref{eq:likeR}) serves as our \emph{``SASI-meter"} to identify the presence of SASI.
To assess the effectiveness of the SASI-meter quantitatively,
we need to find the probability distributions of $\mathcal{L}$ (or, equivalently, $\ln{\mathcal L}$) under the two hypotheses. This was done by simulating (using a Monte Carlo method) $N_{st} = 10^3$ sets $ \mathcal{\tilde P} $ using the KKHT\ model with and without SASI, so we will have $\mathcal{L}_{S}\equiv \mathcal{L}(\mathcal{\tilde P}_{SASI})$ and $\mathcal{L}_{nS}\equiv \mathcal{L}(\mathcal{\tilde P}_{no-SASI})$, and their probability density distributions, $Prob(\mathcal{L}_{S})\simeq Prob(\mathcal{L}|S) $ (where $Prob(\mathcal{L}|S)$ indicates the ``true" probability distribution, which would be obtained in the limit $N_{st} \rightarrow \infty$) and $Prob(\mathcal{L}_{nS})\simeq Prob(\mathcal{L}|nS)$.
A useful way to describe these two distributions, and compare them with one another, is to examine the probabilities that -- under the two hypotheses -- the likelihood ratio exceeds a certain threshold value, $\Lambda$:
\begin{eqnarray}
&P_D=\int_{\mathcal{L}>\Lambda}Prob(\mathcal{L}|S) d\mathcal{L}\label{eq:pd}~,\\
&P_{FI}=\int_{\mathcal{L}>\Lambda}Prob(\mathcal{L}|nS) d\mathcal{L}~. \label{eq:pf}
\end{eqnarray}
$\Lambda$ usually represents a value of the likelihood ratio above which the SASI\ hypothesis is accepted as true (``detection"). Therefore, $P_D$ takes the meaning of SASI \emph{detection probability}, because it represents the probability that the method accepts the SASI\ hypothesis as true when the SASI\ is in fact true. $P_{FI}$ then represents the \emph{false identification probability}, i.e., the probability that the SASI\ hypothesis is accepted when in fact the no-SASI\ hypothesis is the true one.
The formalism discussed in this section becomes clearer in light of the results we have obtained, which are going to be illustrated next.
\subsection{Results: SASI or no-SASI?}
\label{sub:results}
Our main results for hypothesis testing are summarized in fig. \ref{fig:likeR}, for Hyper-K\ and IceCube, and for different distances to the supernova. For each detector and distance, the figure shows the probability distributions of $\ln \mathcal{L}_S$ and $\ln \mathcal{L}_{nS}$.
We observe that, reflecting the expected sensitivity of our SASI-meter, for short distances the two distributions are widely separated, with the distribution for the SASI\ (no-SASI) case peaking at lower (higher) values of the likelihood ratio \footnote{Note that, all cases, the logarithm of the likelihood ratio is positive, meaning that the two parameter template offers a better fit than the zero-parameters one. This is simply a consequence of the larger number of parameters of one template with respect to the other. }.
The separation means that, if the SASI\ hypothesis is true, there is a large probability that the measured value of $\ln \mathcal{L}$ will fall in a region where the no-SASI\ hypothesis is strongly disfavored (i.e., $Prob( \mathcal{L}|nS)\ll Prob( \mathcal{L}|S) $). A similar argument holds if the no-SASI\ hypothesis is true. We conclude, then, that for a relatively close supernova ($D \sim$ few kpc) the two hypotheses are likely to be distinguished with high confidence.
The separation between the two probability distributions decreases as $D$ increases,
until, for $D \sim 10$ kpc, the SASI\ and no-SASI\ curves almost completely overlap, meaning that the two hypotheses are very unlikely to be distinguished. The dependence on the distance is due to how the size of the the statistical fluctuations increases with $D$, eventually overpowering the SASI, which therefore becomes invisible.
The trends shown in Fig. \ref{fig:likeR} are reflected in the behavior of the detection and false identification probabilities, $P_D$ and $P_{FI}$ (Eqs. (\ref{eq:pd}) and (\ref{eq:pf})). These are described by the Receiver Operating characteristic Curve (ROC). The ROC is defined as the curve described in a plane by the points $(P_{FI}(\Lambda),P_D(\Lambda))$, where $\Lambda$ varies in the interval $[0,+\infty]$. Fig. \ref{fig:ROC} shows the ROC for Hyper-K\ and IceCube\ for several distances from the star. The plots show the general features of the ROC: it passes by the points $(0,0)$ and $(1,1)$ (corresponding to $\Lambda \rightarrow +\infty$ and $\Lambda \rightarrow 0$ respectively, see Eqs. (\ref{eq:pd}) and (\ref{eq:pf})). Furthermore, the curve lies in the region $P_D > P_{FI}$, as expected from Fig. \ref{fig:likeR}. A high detectability potential corresponds to a ROC where $P_D$ is as close as possible to $1$ and at the same time $P_{FI}$ is as close as possible to 0. For example, for IceCube\ and $D=5$ kpc, the ROC passes by the point $(P_{FI},P_D)\simeq (0.1, 0.95)$, meaning that, if a 10\% false identification rate is considered acceptable, the likelihood ratio will establish the presence of the SASI in 95\% of the cases. The same situation is realized for Hyper-K\ for $D\simeq 2$ kpc. Naturally, the ROC deteriorates as $D$ decreases, and ultimately (for $D \gtrsim 10$ kpc) it converges to the line $P_D=P_{FI}$, which corresponds to a neutrino\ signal with SASI being completely indistinguishable from a signal without SASI. The ROC curves allow to estimate the range where a fixed $P_D$ is achieved for a desired $P_{FI}$. If, e.g., we require the ROC to have $P_D \geq 0.7$ for $P_{FI}=0.1$, Fig. \ref{fig:ROC} indicates that the largest distance of sensitivity to the SASI is $D\simeq 6$ kpc for IceCube\ and $D\simeq 3$ kpc for Hyper-K.
\begin{figure*}[htp]
\centering
\includegraphics[width=0.45\textwidth]{LikeR10HypKnew}
\includegraphics[width=0.45\textwidth]{LikeR10IceCnew}
\includegraphics[width=0.45\textwidth]{LikeR5HypKnew}
\includegraphics[width=0.45\textwidth]{LikeR707IceCnew}
\includegraphics[width=0.45\textwidth]{LikeR3HypKnew}
\includegraphics[width=0.45\textwidth]{LikeR577IceCnew}
\includegraphics[width=0.45\textwidth]{LikeR2HypKnew}
\includegraphics[width=0.45\textwidth]{LikeR5IceCnew}
\caption{Likelihood ratio probability distribution for SASI and no SASI case in Hyper-K\ (left) and IceCube\ (right), for different values of the distance $D$ to the star (chosen to correspond to integer increments of the number of events, see legends). The likelihoods have been obtained using simulated neutrino\ signal according to the KKHT model.}
\label{fig:likeR}
\end{figure*}
\begin{figure}[htp]
\centering
\includegraphics[width=0.45\textwidth]{ROCHypKPFI}
\includegraphics[width=0.45\textwidth]{ROCIceCPFI}
\caption{Receiver operating characteristic curves based on KKHT model for Hyper-K\ (top panel) and Ice Cube (bottom panel), for several distances to the supernova. See Eqs. (\ref{eq:pd})-(\ref{eq:pf}).}
\label{fig:ROC}
\end{figure}
\section{Parameter estimation}
\label{sec:parameter}
\subsection{Likelihood ratio and best fit parameters}
For the scenarios where the SASI\ hypothesis is accepted as true (${\mathcal L}>\Lambda$), the next step is estimation of the parameters. For definiteness, here we present results for $\Lambda$ that corresponds to $P_{FI}=0.1$ (Eq. (\ref{eq:pf})).
In our method, the best fit values of the SASI\ frequency, $\bar{f_{S}}$, and of the amplitude, $\bar{a}$, are found as the values that maximize the likelihood $L(\tilde {\mathcal P},\Omega)$, within the process of constructing the likelihood ratio (Eq. (\ref{eq:likeR})). From that process, we obtained the probability distributions of $\bar{f_{S}}$, and $\bar{a}$.
We then calculated the mean and standard deviation of $\bar{f_S}$ and $\bar{a}$. The standard deviation gives an estimate of approximately 68\% confidence level error with which an estimate of a given parameter can be obtained.
The results are shown in Fig. \ref{fig:parameterHypIceC} and Tables \ref{tab:sasiPfa10} (for Hyper-K) and \ref{tab:sasiIcePfa10} (for IceCube).
For Hyper-K\ and $D=10$ kpc, where the sensitivity to the SASI\ is poor, the distribution for $\bar f_S$ is very broad, with roughly all values being equally probable. This indicates that, although there might be indication of an oscillatory behavior in the data (such that the likelihood ratio is above the threshold), such outcome is most likely to be due to random statistical fluctuations and not to SASI. An estimate of the frequency would have a large error and might not be physically meaningful. The corresponding distribution for $\bar a$ is similarly broad for $\bar a\gtrsim 0.03$, indicating that, as long as there is indication of an oscillatory pattern in the data, its amplitude can vary widely, and is probably driven by statistical fluctuations.
As $D$ decreases ($D \lesssim 5$ kpc or so) the distributions of both $\bar f_S$ and $\bar a$ start to concentrate around the physical values of the injected SASI\ model, $\bar f_S \sim 120$ Hz and $\bar a \sim 0.05$, indicating a sensitivity to the physical SASI\ signal above statistical fluctuations. This trend appears in Table \ref{tab:sasiPfa10} as well, where one can see the decrease of the standard deviation with the decreasing distance. We note that the width of the distributions for $a$ and $f_S$ depend in part on how the time structure of the neutrino\ signal in the KKHT\ model is only roughly reproduced by the simplified template, Eq. (\ref{eq:mod2}). As a consistency test, we checked that using simulated data drawn from the simplified template has the (expected) effect of producing narrower parameter distributions \footnote{For very high statistics signals, an accurate evaluation of the parameter distributions will require a finer sampling of the parameter space than done here. We checked that the effect of the finite sampling is minor in our results. }.
We caution the reader about the meaning of the multiple peaks that appear in the distributions in Fig. \ref{fig:parameterHypIceC}: these peaks reflect the discrete structure of the power spectrum series $\{\tilde P_k \}$ which is being analyzed, which has a resolution (frequency bin size) of about 20 Hz (see eq. (\ref{eq:tau}) and Fig. \ref{fig:compare}), and therefore do not have a direct physical meaning.
The probability distributions and tabulated values (Table \ref{tab:sasiIcePfa10}) for IceCube\ show a structure and dependence on $D$ similar to those for Hyper-K. A difference is that at $D=10$ kpc, the sensitivity to SASI\ is not completely washed out by the statistical fluctuations, so it might be possible to obtain a (coarse) measurement of $f_S$.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{sasifSmDistriPFA10}
\includegraphics[width=0.45\textwidth]{sasiaSmDistriPFA10}
\includegraphics[width=0.45\textwidth]{sasifSmDistriIcePFA10}
\includegraphics[width=0.45\textwidth]{sasiaSmDistriIcePFA10}
\caption{Probability distribution of the best-fit values of the SASI\ frequency, $\bar f_S$ (left), and oscillation amplitude, $\bar a$ (right), for Hyper-K (top) and for IceCube\ (bottom), for several distances to the supernova. Only cases with sufficient statistical indication of SASI\ activity are considered here, by imposing a threshold on the likelihood ratio (corresponding to $P_{FI}>0.1$, see text). Here, the likelihoods have been obtained using simulated neutrino\ signal according to the KKHT model.}
\label{fig:parameterHypIceC}
\end{figure*}
\subsection{Fisher Information Matrix and minimum uncertainties}
In this section we aim at comparing the standard deviations of the SASI\ parameters obtained using the likelihood ratio method with the theoretical lower bound in the accuracy. The latter is given by the Cramer-Rao\ lower bound \cite{2009fundamentals}, and is derived from the Fisher Information Matrix (FIM).
We begin by summarizing the main formulae of the FIM formalism; these will then be applied to the case at hand.
\begin{table*}
\caption{\label{tab:sasiPfa10}
Mean and standard deviation of the parameter distributions in Fig. \ref{fig:parameterHypIceC}, for Hyper-K. The numbers in the parentheses are the Cramer-Rao\ lower bounds, calculated using Fisher matrix in the time domain. For larger $D$, where the selection effects on ${\mathcal L}$ are strong, a direct comparison is not meaningful and therefore the Cramer-Rao\ bounds are not shown.
}
\begin{ruledtabular}
\begin{tabular}{cccccc}
SASI &$10$ kpc &$5$ kpc&
$3.33$ kpc &$2 $ kpc &$1$ kpc\\
\hline
f(Hz)& 111.77 & 109.2 & 111.67 &116.49
& 119.72 \\
$\delta f$(Hz)& 42.31 & 22.61& 16.7 &9.64(0.08)
&2.35(0.04) \\
a& 0.065 & 0.062& 0.054 &0.049
& 0.049 \\
$\delta a$& 0.017& 0.008 & 0.0059 &0.0053(0.0047)
& 0.0026(0.0023) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}
\caption{\label{tab:sasiIcePfa10}
Mean and standard deviation of the parameter distributions in Fig. \ref{fig:parameterHypIceC}, for IceCube. The numbers in the parentheses are the Cramer-Rao\ lower bounds, calculated using Fisher matrix in the time domain. For larger $D$, where the selection effects on ${\mathcal L}$ are strong, a direct comparison is not meaningful and therefore the Cramer-Rao\ bounds are not shown. }
\begin{ruledtabular}
\begin{tabular}{ccccc}
SASI&$10 $ kpc &$7.07$ kpc &$5.77$ kpc &$5 $ kpc \\
\hline
f(Hz)& 120.45 &115.57&118.33& 119.43\\
$\delta f$(Hz)& 33.36 &10.85 &5.53 (0.078)& 3.34 (0.063)\\
a& 0.057 &0.050&0.048& 0.048\\
$\delta a$& 0.0064 &0.0036&0.0047(0.0041)& 0.0046 (0.0034) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
Let us consider a generic template $R(t_i)$ for the event rate at discrete times, $t_i$ ($i=1, 2,..., N$), which depends on a set of parameters, $\theta_\alpha$ ($\alpha=1,2,3,..,K$) (note that, for our choice of unitary bin size, $\Delta=1$ ms, the event rate and the number of events are numerically the same. Here we omit the factor $\Delta$ to keep the notation compact). The FIM\ is a $K \times K$ matrix, found from the probability distribution. We define the joint probability as:
\begin{equation}
Prob(\tilde{R})=\prod_{i=0}^{N}Prob(\tilde{R}_{i}),
\label{}
\end{equation}
where $\tilde{R}$ is the series of observed neutrino rate $\{\tilde{R}(t_1), \tilde{R}(t_2),...\tilde{R}(t_N)\}$ in time domain. The FIM describes how much each parameter affects the distribution via its second derivatives:
\begin{equation}
\Gamma_{\alpha \beta} = \langle -\frac{\partial^{2} \ln{Prob(\vec{R})}}{\partial \theta_{\alpha} \partial \theta_{\beta}} \rangle~ ,
\label{eq:gamma}
\end{equation}
In the assumption that $Prob(\tilde{R}(t_{i}))$ is a Multivariate Gaussian Distribution in the time domain, the FIM\ reduces to the following expression (see Appendix C):
\begin{equation}
\Gamma_{\alpha \beta} = \mu_{\alpha}^{T} \Sigma^{-1} \mu_{\beta} + \frac{1}{2} Tr[\Tilde{c}_{\alpha} \Tilde{c}_{\beta}]~,
\label{eq:gamma2}
\end{equation}
where $\mu_{\alpha}^{T}$ and $\mu_{\beta}$ are N-dimensional vectors (one component for each value of $t_i$), defined as:
\begin{equation}
\mu_{\alpha} = \frac{\partial \tilde{R}}{\partial \theta_{\alpha}}~,
\label{eq:mualpha}
\end{equation}
and
$\Sigma^{-1}$ is the inverse of the $N\times N$ diagonal covariance matrix:
\begin{equation}
\Sigma^{-1} =
\begin{bmatrix}
{R(t_{1})}^{-1} & 0 & 0 & 0 & 0 \\
0 & {R(t_{2})}^{-1} & 0 & 0 & 0 \\
0 & 0 & {R(t_{3})}^{-1} & 0 & 0 \\
0 & 0 & 0 & \ddots & 0 \\
0 & 0 & 0 & 0 & {R(t_{N})}^{-1} \\
\end{bmatrix}~.
\label{cmatrix}
\end{equation}
Finally $\Tilde{c}_{\alpha}$ is defined as the inverse of the covariance matrix times the partial derivative of the matrix:
\begin{equation}
\Tilde{c}_{\alpha} = \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_{\alpha}}~.
\label{tildealpha}
\end{equation}
The Cramer-Rao\ bound on a parameter $\theta_\alpha$ is given by:
\begin{equation}
\delta \theta_{\alpha} \geq \sqrt{(\Gamma^{-1})_{\alpha \alpha}}
\label{eq:CRbound}
\end{equation}
We can now specialize the FIM\ formalism to our case, where the template is the one in Eq. (\ref{eq:mod2}), and we have two parameters, $\theta_1=a$ and $\theta_2=f_S$. Therefore:
\begin{equation}
R(t_{i}) =R_2(t_i)=(A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n
\end{equation}
\begin{equation}
\mu_{1} = (A-n)\sin{(2 \pi f_{s} t_{i})}
\end{equation}
\begin{equation}
\mu_{2} = 2 \pi t_{i} (A-n) a \cos{(2 \pi f_{s} t_{i})}~.
\end{equation}
The elements of Fisher matrix in time domain can be written analytically as below:
\begin{equation}
\begin{aligned}
\Gamma_{11}&=\sum^{N}_{i=1} \frac{(A-n)^{2} \sin{(2 \pi f_{s} t_{i}))}^{2}}{(A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n}
\\&+
\frac{(A-n)^{2} \sin{(2 \pi f_{s} t_{i}))}^{2}}{2((A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n)^{2}},
\end{aligned}
\label{eq:gamma11}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma_{12}&=\sum^{N}_{i=1} \frac{2a (A-n)^{2} \pi t_{i} \sin{(2 \pi f_{s} t_{i})} \cos{(2 \pi f_{s} t_{i})}}{(A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n}
\\&+
\frac{a (A-n)^{2} \pi t_{i} \sin{(2 \pi f_{s} t_{i})} \cos{(2 \pi f_{s} t_{i})}}{((A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n)^{2}},
\end{aligned}
\label{eq:gamma12}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma_{21}&=\sum^{N}_{i=1} \frac{2a (A-n)^{2} \pi t_{i} \sin{(2 \pi f_{s} t_{i})} \cos{(2 \pi f_{s} t_{i})}}{(A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n}
\\&+
\frac{a (A-n)^{2} \pi t_{i} \sin{(2 \pi f_{s} t_{i})} \cos{(2 \pi f_{s} t_{i})}}{((A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n)^{2}},
\end{aligned}
\label{eq:gamma21}
\end{equation}
and \begin{equation}
\begin{aligned}
\Gamma_{22}&=\sum_{i=1}^{N} \frac{4 a^{2} (A-n)^{2} \pi^{2} t_{i}^{2} \cos{(2 \pi f_{s} t_{i})^{2}}}{(A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n}
\\&+
\frac{2 a^{2} (A-n)^{2} \pi^{2} t_{i}^{2} \cos{(2 \pi f_{s} t_{i})^{2}}}{((A-n)(1+a \sin{(2 \pi f_{s} t_{i})}) + n)^{2}}.
\end{aligned}
\label{eq:gamma22}
\end{equation}
Finally, by combining Eqs. (\ref{eq:gamma11}) to (\ref{eq:gamma22}) with Eq. (\ref{eq:CRbound}), one finds the minimum uncertainties on the parameters: $\delta \theta_1=\delta a$ and $\delta \theta_2=\delta f_S$. These are themselves functions of $a$ and $f_S$, so they have to be estimated at a chosen (best-fit) point in the parameter space.
In Figs. \ref{fig:colterFish}-\ref{fig:colterFishIce}, the relative Cramer-Rao\ uncertainties are shown for selected distances to the star (which determine the widths of the Gaussian probability distributions that enter the calculation), and as functions of one of the parameters, where the other parameter is kept fixed at its best-estimated value (last columns of Tables \ref{tab:sasiPfa10} and \ref{tab:sasiIcePfa10}).
As expected, the uncertainties decrease with decreasing distance. We also note that the dependence on the amplitude $a$ is stronger than that on the frequency.
As a figure of merit, to clarify if our approach is optimal we can
compare the width (error) from the histograms in figure \ref{fig:parameterHypIceC} with
the Cramer-Rao\ lower bound for the SASI\ analytical model we adopt.
In Tables \ref{tab:sasiPfa10} and \ref{tab:sasiIcePfa10} the Cramer-Rao\ uncertainties -- calculated at the points in the parameter space given in the tables themselves -- are listed for the two smallest distances. They can be directly compared to the standard deviations obtained with the likelihood ratio method, because at such distances the selection effects due to the threshold on ${\mathcal L}$ are negligible (nearly all the simulated cases pass the selection). For larger $D$, where the selection effects on ${\mathcal L}$ are strong, a direct comparison is not meaningful and therefore the Cramer-Rao\ bounds are not shown.
It appears that $\delta a$ obtained from the likelihood ratio is close (sightly larger, as expected) to the corresponding Cramer-Rao\ bound, indicating that our method is near optimality for estimating the SASI\ amplitude. In contrast, for $\delta f_S$ the Cramer-Rao\ bound is orders of magnitude more stringent, so in principle, a more effective method than ours for frequency estimation could exist (although an estimator attaining the Cramer-Rao lower bound does not necessarily exist).
\begin{figure*}[htp]
\centering
\includegraphics[width=0.45\textwidth]{RelativeFrequencyPerFrequencyHyperKnew}
\includegraphics[width=0.45\textwidth]{RelativeFrequencyPerAmplitudeHyperKnew}
\includegraphics[width=0.45\textwidth]{RelativeAmplitudePerFrequencyHyperKnew}
\includegraphics[width=0.45\textwidth]{RelativeAmplitudePerAmplitudeHyperKnew}
\caption{Cramer-Rao\ lower bounds based on simplified parametric templates with SASI (see Eq. \ref{eq:mod2}), in the form of relative errors, for the SASI\ frequency (top) and amplitude (bottom), as functions of frequency (left) and amplitude (right), for Hyper-K\ and select distances to the star (see legend). In each curve, the remaining parameter has been fixed at its best-estimated value (the one for $D=1$ kpc) in Table \ref{tab:sasiPfa10}. }
\label{fig:colterFish}
\end{figure*}
\begin{figure*}[htp]
\centering
\includegraphics[width=0.45\textwidth]{RelativeFrequencyPerFrequencyIceCubenew}
\includegraphics[width=0.45\textwidth]{RelativeFrequencyPerAmplitudeIceCubenew}
\includegraphics[width=0.45\textwidth]{RelativeAmplitudePerFrequencyIceCubenew}
\includegraphics[width=0.45\textwidth]{RelativeAmplitudePerAmplitudeIceCubenew}
\caption{Cramer-Rao\ lower bounds based on simplified parametric templates with SASI (see Eq. \ref{eq:mod2}), in the form of relative errors, for the SASI\ frequency (top) and amplitude (bottom), as functions of frequency (left) and amplitude (right), for IceCube\ and select distances to the star (see legend). In each curve, the remaining parameter has been fixed at its best-estimated value (the one for $D=1$ kpc) in Table \ref{tab:sasiIcePfa10}. }
\label{fig:colterFishIce}
\end{figure*}
\section{Summary and discussion}
\label{sec:disc}
We have proposed a novel methodology to do both hypothesis testing and parameter estimation for signatures of SASI\ in the time profile of the neutrino\ event rate from a (galactic) core collapse supernova.
This method is based on the likelihood ratio constructed using the signal power spectrum, for which the effect of statistical fluctuations was modeled, and suitable frequency cuts can be applied.
We quantify the confidence to identify the presence of SASI in terms of receiver operating curves,
a tool which is commonly used in the gravitational wave community to establish the efficiency versus false alarm probability
for gravitational wave signals (see, e.g., \cite{Blackburn:2005qv}).
We have tested the effectiveness of the method, using an injected signal for Hyper-K\ and IceCube\ from a supernova\ numerical simulation by Kuroda, Kotake, Hayama and Takiwaki. Specifically, we have characterized the performance of the method by producing the receiver operating characteristic curves, and by comparing the probability distributions of the best fit parameters (the SASI\ frequency and relative amplitude) with the ultimate minimum uncertainties from the Cramer-Rao\ lower bounds.
For hypothesis testing, our main result are the probability distributions in Fig. \ref{fig:likeR}. Figuratively speaking, these can be considered like a calibrated measurement rod against which we will compare the likelihood ratio from an actual, future supernova\ neutrinos\ detection. We have found that, for a nearby supernova, this ``SASI-meter" is an effective tool: if the experimental likelihood ratio is in the ``red zone" (above a certain threshold for the likelihood ratio, e.g., $\ln {\mathcal L}\gtrsim 30$ for Hyper-K\ and $D\simeq 2$ kpc), then we will be able to confidently claim the presence of SASI. If it is in the ``blue zone" ($\ln {\mathcal L}\lesssim 20$ in the same example), then a model without SASI\ will be favored, and an upper bound on the parameters of possible SASI\ will be established. We obtain that, for the KKHT model, SASI can be identified with high confidence for a distance to the supernova of up to $\sim 6$ kpc for IceCube\ and and up to $\sim 3$ kpc for Hyper-K. The SASI-meter can also be used to identify unusually long periods of SASI, which could help to establish indication of a failed supernova.
For parameter estimation, we find that, for an injected signal with SASI\ and for data sets in the red zone of the SASI-meter, the SASI\ frequency and amplitude can be reconstructed if $D \lesssim 5$ kpc for Hyper-K\ ($D \lesssim 10$ kpc for IceCube), and their uncertainties are consistent with the Cramer-Rao\ lower bounds.
Beyond such distance, the positive response of the SASI-meter, giving indication of an oscillatory pattern in the event rate, is to be attributed to statistical fluctuations and not to the presence of SASI.
The most immediate development of this work will include several three-dimensional supernova simulation results that present SASI and map the performance of the method in different regions of the parameter space. We expect that including several models will result in a blurring of the probability distributions, so the red and blue zones of the SASI-meter will be less clearly separated, or, in other words, the Receiver Operating characteristic Curves will be worse (i.e., closer to the limiting curve $P_D=P_{FI}$). The method will remain valid conceptually, however.
In the long term, our goal is to extend the methodology to joint analyses of neutrino\ and gravitational wave SASI\ signals, for a truly multi-messenger approach \cite{IceCube:2018dnn,Branchesi:2016vef,Kalogera:2019bdd}. Within this goal, the present paper serves to create the foundation of a formalism for neutrinos\ that finds a direct counterpart (using the same tools, like the likelihood ratio and the receiver operating characteristic curve, for example) in existing gravitational wave analysis protocols. Moving forward, new approaches will have to be developed to establish how to most effectively combine the two signals, neutrinos\ and gravitational waves, that have both similarities (e.g., similar SASI\ frequency) and important differences (different sources of noise, for example). Such development work will explore further the territory of multi-messenger astronomy and aid the investigation of a future galactic core collapse supernova.
\subsubsection*{Acknowledgments}
We are thankful to Takami Kuroda who kindly provided the neutrino\ event rates required for the
analysis of this work.
CL and ZL acknowledge funding from the National Science Foundation grant number PHY-1613708.
KK thanks Tomoya Takiwaki for stimulating discussions and acknowledges support by Grant-in-Aid for Scientific Research
(JP17H01130)
from the Japan Society for Promotion of Science (JSPS) and the Ministry of Education, Science and Culture of Japan (MEXT, Nos. JP17H06357,
JP17H06364), and by the Central Research Institute of Stellar Explosive Phenomena (REISEP) at Fukuoka University and the associated projects (Nos.\ 171042,177103).
MZ acknowledges funding from the National Science Foundation grant number PHY-1806885.
|
2,877,628,090,769 | arxiv | \section{Introduction}\label{Sec 1}
\subsection{Mathematical setup and statement of the main results}
In this paper, we are concerned with inverse problems for semilinear parabolic equations. Depending on the form of the nonlinear term, there are two setups for our study, which shall be discussed separately in what follows.
First, we consider the case that the nonlinear term belongs to a $C^1$ class fulfilling a certain growth condition. We begin by introducing the forward model.
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain
with a $C^\infty$-smooth boundary $\Gamma$ for $n \in \N$ and
$\Gamma_0$ be a nonempty relatively open subset of $\Gamma$. For any $T>0$,
we set
$
Q=\Omega\times(0, T)\text{ and }\Sigma=\Gamma \times(0, T).
$
Assume that
$\gamma=\left( \gamma_{ij}(x,t)\right)_{i,j=1}^n \in C^{2,1}(\overline{Q}; \R^{n\times n})$
is a symmetric matrix-valued function in $\overline{Q}$, such that
\[
\rho_0|\xi|^2 \leq \sum_{i,j=1}^n \gamma_{ij}(x,t)\xi_i \xi _j \leq \rho_0^{-1}|\xi|^2, \quad
\forall\ (x,t)\in \overline{Q }\ \ \text{ and }\ \ \xi=(\xi_1,\ldots, \xi_n) \in \R^n,
\]
for some positive constant $\rho_0\in (0, 1)$. Moreover,
we denote by $H^{s, r}(Q)$, $H^{s, r}(\Gamma)$, $C^{k+\alpha}(\overline{\Omega})$ and
$C^{k+\alpha, \frac{k}{2}+\frac{\alpha}{2}}(\overline{Q})$, respectively, the standard Sobolev spaces and
H\"older spaces for $s, r\in\mathbb R$, $k\in \mathbb N$ and $\alpha\in(0, 1)$.
We refer to \cite{Adams2003sobolev} and \cite{evans1998partial} for details of these Banach spaces. Consider the following semilinear parabolic equation:
\begin{eqnarray}\label{eq:parabolic1}
\begin{cases}
u_{t}-\nabla\cdot(\gamma\nabla u)+a(x,t,u)=0 &\text{ in }\ Q,\\
u=f &\text{ on }\ \Sigma,\\
u(x, 0)=g(x) &\text{ in }\ \Omega,
\end{cases}
\end{eqnarray}
where $u_t=\partial_t u= \frac{\p u}{\p t}$, $g \in H^1_0(\Omega)$,
$f\in L^2(\Sigma)$ and $a=a(x,t,y):
Q\times\mathbb R\rightarrow \mathbb R$ is a given function, satisfying
suitable conditions that will be specified later.
For any $g \in H^1_0(\Omega)$ and a suitable function $a: Q\times \R\to \R$,
which guarantees the global well-posedness of (\ref{eq:parabolic1}) (see Section \ref{Sec 2}),
we introduce the following Dirichlet-to-Neumann (DN for short) operator:
\begin{align}\label{eq:DN1}
\begin{split}
\Lambda_{a,g}: \mathcal E & \rightarrow L^2(\Gamma_0\times(0, T)) , \\
f&\mapsto \partial_\nu u_{f}\Big|_{\Gamma_0\times(0, T)}.
\end{split}
\end{align}
Here we use the notation that $\p_\nu u(x):=\nabla u(x)\cdot \nu(x), \text{ for }x\in \Gamma$,
where $\nu=(\nu_1,\ldots, \nu_n)\in\mathbb{S}^{n-1}$ signifies the exterior unit normal to $\Gamma$, and
$u_f$ is the solution to \eqref{eq:parabolic1} associated to the initial
data $g\in H^1_0(\Omega)$ and boundary data $f\in\mathcal E$ with
\begin{eqnarray*}
&&\mathcal{E}=\Big\{ f\in L^2(\Sigma)\
\Big|\ (\ref{eq:parabolic1}) \mbox{ is well-posed associated to } g \mbox{ and }a, \mbox{ such that }
\\
&&\quad\quad\quad\quad\quad\quad\quad\quad \, u\in C([0, T]; L^2(\Omega))
\mbox{ and } \left. \partial_\nu u \right|_{\Gamma_0\times(0, T)} \in L^2(\Gamma_0\times(0, T))
\Big\}.
\end{eqnarray*}
It is known that when $a\in L^\infty(Q; W^{1, \infty}(\mathbb R))$,
$$
\Big\{ f\in H^{\frac{3}{2}, \frac{3}{4}}(\Sigma)\
\Big|\
f(x, 0)=0 \mbox{ on }\Gamma
\Big\}\subseteq \mathcal{E}.
$$
When $f\equiv0$ in $Q$, the DN map is simply denoted by
\begin{align*}
\Lambda_{a,g}^{(0)}:=\Lambda_{a,g}(0),
\end{align*}
In such a case and in the physical situation, the field $u$ is generated by the initial data $g$, acting as a source, which is assumed to be unknown in our inverse problem study. Hence, the boundary measurement encoded in $\Lambda_{a,g}^{(0)}$ is passively taken by the observer, and in the literature, $\Lambda_{a,g}^{(0)}$ is usually referred to as the {\it passive measurement}. In contrast, $\Lambda_{a,g}(f)$ associated with a nontrivial boundary input $f$ is called the {\it active measurement} since the field $u$ is actively induced by the observer by imposing a boundary input.
Associated to the forward model \eqref{eq:parabolic1}--\eqref{eq:DN1}, we are interested in the following two inverse problems:
\begin{itemize}
\item \textbf{Inverse Problem 1.} Can we identify the unknown functions $(a,g)$ by using the passive measurement $\Lambda_{a,g}^0$?
\item \textbf{Inverse Problem 2.} Can we identify the unknown functions $(a,g)$ by using the active measurement $\Lambda_{a,g}$?
\end{itemize}
It is emphasized that the principal coefficient $\gamma=\gamma(x,t)$ of our \textbf{Inverse Problem 1} and \textbf{Inverse Problem 2} can be space-time dependent. Next, we need to introduce certain a priori conditions on the nonlinear term $a$ in order to guarantee the well-posedness of the forward problem as well as the feasibility of the inverse problems. To that end, we assume that $a: Q\times\mathbb R\rightarrow\mathbb R$
with $a(x, t, \cdot)\in C^1(\mathbb R)$ in $Q$, and satisfies the following growth condition:
\begin{align}\label{condition of nonlinear f at infinity data}
\limsup\limits_{y\rightarrow\infty}\displaystyle\frac{\p_y a(x, t, y)}{\mbox{ln}^{\frac{1}{2}}|y|}=0,\quad
\mbox{ uniformly for }(x, t)\in Q.
\end{align}
It is clear that any function in $L^\infty(Q; W^{1, \infty}(\mathbb R))$ satisfies
the condition (\ref{condition of nonlinear f at infinity data}).
For notational clarity, we set
\begin{eqnarray}\label{set A}
\begin{array}{ll}
\displaystyle\mathcal{A}_T=\Big\{ a: Q\times\mathbb R\rightarrow\mathbb R\ \Big| &
a(x, t, \cdot)\in C^1(\mathbb R)\mbox{ in }Q, \ a(\cdot, \cdot, 0)\in L^2(Q),\\
&\displaystyle\mbox{ and the condition }
(\ref{condition of nonlinear f at infinity data})\mbox{ is fulfilled } \Big\}.
\end{array}
\end{eqnarray}
In Section \ref{Sec 2}, we shall show that for any $g \in H^1_0(\Omega)$,
$a\in \mathcal{A}_T$ and $f=0$, \eqref{eq:parabolic1} has a unique solution $u\in H^{2, 1}(Q)$ and therefore, $\partial_\nu u \in L^2(\Sigma).$
We are in a position to state the first recovery result for the inverse problems introduced above.
\begin{thm}[Conditional stability of determining initial data by passive measurement]\label{Main Thm 1}
Given $a \in\mathcal A_T$ and
$g_j \in H_0^1(\Omega)$ $(j=1, 2)$. Let $\Lambda_{a,g_j}^0$ be the
$($\mbox{passive}$)$ DN map associated to the following semilinear parabolic equation:
\begin{align}\label{IBVP for thm 1 for j=1,2}
\begin{cases}
\partial_t u_{j}-\nabla\cdot(\gamma\nabla u_j)+a(x,t,u_j)=0 &\text{ in }\ Q,\\
u_j=0 &\text{ on }\ \Sigma,\\
u_j(x, 0)=g_j(x),
&\text{ in }\ \Omega.
\end{cases}
\end{align}
For any $M>0$, if
\[
\norm{g_1-g_2}_{H^1_0(\Omega)}\leq M,
\]
then there exist positive constants $C$ and $\delta_0\in (0, 1)$,
depending only on $n, T$ and $\Omega$, such that
the following quantitative stability estimate holds:
\begin{align}\label{Stability estimate in Thm 1}
\begin{split}
\norm{g_1 -g_2}^2_{ L^2(\Omega)}
\leq &
\frac{C(1+M)}{\delta_0}\norm{\Lambda^0_{a, g_1}-\Lambda^0_{a, g_2}}_{L^2(\Gamma_0\times(0, T))}\\
&-\frac{CM^2}{\ln\left( \delta_0 \norm{\Lambda^0_{a, g_1}-\Lambda^0_{a, g_2}}
_{L^2(\Gamma_0\times(0, T))}\right) }.
\end{split}
\end{align}
\end{thm}
By Theorem \ref{Main Thm 1}, it is directly verified that if $\Lambda^0_{a, g_1}=\Lambda^0_{a, g_2}$ on $\Gamma_0\times (0, T), then $
$g_1=g_2$ in $\Omega$. Theorem~\ref{Main Thm 1} partially answers {\bf Inverse Problem 1} that if the nonlinear term $a$ belongs to the general class \eqref{set A} and is a-priori known, then the initial data $g$ can be uniquely recovered (in a stable manner) by the passive measurement.
We proceed to consider \textbf{Inverse Problem 2} and introduce
another ``admissible" set on $a$:
\begin{eqnarray}\label{set C}
\begin{split}
\mathcal{B}_{T}=\Big\{ a: Q\times\mathbb R\rightarrow\mathbb R\ \Big|\
&a(x, t, y)=a_0(x, t, y)\chi_{[0, T-\epsilon]}(t)+c(x,t,y)\chi_{[T-\epsilon, T]}(t) \\[2mm]
&\mbox{for some }\epsilon>0\mbox{ and }a_0\in\mathcal{A}_T, \\[2mm]
&\mbox{where }
c\in\mathcal{A}_T\mbox{ and }c(x, t, 0)=0 \mbox{ in } Q \Big\},
\end{split}
\end{eqnarray}
where $\chi_E(t)=\begin{cases}
1 &t\in E,\\
0 &\text{otherwise}
\end{cases}$ signifies the characteristic function of a set $E\subseteq\mathbb R$.
Our main unique recovery result for \textbf{Inverse Problem 2} is stated as follows.
\begin{thm}[Uniqueness of determining initial data by active measurements]\label{Main Thm 2}
Given $a_j\in \mathcal B_{T}$
and $g_j\in H^1_0(\Omega)$ $(j=1, 2)$.
Let $\Lambda_{a_j,g_j}$ be the $($\mbox{active}$)$ DN map of the
semilinear parabolic equation:
\begin{align}\label{IBVP for thm 2 for j=1,2}
\begin{cases}
u_{j, t}-\nabla\cdot(\gamma\nabla u_j)+a_j(x,t,u_j)=0 &\text{ in } Q,\\
u_j=f &\text{ on } \Sigma,\\
u_j(x, 0)=g_j(x) , &\text{ in }\Omega.
\end{cases}
\end{align}
If for any $f\in \mathcal E$ with $\supp(f)\subseteq \Gamma_0\times [0, T]$,
\begin{equation}\label{eq:a1}
\Lambda_{a_1,g_1}(f)=\Lambda_{a_2,g_2}(f) \quad \text{ on }\quad \Gamma_0 \times (0,T),
\end{equation}
then one has that
\begin{equation}\label{eq:a2}
g_1 =g_2 \quad\mbox{ in }\ \Omega.
\end{equation}
\end{thm}
Theorem \ref{Main Thm 2} means that
the map $\Lambda_{a,g}$ uniquely determines the initial data $g$,
independent of functions $a\in \mathcal B_{T}$.
In the second setup of our study, we consider the case that the nonlinear term $a$ belongs to an analytic class. In such a case, we can assume that both the initial data and the nonlinear term are unknown and can simultaneously recover both of them. To that end, we first introduce the analytic class for the nonlinear term.
\begin{defi}[Admissible class]\label{Def: admissible coefficients}
Let $b=b(x,t,y):\overline{Q}\times\R \rightarrow \R$ satisfy the following conditions:
\begin{align}\label{condition of the nonlinear term f(x,t,z)}
\begin{cases}
\mbox{the map $ y\mapsto b(\cdot,\cdot, y)$ is analytic on $\mathbb R$ with values in $C^{2+\alpha,1+\alpha/2}(\overline{Q})$}, \\
b(x,t,0)=0 \mbox{ in } Q,
\end{cases}
\end{align}
for some $\alpha\in (0, 1)$.
This means that $b$ can be written as the Taylor expansion at any $y_0\in
\R$:
\begin{equation}\label{eq:source1}
b(x,t,y)= \sum_{k=0}^\infty b^{(k)}(x,t,y_0) \frac{(y-y_0)^k}{k!},
\end{equation}
where $\dfrac{b^{(k)}(x,t,y_0)}{k!}:=\dfrac{\partial_y^k b(x,t,y_0)}{k!}$ are Taylor's coefficients at $y_0\in \R$ for any $k\in \N$.
\end{defi}
Next, let $\Omega \subset \R^n$ be a bounded domain with a $C^\infty$-smooth boundary $\Gamma$, for $n \geq 2$, and $T>0$. We introduce the forward model by considering the following semilinear parabolic equation:
\begin{align}\label{eq:parabolic simul}
\begin{cases}
\partial_t u- \Delta u +b(x,t,u)=0 &\text{ in }\ Q,\\
u= f &\text{ on }\ \Sigma,\\
u(x, 0)=g(x), &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $b$ is the function given as Definition \ref{Def: admissible coefficients}. It is easily seen that the second condition \eqref{condition of the nonlinear term f(x,t,z)} of $b$ implies that $u=0$ is a trivial solution when the initial and boundary data are both zero.
In Section~\ref{Sec 2}, we shall prove the (local) well-posedness of the forward problem \eqref{eq:parabolic simul} under the assumption that the coefficient $b$, initial data $g$ and the boundary data $f$ fulfil the following compatibility condition:
\begin{align}\label{compatibility conditions semilinear}
g(\cdot)=g_{x_i}(\cdot)=g_{x_i x_j}(\cdot)=f(\cdot, 0)=
f_t(\cdot, 0)=0\quad\mbox{ on }\quad \Gamma, \quad \mbox{ for }i, j=1, \cdots, n.
\end{align}
Next, we introduce the boundary measurement associated with \eqref{eq:parabolic simul} for our inverse problem study. Let $\mathbb{S}^{n-1}$ be the unit sphere of $\R^n$ and fix $\omega_0\in\mathbb{S}^{n-1}$. Define
\begin{align}\label{Gamma plus}
\Gamma_{\pm, \omega_0}=\Big\{x\in\Gamma\ \Big| \ \pm \nu(x)\cdot\omega_0\geq0\Big\}\quad \mbox{and}\quad \Sigma_{\pm, \omega_0}=\Gamma_{\pm, \omega_0}\times(0,T).
\end{align}
Let $\mathcal{U}_{\pm}$ be a neighborhood of $\Gamma_{\pm,\omega_0}$ in $\Gamma$ and set
$$\mathcal{V}_+=\mathcal{U}_{+}\times (0,T)\quad\mbox{ and }\quad\mathcal{V}_{-}=\mathcal{U}_{-}\times (0,T).$$
With these notations and the local well-posedness at hand, the partial DN map $\Lambda^{\mathrm{P}}_{b,g}$
is defined as:
\begin{align}\label{eq:DN2}
\begin{split}
\Lambda^{\mathrm{P}}_{b,g}: C_0^{2+\alpha,1+\alpha/2}(\mathcal{V}_+) & \rightarrow C^{1+\alpha,1+\alpha/2}(\mathcal{V}_{-}), \\
f&\mapsto \p_\nu u_f\Big|_{\mathcal{V}_-},
\end{split}
\end{align}
for sufficiently small $f\in C_0^{2+\alpha,1+\frac{\alpha}{2}}(\mathcal{V}_+)$
and $g\in C_0^{2+\alpha}(\Omega)$,
which satisfy the compatibility conditions \eqref{compatibility conditions semilinear}, where $u_f$ is the unique solution to \eqref{eq:parabolic simul}. Meanwhile, with the (local) well-posedness at hand, the (full) DN map of the initial-boundary value problem \eqref{eq:parabolic simul linear} is given via
\begin{align}\label{eq:DN3}
\begin{split}
\Lambda_{b,g}: C_0^{2+\alpha,1+\frac{\alpha}{2}}(\Sigma) & \rightarrow C^{1+\alpha,1+\frac{\alpha}{2}}(\Sigma), \\
f&\mapsto \p_\nu u_f\Big|_{\Gamma},
\end{split}
\end{align}
for sufficiently small $f\in C_0^{2+\alpha,1+\frac{\alpha}{2}}(\Gamma)$
and $g\in C_0^{2+\alpha}(\Omega)$.
Our third inverse problem is to ask:
\begin{itemize}
\item \textbf{Inverse Problem 3.} Can we determine the unknown functions $(b,g)$ by using active measurements, either $\Lambda_{b,g}$ or $\Lambda^{\mathrm{P}}_{b,g}$?
\end{itemize}
The main result established for {\bf Inverse Problem 3} is stated as follows.
\begin{thm}[Simultaneous recovery for the semilinear parabolic equation]\label{Main Thm:Simultaneous}
Let $\Omega\subset \R^n$ be a bounded connected domain with a $C^\infty$-smooth boundary $\Gamma$ for $n\geq 2$,
$\Gamma_{+,\omega_0}$ be the set given by \eqref{Gamma plus} and $b_j$ $(j=1, 2)$ be admissible.
Then there exists a $\delta>0$, such that for any $g_j \in C_0^{2+\alpha}(\Omega)$ $(j=1, 2)$
with $\left\| g_j\right\|_{C^{2+\alpha}(\Omega)}<\delta/2$, denote by
$\Lambda_{b_j,g_j}$ and $\Lambda_{b_j,g_j}^{\mathrm{P}}$
the full and partial DN maps of the semilinear parabolic equation:
\begin{align}\label{IBVP of simultaneous recovery}
\begin{cases}
u_{t}-\Delta u +b_j(x,t,u)=0 &\text{ in }\ Q,\\
u= f &\text{ on }\ \Sigma,\\
u(x, 0)=g_j(x), &\text{ in }\ \Omega,
\end{cases}
\end{align}
for $j=1,2$, respectively. Then we have the following results:
\begin{itemize}
\item[(a)] (Full data) If
$$\Lambda_{ b_1,g_1}(f)=\Lambda_{ b_2,g_2}(f) \text{ on }\Sigma, $$
for any sufficiently small lateral boundary data $f\in C^{2+\alpha,1+\frac{\alpha}{2}}_0(\Sigma)$, then
$$
g_1=g_2 \text{ in } \Omega\quad \text{ and }\quad b_1=b_2 \text{ in } Q\times \mathbb{R}.
$$
\item[(b)] (Partial data) Given an open connected set $\Omega'\subset \Omega $ satisfying $\Gamma\subset\p\Omega'$, we assume that $b_1=b_2 $ in $\Omega'\times(0,T)\times \R$, if $\Lambda^{\mathrm{P}}_{b_j,g_j}$ are the partial DN map of the semilinear parabolic equation $\eqref{IBVP of simultaneous recovery}$ and $$\Lambda_{ b_1,g_1}^{\mathrm{P}}(f)=\Lambda^{\mathrm{P}}_{ b_2,g_2}(f) \text{ on }\mathcal{V}_-, $$
for any sufficiently small lateral boundary data $f\in C^{2+\alpha,1+\frac{\alpha}{2}}_0(\mathcal{V}_+)$, then
$$
g_1=g_2 \text{ in } \Omega\quad \text{ and }\quad b_1=b_2 \text{ in } Q\times \mathbb{R}.
$$
\end{itemize}
\end{thm}
Theorem~\ref{Main Thm:Simultaneous} states that \textbf{Inverse Problem 3} can be solved under situations. In fact, we can determine $b(x,t,u)$ and $g(x)$ simultaneously by using active measurements with full data. Meanwhile, if we assume $b(x,t,u)$ is known a-priori in a small neighborhood of $\Sigma$, then we can also determine $b(x,t,u)$ and $g(x)$ simultaneously with partial measurements.
\begin{rmk}
We would like to point out that
\begin{itemize}
\item[(a)] The proof of Theorem \ref{Main Thm:Simultaneous} relies on the successive linearization method combining with suitable \emph{complex geometrical optics} (CGO) solutions (see \cite{CY2018logarithmic} or Appendix \ref{Sec: Appendix}) and useful approximation properties (see Section \ref{Sec 4}). We can simply utilize either full or partial DN maps for the semilinear equation \eqref{IBVP of simultaneous recovery} in order to determine both coefficients and initial data uniquely. Moreover, the smallness assumption for both initial and boundary data is needed due to the local well-posedness of the forward problem \eqref{IBVP of simultaneous recovery} (see Section \ref{Sec 2}), but not used to solve the inverse problem.
\item[(b)] In the statement (b) of Theorem \ref{Main Thm:Simultaneous}, the set $\Omega'$ can be chosen as $\Omega'=\Omega \setminus \overline{D}$ with $D\Subset \Omega$ such that $\Omega\setminus \overline{D}$ is connected. Moreover, such a set $D \Subset \Omega$ can be as large as possible so that the set $\Omega'$ is very ``thin". This means, for our partial data result, it is sufficient for us to know the coefficient near the boundary $\Gamma \times (0,T)$ a priori.
\end{itemize}
\end{rmk}
Finally, it would be interesting to consider the linear counterparts of the inverse problems studied in Theorem \ref{Main Thm:Simultaneous}, since to our best knowledge, the simultaneous recovery results are untouched in the literature even in the linear case, namely
$$
b(x,t,y)=q(x,t)y
$$
as a linear function with respect to $y\in \R$. For this linear model, the smallness conditions for initial and boundary data are not required, since the well-posedness for general linear parabolic equations have been well understood (for example, see \cite[Chapter 7]{evans1998partial} or \cite{ladyzhenskaia1988linear}). To proceed, let us consider the linear parabolic equation:
\begin{align}\label{eq:parabolic simul linear}
\begin{cases}
\p _t u - \Delta u +q _j u=0 &\text{ in }\ Q,\\
u= f &\text{ on }\ \Sigma,\\
u(x, 0)=g(x) &\text{ in }\ \Omega.
\end{cases}
\end{align}
In order to derive the well-posedness of strong solutions to \eqref{eq:parabolic simul linear}, we need to impose the following compatibility condition:
\begin{align}\label{compatibility conditions intro}
g(\cdot)=f(\cdot, 0)\quad\mbox{ on } \Gamma.
\end{align}
Via the condition \eqref{compatibility conditions intro}, one has the well-posedness of \eqref{eq:parabolic simul linear} immediately (see \cite[Chapter 7]{evans1998partial}). Therefore, one is able to define the corresponding partial DN map
\begin{align}\label{eq:DN linear}
\begin{split}
\Lambda^{\mathrm{P}}_{q,g}: C^{2+\alpha,1+\alpha/2}_0(\mathcal{V}_+) & \rightarrow C^{1+\alpha,1+\alpha/2}(\mathcal{V}_{-}) , \\
f&\mapsto \p_\nu u \Big|_{\mathcal{V}_-},
\end{split}
\end{align}
and the (full) DN map
\begin{align}\label{eq:DN linear_full}
\begin{split}
\Lambda_{q,g}: C^{2+\alpha,1+\alpha/2}_0(\Sigma) & \rightarrow C^{1+\alpha,1+\alpha/2}(\Sigma) , \\
f&\mapsto \p_\nu u \Big|_{\Sigma}.
\end{split}
\end{align}
Now, the inverse problem is to determine $q(x,t)$ and $g$ by using the measurements either $\Lambda^{\mathrm{P}}_{q,g}$ or $ \Lambda_{q,g}$. The last main unique recovery result is stated as follows.
\begin{thm}[Simultaneous recovery for linear parabolic equations]\label{Main Thm:Simultaneous linear}
Let $\Omega\subset \R^n$ be a bounded domain with $C^\infty$-smooth boundary $\Gamma$.
For any $q_j\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$ and $g_j \in C^{2+\alpha}_0(\Omega)$,
suppose $\Lambda_{q_j,g_j}$ and $\Lambda^{\mathrm{P}}_{q_j,g_j}$ are the full and partial DN maps of the linear parabolic equation:
\begin{align}\label{IBVP of simultaneous recovery linear}
\begin{cases}
(\p_t-\Delta +q_j) u=0 &\text{ in } Q,\\
u= f &\text{ on }\Sigma,\\
u(x, 0)=g_j(x), &\text{ in } \Omega,
\end{cases}
\end{align}
for $j=1,2$, respectively. Then we have
\begin{itemize}
\item[(a)] (Full data) If
\[
\Lambda_{q_1,g_1}(f)=\Lambda_{q_2,g_2}(f) \text{ on }\Sigma,
\]
for any $f\in C^{2+\alpha, 1+\alpha/2}_0(\Sigma)$, then
$$
g_1=g_2 \text{ in } \Omega\quad \text{ and }\quad q_1=q_2 \text{ in }Q.
$$
\item[(b)] (Partial data) Given an open connected subset $\Omega'\subset \Omega$ satisfying $\Gamma \subset \p \Omega'$, we assume that $q_1=q_2$ in $\Omega'\times(0,T)$, where $\Omega'$ is a connected open subset of $\Omega$ such that $\p\Omega\subset\p\Omega'$.
and
$$\Lambda^{\mathrm{P}}_{ q_1,g_1}(f)=\Lambda^{\mathrm{P}}_{ q_2,g_2}(f) \text{ on }\mathcal{V}_-, $$
for any $f \in C^{2+\alpha, 1+\alpha/2}_0 (\mathcal{V}_+),$ then
$$
g_1=g_2 \text{ in } \Omega\quad \text{ and }\quad q_1=q_2 \text{ in }Q.
$$
\end{itemize}
\end{thm}
It is noted that when the initial data $g_1 =g_2=0$ in $\Omega$, the logarithmic stability result for two potentials of the inverse problem associated with the linear parabolic equation with partial data has been investigated in \cite{CY2018logarithmic}.
\subsection{Background and discussion}
In this paper, we are interested in the study of inverse problems for semilinear parabolic equations. A classical result of inverse boundary value problems for semilinear parabolic equations was proposed by Isakov \cite{isakov1993uniqueness_parabolic}, where a first-order linearization technique was exploited to reduce the inverse problem associated with the nonlinear equation into its counterpart associated with a linear equation. Then one can apply some existing results for the linear equations to investigate related inverse problems for the nonlinear equations. In addition, one can also consider the second-order linearization method, which has been successfully adapted in solving some related inverse problems; see \cite{AYT2017direct,CNV2019reconstruction,KN002,sun1996quasilinear,sun1997inverse} and the references cited therein.
In recent years, various inverse problems for nonlinear hyperbolic equations have been proposed and studied. Some works mentioned above are based on solution properties to inverse problems associated with the linearized equations. It turns out that in the inverse problem study associated with nonlinear hyperbolic equations, one finds that the nonlinear interactions bring more information which enables to solve some inverse problems that are still unsolved in the setting associated with linear equations. In \cite{kurylev2018inverse}, the authors investigated inverse problems for hyperbolic equations with a quadratic nonlinearity on a globally hyperbolic $4$-dimensional Lorentzian manifold. For more related works of inverse problems for nonlinear hyperbolic equations, we refer readers to \cite{lassas2017determination,lassas2018inverse,chen2019detection,de2018nonlinear,kurylev2014einstein,wang2016quadartic,LLPT2020uniqueness,LLPT2021stability,LLL2021determining} and references cited therein.
In addition, inverse problems for semilinear elliptic equations have been attracted a lot of attentions in recent years. By utilizing high order linearization approach, it is possible to solve several inverse problems for local and nonlocal nonlinear elliptic equations, and we refer readers to \cite{LLLS2019nonlinear,FO2019semilinear,LLLS2019partial,LLST2020inverse,LL2020inverse,lai2019global,lin2020monotonicity,LZ2020partial,KU2019derivative_partial,KU2019remark,kian2020partial,carstea2020recovery,carstea2020inverse} for more detailed discussions.
The study of inverse problems on simultaneously recovering an unknown source and its surrounding inhomogeneous medium has also received considerable attentions recently in the literature due to its connection to many cutting-edge applications, including the photo- and thermo-acoustic tomography \cite{A1}, magnetic anomaly detection via the geomagnetic monitoring \cite{A3,A4} and quantum mechanics \cite{A5,A6}. Here, in the setup described in the previous section, say e.g. in \eqref{eq:parabolic simul}, the initial data $g$ and $b^{(0)}$ for $b$ in \eqref{eq:source1} represent the source terms, whereas the other terms in \eqref{eq:source1} of $b$ represent the medium effects. In \cite{LLL2021determining}, the simultaneous recovery for inverse problems associated with semilinear hyperbolic systems with unknown sources and nonlinearities was studied. In this paper, we consider the simultaneous recovery for inverse problems associated with semilinear parabolic systems. It is remarked that we develop new strategies which enable us to deal with more general source and medium configurations in the semilinear parabolic setup than the semilinear hyperbolic case. Finally, we would like to mention in passing some related physical applications that can be described by the semilinear parabolic systems in our study, including the heat diffusion \cite{ganji2018nonlinear}, mean-field game theory \cite{cardaliaguet2010notes,gueant2011mean} and phase field theory \cite{boettinger2002phase,karma2001phase}. The inverse problems proposed and studied in this paper can be connected to those practical applications.
The rest of the paper is structured as follows. In Section \ref{Sec 2}, we study the well-posedness of the initial boundary value problems for the semilinear parabolic equations under suitable assumptions. In Section \ref{Sec 3}, we establish the conditional stability estimates, and show the unique determination by utilizing either passive or active measurements. We prove Theorems \ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear} in Section \ref{Sec 4}. Finally, for the sake of completeness, we review some basic properties including CGO solutions and weak maximum principle for linear parabolic equations.
\section{Well-posedness of the forward problems}\label{Sec 2}
This section is devoted to studying
the local and global well-posedness for initial-boundary value problems of semilinear parabolic equations, respectively. Let us consider the following semilinear parabolic equation:
\begin{align}\label{eq:parabolic simul_forward}
\begin{cases}
u_{t}- \nabla\cdot(\tilde\gamma\nabla u) +b(x,t,u)=0 &\text{ in }\ Q,\\
u=\tilde f &\text{ on }\ \Sigma,\\
u(x, 0)=\tilde g(x) &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $\tilde\gamma$ is symmetric and uniformly positive definite with $\tilde\gamma
\in C^{1+\alpha, \alpha/2}(\overline{Q}; \mathbb R^{n\times n})$,
$\tilde g\in C^{2+\alpha}(\overline\Omega)$, $\tilde f\in C^{2+\alpha, 1+\alpha/2}(\overline\Sigma)$ for $\alpha\in (0, 1)$,
and $b$ satisfies the following conditions:
\begin{equation}\label{e9}
b\in C^2(\overline{Q}\times\mathbb R)\quad\mbox{and}\quad b(\cdot, \cdot, 0)=0 \ \mbox{in }Q.
\end{equation}
As a preliminary, we recall the well-posedness result
and the Schauder estimate for linear parabolic equations, which can be found in \cite{ladyzhenskaia1988linear}.
\begin{lem}\label{wellpose1}
For $\alpha\in (0, 1)$, assume that $\tilde\gamma
\in C^{1+\alpha, \alpha/2}(\overline{Q}; \mathbb R^{n\times n})$ and $q\in C^{\alpha,\alpha/2}(\overline Q)$.
For any $\tilde g\in C^{2+\alpha}(\overline\Omega)$, $\tilde f\in C^{2+\alpha, 1+\alpha/2}(\overline\Sigma)$
and $h\in C^{\alpha,\alpha/2}(\overline Q)$ with the compatibility conditions:
\begin{align}\label{compatibility conditions}
\tilde g(x)=\tilde f(x, 0)\ \mbox{and}\
\tilde f_t(x, 0)=\nabla\cdot(\tilde \gamma(x, 0)\nabla \tilde g(x))-q(x, 0)\tilde g(x)+h(x, 0) \ \mbox{ on }\Gamma,
\end{align}
the following linear parabolic equation:
\begin{align}
\begin{cases}
u_{t}- \nabla\cdot(\tilde\gamma \nabla u) +qu= h &\text{ in }\ Q,\\
u=\tilde f &\text{ on }\ \Sigma,\\
u(x, 0)=\tilde g(x) &\text{ in }\ \Omega,
\end{cases}
\end{align}
admits a unique solution $u\in C^{2+\alpha, 1+\alpha/2}(\overline{Q})$. Moreover,
$$\norm{u}_{C^{2+\alpha, 1+\alpha/2}(\overline{Q})}\leq
C\Big(\norm{\tilde f}_{C^{2+\alpha, 1+\alpha/2}(\overline\Sigma)}
+\norm{\tilde g}_{C^{2+\alpha}(\overline\Omega)}
+\|h\|_{C^{\alpha, \alpha/2}(\overline Q)}\Big).$$
\end{lem}
Note that, if $h=0$ in $Q$, $\tilde g\in C^{2+\alpha}(\overline\Omega)$
with $\tilde g=\tilde g_{x_i}=\tilde g_{x_i x_j}=0\ (i, j=1, \cdots, n)$ on $\Gamma$ and $\tilde
f\in C^{2+\alpha, 1+\alpha/2}(\overline\Sigma)$ with $\tilde f(x, 0)=\tilde f_t(x, 0)=0$ on $\Gamma$, it is straightforward
to verify that the compatibility condition \eqref{compatibility conditions} holds.
By Lemma \ref{wellpose1} and the fixed-point method, we have the following local well-posedness for (\ref{eq:parabolic simul_forward}).
\begin{thm}[Local well-posedness]\label{Thm:local well-posedness}
Assume that $\tilde\gamma
\in C^{1+\alpha, \alpha/2}(\overline{Q}; \mathbb R^{n\times n})$ and
$b$ satisfies the condition $(\ref{e9})$. Then there exists a positive constant $\delta$,
such that for any $(\tilde f, \tilde g)\in V_\delta$, the equation \eqref{eq:parabolic simul_forward} has a unique solution $u\in C^{2+\alpha, 1+\alpha/2}(\overline{Q})$,
where
\begin{eqnarray*}
&&V_\delta=\Big\{ (\tilde f, \tilde g)\in C^{2+\alpha, 1+\alpha/2}(\overline{\Sigma}) \times
C^{2+\alpha}(\overline\Omega)
\ \Big|\ \tilde f(x, 0)=\tilde f_t(x, 0)=0 \mbox{ on }\Gamma,\\
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad \quad\quad \quad\quad\quad \quad\quad\quad\tilde g=\tilde g_{x_i}=\tilde g_{x_i x_j}=0, \
i, j=1, \cdots, n \mbox{ on }\Gamma,\\
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad \quad\quad\quad
\quad\quad\quad\mbox{and }\norm{\tilde f}
_{C^{2+\alpha, 1+\alpha/2}(\overline{\Sigma})}+\norm{\tilde g}_{C^{2+\alpha}(\overline\Omega)}\leq \delta \Big\}.
\end{eqnarray*}
\end{thm}
\begin{proof}
The proof can be accomplished by the fixed-point technique. First, we set
$$
K=\Big\{ z\in C^{\alpha, \alpha/2}(\overline{Q}) \ \Big|\ \norm{z}_{C^{\alpha, \alpha/2}(\overline{Q})}\leq 1,
z(\cdot, 0)=\tilde g \mbox{ in }\Omega \mbox{ and }z=\tilde f \mbox{ on }\Sigma \Big\},
$$
where $(\tilde f, \tilde g)\in V_\delta$ for a sufficiently small $\delta>0$. It is straightfoward to show that $K$ is a nonempty convex and compact
subset in $L^2(Q)$. Also, we define
\begin{equation*}
q(x,t,s):=
\begin{cases}
\ \dfrac{b(x,t,s)}{s} &\text{for }s\neq0,\\
\ b_s(x,t,0) &\text{for }s=0.
\end{cases}
\end{equation*}
For any $z\in K$, let us consider the following linear parabolic equation:
\begin{eqnarray}\label{e10}
\begin{cases}
u_t-\nabla\cdot(\tilde\gamma\nabla u)+q_z(x, t)u=0 &\text{ in }\ Q,\\
u=\tilde f &\text{ on }\ \Sigma, \\
u(x,0)=\tilde g(x) &\text{ in }\ \Omega,
\end{cases}
\end{eqnarray}
where $q_z(x, t)=q(x, t, z(x, t))$.
Define the following map:
$$
\Psi(z)=u, \quad\forall \ z\in K,
$$
where $u$ is the solution to (\ref{e10}) associated to $q_z$.
By Lemma \ref{wellpose1}, it follows that $u\in C^{2+\alpha, 1+\frac{\alpha}{2}}(\overline{Q})$. Moreover,
$$\norm{u}_{C^{2+\alpha, 1+\alpha/2}(\overline{Q})}\leq
C(\tilde\gamma, b, n, \Omega, T)\Big(\norm{\tilde f}_{C^{2+\alpha, 1+\alpha/2}(\overline\Sigma)}
+\norm{\tilde g}_{C^{2+\alpha}(\overline\Omega)}
\Big)\leq C(\tilde\gamma, b, n, \Omega, T)\delta,$$
where $C(\tilde\gamma, b, n, \Omega, T)$ denotes a positive constant, depending only on
$\tilde\gamma$, $b$, $n$, $\Omega$ and $T$.
Hence, when $\delta$ is sufficiently small, such that $\|u\|_{C^{2+\alpha, 1+\alpha/2}(\overline{Q})}\leq 1$,
it holds that $\Psi(K)\subset K$. By the Schauder fixed-point theorem, it is ready to show that $\Psi$
has a fixed point in $K$, which is the solution to \eqref{eq:parabolic simul_forward}.
The proof is complete.
\end{proof}
\begin{rmk}\label{remark 2.2}
Regarding the local well-posedness, we make several remarks.
\begin{itemize}
\item[(a)] The condition \eqref{e9} on $b=b(x,t,y)$ is not essential and it is
for convenience to express compatibility conditions.
Also, the admissible condition on $b(x,t,y)$ is not used in the proof of the local well-posedness, but it will be utilized in the proof of our simultaneously recovering inverse problem.
\item[(b)] In \cite{isakov1993uniqueness_parabolic}, it is assumed that the coefficient $b=b(x,y)$, which is independent of $t$ and the condition $\p_yb(x,y)\geq 0$ for any $y\in \R$. In contrast, we provide different time-dependent nonlinearities, and utilize different techniques to study related inverse problems for semilinear parabolic equations.
\item[(c)] In order to apply the higher order linearization method, we need the infinite differentiability of the equation with respect to the given lateral boundary data $f$, which can be shown by applying the implicit function theorem of Banach spaces. To see this, let us define the following spaces.
Set
\begin{eqnarray*}
&&X_1=\Big\{ (f, g)\in C^{2+\alpha, 1+\alpha/2}(\overline{\Sigma}) \!\times\! C^{2+\alpha}(\overline\Omega)
\ \Big|\ f(x, 0)\Big|_{\Gamma}\!=\!f_t(x, 0)\Big|_{\Gamma}\!=\!0,\\
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad g\Big|_{\Gamma}=g_{x_i}\Big|_{\Gamma}=g_{x_i x_j}
\Big|_{\Gamma}=0, i, j=1, \cdots, n \Big\},\\
&&X_2=\Big\{ u\in C^{2+\alpha, 1+\alpha/2}(\overline{Q}) \ \Big|\
u(x, 0)\Big|_{\Gamma}\!=\!u_t(x, 0)\Big|_{\Gamma}\!=\!0,\\
&&\quad\quad\quad\quad\quad u(x, 0)\Big|_{\Gamma}=u_{x_i}(x, 0)\Big|_{\Gamma}=u_{x_i x_j}(x, 0)
\Big|_{\Gamma}=0, i, j=1, \cdots, n,\\
&&\quad\quad\quad\quad\quad \Big[u_{t}(x, 0)- \nabla\cdot(\gamma(x, 0)\nabla u(x, 0))\Big]\Big|_{\Gamma}=0
\Big\},\\
&&\mbox{and }
X_3=\Big\{ h\in C^{\alpha, \alpha/2}(\overline{Q})\ \Big|\ h(x, 0)\Big|_\Gamma=0 \Big\}\times X_1.
\end{eqnarray*}
We can consider the map
$\mathcal G: X_1\times X_2\to X_3$ by
$$\mathcal G(f,g,u)=\Big( u_t-\nabla\cdot(\gamma\nabla u)+b(x,t,u),u\Big|_{\Sigma}-f,u(x,0)-g \Big).$$
Then $\mathcal G(0,0,0)=0$ and $\mathcal G_{u}(0,0,0): X_2\to X_3$ is given by
$$\mathcal G_u(0,0,0)v=\Big( v_t-\nabla\cdot(\gamma\nabla v)+b_u(\cdot,\cdot,0)v,v\Big|_{\Sigma},v(x,0)\Big).$$
It is straightforward to show that $\mathcal G_u(0, 0, 0)$ is a linear isomorphism from $X_2$ to $X_3$ by Lemma $\ref{wellpose1}$.
By the implicit function theorem in Banach spaces, there exists a positive constant $\delta$, and a holomorphic map
$S: V_\delta\to C^{2+\alpha, 1+\alpha/2}(\overline{Q})$, such that
for any $(f,g)\in V_\delta$, we have $G(f,g,S(f,g))=0.$ Set $u=S(f,g)$ and this implies the local well-posedness of \eqref{eq:parabolic simul_forward}.
\item[(d)] Notice that the maps of boundary data to the solution is $C^\infty$-Fr\'echet differentiable, then we can also derive the corresponding DN map is also $C^\infty$-Fr\'echet differentiable.
\end{itemize}
\end{rmk}
Next, for a different nonlinearity, let us consider the global well-posedness of strong solutions to the semilinear parabolic equation:
\begin{equation}\label{e1}
\begin{cases}
u_t-\nabla\cdot(\gamma\nabla u)+a(x,t,u)=0 &\text{ in }\ Q,\\
u=f &\text{ on }\ \Sigma,\\
u(x,0)=g(x) &\text{ in }\ \Omega,
\end{cases}
\end{equation}
where $\gamma$ is symmetric and uniformly positive definite with $\gamma\in C^{1, 0}(\overline{Q}; \mathbb R^{n\times n})$, $g\in H_0^1(\Omega)$,
$f\in H^{\frac{3}{2},\frac{3}{4}}(\Sigma)$ with $f(\cdot, 0)=0$ on $\Gamma$,
and $a: Q\times\mathbb R\rightarrow\mathbb R$ satisfies
\begin{equation}\label{e2}
a(\cdot, \cdot, 0)\in L^2(Q),\quad a(x, t, \cdot)\in C^1(\mathbb R)
\end{equation} and the increasing condition (\ref{condition of nonlinear f at infinity data}).
The global well-posedness result of (\ref{e1}) is stated as follows.
\begin{thm}[Global well-posedness] Assume that $a$ satisfies $(\ref{e2})$ and $(\ref{condition of nonlinear f at infinity data})$.
Then for any $g\in H_0^1(\Omega)$ and
$f\in H^{\frac{3}{2},\frac{3}{4}}(\Sigma)$ with $f(\cdot, 0)=0$ on $\Gamma$, the semilinear parabolic equation $(\ref{e1})$ admits a unique strong solution $u\in H^{2, 1}(Q)$.
\end{thm}
\begin{proof}
First, let us set
\begin{equation*}
q(x,t,s):=
\begin{cases}
\ \dfrac{a(x,t,s)-a(x,t,0)}{s} &\text{for }s\neq0,\\
\ a_s(x,t,0) &\text{for }s=0.
\end{cases}
\end{equation*}
For any $z\in L^{2}(Q)$, consider the following linear parabolic equation:
\begin{eqnarray}\label{e4}
\begin{cases}
u_t-\nabla\cdot(\gamma\nabla u)+a_z(x,t)u+a(x,t,0)=0 &\text{ in }\ Q,\\
u=f &\text{ on }\ \Sigma, \\
u(x,0)=g(x) &\text{ in }\ \Omega,
\end{cases}
\end{eqnarray}
where $a_z(x,t)=q(x,t,z(x,t)).$ By the condition (\ref{condition of nonlinear f at infinity data}), we have that $a_z(\cdot, \cdot)\in L^{n+2}(Q).$
Indeed, there exist positive constants, denoted by $C$, which
may be different in one place or another, such that
\begin{align}\label{e8}
\begin{split}
&\displaystyle\int_Q |a_z(x, t)|^{n+2}\, dxdt
=\int^T_0\norm{a_z(\cdot, t)}^{n+2}_{L^{n+2}(\Omega)}\, dt\\
\displaystyle\leq &C\!+\!C\int^T_0 e^{\norm{a_z(\cdot, t)}^2_{L^{n+2}(\Omega)}}\, dt
\displaystyle
=\!C\!+\!C\int^T_0\sum\limits_{j=0}^{\infty} \frac{1}{j!} \norm{a_z(\cdot, t)}_{L^{n+2}(\Omega)}^{2j} \, dt\\
\displaystyle=&C+\!C\!\int^T_0\sum\limits_{j=0}^{n+2} \frac{1}{j!} \norm{a_z(\cdot, t)}_{L^{n+2}(\Omega)}^{2j} \, dt+
C\!\int^T_0\sum\limits_{j=n+3}^{\infty} \frac{1}{j!} \norm{a_z(\cdot, t)}_{L^{n+2}(\Omega)}^{2j} \, dt\\
\displaystyle\leq & C+C\int^T_0\sum\limits_{j=n+3}^{\infty} \frac{1}{j!} \norm{a_z(\cdot, t)}_{L^{n+2}(\Omega)}^{2j} \, dt\\
\displaystyle= &
C+C\int^T_0\sum\limits_{j=n+3}^{\infty} \frac{1}{j!} \left(\int_\Omega |a_z(x, t)|^{n+2}dx\right)^{\frac{2j}{n+2}}\, dt\\
\displaystyle\leq &C\!+\!C\int^T_0\sum\limits_{j=n+3}^{\infty} \frac{C^j}{j!} \int_\Omega |a_z(x, t)|^{2j}\, dx dt
\leq C\!+\!C\int_Q e^{C|a_z(x, t)|^2}dxdt.
\end{split}
\end{align}
By the condition (\ref{condition of nonlinear f at infinity data}), for any $\epsilon>0$, there always is a positive constant $C_\epsilon$, such that
for any $z\in L^2(Q)$, it holds that
$$
|a_s(x, t, z(x, t))|^2\leq \epsilon \mbox{ln}|z(x, t)|+C_\epsilon.
$$
Hence, for a sufficient small $\epsilon$,
\begin{align}\label{e5}
\begin{split}
& \displaystyle\int_Q e^{C|a_z(x, t)|^2}\, dxdt\leq
\int_Q e^{C[\epsilon\mbox{ln}(1+|z(x, t)|)+C_\epsilon]}\, dxdt\\
\displaystyle\leq &C\int_Q (1+|z(x, t)|)^{C\epsilon}\, dxdt
\leq C\left( 1+\norm{z}^2_{L^2(Q)}\right).
\end{split}
\end{align}
(\ref{e8}) and (\ref{e5}) imply that $a_z\in L^{n+2}(Q)$.
\smallskip
By \cite{ladyzhenskaia1988linear}, the linear parabolic equation (\ref{e4}) admits a unique strong solution
$u\in H^{2, 1}(Q)$. Moreover, by the energy estimate, it is easy to find that
\begin{align}\label{e6}
\begin{split}
&\displaystyle\norm{u}^2_{L^2(0, T; H^1(\Omega))\cap C([0, T]; L^2(\Omega))}+
\norm{u_t}^2_{L^2(0, T; H^{-1}(\Omega))}\\
\displaystyle\leq & Ce^{T\norm{a_z}^2_{L^{n+2}(Q)}}\left(\norm{a(\cdot, \cdot, 0)}_{L^2(Q)}^2+
\norm{g}^2_{L^2(\Omega)}+\norm{f}^2_{H^{1, 0}(\Sigma)}\right).
\end{split}
\end{align}
Define the following map:
$$\mathcal G: L^{2}(Q)\to L^{2}(Q)$$ by
$$\mathcal G(z)=u,$$
where $u$ is the solution to the equation (\ref{e4}) associated to $a_z$.
Obviously, $\mathcal{G}$ is well-posed and compact.
Define
$$
V=\Big\{ z\in L^2(Q)\ \Big|\ \norm{z}_{L^2(Q)}\leq C^* \Big\},
$$
where $C^*$ will be specified later. By (\ref{e8})-(\ref{e6}),
$$
\norm{u}_{L^2(Q)}^2
\leq C\left(\norm{a(\cdot, \cdot, 0)}_{L^2(Q)}^2+
\norm{g}^2_{L^2(\Omega)}+\norm{f}^2_{H^{1, 0}(\Sigma)}\right)\left(1+\norm{z}_{L^2(Q)}\right).
$$
Indeed, we may choose $\epsilon=1/C$ in (\ref{e5}).
It follows that there exists a $C^*>0$, such that
$\mathcal{G}(V)\subseteq V$. By the Schauder fixed point theorem, it is easy to check that
$\mathcal{G}$ has a fixed point in $V$, which is the solution to (\ref{e1}) in $H^{2, 1}(Q)$.
\end{proof}
\section{Unique determination of initial data}\label{Sec 3}
In this section, we present proofs of Theorems~\ref{Main Thm 1} and \ref{Main Thm 2} concerning the first two inverse problems of this paper.
\subsection{Determination by passive measurement}
In order to prove Theorem \ref{Main Thm 1},
we first present two Carleman estimates for the following linear parabolic equation:
\begin{eqnarray}\label{l1}
\begin{cases}
u_t-\nabla\cdot(\gamma\nabla u)+A(x,t)u=F(x, t) &\text{ in }\ Q,\\
u=0 &\text{ on }\ \Sigma, \\
u(x,0)=g(x) &\text{ in }\ \Omega,
\end{cases}
\end{eqnarray}
where $\gamma$ is the same as the one in (\ref{eq:parabolic1}), $A\in L^\infty(0, T; L^{2n}(\Omega))$, $F\in L^2(Q)$ and $g\in H^1_0(\Omega)$.
As preliminaries, for two parameters $\lambda, \mu\geq 1$, we introduce the following functions:
$$
\eta(x, t)=\displaystyle\frac{e^{\mu \psi(x)}-e^{2\mu\norm{\psi}_{C(\overline\Omega)}}}{t^2(T-t)^2},
\quad \varphi(x, t)=\displaystyle\frac{e^{\mu\psi(x)}}{t^2(T-t)^2} \quad \mbox{ and } \quad
\theta_1(x, t)=e^{\lambda \eta(x, t)},
$$
where $\psi(\cdot)\in C^4(\overline{\Omega})$ satisfies that $\psi(x)>0$ in $\Omega$,
$|\nabla \psi(x)|>0$ in $\overline{\Omega}$ and
$$\sum\limits_{i, j=1}^n \gamma_{i j}\psi_{x_i}\nu_j\leq 0\quad\mbox{ on }\quad (\Gamma\setminus\Gamma_0)\times(0, T).$$
Also, for any $L>0$, there exist $t_0\in (0, T)$ and $K>0$, such that
$$
K+t_0<\min\left\{1, \frac{1}{2L}\right\}.
$$
Set $\theta_2(t)=\displaystyle\frac{1}{K+t_0-t}$ for $t\in [0, t_0]$.
The first Carleman estimate is stated as follows.
\begin{lem}\label{lemma1}
There exist positive constants $\lambda_0$, $\mu_0$ and $C$, such that for any $\lambda\geq \lambda_0$
and $\mu\geq \mu_0$, the following estimate holds for any solution to $(\ref{l1})$:
\begin{align}\label{l2}
\begin{split}
&\displaystyle\int_Q \theta_1^2\Big(
\lambda\mu^2\varphi|\nabla u|^2+\lambda^3\mu^4\varphi^3 u^2\Big)\,
dxdt\\
\leq & C\displaystyle\int_Q \theta_1^2F^2\, dxdt+
C\displaystyle\int^T_0\int_{\Gamma_0}
\theta_1^2\lambda\mu \varphi\left.|\p_\nu u \right|^2\, dSdt.
\end{split}
\end{align}
\end{lem}
\begin{proof}
The proof is inspired by \cite[Theorem 2.2]{yuan2017conditional}. In fact, when $A\equiv 0$, the estimate
(\ref{l2}) holds true for any solution to (\ref{l1}). If $A\in L^\infty(0, T; L^{2n}(\Omega))$,
we have that
\begin{align*}
\begin{split}
&\displaystyle\int_Q \theta_1^2\Big(
\lambda\mu^2\varphi|\nabla u|^2+\lambda^3\mu^4\varphi^3 u^2\Big)\,
dxdt\\
\leq &C\displaystyle\int_Q \theta_1^2(F-Au)^2\, dxdt+
C\displaystyle\int^T_0\int_{\Gamma_0} \theta_1^2 \lambda\mu
\varphi\left|\partial_\nu u\right|^2\, dSdt.
\end{split}
\end{align*}
Notice that when $n\geq 3$,
\begin{align*}
&\displaystyle\int_Q \theta_1^2A^2u^2\, dxdt
\leq \int^T_0\norm{A}_{L^{2n}(\Omega)}^2\norm{\theta_1 u}_{L^2(\Omega)}\norm{\theta_1 u}
_{L^{\frac{2n}{n-2}}(\Omega)}\, dt\\
\leq &C\int_0^T\Big( \norm{\theta_1 u}^2_{L^2(\Omega)}+
\norm{\nabla (\theta_1 u)}^2_{L^2(\Omega)}\Big) \, dt
\leq C\int_Q \theta_1^2\Big(|\nabla u|^2+\lambda^2\mu^2\varphi^2 u^2\Big)\, dxdt.
\end{align*}
When $n=1$ and $n=2$, the term $\norm{\theta_1 u}
_{L^{\frac{2n}{n-2}}(\Omega)}$ can be replaced by $\norm{\theta_1 u}
_{L^{\infty}(\Omega)}$ and $\norm{\theta_1 u}
_{L^{4}(\Omega)}$, respectively.
Hence, when $\mu_0$ is sufficiently large, (\ref{l2}) holds for any solution to (\ref{l1}).
\end{proof}
The second Carleman estimate is given as follows.
\begin{lem}\label{lemma2} Assume that $T\in (0, 1)$. Then there exists a positive constant $L_0$, such that
for any $L\geq L_0$, $t_0\in (0, T)$ and $K>0$ with
$$
K+t_0<\min\left\{1, \displaystyle\frac{1}{2L}\right\},
$$
one can always find positive constants $\lambda_0$ and $C$, so that for any $\lambda\geq \lambda_0$,
the following estimate holds for any solution to $(\ref{l1})$:
\begin{align}\label{l3}
\begin{split}
&\displaystyle\int_0^{t_0}\int_\Omega \theta_2^{2\lambda}\Big(
\lambda\theta_2^2u^2+L \sum\limits_{i, j=1}^n \gamma_{i j}u_{x_i}u_{x_j}\Big)\,
dxdt
+\displaystyle\int_\Omega \frac{\lambda}{(K+t_0)^{2\lambda+1}}u^2(x, 0)\, dx
\\
\leq &\displaystyle\int_\Omega\frac{\lambda}{K^{2\lambda+1}}u^2(x, t_0)dx
+\displaystyle\int_\Omega\frac{1}{(K+t_0)^{2\lambda}}\sum\limits_{i, j=1}^n\gamma_{i j}(x, 0)u_{x_i}(x, 0)u_{x_j}(x, 0)\, dx\\
&\quad
+C\displaystyle\int_0^{t_0}\int_\Omega \theta_2^{2\lambda} F^2 \, dxdt.
\end{split}
\end{align}
\end{lem}
\begin{proof}
The proof can be adapted from that of \cite[Theorem 2.4.1]{yu2021PhD}
for the Carleman estimate of stochastic degenerate parabolic
equations. We sketch the necessary modifications in what follows.
First, for any $\lambda\geq 1$, we set $z=\theta_2^\lambda(t) u$. Then it is straightforward to show that
\begin{align*}
&2\theta_2^\lambda \left[
-\lambda\theta_2 z-\sum\limits_{i, j=1}^n (\gamma_{i j}z_{x_i})_{x_j}\right]
\left[
u_t-\sum\limits_{i, j=1}^n (\gamma_{i j}u_{x_i})_{x_j}\right]\\
=&-(\lambda\theta_2 z^2)_t+\lambda \theta_2^2 z^2-
2\sum\limits_{i, j=1}^n(\gamma_{i j}z_{x_i}z_t)_{x_j}
+\sum\limits_{i, j=1}^n(\gamma_{i j}z_{x_i}z_{x_j})_{t}
-\sum\limits_{i, j=1}^n\gamma_{i j, t}z_{x_i}z_{x_j}\\
&\quad
+2\left[\lambda\theta_2 z+\sum\limits_{i, j=1}^n (\gamma_{i j}z_{x_i})_{x_j}\right]^2.
\end{align*}
Integrating the above equality on $\Omega\times (0, t_0)$, we obtain that
\begin{align}\label{l4}
\begin{split}
&\displaystyle\int_0^{t_0} \int_\Omega
\lambda \theta_2^2 z^2\, dxdt+
\int_\Omega\sum\limits_{i, j=1}^n \gamma_{i j}(x, t_0)z_{x_i}(x, t_0)z_{x_j}(x, t_0)\, dx
+\int_\Omega \lambda\theta_2(0) z^2(x, 0)\, dx\\
\leq & \displaystyle\int_\Omega
\sum\limits_{i, j=1}^n \gamma_{i j}(x, 0)z_{x_i}(x, 0)z_{x_j}(x, 0)\, dx
+\int_\Omega \lambda\theta_2(t_0) z^2(x, t_0)\, dx\\
&\quad+\displaystyle\int_0^{t_0} \int_\Omega \left|\sum\limits_{i, j=1}^n
\gamma_{i j, t}z_{x_i}z_{x_j}\right|\, dxdt
+\int_0^{t_0} \int_\Omega \theta_2^{2\lambda} (Au+F)^2\, dxdt.
\end{split}
\end{align}
On the other hand, it is noticed that
$$
\displaystyle 2\theta_2^{2\lambda} u\left[
u_t-\sum\limits_{i, j=1}^n(\gamma_{i j}u_{x_i})_{x_j}
\right]=(\theta_2^{2\lambda}u^2)_t
-2\sum\limits_{i, j=1}^n(\gamma_{i j}z_{x_i}z)_{x_j}
-2\lambda\theta_2^{2\lambda+1}u^2
+2\sum\limits_{i, j=1}^n \gamma_{i j}u_{x_i}u_{x_j}.
$$
This implies that for any $L>0$,
\begin{align}\label{l5}
\begin{split}
&\displaystyle 2L\int^{t_0}_0\int_\Omega
\sum\limits_{i, j=1}^n
\gamma_{i j}z_{x_i}z_{x_j}\, dxdt
+L\int_\Omega
\theta_2^{2\lambda}(t_0)u^2(x, t_0)\, dx\\
\leq &\displaystyle
2L\lambda\int^{t_0}_0\int_\Omega
\theta_2^{2\lambda+1} u^2\, dxdt+
L\int_\Omega \theta_2^{2\lambda}(0)u^2(x, 0)\, dx
+2L\int^{t_0}_0\int_\Omega
\theta_2^{2\lambda}u(Au+F)\, dxdt.
\end{split}
\end{align}
By (\ref{l4}), (\ref{l5}) and the definition of $\theta_2$, it follows that
\begin{align*}
&\int_0^{t_0}\int_\Omega
\Big(\lambda\theta_2^{2\lambda+2}u^2+
2L\theta_2^{2\lambda}
\sum\limits_{i, j=1}^{n} \gamma_{i j} u_{x_i}u_{x_j}
\Big)\, dxdt\\
&\quad+\int_\Omega \frac{\lambda}{(K+t_0)^{2\lambda+1}}u^2(x, 0)\, dx
+\int_\Omega\frac{1}{K^{2\lambda}}
\sum\limits_{i, j=1}^n \gamma_{i j}(x, t_0) u_{x_i}(x, t_0) u_{x_j}(x, t_0)\, dx\\
\leq & L\int_\Omega \frac{1}{(K+t_0)^{2\lambda}}u^2(x, 0)\, dx
+\int_\Omega \frac{1}{(K+t_0)^{2\lambda}}
\sum\limits_{i, j=1}^{n} \gamma_{i j}(x, 0) u_{x_i}(x, 0)u_{x_j}(x, 0)\, dx\\
&\quad+\int_\Omega \frac{\lambda}{K^{2\lambda+1}} u^2(x, t_0)\, dx+\int^{t_0}_0\int_\Omega\Big(
2L\lambda\theta_2^{2\lambda+1}u^2+
\Big|\sum\limits_{i, j=1}^n \gamma_{i j, t}z_{x_i}z_{x_j}\Big|\Big)\, dxdt\\
&\quad+\int^{t_0}_0\int_\Omega
\theta_2^{2\lambda}\Big[
Lu^2+(L+1)(Au+F)^2\Big]\, dxdt.
\end{align*}
Furthermore, we notice that $\theta_2(t)\geq\frac{1}{K+t_0}>2L$. Also, for any $\epsilon>0$,
\begin{align*}
&\int_0^{t_0} \int_\Omega \theta_2^{2\lambda} A^2u^2\, dxdt \\
\leq & \int^{t_0}_0 \theta_2^{2\lambda} \norm{A}^2_{L^{2n}(\Omega)}
\norm{u}_{L^2(\Omega)}\norm{u}_{L^{\frac{2n}{n-2}}(\Omega)}\, dt\\
\leq &\epsilon \int_0^{t_0} \int_\Omega \theta_2^{2\lambda}|\nabla u|^2 \, dxdt+
C\int_0^{t_0} \int_\Omega\theta_2^{2\lambda} u^2 \, dxdt.
\end{align*}
Hence,
for sufficiently large $L$ and $\lambda$,
it follows that
\begin{align*}
&\int_0^{t_0}\int_\Omega
\Big(\lambda\theta_2^{2\lambda+2}u^2+
2L\theta_2^{2\lambda}
\sum\limits_{i, j=1}^{n} \gamma_{i j} u_{x_i}u_{x_j}
\Big)\, dxdt+\int_\Omega \frac{\lambda}{(K+t_0)^{2\lambda+1}}u^2(x, 0)\, dx\\
\leq &\int_\Omega \frac{1}{(K+t_0)^{2\lambda}}
\sum\limits_{i, j=1}^{n} \gamma_{i j}(x, 0) u_{x_i}(x, 0)u_{x_j}(x, 0)\, dx\\
&\quad+\int_\Omega \frac{\lambda}{K^{2\lambda+1}} u^2(x, t_0)\, dx
+C\int^{t_0}_0\int_\Omega
\theta_2^{2\lambda}F^2\, dxdt.
\end{align*}
This implies the desired estimate (\ref{l3}).
The proof is complete.
\end{proof}
Based on Lemmas \ref{lemma1} and \ref{lemma2},
one has the following conditional stability result for the inverse source problem of (\ref{l1}).
\begin{lem}\label{lemma3}
For any $M>0$, if
\[
\norm{g}_{H^1_0(\Omega)}+\norm{F}_{L^2(Q)}\leq M,
\]
there exist positive constants $C$ and $\delta_0\in (0, 1)$,
depending only on $n, T$ and $\Omega$, such that
the following estimate holds for any solution to $(\ref{l1})$:
\begin{equation}\label{l6}
\displaystyle\norm{u(\cdot, 0)}_{L^2(\Omega)}^2
\leq
\frac{C(M+1)}{\delta_0}
\norm{(F, \p_\nu u)}-
\frac{CM^2}{\ln[\delta_0 \norm{(F, \p_\nu u)}]},
\end{equation}
where $\norm{(F, \p_\nu u)}=\left(\norm{F}_{L^2(Q)}^2+
\norm{\partial_\nu u}^2_{L^2(\Gamma_0\times(0, T))}\right)^{1/2}.$
\end{lem}
\begin{proof}
Without loss of generality, we assume that $T<1$. For any $t_1\in
(0, T)\cap(0, \frac{2}{3})\cap(0, \frac{1}{3L})$ with $L$ being the constant in
Lemma \ref{lemma2},
choose $K=\frac{t_1}{2}$ and $t_0\in [\frac{t_1}{2}, t_1]$.
Then,
$$
\displaystyle K+t_0\leq \frac{3}{2}t_1<\min\left\{1, \frac{1}{2L}\right\}
$$
and
$$
\displaystyle \left( \frac{t_1+2t_0}{2}\right)^{-2\lambda}
=(K+t_0)^{-2\lambda} \leq \theta_2^{2\lambda}(t)
\leq \Big(\frac{2}{t_1}\Big)^{2\lambda}, \quad\mbox{for any }\lambda\geq \lambda_0
\mbox{ and }t\in [0, t_0].
$$
By Lemma \ref{lemma2},
\begin{align*}
\begin{split}
&\displaystyle\lambda\int_\Omega
\left( \frac{t_1+2t_0}{2}\right) ^{-2\lambda-1}u^2(x, 0) \, dx\\
\leq & \displaystyle C\int_\Omega
\left( \frac{t_1+2t_0}{2}\right)^{-2\lambda}|\nabla u(x, 0)|^2\, \, dx+C\lambda\Big(\frac{2}{t_1}\Big)^{2\lambda+1}
\left[\int_\Omega u^2(x, t_0)dx+\int_Q F^2(x, t)\, dxdt\right].
\end{split}
\end{align*}
This implies that
\begin{align}\label{l7}
\begin{split}
&\displaystyle\int_\Omega u^2(x, 0)\, dx\\
\leq &\displaystyle\frac{C}{\lambda}\int_\Omega |\nabla u(x, 0)|^2\, dx\\
&\quad+C\left( \frac{t_1+2t_0}{2}\right)^{2\lambda}
\Big(\frac{2}{t_1}\Big)^{2\lambda+1}
\left[\int_\Omega u^2(x, t_0)\, dx+\int_Q F^2(x, t)\, dxdt\right]\\
\leq &\displaystyle\frac{C}{\lambda}\int_\Omega |\nabla u(x, 0)|^2\, dx
+C9^\lambda \norm{(F, t_0)}^2,
\end{split}
\end{align}
where $\norm{(F, t_0)}^2:=\displaystyle\int_\Omega u^2(x, t_0)\, dx+\int_Q F^2(x, t)\, dxdt$.
On the other hand, by Lemma \ref{lemma1}, for fixed parameters $\lambda$ and $\mu$, it holds that
$$
\displaystyle\int^{t_0}_{\frac{t_0}{2}}
\int_\Omega \left( u^2+|\nabla u|^2\right) \, dxdt
\leq C\int_Q F^2dxdt+C\int^T_0\int_{\Gamma_0} \left|\partial_\nu u\right|^2\, dSdt.
$$
Hence, there exists a $\hat t\in (\frac{t_0}{2}, t_0)$, such that
$$
\displaystyle
\int_\Omega \left( u^2(x, \hat t)+|\nabla u(x, \hat t)|^2 \right) \, dx
\leq C\int_Q F^2dxdt+C\int^T_0\int_{\Gamma_0} \left|\partial_\nu u\right|^2\, dSdt.
$$
By the standard energy estimate,
\begin{align}\label{l8}
\begin{split}
&\displaystyle\int_\Omega u^2(x, t_0) \, dx \\
\leq & C\int_\Omega u^2(x, \hat t)\, dx+C\int^{t_0}_{\hat t}\int_\Omega(u^2+ F^2)\, dxdt\\
\displaystyle\leq & C \int_Q F^2\, dxdt+C\int^T_0\int_{\Gamma_0} \left|\partial_\nu u\right|^2\, dSdt.
\end{split}
\end{align}
By (\ref{l7}) and (\ref{l8}), it holds that
\begin{equation}\label{l9}
\int_\Omega u^2(x, 0)\, dx\leq \frac{C}{\lambda}\int_\Omega |\nabla u(x, 0)|^2\, dx+C9^\lambda
\norm{(F, u_\nu)}^2.
\end{equation}
Take
$$
\delta_0\in (0, e^{-\lambda_0\ln 9})\quad\mbox{ and }
\quad \lambda=\frac{1}{\ln 9}\ln \left( \frac{\norm{(F, u_\nu)}+1}{\delta_0 \norm{(F, u_\nu)}}\right),
$$
where $\lambda_0$ is the constant in Lemma \ref{lemma2}. Then, $\lambda\geq \lambda_0$.
Set
$$\hat{u}=\displaystyle\frac{\delta_0}{\norm{(F, u_\nu)}+1}u\quad
\mbox{ and }
\quad\hat F=\displaystyle\frac{\delta_0}{\norm{(F, u_\nu)}+1}F.
$$
Hence, by (\ref{l9}),
\begin{align*}
&\int_\Omega \hat u^2(x, 0)\, dx\\
\leq & \frac{C}{\lambda}\int_\Omega |\nabla \hat u(x, 0)|^2\, dx+
C9^\lambda \frac{\delta_0^2}{(\norm{(F, u_\nu)}+1)^2} \norm{(F, u_\nu)}^2\\
\leq & \frac{C}{\ln \left( \frac{\norm{(F, u_\nu)}+1}{\delta_0 \norm{(F, u_\nu)}}\right)}
\int_\Omega |\nabla \hat u(x, 0)|^2\, dx
+C\frac{\delta_0 \norm{(F, u_\nu)}}{\norm{(F, u_\nu)}+1}.
\end{align*}
It follows that
\begin{align*}
\int_\Omega u^2(x, 0)\, dx\leq
\frac{C}{\ln \left( \frac{\norm{(F, u_\nu)}+1}{\delta_0 \norm{(F, u_\nu)}}\right)}
\int_\Omega |\nabla u(x, 0)|^2\, dx
+C\frac{(\norm{(F, u_\nu)}+1)\norm{(F, u_\nu)}}{\delta_0}.
\end{align*}
For any $M>0$,
$$
\norm{(F, u_\nu)}\leq CM.
$$
Hence,
\begin{align*}
\int_\Omega u^2(x, 0)\, dx\leq
\frac{CM^2}{\ln \left( \frac{\norm{(F, u_\nu)}+1}{\delta_0 \norm{(F, u_\nu)}}\right)}
+C\frac{(M+1)\norm{(F, u_\nu)}}{\delta_0}.
\end{align*}
This implies the desired estimate (\ref{l6}).
\end{proof}
Now, we come back to the proof of Theorem \ref{Main Thm 1}.
\begin{proof}[Proof of Theorem $\ref{Main Thm 1}$]
For any $a\in \mathcal{A}_T$ and two initial values $g_1, g_2\in H^1_0(\Omega)$,
let $\widetilde u=u_1 -u_2$,
where $u_j$ $(j=1, 2)$ are the solutions to \eqref{IBVP for thm 1 for j=1,2}
associated to $g_j$. Then $\widetilde u\in H^{2, 1}(Q)$ is the solution to the
following parabolic equation:
\begin{align}\label{equ proof of thm 1 for j=1,2}
\begin{cases}
\widetilde u_{ t}-\nabla\cdot(\gamma\nabla \widetilde u)+A(x,t)\widetilde u=0 &\text{ in }\ Q,\\
\widetilde u=0 &\text{ on }\ \Sigma,\\
\widetilde u(x, 0)=g_1-g_2,
&\text{ in }\ \Omega,
\end{cases}
\end{align}
with
\begin{align*}
A(x,t)\widetilde u= a(x,t,u_1)-a(x,t,u_2) = \left( \int_0^1 a_u (x,t,su_1 + (1-s)u_2)\, ds \right)\cdot \widetilde u.
\end{align*}
with $A(x,t)= \displaystyle\int_0^1 a_u (x,t,su_1 + (1-s)u_2)\, ds$. Similar to \cite[Theorem 3.2]{LLL2021determining},
we can prove that $A\in L^\infty(0, T; L^{2n}(\Omega))$. By Lemma \ref{lemma3},
for any $M>0$, if
$\norm{g_1-g_2}_{H^1_0(\Omega)}\leq M, $
there exist positive constants $C$ and $\delta_0\in (0, 1)$,
depending only on $n, T$ and $\Omega$, such that
$$
\displaystyle\norm{\widetilde u(\cdot, 0)}_{L^2(\Omega)}^2
\leq
\frac{C(M+1)}{\delta_0}
\norm{\p_\nu \widetilde u }_{L^2(\Gamma_0\times(0, T))}-
\frac{CM^2}{\ln\left( \delta_0 \norm{\p_\nu \widetilde u}
_{L^2(\Gamma_0\times(0, T))}\right)}.
$$
This proves the desired estimate (\ref{Stability estimate in Thm 1}).
\end{proof}
Furthermore, there is a counterexample showing that if $a$ is unknown, the passive measurement cannot uniquely determine all unknowns.
\begin{thm}[Non-uniqueness]\label{thm:2}
Suppose that $\gamma=\left( \gamma^{ij}(x)\right)_{i,j=1}^n \in
C^{2, 1}(\overline{Q}; \R^{n\times n})$ is symmetric uniformly positive definite, $a_j\in\mathcal A_T$ and
$g_j\in H^1_0(\Omega)$ for $j=1, 2$. Denote by
$\Lambda_{a_j, g_j}^0$ the DN map of the following semilinear parabolic equation:
\begin{align}\label{llll}
\begin{cases}
u_{j, t}-\nabla\cdot(\gamma\nabla u_j)+a_j(x,t,u_j)=0 &\text{ in }\ Q,\\
u_j=0 &\text{ on }\ \Sigma,\\
u_j(x,0)=g_j(x), &\text{ in }\ \Omega.
\end{cases}
\end{align}
Then there exist two groups of unknown sources $(g_1, a_1), (g_2, a_2)
\in H^1_0(\Omega)\times\mathcal A_T$, such that
\[
(g_1, a_1) \neq (g_2, a_2),
\]
but
\[
\Lambda^0_{g_1, a_1}=\Lambda^0_{g_2,a_2}.
\]
\end{thm}
\begin{proof} Assume that two functions $u_1, u_2\in C^\infty(\overline{Q})$ satisfy that
\begin{eqnarray*}
&&u_1(\cdot, 0)\neq u_2(\cdot, 0) \mbox{ in a measurable set of }\Omega\mbox{ with positive measure}, \\[2mm]
&&\mbox{and}\ u_1(x, t)=u_2(x, t)=0 \mbox{ in } \Omega_\epsilon\times[0, T],
\end{eqnarray*}
where $\Omega_\epsilon=\Big\{ x\in \Omega\ \Big|\ \dist(x, \Gamma)<\epsilon \Big\}$.
Set
$$
A_j(x, t)=-u_{j, t}(x,t)+\nabla\cdot(\gamma\nabla u_j(x, t)), \quad \text{ for }j=1,2 \mbox{ and }(x, t)\in Q.
$$
It is easy to show that $u_j$ $(j=1, 2)$ are solutions to (\ref{llll}) associated to
\begin{align*}
&g_j(x)=u_j(x, 0)\quad\mbox{ and } \quad a_j(x, t, u_j)=A_j(x, t).
\end{align*}
Then,
\[
(g_1, a_1)\neq (g_2, a_2),
\]
but $$\partial_\nu u_{1}\Big|_{\Gamma_0\times(0, T)}=\Lambda^0_{g_1, a_1}=
\Lambda^0_{g_2,a_2}=\partial_\nu u_{2}\Big|_{\Gamma_0\times(0, T)}=0.$$
\end{proof}
\subsection{Determination by active measurements}
This subsection is devoted to proving Theorem \ref{Main Thm 2} under the condition that
$a\in \mathcal{B}_{T}$ (see (\ref{set C})).
\begin{proof}[Proof of Theorem $\ref{Main Thm 2}$]
For any $a_j\in \mathcal{B}_{T}$ $(j=1, 2)$,
$$a_j(x, t, y)=a_0(x, t, y)\chi_{[0, T-\epsilon]}(t)+c_j(x,t,y)\chi_{[T-\epsilon, T]}(t),
$$
where $\epsilon>0$, $a_0\in \mathcal{A}_T$ and
$c_1, c_2\in \mathcal{A}_T$ with $c_1(x, t, 0)=c_2(x, t, 0)=0$ in $Q$.
By the assumptions in Theorem $\ref{Main Thm 2}$,
for any
$g_j\in H_0^1(\Omega)$ $(j=1, 2)$, it holds that
\begin{equation}\label{eq:p1}
\Lambda_{g_1, a_1}(f)=\Lambda_{g_2, a_2}(f),\quad \text{ for any } f\in
\mathcal E
\mbox{ with }\supp f\subset \Gamma_0\times[0, T].
\end{equation}
Let $u_1$ and $u_2$ be, respectively, the solutions to (\ref{IBVP for thm 2 for j=1,2}) associated to the above
mentioned $(g_1,a_1, f)$ and $(g_2,a_2,f)$.
Set
\[
\widetilde u=u_1-u_2.
\]
It is straightforward to see that
\begin{align}\label{eq:wave2}
\begin{cases}
\widetilde u_{t}-\nabla\cdot(\gamma\nabla \widetilde u)+a_1(x,t,u_1(x, t))- a_2(x, t, u_2(x, t))=0 &\text{ in }\ Q,\\[2mm]
\widetilde u=0 &\text{ on }\ \Sigma,\\
\partial_\nu \widetilde u=0 &\text{ on }\ \Gamma_0\times(0, T),\\
\widetilde u(x, 0)=\widetilde g(x) &\text{ in }\ \Omega,
\end{cases}
\end{align}
where
$\widetilde g=g_1-g_2$.
Notice that
\begin{align*}
&a_1(x,t,u_1(x, t))-a_2(x, t, u_2(x, t))\\[2mm]
=&\Big[a_1(x,t,u_1(x, t))-a_1(x, t, u_2(x, t))\Big]
+\Big[a_1(x,t,u_2(x, t))-a_2(x, t, u_2(x, t))\Big] \\
=&\left( \displaystyle\int^1_0 a_{1, u}(x, t, su_1(x, t)+(1-s)u_2(x, t))\, ds\right)\cdot \widetilde u(x, t) \\[1mm]
&\quad+ a_1(x,t, u_2(x, t))- a_2(x, t, u_2(x, t)).
\end{align*}
By $a_1\in \mathcal{A}_{T}$,
$\displaystyle\int^1_0 f_{1, u}(x, t, su_1(x, t)+(1-s)u_2(x, t))\, ds\in L^\infty(0, T; L^{2n}(\Omega))$.
By Lemma \ref{lemma3}, one has that
\begin{align}\label{l13}
\begin{split}
\displaystyle\norm{g_1-g_2}^2_{L^2(\Omega)}
\displaystyle\leq &C(u_1, u_2)\norm{a_1(\cdot, \cdot, u_2(\cdot, \cdot))
-a_2(\cdot, \cdot, u_2(\cdot, \cdot))}_{L^2(Q)}\\
&\quad\displaystyle-\frac{C(u_1, u_2)}{\ln[\delta_0
\norm{a_1(\cdot, \cdot, u_2(\cdot, \cdot))-a_2(\cdot, \cdot, u_2(\cdot, \cdot))}_{L^2(Q)}]},
\end{split}
\end{align}
where $C(u_1, u_2)$ denotes a constant depending on $u_1$ and $u_2$.
Recall the controllability result for the following semilinear parabolic equation (see \cite{dzz}):
\begin{align}\label{l14}
\begin{cases}
u_{2, t}-\nabla\cdot(\gamma\nabla u_2)+a_2(x,t,u_2)=0 &\text{ in }\ \Omega\times(0, T-\epsilon),\\
u_2=f &\text{ on }\ \Gamma\times(0, T-\epsilon),\\
u_2(x, 0)=g_2(x) , &\text{ in }\ \Omega.
\end{cases}
\end{align}
For any $g_2\in L^2(\Omega)$,
there always exists an $f^*\in L^2(\Gamma\times(0, T-\epsilon))$ with $\supp f^*\subset
\Gamma_0\times(0, T-\epsilon]$,
such that the corresponding solution $u_2$ to (\ref{l14}) satisfies that
$u_2(x, T-\epsilon)=0$ in $\Omega$.
Choose
$$f(x, t)=
\begin{cases}
\displaystyle f^*(x, t) &\mbox{ on }\ \Gamma_0\times[0, T-\epsilon),\\
\displaystyle 0 &\mbox{ on }\ \Gamma_0\times[T-\epsilon, T].
\end{cases}$$
By the fact that $c_1(\cdot, \cdot, 0)=c_2(\cdot, \cdot, 0)\equiv 0$ in $Q$, the solution
$u_2=u_2(\cdot, \cdot; f)$ to (\ref{l14}) associated to the above
$f$ satisfies that
$$
u_2(x, t)\equiv 0\quad\mbox{ in }\ \Omega\times [T-\epsilon, T].
$$
Combining the above result with (\ref{l13}), one readily obtains
\[
g_1=g_2 \quad \mbox{ in }\ \Omega,
\]
which proves the assertion in Theorem $\ref{Main Thm 2}$.
\end{proof}
\section{Simultaneous recovery results for inverse problems}\label{Sec 4}
In this section, we present the proofs of Theorems~\ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear} on the simultaneous recovery results for the inverse problems. We first derive the unique determination of the coefficient for the linear parabolic equation.
To that end, let us prove some useful properties, which will be needed in the proofs of Theorems \ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear}.
\subsection{Approximation and denseness properties}
Let us begin with the Runge approximation properties for linear parabolic equations. The following approximation property will be used in the proof of Theorems \ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear} with full data.
\begin{lem}[Runge approximation with full data]\label{appro}
Let $q\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$. Then for any solutions $v_{\pm}\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ to
\begin{equation}
\begin{cases}\label{to App1}
\p_tv_+-\Delta v_+ +qv_+=0 &\text{ in }\ Q,\\
v_+(x,0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation} and
\begin{equation}\label{to App2}
\begin{cases}
-\p_tv_--\Delta v_- +qv_-=0 &\text{ in }\ Q,\\
v_-(x,T)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}
and any $\eta>0$, there exist solutions $V_{\pm}\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$ to
\begin{equation}\label{app1}
\begin{cases}
\p_tV_+-\Delta V_++ qV_+=0 &\text{ in }\ Q,\\
V_+(x,0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation} and
\begin{equation}\label{app2}
\begin{cases}
-\p_tV_--\Delta V_-+ qV_-=0 &\text{ in }\ Q,\\
V_-(x,T)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}
such that
$$\norm{V_{\pm}-v_{\pm}}_{L^2(Q)}<\eta.$$
\end{lem}
\begin{proof}
We only prove the case for the forward parabolic equation, and the backward one can be proved similarly. Define
$$X=\Big\{V\in C^{2+\alpha,1+\alpha/2}(\overline{Q})\,\Big|\,
V \text{ is a solution to } \eqref{app1} \Big\}$$and
$$Y=\Big\{v\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega)) \,\Big|\, v \text{ is a solution to } \eqref{to App1} \Big\}.$$
We aim to show that $X$ is dense in $Y$. By the Hahn-Banach theorem, it suffices to prove the following statement: If $f\in L^2(Q)$ satisfies
$$\int_{Q}fV\,dxdt=0, \quad \text{ for any } V\in X,$$
then
$$\int_{Q}fv\,dxdt=0, \quad \text{ for any }v\in Y.$$
To this end,
let $f\in L^2(Q)$ and suppose $\int_{Q}fV\,dxdt=0$, for any $V\in X$. Consider
\begin{equation}
\begin{cases}
-\p_t\overline{V}-\Delta \overline{V} +q\overline{V}=f &\text{ in }\ Q,\\
\overline{V}=0 &\text{ on }\ \Sigma,\\
\overline{V}(x,T)=0 &\text{ in }\ \Omega
\end{cases}
\end{equation}
and its solution is in $H^{2, 1}(Q)$. For any $V\in X$, one has
\begin{align*}
0=&\int_{Q}fV\,dxdt=\int_Q (-\p_t\overline{V}-\Delta \overline{V} +q\overline{V}) V\,dxdt=\int_{\Sigma} \p_\nu \overline{V}V\, dSdt.
\end{align*}
Since $V|_{\Sigma}$ can be arbitrary function, which is compactly supported on $\Sigma$, we must have $\p_\nu \overline{V}=0$ on $\Sigma$.
Thus, for any $ v\in Y$,
\begin{align*}
\int_{Q}fv \,dxdt&=\int_Q (-\p_t\overline{V}-\Delta \overline{V} +q\overline{V}) v\,dxdt=\int_{\Sigma}\p_\nu\overline{V}v\,dSdt= 0,
\end{align*}
which verifies the assertion.
\end{proof}
Let $\Omega\subset \R^n$ be a connected domain, and $\Omega'$ be a connected open subset of $\Omega$ such that $\p\Omega\subset\p\Omega'$. Define $Q'=(\Omega\backslash\Omega')\times(0,T)$. Meanwhile, for given $\varepsilon>0$ and $\omega\in\mathbb{S}^{n-1}$, we set
\begin{align*}
&\Gamma_{+,\omega,\varepsilon}:=\Big\{x\in\Gamma\ \Big| \ \nu(x)\cdot\omega>\varepsilon\Big\}, \\
&\Gamma_{-,\omega,\varepsilon}:=\Big\{x\in\Gamma \ \Big| \ -\nu(x)\cdot\omega>\varepsilon\Big\},\\
&\mbox{and }\Sigma_{\pm,\omega,\varepsilon}:=\Gamma_{\pm,\omega, \varepsilon}\times(0,T).
\end{align*}
The following approximation property will be used to prove Theorems \ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear} with partial data.
\begin{lem}[Runge approximation with partial data]\label{appro2}
Let $q\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$. Then for any solutions $W_{\pm}\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ to
\begin{equation}\label{to app3}
\begin{cases}
\p_tW_+-\Delta W_+ +qW_+=0 &\text{ in }\ Q,\\
W_+(x,0)=0 &\text{ in }\ \Omega
\end{cases}
\end{equation}and
\begin{equation}\label{to App4}
\begin{cases}
\p_tW_--\Delta W_- +qW_-=0 &\text{ in }\ Q,\\
W_-(x,T)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}
and any $\eta>0$, there exist solutions $v_{\pm}\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$ to
\begin{equation}\label{app3}
\begin{cases}
\p_tv_+-\Delta v_++ qv_+=0 &\text{ in }\ Q,\\
v_+=0 &\text{ on }\ \Gamma_{-,\omega,\varepsilon}\times(0,T),\\
v_+(x,0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}and
\begin{equation}\label{app4}
\begin{cases}
-\p_tv_--\Delta v_-+ qv_-=0 &\text{ in }\ Q,\\
v_-=0 &\text{ on }\ \Gamma_{+,\omega,\varepsilon}\times(0,T),\\
v_-(x,T)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}
such that
$$\|W_{\pm}-v_{\pm}\|_{L^2(Q')}<\eta.$$
\end{lem}
\begin{proof}
We may only prove the case for forward parabolic equations. Define
$$X'=\Big\{v\in C^{2+\alpha,1+\alpha/2}(\overline{Q}) \,\Big|\, v \text{ is a solution to }
\eqref{app3} \Big\}$$and
$$Y'=\Big\{W\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))\,\Big|\, V
\text{ is a solution to } \eqref{to app3} \Big\}.$$
We aim to show that $X'$ is dense in $Z$. By the Hahn-Banach theorem again, it suffices to claim that if $f\in L^2(Q')$ satisfies
$$\int_{Q'}fv\,dxdt=0, \text{ for any } v\in X',$$
then
$$\int_{Q'}fW\,dxdt=0, \text{ for any } W\in Y'.$$
Let $f\in L^2(Q')$ satisfy that $\int_{Q'}fv\,dxdt=0, \ \forall v\in X'$. We extend $f$ to $Q$ by letting $f=0$ outside $Q'$.
Consider
\begin{equation}
\begin{cases}
-\p_t\overline{v}-\Delta \overline{v} +q\overline{v}=f &\text{ in }\ Q,\\
\overline{v}=0 &\text{ on }\ \Sigma,\\
\overline{v}(x,T)=0 &\text{ in }\ \Omega,
\end{cases}
\end{equation}
and its solution is in $H^{2, 1}(Q).$ Then for any $v\in X'$,
\begin{align*}
&0=\int_{Q}fv\,dxdt=\int_Q (-\p_t\overline{v}-\Delta \overline{v} +q\overline{v}) v\,dxdt=\int_{\Sigma} \p_\nu \overline{v}v\,dSdt.
\end{align*}
Since $v|_{\Sigma}$ can be arbitrary function, which is compactly supported on $\Sigma\backslash(\Gamma_{-,\omega,\varepsilon}\times(0,T))$ and $v=0$ on $\Gamma_{-,\omega,\varepsilon}\times(0,T) $, we have that $\p_\nu \overline{v}=0$ on $\Sigma\backslash(\Gamma_{-,\omega,\varepsilon}\times(0,T))$.
Next, fix a set $\Omega_1$ with
nonempty interior, such that $(\Omega_1 \cap \p\Omega) \subset (\Gamma\backslash\Gamma_{-,\omega,\varepsilon})$. Then $\overline{v}=0$ on $\Omega_1\times(0,T)$. Notice that
$$ -\p_t\overline{v}-\Delta \overline{v} +q\overline{v}=0 \text{ in } (\Omega'\cup\Omega_1)\times(0,T).$$
Since $\Omega'\cup\Omega_1$ is open and connected, by the unique continuation principle for linear parabolic equations (for instance, see \cite{saut1987unique}), we have $\overline{v}=0$ on $\Omega'\times(0,T).$
Hence, $\overline{v}\Big|_{\p\Omega'\times(0,T)}=\p_\nu \overline{v}\Big|_{\p\Omega'\times(0,T)}=0,$ and it follows that $$\overline{v}\Big|_{\p(\Omega\backslash\Omega')\times(0,T)}=\p_\nu \overline{v}\Big|_{\p(\Omega\backslash\Omega')\times(0,T)}=0.$$
Hence, for any $W\in Y'$,
\begin{align*}
\int_{Q'}fW \,dxdt=\int_{Q'} (-\p_t\overline{v}-\Delta \overline{v} +q\overline{v}) W\,dxdt=\int_{\p(\Omega\backslash\Omega')\times(0,T)}\p_\nu\overline{v}W\,dSdt= 0.
\end{align*}
This completes the proof.
\end{proof}
\begin{rmk}
Let us refer readers to some related approximation property for some different diffusion equations, such as \cite[Lemma 5.3]{CK2018determination}. Since the proofs of the global uniqueness results with either full data or partial data are similar, we focus on presenting the arguments for the full data case and remark the necessary modifications for the partial data case, and vice versa.
\end{rmk}
\begin{lem}[Denseness property]\label{Lem: denseness}
Let $q_1,q_2 \in L^\infty(Q)$.
Assume that $f\in L^\infty(Q)$, such that
$$\int_{Q} fv_1v_2\, dxdt =0,$$
for any $v_1$ and $v_2$, which satisfy $v_1v_2\in L^1(Q)$, and are, respectively, solutions to
\begin{align}\label{equ forward CGO}
\begin{cases}
\p_{ t} v_1-\Delta v_1+q_1 v_1=0 &\text{ in }\ Q,\\
v_1(x,0)=0 &\ \text{ in }\Omega,
\end{cases}
\end{align}
and
\begin{align}\label{equ backward CGO}
\begin{cases}
\p_{ t} v_2+\Delta v_2-q_2 v_2=0 &\text{ in }\ Q,\\
v_2(x,T)=0 &\text{ in }\ \Omega.
\end{cases}
\end{align}
Then $f=0$.
In other words, the linear span of products of solutions to forward and backward
parabolic equations are dense in $L^1(Q)$.
\end{lem}
\begin{proof}
Since $q_j\in L^\infty(Q)$ for $j=1,2$, without loss of generality, we may
assume that there exists a positive number $m$, such that $q_1, q_2\in \Big\{ q\in L^{\infty}(Q)\ \Big|\
\norm{q}_{L^{\infty}(Q)}<m \Big\}$.
First, let us fix $\omega\in\mathbb{S}^{n-1}$. Consider $\rho>0$ to be sufficiently large, and $(\xi,\tau)\in M:
=\Big\{(\xi, \tau)\in\mathbb{R}^{n+1} \big| \, \xi\cdot\omega=0 \Big\}$ with $|(\xi,\tau)|^2<\rho-1$.
Then by Proposition \ref{CGO_F}, there is a solution $v_{1,\rho}(\cdot, \cdot; \xi, \tau)$ to \eqref{equ forward CGO} such that $$v_{1,\rho}=\psi_{-, \rho}(\theta_{+, \rho}+z_{+, \rho, q_1})$$ with $\norm{z_{+, \rho, q_1}}_{L^2(Q)}\to 0$ as $\rho\to\infty$.
Similarly, there is a solution $v_{2,\rho}(\cdot, \cdot)$ to the backward parabolic equation \eqref{equ backward CGO} such that $$v_{2,\rho}=\psi_{+, \rho}(\theta_{-,\rho}+z_{-, \rho, q_2})$$ with $\norm{z_{-, \rho, q_2}}_{L^2(Q)}$
tending to $0$, as $\rho\to\infty$.
Then
\begin{align*}
v_{1,\rho}v_{2,\rho}&=\theta_{+, \rho}\theta_{-, \rho}+\theta_{+, \rho} z_{-, \rho, q_2}
+z_{+, \rho, q_1}\theta_{-, \rho}+z_{+, \rho, q_1}z_{-, \rho, q_2}\\
&=\varphi_{\rho}(t)e^{-\mbox{i}(x,t)\cdot(\xi,\tau)}+\theta_{+,\rho}z_{-,\rho, q_2}+z_{+, \rho, q_1}\theta_{-,\rho}+z_{+, \rho, q_1}z_{-, \rho, q_2},
\end{align*}
where $\varphi_{\rho}(t)=1-\exp(-\rho^{3/4}t)-\exp(-\rho^{3/4}(T-t))+\exp(-\rho^{3/4}T)$.
Note that $\theta_{+,\rho}$ and $\theta_{-,\rho}$ are bounded with respect to $\rho>0$.
Hence, letting $\rho\to+\infty$ in $\displaystyle\int_Q fv_{1, \rho}v_{2, \rho}\, dxdt=0$, we have that
\begin{equation}\label{integral id}
\int_Qf e^{-\mbox{i}(x,t)\cdot(\xi,\tau)}\, dxdt=0.
\end{equation}
Therefore, for a fixed $\omega\in\mathbb{S}^{n-1}$, \eqref{integral id} holds in any compact subset of $M$. Clearly, $M$ is an $n$-dimensional subspace of $\mathbb{R}^{n+1}$. Notice that $f$ has compact support as a distribution and its Fourier transform is analytic. The Fourier transform of $f$ is zero in any compact subset of $M$ as shown, and therefore by changing $\omega\in\mathbb{S}^{n-1}$ in a small conic neighborhood,
we can conclude it is zero in $\mathbb{R}^{n+1}$. This implies $f=0$ in $Q$ as desired.
\end{proof}
In the application of the preceding denseness result with full data, we are able to derive the following global uniqueness result as follows.
\begin{cor}[Global uniqueness with full data]\label{fulldata}
Let $q_1,q_2 \in L^\infty(Q)$.
Let $\Lambda_{q_j}$ be the full DN map of the linear heat equation:
\begin{align}\label{equ forward full data}
\begin{cases}
\p_{ t} v_j-\Delta v_j+q_j v_j=0 &\text{ in }\ Q,\\
v_j(x,0)=0 &\ \text{ in }\Omega,
\end{cases}
\end{align}
for $j=1,2$, respectively. Assume that
\begin{align}\label{Same DN full}
\Lambda_{q_1}(f)=\Lambda_{q_2}(f) \text{ on }\Sigma,
\end{align}
for any $f\in H^{-1/2,-1/4}(\Sigma)$, then $q_1=q_2$ in $Q$.
\end{cor}
\begin{proof}
This result can be regarded as an application of \cite{CY2018logarithmic}, and we offer the proof for the sake of completeness. Let $\hat{v}$ be a solution to the backward heat equation:
\begin{align}\label{equ backward full data}
\begin{cases}
\p_{ t} \hat v+\Delta \hat v-q_2 \hat v=0 &\text{ in }\ Q,\\
\hat v(x,T)=0 &\text{ in }\ \Omega.
\end{cases}
\end{align}
Subtracting \eqref{equ forward full data} with $j=1,2$, then we have
\begin{align}\label{equ forward full data difference}
\begin{cases}
\p_{ t} \widetilde v-\Delta \widetilde v +q_2 \widetilde v= (q_2-q_1)v_1 &\text{ in }\ Q,\\
\widetilde v(x,0)=0 &\ \text{ in }\Omega,
\end{cases}
\end{align}
where $\widetilde v=v_1-v_2$ in $Q$. Multiplying \eqref{equ forward full data difference} by the solution $\hat{v}$ of \eqref{equ backward full data}, with the condition \eqref{Same DN full} at hand, it is easy to derive that
\begin{align}
\int_Q (q_2-q_1)v_1 \hat{v}\, dxdt=0.
\end{align}
Therefore, by applying \eqref{Lem: denseness}, one can conclude that $q_1=q_2$ in $Q$ as desired.
\end{proof}
\begin{lem}[Global uniqueness with partial data]\label{Lem:CGO}
Let $\Omega\subset \R^n$ be a bounded domain with $C^\infty$-smooth boundary $\Gamma$.
For any $q_j\in C^{2+\alpha, 1+\frac{\alpha}{2}}(\overline{Q})$ $(j=1, 2)$,
assume that $\Lambda^{\mathrm{P}}_{q_j}$ are the partial DN maps of the linear parabolic equation:
\begin{align}\label{linear term parabolic}
\begin{cases}
(\p_t-\Delta +q_j) u=0 &\text{ in }\ Q,\\
u= f &\text{ on }\ \Sigma,\\
u(x, 0)=0, &\text{ in }\ \Omega,
\end{cases}
\end{align}
and
$$\Lambda^{\mathrm{P}}_{ q_1}(f)=\Lambda^{\mathrm{P}}_{ q_2}(f) \text{ in } \mathcal{V}_- , $$
for any $f \in C_0^{2+\alpha,1+\alpha/2}(\mathcal{V}_+)$.
If $q_1=q_2$ in $\Omega'\times(0,T)$, where $\Omega'$ is an arbitrarily given
connected open subset of $\Omega$ with $\Gamma\subset\p\Omega'$, then
$$
q_1=q_2 \text{ in }Q.
$$
\end{lem}
\begin{proof}
By Proposition \ref{CGO_F}, there is a solution
$$v_1(\cdot, \cdot; \rho, \xi, \tau,\omega)=\psi_{-,\rho}(\theta_{+,\rho}+z_{+, \rho, q_1})\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$$
to the forward parabolic equation (\ref{linear term parabolic}) with respect to $q_1$ such that
$$
\lim\limits_{\rho\to\infty}\norm{z_{+, \rho, q_1}}_{L^2(Q)}=0.
$$
For $j\in\{1,2\}$, let us define
$$
S_j=\Big\{v\in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))\,\Big|\, (\p_t-\Delta +q_j)
v=0 \text{ in } Q, v(x,0)=0 \text{ in }\Omega \Big\},
$$
and the map $\mathcal{M}:S_1\to S_2$ is defined by
$$\mathcal{M}(v_1)=v_2,$$
where $v_2$ is the solution to
\begin{align}
\begin{cases}
(\p_t-\Delta +q_2) v_2=0 &\text{ in }\ Q,\\
v_2= v_1 &\text{ on }\ \Sigma,\\
v_2(x, 0)=0, &\text{ in }\ \Omega.
\end{cases}
\end{align}
By using the trace theorem, $v_1 \big|_{\Sigma}\in L^2(0,T;H^{1/2}(\Gamma)) $ and the map $\mathcal M$ is well-defined.
Now we have
\begin{align}\label{111}
\begin{cases}
(\p_t-\Delta +q_2) (v_1-v_2)=(q_2-q_1)v_1 &\text{ in }\ Q,\\
v_1-v_2= 0 &\text{ on }\ \Sigma,\\
(v_1-v_2)(x,0)=0 &\text { in }\ \Omega.
\end{cases}
\end{align}
Consider a solution $\hat{v}$ to the backward parabolic equation \eqref{cgo2} of the form that we have constructed in Proposition \ref{CGO_F} with $q=q_2$.
Then by Lemma $\ref{appro2}$, there are two sequences of functions
$\left\{v^k_1 \right\}^{\infty}_{k=1}$, $\left\{\hat{v}^k\right\}^{\infty}_{k=1}\in
C^{2+\alpha, 1+\frac{\alpha}{2}}(\overline{Q})$, such that
$v^k_1$ are solutions to \eqref{app3}, $\hat{v}^k$ are solutions to \eqref{app4}, and $v^k_1\to v_1$, $\hat{v}^k\to\hat{v}$ in $L^2(Q')$ as $k\to \infty$. Hence, we have
\begin{align}
\begin{cases}
(\p_t-\Delta +q_2) (v^k_1-\mathcal{M}(v_1^k))=(q_2-q_1)v^k_1 &\text{ in }\ Q,\\
v^k_1-\mathcal{M}(v_1^k)= 0 &\text{ on }\ \Sigma,\\
(v^k_1-\mathcal{M}(v_1^k))(x,0)=0 &\text { in }\ \Omega.
\end{cases}
\end{align}
Let $v^k_2=\mathcal{M}(v_1^k)$. Multiplying by the functions $\hat{v}^k$ on the both sides of the above equation and integration by parts implies
$$\int_{Q} \left( q_2-q_1 \right) v^k_1\hat{v}^k\ dxdt =\int_{\Sigma}
\hat{v}^k\partial_\nu (v^k_1-v_2^k)\, dS dt.$$
Since $\mathcal{U}_{\pm}$ is a neighborhood of $\Gamma_{\pm,\omega_0}$ (recalling $\mathcal{V}_{\pm}=\mathcal{U}_{\pm}\times(0,T)$), there is an $\varepsilon>0$, such that\footnote{We also utilize the same parameter $\varepsilon$ to construct the solution $v_1$.}
\begin{align*}
\Big\{x\in\Gamma \, \Big| \, 0<\omega_0\cdot\nu(x)<2\varepsilon\Big\}\times(0,T)&\subset\mathcal{V}_-,\\
\Big\{x\in\Gamma \, \Big| \, \omega_0\cdot\nu(x)>-2\varepsilon\Big\}\times(0,T)&\subset\mathcal{V}_+.
\end{align*}
Therefore, by choosing
$$
\omega\in \Big\{\omega\in\mathbb{S}^{n-1}\, \Big| \, |\omega-\omega_0|<\varepsilon\Big\},
$$ we get that
\begin{align*}
\text{supp } v^k_1\Big|_{\Sigma}&\subset \Big\{x\in\Gamma \, \Big| \, \omega\cdot\nu(x)\geq -\varepsilon \Big\} \times(0,T) \\ &\subset \Big\{x\in\Gamma \, \Big| \, \omega_0\cdot\nu(x)>-2\varepsilon\Big\}\times(0,T)\subset\mathcal{V}_+,
\end{align*}
and
\begin{align*}
\Big\{x\in\Gamma \, \Big|\, \omega_0\cdot\nu(x)\geq 2\varepsilon \Big\} \subset \Gamma_{+,\omega,\varepsilon} .
\end{align*}
Note that $v_1^k\Big|_{\Sigma}=v_2^k\Big|_{\Sigma}\in C_0^{2+\alpha,1+\alpha/2}(\mathcal{V}_+)$ and recall $\hat{v}^k=0$ on $ \Gamma_{+,\omega,\varepsilon}$. Then we have
\begin{align*}
&\left|\int_{\Sigma}\hat{v}^k\partial_\nu (v^k_1-v^k_2)\, dSdt\right|\\
=&\left|\int_{\{\omega_0\cdot\nu\geq2\varepsilon\}}\hat{v}^k\partial_\nu (v^k_1-v^k_2)\,dSdt\right|+ \left|\int_{\{0<\omega_0\cdot\nu<2\varepsilon\}}\hat{v}^k\partial_\nu (v^k_1-v^k_2)\,dSdt\right|\\
&+\left|\int_{\{\omega_0\cdot\nu\leq 0\}}\hat{v}^k\partial_\nu (v^k_1-v^k_2)\,dSdt\right|\\
=&0
\end{align*}
Then
$$\int_{Q'} \left( q_2-q_1 \right) v^k_1\hat{v}^k\ dxdt+ \int_{Q\backslash Q'} \left( q_2-q_1 \right) v^k_1\hat{v}^k\ dxdt= 0.$$
Since we assume $q_1=q_2$ in $Q \setminus Q'$, it follows that
$$\int_{Q'} \left( q_2-q_1 \right) v^k_1\hat{v}^k\, dxdt=0.$$
Therefore, by similar arguments as in Corollary \ref{fulldata}, letting $\rho\to\infty$, one has that
\begin{equation*}\label{Q'}
\int_{Q'}(q_2-q_1) e^{-\mathrm{i}(x,t)\cdot(\xi,\tau)}\, dxdt=0,
\end{equation*}
where $\mathrm{i}=\sqrt{-1}$.
Since $
\omega\in \Big\{\omega\in\mathbb{S}^{n-1}\, \Big| \, |\omega-\omega_0|<\varepsilon\Big\},
$ it can be changed in a small conic neighborhood. By by using similar arguments as in Corollary \ref{fulldata}, we have
$$
q_1=q_2 \text{ in } Q
$$
as desired.
\end{proof}
\begin{rmk}
For the full data case, we can use Lemma \ref{appro} to get an approximation of CGO solutions instead of Lemma \ref{appro2}. We do not need to assume $q_1=q_2$ in $\Omega'\times(0,T)$, and we also point out that we cannot apply Corollary \ref{fulldata} to get the result for full data because we need to control the trace of solution on $\Sigma$. In other words, Lemma \ref{appro} is necessary even for the full data case in this paper.
\end{rmk}
\subsection{Proof of Theorem \ref{Main Thm:Simultaneous}}
With Lemma \ref{Lem:CGO} at hand, combining with the higher order linearization method, we are able to prove Theorem \ref{Main Thm:Simultaneous}.
\begin{proof}[Proof of Theorem $\ref{Main Thm:Simultaneous}$] Let us first remark that the proofs of (a) and (b) in Theorem \ref{Main Thm:Simultaneous} are similar, so it suffices to show the global uniqueness result with partial data. The whole proof is
divided into five parts.
\medskip
{\it Step 1. Initiation}
\medskip
\noindent Let us introduce the following boundary value
\begin{align}\label{small lateral BCs}
f(x,t;\epsilon)=\sum_{\ell=1}^M \epsilon_\ell f_\ell\quad \text{on} \ \Sigma,
\end{align}
where $M\in \N$, $f_1, \cdots, f_M \in C^{2+\alpha,1+\alpha/2}_0(\mathcal{V}_+)$
and $\epsilon=(\epsilon_1, \ldots,\epsilon_M)$ is a parameter vector in $\mathbb R^M$ with $|\epsilon|=\sum\limits_{\ell=1}^{M} |\epsilon_\ell|$ small enough, such that $\left\|\displaystyle\sum_{\ell=1}^M \epsilon_\ell f_\ell \right\|_{C^{2+\alpha,1+\alpha/2}_0(\overline{\Sigma})}$ is sufficiently small.
For $j=1, 2$, by the local well-posedenss property in Section \ref{Sec 2}, there exist unique solutions $u_j=u_j(x,t;\epsilon)\in C^{2+\alpha,1+\alpha/2}(\overline{Q})$ to
\begin{align}\label{IBVP of simultaneous recovery-eps}
\begin{cases}
u_{j, t}-\Delta u_j+b_j(x,t,u_j)=0 &\text{ in }\ Q,\\
u_j=\displaystyle\sum_{\ell=1}^M \epsilon_\ell f_\ell\ &\text{ on }\ \Sigma,\\
u_j(x, 0)=g_j(x) &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $g_j\in C^{2+\alpha}_0(\Omega)$ with $\norm{g_j}_{C^{2+\alpha}(\Omega)}<\frac{\delta}{2}$ being sufficiently small, and $b_j(x,t,z)$ are admissible coefficients defined in Section \ref{Sec 1}.
For the sake of convenience, when $\epsilon=0 $, let $\widetilde u_j=u_j(\cdot, \cdot; 0)$ be the solutions to
\begin{align}\label{IBVP of simultaneous recovery-eps=0}
\begin{cases}
\widetilde u_{j, t}-\Delta \widetilde u_j +b_j(x,t, \widetilde u_j)=0 &\text{ in }\ Q,\\
\widetilde u_j=0 &\text{ on }\ \Sigma,\\
\widetilde u_j(x, 0)=g_j, &\text{ in }\ \Omega.
\end{cases}
\end{align}
By utilizing the higher order linearization to \eqref{IBVP of simultaneous recovery-eps} around
the solution $\widetilde u_j$ to \eqref{IBVP of simultaneous recovery-eps=0}, we will determine information on $b_j$ for $j=1,2$.
\medskip
{\it Step 2. The first order linearization $(M=1)$}
\medskip
\noindent One can linearize the equation \eqref{IBVP of simultaneous recovery-eps} around $\widetilde u_j$, where $\widetilde u_j$ is the solution to \eqref{IBVP of simultaneous recovery-eps=0}, for $j=1,2$. Due to Remark \ref{remark 2.2}, direct computations demonstrate that for $j=1, 2$ and $\ell=M=1$\footnote{In fact, the arguments hold for all $\ell=1,\ldots,M$, and we will use in steps 2-5.},
$$
v_j^{(\ell)}(x,t)=\lim_{\epsilon \to 0} \frac{u_j(x,t)-\widetilde u_j(x,t)}{\epsilon_\ell}
$$ satisfies the following parabolic equation:
\begin{align}\label{first linearization}
\begin{cases}
v_{j, t} ^{(\ell)}-\Delta v_j^{(\ell)}+q_j v_j ^{(\ell)}=0 &\text{ in }\ Q, \\
v_j ^{(\ell)}= f _\ell& \text{ on }\ \Sigma,\\
v_j^{(\ell)}(x, 0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{align}
where
\[
q_j (x,t):=b_{j, u} (x,t,\widetilde u_j(x,t))\ \text{ in }Q\quad \mbox{ and }\quad q_j\in C^{2+\alpha,1+\frac{\alpha}{2}}(\overline{Q}).
\]
We need to point out that both $\widetilde u_j$ and $v_j^{(\ell)}$ in \eqref{IBVP of simultaneous recovery-eps=0} and \eqref{first linearization} are still unknown, respectively, since they solve parabolic equations with unknown coefficients and initial data. In this step, we will show that
\begin{align}\label{claim1}
q_1 (x,t)=q_2 (x,t)\ \text{ in } Q.
\end{align}
With the same partial DN maps at hand
$$\Lambda^{\mathrm{P}}_{b_1,g_1}(f)=\Lambda^{\mathrm{P}}_{b_2,g_2}(f), \quad \text{ for any sufficiently small }f\in C^{2+\alpha,1+\frac{\alpha}{2}}_0(\mathcal{V}_+),
$$
such that we have
\begin{align}\label{first linearized DN maps agree}
\begin{array}{rll}
&v_1^{(\ell)}(x, 0)=v_2^{(\ell)}(x, 0), \quad \left. v_1^{(\ell)} \right|_{\Sigma}= \left. v_2^{(\ell)}\right|_{\Sigma}, \quad & \left. \p _\nu v_1^{(\ell)} \right|_{\mathcal{V}_{-}}= \left. \p_\nu v_2^{(\ell)}\right|_{\mathcal{V}_{-}},
\end{array}
\end{align}
for $\ell=M=1$.
Now, subtracting \eqref{first linearization} with $j=1,2$, we have
\begin{align}\label{first linearization subtraction}
\begin{cases}
v^{(\ell)}_{t}- \Delta v^{(\ell)} +q_2 v^{(\ell)}= (q_2 -q_1 )v^{(\ell)}_1 & \text{ in }\ Q, \\
v^{(\ell)}=0 &\text{ on }\ \Sigma, \\
v^{(\ell)}(x,0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $v^{(\ell)}:=v^{(\ell)}_1-v^{(\ell)}_2$. Let $\tilde v_2^{(\ell)}$ be a solution to the
following backward parabolic equation:
\begin{align}\label{tilde v_1 equation}
\begin{cases}
\tilde v_{2,t}^{(\ell)}+\Delta \tilde v_2^{(\ell)} - q_2 \tilde v_2^{(\ell)}=0 & \text{ in }\ Q, \\
\tilde v_2(x,T)=0 & \text{ in }\ \Omega.
\end{cases}
\end{align}
Multiplying both sides of the first equation in \eqref{first linearization subtraction} by $\tilde v_2^{(\ell)}$, by \eqref{first linearized DN maps agree}, an integration by parts yields that
\begin{align}\label{integral id of 1st linearized equation1}
\int_{Q} \left( q_2-q_1 \right) v_1^{(\ell)}\tilde v_2^{(\ell)}\, dxdt =\int_{\Sigma}
\tilde v_2^{(\ell)}\partial_\nu v_1^{(\ell)}\, dS dt.
\end{align}
Moreover, with the condition $b_1=b_2 $ in $\Omega'\times(0,T)\times \R$ at hand, by applying Lemma \ref{Lem:CGO}, one can easily see that the claim \eqref{claim1} holds. Furthermore, as $q_1=q_2$ in $Q$, $v_1^{(\ell)}$ and $v_2^{(\ell)}$ satisfy the same parabolic equation \eqref{first linearization}, by the uniqueness of solutions, we obtain that
\begin{align}\label{uniqueness for solutions of the first linearized equation}
v^{(\ell)}:= v_1^{(\ell)}=v_2^{(\ell)} \text{ in }Q.
\end{align}
\medskip
{\it Step 3. The second order linearization $(M=2)$}
\medskip
\noindent For the second linearization ($m=2$), one can differentiate (\ref{IBVP of simultaneous recovery-eps})
with respect to different parameters $\epsilon_1$ and $\epsilon_2$.
A direct computation shows that $w^{(2)}_j$ $(j=1, 2)$ satisfy
\begin{align}\label{second linearization}
\begin{cases}
w_{j, t} ^{(2 )}-\Delta w_j^{(2)}+qw_j^{(2)} +b_{j, uu} (x,t,\widetilde u_j)v^{(1)}v^{(2)}=0 &\text{ in }\ Q, \\
w_j^{(2)} = 0 & \text{ on }\ \Sigma,\\
w_j^{(2)}(x, 0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $q=q_1=q_2$, $b_{j, uu}(\cdot, \cdot, \tilde u_j)\in C^{2+\alpha,1+\frac{\alpha}{2}}(\overline{Q})$
and $v^{(1)}, v^{(2)}\in C^{2+\alpha,1+\frac{\alpha}{2}}(\overline{Q})$ satisfy
\begin{align*}
\begin{cases}
v_{t} ^{(\ell)}-\Delta v^{(\ell)}+q(x,t)v^{(\ell)}=0 &\text{ in }\ Q, \\
v^{(\ell)}= f_\ell& \text{ on }\ \Sigma,\\
v^{(\ell)}(x, 0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{align*}
here $f_1$ and $f_2$ can be arbitrarily chosen.
Next, we will prove that
\begin{align}\label{claim2}
b_{1, uu} (x,t,\widetilde u_1(x, t))=b_{2, uu}(x,t,\widetilde u_2(x, t)) \quad {in}\quad Q.
\end{align}
With the same DN map at hand, by differentiating $\epsilon_1$ and $\epsilon_2$, we have
\begin{align}\label{second linearized DN maps agree}
\begin{array}{rll}
w_1^{(2)}(x, 0)=w_2^{(2)}(x, 0), \quad w_1^{(2)} \Big|_{\Sigma}=w_2^{(2)}\Big|_{\Sigma}, \quad & \p _\nu w_1^{(2)} \Big|_{\mathcal{V}_{-}}=\p_\nu w_2^{(2)}\Big|_{\mathcal{V}_{-} }.
\end{array}
\end{align}
Let $v^{(0)}$ be any solution to the backward parabolic equation:
\begin{eqnarray}\label{AA}
\left\{
\begin{array}{ll}
v_{t}^{(0)}+\Delta v^{(0)} - q v^{(0)}=0 &\text{ in } Q,\\[2mm]
v^{(0)}(x, T)=0 &\text{ in }\Omega.
\end{array}
\right.
\end{eqnarray}
By subtracting the equations \eqref{second linearization} associated to $j=1, 2$, an integration by parts yields
\begin{align}\label{second integral id}
\begin{split}
\int_{Q} \Big[b_{1, uu} (x,t,\widetilde u_1(x, t))-b_{2, uu}(x,t,\widetilde u_2(x, t))\Big] v^{(0)} v^{(1)}v^{(2)} \, dxdt =0
\end{split}
\end{align}
We next choose a nonzero boundary data $f_2$ such that $f_2\geq 0$ on $\Sigma$ and $f_2>0$ on $D_t\times(0, T)$, where $D_t\subset \Gamma$ is a relative open subset for any $t\in (0, T)$. Via the condition $f_2=v^{(2)}|_{\Sigma}\in L^\infty(\Sigma)$ at any time $t\in (0, T)$, by applying the maximum principle for parabolic equation (for example, see \cite[Chapter 7]{evans1998partial} or Appendix \ref{Sec: Appendix}), we have a bounded positive solution $v^{(2)}$ in $Q$.
Now, by selecting $v^{(1)}$ and $v^{(0)}$ as the CGO solutions of forward and backward parabolic equations, via Corollary \ref{fulldata}, we get
\[
\big[b_{1, uu} (x,t,\widetilde u_1(x, t))-b_{2, uu}(x,t,\widetilde u_2(x, t))\big] v^{(2)}=0 \text{ in }Q.
\]
With the positivity of $v^{(2)}$ in $Q$ at hand, we have (\ref{claim2}) as desired.
Furthermore, by the uniqueness of solutions to \eqref{second linearization}, one can immediately obtain
\[
w_1^{(2)}= w_2^{(2)} \quad\text{ in }Q.
\]
\medskip
{\it Step 4. The higher order linearization $(M>2)$}
\medskip
\noindent By utilizing the higher order linearization with the induction hypothesis, we are able to find $M$-th order derivative of \eqref{IBVP of simultaneous recovery-eps} and prove that
\begin{align}\label{claim3}
\p_u^M b_1 (x,t, \widetilde u_1(x, t))=\p_u^M b_2 (x,t, \widetilde u_2(x, t))\quad \text{ in }Q,
\end{align}
for any $M=3, 4, \cdots$.
Let us first assume that
$$
\p_u^k b_1 (x,t, \widetilde u_1(x, t))=\p_u^k b_2 (x,t, \widetilde u_2(x, t))\ \text{ in }Q, \ \text{ for any }k=1,\dots, M-1.
$$
Similar to previous steps, we differentiate \eqref{IBVP of simultaneous recovery-eps} with respect to $\epsilon_1,\ldots, \epsilon_{M-1}$ and $\epsilon_{M}$, then we have
\begin{align*
\int_{Q}\Big[\p_u ^M b_1 (x,t, \widetilde u_1(x, t)) - \p_u^M b_2 (x,t, \widetilde u_2(x, t))\Big] v^{(0)}v^{(1)} \cdots v^{(M)}\, dxdt =0,
\end{align*}
where $v^{(0)}$ is the solution to the backward parabolic equation \eqref{AA}, and $v^{(\ell)}$ $(\ell=1,2, \cdots, M)$ are solutions to the forward parabolic equation \eqref{first linearization}.
Similar to Step 3, let us choose $v^{(0)}$ and $v^{(1)}$ as CGO solutions, and $v^{(2)},\ldots, v^{(M) }$ are bounded positive solutions in $Q$
\begin{equation}\label{RRR}
\p_u ^M b_1 (x,t, \widetilde u_1(x, t))=\p_u^M b_2 (x,t, \widetilde u_2(x, t))\mbox{ in }Q, \quad \text{ for any }M\in \N.
\end{equation}
\medskip
{\it Step 5. The determination of initial data and coefficients}
\medskip
\noindent Recall that $\widetilde u_j$ $(j=1, 2)$ are the solutions to the semilinear parabolic equation:
\begin{align*}
\begin{cases}
\widetilde u_{j, t}-\Delta \widetilde u_j +b_j(x,t, \widetilde u_j)=0 &\text{ in }\ Q,\\
\widetilde u_j=0 &\text{ on }\ \Sigma,\\
\widetilde u_j(x, 0)=g_j, &\text{ in }\ \Omega.
\end{cases}
\end{align*}
As in the proof of \cite[Theorem 1.3]{LLL2021determining}, by the admissible property of $b_1$ and $b_2$,
\begin{eqnarray}\label{LLL}
\begin{array}{ll}
&b_1(x, t, \widetilde u_1(x, t))-b_2(x, t, \widetilde u_2(x, t))\\[2mm]
=&\displaystyle\sum\limits_{k=1}^\infty \frac{\partial_u^k b_2(x, t, \widetilde u_2(x, t))}{k!}\Big[-\widetilde u_2(x, t)\Big]^k
-\sum\limits_{k=1}^\infty \frac{\partial_u^k b_1(x, t, \widetilde u_1(x, t))}{k!}\Big[-\widetilde u_1(x, t)\Big]^k\\
=&\displaystyle\sum\limits_{k=1}^\infty \frac{\partial_u^k b_1(x, t, \widetilde u_1(x, t))(-1)^k}{k!}\Big\{\Big[\widetilde u_2(x, t)\Big]^k-\Big[\widetilde u_1(x, t)\Big]^k\Big\}.
\end{array}
\end{eqnarray}
Since both $\widetilde u_1$
and $\widetilde u_2$ are bounded, set $R=\|\widetilde u_1\|_{L^\infty(Q)}+\|\widetilde u_2\|_{L^\infty(Q)}$. Then, for any $L>0$ and $(x, t)\in Q$,
\begin{align*}
&\left|\frac{b_1(x, t, \widetilde u_1(x, t))-b_2(x, t,\widetilde u_2(x, t))}{\widetilde u_1(x, t)-\widetilde u_2(x, t)}\right|\\
=&\left|\sum\limits_{k=1}^\infty \frac{\partial_u^k b_1(x, t, \widetilde u_1 (x, t))}{k!}(-1)^{k+1}\Big\{\Big[\widetilde u_1(x, t)\Big]^{k-1}+\Big[\widetilde u_1(x, t)\Big]^{k-2}\widetilde u_2(x, t)+\cdots\right.\\
&\quad \left.+\widetilde u_1 (x, t)\Big[\widetilde u_2(x, t)\Big]^{k-2}+\Big[\widetilde u_2(x, t)\Big]^{k-1}\Big\}\right|\\
\leq &\sum\limits_{k=1}^\infty \left|\partial_u^k b_1(x, t, \widetilde u_1(x, t))\right|
\frac{R^{k-1}}{(k-1)!} \\
\leq &\sum\limits_{k=1}^\infty \frac{k R^{k-1} }{L^k} \sup\limits_{|z-\widetilde u_1(x, t)|=L}
|b_1(x, t, z)|.
\end{align*}
Choose $L=2(R+1)$. By the admissibility of $b_1$ and $b_2$,
$$
G(\cdot, \cdot)=\frac{b_1(\cdot, \cdot, \widetilde u_1(\cdot, \cdot))-b_2(\cdot, \cdot, \widetilde u_2(\cdot, \cdot))}{\widetilde u_1(\cdot, \cdot)-\widetilde u_2(\cdot, \cdot)}
\in L^\infty(Q).
$$
Set $w=\widetilde u_1-\widetilde u_2$. It is easy to see that
\begin{align*}
\begin{cases}
w_{t}-\Delta w +Gw=0 &\text{ in }\ Q,\\
w=0 &\text{ on }\ \Sigma,\\
w(x, 0)=g_1-g_2 &\text{ in }\ \Omega.
\end{cases}
\end{align*}
By $\Lambda_{b_1,g_1}(0)=\Lambda_{b_2, g_2}(0)$
and Lemma \ref{lemma3}, we have
$$
g_1=g_2 \text{ in }\Omega \quad \mbox{ and }\quad \widetilde u_1=\widetilde u_2 \text{ in }\ Q.
$$
By (\ref{LLL}),
$$
b_1(x, t, \widetilde u_1(x, t))=b_2(x, t, \widetilde u_2(x, t)) \quad\mbox{ in }\ Q.
$$
In addition, note that for $j=1, 2$ and any $(x, t, z)\in Q\times\mathbb R$,
\begin{align*}
&&b_j(x, t, z)=b_j(x, t, \widetilde u_j(x, t))
+\sum\limits_{k=1}^{\infty}
\frac{\partial_u^k b_j(x, t, \widetilde u_j(x, t))}{k!}\left( z-\widetilde u_j(x, t)\right)^k,
\end{align*}
which implies that $b_1(x, t, z)=b_2(x, t, z)$ in $Q\times\mathbb R$. This proves the assertion.
\end{proof}
\subsection{Proof of Theorem \ref{Main Thm:Simultaneous linear}}
Similar to the proof of Theorem \ref{Main Thm:Simultaneous}, we are ready to prove Theorem \ref{Main Thm:Simultaneous linear}.
\begin{proof}[Proof of Theorem \ref{Main Thm:Simultaneous linear}]
The argument is similar to the proof of Theorem \ref{Main Thm:Simultaneous}, and we prove this result with the full data. Let us divide the proof into two steps.
\medskip
{\it Step 1. Unique determination of coefficients}
\medskip
\noindent Let $u_j=u_j(x,t)$ be the solution to
\begin{align*
\begin{cases}
u_{j, t}-\Delta u_j+q_j u_j=0 &\text{ in }\ Q,\\
u_j=f &\text{ on }\ \Sigma,\\
u_j(x, 0)=g_j(x) &\text{ in }\ \Omega,
\end{cases}
\end{align*}
and let $\widetilde u_j=\widetilde u_j(x,t)$ be the solution to
\begin{align}\label{IBVP of simultaneous recovery-linear2}
\begin{cases}
\widetilde u_{j, t}-\Delta \widetilde u_j+q_j \widetilde u_j=0 &\text{ in }\ Q,\\
\widetilde u_j=0 &\text{ on }\ \Sigma,\\
\widetilde u_j(x, 0)=g_j(x) &\text{ in }\ \Omega,
\end{cases}
\end{align}
for $j=1,2$. With the same DN maps on the lateral boundary at hand, we have
\begin{align}\label{same DNs}
\p_\nu u_1 =\p_\nu u_2 \quad \text{ and }\quad \p_\nu \widetilde u_1 =\p_\nu \widetilde u_2 \quad \text{ on }\quad \Sigma.
\end{align}
We next consider $v_j :=u_j -\widetilde u_j$ for $j=1,2$, then $v_j$ is the solution of
\begin{align}\label{IBVP of simultaneous recovery-linear3}
\begin{cases}
v_{j, t}-\Delta v_j+q_j v_j=0 &\text{ in }\ Q,\\
v_j=f &\text{ on }\ \Sigma,\\
v_j(x, 0)=0 &\text{ in }\ \Omega.
\end{cases}
\end{align}
Subtracting \eqref{IBVP of simultaneous recovery-linear3} with respect to $j=1,2$, we get
\begin{align}\label{IBVP of simultaneous recovery-linear4}
\begin{cases}
v_{ t}-\Delta v+q_2 v= (q_2-q_1)v_1 &\text{ in }\ Q,\\
v=\p_\nu v=0 &\text{ on }\ \Sigma,\\
v(x, 0)=0 &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $v=v_1-v_2$. Moreover, via the condition \eqref{same DNs}, we have $\p_\nu v=0$ on $\Sigma$.
On the other hand, let $\widetilde v_2$ be a solution to the backward parabolic equation
\begin{align*
\begin{cases}
\widetilde v_{2,t}+\Delta \widetilde v_2 +q_2\widetilde v_2 =0 & \text{ in }\ Q,\\
\widetilde v_2(x,T)=0 &\text{ in }\ \Omega.
\end{cases}
\end{align*}
Multiplying \eqref{IBVP of simultaneous recovery-linear4} by the function $\widetilde v_2$, an integration by parts yields that
\begin{align}
\int_{Q}(q_2-q_1)v_1 \widetilde v_2 \, dxdt=0.
\end{align}
By applying the global uniqueness result with full data (Corollary \ref{fulldata}), then we have $q_1=q_2$ as desired.
\medskip
{\it Step 2. Unique determination of initial data}
\medskip
\noindent Recalling that $\widetilde u_j$ is the solution of \eqref{IBVP of simultaneous recovery-linear2}, by using the uniqueness $q_1=q_2$, we can subtract \eqref{IBVP of simultaneous recovery-linear2} with respect to $j=1,2$, then we obtain
\begin{align}\label{IBVP of simultaneous recovery-linear5}
\begin{cases}
u_{ t}-\Delta u+q u=0 &\text{ in }\ Q,\\
u=0 &\text{ on }\ \Sigma,\\
u(x, 0)=g_1-g_2 &\text{ in }\ \Omega,
\end{cases}
\end{align}
where $q=q_1-q_2$ and $u= \widetilde u_1-\widetilde u_2$. Via the condition \eqref{same DNs} again, we have $\p_\nu u=0 $ on $\Sigma$. Finally, by applying the quantitative stability estimate \eqref{Stability estimate in Thm 1}, we can obtain the uniqueness of the initial data $g_1=g_2$ in $\Omega$. This proves the assertion.
\end{proof}
\begin{rmk}
One can find that when the initial and boundary data are small enough, Theorem \ref{Main Thm:Simultaneous linear} can be regarded as a corollary of Theorem \ref{Main Thm:Simultaneous}, where we can simply take $b_j(x,t,u):=q_j(x,t)u$ for $j=1,2$. In order to distinguish the statements of Theorems \ref{Main Thm:Simultaneous} and \ref{Main Thm:Simultaneous linear}, we provide two complete proofs of both theorems.
\end{rmk}
|
2,877,628,090,770 | arxiv | \section*{References}
|
2,877,628,090,771 | arxiv | \section{Introduction}
\label{sec:int}
Experimental neutrino physics has regained great interest in the latest
years, with many new experiments presently taking data or in
preparation for the near future. This is justified because although
the Standard Model has been vigorously tested experimentally and
seems to be a remarkably successful description of nature,
its neutrino sector has yet been poorly scrutinized.
We believe that this still mysterious
area of particle physics may give us some hint on the physics beyond
the Standard Model.
It is a common prejudice in the literature to assume
the conservation of the leptonic number and to think about
neutrinos as Dirac particles much lighter than any of the charged
leptons we know. Nevertheless there are no theoretically compelling
reasons why the leptonic number should be a conserved quantity or
why neutrinos should not have a mass comparable to the charged fermions.
It is clear that only the confrontation of theory
with experimental data will eventually clarify the problem of
neutrino mass and nature.
Many direct limits on neutrino mass have been obtained by different
experimental groups~\cite{pdg98} but are not all accepted without
controversy~\cite{tau4,pdg}. Experiments also have been carried out
to try to measure neutrinoless double-$\beta$ decay which, in general,
is a process that will not occur unless one has a Majorana neutrino
involved as an intermediate particle. Here also experiments have
obtained only limits on the so called effective neutrino mass~\cite{hmbb}.
As a rule experimental analysis are model dependent and cannot be
quoted as a general result.
In the hope of contributing to the understanding of neutrinos physics
we have accomplished a comprehensive study of the constraints imposed
by recent experimental data on lepton decays, pion and kaon leptonic
decays as well as by the $Z^0$ invisible width
measurement performed by the LEP experiments to the simplest model
containing Majorana neutrinos.
We will consider a very simple extension of the standard electroweak
model which consists in adding to its particle content a right-handed
neutrino transforming as a singlet under $\text{SU(2)}_L\otimes \text{U(1)}_Y $.
This will be referred as the Minimal Model with Right-handed Neutrino (MMRN). Next,
by allowing it to mix with all the left-handed neutrinos we obtain
that there are, at the tree level, two massless neutrinos ($m_1$, $m_2$)
and two massive ones ($m_P$, $m_F$)~\cite{cj}.
It is interesting to note that this simple extension of the Standard
Model imposes a mass hierarchy for neutrinos.
The massless neutrinos ($m_1$, $m_2$) can acquire
very small mass by radiative corrections~\cite{babu,gandhi}. This seems to
be consistent with the recent evaluation of the the number of light
neutrino species from big bang nucleosynthesis~\cite{nucs}.
The outline of this work is as follows. In Sec.~\ref{sec:mod} the
model consider is briefly reviewed. In Sec.~\ref{sec:lepmix}
we consider the effects of mixing for the decay width of the muon,
for the partial leptonic decay widths of the tau, pion and kaon
and for the $Z^0$ invisible width. These are the quantities that are calculated
theoretically. In Sec.~\ref{sec:ana} we compare our theoretical
results with recent experimental data and obtain from this comparison
allowed regions for mixing angles and masses. In Sec.~\ref{sec:dbeta}
we investigate the possibility of further constraining our results with
the present best limit from neutrinoless double-$\beta$ decay experiments.
Finally, in the last section we establish our conclusions.
\section{A brief description of the model}
\label{sec:mod}
In the MMRN the most general form of the neutrino mass term is
\begin{equation}
{\cal L}_{\nu}^{M}
=-\sum_{\alpha=e,\mu,\tau} a_\alpha \overline{{\nu_{\alpha}}}_L
N_{R}-\frac{1}{2}M \overline{N^{c}_{R}} N_{R}+H.c.
\label{mass}
\end{equation}
where the left-handed neutrino fields are the usual flavor eigenstates and we
have assumed that the charged leptons have already been diagonalized. In this
model, there are four physical neutrinos $\nu_1,\nu_2,\nu_P$ and
$\nu_F$, the first two are massless ($m_1=m_2=0$) and the last
two are massive Majorana neutrinos with masses
\begin{equation}
m_P=\frac{1}{2}(\sqrt{M^2+4a^2}-M)~ \text{ and } ~ m_F=\frac{1}{2}(\sqrt{M^2+4a^2}+M),
\label{mass2}
\end{equation}
where $a^2=a_e^2+a_\mu^2+a_\tau^2$.
In terms of the physical fields the charged current interactions are
\begin{equation}
{\cal L}^{CC}= \frac{g}{\sqrt{2}}
\begin{array}{cccc}(\overline{\nu_{1}}& \overline{\nu_{2}}& \overline{\nu_{P}}&
\overline{\nu_{F}}\end{array})_L\gamma^\mu \Phi R
\left(\begin{array}{c}
e\\ \mu\\ \tau \\ 0 \end{array}\right)_LW_\mu^++H.c.,
\label{cc}
\end{equation}
where $\Phi=diag(1,1,i,1)$ and $R$ is the matrix
\begin{equation}
\left(
\begin{array}{cccc}
R_{e1} & R_{\mu 1}& R_{\tau 1} & R_{0 1} \\ R_{e2} & R_{\mu 2} &
R_{\tau 2} & R_{0 2} \\ R_{eP} & R_{\mu P} & R_{\tau P} & R_{0 P}
\\ R_{eF} & R_{\mu F} & R_{\tau F} & R_{0 F} \\
\end{array}\right)
=\left(
\begin{array}{cccc}
c_\beta &-s_\beta s_\gamma & -s_\beta c_\gamma & 0 \\ 0 & c_\gamma &
-s_\gamma & 0 \\ c_\alpha s_\beta\; & c_\alpha c_\beta s_\gamma\;
&c_\alpha c_\beta c_\gamma\; &-s_\alpha \\ s_\alpha s_\beta \;&
s_\alpha c_\beta s_\gamma\; & s_\alpha c_\beta c_\gamma\; &
c_\alpha
\end{array}\right).
\label{mm}
\end{equation}
In Eq.~(\ref{mm}) $c$ and $s$ denote the cosine and the sine of the
respective arguments. The angles $\alpha,\beta$ and $\gamma$ lie in
the first quadrant and are related to the mass parameter as follows
\begin{equation}
s_\alpha=\sqrt{m_P/(m_P+m_F)},
\label{angles}
\end{equation}
\begin{equation}
s_\beta=a_e/a,\;c_\beta s_\gamma=a_\mu/a,\;c_\beta
c_\gamma=a_\tau/a.
\label{angles2}
\end{equation}
The choice of parameterization is such that for
$\alpha=\beta=\gamma=0$ , $\nu_1\rightarrow\nu'_e$,
$\nu_2\rightarrow\nu'_\mu$ and $\nu_P\rightarrow\nu'_\tau$.
The neutral current interactions for neutrinos written in the
physical basis of MMRN read
\begin{equation}
{\cal L}^{NC}= \frac{g}{4 \cos\theta_W} \left(\begin{array}{cccc}
\overline{\nu_{1}}&\overline{\nu_{2}}&\overline{\nu_{P}}&\overline{\nu_{F}}\end{array}\right)_L
\gamma^\mu\left(\begin{array}{cccc}
1 \;& 0\; & 0 & 0 \\ 0 \;& 1\; & 0 & 0 \\ 0 \;& 0 \;& c^2_\alpha &
ic_\alpha s_\alpha \\ 0 \;& 0 \;& -ic_\alpha s_\alpha &
s^2_\alpha\end{array}\right)
\left(\begin{array}{c}
\nu_{1}\\ \nu_{2} \\ \nu_{P} \\ \nu_{F}\end{array}\right)_L Z_\mu+H.c.
\label{nc}
\end{equation}
Notice that there are four independent parameters in MMRN.
We will choose them to be the angles $\beta$ and $\gamma$
and the two Majorana masses $m_P$ and $m_F$.
These are the parameters that we will constrain with experimental data.
\section{Four generation mixing in the leptonic sector}
\label{sec:lepmix}
In this section we will present the expressions that will be used in
our analysis for muon and tau leptonic decays, pion and kaon leptonic decays
and the $Z^0$ invisible width. The coupling constant $G$ and the decay
constants $F_\pi$ and $F_K$ used in our theoretical expressions have
not the same values of the standard $G_\mu$, $f_\pi$ and $f_K$ given
in Ref.~\cite{pdg98}, this important point will be discussed at the
end of this section.
\subsection{Lepton decays}
\label{sec:ldec}
We can now write the most general expression for the partial decay
width of a lepton $l'$ into a lepton $l$ and two neutrinos $\bar \nu_{l}
\nu_{l'}$ in the context of MMRN as
\begin{eqnarray}
\Gamma(l'\to l\bar\nu_l\nu_{l'})&=&\frac{G^2m^5_{l'}}{192\pi^3}
{\cal R}^{l'} \left\{
\left(\vert R_{l'1}\vert^2+\vert R_{l'2}\vert^2\right)
\left(\vert R_{l1}\vert^2+\vert
R_{l2}\vert^2\right)\,\Gamma^{l'l}_{11}\right.\nonumber \\ &+&
\left.\left(\vert R_{lP}\vert^2[\vert R_{l'1}\vert^2+\vert R_{l'2}\vert^2]
+\vert R_{l'P}\vert^2[\vert R_{l1}\vert^2+\vert R_{l2}\vert^2]
\right)\,\Gamma^{l'l}_{1P}\right.\nonumber \\
&+&
\left.\left(\vert R_{lF}\vert^2[\vert R_{l'1}\vert^2+\vert R_{l'2}\vert^2]
+\vert R_{l'F}\vert^2[\vert R_{l1}\vert^2+\vert R_{l2}\vert^2]
\right)\,\Gamma^{l'l}_{1F}\right.\nonumber \\
&+&
\left.\left(\vert R_{lF}\vert^2 \vert R_{l'P}\vert^2+
\vert R_{l'F}\vert^2 \vert R_{lP}\vert^2
\right)\,\overline{\Gamma}^{l'l}_{PF}\right.\nonumber \\
&+&\left.\vert R_{l'P}\vert^2 \vert
R_{lP}\vert^2\,\overline{\Gamma}^{l'l}_{PP} + \vert R_{l'F}\vert^2
\vert R_{lF}\vert^2\,\overline{\Gamma}^{l'l}_{FF} \right\},
\label{decay}
\end{eqnarray}
with $l'=\mu,\tau$ and $l=e,\mu$ for the tau decays and $l=e$ for
the muon decay. Notice that $G^2$ in Eq.\ (\ref{decay}) is the
universal constant defined as $G^2/\sqrt2=g^2/8m^2_W$.
In Eq.\ (\ref{decay}) we have used the integrals
\begin{equation}
\Gamma^{l'l}_{11}=2\int^{t_M}_{t_m}(t^2-B)^{\frac{1}{2}}[t(3k-2t)-B]dt,
\label{r0}
\end{equation}
\begin{eqnarray}
\Gamma^{l'l}_{1J} & = &
2\int^{t_M}_{t_m}(t^2-B)^{\frac{1}{2}}\frac{(k-\delta_{J
l'}^2-t)}{(k-t)^3}
\left[(k-\delta_{J l'}^2-t)^2t(k-t) \right. \nonumber \\
& + &\left. [(k-t)^2+\delta_{J l'}^2(k-t)-2\delta_{J l'}^4]
(2kt-t^2-B)\right] \theta(m_{l'}-m_l-m_J) dt,
\label{r3}
\end{eqnarray}
\begin{equation}
\overline{\Gamma}^{l'l}_{JJ'} = \Gamma^{l'l}_{JJ'} + \epsilon_{JJ'} {\Gamma}'^{l'l}_{JJ'},\quad \epsilon_{JJ'}= \left\{ \begin{array}{rl} 1 & (J=J') \\
-1 & (J\neq J') \end{array} \right.
\label{baradd}
\end{equation}
\begin{eqnarray}
\Gamma^{l'l}_{JJ'} & = &
2\int^{t_M}_{t_m}(t^2-B)^{\frac{1}{2}}C_{JJ'}
\left[2(k-t)tC^2_{JJ'}\right. \nonumber \\ &+&\left.(2kt-t^2-B)B_{JJ'}
\right]
\theta((m_{l'}-m_l)-m_J-m_{J'}) dt, \label{rFP}
\end{eqnarray}
\begin{eqnarray}
{\Gamma'}^{l'l}_{JJ'} & = &
-12\int^{t_M}_{t_m}(t^2-B)^{\frac{1}{2}}C_{JJ'}\delta_{J l'}\delta_{J'
l'}
\theta((m_{l'}-m_l)-m_J-m_{J'}) dt, \label{rFPL}
\end{eqnarray}
with
\begin{equation}
k=1+\delta^2_{ll'},\quad B=4(k-1),\quad
\delta_{Jl'}=\frac{m_{J}}{m_{l'}},\quad
\delta_{ll'}=\frac{m_l}{m_{l'}},
\label{def1}
\end{equation}
\begin{equation}
t_m=2\delta_{ll'},\quad t_M=k-\frac{(m_{i}+m_{j})^2}{m^2_{l'}},
\label{def2}
\end{equation}
\begin{equation}
C_{JJ'}=\frac{\left[
(k-t)^2+(\delta_{Jl'}^2-\delta_{J'l'}^2)^2-2(\delta_{Jl'}^2+\delta_{J'l'}^2)(k-t)\right]^\frac{1}{2}}{k-t},
\label{cjj}
\end{equation}
\begin{equation}
B_{JJ'}=\frac{2}{(k-t)^2}\left[
(k-t)^2-2(\delta_{Jl'}^2-\delta_{J'l'}^2)^2+(\delta_{Jl'}^2
+\delta_{J'l'}^2)(k-t)\right],
\label{bjj}
\end{equation}
where $i,j=1,2,P,F$; $J,J'=P,F$; $m_{l(l')}$ are the corresponding lepton
masses; $\Gamma^{l'l}_{11}$ and
$\Gamma^{l'l}_{1J}$ are respectively the phase space
contributions to the $l'\to l\bar\nu_{l}\nu_{l'}$
decays for two massless and one massive
neutrino (for either Dirac or Majorana type neutrinos)~\cite{sharma}.
If the final state neutrinos were two
massive Dirac neutrinos the contribution would be simply
$\Gamma^{l'l}_{JJ'}$, but since here they are Majorana neutrinos
there is an additional contribution ${\Gamma'} ^{l'l}_{JJ'}$.
The quantity ${\cal R}^{l'}$ describes the leading radiative
corrections to the lepton decay process that can be found in the
Appendix.
Explicitly using the parameterization given in Eq.\ (\ref{mm}) and defining
$x=s_\beta^2$, $y=s_\gamma^2$ and $z=s_\alpha^2$ we obtain
\begin{equation}
\Gamma(\mu\to e\nu_\mu\bar\nu_e)=\Gamma^{\mu e}=\frac{G^2m^5_\mu}{192\pi^3}{\cal R}^{\mu}
f^{\mu e}(x,y,\delta_{e\mu},\delta_{P \mu},\delta_{F \mu}),
\label{def3}
\end{equation}
for the partial rate of the muon decay into electron, and
\begin{equation}
\Gamma(\tau\to e\nu_\tau\bar\nu_e)=\Gamma^{\tau e}=\frac{G^2m^5_\tau}{192\pi^3}
{\cal R}^{\tau}f^{\tau e}(x,y,\delta_{e\tau},\delta_{P
\tau},\delta_{F \tau}),
\label{def4}
\end{equation}
\begin{equation}
\Gamma(\tau\to \mu\nu_\tau\bar\nu_\mu)=\Gamma^{\tau \mu}=\frac{G^2m^5_\tau}{192\pi^3}
{\cal R}^{\tau}f^{\tau
\mu}(x,y,\delta_{\mu\tau},\delta_{P\tau},\delta_{F\tau}),
\label{def41}
\end{equation}
for the partial widths of the tau decay into electron and muon, respectively.
The following definitions were used
\begin{eqnarray}
f^{\mu e}(x,y,\delta_{e\mu},\delta_{P \mu},\delta_{F \mu})
&=&\left[ (xy+(1-y))(1-x)\,\Gamma^{\mu
e}_{11} \right . \nonumber \\ &+& \left.
(1-z)(x^2y+x(1-y)+(1-x)^2y)\Gamma^{\mu e}_{1P}\right.\nonumber \\
&+&
\left. z(x^2y+
x(1-y)+(1-x)^2y)\Gamma^{\mu e}_{1F} \right.\nonumber \\ &+& \left.
2((1-z)xz(1-x)y)\overline{\Gamma}^{\mu e}_{PF} \right.\nonumber \\
&+& \left. (1-z)^2y(1-x)x\,\overline{\Gamma}^{\mu e}_{PP}
+z^2y(1-x)x\,\overline{\Gamma}^{\mu e}_{FF}
\right],
\label{mue}
\end{eqnarray}
\begin{eqnarray}
f^{\tau e}(x,y,\delta_{e\tau},\delta_{P
\tau},\delta_{F \tau})&=&\left[(
x(1-y)+y)(1-x)\,\Gamma^{\tau e}_{11}\right.\nonumber \\ &+& \left.
(1-z)(x^2(1-y)+xy+(1-x)^2(1-y))\,\Gamma^{\tau e}_{1P}\right.\nonumber
\\ &+&\left. z(x^2(1-y)+xy+(1-x)^2(1-y))\,\Gamma^{\tau
e}_{1F}\right.\nonumber \\
&+&\left.2((1-z)(1-x)(1-y)zx)\,\overline{\Gamma}^{\tau
e}_{PF}\right.\nonumber \\
&+&\left.(1-y)(1-x)x(1-z)^2\,\overline{\Gamma}^{\tau e}_{PP}
+(1-y)(1-x)xz^2\,\overline{\Gamma}^{\tau e}_{FF}
\right],
\label{taue}
\end{eqnarray}
\begin{eqnarray}
f^{\tau
\mu}(x,y,\delta_{\mu\tau},\delta_{P\tau},\delta_{F\tau})
&=&\left[(x(1-y)+y)(xy+(1-y))\,
\Gamma^{\tau\mu}_{11}
\right.\nonumber
\\ &+&\left. \left(y(x(1-y)+
y)+(1-y)(xy+(1-y)) \right)(1-z)(1-x)\,
\Gamma^{\tau\mu}_{1P}
\right.\nonumber
\\ &+&\left. \left(y(x(1-y)+
y)+(1-y)(xy+(1-y)) \right)z(1-x)\,
\Gamma^{\tau\mu}_{1F}
\right.\nonumber
\\ &+&\left. 2 \left( z(1-x)^2y(1-z)(1-y) \right)\,
\overline{\Gamma}^{\tau\mu}_{PF}
\right.\nonumber
\\ &+&\left.
(1-y) y (1-x)^2 (1-z)^2\,\overline{\Gamma}^{\tau\mu}_{PP} +(1-y)y
(1-x)^2z^2\,\overline{\Gamma}^{\tau\mu}_{FF}\right].
\label{taumu}
\end{eqnarray}
\subsection{Pion and Kaon leptonic decays}
\label{sec:hdec}
We will also consider decays such as $h\to l+\nu_l$; where
$h=\pi , K ~\mbox{ and } ~l=e,\mu$.
The partial width for the leptonic decay of hadrons in MMRN is
\begin{eqnarray}
\Gamma(h \rightarrow l \nu_l) &=& \Gamma^{h l}\nonumber\\
&=& \frac{G^2F^2_h V^2_{KM} m^3_h}{8\pi} {\cal R}_{hl} f^{hl}(x,y,\delta_{h
l},\delta_{P l},\delta_{F l}),
\label{hadw}
\end{eqnarray}
with $m_h$ being the mass of the hadron $h$ and
\begin{eqnarray}
f^{hl}(x,y,\delta_{h l},\delta_{P l},\delta_{F l})
&=&\left[ \left(\vert R_{l1}\vert^2 +\vert R_{l2}\vert^2\right)\Gamma^{h l}_1+
\vert R_{lP}\vert^2 \Gamma^{h l}_P+ \vert R_{lF}\vert^2 \Gamma^{h l}_F
\right],
\label{hadwdef}
\end{eqnarray}
where $\Gamma^{h l}_1$ is the massless neutrino contribution given by
\begin{equation}
\Gamma^{h l}_1=(\delta_{hl}^2-\delta_{hl}^4) \lambda^\frac{1}{2}(1,\delta_{hl}^2,0),
\label{hmassl}
\end{equation}
and $\Gamma^{h l}_J$ are the massive neutrino contributions~\cite{ro}
\begin{equation}
\Gamma^{h l}_J=
\left[\delta^2_{hl}+\delta^2_{Jl}-(\delta^2_{hl}-\delta^2_{Jl})^2\right]
\lambda^{\frac{1}{2}}(1,\delta^2_{lh},\delta^2_{Jl})\theta(m_h-m_l-m_J),
\label{piw}
\end{equation}
$J=P,F$, $\delta_{hl}=m_l/m_h$, $V^2_{KM}$ is the appropriate
Cabibbo-Kobayashi-Maskawa matrix element of the quark sector and
$\lambda$ is the triangular function defined by
\[\lambda(a,b,c)=a^2+b^2+c^2-2(ab+ac+bc).\]
The quantity ${\cal R}_{hl}$ in Eq.(\ref{hadw}) represents the leading
radiative corrections to the hadron $h$ decay given in the Appendix.
In particular when the final state is a muon we have
\begin{eqnarray}
f^{h\mu}(x,y,\delta_{h \mu},\delta_{P \mu},\delta_{F \mu}) &=&\left(\vert
R_{\mu1}\vert^2+ \vert R_{\mu2}\vert^2 \right)\Gamma^{h\mu}_1+
\vert R_{\mu P}\vert^2\Gamma^{h\mu}_P +\vert R_{\mu F}\vert^2\Gamma^{h\mu}_F \nonumber \\&=&
\left(yx+1-y \right)\Gamma^{h\mu}_1+
y(1-x)(1-z)
\Gamma^{h\mu}_P+ y(1-x)z
\Gamma^{h\mu}_F,
\label{pimu}
\end{eqnarray}
and when the final state is an electron
\begin{eqnarray}
f^{h e}(x,y,\delta_{h e},\delta_{P e},\delta_{F e}) &=&\left( \vert
R_{e1}\vert^2 +\vert R_{e2}\vert^2\right)\Gamma^{h e}_1+
\vert R_{eP}\vert^2 \Gamma^{h e}_P+ \vert R_{eF}\vert^2 \Gamma^{h e}_F
\nonumber \\ &=&
(1-x)\Gamma^{h e}_1+ (1-z)x \Gamma^{h e}_P +zx \Gamma^{h e}_F.
\label{pie}
\end{eqnarray}
\subsection{$Z^0$ invisible width}
\label{sec:Zinv}
In this section we will extend and update our previous analysis in
Ref.~\cite{cec}. In the MMRN scheme the $Z^0$ partial invisible width can
be written as~\cite{cj}
\begin{equation}
\Gamma^{\mbox{\scriptsize inv}}(Z\rightarrow
\nu's)=\Gamma_0 (2+( 1-z^2) \chi_{PP}+2 ( 1-z ) z \chi_{PF}
+ z^2 \chi_{FF}),
\label{width}
\end{equation}
where $\Gamma_0$ is given by
\begin{equation}
\Gamma_0=\frac{G M^3_{Z}}{6\sqrt{2}\pi}( \bar g^2_V + \bar g^2_A),
\end{equation}
and the electroweak corrections to the width are incorporated in the
couplings $\bar g_V$ and $\bar g_A$,
\begin{equation}
\chi_{ij}=\frac{\sqrt{\lambda (M^2_Z,m^2_i,m^2_j)}}{M^2_Z} X_{ij}\theta(M_Z-m_i-m_j),
\label{chi}
\end{equation}
here $i,j=P,F$; $\lambda$ is the usual triangular function already
defined and $X_{ij}$ include the mass dependence of the matrix
elements. Explicitly,
\[X_{PP}=1-4\frac{m^2_P}{M^2_Z}, \]
\[X_{FF}=1-4\frac{m^2_F}{M^2_Z}, \]
\begin{equation}
X_{FP}=1-\frac{\Delta m^2_{FP}}{2M^2_Z}-\frac{m^2_P+3m_Fm_P}{M^2_Z}
-\frac{(\Delta m^2_{FP})^2}{4M^4_Z},
\label{x}
\end{equation}
where we have defined $\Delta m^2_{FP}=m^2_F-m^2_P$. Thus,
$\chi_{ij}$ are bounded by unity whereby
\begin{equation}
\Gamma^{\mbox{\scriptsize inv}}(Z\rightarrow \nu's)\leq 3\Gamma_0.
\label{width2}
\end{equation}
\subsection{Comment on $G$ and $F_h$}
\label{sec:const}
It is common to assume that
standard processes will practically not be affected, at tree level,
by the introduction of new physics, and that the most effective way
of constraining new physics is by looking at exotic processes.
This is correct in most situations envisaged in the literature.
For instance in Ref.~\cite{ng} the
emphasis is given to lepton flavor violation processes like
$\mu\rightarrow e\gamma$.
Nevertheless we would like to point out that constants
used in the standard weak decays may take different values as a consequence
of mixing.
The experimental value for the muon decay constant, $G_\mu$, is
obtained by comparing the Standard Model formula for the muon decay width
\begin{equation}
\Gamma^{\text{SM}}(\mu\to e\bar\nu_e\nu_{\mu})=\frac{G^2_{\mu}m^5_{\mu}}{192\pi^3}
{\cal R}^{\mu} \Gamma_{11}^{\mu e},
\label{gu}
\end{equation}
with the measured muon lifetime. As the error obtained in this way is
very small, $G_\mu$ is often used as an input in the calculations of radiative
corrections~\cite{franchioti}.
Now if we have mixing the expression for the
muon decay width is modified as in Eq.~(\ref{def3}). So that
comparing this equation with Eq.~(\ref{gu}), it is clear that
the numerical value of $G_\mu$ is not equal to the numerical value of $G$,
as a general rule, independently of the accuracy of $G_\mu$ determination.
They are related by:
\begin{equation}
G^2 = \frac{\Gamma_{11}^{\mu e} G_\mu^2}{ f^{\mu
e}(x,y,\delta_{e\mu},\delta_{P \mu},\delta_{F \mu})}.
\label{gmug}
\end{equation}
From Eqs.~(\ref{mue}) and (\ref{gmug}) we see that $G \geq G_\mu$.
A consequence of this is that the $Z^0$ invisible decay width
\begin{equation}
\Gamma^{\mbox{\scriptsize inv}}(Z\rightarrow \nu's) \leq 3\Gamma_0
= 3 \frac{G}{G_\mu} \Gamma_0^{\mbox{\scriptsize SM}},
\label{Zlim}
\end{equation}
could, in principle, even exceed $3~ \Gamma_0^{\mbox{\scriptsize SM}}$,
where $\Gamma_0^{\mbox{\scriptsize SM}}$ is the Standard Model width.
In a similar way the experimental value of the pseudoscalar meson decay
constant $f_h$ is obtained by
comparing the Standard Model prediction for the hadron
leptonic decay width
\begin{equation}
\Gamma^{\text{SM}}(h \rightarrow l \nu_l) = \frac{G^2_{\mu}f^2_h V^2_{KM}m^3_h}{8\pi}
{\cal R}_{hl} \Gamma^{h l}_1,
\label{hadw_sm}
\end{equation}
with experimental data.
The values of $f_h$ quoted in PDG depend on the type of
radiative corrections used~\cite{finke1,sirlin}. The extracted values
$f_\pi=130.7\pm0.4$ MeV and
$f_K=159.8\pm1.5$ MeV~\cite{pdg98},
were obtained using the expression of
${\cal R}_{hl}$ as in our Appendix.
Here also the numerical values of $F_\pi$ and $F_K$ are not equal to
the numerical values of $f_\pi$ and $f_K$ given above, since
the constant $F_h$ that appears in Eq.~(\ref{hadw}) is
related to $f_h$ in Eq.~(\ref{hadw_sm}) by
\begin{equation}
\Gamma_1^{h \mu} G^2_\mu f^2_h = G^2 F^2_h f^{h \mu}(x,y,\delta_{h \mu},
\delta_{P l},\delta_{F l}).
\label{Ffhad}
\end{equation}
\section{Experimental constraints on mixing angles and neutrino masses}
\label{sec:ana}
As we explained in the previous section the values of $G^2$ and
$F_h$ are unknown in MMRN. So we will used theoretical ratios to eliminate
the dependence on these parameters to compare our expressions with
experimental results.
We will now write down the theoretical expressions that can be directly
compared to the experimental data found in Table~\ref{tab1}.
Using Eqs.~(\ref{def3}) -- (\ref{def41}) we obtain
\begin{equation}
\left(\frac{m_{\mu}}{m_{\tau}}\right)^5 \frac{\Gamma^{\tau e}}
{\Gamma^{\mu e}}= \frac{{\cal R}^{\tau}f^{\tau
e}(x,y,\delta_{e\tau},\delta_{P\tau},\delta_{F\tau})}{{\cal R}^{\mu}
f^{\mu e}(x,y,\delta_{e\mu},\delta_{P\mu},\delta_{F\mu})}=
\left(\frac{m_{\mu}}{m_{\tau}}\right)^5
\frac{B^{\tau e}\tau_{\mu}}{B^{\mu e}\tau_{\tau}} \equiv
\left(\frac{G_\tau}{G_\mu}\right)^2 ,
\label{tu}
\end{equation}
with $\tau_{\tau}$ and $\tau_{\mu}$ being respectively the tau and the
muon lifetimes, $B^{l'l}$ the branching ratio for the decay
$l' \to l \bar \nu_l \nu_{l'}$ and
\begin{equation}
\frac{\Gamma^{\tau \mu}}
{\Gamma^{\tau e}}=
\frac{f^{\tau
\mu}(x,y,\delta_{\mu\tau},\delta_{P\tau},\delta_{F\tau})}{f^{\tau e}
(x,y,\delta_{e\tau},\delta_{P\tau},\delta_{F\tau})}=
\frac{B^{\tau \mu}}{B^{\tau e}}.
\label{taufra}
\end{equation}
From Eqs.~(\ref{hadw}), (\ref{pimu}) and (\ref{pie}) we obtain for the pion
decays
\begin{equation}
\frac{\Gamma^{ \pi e}}{\Gamma^{\pi \mu}}=
\frac{{\cal R}_{\pi e} f^{\pi e}(x,y,\delta_{\pi e},\delta_{P e},\delta_{F e})}
{{\cal R}_{\pi\mu}
f^{\pi \mu}(x,y,\delta_{\pi \mu},\delta_{P \mu},\delta_{F \mu})}
=\frac{B^{\pi e}}{B^{\pi\mu}},
\label{piemu}
\end{equation}
where $B^{\pi l}$ is the branching ratio for the decay
$\pi \to l \nu_l $ ($l=\mu,e$).
For the kaon decays an alike expression can be derived. Before we give
this expression we would like to make some remarks.
Kaon leptonic decay measurements are not only less precise than the pion
leptonic decay ones but also suffer from an important background
contamination.
The average leptonic width given in PDG is dominated by the result of one
experiment, the CERN-Heidelberg experiment\cite{he1,he2}.
In order to avoid the contamination of
$K_{l2}$ ($K^{+}\rightarrow l^{+}\nu _l$) events
by beta decay $K_{l3}$ $(K^{+}\rightarrow l^{+}\nu _l\pi ^0)$ events,
experimentalists are forced to impose a cut in the measured
momentum of the final charged lepton.
For massless neutrinos in $K_{l2}$ decays
one expects the momentum $p_l$ ($l=e,\mu$), to be monochromatic
i.e., $p_e=247$~MeV for the electron channel and $p_\mu =236$~MeV
for the muon channel.
Based on this, $K_{e2}$ events are experimentally characterizes as
having $240$~MeV $\leq p_e\leq 260$~MeV and
$K_{\mu2}$ events as having $220$~MeV $\leq p_\mu \leq 252$~MeV~\cite{he1,he2}.
If neutrinos produced in these decays are massive we expected
as many lines in the spectrum of charged lepton as the number of
massive neutrinos. For a massive neutrino with mass $m_i$
\[
m_K=\sqrt{p_l(m_i)^2+m_l^2}+\sqrt{p_l(m_i)^2+m_i^2},
\]
which can be solved in terms of the final lepton momentum, $p_l(m_i)$,
giving~\cite{winter}
\begin{equation}
p_l(m_i)=p_l(0)\sqrt{1-\frac{2\left( m_K^2+m_l^2\right) m_i^2-m_i^4}
{4m_K^2p_l(0)^2}},
\label{p_i}
\end{equation}
where $m_l$ is the mass of the charged lepton and
$m_K$ is the mass of the kaon and $p_l(0)$ is the momentum for a
massless neutrino $p_l(0)=\displaystyle\frac{m_K^2-m_l^2}{2m_K}$.
The experimental lower cut in the momentum of the final lepton
together with Eq.(\ref{p_i}) imply a maximum value for the observable
neutrino mass~\cite{ro}. Explicitly for $p_e~>~240$~MeV we have
$m_i<m_{e}^{cut}=82$~MeV and for
$p_{\mu}>220$~MeV, $m_i<m_{\mu}^{cut}=118$~MeV. That means, neutrinos
with a mass greater than $118$~MeV are not visible in either of
these decays.
These restrictions imply that Eqs.~(\ref{pimu}) and (\ref{pie})
will have to be changed for the kaon case :
$f^{K e} \rightarrow \hat f^{K e}$ where
\begin{equation}
\hat f^{K e}(x,y,\delta_{K e},\delta_{P e},\delta_{F e})
=(1-x) \Gamma^{K e}_1+
(1-z) x \Gamma^{K e}_P\theta(m_{e}^{cut}-m_P)
+z x \Gamma^{K e}_F\theta(m_{e}^{cut}-m_F),
\label{ke}
\end{equation}
and also
$f^{K \mu} \rightarrow \hat f^{K \mu}$ where
\begin{eqnarray}
\hat f^{K \mu}(x,y,\delta_{K \mu},\delta_{P \mu},\delta_{F \mu})
&=&(yx+1-y)\Gamma^{K \mu}_1
+(1-z) (1-x) y \Gamma^{K\mu}_P \theta(m_{\mu}^{cut}-m_P) \nonumber \\
&+&y(1-x)z \Gamma^{K \mu}_F \theta(m_{\mu}^{cut}-m_F),
\label{kmu}
\end{eqnarray}
so that finally we have
\begin{equation}
\frac{\Gamma^{ K e}}{\Gamma^{K \mu}}=
\frac{{\cal R}_{K e} \hat f^{K e}(x,y,\delta_{K e},\delta_{P e},\delta_{F e})}
{{\cal R}_{K \mu}
\hat f^{K \mu}(x,y,\delta_{K \mu},\delta_{P \mu},\delta_{F \mu})}
=\frac{B^{K e}}{B^{K \mu}},
\label{kemu}
\end{equation}
where $B^{K l}$ is the branching ratio for the decay
$K \to l \nu_l $ ($l=\mu,e$).
For the $Z^0$ invisible width we use
\begin{equation}
\Gamma^{\mbox{\scriptsize inv}}(Z \rightarrow \nu's)
=\sqrt{\frac{\Gamma_{11}^{\mu e} G_\mu^2}{ f^{\mu
e}(x,y,\delta_{e\mu},\delta_{P \mu},\delta_{F \mu})}}
\Gamma_0^{\mbox{\scriptsize SM}}
(2+(1-z^2) \chi_{PP}+2(1-z)z \chi_{PF}
+z^2 \chi_{FF}).
\label{Zwidth}
\end{equation}
Now to establish the allowed regions for the free parameters of MMRN we have
built the $\chi^2$ function
\begin{equation}
\chi^2(x,y,m_P,m_F) = \sum_{i=1,5}\frac{(F_i-F_i^{\text{exp}})^2}{\sigma_i^2}
\label{Xi2},
\end{equation}
\noindent
where each $F_i$ is the theoretical value calculated using one of the
expressions given in Eqs.\ (\ref{tu}),(\ref{taufra}),(\ref{piemu}),
(\ref{kemu}) and (\ref{Zwidth}), and $F_i^{\text{exp}}$
and $\sigma_i$ are its corresponding
experimental value and error according to Table~\ref{tab1}.
We have minimized this $\chi^2$ function with respect to its four parameters.
The minimum $\chi^2$ found for one d.o.f. (five experimental data points
minus four free parameters) is $\chi^2_{\mbox{\scriptsize min}} = $ 1.29
for $x=.22~10^{-5}$, $y= 0.47$, $m_P=.28$ MeV and $m_F=1.10$ MeV,
this is a bit smaller than $\chi^2_{\mbox{\scriptsize SM}} = $ 1.33, that
we get for $x=y=z=0$. The error matrix corresponding to the result of our
minimization is:
\begin{equation}
\left(
\begin{array}{cccc}
V_{m_P m_P} & V_{m_P m_F} & V_{m_P x} & V_{m_P y} \\
V_{m_F m_P} & V_{m_F m_F} & V_{m_F x} & V_{m_F y} \\
V_{x m_P} & V_{x m_F} & V_{x x} & V_{x y} \\
V_{y m_P} & V_{y m_F} & V_{y x} & V_{y y} \\
\end{array}\right)
=\left(
\begin{array}{cccc}
.69~10^{-7} & .51~10^{-5} & .15~10^{-9} & 0. \\
.51~10^{-5} & .43~10^{-3} & .12~10^{-7} & 0. \\
.15~10^{-9} & .12~10^{-7} & .72~10^{-12} & 0. \\
0. & 0. & 0. & .36~10^{-11} \\
\end{array}\right).
\label{errmat}
\end{equation}
We have computed the 90\% C.L.\ contours determined by
the condition $ \chi^2 = \chi^2_{\mbox{\scriptsize min}} + 7.78 $.
In order to display our results we have fixed the values of $m_F$ and
presented the allowed regions in a $m_P \times y$ plot for several values
of $x$. We have chosen to display the allowed regions for four different
$m_F$ values to give an idea of the general behavior.
This is shown in Figs.\ \ref{fig1}.
We note that our $\chi^2$ function is very sensitive to changes in $x$
and $m_P$ but rather not so sensitive to $y$ or $m_F$. This behavior reflects
on the fact that the maximum possible value of $m_P$ for each contour
we have obtained, reached at $y \to 0$, is very sensitive to $x$ but not so
sensitive to $m_F$.
For $x>10^{-4}$ we see that the maximum allowed $m_P$ depends on $m_F$
but is almost independent of $y$. In fact,
this is expected as all our expressions become independent
of $y$ as $x \rightarrow 1$.
The absolute maximum allowed value of $m_P$,
for $x,y \to 0 $, consistent with the data is $\simeq$ 40 MeV.
This is still true even if $m_F>$ 1 TeV.
We observe that the contours in the $m_P \times y$ plane have basically
the same shape and allow for a lower maximum value of $m_P$ as a function
of $y$ and as $m_F$ decreases. Nevertheless there are two values for $m_F$
that change the behavior of the allowed contours. This is due to the fact
that the presence of massive neutrinos in the considered decays depends
on kinematical constraints.
At $m_F = m_K-m_e$ higher values of $m_P$ as a function of $y$ become
possible, here $m_F$ starts to participate in kaon decays. At
this point the contour curve changes a little bit its shape and becomes less
restrictive. From then on, as $m_F$ decreases, the allowed curves
share once more the same shape and start again to constrain the parameters.
At $m_F = m_\pi-m_\mu$ we have a new change of behavior and higher values of
$m_P$ become allowed since now $m_F$ can participate of all pion
decays. Again after that for smaller values of $m_F$ the curves
will confine even more the parameters.
In Fig.\ \ref{fig1}(a) we see bellow each one of the curves the allowed
regions, at 90\% C.L.,
of $m_P$ as a function of $y$ for $m_F=$ 1 TeV and four different
values of $x$.
In Fig.\ \ref{fig1}(b) we see the same contours for $m_F=$ 1 GeV. We note
that the allowed regions are not much more limited than in the previous
case even though we have decreased $m_F$ by three orders of magnitude.
In Fig.\ \ref{fig1}(c) we see the allowed contours for $m_F=$ .1 GeV. Here
we have already passed by $m_F = m_K-m_e$ where the first change in
behavior occurred.
Finally in Fig.\ \ref{fig1}(d) we see the allowed contours for
$m_F=$ 10 MeV. Some comments are in order here. One can see that the allowed
regions in this case, although $m_F$ is much smaller than in
Fig.\ \ref{fig1}(c) are less restrictive. This is because we have crossed
the value $m_F = m_\pi-m_\mu$ as explained above.
Note also that for the lowest values of $x$ the curves are interrupted
by the condition that $m_P \leq m_F$, this means that for $y \lesssim 0.15$
the only prerequisite is $m_P \leq m_F$.
For $10^{-2} \leq x \leq 1$ the maximum allowed $m_P$ is really independent
of $y$. This case can be subdivided into three regions:
(i) for $m_F >$ 495 MeV, $m_P^{\mbox{\scriptsize max}}$ is also
independent of $m_F$ as can be seen in Table~\ref{tab2};
(ii) for smaller values of $m_F$ the product
$m_P^{\mbox{\scriptsize max}} \times x$ is constant with $m_F$ as shown
in Table~\ref{tab3} and (iii) for $m_F<$ 43 keV there is no restriction
on $x$ and $y$ for $m_P \leq m_F$.
Note that our analysis was done in the context of a specific model and
that we did not impose the ad hoc limit to neutrino masses
used in Ref.~\cite{Ng}.
Some general remarks about our results are in order here.
The $Z^0$ invisible width measurement at LEP along with the pion decay
data were by far the most significant experimental constraints to the model
parameters. The invisible width is today a extremely precise measurement and
as one should expect imposes great restrictions on neutrinos couplings.
The pion decay measurements are also very precise and being phase space
limited two body decays they have great power in constraining neutrino
masses and couplings as long as they can participate in pion decays.
On the other hand the kaon decay and the lepton decay data we have analyzed
have not been so effective in constraining the model. Kaon decays
unfortunately suffer from experimental contamination which makes their data
less useful at the present moment than one should hope it to be.
We would expect that experimental improvements here would affect
our results. The $\mu$ and $\tau$ lepton decays are three body decays
containing two neutrinos in the final state. This explain the fact that
although the experimental measurements are quite accurate the overall
effect of these data is not so constrictive to masses and couplings of
individual neutrinos.
\section{Neutrinoless double-$\beta$ decay}
\label{sec:dbeta}
Besides the experimental limits already imposed by the decays in the previous
section, since our neutrinos have Majorana nature, we can hope to further
restrict the mixing parameters of the model by imposing the constraint
coming from the non observation of neutrinoless double-$\beta$ decays,
i.e. (A,Z)$\to $(A,Z+2) $+~ 2 e^-$ transitions.
This type of process can be analyzed in terms of an effective neutrino
mass $ \langle m_\nu \rangle$ given in MMRN by~\cite{doi}
\begin{equation}
\langle m_\nu \rangle =\sum_{i=P,F} (\Phi R)_{ei}^2 m_i F(m_i,A),
\label{mnu}
\end{equation}
where $F(m_i,A)$ is the matrix element for the nuclear transition which is a
function of the neutrino mass $m_i$.
This has been computed in the literature for a number of different
nuclei as the ratio~\cite{klapdor}
\begin{equation}
F(m_i,A) = \frac{M_{GT}(m_i)-M_F(m_i)}{M_{GT}(0)-M_F(0)}.
\label{rmj}
\end{equation}
The best experimental limit on neutrinoless double-$\beta$ decay
comes from the observation of the nuclear transition
$^{76}$Ge~$\rightarrow ^{76}$Se.
The result of the calculation of the nuclear matrix element $F(m_i,A)$ for
$^{76}$Ge~$\rightarrow ^{76}$Se transitions can be found in
Ref.~\cite{klapdor} and we will now refer to this simply as $F(m_i)$.
This ratio is unity for $m_i \lesssim $ 40 MeV.
For 40 MeV $ < m_i < $ 1 GeV we have used the following parabolic fit
that agrees with Fig.\ 8 of Ref.~\cite{klapdor} up to less than 10 \%
\begin{equation}
\log F(m_i) = -37.96 + 10.1 \log m_i -0.6719 (\log m_i)^2,
\label{pfit1}
\end{equation}
and for $m_i > $ 1 GeV one can use
\begin{equation}
F(m_i) = 3.2 \, (10^8 \mbox{eV}/m_i)^2,
\label{pfit2}
\end{equation}
with $m_i$ in eV in both of the above expressions.
We have used Eqs.\ (\ref{pfit1}) and (\ref{pfit2}) along with the current
best experimental limit $ \left| \langle m_\nu \rangle \right| < 0.6$ eV
at 90\% C.\ L.~\cite{hmbb} to draw our conclusions about the possible
extra constraints that might be imposed to our previous results.
Due to the behavior of the nuclear matrix element $F(m_i)$ in
$^{76}$Ge~$\rightarrow ^{76}$Se transitions and taken into account
our previous results which always exclude $m_P>$ 40 MeV,
we conclude that we have in MMRN three different regions to inspect:
\begin{description}
\item (a) $m_P$,$m_F \leq $ 40 MeV;
\item (b) $m_P<$ 40 MeV and 40 MeV $< m_F< $ 1 GeV;
\item (c) $m_P<$ 40 MeV and $m_F \geq $ 1 GeV.
\end{description}
In case (a) $F(m_P)=F(m_F)=1$ and Eq.(\ref{mnu}) gives
\begin{equation}
\langle m_\nu \rangle = (\Phi R)_{eP}^2m_P+ (\Phi R)_{eF}^2m_F=s_\beta ^2\left(
-c_\alpha ^2m_P+s_\alpha ^2m_F\right) =0;
\label{mnu1}
\end{equation}
here, it is clear, the mixing parameters cannot be further constrained by
the neutrinoless double-$\beta$ decay limit.
In cases (b) and (c) we have $F(m_P)=1$ and
\begin{equation}
\langle m_\nu \rangle =s_\beta ^2s_\alpha ^2m_F\left( F(m_F)-1\right)
=xz m_F\left( F(m_F)-1\right),
\label{mnu2}
\end{equation}
so in these cases extra limits on the mixing parameters can be expected.
Using Eq.\ (\ref{pfit1}) in Eq.\ (\ref{mnu2}) and imposing the current
experimental limit of 0.6 eV one gets the maximum possible value
of the product $xz$ allowed by the data.
In region (c) we use Eq.\ (\ref{pfit2}) in Eq.\ (\ref{mnu2}) and again impose
the experimental limit.
This procedure permits us to compute the maximum allowed value for $m_P$,
$m_P^{\scriptsize \mbox{max}}$, as a function of $x$ for a given $m_F$.
This can be seen in Fig.\ \ref{fig2} for three different values of
$m_F$.
For example in region (c), for $m_F=$ 1 TeV and $x \sim 10^{-5}$,
$m_P \lesssim 0.06$ MeV. In region (b) for $m_F=$ .1 GeV and
$x \sim 10^{-5}$, $m_P \lesssim 0.2$ MeV. Both results are independent of
the values of $y$.
For higher values of $x$ the limits on $m_P$ are even more strict.
We see from this that in regions (b) and (c) the neutrinoless
double-$\beta$ decay limit can severely constrain the parameters of
the model.
\section{Conclusions}
\label{sec:conc}
We have analyzed the constraints imposed by recent experimental data
from $\mu$ decay, $\tau$ , $\pi$ and $K$ leptonic decays, the
$Z^0$ invisible width on the values of the four mixing parameters,
$x$, $y$, $m_P$ and $m_F$, of the MMRN model.
We have found regions allowed by the combined data at 90\% C.\ L.\
in the four parameter space. These allowed regions are very sensitive to
changes in the values of $x$ and not so sensitive to changes in $y$.
We were also able to find that the maximum possible value
for the lightest neutrino mass $m_P$, obtained in the limit $x,y \to 0$,
is about 40 MeV, even if $m_F>$ 1 TeV.
Although this is not so restrictive as the maximum value of $\nu_\tau$
obtained experimentally by ALEPH~\cite{pdg98} it is very interesting to see
that the electroweak data alone can indirectly lead to a value already so
limited.
We also have investigated and found that for $m_F>$ 40 MeV
the most recent neutrinoless double-$\beta$ decay limit can constrain
considerably more the model free parameters, in particularly the maximum
allowed value of $m_P$. For instance if $m_F= 1$ TeV and $x=1$, then
$m_P^{\text{max}} \sim$ 0.6 eV.
After combining the results from the particle decay analysis with the
constraints from neutrinoless double-$\beta$ decay we get finally :
\begin{description}
\item (a) for $m_P$,$m_F \leq $ 40 MeV, the constraints on the free
parameters are simply given by accelerator decay data, such as in
Fig.\ \ref{fig1}(d);
\item (b) for $m_F> $ 40 MeV, the limit from neutrinoless double-$\beta$
decay constrains the maximum value of $m_P$ to much smaller values than what
are still possible with the accelerator data, as shown in Fig.\ \ref{fig2}.
\end{description}
We have not used the available data on charm (or even beauty) meson
leptonic decay modes such as $D_s \rightarrow \mu \nu_\mu$ and
$D_s \rightarrow \tau \nu_\tau$. This data have very large uncertainties
attached to them and would not affect our results at the present moment.
We also have not used the data from $\tau \rightarrow \pi (3\pi) \nu_\tau$
due to the fact that they are experimentally less precise and
theoretically more problematic than $\tau$ leptonic decays. We do not
think these two modes would affect very much, if at all, our conclusions.
\section{Acknowledgments}
This work was supported by DGICYT grant PB95-1077, by the EEC
under the TMR contract ERBFMRX-CT96-0090, by Conselho Nacional de
Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq) and
by Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo (FAPESP).
\section*{APPENDIX : Radiative Correction formulae}
The leading radiative corrections to the lepton decay process
$ {l'} \rightarrow l \bar\nu_l \nu_{l'}$, ${\cal R}^{l'}$,
is give by~\cite{sirlin86}
\begin{equation}
{\cal R}^{l'} = \left[1+\frac{\alpha(m_{l'})}{2\pi}\left(\frac{25}{4}-\pi^2\right)\right]\left(1+\frac{3m^2_{l'}}{5m_W^2}\right),
\label{leprad}
\end{equation}
where $m_{l'}$ is the initial lepton mass, $ m_W$ is the $W$ boson mass
and $\alpha(m_{l'})$ is the running electromagnetic coupling constant.
The leading radiative corrections to hadron leptonic decays ${\cal R}_{hl}$
is given by~\cite{pdg98,sirlin}
\begin{eqnarray}
{\cal R}_{h l} &=&
\left[ 1 + \frac{2 \alpha}{\pi}
\ln\left(\frac{M_Z}{m_\rho}\right)\right]
\left[1+\frac{\alpha}{\pi} F(\delta _{l h}) \right]
\nonumber \\
& & \times \left\{ 1 - \frac{\alpha}{\pi}
\left[\frac{3}{2}\ln\left(\frac{m_\rho}{m_h}\right) +
C_1 +
C_2 \frac{m_l^2}{m_\rho^2} \ln\left(\frac{m_\rho^2}{m_l^2}\right) +
C_3 \frac{m_l^2}{m_\rho^2} + \dots
\right]\right\},
\label{pirc}
\end{eqnarray}
where
\begin{eqnarray}
F(x) &=&
3\ln x + \frac{13-19 x^2}{8(1-x^2)} -
\frac{8-5x^2}{2(1-x^2)^2}x^2\ln x
\nonumber \\
& &
\mbox{} - 2 \left( \frac{1+x^2}{1-x^2}\ln x +1 \right)\ln(1-x^2) +
2\left( \frac{1+x^2}{1-x^2} \right)L(1-x^2).
\end{eqnarray}
Here, $m_\rho = 796$~MeV is
the $\rho$ meson mass, $M_Z$ the $Z^0$ boson mass, $\alpha$ is the fine
structure constant and $m_{l}$ is the final lepton mass.
$C_i$ are structure constants whose numerical value have
large uncertainties and for this reason these terms will be neglected by us~\cite{pdg98}.Also, in the above,
$L(z)$ is defined by
\begin{equation}
L(z) = \int_0^z \frac{\ln(1-t)}{t} dt.
\end{equation}
|
2,877,628,090,772 | arxiv | \section{Introduction}
Organic charge-transfer salts show a wide variety of quantum phases and
represent prominent examples to study correlation effects in low-dimensional
systems. The most celebrated case is given by the TTF-TCNQ salt that has been
primarily regarded as a prototype for testing theories of one-dimensional
conductors.~\cite{solyom} Organic salts may also form crystals in two and
three dimensions, and, in this respect, an increasing attention has been
devoted to a particular family denoted by $\kappa$-(ET)$_2$X, whose
building block is the so-called BEDT-TTF (or ET) molecule and X is a monovalent
anion.~\cite{kanoda} Here, strongly dimerized ET molecules are arranged in a
two-dimensional triangular lattice. Each dimer has a charge state with one
hole and therefore the conducting band is half filled. A sizable effective
Coulomb repulsion is felt by two holes on the same dimer.
A huge variety of phases have been found (by varying temperature,
pressure or the nature of the anion X), ranging from correlated (bad) metals
with superconductivity at low temperatures, to Mott insulators with magnetic
order.~\cite{kanodamag,elsinger,lefebvre,limelette}
Interestingly, by acting with hydrostatic pressure, metal-insulator
transitions have been observed,~\cite{kanodamott1,kanodamott2} with the
remarkable possibility to stabilize a non-magnetic Mott insulating phase in
$\kappa$-(ET)$_2$Cu$_2$(CN)$_3$.~\cite{kanodaliquid} In this material, there
is no evidence of magnetic order down to $T \simeq 30 mK$, which is four
orders of magnitude lower than the estimate of the super-exchange coupling
$J \simeq 250 K$.
It has been argued that $\kappa$-(ET)$_2$X compounds can be described by a
single-band Hubbard model on the anisotropic triangular lattice,~\cite{kino}
where chains described by an hopping $t^\prime$ are coupled together with
zig-zag hoppings $t$, see Fig.~\ref{fig:lattice}. An on-site repulsive term
$U$ is also present in the Hamiltonian. However, a realistic estimate of these
microscopic parameters is not exempt from complications.
Indeed, the values obtained some time ago by extended H\"uckel band structure
calculations~\cite{mckenzie} have been put in doubt by two recent ab-initio
calculations, based upon local-density approximation (LDA) and generalized
gradient approximation (GGA).~\cite{nakamura,valenti}
Interestingly, the new results suggest that these organic salts are less
frustrated than previously assumed, and that $t^\prime/t$ is smaller than one.
Indeed, the frustrating ratio is $t^\prime/t \sim 0.8$ for
$\kappa$-(ET)$_2$Cu$_2$(CN)$_3$ and $t^\prime/t \sim 0.6$ for
$\kappa$-(ET)$_2$Cu$_2$(SCN)$_2$.~\cite{nakamura,valenti}
Other materials, with X=Cu[N(CN)$_2$]Cl or Cu[N(CN)$_2$]Br, have a
substantially smaller frustrating ratio, i.e.,
$t^\prime/t \sim 0.4$.~\cite{valenti}
Unfortunately, an accurate determination of the correlation energy is rather
difficult and these two calculations give a considerably different estimation
of the Coulomb repulsion, namely $U/t \sim 12 \div 15$
(Ref.~\onlinecite{nakamura}) and $U/t \sim 5 \div 7$
(Ref.~\onlinecite{valenti}).
Here, we apply our improved Monte Carlo calculations, based
upon the recently introduced backflow wave function~\cite{tocchio}
in order to analyze the possibility of having a non-magnetic
insulator for large enough frustration and interaction strength.
The paper is organized as follow: in section~\ref{sec:model}, we introduce the
Hamiltonian; in section~\ref{sec:approach}, we describe our variational wave
function; in section~\ref{sec:results}, we present our numerical results and,
finally, in section~\ref{sec:conc} we draw the conclusions.
\section{Model}\label{sec:model}
We consider the Hubbard model described by
\begin{equation}\label{hubbard}
{\cal H}=-\sum_{i,j,\sigma} t_{ij} c^{\dagger}_{i,\sigma} c_{j,\sigma} + H.c.
+U \sum_{i} n_{i,\uparrow} n_{i,\downarrow},
\end{equation}
where $c^{\dagger}_{i,\sigma} (c_{i,\sigma})$ creates (destroys) an electron
with spin $\sigma$ on site $i$,
$n_{i,\sigma}=c^{\dagger}_{i,\sigma}c_{i,\sigma}$, $t_{ij}$ is the hopping
amplitude and $U$ is the on-site Coulomb repulsion. In this work, we focus
our attention on the half-filled case with $N$ electrons on $N$ sites and
consider a square lattice with a nearest-neighbor hopping $t$, along the
$(1,0)$ and $(0,1)$ directions, and a further next-nearest-neighbor hopping
$t^\prime$ along $(1,1)$; this choice of the hopping amplitudes is
topologically equivalent to the anisotropic triangular lattice,~\cite{notet}
see Fig.~\ref{fig:lattice}.
In the last years, an intense effort has been devoted to this problem
by use of a large variety of methods, including exact
diagonalization,~\cite{clay} path-integral renormalization
group,~\cite{imada} variational Monte Carlo,~\cite{watanabe1,watanabe2,zhang}
cluster dynamical mean field theory,~\cite{kyung,ohashi} and dual
Fermions.~\cite{lee}
All these methods give rather different outcomes and there are huge
discrepancies on the phase boundaries and, most importantly, on the expected
nature of the non-magnetic insulator. The aim of this work is to clarify the
ground-state properties for two values of $t^\prime/t=0.6$ and $0.85$,
relevant for materials with X=Cu$_2$(SCN)$_2$ and Cu$_2$(CN)$_3$, respectively.
\begin{figure}
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{\label{fig:lattice}
Illustration of the lattice in the square topology (a) used in this work
and in the equivalent triangular one (b). Solid and dashed lines indicate
hopping amplitudes $t$ and $t^\prime$, respectively.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{\label{fig:accuracy}
(Color online) Results for 18 electrons on 18 sites as a function of $U/t$.
Upper panels: Accuracy of energy $\Delta E=(E_0-E_v)$, $E_v$ and $E_0$ being
the variational and the exact values, respectively. Lower panels: Overlap
between the exact ground state and the variational BCS wave functions. The
state without (with) backflow correlations is denoted by diamonds (squares).}
\end{figure}
\section{Variational approach}\label{sec:approach}
A variational wave function for an insulator with antiferromagnetic (AF) order
can be constructed by considering the ground state $|AF\rangle$ of a
mean-field Hamiltonian containing a band contribution and a magnetic term:
\begin{equation}
{\cal H}_{AF} = \sum_{q,\sigma} \epsilon_q c^{\dagger}_{q,\sigma} c_{q,\sigma}
+ \Delta_{AF} \sum_i {\bf n}_i \cdot {\bf S}_i,
\end{equation}
where ${\bf n}_i$ is a unitary vector that depends upon the lattice site
$i$ and ${\bf S}_i=(S_i^x,S_i^y,S_i^z)$ is the spin operator. Moreover,
$\epsilon_q = -2t(\cos q_x+\cos q_y) -2t_d \cos(q_x+q_y)$ is a variational
band term: $t$ gives the energy scale and $t_d$ can be optimized to minimize
the variational energy. In order to correctly describe spin fluctuations
orthogonal to the plane where the magnetic order lies, we take ${\bf n}_i$
in the $x{-}y$ plane and we include a spin Jastrow factor
${\cal J}_s=\exp [-\frac{1}{2} \sum_{i,j} u_{i,j} S_i^z S_j^z ]$
in the wave function.~\cite{becca} Another density Jastrow factor
${\cal J}=\exp [-\frac{1}{2} \sum_{i,j} v_{i,j} n_i n_j ]$ (that includes the
on-site Gutzwiller term) is considered to adjust electron correlations.
In summary, the correlated wave function is defined by
\begin{equation}
|\Psi_{AF}\rangle = {\cal J}_s {\cal J} |AF\rangle.
\end{equation}
Notice that, in this case, the variational state has not a definite total spin,
which is suitable for a magnetically ordered phase. In fact, both $|AF\rangle$
and the spin Jastrow factor ${\cal J}_s$ break the SU(2) symmetry.
On the other hand, superconducting or metallic phases can be constructed by
considering the ground state $|BCS\rangle$ of a superconducting
Bardeen-Cooper-Schrieffer (BCS) Hamiltonian with both band and pairing
contributions,~\cite{gros,zhang88}
\begin{equation}
{\cal H}_{BCS} = \sum_{q,\sigma} \epsilon_q c^{\dagger}_{q,\sigma} c_{q,\sigma}
+ \sum_{q} \Delta_q
c^{\dagger}_{q,\uparrow} c^{\dagger}_{-q,\downarrow} + H.c.,
\end{equation}
here the band term may also contain a variational chemical potential $\mu$,
since the BCS Hamiltonian does not conserve the particle number, i.e.,
$\epsilon_q = -2t(\cos q_x+\cos q_y) -2t_d \cos(q_x+q_y) -\mu$. In this case,
$t_d$ and $\mu$ can be varied to optimize the variational wave function.
The full correlated state is given by
\begin{equation}
|\Psi_{BCS}\rangle = {\cal J} |BCS\rangle,
\end{equation}
in this case, no spin Jastrow is considered, in order to have a perfect singlet
state, suitable for a non-magnetic phase. Notably, within this kind of wave
function, it is possible to obtain a pure (i.e., non-magnetic) Mott insulator
just by considering a sufficiently strong Jastrow factor, i.e.,
$v_q \sim 1/q^2$ ($v_q$ being the Fourier transform of
$v_{i,j}$).~\cite{capello}
As we recently demonstrated,~\cite{tocchio} the projected BCS state is
not sufficiently accurate for Hubbard-type models, especially in the important
strong-coupling regime, i.e., for $U/t \gtrsim 10$, where the super-exchange
energy scale $J=4t^2/U$ is not correctly reproduced. One efficient way to
overcome this problem is to consider backflow correlations,~\cite{tocchio}
that modify the single-particle orbitals,~\cite{noteph} in the same spirit
of what was put forward long-time ago by Feynman and Cohen.~\cite{feynman}
\begin{figure}
\includegraphics[width=\columnwidth]{fig3.eps}
\caption{\label{fig:energy}
(Color online) Variational energies per site (in unit of $J=4t^2/U$) for the
BCS state with a density Jastrow factor (diamonds) and for the AF wave function
with both density and spin Jastrow terms (circles); The correlated Fermi gas
with Jastrow factor is also reported for $t^\prime/t=0.85$ (squares).
All states have backflow correlations and results are for 100 sites.}
\end{figure}
Following Ref.~\onlinecite{tocchio}, we consider a general definition of
the new ``orbitals'' by taking all the possible virtual hoppings of the
electrons:
\begin{eqnarray}\label{backlattice2}
&&\phi_q^b({\bf r}_{i,\sigma}) \equiv \epsilon \phi_q({\bf r}_{i,\sigma})
+ \eta_1 \sum_j t_{ij} D_i H_j \phi_q({\bf r}_{j,\sigma})
+ \nonumber \\
&& \eta_2 \sum_j t_{ij} n_{i,\sigma} h_{i,-\sigma}
n_{j,-\sigma} h_{j,\sigma} \phi_q({\bf r}_{j,\sigma}) + \nonumber \\
&& \eta_3 \sum_j t_{ij} \left( D_i n_{j,-\sigma} h_{j,\sigma}
+n_{i,\sigma} h_{i,-\sigma} H_j \right) \phi_q({\bf r}_{j,\sigma}),
\end{eqnarray}
where we used the notation
$\phi_q({\bf r}_{i,\sigma})= \langle 0|c_{i,\sigma}|\phi_q\rangle$, being
$|\phi_q\rangle$ the eigenstates of the mean-field Hamiltonian,
$D_i=n_{i,\uparrow}n_{i,\downarrow}$, $H_i=h_{i,\uparrow}h_{i,\downarrow}$,
with $h_{i,\sigma}=(1-n_{i,\sigma})$.
$\epsilon$, $\eta_1$, $\eta_2$, and $\eta_3$ are variational parameters.
As a consequence, already the determinant part of the wave function includes
correlation effects. The backflow corrections of Eq.~(\ref{backlattice2})
(in particular the $\eta_1$ term) make it possible to mimic the effect of the
virtual hopping, which leads to the super-exchange mechanism.
All the parameters of the wave function can be optimized by using the method of
Ref.~\onlinecite{yunoki}.
Finally, the accuracy of the variational calculations can be assessed by using
Lanczos diagonalizations on small lattices and Green's function Monte Carlo
within the so-called fixed-node approximation,~\cite{ceperleyfn} which
gives accurate (but approximate) results on large systems. A detailed
description of the fixed-node approximation can be found
in Ref.~\onlinecite{lugas}. In brief, starting from the original Hamiltonian
${\cal H}$, we define an effective Hamiltonian by adding a perturbation $O$:
\begin{equation}\label{fnham}
{\cal H}_{eff} = {\cal H} + O.
\end{equation}
The operator $O$ is defined through its matrix elements and depends upon
a given {\it guiding function} $|\Psi \rangle$, that is for instance the
variational state itself
\begin{equation}
O_{x^\prime,x} = \left \{
\begin{array}{ll}
-{\cal H}_{x^\prime,x} & {\rm if} \; x\prime \neq x \; {\rm and}\; s_{x^\prime,x} > 0 \nonumber \\
0 & {\rm if} \; x\prime \neq x \; {\rm and}\; s_{x^\prime,x} < 0 \nonumber \\
\sum_{y:s_{y,x}>0} {\cal H}_{y,x} \frac{\Psi_y}{\Psi_x} & {\rm for} \; x^\prime=
x,
\end{array}
\right .
\end{equation}
where $\Psi_x = \langle x|\Psi \rangle$ and $s_{x^\prime,x} = \Psi_{x^\prime} {\cal H}_{
x^\prime,x} \Psi_x$.
Notice that the above operator annihilates the guiding function, namely
$O |\Psi \rangle=0$. Therefore, whenever the guiding function is close
to the exact ground state of ${\cal H}$, the perturbation $O$ is expected to be
small and the effective Hamiltonian becomes very close to the original one.
\section{Results}\label{sec:results}
By allowing the most general singlet and complex BCS pairing in the state
without backflow terms, we find that this quantity has $d_{x^2-y^2}$ symmetry
up to $t^\prime \sim t$, namely the best (nearest-neighbor) pairing function is
$\Delta_q= 2 \Delta (\cos q_x - \cos q_y)$, in agreement with previous
results.~\cite{powell,zhang2,nandini,powell2}
Therefore, within our improved backflow wave function, we only considered
a real BCS pairing. We mention that $\Delta$ is very small (especially in the
presence of backflow correlations) in the conducting phase,
see table~\ref{tab:delta}, and it becomes sizable only in the regime where the
magnetic solution prevails over the BCS state. In this regard, we do not find
a clear signature of superconductivity close to the metal-insulator transition,
as suggested in Ref.~\onlinecite{nandini}.
We also stress that, once the backflow correlations are considered, there is no
energy gain by allowing a translational symmetry breaking (e.g., by considering
a $2 \times 1$ unit cell in the BCS Hamiltonian, suitable for dimerized states)
and the $d_{x^2-y^2}$ solution has always a lower energy than dimerized states.
Finally, we find that the variational band term of the BCS Hamiltonian
$\epsilon_q= -2t(\cos q_x + \cos q_y) -2t_d \cos(q_x+q_y) -\mu$
has $t_d \simeq 0$ for most of the cases considered, except for small $U/t$,
inside the conducting phase, where a finite $t_d$ can be stabilized.
\begin{table}
\caption{\label{tab:delta}
BCS pairing $\Delta$ for various $U/t$ in the metallic region for two sizes
of the lattice: $N=100$ (third column) and $N=196$ (fourth column).
Notice that for $U/t=8$ and $t^\prime/t=0.85$ and for $U/t=6$ and
$t^\prime/t=0.6$ the BCS wave function is still metallic but the AF one
(insulating) has a lower variational energy.}
\begin{tabular}{cccc}
\hline
$U/t$ & $t^\prime/t$ & $\Delta/t$ & $\Delta/t$ \\
\hline \hline
6 & 0.85 & 0.026(1) & 0.018(1) \\
7 & 0.85 & 0.051(1) & 0.025(1) \\
8 & 0.85 & 0.161(1) & 0.037(1) \\
\hline \hline
4 & 0.6 & 0.013(1) & 0.005(1) \\
5 & 0.6 & 0.027(1) & 0.008(1) \\
6 & 0.6 & 0.056(1) & 0.019(1) \\
\hline \hline
\end{tabular}
\end{table}
As far as the magnetic wave function is concerned, both Hartree-Fock and
fixed-node calculations give a clear indication that spin-spin correlations
remain commensurate at $Q=(\pi,\pi)$ for $t^\prime/t \lesssim 0.9$. Therefore,
we use an AF wave function having N\'eel order with pitch vector $Q=(\pi,\pi)$
and we do not consider the implementation of a generic magnetic state with
incommensurate order. Moreover, we verified that, for
$t^\prime/t \lesssim 0.9$, this AF state has a lower energy with respect to
the AF state with 120$^\circ$ order, suitable for $t^\prime=t$.
\begin{figure}
\includegraphics[width=\columnwidth]{fig4.eps}
\caption{\label{fig:nq}
(Color online) Variational results for the density-density correlations $N_q$
divided by $|q|$, along the $(1,0)$ direction, for 100 (red symbols) and 196
(black symbols) sites. Full (empty) symbols refer to the BCS (AF) wave
function. Upper panel: from top to bottom, $U/t=4$, $5$, $6$, $8$, and $10$.
Lower panel: from top to bottom, $U/t=6$, $7$, $8$, $10$, and $20$.}
\end{figure}
\subsection{Quality of the variational states}
In Fig.~\ref{fig:accuracy}, we show the accuracy of the BCS variational state
and its overlap with the exact ground state in a small lattice with 18 sites
(which is tilted by 45 degrees). We report two cases with $t^\prime/t=0.6$
and $0.85$ and different values of $U/t$. As in the case of the frustrated
square lattice studied in Ref.~\onlinecite{tocchio}, the backflow terms highly
improve the quality of the variational wave function that remains very accurate
even for large correlation, i.e., up to $U/t \sim 30$. We would like to mention
that, for this small cluster, the AF state has a slightly lower energy than
the BCS one for both $t^\prime/t=0.6$ and $0.85$. For $t^\prime/t=0.6$, the AF
state has also a better overlap with the exact ground state $|\Psi_0\rangle$
(e.g., $\langle \Psi_0|\Psi_{AF}\rangle = 0.962$ for $U/t=20$) than the BCS
state (e.g., $\langle \Psi_0|\Psi_{BCS}\rangle = 0.958$), while it has a
substantially lower overlap for $t^\prime/t=0.85$ (e.g.,
$\langle \Psi_0|\Psi_{AF}\rangle = 0.904$ against
$\langle \Psi_0|\Psi_{BCS}\rangle = 0.959$).
The accuracy of the variational state remains very high also for large
systems, where the backflow corrections give a sizable and size-consistent
improvement. In Fig.~\ref{fig:energy}, we report the energy per site as a
function of $U/t$ for both $t^\prime/t=0.6$ and $0.85$ for $N=10 \times 10$
(see also table~\ref{tab:energies}).
\begin{table}
\caption{\label{tab:energies}
Our best energies per site for $N=100$: pure variational $E_{vmc}$ and
fixed-node $E_{fn}$ (still variational) results are reported.}
\begin{tabular}{cccc}
\hline
$U/t$ & $t^\prime/t$ & $E_{vmc}/t$ & $E_{fn}/t$ \\
\hline \hline
4 & 0.85 & -1.03029(2) & -1.0315(1) \\
8 & 0.85 & -0.51876(5) & -0.5238(1) \\
12 & 0.85 & -0.36569(5) & -0.3764(1) \\
16 & 0.85 & -0.2834(1) & -0.2910(1) \\
20 & 0.85 & -0.2311(1) & -0.2364(1) \\
\hline \hline
4 & 0.6 & -0.92356(2) & -0.9251(1) \\
8 & 0.6 & -0.51837(3) & -0.5228(1) \\
12 & 0.6 & -0.36550(3) & -0.3689(1) \\
16 & 0.6 & -0.28041(3) & -0.2833(1) \\
20 & 0.6 & -0.22685(3) & -0.2291(1) \\
\hline \hline
\end{tabular}
\end{table}
\subsection{Metal-insulator transition}
The metal-insulator transition can be detected by a direct inspection of the
static density-density correlations
\begin{equation}
N_q = \frac{1}{N} \sum_{j,l} e^{i q (R_j-R_l)} \langle n_j n_l \rangle.
\end{equation}
In fact, this quantity makes it possible to discriminate between gapless
(conducting) and gapped (insulating) phases: a linear behavior $N_q \sim |q|$
for $|q| \to 0$ is typical of a conducting phase, whereas a quadratic behavior
$N_q \sim q^2$ can be associated to an insulating character.~\cite{capello}
The results presented in Fig.~\ref{fig:nq} indicate that a metal-insulator
transition takes place by increasing $U/t$ and it can be placed at
$U_c^{\rm MIT}/t= (5.5 \pm 0.5)$ and $(7.5 \pm 0.5)$ for $t^\prime/t=0.6$
and $0.85$, respectively.
The transition is first order, with a small jump in the linear coefficient
of $N_q$ for small momenta. In fact, for small $U/t$, the best wave function
is the BCS one (with small superconducting pairing), whereas, by increasing
the interaction, the AF one prevails, thus inducing a metal-insulator
transition, see Fig.~\ref{fig:energy}.
\begin{figure}
\includegraphics[width=\columnwidth]{fig5.eps}
\caption{\label{fig:sq}
(Color online) Size scaling of the spin-spin correlations $S_Q/N$ for
$Q=(\pi,\pi)$. Data are for $t^\prime/t=0.6$ with $U/t=10$ (squares) and
$U/t=20$ (circles), and $t^\prime/t=0.85$ with $U/t=10$ (triangles) and
$U/t=20$ (diamonds). Variational and fixed-node results are denoted by empty
and full symbols, respectively. The variational results do not depend
substantially upon $U$ and $t^\prime$. The fixed-node results indicate
long-range order for $t^\prime/t=0.6$ but not for $t^\prime/t=0.85$.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig6.eps}
\caption{\label{fig:phase_diagram}
(Color online) Proposed phase diagram for the two discussed hopping ratios,
$t^\prime/t=0.6$ and $t^\prime/t=0.85$. In the first case, variational (VMC)
and fixed-node results (FN) indicate both a direct transition between a
metal and an insulator with AF N\'eel order at a critical value of the
electron-electron repulsion $U_{c}$. For $t^\prime/t=0.85$, the
variational results predict the existence of three different phases
at increasing $U/t$: a metal, an AF insulator with N\'eel order and a
spin liquid, while, within the fixed-node approximation, the non-magnetic
ground state extends down to the metal-insulator transition.}
\end{figure}
\subsection{Insulating Phase}
In the insulating regime and for small frustrating ratios, the AF wave function
has always a lower energy than the spin-liquid state, and this fact is
particularly evident close to the transition, see Fig.~\ref{fig:energy}.
On the contrary, for the case with $t^\prime/t=0.85$, the BCS state competes
with the AF one and it becomes better in energy for $U/t \gtrsim 13$,
indicating an insulating spin-liquid behavior at large $U$
(notice that in this region $N_q \sim q^2$). In this regime,
the BCS pairing is relevant, since the simple projected Fermi
sea has a much higher energy, see Fig.~\ref{fig:energy}.
Remarkably, the BCS and AF variational energies
are always quite close for $t^\prime/t=0.85$, suggesting that the actual
ground state might be non-magnetic for all $U > U_c^{\rm MIT}$, or at least
down to values lower than expected on the basis of the variational estimate.
This fact is supported by the fixed-node calculations
for the spin-spin correlations
\begin{equation}
S_q = \frac{1}{N} \sum_{j,l} e^{i q (R_j-R_l)} \langle S^z_j S^z_l \rangle.
\end{equation}
In Fig.~\ref{fig:sq}, we report the size scaling of the variational and the
fixed-node results by considering the BCS state as the {\it guiding function}.
We stress the fact that, in the insulating regime, $S_q$ has a peak at the
commensurate momentum $Q=(\pi,\pi)$. Remarkably, the fixed-node approach is
able to recover a finite value of $S_Q/N$ for $Q=(\pi,\pi)$ (i.e., the square
of the magnetic order parameter) in the thermodynamic limit for
$t^\prime/t=0.6$, even though the BCS wave function is non magnetic.
By contrast, $S_Q/N$ tends to zero for $t^\prime/t=0.85$ (both for $U/t=10$
and $20$), supporting the fact that the ground state is non magnetic for this
frustrating ratio, even close to the metal-insulator transition. The resulting
phase diagram is summarized in Fig.~\ref{fig:phase_diagram}.
\section{Discussion}\label{sec:conc}
We have studied the anisotropic triangular lattice at half filling away from
the isotropic point $t^\prime=t$, with $t^\prime<t$, using both a
Gutzwiller-Jastrow variational ansatz including backflow correlations as well
as a Green's function Monte Carlo approach within the fixed node approximation.
We find that the square lattice states persist up to large values of
$t^\prime/t<1$, both in terms of the d-wave superconducting order parameter
as well as for the AF N\'eel ordering.
The main outcome of this work is that, thanks to the improvement given by
backflow correlations, a spin-liquid wave function can be stabilized over
magnetic states, for large but still moderate Coulomb repulsions and close
to the isotropic limit. These variational results are corroborated by fixed-node
calculations. We find, in particular, that for $t^\prime/t=0.85$, which is
relevant for $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$,~\cite{nakamura,valenti}
the insulating phase has a pure Mott character, without magnetic order.
On the other hand, for $t^\prime/t=0.6$, suitable for
$\kappa$-(ET)$_2$Cu$_2$(SCN)$_2$, (or even smaller $t^\prime/t$ values)
the insulating phase always shows N\'eel order with $Q=(\pi,\pi)$.
Let us finish by discussing our results also in comparison to other
calculations and experimental findings. First of all, in various papers, it
has been suggested that the spin-liquid phase can be stabilized by charge
fluctuations that may destabilize a magnetically ordered state. This claim has
been corroborated by calculations on Heisenberg models in presence of a
ring-exchange term $J_4$ (that appears in the strong-coupling expansion in
$t^4/U^3$).~\cite{motrunich} However, it turns out that the actual value of
$J_4$ for stabilizing a disordered phase is rather large and, probably, beyond
the validity of a perturbative expansion. The existence of a direct transition
from a magnetic phase to a disordered one has been also found in the original
Hubbard model, by decreasing the on-site repulsion $U$.~\cite{clay,imada,kyung}
We do not find any evidence of such a possibility and, in our approach, the
magnetic phase is stable in presence of charge fluctuations, even close to
the metal-insulator transition: this is the case of $t^\prime/t=0.6$.
Instead, the spin-liquid phase turns out to be directly connected with the one
at strong coupling, while antiferromagnetic correlations become stronger when
decreasing $U/t$. For example, for $t^\prime/t=0.85$, the variational state
with magnetic order has a slightly lower energy close to $U_c$ and we
need to apply the Green's function Monte Carlo approach to extend the
spin-liquid region down to the metal-insulator transition,
see Fig.~\ref{fig:phase_diagram}.
At this stage, we would also like to mention that the metallic phase is likely
to be not superconducting. In fact, the BCS pairing $\Delta$ in the metallic
region is slightly suppressed when improving the accuracy of the variational
wave function by considering backflow correlations and, moreover, it is reduced
by a factor $2 \div 3$ when the lattice size is increased from $10 \times 10$
to $14\times 14$, see table~\ref{tab:delta}. This fact contrasts the previous
claim of a possible superconducting phase close to the metal-insulator
transition by Liu and collaborators.~\cite{nandini}
Another very important point is to clarify the nature of the low-energy
excitations. Very recently, thermodynamic measurements of the specific
heat suggested the possible existence of a Fermi surface of neutral, $S=1/2$
fermionic spinons.~\cite{yamashita} However, it should be noticed that such
a measurement involves a difficult subtraction of a divergent nuclear
specific heat, and instead the thermal conductivity (which is not affected by
a nuclear contribution) shows an activated behavior with a tiny gap of
$0.46 K$.~\cite{yamashita2} This fact has been associated with the existence
of spinless ``vison'' excitations.~\cite{sachdev}
From our calculations, it appears that the disordered insulating phase cannot
sustain a true spinon Fermi surface, as previously suggested both on
variational calculations~\cite{motrunich} and field-theory
approaches,~\cite{leelee} but it has Dirac points at $(\pm \pi/2,\pm \pi/2)$.
In fact, the projected Fermi-sea has a much higher energy than our best
variational ansatz with BCS pairing, see Fig.~\ref{fig:energy}.
Should our results be correct, either a deeper investigation of the minimal
microscopic model for describing organic charge-transfer salts is needed or
a reinterpretation of the experimental data is required.
We thank R. Valenti for very useful discussions. L.T. and C.G. thank partial
support from the German Science Foundation through the Transregio 49.
|
2,877,628,090,773 | arxiv | \section{Introduction}
An interesting topic in recent years both in the general relativity community and in the String theory community is
that the BH area (or entropy) product formula of all horizons independent of
the ADM (Arnowitt-Deser-Misner) mass of the background space-time
\cite{ah09,cgp11,castro12,chen12,sd12,mv13,castro13,ppepjc,ppplb,ppsen}.
For example, the area product formula for a Kerr BH \cite{ah09} depends only on the angular momentum parameter:
\begin{eqnarray}
{\cal A}_{2} {\cal A}_{1} &=& 64\pi^2J^2 ~.\label{prKN}
\end{eqnarray}
where ${\cal A}_{2}$ and ${\cal A}_{1}$ are area of the inner and outer horizons.
Whenever, we have taken the perturbed space-time with a spinning BH in some non-trivial
environment e.g. a BH surrounded by a ring of matter or a multiple BH space-time the same
formula holds. Hence, the area product formula of outer horizon or event horizon (${\cal H}^{+}$)
and inner horizon or Cauchy horizons (${\cal H}^{-}$) for Kerr BH is of an \emph{universal}
quantity: it holds independently of the environment of the BH.
On the other hand, if we incorporate the BPS (Bogomol'ni-Prasad-Sommerfeld) states, the area product
formula of ${\cal H}^{\mp}$ should read \cite{cgp11}
\begin{eqnarray}
{\cal A}_{2}{\cal A}_{1}
&=& 64 \pi^2 {\ell _{pl}}^4 N , \,\, N\in {\mathbb{N}}, N_{1}\in {\mathbb{N}}, N_{2} \in {\mathbb{N}}
~.\label{ppl}
\end{eqnarray}
where $\ell _{pl}$ is the Planck length. This indicates the area product should be quantized.
Alternatively, the area products independent of mass implies that there should be an important role
of the Cauchy horizon in the BH thermodynamics as well as in BH physics. Now the relevant question is
that the mass independent product formula is generic? It has been shown explicitly by Visser \cite{mv13}that
by incorporating the cosmological constant, the area product of all physical horizons is not mass
independent. But typically, some complicated function of inner and outer horizon area is indeed mass
independent.
Previous studies have not made use of the extended phase space formalism.
Thus in this work, we wish to examine the thermodynamic product formula in \emph{extended
phase space}. Where the ADM mass of an AdS BH could be treated as the enthalpy of
the space-time and the cosmological constant should be treated as the thermodynamic
pressure \cite{kastor09}. Therefore there must exists a conjugate quantity
which is a thermodynamic volume associated with the BH space-time.
Besides area (or entropy) products, it needs to be evaluated whether other thermodynamic
products \cite{ppepjc,ppplb,ppmpla,ppahep,ppjetpl,ppjetp} like BH temperature products, specific heat products, Komar energy
products etc. are provide any universal characterization or not, and here we first introduced
the \emph{thermodynamic volume products} when one must considered the extended phase space thermodynamics. Does it
independent of the ADM mass parameter? We will investigate this issues
in the present work. So, when the cosmological constant treated as a thermodynamic pressure
and its conjugate variable as a thermodynamic volume what happens the \emph{Smarr mass formula},
\emph{Smarr-Gibbs-Duhem} relation and BH \emph{equations of state} in the extended phase space. Additionally, we
find the mass independent volume products relation in parallel with the entropy product relations.
BH thermodynamic properties have been investigated for many decades and still it is going on. In the present study, the
main motivation comes from the seminal work of Hawking and Page \cite{haw83} where the thermodynamic properties of BHs in
Schwarzschild-AdS space has been explicitly studied. The author discussed the phase transition (between small and large BHs for
Schwarzschild-AdS BH) which is called famous Hawking-Page phase transition. The special interest is due to the
application of AdS space-time in gauge/gravity duality via dual conformal field theory (CFT) through AdS/CFT
correspondence \cite{witten}. Several exotic phenomena occurs in the AdS space-time. First example of course
be Hawking-Page phase transition in Schwarzschild-AdS spacetime. The second one is that in charged AdS spacetime, the
gravitational analogue of the liquid/gas phase transition has been observed in the phase diagram which was explicitly
investigated by several authors \cite{chamblin99,chamblin99a,gubser,david12} and the fact that for charged AdS BH the
notion of thermodynamic equilibrium is a straightforward concept. The third one is that Kerr-AdS spacetime admits reentrant
phase transition and showing a tri-critical point in their phase diagram \cite{david13}.
The current interest is involved due to the variation of negative cosmological constant and also it is proportional to the
thermodynamic pressure \cite{dolan10,dolan11,david12}. The thermodynamic products especially area (or entropy)
products in charged AdS BHs were calculated in \cite{mv13} but the author has not been considered there the
extended phase space. Here we shall compute the volume products by considering the thermodynamic pressure ($P$)
is equal to the negative cosmological constant ($\Lambda$) divided by $8\pi$ (where $G=c=k=\hslash$=1) i.e.
\begin{eqnarray}
P &=& -\frac{\Lambda}{8\pi}=\frac{3}{8\pi \ell^2} ~.\label{pr}
\end{eqnarray}
and the corresponding thermodynamic volume can be defined as
\begin{eqnarray}
V &=& \left(\frac{\partial M}{\partial P}\right)_{S,Q,J} ~.\label{vm}
\end{eqnarray}
This volume for charged-AdS BH should read as
\begin{eqnarray}
V_{i} &=& \frac{4}{3}\pi r_{i}^3 ~.\label{vm1}
\end{eqnarray}
where $r_{i}$ is the corresponding horizon radius and $i=1,2,3,4$.
It has been shown that the \emph{Reverse Isoperimetric Inequality} is satisfied for
event horizon \cite{cvetic11}. Here we conjecture that this inequality is valid for
all the horizons i.e.
\begin{eqnarray}
{\cal R}_{i} &=& \left(\frac{3V_{i}}{4\pi} \right)^{\frac{1}{3}}
\left(\frac{4\pi}{{\cal A}_{i}} \right)^{\frac{1}{2}} \geq 1 ~.\label{rip}
\end{eqnarray}
It should be noted that a class of BHs with non-compact event horizons do not satisfy
this inequality \cite{mannprl,mannjhep}.
The structure of the paper is as follows. In Sec.(\ref{tnt}), we have described the thermodynamic properties
of RN-AdS BH. The Sec.(\ref{quint}) describes the thermodynamic properties of the RN-AdS BH surrounded by
quintessence. In Sec. (\ref{fr}), we have given the thermodynamic properties of the
$f(R)$ gravity. Finally, we conclude in Sec.(\ref{dis}). In appendix-A, we have examined the thermodynamic
volume products for axisymmetric space-time and in appendix-B, we have investigated the $P-V$ criticality
of inner horizon for RN-AdS BH.
\section{\label{tnt} Thermodynamic properties of Charged AdS BH:}
Let us begin with the charged-AdS space-time metric can be written as in terms of
Schwarzschild like coordinates\cite{chamblin99,chamblin99a}:
\begin{eqnarray}
ds^2 &= & -{\cal U}(r) dt^2 + \frac{dr^2}{{\cal U}(r)} +r^2 d\Omega_{2}^2 .~\label{metric}
\end{eqnarray}
where,
\begin{eqnarray}
{\cal U}(r) &=& 1-\frac{2M}{r}+\frac{Q^2}{r^2}+\frac{r^2}{\ell^2},\label{h1}
\end{eqnarray}
and $d\Omega_{2}^2$ is the metric on the unit sphere in two dimensions.
The electromagnetic potential one form for the space-time (\ref{metric}) is
\begin{eqnarray}
A=A_{\mu}dx^{\mu}=-\frac{Q}{r}dt.~\label{h2}
\end{eqnarray}
The BH horizons determined by the condition ${\cal U}(r)=0$ i.e.
\begin{eqnarray}
\frac{r^4}{\ell^2}+r^2-2Mr+Q^2 &=& 0 ~.\label{rna1}
\end{eqnarray}
In terms of thermodynamic pressure, this could be rewritten as
\begin{eqnarray}
\frac{8\pi P}{3}r^4+r^2-2Mr+Q^2 &=& 0 ~.\label{rna2}
\end{eqnarray}
To finding the roots we apply the Vieta's rule, we get
\begin{eqnarray}
\sum_{i=1}^{4} r_{i} &=& 0 ~.\label{eq1}\\
\sum_{1\leq i<j\leq 4} r_{i}r_{j} &=& \frac{3}{8\pi P} ~.\label{eq2}\\
\sum_{1\leq i<j<k\leq 4} r_{i}r_{j} r_{k} &=& \frac{3M}{4\pi P} ~.\label{eq3}\\
\prod_{i=1}^{4} r_{i} &=& \frac{3Q^2}{8\pi P} ~.\label{eq4}
\end{eqnarray}
The entropy of the BH can be defined as
\begin{eqnarray}
{\cal S}_{i} &=& \frac{{\cal A}_{i}}{4} ~.\label{eq5}
\end{eqnarray}
where the area of the BH is
\begin{eqnarray}
{\cal A}_{i} &=& 4\pi r_{i}^2 ~.\label{eq6}
\end{eqnarray}
and now the BH temperature reads as
\begin{eqnarray}
T_{i} &=& \frac{{\cal U}'(r)}{4\pi}=
\frac{1}{4\pi r_{i}} \left(1+8\pi P r_{i}^2-\frac{Q^2}{r_{i}^2} \right)~. \label{eq7}
\end{eqnarray}
The electric potential could be defined as
\begin{eqnarray}
\Phi_{i} &=& \frac{Q} {r_{i}} ~.\label{eq8}
\end{eqnarray}
We should be noted that in the extended phase space the ADM mass can be treated as the total
gravitational enthalpy of the system i.e. $M=H=U+PV$. Where $U$ is the thermal energy of the
system\cite{kastor09}. Then the first law of BH thermodynamics in the extended phase space becomes
\begin{eqnarray}
dH &=& T_{i} d{\cal S}_{i} + V_{i} dP +\Phi_{i} dQ ~. \label{eq9}
\end{eqnarray}
and the corresponding Smarr-Gibbs-Duhem relation becomes
\begin{eqnarray}
H &=& 2T_{i} {\cal S}_{i} - 2P V_{i} +Q \Phi_{i} ~. \label{eq10}
\end{eqnarray}
Now we compute the mass(or enthalpy) independent entropy sum and entropy product formula
using Eqs.(\ref{eq1},\ref{eq2},\ref{eq4},\ref{eq5},\ref{eq6}):
\begin{eqnarray}
\sum_{i=1}^{4} \sqrt{{\cal S}_{i}} &=& 0 ~.\label{eq11}\\
\sum_{1\leq i<j\leq 4} \sqrt{{\cal S}_{i}{\cal S}_{j}} &=& \frac{3}{8P} ~.\label{eq12}\\
\prod_{i=1}^{4} \sqrt{{\cal S}_{i}} &=& \frac{3 \pi Q^2}{8P} ~.\label{eq13}
\end{eqnarray}
In terms of two horizons, the mass-independent entropy product formula should read
\begin{eqnarray}
\frac{\left(\frac{3Q^2}{8P} \right)}{\sqrt{{\cal S}_{1}{\cal S}_{2}}}-
\frac{{\cal S}_{1}+{\cal S}_{2}+\sqrt{{\cal S}_{1}{\cal S}_{2}}}{\pi} &=& \frac{3}{8\pi P}
~. \label{eqq1}
\end{eqnarray}
Although it is a complicated function of two horizons but it is explicitly mass independent
function of inner horizon and outer horizons.
Now we turn into another important relations that is the \emph{volume sum} and
\emph{volume product} which are mass independent:
\begin{eqnarray}
\sum_{i=1}^{4} {V_{i}}^{\frac{1}{3}} &=& 0 ~.\label{eq14}\\
\sum_{1\leq i<j\leq 4}(V_{i}V_{j})^{\frac{1}{3}} &=& \left(\frac{3}{32 \pi}\right)^{\frac{1}{3}}
\frac{1}{P} ~.\label{eq15}\\
\prod_{i=1}^{4} (V_{i})^{\frac{1}{3}}&=&
\left(\frac{\pi}{6}\right)^{\frac{1}{3}}\frac{Q^2}{P} ~.\label{eq16}
\end{eqnarray}
Again in terms of two horizons, the mass independent volume product formula should
be
\begin{eqnarray}
\left(\frac{3}{32 \pi}\right)^{\frac{1}{3}} \frac{\left(\frac{Q^2}{P}\right)}{(V_{1}V_{2})^{\frac{1}{3}}}
-\left(\frac{3}{4\pi}\right)^{\frac{2}{3}}\left[{V_{1}}^{\frac{2}{3}}+{V_{2}}^{\frac{2}{3}}+(V_{1}V_{2})^{\frac{1}{3}}
\right] &=& \frac{3}{8\pi P}
~. \label{eqq2}
\end{eqnarray}
These are explicitly mass-independent relations in the extended phase space.
Finally, the equation of state in the extended phase space should read
\begin{eqnarray}
P &=& \frac{T_{i}} {2r_{i}} -\frac{1}{8\pi r_{i}^2}+\frac{Q^2}{8\pi r_{i}^4} ~. \label{eq17}
\end{eqnarray}
where $r_{i}=\left(\frac{3V_{i}}{4\pi}\right)^{1/3}$. Now in terms of specific volume
$v_{i}=2r_{i}$ the above equation could be re-written as
\begin{eqnarray}
P &=& \frac{T_{i}} {v_{i}} -\frac{1}{2\pi v_{i}^2}+\frac{2Q^2}{\pi v_{i}^4} ~. \label{eq18}
\end{eqnarray}
The critical point can be obtained from the following conditions:
\begin{eqnarray}
\frac{\partial P}{\partial v_{i}} &=& \frac{\partial^2 P}{\partial v_{i}^2}=0 ~. \label{eq19}
\end{eqnarray}
The critical values are explicitly computed in \cite{david12}.
Defining further $p=\frac{P}{P_{c}}$, $\nu_{i}=\frac{v_{i}}{v_{c}}$ and $\tau_{i}=\frac{T_{i}}{T_{c}}$,
the law of corresponding states become
\begin{eqnarray}
8 \tau_{i} &=& 3\nu_{i}\left(p+\frac{2}{\nu_{i}^2}\right)-\frac{1}{\nu_{i}^3} ~. \label{eq20}
\end{eqnarray}
\footnote{The critical values for charged-AdS BH determined in \cite{david12} are $P_{c}=\frac{1}{96\pi Q^2}$,
$v_{c}=2\sqrt{6} Q$ and $T_{c}=\frac{\sqrt{6}}{18\pi Q}$}.
In the extended phase space the Gibbs free energy could be defined as
\begin{eqnarray}
G_{i} &=& H-T_{i}S_{i}=M-T_{i}S_{i}=\frac{r_{i}}{4}-\frac{2\pi P}{3} r_{i}^3+\frac{3Q^2}{4r_{i}} ~. \label{eq21}
\end{eqnarray}
and the Helmholtz free energy is given by
\begin{eqnarray}
F_{i} &=& G_{i}-PV_{i}=\frac{r_{i}}{2}-\pi T_{i} r_{i}^2+\frac{Q^2}{2r_{i}} ~. \label{eq22}
\end{eqnarray}
which is very important to determine the behavior of the critical exponents.
\footnote{The critical exponents for the BH system are $\alpha$, $\beta$, $\gamma$ and $\delta$. The numerical values
for charged-AdS BH are calculated in\cite{david12} as $\alpha=0$, $\beta=1/2$, $\gamma=1$ and $\delta=3$. }
One may compute the entropy via the relation:
\begin{eqnarray}
{\cal S}_{i} &=& - \left(\frac{\partial F_{i}}{\partial T_{i}}\right)_{V_{i}}=\pi r_{i}^2 ~. \label{eq23}
\end{eqnarray}
which is exactly same as in Eq. (\ref{eq5}). There are two types of specific heat. The specific heat
at constant thermodynamic volume and the specific heat at constant pressure. They are defined as
\begin{eqnarray}
\left( C_{V} \right)_{i} &=& T_{i} \left(\frac{\partial S_{i}}{\partial T_{i}}\right)_{V} .~\label{cv}
\end{eqnarray}
and
\begin{eqnarray}
\left( C_{P} \right)_{i} &=& T_{i} \left(\frac{\partial S_{i}}{\partial T_{i}}\right)_{P}
.~\label{cp}
\end{eqnarray}
From Eq. \ref{eq23}, we can easily see that the entropy ${\cal S}_{i}$ is independent of $T_{i}$ therefore we get
\begin{eqnarray}
\left( C_{V} \right)_{i} &=& 0 .~\label{cv1}
\end{eqnarray}
and we find
\begin{eqnarray}
\left( C_{P} \right)_{i} &=&
-2\pi r_{i}^2 \frac{\left(1-\frac{Q^2}{r_{i}^2}+8\pi Pr_{i}^2\right)}{\left(1-\frac{3Q^2}{r_{i}^2}-8\pi Pr_{i}^2\right)}
.~\label{cp2}
\end{eqnarray}
The specific heat $C_{P}$ diverges at
\begin{eqnarray}
8\pi P r_{i}^4-r_{i}^2+3Q^2 &=& 0 ~. \label{eq26}
\end{eqnarray}
or i.e. at
\begin{eqnarray}
r_{i} &=& \pm \sqrt{\frac{1\pm\sqrt{1-96\pi Q^2 P}}{16\pi P}} ~. \label{eq27}
\end{eqnarray}
which implies a second order phase transition occurs at that point.
\section{\label{quint} Thermodynamic properties of the RN-AdS BH surrounded by quintessence:}
In this section, we will show how the quintessence dark energy matter does affect on the
thermodynamic product relation in the extended phase space. The metric function of Eq. (\ref{metric})
for RN-AdS BH surrounded by quintessence can be written as \cite{kies03}
\begin{eqnarray}
{\cal U}(r) &=& 1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{a}{r^{3w_{q}+1}}-\frac{\Lambda}{3}r^2
~. \label{eq28}
\end{eqnarray}
where $w_{q}$ is the state parameter and $a$ is the normalization factor related to the density
of quintessence. The ranges for quintessence dark energy is $-1<w_{q}<-\frac{1}{3}$ and for
phantom dark energy: $w_{q}<1$. In terms of $a$, the density of quintessence can be defined as
\begin{eqnarray}
\rho_{q} &=& -\frac{3aw_{q}}{2r^{3w_{q}+1}} ~. \label{eq29}
\end{eqnarray}
In the extended phase space the function can be written as
\begin{eqnarray}
{\cal U}(r) &=& 1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{a}{r^{3w_{q}+1}}+\frac{8\pi P}{3}r^2
~. \label{eq30}
\end{eqnarray}
Now the horizon Eq. can be written as
\begin{eqnarray}
\frac{8\pi P}{3}r^{3w_{q}+3}+r^{3w_{q}+1}-2Mr^{3w_{q}}+Q^2r^{3w_{q}-1} -a &=& 0 ~.\label{eq31}
\end{eqnarray}
Using Vieta's theorem, we find
\begin{eqnarray}
\sum_{i=1}^{3w_{q}+3} r_{i} &=& 0 ~.\label{eq32}\\
\sum_{1\leq i<j\leq (3w_{q}+3)} r_{i}r_{j} &=& \frac{3}{8\pi P} ~.\label{eq33}\\
\sum_{1\leq i<j<k\leq (3w_{q}+3)} r_{i}r_{j} r_{k} &=& \frac{3M}{4\pi P} ~.\label{eq34}\\
\sum_{1\leq i<j<k<l\leq (3w_{q}+3) } r_{i}r_{j} r_{k}r_{l} &=& \frac{3Q^2}{8\pi P} ~.\label{eq35}\\
\prod_{i=1}^{3w_{q}+3} r_{i} &=& a ~.\label{eq36}
\end{eqnarray}
It should be mentioned that $3w_{q}$ is an integer quantity.
The entropy ${\cal S}_{i}$ and electric potential $\Phi_{i}$ are same as in RN-AdS case. Now the mass of
the BH could be expressed in terms of the horizon radius and dynamic pressure:
\begin{eqnarray}
M &=& \frac{r_{i}}{2}\left(1+\frac{Q^2}{r_{i}^2}-\frac{a}{r_{i}^{3w_{q}+1}}+\frac{8\pi P}{3}r_{i}^2\right)
~. \label{eq37}
\end{eqnarray}
Hence the first law of thermodynamics becomes
\begin{eqnarray}
dH &=& T_{i} d{\cal S}_{i} + V_{i} dP +\Phi_{i} dQ + {\cal A}_{i} da~. \label{eq38}
\end{eqnarray}
where ${\cal A}_{i}=\left(\frac{\partial H}{\partial a} \right)_{S_{i},Q,P}=-\frac{1}{2r_{i}^{3w_{q}}}$ is
defined to be a physical quantity conjugate to the state parameter \cite{li}.
The corresponding Smarr relation reads
\begin{eqnarray}
H &=& 2T_{i} {\cal S}_{i} - 2P V_{i} +Q \Phi_{i}+(1+3w_{q}){\cal A}_{i} da ~. \label{eq39}
\end{eqnarray}
Again the mass(or enthalpy) independent entropy sum and entropy product relations are
\begin{eqnarray}
\sum_{i=1}^{(3w_{q}+3)} \sqrt{{\cal S}_{i}} &=& 0 ~.\label{eq40}\\
\sum_{1\leq i<j\leq (3w_{q}+3)} \sqrt{{\cal S}_{i}{\cal S}_{j}} &=& \frac{3}{8P} ~.\label{eq41}\\
\prod_{i=1}^{(3w_{q}+3)} \sqrt{\frac{{\cal S}_{i}}{\pi}} &=& a ~.\label{eq42}
\end{eqnarray}
Similarly, the mass independent volume sum and volume product relations are
\begin{eqnarray}
\sum_{i=1}^{(3w_{q}+3)} {V_{i}}^{\frac{1}{3}} &=& 0 ~.\label{eq43}\\
\sum_{1\leq i<j\leq (3w_{q}+3)}(V_{i}V_{j})^{\frac{1}{3}} &=& \left(4\pi\right)^{\frac{2}{3}} \frac{3}{8\pi P}
~.\label{eq44}\\
\prod_{i=1}^{(3w_{q}+3)} \left(\frac{3V_{i}}{4\pi}\right)^{\frac{1}{3}}&=& a ~.\label{eq45}
\end{eqnarray}
These are explicitly mass-independent relations for RN-AdS BH surrounded by quintessence.
It follows from the above analysis that the entropy product and volume product relations are
strictly dependent on \emph{quintessence dark energy matter}. It is quite interesting to mentioned
that the entropy product is mass-independent but there has been effect of quintessence dark energy matter on
that thermodynamic product relations.
Now the BH temperature for all the horizons could be defined as
\begin{eqnarray}
T_{i} &=& \frac{1}{4\pi r_{i}} \left(1+8\pi P r_{i}^2-\frac{Q^2}{r_{i}^2}+ \frac{3aw_{q}}{r_{i}^{3w_{q}+1}}\right)
~. \label{eq46}
\end{eqnarray}
and the BH equation of state should read
\begin{eqnarray}
P &=& \frac{T_{i}} {2r_{i}} -\frac{1}{8\pi r_{i}^2}+\frac{Q^2}{8\pi r_{i}^4}-
\frac{3aw_{q}}{8\pi r_{i}^{3(w_{q}+1)}} ~. \label{eq47}
\end{eqnarray}
where $r_{i}=\left(\frac{3V_{i}}{4\pi}\right)^{1/3}$. Again in terms of specific
volume $v_{i}=2r_{i}$ the above equation should be rewritten as
\begin{eqnarray}
P &=& \frac{T_{i}} {v_{i}} -\frac{1}{2\pi v_{i}^2}+\frac{2Q^2}{\pi v_{i}^4}-\frac{3aw_{q}2^{4w_{q}}}{\pi v_{i}^{3(w_{q}+1)}}
~. \label{eq48}
\end{eqnarray}
The critical values are explicitly calculated in \cite{li}. So we do not written here.
The Gibbs free energy for all the horizons could be written as
\begin{eqnarray}
G_{i} &=& H-T_{i}S_{i}=M-T_{i}S_{i}
=\frac{r_{i}}{4}-\frac{2\pi P}{3} r_{i}^3+\frac{3Q^2}{4r_{i}}-\frac{3aw_{q}+2a}{4 r_{i}^{3w_{q}}}
~. \label{eq49}
\end{eqnarray}
Again we compute the specific heat at constant thermodynamic pressure to study the local stability of the BH given by
\begin{eqnarray}
\left(C_{P} \right)_{i} &=& -2\pi r_{i}^2 \frac{\left(1-\frac{Q^2}{r_{i}^2}+\frac{3aw_{q}}{r_{i}^{3w_{q}+1}}+8\pi Pr_{i}^2\right)}
{\left(1-\frac{3Q^2}{r_{i}^2}+\frac{3(2+3w_{q})aw_{q}}{r_{i}^{3w_{q}+1}}-8\pi Pr_{i}^2\right)} .~\label{sh1}
\end{eqnarray}
It should be noted that the specific heat diverges at
\begin{eqnarray}
1-\frac{3Q^2}{r_{i}^2}+\frac{3(2+3w_{q})aw_{q}}{r_{i}^{3w_{q}+1}}-8\pi Pr_{i}^2 &=& 0 .~\label{sh2}
\end{eqnarray}
which signals a second order phase transition.
\section{\label{fr} Thermodynamic properties of AdS BH in $f(R)$ gravity:}
This section is dedicated to study the thermodynamic properties of a static, spherically symmetric AdS BH in $f(R)$ gravity.
It is a kind of modified gravity and it is very useful tool for explaining the current and future state of the
accelerating universe. It is also helpful for explaining the inflation and structure formation in the early universe.
The metric \cite{moon11,chen} function for this kind of gravity can be written as
\begin{eqnarray}
{\cal U}(r) &=& 1-\frac{2m}{r}+\frac{q^2}{\alpha r^2}-\frac{R_{0}}{12}r^2 ~. \label{eq50}
\end{eqnarray}
where $\alpha=1+f'(R_{0})$. The quantities $m$ and $q$ are related to the $M$(ADM mass) and $Q$(electric charge) in
this gravity becomes
\begin{eqnarray}
M =m\alpha ,\,\,\,\, Q=\frac{q}{\sqrt{\alpha}}~. \label{eq51}
\end{eqnarray}
As is the thermodynamic pressure in $f(R)$ gravity can be written as $P=-\frac{\Lambda}{8\pi} \alpha$ and the
constant scalar curvature as $R_{0}=-\frac{12}{\ell^2}=4\Lambda$. The corresponding thermodynamic volume can be
defined as
$V_{i} = \frac{4}{3}\pi r_{i}^3$.
Therefore the horizon equation should read
\begin{eqnarray}
\frac{8\pi P}{3}\alpha r^4+\alpha r^2-2m\alpha r+q^2 &=& 0 ~.\label{eq52}
\end{eqnarray}
To finding the roots we again apply the Vieta's rule, we have
\begin{eqnarray}
\sum_{i=1}^{4} r_{i} &=& 0 ~.\label{eq53}\\
\sum_{1\leq i<j\leq 4} r_{i}r_{j} &=& \frac{3}{8\pi P} ~.\label{eq54}\\
\sum_{1\leq i<j<k\leq 4} r_{i}r_{j} r_{k} &=& \frac{3m}{4\pi P} ~.\label{eq55}\\
\prod_{i=1}^{4} r_{i} &=& \frac{3q^2}{8\pi \alpha P} ~.\label{eq56}
\end{eqnarray}
The entropy can be defined as
\begin{eqnarray}
{\cal S}_{i} &=& \pi \alpha r_{i}^2 ~.\label{eq57}
\end{eqnarray}
and the BH temperature\cite{chen} should be
\begin{eqnarray}
T_{i} &=& \frac{1}{4\pi r_{i}} \left(1+\frac{8\pi P}{\alpha} r_{i}^2-\frac{q^2}{\alpha r_{i}^2} \right)
~.\label{eq58}
\end{eqnarray}
Again the electric potential in $f(R)$ gravity could be defined as
\begin{eqnarray}
\Phi_{i} &=& \frac{q}{r_{i}}\sqrt{\alpha} ~.\label{eq59}
\end{eqnarray}
Now the mass(or enthalpy) independent entropy sum and entropy product formula in $f(R)$
gravity should read:
\begin{eqnarray}
\sum_{i=1}^{4} \sqrt{{\cal S}_{i}} &=& 0 ~.\label{eq60}\\
\sum_{1\leq i<j\leq 4} \sqrt{{\cal S}_{i}{\cal S}_{j}} &=& \frac{3\alpha}{8P} ~.\label{eq61}\\
\prod_{i=1}^{4} \sqrt{{\cal S}_{i}} &=& \frac{3 \pi\alpha q^2}{8P} ~.\label{eq62}
\end{eqnarray}
In terms of two horizons, the mass-independent entropy product formula reads as
\begin{eqnarray}
\frac{\left(\frac{3q^2}{8P} \right)}{\sqrt{{\cal S}_{1}{\cal S}_{2}}}-
\frac{{\cal S}_{1}+{\cal S}_{2}+\sqrt{{\cal S}_{1}{\cal S}_{2}}}{\pi \alpha} &=& \frac{3}{8\pi P}
~. \label{eqq3}
\end{eqnarray}
Again the mass independent \emph{volume sum} and \emph{volume product} becomes
\begin{eqnarray}
\sum_{i=1}^{4} {V_{i}}^{\frac{1}{3}} &=& 0 ~.\label{eq63}\\
\sum_{1\leq i<j\leq 4}(V_{i}V_{j})^{\frac{1}{3}} &=& \left(\frac{3}{32 \pi}\right)^{\frac{1}{3}}\frac{1}{P}
~.\label{eq64}\\
\prod_{i=1}^{4} (V_{i})^{\frac{1}{3}}&=&
\left(\frac{\pi}{6}\right)^{\frac{1}{3}} \frac{q^2}{\alpha P} ~.\label{eq65}
\end{eqnarray}
Again in terms of two horizons, the mass independent volume product formula reads
\begin{eqnarray}
\left(\frac{3}{32 \pi}\right)^{\frac{1}{3}} \frac{\left(\frac{q^2}{P}\right)}{\alpha(V_{1}V_{2})^{\frac{1}{3}}}
-\left(\frac{3}{4\pi}\right)^{\frac{2}{3}}\left[{V_{1}}^{\frac{2}{3}}+{V_{2}}^{\frac{2}{3}}+(V_{1}V_{2})^{\frac{1}{3}}
\right] &=& \frac{3}{8\pi P} ~. \label{eqq4}
\end{eqnarray}
Once again these are explicitly mass-independent relation in the extended phase space in $f(R)$ gravity. It should be noted
that in the limit $\alpha=1$, one obtains the result of RN-AdS BH in extended phase space. For our record we should
be noted that the equation of state in $f(R)$ gravity\cite{chen}:
\begin{eqnarray}
P &=& \frac{\alpha T_{i}} {2r_{i}} -\frac{\alpha}{8\pi r_{i}^2}+\frac{q^2}{8\pi r_{i}^4} ~. \label{eq66}
\end{eqnarray}
In terms of specific volume $v_{i}=2r_{i}$, the above Eq. could be rewritten as
\begin{eqnarray}
P &=& \frac{\alpha T_{i}} {v_{i}} -\frac{\alpha}{2\pi v_{i}^2}+\frac{2q^2}{\pi v_{i}^4} ~. \label{eq67}
\end{eqnarray}
From the equation of state we can easily derived the critical constants by applying the appropriate condition.
\footnote{The critical values for $f(R)$ gravity explicitly computed in \cite{chen} are $P_{c}=\frac{\alpha^2}{96\pi q^2}$,
$v_{c}=\frac{2q\sqrt{6}}{\sqrt{\alpha}}$ and $T_{c}=\frac{\sqrt{6 \alpha}}{18\pi q}$}.
Finally, the Gibbs free energy \cite{chen} should read
\begin{eqnarray}
G_{i} &=& \frac{\alpha r_{i}}{4}-\frac{2\pi P}{3} r_{i}^3+\frac{3q^2}{4r_{i}} ~. \label{eq68}
\end{eqnarray}
In this case, the specific heat at constant pressure is found to be
\begin{eqnarray}
\left(C_{P}\right)_{i} &=& -2\pi r_{i}^2 \frac{\left(1-\frac{q^2}{\alpha r_{i}^2}+8\pi Pr_{i}^2\right)}
{\left(1-\frac{3q^2}{\alpha r_{i}^2}-8\pi Pr_{i}^2\right)} .~\label{sh3}
\end{eqnarray}
The specific heat diverges at
\begin{eqnarray}
8\pi \alpha P r_{i}^4-\alpha r_{i}^2+3q^2 &=& 0 ~. \label{eq69}
\end{eqnarray}
or
i.e. at
\begin{eqnarray}
r_{i} &=& \pm \sqrt{\frac{\alpha \pm\sqrt{\alpha^2-96\pi \alpha Pq^2}}{16\pi \alpha P}} ~. \label{eq70}
\end{eqnarray}
Again it signals a second order phase transition.
\section{\label{dis} Conclusion:}
The present study demonstrated that the thermodynamic properties of spherically symmetric charged-AdS black
hole, charged AdS BH surrounded by quintessence and charged AdS BH in $f(R)$ gravity in the extended
phase-space. The extended phase space means where the cosmological constant should be treated as
thermodynamic pressure and its conjugate variable as a thermodynamic volume. We derived various
thermodynamic products particularly entropy products and thernodynamic volume products. In all the three cases,
it has been shown that the mass(or enthalpy) independent properties turn out to be an universal like quantities. It should
be noted that the presence of the quintessence matter does affect on the expression of entropy product
and thermodynamic volume products. The first law of BH thermodynamics and Smarr formula have been studied
for all the horizons. The BH equation of state has been derived for all the horizons. The divergence of the
specific heat indicates that the second order phase transition should occur at a certain condition. In summary, the
thermodynamic relations that we derived provide some universal characterization of the BH which \emph{may} provide
insight into the origin of BH entropy both \emph{inner and outer}.
|
2,877,628,090,774 | arxiv | \section{Introduction and data}
The Space Telescope Imaging Spectrograph (STIS) instrument on the
Hubble Space Telescope (HST) offers new opportunities
for QSO host galaxy studies. The instrument may be used to obtain direct
images with the CCD which are deeper and less noisy than WFCP2, and UV
images with the MAMA detectors. Use of a long slit with STIS enables
spatially resolved spectra of the host galaxy, both through and away from
the nucleus, with a pixel spacing of $\sim$ 0.05" in the visible and 0.025"
in the UV.
This paper presents the results of CCD imaging and spectroscopy of the
QSO 1345+584, which has redshift 2.039. The high redshift moves rest-UV
into the visible, so little observed UV flux is expected and no MAMA
observations were attempted. The QSO was chosen for several reasons:
it has detected extended Ly$\alpha$ flux from ground-based observations (Heckman
et al 1991), with $\sim$1.5" resolution; it is a radio-loud object with
complex compact structure (Lonsdale et al 1993) that might be seen in the
visible; and it lies
in the continuous viewing zone (CVZ) of HST so that long integrations could
be performed.
The QSO has a catalogued visible magnitude of 17.5 and is also known as
4C 58.27 or OP 577 (Hewitt and Burbidge 1993).
Lehnert et al (1993) were unable to detect any resolved continuum flux
around the QSO, in ground-based observations. Barthel et al (1988) noted
that the bend angle of the structure is remarkable and suggests stong interaction
with the surrounding medium. Bremer et al (1992) describe long-slit spectra
in which no definite extended line emission was detected.
The observations were performed on
1997 Dec 19 as shown in Table 1. The short image exposure was taken to
obtain the nucleus without detector saturation. The roll angle was
determined by the CVZ window, and spectra were taken centred on the nucleus
(slit A) and also offset arbitrarily to the north by 1.5" (slit B), through
a 2" wide slit. The wide slit degrades the
spectral resolution but was judged necessary to optimise detection of the
continuum signal from the faint QSO host galaxy. All observations
were performed with 3 CCD readouts, to identify cosmic rays.
Standard wavelength calibration exposures were taken with the spectra.
All the images were processed with standard CALSTIS reduction, but
using a mean dark image from the day of observation to eliminate hot pixels
as well as possible, since the signal levels of interest are low. The
individual reads were combined to eliminate cosmic rays in the standard way.
We discuss below the results and measurements made in the processed data.
\section{Image}
The direct image of the QSO and immediate surroundings is shown in
Figure~\ref{fig1}. The QSO is the brightest object in the field and shows
diffraction spikes and a reflection ring to the E which is caused by a
reflection off the detector window. The STIS read noise, wide bandpass, and
DQE make this image comparable in depth with a WFPC2 broad-band image of
some 10 times the exposure, or 20000 sec. Point sources to 30 mag are
detected with 3$\sigma$ confidence, as surface brightnesses to $\sim$27
mag/arcsec$^2$. While the surface brightness limit is not deep compared
with large ground-based telescopes, it applies over areas significantly
smaller than one arcsec.
The ring to the E of the QSO that looks like a curved arm is the
internal camera reflection, as are some small radial features around the
point. The exact structure of the PSF, particularly the reflection ring,
depends on the location within the image. Suitably placed stellar images were
found in the database of parallel images for use in removing the PSF
from the QSO image. Figure~\ref{fig2} shows the PSF-removed image. The signal level
in the ring is some 10 times higher than the faintest structure detectable
in the clean parts of the image, and the noise and uncertainty of the
subtraction means that we know little of the real structure in this region.
However, there is definite detected structure at distances several times
larger than the ring, mostly on the opposite (W) side. This agrees with the
ground-based result, as shown in Figure~\ref{fig3}. The ground-based result goes to
fainter fluxes, but lacks information in the inner part and has some 30
times lower spatial resolution.
Comparison of the visible structure with the radio shows a high degree
of coincidence, if we align the eastern radio knot with the optical nucleus,
as also shown in Figure~\ref{fig3}.
The correspondence is not exact in position or relative brightness.
With the exception of the inner bright knot, the radio knots are more compact
than the optical. The optical flux at the western-most radio knot is weaker and
lies outside the radio: this is where it curves most strongly. The outer
two knots to the north lie very close to the optical. In addition to the
radio knots, there are other regions of optical flux.
The QSO nucleus has m$_V\sim$ 18.5, which is fainter than the 17.5 quoted in
Hewitt and Burbidge (1993), but consistent with the m$_B$=18.6 quoted by
Heckman et al (1991), and the very blue continuum seen in our spectrum.
The resolved flux from the host galaxy may be measured in
two ways. First, the azimuthally averaged luminosity profile is compared with that
of the PSF normalised to the peak value, and the flux of the difference measured.
Second the flux from the continuum-subtracted image is measured (Fig 2).
In the first instance, the central pixels of the QSO are saturated in the
long exposure, and were replaced by scaled values of the short exposure.
The difference flux down to a level of 26 mag/arcsec$^2$ is 31 times less
than the PSF, or magnitude 22.4. The flux from the PSF-subtracted QSO is
less by 25\%. This reaches to similar flux levels, but is subject to innacuracies
near the bright core, and this region was omitted from the measurement.
Thus a good estimate of the host magnitude is 22.4, which corresponds to absolute
magnitude $\sim$-22 if it is a star-forming galaxy (assuming H$_0$=65 and q$_0$=0.5).
However, we show below that
much of the blue flux is Ly$\alpha$ line emission, and since C IV is the only other
strong line feature in the CCD bandpass (rest wavelength $\sim$1200 to 2900A),
the galaxy continuum may have M$_V$ nearer -21.5.
A luminosity plot (Figure~\ref{fig4})
of a smoothed (gaussian FWHM 2.5 pixels) image shows
the QSO profile continues to fall over 80 pixels (4 arcsec) which is comparable
with the ground-based detection contours (Heckman et al 1991). A contour plot
of the smoothed image (Figure 3) shows the faint flux is not symmetrical, but more
extended to the W, as also found in the ground-based data. Similar faint haloes
appear around the other bright galaxies, so that their outer limits fill a
good fraction ($\sim$25\%) of the sky in this group.
The image also shows a group of remarkable looking galaxies in the field.
They are not evenly distributed, lying almost entirely to the W of the
QSO. Table 2 shows the photometric measures of the principal galaxies, as
labelled in Figure~\ref{fig1}. The bright objects have similar sizes and total
flux, which are similar to the QSO host galaxy. This kind of group is unusual:
in comparing randomly selected images from the parallel exposure program, of 7
other high latitude fields with STIS,
with similar exposures, we find no other case of a concentration within the field,
of galaxies with this small brightness range. (The closest
cases are a group of smooth galaxies associated with a large bright galaxy,
presumably at low redshift, and in a deep
image (2100 sec) a few non-clustered compact knotty galaxies, presumably at high z.
The other random images are devoid of any galaxies of this brightness.)
In the 1345+584 field, the galaxy morphologies are varied, but mostly
irregular and knotty - very like rest UV images of galaxies with active
star-formation. Galaxy 5 is a face-on spiral with a faint nucleus and
bulge but with very bright knots in the spiral arms. Galaxy 1 is edge-on
with no bulge or strong dust lane, and a knot in the plane SW of the nucleus.
Galaxy 2 lies next to a star, but has irregular morphology and several
bright knots, none of which is central. Galaxy 6 has an elliptical shape but
contains 3 bright knots as well as a resolved nuclear region. Galaxy 4 is compact
and bright, irregular, and appears to be associated with a curved string
of knots to the NE, the inner ones of which connect to the main galaxy.
There are numerous other galaxies with a range of size and surface brightness,
and almost all irregular or knotty.
Thus, it seems possible that these galaxies are companions of the QSO at z=2.0, seen
in rest UV wavelengths. Sizes at this redshift are in the range 5 to 20 Kpc,
and the entire group lies within $\sim$150 Kpc projected on the sky.
If they are at the QSO redshift and young, k-corrections will be negative
(-1.5 for mid-B star at z=2.0) and the bright galaxy absolute magnitudes
for H$_0$=65 and q$_0$=0.5 are given in Table 2. These are roughly what we
expect for L* galaxies with active star-formation. The spatial distribution of
the numerous smaller galaxies suggests they are also associated with
the group. Their sizes are as small as fractions of an arcsec, corresponding to
$<$5 Kpc at z=2, and their absolute magnitudes for a young population a z=2 range
from -18 to -20. We note in this respect the group of compact galaxies that
appear to be associated with the z$\sim$2 QSO behind cluster A851 (Dressler et al
1993, Hutchings and Davidge 1997).
\section{Spectra}
Figure~\ref{fig2} shows the placement of the slits, assuming that the nuclear
slit was centred. The absence of detectable scattered nuclear signal is
consistent with this placement, since the nuclear signal is spread over
all wavelengths and the resolved light is almost entirely in Ly$\alpha$.
The off-nuclear signals are very weak and are only seen significantly at
Ly$\alpha$. The faint small galaxy to the SW is excluded from the slit but another
to the W is included. It shows no detectable continuum, which is not
surprising as it is 25 magnitude, but it also shows no line emission in the
wavelength range covered. No other objects are covered by the slits.
The nuclear spectrum (Figure~\ref{fig5})
shows broad line emission from O VI, Ly$\alpha$, N V,
Si IV, C IV, He II and the edge of C III]. There are other weak broad features
(e.g. C III/N IV at 3965A or rest 1307A) and what appear to be several Ly$\alpha$
absorbers. The redshift is measured at
2.037. The new nuclear spectrum is clearly better than the one
(Wills and Wills 1974) on which the published redshift of 2.039 is based.
The off-nuclear spectrum was cleaned carefully of hot pixels and cosmic rays
and smoothed with different functions to detect significant signal. No continuum
source is seen in either slit position, including the position of the bright knot
in the radio jet. Figure~\ref{fig6} shows plots of the spectra from different
positions in the images.
There is faint emission near the position of Ly$\alpha$ extending
several arcsec to the W, and somewhat less to the E of the QSO, in both slit
positions. The 2 arcsec wide slit means that we have essentially slitless
spectra of the emission-line
material: the spectral image is determined by the size, position, and velocity
of the resolved regions within the slit. Thus, the undispersed image is essential
in interpreting the spectral image (see e.g. Hutchings et al 1998).
The slit width corresponds to $\sim$100A or a velocity of 7000 km/s, so the
spectrum is dominated by the spatial structure of the knots, seen in
Figure~\ref{fig2}.
There is a knot of brighter emission at the position along slit A of the
bright radio/optical knot. This Ly$\alpha$ emission is broadened and thus may
indicate a velocity gradient within the knot; however, the signal is weak and
occupies only a few pixels. No extended signal is detected at C IV.
In the spectral image from slit A (centred on the nucleus), there are two
bands of emission, seen on either side of the nuclear peak wavelength
(Figure~\ref{fig6}). Both bands run E-W through the nucleus but the band
at shorter wavelength extends further to the W and less far to the E: it is also
narrower. The slit displaced to the N (slit B) also shows the narrower band, this
time displaced to longer wavelength. The two slits overlap by 0.5 arcsec just where
there is a region of bright flux seen in the image, so that the apparent wavelength
shift is due to its placement at the left and right side of the two slit positions.
Slit B also shows emission some distance (7 - 8 arcsec) to the E, which does not
correspond to any galaxy or continuum feature in the image. This is shown in
Figure~\ref{fig6}.
Thus there seems to be gas at the QSO redshift throughout its general vicinity.
Because of the wide slit, velocities with respect to the nucleus cannot be
estimated accurately. However, using the undispersed image as a template
(Figure~\ref{fig2}),
we do measure velocities consistently within 100 km/s for the clouds in the
slit overlap region. Thus we have some confidence in the values. They indicate
that clouds to the S of the nucleus have velocities of approach of order 1000 km/s,
while those to the N are receding at velocities of 100 km/s near the nucleus
and up to 1000 km/s to the W.
If the emission from the brightest knot is correctly identified by its displacement
along the slit, then it, exceptionally, has high approach velocity, from
-3200 to -3700 km/s. We see no signal from another other knot
within the radio structure, which lies in slit B, although it is as bright as the
other features in the direct image. All this suggests that Ly$\alpha$
is suppressed in or near the radio jet, and such material has different velocity,
apparently of approach. The Ly$\alpha$ gas not associated with the jet has a
systematic velocity field approaching to the S and receding to the N.
Finally, we infer that the jet-associated material has a continuum spectrum,
which may be synchrotron or hot stars.
We may make a rough estimate of the flux budget for the host galaxy.
The ratio between nuclear continuum (excluding the strong Ly$\alpha$ and C IV emission
lines) to total image signal, scaled to the same exposure, is 3. That is, some 2/3
of the image signal comes from the wavelength region from $\sim$5000A to the limit
of the CCD sensitivity near 1 micron. There are no strong emission lines in this
(redshifted) region of the spectrum. The total signal from Ly$\alpha$ extended
line emission is about half the resolved image flux, scaled by this factor 3
(which assumes the SED of the host is the same as the QSO nucleus). Thus,
it appears that about half the resolved flux in our spectral region is in
the form of continuum radiation which is too faint to detect. The signal level
from this flux spread over several arcsec would be below the detection limits of
the data. Summing 100 rows far from the QSO and comparing them with 100 rows next
to the QSO shows the region near the QSO to have slightly bluer SED - i.e. more
far-UV continuum. The difference is significant only at the 1-2$\sigma$ level.
The nuclear spectrum from 3000A to 5700A has total flux of 5.4 x 10$^{-13}$
erg/sec. The measured Ly$\alpha$ flux is $\sim$4 x 10$^{-15}$ erg/sec and from
the image we estimate that about an equal amount lies outside the slits we used,
so that the total Ly$\alpha$ flux is of order 10$^{-14}$ erg/sec. This agrees
reasonably with the value 6.3 x 10$^{-15}$ estimated by Heckman et al (1991).
\section{Conclusion}
The data presented reveals complex structure of material whose brightest features
correspond with radio knots in a curved compact structure around the QSO.
We find Ly$\alpha$ cloud velocities up to 1000 km/s near the nucleus, and possibly
several times that within the radio jet. The Ly$\alpha$ flux is weaker relative
to the total image signal in the knots associated with the radio structure,
indicating a higher ionisation in these regions. This is consistent with the
findings of Axon et al (1998) in NGC 1068, and the high ionisation of the regions
of bent radio structure in PKS 2152-699 reported by Fosbury et al (1998).
The QSO
appears to lie at the edge of a dense group of young galaxies with significant
Ly$\alpha$ emitting gas around it. This suggests that the radio/optical structure
is a tail caused by the QSO's motion through the group. We infer that much of the
structure is UV continuum so that the bright knots may be regions of star-formation
excited by the radio jet through the surrounding gas.
This QSO and its environment appear to be of considerable interest in offering
a snapshot of early formation of galaxies in a dense group, and the activation of
a QSO. Further information on the dynamics and stellar populations present,
require slit spectroscopy of the galaxies and region with a large aperture
telescope.
I am grateful to the following for their help and advice on processing the
STIS data: Keith Feggans, Bob Hill, Jon Gardner, and Don Lindler. I also thank
Colin Lonsdale, Daniel Durand, and Sharon Hanna for providing data for some of
the diagrams.
\newpage
\centerline{References}
Axon D.J., Marconi A., Capetti A., Macchetto F.D., Schreier E., Robinson A.,
1998, ApJ, (letters preprint)
Barthel P.D., Miley G.K., Schilizzi R.T., Lonsdale C.J., 1988, AAS, 73, 515
Bremer M.N., Crawford C.S., Fabian A.C., Johnstone R.M., 1992, MNRAS, 254, 614
Dressler A., Oemler A., Gunn J.E., Butcher H., 1993, ApJ, 404, L45
Fosbury R.A.E., Morganti R., Wilson W., Ekers R.D., di Serego Alighieri S.,
Tadhunter C.N., 1998, MNRAS (preprint astro-ph/9801249)
Heckman T.M., Lehnert M.D., van Breugel W., Miley G.K., 1991, ApJ, 370, 78
Hewitt A., and Burbidge G.R., ApJS, 87, 451
Hutchings J.B., and Davidge T.J., 1997, PASP, 109, 667
Hutchings J.B., et al 1998, ApJ, 492, L115
Lehnert M.D., Heckman T.M., Chambers K.C., Miley G.K., 1992, ApJ, 393, 68
Lonsdale C.J., Barthel P.D., Miley G.K., 1993, ApJS, 87, 63
Wills D., and Wills B.J., 1974, ApJ, 190, 271
\newpage
\centerline{Captions to figures}
\figcaption[hutchings.fig1.ps]{STIS CCD image of QSO 1345+584 and surroundings.
The QSO and its brightest 6
companions are labelled. N is up and E to the left, and the field shown is
40 x 30 arcsec. The bright `arm' to the E of the QSO is part of the PSF. The field
outside the section shown is empty of galaxies on this display. \label{fig1}}
\figcaption[hutchings.fig2.ps]{Detail of QSO image before and after PSF subtraction.
The area shown is 7.5 arcsec on a side. The subtracted image is
smoothed to reduce noise. The two slit positions are shown: they are 2 arcsec wide
and offset by 1.5 arcsec, and thus overlap by 0.5 arcsec. The X marks the nucleus
position. \label{fig2}}
\figcaption[hutchings.fig3.ps]{Contour plots of STIS image matched with
ground-based data.
Upper: smoothed STIS CCD image on left and optical PSF-subtracted image from
Heckman et al (1991). Lower: PSF-subtracted STIS image and 15 GHz image from
Lonsdale et al (1993). The QSO nucleus is marked with X and the radio
knots correspond with optical knots. See text for discussion. \label{fig3}}
\figcaption[hutchings.fig4.ps]{Azimuthally averaged luminosity profile of
1345+584 and star image at same
point on the detector. Typical error bars are shown at different radii, but the
resolved flux is not azimuthally symmetric. \label{fig4}}
\figcaption[hutchings.fig5.ps]{Spectrum of the QSO nucleus with principal
emission lines marked. Note the
unreddened continuum and Ly$\alpha$ absorption lines. \label{fig5}}
\figcaption[hutchings.fig6.ps]{Off-nuclear spectra compared with the nucleus,
which has been scaled by 1/100. The knot spectrum is scaled by 1/3 of the others.
The units are non-fluxed counts. The lowest traces are smoothed sky-subtracted
plots, with approximate linear normalisations sketched in. Note the different
Ly$\alpha$ emission components. Most of the line displacement in wavelength is
due to positions of line emitting regions within the 2" wide slits. \label{fig6}}
\newpage
\begin{deluxetable}{llll}
\tablenum{1}
\tablecaption{STIS observations of QSO 1345+584}
\tablecaption{Position angle -98$^o$, Dec 19 1997}
\tablehead{\colhead{Observation} &\colhead{Mode} &\colhead{Slit}
&\colhead{Exposure} \\ &&\colhead{(sec)}}
\startdata
Image &MIRVIS &50CCD &200\nl
Image &MIRVIS &50CCD &2000\nl
Spectrum on-nucleus &G450L &52x2 &3360\nl
Spectrum 1.5" N of nucleus &G450L &52x2 &3360\nl
\enddata
\end{deluxetable}
\begin{deluxetable}{llll}
\tablenum{2}
\tablecaption{Galaxies in field of QSO 1345+584}
\tablehead{\colhead{Galaxy \#} &\colhead{Comment} &\colhead{Mag}
&\colhead{Abs mag\tablenotemark{a}}}
\startdata
Q &QSO &18.5 &-26.0\nl
Q &Fuzz &22.4 &-22.1\nl
1 &Edge-on disk &21.7 &-22.8\nl
2 &Knotty + star &22.7 &-21.8\nl
3 &LSB smooth with knot &22.1 &-22.3\nl
4 &Compact knotty &22.7 &-21.8\nl
5 &Face-on knotty spiral &22.0 &-22.5\nl
6 &Nucleated E with knot &21.8 &-22.7\nl
&Typical small galaxy &24.5 &-20\nl
\enddata
\tablenotetext{a}{H$_0$=65, q$_0$=0.5}
\end{deluxetable}
\end{document}
|
2,877,628,090,775 | arxiv | \section{Introduction}\label{SIntro}
In \cite{KL1,KL2}, Khovanov and Lauda have introduced a new family of
graded algebras whose representation theory is related to categorification of quantum groups. Similar algebras have been defined by Rouquier \cite{Ro}.
In this note we give an explicit construction of
the irreducible graded representations of simply laced Khovanov-Lauda algebras which are concentrated in one degree. These {\em homogeneous} representations turn out to be similar to seminormal representations of affine Hecke algebras. In type $A$ this can be explained using \cite{BK}.
By-products of our construction are notions of skew shape and standard tableaux for arbitrary simply laced types. Equivalent notions have been considered before by Peterson, Proctor, Stembridge, and Fan \cite{P1,P2,S1,S2,F,N1,N2}. In particular, the Peterson-Proctor hook formula gives dimensions of the homogeneous irreducible modules corresponding to straight shapes.
\subsection*{Acknowledgements}
The first author is grateful to the University of Melbourne for support and hospitality. Both authors are grateful to K. Nakada, J. Stembridge and R. Green for useful comments.
\section{Khovanov-Lauda algebras}
\subsection{Definition}
Let $\Gamma$ be a graph without multiple edges and loops (cycles allowed). Denote the set of vertices of $\Gamma$ by $I$. If $i,j\in I$ are connected by an edge, we will say that $i$ and $j$ are {\em neighbors} (in ${\Gamma}$). We allow for $I$ to be infinite and for $\Gamma$ to contain cycles. To $\Gamma$ we associate a generalized Cartan matrix $(a_{ij})_{i,j\in I}$ as in \cite{Kac}, so that
$$
a_{ij}=
\left\{
\begin{array}{ll}
2 &\hbox{if $i=j$,}\\
-1 &\hbox{if $i$ and $j$ are neighbors,}\\
0 & \hbox{otherwise.}
\end{array}
\right.
$$
We fix an orientation on the edges of $\Gamma$.
Let $Q =\bigoplus_{i \in I} {\mathbb Z}{\alpha}_i$ be a lattice with
a basis $\{{\alpha}_i\}_{i\in I}$ labeled by~$I$. Set $$Q_+ =\bigoplus_{i\in I}{\mathbb Z}_{\geq 0}{\alpha}_i.$$ For ${\alpha}=\sum_{i\in I}m_i{\alpha}_i\in Q_+$
define the {\em height} of ${\alpha}$ as
$$
{\operatorname{ht}}({\alpha}):=\sum_{i\in I}m_i
$$
The symmetric group $S_d$
with basic transpositions $s_1,\dots,s_{d-1}$
acts
on $I^d$ on the left by place
permutations.
We have a decomposition of $I^d$ into $S_d$-orbits:
$$
I^d = \bigsqcup_{\alpha\in Q^+\atop {\operatorname{ht}}(\alpha) = d} I^\alpha,
$$
where
\begin{equation*}
I^\alpha := \{\text{\boldmath$i$}=(i_1,\dots,i_d) \in I^d\:|\:\alpha_{i_1}+\cdots+\alpha_{i_d} = \alpha\}.
\end{equation*}
Fix an arbitrary ground field $F$ and an element ${\alpha}\in Q_+$ of height $d$. The {\em Khovanov-Lauda algebra} $R_{\alpha}$ is an associative ${\mathbb Z}$-graded unital $F$-algebra, given by generators
\begin{equation}\label{EKLGens}
\{e(\text{\boldmath$i$})\mid \text{\boldmath$i$}\in I^{\alpha}\}\cup\{y_1,\dots,y_{d}\}\cup\{\psi_1, \dots,\psi_{d-1}\}
\end{equation}
and the following relations for all $\text{\boldmath$i$},\text{\boldmath$j$}\in I^{\alpha}$ and all admissible $r$ and $s$:
\begin{equation}
e(\text{\boldmath$i$}) e(\text{\boldmath$j$}) = {\delta}_{\text{\boldmath$i$},\text{\boldmath$j$}} e(\text{\boldmath$i$}),
\quad{\textstyle\sum_{\text{\boldmath$i$} \in I^\alpha}} e(\text{\boldmath$i$}) = 1;\label{R1}
\end{equation}
\begin{equation}\label{R2PsiY}
y_r e(\text{\boldmath$i$}) = e(\text{\boldmath$i$}) y_r;
\end{equation}
\begin{equation}
\psi_r e(\text{\boldmath$i$}) = e(s_r\text{\boldmath$i$}) \psi_r;\label{R2PsiE}
\end{equation}
\begin{equation}\label{R3Y}
y_r y_s = y_s y_r;
\end{equation}
\begin{equation}\label{R3YPsi}
y_r \psi_s = \psi_s y_r\qquad (r \neq s,s+1);
\end{equation}
\begin{equation}
(y_{r+1} \psi_r-\psi_r y_r) e(\text{\boldmath$i$}) =
\left\{
\begin{array}{ll}
e(\text{\boldmath$i$}) &\hbox{if $i_r=i_{r+1}$,}\\
0 &\hbox{if $i_r\neq i_{r+1}$;}
\end{array}
\right.
\label{R5}
\end{equation}
\begin{equation}
(\psi_r y_{r+1} -y_r\psi_r)e(\text{\boldmath$i$})
=
\left\{
\begin{array}{ll}
e(\text{\boldmath$i$}) &\hbox{if $i_r=i_{r+1}$,}\\
0 &\hbox{if $i_r\neq i_{r+1}$;}
\end{array}
\right.
\label{R6}
\end{equation}
\begin{equation}
\psi_r^2e(\text{\boldmath$i$}) =
\left\{
\begin{array}{ll}
0 &\hbox{if $i_r=i_{r+1}$,}\\
e(\text{\boldmath$i$}) & \hbox{if $a_{i_r i_{r+1}}=0$,}
\\
(y_r-y_{r+1})e(\text{\boldmath$i$})
&\hbox{if $i_r\rightarrow i_{r+1}$,}
\\
(y_{r+1}-y_{r})e(\text{\boldmath$i$})
&\hbox{if $i_{r+1}\rightarrow i_{r}$;}
\end{array}
\right.
\label{R4}
\end{equation}
\begin{equation}
\psi_r \psi_s = \psi_s \psi_r\qquad (|r-s|>1);\label{R3Psi}
\end{equation}
\begin{equation}
\begin{split}
(\psi_{r+1}\psi_{r} \psi_{r+1}-\psi_{r} \psi_{r+1} \psi_{r}) e(\text{\boldmath$i$})
=
\left\{\begin{array}{ll}
e(\text{\boldmath$i$})&\text{if $i_{r+2}=i_r\rightarrow i_{r+1}$,}\\
-e(\text{\boldmath$i$})&\text{if $i_{r+1}\rightarrow i_r=i_{r+2}$,}\\
0 &\text{otherwise.}
\end{array}\right.
\end{split}
\label{R7}
\end{equation}
The grading on $R_{\alpha}$ is defined by
$$
\deg(e(\text{\boldmath$i$}))=0,\quad \deg(y_re(\text{\boldmath$i$}))=2,\quad\deg(\psi_r e(\text{\boldmath$i$}))=-a_{i_ri_{r+1}}.
$$
\subsection{Basis Theorem}
For each element $w\in S_d$ fix a reduced expression $w=s_{i_1}\dots s_{i_m}$ and set
$$
\psi_w:=\psi_{i_1}\dots \psi_{i_m}.
$$
In general, $\psi_w$ is not independent of the choice of reduced expression of $w$.
\begin{Theorem}\label{TBasis}{\cite[Theorem 2.5]{KL1}}%
{\bf (Basis Theorem)}
The elements
\begin{equation}\label{EBasis}
\{\psi_w y_1^{m_1}\dots y_d^{m_d}e(\text{\boldmath$i$})\mid w\in S_d,\ m_1,\dots,m_d\in{\mathbb Z}_{\geq 0}, \ \text{\boldmath$i$}\in I^{\alpha}\}
\end{equation}
form an $F$-basis for $R_{\alpha}$.
\end{Theorem}
Denote by $P_{\alpha}$ the (commutative) subalgebra of $R_{\alpha}$ generated by $y_1,\dots,y_d$ and all $\{e(\text{\boldmath$i$})\mid \text{\boldmath$i$}\in I^{\alpha}\}$. By the Basis Theorem,
$$
\{y_1^{m_1}\dots y_d^{m_d}e(\text{\boldmath$i$})\mid m_1,\dots, m_d\in{\mathbb Z}_{\geq 0},\ \text{\boldmath$i$}\in I^{\alpha}\}
$$
is a basis of $P_{\alpha}$.
\subsection{Modules, weights, and characters}
If $V=\oplus_{k\in {\mathbb Z}} V[k]$ is a ${\mathbb Z}$-graded vector space, its {\em graded dimension} is
$$
{\operatorname{gdim}\,} V:=\sum_{k\in {\mathbb Z}}(\dim V[k])q^k\in{\mathbb Z}[q,q^{-1}].
$$
Recall that $R_{\alpha}$ is a ${\mathbb Z}$-graded algebra. All $R_{\alpha}$-modules will be assumed {\em graded}, unless otherwise stated. We will work in the category
$$
\mod{R_{\alpha}}=\{\text{finite dimensional graded $R_{\alpha}$-modules}\}.
$$
Since all $y_re(\text{\boldmath$i$})$ are positively graded, the elements $y_r$ act nilpotently on all modules $M\in\mod{R_{\alpha}}$.
For every $\text{\boldmath$i$}\in I^{\alpha}$ and any $M\in\mod{R_{\alpha}}$, the {\em $\text{\boldmath$i$}$-weight space} of $M$ is
$
M_\text{\boldmath$i$}:=e(\text{\boldmath$i$})M.
$
We have a decomposition of (graded) vector spaces
$$
M=\bigoplus_{\text{\boldmath$i$}\in I^{\alpha}}M_\text{\boldmath$i$}.
$$
We say that $\text{\boldmath$i$}$ is a {\em weight of $M$} if $M_\text{\boldmath$i$}\neq 0$, and refer to $I^{\alpha}$, as the set of {\em weights for $R_{\alpha}$}. Note by (\ref{R2PsiE}) that
\begin{equation}\label{EAction}
\psi_r M_\text{\boldmath$i$}\subseteq M_{s_r \text{\boldmath$i$}}.
\end{equation}
Let ${\mathbb Z}[q,q^{-1}][I^{\alpha}]$ be the free ${\mathbb Z}[q,q^{-1}]$-module with basis $\{e^\text{\boldmath$i$}\mid \text{\boldmath$i$}\in I^{\alpha}\}$. The {\em formal character} of the module $M\in\mod{R_{\alpha}}$ is
\begin{equation*
{\operatorname{ch}\:} M:=\sum_{\text{\boldmath$i$}\in I^{\alpha}}({\operatorname{gdim}\,} M_\text{\boldmath$i$}) e^{\text{\boldmath$i$}}.
\end{equation*}
The formal character map ${\operatorname{ch}\:}: \mod{R_{\alpha}}\to {\mathbb Z}[q,q^{-1}][I^{\alpha}]$ factors through to give a ${\mathbb Z}[q,q^{-1}]$-linear map from the Grothendieck group
\begin{equation}\label{EChMap}
{\operatorname{ch}\:}: K(\mod{R_{\alpha}})\to {\mathbb Z}[q,q^{-1}][I^{\alpha}].
\end{equation}
The following result shows that the characters of the irreducible $R_{\alpha}$-modules are linearly independent.
\begin{Theorem}\label{TFCh}{\rm \cite[Theorem~3.17]{KL1}}\
The map (\ref{EChMap}) is injective.
\end{Theorem}
\subsection{Weight graph}
Let $1\leq r<d$ and $\text{\boldmath$i$}\in I^{\alpha}$. We call $s_r$ an {\em admissible transposition} for $\text{\boldmath$i$}$ if $a_{i_r i_{r+1}}=0$.
By (\ref{R4}), if $\text{\boldmath$i$}$ is a weight of $M\in\mod{R_{\alpha}}$ and $s_r$ is an admissible transposition for $\text{\boldmath$i$}$, then ${\operatorname{gdim}\,} M_\text{\boldmath$i$}={\operatorname{gdim}\,} M_{s_r\text{\boldmath$i$}}$. This explains our interest in the following combinatorial object.
Define the {\em weight graph} $G_{\alpha}$ as the graph with the set of vertices $I^{\alpha}$, and with $\text{\boldmath$i$},\text{\boldmath$j$}\in I^{\alpha}$ connected by an edge if and only if $\text{\boldmath$j$}=s_r \text{\boldmath$i$}$ for some admissible transposition $s_r$ for $\text{\boldmath$i$}$.
We want to describe the connected components of $G_{\alpha}$.
Let $\text{\boldmath$i$}\in I^{\alpha}$, and $a,b\in I$ be neighbors in ${\Gamma}$. The \emph{$\{a,b\}$-sequence of
$\text{\boldmath$i$}$} is the sequence of $a$'s and $b$'s obtained by ignoring all entries of $\text{\boldmath$i$}$ different from
$a$ and $b$. For example, the $\{1,2\}$ sequence of $\text{\boldmath$i$}=(1,2,2,3,4,1,2,1)$ is $(1,2,2,1,2,1)$. Note that if $s_r$ is admissible transposition for $\text{\boldmath$i$}$ then the $\{a,b\}$-sequence of $\text{\boldmath$i$}$ is the same as the $\{a,b\}$-sequence of $s_r\text{\boldmath$i$}$ for every pair of neighbors $a,b\in I$. So the $\{a,b\}$-sequences are invariants of connected components of $G_{\alpha}$. It turns out that these invariants are enough to describe the components:
\begin{Proposition}\label{PComb
Let $\text{\boldmath$i$},\text{\boldmath$j$}\in I^{\alpha}$. Then $\text{\boldmath$i$}$ and $\text{\boldmath$j$}$ belong to the same connected component of $G_{\alpha}$ if and only if their $\{a,b\}$-sequences coincide for each pair of neighbors $a,b\in I$.
\end{Proposition}
\begin{proof}
We prove the result by induction on $d={\operatorname{ht}}({\alpha})$.
Assume that $\text{\boldmath$i$}=(i_1,\ldots, i_d)$ and $\text{\boldmath$j$}=(j_1,\ldots, j_d)$ are elements of $I^\alpha$ so that
the $\{a,b\}$-sequences of $\text{\boldmath$i$}$ and $\text{\boldmath$j$}$ coincide for all pairs of neighbors $a,b\in I$.
If $d=1$ then $\text{\boldmath$i$}=\text{\boldmath$j$}$, and so $\text{\boldmath$i$}$ and $\text{\boldmath$j$}$ are in the same connected component of $I^\alpha$.
If $d>1$ let
$b = j_d$ and let $a$ be a neighbor of $b$. Let $k$ be maximal such that $i_k=b$. None of $i_{k+1},\ldots, i_d$ is equal to $a$. Therefore $\text{\boldmath$i$}$ is connected to
$$\text{\boldmath$i$}' = s_{d-1}\cdots s_{k+1}s_k\text{\boldmath$i$} = (i_1,\ldots, i_{k-1}, i_{k+1},\ldots, i_d,b).$$
Now $\text{\boldmath$i$}'$ and $\text{\boldmath$j$}$ are in the same connected component since, by inductive assumption, $(i_1,\ldots, i_{k-1}, i_{k+1},\ldots, i_{d})$ and $(j_1,\ldots, j_{d-1})$ are in the same connected component of $G_{\alpha - \alpha_b}$.
\end{proof}
\subsection{Configurations and standard tableaux}\label{SSConf}
We suggest `geometric' objects called configurations to visualize connected components of $G_{\alpha}$. First, the {\em $\Gamma$-abacus} is $\Gamma\times{\mathbb R}_{\geq 0}$, imagined as the abacus with the runners going up on each vertex of $\Gamma$. We picture the $\Gamma$-abacus in ${\mathbb R}^3$ with the distance
between neighboring runners always equal to $1$. For example,
for $\Gamma =D_4$ and $\Gamma=A_\infty$ the abaci look like this:
$$\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -2 to 2, y from -1 to 3
\put{$1$} at -1 -0.3
\put{$2$} at 0 -0.3
\put{$3$}[l] at 1.3 0.4
\put{$4$}[t] at 0.8 -0.7
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0.05 1.1 0.35 /
\put{$\circ$} at 1.2 0.35
\plot 0.1 -0.05 0.7 -0.5 /
\put{$\circ$} at 0.8 -0.5
\setdots
\plot -1 0.1 -1 3 /
\plot 0 0.1 0 3 /
\plot 1.2 0.45 1.2 3 /
\plot 0.8 -0.4 0.8 2.6 /
\endpicture
\qquad \text{and}\qquad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -4 to 4, y from -1 to 3
\put{$-3$} at -3 -0.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$3$} at 3 -0.5
\put{$\cdots$} at 4 -0.5
\put{$\cdots$} at -4 -0.5
\plot -4 0 -3.1 0 /
\put{$\circ$} at -3 0
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\put{$\circ$} at 3 0
\plot 3.1 0 4 0 /
\setdots
\plot -3 0.1 -3 3 /
\plot -2 0.1 -2 3 /
\plot -1 0.1 -1 3 /
\plot 0 0.1 0 3 /
\plot 1 0.1 1 3 /
\plot 2 0.1 2 3 /
\plot 3 0.1 3 3 /
\endpicture
$$
The `beads' of the abacus have shape depending on the runners. The bead on runner $i$ is `glued' out of isosceles right triangles with hypotenuse of length 2 on the runner, and the $90^\circ$ vertex sticking towards the neighboring runner (and touching it).
Examples of a bead on runner $1$ for type $A_3$, a bead on runner $i$ for type $A_{\infty}$, a bead on runner $2$ for type $D_4$, and a bead on runner $1$ for type $A_1$ are:
$$\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from .5 to 4, y from -1 to 3
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$3$} at 3 -0.5
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\put{$\circ$} at 3 0
\plot 1 0.1 1 2 /
\plot 1 2 2 1 /
\plot 2 1 1 0.05 /
\setdots
\plot 1 0.1 1 3 /
\plot 2 0.1 2 3 /
\plot 3 0.1 3 3 /
\endpicture
\qquad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from 0 to 3.5, y from -1 to 3
\put{$i-1$} at 1 -0.5
\put{$i$} at 2 -0.5
\put{$i+1$} at 3 -0.5
\put{$\cdots$} at 0 -0.5
\put{$\cdots$} at 4 -0.5
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\put{$\circ$} at 3 0
\plot 3.1 0 4 0 /
\plot 2 2 3 1 /
\plot 2 2 1 1 /
\plot 3 1 2 0.05 /
\plot 1 1 2 0.05 /
\setdots
\plot 1 0.1 1 3 /
\plot 2 0.1 2 3 /
\plot 3 0.1 3 3 /
\endpicture
\qquad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -2 to 2, y from -1 to 3
\put{$1$} at -1 -0.3
\put{$2$} at 0 -0.3
\put{$3$}[l] at 1.3 0.4
\put{$4$}[t] at 0.8 -0.7
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0.05 1.1 0.35 /
\put{$\circ$} at 1.2 0.35
\plot 0.1 -0.05 0.7 -0.5 /
\put{$\circ$} at 0.8 -0.5
\plot 0 2 1.2 1.2 /
\plot 1.2 1.2 0.7 0.7 /
\plot 0 2 0.8 0.4 /
\plot 0.8 0.4 0.08 0 /
\plot 0 2 -1 1 /
\plot -1 1 -0.08 0.08 /
\setdots
\plot -1 0.1 -1 3 /
\plot 0 0.1 0 3 /
\plot 1.2 0.45 1.2 3 /
\plot 0.8 -0.4 0.8 2.6 /
\setdashes
\plot 1.2 1.2 0 0 /
\endpicture
\qquad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from 0 to 2, y from -1 to 3
\put{$1$} at 1 -0.5
\put{$\circ$} at 1 0
\plot 1 0.1 1 2 /
\setdots
\plot 1 0.1 1 3 /
\endpicture
$$
Note that if $i$ has no neighbors, the shape of the bead is interpreted as just a segment of length $2$ (`hypotenuse without triangles').
Recall that ${\alpha}=\sum_{i\in I}m_i{\alpha}_i$ is a fixed element of $Q_+$ of height $d$.
A {\em configuration} of type ${\alpha}$ is obtained by placing $d$ beads on the runners of the $\Gamma$-abacus, letting each bead slide down the runner as far as gravity takes it,
so that there are a total of $m_i$ beads on runner $i$ for each $i\in I$.
We note that configurations are essentially the same as heaps defined by Viennot \cite{V}, see also Stembridge \cite{S1,S2}.
Let ${\lambda}$ be a configuration. A {\em tableau} of shape ${\lambda}$ or a {\em ${\lambda}$-tableau} is a bijection
$$
T:\{1,2,\dots,d\}\to\{\text{beads of ${\lambda}$}\}.
$$
A bead $B$ of $\lambda$ is {\em removable} if it can be lifted off its runner without interfering with other beads. If $B$ is on runner $i$, this is equivalent to the requirement that there are no beads on neighboring runners which are above $B$ in $\lambda$.
A $\lambda$-tableaux is called {\em standard} if for each $k$,
the bead $T(k)$ is above the bead $T(m)$ whenever $m<k$ and $T(m)$ is on a neighboring runner.
Equivalently, $T$ is standard, if and only if $T(k)$ is a removable bead for the configuration $\lambda\setminus \{T(k+1),\dots,T(d)\}$ for all $1\leq k\leq d$.
Let $\text{\boldmath$i$}=(i_1,\dots,i_d)\in I^{\alpha}$. Place a bead on the runner $i_1$, then place a bead on the runner $i_2$, and so on, finally placing the last bead on the runner $i_d$. This procedure produces the {\em configuration of $\text{\boldmath$i$}$}, written ${\operatorname{con}}(\text{\boldmath$i$})={\operatorname{con}}_\Gamma(\text{\boldmath$i$})$, and the standard tableaux $T^\text{\boldmath$i$}$ of the corresponding shape. For example:
$$\text{In type $A_\infty$,}\ T^{(0,-2,2)}=\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -4 to 4, y from -1 to 3
\put{$-3$} at -3 -0.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$3$} at 3 -0.5
\put{$\cdots$} at 4 -0.5
\put{$\cdots$} at -4 -0.5
\put{${\mathbf 2}$} at -2 1
\put{${\mathbf 1}$} at 0 1
\put{${\mathbf 3}$} at 2 1
\plot -4 0 -3.1 0 /
\put{$\circ$} at -3 0
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\put{$\circ$} at 3 0
\plot 3.1 0 4 0 /
\plot 2 2 3 1 /
\plot 2 2 1 1 /
\plot 3 1 2 0.05 /
\plot 1 1 2 0.05 /
\plot 0 2 1 1 /
\plot 0 2 -1 1 /
\plot 1 1 0 0.05 /
\plot -1 1 0 0.05 /
\plot -2 2 -1 1 /
\plot -2 2 -3 1 /
\plot -1 1 -2 0.05 /
\plot -3 1 -2 0.05 /
\setdots
\plot -3 0.1 -3 3 /
\plot -2 0.1 -2 3 /
\plot -1 0.1 -1 3 /
\plot 0 0.1 0 3 /
\plot 1 0.1 1 3 /
\plot 2 0.1 2 3 /
\plot 3 0.1 3 3 /
\endpicture
\ ,
$$
$$
\text{in type $A_\infty$,}\ T^{(0,1,-1)}=
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -4 to 4, y from -1 to 3.5
\put{$-3$} at -3 -0.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$3$} at 3 -0.5
\put{$\cdots$} at 4 -0.5
\put{$\cdots$} at -4 -0.5
\put{${\mathbf 1}$} at 0 1
\put{${\mathbf 2}$} at 1 2
\put{${\mathbf 3}$} at -1 2
\plot -4 0 -3.1 0 /
\put{$\circ$} at -3 0
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\put{$\circ$} at 3 0
\plot 3.1 0 4 0 /
\plot 2 2 1 3 /
\plot 0 2 1 3 /
\plot 1 1 2 2 /
\plot 0 2 1 1 /
\plot 0 2 -1 1 /
\plot 1 1 0 0.05 /
\plot -1 1 0 0.05 /
\plot -2 2 -1 1 /
\plot -2 2 -1 3 /
\plot -1 3 0 2 /
\setdots
\plot -3 0.1 -3 3.5 /
\plot -2 0.1 -2 3.5 /
\plot -1 0.1 -1 3.5 /
\plot 0 0.1 0 3.5 /
\plot 1 0.1 1 3.5 /
\plot 2 0.1 2 3.5 /
\plot 3 0.1 3 3.5 /
\endpicture,
$$
$$
\text{in type $D_4$,}\ {\operatorname{con}}(2,1,3,4,2)=\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -2 to 4, y from -1 to 5
\put{$1$} at -1 -0.3
\put{$2$} at 0 -0.3
\put{$3$}[l] at 1.3 0.4
\put{$4$}[t] at 0.8 -0.7
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0.05 1.1 0.35 /
\put{$\circ$} at 1.2 0.35
\plot 0.1 -0.05 0.7 -0.5 /
\put{$\circ$} at 0.8 -0.5
\plot 0 2 0.8 0.4 /
\plot 0.8 0.4 0.08 0 /
\plot 0 2 -1 1 /
\plot -1 1 -0.08 0.08 /
\plot -1 3 0 2 /
\plot -1 1 -1 3 /
\plot 1.2 3.2 0.79 2.79 /
\plot 1.2 1.2 1.2 3.2 /
\plot 0.8 2.6 0 2 /
\plot 0.8 0.4 0.8 2.55 /
\plot 0 4 -1 3 /
\plot 0 4 1.2 3.2 /
\plot 0 4 0.8 2.6 /
\plot .8 1.47 1.2 1.2 /
\plot 0.8 0.8 1.2 1.2 /
\setdots
\plot -1 0.1 -1 5 /
\plot 0 0.1 0 5 /
\plot 1.2 0.45 1.2 5 /
\plot 0.8 -0.4 0.8 4.6 /
\setdashes
\plot 0 2 .8 1.47 /
\plot 0 0 0.8 0.8 /
\plot 0.79 2.79 0 2 /
\endpicture
\ .
$$
The reader might note that in type $A_\infty$ configurations are closely related to the `Russian' notation for Young diagrams, cf. \cite{VK,Ok}.
For any $\lambda$-tableau $T$ we denote by $\text{\boldmath$i$}^T$ the element
$$\text{\boldmath$i$}^T=(i^T_1,\dots,i^T_d)\in I^{\alpha},$$
where $i^T_k$ is the label of the runner occupied by the bead $T(k)$ ($1\leq k\leq d$).
Now note that the maps $T\mapsto \text{\boldmath$i$}^T$ and $\text{\boldmath$i$}\mapsto T^\text{\boldmath$i$}$ are mutually inverse bijections between the set $\mathcal{T}({\lambda})$ of the standard $\lambda$-tableaux and the set of weights $\text{\boldmath$i$}\in I^{\alpha}$ with ${\operatorname{con}}(\text{\boldmath$i$})=\lambda$.
Now we can interpret Proposition~\ref{PComb} as the following statement:
\begin{Proposition}\label{PShape
Two weights $\text{\boldmath$i$},\text{\boldmath$j$}\in I^{\alpha}$ are in the same connected component of $G_{\alpha}$ if and only if ${\operatorname{con}}(\text{\boldmath$i$})={\operatorname{con}}(\text{\boldmath$j$})$. Moreover, the maps $T\mapsto \text{\boldmath$i$}^T$ and $\text{\boldmath$i$}\mapsto T^\text{\boldmath$i$}$ are mutually inverse bijections between the set of the standard $\lambda$-tableaux and the set of all weights $\text{\boldmath$i$}\in I^{\alpha}$ with ${\operatorname{con}}(\text{\boldmath$i$})=\lambda$.
\end{Proposition}
\section{Homogeneous representations}
We continue working with a fixed graph $\Gamma$ and a fixed ${\alpha}=\sum_{i\in I}m_i{\alpha}_i\in Q_+$ of height $d$.
A module $M\in \mod{R_{\alpha}}$ is called \emph{homogeneous} if it is concentrated in one degree,
i.e. $M=M[k]$ for some $k\in {\mathbb Z}$.
The homogeneous irreducible modules are especially easy to understand. They are labeled by `skew shapes', and their formal characters are `sums of standard tableaux' of that shape.
\subsection{Calibrated representations}
First, we consider a seemingly different class of modules. A module $M\in\mod{R_{\alpha}}$ is called {\em calibrated} if $y_1,\dots,y_d$ act as zero on $M$.
Other authors might use different terminology here, for example {\em Gelfand-Zetlin} \cite{Ch,OV}, {\em completely splittable} \cite{KCS,Kbook, R}, {\em skew} \cite{Mo}, {\em seminormal} \cite{Ma}, etc.
Our goal is to classify irreducible calibrated modules following the approach of \cite{KrR}.
\begin{Proposition}\label{PCal1
Let $M\in\mod{R_{\alpha}}$ be an irreducible calibrated module, and $\text{\boldmath$i$}$ be a weight of $M$. Then:
\begin{enumerate}
\item[{\rm (i)}] there is no $r$ with $i_r= i_{r+1}$;
\item[{\rm (ii)}] there is no $r$ such that $i_r,i_{r+1}$ are neighbors and $i_{r+2}=i_r$;
\item[{\rm (iii)}] $\dim M_\text{\boldmath$i$}=1$;
\item[{\rm (iv)}] the weights of $M$ form one connected component of $G_{\alpha}$.
\end{enumerate}
\end{Proposition}
\begin{proof}
(i) Assume $i_r = i_{r+1}$ and let $v\in M_\text{\boldmath$i$}$ be nonzero. Since $M$ is calibrated, $y_r$ and $y_{r+1}$ act as $0$, and \eqref{R5} leads to a contradiction:
$$0 = (y_{r+1}\psi_r - \psi_r y_r)e(\text{\boldmath$i$})v= e(\text{\boldmath$i$})v=v.$$
(ii) Assume $(i_r,i_{r+1},i_{r+2}) = (a,b,a)$, $a$ and $b$ are neighbors, and
$v\in M_\text{\boldmath$i$}$ is nonzero. By \eqref{R2PsiE}, $\psi_{r+1}v\in M_{s_{r+1}\text{\boldmath$i$}}$ and
$\psi_r v\in M_{s_r\text{\boldmath$i$}}$. So, by (i), we have $\psi_{r+1}v = 0$ and $\psi_rv=0$.
Using \eqref{R7}, we get a contradiction:
$$0 = (\psi_{r+1}\psi_r\psi_{r+1} - \psi_r \psi_{r+1}\psi_r)v =\pm v.
$$
(iii) Assume for a contradiction that $v,w$ are two linearly independent elements of $M_\text{\boldmath$i$}$. As $M$ is irreducible and calibrated, we may assume (up to rescaling) that $v=\psi_{r_1}\psi_{r_2}\dots\psi_{r_k}w$ and that $k$ is minimal possible. It follows from (\ref{R2PsiE}) and (i) that $s_{r_1}s_{r_2}\dots s_{r_k}=1$ in $S_d$. So we can use braid relations to rewrite
$$
s_{r_1}\dots s_{r_k}=s_{t_1}\dots s_{t_{m-2}}s_{t}s_{t}s_{t_{m+1}}\dots s_{t_k}.
$$
By (ii) and (\ref{R7}), $\psi_r$'s acting on $M$ also satisfy braid relations, so we can rewrite, using also (\ref{R4}),
\begin{align*}
\psi_{r_1}\dots \psi_{r_k}w
&=\psi_{t_1}\dots \psi_{t_{m-2}}\psi_{t}\psi_{t}\psi_{t_{m+1}}\dots \psi_{t_k}w
\\
&= c\psi_{t_1}\dots \psi_{t_{m-2}}\psi_{t_{m+1}}\dots \psi_{t_k}w
\end{align*}
for some constant $c$, which must be non-zero, and hence $c=1$.
This contradicts the minimality of $k$.
(iv) If $\text{\boldmath$i$}$ is a weight of $M$, and $s_r$ is an admissible transposition for $\text{\boldmath$i$}$, then $s_r\text{\boldmath$i$}$ is also a weight of $M$, thanks to (\ref{R2PsiE}) and (\ref{R4}). So all weights in the connected component of $\text{\boldmath$i$}$ in $G_{\alpha}$ appear in $M$. To see that there are no other weights, it suffices to show that if $\text{\boldmath$j$}$ and $s_r\text{\boldmath$j$}$ are weights of $M$ then $s_r$ is an admissible transposition for $\text{\boldmath$j$}$.
So let $v\in M_\text{\boldmath$j$}$, $w\in M_{s_r\text{\boldmath$j$}}$ be non-zero vectors. After rescaling, we may assume that $w=\psi_{r_1}\dots \psi_{r_k}v$, and let $k$ be minimal possible. By (\ref{R2PsiE}) and (i), $s_{r_1}\dots s_{r_k}=s_r$ in $S_d$. As in the proof of (iii), we deduce from the minimality of $k$ that $k=1$ and $r_1=r$, i.e. $w=\psi_r v$. Similarly, we can write $cv=\psi_rw$ for a non-zero constant $c$. So $\psi_r^2v\neq 0$. In view of (\ref{R4}), $j_r$ and $j_{r+1}$ are not neighbors, whence $s_r$ is an admissible transposition for $\text{\boldmath$j$}$.
\end{proof}
\begin{Corollary}\label{CGZ
Let $M\in\mod{R_{\alpha}}$ be an irreducible module. Then $M$ is homogeneous if and only if $M$ is calibrated.
\end{Corollary}
\begin{proof}
If $M$ is homogeneous, then $y_1,\dots,y_d$ act on $M$ as zero since they have positive degrees. Conversely, if $M$ is calibrated, it follows from Proposition~\ref{PCal1} that $M$ is a span of some $\psi_{r_1}\dots\psi_{r_k}v$ where $v\in M_\text{\boldmath$i$}$ for some $\text{\boldmath$i$}$, and $s_{r_m}$ is an admissible transposition for $s_{r_{m+1}}\dots s_{r_k}\text{\boldmath$i$}$, for all $m=1,\dots,k$. It follows that the degree of each $\psi_{r_1}\dots\psi_{r_k}v$ is the same as the degree of $v$, so $M$ is homogeneous.
\end{proof}
\subsection{Construction of homogeneous modules}
We now give an explicit construction of the homogeneous representations, which can be thought of as a generalization of Young's seminormal form \cite{Ch} from type $A_\infty$ quiver to an arbitrary quiver without loops and multiple edges.
Let $C$ be a connected component of $G_{\alpha}$. We say that $C$ is {\em homogeneous} if for each $\text{\boldmath$i$}\in C$ the following condition holds:
\begin{equation}\label{ENC}
\begin{split}
\text{if $i_r=i_s$ for some $r<s$ then there exist $t,u$ with}
\\
\text{$r<t<u<s$ such that
$a_{i_ri_t}=a_{i_r,i_u}=-1$.}
\end{split}
\end{equation}
\begin{Lemma}\label{equivconds
Let $C$ be a connected component of $G_{\alpha}$.
\begin{enumerate}
\item[{\rm (i)}] $C$ is homogeneous if and only if the condition (\ref{ENC}) holds for {\em some} $\text{\boldmath$i$}\in C$.
\item[{\rm (ii)}] $C$ is homogeneous if and only if
the conditions (i) and (ii) of Proposition~\ref{PCal1} hold for each $\text{\boldmath$i$}\in C$.
\end{enumerate}
\end{Lemma}
\begin{proof}
(i) Condition (3.1) is a condition on the $\{a,b\}$-sequences of $\text{\boldmath$i$}$ which requires that
$$\text{\boldmath$i$} = \cdots a \cdots a\cdots
\quad\hbox{only if}\quad
\text{\boldmath$i$} = \cdots a \cdots b \cdots c\cdots a\cdots
$$
with $b$ and $c$ distinct neighbors of $a$. If this condition holds for one $\text{\boldmath$i$}\in C$ then,
by Proposition~\ref{PComb}, it holds for all $\text{\boldmath$i$}\in C$.
(ii) `$\Rightarrow$': If Proposition 3.1 (i) or (ii) is violated then there exists $\text{\boldmath$i$}\in C$ with
$$\text{\boldmath$i$} = \cdots aa \cdots\quad\hbox{or}\quad
\text{\boldmath$i$} = \cdots aba\cdots,
$$
with $b$ a neighbor of $a$. In either case $\text{\boldmath$i$}$ violates the condition in (3.1).
`$\Leftarrow$': If condition (3.1) is violated then there exists $\text{\boldmath$i$} \in C$ such that $\text{\boldmath$i$}$ looks like
$$\hbox{Case 1:}\quad \text{\boldmath$i$} = \cdots a \cdots a\cdots,$$
with $a=i_r=i_s$ and no neighbors of $a$ in between, or
$$\hbox{Case 2:}\quad \text{\boldmath$i$} = \cdots a \cdots b\cdots a\cdots,$$
with $a = i_r=i_s$, $b=i_t$ a neighbor of $a$ and no other neighbors of $a$ in between $i_r$ and $i_s$.
In Case 1, $\text{\boldmath$i$}$ is connected to
$$\text{\boldmath$j$} = s_{i_s-1}\cdots s_{i_r+1}s_{i_r}\text{\boldmath$i$} = \cdots aa\cdots,$$
which violates Proposition 3.1(i). In Case 2, $\text{\boldmath$i$}$ is connected to
$$\text{\boldmath$j$} = (s_{i_t-1}\cdots s_{i_r+1}s_{i_r})(s_{i_t+1}\cdots s_{i_s-2}s_{i_s-1})\text{\boldmath$i$} = \cdots aba\cdots ,$$
which violates Propositions 3.1 (ii).
\end{proof}
\begin{Theorem}\label{THomog
Let $C$ be a homogeneous connected component of $G_{\alpha}$, and let us consider a vector space $S(C)$ with a homogeneous basis $\{v_\text{\boldmath$i$}\mid \text{\boldmath$i$}\in C\}$ labeled by the elements of $C$.
The formulas
\begin{align*}
e(\text{\boldmath$j$})v_\text{\boldmath$i$}&={\delta}_{\text{\boldmath$i$},\text{\boldmath$j$}}v_\text{\boldmath$i$} \qquad (\text{\boldmath$j$}\in I^{\alpha},\ \text{\boldmath$i$}\in C),\\
y_r v_\text{\boldmath$i$}&=0\qquad (1\leq r\leq d,\ \text{\boldmath$i$}\in C),\\
\psi_rv_{\text{\boldmath$i$}}&=
\left\{
\begin{array}{ll}
v_{s_r\text{\boldmath$i$}} &\hbox{if $s_r\text{\boldmath$i$}\in C$,}\\
0 &\hbox{otherwise;}
\end{array}
\right.
\quad(1\leq r<d,\ \text{\boldmath$i$}\in C)
\end{align*}
define an action of $R_{\alpha}$ on $S(C)$, under which $S(C)$ is a homogeneous irreducible $R_{\alpha}$-module. Moreover, $S(C)\not\cong S(C')$ if $C\neq C'$, and every homogeneous irreducible $R_{\alpha}$-module is isomorphic to one of the modules $S(C)$.
\end{Theorem}
\begin{proof}
It is straightforward to verify that the formulas above define operators which satisfy the defining relations of $R_{\alpha}$, and so $S(C)$ is a well defined $R_{\alpha}$-module. It is also clear that it is concentrated in one degree, i.e. is homogeneous. The irreducibility of $S(C)$ follows from the definition of $C$ as a connected component of $G_{\alpha}$. If $C\neq C'$ then of course $S(C)$ is not isomorphic to $S(C')$ since they have different weights. Finally, if $S$ is an irreducible homogeneous $R_{\alpha}$-module then by Corollary~\ref{CGZ} and Proposition~\ref{PCal1} the formal character of $S$ equals ${\operatorname{ch}\:} S(C)$ for some homogeneous connected component $C$, and so $S\cong S(C)$ thanks to Theorem~\ref{TFCh}.
\end{proof}
\subsection{Skew shapes}\label{SSSS}
By Theorem~\ref{THomog}, the homogeneous connected components correspond to the homogeneous representations of $R_{\alpha}$. The homogeneous connected components are characterized by the properties (i) and (ii) from Proposition~\ref{PCal1}. The corresponding configurations can be characterized as follows:
\begin{Definition}\label{skshapes
{\rm
A configuration $\lambda$ is called {\em skew shape} if whenever $B_1$ and $B_2$ are two beads of $\lambda$ on the same runner then there are at least two beads on different neighboring runners separating $B_1$ from $B_2$.
}
\end{Definition}
For example, in type $A_\infty$,
$$\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -2 to 2.5, y from -1 to 4.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$\cdots$} at 1.7 -0.5
\put{$\cdots$} at -1.7 -0.5
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 0 2 1 3 /
\plot 0 4 1 3 /
\plot 0 4 -1 3 /
\plot 0 2 1 1 /
\plot 0 2 -1 1 /
\plot 1 1 0 0.05 /
\plot -1 1 0 0.05 /
\plot -1 3 0 2 /
\setdots
\plot -2 0.1 -2 4.5 /
\plot -1 0.1 -1 4.5 /
\plot 0 0.1 0 4.5 /
\plot 1 0.1 1 4.5 /
\plot 2 0.1 2 4.5 /
\endpicture
\quad
\text{and}
\quad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -3.3 to 3.1, y from -1 to 4.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$\cdots$} at 2.7 -0.5
\put{$\cdots$} at -2.7 -0.5
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\plot 0 2 1 3 /
\plot 0 4 1 3 /
\plot 0 4 -1 3 /
\plot 0 2 1 1 /
\plot 0 2 -1 1 /
\plot 1 1 0 0.05 /
\plot -1 1 0 0.05 /
\plot -2 2 -1 1 /
\plot -2 2 -1 3 /
\plot -1 3 0 2 /
\setdots
\plot -2 0.1 -2 4.5 /
\plot -1 0.1 -1 4.5 /
\plot 0 0.1 0 4.5 /
\plot 1 0.1 1 4.5 /
\plot 2 0.1 2 4.5 /
\endpicture
$$
are not skew shapes, while
$$\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -4 to 3, y from -1 to 3.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$\cdots$} at 2.7 -0.5
\put{$\cdots$} at -2.7 -0.5
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\plot 0 3 1 2 /
\plot 0 3 -1 2 /
\plot 1 2 0 1 /
\plot -1 2 0 1 /
\plot 0 1 -1 0.05 /
\plot -1 0.05 -2 1 /
\plot -2 1 -1 2 /
\setdots
\plot -2 0.1 -2 3.5 /
\plot -1 0.1 -1 3.5 /
\plot 0 0.1 0 3.5 /
\plot 1 0.1 1 3.5 /
\plot 2 0.1 2 3.5 /
\endpicture
\quad
\text{and}
\quad
\beginpicture
\setcoordinatesystem units <.75cm,.75cm>
\setplotarea x from -3.3 to 3.1, y from -1 to 4.5
\put{$-2$} at -2 -0.5
\put{$-1$} at -1 -0.5
\put{$0$} at 0 -0.5
\put{$1$} at 1 -0.5
\put{$2$} at 2 -0.5
\put{$\cdots$} at 2.7 -0.5
\put{$\cdots$} at -2.7 -0.5
\plot -2.9 0 -2.1 0 /
\put{$\circ$} at -2 0
\plot -1.9 0 -1.1 0 /
\put{$\circ$} at -1 0
\plot -0.9 0 -0.1 0 /
\put{$\circ$} at 0 0
\plot 0.1 0 0.9 0 /
\put{$\circ$} at 1 0
\plot 1.1 0 1.9 0 /
\put{$\circ$} at 2 0
\plot 2.1 0 2.9 0 /
\plot 0 2 1 3 /
\plot 0 4 1 3 /
\plot 0 4 -1 3 /
\plot 0 2 1 1 /
\plot 0 2 -1 1 /
\plot 1 1 0 0.05 /
\plot -1 1 0 0.05 /
\plot -2 2 -1 1 /
\plot -2 2 -1 3 /
\plot -1 3 0 2 /
\plot 1 3 2 2 /
\plot 2 2 1 1 /
\setdots
\plot -2 0.1 -2 4.5 /
\plot -1 0.1 -1 4.5 /
\plot 0 0.1 0 4.5 /
\plot 1 0.1 1 4.5 /
\plot 2 0.1 2 4.5 /
\endpicture
$$
are skew shapes. Note that, up to a horizontal shift, skew shapes in type $A_\infty$ are obtained by considering all usual skew shapes in the Russian notation and allowing all beads to slide down as far as gravity will take them.
If $\lambda$ is a configuration, then $S_d$ acts on the set of $\lambda$-tableaux by permutations of $\{1,2,\dots,d\}$.
Theorem~\ref{THomog} can now be restated as follows:
\begin{Theorem}\label{tabconst
Let $\lambda$ be a skew shape, and $\mathcal{T}(\lambda)$ be the set of all standard $\lambda$-tableaux. Consider a vector space $S(\lambda)$ with a homogeneous basis $\{v_T\mid T\in \mathcal{T}(\lambda)\}$.
The formulas
\begin{align*}
e(\text{\boldmath$j$})v_T&={\delta}_{\text{\boldmath$i$}^T\text{\boldmath$j$}}v_T \qquad (\text{\boldmath$j$}\in I^{\alpha},\ T\in \mathcal{T}(\lambda)),\\
y_r v_T&=0\qquad (1\leq r\leq d,\ T\in \mathcal{T}(\lambda)),\\
\psi_rv_T&=
\left\{
\begin{array}{ll}
v_{s_rT} &\hbox{if $s_rT$ is standard,}\\
0 &\hbox{otherwise;}
\end{array}
\right.
\quad(1\leq r<d,\ T\in \mathcal{T}(\lambda))
\end{align*}
define an action of $R_{\alpha}$ on $S(\lambda)$, under which $S(\lambda)$ is a homogeneous irreducible $R_{\alpha}$-module. Moreover, $S(\lambda)\not\cong S(\lambda')$ if $\lambda\neq \lambda'$ and every homogeneous irreducible $R_{\alpha}$-module is isomorphic to one of the modules $S(\lambda)$.
\end{Theorem}
\subsection{Characters and the Littlewood-Richardson rule}
Let $\lambda$ be a skew shape and let $S(\lambda)$ be the corresponding
irreducible homogeneous $R_\alpha$-module constructed in Theorem \ref{tabconst}.
Recall the maps $\text{\boldmath$i$}\mapsto T^\text{\boldmath$i$}$ and $T\mapsto \text{\boldmath$i$}^T$ from \S\ref{SSConf}.
Since $v_T$ is in the $\text{\boldmath$i$}^T$-weight space, and this weight space is one dimensional,
the formal character of $R^\lambda_\alpha$
is
\begin{equation}\label{skewschur}
{\operatorname{ch}\:} S(\lambda) = \sum_{T\in \mathcal{T}(\lambda)} e^{i^T},
\end{equation}
where the sum is over all standard tableaux $T$ of shape $\lambda$.
Let $\beta,\gamma\in Q_+$ such that $\beta+\gamma = \alpha$. The product
$R_\beta\otimes R_\gamma$ is naturally a subalgebra of $R_\alpha$, cf. \cite[\S2.6]{KL1}. If $M$ is a
homogenous $R_{\alpha}$-module then its restriction to $R_\beta\otimes R_\gamma$ is homogenous.
It follows from \eqref{skewschur} that
\begin{equation}\label{LRrule}{\operatorname{res}\:}^{R_\alpha}_{R_\beta\otimes R_\gamma}S(\lambda)
= \sum_{\mu\subseteq \lambda} S(\mu)\otimes S(\lambda/\mu),
\end{equation}
where the sum is over all configurations $\mu$ of type $\beta$ which are obtained by consecutive removals of removable beads from
$\lambda$,
and
$\lambda/\mu$ is the configuration determined by the beads of $\lambda$ that are
not in $\mu$. The formula \eqref{LRrule} is a generalization of the skew Schur function formula from \cite[(5.10)]{Mac}:
$$
s_{\lambda/\mu}(x,y) = \sum_{\lambda \supseteq \nu\supseteq \mu}
s_{\lambda/\nu}(x)s_{\nu/\mu}(y)
$$
\subsection{Minuscule elements and hook formula} Finally, we explain a connection
between skew shapes and the fully commutative elements in Coxeter groups studied by Stembridge \cite{S2} and Fan \cite{F}. A special class of fully commutative elements called dominant minuscule elements will allow us to select straight shapes from the class of skew shapes.
Using notation of \cite{Kac}, let $\Phi_+$ be
the set of positive roots, $<$ the dominance order, $P_+$ the set of {\em dominant weights},
and $W$ be the Weyl group with simple reflections $r_i$ for $\text{\boldmath$i$}\in I$, so that $W$ is the Coxeter group
with Coxeter graph $\Gamma$.
An element $w\in W$ is {\em fully commutative} if for every pair of
non-commuting generators $r_i$ and $r_j$ there is no reduced expression for $w$
containing a subword of the form $r_i r_j r_i$.
An element $w\in W$ is {\em dominant minuscule} if there is ${\Lambda}\in P_+$ and a reduced expression
$w = r_{i_1} \dots r_{i_d}$ such that
$$
r_{i_k}r_{i_{k+1}}\dots r_{i_d}{\Lambda}={\Lambda}-{\alpha}_{i_k}-{\alpha}_{i_{k+1}}-\dots-{\alpha}_{i_d}\qquad(1\leq k\leq d).
$$
Using the terminology of \S\ref{SSSS}, let ${\lambda}$ be a skew shape and $\mathcal{T}({\lambda})$
the set of standard ${\lambda}$-tableaux. If $T\in \mathcal{T}({\lambda})$ and $\text{\boldmath$i$}^T=(i_1,\dots,i_d)$, set
\begin{equation}\label{ESt}
w^{\lambda}:=r_{i_d}r_{i_{d-1}}\dots r_{i_1}\in W.
\end{equation}
In view of Lemma \ref{equivconds} and Definition \ref{skshapes}, skew shapes and standard tableaux can now be interpreted as follows.
\begin{Proposition}\label{PSt
The element $w^{\lambda}$ depends only on ${\lambda}$ and does not depend on $T\in\mathcal{T}({\lambda})$.
Moreover:
\begin{enumerate}
\item[{\rm (i)}] the right hand side of (\ref{ESt}) is a reduced decomposition of $w^{\lambda}$;
\item[{\rm (ii)}] ${\lambda}\mapsto w^{\lambda}$ is a bijection between the skew shapes with $d$ boxes and the fully commutative elements of $W$ of length $d$;
\item[{\rm (iii)}] for a fixed skew shape ${\lambda}$, the assignment (\ref{ESt}) is a bijection between the standard ${\lambda}$-tableaux and the reduced decompositions of $w^{\lambda}$.
\end{enumerate}
\end{Proposition}
Dominant minuscule elements are known to be fully commutative, see e.g. \cite[Proposition 2.1]{S2}, and can be characterized in terms of their reduced expressions as follows.
\begin{Proposition}\label{PDM}{\rm \cite[Proposition 2.5]{S2}}
If $w = r_{i_1} \dots r_{i_d} \in W$ is a reduced expression, then $w$
is dominant minuscule if and only if the following two conditions are satisfied:
\begin{enumerate}
\item[{\rm (i)}] between every pair of occurrences of a generator $r_i$
(with no other occurrences of $r_i$ in between) there are exactly two terms (possibly equal to each other) that do not commute with $r_i$;
\item[{\rm (ii)}] the last occurrence of each generator $r_i$ is followed by at most
one generator that does not commute with $r_i$.
\end{enumerate}
\end{Proposition}
Now it is easy to see that in type $A_\infty$, skew shapes ${\lambda}$ with
$w^{\lambda}$ dominant minuscule are (disjoint unions of) `straight' shapes in the usual sense, i.e.
partitions drawn in the Russian notation. This motivates the following
definition.
A skew shape ${\lambda}$ is a {\em straight shape} if $w^{\lambda}$ is dominant minuscule. Proposition~\ref{PDM} yields the following explicit characterization of the straight shapes.
\begin{Lemma
Let ${\lambda}$ be a configuration. Then ${\lambda}$ is a straight shape if and only if
the following conditions are satisfied:
\begin{enumerate}
\item[{\rm (i)}] between every pair of beads $A$, $B$ on a runner $i$
(with no beads on the runner $i$ between $A$ and $B$) there are exactly two beads between $A$ and $B$, which lie on runners neighboring $i$ (possibly on the same runner);
\item[{\rm (ii)}] the bottom bead on a runner $i$ has at most
one bead below it on runners neighboring $i$.
\end{enumerate}
\end{Lemma}
Peterson and Proctor have given a hook-type formula for the number of standard
tableaux of a straight shape.
The proof of this hook formula, and generalizations of it,
can be found e.g. in Nakada in \cite{N2}.
In view of Proposition~\ref{PSt}(iii)), the Peterson-Proctor hook
formula can be stated, in our context, as follows.
\begin{Theorem
{\bf (Peterson-Proctor Hook Formula)}
Let ${\lambda}$ be a straight shape with $d$ beads. Using notation as in Theorem~\ref{tabconst},
the dimension of the corresponding representation of the Khovanov-Lauda algebra is
$$
\dim S(\lambda) = \mathrm{Card}(\mathcal{T}({\lambda}))
= \frac{d!}{\prod_{\beta\in \Phi(w^{\lambda})}{\operatorname{ht}}(\beta)},
$$
where
$
\Phi(w):=\{{\beta}\in \Phi_+\mid w^{-1}({\beta})<0\},
$
and $\mathrm{Card}(\mathcal{T}({\lambda}))$ is the number of standard tableaux of shape $\lambda$.
\end{Theorem}
|
2,877,628,090,776 | arxiv | \section{INTRODUCTION}
Radiation resistance of graphite has been one of the major concerns
of the nuclear industry.\cite{Simmons65,Telling03}
Nowadays, radiation treatment by high-energy
electrons or ions is also viewed as a versatile tool for the design of
new materials. The formation of irradiation-induced defects in graphite-like
layered carbon nanostructures (multiwalled and bundled carbon
nanotubes, nanoonions, etc.)
changes their mechanical \cite{Kis04} and electronic
properties \cite{Miko03,Han03} and may even trigger
dramatic structural changes.\cite{Banhart97,Terrones00}
However, the structure and dynamics of defects in
graphite and carbon nanostructures as well as the mechanisms underlying
their creation and transformation remain elusive.
This knowledge is crucial for a defect-assisted engineering of
nanostructures with applications in, e.g.,
manufacturing of nanoelectromechanical systems.\cite{NEMS}
Radiation damage of matter is governed by the displacement of atoms
from their equilibrium positions due to electronic excitations and direct
collisions of high-energy particles with the nuclei. In metals and narrow
band gap semiconductors electronic excitations quench instantaneously,
leaving collisions with nuclei as the sole mechanism responsible for
the creation of defects in graphite and related carbon materials.\cite{Banhart99}
If the kinetic energy transferred from a high-energy electron or ion to the nucleus
is higher than the displacement threshold $T_d$, a carbon atom can leave
its initial position to form a metastable defect structure on
a sub-picosecond timescale. Such events are called knock-on displacements.
For highly anisotropic layered carbon materials the threshold of the off-plane
displacement is $T^\bot_d$$\approx$15-20~eV \cite{Banhart97}
while a creation of defect due to the in-plane knock-on collision requires higher transferred energies, $T^{||}_d$$\geq$30~eV.
Possible defects produced by radiation damage include separated and intimate \cite{Ewels03} pairs of interstitial atoms and vacancies,
and in-plane topological defects involving non-sixmembered rings, e.g.
Stone-Wales defect.\cite{Stone86,Kaxiras88}
The existence of defects in carbon nanostructures has been
confirmed by direct observations.\cite{Hashimoto04,Urita05}
Upon knock-on events a large amount of energy is transferred to
only a few degrees of freedom. The resulting defect structures
formed on a picosecond timescale depend on the magnitude and on
the direction of the transferred momentum and determine the fate
of the system at longer timescales. Therefore gaining control
over the early stages of defect formation by tuning the irradiation
conditions will make the paradigm of the defect-assisted engineering
feasible. Molecular dynamics (MD) simulations performed with empirical
potentials \cite{Nordlund96} or tight-binding models \cite{Crespi96,Ajayan98,Krasheninnikov05}
have been used for the studies of radiation damage of various carbon materials.
In this work, we report a {\it systematic first principles} study of
the early stages of radiation damage of graphite,
a general model for closely related layered carbon nanostructures.
The paper is organized in the following way: In Section~\ref{sec:methods}
we provide a description of the computational methods used in this work.
The observed defect structures, mechanisms of their formation, and
practical implications are discussed in Section~\ref{sec:results}.
Section~\ref{sec:conclusions} briefly concludes our work.
\section{\label{sec:methods}COMPUTATIONAL METHODS}
By using \textit{ab initio} molecular dynamics
we simulate the process of defect formation after the initial transfer
of a momentum $\vec{T}$ to one of the carbon atoms in the system.
The periodic model system consists of a unit cell with 108
carbon atoms, which contains two graphene sheets with stacking ABAB.
The dimension of the unit cell in the direction perpendicular to
the graphene planes was fixed to 6.7~\AA\ in accordance with
the experimental inter-layer distance 3.35~\AA.\cite{Hanfland89}
This distance shows only weak variation among different layered
carbon nanostructures. Our computational methodology
is based on density functional theory (DFT), which lacks a correct
description of weak van der Waals interactions between graphene planes.
However, by fixing the unit cell dimension in the direction perpendicular to
the graphene planes we provide a realistic description of layered carbon
nanostructures without any explicit inclusion of van der Waals
forces.
The in-plane distance between two periodic images is 12.7~\AA\,
which is large enough to ensure localization of the defect within
the unit cell.
A coarse sampling of the irreducible wedge of the
space spanned by the magnitude of transferred energy
$T$ and the pair of angles $\phi$$\in$$[0^\circ; 90^\circ]$
and $\theta$$\in$$[0^\circ; 60^\circ]$ (Fig.~\ref{fig:char1}, inset)
has been performed.
The \textit{ab initio} MD simulations were
carried out using the \texttt{CPMD} plane wave DFT code\cite{CPMD}
and the Perdew, Burke, and Ernzerhof exchange-correlation density
functional.\cite{Perdew96} A plane wave kinetic energy cutoff of 60~Ry and
norm-conserving pseudopotentials \cite{Troullier91} have been used.
The simulations were performed within the spin-unrestricted
formulation of DFT starting from an initial guess
asymmetric with respect to the spin components. Such a starting configuration
is required in order to ensure
a broken-symmetry path of bond breaking events.\cite{Gunnarsson76}
The first 100~fs of each MD simulation
were performed using the Born-Oppenheimer scheme.
The MD timestep was set to 0.5~fs.
In our simulations we observed that during the first 100~fs the transferred
kinetic energy was well dissipated over the entire system.
The initial simulation was followed by a Car-Parrinello simulation \cite{Car85}
carried out using a Nos\'e-Hoover thermostat \cite{Nose84a}
(350~K) until a stable defect structure was reached (about 1~ps).
This thermal coupling methodology provides a realistic
description of the excess kinetic energy dissipation after knock-on
collisions of reasonably low transferred energies.
The Car-Parrinello equations of motion were integrated with a
time step of 0.1~fs using a fictitious electron mass
of 400~a.u.
Finally, the obtained defect structures were relaxed by slow
annealing of both ionic and electronic degrees of freedom.
The formation energies were evaluated using the
\texttt{SIESTA} code \cite{Soler02} by relaxing the
ionic coordinates and the in-plane cell dimensions.
The same norm-conserving pseudopotentials and density
functional as in the plane wave calculations together with an optimized
double-$\zeta$ plus polarization function (DZP) basis set were used.
A 2$\times$2$\times$2 k-point grid (including the $\Gamma$ point)
was employed in order to
obtain accurate defect formation energies.\cite{VandeWalle04}
\section{\label{sec:results}RESULTS AND DISCUSSION}
\begin{figure}
\includegraphics[width=8.6cm]{fig1.eps}
\caption{\label{fig:char1}
Polar coordinates
representation of the simulation outcomes as a function of the parameters $T$, $\phi$, and $\theta$.
The defect structures at each parameter set are given according to the nomenclature shown in Figs.~\ref{fig:eps1}
and \ref{fig:eps2}.
Other labels correspond to:
'n' -- no defect formation,
'c' -- displacement cascade, and '-' -- simulation not performed.
The different outcomes corresponding to the off-plane displacements of $\alpha$/$\beta$ carbon atoms
(see left inset) are shown separately.
For each pair of parameters ($\phi$, $\theta$) the outcomes at different values of $T$
are listed according to the values given in the right inset.
The left inset shows the definition of the parameters determining the knock-on displacements.
}
\end{figure}
\subsection{Off-plane recoils}
The outcomes of our simulations are summarized in Figure~\ref{fig:char1}
(movies of selected MD trajectories are available online
\cite{EPAPS-note}).
We first discuss the simulation results for the off-plane displacements
($\phi$$\in$$\left\{0^\circ; 30^\circ; 60^\circ \right\}$) of carbon atoms in inequivalent positions $\alpha$ and $\beta$.
The outcomes can be divided into four major classes: (i) no defect formation due
to insufficient transferred momentum or due to instantaneous recombination of
the recoil atom with the vacancy ('n');
(ii) separated interstitial-vacancy pairs
(I-V); (iii) intimate interstitial-vacancy
pairs (iIV); (iv) displacement cascades ('c') in which the recoil atom
is able to displace other atoms in the lattice.
The latter case can be viewed
as a series of elementary events of types (i)--(iii).
The simulation of displacement cascades is beyond the scope of this study
and would require a larger unit cell than the one used here.
\begin{figure}
\includegraphics[width=8.6cm]{fig2.eps}
\caption{\label{fig:eps1}
Perspective views of the atomic structures
of the vacancy (top, left), interstitial (top, right), and
intimate Frenkel pair (bottom) defects observed in our simulations.
The formation energies are given in parentheses. The values given for interstitial
defects refer to the formation energies of corresponding Frenkel pairs.
For iIV defects the created vacancy is situated in the upper graphene layer.}
\end{figure}
The formation of well-separated Frenkel pairs was observed
for atoms in both $\alpha$ and $\beta$ positions at $T$$\ge$25~eV.
Surprisingly, the interstitial defects were produced only
in the form of a symmetric ``dumbell'' structure (I$_2$) \cite{Li05,Ma05}
where the two carbon atoms are symmetrically displaced
from the graphene plane (Fig.~\ref{fig:eps1}, top).
Despite the highly distorted coordination sphere of these atoms,
the C--C distance of 1.58~\AA\ is close to the one of a typical $\sigma$ bond.
The core atomic structure is the same as for the
[1.1.1]propellane molecule for which a very similar C--C bond length
(1.60$\pm$0.02~\AA) has been observed experimentally.\cite{Wiberg85}
No single off-plane recoil led to the ``bridge''
structure (I$_1$) \cite{Nordlund96,Lehtinen03,Li05} with the interstitial
atom situated between two graphene planes.
The formation energy of I$_2$ ($E_f$=14.3~eV, the value refers to the formation energy of the corresponding I-V pair) is only 0.5~eV lower than the one of I$_1$ ($E_f$=14.8~eV) where the ``bridge'' interstitial defect is bonded only to
the neighbor atom in the same layer.
In this case, a steric repulsion with the opposite
graphene layer contributes to the destabilization of the I$_1$ defect.
However, bonding to the opposite layer leads to more stable
shared interstitial defect structures.\cite{Telling03,Li05}
In the ylid ($E_f$=14.1~eV) and spiro ($E_f$=13.1~eV) configurations,
the shared interstitial atom is additionally
bound to one and, respectively, to two carbon atoms of the
adjacent layer.
These structures have not been observed in our MD simulations.
The observed preference for the I$_2$ configuration in graphite
may have the following origin.
In the ``dumbell'' configuration the recoil atom is able to
transfer its excess kinetic energy to the other atoms more efficiently
than in the case of the ``bridge'' configuration.
At the same time, the formation of shared interstitials
requires the improbable collective motion of a number of atoms in the two
adjacent graphene layers in the direction of the recoil atom.
This explains the observed high probability for the formation of the
I$_2$ defect structure in the early stages of the
radiation-induced defect formation.
For the isolated graphene sheet, I$_1$ is 0.2~eV
more stable than I$_2$ due to the absence of the steric repulsion with
the adjacent graphene layer.
In curved graphenic structures, like carbon nanotubes,
the ``bridge'' interstitial defect undergoes further stabilization.
Our first principles calculations predict that
the transition from the defect structure I$_2$ to
the structure I$_1$ in graphite
is characterized by an activation barrier of 0.9~eV.
The ``dumbell'' interstitial
can also be viewed as a stable intermediate of the self-diffusion process
in graphite along the $c$-axis, occurring via
the substitution of a carbon atom in the graphene layer.\cite{Xu93}
The energy diagram for the diffusion process along the $c$-axis is shown in Figure~\ref{fig:eps4}.
\begin{figure}
\includegraphics[width=8.6cm]{fig4.eps}
\caption{\label{fig:eps4}
The scheme of the self-diffusion process in graphite along the $c$-axis. The diffusing
carbon adatom (highlighted in the figure) substitutes one of the carbon atoms in the graphite layer.
The relative energies of local minima and transition states are shown.}
\end{figure}
Formation of intimate interstitial-vacancy pairs (iIV) requires lower
transferred kinetic energies.
At $T$=20~eV we observed the formation of two low energy iIV pairs, iIV$_1$ ($E_f$=10.5~eV) and iIV$_2$
($E_f$=11.0~eV) (Fig.~\ref{fig:eps1}, bottom).
The displaced atom bridges the defect vacancy with two, respectively, three
neighbor atoms in the opposite layer, which undergo rehybridization.
A fine scan of the transferred
momentum space indicates a $T_d$ value of 18~eV
for graphite, in agreement with other reported values.\cite{Banhart97,Smith01}
As a consequence, the use of a particle beam energy capable of achieving a maximum
kinetic energy transfer just above $T_d$ will {\it selectively} create iIV defects.
This value for $T_d$ would correspond to the maximum kinetic energy transferred by an
electron beam of 90~keV.\cite{Smith01}
In the case of carbon nanotubes, $T_d$ is expected to be lower due
to curvature effects.\cite{Urita05}
This proves the crucial role of the iIV defects
in the reinforcement of carbon nanotube bundles \cite{Kis04,daSilva05}
produced by 80~keV electron irradiation.
Our results suggest the optimal conditions for the modification of
mechanic and electronic properties of carbon-based layered
nanostructures by means of the formation of iIV defects.
For graphite and closely related nanostructures, electron beam acceleration voltages of 90--110~kV
can be used.
Such modifications are non-destructive since iIV defects tend to self-recombine
without producing extensive damage of the nanostructure.\cite{Urita05}
This is also supported by the fact that the barriers for iIV$_1$
defect recombination \cite{Ewels03} and for the transformation of iIV$_1$ into iIV$_2$
(0.9~eV in this study) lie below the formation energies of I-V pairs.
Our computed formation energy for the previously proposed iIV$_1$ structure \cite{Ewels03}
is in good agreement with the values reported in
other studies.\cite{Ewels03,Li05,daSilva05}
However, MD simulations on a longer time scale
indicate that the asymmetric iIV$_1$ defect in graphite
is not stable against recombination at 350~K if the shear of
neighboring graphite layers is allowed.
By contrast, the symmetric iIV$_2$ defect is
stable throughout our MD simulations.
Two other intimate Frenkel pairs, iIV$_3$ ($E_f$=11.6~eV) and iIV$_5$ ($E_f$=12.2~eV)
have been obtained upon off-plane recoils caused by
larger transferred momenta.
In both structures the displaced carbon atom
is linked to two carbon atoms in its host layer and one atom in
the neighboring layer.
It is notable that the formation of
iIV defects has been observed {\it only} upon recoil of the $\beta$ carbon atom.
We explain this observation by the large probability of
instantaneous recombination of the $\alpha$ atom recoils due to the local
arrangement of atoms in the adjacent layer.
\subsection{In-plane recoils}
\begin{figure}
\includegraphics[width=8.6cm]{fig3.eps}
\caption{\label{fig:eps2}
Top: The mechanism of formation of a Stone-Wales defect
upon in-plane knock-on displacement ($T$=30~eV).
The carbon atoms involved in the rearrangement are marked with letters.
Bottom: Atomic structures of
7-5-4-8 and 5-7-7-4-4-9 topological defects (arabic numbers indicate non-sixmembered rings).}
\end{figure}
The formation of defects after displacement in the graphene
plane ($\phi$=90$^\circ$) requires higher transferred energies $T$$\ge$30~eV.
At $T$=30~eV ($\theta$=30$^\circ$) we observed the formation of a Stone-Wales (SW)
defect,\cite{Stone86}
which is the lowest energy ($E_f$=4.8~eV) defect in graphite.
The mechanism of its formation involves the cyclic permutation
of three carbon atoms occurring during the first 100~fs after the knock-on
collision (Fig.~\ref{fig:eps2}, top).
A much lower activation barrier of $\approx$10~eV is required
when the SW defect is formed upon simultaneous in-plane rotation of two neighboring carbon atoms.\cite{Kaxiras88}
However, this mechanism {\it cannot} be realized upon knock-on collisions because in this case
the kinetic energy is transferred to a single atom.
Irradiation of graphene-based materials, using an electron beam of energy just above
150~keV and oriented along the graphene plane, will result in an {\it increase}
of yield of SW defects.
This can be used for tuning electronic properties of materials.\cite{Crespi97}
However, because of the high energy transfer required for their
formation, SW defects will be accompanied by the formation
of Frenkel pairs, which form upon low energy ($T$$<$30~eV) off-plane recoils.
For $T$$>$30~eV two possible general mechanisms of defect
formation have been identified. The first
one involves the formation of strained structures containing
non-sixmembered rings, which have formation energies higher
than the formation energy of the SW defect.
Two such structures, 7-5-4-8 ($E_f$=11.3~eV) and 5-7-7-4-4-9 ($E_f$=12.2~eV)
have been observed in our simulations (Fig.~\ref{fig:eps2}, bottom).
The second mechanism involves the expulsion of one carbon atom
from the graphene plane shortly after the collision.
In this case interstitial-vacancy pairs are formed.
We observed formation of the ``bridge'' interstitial defect
I$_1$ caused by the expulsion of a carbon atom with low kinetic energy.
In addition, two new intimate
interstitial-vacancy pairs, iIV$_4$ ($E_f$=12.1~eV) and iIV$_6$ ($E_f$=13.0~eV)
have been characterized.
In the iIV$_4$ structure the defect is localized in the graphene
layer where the collision took place.
On the contrary,
the displaced carbon atom in the iIV$_6$ structure bridges three atoms of its
host layer with one of the neighboring layers. The formation
energies of all six iIV structures found in our
simulations lie within a narrow interval of 2.5~eV,
and they are all below the formation energies of separated I-V pairs.
These defects should be stable at long time scales and at moderate temperatures.
\section{\label{sec:conclusions}CONCLUSIONS}
In conclusion, we performed an \textit{ab initio} molecular dynamics study
of radiation-induced defect formation in graphite.
A variety of different defects, including structures which have
never been discussed previously, were observed in our simulations.
The produced defects depend strongly upon the direction and magnitude of the transferred momentum, resulting in the selective formation
of certain defect structures.
We showed the crucial role played by the early stage dynamics in the defect formation process,
and we identified the conditions at which selective creation of defects can be achieved.
In particular, we identified an interval of electron beam energies
at which only low-energy intimate Frenkel pair defects bridging adjacent graphene layers
are produced. We also conclude that Stone-Wales defects, characterized by the lowest formation energy,
cannot be produced selectively upon irradiation.
Our results are of practical importance for radiation-assisted
manufacturing of carbon materials and nanostructures with new desired properties
and functions.
\section*{ACKNOWLEDGMENTS}
The authors acknowledge L.~Forr\'o, A.~Kis, A.~Kulik, S.~Reich,
B.~I.~Yakobson, and A.~Zettl for discussions.
O.~Y. thanks the Swiss NSF for financial support.
The computational resources were provided by the CSCS and the DIT-EPFL.
|
2,877,628,090,777 | arxiv | \section{Introduction}
Solving a generic family of quantum many-body problems and ultimately predicting their phase diagram is a challenging task~\cite{Vojta2003,Sachdev2011}. The exponential growth of the Hilbert space with the system size, especially
for high dimensional systems, makes most realistic models intractable in practice. Some problems, such as the transverse-field Ising model in one dimension, can be solved analytically~\cite{Pfeuty1970}. However, more generally, obtaining the phase diagram of an interacting quantum many-body system is a critical open problem. To this end, several numerical tools have been developed, including Monte Carlo simulations~\cite{foulkes2001quantum}, and tensor-network algorithms~\cite{Schollwck2011thedensity}. Nevertheless, despite considerable progress, the phase diagram of many quantum systems in two and three dimensions remain unknown~\cite{Arovas2022,Qin2022}.
\begin{figure}[b!]
\centering
\vspace{-25pt}
\includegraphics[width = 0.48\textwidth]{introfig_edit_3.pdf}
\vspace{-25pt}
\caption{\label{fig:intro} Neural network approach to quantum phase transitions. (a) Cubic Ising lattice of interacting spins in a transverse magnetic field, here a system of size $3\times 3 \times 3$. (b) A neural network takes a configuration of the spins, encoded in the vector $\vec \sigma = (\sigma_1, ..., \sigma_N)$, and outputs the corresponding value of the wave function, $\psi_{\vec \theta}(\vec \sigma) = \langle\vec \sigma|\psi\rangle$, which depends on the variational parameters in $\vec \theta$. (c) From the fluctuations of the magnetization, we extract the zeros of the moment generating function of the magnetization and investigate their motion in the complex plane as we increase the system size. (d) Above the critical field, $h>h_c$, the zeros remain complex in the thermodynamic limit, and the system is in the paramagnetic phase (PM). At $h=h_c$, the zeros reach the real-axis, signaling a quantum phase transition. For $h<h_c$, the system is in the ferromagnetic phase (FM) with finite magnetization.}
\end{figure}
Neural network quantum states
are a recently developed class of variational states~\cite{Carleo2017} that have shown
great potential for parametrizing and finding the ground state of interacting quantum many-body
systems~\cite{sharir2020deep, zen2020badcritical, wu2021unbiased, sharir2021neural, zhang2022ground, hibatallah2020recurrent,roth2021group, choo2019twodimensional, Nomura2021helping, Westerhout2020generalization, park2022expressive, szabo2020neural, roth2022high,Choo2020fermionic,Barrett2022autoregressive,adams2021variational, rigo2022solving, lovato2022hidden}. Neural network quantum states represent the wave function of a quantum many-body system as a neural network. Specifically, the neural network is a parametrized function that takes the configuration of a many-body system as the input and outputs the corresponding amplitude and phase of the wave function. By optimizing the parameters of the neural network, so that the energy is minimized, an accurate approximation of the ground state can be found. Neural network quantum states exploit the fact that neural networks can faithfully represent many complex functions~\cite{Hornik1989multilayer}, including a variety of quantum many-body wave functions.
They have already been applied to find the wave functions of several spin models~\cite{sharir2020deep, zen2020badcritical, wu2021unbiased, sharir2021neural, zhang2022ground, hibatallah2020recurrent,PhysRevX.11.041021}, including the $J_1-J_2$ Heisenberg model~\cite{roth2021group, choo2019twodimensional, Nomura2021helping, Westerhout2020generalization, park2022expressive, szabo2020neural, roth2022high}. Moreover, their use has been extended to fermionic~\cite{stokes2020phases, Choo2020fermionic} and bosonic~\cite{Saito2018machine, han2021neural, choo2018symmetries} systems, as well as to molecules~\cite{Choo2020fermionic,Barrett2022autoregressive} and nuclei~\cite{adams2021variational, rigo2022solving, lovato2022hidden}.
In the context of critical behavior, a rigorous foundation of phase transitions was established by Lee and Yang, who considered the zeros of the partition function in the complex plane of the control parameters, for example an external magnetic field or the inverse temperature~\cite{leeYang1952StatisticalI,leeYang1952StatisticalII,blythe2003theleeyang,bena2005statistical}. This approach relies on the fact that for systems of finite size, the partition function zeros are all complex. However, if a system exhibits a phase transition, the zeros will approach the critical value on the real axis in the thermodynamic limit of large system sizes, giving rise to a non-analytic behavior of the free energy density~\cite{kim1998fisher,mulken2001classification,yamamoto2009dense,deger2018leeYang,deger2019determination, fodor2019trying,wakayama2019lee,liu2019fisher,deger2020leeYang,deger2020leeYang2,dimopoulos2022contribution,Matsumoto2022}.
Lee-Yang zeros are not just a theoretical concept, but they can also be determined experimentally~\cite{Binek1998,Wei2012,Peng2015,Flindt2013,Brandner2017}. In recent years, applications of Lee-Yang theory have been expanded to dynamical quantum phase transitions in quantum many-body systems after a quench~\cite{heyl2013dynamical,peotta2020determination,Brange2022} and to quantum phase transitions in systems at zero temperature~\cite{kist2021leeyang,vecsei2022leeyang}.
Here, we combine neural network quantum states with a Lee-Yang theory of quantum phase transitions to predict the critical behavior of interacting spin lattices in one, two, and three dimensions. As illustrated in Fig.~\ref{fig:intro}(a), we consider the transverse-field Ising model in
different dimensions and lattice geometries. We then find the ground state of the system as well as the fluctuations of the magnetization using neural network quantum states, Fig.~\ref{fig:intro}(b). From these fluctuations, we determine the complex zeros of the moment generating function of the magnetization and follow their motion as the system size is increased. As illustrated in Fig.~\ref{fig:intro}(c), the zeros remain complex in the thermodynamic limit in case there is no phase transition. On the other hand, if the magnetic field is tuned to its critical value, the zeros of the moment generating function will reach the real axis, signaling a phase transition. Thus, by investigating the positions of the zeros for different magnetic fields, we can map out the phase diagram of the system, Fig.~\ref{fig:intro}(d).
Our manuscript is organized as follows: In Sec.~\ref{sec:methods}, we describe the methods that we use throughout this work. In particular, we introduce the transverse-field Ising model, we discuss our calculations of the magnetization cumulants in the ground state using neural network quantum states, and we provide the details of the Lee-Yang theory that we use to predict the critical magnetic field for a given lattice geometry. In Sec.~\ref{sec:results}, we present the results of our calculations. As examples, we first discuss our procedure for the transverse-field Ising model on a one-dimensional chain, a two-dimensional square lattice, and a cubic lattice in three dimensions. We then provide predictions of the critical fields for several other lattice geometries. In Sec.~\ref{sec:discussion}, we discuss our results and the role of the coordination number and dimensionality of a given lattice. We also compare our predictions with mean-field theory, which becomes increasingly accurate in higher dimensions. Finally, in Sec.~\ref{sec:conclusion}, we summarize our conclusions. Technical details of our neural network calculations are provided in Appendix~\ref{app:details}.
\section{Methods}
\label{sec:methods}
\subsection{Transverse-field Ising model}
We consider the transverse-field Ising model on a lattice of spin-$1/2$ sites as described by the Hamiltonian
\begin{equation}
\hat \mathcal{H} = - J \sum_{\langle i,j \rangle} \hat \sigma_i^z \hat \sigma_j^z - h \sum_i \hat \sigma_i^x.
\end{equation}
Here, the first sum runs over all nearest neighbors, denoted by $\langle i,j \rangle$, the coupling between them is $J$, and $h$ is the transverse magnetic field.
The one-dimensional version of this model can be solved analytically and it is known to exhibit a continuous phase transition at the critical field $h_c=J$~\cite{Pfeuty1970}. Above the critical field, the system is in a paramagnetic phase with vanishing magnetization. Below it, the system exhibits spontaneous symmetry-breaking and enters a ferromagnetic phase with a non-vanishing magnetization. In the following we will investigate the model in different dimensions and geometries.
The two-dimensional systems we consider are square, honeycomb, Kagome, and triangular lattices. In three dimensions, we consider cubic, face-centred cubic, body-centred cubic, and diamond lattices. In all of these cases, we impose periodic boundary conditions, and we compare our predictions with earlier results based on large-scale quantum Monte Carlo simulations~\cite{bloete2002cluster}.
\begin{figure*}
\centering
\includegraphics[width = 1.0\textwidth]{plot_lyzs_func_of_h_linear_square_cubic.pdf}
\caption{\label{lyzplot} Extraction of zeros from the cumulants of the magnetization. (a) Extracted zeros for a linear Ising chain in different magnetic fields, $h=0.6, 0.7, 0.8, 0.9, 0.95, 1.0, 1.05, 1.1, 1.15, 1.2J$ (starting from the lower curve), as a function of the inverse system size, $1/L$. The solid lines are the finite-size scaling ansatz in Eq.~(\ref{eq:extrapolationFunction}), which allows us to determine the value in the thermodynamic limit, where $1/L$ approaches zero. (b) Similar results for a two-dimensional square lattice with the following values of the magnetic field, $h=0.5, 1.0, 1.5, 2.0, 2.5, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5J$ (starting from the lower curve). (c) Results for a cubic lattice in three dimensions with $h=0.0,1.0,2.0,4.0,5.0,5.2, 5.4, 5.6, 5.8, 6.0J$ (starting from the lower curve).}
\end{figure*}
\subsection{Neural network quantum states}
To find the ground state of the system together with the moments and cumulants of the magnetization, we use neural network quantum
states. The neural network quantum states are variational states of the form
\begin{equation}
\psi_{\vec \theta}(\vec \sigma) = \langle \vec \sigma |\psi_{\vec \theta}\rangle,
\end{equation}
where the vector $\vec \theta$ contains the variational parameters that we need to determine to minimize the energy and thereby find the ground state. The neural network provides a compressed algorithmic representation of the coefficients of the wavefunction, and it takes a spin configuration in the computational basis as the input, and outputs the wave function in response. The energy is minimized using stochastic reconfiguration, which is an approximate imaginary time-evolution within the variational space of the neural network. Neural network state methodologies have been extended to the time-evolution of quantum systems~\cite{Carleo2017, Lin2022scaling, Gutirrez2022realTime}, quantum state tomography~\cite{Torlai2018neuralnetwork, koutny2022neural, neugebauer2020neural}, as well as finite-temperature equilibrium physics~\cite{irikura2020neural, hendry2020neural, nomura2021purifying}.
Importantly, while many other approaches are not able to exploit the computational power of massive parallel computing, neural network quantum states can be implemented with modern graphics processing units.
The energy is evaluated by sampling over the wave function as
\begin{equation}
\langle \hat \mathcal{H} \rangle = \frac{\sum_{\vec \sigma \vec \sigma'} \psi^*(\vec \sigma) \langle \vec \sigma |\hat \mathcal{H} |\vec \sigma'\rangle \psi(\vec \sigma') }{ \sum_{\vec \sigma'} |\psi(\vec \sigma')|^2} = \sum_{\vec \sigma} P_\psi (\vec \sigma) \mathcal{H}_{\text{loc}}(\vec \sigma),
\label{eq:aveHam}
\end{equation}
where we have defined the probability
\begin{equation}
P_\psi(\vec \sigma) = \frac{|\psi(\vec \sigma)|^2}{ \sum_{\vec \sigma'} |\psi(\vec \sigma')|^2}
\end{equation}
and the local spin Hamiltonian
\begin{equation}
\mathcal{H}_{\text{loc}}(\vec \sigma) = \sum_{\vec \sigma'} \langle\vec \sigma |\hat \mathcal{H} |\vec \sigma'\rangle \frac{\psi(\vec \sigma')}{\psi(\vec \sigma)}.
\label{eq:locHam}
\end{equation}
Since Eq.~(\ref{eq:aveHam}) is just an average with respect to a normalized probability distribution, Markov-chain Monte Carlo can be used for evaluating the energy and the gradients~\cite{Carleo2019Netket}. It is worth noting that the spin Hamiltonian in Eq.~(\ref{eq:locHam}) is given by only a few terms in the sum, since only nearest neighbors are coupled. We will also need the expectation value of the total magnetization and its moments, which we express as
\begin{equation}
\langle \hat M_z^n \rangle = \sum_{\vec \sigma} P_\psi(\vec \sigma) M_z^n(\vec \sigma),
\end{equation}
since $\hat M_z$ is diagonal in the computational basis, such that $M_z^n(\vec \sigma) = ( \langle \vec \sigma |\hat M_z|\vec \sigma\rangle ) ^n= \langle \vec \sigma |\hat M_z^n|\vec \sigma\rangle$. Additional details of these calculations are provided in
Appendix~\ref{app:details}.
\begin{figure*}[t!]
\centering
\includegraphics[width = \textwidth]{plot_lyzextrapol_func_of_h_linear_square_cubic.pdf}
\caption{\label{plotOfLee-YangextrapolChainSquareCube} Convergence points of the zeros in the thermodynamic limit. (a) Convergence points for a linear Ising chain as a function of the magnetic field. A quantum phase transition occurs at $h_c=1.00J$, where the curve exhibits a kink, and the zeros reach the real-axis. Above the critical field, the system is in the paramagnetic phase, while it is in the ferromagnetic phase below it. (b,c) Similar results for the two-dimensional square lattice (b) and the cubic lattice in three dimensions (c).
}
\end{figure*}
\subsection{Lee-Yang theory}
\label{subsec:leeYang}
The classical Lee-Yang theory of phase transitions considers the zeros of the partition function in the complex plane of the control parameter, for instance magnetic field or inverse temperature~\cite{leeYang1952StatisticalI,leeYang1952StatisticalII,blythe2003theleeyang,bena2005statistical}.
For finite systems, the partition function zeros are situated away from the real axis. However, in case of a phase transition, they will approach the critical value on the real axis in the thermodynamic limit. One may thereby predict the occurrence of a phase transition by investigating the position of the zeros as the system size is increased. The Lee-Yang theory of phase transitions has found applications in condensed matter physics~\cite{deger2020leeYang, deger2019determination, deger2020leeYang2, deger2018leeYang, kim1998fisher,liu2019fisher}, atomic physics~\cite{mulken2001classification} and particle physics~\cite{barbour1992complex, fodor2002lattice, ejiri2006leeYang, yamamoto2009dense, nagata2015lee, fodor2019trying, wakayama2019lee, dimopoulos2022contribution}.
Recently, it has been extended to the zeros of the moment generating function that describes the fluctuations of the order parameter~\cite{kist2021leeyang,vecsei2022leeyang} and thereby allows for the detection of quantum phase transitions. Following this approach, we define the moment generating function
\begin{equation}
\chi(s) = \langle e^{s \hat M_z}\rangle= \frac{1}{g} \sum_{k=1}^g \langle\psi^{(0)}_{k}|e^{s \hat M_z}|\psi^{(0)}_{k}\rangle,
\end{equation}
where $\hat M_z$ is the total magnetization, and $s$ is referred to as the counting field. Here, we have included the possibility that the system may have $g$ degenerate and normalized ground states that we denote by $|\psi^{(0)}_{k}\rangle$, $k=1,\ldots,g$. Within this framework, the moment generating function plays the role of the partition function in the classical Lee-Yang theory, and the cumulant generating function, $\Theta(s) = \ln \chi(s)$, becomes the corresponding free energy. The moments and cumulants of the magnetization are given by derivatives with respect to the counting field as
\begin{equation}
\langle \hat M_z^n\rangle = \partial_s^n \chi(s)|_{s=0}
\end{equation}
and
\begin{equation}
\langle\!\langle \hat M_z^n\rangle\!\rangle = \partial_s^n \Theta(s)|_{s=0}.
\end{equation}
Importantly, away from a phase transition, the cumulants are expected to grow linearly with the system size, such that the normalized cumulants $\langle\!\langle \hat M_z^n\rangle\!\rangle/N$ converge to finite values as the number of spins $N$ approaches infinity. By contrast, at a phase transition, a different scaling behavior is expected due to as non-analytic behavior of the cumulant generating function at $s=0$ \cite{Karzig2010,deger2018leeYang}. This non-analytic behavior emerges in the thermodynamic limit, if the complex zeros of the moment generating function approach $s=0$.
To determine the position of the zeros that are closest to $s=0$, we use the cumulant method that was developed in Refs.~\cite{Flindt2013,deger2018leeYang,kist2021leeyang,vecsei2022leeyang,Brandner2017}. In this approach, the zeros of the moment generating function can be determined from the high cumulants of the order parameter. By doing so for different system sizes, we can then find the convergence points in the thermodynamic limit using finite-size scaling~\cite{Flindt2013,deger2018leeYang,kist2021leeyang, vecsei2022leeyang}. The cumulant method allows us to express the zeros in terms of the high cumulants of the magnetization. Moreover, for the transverse-field Ising model, the symmetry, $\hat U^\dag\hat H\hat U= \hat H$, with respect to the unitary operator $\hat U = \prod_i \hat\sigma_i^x$ that flips all spins, implies that all odd cumulants vanish, and in this model the zeros are purely imaginary~\cite{kist2021leeyang, vecsei2022leeyang}. In that case, the zeros that are closest to $s=0$ can be approximated as~\cite{vecsei2022leeyang}
\begin{equation}
\text{Im}(s_0) \simeq \sqrt{2n (2n + 1) |\llangle \hat M_z^{2n} \rrangle / \llangle \hat M_z^{2n+2} \rrangle|}
\label{eq:cumulantmeth}
\end{equation}
for large enough cumulant orders, $n\gg1$. Thus, in the following, we find the zeros from the high magnetization cumulants, which we calculate using neural network quantum states, and we ensure that the results from Eq.~(\ref{eq:cumulantmeth}) are unchanged if we increase the cumulant order. We then use the scaling ansatz ~\cite{kist2021leeyang,vecsei2022leeyang}
\begin{equation}
\text{Im}(s_0) \simeq \text{Im}(s_{0,c}) + \alpha L^{-\gamma}
\label{eq:extrapolationFunction}
\end{equation}
to predict the convergence point, $\text{Im}(s_{0,c})$, in the thermodynamic limit, where $L\rightarrow\infty$ is the linear system size. We carry out this procedure for different magnetic fields to find the critical field, where the zeros reach $s=0$, and the system exhibits a phase transition.
\begin{figure*}[t!]
\centering
\includegraphics[width = 1.0\textwidth]{plot_lyzextrapol_func_of_h_honeycomb_kagome_diamond.pdf}
\caption{\label{plotOfLee-YangextrapolHoneycombKagomeDiamond} Convergence points of the zeros in the thermodynamic limit. (a) Convergence points for honeycomb lattice as a function of the magnetic field. A quantum phase transition occurs at $h_c\approx 2.14J$, where the curve exhibits a kink, and the zeros reach the real-axis. Above the critical field, the system is in the paramagnetic phase, while it is in the ferromagnetic phase below it. (b,c) Similar results for the Kagome lattice (b) and the diamond lattice (c). }
\end{figure*}
\section{Results}
\label{sec:results}
\subsection{Extracted zeros}
Figure~\ref{lyzplot} shows zeros obtained for the transverse-field Ising model in one (chain), two (square), and three (cube) dimensions. In each case, we have determined the zeros from Eq.~(\ref{eq:cumulantmeth}) using magnetization cumulants of up to order $n=10$ for a fixed magnetic field and a given system size. We then obtain the imaginary part of the zeros, and using the finite-size scaling ansatz from Eq.~(\ref{eq:extrapolationFunction}), we find the convergence point in the thermodynamic limit as illustrated in the figure. As an example, we see in Fig.~\ref{lyzplot}a how the zeros eventually reach $s=0$ as we decrease the magnetic field from above to $h\simeq J$, where the system exhibits a quantum phase transition.
In Figs.~\ref{lyzplot}b and~\ref{lyzplot}c, we show similar results for the two-dimensional square lattice and for the three-dimensional cubic lattice. For increased dimensionality, we observe that the quantum phase transitions occurs at higher magnetic fields, as expected for an increasing number of nearest neighbors. In one dimension, we use chains of up to a length of $L=100$. For the two-dimensional square lattices, we consider systems of sizes up to $L\times L= 10 \times 10$, while in three dimensions, the biggest lattice is of size $L\times L\times L =4\times 4 \times 4$. The figure includes small error bars that represent sampling errors in the neural network quantum states. We note that
additional errors could potentially arise from small inaccuracies in the variational ground state.
The results for the three different geometries are combined in Fig.~\ref{plotOfLee-YangextrapolChainSquareCube}, where we show the extracted convergence points as a function of the transverse magnetic field. The extrapolation is performed by a
constrained minimization of $\text{Im}(s_{0,c})$, imposing that the imaginary part is not negative. At large magnetic fields, the systems are in the paramagnetic phase with the spins mostly pointing along the direction of the field. In that case, the zeros of the moment generating function do not converge to $s=0$ in the thermodynamic limit. By contrast, as the magnetic field is lowered, the zeros eventually reach $s=0$, signaling a quantum phase transition. Based on our calculations, we estimate the critical fields to be $h_c = 1.00J$ for the one-dimensional chain, $h_{c} = 3.05J$ for the two-dimensional square lattice, and $h_{c} = 5.16J$ for the three-dimensional cubic lattice. These values are all within less than 1\% difference from other numerical results~\cite{bloete2002cluster}. Below the critical field, the zeros also reach $s=0$, since the system is in the ferromagnetic phase with spontaneous magnetization. In that case, the ground state is two fold-degenerate, and the system will exhibit an abrupt change if a small magnetic field is applied in the $z$-direction.
\subsection{Critical magnetic fields}
We have considered other geometries in two and three dimensions as illustrated in
Fig.~\ref{plotOfLee-YangextrapolHoneycombKagomeDiamond}, where we show results for a honeycomb lattice, a Kagome lattice, and a diamond lattice. The honeycomb lattice has two sites per unit cell, and we restrict ourselves to a linear dimension of $L = 8$, which corresponds to $2\times L^2 = 128$ sites. Similarly, for the Kagome lattice, we go up to $L = 6$, while for the diamond lattice, we consider systems of linear size up to $L=4$, which corresponds to $2\times L^3 = 128$ sites. The results in Fig.~\ref{plotOfLee-YangextrapolHoneycombKagomeDiamond} are qualitatively similar to those in Fig.~\ref{plotOfLee-YangextrapolChainSquareCube}, but with different critical fields. In particular, we find $h_c = 2.14J$ for the honeycomb lattice, $h_c = 2.95J$ for the Kagome lattice, and $h_c = 3.20J$ for the diamond lattice.
The predictions of the critical fields are summarized in Table~\ref{tab:overViewResults}, where we also show results for triangular lattices in two dimensions and face-centred cubic (FCC) and body-centred cubic (BCC) lattices in three dimensions. The results are ordered according to the dimension $D$ as well as the number of nearest neighbors,
the coordination number $C$. In addition, we indicate the maximum linear dimension that we have used, $L_\text{max}$, and the number of sites in a unit cell, $N_\text{cell}$. Those parameters control the maximum number of spins in the lattice that we have considered, $N_\text{max}$. The last column contains the critical magnetic fields that we predict with the combination of Lee-Yang theory and neural network quantum states. We note that our methodology provides accurate predictions even with a rather low number of lattice sites.
\section{Discussion}
\label{sec:discussion}
\subsection{Dimensionality and lattice geometry}
The importance of the lattice geometry and the dimension of the system can be understood from the results in Table~\ref{tab:overViewResults}. The chain and the honeycomb lattice, which have the lowest coordination numbers, also have the lowest critical fields. The coordination numbers are larger for the Kagome and the square lattices, where each spin has four nearest neighbors, as well as for the triangular lattice with six nearest neighbors, and we see that the critical fields increase accordingly. For the lattices in three dimensions, the coordination numbers and the critical fields are even larger. Despite this general behavior, we also see that lattices with the same dimension and coordination number (the square and Kagome lattices) still have different critical fields, which are directly related to their specific lattice geometries.
\begin{figure*}
\centering
\includegraphics[width = 1.0 \textwidth]{coordnum_2panels.pdf}
\caption{\label{coordnum} Comparison with mean-field theory. (a) The critical fields are shown as functions of the coordination number, $C$. The dashed line is a simple mean-field approximation that directly links the critical field to the coordination number as $h_c^{\text{MF}}=CJ$. (b) The ratio of the critical fields over the mean-field approximation as functions of the coordination number, $C$. For large coordination numbers and dimensions, the critical fields approach the mean-field approximation indicated with a dashed line.}
\end{figure*}
\subsection{Mean-field approximation}
To better understand the role of the coordination number, we show in Fig.~\ref{coordnum} the critical fields as a function of the coordination number. In Fig.~\ref{coordnum}a, we see the clear trend that the critical fields increase with the coordination number. Indeed, within a simple mean-field approximation, we would expect that the critical field is directly related to the coordination number as $h^{\text{MF}}_c = C J$~\cite{strecka2015brief}. We show this mean-field approximation with a dashed line in the figure and find good qualitative agreement with our predictions. We also see that our results come closer to the mean-field approximation as the dimension of the system is increased. In particular, it is clear that the critical field for the one-dimensional chain is furthest away
from the mean-field approximation, while the results for the three-dimensional lattices are much closer.
To further support these observations, we show in Fig.~\ref{coordnum}b the ratio of the critical fields over the mean-field approximation. This ratio allows us to characterize how the relative deviations from the mean-field prediction decrease for larger coordination numbers. Still, we see that the critical fields are all smaller than the mean-field approximation, which ignores quantum fluctuations. The results for the critical fields in three dimensions are closer to the mean-field approximation as compared with one and two dimensions. This observation is in line with the expectation that mean-field theory becomes more accurate in higher dimensions.
\begin{table}
\begin{tabular}{ |c | c | c | c | c | c | c |}
\hline
Lattice & $D$ & $C$ & $L_\text{max}$ & $N_\text{cell}$ & $N_\text{max}$ & $h_c/J$\\
\hline
Chain & 1 & 2 & 60 & 1 & 60& 1.00\\
Honeycomb & 2 & 3 & 8 & 2& 128& 2.14\\
Kagome & 2 & 4 & 6 & 3 & 108& 2.95\\
Square & 2 & 4 & 10 & 1 & 100& 3.05\\
Triangular & 2 & 6 & 10 & 1 & 100& 4.78\\
Diamond & 3 & 4 & 4 & 2 & 128& 3.20\\
Cubic & 3 & 6 & 4 & 1 & 64& 5.16\\
BCC & 3 & 8 & 4 & 1 & 64& 7.10\\
FCC & 3 & 12 & 4 & 1 & 64& 10.8\\
\hline
\end{tabular}
\caption{\label{tab:overViewResults} Summary of critical fields. For each lattice, we indicate the dimension, $D$, and the coordination number, $C$. We also show the maximum linear dimension, $L_\text{max}$, and the number of sites per unit cell, $N_\text{cell}$, which give the maximum number of sites that we have used as $N_\text{max} = N_\text{cell} \times L_\text{max}^D$. The last column contains our predictions of the critical field.}
\end{table}
\section{Conclusions}
\label{sec:conclusion}
We have combined a Lee-Yang theory of quantum phase transitions with neural network quantum states to predict the critical field of the transverse-field Ising model in different dimensions and lattice geometries. Specifically, we have used neural network quantum states to find the ground state of the interacting spin system, which further
makes it possible to extract the cumulants of the magnetization. From these cumulants, we determine the complex zeros of the moment-generating function, which reach the real-axis in the thermodynamic limit if the system exhibits a phase transition. Our method works with rather small systems, which in turn allows us to treat lattices in two and three dimensions. Our predictions agree well with results that were obtained using large-scale quantum many-body methods.
We have also analyzed the differences between our predictions and a simple mean-field approximation, which becomes increasingly accurate for higher coordination numbers and dimensions.
Thanks to the flexibility of neural network quantum states, the method can potentially treat frustrated problems,
in stark contrast to quantum Monte Carlo approaches that suffer from sign-problems.
Our results show that the combination of Lee-Yang theories of phase transitions with neural network quantum states provides a viable way forward to predict the phase behavior of complex quantum many-body systems such as Heisenberg models and fermionic Hubbard models.
\acknowledgements
We acknowledge the computational resources provided by the Aalto Science-IT project and the support from the Finnish National Agency for Education (Opetushallitus), the Academy of Finland through grants (Grants No.~331342 and No.~336243) and the Finnish Centre of Excellence in Quantum Technology (Projects No.~312057 and No.~312299),
and from the Jane and Aatos Erkko Foundation.
|
2,877,628,090,778 | arxiv |
\section{Introduction}
\label{sec:intro}
Machine translation is shifting to an end-to-end approach based on deep neural networks. Recent studies in neural machine translation (NMT) such as \cite{Vaswani:2017, Bahdanau:2014, Wu:2016,Gehring:2017} have produced impressive advancements over phrase-based systems while eliminating the need for hand-engineered features. Most NMT systems are based on the encoder-decoder architecture which consists of two neural networks. The encoder compresses the source sequences into a real-valued vector, which is consumed by the decoder to generate the target sequences.
The process is done in an end-to-end fashion, demonstrated the capability of learning representation directly from the training data.
The typical sequence-to-sequence machine translation model consists of two recurrent neural networks (RNNs) and an attention mechanism \cite{Bahdanau:2014, Luong:2015b}. Despite great improvements over traditional models \cite{Wu:2016, Sennrich:2016, Luong:2015a} this architecture has certain shortcomings, namely that the recurrent networks are not easily parallelized and limited gradient flow while training deep models.
Recent designs such as ConvS2S \cite{Gehring:2017} and Transformer \cite{Vaswani:2017} can be better parallelized while producing better results on WMT datasets. However, NMT models take a long time to train and include many hyper-parameters. There is a number of works that tackle the problem of hyper-parameter selection \cite{Britz:2017, Popel:2018} but they mostly focus on high-resource language pairs data, thus their findings may not translate well to low-resource translation tasks such as English-Vietnamese. Unlike in Computer Vision \cite{Huang:2016, Kandaswamy:2014}, the task of adapting parameters spaces from one NMT model to other NMT models is nearly impossible \cite{Britz:2017}. This reason limits researchers and engineers to reach good-chose hyper-parameters and well-trained models.
To date there are several research works on English-Vietnamese machine translation such as \cite{Le:2003, Dinh:2003, Ho:2009, Nguyen:2016}, using traditional methods with modest BLEU scores. Some newer works such as \cite{Luong:2015c, Huang:2017} experimented on the IWSLT English-Vietnamese dataset \cite{IWSLT:2015} and showed great potential to improve English-Vietnamese translation tasks using more data and more complex models.
In \cite{Phan:2017} the authors introduced datasets for bilingual English-Vietnamese translation and attained state-of-the-art BLEU scores using sequence-to-sequence models and vanilla preprocessing. In this work we perform extensive experiments on large-scale English-Vietnamese datasets with the latest NMT architectures for further improvements in BLEU scores and report our empirical findings.
Our main contributions are as follows: (1) A brief survey of current state of the art in NMT. (2) The construction of a large parallel corpus for English-Vietnamese translation, which will be publicly available. (3) Implementation and experimentation of the newest models, and our source code will also be shared. (4) Empirical findings on tuning the aforementioned models.
\section{Latest NMT architectures}
\subsection{Sequence-to-Sequence RNNs}
Here we introduce the sequence-to-sequence model based on an encoder-decoder architecture with attention mechanism \cite{Luong:2015b}. Let $(X, Y)$ be the pair of source and target sentences, where $X = x_1, \ldots, x_m$ is a sequence of $m$ symbols and $Y=y_1, \ldots, y_n$ a sequence of $n$ symbols. The encoder function $f_{enc}$ maps the input sequence $X$ to a fixed size vector, which the decoder function $f_{dec}$ uses to generate the output sequence $Y$.
While $f_{dec}$ is usually a uni-directional RNN, $f_{enc}$ can be a uni-directional, bi-directional or hybrid RNN. In this work we consider bi-directional encoders. Each state of $f_{enc}$ has the form $\overline{h}_i = [\overrightarrow{h_i},\overleftarrow{h_i}]$ where the components encode $X$ in forward and backward directions. The auto-regressive decoder $f_{dec}$ then predicts each output token $y_i$ from the recurrent state $s_i$, the previous tokens $y_{<i}$ and a context vector $c_i$.
The context vector $c_i$ is also called attention vector and depends on encoder states together with the current decoder state. Among known attention architectures, in this work we use the most efficient as described in \cite{Luong:2015b}. At the decoding step $t$, an alignment vector $a_t$ is derived from the current decoder hidden state $h_t$ and each encoder hidden state $\overline{h}_s$. The context vector $c_t$ is a weighted average over all encoder states with weights $a_t$.
\begin{align}
a_t(s) &= align(h_t, \overline{h}_s) \\
c_t &= \sum a_th_s
\end{align}
The context vector $c_t$ is concatenated with the current hidden decoder state $h_t$ to produce an attentional state $\tilde{h}$, which is fed through a softmax layer to produce the predicted distribution.
\begin{align}
\tilde{h} &= tanh(W_c[c_t; h_t]) \\
p(y_t|y_{<t}, x) &= softmax(W_s\tilde{h}_s)
\end{align}
\subsection{The Convolutional Sequence-to-Sequence Model}
\begin{figure}[h!]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.94\textwidth]{ConvS2SOverall.pdf}
\caption{\small The Convolution Sequence-to-Sequence model architecture, adapted from \cite{Gehring:2017}}
\label{convs2s_overal}
\end{minipage}~
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.94\textwidth]{TransformerOveral.pdf}
\caption{\small Overall architecture of the Transformer}
\label{tf_overal}
\end{minipage}
\end{figure}
The Convolutional Sequence-to-Sequence Model (ConvS2S) \cite{Gehring:2017} is a sequence-to-sequence model that uses a fully convolutional architecture. The model is equipped with gated linear units \cite{Dauphin:2016} and residual connections \cite{He:2015}.
\subsubsection{Position Embeddings}
Because the CNN itself can not convey positional information, ConvS2S uses position embeddings to tackle this problem. The input element $x=(x_1,\ldots, x_m)$ is represented as a vector $z = w + p$ where $w = (w_1,\ldots, w_m)$ embeds the symbols $x_i$ into an Euclidean space $ \mathbb{R}^{f}$ and $p=(p_1, \ldots, p_m)$ embeds the positions of the $x_i$ into $\mathbb{R}^{f}$. The same process is applied to the output elements generated by the decoder network, and the resulting representations are fed back into the decoder.
\subsubsection{Convolutional Layer Structure}
We denote the output of the $i^{th}$ layer by $e^{i} = (e_1^i, \ldots, e_n^i)$ for the encoder network and $d^{i} = (d_1^i, \ldots, d_o^i)$ for the decoder network. In the model, each layer contains a one dimensional convolution followed by a non-linearity.
Each convolution kernel is parameterized as a weight $W \in \mathbb{R}^{2s \times ks}$ and a bias $b_w \in \mathbb{R}^{2s}$. The kernel's input is a matrix $X \in \mathbb{R}^{k\times s}$ which is a concatenation of $k$ input elements embedded in $s$ dimensions, the kernel's output is a vector $Y \in \mathbb{R}^{2s}$ that has twice the dimensionality of the input elements. Each group of $k$ output elements of the previous layer are operated by a subsequence layer. The non-linearity is the gated linear unit (GLU:\cite{Dauphin:2016}) which implements a gating mechanism over the output of the convolution $Y = [A\ B] \in \mathbb{R}^{2s}$:
\begin{equation}
v([A\ B]) = A \otimes \sigma(B)
\end{equation}
where $A, B \in \mathbb{R}^{s}$ are the non-linearity input, $\otimes$ is the point-wise multiplication and the output $v([A\ B]) \in \mathbb{R}^s$ has half size of $Y$. The gates $\sigma(B)$ control which inputs A of the current context are relevant \cite{Gehring:2017}.
Residual connections from the input of each convolution to the output are applied, similar to \cite{He:2015}
\begin{equation}
d^i_j = v(W^i[d^{i-1}_{ j-k/2}, \ldots, d^{i-1}_{j+k/2}] + b^i_w) + d_j^{i-1}
\end{equation}
The convolution outputs that are of size $2s$ are mapped to the embedding of size $f$ by linear projections. These linear mappings are applied to $w$ while feeding embeddings to the encoder network, to the encoder output $e^i_j$, to the final layer of the decoder just before the $softmax$ $d^L$ and to all decoder layers $d^i$ before computing the scores the attentions.
Finally, a distribution over the $T$ possible next target elements $y_{j+1}$ is computed by transforming the top decoder output $d^L_j$ via a linear layer with weights $W_o$ and bias $b_o$:
\begin{equation}
p(y_{j+1} | y_1,\ldots, y_j, x) = softmax(W_od_j^L + b_o) \in \mathbb{R} ^ T
\end{equation}
\subsubsection{Multi-step Attention}
In ConvS2S, the attention mechanism is applied separately for each encoder layer. The attention mechanism works as multiple ``hops'' \cite{Sukhbaatar:2015} compared to single step attention \cite{Bahdanau:2014}, \cite{Luong:2015b}, \cite{Zhou:2016}, \cite{Wu:2016}. At the decoder layer $i$, the attention $a_{kj}^i$ of state $k$ and the source element $j$ are computed as a dot-product between the decoder state summary $v^i_k$ and each output $e_j^u$ of the last encoder layer $u$:
\begin{equation}
a^i_{kj} = \frac{\exp(v_k^i \cdot e_j^u)}{\sum_{t=1}^m\exp(v^i_k \cdot e_j^u)}
\end{equation}
where $v_k^i$ is combined of the current decoder state $d_k^i$ and the embedding of the previous target element $g_k$:
\begin{equation}
v^i_k = W_v^i d_k^i + b_v^i + g_k
\end{equation}
The conditional input vector $c_k^i$ to the current decoder layer is a weighted sum of the encoder output as well as the input embeddings $z_j$:
\begin{equation}
c^i_k = \sum_{j=1}^m a_{kj}^i(e_j^u + z_j)
\end{equation}
After that, $c_k^i$ is added to the output of the corresponding decoder layer $d_k^i$. This attention mechanism can be seen as determining useful information from the current layer to feed to the subsequent layer. The decoder can easily access the attention history of $k-1$ previous time steps. Therefore, the model can take into account which previous inputs have been attended more easily than recurrent networks \cite{Gehring:2017}.
\subsection{The Transformer Model}
Unlike other transduction models, Transformer does not use RNNs or CNNs for modeling sequences. It has been claimed by authors to be the first transduction model to rely entirely on self-attention to compute representations of its input and output \cite{Vaswani:2017}. Like other competitive sequence transduction models, Transformer has an encoder and a decoder. The model is auto-regressive, consuming at each step the previous generated symbols as additional input to emit the next symbol. Compared to RNNs the proposed self-attention mechanism allows for a high degree of parallelization in training, while relying on positional embeddings to capture global dependencies within each sequence.
\subsubsection{Overall Structure}
Like ByteNet \cite{Kalchbrenner:2016} or ConvS2S \cite{Gehring:2017}, the decoder is stacked directly on top of the encoder. Without the recurrence or the convolution, Transformer encodes the positional information of each input token by a \textit{position encoding} function. Thus the input of the bottom layer for each network can be expressed as $Input = Embedding + Positional Encoding$.
The encoder has several identical layers stacked together. Each layer consists of a \textit{multi-head self-attention mechanism} and a \textit{position-wise fully connected feed-forward network}. Each of these sub-layers has a residual connection around itself, followed by layer normalization \cite{LeiBa:2016} (Figure \ref{tf_overal}). The output of each sub-layer is $LayerNorm(x + Sublayer(x))$ where $x$ is the sub-layer input and $Sublayer$ is the function implemented by the sub-layer itself. The outputs of all sub-layers and the embedding layers in the model are vectors of dimension $d_{model}$.
The decoder is also a stack of identical layers, each layer comprising three sub-layers. At the bottom is a masked multi-head self-attention, which ensures that the predictions for position $i$ depend only on the known outputs at the positions less than $i$. In the middle is another multi-head attention which performs the attention over the the encoder output. The top of the stack is a position-wise fully connected feed-forward sub-layer.
The decoder output finally goes through a linear transform with softmax activation to produce the output probabilities. The final linear transform shares the same weight matrix with the embedding layers of the encoder and decoder networks, except that the embedding weights are multiplied by $\sqrt{d_{model}}$.
\subsubsection{Attention}
The attention is crucial in NMT. It maps a \textit{query} and a set of \textit{key-value} pairs to an output. The output of the attention is a weighted sum of the \textit{values} whose weights show the correlation between each \textit{key} and \textit{query}. The novelty is that the Transformer's attention is a \textit{multi-head self-attention}. In the Transformer's architecture, the $query$ is the decoder's hidden state, the $key$ is the encoder's hidden state and the $value$ is the normalized weight measuring the ``attention'' that each \textit{key} is given. It is assumed that the queries and the keys are of dimension $d_k$ and the values are of dimension $d_v$.
\begin{itemize}
\item Scaled dot-product attention:
Let $Q$ be the matrix of queries, $K$ be the matrix of keys and $V$ be the matrix of values. The attention is calculated as follows:
\begin{equation}
Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})V.
\end{equation}
Instead of using a single attention function, Transformer uses multi-head attentions. The multi-head attention consists of $h$ layers (heads). The queries, keys and values are linearly projected to $d_k$ and $d_v$ dimensions. Each head receives a set of projections and performs a separate attention function yielding $d_v$-dimensional output values. The heads' outputs are concatenated and projected, resulting in the final multi-head attention output.
\begin{equation}
MultiHead(Q, K, V) = Concat(head_{1}, head_{2},\dots, head_{h})W^{O}
\end{equation}
where $head_{i} = Attention(QW_{i}^{Q}, KW_{i}^{K}, VW_{i}^{K})$.
The projections are parameter matrices $W_i^Q \in \mathbb{R}^{d_{model} \times d_k}$, $W_i^K \in \mathbb{R}^{d_{model} \times d_k}$, $ W_{i}^{V} \in \mathbb{R}^{d_{model} \times d_v}$ and $W_{i}^{O} \in \mathbb{R}^{hd_{v} \times d_{model}}$.
\end{itemize}
If we set $d_k = d_v = d_{model}/h$, the multi-head attention then has the same computational cost as a single full-dimensionality attention.
The Transformer's attention mechanism imitates the classical attention mechanism where the attention queries are previous decoder layer outputs, the keys and the values (memory) are the encoder layer outputs.
\subsubsection{Position-wise Feed-forward Networks}
The fully connected feed-forward network (FFN) at the top of each layer is applied to each input position separately and identically. Each FFN here consists of two linear transformations with a ReLU activation in between, acting like a stack two convolutions with kernel size 1:
\begin{equation}
FFN(x) = ReLU(xW_1 + b_1)W_x + b_2.
\end{equation}
\subsubsection{Positional Encoding}
There are many types of positional encodings, including both learned and fixed variants \cite{Gehring:2017}. Here the positional encodings are chosen as follows:
\begin{align}
PE(pos, 2i) &= sin(\frac{pos}{10000^{2i/d_{model}}}) \\
PE(pos, 2i+1) &= cos(\frac{pos}{1000^{2i/d_{model}}}),
\end{align}
where $pos$ is the position and $i$ is the dimension. The authors hypothesized this function would allow the model to easily learn to attend by relative positions \cite{Vaswani:2017}. Their experiments showed that these encodings have the same performance as learned positional embedding. Furthermore they allow the model to extrapolate to sequences longer than the training sequences.
\section{Parallel corpus construction from public sources}
\subsection{Data Crawling}
An essential component of any machine translation system is the parallel corpus. A good system requires a parallel corpus with a substantial number of qualified sentence pairs. There are various projects building English-Vietnamese corpora for specific tasks such as word-sense disambiguation~\cite{Dinh:2002:BTC:1118794.1118801} \cite{Dien:2003:PEB:1118905.1118921}, VLSP project~\cite{22:VLSP}, web mining \cite{conf/rivf/DangH07}, etc. EVBCorpus \cite{Ngo:2013} is a multi-layered English-Vietnamese Bilingual Corpus containing over 10,000,000 words.
However since corpora such as EVBCorpus or VLSP are not openly published, we first needed to build a high-quality large-scale English - Vietnamese parallel corpus. We developed a web crawler to collect English - Vietnamese sentences from 1,500 movie subtitles from the Internet. We also use the TED Talk subtitles collected in \cite{24:cettoloEtAl:EAMT2012}.
We use the Scrapy framework \cite{Scrapy:2016} to build our own crawler which consists of the following components (Figure~\ref{tf_scrapy})
\begin{figure}[h]
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[scale=0.4]{scrapy_architecture.png}
\caption{Scrapy-based crawler engine architecture. Source: http://scrapy.org}
\label{tf_scrapy}
\end{minipage}~
\begin{minipage}{.45\textwidth}
\begin{tabular}{c|cc}
dataset & number of lines & number of tokens \\
\hline
train.en & 886224 & 10151378 \\
train.vi & 886224 & 11454886 \\
tst2012.en & 1553 & 28723\\
tst2012.vi & 1553 & 34345\\
tst2013.en & 1268 & 27317\\
tst2013.vi & 1268 & 33764\\
tst2015.en & 1080 & 21332\\
tst2015.vi & 1080 & 25341\\
\end{tabular}
\captionof{table}{Details of experiment dataset.}
\label{table:data}
\end{minipage}
\end{figure}
\begin{description}[style=multiline,leftmargin=3cm,font=\normalfont]
\item[\rm Scrapy Engine] This component has the responsibility to control the data flow between components for coordination and triggering system events.
\item[Scheduler] The Scheduler implements strategies to order URL crawling requests received from the Engine.
\item[Downloader] The Downloader is responsible for fetching web pages and return crawled data to the Scrapy Engine.
\item[Spider] Spiders is responsible to parse responses and extract items from them or to perform additional requests to follow. We had to write our-own spiders classes to extract parallel English-Vietnamese sentences from HTML contents, based on CSS selectors and XPath expressions.
\item[Item pipelines] The Item Pipeline processes the items after being extracted by the spiders. We defined our pipeline module to store scrapped items into a Mongo database instance.
\item[Downloader middleware] Downloader middlewares hook between the Engine and the Downloader to intercept requests and responses. We had to write several downloader middlewares to rotate proxies, user-agents in order to improve our crawlers stability.
\item[Spider middleware] Spider middlewares process spider input and output.
\end{description}
\vspace{0.5cm}
Built on Scrapy, we do not have to implement all the above components. We instead implemented only Spider, Item pipelines, and Downloader middlewares. Our web crawler collected around 1.2 millions parallel English - Vietnamse sentences.
\subsection{Data Cleaning and Preprocessing}
The following steps were conducted to clean the dataset:
\begin{itemize}
\item Detecting and removing incomplete translations: A big part of our dataset is movie subtitles, .where we found many partially translated examples. In order to detect and remove such subtitles, we use Princeton WordNet \cite{Miller:1995:WLD:219717.219748} to extract an English vocabulary. We then scan each subtitle for tokens found in the vocabulary. If a half of all tokens match this criteria, the subtitle is considered untranslated. We also use $langdetect$ package \footnote{https://pypi.python.org/pypi/langdetect} to filter out sentences which are not in Vietnamese. Manual observation on a random subset of removed subtitles shows that this heuristic filter works sufficiently for our purpose.
\item Removing low quality translations: There are many of low quality translations in our collected data, which we had to remove manually.
\end{itemize}
After filtering we obtained 886,224 sentences pairs for training. We use tst2012 for validation; tst2013, tst2015 for testing; all the thress are from IWSLT as provided in \cite{24:cettoloEtAl:EAMT2012}. The sizes of the datasets are shown in Table \ref{table:data}.
Following\cite{Phan:2017} we only use subword for our experiments. In particular we created a shared subword code file using Byte Pair Encoding (BPE) \cite{Sennrich:2016} using 32,000 merge operations. This shared subword code file was then used to transform the train, validation and test datasets to sub-words with a vocabulary size of approximately 20,000.
\section{Experiments and discussions}
\subsection{Overview of Training Configurations}
For authenticity the experiments with each model are performed on original software provided by the authors. Specifically Sequence to Sequence RNN experiments are performed using \cite{Luong:2017}, the Transformer experiments are performed using Tensor2Tensor (T2T) software \cite{Vaswani:2018} and the experiments on ConvS2S are performed using Facebook AI Research Sequence-to-Sequence Toolkit \cite{Gehring:2017}.
Training is performed on a single Nvidia Geforce Titan X. We run each experiment 3 times with random initializations and save one model checkpoint every 1000 steps. The checkpoint for reporting results is selected based on BLEU score for the validation set. We train and report the model's performance at the maximum of 64th epoch due to our computing resource constraints. For the sake of brevity, we only report mean BLEU on our result tables.
In all our experiments, there are some common terms in all the models, which are specified as follow:
\begin{itemize}
\item \textbf{Maximum input length} (\texttt{max\_length}): specifies the maximum length of a sentence in tokens (sub-words in our case). Sentences longer than \texttt{max\_length} are either excluded from the training (T2T) or cut to match the \texttt{max\_length} (RNN). Lowering \texttt{max\_length} allows us to use a higher batch size and/or bigger model but biases the translation towards shorter sentences. Since $99\%$ of the training sentences are not longer than 70, we set \texttt{max\_length} to 70.
\item \textbf{Batch size} (\texttt{batch\_size}) For T2T \texttt{batch\_size} is the approximate number of tokens (subwords) consumed in one training step, while for ConvS2S and RNN \texttt{batch\_size} is the number of sentence pairs consumed in one training step. Hence for consistency we define \texttt{batch\_size} as the approximate average number of tokens consumed in one training step. In fact the number of tokens in a sentence is the maximum of source and target subwords from the pair of training sentences. During training this allows us to put as many training tokens per batch as possible while ensuring that batches with long sentences still fit in GPU memory. In contrast, if we fixed the number of sentence pairs in a training batch, the model can run out of memory if a batch has many long sentences.
\item \textbf{Training epoch} is one complete pass through the whole training set. The number of training steps can be converted to epochs by multiplying by the batch size and dividing by the number of subwords in the training data.
\item \textbf{Model size} is number of trainable parameters of each model. Because of the difference in model structures, it is almost certain that two models with the same model size will not have the same training time.
\end{itemize}
Human judgment is always the best evaluation of machine translation systems; but in practice it is prohibitively expensive in time and resources. Therefore automatic scoring systems that evaluate machine translations against standard human translations are more commonly used. The most popular automatic metric in use is undoubtedly the BLEU score \cite{Papineni:2002}. BLEU has a high correlation with human judgments of quality and is easy to compute. Even though there are some acknowledged problems with BLEU and others better-performing metrics \cite{Bojar:2017}, we still stick to BLEU for its simplicity. In this work, we use the case-insensitive
sacréBLEU \footnote{https://github.com/awslabs/sockeye/tree/master/contrib/sacrebleu} version which uses a fixed tokenization.
\subsection{Sequence-to-Sequence RNN}
Based on previous literature in \cite{Phan:2017}, \cite{Britz:2017}, we build a baseline which is set reasonably large for our dataset.
We use LSTM \cite{Hochreiter:1996} for the two models as suggested in \cite{Britz:2017}. The embedding dimension in the baseline model is set to be equal to the number of cells in each layer. We use two layers of 1024-unit LSTMs for both the encoder and the decoder, whereas the encoder's first layer is a bi-directional LSTM network and each layer is equipped with a residual connection and a dropout of 0.15 is applied to the input of each cell.
We use Stochastic Gradient Descent (SGD) as the optimization algorithm with the batch size set to approximately 1280 tokens per step. The learning rate is set to 1.0; after 10 epochs we begin to halve the learning rate every single epoch. To prevent gradient explosion we enforce a hard constraint on the norm of the gradient by scaling it when its norm exceeds a threshold. In our two models the threshold is set to 5.0. For each training batch we compute $s = ||g||_2$ where $g$ is the gradient divided by the batch size. If $s > 5.0$, we set $g = \frac{5g}{s}$.
\begin{table}[h!]
\begin{center}
\caption{\small The \textit{baseline} system's performance with approximate 98 millions parameters}
\begin{tabular}{c|>{\centering\arraybackslash}m{1.5cm}>{\centering\arraybackslash}m{1.5cm}>{\centering\arraybackslash}m{2.5cm}|ccc}
task & batch size & training epochs & training time (days) & tst2012 & tst2013 & tst2015 \\
\hline
\textit{En-Vi} & 1300 & 32 & 3.5 & 34.77 & 37.00 & 31.03 \\
\textit{Vi-En} & 1300 & 20 & 2.5 & 35.21 & 38.83 & 30.29 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Convolution Sequence to Sequence}
\begin{table}
\caption{\small The hyper-parameters set of ConvS2S model}
\label{convs2s:models_config}
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\begin{small}
\begin{tabular}{c>{\centering\arraybackslash}m{2.8cm}>{\centering\arraybackslash}m{2.8cm}>{\centering\arraybackslash}m{1cm}cc|>{\centering\arraybackslash}m{1.5cm}>{\centering\arraybackslash}m{1cm}}
& encoder & decoder & emb size & $lr$ & $p_{drop}$ & training times (hours) & params $\times 10^6$ \\
\hline
$B_{base}$ & $4 \times [256 \times (3\times 3)]$ & $3 \times [256 \times (3\times 3)]$ & 256 & 0.5 & 0.1 & 4 & 10 \\
\hline
$B_1$ & $4 \times [512 \times (3\times 3)] \newline 2 \times [1024 \times (3\times 3)] \newline 1 \times [2048 \times (1\times 1)]$ & $4 \times [512 \times (3\times 3)] \newline 2 \times [1024 \times (3\times 3)] \newline 1 \times [2048 \times (1\times 1)]$ & 384 & 0.5 & 0.15 & 20 & 62 \\
\hline
$B_2$ & $9 \times [512 \times (3 \times 3)] \newline 4 \times [1024 \times (3 \times 3)] \newline 2 \times [2048 \times (1 \times 1)]$ & $9 \times [512 \times (3 \times 3)] \newline 4 \times [1024 \times (3 \times 3)] \newline 2 \times [2048 \times (1 \times 1)]$ & 768 (512) & 0.5 & 0.15 & 48 & 144 \\
\hline
$B_3$ & $8 \times [512 \times (3 \times 3)] \newline 4 \times [1024 \times (3 \times 3)] \newline 2 \times [2048 \times (1 \times 1)] \newline 1 \times [4096 \times (1 \times 1)]$ & $8 \times [512 \times (3 \times 3)] \newline 4 \times [1024 \times (3 \times 3)] \newline 2 \times [2048 \times (1 \times 1)] \newline 1 \times [4096 \times (1 \times 1)]$ & 768 & 0.5 & 0.15 & 78 & 199 \\
\hline
\end{tabular}
\end{small}
\end{adjustbox}
\end{center}
\end{table}
We introduce four different models for each direction of translation. The hyper-parameters for each experiment are shown in Table \ref{convs2s:models_config}:
\begin{itemize}
\item We used Nesterov's accelerate gradient (NAG) \cite{Sutskever:2013} with a fixed learning rate: 0.25 for the $B_{base}$ model and 0.5 for the rests. After a certain number of epochs, we force-anneal the learning rate (\texttt{lr}) by a \texttt{lr\_shrink} factor: \texttt{lr\_new} = \texttt{lr} * \texttt{lr\_shrink}. We start annealing the learning rate at the 24th epoch with a width \texttt{lr\_shrink} of 0.1 for $B_{base}$ model and at the 50th epoch with the width \texttt{lr\_shrink} set to 0.2 for the rest. Once the learning rate falls below $10^{-5}$ we stop the training process.
\item In the $B_{2}$ model, we use embedding size of 768 for all internal embeddings except the decoder output embedding (pre-softmax layer) which is set to 512.
\item The effective context size of $B_{base}$, $B_1$, $B_2$ and $B_3$ are 9, 13, 27 and 25, respectively.
\item We apply label smoothing of $\epsilon_{ls} = 0.1$ for all 4 models. This makes training perplexity fluctuate in a small interval but improves accuracy and BLEU score \cite{Vaswani:2017}.
\end{itemize}
We use cross-validation's BLEU score to decide which checkpoint to select for evaluation: the $B_{base}$ model is evaluated after 32 training epochs, the rest are evaluated after 64 training epochs. We found that the best perplexity of the validation dataset does not correspond to the best BLEU score on the test set but the BLEU score on the validation dataset does.
\begin{table}[h!]
\caption{ BLEU score of English-Vietnamese and Vietnamese-English translation task on tst2012, tst2013, tst2015 of $B_{base}$, $B_1$, $B_2$ and $B_3$ model using beam-search with length penalty set to 1, $b$ is the beam size}
\label{convs2s-envi}
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|l|ccc|ccc|}
\hline
model & \multicolumn{3}{c|}{English-Vietnamese} & \multicolumn{3}{c|}{Vietnamese-English} \\
& tst2012 & tst2013 & tst2015 & tst2012 & tst2013 & tst2015 \\
\hline
$B_{base}$, $b = 1$ & 24.31 & 25.34 & 23.98 & 25.18 & 27.76 & 23.89 \\
$B_{base}$, $b = 2$ & 26.40 & 28.02 & 26.56 & 26.09 & 27.42 & 24.89\\
$B_{base}$, $b = 5$ & 26.92 & \textbf{28.75} & 27.86 & 26.74 & 28.60 & 25.36 \\
$B_{base}$, $b = 10$ & 27.09 & 28.64 & 27.87 & \textbf{26.97} & \textbf{29.21} & 25.59\\
$B_{base}$, $b = 20$ & 27.08 & 28.66 & 28.09 & 26.86 & 29.46 & \textbf{25.61} \\
$B_{base}$, $b = 100$ & \textbf{27.22} & 28.55 & \textbf{28.18} & 26.83 & 29.31 & 25.60\\
\hline
$B_1$ , $b = 1$ & 25.29 & 27.01 & 24.99 & 26.08 & 28.91 & 24.57\\
$B_1$ , $b = 2$ & 27.97 & 29.01 & 27.15 & 27.67 & 30.07 & 26.38\\
$B_1$ , $b = 5$ & 29.39 & 31.77 & 28.88 & 28.35 & 31.24 & 27.16\\
$B_1$ , $b = 10$ & 29.86 & \textbf{32.26} & 29.31 & 28.40 & 31.63 & 27.32\\
$B_1$ , $b = 20$ & \textbf{29.94} & 32.25 & 29.41 & 28.44 & 31.86 & 27.24\\
$B_1$ , $b = 100$ & 29.84 & 32.15 & \textbf{29.75} & \textbf{28.58} & \textbf{31.87} & \textbf{27.39}\\
\hline
$B_2$ , $b = 1$ & 34.87 & 36.57 & 29.13 & 37.90 & 39.11 & 28.33\\
$B_2$ , $b = 2$ & 36.36 & 37.61 & 30.42 & 39.85 & 41.78 & 29.99\\
$B_2$ , $b = 5$ & 37.20 & \textbf{38.53} & 31.10 & 41.07 & 42.85 & 30.52\\
$B_2$ , $b = 10$ & 37.19 & 38.48 & 31.23 & 41.19 & 43.03 & 30.60\\
$B_2$ , $b = 20$ & 37.36 & 38.36 & 31.25 & 41.44 & 43.32 & \textbf{30.74}\\
$B_2$ , $b = 100$ & \textbf{37.49} & 38.42 & \textbf{31.42} & \textbf{41.49} & \textbf{43.36} & 30.71\\
\hline
$B_3$ , $b = 1$ & 40.38 & 40.81 & 31.40 & 42.63 & 44.17 & 32.68\\
$B_3$ , $b = 2$ & 41.87 & 42.62 & 32.04 & 43.09 & 45.61 & 33.41\\
$B_3$ , $b = 5$ & 42.32 & 42.49 & 33.56 & 44.48 & 46.13 & 34.01\\
$B_3$ , $b = 10$ & 42.40 & 43.51 & \textbf{33.50} & 44.32 & 46.31 & \textbf{34.11}\\
$B_3$ , $b = 20$ & \textbf{42.51 }& 43.56 & 33.39 & 44.31 & 46.42 & 34.10\\
$B_3$ , $b = 100$ & 42.26 & \textbf{43.60} & 33.48 & \textbf{44.52}& \textbf{46.45} & 33.99\\
\hline
\end{tabular}
\label{convs2s-results}
\end{adjustbox}
\end{center}
\end{table}
We did not observe over-fitting with the large number of parameters from the results in Table \ref{convs2s-results}, that suggests the training data is fairly good and the model's dropout probability is suitable. With the hypothesize that the models' beam size and length penalty parameters are independence, we found that all the model's BLEU scores are improved a lots when beam size is increased from $b$ = 1 to $b$ = 10 and is only improved by a small margin (or even worst) when we keep increasing the beam size further. Because the decoding speed would slow down when we increase the beam size, we can conclude that the beam size of the ConvS2S models should set to 10.
\begin{figure}[h!]
\begin{subfigure}{.5\textwidth}
\pgfplotsset{width=0.94\textwidth,compat=1.14}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin=1,
xmax=101,
xlabel= beam size ($b$),
ylabel= decoding speed]
\addplot[color=black, mark=*] coordinates {
(1 , 2660)
(2 , 3066)
(5 , 2981)
(10 , 2420)
(20 , 1890)
(100 , 554)
};
\addlegendentry{$B_{base}$}
\addplot[color=green, mark=square*] coordinates {
(1 , 1468)
(2 , 1400)
(5 , 1320)
(10 , 1020)
(20 , 640)
(100 , 190)
};
\addlegendentry{$B_1$}
\addplot[color=blue, mark=triangle*] coordinates {
(1 , 1260)
(2 , 1180)
(5 , 960)
(10 , 690)
(20 , 430)
(100 , 109)
};
\addlegendentry{$B_2$}
\addplot[color=red, mark=diamond*] coordinates {
(1 , 1070)
(2 , 890)
(5 , 680)
(10 , 490)
(20 , 300)
(100 , 60)
};
\addlegendentry{$B_3$}
\end{axis}
\end{tikzpicture}
\caption{}
\end{subfigure}~
\begin{subfigure}{.5\textwidth}
\pgfplotsset{width=0.94\textwidth,compat=1.14}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin=1,
xmax=101,
xlabel= beam size ($b$),
ylabel= decoding speed]
\addplot[color=black, mark=*] coordinates {
(1 , 3500)
(2 , 1170)
(5 , 840)
(10 , 410)
(20 , 124)
(100 , 35)
};
\addlegendentry{$C_{base}$}
\addplot[color=green, mark=square*] coordinates {
(1 , 890)
(2 , 513)
(5 , 224)
(10 , 116)
(20 , 56)
(100 , 12)
};
\addlegendentry{$C_1$}
\addplot[color=blue, mark=triangle*] coordinates {
(1 , 260)
(2 , 137)
(5 , 85)
(10 , 29)
(20 , 13)
(100 , 5)
};
\addlegendentry{$C_2$}
\end{axis}
\end{tikzpicture}
\caption{}
\end{subfigure}
\caption{\small The impact of beam size to the decoding speed of ConvS2S models (a) and Transformer models (b). The decoding speed of ConvS2S models is often higher and decrease slower when increase the beamsize than the decoding speed of Transformer models with same model size.}
\end{figure}
\subsection{Transformer}
In the Transformer architecture there are many hyper-parameters to be configured such as the number of layers in the encoder and the decoder, the number of attention heads or the size of the FFN weight matrix etc. In this work, we introduce three models based on their number of parameters. Each model's hyper-parameters are shown Table \ref{tf-hp}:
\begin{itemize}
\item We used the Adam optimizer with $\beta_1 = 0.9$, $\beta_2 = 0.98$ and $\epsilon = 10^{-9}$. The learning rate is varied over the training processes according to the following formula: $lr = d_{model}^{-0.5} \cdot min(step_{num} ^ {-0.5}, step_{num} ^ {-0.5} \cdot warmup_{steps}^ {-1.5})$, where \texttt{warmup\_steps} is set to 4000 from $C_{base}$ and \texttt{warmup\_steps} is set to 16000 for the rests.
\item In the course of training we found that if \texttt{batch\_size} is too big, the model can sometimes run out of GPU memory after a long training time. Therefore while $C_2$ can be trained with a \texttt{batch\_size} of 3000 and the rest can be trained with a \texttt{batch\_size} larger than 6500, we recommend a batch size of 2048 for the $C_2$ and 4096 for the rests in order to keep the training stable. Another reason is our observation that the time to convergence does not change significantly once the batch size gets sufficiently large.
\item The learning rate are chosen based on \texttt{batch\_size}. Specifically, we set the learning rate to 0.0001 for the largest model with beam size of 2048 and scale it by $\sqrt{k}$ when multiplying the \texttt{batch\_size} by $k$.
\item We also observed that when the \texttt{batch\_size} is too small (i.e. $< 512$ for the biggest model), the model can only converge when the learning rate is smaller than 0.00005. Even then the model's BLEU is much lower than with a large \texttt{batch\_size}. This is due to the fact that the \textit{gradient noise scale} is proportional to the learning rate divided by the batch size. Thus, lowering the batch size increases the noise scale \cite{Smith:2017}. Therefore, we would rather reduce the model's complexity than reduce the batch size.
\item The performance of the $C_2$ model kept improving epoch by epoch and could potentially be better than reported. However due to resource constraints we report the model's performance at the $64^{th}$ checkpoint.
\end{itemize}
\begin{table}[h!]
\caption{Transformer hyper-parameters}
\label{tf-hp}
\begin{center}
\begin{tabular}{l|ccccp{0.8cm}|>{\centering\arraybackslash}m{1.5cm}>{\centering\arraybackslash}m{1.5cm}}
& $N$ & $d_{model}$ & $h$ & $d_{ff}$ & $p_{drop}$ & training time (hours) & params $\times 10^6$\\
\hline
$C_{base}$ & 2 & 256 & 4 & 1024 & 0.1 & 5 & 9M \\
\hline
$C_1$ & 6 & 512 & 16 & 2048 & 0.15 & 36 & 54M \\
\hline
$C_2$ & 8 & 1024 & 16 & 4096 & 0.15 & 72 & 197M\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\caption{BLEU score of English-Vietnamese translation task on tst2012, tst2013, tst2015 of $C_{base}$, $C_1$ and $C_2$ model, $b$ is the beam size with a default length penalty function}
\label{tf-results}
\begin{center}
\begin{tabular}{|l|ccc|ccc|}
\hline
model & \multicolumn{3}{c|}{English-Vietnamese} & \multicolumn{3}{c|}{Vietnamese-English}\\
\hline
$C_{base}$, $b = 1$ & 31.88 & 33.70 & 31.16 & 27.71 & 29.32 & 25.35\\
$C_{base}$, $b = 2$ & 31.99 & 34.44 & 31.54 & 28.63 & 30.02 & 25.94\\
$C_{base}$, $b = 5$ & \textbf{31.19} & \textbf{34.61} & \textbf{32.06} & \textbf{28.86} & \textbf{30.48} & 26.17\\
$C_{base}$, $b = 10$ & 31.93 & 34.44 & 31.79 & 28.74 & 30.01 & \textbf{26.28}\\
$C_{base}$, $b = 20$ & 31.86 & 34.18 & 30.79 & 28.55 & 30.95 & 26.14\\
$C_{base}$, $b = 100$ & 30.96 & 33.85 & 30.21 & 25.11 & 27.42 & 26.09\\
\hline
$C_1$, $b = 1$ & 36.85 & 39.88 & 33.62 & 33.31 & 35.71 & 29.58\\
$C_1$, $b = 2$ & 37.46 & \textbf{40.99} & 30.88 & 33.27 & 35.77 & 30.28\\
$C_1$, $b = 5$ & 37.61 & 40.88 & 33.48 & \textbf{33.32} & \textbf{36.06} & 30.33\\
$C_1$, $b = 10$ & 37.51 & 40.81 & 33.59 & 33.28 & 35.88 & \textbf{30.36}\\
$C_1$, $b = 20$ & \textbf{37.65} & 40.66 & \textbf{33.73} & 33.18 & 35.83 & 30.30\\
$C_1$, $b = 100$ & 37.43 & 40.02 & 33.31 & 32.62 & 34.99 & 30.17\\
\hline
$C_2$, $b = 1$ & 52.37 & 54.70 & 38.01 & 41.61 & 44.31 & 33.19\\
$C_2$, $b = 2$ & 52.89 & 55.31 & 38.81 & 43.16 & 45.41 & 33.83\\
$C_2$, $b = 5$ & 53.32 & \textbf{55.89} & \textbf{39.14} & \textbf{43.41} & 46.26& 33.94\\
$C_2$, $b = 10$ & \textbf{53.64} & 55.85 & 39.01 & 43.32 & 46.27 & 34.05\\
$C_2$, $b = 20$ & 53.50 & 55.76 & 39.05 & 43.10 & \textbf{46.49} & 34.20\\
$C_2$, $b = 100$ & 53.36 & 55.31 & 38.92 & 42.86 & 45.71 & \textbf{34.24}\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Length normalization for beam search}
\begin{figure}[h!]
\pgfplotsset{width=7.5cm,compat=1.14}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin=0,
xmax=3.1,
xlabel=$\alpha$,
ylabel=BLEU score]
\addplot[color=green, mark=*] coordinates {
(0.1 , 36.21)
(0.2 , 36.32)
(0.3 , 36.41)
(0.4 , 36.49)
(0.5 , 36.64)
(0.6 , 36.69)
(0.7 , 36.78)
(0.8 , 36.82)
(0.9 , 36.94)
(1.0 , 37.07)
(1.1 , 37.11)
(1.2 , 37.26)
(1.3 , 37.28)
(1.4 , 37.36)
(1.5 , 37.39)
(1.6 , 37.36)
(1.7 , 37.37)
(1.8 , 37.45)
(1.9 , 37.45)
(2.0 , 37.46)
(2.1 , 37.42)
(2.2 , 37.45)
(2.3 , 37.46)
(2.4 , 37.47)
(2.5 , 37.47)
(2.6 , 37.51)
(2.7 , 37.52)
(2.8 , 37.51)
(2.9 , 37.54)
(3.0 , 37.55)
};
\addlegendentry{$f_1$}
\addplot[color=red, mark=square*] coordinates {
(0.1 , 36.21)
(0.2 , 36.3)
(0.3 , 36.38)
(0.4 , 36.59)
(0.5 , 36.51)
(0.6 , 36.76)
(0.7 , 36.84)
(0.8 , 37.03)
(0.9 , 37.09)
(1.0 , 37.23)
(1.1 , 37.30)
(1.2 , 37.30)
(1.3 , 37.31)
(1.4 , 37.33)
(1.5 , 37.30)
(1.6 , 37.41)
(1.7 , 37.50)
(1.8 , 37.52)
(1.9 , 37.52)
(2.0 , 37.50)
(2.1 , 37.49)
(2.2 , 37.48)
(2.3 , 37.53)
(2.4 , 37.52)
(2.5 , 37.51)
(2.6 , 37.48)
(2.7 , 37.42)
(2.8 , 37.41)
(2.9 , 37.38)
(3.0 , 37.25)
};
\addlegendentry{$f_2$}
\end{axis}
\end{tikzpicture}
\caption{The effect of length penalty factor $\alpha$ on BLEU of $B_2$ on tst2012 with beam size fixed to 10.}
\label{lenpen}
\end{figure}
Beam search is a widespread technique in NMT, which finds the target sequence that maximizes some scoring function by a tree search. In the simplest case the score is the log probability of the target sequence. This simple scoring favors shorter sequences over longer ones on average since a negative log-probability is added at each decoding step.
Recently, length normalization \cite{Wu:2016} have been shown to improve decoding results for RNN based models. However, there is no guarantee that this strategy works well for other models. In this work, we experiment on two normalization functions described below:
\begin{align}
f_1 &= \frac{(5+|Y|)^\alpha}{6^\alpha} \\
f_2 &= (1+|Y|)^\alpha
\end{align}
We found that the length penalty can help improve the model's performance up to 2 in BLEU scale. The length penalty should be chosen between 2.0 to 3.0.
\subsection{Ensembling}
Ensemble methods combine multiple individual methods to create a learning algorithm that is better than any of its individual parts \cite{Dietterich:2000}. They are widely used to boost machine learning models' performance \cite{Krogh:1994, Dietterich:2000}. In neural machine translation, the most popular ensemble method is checkpoint ensemble., in which the ensembled models are created by combining (averaging) multiple model checkpoints together \cite{Vaswani:2017,Wu:2016}. This method does not require training multiples model and the ensembled model has the size as same as the constituent models.
In many experiments the authors suggest to average checkpoints based on training time \cite{Vaswani:2017,1:Sutskever:2014:SSL:2969033.2969173}, which depends on hardware and hard to reproduce. In this work we experiment with checkpoint ensembling based on training epoch, which can be easily adapted to different platforms.
\begin{table}[h!]
\caption{Effect of checkpoint ensembling ($n$ is number of checkpoint to be averaged) on the $B_2$ (a) and $C_1$ (b) model for English-Vietnamese translation task on tst2015}
\begin{center}
\begin{subfigure}{0.5\textwidth}
\caption{}
\begin{tabular}{c|cccc}
$n$ & \multicolumn{4}{c}{interval (\% of a epoch)}\\
& 1.5 & 3 & 4.5 & 6 \\
\hline
8 & 32.26 & 32.74 & \textbf{32.82} & 32.68 \\
16 & 31.58 & 31.75 & 31.55 & 31.68\\
\end{tabular}
\end{subfigure}~
\begin{subfigure}{0.5\textwidth}
\caption{}
\begin{tabular}{c|cccc}
$n$ & \multicolumn{4}{c}{interval (\% of a epoch)}\\
& 1.5 & 3 & 4.5 & 6 \\
\hline
8 & 30.63 & \textbf{30.90} & 30.77 & 30.81 \\
16 & 30.61 & 30.80 & 30.85 & 30.89 \\
\end{tabular}
\end{subfigure}
\end{center}
\end{table}
According to our experiments checkpoint ensembling always improves the model's performance. ConvS2S benefits the most (up to 7\% on the $B_2$ model) while ensembling has a smaller effect on the Transformer models (at most 5\% on the $C_3$ model) . We observed that taking 8 checkpoints for ensembling often yielded better results than 16 checkpoints. This also has the advantage that less time is spent on checkpoint saving.
Ensembling can also applied by training several new models starting form the same checkpoint . Each model is trained at a random position in the training data. In this setup, these models are semi-independent because they are rooted in the same source checkpoint. These semi-independent models can be averaged as described above, resulting a boost in the result, but in a smaller margin.
\section{Result and Empirical studies}
From the above experiments we observed that the training data is well correlated with the test data and training does not suffer from overfitting. However this makes it hard to tell if the model is general enough.
For new experiments we can always choose RNNs as a reliable base line model, that does not take much effort to achieve good results. The Transformer model has the highest converge speed while RNNs have the lowest converge speed. The Transformer model has showed its superiority in terms of achieving state-of-the-art results when given a suitable batch size and learning rate. Interestingly, even a very simple Transformer model with only 5 training hours can achieve a comparable score.
In our experiments we showed that a well-tuned beam search with length penalty is crucial, which can help the boost the model's score by 1.5 to 3 BLEU point (Table \ref{convs2s-results}, \ref{tf-results}, Figure \ref{lenpen} ). The most effective beam-size is 10 for ConvS2S and 5 for Transformer. The length penalty has a high impact on the final result, which should be set from 2 to 3.
Finally from our experiment results we compared our best performing hyper-parameter sets across all models and combined to a final model with the state-of-the-art results (Table \ref{combined-model} ). This shows that careful hyper-parameter tuning can greatly improve performance.
\begin{table}[h!]
\caption{\small Hyper-parameter settings for our final combined model for bidirectional English-Vietnamese translation}
\label{combined-model}
\begin{center}
\begin{tabular}{|r|l|}
\hline
Hyper-parameters & Value \\
\hline
$N$ & 8 \\
$d_{model}$ & 1024 \\
$h$ & 16 \\
$d_{ff}$ & 4096 \\
$p_{drop}$ & 0.15 \\
\texttt{batch\_size} & 2048 \\
\texttt{length\_penalty} & 1.5 \\
\texttt{beam\_size} & 5 \\
checkpoint to ensemble & 8 \\
ensemble interval & 3\% epoch \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h!]
\caption{\small BLEU results for our final combined model for bidirectional English-Vietnamese translation}
\label{combined-model}
\begin{center}
\begin{tabular}{c|ccc|ccc|}
& \multicolumn{3}{c|}{English-Vietnamese} & \multicolumn{3}{c|}{Vietnamese-English} \\
& tst2012 & tst2013 & tst2015 & tst2012 & tst2013 & tst2015\\
\hline
Combined model & 55.04 & 56.88 & 40.01 & 46.36 & 49.23 & 35.81 \\
\hline
$C_2$ & 52.37 & 54.70 & 38.01 & 42.63 & 44.17 & 32.68\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}
\label{sec:conclusions}
We conducted a broad range of experiments with the RNN sequence-to-sequence, ConvS2S and Transformer models for English-Vietnamese and Vietnamese-English translation, pointing out key factors to achieving state-of-the-art results. In particular we performed extensive exploration of hyper-parameters settings, which can be useful for other research works. In sum, our experiments took about 2,000 GPU hours.
We highlighted several important points: efficient use of batch size, the importance of beam search and length penalty, the importance of initial learning rate, the effectiveness of checkpoint ensembling, and the model's complexity. Along with these contribution we also make our dataset publicly available at (location withheld for review).
We hope our findings can help accelerate the pace of research on and application of English-Vietnamese and Vietnamese-English translation.
\section{Acknowledgement}
This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2017-PC-078.
{\small
\bibliographystyle{IEEEtranS}
|
2,877,628,090,779 | arxiv | \section{Introduction}
Many years ago, superconductivity was discovered in heavy-fermion compounds \cite{steglich, ott, stewart}. It was suggested \cite{cmv1} that the superconductivity was due to collective electronic fluctuations and not due to electron-phonon interactions. Transport properties in the superconducting state were analyzed \cite{ssr, woelfle} to show that superconductivity was in the D-wave symmetry. It was also suggested that the
D-wave symmetry is promoted by Antiferromagnetic fluctuations \cite{miyake} with long enough correlation lengths. This promotes scattering of fermions near the fermi-surface predominantly through angles around $\pm \pi/2$, which is essential for superconducting instability in the "D-wave" channel for a suitable fermi-surface \cite{cmv-review}. The idea of long enough AFM correlation lengths as essential for this mechanism is supported by the fact that in heavy-fermions, superconductivity occurs generally in the regime near the AFM quantum critical point where the correlation lengths are long but the competing AFM phase has lower condensation energy.
At the same time, Random phase approximation on the Hubbard model was used to calculate the spin-fluctuation spectra and to suggest that D-wave superconductivity is promoted by such fluctuations \cite{scalapino}. The properties of the Hubbard model have proven controversial in more elaborate calculations; there are calculations which suggest that the ratio of the transition temperature $T_{\rm c}$ to the typical electronic kinetic energy parameters $t$ is more than $O(10^{-2})$ \cite{DMFT} to less than $O(10^{-3})$ \cite{imada}. Since heavy-fermion properties require Kondo effect of the f-orbital local moments and their magnetic interactions using the wide-band electrons, a multi-orbital model is obviously required \cite {varma-yafet}. The Hubbard model was proposed as a sufficient model for the cuprate compounds \cite{anderson}. But the discovery in under-doped cuprates \cite{bourges} of the predicted time-reversal breaking order parameter \cite{cmv2} on the basis of a multi-orbital model raises doubts on the validity of the Hubbard model for the cuprates. For pnictides, generalization of the Hubbard model to multi-orbital situations and inclusions of Hund's rule couplings appears essential.
We have a more modest goal in this paper than calculating spin-fluctuations from microscopic theory and using it to calculate properties of the superconductor. In recent years inelastic neutron scattering in the heavy fermion compounds CeCu$_2$Si$_2$ \cite{stockert1, stockert2, arndt} and CeIrIn$_5$ \cite{kambe} have provided details of the AFM fluctuation spectrum in the normal state. The primary aim of this paper is to estimate the superconducting transition temperature using the parameters provided by the experiments in these compounds. To do so, we solve the Eliashberg equations for d-wave superconductivity using a phenomenological AFM spectral function with which the experimental data is in good accord. The use of the Eliashberg equations for quantitative calculations may be open to question because the Migdal expansion parameter, which is of $O(10^{-2})$ for the electron-phonon problem is of $O(1)$ for such compounds if one assumes that the scale of the AFM fluctuations extends to the order of the electronic bandwidth. However, when the AFM correlation length $\xi$ is large compared to the lattice constant $a$ or $(2k_{\rm F})^{-1}$, the scale of the AFM fluctuations is reduced correspondingly to $O((a/\xi)^2) t$. But in the limit of large correlation lengths, new questions arise \cite{cmv-review} which are not important in the electron-phonon problem. The most prominent among them are the role of inelastic scattering in depressing $T_{\rm c}$ on the one hand \cite{MSV}, and the fact that the BCS type coupling constant $\lambda$ appears to diverge when the characteristic fluctuation frequency $\to 0$ and the BCS prefactor appears to go to $0$. An answer to these questions and various considerations which determine $T_{\rm c}$ from AFM interactions is possible from the numerical solution of the Eliashberg equations.
We find that it is reasonable to conclude from a comparison of the calculated $T_{\rm c}$ with experiments that AFM fluctuations are responsible for D-wave superconductivity in the heavy fermion compounds. Very importantly, with similar parameters we calculate the {\it measured} coefficient of the anomalous $\propto T^{3/2}$ contribution to the resistivity in these compounds. A claim to quantitative accuracy on both these quantities can however be made only to factors of $O(2)$.
We note here that if one adopts that the dimensionless measure $T_{\rm c}/E_{\rm F}$ for how high is the electronic fluctuation induced superconducting, the heavy fermions may be said to do very well indeed. For example, in many cases, including the compounds studies here, this ratio is $O(10^{-2})$, similar to that of the cuprates.
Following the proposals that AFM fluctuations may also promote superconductivity in the cuprate compounds \cite{various}, there have been many discussions of the mechanism and many calculations based on the Eliashberg equations. A partial list includes the following \cite{references}. The most complete of these calculations appear to us to be those carried out by Monthoux and Lonzarich (ML) \cite{ml1,ml2}, both for 2 and 3 dimensional models. We present below calculations for the 2 dimensional square lattice model with a phenomenological spin-fluctuation spectrum, whose results are no different from those of ML for the range of parameters examined that are common. A difference in the calculations is that we vary the parameters in the two dimensional model so that "nesting" at the AFM wave-vector quantitatively changes. The amount of nesting does have a significant effect on the results. More important is that now that the AFM fluctuation spectra is available, we can use the experimental parameters to test the ideas quantitatively. We also discuss how to put limits on the parameters used based on sum-rule for the fluctuation spectra and show that they are inter-related. Results for the range of physical parameters that we find relevant for the heavy fermions is not available in the published results of ML. This has bearing also on general conditions to determine the extent to which AFM fluctuations give significant $T_{\rm c}$ for relevant parameters in other compounds.
This paper is organized as follows: We present in Sec. (II) the models for fermi-surface and for the spin-fluctuations which we have investigated using the linearized Eliashberg equations. We discuss there the change of effective coupling constants with the AFM correlation length using sum-rules so that the results for numerical solutions of the Eliashberg equations presented later are presaged. We present the results of the calculations in Sec. (III) and discuss the important conclusions immediately after the description of the Models. We also present, in an Appendix, the explicit derivation of the coefficient of the $T^{3/2}$ resistivity from the measured form of the AFM critical fluctuation spectra. This is used in the text to estimate independently the value of a coefficient $\lambda$, which is important for the calculation of $T_{\rm c}$. We give the parameters that have been deduced by inelastic neutron scattering for the heavy fermion compounds CeCu$_2$Si$_2$ \cite{stockert1, stockert2, arndt} and CeIrIn$_5$ \cite{kambe} and compare the measured $T_{\rm c}$ with the calculations. We should emphasize that such a comparison is meant to be only illustrative of the physical principles involved; no detailed quantitative agreement is to be expected, especially given that the electronic structure of these compounds is far more complicated than assumed in the models studied. However, enough details can be provided so that one can conclude that the idea of AFM fluctuations near the quantum critical point in these compounds as the source of D-wave superconductivity is well supported. For example using measured properties, different levels of assumed nesting in the band-structure need a coupling constant $\lambda$ between 1.5 and 3 to get the measured $T_{\rm c}$. In this range of $\lambda$ and for the measured AFM correlation length $T_{\rm c}$ is close to being linear in $\lambda$. This range of values is compared with the value of $\lambda \approx 1.6$ needed to get the measured coefficient of the $T^{3/2}$ resistivity, which is relatively insensitive to nesting. One can assess the results from the fact that in the range of $\lambda$ deduced, $T_c$ is found to be approximately linear in $\lambda$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.45\textwidth]{FS.eps}
\end{center}
\caption{The fermi-surfaces given by the tight binding spectrum with four values of the next nearest hopping $t'$ with filling=1.05 and t=0.3meV. }
\label{fig:01}
\end{figure}
\section{Models and Results for $T_{\rm c}$}
\subsection{Fermi-surface}
In our calculations we will consider two types of fermi-surfaces, a free electron fermi-surface and the others given by the tight binding spectrum in a two dimensional square lattice with nearest neighbor and next nearest neighbor hopping $t$ and $t'$ respectively:
\begin{equation}
\varepsilon_{\vec{k}} = -2t( \cos k_x a + \cos k_y a) + 4 t^{'} \cos(k_x a) \cos(k_y a)
\label{dis}
\end{equation}
The fermi-surface with the tight binding spectrum are shown in Fig. \ref{fig:01} for four values of the next nearest hopping $t'/t$ and the AFM wave-vector.
The nesting in the model changes as $t'$ increases.
We will show detailed result for three fermi-surfaces, the free electron fermi-surface, the fermi surface (FS1) with tight binding spectrum with $t'=0.4t$ and the fermi surface (FS2) with tight binding spectrum with $t'=0.1t$. Of the four Fermi-surfaces shown in Fig. \ref{fig:01}, the one with $t'=0.4t$ has the worst nesting and the one with $t'=0.1t$ has the best nesting.
Fig. \ref{fig:02} shows the circular fermi-surface, FS1, FS2 and the corresponding AFM wave-vectors.
We will discuss using the results of ML together with ours, that if properly normalized density of states and fluctuation spectra are used, two dimensional and three dimensional models give similar results for $T_{\rm c}$ provided one adjusts the ratio of the region of fermi-surface nesting to the total fermi-surface. This is in general is always lower in three than in two dimensions. It is also important to note, as discovered long ago \cite{mcmillan}, \cite{Allen-Dynes} for the case of s-wave superconductors that $T_c$ is a rather gross quantity which depends to a very good approximation on the average density of states near the chemical potential only and not on details such as the number of fermi-surface sheets and shapes. For d-wave superconductors, we we show below, it is important to also include effects of nesting of the fermi-surface near the AFM wave-vectors.
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.45\textwidth]{fs3.eps}
\end{center}
\caption{Three types of Fermi surfaces and $\vec{Q}$-vector.
The blue line shows the circular fermi-surface, the black line shows FS1 which is given by the tight binding spectrum in two dimensional square lattice with filling=1.05 and $t'=0.40t$.
The red line shows FS2 which is also given by the tight binding spectrum in two dimensional square lattice but with filling=1.05, and $t'=0.10t$.}
\label{fig:02}
\end{figure}
\subsection{Dynamical Spin Susceptibility for AFM Fluctuations, Correlation lengths, \\ Partial Sum rules and Coupling Constants}
The dynamical spin-susceptibility for itinerant fermions with AFM correlations may be usefully divided into two parts: The normal contribution of non-interacting fermions $\chi_0(\vec{q},i\omega)$ and the part $\chi_{\rm AFM}(\vec{q},i\omega)$ affected by AFM correlations. The two together obey the total magnetic moment sum-rule. The non-interacting susceptibility can play only an insignificant role in promoting superconductivity \cite{cmv-review} and should be ignored. The two contributions to the susceptibility can be distinguished by their momentum dependence. The characteristic momentum dependence of the non-interacting spin fluctuations is on a scale of $O(2k_f^{-1})$ while that of the AFM spin-fluctuations is much shorter. A partial sum-rule on $\chi_{\rm AFM}(\vec{q},i\omega)$ in terms of the ordered moment in the AFM phase can be used to relate the integrated fluctuations to the AFM correlation length. Double counting by using the sum-rule on the total susceptibility for the fluctuations and yet having (free) fermions interacting with such spin-fluctuations is incorrect as it over-counts the total degrees of freedom. Such considerations have usually been blithely ignored in most of the previous phenomenological work on this problem.
The fermions interact with spin-fluctuations with a phenomenological Action
\begin{eqnarray}
S_{\rm int} = g^2\sum_{{\bf q}, {\bf k} {\bf k'}, i, \alpha, \beta, \gamma, \delta} \sum_{\omega_n} \chi( \vec{q}, i \omega_n) \psi_{{\bf k'-q}, \gamma}^+ {\bf \sigma}_{\gamma, \delta}^i \psi_{{\bf k'}, \delta}\psi_{{\bf k+q}, \alpha}^+ {\bf \sigma}_{\alpha, \beta}^i \psi_{{\bf k}, \beta} + H.C.
\end{eqnarray}
$\chi$ will be chosen to have dimensions of inverse of energy (after subsuming a factor of $4\mu_B^2$ in its definition). So $g$ is a coupling function of dimension of energy. $g$ for heavy fermions is the exchange energy between the conduction electrons and the $f$- local moments. Its meaning for d-band problems is more ambiguous, and may be best inferred from independent experiments, for example the resistivity above $T_c$.
A suitable phenomenological form for the dynamical spin-fluctuations due to AFM correlations, with which experimental results \cite{stockert1, stockert2} can be fitted, is
\begin{eqnarray}
\chi( \vec{q}, i \omega) &=& \frac{\bar{\chi_0} \Gamma_{\rm AFM} }{ \psi_{\vec{q}} + \vert \omega \vert },\\
\psi_{\vec{q}} &\equiv& \Gamma_{\rm AFM} \left[ (\xi/a)^{-2} + a^2\left( \vec{q} - \vec{Q} \right)^2 \right].
\label{chi}
\end{eqnarray}
where $\Gamma_{\rm AFM}$ is the damping rate of the fluctuations, $Q$ is the antiferromagnetic vector.
The correlation length $\xi$ is related to the deviation from the Quantum Critical Point (QCP) by variation in pressure, doping, magnetic field, etc. as well as by temperature. $\Gamma_{\rm AFM}$, ${\bf Q}$ and $\xi$ may all be determined from experiments. The temperature dependence of $\xi$ has been studied by renormalization group (RG) \cite{hertz,millis} and by the self-consistent renormalization (SCR) methods \cite{moriya1, moriya2}.
$\xi^{-2} \propto T^{3/2}$ near AFM QCP in the 3 dimensions and dynamical critical exponent $=2$.
SCR derives using the same dynamical critical exponent that near the magnetic QCP, $\xi^{-2}(T) \sim \chi_{\vec{Q}} \propto T + \theta$.
\par
In our calculations in two dimensions $\xi$ will be assumed to be of the form
\begin{eqnarray}
\label{corr2}
(\xi/a)^{-2}(T) &=& (\xi^{*}/a)^{-2} + \gamma \frac{T}{\Gamma_{\rm AFM}},
\label{xi}
\end{eqnarray}
where $\xi^{*}$ is the asymptotic $T=0$ value of the correlation length. One of our results is that the temperature dependence of $\xi$ is of insignificant consequence in determining $T_{\rm c}$.
The linearized Elisahberg equations give that the kernel for Cooper pair coupling in the d-wave channel in a square lattice is proportional to the projection of
$|g({\bf k,k}')|^2 \chi({\bf k-k}', \omega)$
to $(\cos k_x-\cos k_y) (\cos k_x'-\cos k_y')$. In spin-fluctuations theories, $|g({\bf k,k}')|^2$ has only a smooth momentum dependence. So, Cooper-pair coupling prominently depends depends only on (i) the momentum dependence of $\chi({\bf k-k}', \omega)$ determined by the correlation length $\xi$ (ii) the integrated weight in the momentum dependent part and (iii) the energy scale of the momentum dependent fluctuations. The first is qualitatively obvious from the fact that a q-independent spin - fluctuation contributes zero to the Cooper channel in the d-wave channel. It is not possible to make quantitative statements on these effects without detailed calculations because the results also depend on the nesting in the band-structure near the AFM ${\bf Q}$. We will show that the three ingredients in $\chi({\bf k-k}', \omega)$ are not mutually independent.
To gain physical insight, the effect of $\xi$ on the integrated spectral weight may be discussed before detailed calculations through the the partial sum-rule on $\chi_{\rm AFM}(\vec{Q}, \omega)$, which determines the effective coupling constant for superconductivity:
\begin{align}
\notag \sum_i \langle S_i^2 \rangle_{\rm AFM} &= \frac{1}{\pi} \sum_{\vec{q}} \int_0^{\omega_c} {\rm d} \omega {\rm Im} \chi ( \vec{q}, \omega)\\
&= \frac{\omega_c }{\pi^2} \bar{\chi^0} \left[ \frac{\pi}{2} - {\rm Tan}^{-1}\left( \frac{\Gamma_{\rm AFM}(\xi/a)^{-2} }{ \omega_c} \right) - \frac{1}{2} \frac{\Gamma_{\rm AFM}( \xi/a)^{-2} }{\omega_c} \log \left[ 1 + \left( \frac{\Gamma (\xi/a)^{-2} }{\omega_c} \right)^{-2} \right] \right].
\label{tmsr}
\end{align}
With the assumed
Lorentzian form, it is necessary to introduce an upper cut-off $\omega_c$ in the frequencies $\omega$ up to which the fluctuations extend. Actually, spin fluctuation are actually quite suppressed for $\omega \sim \Gamma_{\rm AFM} $ and we can simply use $\omega_c \approx \Gamma_{\rm AFM}$ in calculations of Eliashberg equations. It is important to take into account that there are four equivalent AFM-vector for the two dimensional problem in the paramagnetic regime of the model, however strongly fluctuating it may be.
This has been taken into account in the sum-rule by multiplying the measured Im$\chi(\vec{q}, \omega)$ by 4. For $d=3$, the number of equivalent AFM vectors is larger and a correspondingly larger multiplicative factor should be used.
In the regime of very long correlation lengths, $(\xi/a)^{2} \gg 1$, i.e. close to the quantum-critical point, the sum rule simply gives
\begin{equation}
\sum_i \langle S_i^2 \rangle_{\rm AFM} \approx \frac{\omega_c }{2\pi} \bar{\chi^0} + O(a/\xi)^2
\end{equation}
$\langle S_i^2 \rangle_{\rm AFM}$ may to a first approximation be estimated from the ordered moment $\langle S \rangle $ in nearby AFM phase but more properly from integration of the relevant momentum and frequency range of the measured fluctuations in absolute units using polarized neutrons.
Fig. \ref{fig:00} shows the $(\xi/a)^{-2}$-dependence of $\displaystyle \sum_i \langle S_i^2 \rangle_{\rm AFM}/ \omega_c (\bar{\chi}_0/2 \pi)$ for $\Gamma_{\rm AFM}/\omega_c$=1.0, 2.0 and 10.0.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width = 0.60\textwidth]{srule.eps}
\caption{The dependence of the quantity $\displaystyle \sum_i \langle S_i^2 \rangle_{\rm AFM}/ \omega_c (\bar{\chi}_0/2 \pi)$ which is shown in the text to be approximately proportional to the effective coupling constant $\lambda_{eff}$ on the correlation length $\xi/a$ is exhibited for various values of $\Gamma_{\rm AFM}/\omega_c$ shown.}
\label{fig:00}
\end{center}
\end{figure}
Let us now consider the sum-rule in the opposite limit, that the correlation length is small compared to the lattice constant, i.e. the system is very far from the quantum critical point. Then
\begin{equation}
\sum_i \langle S_i^2 \rangle_{\rm AFM} \approx \frac{\omega_c }{2\pi} \bar{\chi^0} \frac{1}{2}[ (\xi/a)^2+ O(\xi/a)^4]
\end{equation}
As already shown by ML and further elaborated below, for a given band-structure, the results of the Eliashberg calculations for $T_{\rm c}/\Gamma_{\rm AFM}$ may be parametrized in terms of a dimensionless "bare" coupling constant $\lambda$ and a correlation length $\xi$,
\begin{equation}
\label{lambda}
\lambda = g^2 N_{\rm F} \bar{\chi}_0
\end{equation}
$\bar{\chi}_0$ may be determined in terms of $\langle S_i^2 \rangle_{\rm AFM}$ and therefore (approximately) to the ordered moment through the sum-rule. We may define an effective coupling constant $\lambda_{\rm eff}$ to incorporate the effect of the correlation length. Using that the sum-rule becomes the total moment sum-rule in the limit of infinite correlation length and the maximum possible ordered moment, i.e. that of the AFM insulator (ignoring the zero-point effects), one concludes that in the limit of very large correlation lengths
\begin{equation}
\label{lambdainfty}
\lambda_{\rm eff} \to \lambda_{\infty} = g^2 N_{\rm F} f^2\langle S_i ^2\rangle_{\rm max} \frac{2\pi}{\omega_c},
\end{equation}
where $f$ is the fraction of the maximum possible ordered moment. From Fig. (\ref{fig:00} and from the detailed calculations presented in the next section, one deduces that the limit for $\lambda_{\infty}$ is reached for $\xi/a \gtrsim 10$, below which there is an exponential fall off of $T_{\rm c}/\Gamma_{\rm AFM}$.
For smaller correlation-lengths, Fig. (\ref{fig:00}) shows that the $\lambda_{\rm eff}$ for $T_{\rm c}$ decreases with decreasing correlation length.
In the work of ML, $\lambda$ values from about 5 to about 50 are used in the calculations with varying correlation lengths. Actually, one obtains for the considerations of the sum-rules above that for spin-(1/2) problems, even the coupling constant $\lambda_{\infty}$ is only of $O(1)$, because $g N_{\rm F}$ is of $O(1)$ and so is the upper limit on the ratio $\Gamma_{\rm AFM}/\omega_c$.
An independent estimate of $\lambda_{\rm eff}$ may be obtained from the normal state properties, for example the coefficient of the temperature dependence of the resistivity of non-fermi-liquid form in the quantum critical region. Again only $\lambda$ of $O(1)$ will be found consistent.
It is also important to note that these BCS type coupling constants do not carry information on the retardation effects due to the frequency dependence of the interaction; these as well as the effects of inelastic scattering which are particularly important for anisotropic superconductors are properly treated through the numerical solution of the Eliashberg equations. The difference from electron-phonon induced superconductivity where a single parameter $\lambda$ need by introduced \cite{mcmillan} should also be noted.
It should also be pointed out that in some heavy fermions, the quantum-critical fluctuations do not have the functional form given by the simple RG or SCR approximations as above, but displays "local criticality" \cite{lqcp} as suggested for the cuprates \cite{mfl}. In this paper, we only consider fluctuations which are well specified by the form given above.
\section{Resistivity in the Quantum-critical Region}
The temperature dependence of the resistivity near the quantum-critical points has been derived several times \cite{Rosch}. Here, we rederive it paying special attention to the coefficient in front of the anomalous temperature dependent part. An expression of the resistivity $\rho(T)$ in the antiferromagnetic quantum critical region suitable for heavy fermions may be derived with the following formula derived from the Boltzmann equation.
\begin{equation}
\rho^{-1}(T) = \frac{1}{ 4\pi^3} \frac{e^2 v_{\rm F} }{\hbar} \frac{1}{3} \int \tau_{\vec{k}} dS_{\rm FS},
\label{aq1}
\end{equation}
where the integration is taken over the fermi-surface, and $v_{\rm F}$ the fermi velocity.
This assumes that the {\it actual} electronic structure near the chemical potential is sufficiently complicated that in the temperature region of interest, vertex corrections which lead to emphasis on large momentum scattering for resistivity are unimportant. In that case the scattering rate which determines the resistivity is the same as the single-particle scattering rate averaged over the fermi-surface. This is true in a multi-sheeted fermi-surface and is suitable for heavy fermions. This is similar to the case of transition metals where the resistivity from electron-phonon scattering at low temperatures is $\propto T^5$ in contrast to the nearly free-electron metals where it is $\propto T^3$. For weakly anisotropic single band scattering, as in the cuprates, the resistivity for large AFM correlation lengths is close to the Fermi-liquid temperature dependence although near the hot spots the scattering rate is nearly $\propto T$ \cite{Hlubina-Rice}.
Equation (\ref{aq1}) can also be expressed as follows.
\begin{equation}
\rho^{-1} = \frac{ n e^2 }{ m^{*}} \langle \tau_{\vec{k}} \rangle_{\rm FS}.
\label{aq2}
\end{equation}
where $\langle \cdots \rangle_{\rm FS} \equiv \frac{1}{4 \pi k_{\rm F}^2} \int \cdots dS_{\rm FS}$ means the average over the fermi-surface, $m^{*}$ the renormalized effective mass.\par
Here, $\tau_{\vec{k}}$ can be derived from the imaginary part of the self-energy.
\begin{equation}
\frac{ \hbar}{ 2 \tau_{\vec{k}} } = - {\rm Im} \Sigma ( \vec{k}, \varepsilon + i\delta)\vert_{\varepsilon \rightarrow 0},
\label{aq3}
\end{equation}
where the self-energy due to the antiferromagnetic quantum fluctuations is given as follows.
\begin{equation}
\Sigma ( \vec{p}, i \varepsilon_n)= g^2 k_{\rm B} T \sum_{\omega_m} \sum_{\vec{q}} G( \vec{p}- \vec{q}, i\varepsilon_n - i\omega_m) \chi( \vec{q}, i\omega_m)
\label{aq4}
\end{equation}
The result for the resistivity in the limit $\xi/a \to \infty$ is derived in an Appendix A. by explicitly calculating the self-energy given by eq. (\ref{aq4}) is
\begin{equation}
\rho(\xi/a = \infty) = \lambda \frac{3}{4 e^2} \frac{a \hbar}{ (\varepsilon_{\rm F}/k_{\rm B}) \sqrt{ \Gamma_{\rm AFM}/k_{\rm B} }} T^{3/2}.
\label{rho}
\end{equation}
\section{Solution of the Linearized Eliashberg Equations}
The superconducting transition temperature is given by the linearized version of the Eliashberg Equations for the normal self-energy $- i \omega_nZ(\theta_{\vec{k}},i \omega_n)$ and the anomalous or pairing self-energy $W( \theta_{\vec k}, i \omega_n )$.
\begin{align}
\label{leliash1}
\left[ 1 - Z(\theta_{\vec{k}},i \omega_n) \right] i \omega_n &= - \int_{\rm FS} \frac{ {\rm d}^{d} S_{\vec{p}} }{ (2 \pi)^d v_{\vec{p}} } \pi T \sum_{\Omega_m} i\ {\rm sgn}(\Omega_m) g^2 \chi ( \vec{k} - \vec{p},i \omega_n - i\Omega_m),\\
\label{leliash2}
W( \theta_{\vec k}, i \omega_n ) &= - \int_{\rm FS} \frac{ {\rm d}^{d} S_{\vec{p}} }{ (2 \pi)^d v_{\vec{p}} } \pi T \sum_{\Omega_m} \frac{ W( \theta_{\vec p}, i \Omega_m )}{ \vert \Omega_m Z( \theta_{\vec{p}}, i \Omega_m) \vert} g^2 \chi ( \vec{k} - \vec{p},i \omega_n - i\Omega_m).
\end{align}
Here $\omega_n$ are the Matsubara frequencies; $g$ is a momentum-independent coupling matrix element, which has already been defined , $\theta_{\vec{k}}$ is an angle parameterizing the Fermi surface, $N(\theta_{\vec{k}})$ is the density of states at angle $\theta_{\vec{k}}$. The $\vec{p}$-integral is over the Fermi surface, $v_{\vec{p}}= \partial \varepsilon_{\vec{p}}/ \partial \vec{p}$ is the unrenormalized velocity.
\subsection{Results for variation of $T_{\rm c}$ with Parameters in the Models}
Our principal general results for $T_{\rm c}$ on the basis of solution of the linearized Eliashberg equations in terms of $\lambda$ and the parameters in $\chi_{\rm AFM}(\vec{Q}, \omega)$ are given in this section. The numerical evaluation is done by first simplifying the Eliashberg equations (\ref{leliash1}) and (\ref{leliash2}), as far as possible analytically. The final expressions for the numerical evaluation, both for the circular Fermi-surface and the tight-binding Fermi-surfaces are given in Appendix B.\par
Figure. \ref{fig:03a} show the $(\xi^{*}/a)^{-2}$-dependences of $T_{\rm c}/\Gamma_{\rm AFM}$ for the circular fermi-surface on the bare coupling constant $\lambda$ in the large correlation length limit on the right. For small bare coupling $\lambda$, the latter does have the BCS form while for $\lambda \gtrsim 1$, the dependence is approximately linear. Consistent with earlier discussions \cite{ml1,ml2}, $T_{\rm c}/\Gamma_{\rm AFM}$ shows a very shallow peak at around $(\xi^{*}/a)^{-2}\sim 5 \times 10^{-3}$.
$T_{\rm c}/\Gamma_{\rm AFM}$ shows a drop-off as the correlation length decreases, while it also shows moderate decreases as the correlation length increases.
ML pointed out that this moderate decrease is caused by the rapid diverges of $Z$ as the correlation length increases. We note also that the quantum-classical crossover correction to the correlation length proportional to the factor $\gamma$ in Eqs. (\ref{corr2}) has a negligible effect on $T_{\rm c}$. This will not be considered in any further calculations.
The principal message from Fig. (\ref{fig:03a}) is that the infinite correlation length result for $T_{\rm c}$ is well obeyed up to $\xi/a \approx 10$ with a very sharp fall off thereafter which will be seen later to be exponential. For large $\xi/a$, no BCS type approximation for $T_{\rm c}$ is valid. The limit of very large correlation length is equivalent to the effective frequency of fluctuations $\to 0$, as may be seen from Eq.(\ref{chi}). If we use the McMillan \cite{mcmillan} type approximation, in which $\lambda_{\rm M} \propto \langle \omega^2 \rangle^{-1}$, the inverse of the average squared frequency of fluctuations, we get a divergent coupling. Fig. (\ref{fig:03a}) gives a finite limit to $T_{\rm c}/\Gamma_{\rm AFM}$, which depends on the bare coupling constant $\lambda$. One may understand this result from the calculations of Allen and Dynes \cite{Allen-Dynes}, deduced for s-wave Eliashberg equation, that in the limit of a diverging coupling constant $T_{\rm c} \propto \sqrt{\lambda_{\rm M}} \langle \omega \rangle$, where $\langle \omega \rangle$ may be taken approximately to be the square-root of $\langle \omega^2 \rangle$.
-
\begin{figure}[ptb]
\begin{center}
\includegraphics[width = 0.75\textwidth]{newTccircFS.eps}
\caption{ The transition temperature normalized to $\Gamma_{\rm AFM}$ vs $(\xi^{*}/a)^{-2}$ for a circular fermi-surface and $\lambda=1$ is shown on the left, and
the transition temperature normalized to $\Gamma_{\rm AFM}$ vs $\lambda$ for a circular fermi-surface and $(\xi^{*}/a)^{-1} = 0$ is shown on the right. }
\label{fig:03a}
\end{center}
\end{figure}
Next we show in Fig. (\ref{fig:04a}) the $(\xi^{*}/a)^{-2}$-dependence of $T_{\rm c}/\Gamma_{\rm AFM}$ for the four fermi surface shown in Fig. (\ref{fig:01}) .
For $t'=0.4t$, the worst nesting case, $T_{\rm c}/\Gamma_{\rm AFM}$ is similar to that for the circular fermi-surface.
Improving the nesting condition increases , $T_{\rm c}/\Gamma_{\rm AFM}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.65\textwidth]{comTcFs.eps}
\caption{The transition temperature normalized to $\Gamma_{\rm AFM}$ vs $(\xi^{*}/a)^{-2}$ for the four fermi-surfaces shown in Fig. (\ref{fig:01}).}
\label{fig:04a}
\end{center}
\end{figure}
We show in (\ref{fig:05a}) $T_c$ as a function of $\lambda$ for the worst nesting fermi-surface and the best nested fermi-surface of those in Fig. (\ref{fig:01}). A increase of $O(2)$ in $T_{\rm c}$ for the similar values of $\lambda$ is discerned from the worst to the best nesting conditions.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width = 0.75\textwidth]{goodbadlam.eps}
\caption{Left: The transition temperature normalized by $\Gamma_{\rm AFM}$ vs $\lambda$ for the worst nested fermi-surface (FS1) of
Fig (\ref{fig:01}); Right: The transition temperature normalized by $\Gamma_{\rm AFM}$ vs $\lambda$ for the best nested fermi-surface (FS4) of
Fig (\ref{fig:01})}.
\label{fig:05a}
\end{center}
\end{figure}
ML have also presented detailed results for calculations on a 3 d electronic dispersion with the symmetry of a cubic lattice.
They remark that other parameters being the same 2 dispersion gives higher $T_{\rm c}$ than a three dimensional dispersion.
Based on our results for changes in $T_{\rm c}$ in the 2 d problem, we conclude that this is because of the much better nesting that is obtainable in model 2 d systems compared to the 3d systems for a given ${\bf Q}$ which spans the Fermi-surface in some (usually symmetry) direction. In fact, we can place our 2 d-results for the very weakly nested fermi-surface over the ML results for the 3d Fermi-surface and find for other parameters the same that the systematics of the results for $T_{\rm c}$ as well as its value is very similar.
\section{Comparison with Experiments in Heavy Fermions}
In this section, we compare the estimates of $T_{\rm c}$ from the calculations with the experimental result in CeCu$_2$Si$_2$ and CeIrIn$_5$. For convenience, we show the measured intensity \cite{arndt} proportional to the dynamic structure factor $S({\bf Q}, \omega) = \coth (\omega/2T) {\rm Im} \chi({\bf Q}, \omega)$ in Fig. (\ref{fig:S(q)}) for ${\bf Q } $ near ${\bf Q}_{\rm AFM}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.75\textwidth]{Sq.eps}
\caption{The measured dynamic structure factor at the antiferromagnetic Bragg vector as a function of energy for various temperatures in CeCu$_2$Si$_2$. From Ref. (\onlinecite{arndt}). The variation with ${\bf Q}$ to get the correlation length is also available in (\onlinecite{arndt}) and references therein.}.
\label{fig:S(q)}
\end{center}
\end{figure}
Although the magnetic fluctuation spectrum found through inelastic scattering in CeCu$_2$Si$_2$ is well represented by the form of Eq. (\ref{chi}), the electronic structure is far more complicated than assumed here or in the 3 d calculations of ML.
We have seen that $T_{\rm c}$, especially in the limit of large magnetic correlation lengths depends only on gross parameters like $\lambda$ and secondarily on the amount of nesting. The comparison can only be very limited and can only give insight into the orders of magnitudes expected and to the physics involved.
\paragraph{CeCu$_2$Si$_2$}\cite{stockert1,arndt}\\
\noindent
To fit the phenomenological susceptibility to these results, the parameters take the following values: \\
\noindent
{\it AFM wave vector}:
$Q_{\rm AFM}$= (0.22, 0.22, 1.46)\\
{\it Ordered Moment in the AFM phase} :
$\langle S \rangle \approx 0.2 \mu_B$/Ce
{\it Correlation length:}
$\xi \simeq$ 25 \AA;\\
{\it Characteristic energy scales } :\\
$\Gamma_{\rm AFM} \simeq $ 1.5 [meV]; \\
{\it Density of States}: From the measured uniform $(\bf q =0)$ Paramagnetic susceptibility in the normal state, one deduces the $N(E_{\rm F}) \approx$ (1/7) [meV]$^{-1}/$unit-cell.\\
{\it AFM spin-fluctuations parameter}: $\bar{\chi}_0$: \\
The experimental results for $\chi(q,\omega)$ in Ref.(\onlinecite{stockert1}) are parametrized in terms of three quantities $\xi, \chi_0$ and $\Gamma$. The correlation length $\xi$ in Ref.(\onlinecite{stockert1}) is the same quantity used by us. For clarity we give here the relation of the other two parameters to the parameters used by us. The conversion from the quantity $\chi_0$, which we will call $\chi_{0,S}$ to our $\bar{\chi}^0$ is obtained by equating the integral over all $q,\omega$ of Equation S4 in Ref.(\onlinecite{stockert1}) to the integral of the same physical quantity given in Eq. (\ref{tmsr}). In the limit of $(\xi/a)^2 >> 1$, one gets $\bar{\chi}^0 \approx \chi_{0,S} (a/\xi)^2$. The experimental result is $\chi_{0,S} = 15.64 \mu_B^2/meV$. This then gives $\bar{\chi}^0 \approx 0.4 \mu_B^2/unit-cell/meV$ . \\
The quantity $\Gamma$ is related to $\Gamma_{AFM}$ by $\Gamma_{AFM} = \Gamma (a)^{-2}$.\\
{\it Transition Temperature} :
$T_{\rm c} \sim$ 0.6[K] \\
CeCu$_2$Si$_2$ has a very anisotropic fermi-surface with very little dispersion along the tetragonal axis. The fermi-surface in the plane is very complicated but we assume that just as in s-wave superconductivity \cite{mcmillan}, $T_{\rm c}$ depends only on the average density of states at the Fermi-surface, supplemented by knowledge of nesting of the fermi-surface near ${\bf Q}_{AFM}$. Among other things, our results below may be taken to be test of this assumption.
\paragraph{CeIrIn$_5$}\cite{kambe,yashima}\\
\noindent
Although long-range magnetic order competing with superconductivity in CeIrIn$_5$ has not been accessed in this stoichiometric compound, there are strong experimental results indicating that the compound lies in the vicinity to an AFM quantum-critical point. The resistivity of this material exhibits a non-Fermi liquid behavior similar to that observed in CeCoIn$_5$, which is known to lie in the vicinity of an AFM quantum-critical point which has been accessed by doping the compound. Moreover, the nuclear spin relaxation rate of CeIrIn$_5$ is also similar to that of CeCoIn$_5$.
The dynamical susceptibility has been recently deduced by NMR experiments \cite{kambe} in agreement with this conclusion.\\
To fit the susceptibility to these experimental result, the parameters take the following values.\\
{\it AFM wave vector}:
$Q_{\rm AFM}$= (0.5, 0.5, 0.5)\\
The chosen $Q_{\rm AFM}$ and ordered moment are taken to be that of the related compound CeCoIn$_5$ \cite{stock} \\
{\it Ordered Moment in the AFM phase}:
$\langle S \rangle \approx 0.15 \mu_{\rm B}$\\
{\it Correlation length}\\
$(\xi^{*}/a) \simeq 10$ at $T=1$[K]\cite{kambe}
{\it Characteristic energy scales of AFM:}
$\Gamma_{\rm AFM} \simeq$ 1.5[meV]; \\
{\it Transition Temperature}:
$T_{\rm c}$ = 0.4[K]
\ \par
The experimental results show that both CeCu$_2$Si$_2$ and CeIrIn$_5$ lie not far from the asymptotic large correlation length limit and that their $T_{\rm c}/\Gamma_{\rm AFM}$ are both about 0.03. For a circular fermi-surface, and using the measure value of $\xi/a$ in the former, we may refer then simply to Fig. (\ref{fig:03a}) and find that $\lambda \approx 3$ gives the right value of $T_{\rm c}$. For the best nested Fermi-surface, however, a $\lambda \approx 1$ is sufficient as shown in Fig. (\ref{fig:05a}).
We may now try to estimate $\lambda$ to see if these values are reasonable. We do this in two different ways. To utilize the neutron scattering results for this purpose, we need to know $g$ besides the directly measured properties listed above. The renormalizations in the heavy fermion problem are such that near the critical point the AFM interaction between magnetic moments is of the same order as the heavy fermion bandwidth. Then $g$ is of the order of the effective fermi-energy, i.e $gN_{\rm F}\sim 1$.
Then we may use the experimental values of $N_{\rm F}$ and $\bar{\chi_0}$ deduced from experiments above in Eq.(\ref{lambda}) to get $\lambda \approx 2$.
The above manner of estimation has forced us to guess the value of $g$. We can estimate the value of $\lambda$ much better and independently from the non-fermi-liquid resistivity proportional to $T^{3/2}$ observed in the quantum critical regime of CeCu$_2$Si$_2$, whose coefficient is proportional to $\lambda$.
The resistivity $\rho$ in the quantum-critical region for $\xi/a \to \infty$ is given by Eq.(\ref{rho}).
Using the values of CeCu$_2$Si$_2$ mentioned above, the resistivity is estimated $\rho = 3.02 \times 10^{-8} \lambda T^{3/2}$ [$\Omega {\rm m}$] from eq. (\ref{rho}).
The non-fermi liquid resistivity observed in CeCu$_2$Si$_2$ takes the form: $\rho(T)/\rho_{\rm 300K}=0.151 + 0.071 T^{3/2}$ \cite{gegenwart} where $\rho_{\rm 300K} \sim 70\mu \Omega {\rm cm}$ \cite{schneider}.
From the comparison of the coefficient of $T^{3/2}$-term in the resistivity between theoretical and experimental results, $\lambda$ is estimated as $\lambda \sim 1.6$. This should be considered an important evidence for the rather obvious idea that fluctuations that determine the normal state scattering also determine $T_{\rm c}$, and of the consistency of the present calculations. The extent to which the calculations correctly estimate $T_{\rm c}$ may be judged from the fact that in the range of $\lambda$ from the different estimates for it, $T_{\rm c} \propto \lambda$. \\
We comment briefly on an estimation of the condensation energy due to superconductivity and its comparison with the increase in energy of AFM fluctuations on entering superconductivity \cite{stockert1}. The latter has been estimated to be almost a factor of 20 larger than the superconducting condensation energy. The suggestion has been offered that this factor of 20 may be the increase in kinetic energy. In BCS theory for electron-phonon interactions, the absolute magnitude of the change in kinetic and in potential energy are both of the same order as the condensation energy. So, a good reason has to be found for this factor of 20. We do not have a solution to this enigma.
\section{Summary}
We have presented a solution to the linearized Eliashberg equations using a phenomenological spin-fluctuation spectrum and simple fermi-surfaces to highlight the important parameters that determine $T_{\rm c}$ for d-wave symmetry. Careful attention has been paid to the partial sum-rule on the q-dependent part of the spin-fluctuation spectra to estimate the effective coupling constant which depends on parameters such as the total partial spectral weight, the correlation length and the upper frequency cut-off of the q-dependent spin-fluctuations. These parameters are not independent and we show their relationship in the simple model studied. With regard to the electronic structure, a knowledge of the average density of states at the fermi-surface is sufficient for determining $T_{\rm c}$ in the s-wave channel \cite{mcmillan}. But for d-wave superconductivity through exchange of well correlated spin-fluctuations, this must be supplemented by a knowledge of nesting. The results for the general solutions are employed for two heavy fermion compounds using their measured spin-fluctuation spectra and other quantities such as specific heat and magnetic susceptibility. Correct estimates for $T_{\rm c}$ to factors of
$O(2)$ are obtained. Confidence in these results is bolstered by getting the correct observed temperature dependence of the anomalous $T^{3/2}$ resistivity with a coefficient using the same parameters, again correct to factors of $O(2)$. This puts a semi-quantitative backbone to the surmise made long ago that d-wave superconductivity in such heavy fermions is promoted by large amplitude spin-fluctuations with large correlation lengths such as occur near some AFM quantum critical points.\\
{\it Acknowledgements:} Part of the work by SN was done while visiting University of California, Riverside with the aid of the Global COE program (G10) from the Japan Society for the Promotion of Science. The work of KM is partially supported by a Grand-in-Aid for Scientific Research on Innovative Area ``Heavy Electrons'' (No. 20102008) and a Grant-in-Aid for Specially Promoted Research (No. 20001004) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The work of CMV is partially supported by NSF under grant DMR-1206298.\\
\noindent
{\bf Appendix A: Derivation of Resistivity Near the Antiferromagnetic Quantum Critical Point}\\
An expression for the resistivity under assumptions suitable for heavy fermions with a multi-sheeted fermi-surface and/or sufficient impurity scattering \cite{Maebashi} is given by eq. (\ref{rho}) in terms of the self-energy function Eq. (\ref{aq4}).
Here, we derive the relation (\ref{rho}) explicitly.
Substituting $\chi( \vec{q}, i \omega_m)$ in the spectral representation into eq. (\ref{aq4}) and carrying out the $\omega_m$-summation, one gets
\begin{equation}
\Sigma ( \vec{p}, i \varepsilon_n)= - \frac{g^2}{2} \sum_{\vec{q}} \int_{ -\infty}^{\infty} \frac{dx}{ \pi} \frac{ {\rm Im} \chi( \vec{q},x) }{x - i\varepsilon_n + \xi_{\vec{p}-\vec{q}}} \left( \tanh \frac{\xi_{\vec{p} -\vec{q} }}{2k_{\rm B}T} + \coth \frac{x}{2k_{\rm B}T} \right).
\label{aq5}
\end{equation}
Taking the analytic continuation of $\Sigma(\vec{p}, i\varepsilon_n)$, the imaginary is given as
\begin{align}
\notag {\rm Im} \Sigma ( \vec{p}, \varepsilon + i\delta) &= - \frac{g^2}{2} \sum_{\vec{q}} \int_{ -\infty}^{\infty} \frac{dx}{ \pi} {\rm Im} \chi( \vec{q},x) \pi \delta(x - \varepsilon + \xi_{\vec{p} -\vec{q}}) \left( \tanh \frac{\xi_{\vec{p}-\vec{q}}}{2k_{\rm B}T} + \coth \frac{x}{2k_{\rm B}T} \right)\\
\notag &=- \frac{g^2}{2} \sum_{\vec{q}} {\rm Im} \chi( \vec{q}, \varepsilon - \xi_{\vec{p}-\vec{q}} ) \left( \tanh \frac{\xi_{\vec{p}-\vec{q}}}{2k_{\rm B}T} + \coth \frac{\varepsilon - \xi_{\vec{p} -\vec{q}} }{2k_{\rm B}T} \right)\\
&=- \frac{g^2}{2} \sum_{\vec{q}} \frac{ \bar{\chi_0}\Gamma_{\rm AFM}^{-1} (\varepsilon - \xi_{\vec{p} - \vec{q}} ) }{ [(\xi/a)^{-2} + a^2 (\vec{q} - \vec{Q} )^2 ]^2 + \left( \frac{ \varepsilon - \xi_{\vec{p} - \vec{q}}}{ \Gamma_{\rm AFM}} \right)^2 } \left( \tanh \frac{\xi_{\vec{p-q}}}{2k_{\rm B}T} + \coth \frac{\varepsilon - \xi_{\vec{p-q}} }{2k_{\rm B}T} \right).
\label{aq6}
\end{align}
We now consider the behavior at around the antiferromagnetic quantum critical point, i.e., $ (\xi/a)^{-1} \sim 0$.
In a low temperature region where the non-fermi liquid behavior appears, $\varepsilon \sim 0$ gives the dominant contribution for eq. (\ref{aq6}).
Moreover, using the following relation,
\begin{equation}
\tanh \frac{x}{2} - \coth \frac{x}{2} = \frac{-2}{\sinh{x}} ,
\label{aq7}
\end{equation}
eq. (\ref{aq6}) is transformed as
\begin{align}
\notag {\rm Im} \Sigma ( \vec{p}, 0 + i\delta) &= - g^2 \bar{\chi_0}\Gamma_{\rm AFM}^{-1} \sum_{\vec{q}} \frac{ \xi_{\vec{p} - \vec{q}} }{ a^4 (\vec{q} - \vec{Q} )^4 + \left( \frac{ \xi_{\vec{p} - \vec{q}}}{ \Gamma_{\rm AFM}} \right)^2 } \frac{1}{\sinh( \frac{\xi_{\vec{p}-\vec{q}}}{k_{\rm B}T}) }\\
&= - g^2 \bar{\chi_0}\Gamma_{\rm AFM}^{-1} \sum_{\vec{q}^{'}} \frac{ \xi_{\vec{p} - \vec{Q} - \vec{q}^{'} } }{ a^4 \vec{q}^{'4} + \left( \frac{ \xi_{\vec{p} - \vec{Q} - \vec{q}^{'} }}{ \Gamma_{\rm AFM}} \right)^2 } \frac{1}{\sinh( \frac{\xi_{\vec{p}- \vec{Q} - \vec{q}^{'} }}{k_{\rm B}T}) }.
\label{aq8}
\end{align}
Next, we consider the $\vec{q}$-integration in eq. (\ref{aq8}).
Because the denominator in eq. (\ref{aq8}) has $\vec{q}^{'4}$ term, $q^{'} \sim 0$ gives the dominant contribution in the $\vec{q}^{'}$-integration.
Therefore, one gets
\begin{align}
{\rm Im} \Sigma ( \vec{p}, 0 + i\delta) &= - g^2 \bar{\chi_0}\Gamma_{\rm AFM}^{-1} \frac{1}{2 \pi^2} \int_0^{q_c} d q^{'} \frac{ \xi_{\vec{p} - \vec{Q}} }{ a^4 q^{'4} + \left( \frac{ \xi_{\vec{p} - \vec{Q} }}{ \Gamma_{\rm AFM}} \right)^2 } \frac{1}{\sinh( \frac{\xi_{\vec{p}- \vec{Q} }}{k_{\rm B}T}) }.
\label{aq9}
\end{align}
Since the integrated function in eq. (\ref{aq9}) rapidly decays as $q^{'}$ increases, we take $q_c$ as $\infty$ and obtain following result by easy calculation.
\begin{equation}
{\rm Im} \Sigma ( \vec{p}, 0 + i\delta) = - g^2 \bar{\chi_0}\Gamma_{\rm AFM}^{-1/2} \frac{1}{8 \sqrt{2} \pi a^3 } \frac{\vert \xi_{\vec{p} - \vec{Q}} \vert^{1/2} }{\sinh( \frac{\xi_{\vec{p}- \vec{Q} }}{k_{\rm B}T}) }.
\label{aq10}
\end{equation}\par
Substituting eqs. (\ref{aq10}) and (\ref{aq3}) into eq. (\ref{aq2}), $\rho$ is given as
\begin{equation}
\rho \simeq \lambda \frac{a \hbar}{2 \sqrt{2} \Gamma_{\rm AFM}^{1/2} e^2} \langle \frac{ \vert \xi_{\vec{p} - \vec{Q}} \vert^{1/2} }{ \sinh \frac{\xi_{\vec{p} - \vec{Q}}}{k_{\rm B} T} } \rangle_{\rm FS},
\label{aq11}
\end{equation}
where we use $na^3 \sim 1$ and $N_{\rm F} = m^{*} k_{\rm F}/(2 \pi^2 \hbar^2)$.\par
Here, we estimate the average over the fermi-surface in eq. (\ref{aq11}) assuming that the fermi-surface is spherical.
\begin{equation}
\langle \frac{ \vert \xi_{\vec{p} - \vec{Q}} \vert}{ \sinh \frac{\xi_{\vec{p} - \vec{Q}}}{k_{\rm B} T} } \rangle_{\rm FS} = \frac{1}{ 4 \pi k_{\rm F}^2 } \int \frac{ \vert \xi_{\vec{p} - \vec{Q}} \vert^{1/2} }{ \sinh \frac{\xi_{\vec{p} - \vec{Q}}}{k_{\rm B} T} } d S_{\rm FS}.
\label{aq12}
\end{equation}
The dominant contribution in eq. (\ref{aq12}) comes from ``hot'' line where the relation $\vert \vec{p} \vert = \vert \vec{p} - \vec{Q} \vert = k_{\rm F} $ is satisfied as shown in Fig. \ref{fig:11}
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.5\textwidth]{drawing.eps}
\end{center}
\caption{The cross-sectional circular fermi-surface and the wave vector $\vec{p}$ which satisfies the relation $\vert \vec{p} \vert = \vert \vec{p} - \vec{Q} \vert = k_{\rm F} $.}
\label{fig:11}
\end{figure}
Assuming that the dispersion near the fermi-surface is given by linear dispersion, we obtain $\xi_{\vec{p} - \vec{Q}} \simeq v_{\rm F} k_{\perp} \cos( 2 \sigma - \pi/2) = k_{\rm F} \sin 2 \alpha$, where $k_{\perp} $ is the deviation from the ``hot'' line.
For one ``hot'' spot, the integration is estimated as
\begin{equation}
\frac{1}{ 4 \pi k_{\rm F}^2 } k_{\rm F} \sin \alpha \int_{-k_c}^{k_c} d k_{\perp} \frac{ \sqrt{ \vert v_{\rm F} k_{\perp} \sin 2 \alpha \vert} }{ \sinh \frac{ v_{\rm F} k_{\perp} \sin 2 \alpha }{k_{\rm B} T} }.
\label{aq13}
\end{equation}
Changing the integration variable as $x \equiv v_{\rm F} k_{\perp} \sin 2\alpha/( k_{\rm B} T)$, eq. (\ref{aq13}) is transformed as
\begin{equation}
\frac{ (k_{\rm B} T)^{3/2} }{ 8 \pi v_{\rm F} k_{\rm F} \cos \alpha } \int_{0}^{ \frac{v_{\rm F} k_c \sin 2 \alpha}{ k_{\rm B}T }} d x \frac{ \sqrt{x}}{ \sinh x}.
\label{aq14}
\end{equation}
Now, we take the upper limit of the integration as $\infty$ because we consider the low temperature region, and eq. (\ref{aq14}) can be calculated as
\begin{equation}
\frac{ (k_{\rm B} T)^{3/2} }{ 8 \pi v_{\rm F} k_{\rm F} \cos \alpha } \frac{ 2\sqrt{2} - 1}{ \sqrt{2}} \zeta \left( \frac{3}{2} \right) \Gamma \left( \frac{3}{2} \right)
\simeq \frac{ 3 (k_{\rm B} T)^{3/2} }{ 4 \pi v_{\rm F} k_{\rm F} \cos \alpha }
\label{aq15}
\end{equation}\par
Since such a ``hot'' point makes two rings whose total length is equal to $4 \pi$ in the sphere fermi-surface, eq. (\ref{aq12}) is given by
\begin{equation}
\langle \frac{ \vert \xi_{\vec{p} - \vec{Q}} \vert}{ \sinh \frac{\xi_{\vec{p} - \vec{Q}}}{k_{\rm B} T} } \rangle_{\rm FS} = \frac{ 3 (k_{\rm B} T)^{3/2} }{ 2 \varepsilon_{\rm F} \cos \alpha }.
\label{aq16}
\end{equation}\par
Substituting eq. (\ref{aq16}) into eq. (\ref{aq11}), we obtain
\begin{equation}
\rho \simeq \lambda \frac{3 a \hbar}{4 \sqrt{2} \cos \alpha } \frac{1}{ (\varepsilon_{\rm F}/k_{\rm B}) (\Gamma_{\rm AFM}/k_{\rm B}) ^{1/2} e^2} T^{\frac{3}{2}}.
\label{aq17}
\end{equation}
In this calculation, the $Q$-vector is given by $2k_{\rm F} \sin \alpha$.
The $Q$-vector of the CeCu$_2$Si$_2$ is observed as (0.215, 0.215, 0.1458) giving $\vert \vec{Q} \vert = 1.49 a/\pi$.
Therefore, $\alpha$ is estimated as $ \alpha \sim \pi/4$, and we obtain the result used for the estimation of $\lambda$:
\begin{equation}
\rho \simeq \lambda \frac{3 }{4e^2 } \frac{ a \hbar}{ (\varepsilon_{\rm F}/k_{\rm B}) (\Gamma_{\rm AFM}/k_{\rm B}) ^{1/2} } T^{\frac{3}{2}}.
\label{aq18}
\end{equation}
\noindent
{\bf Appendix B: Final Expressions for Evaluation of $T_{\rm c}$}\\
\noindent
{\it Circular Fermi-surface}\\
For a circular Fermi-surface, it is possible to do the momentum integrals in the Eliashberg equations (\ref{leliash1}) and (\ref{leliash2}) analytically so that only a diagonalization in discrete frequency space needs to be done numerically. The final expressions used for numerical evaluation for the normal and the anomalous self-energy are:
\begin{align}
\label{f1}
Z(\theta_{\vec{k}},i \omega_n) &= 1 + \frac{ \lambda}{ \omega_n/(\pi T)} \sum_{\Omega_m} \frac{{\rm sgn}(\Omega_m)}{\sqrt{ \alpha^2 - \beta^2}} \\
\label{f2}
W_{2}(i \omega_n) &= \pi T \sum_{\Omega_m} K( \omega_n, \Omega_m) \frac{ W_{2} (i\Omega_m)}{ \vert \Omega_m \vert },\\
\label{f3}
K(\omega_n, \Omega_m) &= - \lambda \int_{0}^{2\pi} \frac{ {\rm d} \theta_{\vec p}}{2 \pi}\frac{ \cos 2 \theta_{\vec{p}} \cos2x }{ \vert Z( \theta_{\vec{p}}, \Omega_m) \vert} \frac{ A - \sqrt{ A^2 - B^2 - C^2} }{ (B^2 + C^2) \sqrt{ A^2 - B^2 - C^2}},
\end{align}
where
\begin{eqnarray}
\alpha &=& \frac{\vert \omega_n - \Omega_m \vert}{ \Gamma_{\rm AFM} } + (\xi/a)^{-2} + a^2 ( \vert \vec{k} \vert^2 + \vert \vec{p} \vert^2 + \vert \vec{Q} \vert^2) - 2 a^2 \vert \vec{k} \vert \vert \vec{Q} \vert \cos(\theta_{\vec{k}} - \theta_{\vec{Q}} ), \\
\beta &=& 2 a^2 \vert \vec{p} \vert \sqrt{ \vert \vec{Q} \vert^2 + \vert \vec{k} \vert^2 - 2 \vert \vec{Q} \vert \vert \vec{k} \vert \cos ( \theta_{\vec{k}} - \theta_{\vec{Q}} ) },\\
A &=& \frac{\vert \omega_n - \Omega_m \vert}{a^2 \Gamma_{\rm AFM} } + (\xi/a)^{-2} + a^2 ( \vert \vec{k} \vert^2 + \vert \vec{p} \vert^2 + \vert \vec{Q} \vert^2) + 2a^2 \vert \vec{p} \vert \vert \vec{Q} \vert \cos(\theta_{\vec{p}} - \theta_{\vec{Q}} ), \\
B^2 + C^2 &=& 4 a^4 \vert \vec{k} \vert^2 \left[ \vert \vec{p} \vert^2 + \vert \vec{Q} \vert^2 + 2\vert \vec{p} \vert \vert \vec{Q} \vert \cos ( \theta_{\vec{p}} - \theta_{\vec{Q}} )\right],\\
x &=& \tan^{-1} \left[ \frac{ \vert \vec{Q} \vert \sin \theta_{\vec{Q}} + \vert \vec{p} \vert \sin \theta_{\vec{p}}} { \vert \vec{Q} \vert \cos \theta_{\vec{Q}} + \vert \vec{p} \vert \cos \theta_{\vec{p}}} \right].
\label{f4}\\
\end{eqnarray}\par
\noindent
{\it Tight-Binding Fermi-surfaces}\\
With tight binding approximation, only some simplifications in the momentum integrals in the Eliashberg equations (\ref{leliash1}) and (\ref{leliash2}) can be done analytically. The final expressions used in this paper for numerical evaluation are
\begin{align}
\label{f5}
Z(\theta_{\vec{k}},i \omega_n) &= 1 + \frac{1}{ \omega_n/(\pi T)} \sum_{\Omega_m} {\rm sgn}(\Omega_m) \int_{FS} \frac{ {\rm d}^2 S_{\vec{p}} }{(2 \pi)^2 v_{\vec{p}}} \frac{ \lambda }{ \frac{\vert \omega_n - \Omega_m \vert}{\Gamma_{\rm AFM}} + (\xi/a)^{-2} + a^2 ( \vec{k} - \vec{p} - \vec{Q} )^2 }, \\
W_{2}(i \omega_n) &= \pi T \sum_{\Omega_m} K( \omega_n, \Omega_m) \frac{ W_{2} (i\Omega_m)}{ \vert \Omega_m \vert },\\
K( \omega_n,\Omega_m ) &\equiv - 2\lambda \int_{FS} \frac{ {\rm d}^2 S_{\vec{k}} }{(2 \pi)^2 v_{\vec{p}}} \int_{FS} \frac{ {\rm d}^2 S_{\vec{p}} }{(2 \pi)^2 v_{\vec{p}}} \frac{ 1 }{ \vert Z(\theta_{\vec{p}} ,i\Omega_m ) \vert }\frac{ \big(\cos (k_{Fx}a) - \cos (k_{Fy}a)\big)\big(\cos (p_{Fx}a) - \cos (p_{Fy}a)\big)}{ \vert \omega_n - \Omega_m \vert/\Gamma_{\rm AFM} + (\xi/a)^{-2} + a^2( \vec{k} - \vec{p} - \vec{Q} )^2 }.
\label{f6}
\end{align}
For both circular and tight binding Fermi-surfaces, the best numerical strategy to evaluate $T_{\rm c}$ is to cast Eqs. (\ref{f3}) and (\ref{f6}) in the form of an eigenvalue equation for the eigenvector $W/ \vert \omega_n \vert $
\begin{align}
\sum_{\Omega_m} \left[ K(\omega_n,\Omega_m) - \frac{ \vert \omega_n \vert}{ \pi T} \delta_{n,m} \right] \left[ \frac{W(i \Omega_m) }{ \vert \Omega_m \vert } \right] = 0.
\label{a9}
\end{align}
\par
It should be noted that the Matrix of eq. (\ref{a9}) is not Hermitian because $K(\omega_n, \Omega_n)$ includes the renormalization factor $Z(\omega_n, \theta_{\vec{k}})$.
If the angle dependence of $Z$ can be neglected, we can define $K$ in a form which does not include $Z$, and we obtain the eigenvalue equation with a Hermitian Matrix for the eigenvector $W/ \vert \omega_n Z(\omega_n) \vert$.
In s-wave superconductor case, such a situation, namely angle-independent self-energy appears.
However, in the d-wave case, $Z(i \omega_n, \theta_{\vec{k}})$ strongly depends on $\theta_{\vec{k}}$.
On including $Z$ in the kernel $K$, the latter is no longer symmetric for the frequency exchange, $\omega_n$ and $\Omega_m$.\par
At high temperatures the eigenvalues of eq. (\ref{a9}) are close to the negative odd integers.
As the temperature decreases, the largest eigenvalue increases and crosses zero at transition temperature $T = T_{\rm c}$.\\
\par
|
2,877,628,090,780 | arxiv | \section{Introduction}
Determining the rank of an elliptic curve is a difficult problem, and there is currently
no known unconditional algorithm for determining the rank of a given curve. The basic
method for rigorously determining the rank of a curve is to find an upper bound for
the rank by computing the size of some Selmer groups and to find a lower bound for
the rank by finding enough independent rational points. In theory, if one continues
this process long enough, and the Shafarevich-Tate group of the curve is finite, the
upper and lower bounds should eventually coincide and the rank will be determined
exactly.
In practice, things are not so simple. Finding points on the curve is sometimes not too bad,
but the upper bounds for the rank are more problematic. Even the computation of the $2$-Selmer
rank is difficult, and it becomes prohibitively time consuming as the coefficients of
the elliptic curve grow; it is easy to write down a curve for which the state of the art
program for computing the $2$-Selmer group, John Cremona's mwrank \cite{mwrank},
will effectively take ``forever.''
If one is willing to accept the Birch and Swinnerton-Dyer conjecture that the rank of an
elliptic curve is the same as the order of vanishing of its $L$-function at the central
point, then it is possible to use the $L$-function to get information about the
rank. In fact, when the order of vanishing is between $0$ and $3$, it can be possible
to compute the $L$-function to enough precision and use some extra information about
the curve to determine the analytic rank exactly, as is done in \cite{rank3-bsd}, for example.
When the rank is larger than this, though, currently the best one can do is determine
that the first $r$ derivatives of the $L$-function are very close to $0$, and the
$(r+1)$-st is not, which will provide a very good guess for the rank and a rigorous upper
bound, assuming BSD.
This approach has its own problems, as it is much easier to write down a curve
of large conductor than it is to compute the $L$-function of such a curve.
For example, the known curve of rank at least $28$ \cite{rank28-curve}, which we will
write down later,
has conductor $N \approx 3.5 \times 10^{141}$,
and current methods (such as those described in \cite{rubinstein-methods-and-experiments})
typically
require summing on the order of $\sqrt{N}$ terms to compute the central value of the $L$-function.
(It would take a compute about $10^{53}$ cpu-years just to add $1$ to itself $10^{70}$ times.)
We present here a third method which is rather effective at bounding the rank, especially
when the rank is large compared to the conductor, as long as one is willing
to assume both the Birch and Swinnerton-Dyer
conjecture and the Riemann Hypothesis for the $L$-function of the curve.
This method is not completely new. It is based on
Mestre's method \cite{mestre} for (conditionally) bounding the rank of an elliptic curve based
only on its conductor, and it was used by Fermigier \cite{fermigier} to study ranks of
elliptic curves in certain families. However, it does not seem to have gained much traction
and does not seem to have been used much, if at all, since.
The idea, in brief, is as follows. Take $f(x)$ to be a function such that
$f(0) = 1$ and $f(x) \ge 0$ for all real $x$. Then, assuming the Riemann hypothesis,
the sum $\sum f(\gamma)$, where $1/2 + i\gamma$ runs over the nontrivial zeros of $L(s, E)$,
will be an upper bound for the analytic rank of $E$. Moreover,
for certain choices of $f(x)$ this sum may be efficiently evaluated using the
explicit formula for the $L$-function attached to $E$.
This method has recently been implemented by the author, and is available as part
of William Stein's \psage{ }\cite{psage} add-ons to \sage{ }\cite{sage-4.7.2}. As an example,
of what it can do, we will examine $6$ curves that are known to have rather large rank. We denote
these curves as $E_n$, $n = 20, 21, 22, 23, 24, 28$, where $n$ is a known lower bound for
the rank. We will write down these curves later (they are all taken from A. Dujella's
website \cite{rank-records}, and
at the time of discovery each held the record for the curve with largest number of known
independent rational points). The exact rank is not known for any of these curves. However,
conditionally we may claim
\begin{theorem}\label{main-theorem}
Assuming BSD and GRH, $E_n$ has rank exactly $n$ for $n = 20, 21, 22, 23,$ and $24$, while
$E_{28}$ has rank $28$ or $30$.
\end{theorem}
\subsection{Acknowledgements}
Most of the computations in this paper run in a short amount of time,
and were done on the author's personal computer. Some longer computations
were run on the sage cluster at the University of Washington, supported
by NSF grant DMS-0821725, and the riemann cluster at the University
of Waterloo, funded by the Canada Foundation for Innovation, the Ontario
Innovation Trust, and SGI.
The source code for our implementation is available as part of PSAGE \cite{psage}.
It uses Sage \cite{sage-4.7.2}, and hence PARI \cite{pari-2.4.3}, to compute $a_p$ for
bad primes, and uses Andrew Sutherland's smalljac \cite{smalljac} to compute
all other values of $a_p$.
Parts of this work began while the author was in residence at the
Mathematical Sciences Research Institute during the Arithmetic
Statistics program, Spring 2011, during which time the author was
partially supported by NSF grant DMS-0441170, administered by MSRI.
Discussions during the informal ``explicit formula seminar,'' especially
with David Farmer and Michael Rubinstein, were influential in encouraging
this work.
Currently the author is supported by NSF grant DMS-0757627, administered
by the American Institute of Mathematics.
\section{Bounding ranks}
\subsection{The method}
Let
\[
L(s, E) = \sum_{n=1}^\infty \frac{a_n}{n^s} = \prod_p L_p(s, E)^{-1}
\]
be the $L$-function of an elliptic curve, normalized so that the completed $L$-function
$\Lambda(s, E) = \epsilon \Lambda(1 - s, E)$, and let $c_n$ be defined by
\[
-\frac{L'(s, E)}{L(s, E)} = \sum_{n=1}^\infty \frac{c_n}{n^s}.
\]
More explicitly, if we define $\alpha(p)$ and $\beta(p)$ by
\[
L_p(s, E) = (1 - \alpha(p)p^{-s})(1 - \beta(p)p^{-s}),
\]
(note that $\alpha$ and $\beta$ are only well defined up to permutation, and that at least one
of them will be $0$ when $p$ is a prime of bad reduction), then
\begin{equation}
c_{p^m} = \big(\alpha(p)^m + \beta(p)^m\big)\log p,
\end{equation}
and $c_n = 0$ when $n$ is not a prime power.
Our main tool will be the explicit formula for $L(s, E)$, which we state in a
friendly form in the following lemma.
\begin{lemma}
Suppose that $f(z)$ is an entire function with $f(x + iy) \ll x^{-(1 + \delta)}$ for $\abs{y} < 1 + \epsilon$,
for some $\epsilon > 0$, and that the Fourier transform of $f$
\[
\hat f(y) = \int_{-\infty}^\infty f(x)e^{-2 \pi i x y} \dx
\]
exists and is such that
\[
\sum_{n=1}^\infty \frac{c_n}{n^{1/2}} \hat f\left(\frac{\log n}{2\pi}\right)
\]
converges absolutely. Then
\begin{multline}\label{eq-xxx-general}
\sum_{\gamma} f(\gamma) = \hat f(0) \frac{\log N}{2\pi} - \hat f(0) \frac{\log 2\pi}{\pi}
+ \frac{1}{\pi} \Re \left\{\int_{-\infty}^\infty \frac{\Gamma'}{\Gamma}(1 + it)f(t) \ud t\right\} \\
- \frac{1}{2\pi}\sum_{n = 1}^\infty \frac{c(n)}{n^{1/2}}\left(\hat f\left(\frac{\log n}{2\pi}\right)
+ \hat f\left(-\frac{\log n}{2\pi}\right) \right),
\end{multline}
where $1/2 + i\gamma$ runs over the nontrivial zeros of $L(s, E)$, where $E$ is an elliptic
curve with conductor $N$.
\end{lemma}
\begin{proof}
A proof of the explicit formula in this form, or in a similar form, can be found in various
sources, e.g. \cite[Theorem 5.12]{iwaniec-kowalski}, so we give only a brief sketch. The
idea is to integrate the function
\[
F(s) \frac{L'(s, E)}{L(s, E)},
\]
where $F(1/2 + is) = f(s)$, on a vertical line to the right of the critical strip and, in the
reverse direction, on a vertical line to the left of the critical strip. By the residue
theorem, this integral will be equal to $2\pi \sum_\gamma f(\gamma)$. One now applies the
functional equation to write the integral in the left half-plane as an integral in the right
half-plane.
The sum over the Fourier coefficients of $f$ arises from shifting contours to the region
of absolute convergence and using the Dirichlet series for $L'(s)/L(s)$, while the other
terms arise from shifting the remaining integrals to the line $\Re(s) = 1/2$.
The conditions on $f(z)$ are exactly those needed to make sure that this process can go
through without trouble. Of course, it is also important that $L(s, E)$ is entire
and that it satisfies a functional equation \cite{W, TW, BCDT}.
\end{proof}
A convenient function to use in an application of the explicit formula is
\[
f(z) = f(z; \Delta) = \left(\frac{\sin(\Delta \pi z)}{\Delta \pi z}\right)^2,
\]
which has the simple Fourier transform
\[
\hat f(x; \Delta) = \left(\frac{1}{\Delta}\right)\left(1 - \abs{\frac{x}{\Delta}}\right), \abs{x} < \Delta.
\]
With this choice of $f$, equation \eqref{eq-xxx-general} takes the form
\begin{multline}\label{eq-xxx-specific}
\sum_{\gamma} f(\gamma; \Delta) = \frac{\log N}{\Delta 2\pi} - \frac{\log 2\pi}{\Delta \pi}
+ \frac{1}{\pi} \Re\left\{\int_{-\infty}^\infty \frac{\Gamma'}{\Gamma}(1 + it)f(t; \Delta)\ud t\right\} \\
- \frac{1}{\Delta \pi} \sum_{p \le \exp(2\pi\Delta)} \log p \sum_{k=1}^{\floor{2\pi\Delta/\log p}} \frac{k}{p^{k/2}}\big(\alpha(p)^k + \beta(p)^k\big)\left(1 - \frac{k\log p}{2\pi\Delta}\right).
\end{multline}
Since $f(\gamma; \Delta) \ge 0$ as long as $\gamma$ is real, and $f(0; \Delta) = 1$,
equation \eqref{eq-xxx-specific} will give an upper bound for the order
of vanishing of $L(s, E)$ at $s = 1/2$, as long as the Riemann Hypothesis holds
for $L(s, E)$. And if
$\Delta$ is not too large, we can quickly evaluate the right hand side of equation
\eqref{eq-xxx-specific} to calculate this upper bound. It is also worth
noting that, assuming RH,
\begin{multline*}
-\lim_{\Delta \rightarrow \infty} \frac{1}{\Delta \pi} \sum_{p \le \exp(2\pi\Delta)} \log p \sum_{k=1}^{\floor{2\pi\Delta/\log p}} \frac{k}{p^{k/2}}\big(\alpha(p)^k + \beta(p)^k\big)\left(1 - \frac{k\log p}{2\pi\Delta}\right) \\
= \mathrm{ord}_{s = 1/2} L(s, E)
\end{multline*}
so that, in principle, we should be able to get as good a bound for the rank as we like
through this method. However, as the length of the prime sum grows exponentially in $\Delta$,
this method quickly becomes infeasible once $\Delta$ gets a little large than $4$.
\subsection{Some curves}
As an example, we examine $6$ elliptic curves from Dujella's online tables. They are
\begin{multline*}
E_{20}: y^2 + xy = x^3 - 431092980766333677958362095891166x \\
+ 5156283555366643659035652799871176909391533088196,
\end{multline*}
\begin{multline*}
E_{21}: y^2 + xy + y = x^3 + x^2 - 215843772422443922015169952702159835x \\
- 19474361277787151947255961435459054151501792241320535,
\end{multline*}
\begin{multline*}
E_{22}: y^2 + xy + y = x^3 - 940299517776391362903023121165864x \\
+ 10707363070719743033425295515449274534651125011362,
\end{multline*}
\begin{multline*}
E_{23}: y^2 + xy + y = x^3 - 19252966408674012828065964616418441723x \\
+ 32685500727716376257923347071452044295907443056345614006,
\end{multline*}
\begin{multline*}
E_{24}: y^2 + xy + y = x^3 - 120039822036992245303534619191166796374x \\
+ 504224992484910670010801799168082726759443756222911415116,
\end{multline*}
and
\begin{multline*}
E_{28}: y^2 + xy + y = x^3 - x^2 -
{20067762415575526585033208 \times 10^{30} \choose +\ 209338542750930230312178956502}x \\
+ {3448161179503055646703298569039072037485594 \times 10^{40} \choose
+\ 4359319180361266008296291939448732243429}.
\end{multline*}
Each $E_n$ has $n$ known independent rational points of infinite order, so thus has at least
rank $n$.
(See \cite{rank20-curve, rank21-curve, rank22-curve, rank23-curve, rank24-curve, rank28-curve},
or \cite{rank-records} for quick reference.)
Using the methods described above, we compute rank
bounds for each of these curves. These are listed in Table \ref{table1}. The global root number
can be computed for each curve. (In \sage, {\tt E.root\_number()}, which
uses \pari{ }\cite{pari-2.4.3}, will finish quickly for
$E_{20}$, $E_{21}$, and $E_{22}$ and within a few hours for $E_{23}$ and $E_{24}$. For $E_{28}$
it is best to see the mailing list discussion which gives the factorization of the
discriminant \cite{rank28-discussion}.) In each case the root number agrees with the parity
of the known number of independent points, so to get a tight upper bound for the rank
we only need to get within $2$ of the number of known independent points, and so
the computation in Table \ref{table1} gives the proof of Theorem \ref{main-theorem}.
\begin{table}
\begin{tabular}{c|c|c|c|c}
Curve & $\log N_E$ & $\Delta$ & $\sum_{\gamma} f(\gamma; \Delta)$ & $\frac{\log N_E}{2\pi\Delta}$\\\hline
$E_{20}$ & $170.09$ & $2.0$ & $21.70$ & $13.54$ \\
$E_{21}$ & $196.68$ & $2.5$ & $22.68$ & $12.52$ \\
$E_{22}$ & $182.72$ & $2.0$ & $23.71$ & $14.54$ \\
$E_{23}$ & $205.06$ & $2.5$ & $24.49$ & $13.05$ \\
$E_{24}$ & $219.93$ & $2.5$ & $25.57$ & $14.00$ \\
$E_{28}$ & $325.90$ & $3.2$ & $31.30$ & $16.21$
\end{tabular}
\vspace{.1in}
\caption{Computed upper bounds for the ranks of some curves, along with a heuristic
guess of what these bounds should for a typical elliptic curve. The sum over the zeros here is
rounded up; other numbers are rounded to nearest.} \label{table1}
\end{table}
\subsection{Curves of small conductor}\label{section-small-conductor}
For further testing, this method was also run
on all elliptic curves up with conductor below $180000$
(from Cremona's tables \cite{cremona-online})
using $\Delta = 2.0$, a computation
which ran in under a day on a fast $8$ core computer. In this range there are $790677$
isogeny classes of elliptic curves, and for all but $9882$ isogeny classes it
turns out that $\floor{\sum_{\gamma} f(\gamma; 2.0)} = \mathrm{rank}(E)$; in the remaining
cases, $\floor{\sum_{\gamma} f(\gamma; 2.0)} = \mathrm{rank}(E) + 1$, so consideration
of the root number of the curve gives the exact rank.
\section{Further comments}
\subsection{Some evidence towards BSD}
There is a way in which these computations can be seen as giving mild evidence in support of the
Birch and Swinnerton-Dyer conjecture. The upper bound computed for a curve $E$ is the
value of the sum $\sum_\gamma f(\gamma; \Delta)$, and as $f(\gamma; \Delta)$
decays fairly rapidly as $\gamma$ grows, one does not expect this sum to be very large
for a typical elliptic curve.
To obtain a crude approximation to what we might expect the value of this sum to be, consider
that the local zero density of a typical $L(s, E)$ near the central point is approximately
$\frac{2\pi}{\log N_E}$. Then, if the zeros are spaced uniformly at random (an assumption
that is not really correct, but is close enough to true for our crude purposes), we might expect
that
\[
\sum_{\gamma} f(\gamma, \Delta) \approx \frac{\log N_E}{2\pi} \int_{-\infty}^\infty f(t; \Delta) \ud t
= \frac{\log N_E}{2\pi\Delta},
\]
possibly with a small adjustment to take into account the parity of the rank. (More precisely,
we might expect that if we average this sum over all elliptic curves of conductor close to $N_E$,
the answer will not be too far from this integral.) Thus, when this sum is significantly larger
than this estimate, it indicates an extreme concentration of zeros near the central point. (It
is also possible to arrive at more refined version of this heuristic by considering the explicit
formula. In such a case, it is necessary to assume that the family of elliptic curves considered
is large enough that $a_p(E)$ averages to zero for each $p$, and we notice that the integral
of the $\Gamma$-factor plays a small role as well.)
As some further small evidence for this heuristic, we note that the average of
\[
\frac{4 \pi}{\log N} \sum_{\gamma} f(\gamma; 2.0)
\]
over all isogeny classes up to $180000$ is approximately $.9638$. The small difference from
$1$ should be accounted for by the $\Gamma$-factor, which tends to push zeros away from the
central point.
It should also be possible to refine this heuristic somewhat to make a guess as to what
the sum should be for a high rank curve by making the assumption that a zero of high
order at the central point will push other zeros away.
\subsection{Correctness tests}
The method described here is simple enough that it is easy to implement, which reduces the
likeliness of bugs. It is still important to test it where possible, however, in order to
have more confidence in its correctness.
As described in Section \ref{section-small-conductor}, this code was run on every isogeny
class up to conductor $180000$, and
the results there suggest a high degree of confidence in the results elsewhere. As a further test,
one can also compute many zeros for the $L$-function of an elliptic curve of small conductor,
compute the sum over zeros directly, and verify that it agrees with our explicit formula
implementation. This was done with the elliptic curve ``11a1'' for a few values of $\Delta$,
and little over $200000$ zeros (computed using M. Rubinstein's lcalc package \cite{lcalc}),
and the agreement is generally to within about $10^{-6}$, which
is in line with what is expected using only $200000$ zeros, and which is roughly the precision
to which the integral in the explicit formula was calculated. Similar tests have also been done
with a smaller number of zeros for other $L$-functions.
|
2,877,628,090,781 | arxiv | \section{Introduction}
Since its introduction by Gale and
Shapely~\cite{GaleS62}, the Hospital
Residents problem, which is also known as the College Admission problem,
has
attracted a lot of attention. Besides a rich body of theoretical
work~\cite{DBLP:books/ws/Manlove13}, also
many
practical applications have been
identified~\cite{DBLP:journals/ijgt/Roth08}.
Applications include various centralized assignment problems, for example,
in
the context of education
\cite{10.1257/000282805774670167,biroGS,Zhang2010AnalysisOT} or in the
context of assigning
career starters to their first work
place~\cite{RePEc:inm:orinte:v:33:y:2003:i:3:p:1-11,RePEc:ucp:jpolec:v:92:y:1984:i:6:p:991-1016}.
In the classical
Hospital Residents problem (\textsl{HR-$\text{Q}^\text{U}$}\xspace), we are given a set of residents, each
with strict
preferences over hospitals, and a set of hospitals,
each
with an upper quota and strict preferences over residents.
In a
feasible matching of residents to hospitals, the number of residents that
are assigned to a hospital is at most its
upper quota. A hospital-resident pair~$(h,r)$ blocks a matching~$M$ if
resident~$r$
prefers hospital~$h$ to the hospital to which $r$ is matched by~$M$ and
the number of residents
matched to $h$ is below its upper quota or $h$
prefers~$r$
to one of the residents
matched to it.
The task in the Hospital Residents problem is to find a stable
matching, i.e., a feasible matching that does not admit a blocking pair.
Gale and Shapely~\cite{GaleS62} presented
a linear-time
algorithm that always finds a stable matching in a Hospital Residents instance.
In practice, some hospitals may also
have a lower quota, i.e., a minimum number of assigned residents such that
the hospital can open and
accommodate them. For example, due to economical or
political reasons, a university might have a
lower quota on the number of students for each course of study. Moreover,
practical considerations might impose lower quotas as well, for instance, if
residents assigned to a hospital need to perform certain tasks together
which require at least a given number of participating residents.
Bir{\'{o}} et
al.~\cite{DBLP:journals/tcs/BiroFIM10}
captured these considerations by extending the Hospital
Residents problem such that each hospital has a lower and upper quota (\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace).
Here, feasibility additionally requires that the number of residents
assigned to a hospital is either zero or at least its lower quota, while
stability additionally requires that there does not exist a blocking
coalition, i.e., a sufficiently large subset of
residents that
want to open a currently closed hospital together.
Bir{\'{o}} et
al.~\cite{DBLP:journals/tcs/BiroFIM10} proved that deciding the existence
of a stable
matching in an~\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance is
NP-complete. We
complement their work with a thorough parameterized complexity analysis of
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace and closely related problems, considering various problem-specific
parameters. Moreover, we
study the special case of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace where hospitals have only a lower quota (\textsl{HR-$\text{Q}_\text{L}$}\xspace), which
has not been considered before.
Lower and upper quotas have also been
applied to
the House Allocation problem~(\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace)
where the goal is to match a set of applicants to a set of houses
\cite{DBLP:journals/mss/CechlarovaF17,DBLP:journals/orl/Kamiyama13,MONTE201314}.
In~\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace,
houses have a lower and upper quota but no preferences over
applicants, while applicants have preferences over houses. One
possible application
of this model is the assignment of kids to
different
activities, where lower quotas could arise due to
economical or practical constraints,
for
instance, playing soccer with only three kids is
less fun.
So far, literature on
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace mainly focused on finding Pareto optimal matchings.
However, in
contrast to the
classical House Allocation problem,
Pareto optimality in \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace does
not imply stability. Thus, finding stable matchings here is an interesting
problem on its own. For notational convenience, we also refer to houses as
hospitals and to applicants as residents in context of \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace.
\subsection{Our Contributions}
We provide an extensive parameterized complexity analysis of the Hospital
Residents
problem with lower and upper quotas (\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace) and of the two closely related
problems~\textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace. We also (briefly)
consider a generalization
of these models where we allow for ties in the preferences (\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace,
\textsl{HR-$\text{Q}_\text{L}$-T}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace, respectively).
By applying the framework of parameterized
complexity, we
analyze the influence of various problem-specific parameters such as the number of residents or the number of hospitals on the
complexity of these problems.
Motivated by
the observation that there might exist stable matchings opening a different
set of hospitals
(of possibly different sizes)~\cite{DBLP:journals/tcs/BiroFIM10}, we also
consider the problem of
deciding whether there exists a stable matching where exactly a given set
of hospitals~$H_{\open}$ is open and the problem of deciding whether there
exists a
stable matching with exactly $m_{\open}$ ($m_{\text{closed}}$) open
(closed)
hospitals parameterized by $m_{\open}$ ($m_{\text{closed}}$).
We present an overview of our results in \Cref{ta:sum}. Our most
important technical contribution
is the design of a polynomial-time algorithm for \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace (and therefore also
for \textsl{HR-$\text{Q}_\text{L}$}\xspace) instances
where all hospitals have lower quota at most two. This answers an open
question raised by Bir\'{o} et al. \cite{DBLP:journals/tcs/BiroFIM10} and by Manlove
\cite[p.~298]{DBLP:books/ws/Manlove13}. Such \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instances are of
special
theoretical interest, as they, for example, subsume a variant of
three-dimensional \textsc{Stable Marriage}, where,
given two sets
of agents each with preferences over the agents from the other set,
the
goal
is to find a stable set of triples, each consisting of two agents from the
first and one
agent from the second set. Moreover, there also exist several
applications where
a lower quota of two is of particular interest, for example, assuming
that hospitals correspond to (tennis) coaches and residents to (tennis)
players, a coach may require that at least two players are assigned to her
(as she does not always want to play herself).
\begin{table}[t!]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\setlength\tabcolsep{4pt}
\resizebox{\textwidth}{!}{\begin{tabular}{c|c|c|c|c|c|c|c|c} \hline
& $q_l\leq 2$ & $q_l = 3$ & $H_{\open}$ &$n$ & $m$ &
$m_{\quota}$ &
$m_{\open}$ & $m_{\text{closed}}$ \\ \hline\hline
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace & \multirow{1}{*}{P} (T. \ref{t:q2}) &
\multirow{6}{*}{\shortstack{NP-c. \\ (T. \ref{th::NP-compl})}}
&P (P. \ref{pr:HRULQ-H'})& \multirow{6}{*}{\shortstack{W[1]-h.
\\ (T.
\ref{th:W-n})}} &
\multicolumn{2}{c|}{FPT (C. \ref{co:mFPT})}
&
W[1]-h. (C. \ref{c:op}) &
\multirow{4}{*}{\shortstack{W[1]-h. \\(T.
\ref{thm:ha-m-closed})}}
\\
\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace & NP-c. (P. \ref{p:q2T})&&NP-c. (T.
\ref{th:HRLUQIH'})&& \multicolumn{2}{c|}{W[1]-h. (P.
\ref{pr:HRT-W})}& para-h. (C. \ref{c:HAco})& \\
\cline{1-2} \cline{4-4}\cline{6-8}
\textsl{HR-$\text{Q}_\text{L}$}\xspace & P (T. \ref{t:q2}) & &
P (O. \ref{ob:HRLQ-H'})& &
\multicolumn{2}{c|}{FPT (C. \ref{co:mFPT})}
& \multirow{2}{*}{\shortstack{W[1]-h.\\ (C. \ref{c:op})}}
&
\\
\textsl{HR-$\text{Q}_\text{L}$-T}\xspace & NP-c. (P. \ref{p:q2T})&&P (P.
\ref{pr:HRLQT-H'})&&\multicolumn{2}{c|}{FPT (P.
\ref{pr:LTies})}&& \\ \cline{1-2}
\cline{4-4}
\cline{6-9}
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace & \multirow{2}{*}{\shortstack{NP-c. \\(P.
\ref{pr:HRLUQI-NP-const})}} &&
\multirow{2}{*}{\shortstack{NP-c.\\
(T.
\ref{th:HRLUQIH'})}} & &
FPT (P.
\ref{pr:HRLUQI-FPTM}) &
\multirow{2}{*}{\shortstack{para-h. \\(P.
\ref{pr:HRLUQI-NP-const})}}&
\multirow{2}{*}{\shortstack{para-h. \\(C. \ref{c:HAco})}} &
\multirow{2}{*}{\shortstack{W[1]-h. \\(C. \ref{c:HAco})}} \\
\textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace & && & & FPT (C.
\ref{co:HRLUQIT-m})
&&
& \\ \hline
\end{tabular}}
\caption{Overview of our results.
The maximum lower quota of a hospital is denoted by~$q_l$, the
number of residents by~$n$, the number of hospitals by~$m$, the
number of hospitals with lower quota larger than 1 by~$m_{\quota}$,
the number of open hospitals by $m_{\open}$, and the number of
closed hospitals by $m_{\operatorname{closed}}$.
``P'' stands for polynomial-time solvability, ``NP-c.'' for NP-completeness, ``FPT'' for fixed-parameter tractability, ``W[1]-h.'' for W[1]-hardness, and ``para-h.'' for paraNP-hardness with respect to the corresponding parameter.
With the exception of the parameter
$m_{\text{closed}}$ for \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace, and \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace, we
present XP algorithms for
all
W[1]-hard cases.
}
\label{ta:sum}
\end{center}
\end{table}
Our rich set of tractability
and intractability results allows us to draw several high-level conclusions
about the considered problems.
First, our results highlight the differences between the three considered
models from a computational perspective: While~\textsl{HR-$\text{Q}_\text{L}$}\xspace is very
similar to \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace,
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace is computationally more demanding than \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace. The first observation
suggests that the complexity of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace comes solely from the
lower quotas of hospitals. The second observation
indicates that the hospitals' preferences in the lower and upper quotas
setting make the problem easier, as
they may act as a ``tie-breaker'' to decide which resident deserves a
better spot in a stable matching.
Second, our results identify the ``difficult parts'' of the
considered
problems. Considering that for \textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, we can decide
in
polynomial time whether there exists a stable matching opening exactly a
given set of
hospitals, the complexity of \textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace comes purely from
deciding which hospitals to open and not from the
task of assigning residents to hospitals. This
finding is strengthened by the observation that most of our
hardness reductions
also work if we ignore blocking pairs, i.e., for
the problem of assigning residents
to hospitals such that the lower quota of each hospital is respected and no
blocking coalition of residents to open a closed hospital exists.
Third, our results identify the fine line between
tractability and intractability. For example, parameterizing the three
problems by the number of hospitals leads to fixed-parameter tractability,
while only considering the number of open or closed hospitals in a stable
matching as a parameter results in W[1]-hardness.
A similar contrast arises when considering the
influence of the total
number of hospitals and the number of hospitals with non-unit lower quota
on~\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace.
Fourth, we analyze what happens if we allow for ties in the preferences. In
this case, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace generalizes both \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace and \textsl{HR-$\text{Q}_\text{L}$-T}\xspace. This is also
reflected in the
complexity of the three problems parameterized by the number $m$ of
hospitals: While for \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace and \textsl{HR-$\text{Q}_\text{L}$-T}\xspace it is possible to construct a
fixed-parameter tractable algorithm by bounding the number
of resident types in a function of $m$ (hospitals are indifferent
between residents), \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace is W[1]-hard parameterized by $m$. To prove
this, as a side
result, we establish
that the problem of deciding whether a \textsl{HR-$\text{Q}^\text{U}$-T}\xspace instance admits a stable
matching which matches all residents is W[1]-hard parameterized by $m$.
\subsection{Related Work}
After the work of Bir{\'{o}} et al.~\cite{DBLP:journals/tcs/BiroFIM10},
only few papers revisited computational problems related to the NP-hard
Hospital
Residents problem with lower and
upper quotas.
A notable exception is the work of
Agoston et al.~\cite{DBLP:journals/jco/AgostonBM16} who proposed an ILP
formulation to find stable matchings and several
preprocessing rules to decide which hospitals must be open in a stable
matching.
Apart from this, most of the
follow-up work applied the idea of lower and upper quotas to other
settings, such as the House Allocation problem
\cite{DBLP:journals/mss/CechlarovaF17,DBLP:journals/orl/Kamiyama13,MONTE201314} or maximum-weight many-to-one matchings in bipartite graphs~\cite{DBLP:journals/algorithmica/ArulselvanCGMM18},
or interpreted it differently.
For example, Arulselvan et
al.~\cite{DBLP:journals/algorithmica/ArulselvanCGMM18}
studied finding maximum-weight many-to-one matchings in bipartite
graphs where vertices on one side of the bipartition have a lower and upper quota. They
conducted a parameterized
analysis of the resulting computational problems and studied quota-
and degree-restricted cases.
Hamada et al.~\cite{DBLP:journals/algorithmica/HamadaIM16} introduced
an alternative version of the Hospital Residents problem with lower and
upper quotas. In their model, hospitals have lower and upper quotas,
but are not allowed to be closed. Thus, in a feasible matching, the
lower and upper quota of each hospital
needs to be respected. As deciding whether a stable matching
exists is polynomial-time solvable for this model, their main focus lied on
finding a feasible matching
minimizing the number of blocking pairs. Mnich et
al.~\cite{DBLP:journals/algorithmica/MnichS20} studied the \textsc{Stable
Marriage with Covering Constraints} problem, which corresponds to the special case
of Hamada et al.'s model where each hospital
has unit upper quota, from a parameterized perspective considering parameters
such as the number of
blocking pairs and the
number of hospitals with non-zero lower quota. To capture stable matching
problems
with
diversity or distributional constraints, the model of Hamada et
al.~\cite{DBLP:journals/algorithmica/HamadaIM16} has been adapted and
further developed in various directions, for example, by assuming that
residents belong to different types and each hospital has type-specific lower and upper
quotas
\cite{DBLP:conf/atal/0001GSW19,DBLP:journals/jet/EhlersHYY14,DBLP:journals/jair/KurataHIY17}.
Another popular stable matching problem is the Hospital Residents problem
with
couples~(\textsl{HRC})~\cite{DBLP:conf/wea/BiroMM14,DBLP:journals/disopt/MarxS11},
where some of the
residents are grouped in pairs and submit their preferences together.
The \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace problem where all hospitals have upper quota at most
two is closely related to the special case of \textsl{HRC}
where all hospitals have upper quota one:
Switching the roles of residents and hospitals and interpreting couples as
hospitals with lower quota two, the only difference between the two problems is
that the preferences of couples are over pairs of hospitals, while the
preferences of quota-two hospitals are over single residents. Notably,
\textsl{HRC} is already NP-hard in the
described special case \cite{DBLP:conf/wea/BiroMM14}, which is in sharp
contrast to our polynomial-time algorithm which also applies for \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace where
all hospitals have upper quota at most two.
From a technical perspective, our work falls in line with previous work on
the parameterized complexity of stable matching problems
\cite{DBLP:journals/tcs/AdilGRSZ18,DBLP:conf/sagt/BoehmerBHN20,DBLP:conf/isaac/BredereckHKN19,DBLP:journals/algorithmica/MarxS10,DBLP:journals/disopt/MarxS11,MeeksR20,DBLP:journals/algorithmica/MnichS20}.
\subsection{Structure of the Paper}
The paper is structured as follows.
We start with the preliminaries in \Cref{se:pre}.
In \Cref{sec:strict}, we analyze the parameterized complexity of the
considered Hospital Residents problems with lower and upper quotas with
respect to several parameters related to the number of residents or
hospitals.
In \Cref{sec:q2}, we present a polynomial-time algorithm for the Hospital Residents problem with lower and upper quotas if the lower quota of each hospital is at most two.
Afterwards, we extend our model by allowing ties in the preferences in
\Cref{se:ties}.
\section{Preliminaries} \label{se:pre}
We consider different models of stable bipartite many-to-one
matchings. For the sake of readability, we refer to all of them as
different
variants of the Hospital Residents problem with lower and upper quotas
(\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace).
In \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, we are given a set~$R=\{r_1,\dots r_n\}$ of
\emph{residents} and a set $H=\{h_1,\dots,
h_m\}$ of \emph{hospitals}, each with a lower and upper quota.
Throughout the paper, $n$ denotes the number of residents and~$m$ the
number of hospitals.
We refer to the joint set of
residents and hospitals as
\emph{agents}.
Each resident $r\in R$ \emph{accepts} a subset of hospitals $A(r)\subseteq
H$ and
each hospital $h\in H$ \emph{accepts} a subset of residents~$A(h) \subseteq
R$. We assume that acceptability is symmetric,
that is, $h\in A(r)$ if and only if $r\in A(h)$.
Each agent $a\in R\cup H$ has a preference list $\succ_a$ in which all
agents
from~$A(a)$ are ranked in strict order. For three agents $a$, $a_1$, and
$a_2$,
we say that $a$ \emph{prefers}~$a_1$ to $a_2$ and write $a_1\succ_a a_2$ if
$a_1,a_2\in A(a)$ and $a$ ranks $a_1$ above $a_2$.
A \emph{matching} $M$ is a subset of $R\times H$ where each resident
is contained in at most one pair and for each pair $(r,h)\in M$,
agents $r$ and $h$ accept each other. For a matching~$M$ and a resident~$r\in
R$, we denote by $M(r)$ the hospital to which $r$ is
matched
in~$M$,~i.e.,~$M(r)=h$ if $(r,h)\in M$, and we set $M(r):=\square$
if $r$ is not matched, i.e., $r\neq r'$ for all $(r', h) \in M$. All residents $r$ prefer each
hospital~$h\in A(r)$ to being unmatched, i.e., to
$M(r)=\square$.
Further, for a hospital~$h\in H$, we denote
by $M(h)$ the set of residents
that are matched to $h$, i.e., $r\in M(h)$ if $(r,h)\in M$. We
sometimes
write $M$ as a set of pairs of
the form $(h,\{r_1,\dots, r_k\})$ which denotes that the residents
$r_1,\dots, r_k$ (and possibly other residents if $h$ appears in more than
one
tuple) are matched to hospital $h$ in~$M$.
In \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, each hospital $h\in H$ has an upper quota
$u(h)$ and a lower quota $l(h)$ with $1\leq l(h)\leq u(h)$. We call a
matching $M$ \emph{feasible} if,
for all hospitals $h\in H$, it either holds that $|M(h)|=0$ or
$l(h)\leq|M(h)|\leq u(h)$. We say that a hospital $h\in H$ is \emph{closed}
in $M$ if $|M(h)|=0$ and we say that it is \emph{open} otherwise. Moreover,
we call an open hospital $h\in H$ \emph{full} if $|M(h)|=u(h)$ and an open
hospital~$h\in H$ \emph{undersubscribed} if~$|M(h)|<u(h)$.
In a matching~$M$, a hospital-resident pair~$(r, h)\in R\times H$ is a
\emph{blocking pair} if
$h$ is open in $M$, both~$r$ and $h$ find each other acceptable, $r$
prefers $h$ to $M(r)$, and~$h$ is
either undersubscribed or prefers~$r$ to
at least one resident
from $M(h)$.
Moreover, we call $(h,\{r_1,\dots, r_k\})$ with $k=l(h)$ a \emph{blocking
coalition} if
$h$ is closed in $M$ and, for all~$i\in [k]$, resident $r_i$
prefers $h$ to $M(r_i)$.
In this case, we also write that $\{r_1,\dots,
r_k\}$ forms a blocking coalition to open $h$. A
feasible matching is called \emph{stable} if it neither admits a blocking
pair nor a blocking coalition.
It is possible to express different problems related to the classical
\textsc{Stable Marriage} problem~\cite{GaleS62} in this framework.
For example, the
\textsc{Stable Marriage with Incomplete Lists} problem corresponds
to finding
a stable matching in an \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance where all hospitals have upper
quota one.
We also consider the
Hospital Residents problem with upper and lower quotas and ties (\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace).
In this case, the preference list $\succsim_{a}$ of each agent $a\in R\cup
H$ is a
weak order
over the set of agents they accept. We say that an agent~$a$
is \emph{indifferent} between two agents $a_1$ and $a_2$ and write
$a_1\sim_a a_2$ if both $a_1 \succsim_a a_2$ and $a_2 \succsim_a a_1$.
Further, we say that an agent $a$ \emph{prefers} agent~$a_1$ to
agent $a_2$ and write $a_1\succ_a a_2$ if $a_1\succsim_{a} a_2$ and $a$ is
not indifferent between $a_1$ and $a_2$. Note
that the resulting stability notion is known as weak stability in the
literature \cite[Chapter 3]{DBLP:books/ws/Manlove13}.
We now describe how the other two models considered in this paper can be
formulated as variants of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace.
\paragraph{House Allocation problem with lower and upper quotas.}
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace corresponds to \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace with one-sided
preferences,
i.e., all hospitals are indifferent among all
residents and residents have strict preferences over hospitals. While the definition of a blocking coalition still applies in this setting, a hospital-resident pair $(r, h)\in R\times H$ is only
\emph{blocking} if
$h$ is open in $M$, $r$ accepts~$h$, resident $r$
prefers $h$ to~$M(r)$, and~$h$ is
undersubscribed. Note that
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace does not subsume \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, as, in
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, no ties in the preferences are allowed.
However, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace subsumes \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace by setting the preference of each hospital to be a tie involving all acceptable residents.
\paragraph{Hospital Residents problem with lower quotas.} \textsl{HR-$\text{Q}_\text{L}$}\xspace is
the special case of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace where each hospital has upper quota $n+1$.
Thereby, no hospital can be full in a matching. Consequently, in a
matching~$M$, a resident $r$ forms a blocking pair with each open
hospital~$h$
she prefers to $M(r)$. Thus, in every stable matching, all residents
need to be matched to their most preferred open hospital. This in
turn implies that the preferences of hospitals over residents can be
omitted, as they have no
influence on the stability of a matching.
Hence, \textsl{HR-$\text{Q}_\text{L}$}\xspace is equivalent to the House
Allocation problem with lower quotas $(=\textsl{HA-$\text{Q}_\text{L}$}\xspace)$ and thus lies in the ``intersection'' of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace.
\paragraph{First observations.}
\label{se:obs}
As already observed, \textsl{HR-$\text{Q}_\text{L}$}\xspace instances can be expressed both as~\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace and
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instances.
Notably, most instances constructed in our reductions fulfill an additional
property which directly transfers the hardness results to a variant
of~\textsl{HR-$\text{Q}_\text{L}$}\xspace
where only blocking coalitions may make a matching unstable.
\begin{observation} \label{ob:equiv}
In \textsl{HR-$\text{Q}_\text{L}$}\xspace instances where for each hospital $h\in H$, the
number of
residents
accepting $h$ is equal to its lower quota $l(h)$, no feasible matching admits a blocking pair,
while a feasible matching may admit blocking coalitions.
\end{observation}
Unfortunately, a stable matching may fail to exist in \textsl{HR-$\text{Q}_\text{L}$}\xspace instances (and
therefore also in \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instances),
even if all hospitals have lower quota
at most two:
\begin{observation}\label{ob:counter}
Let $R=\{r_1,r_2,r_3\}$ and $H=\{h_1,h_2,h_3\}$ be a \textsl{HR-$\text{Q}_\text{L}$}\xspace instance
with each hospital having lower quota two and the following
preferences: $r_1: h_1 \succ h_2;$ $r_2: h_2 \succ h_3;$ $r_3: h_3
\succ h_1.$ This instance does not admit a stable matching.
\end{observation}
Note that this example resembles the Condorcet paradox.
In the following hardness reductions, we
will frequently use this construction
as a penalizing component to ensure that certain residents
are matched to some designated set of hospitals in a stable matching.
For a visualization of the example, see \Cref{fig:counter}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node[vertex, label=180:$r_1$] (r1) at (0, 2) {};
\node[vertex, label=180:$r_2$] (r2) at (0, 1) {};
\node[vertex, label=180:$r_3$] (r3) at (0, 0) {};
\node[squared-vertex, label=90:$h_3$, label=0:{$[2,\infty]$}]
(h3) at (2.5,0) {};
\node[squared-vertex, label=90:$h_2$, label=0:{$[2,\infty]$}]
(h2) at (2.5,1) {};
\node[squared-vertex, label=90:$h_1$, label=0:{$[2,\infty]$}]
(h1) at (2.5,2) {};
\draw (h1) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $1$} (r1);
\draw (h2) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $2$} (r1);
\draw (h2) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $1$} (r2);
\draw (h3) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $2$} (r2);
\draw (h3) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $1$} (r3);
\draw (h1) edge node[pos=0.78, fill=white, inner sep=2pt]
{\scriptsize $2$} (r3);
\end{tikzpicture}
\end{center}
\caption{Visualization of a \textsl{HR-$\text{Q}_\text{L}$}\xspace instance without a stable matching
(\Cref{ob:counter}).}
\label{fig:counter}
\end{figure}
In this and all
following
figures, each resident is represented by a dot and each hospital by a
square. The quotas of each hospital $h\in H$ are depicted next to the
corresponding square as $[l(h),u(h)]$. The preferences of the agents are
encoded on the edges:
The number on an edge between agent $a$ and agent $a'$ that is closer to
agent
$a$ indicates the rank of $a'$ in $a$'s preference relation, that is, the
number of agents $a$ prefers to $a'$ plus one. Note that in this
example, hospitals do not have preferences.
Moreover, for all
three models, there might exist stable matchings with a different number of
open/closed hospitals and a different number of assigned residents.
Consider an \textsl{HR-$\text{Q}_\text{L}$}\xspace instance consisting of one
hospital~$h_1$ with lower quota
one, one hospital $h_2$ with lower quota two, and one hospital~$h_3$ with
lower quota four together with four residents with
the following preferences:
$
r_1: h_3 \succ h_1;$ $
r_2: h_2 \succ h_3;$ $
r_3: h_3 \succ h_2;$ $r_4:h_3.
$
This instance admits two stable
matchings~$M_1=\{(h_3,\{r_1,r_2,r_3,r_4\})\}$
and
$M_2=\{(h_1,\{r_1\}),(h_2,\{r_2,r_3\})\}$.
\paragraph*{Basic concepts of parameterized complexity.}
A \emph{parameterized problem}~$P\subseteq \{0,1 \}^* \times \mathbb{N}$ is \emph{fixed-parameter tractable} if there exists an algorithm running in
time~$f(k)|\mathcal{I}|^{O(1)}$ for some computable function~$f$ which decides every instance~$(\mathcal{I}, k)$ of $P$. Moreover, a parameterized problem~$P$ is in XP if there exists an algorithm running in time $|\mathcal{I}|^{f(k)}$ for some computable function~$f$ which decides every instance~$(\mathcal{I}, k)$ of~$P$.
There is also a theory of hardness of parameterized
problems that includes the notion of W$[1]$-hardness. If a parameterized problem is
W[$1$]-hard, then it is widely believed
not to be
fixed-parameter tractable. The usual approach to prove W[$1$]-hardness of
a given parameterized problem is
to reduce a known W[$1$]-hard problem to it, using a
parameterized reduction. In this paper, we do not use parameterized reductions
in full generality but only a special case of
parameterized reductions, that is, standard many-one
reductions which run in polynomial time and fulfill that the parameter of the
output
instance is upper-bounded by a computable function of the parameter of the input
instance.
A parameterized problem is \emph{paraNP-hard} if it is NP-hard for a constant parameter value.
For more details on parameterized complexity, we refer to standard textbooks~\cite{DBLP:books/sp/CyganFKLMPPS15,DBLP:series/txcs/DowneyF13}.
\section{Parameterized Complexity}
\label{sec:strict}
In this section, we analyze the parameterized
computational complexity of \textsl{HR-$\text{Q}_\text{L}$}\xspace, \mbox{\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace},
and \mbox{\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace}.
We start by proving in \Cref{sec:NP-hard} that all three problems are
NP-complete.
Then, in \Cref{sec:resi}, we
analyze the influence of the number of residents on the complexity of the
three problems.
Lastly, in \Cref{se:hosp}, we consider the number of hospitals and various
parameters related to it such as the number of hospitals with non-unit
lower quota and the number of open (or closed) hospitals in a stable
matching.
\subsection{An NP-Completeness Result}\label{sec:NP-hard}
Bir{\'{o}} et al.~\cite{DBLP:journals/tcs/BiroFIM10} proved that
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace is
NP-complete, even if each hospital has upper quota at most three.
However, their reduction
does not settle the computational
complexity of~\textsl{HR-$\text{Q}_\text{L}$}\xspace or \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace.
To answer this question, note that \textsl{HR-$\text{Q}_\text{L}$}\xspace and therefore also \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace and
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace subsume hedonic games
(see
\cite{DBLP:reference/choice/AzizS16} for definitions):
We introduce a resident for
each agent in the given hedonic game and a hospital for each possible
coalition with lower quota equal
to the size of the coalition. We replace the coalitions in the agents'
preferences by the corresponding hospitals.
Core stable outcomes in the hedonic game then correspond
to
stable matchings in the constructed \textsl{HR-$\text{Q}_\text{L}$}\xspace instance, which notably falls
under
\Cref{ob:equiv}. As deciding the existence of a core stable outcome is
NP-complete, even if all coalitions
have size three
\cite{DBLP:journals/siamdm/NgH91}, this
implies that all three problems are NP-complete, even if each hospital has
lower quota (and upper
quota) at most three.
By slightly adopting the reduction from Ng and Hirschberg~\cite{DBLP:journals/siamdm/NgH91},
one can also bound the number of residents acceptable to a hospital and the
number of hospitals acceptable to a resident. For the sake of completeness
and as a warm-up to
illustrate the basic features of the three problems we consider, we present
here
an alternative reduction from
\textsc{Satisfiability}.
\begin{theorem}
\label{th::NP-compl}
\textsl{HR-$\text{Q}_\text{L}$}\xspace, \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace are NP-complete, even if each resident accepts
at most four hospitals, each hospital accepts at most three
residents, and the lower (and upper) quota of every hospital
is at most
three.
\end{theorem}
\begin{proof}
We reduce from the NP-hard variant of \textsc{Satisfiability} where
each
clause
contains
exactly three literals and each variable occurs exactly twice
positively
and twice negatively~\cite{DBLP:journals/eccc/ECCC-TR03-049}. An
instance of \textsc{Satisfiability} consists of
a
set
$X=\{x_1,\dots x_q\}$ of variables and a set $C=\{c_1,\dots c_p\}$ of
clauses. For
each
$i\in [q]$, we denote as $c^\text{pos}_{i,1}$ and
$c^\text{pos}_{i,2}$
the two clauses in which variable $x_i$ appears positively and as
$c^\text{neg}_{i,1}$ and $c^\text{neg}_{i,2}$ the two clauses in which
variable $x_i$ appears negatively. From this, we construct an instance
of \textsl{HR-$\text{Q}_\text{L}$}\xspace.
\textbf{Construction:}
For each $i\in [q]$, we introduce two hospitals $h_{i}$ and
$\bar{h}_i$ with lower quota three and two hospitals $h_{i}^*$ and
$\bar{h}_{i}^*$
with
lower quota two. Moreover, we add three hospitals~$h^1_i$, $h^2_i$, and~$h^3_i$ with lower quota two that will help us to build a penalizing
component. Finally, we create one hospital $h_c$ for each clause $c\in
C$
with lower quota three, which is supposed to be closed in every stable
matching.
Turning to the residents, for each $i\in [q]$, we add two
\emph{variable residents} ($r_{i}$
and~$\bar{r}_i$),
two \emph{dummy residents} ($d^1_i$ and
$d^2_i$), and three \emph{penalizing residents} ($s^*_i,s^1_i,s^2_i$).
The
assignment of the variable residents encode the truth assignments and
their
preferences are as follows:
$$r_i: h_{i} \succ h_{c^\text{pos}_{i,1}} \succ
h_{c^\text{pos}_{i,2}} \succ h_{i}^*, \qquad \bar{r}_i:
\bar{h}_i \succ h_{c^\text{neg}_{i,1}} \succ
h_{c^\text{neg}_{i,2}} \succ \bar{h}_{i}^*.$$
The two dummy residents enable us to choose for each $i\in [q]$ which
of
the two hospitals~$h_{i}$ and $\bar{h}_i$ should be open and
thereby which of the two residents $r_i$ and $\bar{r}_i$
are
matched to her top-choice. The preferences of the dummy residents are as
follows:
$$d^1_i: h_{i} \succ \bar{h}_i, \qquad d^2_i:
\bar{h}_i \succ h_{i}.$$
Lastly, we construct a penalizing component for each variable $i\in
[q]$
consisting of three agents whose preferences are as follows:
$$s^*_i: h_{i}^*\succ \bar{h}_{i}^* \succ h_i^1 \succ h_i^2,
\qquad s_i^1:
h_{i}^2 \succ h_{i}^3, \qquad s_i^2:
h_{i}^3 \succ h_{i}^1.$$ Note that if $s^*_i$ is not matched to
$h_{i}^*$ or $\bar{h}_{i}^*$, then no stable matching of these
three residents~$s_i^*$, $s_i^1$, and $s_i^2$ to their acceptable
hospitals $h_i^1$, $h_i^2$, and $h_i^3$ exists. Using this, we prove in
the following that in a stable matching, for
all $i\in [q]$, either $r_i$ is assigned to $h_i^*$ and
$\bar{r}_i$ is assigned to $\bar{h}_i$ (which
corresponds to setting $x_i$ to false) or
$\bar{r}_i$ is assigned to $\bar{h}_i^*$ and $r_i$
is assigned to~$h_i$ (which corresponds to setting $x_i$ to
true). The resulting assignment needs to satisfy every clause~$c\in
C$, as otherwise the residents corresponding to the literals from $c$
form a blocking coalition to open $h_c$. Note that
in the
constructed instance, for all hospitals, the number of residents
accepting it is equal to its lower quota.
As every variable appears exactly twice positively and twice negatively, every resident accepts at most four hospitals.
As every clause contains exactly three literals, every hospital accepts at most three residents.
We now prove that
the
given \textsc{Satisfiability} instance has a satisfying assignment if
and only if
there exists a stable matching in the constructed~\textsl{HR-$\text{Q}_\text{L}$}\xspace instance.
{\bfseries ($\Rightarrow$)} Let $Z$ be the set of variables that are
set
to
true in a satisfying assignment of the given \textsc{Satisfiability} instance.
From
this we construct a stable matching $M$:
\begin{align*}
\{(h_{i},\{r_i, d_i^1, d_i^2\}),(\bar{h}_{i}^*,\{\bar{r}_i,
s^*_i\}) \mid x_i\in Z\} & \cup
\{(\bar{h}_i,\{\bar{r}_i,
d_i^1, d_i^2\}),(h_{i}^*,\{r_i,
s^*_i\}) \mid x_i\notin Z\}\\ & \cup
\{(h_i^3, \{s_i^1, s_i^2\})\mid i\in [q] \}.
\end{align*}
As the constructed instance falls under \Cref{ob:equiv}, no blocking
pair exists.
We now iterate over all closed hospitals and argue why there
does not exist a blocking coalition to open this hospital. For each $i\in
[q]$, one of $h_{i}$ and $\bar{h}_i$ is closed. However,
as
one of the three residents (either $d_i^1$ or~$d_i^2$) that find such a
hospital acceptable is matched
to her top-choice in $M$, no blocking coalition to open $h_i$ or $\bar{h}_i$ exists.
Moreover, for each $i\in [q]$, one of $h_{i}^*$ and $\bar{h}_i^*$
is closed. However,
as
one of the two residents (either $r_i$ or $\bar{r}_i$) that find
such a
hospital acceptable is matched
to her top-choice in~$M$, no blocking coalition to open $h_i^*$ or $\bar{h}_i^*$ exists.
Next we consider a clause hospital~$h_c$ for some $c\in C$.
As~$Z$ induces a satisfying assignment, for at
least
one
literal occurring in~$c$, the corresponding resident is matched to its
top-choice.
Thus, there is no blocking coalition to open~$h_c$.
Finally, there does not exist a blocking coalition to open~$h_i^1$ and~$h_i^2$, as both or one of the two residents that find one
of
these hospitals acceptable are matched better in $M$.
{\bfseries ($\Leftarrow$)} Assume that there exists a stable matching
$M$
of
residents to hospitals. First of all, note that every resident $s_i^*$
needs
to be matched to~$h_i^*$ or $\bar{h}_i^*$, as otherwise no stable
matching of the residents~$s_i^*$, $s_i^1$, and~$s_i^2$ to the
remaining acceptable hospitals $h_i^1$, $h_i^2$, and~$h_i^3$ can exist
(see also \Cref{ob:counter}).
Consequently, for each $i\in [q]$,
either $h_{i}^*$ or~$\bar{h}_{i}^*$ needs to be open and at
least one of the residents
$r_i$ and $\bar{r}_i$ needs to be matched to one of the two.
Moreover,
no clause hospital can be open in a stable matching: Let us assume
that
some clause hospital is open and, without loss of generality, some
resident $r_i$ is matched to
it. Then, as argued above, $\bar{r}_i$ needs to be assigned to
$\bar{h}_{i}^*$. However, such an assignment is blocked by the two
unassigned
dummy residents~$d_i^1$ and~$d_i^2$ and~$\bar{r}_i$ to
open~$\bar{r}_i$'s top-choice~$\bar{h}_i$.
Thus, for each~$i \in [q]$, either~$r_i$ is matched to~$h_{i}$
and~$\bar{r}_i$ is matched to~$\bar{h}_i^*$, or~$r_i$ is
matched
to~$h_{i}^*$ and~$\bar{r}_i$ is matched to
$\bar{h}_i$.
Let $Z=\{
x_i\in X \mid M(r_{i})\neq h_{i}^*\}$. We claim that setting all
variables in
$Z$
to true and all others to false induces a satisfying assignment of the
given propositional formula. For the sake of contradiction, let us
assume
that there exists a clause $c=\{\ell_1,\ell_2,\ell_3\}\in C$ which is
not
fulfilled. However, this implies that all three corresponding variable
residents are all matched to their least preferred acceptable
hospital. From this is follows that $M$
cannot be stable, as these three variable residents then form a
blocking coalition to open~$h_c$.
\end{proof}
Note that the \textsl{HR-$\text{Q}_\text{L}$}\xspace instance constructed in the reduction falls under
\Cref{ob:equiv}. This implies that all three models we consider remain
computationally hard for lower quota at most three if stability only
requires that no
blocking coalition exists (but blocking pairs might exist).
\subsection{Parameterization by Number of Residents}\label{sec:resi}
After establishing the NP-hardness of \textsl{HR-$\text{Q}_\text{L}$}\xspace, \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, we now analyze
their computational complexity parameterized by the number of residents.
While there
exists a
straightforward XP algorithm for this parameter that guesses for each
resident the hospital she is assigned to, all three problems are W[1]-hard.
\begin{theorem}
\label{th:W-n}
Parameterized by the number $n$ of residents, \textsl{HR-$\text{Q}_\text{L}$}\xspace,
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, and~\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace are W[1]-hard, even if every hospital has lower
(and upper) quota at most four and accepts at most four residents.
\end{theorem}
\begin{proof}
We prove hardness by a parameterized reduction from
\textsc{Multicolored
Independent Set}. In an instance of \textsc{Multicolored
Independent Set}, we are
given an
undirected graph~$G=(V=\{v_1,\dots, v_n\},E)$ together with a
partition
$(V^1,\dots, V^k)$ of
$V$ into $k$ different colors and the question is whether there
exists an independent set of size~$k$ containing
exactly one vertex from each color,
i.e.,
a
subset $V'\subseteq V$ with $|V'|=k$ and $|V'\cap V^c|=1$ for all
$c\in
[k]$ such that no two vertices in $V'$ are adjacent.
\textsc{Multicolored Independent Set} parameterized by~$k$ is
W[1]-hard~\cite{Pietrzak03}.
To
simplify
notation, we reduce from a restricted still W[1]-hard version of
\textsc{Multicolored Independent Set} where there exist some
(arbitrary) integers~$p$ and~$q$
such
that each vertex $v$ is adjacent to exactly~$p$ vertices, and
each color $c\in [k]$ contains exactly~$q$ vertices.
Moreover, without loss of generality, we assume that there are no edges between
vertices of the same color.
For each vertex~$v\in V$, let $u_1^{v},\dots,u_{p}^{v}$ be a list
of all
vertices
incident to $v$. For a vertex $v\in V$ and an integer $i\in
[p]$, we write $\text{z}(v,i)$ to denote the $i$-th vertex that is
incident to $v$, i.e., $\text{z}(v,i):=u_i^{v}$.
Moreover, for each
color $c\in [k]$, let $v_1^c,\dots, v_{q}^c$ be a list of all
vertices
with this
color. From an instance of \textsc{Multicolored Independent Set},
we now construct an instance
of \textsl{HR-$\text{Q}_\text{L}$}\xspace.
\textbf{Construction:} For each vertex
$v\in V$, we introduce a \emph{vertex hospital}
$h_v$ with lower quota
three. Moreover,
for each edge $\{v,v'\}\in V$,
we introduce an \emph{edge hospital}
$h_{\{v,v'\}}$ with lower quota
four. Furthermore, we
introduce for each color $c\in [k]$ a penalizing component
consisting
of
three \emph{penalizing hospitals} $h_1^c$, $h_2^c$, and $h_3^c$
with lower
quota two.
Turning to the residents, we introduce for each color $c\in [k]$
two
\emph{color residents} $r^c_1$ and~$r^c_2$.
One of the color residents ranks the vertex
hospitals corresponding to vertices of color $c$ in some ordering
and
the
other color resident ranks them in reversed order. Both residents
rank
directly
in front of each vertex hospital all edge hospitals involving
the
corresponding vertex in an arbitrary order. That
is:
\begin{align*}
r^c_1:\mkern4mu & h_{\{v_1^c, \text{z}(v_1^c,1)\}}\succ \dots \succ
h_{\{v_1^c,
\text{z}(v_1^c,p)\}}\succ h_{v_1^c}\succ \dots
\succ
h_{\{v_{2}^c, \text{z}(v_{2}^c,1)\}}\succ \dots \succ
h_{\{v_{2}^c,
\text{z}(v_{2}^c,p)\}}
\succ h_{v_{2}^c} \succ \\
& h_{\{v_{q-1}^c, \text{z}(v_{q-1}^c,1)\}}\succ \dots \succ
h_{\{v_{q-1}^c,
\text{z}(v_{q-1}^c,p)\}}\succ h_{v_{q-1}^c}\succ \dots
\succ
h_{\{v_{q}^c, \text{z}(v_{q}^c,1)\}}\succ \dots \succ
h_{\{v_{q}^c,
\text{z}(v_{q}^c,p)\}}
\succ h_{v_{q}^c},\\
r^c_2:\mkern4mu & h_{\{v_{q}^c, \text{z}(v_{q}^c,1)\}}\succ \dots
\succ
h_{\{v_{q}^c,
\text{z}(v_{q}^c,p)\}}
\succ h_{v_{q}^c}\succ \dots \succ h_{\{v_{q-1}^c,
\text{z}(v_{q-1}^c,1)\}}\succ \dots \succ h_{\{v_{q-1}^c,
\text{z}(v_{q-1}^c,p)\}}\succ h_{v_{q-1}^c} \succ \\
& h_{\{v_{2}^c, \text{z}(v_{2}^c,1)\}}\succ \dots \succ
h_{\{v_{2}^c,
\text{z}(v_{2}^c,p)\}}
\succ h_{v_{2}^c}\succ \dots \succ h_{\{v_1^c,
\text{z}(v_1^c,1)\}}\succ \dots \succ h_{\{v_1^c,
\text{z}(v_1^c,p)\}}\succ h_{v_1^c}.
\end{align*}
Moreover, for each color $c\in [k]$, we introduce a penalizing
component ensuring that no edge hospital can be open in a stable
matching. The penalizing component consists of three
\emph{penalizing residents}
$s^c_*$, $s^c_1$,
$s^c_2$:
$$s^c_*:h_{v_1^c}\succ\dots \succ h_{v_{q}^c}\succ h_1^c\succ
h_2^c,
\qquad s^c_1: h_2^c\succ h_3^c, \qquad s^c_2: h_3^c\succ h_1^c.$$
See \Cref{fig:residents-hardness} for an example of the construction.
The penalizing component enforces that for each color at least
(and, in
fact, exactly one) hospital needs to be open. The preferences of
the
color residents are constructed in a way such that no two
hospitals that correspond to adjacent vertices can be open. Note
that for each constructed hospital, the number of residents
accepting it is equal its lower quota.
Thereby, the set of open vertex hospitals corresponds to a
multicolored independent set.
We now prove that there exists a solution to the given
\textsc{Multicolored
Independent Set} instance if and only if there exists a stable
matching
in
the
constructed \textsl{HR-$\text{Q}_\text{L}$}\xspace instance.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node (ver-dist) at (0, 1.5) {};
\node (hor-dist) at (5, 0) {};
\node (hd) at (3, 0) {};
\node[vertex, label=90:$r^c_1$] (r1) at (0, 0) {};
\node[vertex, label=90:$r^c_2$] (r2) at ($(r1) + 1.5*(ver-dist)$) {};
\node[vertex, label=90:$s^c_*$] (r3) at ($(r2) + 1.5*(ver-dist)$) {};
\node[vertex, label=90:$s^c_1$] (r4) at ($(r3) + (ver-dist)$) {};
\node[vertex, label=90:$s^c_2$] (r5) at ($(r4) + (ver-dist)$) {};
\node[vertex, label=90:$r^d_1$] (s1) at ($(r1) + (hor-dist)$) {};
\node[vertex, label=90:$r^d_2$] (s2) at ($(s1) + 1.5*(ver-dist)$) {};
\node[vertex, label=90:$s^d_*$] (s3) at ($(s2) + 1.5*(ver-dist)$) {};
\node[vertex, label=90:$s^d_1$] (s4) at ($(s3) + (ver-dist)$) {};
\node[vertex, label=90:$s^d_2$] (s5) at ($(s4) + (ver-dist)$) {};
\node[squared-vertex, label=90:$h_1^c$, label=270:{$[2,\infty]$}] (hc1) at ($(r3) - (hd)$) {};
\node[squared-vertex, label=90:$h_2^c$,
label={[yshift=-0.05cm]270:{$[2,\infty]$}}] (hc2) at ($(hc1)
+ (ver-dist)$) {};
\node[squared-vertex, label=90:$h_3^c$,
label={[yshift=-0.05cm]270:{$[2,\infty]$}}] (hc3) at ($(hc2)
+
(ver-dist)$) {};
\node[squared-vertex, label=90:$h_1^d$, label=270:{$[2,\infty]$}] (hd1) at ($(s3) + (hd)$) {};
\node[squared-vertex, label=90:$h_2^d$,
label={[yshift=-0.05cm]270:{$[2,\infty]$}}] (hd2) at ($(hd1)
+ (ver-dist)$) {};
\node[squared-vertex, label=90:$h_3^d$,
label={[yshift=-0.05cm]270:{$[2,\infty]$}}] (hd3) at ($(hd2)
+ (ver-dist)$) {};
\node[squared-vertex, label={[yshift=0.15cm]90:$h_{v_1^c, v_1^d}$}, label=270:{$[4,\infty]$}] (ve1) at ($0.5*(hor-dist) -0.5*(ver-dist)$) {};
\node[squared-vertex, label=90:$h_{v_1^c, v_2^d}$, label=270:{$[4,\infty]$}] (ve2) at ($(ve1) + (ver-dist) $) {};
\node[squared-vertex, label=90:$h_{v_2^c, v_3^d}$,
label={[yshift=-0.13cm]270:{$[4,\infty]$}}] (ve3) at ($(ve2)
+(ver-dist) $) {};
\node[squared-vertex, label=90:$h_{v_3^c, v_3^d}$,
label={[xshift=0.1cm]0:{$[4,\infty]$}}] (ve4) at ($(ve3)
+(ver-dist) $) {};
\node[squared-vertex, label={[xshift=-0.1cm]90:$h_{v_1^c}$}, label=270:{$[3,\infty]$}] (vc1) at ($(r1) - (hd) - 0.5*(ver-dist)$) {};
\node[squared-vertex, label={[xshift=-0.1cm]90:$h_{v_2^c}$}, label=270:{$[3,\infty]$}] (vc2) at ($(vc1) + (ver-dist)$) {};
\node[squared-vertex, label={[xshift=-0.1cm]90:$h_{v_3^c}$},
label={[yshift=-0.1cm]270:{$[3,\infty]$}}] (vc3) at ($(vc2)
+ (ver-dist)$) {};
\node[squared-vertex, label={[xshift=0.05cm]90:$h_{v_1^d}$},
label=270:{$[3,\infty]$}] (vd1) at ($(s1) + (hd) -
0.5*(ver-dist)$) {};
\node[squared-vertex, label=90:$h_{v_2^d}$, label=270:{$[3,\infty]$}] (vd2) at ($(vd1) + (ver-dist)$) {};
\node[squared-vertex, label=90:$h_{v_3^d}$,
label={[yshift=-0.1cm]270:{$[3,\infty]$}}] (vd3) at ($(vd2)
+ (ver-dist)$) {};
\draw (vc1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (r1);
\draw (vc1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $7$} (r2);
\draw (vc2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (r1);
\draw (vc2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (r2);
\draw (vc3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $7$} (r1);
\draw (vc3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r2);
\draw (vd1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (s1);
\draw (vd1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $7$} (s2);
\draw (vd2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (s1);
\draw (vd2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (s2);
\draw (vd3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $7$} (s1);
\draw (vd3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (s2);
\draw (vc1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r3);
\draw (vc2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r3);
\draw (vc3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (r3);
\draw (hc1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (r3);
\draw (hc2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (r3);
\draw (hc2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r4);
\draw (hc3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r4);
\draw (hc3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r5);
\draw (hc1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r5);
\draw (ve1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r1);
\draw (ve1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$5$} (r2);
\draw (ve2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r1);
\draw (ve2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$6$} (r2);
\draw (ve3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (r1);
\draw (ve3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (r2);
\draw (ve4) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $6$} (r1);
\draw (ve4) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r2);
\draw (vd1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (s3);
\draw (vd2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (s3);
\draw (vd3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (s3);
\draw (hd1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (s3);
\draw (hd2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (s3);
\draw (hd2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (s4);
\draw (hd3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (s4);
\draw (hd3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (s5);
\draw (hd1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (s5);
\draw (ve1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (s1);
\draw (ve1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $6$} (s2);
\draw (ve2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (s1);
\draw (ve2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (s2);
\draw (ve3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$6$} (s1);
\draw (ve3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (s2);
\draw (ve4) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$5$} (s1);
\draw (ve4) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (s2);
\end{tikzpicture}
\end{center}
\caption{An example for the reduction showing W[1]-hardness of
\textsl{HR-$\text{Q}_\text{L}$}\xspace, \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace from \Cref{th:W-n}.
Let $G=(\{v_1^c,v_2^c,v_3^c,v_1^d,v_2^d,v_3^d\},\{\{v_1^c,
v_1^d\},\{v_1^c, v_2^d\},\{v_2^c, v_3^d\},\{v_3^c,
v_3^d\}\})$ and
$(V^c=\{v_1^c,v_2^c,v_3^c\},V^d=\{v_1^d,v_2^d,v_3^d\})$.
The picture shows the output of the reduction on this instance.}
\label{fig:residents-hardness}
\end{figure}
{\bfseries ($\Rightarrow$)} Assume that $V'=\{v_{i_1},\dots,
v_{i_{k}}\}$ is an
independent set
in $G$ with $v_{i_c}\in V^c$ for all~$c\in [k]$. From this we
construct a stable matching $M$ as follows:
$$M=\{(h_{v_{i_c}},\{r^c_1,r^c_2, s^c_*\}),
(h^{c}_3,\{s^c_1,s^c_2\})\mid
c\in [k]\}.$$
As the constructed instance falls under \Cref{ob:equiv}, no
blocking pair can exist.
It remains to argue that for no closed hospital $h$
there exists a
coalition of residents to open it in $M$. Note that
there does not exist a coalition to open $h^c_1$ or $h^c_2$ for any
$c\in
[k]$, as $s^c_*$ is
matched
to a hospital she prefers to both $h^c_1$ and $h^c_2$. Note further
that for
each color $c\in [k]$, the only hospitals that both $r^c_1$ and
$r^c_2$
prefer
to the hospital~$h_{v_{i_c}}$ (which is the hospital $r^c_1$ and $r^c_2$ are
matched to in $M$) are the
edge
hospitals
$h_{\{v_{i_c},
\text{z}(v_{i_c},1)\}},\dots, h_{\{v_{i_c},
\text{z}(v_{i_c},p)\}}$
corresponding to
$v_{i_c}$ and the vertices that are adjacent to~$v_{i_c}$. Hence,
as
no two color residents
prefer the
same vertex hospital, there cannot exist a blocking coalition to
open a
vertex hospital. Moreover, as $V'$ is an independent set, there
do not exist
two
adjacent vertices in $V'$ and thereby also no edge
hospital
corresponding to a vertex pair from $V'$. Thus, there does not
exist an edge hospital that is preferred by four color residents to
the
hospital
they are matched to in $M$. Thus, $M$ is stable.
{\bfseries ($\Leftarrow$)} Assume that there exists a stable
matching $M$ in
the
constructed \textsl{HR-$\text{Q}_\text{L}$}\xspace instance. First of all note that, for each color
$c\in [k]$, resident $s^c_*$
needs to be
matched
to a vertex hospital in~$M$, as otherwise there does not exist a
stable matching of the residents $s_*^c$,~$s_1^c$, and~$s_2^c$ to
the hospitals~$h^c_1$, $h^c_2$, and $h^c_3$ (see
\Cref{ob:counter}). Thus, for each color $c\in [k]$,
there
exists
exactly one
vertex $v_{i}^c\in V^c$ such that the
hospital~$h_{v_{i}^c}$ is open in $M$. We claim that $V'=\{
v_{i}^c
\mid c\in[k] \wedge i\in [q] \wedge \text{ $h_{v_{i}^c}$ is open in
}M\}$ forms an
independent set in $G$. For the sake of
contradiction,
let us assume that there exists a pair of vertices~$v,v'\in V'$
with
$v\in
V^{c}$ and $v'\in V^{d}$ for two $c\neq d\in [k]$ that are
adjacent. By
construction,
$r^{c}_1$ and $r^{c}_2$ are matched to $h_{v}$
and
$r^{d}_1$ and $r^{d}_2$ are matched to~$h_{v'}$ in $M$.
However, as
$v$
and $v'$ are adjacent, this implies that all four
residents~$r^{c}_1$,~$r^{c}_2$,~$r^{d}_1$, and $r^{d}_2$
prefer the
edge hospital
$h_{\{v,v'\}}$ to the hospital they are matched to in~$M$, which
contradicts our assumption that $M$ is a stable matching.
\end{proof}
Again, the \textsl{HR-$\text{Q}_\text{L}$}\xspace instance constructed in the reduction falls under
\Cref{ob:equiv}, which implies that the three models are W[1]-hard
parameterized by $n$ even if stability only forbids the existence of
blocking coalitions.
Compared to \Cref{th::NP-compl}, the hardness statement from \Cref{th:W-n} does not
bound the number of hospitals accepted by each resident.
In
fact, all three problems are
fixed-parameter tractable by the combined parameter number of residents plus maximum number of hospitals accepted by a resident,
as the size of the instance (ignoring hospitals which no resident accepts)
is bounded in a function of this parameter.
In the following \Cref{se:hosp}, among others, we address the computational
complexity of deciding whether
there exists a stable matching where exactly a given subset of hospitals is
open and the parameterized complexity of deciding whether there exists a
stable matching
opening~$m_{\text{open}}$~hospitals parameterized by $m_{\text{open}}$.
Obviously, both questions can be analogously asked for residents. However,
the reduction from \Cref{th:W-n} proves that deciding the existence of a
stable matching assigning all residents is NP-hard and W[1]-hard
parameterized by the number of residents. Thus, both questions asked for
the residents are computationally intractable for all our three models.
\subsection{Influence of Hospitals}\label{se:hosp}
After studying the parameterization by the number of residents,
we turn to the number of hospitals and several closely related parameters.
We start by considering
the problem of finding a stable matching opening exactly a given
set of hospitals.
\subsubsection{Which hospitals should be open?} \label{se:hopen}
It is possible
to think of finding a stable matching as a two-step
process. First,
decide which hospitals are open and second,
compute a stable matching between the residents and the selected set of open hospitals
respecting all quotas.
This observation leads to the question what happens if the first step
has been already done, e.g., by an
oracle or by some
authority, and we are left with the task of finding a stable matching where
exactly a given set
of hospitals is open. We show that while for \textsl{HR-$\text{Q}_\text{L}$}\xspace
and
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace
this problem is solvable in
polynomial time, it is NP-hard for \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace.
As already observed in \Cref{se:obs}, in an \textsl{HR-$\text{Q}_\text{L}$}\xspace instance $(H,R)$, every stable matching assigns all
residents to their most preferred open hospital.
Thereby, deciding whether there exists a stable matching where exactly a
given set $H_{\open}\subseteq H$ of hospitals is open reduces to
assigning each
resident to her most preferred hospital in $H_{\open}$ and checking
whether the
resulting matching is stable in $(H,R)$.
\begin{observation} \label{ob:HRLQ-H'}
Given a subset of hospitals $H_{\open}\subseteq H$, deciding
whether there
exists a stable matching in an \textsl{HR-$\text{Q}_\text{L}$}\xspace instance $(H,R)$ in which exactly
the hospitals from $H_{\open}$ are open is solvable in
$\mathcal{O}(nm)$ time.
\end{observation}
For \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, a slightly more involved reasoning is needed, which utilizes
the famous Rural Hospitals Theorem
\cite{RePEc:ucp:jpolec:v:92:y:1984:i:6:p:991-1016,10.2307/1913160}:
\begin{proposition} \label{pr:HRULQ-H'}
Given a subset of hospitals $H_{\open}\subseteq H$, deciding
whether there
exists a stable matching in an \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance $(H,R)$ in which exactly
the hospitals from $H_{\open}$ are open is solvable in
$\mathcal{O}(nm)$ time.
\end{proposition}
\begin{proof}
Given an \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance $\mathcal{I}=(H,R)$, we calculate the
resident-optimal stable matching~$M$ of the residents $R$ to the
hospitals
$H_{\open}$ ignoring their lower quota
by applying the Gale and Shapely algorithm~\cite{GaleS62}. We return YES if $M$ is
stable in $\mathcal{I}$ and obeys the
quotas and otherwise NO.
If the algorithm returns YES, then this
answer is clearly correct.
It remains to show that if there exists a stable matching, then the
algorithm returns YES.
Assume that there exists a stable
matching $M'$ in $\mathcal{I}$ that opens exactly the hospitals
$H_{\open}$, and let $M$ be the computed matching.
As $M'$ is a stable
matching that opens exactly the hospitals from $H_{\open}$, matching
$M'$
is also a stable matching
in
the instance $\mathcal{I}' :=(H_{\open},R)$ without lower quotas.
By the Rural Hospitals Theorem, it follows that every stable matching
in $\mathcal{I}'$---and therefore also $M$---matches the same
number of residents as $M'$ to each hospital and thus is a feasible
matching in
$\mathcal{I}$.
Matching $M$ does not admit a blocking pair in $\mathcal{I}$ by the
definition of $M$.
Lastly, due to the resident-optimality of $M$, it follows that any
blocking coalition for $M$ in $\mathcal{I}$ is also a blocking
coalition for $M'$ in $\mathcal{I}$. Thus, as $M'$ is stable in
$\mathcal{I}$, matching $M$ is also stable in~$\mathcal{I}$.
\end{proof}
So far, our hardness results for all three models including
\textsl{HR-$\text{Q}_\text{L}$}\xspace indicated
that the complexity of our problems mainly comes from the hospital's lower
quotas. In fact, for all questions we consider in this and the following
section, \textsl{HR-$\text{Q}_\text{L}$}\xspace and
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace are tractable in the same cases. Moreover, we have shown in
\Cref{ob:HRLQ-H'}
and \Cref{pr:HRULQ-H'} that the difficult part of \textsl{HR-$\text{Q}_\text{L}$}\xspace and
\textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace is to
decide which hospitals to open and not how to assign the residents to the
open
hospitals.
In contrast to this, as proven in the following theorem, for \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, by
exploiting the upper quotas of the hospitals, the problem of assigning the
residents to the open hospitals becomes computationally hard. Generally
speaking, the reason for this is that we have a lot of flexibility how to
assign the residents here, as blocking pairs need to involve an
undersubscribed hospital. For instance, as done in the following reduction,
there may exist a hospital that every resident wants to be matched to. In
this case, by enforcing an upper quota on the number of residents matched
to such a ``popular'' hospital and as this popular hospital does not have
preferences that
could act as a ``tie-breaker'', there exists, in principle, an exponential
number
of
possibilities
which residents to assign to the popular hospital.
Thus, in contrast to the Hospital Residents problem with lower quotas where
upper quotas do not really seem to add
complexity to the problem, as the preferences of hospitals already provide
some information which residents to assign to a hospital, upper quotas add
complexity to the House Allocation problem with lower quotas. We show this
in the following
theorem by proving
that \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace remains NP-complete even if we know which hospitals are open in
a stable matching:
\begin{theorem}
\label{th:HRLUQIH'}
Given a subset of hospitals $H_{\open}\subseteq H$, deciding
whether there
exists a stable matching in an \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance $(H,R)$ in which
exactly
the hospitals from $H_{\open}$ are open
is NP-complete, even if
$|H_{\open}|=4$, each resident
accepts at most five hospitals, all hospitals have lower quota at most
two, and we know that if there exists a stable matching, then it opens
exactly the hospitals from~$H_{\open}$.
\end{theorem}
\begin{proof}
We reduce from the NP-hard \textsc{Independent Set} problem on
3-regular graphs, that is,
graphs were all vertices have exactly three neighbors
\cite{DBLP:books/fm/GareyJ79}.
\begin{figure}
\begin{minipage}{0.25\textwidth}
\begin{center}
\begin{tikzpicture}
\node (ver-dist) at (0, 1.5) {};
\node (hor-dist) at (5, 0) {};
\node (hd) at (3, 0) {};
\node[vertex, label=180:$v_1$] (r1) at (0, 0) {};
\node[vertex, label=90:$v_2$] (r2) at ($(r1) + (1,2)$) {};
\node[vertex, label=0:$v_3$] (r3) at ($(r1) + (2, 0)$) {};
\node[vertex, label=270:$v_4$] (r5) at ($(r1) + (1, -2)$)
{};
\draw (r1) -- (r2) -- (r3) -- (r1);
\draw (r2) -- (r3) -- (r5);
\draw (r5) -- (r1);
\end{tikzpicture}
\end{center}
\end{minipage}\vline
\begin{minipage}{0.65\textwidth}
\begin{center}
\begin{tikzpicture}
\node (ver-dist) at (0, 1.5) {};
\node (hor-dist) at (5, 0) {};
\node (hd) at (3, 0) {};
\node[vertex, label=180:$r_{v_1}$] (r1) at (0, 0) {};
\node[vertex, label=90:$r_{v_2}$] (r2) at ($(r1) + (1,2)$)
{};
\node[vertex, label={[xshift=0.15cm]90:$r_{v_3}$}] (r3) at
($(r1) + (2, 0)$) {};
\node[vertex, label=180:$r_{v_4}$] (r4) at ($(r1) +
(1,-2)$) {};
\node[squared-vertex, label=180:\scriptsize$h_{\{v_1,
v_2\}}$] (e12) at ($0.5*(r1)+ 0.5*(r2)$) {};
\node[squared-vertex,
label={[yshift=-0.1cm]90:\scriptsize$h_{\{v_1, v_3\}}$}]
(e13) at ($0.5*(r1)+ 0.5*(r3)$) {};
\node[squared-vertex,
label={[xshift=0.5cm,yshift=0.2cm]180:\scriptsize$h_{\{v_2,
v_3\}}$}] (e23) at ($0.5*(r3)+ 0.5*(r2)$) {};
\node[squared-vertex,
label={[xshift=0.1cm]180:\scriptsize$h_{\{v_1, v_4\}}$}]
(e15) at ($0.5*(r5)+ 0.5*(r1)$) {};
\node[squared-vertex,
label={[xshift=0.18cm,yshift=0.15cm]180:\scriptsize$h_{\{v_3,
v_4\}}$}]
(e34) at ($0.5*(r3)+ 0.5*(r4)$) {};
\draw (e12) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $2$} (r1);
\draw (e12) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $2$} (r2);
\draw (e23) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $3$} (r2);
\draw (e23) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $2$} (r3);
\draw (e13) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $3$} (r1);
\draw (e13) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $3$} (r3);
\draw (e15) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $4$} (r1);
\draw (e15) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $2$} (r4);
\draw (e34) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $4$} (r3);
\draw (e34) edge node[pos=0.6, fill=white, inner sep=2pt]
{\scriptsize $3$} (r4);
\node[red, squared-vertex, label=90:$h^+$,
label={[yshift=-0.1cm,
xshift=0.1cm]270:{$[1, 4-k]$}}] (hg) at (4, -1) {};
\node[red, squared-vertex, label=90:$h^-$,
label={[yshift=-0.2cm, xshift=0.1cm]270:{$[1, k]$}}] (hb)
at (4, 1) {};
\draw (hb) edge[bend right=25] node[pos=0.9, fill=white,
inner sep=2pt] {\scriptsize $5$} (r1);
\draw (hb) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $4$} (r2);
\draw (hb) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $5$} (r3);
\draw (hb) edge node[pos=0.85, fill=white, inner sep=2pt]
{\scriptsize $4$} (r4);
\draw (hg) edge[bend left = 20] node[pos=0.9, fill=white,
inner sep=2pt] {\scriptsize $1$} (r1);
\draw (hg) edge node[pos=0.85, fill=white, inner sep=2pt]
{\scriptsize $1$} (r2);
\draw (hg) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (r3);
\draw (hg) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (r4);
\node[vertex, label=90:$r^*$] (rs) at (5, 1) {};
\node[vertex, label=180:$r_3$] (re3) at (6, -2) {};
\node[vertex, label=90:$r_1$] (re1) at (6, 2) {};
\node[vertex, label=90:$r_2$] (re2) at (6, 0) {};
\node[red, squared-vertex, label=90:$h_1$,
label=0:{$[2,2]$}] (h1) at (7, 2) {};
\node[squared-vertex, label=90:$h_2$, label=270:{$[2,2]$}]
(h2) at (7, 0) {};
\node[red, squared-vertex, label=90:$h_3$,
label=270:{$[2,2]$}] (h3) at (7, -2) {};
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (re1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $2$} (re1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (re2);
\draw (h3) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $2$} (re2);
\draw (h3) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (re3);
\draw (h1) edge node[pos=0.9, fill=white, inner sep=2pt]
{\scriptsize $2$} (re3);
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $2$} (rs);
\draw (hb) edge node[pos=0.76, fill=white, inner sep=2pt]
{\scriptsize $1$} (rs);
\end{tikzpicture}
\end{center}
\end{minipage}
\caption{An example for the reduction showing NP-hardness of
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace with four open hospitals from \Cref{th:HRLUQIH'}.
For the sake of illustration, the input graph~$T$ (depicted in
the left picture) is not 3-regular.
The output of the reduction is depicted in the right picture.
The lower and upper quota two for each edge hospital is not
drawn
for the sake of readability.
The set $H_{\open}$ is marked in red.
}
\label{fig:hardness-open-hospitals}
\end{figure}
\textbf{Construction:} Let $(G=(V,E),k)$ be an instance of \textsc{Independent
Set}, where $G$ is a 3-regular graph.
For each~$v\in V$, let $e^v_1$, $e^v_2$, and $e^v_3$ be a list of all
edges incident to $v$. We introduce a \emph{good hospital}~$h^+$ with
lower
quota one and upper quota $n-k$ and a \emph{bad hospital}~$h^-$ with
lower
quota one and upper quota $k$. Moreover, we introduce for each edge
$e\in E$ an \emph{edge hospital} $h_e$ with lower and upper quota two.
Turning to the residents, we introduce for each vertex $v\in V$ a
\emph{vertex resident} $r_v$ with the following preferences:
$$r_v:h^+\succ h_{e^v_1}
\succ h_{e^v_2} \succ h_{e^v_3} \succ h^-.$$
Finally, we introduce a
penalizing component ensuring that no vertex resident can be
matched to an edge hospital. The penalizing component consists of three
hospitals $h_1$, $h_2$,
$h_3$, each with lower and upper quota two, and four
residents $r^*$, $r_1$, $r_2$, and $r_3$:
$$r^*:h^-\succ h_1, \qquad r_1:h_1\succ h_2, \qquad r_2:h_2\succ h_3,
\qquad r_3:h_3\succ h_1.$$
Note that the penalizing component ensures that $k$ vertex residents
need to be matched to the bad hospital.
We set $H_{\open}:=\{h^+, h^-, h_1,h_3\}$.
See \Cref{fig:hardness-open-hospitals} for an example.
Intuitively, the vertex
residents assigned to the bad hospital in a stable
matching form an independent set, as two
vertex residents corresponding to adjacent vertices matched to the bad
hospital would form a blocking
coalition to open the respective edge hospital together.
We now prove that there exists an independent set $V'$ of size $k$ in
the given graph~$G$ if and only if the constructed \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance admits a
stable matching opening exactly the hospitals from $H_{\text{open}}$.
{\bfseries ($\Rightarrow$)} Let $V'\subseteq V$ be an independent set
of size $k$
in
the given graph $G$. From this, we construct a stable matching $M$
opening $H_{\open}$ in
the
constructed instance by assigning all vertex residents corresponding to
vertices in $V'$ to the bad hospital, all other vertex residents to
the good hospital, $(h_1,\{r^*,r_1\})$, and
$(h_3,\{r_2,r_3\})$.
As no hospital is undersubscribed, it remains to
argue why there does not exists a blocking coalition to
open a closed hospital: As $r_1$ is matched to her top-choice, there does not exist a
blocking coalition to open $h_2$. Moreover, for no edge are both
residents
corresponding to the two endpoints matched to the bad hospital~$h^-$
as
$V'$ is an independent set. Thus, no blocking coalition to open an edge
hospital exists.
{\bfseries ($\Leftarrow$)} Let $M$ be a stable matching in the constructed
instance. First of all note that~$r^*$ needs to be matched to $h_1$, as
otherwise~$r_1$,~$r_2$, and~$r_3$ cannot be assigned to the
hospitals~$h_1$,~$h_2$, and~$h_3$ in a stable way (see
\Cref{ob:counter}).
This implies that $k$ vertex residents need to be matched to the bad
hospital~$h^-$, implying that the remaining $n-k$ vertex residents are matched to the good hospital~$h^+$.
Thus, all edge hospitals are closed, since the residents which accept
them need to be either matched to the good or bad hospital as discussed
above.
The
$k$~vertices~$\{v_1, \dots, v_k\}$ corresponding to the residents
matched to $h^-$ in $M$ form an
independent set, as a pair of vertex residents $r_{v_i}$ and
$r_{v_{j}}$ with~$\{v_i,v_j\}\in E$ both matched to the bad hospital
forms a blocking coalition to open $h_{
\{v_i,v_j\}}$ in~$M$.
\end{proof}
Note that this reduction can be easily adapted to yield NP-completeness
for \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace restricted to instances with upper quota at most two by
splitting the hospitals $h^+$ and~$h^-$ into multiple hospitals with
upper and lower quota one.
However, in this case, the bounds on the size of $H_{\open}$ and
the number
of hospitals acceptable to a single resident do not hold any more.
\subsubsection{Parameterization by the number of hospitals (with non-unit
lower quota)}
Together
with the number $n$ of residents, the number $m$ of hospitals is a very
important and straightforward structural parameter of the studied problems.
As in some applications this
parameter is much smaller than the number of residents, checking for
fixed-parameter tractability is of special interest here.
For both \textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, it is possible to iterate over all possible
subsets $H_{\open}\subseteq H$ of hospitals and use
\Cref{ob:HRLQ-H'} and
\Cref{pr:HRULQ-H'}, respectively, to decide whether there exists a stable
matching in which exactly the hospitals from $H_{\open}$ are open.
Let
$H^{\quota}\subseteq H$ denote the set of hospitals with non-unit
lower quota.
In fact, it is only necessary to iterate over all possible
subsets~$H_{\open}\subseteq H^{\quota}$ with non-unit lower quota and
subsequently add all hospitals
with
lower quota one to $H_{\open}$. For each resulting set $H_{\open}$, we apply
the procedures
described in
\Cref{ob:HRLQ-H'} and \Cref{pr:HRULQ-H'} to compute a matching which we
then check for stability and feasibility.
\begin{corollary} \label{co:mFPT}
\textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace are solvable in
$\mathcal{O}(n m \cdot 2^{m_{\quota}})$ time, where $m_{\quota}$ is the
number of hospitals with non-unit lower quota.
\end{corollary}
Turning to \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, despite the fact that it is NP-complete to
decide whether there exists a stable matching even if the set of open
hospitals is given,
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace parameterized by the number~$m$ of hospitals turns out to be
fixed-parameter tractable. The algorithm utilizes that the number of
different resident types in a \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance can be bounded in a function
of
$m$, as a resident is fully
characterized by her
preferences over hospitals. This observation can be used to construct an
integer linear program (ILP) where the
number of variables is bounded in a function of $m$.
Employing
Lenstra's algorithm
\cite{DBLP:journals/mor/Kannan87,DBLP:journals/mor/Lenstra83}
shows that the problem is fixed-parameter tractable parameterized by
the number of hospitals.
\begin{proposition}
\label{pr:HRLUQI-FPTM}
Parameterized by the number $m$ of hospitals, \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace is
fixed-parameter
tractable.
\end{proposition}
\begin{proof}
Let $(H,R)$ be a given \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance.
Note that as hospitals are indifferent among all residents, a
resident $r\in R$ is fully characterized by her preferences over
hospitals from~$H$. Thereby, the number of resident types is bounded by
the
number of ordered subsets of $m$ elements which is~$\mathcal{O}(m \cdot m!)$ (one can create each possible preference by taking the first $\ell$ hospitals from some permutation of the hospitals).
Let
$t_1$, \dots, $t_q$ be a list of all resident types and, for $i\in
[q]$, let $A(t_i)$ denote the set of hospitals which residents of type $t_i$
accept. For two hospitals
$h\neq h'\in H$, we write $h\succ_{t_i} h'$ if residents of type $t_i$ prefer
$h$
to $h'$. For each~$i\in [q]$, let $n_i$ denote the number of residents in the given
instance of type $t_i$. We design an ILP
solving the given \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance as follows.
We introduce a variable
$x_{i,h}$ for each hospital~$h\in H$ and
each $i\in [q]$ representing the number of residents of type
$t_i$ assigned to hospital~$h$. Moreover, for each hospital~$h\in H$, we introduce a binary
variable~$o_h$ which is $1$ if $h$
is
open and $0$ otherwise.
Furthermore, to prevent blocking pairs, we also use an additional
binary variable $y_h$ for each hospital $h\in H$ which shall be $1$ if
and only if $h$ is undersubscribed.
In the following, we extend our notation by also writing~$h\succ_{t_i}
h'$ if $h'\notin A(t_i)$ and
$h\in A(t_i)$ for a
resident type~$t_i$. Using this notation, the problem can be
solved using the
following ILP:
\begin{align}
\sum_{i\in [q]} x_{i,h} +y_hn\geq o_hu(h), \qquad & \forall h \in H \label{cond:undersubscribed}\\
x_{i,h}+\sum_{\substack{h'\in H \\ h'\succ_{t_i} h}} x_{i,h'} + (1
- y_h) n\geq
n_i, \qquad & \forall i \in
[q], h\in A(t_i) \label{ILP:no-bp}\\
\sum_{\substack{i\in [q], h'\in H: \\ h\succ_{t_i} h'}}
x_{i,h'}\leq
l(h)+o_hn, \qquad & \forall h\in H \label{ILP:no-bc}\\
o_h l(h)\leq \sum_{i\in [q]} x_{i,h} \leq o_h u(h), \qquad&
\forall h\in
H \label{ILP:quotas}\\
\sum_{h\in H} x_{i,h} \leq n_i, \qquad& \forall i\in [q] \label{ILP:num-residents}
\\
x_{i,h} = 0, \qquad &\forall i \in [q], h\in H\setminus
A(t_i)
\label{ILP:acceptablitiy}\\
x_{i,h}\in\{0,1,\dots,n_i\}, \qquad o_h \in \{0, 1\}, \qquad y_h \in \{0, 1\}, \qquad &\forall i\in [q],
\forall h \in H
\label{ILP:integrality}
\end{align}
Condition (\ref{cond:undersubscribed}) ensures $y_h = 1 $ holds for every undersubscribed hospital.
Condition (\ref{ILP:no-bp}) ensures that no blocking pair between a
resident of type~$t_i$ and hospital~$h\in H$ exists:
If $y_h = 0$ (i.e.\, $h$ is closed or full), then $h$ is not part
of a blocking pair and the inequality is fulfilled.
Otherwise the inequality
enforces that for each resident type $t_i$ and hospital $h\in H$,
all residents of type~$t_i$ are assigned to $h$ or
hospitals they prefer to $h$.
Condition (\ref{ILP:no-bc}) ensures that no blocking coalition exists by
enforcing that for all closed hospitals $h\in H$ the number of
residents that
are assigned to hospitals they find worse than $h$ is below $l(h)$.
Conditions (\ref{ILP:quotas})-(\ref{ILP:integrality}) ensure that
the matching encoded in the
variables is a feasible assignment by checking whether the number
of resident that are assigned to a hospital is either zero or
between its lower and upper quota, by enforcing that for each
resident type the number of assigned residents of this type is
smaller or equal to the number of residents of this type in the
instance and by enforcing that no resident is
assigned to a hospital that she does not accept.
Thus, the ILP admits a feasible solution if and only if the given
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance admits a stable matching. As the number of variables used in
the ILP lies in $\mathcal{O}(m^2 \cdot m!)$, it is
possible to apply Lenstra's algorithm
\cite{DBLP:journals/mor/Kannan87,DBLP:journals/mor/Lenstra83}
to solve the
problem in $\mathcal{O}(f(m)\cdot n)$ time for some computable
function~$f$.
\end{proof}
However, it is not
possible to follow a similar approach to construct a fixed-parameter
tractable algorithm for the the number $m_{\text{quota}}$ of hospitals with
non-unit lower quota. In
fact,~\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace is NP-complete even for only three
hospitals
with non-unit lower quota.
\begin{proposition}\label{pr:HRLUQI-NP-const}
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace is NP-complete, even if only
three hospitals have
lower and
upper quota two and all other hospitals have upper quota one.
\end{proposition}
\begin{proof}
We reduce from the
NP-hard
\textsc{Clique} problem \cite{DBLP:books/fm/GareyJ79}, where given
a graph $G$ and an integer~$k$, the
task is to decide whether there exists a subset of vertices of size
$k$ in $G$ that are all pairwise adjacent. Given an
instance~$((V=\{v_1,\dots v_n \},E),k)$ of
\textsc{Clique}, we construct an
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance as
follows. For each vertex $v\in V$, let $e^v_1,\dots, e^v_{p_v}$ be
a
list of all edges incident to $v$.
\textbf{Construction:} For each vertex $v\in V$, we add a \emph{vertex
hospital}
$h_v$ with
lower and
upper quota one. Similarly, for each edge $e\in E$, we add an
\emph{edge
hospital} with lower and upper quota one. Moreover, we add $k$
\emph{vertex
selection hospitals} $h^\text{vert}_1,\dots, h^\text{vert}_{k}$
and
$\binom{k}{2}$ \emph{edge selection hospitals}
$h^\text{edge}_1,\dots,
h^\text{edge}_{\binom{k}{2}}$, all with lower and upper quota one.
Turning to the residents, for each vertex $v\in V$, we introduce a
\emph{vertex resident} $r_v$ with the following preferences:
$$r_v : h^\text{vert}_1 \succ \dots \succ h^\text{vert}_{k}\succ
h_{e^v_1} \succ \dots \succ h_{e^v_{p_v}} \succ h_v.$$
Moreover, we introduce an \emph{edge resident} $r_e$ for
each edge
$e\in E$ with the following preferences:
$$h^\text{edge}_1\succ \dots \succ
h^\text{edge}_{\binom{k}{2}}\succ
h_e.$$
Notably, in every stable matching, all vertex selection hospitals
will be filled with vertex residents and all edge selection
hospitals
will be
filled with edge residents.
In addition, we add for each~$i \in [k]$ a \emph{filling agent}
$r_i^\text{fill}$ with
the
following
preferences:
$$r_i^\text{fill} : h_{v_1}\succ \dots \succ h_{v_n} .$$
Finally, we introduce a penalizing component consisting of three
hospitals $h_1$, $h_2$, and~$h_3$ all with lower and upper quota
two and four residents $r^*$,
$r_1$, $r_2$, and $r_3$:
$$r^*:h_{v_1}\succ \dots \succ h_{v_n}\succ h_1, \qquad
r_1:h_1\succ
h_2,
\qquad r_2:h_2\succ h_3,
\qquad r_3:h_3\succ h_1.$$
See \Cref{fig:hardness-few-hospitals} for an example of the reduction.
The general idea behind the reduction is that the penalizing
component enforces that a vertex resident or a filling resident is
assigned to every vertex hospital.
Thus, no vertex resident can be matched to an edge hospital, and therefore,
$\binom{k}{2}$
edge hospitals need to be closed in any stable matching.
The vertex residents
corresponding to the
end points of the closed edge hospitals
need to be matched to a vertex selection hospital (since otherwise the edge hospital and vertex resident form a blocking pair), implying that
the vertices matched to vertex selection hospitals form a
clique.
We now show that there exists a clique of size~$k$ in the given
graph if and only if there exists a stable matching in the constructed \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace
instance.
\begin{figure}
\begin{minipage}{0.2\textwidth}
\begin{center}
\begin{tikzpicture}
\node[vertex, label=180:$v_1$] (r1) at (0, 0) {};
\node[vertex, label=90:$v_2$] (r2) at ($(r1) + (1,2)$) {};
\node[vertex, label=0:$v_3$] (r3) at ($(r1) + (2, 0)$) {};
\node[vertex, label=270:$v_4$] (r5) at ($(r1) + (1, -2)$) {};
\draw (r1) -- (r2) -- (r3) -- (r1);
\draw (r2) -- (r3) -- (r5);
\draw (r5) -- (r1);
\end{tikzpicture}
\end{center}
\end{minipage} \hspace*{0.6cm}\vline \hspace*{0.15cm}
\begin{minipage}{0.65\textwidth}
\begin{center}
\begin{tikzpicture}
\node (ver-dist) at (0, 1.5) {};
\node (hor-dist) at (1.5, 0) {};
\node (hd) at (3, 0) {};
\node[vertex, label=270:$r_{v_1}$] (r1) at (0, 0) {};
\node[vertex, label=270:$r_{v_2}$] (r2) at ($(r1) + (ver-dist)$) {};
\node[vertex, label={[yshift=0.15cm]90:$r_{v_3}$}] (r3) at
($(r2) + (ver-dist)$) {};
\node[vertex, label=90:$r_{v_4}$] (r4) at ($(r3) + (ver-dist)$) {};
\node[vertex,
label={[xshift=-0.15cm,yshift=-0.1cm]90:$h_{v_1}$}] (s1) at
($(r1) + ( hor-dist)$) {};
\node[vertex, label=90:$h_{v_2}$] (s2) at ($(s1) + (ver-dist)$) {};
\node[vertex,
label={[xshift=0.15cm,yshift=-0.1cm]90:$h_{v_3}$}] (s3) at
($(s2) + (ver-dist)$) {};
\node[vertex, label={[yshift=-0.1cm]90:$h_{v_4}$}] (s4) at
($(s3) + (ver-dist)$) {};
\node[squared-vertex, label=0:$h_1^{\ver}$] (hv1) at ($(r4) +
(hor-dist) + (ver-dist)$) {};
\node[squared-vertex, label=0:$h_2^{\ver}$] (hv2) at ($(r1) -
(ver-dist) + (hor-dist)$) {};
\draw (hv1) edge node[pos=0.92, fill=white, inner sep=2pt] {\scriptsize $1$} (r1);
\draw (hv2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r1);
\draw (hv1) edge node[pos=0.92, fill=white, inner sep=2pt] {\scriptsize $1$} (r2);
\draw (hv2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (r2);
\draw (hv1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r3);
\draw (hv2) edge node[pos=0.92, fill=white, inner sep=2pt] {\scriptsize $2$} (r3);
\draw (hv1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r4);
\draw (hv2) edge node[pos=0.92, fill=white, inner sep=2pt] {\scriptsize $2$} (r4);
\node[squared-vertex,
label={[xshift=-0.1cm,yshift=0.1cm]270:$h_{\{v_1, v_2\}}$}]
(e12) at ($(r1) - (hor-dist)$) {};
\node[squared-vertex, label={[xshift=-0.175cm]270:$h_{\{v_1,
v_3\}}$}] (e13) at ($(e12) + (ver-dist)$) {};
\node[squared-vertex, label={[xshift=-0.175cm]270:$h_{\{v_2,
v_3\}}$}] (e23) at ($(e13) + (ver-dist)$) {};
\node[squared-vertex,
label={[xshift=-0.1cm,yshift=-0.05cm]90:$h_{\{v_1, v_4\}}$}]
(e14) at ($(e23) + (ver-dist)$) {};
\node[squared-vertex,
label={[xshift=-0.1cm,yshift=-0.05cm]90:$h_{\{v_3, v_4\}}$}]
(e34) at ($(e14)+ (ver-dist)$) {};
\node[vertex, label={[yshift=0.05cm]270:$r_{\{v_1, v_2\}}$}]
(r12) at ($(e12) - (hor-dist)$) {};
\node[vertex, label={[yshift=0.05cm]270:$r_{\{v_1, v_3\}}$}]
(r13) at ($(e13) - (hor-dist)$) {};
\node[vertex,
label={[xshift=0.1cm,yshift=-0.05cm]90:$r_{\{v_2, v_3\}}$}]
(r23) at ($(e23) - (hor-dist)$) {};
\node[vertex, label={[yshift=-0.05cm]90:$r_{\{v_1, v_4\}}$}]
(r14) at ($(e14) - (hor-dist)$) {};
\node[vertex,
label={[xshift=0.1cm,yshift=-0.05cm]90:$r_{\{v_3, v_4\}}$}]
(r34) at ($(e34) - (hor-dist)$) {};
\node[squared-vertex,
label={[xshift=-0.1cm]90:$h_1^{\edge}$}] (he1) at ($(r23) -
(hor-dist)$) {};
\draw (he1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r12);
\draw (he1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r13);
\draw (he1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r23);
\draw (he1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r34);
\draw (he1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (r14);
\draw (e12) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r12);
\draw (e13) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r13);
\draw (e23) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r23);
\draw (e14) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r14);
\draw (e34) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r34);
\node[vertex, label={[xshift=-0.1cm]45:$r^*$}] (rs) at ($(s3) + 0.5*(hor-dist) - 0.5*(ver-dist)$) {};
\node[vertex, label=180:$r_3$] (re3) at ($(s3) + (hor-dist) + 0.5*(ver-dist)$) {};
\node[vertex, label=90:$r_2$] (re2) at ($(re3) - (ver-dist)$) {};
\node[vertex, label=90:$r_1$] (re1) at ($(re2) -(ver-dist)$) {};
\node[blue, squared-vertex, label=90:$h_1$] (h1) at ($(re1)+
(hor-dist)$) {};
\node[blue, squared-vertex, label=90:$h_2$] (h2) at ($(re2)+
(hor-dist)$) {};
\node[blue, squared-vertex, label=90:$h_3$] (h3) at ($(re3)+
(hor-dist)$) {};
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (re1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (re1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (re2);
\draw (h3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (re2);
\draw (h3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (re3);
\draw (h1) edge node[pos=0.9, fill=white, inner sep=2pt] {\scriptsize $2$} (re3);
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (rs);
\draw (s1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $1$} (rs);
\draw (s2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $2$} (rs);
\draw (s3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (rs);
\draw (s4) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (rs);
\draw (e12) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $3$} (r1);
\draw (e12) edge node[pos=0.82, fill=white, inner sep=2pt] {\scriptsize
$3$} (r2);
\draw (e23) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (r2);
\draw (e23) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$4$} (r3);
\draw (e13) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $4$} (r1);
\draw (e13) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (r3);
\draw (e34) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize $5$} (r3);
\draw (e34) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$4$} (r4);
\draw (e14) edge node[pos=0.9, fill=white, inner sep=2pt] {\scriptsize
$5$} (r1);
\draw (e14) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (r4);
\draw (s1) edge node[pos=0.85, fill=white, inner sep=2pt] {\scriptsize $6$} (r1);
\draw (s2) edge node[pos=0.85, fill=white, inner sep=2pt] {\scriptsize $5$} (r2);
\draw (s3) edge node[pos=0.85, fill=white, inner sep=2pt] {\scriptsize $6$} (r3);
\draw (s4) edge node[pos=0.85, fill=white, inner sep=2pt] {\scriptsize $5$} (r4);
\end{tikzpicture}
\end{center}
\end{minipage}
\caption{An example for the reduction showing NP-hardness of
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace with four open hospitals from \Cref{pr:HRLUQI-NP-const}.
The input instance is~$(T,2)$, where~$T$ is depicted in the left
picture.
The output of the reduction is depicted in the right picture.
Hospitals with lower (and upper) quota two are marked in blue, while all other hospitals have lower (and upper) quota one.
}
\label{fig:hardness-few-hospitals}
\end{figure}
{\bfseries ($\Rightarrow$)} Let $V'=\{v_{i_1},\dots
v_{i_k}\}\subseteq V$ be a
clique of
size $k$ in $(V,E)$ and $E'=\{e_{j_1}, \dots,
e_{j_{\binom{k}{2}}}\}$
the set of all edges lying in
the clique. We claim that the following matching $M$ is stable:
\begin{align*}
M= & \{(h^\text{vert}_\ell,\{r_{v_{i_\ell}}\})\mid \ell\in
[k] \}\cup \{(h^\text{edge}_\ell,\{r_{e_{j_\ell}}\})\mid
\ell\in [{{k}\choose{2}}] \}\ \\
& \cup \{(h_v,\{r_v\})\mid v\in V\setminus V' \} \cup
\{(h_e,\{r_e\})\mid
e\in E\setminus E' \} \\
& \cup \{(h_{v_{i_\ell}}, \{r_\ell^\text{fill}\}) \mid \ell \in
[k]\}
\cup \{(h_1,\{r^*, r_1\})\} \cup \{(h_3,\{r_2, r_3\})\}
\end{align*}
As no hospital is undersubscribed, only blocking coalitions can
block
$M$. Consequently, it is enough to iterate over all closed hospitals
and
argue why there does not exist a blocking coalition to open them.
The
only closed hospitals are $h_2$ and the edge hospitals corresponding
to
edges lying in the clique. As $r_1$ is matched to its most
preferred
hospital, there does not exist a blocking coalition to open $h_2$.
Moreover, as all edges~$e\in E$ such that $h_e$ is closed have both
their endpoints in~$V'$,
both vertex residents that find $h_e$ acceptable are
matched to
vertex selection hospitals and therefore do not want to open this
hospital.
{\bfseries ($\Leftarrow$)} Let $M$ be a stable matching in the
constructed
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace
instance. First of all note that $r^*$ needs to be matched to
$h_1$, as
otherwise no stable matching of the three
residents~$r_1$,~$r_2$, and~$r_3$ to the three hospitals $h_1$,
$h_2$,
and $h_3$
can exist (see \Cref{ob:counter}). Thus, all vertex hospitals need
to
be
full, which is only possible if all vertex residents are matched
to
vertex hospitals or vertex selection hospitals. From this it
follows that
exactly
${k}\choose{2}$ edge hospitals need to be closed, as all edge
residents
prefer every edge selection hospital to their designated edge hospital
and no vertex resident can be matched to an edge hospital.
Because $M$ is stable, no vertex resident forms a blocking
coalition
to open one of the ${k}\choose{2}$ closed edge hospitals.
Therefore, all vertex residents corresponding to endpoints of the
corresponding ${k}\choose{2}$ edges are assigned to one of
the~$k$
vertex selection hospitals. This implies that the vertices
corresponding to vertex residents assigned
to
vertex selection hospitals form a clique.
\end{proof}
As all hospitals from the \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance constructed in the above
reduction have lower
quota at most two, this result also strengthens the NP-hardness of \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace
from
\Cref{th::NP-compl} with respect to the lower and upper quota of the
hospitals.
\subsubsection{Number of open (or closed) hospitals in a stable matching}
\label{ss:oc}
As there may exist stable matchings of different sizes in the studied
many-to-one
matching problems, one might want to find a matching with the
lowest/highest number of open hospitals. For instance, this could be
useful in applications where opening a hospital comes at some cost or
where organizers get money or receive points for each open hospital.
The associated decision problem is
to decide for some given number $m_{\open}$ whether
there exists a stable matching in which exactly
$m_{\open}$ hospitals are
open. Obviously, it is also possible to ask the question for the dual
parameter, i.e., the number~$m_{\text{closed}}$ of closed hospitals in a
stable matching.
For the \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace model, \Cref{th:HRLUQIH'} and the proof of
\Cref{pr:HRLUQI-NP-const}
already imply parameterized hardness results for both parameters: By \Cref{th:HRLUQIH'}, it is NP-complete to decide whether there exists a stable matching opening four hospitals, implying paraNP-hardness for the parameter~$m_{\open}$. Moreover,
W[1]-hardness for the parameter $m_{\text{closed}}$ follows from the
reduction from \Cref{pr:HRLUQI-NP-const} (as only ${k}\choose{2}$ edge
hospitals and one hospital from the penalizing component are closed in a
stable matching in the constructed instance).
\begin{corollary} \label{c:HAco}
Deciding whether there exists a stable matching with four open
hospitals in a \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance is NP-hard. Deciding whether there
exists a stable matching with~$m_{\text{closed}}$ closed hospitals in a
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace instance is W[1]-hard parameterized by~$m_{\text{closed}}$,
even
if one knows that otherwise no stable matching exists and only three
hospitals have non-unit upper quota.
\end{corollary}
It remains open whether $\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace$
parameterized by $m_{\text{closed}}$ is in XP or paraNP-hard.
For the other two models, the parameterized hardness for the parameter
$m_{\open}$ follows from
\Cref{th:W-n} (which shows W[1]-hardness
parameterized by the number of residents), as every stable matching
can open at most one hospital per resident.
\begin{corollary} \label{c:op}
Deciding whether there exists a stable matching with $m_{\open}$
open hospitals in a \textsl{HR-$\text{Q}_\text{L}$}\xspace or \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance is W[1]-hard parameterized
by~$m_{\text{open}}$.
\end{corollary}
Since the created instance in the reduction in the proof of \Cref{th:W-n}
fulfills \Cref{ob:equiv}, these hardness results still
hold
even
if we relax the stability condition to allow
blocking pairs in a stable matching (but no blocking coalitions).
Turning now to the dual parameter $m_{\text{closed}}$,
none of the previous reductions has any implication on the complexity of
\textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace with respect to this parameter, as we have always used the
closed hospitals to encode some
constraints on the solution. This may lead to the initial hypothesis
that restricting the
number of closed hospitals makes the problem tractable.
Unfortunately, it turns out that this is not the case, as we show W[1]-hardness for this parameter by
somewhat switching the roles of open and closed hospitals.
\begin{proposition}
\label{thm:ha-m-closed}
Deciding whether there exists a
stable matching
with $m_{\text{closed}}$ closed hospitals in a \textsl{HR-$\text{Q}_\text{L}$}\xspace or \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace
instance
is
W[1]-hard parameterized by $m_{\text{closed}}$, even if one knows that
otherwise
there does not exist any stable matching.
\end{proposition}
\begin{proof}
We reduce from \textsc{Multicolored Clique}. Let $G=(V,E)$ be an
undirected graph with a partitioning of the vertices in $k$ different
colors $(V^1,\dots, V^k)$. \textsc{Multicolored Clique} asks whether there exists a clique
$V'\subseteq V$ of size $k$ containing one vertex from each color.
Without loss of generality, we assume that
there do not exist any edges between vertices of the same color.
Moreover, for each $c\neq d\in [k]$, let
$E^{c,d}$ denote the set of edges with one endpoint colored
in $c$ and the other endpoint colored in $d$, i.e., $e=\{u,v\}\in
E^{c,d}$ if~$(u\in V^c\wedge v\in V^{d})\vee (v\in V^c\wedge u\in
V^{d})$.
Parameterized by $k$, \textsc{Multicolored Clique} is
W[1]-hard~\cite{Pietrzak03}. Given an instance of \textsc{Multicolored
Clique}, we now construct an instance of \textsl{HR-$\text{Q}_\text{L}$}\xspace as follows.
\textbf{Construction:} We start by introducing a penalizing
component consisting of four hospitals: $h^*$ with lower quota
$\binom{k}{2}+2$
and $h_1$, $h_2$, and $h_3$ each with lower quota two. Moreover, we
introduce four penalizing residents $r^*$, $r_1$, $r_2$, and $r_3$ with
the following preferences:
$$r^*: h^*\succ h_1, \quad r_1:h_1\succ h_2, \quad r_2: h_2\succ h_3,\quad
r_3:h_3\succ h_1.$$
For each color $c\in [k]$, we insert a vertex
hospital~$h_v$ with lower quota $n^3|V^c|$ for each vertex of this
color $v\in
V^c$.
For each pair~$(v, v')$ of different vertices of the same color (i.e., $v \neq v'$ and $v, v'\in V^c$ for some $c\in [k]$), we introduce $n^3$ residents~$r_{v,v'}^i$ ($i\in [n^3]$) with the following preferences:
$$r_{v,v'}^i: h_v\succ h_{v'}, \qquad \forall i\in [n^3].$$
Note that for every vertex hospital $h_v$ for $v\in V^c$, there exist
$n^3(|V^c|-1)$ residents with $h_v$ as their top-choice.
In addition, for each edge
$e=\{u,v\}\in
E^{c,d}$ for some $c<d\in [k]$, we introduce an edge hospital $h_e$
with lower quota
$|E^{c, d}|+1$. We add a penalizing resident~$r^*_e$ with
the following preferences:
$$r_e^*:h_e\succ h^*.$$
The penalizing component ensures that at most $\binom{k}{2}$ edge
hospitals are closed in a stable matching.
Moreover, for each edge
$e=\{u,v\}\in
E^{c,d}$ for some $c<d\in [k]$ and each $e'\in
E^{c,d}\setminus\{e\}$,
we introduce an edge resident
$r_{e,e'}$ with the following preferences: $$r_{e,e'}: h_e\succ
h_u\succ
h_v\succ h_{e'}.$$
Hence, for every edge hospital $h_e$ with $e\in E^{c,d}$, there exist
$|E^{c,d}|-1$ residents with $h_e$ as top-choice.
See \Cref{fig:closed-hospitals} for an example.
We set $m_{\text{closed}}:=k+\binom{k}{2}+1$.
In a stable matching, all but one
hospital corresponding to vertices of one color should be open. The
closed hospital
corresponds to the selected vertex from this color for the clique.
Similarly, for each color combination $c < d\in [k]$, we
introduced
a gadget consisting of one hospital for each edge with this color
combination. In a stable matching, all but one hospital in this gadget
should be open and the closed hospital corresponds to the edge with
this color combination lying in the constructed clique.
We show that there exists a multicolored clique in the given graph if
and only if there exists a stable matching with $m_{\text{closed}}$ hospitals
in the constructed~\textsl{HR-$\text{Q}_\text{L}$}\xspace instance.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node[star, draw, label=180:$r_{v,w}$] (rvw) at (-4, 0) {};
\node[star, draw, label=180:$r_{w, v}$] (rwv) at ($(rvw) - (0, 2)$) {};
\node[star, draw, label=0:$r_{x, y}$] (rxy) at (6, 0) {};
\node[star, draw, label=0:$r_{y, x}$] (ryx) at ($(rxy) + (0 , -2)$) {};
\node[squared-vertex, label=90:$h_{v}$,label={[yshift=0.35cm]90:{$[128,
\infty]$}}] (hv) at ($(rvw) + (1, 0)$) {};
\node[squared-vertex,
label=270:$h_{w}$,label={[yshift=-0.35cm,xshift=-0.45cm]270:{$[128,
\infty]$}}] (hw) at ($(hv) + (0, -2)$) {};
\node[squared-vertex,
label=90:$h_{x}$,label={[yshift=0.35cm,xshift=0.1cm]90:{$[128,
\infty]$}}] (hx) at ($(rxy) - (1, 0)$) {};
\node[squared-vertex,
label=270:$h_{y}$,label={[yshift=-0.45cm]270:{$[128,
\infty]$}}] (hy) at ($(hx) + (0, -2)$) {};
\node[squared-vertex, label=90:$h_{\{v,
x\}}$,label={[yshift=0.45cm]90:{$[4,
\infty]$}}] (hvx) at ($(rvw) + (3, 1)$) {};
\node[squared-vertex, label=180:$h_{\{v,
y\}}$,label={[yshift=0.05cm,xshift=-0.1cm]90:{$[4,
\infty]$}}] (hvy) at ($(hvx) - (0, 2)$) {};
\node[squared-vertex, label=315:$h_{\{w,
y\}}$,label={[yshift=.cm,xshift=1.65cm]270:{$[4,
\infty]$}}] (hwy) at ($(hvy) - (0, 2)$) {};
\node[vertex, label=90:$r_{e_1, e_2}$] (rvxvy) at (3, 2) {};
\node[vertex, label=90:$r_{e_1, e_3}$] (rvxwy) at ($(rvxvy) - (0, 1)$)
{};
\node[vertex, label=270:$r_{e_2, e_1}$] (rvyvx) at ($(rvxwy) - (0, 1)$)
{};
\node[vertex, label=270:$r_{e_2, e_3}$] (rvywy) at ($(rvyvx) - (0, 1)$)
{};
\node[vertex, label=270:$r_{e_3, e_1}$] (rwyvx) at ($(rvywy) - (0, 1)$)
{};
\node[vertex, label=270:$r_{e_3, e_2}$] (rwyvy) at ($(rwyvx) - (0, 1)$)
{};
\node[vertex, label={[xshift=0.18cm]180:$r_{\{v, x\}}^*$}] (rvx) at
($(hvx) + (-1.5, -2)$) {};
\node[vertex, label=0:$r_{\{v, y\}}^*$] (rvy) at ($(hvy) + (-1.5, -2)$)
{};
\node[vertex, label=270:$r_{\{w, y\}}^*$] (rwy) at ($(hwy) + (0, -1)$)
{};
\draw (hv) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvw);
\draw (hw) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvw);
\draw (hv) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rwv);
\draw (hw) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rwv);
\draw (hx) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rxy);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rxy);
\draw (hx) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (ryx);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (ryx);
\draw (hvx) edge node[pos=0.67, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvxvy);
\draw (hvx) edge node[pos=0.6, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvxwy);
\draw (hvx) edge node[pos=0.7, fill=white, inner sep=2pt] {\scriptsize
$4$} (rvyvx);
\draw (hvx) edge node[pos=0.9, fill=white, inner sep=2pt] {\scriptsize
$4$} (rwyvx);
\draw (hvy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvyvx);
\draw (hvy) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvywy);
\draw (hvy) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$4$} (rvxvy);
\draw (hvy) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$4$} (rwyvy);
\draw (hwy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rwyvx);
\draw (hwy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rwyvy);
\draw (hwy) edge node[pos=0.9, fill=white, inner sep=2pt] {\scriptsize
$4$} (rvxwy);
\draw (hwy) edge node[pos=0.7, fill=white, inner sep=2pt] {\scriptsize
$4$} (rvywy);
\draw (hvx) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvx);
\draw (hvy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rvy);
\draw (hwy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rwy);
\draw (hv) edge node[pos=0.6, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvxvy);
\draw (hx) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rvxvy);
\draw (hv) edge node[pos=0.83, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvxwy);
\draw (hx) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rvxwy);
\draw (hv) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvyvx);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rvyvx);
\draw (hv) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvywy);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rvywy);
\draw (hw) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rwyvx);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rwyvx);
\draw (hw) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rwyvy);
\draw (hy) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$3$} (rwyvy);
\node[squared-vertex,
label=270:$h^*$,label={[yshift=.cm,xshift=0.6cm]270:{$[3,
\infty]$}}] (hs) at ($(hw) - (0, 2)$) {};
\node[squared-vertex, label=0:$h_1$] (h1) at ($(hs) - (0, 1)$) {};
\node[squared-vertex, label=0:$h_2$] (h2) at ($(h1) - (0, 1)$) {};
\node[squared-vertex, label=0:$h_3$] (h3) at ($(h2) - (0, 1)$) {};
\draw (hs) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvx);
\draw (hs) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rvy);
\draw (hs) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rwy);
\node[vertex, label=180:$r^*$] (rs) at ($(hs) - (1, 0)$) {};
\node[vertex, label=180:$r_1$] (r1) at ($(rs) - (0, 1)$) {};
\node[vertex, label=180:$r_2$] (r2) at ($(r1) - (0, 1)$) {};
\node[vertex, label=180:$r_3$] (r3) at ($(r2) - (0, 1)$) {};
\draw (hs) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (rs);
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (rs);
\draw (h1) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (r1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$2$} (r1);
\draw (h2) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (r2);
\draw (h3) edge node[pos=0.8, fill=white, inner sep=2pt] {\scriptsize
$2$} (r2);
\draw (h3) edge node[pos=0.76, fill=white, inner sep=2pt] {\scriptsize
$1$} (r3);
\draw (h1) edge node[pos=0.86, fill=white, inner sep=2pt] {\scriptsize
$2$} (r3);
\end{tikzpicture}
\end{center}
\caption{An example of the reduction in the proof of \Cref{thm:ha-m-closed}
with input graph $(\{v, w, x, y\}, \{e_1 = \{v, x\}, e_2= \{v, y\}, e_3
=
\{w, y\}\})$ and color distribution $(V^c=\{v,w\},V^d=\{x,y\})$.
Residents depicted by a star represent $4^3\cdot 1 = 64$ residents with
these
preferences.}\label{fig:closed-hospitals}
\end{figure}
{\bfseries ($\Rightarrow$)} Let $V'\subseteq V$ be a clique in $G$ with
$|V'\cap
V^c|=1$ for all $c\in [k]$.
We denote by $v^c_*$ the vertex from~$V' \cap V^c$.
Let $E'$ denote the set of all edges
lying in the clique, i.e., all edges where both endpoints belong to
$V'$. Note that for each two different colors $c< d\in [k]$ it holds
that
$|E'\cap
E^{c,d}|=1$. We construct a stable matching
$M$ as follows. For each $c\in [k]$ and each~$v\in
V^c\setminus V'$, we match to $h_v$ all residents with $h_v$ as top
choice, i.e., $\bigcup_{v'\neq v\in
V^c\wedge i\in [n^3]} \{r_{v,v'}^i\}$, and all residents which have
$h_{v_*^c}$ as their top-choice and $h_v$ as their second choice, i.e.,
$\bigcup_{ i\in [n^3]} \{r_{v_*^c,v}^i\}$. Moreover, for each $c<d\in
[k]$ and
each edge $e\in E^{c, d}\setminus E'$, we
match to $h_e$ all edge
residents~$r_{e,e'}$ with $h_e$ as top choice, i.e.,
$\bigcup_{e'\neq e\in E^{c, d}}
\{r_{e,e'}\}$, the penalizing resident $r^*_e$, and the
resident~$r_{e^*, e}$ where~$\{e^*\}=E^{c, d}\cap E'$.
Finally, we match $r^*$ and $r_1$ to hospital $h_1$ and $r_2$ and $r_3$ to
hospital $h_3$.
Since there are exactly $n^3 |V^c|$ residents matched to each open vertex hospital $h_v$ with $v\in V^c$, exactly $|E^{c, d}| + 1$ residents matched to each open edge hospital $h_e$ with $e\in E^{c, d}$, and two residents matched to $h_1$ and $h_3$, it follows that $M$ is feasible.
We now argue that $M$ is stable.
Since all residents are matched to their most-preferred open hospital, it follows that there is no blocking pair.
Next, we show that there exists no blocking coalition to open a closed
vertex hospital~$h_v$ for a vertex $v\in V^c$ for some $c\in [k]$.
Since $h_v$ is closed, it follows that $v\in V'$.
The only vertex
residents which prefer to be matched to $h_v$ to their hospital in $M$ are
the $n^3 (|V^c| - 1)$ vertex residents of the form $r^i_{v, v'}$ for $i\in
[n^3]$ and $v' \in V^c\setminus \{v\}$.
Moreover, there exist at most
$(n-1)n^2$ edge residents that find $h_v$ acceptable (number of edges
incident to $v$ times two). As $(n-1)n^2 <
n^3$, there
cannot exist a blocking coalition of $n^3|V^c|$ residents to open~$h_v$.
We now turn to a closed edge hospital~$h_{e^*}$.
Since $h_{e^*}$ is closed, it follows that both endpoints of~${e^*}$ are contained in $V'$.
Let ${e^*}\in E^{c, d}$ for some $c< d\in [k]$.
The residents which prefer~$h_{e^*}$ to their hospital in~$M$ are the
edge residents~$r_{e^*, e}$ for $e\in E^{c, d}\setminus\{e^*\}$ and $r^*_{e^*}$. However,
these are only $|E^{c, d}| < l (h_{e^*})$ residents.
Thus, there
does not exist a blocking coalition to open $h_{e^*}$.
Finally,
there exist $\binom{k}{2}$ penalizing residents~$r_e^*$ that are
unmatched. However,
as $h^*$ has lower quota~$\binom{k}{2}+2$ and is only preferred by those $\binom{k}{2}$
unmatched
penalizing residents and $r^*$, there does not exist a blocking coalition
to open~$h^*$.
{\bfseries ($\Leftarrow$)} Assume that there exists a stable matching $M$
in the
constructed \textsl{HR-$\text{Q}_\text{L}$}\xspace instance. Note first of all that every stable matching
matches
$r^*$ to $h_1$, as otherwise two residents from~$r_1$, $r_2$, and~$r_3$
form a blocking coalition to open one of $h_1$, $h_2$, or $h_3$ (see
\Cref{ob:counter}). This implies that~$h^*$ needs to be closed
which in turn implies that at most $\binom{k}{2}$ edge hospitals can be
closed (because otherwise $r^*$ together with the at least $\binom{k}{2}$
residents $r_e^*$ such that $h_e$ is closed form a blocking coalition to open
$h^*$). For each of the $\binom{k}{2}$ color combinations $c< d\in [k]$,
there are $|E^{c, d}|^2$ residents that find at least one edge
hospital~$h_e$ with $e\in E^{c,d}$ acceptable.
Thus, the
residents only suffice to open $|E^{c, d}|-1$ of the edge hospitals
corresponding to edges from $E^{c,d}$. Note further that it is
only possible to open $|E^{c, d}|-1$ edge hospitals if for all $e\in E^{c, d}$ edge resident $r_e$ is matched to an edge hospitals $h_{e'}$ for some $e' \in E^{c, d}$.
This implies that
all edge residents need to be matched to edge hospitals.
Note further that for each color~$c\in [k]$, at least $|V^c|-1$ vertex
hospitals corresponding to vertices from $V^c$ are open (if there are two
closed vertex hospitals~$h_v$ and $h_{v'}$ for $v\neq v'\in V^c$, then
$\{r_{v, w}^i : i\in [n^3], w \in V^c \setminus \{v\}\} \cup
\{r_{v', v}^i : i \in [n^3]\}$ form a blocking coalition to open
$h_{v}$. Consequently, at most one vertex
hospitals from each color is closed in every stable matching.
If less than $k$ vertex hospitals are closed or the $k$ closed vertex
hospitals do not form a clique, then there
exists an edge~$e = \{v, w\}\in E^{c, d}$ for some~$c< d\in [k]$ such
that $h_e$ is closed and $h_v$ or $h_w$ is open in~$M$ (as for each
$c<d\in [k]$ there exists a hospital $h_e$ with $e\in E^{c,d}$ that is
closed). We assume without loss of generality that $h_v$ is open in~$M$.
As $h_e$ is
closed and all edge residents are matched to edge hospitals, for every edge $e' \in E^{c, d}\setminus \{e\}$, resident $r_{e,e'}$ together with $h_v$
forms a blocking pair, as
$r_{e, e'}$ prefers being matched to $h_v$ to being matched to~$M(r_{e,e'})=h_{e'}$.
Thus, the $k$ vertices corresponding to the closed
vertex hospitals form a clique.
\end{proof}
While we have seen in this section that for both \textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace deciding
whether a stable matching with $m_{\open}$ open ($m_{\text{closed}}$
closed)
hospitals exists is W[1]-hard parameterized by $m_{\open}$
($m_{\text{closed}}$),
the two problems still lie in XP. For both parameters, we can iterate over
all possible subsets of open hospitals $H_{\open}\subseteq H$ of allowed
size and
subsequently employ the algorithms from \Cref{ob:HRLQ-H'} and
\Cref{pr:HRULQ-H'}.
\section{A Restricted Case: Lower Quota Two} \label{sec:q2}
In this section, we consider the special case of our models where all
hospitals have lower quota at most two. For \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, we have already seen in
\Cref{pr:HRLUQI-NP-const} that the problem remains NP-hard even in this
case. However, for the other two models, all our hardness proofs
used hospitals with lower quota three.
In the following, let \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace denote
the restriction of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace to instances in which all hospitals
have
lower quota one or two.
We present a
polynomial-time algorithm for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace, which computes a stable matching
if one
exists. As \textsl{HR-$\text{Q}_\text{L}$}\xspace is a special
case
of \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, this algorithm also applies to \textsl{HR-$\text{Q}_\text{L}$}\xspace instances where all
lower quotas are at most two. The main result of this section is the
following:
\begin{restatable}[]{theorem}{qq}
\label{t:q2}
If the lower quota of each hospital is at most two, then \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace (and
thereby also
\textsl{HR-$\text{Q}_\text{L}$}\xspace)
is solvable in $\mathcal{O}(n^3 m)$ time.
\end{restatable} We split the proof of this result in two sections. In the
first section, we give a description of our algorithm. In the second section,
we prove its correctness.
At the end of the section, we also derive a weakened version of the Rural
Hospitals
Theorem~\cite{RePEc:ucp:jpolec:v:92:y:1984:i:6:p:991-1016,10.2307/1913160}
for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace, showing that every stable matching in a \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace instance
matches the same set of
resident and opens the same number of hospitals.
In the following, we refer to all
hospitals with lower quota one as \emph{quota-one hospitals} and to all
hospitals
with lower quota two as \emph{quota-two hospitals}.
\subsection{Warm-up: Irving's Algorithm for Stable Roommates}
\label{sec:irving}
As a warm-up, we first consider the case in which each hospital with lower
quota two finds at most two residents acceptable and each quota-one
hospital also has upper quota one.
Such an instance of \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace can be reduced to a \textsc{Stable Roommates}
instance.
The \textsc{Stable Roommates} problem takes as input a set of agents, where
each agent has preferences over an acceptable subset of the other agents,
and the goal is to find a stable matching where each agent is matched to at
most one other agent and stability is defined as the absence of a pair of
agents preferring each other over their partner in the matching.
The reduction works as follows.
We insert an agent~$a_r$ for each resident $r\in R$
and an agent $a_h$ for each quota-one hospital $h\in H$. For resident~$r\in R$, we construct the preferences of $a_r$ from the preferences of
$r$ by replacing each quota-one hospital $h\in H$ by $a_h$ and each
quota-two hospital $h\in H$ which accepts $r$ and one other resident~$r'$
by the agent $a_{r'}$ corresponding to resident $r'$. Moreover, for each
quota-one hospital~$h\in H$, we construct the preferences of $a_h$ from
the preferences of $h$ by replacing each resident~$r\in R$ by the
corresponding agent $a_r$ (note that parts of this reduction are
``inverse'' to
the one sketched at the beginning of \Cref{sec:NP-hard}).
It is easy to see that any stable matching for the \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace instance
induces a stable matching for the \textsc{Stable Roommates} instance and
vice versa.
Thus, \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace is a generalization of \textsc{Stable Roommates}, and our algorithm to solve \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace will be a generalization of Irving's algorithm~\cite{DBLP:journals/jal/Irving85} to solve \textsc{Stable Roommates}.
In the following, we give a brief description of Irving's algorithm as a warm-up.
The algorithm consists of two phases:
In the first phase of Irving's algorithm, every agent proposes to the first
agent on its preference list.
Whenever an agent receives multiple proposals, it rejects all proposals but the best.
If agent $a$ rejects the proposal of $a'$, then the acceptability of $a$
and $a'$ is deleted, and $a'$ proposes to its most preferred agent that is
still contained in its preferences unless its preferences got empty.
This phase ends when every agent either holds and receives exactly one proposal or has empty preferences.
In the second phase of Irving's algorithm, the algorithm searches for a
certain substructure called \emph{rotation}.
A rotation is a sequence of pairs of agents $(a_1, b_1), \dots, (a_k ,
b_k)$ such that $a_i$ is the last agent on $b_i$'s preferences, and
$b_{i+1}$ is the second agent on $a_i$'s preferences (where all indices are
taken modulo~$k$).
Note that, by the definition of Phase 1, one can show that $b_i$ is the
first agent in $a_i$'s preferences.
Whenever such a rotation is found, it is \emph{eliminated}, meaning that $b_i$ deletes the mutual acceptability to all agents which are after~$a_{i-1}$ in its preferences (note that this always includes $a_i$).
The algorithm successively eliminates rotations.
If the preferences of an agent become empty through the elimination of a rotation, then the algorithm concludes that there does not exist a stable matching.
Otherwise, at some point the preferences of each agent contain at most one
agent inducing a stable matching.
Note that the elimination of a rotation can also be interpreted as deleting
the mutual acceptabilities of $a_i$ and $b_i$ for each $i\in [k]$ and
restarting Phase~1 afterwards.
This interpretation is closer to our algorithm for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace.
Our algorithm generalizes Irving's algorithm as follows.
In the first phase, residents propose and reject proposals as in Irving's algorithm.
However, a hospital~$h$ proposes to the first $u (h)$ hospitals on its
preferences, and only rejects a proposal if it already received $u(h)$
proposals.
Furthermore, quota-two hospitals only start proposing after they received a proposal.
In the second phase, we search for some substructures (which are
generalizations of rotations).
However, we have to take into account that quota-two hospitals are not
necessarily open if they received only one proposal.
Therefore, we avoid that possibly closed quota-two hospitals are
contained in
a rotation by replacing quota-two
hospitals occurring in a rotation by the first or second resident on their
preferences.
Between the first and second phase (we call this later Phase~1b), we replace each quota-two hospital which has to be open (i.e., which received at least two proposals) by several quota-one hospitals to simplify the presentation of Phase~2.
We now describe our algorithm for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace\ in detail.
\begin{algorithm}[t]
\caption{Algorithm for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace (high-level
description)}\label{alg:euclid}
\begin{algorithmic}[1]
\Input{An \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace instance $\mathcal{I}$}
\Output{A stable matching in $\mathcal{I}$ or NO if $\mathcal{I}$
does not
admit a stable matching.}
\State Apply \textbf{Phase 1a - Propose\&Reject}
\State $S \gets \{\text{residents with non-empty
preferences}\}$\Comment{Initialization}
\While{there exists some resident $r$ with at least two hospitals
on her
preferences}
\State Apply \textbf{Phase 1a - Propose\&Reject}
\While{a hospital holding at least two proposals
exists}
\For{\textbf{each} hospital $h$ holding at least two proposals}
\State Split $h$ into
$u(h)$ hospitals $h^1$, \dots, $h^{u(h)}$ \Comment{Phase 1b}
\EndFor
\State Apply \textbf{Phase 1a - Propose\&Reject}
\EndWhile
\If{there exists some resident $r$ with at least two hospitals on
her
preferences}
\State Find a generalized rotation $R$. \Comment{Phase 2}
\State Eliminate $R$.
\EndIf
\EndWhile
\If{all residents from $S$ have exactly one hospital on their
preferences
left}
\State \textbf{return} matching $M$ that matches every resident $r$
with
non-empty preferences to the
hospital from $\mathcal{I}$ corresponding to the remaining hospital
on her preferences.
\EndIf
\State \textbf{return} NO
\end{algorithmic}
\label{alg}
\end{algorithm}
\subsection{Description of the Algorithm for
\textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace
}
Similar to Irving's algorithm, our algorithm is split into two phases,
where
the first
phase is again split into Phase 1a and Phase 1b.
Phase 1a identifies hospital-resident pairs which cannot be part of a
stable matching using a propose-and-reject approach.
Subsequently, for each such hospital-resident pair $(r,h)$, hospital $h$ is deleted
from the preferences of $r$ and vice versa.
Furthermore, Phase 1a identifies some quota-two hospitals which
are open in every stable matching.
Phase 1b further simplifies the instance by replacing quota-two hospitals
that
are open in every stable matching by multiple copies of this hospital with lower quota one.
Phase~1a and Phase 1b are applied repeatedly until no hospital from which
we know that it is open in every stable matching exists. After that, in
Phase 2, we identify
substructures which we call ``generalized
rotations'' and subsequently eliminate them by deleting the acceptability
of some hospital-resident pairs.
\Cref{alg} gives a high-level description of our algorithm.
While Phase~1 keeps the number of stable matchings identical,
Phase 2 may
reduce the number of stable matchings in the instance, but still guarantees
that at least one stable matching survives (if there exists one in the
original instance).
The algorithm applies Phase 1 and Phase 2 alternately
until every resident has at most one hospital on her preferences.
The algorithm returns NO if, after the initialization, the
preferences of a resident got empty, as one can show that
all residents having non-empty preferences after the initialization are matched
in every
stable
matching. Otherwise, the algorithm
constructs a stable matching where all
residents with empty preferences are unmatched and all residents with
non-empty preferences are matched to the hospital on their preference
list (if this hospital was created by splitting a hospital $h$, then the resident is matched to $h$).
We now describe Phases 1a, 1b, and 2 in more detail.
We start with an
observation that allows us to assume that quota-one hospitals have also upper
quota one:
\begin{observation}
Given an \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace instance, replacing all quota-one
hospitals~$h\in H$ by $u(h)$ copies
$h^1,\dots, h^{u(h)}$ each with lower and upper quota one and with
the same preferences as $h$ and replacing $h$ by $h^1\succ \dots \succ
h^{u(h)}$ in the preferences of all residents results in an equivalent
instance.
\end{observation}
\subsubsection{Phase 1a - Propose\&Reject}
In Phase 1a, residents and hospitals propose
to one another. Residents always propose to hospitals and hospitals always
to residents. If a resident $r\in R$ proposes to a hospital $h\in H$, then $h$ can either \emph{accept} or \emph{reject} the proposal from~$r$. We
say that a hospital~$h$ \emph{holds} a proposal $r$ if~$r$ proposed to $h$
and $h$
did not reject the proposal (until now). We say that a resident~$r$
(currently)
\emph{issues} a proposal if there exists a hospital~$h$ that holds
the
proposal
$r$. The notation also applies if the roles of residents and hospitals are
swapped. Considering quota-two hospitals, we distinguish between
\emph{activated} and \emph{deactivated} hospitals.
Activated hospitals propose to residents, while deactivated hospitals do not.
Initially, all quota-two
hospitals
are deactivated.
\paragraph{Algorithm (Phase 1a).} We proceed in multiple rounds. In
each
round, a resident or quota-one hospital with non-empty preferences that
does not
currently issue a proposal or an activated quota-two hospital is selected.
If a resident or quota-one hospital is selected, then it
proposes to the first hospital or resident on its preference list. If an
activated quota-two hospital~$h$ is selected, then the
hospital
proposes to the first $u(h)$ residents on its preference list
unless $h$ received exactly one proposal and this one proposal comes from a
resident~$r$ which is among the first $u(h)$ residents in $h$'s
preferences: In this case,~$h$ only proposes to the first $u(h)-1$
residents that are not~$r$.
If a resident or a quota-one hospital receives a proposal, then it
\emph{accepts} the proposal if it does not hold a proposal or if it prefers the new
proposal to the one it currently holds. Similarly, it \emph{rejects} a
proposal if it
either already holds or later receives a better proposal. A quota-two
hospital $h$ \emph{accepts} a proposal $r$ if it does not hold
$u(h)$ proposals it prefers to $r$. It \emph{rejects} a proposal $r$ if it
holds or at some point receives $u(h)$ proposals it prefers to $r$, or if
the hospital has been rejected by all but one resident on its preference
list. If
an agent $a$ proposes to an agent $a'$ and $a'$ rejects the proposal, then we
delete $a'$ from the preference list of $a$ and $a$ from the
preference list of $a'$.
A quota-two hospital $h$ gets activated if it receives a proposal or if one
of
its
proposals gets rejected. If $h$ currently
holds exactly one proposal by one of its~$u(h)$ most preferred residents
$r$, then it gets deactivated if it currently issues
$u(h)-1$ proposals
or has proposed to all residents on its preference list except $r$.
Otherwise, it gets deactivated if it currently issues $u(h)$ proposals or
has proposed to all residents on its preference list.
At the end of Phase 1a, we delete from the
preferences of all
quota-one hospitals and residents holding
a proposal all agents to which they
prefer the held proposal. Subsequently, we restore the mutual acceptability
of agents
by deleting for each agent $a$ an agent $a'$ from its preference list
if $a$ does not appear on the preference list of $a'$. Finally, we delete
all
quota-two hospitals with at most one resident on their preference list from
the instance and the preferences of all~agents.
\medskip
The intuitive
reasoning behind Phase 1a is the following. If an agent
rejects the proposal of another agent, then the two can never be matched to
each other in a stable matching. Thereby, no agent can be matched better
than the agent it proposes to.
Thus, any agent receiving a proposal can be sure that it does not end up
worse than the proposal it currently holds in a stable matching, since it
forms a blocking pair with the agent issuing its proposal otherwise.
After Phase 1a, each resident and quota-one hospital issues a
proposal to the first agent on its preference list and holds a proposal
from the last agent on its preference list.
The formal correctness proof of this phase presented in \Cref{a:p1a}
consists of proving that, in a stable matching, no agent can be matched to
an agent which was
deleted from its preferences during Phase 1a and that
all matchings that are stable after applying Phase 1a are also stable
before applying Phase 1a.
\begin{figure}[t]
\begin{minipage}{0.49\textwidth}
\begin{tikzpicture}
\node (I) at (0, 0) {\textbf{I}};
\node (xshift) at (2.5,0) {};
\node (yshift) at (0, -0.5) {};
\node (anchor1) at (0.5,-0.) {};
\node (anchor2) at ($(anchor1) + (yshift)$) {};
\node (anchor3) at ($(anchor2) + (yshift)$) {};
\node[anchor=west] (h1) at (anchor1) {$h_1 : r_3 \succ r_1$};
\node[anchor=west] (r1) at ($(anchor1) + (xshift)$) {$r_1 : h_1 \succ h_2$};
\node[anchor=west] (h2) at ($(anchor1) + (yshift)$) {$h_2 : r_1 \succ r_2$};
\node[anchor=west] (r2) at ($(anchor2) + (xshift)$) {$r_2 : h_4 \succ h_2 \succ h_3$};
\node[anchor=west] (h3) at ($(anchor2) + (yshift)$) {$h_3 : r_2 \succ r_3$};
\node[anchor=west] (r3) at ($(anchor3) + (xshift)$) {$r_3 : h_3 \succ h_1 \succ h_4$};
\node[anchor=west] (h4) at ($(anchor3) + (yshift)$) {$h_4 : r_2 \succ r_3$};
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\begin{tikzpicture}
\node (I) at (0, 0) {\textbf{II}};
\node (xshift) at (2.5,0) {};
\node (yshift) at (0, -0.5) {};
\node (anchor1) at (0.5,-0.) {};
\node (anchor2) at ($(anchor1) + (yshift)$) {};
\node (anchor3) at ($(anchor2) + (yshift)$) {};
\node[anchor=west] (h1) at (anchor1) {$h_1 : r_3 \succ r_1$};
\node[anchor=west] (r1) at ($(anchor1) + (xshift)$) {$r_1 : h_1 \succ h_2$};
\node[anchor=west] (h2) at ($(anchor1) + (yshift)$) {$h_2 : r_1 \succ r_2$};
\node[anchor=west] (r2) at ($(anchor2) + (xshift)$) {$r_2 : h_2 \succ h_3$};
\node[anchor=west] (h3) at ($(anchor2) + (yshift)$) {$h_3 : r_2 \succ r_3 $};
\node[anchor=west] (r3) at ($(anchor3) + (xshift)$) {$r_3 : h_3 \succ h_1$};
\node[anchor=west] (h4) at ($(anchor3) + (yshift)$) {$h_4 :$};
\end{tikzpicture}
\end{minipage}
\caption{An example for Phase 1.
Hospital $h_1 $ is a quota-one hospital, while the other three
hospitals are quota-two hospitals with upper quota two.
In the beginning (see instance I), each resident and $h_1$ propose to the
first agent on their preferences. All agents accept their received
proposal.
Since $h_4$ receives a proposal from $r_2$ and $h_3$ receives a proposal
from $r_3$, they get activated and
propose to $r_3$ respectively $r_2$.
Resident $r_3$ rejects the proposal from $h_4$, as she holds the proposal
of $h_1$, while $r_2$ accepts the proposal of $h_3$.
Then, the preferences of $h_4$ contain only one resident, and thus $h_4 $
rejects the proposal from $r_2$.
Consequently, $r_2$ proposes to $h_2$, which activates $h_2$.
Subsequently, $h_2$ proposes to $r_1$, who accepts the proposal. As no
quota-two
hospital received two proposals, no hospital gets split and Phase 1 ends.
The resulting instance is depicted as instance II.
}
\label{fig:example-p1-small}
\end{figure}
\subsubsection{Phase 1b - Split Hospitals} In this phase, we identify
quota-two hospitals that are open in every stable matching
and replace them by quota-one hospitals:
\paragraph{Algorithm (Phase 1b).} We replace each quota-two hospital $h$
holding at least two proposals by~$u(h)$ hospitals $\ensuremath{h^1}$,
\dots, $h^{u (h)}$ with lower and upper quota one
with the same preferences
as~$h$. In the preferences of all residents, $h$ is replaced by $h^1 \succ
\dots \succ h^{u (h)}$.
\medskip
Phase~1b allows us to replace quota-two hospitals with at least two
proposals by quota-one hospitals.
This makes Phase~2 simpler, as we do not have to distinguish between quota-two hospitals receiving at most one proposal and quota-two hospitals receiving at least two proposals.
We show in \Cref{a:p1b} that there exists
a one-to-one correspondence between stable matchings before and after the
application of Phase 1b.
To summarize, Phase
1 consists of
applying Phase 1a and Phase 1b as long as at least one hospital was split
in the last execution of Phase 1b.
An example for the execution of Phase 1 can be found in
\Cref{fig:example-p1-small}.
\subsubsection{Phase 2 - Eliminate Generalized Rotations}
We introduce some notation for the definition of a generalized rotation. We
call a quota-two hospital with more than two residents on its preferences
\emph{flexible\xspace}
and all other
quota-two hospitals
\emph{inflexible\xspace}.
Note that while we already know which
residents will be assigned to an open inflexible\xspace hospital (as the number of
residents on its preferences is equal to its lower quota), this is not
clear for
open flexible\xspace hospitals.
Given a resident $r$, we denote by~$h(r) $ the first hospital on $r$'s
preferences.
If $h(r)$ is flexible, then we define $g(r):= h(r)$.
Otherwise, we define $g(r)$ to be the second hospital on $r$'s preference
list.
A \emph{generalized rotation} is a sequence
$(a_1, b_1), \dots,
(a_k, b_k)$ consisting of residents and quota-one hospitals with $a_i\neq
a_j$ for all $i\neq j$
such that (all following indices are taken modulo $k$):
\medskip
\noindent \textbf{Relationship between $a_i$ and $b_{i+1}$}:
\begin{description}
\setlength\itemsep{-0.3em}
\item[\ab{1}] If $a_i$ is a quota-one hospital, then $b_{i+ 1}$ is
the
second resident on $a_i$'s preferences.
\item[\ab{2}] If $a_i$ is a resident and $h(a_i)$ is a flexible\xspace hospital,
then
$b_{i+1}$ is the second-most preferred
\item[] \hspace*{1.4cm} resident on $h (a_i)$'s preferences who is not $a_i$. \smallskip
\item[]\hspace*{-0.3cm} If $a_i$ is a resident and $h(a_i)$ is an
inflexible\xspace
hospital or a quota-one hospital and \smallskip
\item[\ab{3a}] \hspace*{0.5cm} if
$g(a_i)$ is a quota-one hospital, then $b_{i+1}:=g(a_i)$.
\item[\ab{3b(i)}] \hspace*{0.000cm} if $g(a_i)$ is a quota-two
hospital holding a
proposal $r$, then $b_{i+1} :=r$.
\item[\ab{3b(ii)}] if $g(a_i)$ is a quota-two
hospital which does
not hold a proposal, then $b_{i+1}$ is~$g(a_i)$'s
\item[] \hspace*{2.25cm} most preferred
resident who is not $a_i$.
\end{description}
\noindent \textbf{Relationship between $b_i$ and $a_i$}:
\begin{description}
\setlength\itemsep{-0.3em}
\item[\ba{1}] If $b_i$ is a quota-one hospital, then $a_{i}$ is the
last resident on $b_i$'s preferences.
\item[\ba{2}] If $b_i$ is a resident and the last hospital $h$ on
$b_i$'s
preferences is of quota one, then $a_{i} := h$.
\item[\ba{3}] If $b_i$ is a resident and the last hospital $h$ on
$b_i$'s
preferences is of quota two, then $a_{i}$ is
\item[]\hspace*{1.1cm} the resident with $h$ as top-choice,
i.e., the resident proposing to
$h$.
\end{description}
Note that residents and quota-one hospitals are treated the same as ordinary agents in
Irving's algorithm, while inflexible\xspace hospitals are treated as ``edges'' between
two residents.
\begin{figure}[t]
\begin{minipage}{0.49\textwidth}
\begin{tikzpicture}
\node (I) at (0, 0) {\textbf{I}};
\node (xshift) at (2.5,0) {};
\node (yshift) at (0, -0.5) {};
\node (anchor1) at (0.5,-0.) {};
\node (anchor2) at ($(anchor1) + (yshift)$) {};
\node (anchor3) at ($(anchor2) + (yshift)$) {};
\node[anchor=west] (h1) at (anchor1) {$h_1 : r_3 \succ r_1$};
\node[anchor=west] (r1) at ($(anchor1) + (xshift)$) {$r_1 : h_1 \succ h_2$};
\node[anchor=west] (h2) at ($(anchor1) + (yshift)$) {$h_2 : r_1 \succ r_2$};
\node[anchor=west] (r2) at ($(anchor2) + (xshift)$) {$r_2 : h_2 \succ h_3$};
\node[anchor=west] (h3) at ($(anchor2) + (yshift)$) {$h_3 : r_2 \succ r_3$};
\node[anchor=west] (r3) at ($(anchor3) + (xshift)$) {$r_3 : h_3 \succ h_1$};
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\begin{tikzpicture}
\node (I) at (0, 0) {\textbf{II}};
\node (xshift) at (2.5,0) {};
\node (yshift) at (0, -0.5) {};
\node (anchor1) at (0.5,-0.) {};
\node (anchor2) at ($(anchor1) + (yshift)$) {};
\node (anchor3) at ($(anchor2) + (yshift)$) {};
\node[anchor=west] (h1) at (anchor1) {$h_1 : r_3$};
\node[anchor=west] (r1) at ($(anchor1) + (xshift)$) {$r_1 : h_2$};
\node[anchor=west] (h2) at ($(anchor1) + (yshift)$) {$h_2 : r_1 \succ r_2$};
\node[anchor=west] (r2) at ($(anchor2) + (xshift)$) {$r_2 : h_2$};
\node[anchor=west] (h3) at ($(anchor2) + (yshift)$) {$h_3 : r_3 $};
\node[anchor=west] (r3) at ($(anchor3) + (xshift)$) {$r_3 : h_3 \succ h_1$};
\end{tikzpicture}
\end{minipage}
\caption{An example for Phase 2.
Hospital $h_1 $ is a quota-one hospital, while the other two hospitals are
quota-two hospitals with upper quota two. Initially (see
instance I), $h_1$ holds the proposal of $r_1$, $h_2$ the proposal
of $r_2$, and~$h_3$ the proposal of $r_3$.
The instance admits the following generalized rotation:
$(r_1, h_1), (r_3, r_2)$. Note that for $b_1=h_1$ case BA-1 applies, for
$a_1=r_1$ case AB$^+$-3b(i) applies, for $b_2=r_2$ case BA-3 applies, and
for~$a_2=r_3$
case AB$^+$-3a applies.
Eliminating this generalized rotation results in the instance II.
Afterwards, Phase~1 is applied, and $h_3 $ rejects $r_3$ because $h_1$ has only $r_3$ in its preferences.
The algorithm ends with the stable matching $\{(h_1 , r_3), (h_2, \{r_1, r_2\})\}$.
}
\label{fig:example-p2-small}
\end{figure}
\paragraph{Algorithm (Phase 2).}
If the preference list of every resident contains at most one agent, then these preference lists correspond to a stable matching.
Otherwise, Phase 2 computes a generalized rotation by starting with an
arbitrary
resident whose
preference list has length at least two as $a_1$ and subsequently applying
the
relationships depicted above to find $b_2$, $a_2$, \dots~until this
procedure cycles and a generalized rotation has been found.
Subsequently, we \emph{eliminate} the found rotation by deleting, for
all $i\in
[k]$, the mutual
acceptability of $a_i$ and $b_i$ if one of them is a hospital, and
otherwise the mutual acceptability of hospital $h(a_i)$ and $b_i$.
After that, Phase 1 is applied again to the resulting instance.
\medskip
An
example for
Phase 2 can
be found in \Cref{fig:example-p2-small}.
The correctness proof of this phase presented in \Cref{a:p2} starts with
showing that if there exists a stable matching in
the instance before the elimination of a generalized rotation, then there also exists a
stable matching after its elimination (\Cref{lem:rotation}). For the other
direction, we show
that each matching that is stable after the elimination of a generalized
rotation, and matches all residents with non-empty preferences (for whom
it can be proven that they have to be matched in any stable matching), is
also stable before its elimination (\Cref{lem:forward}).
In a ``classical'' rotation $(a_1, b_1), \dots,
(a_k, b_k)$ for \textsc{Stable
Roommates}~\cite{DBLP:journals/jal/Irving85}, for all $i\in [k]$,
agent~$b_{i+1}$ is
the second
agent on the preference list of~$a_{i}$ and~$a_i$ is the last agent on
the preference list of~$b_i$, which implies that $b_i$ is the top-choice of
$a_i$. Here, eliminating a rotation consists of deleting the mutual
acceptability of $a_i$ and~$b_i$ for all $i\in [k]$ and results in an
instance that admits a stable matching if the original instance admits one.
Part of the reason for this is that if we assume
that
there exists a stable matching $M$ which contains the pairs $(a_i, b_i)$
for all
$i
\in [ k]$, then the matching~$M'$ arising by replacing these pairs by
the pairs $(a_i,
b_{i+1})$ is also stable:
Each agent~$b_i$ prefers~$M'$ to~$M$, and agent $a_i$ can only form a
blocking pair with $b_i$, implying that no blocking pair has been
introduced.
However, applying this classical definition to \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace, the
observation from above does not longer hold (for example, if a inflexible\xspace
quota-two hospital would appear as $a_i$, then a feasible matching must
match it to both $b_i$ and $b_{i+1}$ or close it).
Therefore, we generalize the definition of a rotation in a way such that no
quota-two
hospital appears in a generalized rotation, while keeping the intuition
that $a_i$ still corresponds to $b_i$'s ``least preferred option'' and
$b_{i+1}$ to $a_i$'s ``second-most preferred option'':
For~\ba{1} and~\ba{2}, $a_i$ is still $b_i$'s least preferred agent, while
for~\ba{3} following the classical
definition, it would be necessary to set $a_i$ to a quota-two hospital~$h$.
Instead, we set $a_i$ to be the resident proposing to~$h$, which can be
interpreted as matching $b_i$ to~$h$ together with $a_i$. For the
relationship between~$a_i$ and $b_{i+1}$, for \ab{1}, $b_{i+1}$ is still
the second agent in the preferences of $a_i$. For \ab{2}, it
is necessary to recall that for a
flexible\xspace hospital $h$ there exist multiple possibilities which residents
are matched to~$h$ in a stable matching. The ``most preferred option'' of
a resident $r$ is to be
matched to $h(r)$ together with $h(r)$'s most preferred remaining resident,
while
her ``second-most preferred option'' is to be matched to $h(r)$ with
$h(r)$'s
second-most preferred remaining resident. If~$h(r)$ is inflexible\xspace, then
there
exists only
one possibility for a resident $r$ to be matched to~$h(r)$ so her
second-most
preferred alternative is to be matched to $g(r)$ with the necessary case
distinctions made in \ab{3a}, \ab{3b(i)}, and \ab{3b(ii)}, which
corresponds to treating~$h(r)$ like an edge between $r$ and the other
resident acceptable to $h(r)$.
\subsection{Proof of Correctness of the Algorithm}
In the following section, we prove the correctness of the presented
algorithm. We start by proving the correctness for each phase separately, meaning
that there exists a stable
matching in the instance after the execution of a phase if and only if
there exists a stable matching before the execution of the phase.
The correctness of both phases together then proves the correctness of
the full algorithm.
\subsubsection{Phase 1a}\label{a:p1a}
We start proving the correctness of Phase 1a by showing that if an agent deletes another agent from its
preference list, then they cannot be matched in a stable matching. To do
so, we iterate over all possible cases in Phase 1a where the
preferences of an agent get modified.
In the following, we slightly abuse notation by calling a blocking
coalition $(h, \{r\})$ to open a quota-one hospital $h$ also a blocking
pair.
\begin{restatable}{lemma}{firstphase}
\label{lem:first-phase}
If an agent $a^*$ rejects the proposal of another agent $a'$, then
$a^*$ cannot be matched to $a'$ in any stable matching.
\end{restatable}
\begin{proof}
Assume that the lemma does not hold, and let $r$ be the first resident
and $h$ the first hospital such that one of them rejected the other,
but $M(r) = h$ in some a stable matching~$M$. There are
two possible reasons for this deletion: Either $r$ rejected a proposal
from $h$ or $h$ rejected a proposal from $r$.
{\bfseries Case 1: } $r$ rejected $h$.
Then $r$ received a proposal from a hospital $h'$ it prefers to $h$.
{\bfseries Case 1.1: } $h'$ is a quota-one hospital.
Then $h'$ has been rejected from all residents it prefers to $r$.
By the choice of $r$ and~$h$, it follows that $h'$ cannot be matched to
a resident it prefers to $r$ in $M$.
Thus, $(r, h')$ is a blocking pair for $M$, a contradiction.
{\bfseries Case 1.2: } $h'$ is a quota-two hospital.
Then $h'$ has received a proposal from some resident $r'\neq r$ before
proposing to $r$.
By the choice of~$h$ and~$r$, it follows that $r'$ cannot be matched
to a hospital it prefers to $h'$.
It follows that if $h'$ is closed in $M$, then $(h',\{r,r'\})$ is a
blocking coalition in $M$, a contradiction.
Furthermore, as $r$ received a proposal from $h'$, $r$ is one of the
first $u (h')$ residents on the
preferences of $h'$ which did not reject $h'$.
By the choice of~$h$ and $r$, it follows that $h'$ is not matched to
$u (h')$ resident it prefers over $r$.
Thus, if~$h'$ is open, then $(h',r)$ is a blocking pair in $M$, a
contradiction.
{\bfseries Case 2: } $h$ rejected $r$.
{\bfseries Case 2.1: } $h$ is a quota-one hospital.
Then before rejecting $r$, hospital $h$ received a proposal from a resident $r'$
it prefers over $r$.
By the choice of $h$ and $r$, it follows that $r'$ prefers $h$ to
$M(r')$.
Consequently,~$(h,r')$ is blocking pair in $M$, a contradiction.
{\bfseries Case 2.2: } $h$ is a quota-two hospital.
The hospital $h$ cannot reject $r$ because it has been rejected by all
other residents on its preference list, as otherwise $r'$ with $r\neq
r'\in M(h)$ has been rejected by~$h$ before $h$ rejected~$r$,
contradicting the choice of $h$ and $r$.
Thus before rejecting $r$, hospital $h$ received proposals from~$u (h)$
residents $r_1$, \dots,
$r_{u (h)}$ which $h$ prefers over $r$.
By the choice of $h$ and~$r$, it follows that $r_i$ cannot be matched
to a hospitals it prefers over $h$ for all~$i \in [u (h)]$. As $r$ is
matched to~$h$ in $M$, there needs to exists at least one $j\in [u (h)]$
with~$M(r_j)\neq h$. Then $(h,r_j)$ forms a blocking pair in $M$, a
contradiction.
\end{proof}
Recalling that agents propose to other agents in order of their
preference list, the
following observation directly follows from \Cref{lem:first-phase}.
\begin{observation}\label{ob:ma}
No resident or quota-one hospital $a^*$ which issues a proposal to an
agent~$a'$ can be
matched to an agent $\widetilde{a}$ which she prefers to $a'$ in a
stable matching. No
quota-two hospital~$h$ which issues a proposal to some residents
$r_1,\dots r_{u(h)}$ can be matched to a resident which it prefers to
$r_i$
for all $i\in [u(h)]$ and which does not propose to $h$ in a stable
matching.
\end{observation}
Using this observation, it is possible to prove that an agent cannot be
matched worse than the proposal it holds in a stable matching.
\begin{lemma} \label{le:deletion2}
If a resident or a quota-one hospital~$a^*$ holds the proposal of
another
agent~$a'$, then there
cannot exist a stable matching in which $a^*$ is matched to an agent
$\widetilde{a}$ to
which it prefers $a'$.
\end{lemma}
\begin{proof}
For the sake of contradiction, let us assume that there exists a stable
matching~$M$ in which a resident or quota-one hospital~$a^*$ is matched
to an agent~$\widetilde{a}$ while $a^*$ holds a
proposal from an agent $a'$ it prefers to $\widetilde{a}$.
If $a^*$ is a quota-one hospital, then $\widetilde{a}$ and
$a'$ are both residents and by \Cref{ob:ma}, resident~$a'$ needs to
prefer~$a^*$ to $M(a')$. Thus, $(a^*,a')$ forms a blocking pair.
If $a^*$ is a resident and $a'$ is a quota-one hospital,
then by \Cref{ob:ma}, hospital~$a'$ needs to prefer~$a^*$ to~$M(a')$.
Thus,
$(a',a^*)$ forms a blocking pair.
If $a^*$ is a resident and $a'$ is a quota-two hospital
which is closed in $M$, then as $a'$ issues a proposal there
needs to exists a resident $r$ issuing a proposal to $a'$. By
\Cref{ob:ma}, resident~$r$ needs to prefer $a'$ to~$M(r)$. Thus, $a^*$
and $r$
form together a blocking coalition to open~$a'$.
If $a^*$ is a resident and $a'$ is a quota-two hospital
which is open in $M$, then by \Cref{ob:ma}, hospital~$a'$ cannot be matched to
$u(h)$ residents which $a'$ all prefers to $a^*$. Thus, $(a',a^*)$ is a
blocking pair for~$M$.
\end{proof}
Using the two proceeding lemmas, we are now able to prove that all changes
made to the
preferences during Phase 1a do not delete any stable matchings, i.e., the
mutual acceptability of no hospital-residents pair occurring in any stable
matching is deleted in Phase 1a.
\begin{restatable}{lemma}{firstphasee}
\label{lem:first-phase2}
If an agent $a^*$ deletes another agent $a'$ from its preference list,
then
$a^*$ cannot be matched to $a'$ in any stable matching.
\end{restatable}
\begin{proof}
There are four types of modifications applied to the agents'
preferences in Phase~1a.
First, an agent might delete another agent from its preferences
because one of them rejected the proposal of the other. Here, the
correctness follows directly from \Cref{lem:first-phase}.
Second, a resident or quota-one hospital might delete an agent
from its preference list because it holds a proposal it prefers
to the deleted agent at the end of Phase~1a. Here, the correctness follows directly from
\Cref{le:deletion2}.
Third, $a^*$ might delete $a'$ from its preference list because it does
not appear on the preference list of~$a'$. This implies that $a'$ has
deleted $a^*$ from its preferences because of one of the two
proceeding cases. By \Cref{lem:first-phase} and \Cref{le:deletion2},
this implies that $a^*$ and $a'$ cannot be matched in any stable
matching.
Fourth, a quota-two hospital~$a^*$ is deleted from the instance and
thereby also from the the preferences of $a'$, if $a'$ is the only
remaining agent the preference list
of~$a^*$.
As $a^*$ has only $a'$ on its preference list left, the
proceeding three observation imply that $a^*$ cannot be open in
any stable matching and thereby that $a'$ can never be matched to $a^*$
in a stable matching.
\end{proof}
Now, we show that the modifications of the preferences made in Phase 1a do
not
delete any blocking coalitions or pairs, i.e., each matching that is
stable after applying Phase~1a is also stable before applying Phase 1a:
\begin{restatable}{lemma}{firstphaseback}
\label{lem:first-phase-backward}
Any matching $M$ that is stable in the instance $\ensuremath{\mathcal{I}^\mathrm{after}}$ after
applying Phase~1a is also stable in the instance $\ensuremath{\mathcal{I}^\mathrm{before}}$ before
applying
Phase~1a.
\end{restatable}
\begin{proof}
Let $h$ and $r$ be the first hospital and resident pair such
that the
mutual acceptability of~$h$ and $r$ is deleted and both $h$ and $r$
appear in a blocking coalition $(h, \{r,r_1\})$ for
some $r_1\in R$ or
as a
blocking pair $(h,r)$ for some
matching $M$ in $\ensuremath{\mathcal{I}^\mathrm{before}}$ that is stable in $\ensuremath{\mathcal{I}^\mathrm{after}}$.
We now iterate over all possible cases in which the mutual
acceptability of two agents is deleted.
Note that all cases which lead to a deletion of a mutual acceptability
reduce
to only four possibilities that trigger a deletion of a mutual
acceptability, namely
\begin{itemize}
\setlength\itemsep{0em}
\item that $r$ received a better proposal than $h$ (which may lead
to a
deletion because $r$ rejects the proposal from $h$ or at the end of
Phase 1a because $r$ holds a proposal it prefers to $h$), or
\item that $h$ is a quota-one hospital and received a better
proposal than $r$ (which may lead to a deletion because $h$ rejects
the proposal from $r$ or at the end of Phase 1a because~$h$ holds a
proposal it prefers to $r$), or
\item that $h$ is a quota-two hospital and received $u(h)$
proposals it prefers to $r$ (which may lead to a deletion because
$h$ rejects the proposal of $r$), or
\item that $h$ is a quota-two hospital and has only $r$ left on
its preferences (which leads to a deletion because $h$ rejects
the proposal of $r$ or because $h$ is deleted from the instance).
\end{itemize}
{\bfseries Case 1: } $r $ received a proposal $h'$ it prefers to $h$.
Thus, at the end of Phase 1a, resident~$r$ holds a proposal from a hospital
$h''$ it prefers to $h$.
However, as $r$ forms a blocking pair or coalition with $h$, resident $r$
needs to be matched worse than~$h$ in $M$ and therefore also worse than
$h''$.
{\bfseries Case 1.1: } $h''$ is a quota-one hospital.
Since $r$ is the first resident on the preferences of $h''$ in
$\ensuremath{\mathcal{I}^\mathrm{after}}$, hospital $h''$ is matched worse than~$r$ or is not matched
at all in $M$, implying
that $(h'', r)$ is
a blocking pair or $(h'',\{r\})$ a blocking coalition for $M$
in~$\ensuremath{\mathcal{I}^\mathrm{after}}$
(since $r$ holds a proposal of
$h''$, the mutual acceptability has not been deleted and they both
accept each other in $\ensuremath{\mathcal{I}^\mathrm{after}}$.
{\bfseries Case 1.2: } $h''$ is a quota-two hospital.
Before proposing to $r$, hospital $h''$ received a proposal from a
resident $r' \neq r$.
Since $h''$ did not reject~$r$, it still holds at least one proposal
from some resident~$r''$.
Therefore $r''$ cannot be matched better
than
$h''$ in $M$ (as $h''$ is the first hospital in the preferences of
$r''$ in \ensuremath{\mathcal{I}^\mathrm{after}}), and as $h''$ proposes to~$r$, hospital $h''$ can
have at most $u (h'') -1$ agents it prefers to $r$ on its
preferences in \ensuremath{\mathcal{I}^\mathrm{after}}.
If~$h''$ is closed in $M$, then $(h'', \{r, r''\})$ is a blocking
coalition for $M$ in~$\ensuremath{\mathcal{I}^\mathrm{after}}$.
Otherwise, $(h'', r)$ is a blocking
pair for $M$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$.
{\bfseries Case 2: } $h$ is a quota-one hospital and received a
proposal it prefers to $r$.
Then, $h$ holds a proposal from a resident $r'$ it prefers to $r$ at
the end of Phase~1a.
Moreover, hospital~$h$ is the first hospital in the preferences of $r'$ in
\ensuremath{\mathcal{I}^\mathrm{after}}.
Thus, as $(h,r)$ is a blocking pair for~$M$ in~$\ensuremath{\mathcal{I}^\mathrm{before}}$, hospital $h$ cannot
be matched to $r'$. Thus, $(h,
r')$ is a blocking pair for $M$ in~$\ensuremath{\mathcal{I}^\mathrm{after}}$, a contradiction.
{\bfseries Case 3: } $h$ is a quota-two hospital and received $u (h)$
proposals from residents it
prefers to~$r$.
Let $s_1, \dots, s_{u (h)}$ be the residents whose proposal $h$ holds
at the end of Phase 1a.
Since~$h$ is the first hospital in the preferences of $s_i$ for all
$i\in [u (h)]$, it follows that none of~$s_1, \dots, s_{u
(h)}$ prefers~$M ( s_i)$
to $h$. However, as $(h,r)$ is a blocking
pair for $M$ in~$\ensuremath{\mathcal{I}^\mathrm{before}}$, hospital~$h$ is undersubscribed or there exist one
resident that is
matched to~$h$ to which~$h$ prefers~$r$. Thereby, for at least one
$j\in [u
(h)]$, resident~$s_j$ is not matched to $h$ in $M$ which implies that
$(h,s_j)$
is a blocking pair for $M$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$, a contradiction.
{\bfseries Case 4: } $h$ is a quota-two hospital with only $r$ on its
preference list.
Then $h$ needs to be closed in $M$ (as it has been deleted from the
instance $\ensuremath{\mathcal{I}^\mathrm{after}}$/ its preferences are empty in $\ensuremath{\mathcal{I}^\mathrm{after}}$).
Thus, by our assumption, $(h, \{r, r_1\})$ is a blocking coalition to
open~$h$ in $\ensuremath{\mathcal{I}^\mathrm{before}}$.
However, for $h$ to have only $r$ on its preference list left, $r_1$
needs to reject $h$. Then, we are again in Case 1 and can thereby
conclude that $r_1$ and $h$ together cannot be part of a blocking coalition in~$M$.
\end{proof}
We finish the description and correctness proof of Phase 1a by drawing
two
conclusions about the set of agents matched in a stable matching:
\begin{corollary}
\label{cor:first-phase}
After applying Phase 1a,
\begin{itemize}
\setlength\itemsep{0em}
\item[1.] all residents holding a proposal are
matched in all
stable
matchings, and
\item[2.] if a hospital $h$
holds
$\ell (h)$ proposals, then $h$ is open in all stable matchings.
\end{itemize}
\end{corollary}
\begin{proof}
1. Assume for the sake of contradiction that there exists a stable
matching $M$ in which a resident $r$ currently holding proposal $h$ is
unmatched. If $h$ is a quota-one hospital, then we know by
\Cref{ob:ma} that $h$ cannot be matched to a resident it
prefers to $r$. Thus, $(h,r)$ blocks~$M$. If $h$ is a quota-two
hospital, let $r'\neq r$ be a resident that activated~$h$ by proposing
to it. By \Cref{ob:ma}, resident~$r'$ cannot be matched to a hospital
she prefers to $h$. Thus, if $h$ is closed in $M$, then $(h, \{r,r'\})$ forms a
blocking coalition to open~$h$. Now assume that $h$ is open. As $h$
proposed to $r$, by
\Cref{ob:ma}, there cannot
exist $u(h)$ residents that $h$ prefers to~$r$ and that are matched
to $h$ in a stable matching. Consequently, if $h$ is open, $(h,r)$ is a
blocking pair.
2. Assume for the sake of contradiction that there exists a stable
matching $M$ in which a hospital~$h$ which received $\ell (h)$
proposals from residents $r_1, \dots, r_{\ell (h)}$ is closed. From
\Cref{ob:ma} it follows that $r_1, \dots, r_{\ell(h)}$ cannot be
matched to a hospital they prefer to $h$. Thus, $\{r_1,\dots,r_{\ell (h)}\}$ form a
blocking coalition to open $h$.
\end{proof}
\subsubsection{Phase 1b} \label{a:p1b}
Recall that in Phase 1b, we replace each quota-two hospital $h$
holding at least two proposals by~$u(h)$ hospitals $\ensuremath{h^1}$,
\dots, $h^{u (h)}$ with lower and upper quota one
with the same preferences
as~$h$. In the preferences of all residents, $h$ is replaced by $h^1 \succ
\dots \succ h^{u (h)}$.
We now prove that there exists a one-to-one
mapping between the set of stable matchings before and after the
application
of Phase 1b:
\begin{restatable}{lemma}{splitting}
\label{lem:splitting}
There exists a one-to-one mapping between the stable matchings in the
instance~$\ensuremath{\mathcal{I}^\mathrm{before}}$ before applying Phase 1b and in the instance
$\ensuremath{\mathcal{I}^\mathrm{after}}$
after applying Phase 1b.
\end{restatable}
\begin{proof}
Let $h$ be a quota-two hospital holding two proposals, and let
$\ensuremath{\mathcal{I}^\mathrm{before}}$ be the instance before splitting hospital~$h$ and $\ensuremath{\mathcal{I}^\mathrm{after}}$ the
instance after splitting $h$.
We define a function $\sigma$ by mapping a matching~$M$ with $M(h) = \{r_1,
\dots, r_k\}$ and $h$ preferring $r_i$ to $r_{i+1}$ for all $i\in [k]$
in~$\ensuremath{\mathcal{I}^\mathrm{before}}$ to
the matching~$\sigma (M) := (M\setminus \{(h , \{r_1, \dots, r_k\})\})
\cup \{ (h^1, r_1), \dots, (h^k, r_k) \}$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$.
We now show that $\sigma$
is a bijection
from the set
of stable matchings in $\ensuremath{\mathcal{I}^\mathrm{before}}$ to the stable matchings in $\ensuremath{\mathcal{I}^\mathrm{after}}$, proving the lemma.
Let $M$ be a stable matching in \ensuremath{\mathcal{I}^\mathrm{before}}.
By \Cref{cor:first-phase}, $h$ is open in $M$.
We claim that~$M' :=\sigma (M)$ is a stable matching in $\ensuremath{\mathcal{I}^\mathrm{after}}$.
Any blocking coalition or pair must contain one resident from~$r_1, r_2,
\dots, r_k$ or a hospital from \ensuremath{h^1}, \dots, $h^{u (h)}$, as all other agents are matched the same in~$M$ and~$M'$.
If a blocking coalition or pair involves $r_i$ for some $i\in [k]$,
then it must also involve a
hospital $h^j$ for some~$j\in [u (h)]$, as there does not exist a hospital
that $r_i$ prefers
to~$h^1$, \dots, $h^k$ which she does not prefer to $h$.
However, since $h^\ell$ and $r_\ell$ prefer each other to $r_p$ and
$h^p$ for $\ell < p$, there is no such blocking coalition or pair.
Next, assuming that for some $r\in R\setminus\{r_1,\dots, r_k\}$
and~$i\in
[u (h)]$, pair~$(h^i,r)$ is a blocking pair for $M'$, then $r$
prefers $h^i$ to $M'(r)$ and therefore also $h$ to $M(r)$.
If $ i \le k$, then, as $(h^i,r)$ blocks $M'$, $h^i$ (and thus $h$)
prefers $r$ to $r_i $.
If $i > k$, which implies that $h^i$ is closed in~$M'$, then $h$ is
undersubscribed in~$M$.
In both cases, $(h,r)$
is a blocking pair for $M$,
a contradiction.
Note that $\sigma$ is obviously injective.
Vice versa, let $M'$ be a stable matching in \ensuremath{\mathcal{I}^\mathrm{after}}.
Note that by our initial assumption, $h$ holds at least two proposals
in~$\ensuremath{\mathcal{I}^\mathrm{before}}$ and that $h^1$
and~$h^2$ are the first choices of the two residents proposing to $h$.
Therefore, hospitals~$\ensuremath{h^1}$ and~$\ensuremath{h^2}$ are both matched in $M'$.
Note that due to the stability of $M'$,
there exists some $2\le k\le u (h)$ such that $h^i$ is matched to an agent $r_i$ for $i \le k$ and $h^i$ is unmatched for all $i > k$.
We claim that $M := \sigma^{-1} (M') =\bigl(M' \setminus \{(\{h^i, \{r_i\}) : i\in
[k]\}\bigr) \cup \{(h, \{r_1, \dots, r_k\})\}$ is a stable matching in~\ensuremath{\mathcal{I}^\mathrm{before}}.
Any blocking coalition or pair for $M$ in $\ensuremath{\mathcal{I}^\mathrm{before}}$ must involve $h$
and a resident from~$R\setminus\{r_1, \dots, r_k\}$. As $h$ is open,
there
cannot exist a blocking coalition in $M'$.
If there exists a blocking pair $(h, s)$ for some resident~$s\in
R\setminus\{r_1, \dots, r_k\}$, then $s$
prefers $h$ to $M(s)$ and $h$ is undersubscribed or for some $j\in
[k]$, hospital~$h$ prefers $s$ to $r_j$.
In the first case, there exists some $h^j$ such that $h^j$ is unmatched,
implying that $(h^j, \{s\})$ is a blocking coalition for $M'$ in~$\ensuremath{\mathcal{I}^\mathrm{after}}$,
contradicting the stability of $M'$.
In the later case, $(h^j, s)$ is a blocking pair for $M'$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$,
contradicting the stability
of $M'$.
Note further that $\sigma(M)=M'$ which implies that $\sigma^{-1}$ is a
right inverse of $\sigma$ and thus that $\sigma$ is surjective. To see
this, recall that all hospitals $h^i$ for $i\in [u(h)]$ rank all
residents in the same order and that all residents accepting them prefer
$h^i$ to $ h^{i+1}$ for all $i\in [u(h)-1]$. Thus, by the stability of
$M'$
it needs to hold that $h$ prefers $r_i$ to $r_{i+1}$ for all $i\in
[k-1]$
and thereby that $\sigma(M)=M'$.
\end{proof}
We conclude with several observations about the situation after the
excessive
application of Phase 1 that we will use for the definition of a generalized rotation
in Phase~2:
\begin{lemma}
\label{lem:first-phase-rotation}
After the application of Phase 1,
\begin{enumerate}
\setlength\itemsep{0em}
\item[(1)] each agent either holds and issues exactly
one proposal or neither holds nor issues a proposal.
\item[(2)] each quota-two hospital
\begin{itemize}
\setlength\itemsep{-0.1em}
\item has upper quota two, and holds the proposal
of one of its first two residents,
\item holds a proposal and has only two agents left on its
preferences, or
\item holds no proposal.
\end{itemize}
\item[(3)] if an agent $a^*$ appears on the preference list of
another
agent $a'$ and $a'$ has more than one agent on its preference
list, then
$a^*$ also has more than one agent on its preference list.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) No quota-two hospital holds more than one proposal, as a hospital
holding two proposals gets split into multiple quota-one hospitals
in Phase
1b. Thereby, every agent can hold at most one proposal.
We claim that every agent that holds a proposal also issues a proposal.
If a resident or quota-one hospital does not issue a proposal, then its preferences are empty.
Consequently, it cannot hold a proposal.
For quota-two hospitals
$h$ holding a proposal from a resident $r$, the claim holds, as in the
case where all residents except $r$ reject the proposal $h$ and thereby
$h$ does not
issue a proposal, $h$ rejects the proposal~$r$.
Consequently, as no agent can hold more than one proposal and each
agent that holds a proposal also issues one, every agent issues exactly
one proposal and holds exactly one proposal or neither holds nor issues
a proposal.
(2) The second statement directly follows from the first one by
observing that a quota-two hospital only issues one
proposal in the case where its upper quota is two and it holds the
proposal of one of its first two residents or if it only has two
residents on its preferences left.
(3)
If the preferences of $a^*$ only contain $a'$, then $a^*$ is not a
quota-two hospital, $a^*$ proposes to $a'$ and by (1) $a^*$ also
receives a
proposal from $a'$.
Thus, $a'$ is not a quota-two hospital (as any quota-two hospital receiving exactly one proposal~$r$ does not propose to~$r$, and there are no agents receiving multiple proposals by (1)).
As $a'$ proposes to~$a^*$, agent~$a^*$ is the first agent in the
preferences of $a'$, and as $a'$ receives a proposal from $a^*$, all
agents after $a^*$ are deleted from the preferences of $a'$.
Thus, the preferences of $a'$ only contain $a^*$.
\end{proof}
\subsubsection{Phase 2}\label{a:p2}
We start by proving that as
long as there exists a resident
with more
than one hospital on her preference list, a generalized rotation is
guaranteed to exist and can be found in linear time.
\begin{lemma}
\label{lem:one-proposal}
Unless the preference list of every resident contains at most one
hospital, a generalized rotation always exists and can be found in
$\mathcal{O} (n)$ time, where $n$ is the number of residents.
\end{lemma}
\begin{proof}
If there exists a resident $r$ with at least two hospitals on her
preference list, it is always possible to find a generalized rotation by starting with
setting $a_1:=r$ and then use \ab{1} to \ab{3b(ii)}
as defined above to find $b_2$ and subsequently \ba{1} to \ba{3} to find $a_2$.
We continue doing so until the process cycles, i.e., we have found some $i\neq
j$
with $a_i=a_j$. Note that it is also enough to find some $i\neq j$ with
$b_i=b_j$ as this directly implies that $a_i=a_j$.
As each computed pair~$(a_i, b_i)$ contains at least one resident, it follows
that the running
time for finding a generalized rotation lies in $\mathcal{O}(n)$.
It remains to argue
that for residents and quota-one hospitals $s$ with at least two agents on
their preference list, if we set $a_i=s$, then $b_{i+1}$ always exists, has
at least two agents on its preference list and is unique. Moreover, we need to
prove the same for $a_i$ if we set $b_{i}=s$.
Note that for all \ab{1} to \ab{3b(ii)} and \ba{1} to \ba{2}, the successor
clearly always exists and is unique. For \ba{3}, it needs to holds that resident
$b_i$ holds a proposal of the last hospital $h$ on her preference
list. By \Cref{lem:first-phase-rotation}, this implies that $h$ also receives
exactly one proposal. Thus $a_i$ is well-defined and unique.
Furthermore, all computed successors appear on the
preferences of agents whose preference list has length at least two and, by
\Cref{lem:first-phase-rotation},
also have at least two agents on their preference list.
\end{proof}
Given an agent $a_i$ appearing in a generalized rotation, we say that
\emph{\ab{$x$}
applies to $a_i$} for~$x\in \{1, 2, \text{3a}, \text{3b(i)}, \text{3b(ii)}\}$
if
case \ab{$x$} needs to be applied to compute $b_{i+1}$ from~$a_i$.
Similarly, for an agent $b_i$ appearing in a generalized rotation, we say
that
\emph{\ba{$x$} applies to $b_i$} for~$x\in \{1,2,3\}$ if case \ba{$x$} needs to
be applied to compute $a_{i}$ from $b_i$. Moreover, in the following, for two
agents $a,a'$ of which one is a resident and the other is a quota-one hospital,
we write $\{a,a'\}\in M$ to denote that $a$ and $a'$ are matched to each other
in~$M$. We now show a list of statements needed to prove the correctness of
Phase~2.
We start by considering whether and where agents might repeatedly appear in a
single generalized rotation:
\begin{observation}\label{obs:b}
In any generalized rotation, $b_i \neq b_j$ holds for $i \neq j$.
\end{observation}
\begin{proof}
If $b_i = b_j$ holds, then also $a_{i} = a_{j}$ holds, which
implies $i =
j$.
\end{proof}
\begin{lemma}\label{lem:no-same-g}
Let $(a_1, b_1)$, \dots, $(a_k, b_k)$ be a generalized rotation.
If a stable matching $M$ contains $\{a_i, b_i\}$ or $(h (a_i),
\{a_i, b_i\})$ for all $i \in [k]$, then it holds for all $i,j \in
[k]$ that
\begin{itemize}
\setlength\itemsep{0em}
\item[(i)] $a_i \neq b_j$,
\item[(ii)] $g (a_i)\neq g (a_j)$ for $i\neq j$ whenever $a_i$ and $a_j$ are
residents, and
\item[(iii)] $h(a_i) \neq g(a_j)$ for $i\neq j$ whenever $a_i$ and $a_j$ are
residents.
\end{itemize}
\end{lemma}
\begin{proof}
To prove (i), for the sake of contradicting, let us assume that
$a_i = b_j$ for some~$i,j \in [k]$.
By \ba{1} to \ba{3} it follows that $i \neq j$ (for \ba{3}, note
that since $b_j$ is part of a rotation, it follows that $b_j$ has at least two
hospitals in its preferences, and thus does not propose to $h$, the last
hospital on its preferences).
If $a_i$ is a quota-one hospital or $a_i$ is a resident and
$h(a_i)$ is a quota-one hospital, then it also holds that $b_i =
a_j$, since $M$ contains $\{a_\ell, b_\ell\}$ for $\ell \in \{i,j\}$.
Thus, $a_i$ and $b_i$ propose to each other, implying that their
preference lists contain only one agent.
It follows that they cannot be part of a generalized rotation.
Otherwise $a_i$ is a resident and $h(a_i)$ is a quota-two hospital.
Thus, $M$ contains both $(h(a_i), \{a_i, b_{i}\})$ and $(h(a_{j}),
\{a_{j}, a_i\})$, implying that $h(a_i) = h(a_{j})$.
It follows that both $a_i$ and $a_{j}$ issue a proposal to
$h(a_i)$.
Since every hospital with at least two proposals got split, it
follows that
$a_i = a_{j}$, a contradiction.
Thus, $a_i \neq b_j$ for all $i,j \in [k]$ holds.
To prove (ii), assume that $g:= g (a_i) = g (a_j)$ for some $i \neq
j\in [k]$ such that $a_i$ and $a_j$ are residents.
Note that $g$ cannot be a quota-one hospital, as from this it would
follow that \ab{3a} applies for both~$a_i$ and~$a_j$ which would
imply
that $b_{i+1} = b_{j+1}$, contradicting \Cref{obs:b}. Thus, we
assume that
$g$ is a quota-two hospital and distinguish between the case where
$g$ received
a proposal and the case where $g$ did not receive a proposal.
We will find for both
cases a
resident $r$ appearing both as $a_i$ and $b_{j+1}$, contradicting
(i).
{\bfseries Case 1: } $g$ did not receive a proposal.
Then $g$ cannot appear as $h (a_p)$ for any $p\in [k]$, and
\ab{3b(ii)} applies for both $a_i$ and~$a_j$.
Let $r $ be the first resident on $g$'s preference list.
Then $b_{\ell+1} = r$ if and only if $a_\ell \neq r$ for $\ell \in
\{i,j\}$.
Since $a_i \neq a_j$ by the definition of a generalized rotation and $b_{i +
1} \neq
b_{j+1}$ by \Cref{obs:b}, it follows without loss of generality
that $r = a_i$ and~$r\neq a_j$.
Since $r\neq a_j$, it follows that $b_{j+1} = r = a_i$.
{\bfseries Case 2: } $g$ received a proposal from a resident $r$.
Then $b_{\ell+1} = r$ if and only if $a_\ell \neq r$ for $\ell \in
\{i,j\}$ (note that \ab{2} only applies if $a_\ell = r$).
It follows by \Cref{obs:b} that w.l.o.g.\ $r = a_i$ and $r\neq
a_j$. Since $r\neq a_j$, it follows that $b_{j+1} = r = a_i$.
To see (iii), assume that there exists some $i\neq j$ such that $h(a_i) = g (a_j)$.
By (ii), we have that $g(a_j) = h(a_i) \neq g(a_i)$.
Therefore, $h(a_i)$ is inflexible\xspace.
Thus, it follows that $a_i$ and $b_i$ are the only agents in the preferences of $h(a_i)$ and thus $\{a_i, b_i\} = \{a_j, b_{j+1}\}$.
By (i), it follows that $a_i = a_j$, a contradiction.
\end{proof}
The definition of a generalized rotation and \Cref{lem:first-phase-rotation} imply the following
observations.
\begin{observation} \label{ob:ab2}
If for some $a_i$ case \ab{2} applies, i.e., $a_i$ is a resident and
$h(a_i)$
is a flexible hospital, then
\begin{enumerate}
\setlength\itemsep{-0.2em}
\item $b_i$ and $b_{i+1}$ are also residents, and
\item $a_i$ and $b_i$ are the two first residents on $h(a_i)$'s
preference list and $b_{i+1}$ is the third resident on
$h(a_i)$'s
preference list.
\end{enumerate}
\end{observation}
\begin{proof}
As \ab{2} applies for $a_i$, agent~$b_{i+1}$ is the second-most preferred
resident
on the preferences of $h(a_{i})$
which is not $a_{i}$.
By the definition of~$h(a_i)$, resident $a_i$ proposes to~$h(a_i)$,
and is therefore by \Cref{lem:first-phase-rotation} among the first two
residents in the preferences of~$h(a_i)$.
We claim that \ba{3} applies for $b_i$.
If \ba{1} applies for~$b_{i}$, then~$a_i$ is the last choice of
quota-one hospital~$b_i$ and therefore proposes to $b_i$, but we have $b_i \neq
h(a_i)$ since $h(a_i)$ is a quota-two hospital. If \ba{2} applies for
$b_i$, then $a_i$ is a quota-one hospital, a contradiction.
Thus, \ba{3} applies for~$b_i$.
As the preferences of resident $b_i$ are non-empty, it holds a
proposal from the last hospital on its preferences.
Thus, by \ba{3}, it holds that $h(a_i)$ proposes to $b_i$.
By \Cref{lem:first-phase-rotation}, $h (a_i)$ has upper quota two.
From this, it follows
that $b_i$ and $a_i$ are among the first two residents in the preferences of $h(a_i)$.
\end{proof}
From \Cref{lem:first-phase-rotation}, we can conclude the following
observation:
\begin{observation} \label{ob:hai}
Let $(a_1, b_1), \dots, (a_k, b_k)$ be a generalized rotation. For all $i\in
[k]$ such that $a_i$ is a resident and $h(a_i)$ is a quota-two hospital, hospital
$h(a_i)$ holds the proposal of $a_i$ and if $h(a_i)$ is open in a
stable matching $M$, then exactly two residents are matched to
$h(a_i)$,
one of which must be $a_i$.
\end{observation}
\begin{proof}
Let $a_i$ be a resident and $h(a_i)$ a quota-two hospital for
some $i\in [k]$. Then, by the definition of Phase 1a, $h(a_i)$
holds the proposal of $a_i$. There exist two possibilities. If
$h(a_i)$ has upper quota two, then $a_i$ is one of the fist two
residents on the preference list of $h(a_i)$ by
\Cref{lem:first-phase-rotation}. Thus, if $h(a_i)$ is open in a
stable matching, then exactly two residents need to be matched to
$h(a_i)$. One of these two needs to be $a_i$, as otherwise
$(a_i,h(a_i))$ is a blocking pair.
Otherwise, by \Cref{lem:first-phase-rotation}, $h(a_i)$ has only
two residents on its preferences left one of which is $a_i$. From
this, it directly follows that if $h(a_i)$ is open, then $a_i$ and
the other resident left on the preferences of $h(a_i)$
need
to be matched to it.
\end{proof}
As the final step before proving the correctness of Phase 2, we prove that,
similar as for classical rotations in the \textsc{Stable Roommates}
problem, in a stable matching,
either each agent pair in a generalized rotation
is matched (either to each other or to a specific hospital) or none of
them.
\begin{lemma}\label{le:rotation-eli}
Let $(a_1, b_1), \dots, (a_k, b_k)$ be a generalized rotation.
If a stable matching~$M$ contains $\{a_i, b_i\}$ or $(h (a_i),
\{a_i, b_i\})$ for some $i\in [k]$, then $M$ contains $\{a_i,
b_i\}$ or~$(h (a_i), \{a_i, b_i\})$ for all~$i \in [k]$.
\end{lemma}
\begin{proof}
Note that by \Cref{ob:hai}, the hospital $h(a_i)$
can be matched to at most two residents for every $i \in [k]$.
Assume, for the sake of contraction, that for some $i\in [k]$, $M$
contains $\{a_i, b_i\}$ or $(h
(a_i), \{a_i,
b_i\})$ and
that the following two assumptions hold:
\begin{align}
\{a_{i-1}, b_{i-1}\}\notin M \tag{1}\label{a1}\\
( h (a_{i-1}), \{a_{i-1}, b_{i-1}\}) \notin M \tag{2}\label{a2}
\end{align}
We will now make a case distinction over all possible assignments for $a_i$, $b_i$,
$a_{i-1}$, and~$b_{i-1}$ and argue that $\{a_i,
b_i\}\in M$ or $(h (a_i), \{a_i, b_i\}) \in M$ together with
Assumption~(1) and
Assumption~(2) implies a blocking pair or coalition for~$M$, a contradiction to the stability of~$M$.
{\bfseries Case 1: } $a_{i-1} $ is a quota-one hospital.
Then $b_i$ is a resident.
Since $a_i$ or $h(a_i)$ is the last hospital in the preferences of~$b_i$, matching~$M$ matches $b_i$ to the last hospital on her preferences, and thus $b_i$ prefers~$a_{i-1}$ to $M(b_i)$.
Note that $b_{i-1}$ is the best resident on $a_{i-1}$'s preferences and $b_i$ is the second-best resident on $a_{i-1}$'s preferences.
As $M(a_{i-1} ) \neq b_{i-1}$ by Assumption~(1), it follows that $a_{i-1}$ prefers $b_i$ to~$M(a_{i-1})$.
Thus, $\{a_{i-1}, b_i\}$ is a blocking pair, a contradiction.
{\bfseries Case 2: } $a_{i-1} $ is a resident.
\textbf{Case 2.1: } $b_i$ is a quota-one hospital.
Then $a_i$ is the last resident on the preferences of $b_i$ and thus $b_i$ prefers $a_{i-1}$ to $a_i$.
As $b_i$ is a hospital, hospital~$h(a_{i-1})$ is not flexible\xspace.
Thus, Assumptions~(1) and~(2) imply that $a_{i-1}$ is not matched to~$h (a_{i-1})$.
As $a_{i-1}$ is not matched to the first hospital~$h(a_{i-1})$ in its preferences and also not to the second hospital~$b_i$, it follows that $\{a_{i-1}, b_i\}$ is a blocking pair.
\textbf{Case 2.2: } $b_i$ is a resident.
Then $a_i$ or $h(a_i)$ is the last hospital on $b_i$'s preferences, and thus $b_i$ prefers $g(a_{i-1})$ to~$M (b_i)$.
If $g(a_{i-1}) = h(a_{i-1})$, then $g(a_{i-1})$ is the first
hospital in $a_{i-1}$'s preferences. Otherwise, $h (a_{i-1})$ is
inflexible\xspace or a quota-one hospital, and thus by Assumptions~(1) and~(2)
$M(a_{i-1}) \neq h(a_{i-1})$.
In both cases, it follows that $a_{i-1}$ does not prefer $M(a_{i-1})$ to $g(a_{i-1})$.
If $g(a_{i-1}) $ is closed, then $(g(a_{i-1}), \{a_{i-1}, b_i\})$ is a blocking coalition to open $g(a_{i-1})$.
Otherwise $b_i$ is among the first two residents in the preferences
of $g(a_{i-1})$, and thus, $\{g(a_{i-1}), b_i\}$ is a blocking pair.
\end{proof}
Proving the correctness of Phase 2, we start by showing that if there
exists a stable matching before the elimination of a generalized rotation, there also
exists a stable matching after the elimination which matches the same set of residents.
\begin{restatable}{lemma}{rot}
\label{lem:rotation}
Let $(a_1, b_1), \dots, (a_k, b_k)$ be a generalized rotation.
If an instance admits a stable matching~$M$, then it still admits a
stable matching~$M'$ which matches the same residents and quota-one
hospitals
after the elimination of this generalized rotation.
Furthermore, $M$ and $M'$ open the same number of hospitals.
\end{restatable}
\begin{proof}
Let $M$ be a stable matching. We distinguish two cases.
{\bfseries Case 1: } $M$ contains neither $\{a_i, b_i\}$ nor $(h (a_i),
\{a_i,
b_i\})$ for all $i\in [k]$.
We claim that $M$ is also contained in the reduced preferences after
the elimination of the generalized rotation. For the sake of contradiction, we
assume that the mutual acceptability of a hospital-residents pair that
is part of $M$ was deleted in the elimination of the generalized rotation and show
that this leads to a contradiction.
Assume that the mutual acceptability of a resident $r $ and a
hospital $h$ with $M(r) = h$ was deleted.
If $h$ is a quota-one hospital, then the only residents that were
deleted from the preferences of $h$ are residents $r$ which appear
together with $h$ in a
generalized rotation pair, i.e., $(h,r)=(a_i,b_i)$ or $(r,h)=(a_i,b_i)$ for some
$i\in [k]$. However, by our initial assumption that $M$ does not
contain~$\{a_i,b_i\}$ for $i\in [k]$, this implies that~$M(r)\neq h$,
a
contradiction.
Thus $h$ is a quota-two hospital. A resident~$r$ is only deleted
from the preferences of $h$ if $h= h (a_i)$ and $r= b_i$ for some $i
\in [k]$. As~$h(a_i)$ is the top-choice of~$a_i$, hospital $h(a_i)$
holds the proposal of $a_i$. This implies by
\Cref{lem:first-phase-rotation} that $a_i$ is ranked among the two best
residents in the
preferences of $h$, and that $M$ has either upper quota two or
exactly two residents on its preference list. As we have assumed
that~$M(b_i)=h(a_i)$ and~$(h(a_i),\{a_i,b_i\})\notin M$, it needs to
hold
that
$M(h) = \{b_i, r'\}$ for some~$r'\neq a_i$.
As~$h(a_i)$ is the top-choice of $a_i$ and $a_i$ is among the two most
preferred residents of $h(a_i)$, pair~$(h, a_i)$
blocks $M$, a contradiction.
\medskip
{\bfseries Case 2: } $M$ contains $\{a_i, b_i\}$ or $(h (a_i), \{a_i, b_i
\})$ for all $i\in [k]$.
\paragraph{Construction of new matching.} We claim that replacing, for
all $i\in [k]$,
$\{a_i, b_i\}$ or $(h
(a_i), \{a_i, b_i\})$ by~$(g(a_i), \{a_i,
b_{i+1}\})$ if $a_i$ and $b_{i+1}$ are both residents and by
$\{a_i,
b_{i+1}\}$ otherwise, results in a stable matching $M'$. Note that in
the former case $g(a_i)$ is a quota-two hospital, as otherwise~$b_{i+1}$
would be a hospital instead of a resident.
Clearly, $M'$ matches the same set of residents and quota-one hospitals.
\paragraph{Feasibility.} We first show that $M'$ does not violate any
quotas. First of all note
that by \Cref{ob:hai} matching $M$ matches only $a_i$ and $b_i$ to
$h(a_i)$.
All hospitals $h(a_i)$ are either closed or respect their
lower quota in $M'$:
If $h(a_i)$ is flexible\xspace, then it is also open in $M'$, as in this case,
$g(a_i) = h(a_i)$ and $(g(a_i), \{a_i, b_{i+1}\})\in M'$.
Otherwise $h(a_i)$ is inflexible\xspace, and then $h(a_i) \neq g(a_j)$ for all $j\in [k]$ by
\Cref{lem:no-same-g}.
Thus, $h (a_i)$ is closed in $M'$.
Next, we show that every hospital obeys its upper quota in $M'$.
Matching~$M'$ can only violate the upper
quota of a hospital~$h = g(a_i)$ that is also open in $M$, as all replacements
described above do not increase the number of residents matched to all other hospitals.
So assume for a contradiction that there exists some hospital~$h = g(a_i)$ such that the upper quota of $h$ is violated in~$M'$.
If $h (a_i)$ is flexible\xspace, then, as $g (a_i) \neq g(a_{i'})
$ for all $i \neq i' \in [k]$ by \Cref{lem:no-same-g}, it follows that $M' (h) = (M(h) \setminus\{b_i\}) \cup \{b_{i+1}\}$ and $h$ obeys its upper quota also in~$M'$.
Otherwise $h(a_i)$ is inflexible\xspace.
We will now show that $h$ is closed in $M'$, implying that $h$
obeys its upper quota in $M'$.
By \Cref{lem:no-same-g}, it holds that $h \neq h(a_j)$ for all $j\in [k]$.
This
implies that either \ab{3b(i)} or \ab{3b(ii)} applies for~$a_i$. If
\ab{3b(i)} applies, then $h$ holds the proposal of $b_{i+1}$ which
implies by \Cref{lem:first-phase-rotation} that $b_{i+1}$ is among the
first two residents on the preference list of $h$. If \ab{3b(ii)}
applies, then by definition, $b_{i+1}$ is one of the two most
preferred residents on the preference list of $h$.
In both cases, $h$ prefers $b_{i+1}$ to one of
the residents it is matched to in $M$.
Since $h$ is on $b_{i+1}$'s preference list, it follows that
$b_{i+1}$ prefers $h$ to~$M$, as we have assumed that
either $\{a_{i+1}, b_{i+1}\}$ or $(h (a_{i+1}), \{a_{i+1}, b_{i+1}\})$
is part of~$M$
which in both cases implies that $b_{i+1}$ is matched to its least
preferred hospital in $M$.
Therefore, $h$ is closed in $M$, as otherwise $(h,b_{i+1})$ forms a blocking pair in
$M$.
It follows that $h$ does not violate its upper quota.
Therefore $M'$ is a feasible matching.
Next we show that for any pair $(r, h)\in M'$ the acceptability of $r$ and $h$ has not been deleted.
Since $a_i \neq b_j $ for all $i,j \in [k]$ by \Cref{lem:no-same-g}, this can only occur if $g(a_{i-1}) = h(a_i)$.
Thus, $a_i$ issues a proposal to $g(a_{i-1}) = h (a_i)$.
Since $a_i\neq a_{i-1}$ and every hospitals holds at most one proposal,
this implies that resident $a_{i-1}$ cannot issue a proposal to
$g(a_{i-1})$ and thereby that $h(a_{i-1})\neq g(a_{i-1})$ needs to
hold.
It follows that \ab{3b(i)} applies for $a_{i-1}$, and thus, $b_i= a_i$,
contradicting \Cref{lem:no-same-g}.
To see that the number of open hospitals in $M'$ equals the number of open hospitals in $M$, first note that the set of open quota-one hospitals stays the same.
For every flexible\xspace hospital~$h (a_i)$, we have that $h(a_i) = g(a_i)$ is also open in $M'$.
For every inflexible\xspace hospital $h(a_i)$, we have that $h(a_i) \neq g(a_j)$ for all $j\in [k]$ by \Cref{lem:no-same-g}.
It follows that $h(a_i)$ is closed in $M'$, but $g(a_i)$ is open in~$M'$.
In $M$, hospital $h(a_i)$ is open, while we have already seen that $g(a_i)$ is closed in~$M$.
It follows that $M$ and $M'$ open the same number of hospitals.
It remains to show that $M'$
is
stable.
\paragraph{Stability.} Any blocking pair or coalition for $M'$ obviously needs to
include an
agent that is matched differently in $M'$ than in $M$, i.e., $a_i$,
$b_i$, $h(a_i)$, or $g(a_i)$ for some $i\in [k]$. We now show one after
each other for each of these four types of agents that they cannot be
part of a blocking pair or coalition in $M'$. However, we start by
observing the following:
{\bfseries Claim 1: } For all $i\in [k]$, agent $b_i$ is matched differently
in $M'$ than in $M$ and $b_i$ prefers~$M'$ to $M$.
By the definition of \ba{1} to
\ba{3}, $\{a_i,b_i\}\in M$ or $(h(a_i), \{a_i,b_i\})\in M$ implies that
$M(b_i)$ is the last agent in the preferences of~$b_i$.
Thus, it is enough to show that $b_i$ is matched differently in~$M$ and $M'$.
If $b_i$ is a hospital, then $M(b_i) = a_i$ and $M' (b_i) = a_{i-1}$ are both residents, and $a_i \neq a_{i-1}$ implies that $M(b_i)\neq M' (b_i)$.
If $b_i$ is a resident and~$a_i$ is a hospital, then again $M(b_i) = M'
(b_i)$ implies $a_{i} = a_{i-1}$, a contradiction.
If $b_i$ is a resident and $a_i$ is also a resident, then $M(b_i) = h(a_i)$.
If $M(b_i) = M' (b_i)$ holds, then, since $M(b_i)$ is a quota-two
hospital, it holds that $M' (b_i) = g (a_{i-1})$. For the sake of
contradiction, assume that $h(a_i)=M(b_i)=M'(b_i)=g (a_{i-1})$. This
leads to a contradiction as argued in the last paragraph in the part on
feasibility in this proof.
{\bfseries Claim 2: } For all $i\in [k]$, agent $a_i$ is neither part
of a blocking
pair nor of a blocking coalition in~$M'$.
If $a_i$ is a quota-one hospital, then \ba{2} and \ab{1}
apply for $b_i$ and $a_i$ which implies that $a_i$ is the last hospital on
$b_i$'s preferences and $b_i$ is the first resident and $b_{i+1}$ is
the second resident on the preferences of $a_i$. Thereby, $a_i$ is
matched to its second-most preferred resident in $M'$. This implies
that the only possible blocking pair including
$a_i$ in $M'$ is $(a_i,b_i)$. However, as by Claim 1, resident~$b_i$ prefers
$M'$ to $M$,
this cannot be a blocking pair.
If $a_i$ is a resident, then $a_i$ can be either part
of a blocking pair $(a_i, h)$ with some hospital~$h\in H$ or part of a
blocking coalition to open some hospital $h\in H$.
If $h(a_i)$ is flexible\xspace, then $a_i$ is matched to her first choice in $M'$
and thus is neither part of a blocking coalition nor a blocking pair.
Hence, we assume that $h(a_i)$ has quota one or is inflexible\xspace.
As $a_i$ is matched
to her second-most preferred hospital $g(a_i)$ in $M'$, the hospital in any blocking coalition or pair containing~$a_i$ is $h := h(a_i)$.
We distinguish
two cases based on whether $h(a_i)$ is a quota-one or quota-two hospital.
If $h(a_i)$ is a quota-one hospital, then \ba{1} applies for $b_i = h(a_i)$, and $a_i$ is $b_i$'s least preferred
resident. However, as $b_i$ prefers $M'(b_i)$ to $a_i$ as shown in
Claim 1, $(h(a_i), a_i)$ cannot be a blocking pair for $M'$.
If $h(a_i)$ is a quota-two hospital, then $h(a_i)$ is an inflexible\xspace hospital as $h(a_i) \neq g(a_i)$.
Thus, the preferences of $h(a_i)$ contain only two
residents, namely $a_i$ and~$b_i$. As $a_i$
cannot be matched to~$h(a_i)$ in $M'$, hospital~$h(a_i)$ is closed in $M'$.
However, we have observed above that $b_i$ prefers~$M'(b_i)$ to~$M(b_i)=h(a_i)$ and thereby~$b_i$ is not part of a blocking
coalition.
Thus, no blocking coalition to open $h(a_i)$ exists, and~$a_i$ is
neither part of a blocking pair nor a blocking coalition.
{\bfseries Claim 3: } For all $i\in [k]$ such that $a_i$ is a resident, hospital $h(a_i)$ is neither part of a
blocking pair nor a blocking coalition in $M'$.
If $h (a_i)$ is a quota-one hospital, then $h(a_i) = b_i$ and Claim 1
implies
that $h (a_i)$ prefers~$M'$ to~$M$. Moreover, Claim 2 implies that
all residents that prefer $M$ to $M'$ are not part of a
blocking pair or coalition. Thus, any blocking pair or coalition in $M'$ involving~$h(a_i)$ also blocks~$M$ and thus, such blocking pairs and coalitions do not exist.
Thus, $h(a_i)$ is a quota-two hospital. Note that this implies that
\ba{3}
applies for $b_i$.
Thereby $b_i$ is a resident holding a
proposal from $h(a_i)$ and by \Cref{lem:first-phase-rotation} that $a_i$
and $b_i$ are the first two residents on the preferences of $h(a_i)$.
If $h(a_i)$ is an inflexible\xspace hospital, then $h(a_i)$ is closed in $M'$, as
by Claim 1, $b_i$ is not matched to~$M(b_i)=h(a_i)$ in $M'$. The only
possible
blocking coalition to open $h(a_i)$ is~$(h,\{a_i,b_i\})$. However, as
$b_i$
prefers $M'(b_i)$ to $M(b_i)=h(a_i)$, resident $b_i$ cannot be part of this
coalition.
Thus, $h(a_i)$ is contained neither in a blocking coalition nor a blocking pair.
If $h(a_i)$ is a flexible hospital, then by
\Cref{ob:ab2},
$h(a_i) $ has upper quota two, $b_{i+1}$ is a resident and $a_i$,
$b_i$, and $b_{i+1}$ are the first three residents in the preferences
of $h(a_i)$. Moreover, it holds that
$h(a_i) = g(a_i)$. Thus, $h(a_i)$ is matched to $a_i$ and $b_i$
in $M$ and to $a_i$ and $b_{i+1}$ in~$M'$. Consequently, the only
possible blocking pair involving $h(a_i)$ is $(h(a_i), b_i)$.
As by Claim~1, resident~$b_i$ prefers $M'(h)$ to $M(h)=h(a_i)$, this pair cannot
be blocking.
{\bfseries Claim 4: } For all $i\in [k]$ such that $a_i$ is a resident,
hospital $g(a_i)$ is neither part of a blocking pair nor a blocking
coalition in $M'$.
Since $g(a_i)$ is open in $M'$, it cannot be part of a blocking coalition.
Assume that there exists a blocking pair $(r, g(a_i))$ in $M'$. Note
that it needs to hold that $r\neq b_{i+1}$, as $b_{i+1}$ is matched to~$g(a_i)$ in~$M'$.
Furthermore, $r\neq a_j$ for all $j\in [k]$ since by Claim~2 agents~$a_j$
are not part of a blocking pair in $M'$.
If $g(a_i)$ is closed in $M$, then $(g(a_i), \{b_{i+1}, r\})$ is a
blocking coalition in $M$: By Claim~1, resident~$b_{i+1}$ prefers $g(a_i)= M' (b_{i+1})$ to
$M(b_{i+1})$.
If $r=b_i$ for some
$i\in [k]$, then~$r$ is matched better in $M'$ than in $M$, which
implies that $(g(a_i), \{b_{i+1}, r\})$ blocks $M$.
Otherwise we have~$r\neq a_i,b_i$
for all $i\in [k]$, and then $r$ is matched to the same hospital in~$M$ and
$M'$, which again implies that $(g(a_i), \{b_{i+1}, r\})$ blocks $M$, a contradiction.
Therefore, $g(a_i)$ is open in $M$.
Recall that we have proven in the first
part of this proof in the paragraph on feasibility that if a hospital
$g(a_i)$ is open in $M$, then it needs to hold that $g(a_i) = h(a_j)$
for some $j\in [k]$.
Thus, by Claim 3, $g(a_i) = h(a_j) $ is not part of a blocking coalition or pair, a contradiction.
{\bfseries Claim 5: } For all $i\in [k]$, agent $b_i$ cannot be part of
a blocking pair or a blocking coalition in $M'$.
For all $i\in [k]$, by Claim 2-4, we know that $b_i$ cannot form a
blocking pair or coalition with any agents appearing as $a_j$,
$h(a_j)$, or $g(a_j)$ for some $j\in [k]$. Thus, $b_i$ needs to form a
blocking pair or coalition with agents that are matched to the
same partners in~$M$ and in $M'$. However, as by Claim 1, agent~$b_i$ prefers
$M'(b_i)$ to $M(b_i)$, this implies that the such a blocking pair or
coalition would also block $M$, a contradiction.
\end{proof}
Before we prove that no stable matching is created by the elimination of a
generalized rotation, we identify a sufficient criterion for the non-existence of a
stable matching, which directly follows from \Cref{cor:first-phase} and
\Cref{lem:rotation}.
\begin{corollary}\label{cor:empty-preferences}
If the preference list of a resident gets empty during Phase 2
or contains only quota-two hospitals which have only one resident on their
preferences, or
the preference list of a quota-one hospital gets empty, then the instance
does not admit a stable matching.
\end{corollary}
Thus, we reject an instance as soon as the preference list of a resident
becomes empty by eliminating a rotation. This also implies that in the
following we can assume that the set of residents
with non-empty preferences is the same before and after eliminating a
rotation.
\begin{restatable}[]{lemma}{forward}
\label{lem:forward}
Let $\ensuremath{\mathcal{I}^\mathrm{before}}$ be the instance before the elimination of a generalized
rotation (but after applying Phase 1) and $\ensuremath{\mathcal{I}^\mathrm{after}}$ the instance after
eliminating a generalized rotation from~$\ensuremath{\mathcal{I}^\mathrm{before}}$.
Any stable matching $M$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$ which matches all residents
with non-empty preferences in~$\ensuremath{\mathcal{I}^\mathrm{before}}$ is also stable in
$\ensuremath{\mathcal{I}^\mathrm{before}}$.
\end{restatable}
\begin{proof}
Let $M$ be a stable matching in $\ensuremath{\mathcal{I}^\mathrm{after}}$ matching all residents
with non-empty preferences in \ensuremath{\mathcal{I}^\mathrm{before}}. For the sake of
contradiction, let us assume that $M$ is not stable in~$\ensuremath{\mathcal{I}^\mathrm{before}}$.
Let $C$ be a blocking coalition or blocking pair involving some
hospital $h$ for~$M$ in~$\ensuremath{\mathcal{I}^\mathrm{before}}$. The only possibility that $C$ is
not blocking in $\ensuremath{\mathcal{I}^\mathrm{after}}$ is that the mutual acceptability of $h$ and
some resident occurring in $C$ has been deleted. We now make a case
distinction whether $h$ is a quota-one or a quota-two hospital and
argue for both cases that $C$ cannot block $M$ in $\ensuremath{\mathcal{I}^\mathrm{before}}$.
{\bfseries Case 1: } Hospital $h$ is a quota-one hospital.
Let $r$ be a resident such that $h$ and $r$ together form a blocking
pair or coalition for $M$ in~$\ensuremath{\mathcal{I}^\mathrm{before}}$. For the mutual acceptability
of $h$ and $r$ to be
deleted, it needs to hold that $\{h, r\} = \{a_i, b_i\}$ for some $i\in
[k]$.
Agent~$a_i$ is then the last agent on the preferences of~$b_i$.
Therefore, $\{h, r\}$ can only block~$M$ if $b_i$ is unmatched, which
we assume in the following.
Since every resident with non-empty preferences in~$\ensuremath{\mathcal{I}^\mathrm{before}}$ is matched in $\ensuremath{\mathcal{I}^\mathrm{after}}$, it follows that $b_i = h$ and $a_i = r$.
We claim that $a_{i-1}$ prefers $b_i
$ to
$M(a_{i-1})$:
The preferences of resident $a_{i-1}$ start with a quota-one or inflexible\xspace
hospital
$h (a_{i-1})$, followed by $g(a_{i-1}) = b_i$, as \ab{3a} has to apply
for $a_{i-1}$.
Furthermore, $a_{i-1}$ is not matched to~$h(a_{i-1})$ in $M$ because
either $h(a_{i-1})$ is inflexible\xspace, implying that $h(a_{i-1})$ has lower
quota two but only $a_{i-1}$ on its preferences in~$\ensuremath{\mathcal{I}^\mathrm{after}}$ after the
elimination of the generalized rotation, or
$h(a_{i-1})$ has lower quota one and the mutual acceptability of
$a_{i-1}$ and $h(a_{i-1}) = b_{i-1}$ has been deleted by the
elimination of the generalized rotation.
Thus, as $b_i$ is unmatched in $M$, $(a_{i-1}, b_i)$ is a blocking pair
for $M $ in \ensuremath{\mathcal{I}^\mathrm{after}}, a
contradiction to the stability of $M$.
{\bfseries Case 2: } Hospital $h$ is a quota-two hospital.
Then, it either holds that $C = (h, r_1)$ is a blocking pair or $C =
(h, \{ r_1, r_2\})$ is a blocking coalition for two residents $r_1,
r_2\in R$ for $M$ in $\ensuremath{\mathcal{I}^\mathrm{before}}$.
Assume without loss of generality that the acceptability of $r_1$ and
$h$ has been deleted. This implies that $h=h(a_i)$ and $r_1=b_i$ for
some~$i\in[k]$. Note that $r_1$
prefers all hospitals on her preference list to $h$.
As $h(a_i)$ is contained in the preferences of $r_1$ in $\ensuremath{\mathcal{I}^\mathrm{after}}$, the
preferences of $r_1 $ are non-empty and thus, by our assumption on $M$,
resident $r_1$ is matched in $M$.
This implies that
$r_1$ does not prefer $h$ to her partner in~$M$, a contradiction.
\end{proof}
\subsubsection{Proof of \Cref{t:q2}}
We are now ready to prove that the full algorithm works correctly.
\qq*
\begin{proof}
Each application of Phase 1a or 2 removes the mutual
acceptability of at least one resident and at least one hospital, while
each execution of Phase 1b reduces the number of quota-two hospitals.
Since we may assume that each hospital has upper quota at most $n$, it
follows that there are at most $\mathcal{O}(mn)$ hospitals at any stage
of the algorithm.
Thus, there are at most $\mathcal{O}(n \cdot mn)=
\mathcal{O}(n^2 m)$ mutual acceptabilities at any stage of the
algorithm, and all executions of Phase 1b together can be performed in
$\mathcal{O}(n^2 m)$.
Since all except the first proposal an agent receives results in a
rejection and reduces the number of mutual acceptabilities, it follows
that all executions of Phase 1a can be performed in $\mathcal{O}(n^2
m)$ time.
A generalized rotation can be found in $\mathcal{O}(n)$ time by \Cref{lem:one-proposal}.
Thus, all executions of Phase 2 can be executed in $\mathcal{O}(n\cdot n^2 m)$ time.
Thus, the runtime $\mathcal{O}(n^3 m)$ follows.
If \Cref{alg} returns a matching $M$, then $M$ matches all agents from
$S$.
Thus, the preferences of a resident can only get empty in the first
application of Phase 1a, which implies that the set of residents with
empty preferences is the same before eliminating the first generalized
rotation and after eliminating the last generalized rotation.
Consequently,
\Cref{lem:first-phase-backward,lem:splitting,lem:forward} imply that
$M$ is also stable in the input instance.
Vice versa, if the input instance contains a stable matching $M^*$,
then
\Cref{cor:first-phase} implies that $M^*$ matches all agents from $S$.
Consequently,
\Cref{lem:splitting,lem:rotation,lem:first-phase2} ensure that there
also exists a
stable matching covering $S$ after all modifications performed by the algorithm.
As the algorithm returns the only matching~$M$ which covers~$S$ and is still present at the termination of the algorithm, matching~$M$ is stable, and the correctness of the algorithm follows.
\end{proof}
From the correctness of our algorithm, we can also translate (a weaker
version of) the so-called Rural Hospitals
Theorem~\cite{RePEc:ucp:jpolec:v:92:y:1984:i:6:p:991-1016,10.2307/1913160}
to \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace.
The Rural Hospitals Theorem states that in every Hospital Residents instance (without lower quotas), every stable matching matches the same residents and for each hospital, the same number of residents is matched to it in every stable matching.
The latter part of this statement does not hold for \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, as the set of
open hospitals can differ for two stable matchings.
For instance, consider the following instance with two residents $r_1$ and $r_2$ as well as two hospitals $h_1 $ and $h_2$ with lower and upper quota two, and the following preferences: $r_1 : h_1 \succ h_2 $ and $r_2 : h_1 \succ h_2$.
Then both $(h_1, \{r_1 , r_2\})$ and $( h_2, \{r_1, r_2\})$ are stable matchings.
However, we can still show that every stable matching in an \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace
instance matches the same set of residents and opens the same number of
hospitals:
\begin{proposition}
Given an instance of \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace, every stable matching matches the same set of residents and opens the same number of hospitals.
\end{proposition}
\begin{proof}
\Cref{cor:first-phase} implies that the set of matched residents is the same in every stable matching.
Assume for a contradiction that there exist two stable matchings which
open a different number of hospitals.
Then \Cref{lem:first-phase2,lem:splitting,lem:rotation} imply that the
algorithm preserves that there are two stable matchings opening a
different number of hospitals.
However, at the termination of the algorithm, there is only one stable matching, a contradiction.
\end{proof}
\section{Introducing Ties} \label{se:ties}
It is also possible to extend the models that we have considered so far by
allowing for ties in the preferences of residents
and hospitals (if they have preferences). We call the resulting problems
\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace, \textsl{HR-$\text{Q}_\text{L}$-T}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace. Notably, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace now also generalizes
\textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace, as it is possible to make the hospitals indifferent among all
residents. In the following, we start by considering \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace before turning
to \textsl{HR-$\text{Q}_\text{L}$-T}\xspace and finally to \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace.
Note that the stability concept we apply here, that is, all agents need to
be better off after a deviation and not just indifferent, is usually called
\emph{weak} stability in the literature (opposed to \emph{strong} and
\emph{super-}stability, where one or both agents of a blocking pair might
be indifferent between the blocking pair and the matching).
All hardness results for
\textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace clearly also hold for \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace. Moreover, the only positive result
for \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, that is, the FPT result for the parameterization by the number
$m$ of hospital, still holds, as it is
possible to adapt the ILP constructed in \Cref{pr:HRLUQI-FPTM} in a
straightforward way (residents are still fully characterized by their
preference list and the number of different preference lists can be bounded
in a function of
$m$):
\begin{corollary} \label{co:HRLUQIT-m}
Parameterized by the number $m$ of hospitals, \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace is
fixed-parameter tractable.
\end{corollary}
We now show that that both \textsl{HR-$\text{Q}_\text{L}$-T}\xspace and
\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace are NP-complete even if all hospitals
have lower (and upper) quota two (in contrast to our polynomial-time algorithm for \textsl{HR-$\text{Q}_{\text{L}\le 2}^\text{U}$}\xspace).
To do so, we reduce from \textsc{Complete Stable Marriage with Ties and
Incomplete Preferences}.
In this problem, we are given a set $U$ of $t$ men and a set $W$ of $t$
women, where each man has preferences over a subset of women acceptable to him
and each woman has preferences over a subset of men acceptable to her and
preferences may contain ties. We refer to the combined
set of men and women as agents. A matching~$M$ is called \emph{complete} if
each man is matched to a woman acceptable to him and is called
\emph{stable} if no man-woman pair~$(m,w)\in U\times W$ exists such that both $m$
and $w$ find each
other acceptable and prefer each other to the agent they are
matched to in $M$. The goal is to decide the existence of a
complete and stable matching.
Notably, \textsc{Complete Stable Marriage with Ties and Incomplete
Preferences} is equivalent to \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace if all hospitals have upper quota one
and the task is to decide whether the instance admits a stable matching
which matches all residents.
Reducing from this problem, we now show that \textsl{HR-$\text{Q}_\text{L}$-T}\xspace is already NP-complete,
even
if all hospitals have lower quota at most two. The general idea of the
reduction is similar to the one sketched at the beginning of
\Cref{sec:strict} combined with an appropriate penalizing component.
\begin{proposition}
\label{p:q2T}
\textsl{HR-$\text{Q}_\text{L}$-T}\xspace, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace, and \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace are NP-complete, even if all hospitals
have lower
quota two, each hospital is accepted by at most two residents and each
resident accepts at most four hospitals.
\end{proposition}
\begin{proof}
Consider an instance of \textsc{Complete Stable Marriage with Ties and
Incomplete Preferences} where each agent accepts at most three other
agents.
This problem is known to be NP-complete~\cite{IrvingMO09}.
\textbf{Construction:} We construct an instance of \textsl{HR-$\text{Q}_\text{L}$-T}\xspace as follows. For
each man $m\in U$, we add a resident $r_m$, and for each woman $w\in W$,
we add a resident $r_w$.
For each man-woman pair~$(m, w)\in U\times W$ where both $m$ and $w$ find
each other acceptable, we add a hospital~$h_{(m, w)}$ (matching $(m,w)\in
U\times W$ in the given instance corresponds to matching both $m$ and $w$
to~$h_{(m, w)}$).
Furthermore, for each man~$m\in U$, we add a penalizing component
consisting of four hospitals $h^*_m$, $h_m$, $h_m'$, and $h_m''$ with
lower quota
two and four residents~$r^*_m$, $r_m'$, $r_m''$, and~$r_m'''$.
The preferences of the residents are as follows.
For each man~$m\in U$, the preferences of resident~$r_m$ arise from the
preferences of $m$ by replacing each woman~$w\in W$ by the
hospital~$h_{(m, w)}$.
At the end, we append the hospital~$h^*_m$.
For each woman~$w\in W$, the preferences of resident~$r_w$ arise from the
preferences of~$w$ by replacing each man~$m\in U$ by the
hospital~$h_{(m,w)}$.
For each~$m\in U$, the preferences of the residents in the penalizing
components are: $$r^*_m
: h^*_m \succ h_m, \qquad r_m' : h_m \succ h_m', \qquad r_m'' : h_m'
\succ h_m'', \qquad r_m''' : h_m'' \succ h_m.$$
Note that the residents $r_m'$, $r_m''$, and $r_m'''$ together with the
hospitals $h_m$, $h_m'$, and $h_m''$
correspond to the instance from
\Cref{ob:counter} not admitting a stable matching. This directly implies
that all hospitals~$h^*_m$ need to be closed in a stable matching and
thus all men $m\in U$ need to be matched to a hospital $h_{(m, w)}$ on
their preferences.
Since each man~$m\in U$ and each woman~$w\in W$ in the given instance has
at most three agents
on its preferences, it follows that each resident $r_m$ has at most four
hospitals on her preferences, and each resident $r_w$ has at most three
hospitals on her preferences.
{\bfseries ($\Rightarrow$)} Given a complete stable matching~$N$ in the
\textsc{Complete Stable Marriage with Ties and
Incomplete Preferences} instance, we get a stable
matching $M$ in the constructed \textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance by taking $(\{r_m, r_w\},
h_{(m, w)})$ for each $(m, w)\in N$
and $(\{r^*_m, r_m'\}, h_m)$ and $(\{r_m'', r_m'''\}, h_m'')$ for each
man~$m\in U$.
Since every open hospital is matched to all resident it accepts, there
cannot be blocking pairs.
Moreover, there cannot be a blocking coalition of the form $(\{m, w\},
h_{(m, w)})$ for some $(m,w)\in U\times W$, since otherwise $(m, w)$
would be
a blocking pair for $N$.
Thus, any blocking coalition must involve a hospital from a penalizing component.
However, since each man~$m\in U$ is matched in $N$, it follows that $r_m$
prefers the hospital she is matched to over $h^*_m$.
Thus,
there cannot be a blocking coalition to open $h^*_m$. Moreover, as $r_m'$
is
matched to her top-choice, there is no blocking coalition to open $h_m'$.
{\bfseries ($\Leftarrow$)} Vice versa, given a stable matching~$M$ in the
constructed
\textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance, first obverse
that for each $m\in U$, matching~$M$ matches resident~$r_m$ to a hospital~$h_{(m,
w)}$ for a
woman~$w\in W$, as otherwise $r_m$ is matched to $h^*_m$ together with
$r^*_m$ and there is a blocking coalition in the corresponding
penalizing component (see \Cref{ob:counter}).
Thus, the matching $N:= \{(m, w)\mid (\{r_m, r_w\}, h_{(m,w)})\in M\}$
matches all men, and it remains to show that it is stable.
Assume for a contradiction that there exists a blocking pair~$(m, w)\in U
\times W$ for $N$.
Then, both $r_m$ and $r_w$ prefer $h_{(m, w)}$ over their assigned
hospital in $M$, and it follows that $(\{r_m, r_w\}, h_{(m, w)})$ is a
blocking coalition in $M$.
\end{proof}
Notably, the construction falls under \Cref{ob:equiv}, implying that all
three problems are still hard if stability only requires that no blocking
coalition exists.
Recall that for both \textsl{HR-$\text{Q}_\text{L}$}\xspace and \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace, it is possible to decide whether
there exists a stable matching in which exactly
a given set of hospital is open in polynomial time. This result also
implies fixed-parameter
tractability with respect to the number
of hospitals. In the following, we investigate whether these positive results can be extended
if ties in the preferences are allowed. In fact, for \textsl{HR-$\text{Q}_\text{L}$-T}\xspace, it is still
possible to decide
whether there exists a stable matching opening a given set of hospitals
$H_{\open}\subseteq H$ in polynomial time. The underlying idea is that each
resident needs to be
matched to
one of her most preferred hospitals from $H_{\open}$ in every stable
matching~$M$. Thus, we already know for each resident the set of hospitals
she prefers to the hospital she is matched to in $M$. Using this, one
can easily check whether $M$ admits a
blocking coalition. The remaining task is then to find an assignment of
residents to one of their most preferred hospitals from $H_{\open}$ that
respects the lower
quota of each hospital from $H_{\open}$. This task can be formulated as an
instance of bipartite maximum matching.
\begin{proposition}\label{pr:LTies}
\label{pr:HRLQT-H'}
Given a subset of hospitals $H_{\open}\subseteq H$, deciding whether
there exists a stable matching in an \textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance $(H,R)$ in which
exactly the hospitals from $H_{\open}$ are open is solvable in
$\mathcal{O}(nm + n^{2.5})$ time.
Parameterized by the number $m_{\quota}$ of hospitals with non-unit
lower quota,
\textsl{HR-$\text{Q}_\text{L}$-T}\xspace is solvable in
$\mathcal{O}\big((nm+n^{2.5})2^{m_{\quota}}\big)$
time.
\end{proposition}
\begin{proof}
Let $(H,R)$ be the given \textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance and let
$H_{\open}$ be the set of hospitals to open. For every resident~$r\in R$, let $H_r \subseteq H_{\open}$
denote the
set of hospitals from $H_{\open}$ to which $r$ does not prefer
any
hospital
from $H_{\open}$ (the set of her top-choices in $H_{\open}$). Note that
$r$ needs to be assigned to one of the hospitals
from $H_r$ in every stable matching, as otherwise she forms a blocking
pair with any open hospital from $H_r$.
As $r$ is indifferent
among the hospitals in~$H_r$, for the stability of the
resulting matching, it does
not matter to which of the hospitals from $H_r$ she is assigned. Thus,
we
return NO if there exists a hospital~$h\in H \setminus H_{\open}$ and a
set of $l(h)$ residents~$r_1,\dots r_{l(h)}$ with $r_i$
preferring~$h$ to the hospitals in $H_{r_i}$ for all~$i\in [l(h)]$.
This can be done in $\mathcal{O} (mn)$.
After that, the problem reduces to finding a feasible matching where
each
resident $r$ is assigned to a hospital from $H_r$ and each hospital
from~$H_{\open}$ meets its lower quota. Note that if such a matching
exists, then this matching is stable, as the fact that we have not
rejected $H_{\open}$ before implies
that there does not exist a blocking coalition.
The task of finding a feasible matching can be solved by computing a
maximum matching in a bipartite graph~$G$ which we construct as
follows: We add one vertex for each resident $r\in R$ and $\ell (h)
$ vertices for each hospital~$h\in H_{\open}$ and an edge between a
vertex corresponding to a resident $r\in R$ and a vertex corresponding
to a hospital $h\in H$ if $h \in H_r$.
Note that we may assume that $\sum_{h\in H_{\open}} \ell (h) \le n$ (otherwise we have a trivial No-instance).
Therefore, $G$ has $\mathcal{O} (n)$ vertices and $\mathcal{O} (n^2)$
edges, and thus, using Hopcroft and Karp's
algorithm~\cite{DBLP:journals/siamcomp/HopcroftK73}, a matching in $G$
can be computed in
$\mathcal{O} (n^{2.5})$ time. There exists a bipartite matching which
matches all vertices corresponding to hospitals if and only if there
exists a feasible matching of the residents to the hospitals where each
resident $r\in R$ is matched to a hospital from $H_r$.
By iterating over all possible subsets of~$H$ and subsequently
employing the algorithm described before, it is possible to check
whether there exists a stable matching in a \textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance
in~$\mathcal{O}((n m + n^{2.5})2^m)$~time.
Let $H^{\quota}\subseteq H$ be the set of hospitals with non-unit
lower quota.
It is possible to improve the FPT algorithm from above by instead
iterating
over all subsets $H_{\open}^{\quota}\subseteq H^{\quota}$ and
subsequently deciding whether there exists a stable matching where the
set of open hospitals with a non-unit lower quota is exactly
$H^{\quota}_{\open}$. To answer this question, we need to modify the
algorithm from above slightly. That is, at the beginning of the
algorithm, we assign a resident~$r\in R$ to a quota-one hospital~$h\in
H$ (and delete her from the set of residents) if $r$ prefers $h$ to all
all hospitals from $H^{\quota}_{\open}$. The rest of the algorithm
remains unchanged leading to an overall running time
of~$\mathcal{O}((nm+n^{2.5})2^{m_{\quota}})$.
\end{proof}
Note that this result implies that deciding whether there exists a stable
matching with~$m_{\open}$ open ($m_{\text{closed}}$
closed)
hospitals in a \textsl{HR-$\text{Q}_\text{L}$-T}\xspace instance lies still in XP with respect
to~$m_{\open}$~($m_{\text{closed}}$).
As \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace is a generalization of \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace, by
\Cref{th:HRLUQIH'}, it is NP-hard to decide whether there exists a stable
matching in a \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace instance where exactly some given set of hospitals is
open. Turning to the
complexity of \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace parameterized by the number of $m$ hospitals, recall
that the basic idea of our ILP in \Cref{pr:HRLUQI-FPTM} for \textsl{HA-$\text{Q}_\text{L}^\text{U}$}\xspace was to
bound the number of
residents types in $m$.
However, this does not work for \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace, as different hospitals may
rank residents differently.
In fact, it turns out that, in contrast to \textsl{HR-$\text{Q}_\text{L}$-T}\xspace and \textsl{HA-$\text{Q}_\text{L}^\text{U}$-T}\xspace, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace is W[1]-hard with respect to the number of
hospitals. To prove this result, we first show that deciding whether there
exists a stable matching in a \textsl{HR-$\text{Q}^\text{U}$-T}\xspace instance (all hospitals only have an
upper quota they need to obey) which matches all residents
(\textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace) is
W[1]-hard with respect to the number of hospitals, a result which may also
be of
independent~interest.
In particular, Meeks and Rastegari~\cite{MeeksR20} asked for the
computational complexity of \textsc{Complete Stable Marriage with Ties and
Incomplete Lists} when men are of a few different ``types'' and each man of the
same type has the same preferences and women are indifferent between two
men of the same type.
This corresponds to \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace, as one hospital can be seen as its
upper quota many identical agents. Thus, the number of different types
in Meeks and Rastegari's model corresponds to the number of hospitals in
our model. The following result shows that parameterized by the number of
different types of men, \textsc{Complete Stable Marriage with Ties and
Incomplete Lists} is W[1]-hard.
\begin{proposition}
\label{pr:COMHRT}
Parameterized by the number $m$ of hospitals, \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace is
W[1]-hard.
\end{proposition}
\begin{proof}
We reduce from \textsc{Multicolored Clique} which we introduced in
\Cref{thm:ha-m-closed} and
which is W[1]-hard parameterized by the number $k$ of different colors~\cite{Pietrzak03}.
Let
$G=(V,E)$ be an
undirected graph with a partitioning of the vertices in $k$ different
colors $(V^1,\dots V^k)$.
Moreover, for each $c< d\in [k]$, let
$E^{c, d}$ denote the set of edges with one endpoint colored
with color $c$ and the other endpoint colored with color $d$. Without
loss of
generality and to simplify notation, we assume that all colors contain
$\ell$ vertices and for
all color combinations~$c<d\in [k]$, the number of edges with one
endpoint colored in $c$ and one endpoint colored in $d$ is $p$.
For~$c\in [k]$, we write~$V^c=\{v^c_1,\dots v^c_\ell\}$.
For a
vertex $v\in V$, let $\delta(v)$ denote the set of edges that are
incident to~$v$ in~$G$. Moreover, for a set of residents $R'$, let
$[R']$
denote an arbitrary strict linear ordering of all residents in $R'$
and $(R')$ a single tie containing all residents from $R'$.
\textbf{Construction:} We introduce a vertex selection
gadget for each color $c\in [k]$ and an edge selection gadget
for each pair of colors $c< d\in [k]$. For each $c\in
[k]$, the vertex selection gadget consists of two
hospitals $h^c$ and $\bar{h}^c$ both with upper quota $\ell$ and
$2\ell$
\emph{selection residents}~$r^c_1,\dots, r^c_\ell$ and $s^c_1,\dots,
s^c_\ell$. In
such a gadget, the vertex with color $c$ of the clique should be
selected. For
each~$c< d\in [k]$, the edge gadget should pick the edge between
the selected vertex of color $c$ and that selected vertex of color
$d$. For this, we introduce one
hospital $g^{c,d}$ with upper quota~$p-1$ and one hospital
$\bar{g}^{c,d}$ with upper quota $1$ together with an edge
resident
$t_e$ for each edge $e\in E^{c, d}$.
The
preferences of all residents and hospitals are as follows.
\begin{align*}
\intertext{Vertex gadgets:}
\bar{h}^c \colon & r^c_1 \succ \dots \succ r^c_\ell \succ
[t_e]_{e\in \delta (v^c_\ell)}\succ s^c_\ell
\succ
[t_e]_{e\in \delta
(v^c_{\ell-1})}\succ s^c_{\ell-1} \succ \dots \succ
[t_e]_{e\in \delta
(v^c_1)}\succ s^c_{1}, &
\forall c\in
[k]
\\
h^c \colon & r^c_1 \succ\dots \succ r^c_\ell \succ s^c_1 \succ
[t_e]_{e\in \delta (v^c_1)} \succ s^c_2
\succ [t_e]_{e\in \delta (v^c_2)}
\succ \dots \succ s^c_\ell \succ [t_e]_{e\in \delta
(v^c_\ell)}, &
\forall c\in
[k]\\
r^c_i \colon & (h^c, \bar{h}^c),\qquad s^c_i \colon (h^c,
\bar{h}^c), \qquad \forall i\in [\ell], \forall c\in [k]\\
\intertext{Edge gadgets:}
g^{c,d} \colon & (t_e)_{e\in E^{c, d}},\qquad
\bar{g}^{c,d} \colon (t_e)_{e\in E^{c, d}}, \qquad \forall c< d \in
[k]\\
t_e \colon & g^{c,d}\succ h^c\succ h^{d} \succ \bar{h}^c
\succ \bar{h}^{d}
\succ \bar{g}^{c,d}, \qquad \forall c<d \in [k],\forall e\in
E^{c, d}
\end{align*}
See \Cref{fig:w-hard-com-hrt} for an example.
For each color $c\in [k]$, selecting vertex $v^c_i\in V^c$ corresponds
to
matching $s^c_1,\dots, s^c_{i}$ and $r^c_{i+1}, \dots, r^c_{\ell}$ to
$h^c$
and $r^c_1, \dots, r^c_i$ and
$s^c_{i+1},\dots, s^c_{\ell}$ to $\bar{h}^c$. For each color
pair~$c< d\in [k]$, selecting edge $e\in E^{c, d}$ corresponds to
matching
$t_e$ to $\bar{g}^{c, d}$.
\begin{figure}
\begin{minipage}{0.2\textwidth}
\begin{center}
\begin{tikzpicture}
\node (hor-dist) at (1, 0) {};
\node (ver-dist) at (0, 1) {};
\node[vertex, label=180:$v_1^c$] (v11) at (0,0) {};
\node[vertex, label=180:$v_2^c$] (v21) at ($(v11) + (ver-dist)$) {};
\node[vertex, label=0:$v_1^d$] (v12) at ($(v11) + (hor-dist)$) {};
\node[vertex, label=0:$v_2^d$] (v22) at ($(v21) + (hor-dist)$) {};
\draw (v11) -- (v12);
\draw (v11) -- (v22);
\draw (v21) -- (v22);
\draw (v21) -- (v12);
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{0.7\textwidth}
\begin{center}
\begin{tikzpicture}
\node (hor-dist) at (1.5, 0) {};
\node (ver-dist) at (0, 1.5) {};
\node[vertex, label=180:$r_{1}^c$] (r11) at (0, 0) {};
\node[vertex, label=180:$r_{2}^c$] (r21) at ($(r11) +
(ver-dist)$) {};
\node[vertex, label=180:$s_{1}^c$] (s11) at ($(r21) + (
ver-dist)$) {};
\node[vertex, label=180:$s_{2}^c$] (s21) at ($(s11) +
(ver-dist)$) {};
\node[vertex, label=0:$r_{1}^d$] (r12) at ($(r11) +
6*(hor-dist)$) {};
\node[vertex, label=0:$r_{2}^d$] (r22) at ($(r12) +
(ver-dist)$) {};
\node[vertex, label=0:$s_{1}^d$] (s12) at ($(r22) + (
ver-dist)$) {};
\node[vertex, label=0:$s_{2}^d$] (s22) at ($(s12) +
(ver-dist)$) {};
\node[vertex, label=270:$t_{v_1^c, v_1^d}$] (e1) at
($0.5*(r11) + 0.5*(r12)$) {};
\node[vertex, label=270:$t_{v_1^c, v_2^d}$] (e2) at ($(e1) +
(ver-dist)$) {};
\node[vertex, label={[xshift=0.cm]90:$t_{v_2^c, v_1^d}$}]
(e3) at ($(e2) + (ver-dist)$) {};
\node[vertex, label=90:$t_{v_2^c,v_2^d}$] (e4) at ($(e3) + (
ver-dist)$) {};
\node[squared-vertex, label=270:$h^c$,
label={[yshift=-0.4cm]270:${[1,2]}$}] (h1) at ($(r11)+
(hor-dist) - 0*(ver-dist)$) {};
\node[squared-vertex, label={[yshift=0.4 cm]90:$\bar{h}^c$},
label=90:${[1,2]}$] (h1b) at ($(h1) + 3*(ver-dist)$) {};
\draw (h1) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r11);
\draw (h1) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $2$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r21);
\draw (h1) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s11);
\draw (h1) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s21);
\draw (h1b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r11);
\draw (h1b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $2$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r21);
\draw (h1b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $8$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s11);
\draw (h1b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $5$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s21);
\draw (h1) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $4$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $2$} (e1);
\draw (h1) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $4$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $2$} (e2);
\draw (h1) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $7$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $2$} (e3);
\draw (h1) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $7$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $2$} (e4);
\draw (h1b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $4$} (e1);
\draw (h1b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $4$} (e2);
\draw (h1b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $4$} (e3);
\draw (h1b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $4$} (e4);
\node[squared-vertex, label=270:$h^d$,
label={[yshift=-0.4cm]270:$[1,2]$}] (h2) at ($(r12) -
(hor-dist)$) {};
\node[squared-vertex, label={[yshift=0.4cm]90:$\bar{h}^d$},
label={90:$[1,2]$}] (h2b) at ($(h2) + 3*(ver-dist)$) {};
\draw (h2) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r12);
\draw (h2) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $2$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r22);
\draw (h2) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s12);
\draw (h2) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s22);
\draw (h2) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $4$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $3$} (e1);
\draw (h2) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $7$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $3$} (e2);
\draw (h2) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $4$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $3$} (e3);
\draw (h2) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $7$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $3$} (e4);
\draw (h2b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r12);
\draw (h2b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $2$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (r22);
\draw (h2b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $8$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s12);
\draw (h2b) edge node[pos=0.3, fill=white, inner sep=2pt]
{\scriptsize $5$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (s22);
\draw (h2b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $5$} (e1);
\draw (h2b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $5$} (e2);
\draw (h2b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $6$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $5$} (e3);
\draw (h2b) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $3$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $5$} (e4);
\node[squared-vertex, label=90:$g^{c,d}$,
label={[yshift=-0.1cm]270:${[1,3]}$}] (g) at
($0.5*(h1)+0.5*(h1b)$) {};
\node[squared-vertex, label=90:$\bar{g}^{c,d}$,
label={[yshift=-0.1cm]270:${[1,1]}$}] (gb) at ($0.5*(h2) +
0.5*(h2b)$) {};
\draw (g) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (e1);
\draw (g) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (e2);
\draw (g) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (e3);
\draw (g) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $1$} (e4);
\draw (gb) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $6$} (e1);
\draw (gb) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $6$} (e2);
\draw (gb) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.84, fill=white, inner sep=2pt]
{\scriptsize $6$} (e3);
\draw (gb) edge node[pos=0.2, fill=white, inner sep=2pt]
{\scriptsize $1$} node[pos=0.8, fill=white, inner sep=2pt]
{\scriptsize $6$} (e4);
\end{tikzpicture}
\end{center}
\end{minipage}
\caption{An example for the reduction showing W[1]-hardness of
\textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace parameterized by the number of hospitals from
\Cref{pr:COMHRT}.
The input graph with coloring $(V^c=\{v_1^c, v_2^c\}, V^d=\{v_1^d,
v_2^d\})$ is depicted on the left, while the output is depicted on
the right.}
\label{fig:w-hard-com-hrt}
\end{figure}
{\bfseries ($\Rightarrow$)}
Assume that $V'=\{v^1_{i_1}, \dots, v^k_{i_k}\}$ is a clique of
size
$k$ in $G$. We claim that the following matching $M$ is
a stable matching which matches all residents in the constructed
instance:
\begin{align*}
M= & \{(g^{c,d}, \{t_e \mid e\in
E^{c, d}\setminus\{\{v^c_{i_c},v^{d}_{i_{d}}\}\}\})\mid c<d \in [k]
\}\cup \{(\bar{g}^{c,d},t_{\{v^c_{i_c},v^{d}_{i_{d}}
\}})\mid c< d \in
[k]\} \\
&\cup \{(\bar{h}^c,\{r^c_1, \dots,
r^c_{i_c},s^c_{i_c + 1},\dots, s^c_\ell \})\mid c\in
[k]\}
\\
& \cup \{(h^c,\{r^c_{i_c+1}, \dots, r^c_\ell,s^c_1, \dots, s^c_{i_c}
\})\mid
c\in [k]\}
\end{align*}
The matching $M$ is well defined, since $V'$ is a clique and clearly a
complete matching which respects the upper quota of all hospitals. To
show
stability, note that all residents except the edge
residents~$t_{\{v_{i^c_c},v_{i^{d}_{d}}\}}$ for all $c< d \in [k]$,
i.e., those
corresponding to the edges lying in the clique,
are
matched to their top-choice.
Thus, the only possible blocking pairs are
$(h^c,t_{\{v_{i^c_c},v_{i^{d}_{d}}\}})$,
$(h^{d},t_{\{v_{i^c_c},v_{i^{d}_{d}}\}})$,
$(\bar{h}^{c},t_{\{v_{i^c_c},v_{i^{d}_{d}}\}})$,
$(\bar{h}^{d},t_{\{v_{i^c_c},v_{i^{d}_{d}}\}})$, and $(g^{c, d},
t_{\{v_{i^c_c},v_{i^{d}_{d}}\}})$ for some $c< d \in [k]$.
However, by the construction of the matching~$M$, for each $c\in [k]$
both $h^c$ and $\bar{h}^c$ prefer all residents assigned to them over
all residents corresponding to the edges that are incident to
$v^c_{i_c}$.
Furthermore, $g^{c, d}$ is full and is indifferent among all residents
it accepts and thus cannot be part of a blocking pair.
{\bfseries ($\Leftarrow$)}
Let $M$ be a complete stable matching.
Since the sum of the capacities of all hospitals equals the number of
residents, it follows that for each $c< d\in [k]$, one edge resident
$t_e$ with $e=\{v^c_i,v^{d}_j\}$ for some $i,j\in [p]$ has to be
matched to $\bar{g}^{c,d}$.
We claim that from this it follows that the vertex-selection gadget for
$V^c$ has to select
$v^c_i$ and the vertex-selection gadget for $V^{d}$ has to select
$v^{d}_j$, from which the correctness of the reduction easily follows.
Without loss of generality, let us look at~$v^c_i$.
Due to the stability of $M$, hospitals $h^c$ and $\bar{h}^c$ have to
prefer all
residents matched to them to $t_{\{v^c_i,v^{d}_j\}}$. Thereby,
$\ell$ residents from $\{r^c_j : j\in [\ell]\}
\cup \{s^c_j : j\in [i]\}$ have to be matched to $h^c$, and
$\ell$
residents from $\{r^c_j : j\in [\ell]\} \cup \{s^c_j: j\in [i + 1,
\ell]\}$ have
to be matched to~$\bar{h}^c$.
Thus, the agents~$\{s^c_j: j \in [i]\}$ are matched to $h^c$, and
$\{s^c_j:
j\in [i+1, \ell]\}$ are matched to $\bar{h}^c$ which corresponds to
selecting $v^c_i$ as the vertex of color $c$ in the clique.
\end{proof}
It is possible to reduce \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace to \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace by introducing a
penalizing component that ensures that each resident needs to be matched to
a hospital from the original instance in every stable matching. This
implies the following result:
\begin{proposition}
\label{pr:HRT-W}
Parameterized by the number $m$ of hospitals, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace is W[1]-hard,
even
if only four hospitals have non-unit lower quota.
\end{proposition}
\begin{proof}
We reduce from \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace, which is W[1]-hard with respect to
the number
of hospitals as proven in \Cref{pr:COMHRT}.
\textbf{Construction:} From an \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace instance,
we construct a \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace instance by adding a lower quota of one to each
of the hospitals. Moreover, we insert a hospital $h^*$ with lower quota
and upper quota two which we insert after all other hospitals in the
preference of
each
resident. Furthermore, we add a penalizing component consisting of
three hospitals $h_1$, $h_2$, and $h_3$ with lower quota two and four
residents $r^*$,
$r_1$, $r_2$, and $r_3$ with the following preferences:
$$r^*: h^*\succ h_1, \quad r_1:h_1\succ h_2, \quad r_2: h_2\succ h_3,
\quad
r_3:h_3\succ h_1.$$ Finally, we set the preferences of $h^*$ such that
$h^*$ accepts all residents and has $r^*$ as its unique top-choice.
Note that the residents $r_1$, $r_2$, $r_3$ together with the hospitals
$h_1$, $h_2$, $h_3$ correspond to the instance from \Cref{ob:counter}
not
admitting a stable matchings. This implies that the
hospital $h^*$ is closed in all stable matchings.
{\bfseries ($\Rightarrow$)}
If there exists a stable matching in the given \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace
instance matching
all residents, then adding $(h_1,\{r^*, r_1\})$ and
$(h_3,\{r_2, r_3\})$ to the matching results in a stable matching in
the constructed \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace instance.
{\bfseries ($\Leftarrow$)}
If there exists a stable matching $M$ in the constructed \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace
instance, then $h^*$ needs to be closed in $M$,
as
otherwise $r^*$ is matched to it and consequently $r_1$, $r_2$, and
$r_3$ cannot be matched in a stable way. This implies that all
residents from the given \textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace instance need to be
matched to hospitals
from the original instance. Thus, $M$ restricted to the
original hospitals and residents is a stable matching in the given
\textsc{Com} \textsl{HR-$\text{Q}^\text{U}$-T}\xspace instance where all residents are matched.
\end{proof}
On the positive side, by guessing for each hospital the least preferred
resident assigned to it in a stable matching, it is again possible to
bound the number of
different resident types in a function of $m$ and subsequently solve
the problem using an ILP. This approach results in an XP algorithm for
\textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace:
\begin{proposition}
\label{pr:HRT-XP}
Parameterized by the number $m$ of hospitals, \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace lies in XP.
\end{proposition}
\begin{proof}
Let $(H,R)$ be the given \textsl{HR-$\text{Q}_\text{L}^\text{U}$-T}\xspace instance.
Similar as in \Cref{pr:HRLUQI-FPTM}, we solve this problem using an
ILP. Let $\succsim_{t_1}$, $\succsim_{t_2}$, \dots,
$\succsim_{t_q}$ be a list of all
weak incomplete orders over $H$.
For each $i\in [q]$, denote by $A(i)$ the set of hospitals contained in
the weak incomplete order~$\succsim_{t_i}$.
The number $q$ lies in
$\mathcal{O}(m\cdot
m!\cdot 2^{m})$, as each weak incomplete order can be created by
picking a strict ordering of the $m$ elements, deleting the last~$i\in
[m]$
elements and subsequently dividing the remaining elements into
equivalence
classes (by guessing the set of residents which are for some tie the first residents of this tie in the ordering).
Unfortunately, it is not possible to
directly bound the number of
different resident types by the number of different preference
relations because different hospitals might rank the
residents differently and this information is relevant when checking for
blocking
pairs. Therefore, we start by guessing the subset
of open hospitals $H_{\open}\subseteq H$ and for each such
hospital $h\in H_{\open}$
the least preferred resident~$r_h\in R$
that is matched to
$h$; in fact, we do not enforce that $r_h$ is matched to $h$. Instead,
we only enforce that either $r_h$ or a resident $r\neq r_h$ for which
it holds that $h$ is indifferent between $r$ and $r_h$ is matched to
$h$.
Furthermore, we guess the set~$H_{\full} \subseteq H_{\open}$ of full hospitals.
For each resident~$r\in R$ and hospital $h\in H_{\open}$,
define $z_r^h$ to be~$1$ if $h$ prefers $r$ to
$r_h$, to be
$0$
if $h$ is
indifferent between $r$ and $r_h$, and to be $-1$ if $h$ strictly
prefers
$r_h$ to $r$.
We call the tuple~$(z_r^h)_{h\in H_{\open}}$ the \emph{hospital
signature} of
resident~$r\in R$.
For the sake of finding a stable matching, a resident is fully
characterized by her preference relation and her hospital signature.
For each $h\in H_{\open}$, $i\in [q]$, and $\mathbf{z}\in
\{-1,0,1\}^m$, we
introduce a variable $x_{i,h}^\mathbf{z}$ denoting the number of
residents with preference relation $\succsim_{t_i}$ and hospital
signature
$\mathbf{z}$ that are assigned to hospital $h$. Further, let
$n_i^\mathbf{z}$
denote the number of resident with preference relation $\succsim_{t_i}$
and
hospital signature~$\mathbf{z}$ in the given instance.
It is possible to check
whether there
exists a stable matching respecting the current guess by solving the
following ILP (below the ILP, we explain the purpose of the different
constraints):
\begin{align*}
\sum_{\substack{h'\in H_{\open}: \\ h'\succsim_{t_i} h}}
x^{\mathbf{z}}_{i,h'}\geq
n^\mathbf{z}_i, \qquad & \forall h \in H_{\open}, \forall i \in
[q], \forall \mathbf{z}\in \{-1,0,1\}^m\text{ with } z_h=1
\tag{1a}\label{ILP:no-bpa2} \\
\sum_{\substack{h'\in H_{\open}: \\ h'\succsim_{t_i} h}}
x^\mathbf{z}_{i,h'}
\geq
n_i^\mathbf{z}, \qquad & \forall h\in H_{\open} \setminus H_{\full} , \forall i \in
[q] \text{ with } h\in A(i), \forall \mathbf{z} \in \{1,0,-1\}^m
\tag{1b}\label{ILP:no-bpb2}\\
\sum_{\substack{i\in [q], h'\in H_{\open}, \\ \mathbf{z}\in
\{-1,0,1\}^m: h \succ_{t_i} h'}}
x^\mathbf{z}_{i,h'}\leq
l(h), \qquad & \forall h\in H\setminus H_{\open} \tag{2}
\label{ILP:no-bc2}\\
l(h)\leq \sum_{\substack{i\in [q], \\ \mathbf{z}\in
\{-1,0,1\}^m}} x^\mathbf{z}_{i,h} \leq u(h), \qquad&
\forall h\in
H_{\open} \tag{3}\label{ILP:quotas2}\\
\sum_{h\in H_{\open}} x^\mathbf{z}_{i,h} \leq n_i^\mathbf{z},
\qquad& \forall
i\in
[q],\forall \mathbf{z}\in \{-1,0,1\}^m \tag{4}
\label{ILP:num-residents2}
\\
x^\mathbf{z}_{i,h} = 0, \qquad &\forall i \in [q], \forall h\in
H_{\open}\setminus
A(t_i), \forall \mathbf{z}\in \{-1,0,1\}^m
\tag{5} \label{ILP:acceptablitiy2}\\
\sum_{i\in [q], \mathbf{z}\in \{-1,0,1\}^m } x^\mathbf{z}_{i,h}
= u(h), \qquad & \forall h\in H_{\full} \tag{6} \label{ILP:full} \\
\sum_{i\in [q], \mathbf{z}\in \{-1,0,1\}^m } x^\mathbf{z}_{i,h}
\le u(h) - 1, \qquad & \forall h\in H_{\open} \setminus H_{\full} \tag{7} \label{ILP:undersubscribed} \\
x^\mathbf{z}_{i,h} = 0, \qquad &\forall i \in [q],\forall h \in
H_{\open}, \forall
\mathbf{z}\in \{-1,0,1\}^m\text{ with } z_h=-1
\tag{8} \label{ILP:guess1}\\
\sum_{\substack{i\in [q], \mathbf{z}\in
\{-1,0,1\}^m\\ \text{ with } z_h=0}}
x^\mathbf{z}_{i,h} \geq 1, \qquad &\forall h \in H_{\open}
\tag{9} \label{ILP:guess2}\\
x^\mathbf{z}_{i,h}\in\{0,1,\dots,n^\mathbf{z}_i\}, \qquad &\forall i\in [q] ,
\forall h \in H_{\open}, \forall \mathbf{z}\in \{-1,0,1\}^m
\tag{10} \label{ILP:integrality2}
\end{align*}
Conditions (\ref{ILP:no-bpa2}) and (\ref{ILP:no-bpb2}) ensure
that no blocking pair exists. Condition (\ref{ILP:no-bpa2}) checks for
all~$h\in H_{\open}$ that
all residents that are preferred by $h$ to $r_h$ are matched
to a hospital that they find as least as good as $h$.
All other residents only form a blocking pair with $h$ if $h$ is
undersubscribed and they are matched to a hospital to which they
strictly prefer $h$. The non-existence of such a pair is enforced by
Condition (\ref{ILP:no-bpb2}).
Condition (\ref{ILP:no-bc2}) ensures that no blocking coalition to open
a hospital in $H\setminus H_{\open}$ exists, while Condition~(\ref{ILP:quotas2}) ensures that all hospitals in $H_{\open}$ respect their lower and upper quota.
Condition (\ref{ILP:num-residents2}) enforces that for each resident
type the number of matched residents of this type does not exceed the
number of
residents of this type from the given instance, while Condition~(\ref{ILP:acceptablitiy2}) enforces that no resident is matched to a
hospital she does not accept.
Conditions~(\ref{ILP:full}) and~(\ref{ILP:undersubscribed}) ensure that each hospital in~$H_{\full}$ is indeed full, while all other hospitals from~$H_{\open} \setminus H_{\full}$ are undersubscribed.
Finally, Conditions~(\ref{ILP:guess1})
and~(\ref{ILP:guess2}) ensure that for all hospitals~$h\in
H_{\open}$, the guess for the worst matched resident $r_h$ is
correct by enforcing that no resident $r\in R$ with
$r_h\succ_{h} r$ is matched to $h$ (Condition~(\ref{ILP:guess1})) and by enforcing that at least one
resident $r\in R$ with $r\sim_h r_h$ is matched to $h$
(Condition~(\ref{ILP:guess2})).
For each guess, we solve the ILP from above and return YES if it is
feasible. Otherwise, we reject the current guess and continue with the
next guess and return NO
after all guesses have been rejected.
Note that the total number of guesses is $3^m\cdot n^m$, and for each
guess, we need to solve the above~ILP. As the number of variables in
the ILP lies
in $\mathcal{O}(m^2\cdot m!\cdot 2^{m}\cdot 3^m)$, by employing
Lenstra's algorithm~\cite{DBLP:journals/mor/Kannan87,DBLP:journals/mor/Lenstra83}, it is
possible to
solve the problem in $\mathcal{O}(f(m)\cdot n^m)$ for some
computable function in $f$.
\end{proof}
\section{Conclusion}
\label{sec:conclusion}
We conducted a thorough parameterized complexity
analysis of the
Hospital
Residents problem with lower and upper quotas. We have shown that the
hardness of this problem arises from choosing the set of open hospitals such
that no blocking coalition exists, as the problem remains hard even if all
hospitals have only lower quotas and pairs cannot block an outcome, but it
becomes easy as soon as the set of open hospitals is given. We have
also
analyzed two variants of this problem.
For future work, one could generalize \textsl{HR-$\text{Q}_\text{L}^\text{U}$}\xspace by having only hospitals, and
hospitals having preferences over other hospitals (similar to the
\textsc{Capacitated Stable Roommates} problem introduced by
Cechl{\'{a}}rov{\'{a}} and Fleiner~\cite{DBLP:journals/talg/CechlarovaF05}).
Then, a feasible matching would be a set of pairs of hospitals where each hospital appears in either no or between its upper and lower quota many pairs.
A blocking pair for a matching~$M$ would be a pair~$\{h_1, h_2\}$ of open
hospitals such that, for $i\in \{1,2\}$, $h_i $ is undersubscribed or
prefers $h_{3-i}$ to one hospital assigned to it by~$M$, while a blocking
coalition would be a closed hospital $h$ together with $\ell (h)$
hospitals $h_1, \dots, h_{\ell (h)}$ such that, for $i\in [l(h)]$, $h_i
$ is undersubscribed or prefers $h$ to one of its partners in $M$.
While all our hardness results clearly carry over to this setting, it would
be interesting to see whether this generalization is also solvable in
polynomial time if all lower quotas are at most two.
Further, we believe that it would be
interesting to analyze other stable many-to-one
matching problems using a similar fine-grained parameterized approach as
taken in this paper to enrich our understanding of the
complexity of these problems.
\bibliographystyle{splncs04.bst}
|
2,877,628,090,782 | arxiv | \section{Introduction}
The study of symplectic embeddings is at the core of symplectic geometry. One of the most important tools in this study are symplectic capacities. A {\bf symplectic capacity} is a function which assigns to each symplectic manifold $(X,\omega)$ of a fixed dimension $2n$, possibly in some restricted class, a number $c(X,\omega)$ satisfying the following conditions:
\begin{enumerate}
\item If there exists an embedding $\varphi:X_1\hookrightarrow X_2$ such that $\varphi^*\omega_2=\omega_1$, then
\[c(X_1,\omega_1)\le c(X_2,\omega_2).\]
\item If $r>0$, then
\[c(X,r\cdot \omega)=r \cdot c(X,\omega).\]
\end{enumerate}
After Gromov's seminal work on symplectic embeddings \cite{Gromov}, many capacities were defined. The majority of these satisfy a normalization condition based on Gromov's non-squeezing. More precisely, let $B^{2n}(r)\subset\C^n$ denote the ball of radius $r$ and let $Z^{2n}(r)=B^{2}(r)\times\C^{n-1}$. As usual, the standard symplectic form on $\C^n (=\R^{2n})$ is defined by
\[\omega_0=\sum_{i=1}^n dx_i\wedge dy_i.\]
We say that a symplectic capacity is {\bf ball normalized}\footnote{A capacity satisfying condition 3) is usually called {\bf normalized} in the literature. We add the word ``ball'' in this paper because we will define another normalization condition below.} if
\begin{itemize}
\item[(3)] $c(B^{2n}(r),\omega_0)=c(Z^{2n}(r),\omega_0)=\pi r^2$.
\end{itemize}
The central question about ball normalized capacities is the following conjecture, which apparently has been folkore since the 1990s.
\begin{conjecture}[strong Viterbo conjecture]
\label{conj:V}
If $X$ is a convex domain in $\R^{2n}$, then all normalized symplectic capacities of $X$ are equal.
\end{conjecture}
We refer to \cite{GHR} for a presentation of known results around Conjecture \ref{conj:V}. The strong Viterbo conjecture is, in particular, proven for all monotone toric domains in dimension 4.
Some of the main examples of symplectic capacities that do not satisfy this ball normalization 3) come in sequences, see \cite{EH2,qech}. For all of these sequences, the first capacity is still ball normalized. Two other capacities stand alone not satisfying 3), namely the {\bf Lagrangian capacity} and the {\bf cube capacity}, defined by Cieliebak--Mohnke \cite{CM} and Gutt--Hutchings \cite{GH}, respectively. In this paper we will introduce a new normalization condition (which we call cube normalization) which is satisfied by these latter capacities. Our main result is an equivalent of the strong Viterbo conjecture for cube normalized capacities.
\begin{theorem}\label{thm:normalized}
All cube normalized symplectic capacities coincide on all monotone toric domain in any dimension.
\end{theorem}
This paper is organized as follows. In Section 2, we define the cube normalization and prove Theorem \ref{thm:normalized}. In Section 3, we provide an explicit formula for the Lagrangian capacity on a large class of toric domains encompassing monotone toric domains. In Section 4, we study cube normalized capacities of an interesting class of examples of non-monotone toric domains and we show that for some parameters, ball normalized capacities coincide while cube normalized do not. Finally, in Section 5, we find an upper bound for the cube capacity of a large class of weakly convex toric domains, which is used in Section 4.
\section{A new normalization condition}
Given a domain\footnote{In this article, a domain is the closure of a non-empty open set.} $\Omega\subset\R^n_{\ge 0}$, define the toric domain
\[X_\Omega=\mu^{-1}(\Omega)=\left\{(z_1,\dots,z_n)\in\C^n\mid (\pi|z_1|^2,\dots,\pi|z_n|^2)\in\Omega\right\}\]
where the map $\mu:\C^n\to[0,+\infty)^n\,:\,(z_1,\ldots, z_n)\mapsto(\pi|z_1|^2,\dots,\pi|z_n|^2)$ is the periodic moment map.
We let \[\partial_+\Omega=\left\{p=(p_1,\dots,p_n)\in\partial \Omega\mid p_i>0 \text{ for }i=1,\dots,n.\right\}.\] Recall from \cite{GHR} that a {\bf monotone toric domain} is a compact toric domain with smooth boundary such that for every $p\in\partial_+\Omega$, the outward pointing normal vector at $p$, $\nu=(\nu_1,\dots,\nu_n)$ satifies $\nu_i\ge 0$ for $i=1,\dots,n$. Note that a monotone toric domain is the limit of toric domains $X_{\Omega'}$ where $\Omega'$ is bounded by the coordinate hyperplanes and the graph of a function whose partial derivatives are all negative, see the proof of \cite[Lemma 3.2]{GHR}.
Consider the following examples of toric domains:
\begin{IEEEeqnarray*}{lrClCrCl}
\textrm{The Ball} \quad & B_n(a) & \coloneqq & \mu^{-1}(\Omega_{B_n(a)}), & \quad & \Omega_{B_n(a)} & \coloneqq & \{ x \in \R^n_{\geq 0} \mid x_1 + \cdots + x_n \leq a \}, \\
\textrm{The Cylinder} \quad & Z_n(a) & \coloneqq & \mu^{-1}(\Omega_{Z_n(a)}), & \quad & \Omega_{Z_n(a)} & \coloneqq & \{ x \in \R^n_{\geq 0} \mid x_1 \leq a \}, \\
\textrm{The Cube} \quad & P_n(a) & \coloneqq & \mu^{-1}(\Omega_{P_n(a)}), & \quad & \Omega_{P_n(a)} & \coloneqq & \{ x \in \R^n_{\geq 0} \mid \forall i = 1, \ldots, n \colon x_i \leq a \}, \\
\textrm{The NDUC} \quad & N_n(a) & \coloneqq & \mu^{-1}(\Omega_{N_n(a)}), & \quad & \Omega_{N_n(a)} & \coloneqq & \{ x \in \R^n_{\geq 0} \mid \exists i = 1, \ldots, n \colon x_i \leq a \}.
\end{IEEEeqnarray*}
Here, NDUC stands for non-disjoint union of cylinders.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\fill[orange!50](0,0)--(0,1.5)--(1.5,0)--(0,0);
\draw [thick](0,0)--(0,1.8) ;
\draw [thick] (0,0)--(1.8,0) ;
\draw (0,1.5) node[left]{$a$};
\draw (1.5,0) node[below]{$a$};
\draw (0.5,0.5) node[above right]{$\Omega_{B_2(a)}$};
\fill[orange!50](3,0)--(3,1.5)--(4.5,1.5)--(4.5,0)--(3,0);
\draw [thick](3,0)--(3,2) ;
\draw [thick] (3,0)--(5,0) ;
\draw (3,1.5) node[left]{$a$};
\draw (4.5,0) node[below]{$a$};
\draw (3.8,0.8) node{$\Omega_{P_2(a)}$};
\fill[orange!50](6,0)--(6,2.1)--(7,2.1)--(7,0)--(6,0);
\draw [thick](6,0)--(6,2) ;
\draw [thick] (6,0)--(8,0) ;
\draw (7,0) node[below]{$a$};
\draw (6.8,0.8) node{$\Omega_{Z_2(a)}$};
\fill[orange!50](9,0)--(9,2.1)--(9.7,2.1)--(9.7,0.7)--(11.1,0.7)--(11.1,0)--(9,0);
\draw [thick](9,0)--(9,2) ;
\draw [thick] (9,0)--(11,0) ;
\draw (9.7,0) node[below]{$a$};
\draw (9,0.7) node[left]{$a$};
\draw (9.7,0.7) node[above right]{$\Omega_{N_2(a)}$};
\end{tikzpicture}
\caption{The domains $\Omega$ for the aforementioned domains for $n=2$}
\end{figure}
Within those toric domains, the ball normalization condition reformulates as
\[
c\big(B_n(1)\big)=c\big(Z_n(1)\big)=1.
\]
This normalization stemmed out of Gromov's non-squeezing theorem \cite{Gromov} asserting that there exists a symplectic embedding $B_n(a)\hookrightarrow Z_n(b)$ if and only if $a\leq b$. The first examples of normalized capacities are the
\textbf{Gromov width} $c_B$ and the {\bf cylindrical capacity} $c^Z$ defined for any symplectic manifold $(X, \omega)$.
\begin{IEEEeqnarray*}{rCll}
c_B(X,\omega) & \coloneqq & \sup & \{ a \mid \text{ there exists a symplectic embedding } B_n(a) \longrightarrow X \}, \\
c^Z(X,\omega) & \coloneqq & \inf & \{ a \mid \text{ there exists a symplectic embedding } X \longrightarrow Z_n(a) \},
\end{IEEEeqnarray*}
Additional examples of normalized symplectic capacities are the Hofer-Zehnder capacity $c_{\textrm{HZ}}$ defined in \cite{HZ} and the Viterbo capacity $c_{\textrm{SH}}$ defined in \cite{V}. There are also useful families of symplectic capacities parametrized by a positive integer $k$ including the Ekeland-Hofer capacities $c_k^{\textrm{EH}}$ defined in \cite{EH,EH2} using calculus of variations; the ``equivariant capacities'' $c_k^{\textrm{GH}}$ defined in \cite{GH} using positive equivariant symplectic homology; and in the four-dimensional case, the ECH capacities $c_k^{\textrm{ECH}}$ defined in \cite{qech} using embedded contact homology. For each of these families, the $k=1$ capacities $c_1^{\textrm{EH}}$, $c_1^{\textrm{CH}}$, and $c_1^{\textrm{ECH}}$ are normalized. For more about symplectic capacities in general we refer to \cite{chls, schlenk} and the references therein.
We now introduce a new normalization based on a ``non-squeezing theorem'' for the cube.
\begin{theorem}[{\cite[Proposition 1.20]{GH}}]
\label{thm:gh cube capacity}
There exists a symplectic embedding $P_n(a)\hookrightarrow N_n(b)$ if and only if $a\leq b$.
\end{theorem}
This theorem, together with the previous discussion, motivates the following definition.
\begin{definition}
We say that a symplectic capacity $c$ is \textbf{cube normalized} if
\begin{IEEEeqnarray*}{c+x*}
c(P_n(1)) = c(N_n(1)) = 1.
\end{IEEEeqnarray*}
\end{definition}
We now wish to present examples of cube normalized symplectic capacities.
The first example is the {\bf cube capacity} $c_P$ \cite{GH}
\[
c_P(X,\omega) := \sup\{ a \mid \text{ there exists a symplectic embedding } P_n(a) \longrightarrow X \},
\]
A second example is the {\bf NDUC capacity} $c^N$
\[
c^N(X,\omega) := \inf\{ a \mid \text{ there exists a symplectic embedding } X \longrightarrow N_n(a) \}
\]
The first non immediate example of a cube normalized symplectic capacity was introduced by Cieliebak and Mohnke \cite{CM} and proved to be cube normalized by the second author in his PhD \cite{Per}.
Let $(X, \omega)$ be a symplectic manifold and let $L \subset X$ be a Lagrangian submanifold. The \textbf{minimal area} of $L$ is given by
\begin{IEEEeqnarray*}{c+x*}
A_{\mathrm{min}}(L) \coloneqq \inf
\Big\{ \int_\sigma \omega \ \Big|\ \sigma \in \pi_2(X, L),\, \int_\sigma\omega > 0 \Big\}.
\end{IEEEeqnarray*}
The \textbf{Lagrangian capacity} of $(X,\omega)$ is defined as
\begin{IEEEeqnarray*}{c+x*}
c_L(X,\omega) \coloneqq \sup \{ A_{\mathrm{min}}(L) \mid L \text{ is an embedded Lagrangian torus} \}.
\end{IEEEeqnarray*}
\begin{theorem}[\cite{Per}]
\[
c_L(P_n(1)) = c_L(N_n(1)) = 1.
\]
\end{theorem}
The second author actually proved a stronger result. For any toric domain $X_{\Omega} \subset \C^n$, define its \textbf{diagonal} to be
\begin{IEEEeqnarray*}{c+x*}
\delta_{\Omega} \coloneqq \sup \{ a \mid (a, \ldots, a) \in \Omega \}.
\end{IEEEeqnarray*}
\begin{theorem}[{\cite[Theorem 7.65]{Per}}]
\phantomsection\label{thm:lag cap convex concave}
If $X_\Omega$ is a convex or concave toric domain the
\begin{IEEEeqnarray*}{c+x*}
c_L(X_{\Omega}) = \delta_\Omega.
\end{IEEEeqnarray*}
\end{theorem}
\begin{remark}
The proof of \cref{thm:lag cap convex concave} uses linearized contact homology, and this result is stated under some assumptions about this theory. For a more detailed discussion on these assumptions see \cite[Disclaimer 1.11]{Sie} and \cite[Section 7.1]{Per}.
\end{remark}
\begin{remark}
\label{exa:other are cube normalized}
The proof of \cref{thm:lag cap convex concave} uses other symplectic capacities, namely
\begin{enumerate}
\item the \textbf{Gutt--Hutchings capacities} from \cite{GH}, denoted by $c^{\mathrm{GH}}_k$;
\item the \textbf{higher symplectic capacities} from \cite{Sie}, denoted by $\mathfrak{g}_k^{\leq 1}$;
\item the \textbf{McDuff--Siegel capacities} from \cite{MS}, denoted by $\tilde{\mathfrak{g}}_k^{\leq 1}$.
\end{enumerate}
Inspecting the proof of this theorem, one sees that the proof extends word for word for any monotone toric domain, and that moreover
\begin{IEEEeqnarray*}{c+x*}
c_L(X_{\Omega})
= \lim_{k \to +\infty} \frac{\tilde{\mathfrak{g}}_k^{\leq 1}(X_{\Omega})}{k}
= \lim_{k \to +\infty} \frac{{\mathfrak{g}}_k^{\leq 1}(X_{\Omega})}{k}
= \lim_{k \to +\infty} \frac{c^{\mathrm{GH}}_k(X_{\Omega})}{k}
= \delta_{\Omega}
\end{IEEEeqnarray*}
for any monotone toric domain $X_{\Omega}$.
\end{remark}
One can therefore define cube normalized symplectic capacities as follows.
\begin{definition}
For a nondegenerate Liouville domain $(X,\lambda)$, let
\begin{IEEEeqnarray*}{rCls+x*}
c^{\mathrm{GH}}_{\inf}(X) & \coloneqq & \liminf_{k} \frac{c_k^{\mathrm{GH}}(X)}{k}, \\
\mathfrak{g}_{\inf}^{\leq 1}(X) & \coloneqq & \liminf_{k} \frac{\mathfrak{g}_{k}^{\leq 1}(X)}{k}, \\
\tilde{\mathfrak{g}}_{\inf}^{\leq 1}(X) & \coloneqq & \liminf_{k} \frac{\tilde{\mathfrak{g}}_{k}^{\leq 1}(X)}{k}.
\end{IEEEeqnarray*}
By Remark \ref{exa:other are cube normalized} the symplectic capacities $c^{\mathrm{GH}}_{\inf}$, $\mathfrak{g}_{\inf}^{\leq 1}$ and $\tilde{\mathfrak{g}}_{\inf}^{\leq 1}$ are cube normalized.
\end{definition}
Using the main result of \cite{GR} asserting that for all $k\geq1$ $c_k^{\mathrm{GH}}=c_k^{\mathrm{EH}}$, we have another cube normalized symplectic capacity
\[
c^{\mathrm{EH}}_{\inf}(X) := \liminf_{k} \frac{c_k^{\mathrm{EH}}(X)}{k}.
\]
Note that the main result of \cite{GR} together with Remark \ref{exa:other are cube normalized} shows that for any monotone toric domain $X_\Omega$
\[
c_L(X_\Omega)=\lim_{k \to +\infty} \frac{c^{\mathrm{EH}}_k(X_{\Omega})}{k}.
\]
This answers (for the monotone toric case) a Question by Cieliebak-Mohnke \cite{CM} who asks whether this equality holds for all convex domains in $\R^{2n}$.
The following theorem, which is an analogue of Viterbo's strong conjecture is our main result:
\begin{theorem}
\label{thm:cube normalized}
All cube normalized capacities coincide on monotone toric domains in $\R^{2n}$.
\end{theorem}
\begin{proof}
Let $c$ be a cube normalized symplectic capacity and let $X_{\Omega}$ be a monotone toric domain in $\R^{2n}$. We are going to show that then the value of $c(X_\Omega)$ is determined.
The monotonicity of $X_{\Omega}$ ensures that
\[P_n(\delta_{\Omega}) \subset X_{\Omega} \subset N_n(\delta_{\Omega}).\] Then,
\begin{IEEEeqnarray*}{rCls+x*}
\delta_{\Omega}
& = & c(P_n(\delta_{\Omega})) & \quad [\text{since $c$ is cube normalized}] \\
& \leq & c(X_{\Omega}) & \quad [\text{by monotonicity}] \\
& \leq & c(N_n(\delta_{\Omega})) & \quad [\text{by monotonicity}] \\
& = & \delta_{\Omega} & \quad [\text{since $c$ is cube normalized}]. & \qedhere
\end{IEEEeqnarray*}
\end{proof}
As a corollary of \cref{thm:cube normalized}, we have the following formula for the value of cube normalized symplectic capacities on monotone toric domains.
\begin{theorem}
Let $c$ be a cube normalized symplectic capacity and let $X_{\Omega}$ be a monotone toric domain in $\R^{2n}$. Then
\[
c(X_{\Omega}) = \delta_{\Omega}.
\]
\end{theorem}
In view of \cref{thm:cube normalized}, it is reasonable to conjecture the following:
\begin{conjecture}
All cube normalized capacities coincide on convex domains in $\C^n$.
\end{conjecture}
We wish now to make a few comments on what precedes:
\begin{remark}
The link between monotone toric and convex is studied intensively and is, at the moment, unclear. All monotone toric domains are dynamically convex\footnote{Convexity is not a symplectically invariant property. This was already pointed out a long time ago but only a few symplectic substitutions have been suggested. The most prominent one is \textbf{dynamical convexity}, introduced in \cite{HWZ2}, where they show that strict convexity guarantees dynamical convexity.} toric domains; however the converse is only true in $\R^4$. Examples of monotone toric domains not symplectomorphic to a convex domain where produced recently \cite{DGZ, CE}.
\end{remark}
\begin{remark}
If $c$ is a cube normalized symplectic capacity, then $c$ is not normalized in the usual sense. Indeed, by \cref{thm:cube normalized}, if $c$ is cube normalized then $c(B_n(1)) = 1/n$ and $c(Z_n(1)) = 1$.
We have the following inequalities (for any 2n-dimensional symplectic manifold $(X,\omega)$):
\[
c_P(X,\omega)\leq c_B(X,\omega)\leq nc_P(X,\omega).
\]
Those inequalities come from the optimal embeddings
\[
B_n(a)\subset P_n(a)\subset B_n(na)
\]
We also have
\[
c^N(X,\omega)\leq c^Z(X,\omega)
\]
coming from the inclusion $Z_n(a)\subset N_n(a)$.
\begin{conjecture}
\[c^Z(X,\omega)\leq nc^N(X,\omega).\]
\end{conjecture}
The conjecture is true for $n=2$. This is the main technical point of \cite{GHR}. This amounts to prove that there exists a symplectic embedding
\[
N_n(a)\hookrightarrow Z_n(na).
\]
\end{remark}
\begin{remark}
The minimal area of a Lagrangian torus, $A_{\mathrm{min}}(L)$, is not continuous in $L$. Indeed on a toric domain $X_\Omega$, $\mu^{-1}(x)$ is a Lagrangian torus for $x=(x_1,\ldots,x_n)\in(\mathrm{int}\Omega\cup\partial_+\Omega)$.
By \cref{lem:a min with exact symplectic manifold},
\begin{equation}\label{eq:Amin}
A_{\mathrm{min}}\big(\mu^{-1}(x)\big)=\inf\{k_1x_1+\cdots+k_nx_n\,|\,k_1,\ldots k_n\in\Z\}.
\end{equation}
\end{remark}
\section{Computing of the Lagrangian capacity for a more general family of toric domains}
In this section, we will see how one can use \cref{thm:lag cap convex concave} to compute the Lagrangian capacity for a larger class of toric domains which are not necessarily monotone (see \cref{thm:lag cap any toric} below). For a toric domain $X_{\Omega}$, define
\begin{IEEEeqnarray*}{c+x*}
\eta_{\Omega} \coloneqq \inf \{ a \mid X_{\Omega} \subset N_n(a) \}.
\end{IEEEeqnarray*}
Notice that if $X_{\Omega}$ is convex or concave, then $\delta_{\Omega} = \eta_{\Omega}$. To prove \cref{thm:lag cap any toric}, we will make use of the following lemma:
\begin{lemma}[{\cite[Lemma 6.16]{Per}}]
\label{lem:a min with exact symplectic manifold}
Let $(X,\lambda)$ be an exact symplectic manifold and $L \subset X$ be a Lagrangian submanifold. If $\pi_1(X) = 0$, then
\begin{IEEEeqnarray*}{c+x*}
A _{\mathrm{min}}(L) = \inf \left\{ \lambda(\rho) \ | \ \rho \in \pi_1(L), \ \lambda(\rho) > 0 \right\}.
\end{IEEEeqnarray*}
\end{lemma}
\begin{proof}
The diagram
\begin{IEEEeqnarray*}{c+x*}
\begin{tikzcd}
\pi_2(X,L) \ar[dr, swap, "\omega"] \ar[r, two heads,"\partial"] & \pi_1(L) \ar[d, "\lambda"] \ar[r, "0"] & \pi_1(X) \\
& \R
\end{tikzcd}
\end{IEEEeqnarray*}
commutes, where $\partial([\sigma]) = [\sigma|_{S^1}]$, and the top row is exact.
\end{proof}
\begin{theorem}
\label{thm:lag cap any toric}
Let $X_{\Omega}$ be a toric domain. If $(\eta_{\Omega},\ldots,\eta_{\Omega}) \in \partial \Omega$ then%
\begin{IEEEeqnarray*}{c+x*}
c_L(X_{\Omega}) = \eta_{\Omega}.
\end{IEEEeqnarray*}
\end{theorem}
\begin{proof}
By definition of $\eta_{\Omega}$, we have $X_{\Omega} \subset N_n(\eta_{\Omega})$. Define $T \coloneqq \mu^{-1}(\eta_{\Omega},\ldots,\eta_{\Omega})$. Then $T$ is an embedded Lagrangian torus in $X_{\Omega}$ (see \cref{fig:main} for an illustration of $\eta_{\Omega}$, $T$, $\Omega$ and $\Omega_{N_n(\eta_{\Omega})}$). Therefore,
\begin{IEEEeqnarray*}{rCls+x*}
\eta_{\Omega}
& = & A_{\mathrm{min}}(T) & \quad [\text{by \cref{lem:a min with exact symplectic manifold}}] \\
& \leq & c_L(X_{\Omega}) & \quad [\text{by definition of $c_L$}] \\
& \leq & c_L(N_n(\eta_{\Omega})) & \quad [\text{by monotonicity}] \\
& \leq & \eta_{\Omega} & \quad [\text{by \cref{thm:lag cap convex concave}}]. & \qedhere
\end{IEEEeqnarray*}
\end{proof}
Note that \cref{thm:lag cap any toric} extends mutatis mutandis, using \cref{eq:Amin}, to the following
\begin{theorem}\label{thm:generalformulacL}
Let $X_\Omega\subset N_n(\eta_{\Omega})$ be a toric domain in $\R^{2n}$ such that there exist a point $x\in\overline{\partial_+\Omega}\cap \partial_+N_n(\eta_{\Omega})$ of the form $x=(k_1\eta_{\Omega},\ldots,k_n\eta_{\Omega})$ where the $k_i\in\N$ (see \cref{fig:main}).
Then,
\[
c_L(X_{\Omega}) = \eta_{\Omega}.
\]
\end{theorem}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\draw[->,color=black] (-0.5,0) -- (7,0);
\draw[->,color=black] (0,-0.5) -- (0,5);
\draw[-,color=blue, thick] (2,2) -- (2,4.5);
\draw[-,color=blue, thick] (2,2) -- (6.5,2);
\draw[dotted,color=black] (2,2) -- (0,2);
\draw[dotted,color=black] (2,2) -- (2,0);
\draw[color=black] (2,0) node[below] {\footnotesize{$\eta_\Omega$}};
\draw[color=black] (0,2) node[left] {\footnotesize{$\eta_\Omega$}};
\draw [black, thick] plot [smooth, tension=1] coordinates { (0,3) (1,4) (2,1.5) (4.1,1.96) (3.5,0.5) (4.5,0.4) (5.5,0.4) (6,0)};
\draw[color=black] (1,1) node {\footnotesize{$\Omega$}};
\draw[color=blue] (2,3) node[right] {\footnotesize{$\Omega_{N_n(\eta_\Omega)}$}};
\draw[dotted,color=black] (4,2) -- (4,0);
\draw[color=black] (4,0) node[below] {\footnotesize{$2\eta_\Omega$}};
\node [red] at (4,2) {\footnotesize{\textbullet}};
\draw[color=red] (4,2) node[above] {\footnotesize{$\Omega_T$}};
\end{tikzpicture}
\caption{Example of $X_{\Omega}$ satisfying the assumption in \cref{thm:generalformulacL}}
\label{fig:main}
\end{figure}
\section{An interesting nonexample}
We now study a family of examples coming from \cite{GHR} of non-monotone toric domains, and we determine when they satisfy the conclusion of \cref{thm:cube normalized}.
For $0<a<1/2$, let $\Omega_a$ be the convex polygon with corners $(0,0)$, $(1-2a,0)$, $(1-a,a)$, $(a,1-a)$ and $(0,1-2a)$, and write $X_a=X_{\Omega_a}$; see \cref{fig:example}. Then $X_a$ is a weakly convex (but not monotone) toric domain.
\begin{figure}[ht]
\centering\label{fig:example}
\begin{tikzpicture}[scale=2]
\fill[orange!50](0,0)--(0,1.8)--(0.2,2)--(2,0.2)--(1.8,0)--(0,0);
\draw[orange!50, dashed] (0,2.2)--(0.2,2);
\draw[orange!50, dashed] (2,0.2)--(2.2,0);
\draw [thick](0,0)--(0,2.3) ;
\draw [thick] (0,0)--(2.3,0) ;
\coordinate (A) at (1.6,0);
\draw (A) node[below] {{$1-2a$}};
\coordinate (B) at (2.2,0);
\draw (B) node[below] {$1$};
\filldraw (1.8,0) circle (1pt);
\filldraw (2.2,0) circle (1pt);
\filldraw (0,2.2) circle (1pt);
\coordinate (C) at (0,1.8);
\draw (C) node[left] {{$1-2a$}};
\coordinate (D) at (0,2.2);
\draw (D) node[left] {$1$};
\filldraw (0,1.8) circle (1pt);
\end{tikzpicture}
\caption{The domain $\Omega_a$}
\end{figure}
\begin{proposition}\label{prop:cpnXa}
The cubic, Lagrangian and NDUC capacities of $X_a$ are given as follows.
\[\begin{aligned}c_P(X_a)&=\min\left(1-2a,\frac{1}{2}\right),\\
c_L(X_a)=c^N(X_a)&=\frac{1}{2}.\end{aligned}
\]
\end{proposition}
\begin{remark}
It follows from Proposition \ref{prop:cpnXa} that $c_P(X_a)\neq c^N(X_a)$ for $a> 1/4$. But in \cite{GHR} it was shown that $c_B(X_a)=c^Z(X_a)$ for all $a\le 1/3$. So for $1/4<a\le 1/3$, the Gromov and cylindrical capacities of $X_a$ coincide, but not the cubic and NDUC capacities.
\end{remark}
\begin{proof}
We note that $\eta_{\Omega_a}=1/2$ for all $a\le 1/2$ and that $(1/2,1/2)\in\Omega_a$. So it follows from Theorem \ref{thm:lag cap any toric} that $c_L(X_a)=1/2$. Since $X_a\subset N_2(a)$, it follows that
\[\frac{1}{2}=c_L(X_a)\le c^N(X_a)\le \frac{1}{2}.\]
So $c^N(X_a)=1/2$.
To compute the cubic capacities, we first observe that \[\begin{aligned}P_n\left(\frac{1}{2}\right)\subset X_a,&\text{ for }&0<a\le 1/4,\\ P_n(1-2a)\subset X_a,&\text{ for }&1/4 \le a <1/2.\end{aligned}\]
So $c_P(X_a)\ge \min(1-2a,1/2)$. Since $c_P(X_a)\le c^N(X_a)=1/2$, it follows that $c_P(X_a)=1/2=\min(1-2a,1/2)$ for $0<a\le 1/4$.
The fact that $c_P(X_a)\le 1-2a$ for $1/4<a<1/2$ follows from Theorem~\ref{thm:cubewc} below.
\end{proof}
\section{The cubic capacity of some weakly convex toric domains}
In this section we obtain an upper bound for the cubic capacity of some non-monotone toric domains, which will not in general coincide with their NDUC capacity.
A four-dimensional toric domain $X_\Omega$ is said to be weakly convex\footnote{Cristofaro-Gardiner defined this to be a convex toric domain in \cite{concaveconvex}, but usually a convex toric domain is defined to be a particular case of this, see \cite{GHR}, for example.} if $\Omega\subset\R^2_{\ge 0}$ is convex and $\partial_+\Omega$ is a piecewise smooth curve connecting the two coordinate axes, see Figure \ref{fig:wtc}. With an extra assumption, we can compute an upper bound for the cubic capacity of $X_\Omega$.
\begin{figure}\label{fig:wtc}
\centering
\input{wconvextoric.pdf_tex}
\caption{A weakly convex toric domain $X_\Omega$}
\end{figure}
\begin{theorem}\label{thm:cubewc}
Let $X_\Omega$ be a weakly convex toric domain, where $\partial_+\Omega$ is parametrized by the curve $(x,y):[0,1]\to\R^2_{\ge 0}$ such that $y(0)=0$ and $x(1)=0$. Suppose that \[\max\left(\frac{x'(0)}{y'(0)},\frac{y'(1)}{x'(1)}\right)\le 1.\]
Then \[c_P(X_\Omega)\le \frac{x(0)+y(1)}{2}.\]
\end{theorem}
The proof of Theorem \ref{thm:cubewc} uses embedded contact homology. Namely, we need a version of \cite[Theorem 1.20]{beyond} for weakly convex toric domains. We now explain the context and the modifications that need to be made in the proof of \cite[Theorem 1.20]{beyond} for our purposes here.
We need some definitions to state a more general version of \cite[Theorem 1.20]{beyond}. Let $X_\Omega$ be a weakly convex toric domain. We define a combinatorial Reeb orbit to be a pair $(v,s)$, where $v=(x_v,y_v)$ is a primitive vector in $\Z^2$ and $s=\{0,1\}$ such that $x_v\ge 0$ or $y_v\ge 0$. A combinatorial orbit set is a finite formal product
\[\alpha=\prod_{i=1}^k (v_i,s_i)^{m_i},\]
where $(v_i,s_i)$ are distinct combinatorial Reeb orbits and $m_i\in\Z_{\ge 1}$ such that $m_i=1$ whenever $s_i=0$. We define the following numbers.
\begin{align}
x(\alpha)&=\sum_{i=1}^k m_ix_{v_i},\label{eq:x}\\
y(\alpha)&=\sum_{i=1}^k m_iy_{v_i},\label{eq:y}\\
I(\alpha)&=x(\alpha)+y(\alpha)+\sum_{i,j=1}^k m_i m_j \max(x_{v_i} y_{v_j},x_{v_j} y_{v_i})+\sum_{i=1}^k s_i m_i,\label{eq:i}\\
m(\alpha)&=\sum_{i=1}^k m_i,\label{eq:m}\\
h(\alpha)&=\sum_{i=1}^k (1-s_i).\label{eq:h}
\end{align}
We note that none of those numbers depend on $\Omega$. The number $I(\alpha)$ is called the combinatorial ECH index of $\alpha$. We define the combinatorial action of $\alpha$ to be
\begin{equation*}
A_\Omega (\alpha)=\sum_{i=1}^k m_i \max\{v_i\cdot p\mid p\in\partial_+\Omega\}.
\end{equation*}
We now state a version of \cite[Definition 1.18]{beyond} for weakly convex toric domains.
\begin{definition}\label{def:beyond}
Let $X_{\Omega}$ and $X_{\Omega'}$ be weakly convex toric domains and let $\alpha$ and $\alpha'$ be combinatorial orbit sets. We write $\alpha\le_{\Omega,\Omega'} \alpha'$ if the following conditions hold:
\begin{itemize}
\item[(i)] $I(\alpha)=I(\alpha')$,
\item[(ii)] $A_\Omega(\alpha)\le A_{\Omega'}(\alpha')$,
\item[(iii)] $x(\alpha)+y(\alpha)-h(\alpha)/2\ge x(\alpha')+y(\alpha')+m(\alpha')-1$.
\end{itemize}
\end{definition}
The version of \cite[Theorem 1.20]{beyond} that we need is the following result.
\begin{theorem}\label{thm:beyond}
Let $X_\Omega$ and $X_{\Omega'}$ be weakly convex toric domains such that $X_\Omega\hookrightarrow X_{\Omega'}$. Let $\alpha'$ be an orbit set such that $I(\alpha')>0$ and $h(\alpha')=0$. Then there is an orbit set $\alpha$ with $I(\alpha)=I(\alpha')$ and product decompositions \[\alpha=\prod_{j=1}^l \alpha_j,\quad\alpha'=\prod_{j=1}^l \alpha_j',\] such that:
\begin{itemize}
\item[(a)] $\alpha_j\le_{\Omega,\Omega'} \alpha_j'$,
\item[(b)] Given $i,j$, if $\alpha_i=\alpha_j$ or $\alpha_i'=\alpha_j'$, then $\alpha_i$ and $\alpha_j$ have no combinatorial Reeb orbits in common with $s=1$.
\item[(c)] For any $\emptyset\neq S\subset\{1,\dots,l\}$, \[I\left(\prod_{j\in S} \alpha_j\right)=I\left(\prod_{j\in S} \alpha_j'\right)>0.\]
\end{itemize}
\end{theorem}
\begin{proof}
The proof is essentially the same as the one of \cite[Theorem 1.20]{beyond}. As in the proof of \cite[Theorem 5.6]{GHR}, we first approximate $\Omega$ by a domain $\widetilde{\Omega}\subset \Omega$ such that $\partial_+\widetilde{\Omega}$ is a smooth curve and the slopes of the tangent lines at the intersections with the $x$-axis and $y$-axis are $\varepsilon$ and $\varepsilon^{-1}$. We observe that for a given orbit set $\alpha$ and $\delta>0$, we can define $\widetilde{\Omega}$ so that $|A_{\Omega}(\alpha)-A_{\widetilde{\Omega}}(\alpha)|<\delta$. We define $\widetilde{\Omega}'\supset \Omega'$ satisfying the same properties as above, c.f. \cite[Lemma 5.4]{beyond}. In particular $X_{\widetilde{\Omega}}\hookrightarrow X_{\widetilde{\Omega}'}$.
We now briefly recall the embedded contact homology (ECH) chain complex. Let $(x,y):[0,1]\to\R^2$ be a parametrization of $\partial_+\widetilde{\Omega}$ such that $y(0)=x(1)=0$. So $y'(0)/x'(0)=x'(1)/y'(1)=\varepsilon$. We assume that $\varepsilon$ is a small irrational number and that $(x''(t),y''(t))\neq 0$ for $t\in[0,1]$. Then the standard Liouville form on $\R^4$ restricts to a contact form $\lambda_0$ on $\partial X_\Omega$ whose Reeb flow foliates $\mu^{-1}((x(t),y(t))$ for each $t\in[0,1]$. Then for each $t\in]0,1[$ such that $x'(t)/y'(t)\in\Q\cup\{\infty\}$, there is a unique $(p,q)\in\Z^2$ such that $p$ and $q$ are relatively prime and
\[(x'(t),y'(t))=c\cdot(p,q), \quad\text{for }c>0.\] So
the torus $T_{p,q}:=\mu^{-1}((x(t),y(t))$ is foliated by closed Reeb orbits. Note that $T_{(p,q)}$ is uniquely determined by $(p,q)$ since $X_\Omega$ is weakly convex. For a Reeb orbit $\gamma\in T_{p,q}$, its symplectic action is defined by
\[A_{\widetilde{\Omega}}(\gamma):=\int_\gamma \lambda_0.\]
It is straight-forward to check that this action doesn't depend on $\gamma$. Indeed for every $\gamma\in T_{p,q}$, it follows from a simple calculation that
\[A_{\widetilde{\Omega}}(\gamma)=\max\{(p,q)\cdot x\mid x\in\partial_+ \widetilde{\Omega}\}=A_{\widetilde{\Omega}}((p,q),1).\] The only other Reeb orbits of $\lambda_0$ are the two circles $\mu^{-1}((x(0),y(0))$ and $\mu^{-1}((x(1),y(1))$. One can check that $\lambda_0$ is Morse--Bott. Given $L>0$, we can perturb the contact form in neighborhoods of the tori $T_{p,q}$ for which $A_{\widetilde{\Omega}}(\gamma)<L$ for $\gamma\in T_{p,q}$,
thus obtaining an elliptic and a hyperbolic Reeb orbit, denoted by $e_{(p,q)}$ and $h_{(p,q)}$, respectively. This is explained in more detail in \cite{qech} and \cite{concave}, for example. Let $\widetilde{\lambda}$ denote the pertubed contact form. The only other closed Reeb orbits of $\lambda$ with action less than $L$ are the two circles fibering above $(x(0),0)$ and $(0,y(1))$, which are elliptic. We denote them by $e_0$ and $e_1$.
An orbit set is a finite formal product
$\alpha=\prod_i \alpha_i^{m_i}$, where $\alpha_i$ is a simple Reeb orbit and $m_i$ is positive integer. We always assume that $\alpha_i\neq \alpha_j$ if $i\neq j$ and $m_i=1$ if $\alpha_i$ is hyperbolic. The action of an orbit set is defined by
\[A_{\widetilde{\Omega}}(\alpha)=\sum_i m_i A_{\widetilde{\Omega}}(\alpha_i).\] The filtered ECH chain complex $ECC^L(\partial X_{\widetilde{\Omega}},\widetilde{\lambda})$ is the $\Z/2$ vector space generated by all orbit sets $\alpha$ such that \[A_{\widetilde{\Omega}}(\alpha)<L.\]
Under the indentification $e_{p,q}=((p,q),1)$ and $h_{p,q}=((p,q),1)$, we can see orbit sets as combinatorial orbit sets and their symplectic actions coincide\footnote{To be precise, the symplectic actions with respect to the perturbed contact form is bounded from the combinatorial action by a small constant which can be as small as desired for a given $L$}.
The differential of $ECC^L(\partial X_{\widetilde{\Omega}},\widetilde{\lambda})$ is obtained by counting pseudo-holomorphic curves in $\R\times \partial X_{\widetilde{\Omega}}$ whose ECH index is 1. We will not define the ECH index here. Instead it suffices to recall that in this setting the ECH index gives rise to an absolute index such that for each orbit set $\alpha$, $I(\alpha)$ is simply the combinatorial ECH index defined in \eqref{eq:i}. The fact that the original definition and the combinatorial definition coincide follows from very similar calculations to the one in the proof of \cite[Lemma 5.4]{beyond}, which uses previous calculations from the proof of \cite[Lemma 3.3]{concave}. Here we have a max instead of a min, because of the opposite concavity, as in \cite[Lemma 5.4]{beyond}. It is worth noting that the calculation of the first Chern class \cite[(3.14)]{concave} is almost identical and in our case it gives \begin{equation}\label{eq:ctau}c_\tau(\alpha)=x(\alpha)+y(\alpha)\end{equation} as defined in \eqref{eq:x} and \eqref{eq:y}.
The rest of the argument uses the cobordism map in ECH and the $J_0$-invariant. It is identical to the proof of \cite[Theorem 1.20]{beyond} using \eqref{eq:ctau}, where we note that the original and the combinatorial definitions of $h$ and $m$ coincide.
\end{proof}
We can now prove Theorem \ref{thm:cubewc}.
\begin{proof}[Proof of Theorem {\ref{thm:cubewc}}]
Suppose that $P_2(a)\hookrightarrow X_\Omega$. We can find a weakly convex toric domain $X_{\Omega'}\supset X_\Omega$ such that the tangent lines to the curve $\partial_+ \Omega'$ at the $x$ and $y$ axes have slopes $1-\delta$ and $1+\delta$ for some small $\delta>0$, respectively. For each $L>0$ sufficiently large and $\varepsilon>0$, we can choose $X_{\Omega'}$ so that \begin{equation}\label{eq:ax0y1}|A_{\Omega'}(e_{1,-1})-x(0)|<\varepsilon\quad\text{ and }\quad |A_{\Omega'}(e_{-1,1})-y(1)|<\varepsilon,\end{equation} and that\[|A_{\Omega}(e_{p,q})-A_{\Omega'}(e_{p,q})|<\varepsilon,\]
for all $(p,q)$ such that $A_{\Omega}(e_{p,q})<L$.
Now let $\alpha'=e_{1,-1}^d e_{-1,1}^d e_{1,1}^2.$ It follows from Theorem \ref{thm:beyond} that there exists an orbit set $\alpha$ and factorizations
\[\alpha=\prod_{j=1}^l \alpha_j,\quad\alpha'=\prod_{j=1}^l \alpha_j',\]
satisfying (a), (b) and (c). Using (b) and (c), we conclude that $l\le 3$ and that $\alpha_i=e_{1,-1}^{d_i} e_{-1,1}^{d_i} e_{1,1}^k$ for some $k\in\{0,1,2\}$ such that $d_i\ge d/3$. Using (a), it follows from properties (ii) and (iii) from Definition \ref{def:beyond} that
\begin{equation*}
\begin{aligned}
3k+2d_i-1&=x(\alpha_i')+y(\alpha_i')+m(\alpha_i')-1\le x(\alpha_i)+y(\alpha_i)\\&=\frac{A_{P_2(a)}(\alpha_i)}{a}\le \frac{A_{\Omega'}(\alpha_i)}{a}<\frac{(d_i(x(0)+y(1))+k)(1+\varepsilon)}{a}.
\end{aligned}
\end{equation*}
Hence \[a<\frac{(d_i(x(0)+y(1))+k)(1+\varepsilon)}{2d_i+3k-1}.\]
Taking the limit as $d\to\infty$ and then as $\varepsilon\to 0$, it follows that
\[a\le \frac{x(0)+y(1)}{2}.\]
Therefore
\[c_P(X_\Omega)\le \frac{x(0)+y(1)}{2}.\]
\end{proof}
\bibliographystyle{alpha}
|
2,877,628,090,783 | arxiv | \section{Introduction}
\label{sec:introduction}
The Linked Open Data (LOD) movement implements Tim Berner-Lee's vision of a \emph{Web of Data}.
The principles underlying Linked Data enable data providers to model, interlink and publish their data on the Web in a distributed and decentralised way.
On the consumer's side the Web-oriented technological basis (i.e. using RDF and HTTP) permits to easily make use of the data and integrate it into various applications.
These benefits and advantages of the Linked Data idea lead to, on the one hand, more and more data providers to contribute to the Web of Data and, on the other hand, more and more developers to spawn new applications making use of this data.
However, the growth and conceptual nature of Linked Data poses some technological challenges.
The distributed nature of Linked Data calls for search engines and indices to provide data catalogues of what kind of data is available and where it can be found.
Furthermore, some applications use data caches to avoid communication overhead when accessing data on the Web.
Such indices and caches correspond to local views on the data.
These local views might be inaccurate in the sense that they do not reflect the state of the data at the original data sources.
There are two main reasons for indices constituting an inaccurate view on the data.
The first reason is rooted in the decentralised and dynamic nature of Linked Data.
Changes at the original data sources are not automatically propagated to the applications and their indices and data caches.
Accordingly a task which needs to be addressed in the context of Linked Data applications is the active maintenance of indices.
This means, that applications have to synchronise their local view on the data with the data at the origin~\cite{P:LDOW:2010:UmbrichHHP,P:ESWC:2014:DividinoKG}.
A second reason are applications which build their indices only in an approximate way~\cite{J:JWS:2012:KonrathGSS}.
This happens, for instance, to achieve scalability and to be able to efficiently process large volumes of Linked Data.
Figure~\ref{fig:concept} illustrates these two paths which can lead from a Linked Data set $R_{\textsc{gs}}$ and an older version of this data set $R$ to an index $I$ which is inaccurate compared to a \emph{perfect} index $I_{\textsc{gs}}$.
In both cases, there is a certain tradeoff between index accuracy and the commitment of limited resources.
When updating and maintaining an index, the limited resource is typically network bandwidth.
Spending more effort on updating an index requires more bandwidth but will yield indices of higher accuracy.
Likewise, the approximate computation of indices can usually be influenced by a parameter which trades index accuracy for required computational resources.
Hence, given the limitations of available resources the providers of LOD indices will need to strike a balance between the effort for building and maintaining an index and the quality of service, i.e. the accuracy of their index.
\begin{figure}[btp]
\centering
\includegraphics[width=100mm]{concept2}
\caption{Inaccurate indices $I$ can occur when computed over outdated data $R$ or when computed in an approximative way over current data $R_{\textsc{gs}}$. A measure of accuracy needs to compare such an index $I$ to a perfect, gold standard index $I_{\textsc{gs}}$ computed over $R_{\textsc{gs}}$ in a lossless way.}
\label{fig:concept}
\end{figure}
While measuring the costs for computation and network bandwidth has been addressed in many other fields, measures for the accuracy of a Linked Data index are less established.
In related work, different methods for measuring index accuracy have been described and applied.
But, rarely more than one approach is used.
This makes it difficult to compare results.
Thus, the idea of this paper is to compare different measures for index accuracy on a theoretical and practical level.
This should help in judging and comparing existing results as well as to support the choice of measures to use in future research work in this field.
In more detail, this paper will make three contributions:
\begin{description}
\item[Survey of Measures:] Following a review of related work, this paper provides an extensive overview of approaches and methods used to measure the accuracy of index structures over Linked Data.
All methods are presented in a unified and formal way for ease of comparison.
\item[Theoretical Comparison:] Considering the general task of RDF indices and the specific Web based setting of Linked Data, this paper compares approaches and methods to measure the accuracy of LOD indices w.r.t their theoretical limitations and advantages.
\item[Practical Comparison:] By using an established data set of evolving Linked Data the paper analyses the approaches and methods to measure the accuracy of LOD indices w.r.t their behaviour in practice.
In particular it presents an analysis of the correlation of the measures to understand how far similar or different notions of index accuracy are captured by the discussed measures.
\end{description}
As a next step we look at related work in Section~\ref{sec:related_work} to get an overview of the context in which LOD indices are used and evaluated for their accuracy.
This section will also provide a first and brief review of the methods used to measure index accuracy.
Subsequently, Section~\ref{sec:index} will present an abstract and formal representation for Linked Data indices.
This formalisation allows to abstract from concrete implementations and serves as basis for the unified definition of index accuracy measures in Section~\ref{sec:measures}.
The theoretic analysis and comparison of the measures is presented in Section~\ref{sec:theory} and the practical and empirical comparison in Section~\ref{sec:empirical}.
The findings are discussed in~\ref{sec:discussion} and the paper is concluded with a summary and an outlook at future work.
\section{Related Work}
\label{sec:related_work}
In recent years, various index models over LOD have been proposed.
Many of them focus on specific aspects of the data or are dedicated to support application specific tasks.
When looking at the RDF basis of LOD, one has to consider also the works on indices for RDF triple stores, such as Hexastore~\cite{J:VLDB:2008:WeissKB} or RDF3X~\cite{J:VLDB:2010:NeumannW}.
These indices are intended for optimising access to a single, centrally managed data storage solution.
In this case, accuracy of the index is not an issue as all changes in the data are under control of the storage solution and are reflected in the index immediately.
More specific to LOD are indices for optimising federated queries~\cite{P:SIGMOD:2009:NeumannW}, on-demand queries on the Web~\cite{P:WWW:2010:HarthHKP} or looking up data sources relevant to particular schema patterns~\cite{J:JWS:2012:KonrathGSS}.
However, most of these approaches do not deal with index accuracy either.
Their focus is more on how to implement or make use of the index in specific scenarios.
So far, only few publications address the issue of Linked Data index accuracy.
This only happens in a context where the dynamics of data is explicitly addressed~\cite{P:ESWC:2013:KaeferAUO,P:COLD:2013:DividinoSGG,P:PROFILES:2014:DividinoGSG}, where implementations trade a loss of index accuracy for an efficient and scalable index computation~\cite{J:JWS:2012:KonrathGSS,P:CSWS:2012:GottronP} or where the accuracy of indices is under investigation itself~\cite{P:ESWC:2014:GottronG,P:PROFILES:2014:Gottron,P:CSWS:2012:GottronP}.
These methods used will be reviewed and explained in more detail in Section~\ref{sec:measures}.
In the context of the ``classical'' Web of documents, there is some work on index accuracy of search engines~\cite{J:Nature:1999:LawrenceG}.
Such analysis is performed by comparing the coverage of relevant web documents for specific queries.
The results of several individual search engines can be compared to the union of their results as well as to a separately obtained collection of relevant documents which serves as ground truth.
Other works attempt to measure age and freshness of an index~\cite{P:SGIMOD:2000:ChoG}.
Incorporating time information into a change prediction allows for more efficient and effective synchronisation plans between Web based data sources.
Yet another direction of research addressed the question of guarantees of freshness of cached copies for web documents~\cite{J:CN:2000:BrewingtonC}.
However, measuring the freshness and accuracy of search indices for Web documents is different from the accuracy of Linked Data indices insofar as the indices are all of the same type, namely mapping keywords to documents.
In the context of Linked Data indices can be of different types and address specific information needs which are directly encoded in the index structure~\cite{P:ESWC:2014:GottronG}.
\section{Abstract Index Models for Linked Data}
\label{sec:index}
As indicated in the previous section, there is a wide range of different index models over LOD.
In this paper we will not look at specific implementations of indices, but rather operate on an abstract level.
Therefore, we now briefly recall a formalisation of abstract index models over Linked Data~\cite{P:ESWC:2014:GottronG}.
This formalisation will serve as basis for a generic and unified definition of accuracy measures in Section~\ref{sec:measures}.
On the LOD cloud we can assume data items to be in the form of NQuads~\cite{w3cnquads}.
In an NQuad $(s,p,o,c)$ the entries $s$, $p$, and $o$ correspond to the subject, predicate and object of the RDF triple statement.
The entry $c$ provides the context, i.e. the data source on the Web where this information has been published.
Thus, we define an index model for LOD over a data set $R$ of $(s,p,o,c)$ NQuads.
Depending on the application scenario an index will typically not serve to store all information contained in the NQuads.
Rather it will define a derived set $D$ of managed data items which are of interest in the scenario and typically constitute a restriction of the quads to smaller tuples.
Such restrictions can be, for instance, the RDF triples or even single entries, e.g. the subject or the context URIs.
Furthermore, an index model has to define a set $\mathcal{K}$ of key elements which are used to lookup and retrieve data items.
These key elements are used as domain for a selection function $\sigma : \mathcal{K} \rightarrow \mathcal{P}(D)$ to select a subset of the data items in the index.
Eventually, an abstract index model is defined as a tuple $(D,\mathcal{K},\sigma)$ of the stored data items $D$, the key elements $\mathcal{K}$ used for the lookup index and the selection function $\sigma$ to retrieve data from the index.
Figure~\ref{fig:abstractindex} illustrates the elements of an abstract index model $I$ computed over a data set $R$.
\begin{figure}[btp]
\centering
\includegraphics[width=70mm]{abstract-index}
\caption{A data set $R$ of NQuads and the elements of an index model over this data.}
\label{fig:abstractindex}
\end{figure}
\section{Measuring Accuracy}
\label{sec:measures}
The formal definition of abstract index models in Section~\ref{sec:index} enables us to now formalise measures for index accuracy in a unified way.
As already indicated in Figure~\ref{fig:concept}, measuring the accuracy of an index will be based on a \emph{perfect} index which is entirely accurate.
In the following, this \emph{gold standard} index will be referred to as $I_\textsc{gs} = (D_\textsc{gs},\mathcal{K}_\textsc{gs},\sigma_\textsc{gs})$ which is built over a data set $R_\textsc{gs}$.
The potentially inaccurate index for which we want to determine its accuracy will be referred to as $I = (D,\mathcal{K},\sigma)$.
Note, that it is not necessary to distinguish between $I$ being inaccurate because it was built over an outdated data set $R$ or because it was computed in an approximative way over the gold standard data set $R_\textsc{gs}$.
The presented measures are divided into four families, based on their methods and underlying ideas for measuring accuracy: (1) index agnostic measures, (2) measures based on the overlap of key elements, (3) distribution based measures and (4) retrieval based measures.
\subsection{Index Agnostic Measures}
\label{subsec:agnostic}
\begin{figure}[btp]
\centering
\includegraphics[width=70mm]{mIndexAgnostic}
\caption{Index agnostic accuracy measures consider only the original data set.}
\label{fig:mAgnostic}
\end{figure}
A direct way to measure the potential impact of data changes on indices is to simply compare the differences in the data itself.
As indicated in Figure~\ref{fig:mAgnostic} such an accuracy measure would accordingly ignore the index and operate only on the original input data.
One established metric for comparing the data sets is the \emph{Jaccard similarity}.
Applying it to $R_\textsc{gs}$ and $R$, it is defined as:
\begin{equation}
\textit{Jaccard}(R_\textsc{gs},R) = \frac{|R_\textsc{gs} \cap R|}{|R_\textsc{gs} \cup R|}
\end{equation}
A higher similarity value indicates a larger overlap between the data sets while a low value indicates a stronger deviation, i.e. change in the data.
Such an index agnostic measures can always be computed over two versions of a data set at different points in time.
Therefore, it is independent of the index model applied.
While this might be suitable to measure the evolution of the data in general, it does not make any statement about the impact on specific index models used in specific use case scenarios.
Furthermore, it requires the availability of the full raw data sets and is by design not applicable in settings were indices are computed in an approximative way over the gold standard data set $R_\textsc{gs}$.
\subsection{Overlap of Key Elements}
\label{subsec:overlap}
A relatively simple approach for comparing indices themselves is to look at the key elements (cf. Figure~\ref{fig:mKeys}).
These are the elements used to retrieve information from the index and, thus, provide the primary access point to the contained data.
Hence, the degree of how far an index reflects the right key elements can already give some insights into its accuracy.
Note, that this type of measure will not consider at all the data elements.
In particular, it might assert an index a perfect accuracy even if the index provides wrong results.
\begin{figure}[btp]
\centering
\includegraphics[width=70mm]{mKeySet}
\caption{Accuracy measures based on the key elements do not make use of any information about the data items.}
\label{fig:mKeys}
\end{figure}
Technically, this approach comes down to comparing two sets: $\mathcal{K}_\textsc{gs}$ and $\mathcal{K}$.
There are various measures for comparing sets.
In the context of measuring the accuracy of LOD indices, also here the \emph{Jaccard similarity} has been used~\cite{P:ESWC:2014:GottronG}:
\begin{equation}
\textit{Jaccard}(\mathcal{K}_\textsc{gs},\mathcal{K}) = \frac{|\mathcal{K}_\textsc{gs} \cap \mathcal{K}|}{|\mathcal{K}_\textsc{gs} \cup \mathcal{K}|}
\end{equation}
Again, a higher similarity value indicates a larger overlap between the sets of key elements while a low value indicates a stronger deviation.
In this way we can get an impression of how stable is the set of elements used for indexing in the different indexing approaches.
A further measures to operate on the set of key elements is the asymmetric \emph{recall}, which measures which fraction of the gold standard key set is covered in an inaccurate index.
\begin{equation}
r(\mathcal{K}_\textsc{gs},\mathcal{K}) = \frac{|\mathcal{K}_\textsc{gs} \cap \mathcal{K}|}{|\mathcal{K}_\textsc{gs}|}
\end{equation}
The rational for using this variation is that key elements in an inaccurate index which are not available in the gold standard might not be used to retrieve information as well. However, this rational is debatable as a query with one of this key elements missing in the gold standard will lead to retrieving false positive information from the inaccurate index, a fact which is neglected by recall.
\tg{How to implement: SPARQL ask queries? }
\subsection{Distribution Based Measures}
\label{subsec:distribution}
One option to include some more information in the evaluation of index accuracy is pursued by distribution based approaches~\cite{P:ESWC:2014:GottronG,P:PROFILES:2014:Gottron}.
The advantage is that these approaches take into consideration the volume of data items retrieved for a given key element.
This idea is visualised in Figure~\ref{fig:mDistrib} by omitting the concrete data items assigned to a key element.
The volume of the data is used to model probabilistic distributions for which there are several well established and well understood metrics for comparison.
However, they do not distinguish the data items but merely use their count information.
Hence, also here there is a certain risk of asserting a perfect index accuracy while the index actually is not accurate.
\begin{figure}[btp]
\centering
\includegraphics[width=70mm]{mDistrib}
\caption{Distribution based measures compare distributions of the data over key elements.}
\label{fig:mDistrib}
\end{figure}
Estimating a distribution over an index is based on determining how probable it is for an element to belong to one specific index key $k$ and---conversely---the amount of data obtained when querying the index for this key element $k$.
If we consider the distribution over an index $I=(D,\mathcal{K},\sigma)$, this effectively corresponds to modelling a random variable $X$ taking values of the key elements $\mathcal{K}$.
The estimated density gives the distribution of this random variable $X$.
This means we determine the probability $P(X=k)$ for each entry $k\in \mathcal{K}$ to be associated with a data item.
To estimate the densities we can use the count information of data elements associated with the key elements in an index.
This corresponds to using a maximum likelihood estimation to derive the probability, i.e.
\begin{equation}
P(X=k) = \frac{|\sigma(k)|}{\sum_{k'\in\mathcal{K}}|\sigma(k')|}
\end{equation}
where $\sigma(k)$ indicates the result set obtained from an index when querying for a specific key element $k$.
As we consider inaccurate or outdated indices it is likely that the set of key elements does not match (see also Section~\ref{subsec:overlap}).
Thus, it can happen that certain key elements might be available in the perfect gold standard index, but not in an outdated or approximated index.
When comparing densities over indices we need to consider the effect of such zero-size entries.
Using a maximum likelihood estimation for the densities would lead to zero probabilities for certain events which renders comparison of densities impractical.
We apply smoothing to overcoming zero probabilities.
We make use of Lidstone smoothing which adds a small constant value of $\lambda$ to all counts obtained for the number of results $|\sigma(k)|$.
The parameter $\lambda$ is set to $0.5$ in our experiments which has shown to provide good results in prior work~\cite{P:PROFILES:2014:Gottron}.
The main idea of measuring index accuracy on the basis of data distributions is to compare their density function.
Common metrics to compare density functions are cross entropy, Kullback-Leibler divergence and perplexity.
Let us briefly review the definitions of these metrics and explain their interpretation.
Assuming two probability distributions $P(X)$ and $P_{\textsc{gs}}(X)$ for the inaccurate index $I$ and the gold standard index $I_{\textsc{gs}}$.
Then \emph{cross entropy} $H(P_{\textsc{gs}},P)$ is defined as:
\begin{equation}
H(P_{\textsc{gs}},P) = - \sum_{k\in \mathcal{K}\cup\mathcal{K}_{\textsc{gs}}} P_{\textsc{gs}}(X=k) \log (P(X=k))
\end{equation}
In the context of compression theory, cross entropy can be interpreted as the average number of bits needed to encode events following the distribution $P_{\textsc{gs}}$ based on an optimal encoding scheme derived from $P$.
If the two distributions are equivalent, then cross entropy corresponds to the normal entropy $H(P_{\textsc{gs}})$.
The entropy of $P_{\textsc{gs}}$ also provides a lower bound for cross entropy.
Based on this interpretation, the \emph{Kullback-Leibler divergence} gives the deviation in entropy (or overhead in encoding) relative to the entropy for $P_{\textsc{gs}}$ and is defined as:
\begin{equation}
D_{\subtxt{KL}}(P_{\textsc{gs}},P) = H(P_{\textsc{gs}},P) - H(P_{\textsc{gs}})
\end{equation}
Therefore, if two distributions are equivalent, they have a Kullback-Leibler divergence of zero.
\emph{Perplexity}, instead, provides an evaluation of a distribution by giving the number of events (in our case key elements) which under a uniform distribution would yield the same entropy value.
As such it is considered to be more easily interpretable by humans than the somewhat abstract entropy values.
Perplexity itself is defined over entropy values, though.
Here we formulate it directly on the basis of cross entropy:
\begin{equation}
\textit{PP}(P_{\textsc{gs}},P) = 2^{H(P_{\textsc{gs}},P)}
\end{equation}
Perplexity is a standard metric for evaluating probabilistic models.
The lower the perplexity is, the better a model explains observed data and the more truthful are its estimates of the probabilities.
Furthermore, the interpretation of perplexity relative to the event space of key elements allows for a normalisation.
The normalised perplexity $\textit{PP}_{\subtxt{norm}}$ is defined as:
\begin{equation}
\textit{PP}_{\subtxt{norm}}(P_{\textsc{gs}},P) = \frac{\textit{PP}(P_{\textsc{gs}},P)}{|\mathcal{K}_{\textsc{gs}}|}
\end{equation}
\subsection{Retrieval Based Measures}
The third type of measures are directly evaluating the results of queries posed towards an index.
\emph{Precision} and \emph{recall} are two such measures which are typically employed in Information Retrieval.
While in general, this approach can be based on a fixed and predefined set of queries, such a set of queries might not be available in most cases.
Furthermore, a fixed set of queries will always address only a certain fraction of the index.
Thus, an alternative approach which has been followed in related work is to pose virtually all possible queries which could be asked over the gold standard as well as the inaccurate index.
In this way, the measures make use of all information stored in the index as indicated in Figure~\ref{fig:mRetrieval}.
\begin{figure}[btp]
\centering
\includegraphics[width=70mm]{mRetrieval}
\caption{Retrieval based measures include the entire index in the evaluation.}
\label{fig:mRetrieval}
\end{figure}
For one specific query, i.e. a key item $k$, precision and recall are defined as follows:
\begin{equation}
p(k) = \frac{|\sigma(k) \cap \sigma_\textsc{gs}(k)|}{|\sigma(k)|}
\end{equation}
\begin{equation}
r(k) = \frac{|\sigma(k) \cap \sigma_\textsc{gs}(k)|}{|\sigma_\textsc{gs}(k)|}
\end{equation}
There is, however a distinction to be made with respect to the way of aggregating results over the set of all queries.
Namely, there are micro- and macro-averages which can be computed.
Macro-average builds an average over all precision or recall values for all of the queries.
Formally this gives (here for the example of precision):
\begin{equation}
p_\subtxt{macro} = \frac{1}{\left | \mathcal{K} \cup \mathcal{K}_{\textsc{gs}} \right |} \sum_{k \in \mathcal{K} \cup \mathcal{K}_{\textsc{gs}}} \frac{|\sigma(k) \cap \sigma_\textsc{gs}(k)|}{|\sigma(k)|}
\end{equation}
Micro-average, instead, aggregates the set sizes of the observed result sets and their intersection.
Precision and recall are then computed on these aggregated count information.
Formally, it is defined as:
\begin{equation}
p_\subtxt{micro} = \frac{ \sum_{k \in \mathcal{K}\cup\mathcal{K}_{\textsc{gs}}} |\sigma(k) \cap \sigma_\textsc{gs}(k)|}{\sum_{k \in \mathcal{K}\cup\mathcal{K}_{\textsc{gs}}} |\sigma(k)|}
\end{equation}
The difference of the two ways for computing averages can be seen as follows.
Macro-average gives equal weight to each query.
Hence a lot of good or bad performing queries which provide small result sets will strongly affect the overall score.
Computing a micro-average, instead, considers also the size of the results set in the overall aggregated score.
Larger result sets and their performance have a stronger impact on the overall result.
\section{Theoretic Analysis of Measures}
\label{sec:theory}
Let us consider the measures introduced above in Section~\ref{sec:measures} and compare them on a theoretic level.
To this end we will consider the following four criteria and argue for and against the individual approches, independent of a concrete scenario, index model or data set.
\begin{description}
\item[Sensitivity to Data Changes]
Not all measures perfectly reflect the accuracy of an index.
In some cases a change in the data and a loss of accuracy of an outdated index will not be captured by an accuracy measure.
The higher the sensitivity of a measure the more precise and reliable it is in evaluating an index's ability to provide truthful results.
In the context of this paper the theoretic sensitivity is judged in a relative and categorial way, by assigning the measures a low ({\color{red}--}\xspace), medium ({\color{gray}o}\xspace) or perfect ({\color{darkgreen}+}\xspace) sensitivity.
A perfect sensitivity implies that each error in an inaccurate index will be detected.
\item[Data Volume]
When computing the measures, one question is how much data needs to be obtained from a perfect index at the original data source.
A smaller volume of data which needs to be transfered from separate indices makes it easier to compute the measures remotely and only request the needed information from the actual computing nodes running the index.
Furthermore, the computational complexity of the presented measures is linear in the amount of data items to consider.
Thus, the data volume indicates how long the computation of the measures will take.
For this reason, the data volume will be given in big $O$ notation based on the elements of abstract index models.
\item[Normalised Value Range]
A normalised value range is favourable in an index accuracy measure.
If a measure provides a normalised value range it is easier to compare its values across experiments.
This can affect both: the comparison of accuracy of the same index over different data sets as well as the comparison of different indices over the same data set.
Having a normalised value range is a binary feature of an accuracy measure, i.e. a measure can either have a normalised range ({\color{darkgreen}$\surd$}\xspace) or not ({\color{red}$\times$}\xspace).
\item[Index Agnostic] If a measure is index agnostic it can be computed independently of a concrete index model or implementation.
The advantage is, that the computation of one single value over the data is sufficient to judge a (potential) impact on different types of indices.
A conceptual disadvantage is, that index agnostic measures cannot be used to evaluate approaches performing an approximate index computation.
Also being index agnostic is a binary feature, i.e. the measure is index agnostic ({\color{darkgreen}$\surd$}\xspace) or not ({\color{red}$\times$}\xspace).
\end{description}
Table~\ref{tab:theory} summarises the theoretic evaluation of the measures from Section~\ref{sec:measures} w.r.t the above mentioned criteria.
Within each family of measures, the characteristics are mostly the same.
Typically, the measures have the same sensitivity, require the same volume of data and are all either index specific or index agnostic.
Judging from this theoretic point of view, the retrieval based measures seem favourable.
After all they directly address the purpose of an index and measure the accuracy using the core functionality.
Furthermore, they provide normalised values which renders them suitable for comparisons across data sets and types of indices.
However, they require a large volume of data.
Basically, all information of the entire index has to be considered for a comparison, which---depending on the type of index---might be as costly as comparing the entire underlying RDF data set.
Under this aspect, the measures using the overlap of key elements and the distribution based measures need less information.
They merely operate on the set of key elements and---in the case of the distribution based measures---a size estimation of how many elements are associated with a key element.
Such size estimations can be realised using count operators over the existing indices, causing only a slight computational overhead.
Their drawback is that they might not always capture perfectly the accuracy of an index.
\begin{table}[tb]
\centering
\caption{Theoretic analysis of measures for index accuracy}
\label{tab:theory}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{l | c c c c }
\toprule
Measure & Sensitivity & Normalised range & Volume & Index agnostic\\
\midrule
{\bf Index agnostic measures} \\
Jaccard (RDF triples) & {\color{darkgreen}+}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|R| + |R_\textsc{gs}|)$ & {\color{darkgreen}$\surd$}\xspace \\
\midrule
{\bf Overlap of key elements} \\
Jaccard (Key elements) & {\color{red}--}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Recall (Key elements) & {\color{red}--}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
\midrule
{\bf Distribution based measures} \\
Cross-Entropy & {\color{gray}o}\xspace & {\color{red}$\times$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
KL-Divergence & {\color{gray}o}\xspace & {\color{red}$\times$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Perplexity & {\color{gray}o}\xspace & {\color{red}$\times$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Normalised Perplexity & {\color{gray}o}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{K} | + | \mathcal{K}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
\midrule
{\bf Retrieval based measures} \\
Recall (macro-avg) & {\color{darkgreen}+}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{D} | + | \mathcal{D}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Precision (macro-avg) & {\color{darkgreen}+}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{D} | + | \mathcal{D}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Recall (micro-avg) & {\color{darkgreen}+}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{D} | + | \mathcal{D}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
Precision (micro-avg) & {\color{darkgreen}+}\xspace & {\color{darkgreen}$\surd$}\xspace & $O(|\mathcal{D} | + | \mathcal{D}_\textsc{gs}|)$ & {\color{red}$\times$}\xspace \\
\bottomrule
\end{tabular}%
}
\end{table}
\section{Empirical Comparison of Measures}
\label{sec:empirical}
Following the theoretical evaluation of the measures we now consider how they perform in practice.
The question we want to answer is which measures are closely correlated to each other in practice.
The aim is to find out which measures seem to be redundant and do not need to be compared.
This might help in reducing overhead in practical evaluation and in identifying efficient measures to approximate or even substitute more complex measures.
In particular, a correlation analysis can also address the question how large is the effect of data changes in practice on the less sensitive key element and distribution based measures, i.e. how far they still agree with the measures having a perfect sensitivity.
To perform such an evaluation we need a set of indices (for the index specific measures) and a data set which undergoes realistic changes.
As data set we can employ the weekly snapshots of a well defined part of the Linked Data cloud provided by the Dynamic Linked Data Observatory (DyLDO)~\cite{P:ESWC:2013:KaeferAUO}.
The range of index models is wide and it is beyond the scope of this paper to evaluate all types of index models.
Thus, let us chose a few selected index models which cover a good range of index granularities and scope.
The indices are taken vom related work and operate on different types of data they return: (1) full triple statements ($D_{\subtxt{triple}}$), (2) single URIs ($D_{\subtxt{URI}}$) or (3) solely the source where relevant information can be found on the Linked Data cloud ($D_{\subtxt{context}}$).
In the following, the index models are described briefly along with a formal definition for clarity.
More details can be found in original publications.
\noindent\begin{samepage}
{\bf Subject Index:} This type of index simply uses URIs as key elements and its selection function returns all triple statement containing a given URI in the subject position.
Using the abstract notation introduced in Section~\ref{sec:index}, such an index can be formalised as $I_{\subtxt{S}} := (D_{\subtxt{triple}}, \mathcal{K}_{\subtxt{S}}, \sigma_{\subtxt{S}})$ where:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{S}} := \{ s \in U\mid \exists p,o,c : (s,p,o,c) \in R\}$
\item Selection function: $\sigma_{\subtxt{S}}(k) := \{(s,p,o) \mid s = k \}$
\end{itemize}
\end{samepage}
\noindent\begin{samepage}
{\bf RDF Type Index:} Making use of the specific semantics behind \code{rdf:type} statements, this index is used to look up all entities (i.e. URIs) which are of a particular RDF class type~\cite{P:ESWC:2014:GottronG}.
Formally it can be defined as $I_{\subtxt{T}} := (D_{\subtxt{URI}}, \mathcal{K}_{\subtxt{T}}, \sigma_{\subtxt{T}})$, with:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{T}} := \{ o\mid \exists s,c : (s,\code{rdf:type},o,c) \in R\} \cup \allowbreak \{ s\mid \exists c : (s,\code{rdf:type},\allowbreak\code{rdfs:Class},c) \in R\}$
\item Selection function: $\sigma_{\subtxt{T}}(k) := \{s \mid \exists c : (s,\code{rdf:type},k,c) \in R \}$
\end{itemize}
\end{samepage}
\noindent\begin{samepage}
{\bf RDF Type Set (TS) Index:} Extending the RDF type class index, this index gives all entities for a specific set of class types, i.e. all entities which satisfy being of exactly all types given in the set (and of no other type)~\cite{J:JWS:2012:KonrathGSS}.
The formal definition is given by $I_{\subtxt{TS}} := (D_{\subtxt{URI}}, \mathcal{K}_{\subtxt{TS}}, \sigma_{\subtxt{TS}})$, where:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{TS}} := \mathcal{P}(\mathcal{K}_{\subtxt{T}})$
\item Selection function: $\sigma_{\subtxt{TS}}(k) := \{s \mid (\forall t \in k : (\exists c : (s,\code{rdf:type},t,c) \in R)) \wedge \forall (s,\code{rdf:type},o,c) \in R : (o \in k)\}$
\end{itemize}
\end{samepage}
\noindent\begin{samepage}
{\bf Property Set (PS) Index:} Using sets of predicates---also known as characteristics sets---this index retrieves entities based on the properties with which they have been described in RDF~\cite{P:SIGMOD:2009:NeumannW}.
Formally it can be defined as $I_{\subtxt{PS}} := (D_{\subtxt{URI}}, \mathcal{K}_{\subtxt{PS}}, \sigma_{\subtxt{PS}})$, with:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{PS}} := \mathcal{P}(\mathcal{K}_{p})$
\item Selection function: $\sigma_{\subtxt{PS}}(k) := \{s \mid (\forall p \in k : (\exists o,c : (s,p,o,c) \in R)) \wedge \forall (s,p,o,c) \in R : (p \in k)\}$
\end{itemize}
\end{samepage}
\noindent\begin{samepage}
{\bf Extended Characteristic Set (ECS) Index:} This index essentially combines the TS and PS index and characterises Linked Data entities by both: their types and properties~\cite{P:COLD:2013:DividinoSGG}.
Formally it is defined by $I_{\subtxt{ECS}} := (D_{\subtxt{URI}}, \mathcal{K}_{\subtxt{ECS}}, \sigma_{\subtxt{ECS}})$, where:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{ECS}} := \mathcal{P}(\mathcal{K}_{\subtxt{P}}\cup \mathcal{K}_{\subtxt{T}})$
\item Selection function: $\sigma_{\subtxt{ECS}}(k) := \{s \mid ( \forall p \in k \cap \mathcal{K}_{\subtxt{P}} : (s \in \sigma_{\subtxt{PS}}(p) )) \wedge (\forall t \in k \cap \mathcal{K}_{\subtxt{T}} : (s \in \sigma_{\subtxt{TS}}(t))) \}$
\end{itemize}
\end{samepage}
\noindent\begin{samepage}
{\bf SchemEX Index over data sources:} The SchemEX index combines induced schematic information about the types of subject URIs, their properties and the types of object URIs they are linked to~\cite{J:JWS:2012:KonrathGSS}.
To this end it re-uses the concepts of type clusters for the domain and range of observed property sets.
Those concepts are combined using a restricted bi-simulation over property sets and a stratification of the entities in equivalence classes of type sets.
Its formal definition is given by $I_{\subtxt{SchemEX}} := (D_{\subtxt{context}}, \mathcal{K}_{\subtxt{SchemEX}}, \sigma_{\subtxt{SchemEX}})$, where:
\vspace{-2mm}
\begin{itemize}
\item Key elements: $\mathcal{K}_{\subtxt{SchemEX}} := \mathcal{P}( \mathcal{K}_{\subtxt{TS}} \times \mathcal{P}( \mathcal{K}_{\subtxt{PS}} \times \mathcal{K}_{\subtxt{TS}}))$
\item Selection function: $\sigma_{\subtxt{SchemEX}} (k=(\textit{ts},E)) := \{c \mid \exists s \in \sigma_{\subtxt{TS}}(\textit{ts}) \wedge \forall (\textit{ps},\textit{ts}_{2}) \in E : (s \in \sigma_{\subtxt{PS}}(\textit{ps}) \wedge \exists o \in \sigma_{\subtxt{TS}}(\textit{ts}_2) : (\forall p \in \textit{ps}: ((s,p,o,c) \in R) )\}$
\end{itemize}
\end{samepage}
For the empirical evaluation we computed these indices over 77 snapshots provided by the DyLDO data set.
Assuming an index has been computed over the initial snapshot, we evaluate it accuracy compared to the later on versions of the data sets.
This corresponds to fixing an index over the initial data set and comparing it to gold standard indices over later data sets.
All measures have been computed for all of the evaluated index models.
This gives us for each update of the data set one observation of the accuracy measures.
The results of a spearman rank correlation analysis between all pairs of measures over this data is visualised in Figure~\ref{fig:correl}.
\begin{figure}[btp]
\centering
\subfigure[Subject index]{
\label{sfig:s}
\includegraphics[width=60mm]{plot-s}
}
\subfigure[RDF type index]{
\label{sfig:type}
\includegraphics[width=60mm]{plot-type}
}
\subfigure[Type set index]{
\label{sfig:ts}
\includegraphics[width=60mm]{plot-ts}
}
\subfigure[Property set index]{
\label{sfig:ps}
\includegraphics[width=60mm]{plot-ps}
}
\subfigure[Extended characteristic set index]{
\label{sfig:ecs}
\includegraphics[width=60mm]{plot-ecs}
}
\subfigure[SchemEX index]{
\label{sfig:schemex}
\includegraphics[width=60mm]{plot-schemex}
}
\caption{Correlation of accuracy measures on different indices over an evolving data set.}
\label{fig:correl}
\end{figure}
There are several interesting observations to be made.
First of all we observe two index models where the correlations are much stronger in general: the subject index and SchemEX.
All other analysed index models exhibit lower correlation values.
The explanation for this behaviour is the granularity of the indices.
Both, the subject index and SchemEX, generate a relatively large number of key elements over the analysed data set.
Furthermore, they have many key elements in the index which provide only one element entry upon request.
This means on the one hand, that measuring the accuracy of a specific query is often reduced to a binary task: either the index provides the correct element or not.
Therefore, the retrieval based and key element based measures are strongly correlated in a positive way.
On the other hand, as the change of a data element being assigned to a different key element is typically caused by a single triple, also the Jaccard index over triples correlates strongly with the measures.
Also the changes in the distribution are affected by this.
Changes for key elements with one entry typically leads to a strong shift in the overall distribution, reducing a MLE probability to zero\footnote{Effectively the probability has a small value, slightly higher than zero due to the use of Lidstone smoothing as explained in Section~\ref{subsec:distribution}.}.
This shift in probability mass is recognised by the distribution based measures.
The observation that the correlation here is highly negative comes from the fact that for those measures a high value indicates a strong change, while the other measures use high values to indicate no or only weak changes.
For all other analysed index models, we have a different overall behaviour.
Typically, the measures within each family are correlated (though to a lower extent than for the subject index or SchemEX).
Furthermore, the key element based measures correlate strongly with the retrieval based measures and the Jaccard index over the sets of RDF triples.
The distribution based measures, instead seem to measure something else.
Beyond a few exception, no strong correlation can be observed here.
Looking at individual measures we can also observe some interesting behaviour:
Across all types of index models macro-average precision seems to be not or only weakly correlated to most other measures.
Hence, the different way of aggregating precision seems to have a strong impact.
The micro-average show a better sensitivity of the accuracy of queries with large results sets.
This seems to make a distinction with other evaluation measures over most analysed index models.
The perfect Spearman correlation between Cross-Entropy and Perplexity, instead, is conceptual as Perplexity is only a rank-preserving non-linear scaling of Cross-Entropy.
\section{Discussion}
\label{sec:discussion}
Given the analysis in the previous two section, we can observe that a look at the key elements might already give a good indication for the accuracy of an index.
While from a theoretic point of view they might not be very sensitive to changes they correlated well with other, more sensitive measures in the empiric evaluation.
In particular they reflect quite well the retrieval based accuracy measures.
This observations holds for all analysed index models.
Additionally, in many cases the distribution based measures provide further insights.
Given that the volume of data needed to compute these measures is of the same order as for the key elements, such an analysis causes little overhead.
In general, the empirical observations in the previous section were made only over a single data set.
However, given that this data set has been carefully designed to give a representative excerpt of the LOD cloud, we might want to conjecture that density based measures as well as the measures based on key elements suffice to judge the quality of Linked Data indices.
As information on the distributions over key elements in Linked Data index structures is sufficient to compute both types of measures, they are also attractive from a computational point of view.
\section{Summary and Conclusions}
\label{sec:summary}
In this paper I investigated the question of how to measure the accuracy of index structures and data caches which are built in an approximative way or over evolving Linked Data sets.
I gave an overview of different approaches for such measures, compared them on a theoretical and empirical level.
One observation is that information about the data distribution over key elements seems to provide good insights into index accuracy without the need to access all data in an index.
This observation also motivates a roadmap for future work.
Operating on samples of Linked Data can provide good estimates of these distributions~\cite{P:PROFILES:2014:Gottron}.
Accordingly an interesting question is how suitable are sampling based strategies for evaluating index accuracy.
This means, that instead of asking all possible queries, we consider only a random sample of queries to obtain distributions.
This might speed up computation or can at least provide certain confidence intervals about what the true accuracy of an index might be.
\vspace{1em}
|
2,877,628,090,784 | arxiv | \subsection*{Abstract}
Symmetries play a major role in
physics, in particular since the work by E.~Noether and H.~Weyl in the
first half of last century. Herein, we briefly
review their role by recalling how symmetry changes allow to
conceptually move from classical to relativistic and quantum physics.
We then introduce our ongoing theoretical analysis in biology and show
that symmetries play a radically different role in this discipline,
when compared to those in current physics. By this comparison, we
stress that symmetries must be understood in relation to conservation
and stability properties, as represented in the
\emph{theories}. We posit that
the dynamics of biological organisms, in their various levels of
organization, are not ``just'' processes, but
permanent (extended, in our terminology) critical transitions and,
thus, symmetry changes. Within the limits of a relative structural
stability (or interval of viability), variability is at the core of
these transitions.
\paragraph{Keywords: } symmetries, systems biology, critical transitions,
levels of organization, hidden variables, coherent structures, downward
causation.
\section{Introduction and summary }
A synthetic understanding of the notion of organism requires drawing
strong correlations between different levels of organization as well as
between the global structure and the local phenomena within the
organism. These issues should govern any systemic view on biology. Here,
we sketch an approach in which the living state of matter is
interpreted as a permanent ``transition'', conceived as an ongoing or
\emph{extended} and \emph{critical} transition. A large amount of
very relevant work pertaining to the Theories of Criticality in physics
has been successfully applied to biology (see below). The mathematical
core of these theories rests upon the idea that a ``phase transition,''
which can be either critical or not, may be described as a
\emph{point} along the line where the intended control parameter
runs. For example, the ferromagnetic / paramagnetic transition takes
place for a precise value of the temperature, the Curie temperature.
Mathematically, this is expressed by the ``pointwise'' value of this
temperature, i.e., one mathematical point in this parameter’s space.
When the temperature decreases and passes through that point, the
magnetic orientation organizes along one direction and magnetism
appears. When the temperature increases through that point, disorder
prevails and magnetism disappears. A (phase) transition is critical
when some observables, or their first or second derivatives, diverge.
This corresponds to the appearance of a ``coherent structure'', that is
to say space and/or time correlations at all scales, which at the
transition point give a ``global'' aspect to the new physical object.
These ideas are relevant to the analysis of biological organisms.
In contrast to known critical transitions in physics, biological
entities should not be analyzed just as transient over a point of a
phase change; instead, they permanently sustain criticality over a
non-zero interval and this with respect to many control parameters
(time, temperature, pressure). This represents a crucial change of
perspective. First, the mathematical tools used in physics for the
analysis of criticality, i.e, the renormalization methods, essentially
use the pointwise nature of the critical transitions. Secondly,
\emph{symmetries} and \emph{symmetry breakings} radically change
when enlarging the mathematical locus of criticality from one point to
a non-zero interval. These symmetry changes make a key theoretical
difference with respect to the few cases in physics where the
transition seems extended (see footnote 10, below). Our approach may be
seen as a move from physics to biology by an analysis of the radically
different symmetries and symmetry breakings at play in their respective
theoretical frames. Thus, we will mostly focus on physical vs
biological criticality in terms of symmetries and then apply this
method to the analysis of the difference between physical and
biological ``objects'' as well as of physical vs biological
``trajectories''.
Living entities are not ``just'' processes, but something more: they are
lasting, \emph{extended critical transitions}, always
transient toward a continually renewed structure. In general, physical
processes do not change fundamental symmetries: to the contrary, they
are mostly meant to preserve them. Typically, conservation properties
(of energy, of momentum) are symmetries in the equations of movement.
Critical transitions are an exception to the preservation of symmetries
in physics; their ``extension'' radically changes the understanding of
what biological processes are. This perspective also proposes a
possible way of overcoming a key issue in the analysis of the
complexity of the living state of matter. As for the construction of
physico-mathematical or computational models, it is difficult to take
the global structure of an organism into consideration, with its
correlations between all levels of organization and in all lengths,
including the many forms of integration and regulation. Thus, the
complexity of the living unity is often modeled by the stacking of
many but \emph{simple }elementary processes. Typically, these formal
systems deal with many observables and parameters. Since the framework
is classical in a physical sense, these variables are local, i.e. they
depend on pointwise values of the intended phase space. Instead,
conceptual and mathematical dependencies in biology should be dealt
with as ``global'' ones, where variables may depend on systemic or
\emph{non-local} effects. In physics, these dependencies are a
relevant aspect of critical transitions, and they are even more so in
biology, where criticality is extended.
\subsection{Hidden variables in biology? }
In classical and relativistic physics, once the suitable ``phase
space'' and the equations that mathematically determine
the system are given, the knowledge of the pointwise position-momentum
of the intended object of analysis allows to describe \emph{in
principle} the subsequent dynamics. This is ``in
principle'' since physical measurement, which is always
approximated, may produce the phenomenon of \emph{deterministic
unpredictability}, in particular in the presence of non-linear
mathematical determination\footnote{ More generally,
unpredictability may appear when the dynamics is determined by an
evolution function or equations that mathematically represent ``rich''
interactions. Non-linearity is a possible mathematical way to express
them.}. Moreover, not all ``forces'' in the game may be known and there
may be ``hidden variables'' (like the frictions along the trajectory of
bouncing dice). Yet, these theories are deterministic and, once all
pertinent variables and forces are assumed to be known, it is the
\emph{epistemic} lack of knowledge which yields classical randomness.
\emph{Per se}, a dice follows a ``geodesic''. This is a unique, optimal
and ``critical'' path, completely determined by the Hamiltonian and may
be computed as an optimum of a Lagrangian
functional.\footnote{ These are mathematical
operators, that is, functions acting on functions that contain all
known physical information concerning the energy state of the
system.}. This very beautiful paradigm, which may be summarized as the
``geodesic principle'', may be further grounded on \emph{symmetries} by
an analysis of conservation principles (see \nptextcite{bailly2011} for
a recent synthesis and references).
In order to compare this situation with other fields of physics and
subsequently to biology, we refer to the pointwise or local nature of
the mathematical variables. Cantorian (and Euclidian) points are
\emph{limit} conceptual constructions; that is, they are the limit of
a physical access to space and time by an always approximated
measurement, i.e., an ``arbitrarily small'' interval. Yet,
their perfect theoretical ``locality'' makes all classical dynamics
intelligible (in principle). So, if something is unknown, one expects
that by adding enough observables and/or more variables with definite
values at any given time, one could increase knowledge, since the
values of these observables are intrinsic and independent of the
context.
The situation is rather different in Quantum Mechanics. The
simultaneous, perfect, pointwise knowledge of position \emph{and}
momentum (or energy \emph{and} time) are, in
principle, forbidden because indeterminacy is
intrinsic to the theory. Moreover, suppose that two quanta interact and
form one system and that they later separate in space. Then acquiring
knowledge regarding an observable quantity by performing a measurement
on one of these quanta produces an instantaneous knowledge of the value
of the measurement made on the other, i.e., the two quanta are
``entangled'' \parencite{EPR}. These features of the theory have
several consequences: for instance, variables cannot always be
associated to separated points and quantum randomness is intrinsic
(under the form of Schrödinger equation, the ``determination'' gives the
\emph{probability} to obtain a value by measurement). Within this
theoretical framework, quantum randomness differs from the classical
one: two interacting dice which later separate obeying independent
statistics, while the probability values of an observable of two
previously interacting quanta are correlated. This is the so called
``violation of Bell inequalities'', which has been empirically verified
repeatedly since the experiments described in \textcite{Aspect}.
Quantum entanglement requires considering some phenomena as being
``non-local'' and unseparable by any physical measurement
(``non-separability'').
Since the ’30s, some have found this situation unsatisfactory and have
searched for ``hidden variables'' like in the epistemic approach to
randomness and determination of classical and relativistic physics. The
idea is that these hidden variables corresponding to quantum mechanical
observables have definite (pointwise/local) values at any given time,
and that the values of those variables are intrinsic and independent of
the device used to measure them. A robust result has instead shown that
these assumptions contradict the fundamental fact that quantum
mechanical observables need not be commutative \parencite{Kochen}. Moreover, even when assuming the existence of, or the need for,
hidden variables, these would be ``non-local'' and thus, far from the
pointwise/local dependence of set-theoretic variables.
The difference between the classical and quantum frameworks has the
following consequence: quantum systems may have a proper systemic unity
for at least two reasons. Conjugated observables (position and
momentum) are ``linked'' by joint indetermination. Entangled quanta
remain a ``system'', in the sense of their non-separability by
measurement\footnote{ Superposition should also be
mentioned, see \textcite{Silverman}.}.
Can this perspective help us in biology? On technical grounds, surely
not, or rather not yet. Perhaps, ``entangled molecular phenomena'' or
``tunnel effects \dots\ in the brain'' may clarify fundamental issues in the
future. However, theoretical ideas in Quantum Mechanics may at least
inspire our attempts in system biology, in particular by considering
the methodological role of symmetries and symmetry breakings in this
area of physics.
A living organism is a system. And entanglement, non locality,
non-separability, superposition, whatever these concepts may mean in
biology, may present themselves both at each specific level of
organization and in the interactions between levels of organization.
Physiological interactions among molecules, cells, tissues, organs~ do
not simply sum each other up: they are ``entangled'', ``non-local'',
``non-separable'' \dots\ they are ``superposed'' (see examples described by
\nptextcite{noble2006,soto2008}). Thus, the theoretical and mathematical
approaches to biology cannot be based only on a continual enrichment of
``local'' views: mathematical models cannot work just by assuming the
need for more and more variables (possibly hidden to the previous
models). A global view of the system and of its symmetries is required.
In this context, the differences in symmetries and their breakings will
help in clarifying and facilitating the passage from physics to
biology.
\section{Symmetry and objectivation in physics}
In Physics, objectivity is obtained by the co-constitutive use of
experiments and mathematized theories. So far, however, there is little
mathematics for a ``theory of the biological organisms'' despite the
large amount of data collected and of theories proposed within specific
levels of organization. These include the geometric analysis of the
fractal structures of lungs, of vascular systems, of various plant
organs, of networks of neural cells, of tumor shapes, to name but a
few. To make further progress towards mathematizing theories in
biology, in particular towards theories of the ``living object'' or of
the organism as a system, it would help first to understand how such a
feat was achieved in physics. Physical theories have very general
characteristics in their constitution of objectivity, and in particular
in their relationship with mathematics. In order to define space and
time, as well as to describe physical objects, physicists ultimately
use the notion of symmetry. Physical symmetries are the transformations
that do not change the intended physical aspects of a system in a
theory. As we shall see, they allow to define these aspects in a
non-arbitrary way.
Galileo’s theory provides a simple and historical example of this role
of symmetries. For scholastic physics, the speed at which a body falls
is proportional to the space traveled. Galileo instead proposed that it
is proportional to the time of the fall and that it is independent of
the nature (including the mass) of the empirical object considered
(Galileo’ law of gravitation). This idea together with the ``principle
of inertia'' has been a starting point for the constitution of
\emph{space} and \emph{time} in classical physics. More precisely,
as a consequence of the analysis of inertia and gravitation, the
geometry of space and time was later described by the Galilean
group\footnote{ Symmetries form a set of
transformations that have a group structure; that is, two symmetries
applied successively yield a symmetry and a symmetry can be inverted.
Galileo’s group is the group of transformations that allows to
transform a Galilean space-time reference system into another. It is
interesting to notice that Galileo measured time by heartbeat, a
biological rhythm; the subsequent theoretical and more ``physical''
measurement of time were
precisely provided by classical mechanics, his invention.}.
A change of this symmetry group, for example by adopting the Poincaré
group\footnote{ The symmetry group of a Euclidean
space is the Euclidean group of automorphisms, while Poincaré’s group
corresponds to the automorphisms defining Minkowski’s spaces.}, can
lead to a very different physical situation, that of special relativity
involving massive conceptual and physical changes. The ``principle of
relativity'' states that the fundamental laws of physics do not depend
on the reference system; they are actually obtained as invariants with
respect to the change of reference system. A specific speed (the speed
of light in the void) appears in the equations of electromagnetism.
Einstein modified Galileo’s group in order to transform this speed into
an invariant of mechanics, which turned time-simultaneity into a
relative notion.
As a result of the role and implications of symmetries, most
contemporary physical challenges lead to the search for the right
symmetries and symmetry changes, such as the work aiming at the
unification of relativistic and quantum theories. In moving from
physics to biology we suggest here to apply a similar approach
(symmetry changes).
Since the 1920s, due to Noether’s theorems, symmetries lead to the
mathematical intelligibility of key physical invariant quantities. For
example, symmetries by time translations are associated with
energy-conservation, and symmetries by space rotations are associated
with the conservation of angular momentum. Thus, conservation laws and
symmetries are in a profound mathematical relation. Consequently, the
various \emph{properties} that define an object (mass, charge,~etc.)
or its \emph{states} (energy, momentum, angular momentum,~etc.) are
associated to specific symmetries which allow these quantities to be
defined. Depending on the theory adopted, this conceptualization
allowed to understand why certain quantities are conserved or not: for
example, there is no local energy conservation in general relativity.
This explicit reference to the theory adopted is required in order to
produce ``scientific objectivity'', \emph{independently}
of the arbitrary choices made by the observer, such as, the choice of
time origin, the unit of measurement, etc, but \emph{relatively} to
the intended theory. Thus, we say that symmetries provide ``objective
determinations'' in physics \parencite{bailly2011}.
The symmetries that define physical properties allow us to understand
the physical object as \emph{generic}, which means that any two
objects that have the same properties can be considered as physically
\emph{identical}; in a sense, they are symmetric or invariant
(interchangeable) in experiments and in pertinent mathematical
framework (typically, the equations describing movement). For example,
for Galileo, all objects behave the same way in the case of free fall,
regardless of their nature. Moreover, symmetries allow the use of the
\emph{geodesic principle,} whereby the local determination of
trajectories leads to the determination of the full trajectory of
physical objects through conservation laws. For example, the local
conservation of the ``tangent'' (the momentum) of movement, typically
yields the global ``optimal'' behavior of the moving object; that is, it
goes along a geodesic. Thus, in classical or relativistic mechanics, a
trajectory is unique and fully deterministic (formally determined). In
quantum mechanics the evolution of the state or wave function (roughly,
a \emph{probability distribution}) is fully deterministic as well –
and determined by Schrödinger’s equation – while measurement follows
this probability distribution (and here appears the indeterministic
nature of quantum mechanics). In conclusion, by symmetries, the
trajectory of a generic classical or quantum physical ``object''
corresponds to a critical path: physical trajectories are
\emph{specific}.
To better understand the problem of \emph{general} mathematical
theorizing in biology, let’s further analyze how, in physics, a
concrete problem is turned into robust models and mathematics. To begin
with, physicists try to choose the right theoretical framework and the
relevant physical quantities (properties and states) which are
constituted by proper symmetries. As a result, typically, a
mathematical framework is obtained, where one can consider a generic
object; in classical mechanics, a pointwise object of mass~$m$,
speed~$v$ and position~$x$, where these quantities are
generic. Now, a generic object will follow a specific trajectory
determined by its invariants obtained by calculus. A measurement is
then made on the experimental object to determine the quantities
necessary to specify where this object is in this mathematical
framework, namely, what is its mass, initial position and speed. And
finally, what specific trajectory will the object follow \dots\ at least
approximately. In classical or relativistic physics, to a specific
measurement will correspond generic objects localized near the
measurement due to the limited precision of this measurement. This
value may have, in principle, an arbitrary high precision. In quantum
mechanics, as we recalled above, the equational determination
(Schroedinger’s equation) yields the dynamics of a probability
law\footnote{In quantum physics, ``objects'' do not
follow trajectories in ordinary space-time, but they do it in a
suitable, very abstract space, a Hilbert space (a space of mathematical
functions); what ``evolves'' is a probability distribution.}.
In classical dynamics, we face a well-known problem: the specific
trajectories can either stay close or diverge very rapidly. The linear
situation corresponds to the first case, whereas the second situation
is called ``sensitive to initial conditions'' (or chaotic, according to
various definitions). Note that even the latter situation leads to the
definition of new invariants associated to the dynamics: in other
words, the attractors that have a precise geometrical structure. In
both cases, these trajectories have robust properties with respect to
the measurement. In quantum physics, the situation is more complex
because the measurement is not deterministic. Yet, when approximations
on the state function are performed, it leads to usually stable. robust
statistics. In all cases, ``robust'' means invariant or approximately
invariant in a definite mathematical sense, as concerns the measurement
of states and properties of generic objects along specific
trajectories. Thus, we can finally say that generic objects, which lead
to a specific measurement, \emph{behave} in the same way or
approximately so. Notice that this property of robustness, allowed by
the genericity of the object, is mandatory for the whole framework to
be relevant. We insist that both genericity for objects and specificity
for trajectories (geodesics) are mathematically understood in terms of
symmetries.
In conclusion, in the broadest sense, symmetries are at the foundation
of physics, allowing objective definitions of space and time and the
constitution of objects and trajectories. In their genericity, these
objects follow specific trajectories associated with invariants that
are robust with respect to measurement.
\section{ Symmetry breakings and criticality in physics }
The physics of criticality is a relatively novel discipline which
analyzes, typically by the renormalization techniques, some peculiar
phase transitions, i.e., state changes (see \nptextcite{toulouse1977introduction,Binney}). This theoretical framework has also been applied
to a possible understanding of life phenomena (see for example, \nptextcite{Bak88,Jensen}, as for ``self-organized criticality''; or,
\nptextcite{Kauffman93}, as for criticality in networks). We will next move
towards biology through a different insight into the symmetries in
criticality.
Since symmetries are at the core of the definition of the physical
objects by their properties and states, a \emph{symmetry change}
(that is, the breaking of some symmetries and the formation of new
ones) means a qualitative change of the object considered, or a change
of physical object, understood as co-constituted by theory and
empiricity. For example, a research project in cosmology is to consider
a single force to have existed in the universe right after the big
bang. Then, the four fundamental forces may have appeared by successive
symmetry breakings, whereby some transformations, which were
symmetries\footnote{The Higgs mechanism is an example
of this phenomenon; in this case, the symmetry breaking in the abstract
electroweak space leads in particular to different masses of bosons and
as a consequence to a very short range for weak interaction and a long
range for electromagnetism.}, did not preserve the object invariance
anymore. In other words, with the cooling of the universe, the system
moved to a smaller symmetry group. Closer to the scale of biology,
materials like water or iron were able to show different properties in
different situations. Depending on the temperature and pressure, water
may be a solid, a liquid, or a gas. When liquid, there is no
privileged direction (the system is isotropic, that is to say symmetric
by rotations), whereas ice has a crystalline structure with spatially
periodic patterns. This implies that the system is no longer symmetric
by continuous rotations: it has a few privileged directions determined
by its crystalline structure and a smaller symmetry group. Similarly,
iron can have paramagnetic behavior (the system is not) or
ferromagnetic behavior (it is magnetized). In most cases, one can
distinguish a more disordered phase at high temperature, where entropy
dominates, and a more ordered phase, where energy dominates. These
situations can be characterized by an \emph{order parameter} which is
$0$ in the disordered phase and different from $0$ in the ordered
phase\footnote{ Here, order means low entropy (or
less symmetries) and disorder means high entropy (and more symmetries,
when symmetries are computed in terms of ``microstates'').}.
Now, in physics, the change of state, or \emph{phase transition},
occurs always mathematically at a point of the parameters’ space. This
point, called the \emph{critical point}, is intuitively associated
with a sudden change of behaviour due to a change of symmetry, and
ultimately to singularities of the state functions (for example, the
order parameter is non-analytical because it goes from a
\emph{constant} 0 to a finite quantity, \emph{by a finite change}).
More technically, the \emph{critical point} represents a singularity
in the partition function describing the
system\footnote{ This function is non-analytical at
the critical point, which means that the usual Taylor expansions,
linearizations or higher order approximations do not actually provide
an increasing approximation.}. In the case of iron’s
paramagnetic-ferromagnetic transition, this allows to deduce the
divergence of some physical observables, such as magnetic
susceptibility. It should be remembered that this notion of
\emph{singularity,} which is associated with infinite quantities at
the critical \emph{point}, is a core notion for physical criticality.
This peculiar situation leads to a very characteristic behaviour at the
critical point \parencite{Jensen}:
\begin{enumerate}
\item Correlation length tends to infinity, and follow a power law, as
for continuous phase transitions (i.e., for a vector~$x$ and an
observable~$N$, if we note by~$<.>_r$ the average over point~$r$ in
space, then $< N(r + x)N(r)>_r-<N(r)>_r^2\sim \|x\|^\alpha$. This is
associated with fluctuations at all scales leading in particular to the
failure of mean field approaches. Following this approach, the value of
an observable at a point is given by the mean value in its
neighbourhood or, more precisely, its mathematical distribution is
uniform.
\item Critical slow down: the time of return to equilibrium of the
system after a perturbation tends to infinity \parencite{suzuki1982critical,tredicce2004critical}.
\item Scale invariance: the system has the same behavior at each scale.
This property leads to fractal geometry and means that the system has a
specific symmetry (scale invariance itself).
\item The determination of the system is global and no longer local.
\end{enumerate}
These properties are the key motivations for the biological interest of
this part of physics. The global ``coherence structure'' that is often
formed at critical transitions provides a possible understanding, or at
least, an analogy for the unity of an organism (in current terminology,
its ``global determination or causation''). Also, power laws, so frequent
in biology, are ubiquitous in critical phenomena. They are
mathematically well-behaved functions (e. g. $f(x) =
x^\alpha$) with respect to the change of scale
[typically, ${\lambda}$ is the scale change in $f(\lambda x) =
\lambda^\alpha f(x) =
\lambda^\alpha x^\alpha$,
a power law in $\alpha$], and they yield \emph{scale symmetries.
}In our example, scale change just multiplies the function $f$ by a
constant $\lambda^\alpha$. Now, a power law
depends on a quantity without physical dimension ($\alpha$ in the
notation above). These quantities involved in critical transitions are
called \emph{critical exponents} and describe how the change of scale
occurs. In our terminology, they describe the properties due to the
objective determination of a phase transition because they are the
invariants associated with the scale symmetry.
Specific analytical methods, called renormalization methods, are used to
find these quantities \parencite{delamotte2004hint}. These methods consist in
analyzing how scale changes transform a model representing the system,
and this analysis is made ``asymptotically'' toward large
scales. One may deduce the critical exponents from the mathematical
operator representing the change of scale. The key point is that a
variety of models ultimately lead to the same quantities, which means
that they have the same behavior at macroscopic scales. Thus, they can
be grouped in so-called \emph{universality classes}. This analytical
feature is confirmed empirically, both by the robustness of its results
for a given critical point and more stunningly by the fact that very
different physical systems happen to undergo the same sort of phase
transitions; that is, they are associated with the same critical
exponents, thus with the same symmetries. Finally, there exist
fluctuations at all scales, which means, in particular, that small
perturbations can lead to very large fluctuations.
To conclude, the transition through a specific point of the parameters’
space, i.e., a transition between two very different kinds of behavior
is associated in physics to a change of symmetries. At this point, the
system has very peculiar properties and symmetries. Symmetries by
dilation (by a coefficient $\lambda$ as above) yield a scale
invariance. This latter invariance is associated to a global
determination of the system and the formation of a ``structure of
coherence''. As observed above, this allows to describe a global
determination of local phenomena and a unity that by-passes the idea of
understanding the global complexity as the sum of many local behaviors
by adding more and more local, possibly hidden, variables. For some
physical phenomena this theoretical framework presents peculiar and
very relevant forms of ``systemic unity''.
\section{ Symmetry breaking and the biological object: extended criticality}
We have presented a picture of the situation in physics, but what about
biology? We need to propose one or several specific frameworks
relevant to the unity and coherence of biological entities, because, to
our knowledge, there are no formalized theories of the ``organism''. To
do so, it may be worthwhile to look at the symmetries which may be
involved in biological theorizing. Here, the concept of symmetry is
used in a more fundamental context than when used, for example, for
``bauplans'', the latter being the main biological research subjet where
the concept is explicitly applied. In physics, one mostly deals with
\emph{fundamental} or \emph{theoretical} symmetries as typically
given by the equations. For example, the already mentioned fundamental
principle of energy conservation corresponds to a time translation
symmetry in the equations of movement. This use of symmetries also
justifies the soundness of empirical results: Galilean inertia is a
special case of conservation of energy and it may be empirically
verified. In biology, as in any science, a missing analysis of
invariants may give unreliable results and data. For example, early
measurements of membrane surfaces gave very different results, since
their measure is not a scale invariant property: as in fractal
structures, it depends on the scale of
observation\footnote{In \textcite{weibel1994}, another
``historical'' example is given as for the different results that are
obtained according to different experimental scales (microscope
magnifications). One team evaluated the surface density of the liver’s
endoplasmic reticulum at \SI[per-mode=symbol]{5.7}{\square\metre\per\cubic\centi\metre}
the other at \SI[per-mode=symbol]{10.9}{\square\metre\per\cubic\centi\metre}
(!). }. In other words, in physics, both the
generality of equations and the very objectivity of measures depend on
theoretical symmetries and their breakings, such as scale invariants
and scale dependencies.
As mentioned above, critical transitions in physics are mathematically
analyzed as isolated points\footnote{The
Kosterlitz-Thouless transition in statistical physics presents a
marginally critical interval; that is, it is a limit case between
critical and not critical. It presents correlations at all scales, as
critical features, but with no symmetry changes. Thus, this particular
situation is not a counter-example to our statement (the essentially
pointwise nature of the proper physical transitions),
in view of a lack of symmetry changes that are essential to our notion
of extended criticality.}. In our approach to biological processes as
``\emph{extended} critical transitions'', ``extended'' means that
\emph{every point} of the evolution/development space is near a
critical point. More technically, the critical points form a
dense\footnote{ Here, dense means that for every
small volume of the intended phase space being considered, there is a
critical point in such volume.} subset of the multidimensional space
of viability for the biological process. Thus, criticality is extended
to the space of all pertinent parameters and observables (or phase
space), within the limits of viability (tolerated temperature, pressure
and time range, or whatever other parameter, say for a given animal), see
\textcite{bailly1993,bailly2008,bailly2011}. In
terms of symmetries, such a situation implies that biological objects
(cells, multicellular organisms, species) are in a \emph{continual
transition between different symmetry groups}; that is, they are in
transition between different phases, according to the language of
condensed matter\footnote{The dense set of symmetry groups may be
potentially infinite, but, of course, an organism (or a species)
explores only finitely many of them in its life span, and only viable
ones. }. These phases swiftly shift between different critical points
and between different \emph{physical determinations} through symmetry
changes.
Our perspective provides an approach concerning the mathematical nature
of biological objects as a \emph{limit} or asymptotic case of
physical states: the latter may yield the dense structure we attribute
to extended criticality only by an asymptotic accumulation of critical
points in a non-trivial interval of viability --- a situation not
considered by current physical theories. In a sense, it is the very
principles grounding physical theories that we are modifying through an
``actual'' limit. Thus, a biological object is mathematically and
fundamentally different from a physical object because it may be
characterized in terms of partial but continual changes of symmetry
within an interval of viability, as an extended locus of critical
transitions. In particular, this mathematical view of ``partial
preservation through symmetry changes'' is a way to characterize the
joint dynamics of \emph{structural stability} and
\emph{variability} proper to life. We thus consider this
characterization as a tool for the mathematical intelligibility of
fundamental biological principles: the global/structural stability is crucially associated
with variability.
A first consequence of these permanent symmetry changes is that there
are very few invariants in biology. Mathematically, invariants depend
on stable symmetries. Structural stability in biology, thus, should be
understood more in terms of \emph{correlations of symmetries within
an interval of the extended critical transition,} rather than on their
identical preservation. It is clear that the \emph{bauplan} and a few
more properties may be ``identically'' preserved. Yet, in biology,
theoretical\emph{ }invariants are continually broken by these
symmetry changes. A biological object (a cell, a multicellular
organism, a species) continually changes symmetries, with respect to
all control parameters, including time. Each mitosis is a symmetry
change because the two new cells are not identical. This variability,
under the mathematical form of symmetry breaking and constitution of
new symmetries, is essential both for evolution and embryogenesis. The
interval of criticality is then the ``space of viability'' or locus of
the possible structural stability.
The changes of symmetries in the dense interval of criticality, which
provide a mathematical understanding of biological variability, are a
major challenge for theorizing. As a matter of fact, we are
accustomed to the theoretical stability warranted by the mathematical
invariants at the core of physics. These invariants are the result of
symmetries in the mathematical (equational) determination of the
physical object. This lack of invariants and symmetries corresponds to
the difficulties in finding equational determinations in
biology\footnote{In a rather naive way, some say
this by observing that any (mathematized) theory in biology has a
``counterexample''. This instability of the
determination goes together with the ``structural
stability'' of biological entities. This is largely due to
the stabilizing role of integration and regulation effects between
different levels of organization. The mathematics of extended
criticality and of variants of the renormalization methods are yet to
be developed.}.
As a further consequence of our approach, phylogenetic or ontogenetic
trajectories cannot be defined by the geodesic principle, since they
are not determined by invariants and their associated symmetries. These
latter are continually changing in a relatively minor but extended way.
Biology may be considered to be in an opposite situation with respect to
physics: in contrast to physics, in biology, \emph{trajectories
}are\emph{ generic }whereas\emph{ objects }are \emph{
specific} \parencite{bailly2011}. That is, a rat, a monkey
or an elephant are the \emph{specific} results of \emph{possible}
(generic) evolutionary trajectories of a common mammal ancestor --- or
each of these individuals is \emph{specific}. They respectively are
the result of a unique constitutive history, yet a possible or
\emph{generic} one \parencite{bailly1993,bailly2011}.
The evolutionary or ontogenetic trajectory of a cell, a multicellular
organism or a species is just a \emph{possible} or
\emph{compatible} path within the ecosystem. The genericity of the
biological trajectories implies that, in contrast to what is common in
physics, we cannot mathematically and \emph{a priori} determine the
ontogenetic and phylogenetic trajectory of a living entity be it an
individual or a species. In other words, in biology, we should consider
\emph{generic} trajectories (or possible paths) whose only
constraints are to remain compatible with the survival of the intended
biological system. Thus, phylogenesis and embryogenesis are
\emph{possible} paths subject to various constraints, including of
course the inherited structure of the \textsc{dna}, of the cell and the
ecosystem. The \emph{specificity} of the biological object, instead,
is the result of critical points and of symmetry \emph{changes} of
the system considered \emph{along its past history} (evolutive and
ontogenetic). These constitute the specific ``properties'' of this
object, which allow to define it. A rat, a monkey or an elephant or
their species are \emph{specific} and cannot be interchanged either
as individuals nor as species. A living entity is the result of its
history and cannot be defined ``generically'' in terms of invariants and
symmetries as it is done for physical objects.
This situation has a particular meaning when we consider time
translation and time reversal symmetries. In physics, time symmetries
correspond to the maintaining of the system’s invariant quantities that
define the geodesics, as for example, conservation of energy. In
biology both symmetries are broken. In particular, evolutionary and
ontogenetic paths are both irreversible and non-iteratable; there is no
way to identically ``rewind'' nor ``restart'' evolution or
ontogenesis. This corresponds to the breaking of time translation and
reversal symmetries. In particular, this lack of time symmetries is
associated with the process of \emph{individuation}, understood here
as the specificity of cells, organisms and species (as much as this
latter notion is well defined). It is crucial to understand that time
plays a key role in this framework, since the \emph{history} of all
the changes in symmetry are not reducible to a specific trajectory in a
given space of the dynamics. Thus,
{\centering
\emph{The sequence of symmetry changes defines the historical
contingency of a living object’s phylogenetic or ontogenetic
trajectory}.
\par}
Biological processes are more ``history based'' than physical processes.
Usual physical processes preserve invariants, whereas extended critical
transitions are a permanent reconstruction of organization and
symmetries, i.e., of invariants. This situation also points to a lack
of symmetry by permutation. For example, even in a clonal population of
bacteria, different bacteria are not generic, because they are in
general not interchangeable, i.e., they cannot be permuted. This allows
to understand biological variability in a deeper way than the usual
Gaussian (or combination of Gaussians) as random distribution of a set
of observables. Now, let us consider organs (and organelles). Some
organs have a functional role that can be expressed in a physical
framework, particularly as far as energy transfer is concerned. This
functional role can lead to restrictions on the variability of the
cells that constitute the organ, while the same could be said for
individual organisms in populations. At least for certains aspects of
their behaviour and on average, these restrictions make cells behave
symmetrically. In other words, those cells behave, in part and
approximately, like generic objects with specific trajectory
(geodesics). They may be interchangeable, like physical objects.
The simple case of cells secreting a protein such as erythropoietin
(\textsc{epo}) under specific conditions indicate that on average, a sufficient
amount of the protein must be produced, independently of the individual
contribution of each cell (which become ``relatively'' generic). Since
the result of these cells’ production is additive (linear), its
regulation does not need to be sharp. Even if some cells do not produce
\textsc{epo} there is no functional problem as long as a sufficient quantity of
this protein is secreted at the tissue level. However, when cells
contribute to a non-linear framework as part of an organ, the
regulation may need to be sharper. This is the case, for example, for
neuronal networks or for cell proliferation where non-linear effects
may be very important. In the latter case, regulation by the tissue and
the organism seems to hold back pathological developments, like cancer,
see \textcite{Society}. This point of view can possibly be
generalized in order to understand the robustness of development.
The role of physical processes in shaping organs is crucial; for
example, exchanges of energy (or matter) force/determine the optimal
(geodesic) fractal structure of lungs and vascular systems. Organs in
an organism may even be replaced by man-made artifacts (as for kidneys,
heart, limbs, etc.). As biological entities, organisms and even cells
are specific or, at most, weakly generic given that they can be
interchanged only within a given population or tissue and occasionally.
In general, they are not generic, and by their specificity they cannot
be replaced by an artifact --- structurally.
In summary, in critical transitions one may consider variables depending
on global processes because of the formation of coherent structures.
For example, there may be functional dependencies on a network of
interactions, which cannot be split into a sum of many local
dependencies (local variables). Thus, the search for more variables
would not take into account this fundamental property of biological
systems, considered as extended critical transitions. Moreover,
symmetries in physics allow to define generic objects which follow
specific trajectories (the latter allowing to find invariants in terms
of symmetries, which are robust regarding measurement). On the
contrary, in biology, the continual symmetry changes lead to generic
trajectories that remain compatible with the survival of the system.
The generic/specific duality with respect to physics helped us
understand this key issue, in relation to extended criticality --- which
is a form of ``relatively stable instability.'' In other words, this is
stability under changes of symmetries in an interval of viability. In a
sense, the biological object is also defined by its symmetries but in a
very different way: it is the \emph{specific} result of a history,
where its dynamics is punctuated by symmetry changes. This makes it
``historical'' and \emph{contingent}.
\section{Additional characteristics of extended criticality}
In physics, criticality implies more than a pointwise symmetry change;
that is, it requires a change on a mathematical point, as it leads to
peculiar behaviors that are relevant to biology. The first of these
properties is that criticality implies a global determination, instead
of a simply local one. More precisely, the singularities involved in
criticality lead to a change of the level of organization in a very
strong sense. Also in physics, in view of the mathematical divergence
of some observables, the singularities break the ability of the ``down
level'' to provide a causal account of the phenomena and they lead to
the need for a ``top level'' to overcome this difficulty. In mathematical
physics, this upper level can be found in the renormalization operator
(it is the abstract level of \emph{changing scale}). In biology,
instead, the upper level is the functional unity of an organism. As a
result, the existence of different levels of organization is a
component of our notion of extended critical transition. ``Downward
causation'' may find the right frame of analysis in this theoretical
context.
The permanent reconstruction of these levels of organization is
mathematically represented by the density of the critical points and by
the continual change of determination (symmetry change) in the passage
between these points within the interval of extended criticality.
The second property is the presence of power laws which seem to be
ubiquitous in biology. They appear regularly especially when
regulation is concerned, such as in
cardiac rhythms \parencites{Makowiec2006,pikkujamsa1999cardiac}, blood
cell number regulation \parencite{Perazzo00}, blood pressure \parencite{wagner1996chaos}, in brain activities \parencite{GerhardWerner07}, sensory cells
\parencite{SebastienCamalet99}, mitochondrial networks \parencite{aon2004percolation}, in ecology \parencite{Sole99} and
gene networks \parencite{IlyaShmulevich05,nykter2008gene}.
Extended critical transitions also concern the relevant lengths of local
and global exchanges, the temporalities mobilized for such exchanges
and biological rhythms. To summarize, the extended critical situation
has at least the following characteristics \parencite{bailly2008,bailly2011}:
\begin{enumerate}
\item A spatial volume enclosed within a semi-permeable membrane;
\item Correlation lengths of the order of magnitude of the greatest
length of the above referred volume;
\item A metabolic activity that is far from equilibrium and
irreversible, involving exchanges of energy, of matter and of entropy
with the environment, as well as the production of entropy due to all
these irreversible processes, see \textcite{bailly2009};
\item An anatomo-functional structuralization into levels of
organization that can be autonomous but also coupled to each other.
They are ``entangled'' in the sense defined by \textcite{bailly2009,soto2008}. These
levels are likely to be distinguished by the existence of fractal
geometries (membranous or arborescent), where the fractal geometries
can be considered as the trace (or a ``model'') of effective passages to
the infinite limit of an intensive magnitude of the system (for
example, local exchanges of energy\footnote{The
fractal dimension of some organs may be calculated by optimizing the
purely physical exchanges within the intended topological dimension
(for example, the maximization, within a volume, of surfaces for lungs,
or of volumes for the vascular system, \nptextcite{west1997}), and it may be
subjected to constraints in terms
of stericity and homogeneity, as in the cases mentioned (lung, vascular
system, kidney, etc).}). The different levels of organization induce,
and are a consequence of, the alternation of ``organs'' and ``organisms'',
such as organelles in cells, which, in turn, make up the organs in
multicellular organisms. Organisms stay in
an extended critical transition, while organs are partially ``optimally shaped'' by
the exchange of physical energy and matter. For example, fractal
geometries essentially manifest in organs that are also the privileged
loci of endogenous rhythms (see below). Correlation lengths are
manifested both \emph{in} and \emph{between} these
levels\footnote{ The term ``entanglement'' in \textcite{soto2008} does not correspond, of course, to the physical meaning of
``quantum entanglement'' as expressed by Schrödinger’s treatment of the
state function and the inseparability of quantum measure, yet it may be
appropriate because there is no
way to isolate one of the organs mentioned above (e.g. put a brain in a
flowerpot) and perform any reasonable physiological measure on it.}.
Likewise, the various biological ``clocks'' are coupled, and in some
cases even synchronized, within and between these levels.
\end{enumerate}
With the purpose of providing biological temporality with a structuring
of the mathematical type, we will consider two other aspects as being
specific to extended criticality.
\begin{itemize}
\item The two-dimensionality of time, proposed in \parencite{bailly2011b}:
\begin{enumerate}
\item One dimension is classical and is parametrized according to the
line of real numbers limited by fertilization on one side, and death on
the other. This dimension is linked to the bio-physicochemical
evolution of the organism in relation to an environment.
\item The other dimension is compactified, i. e. it is parametrized on a
circle. This second dimension is linked to the organism’s endogenous
physiological rhythm that is manifested through \emph{numeric
quantities without dimension} such as the mean total number of
heartbeats and respirations during the lifetime of mammals. These are
the interesting interspecific invariants and they are ``pure'' numbers,
\emph{not frequencies} (they have no dimension; they are the ``total
number of \dots''). They become frequencies (with the inverse of time as a
dimension), according to the average lifespan. The extra dimension is
needed exactly because the invariant phenomenon is not defined by a
period which has the dimension of time, but by this new invariant
observable. For example, on average, the identical (invariant) number
of total heartbeats give different frequencies according to the
different lifespans of an elephant or of a mouse.
\end{enumerate}
Moreover, the temporality of extended criticality involve protention
(i.e. pre-conscious expectation) and retention (i. e. pre-conscious
memory) \parencite{longo2011}, which seems to lead to a breaking of conservation of
information in cognition.
\item The confinement within a volume of a parameter space (such as
temperature, pressure, etc) of $n$ dimensions of which $3$ are
spatial and $2$ temporal and whose measure is different from $0$ (see
above).
\end{itemize}
\section{ Conclusion}
Since ancient Greece (Archimedes’ principle on equilibria) up to
Relativity Theory (and Noether’s and Weyl’s work) and Quantum Mechanics
(from Weyl’s groups to the time-charge-parity symmetry), symmetries
have provided a unified view of the principles of theoretical
intelligibility in physics. We claimed here that some major challenges
for the proposal of mathematical and theoretical ideas in biology
depend, in principle, on the very different roles that symmetries play
in biology when compared to physics. The unifying theoretical framework
in biology is neither associated to invariants nor to transformations
preserving invariants like in (mathematical/theoretical) physics. It
focuses, instead, on the permanent change of symmetries that
\emph{per se} modify the analysis of the internal and external
processes of life, both in ontogenesis and evolution.
In a sense, variability may be considered as the main invariant of the
living state of matter. In order to explain it, we proposed to consider
the role played by local and global symmetry changes along extended
critical transitions. In extended criticality, dynamically changing
coherent structures as global entities provide an understanding of
variability within a global, extended stability. The coherent
structure of critical phenomena also justifies the use of variables
depending on non-local effects. Thus, an explicitly systemic approach
may help in avoiding the accumulation of models and previously hidden
variables. In conclusion, the notion of extended criticality provides a
conceptual framework, to be further mathematized, where the dynamics of
symmetries and symmetry breakings provide a new, crucial role for
symmetries in biology with respect to physics.
\paragraph*{Aknowledgement: } We warmly thank the editors of the published version of this chapter for several and very close preliminary revisions of this conceptually difficult text.
\printbibliography[heading=bibintoc]
\end{document}
|
2,877,628,090,785 | arxiv | \section{Introduction}
Pioneered by Batch Normalization (BN)~\cite{BatchNorm}, feature normalization has become ubiquitous in the development of deep learning. Feature normalization consists of two components: \textit{feature standardization} and \textit{channel-wise affine transformation}. The latter is introduced to provide the capability of undoing the standardization (by design), and can be treated as \textit{feature re-calibration} in general. Many variants of BN have been proposed for practical deployment in terms of variations of training and testing settings with remarkable progress obtained. They can be roughly divided into two categories:
\begin{figure*} [t]
\centering
\includegraphics[width=0.9\linewidth]{Fig/AN.pdf}
\caption{Illustration of the proposed Attentive Normalization (AN). AN aims to harness the best of a base feature normalization (e.g., BN or GN) and channel-wise feature attention in a single light-weight module. See text for details.}
\label{fig:AN}
\end{figure*}
\textit{i) Generalizing feature standardization.} Different methods are proposed for computing the mean and standard deviation or for modeling/whitening the data distribution in general, within a min-batch. They include Batch Renormalization~\cite{BatchReNorm}, Decorrelated BN~\cite{DecorBN}, Layer Normalization (LN)~\cite{LayerNorm}, Instance Normalization (IN)~\cite{InstNorm}, Instance-level Meta Normalization~\cite{InstMetaNorm}, Group Normalization (GN)~\cite{GroupNorm}, Mixture Normalization~\cite{MixtureNorm} and Mode Normalization~\cite{ModeNorm}. Switchable Normalization (SN)~\cite{SwitchNorm} and its sparse variant (SSN)~\cite{SSN} learn to switch between different vanilla schema. These methods adopt the vanilla channel-wise affine transformation after standardization, and are often proposed for discriminative learning tasks.
\textit{ii) Generalizing feature re-calibration.} Instead of treating the affine transformation parameters directly as model parameters, different types of task-induced conditions (\emph{e.g.}, class labels in conditional image synthesis using generative adversarial networks) are leveraged and encoded as latent vectors, which are then used to learn the affine transformation parameters, including different conditional BNs~\cite{CBatchNorm1,CBatchNorm2,ConditionNorm,CGAN,BigGAN}, style-adaptive IN~\cite{StyleGAN} or layout-adaptive IN~\cite{SPatialAdaNorm,ISLANorm}. These methods have been mainly proposed in generative learning tasks, except for the recently proposed Instance-level Meta Normalization~\cite{InstMetaNorm} in discriminative learning tasks.
In the meanwhile, \textit{feature attention} has also become an indispensable mechanism for improving task performance in deep learning. For computer vision, spatial attention is inherently captured by convolution operations within short-range context, and by non-local extensions~\cite{NonlocalNet,CrissCross} for long-range context. Channel-wise attention is relatively less exploited. The squeeze-and-excitation (SE) unit~\cite{SENet} is one of the most popular designs, which learn instance-specific channel-wise attention weights to re-calibrate an input feature map. Unlike the affine transformation parameters in feature normalization, the attention weights for re-calibrating an feature map are often directly learned from the input feature map in the spirit of self-attention, and often instance-specific or pixel-specific.
Although both feature normalization and feature attention have become ubiquitous in state-of-the-art DNNs, they are usually studied as separate modules. Therefore, in this paper we address the following problem:
\textit{How to learn to re-calibrate feature maps in a way of harnessing the best of feature normalization and feature attention in a single light-weight module?}
And, we present \textbf{Attentive Normalization (AN)}: Fig.~\ref{fig:AN} illustrates the proposed AN. The basic idea is straightforward. Conceptually, the affine transformation component in feature normalization (Section~\ref{sec:featnorm}) and the re-scaling computation in feature attention play the same role in learning-to-re-calibrate an input feature map, thus providing the foundation for integration (Section~\ref{sec:featattn}). More specifically, consider a feature normalization backbone such as BN or GN, our proposed AN keeps the block-wise standardization component unchanged. Unlike the vanilla feature normalization in which the affine transformation parameters ($\gamma$'s and $\beta$'s) are often frozen in testing, we want the affine transformation parameters to be adaptive and dynamic in both training and testing, controlled directly by the input feature map. The intuition behind doing so is that it will be more flexible in accounting for different statistical discrepancies between training and testing in general, and between different sub-populations caused by underlying inter-/intra-class variations in the data.
To achieve the dynamic and adaptive control of affine transformation parameters, the proposed AN utilizes a simple design (Section~\ref{sec:AN}). It learns a mixture of $K$ affine transformations and exploits feature attention mechanism to learn the instance-specific weights for the $K$ components. The final affine transformation used to re-calibrate an input feature map is the weighted sum of the learned $K$ affine transformations. We propose a general formulation for the proposed AN and study how to learn the weights in an efficient and effective way (Section~\ref{sec:learn_wts}).
\section{Related Work}
\textbf{Feature Normalization.}
There are two types of normalization schema, feature normalization (including raw data)~\cite{BatchNorm,BatchReNorm,LayerNorm,InstNorm,GroupNorm,SwitchNorm,SSN,MixtureNorm,ModeNorm} and weight normalization~\cite{WN,OrthWtsNorm}. Unlike the former, the latter is to normalize model parameters to decouple the magnitudes of parameter vectors from their directions.
We focus on feature normalization in this paper.
Different feature normalization schema differ in how the mean and variance are computed. BN~\cite{BatchNorm} computes the channel-wise mean and variance in the entire min-batch which is driven by improving training efficiency and model generalizability. BN has been deeply analyzed in terms of how it helps optimization~\cite{HowBNWorks}. DecorBN~\cite{DecorBN} utilizes a whitening operation (ZCA) to go beyond the centering and scaling in the vanilla BN. BatchReNorm~\cite{BatchReNorm} introduces extra parameters to control the pooled mean and variance to reduce BN's dependency on the batch size. IN~\cite{InstNorm} focuses on channel-wise and instance-specific statistics which stems from the task of artistic image style transfer. LN~\cite{LayerNorm} computes the instance-specific mean and variance from all channels which is designed to help optimization in recurrent neural networks (RNNs). GN~\cite{GroupNorm} stands in the sweet spot between LN and IN focusing on instance-specific and channel-group-wise statistics, especially when only small batches are applicable in practice. In practice, synchronized BN~\cite{SyncBN} across multiple GPUs becomes increasingly favorable against GN in some applications. SN~\cite{SwitchNorm} leaves the design choices of feature normalization schema to the learning system itself by computing weighted sum integration of BN, LN, IN and/or GN via softmax, showing more flexible applicability, followed by SSN~\cite{SSN} which learns to make exclusive selection. Instead of computing one mode (mean and variance), MixtureNorm~\cite{MixtureNorm} introduces a mixture of Gaussian densities to approximate the data distribution in a mini-batch. ModeNorm~\cite{ModeNorm} utilizes a general form of multiple-mode computation. Unlike those methods, the proposed AN focuses on generalizing the affine transformation component. Related to our work, Instance-level Meta normalization(ILM)~\cite{InstMetaNorm} first utilizes an encoder-decoder sub-network to learn affine transformation parameters and then add them together to the model's affine transformation parameters. Unlike ILM, the proposed AN utilizes a mixture of affine transformations and leverages feature attention to learn the instance-specific attention weights.
On the other hand, conditional feature normalization schema~\cite{CBatchNorm1,CBatchNorm2,ConditionNorm,BigGAN,StyleGAN,SPatialAdaNorm} \cite{ISLANorm} have been developed and shown remarkable progress in conditional and unconditional image synthesis. Conditional BN learns condition-specific affine transformations in terms of conditions such as class labels, image style, label maps and geometric layouts. Unlike those methods, the proposed AN learns self-attention data-driven weights for mixture components of affine transformations.
\textbf{Feature Attention.}
Similar to feature normalization, feature attention is also an important building block in the development of deep learning. Residual Attention Network~\cite{ResAttention} uses a trunk-and-mask joint spatial and channel attention module in an encoder-decoder style for improving performance. To reduce the computational cost, channel and spatial attention are separately applied in~\cite{CBAM}. The SE module~\cite{SENet} further simplifies the attention mechanism by developing a light-weight channel-wise attention method. The proposed AN leverages the idea of SE in learning attention weights, but formulates the idea in a novel way.
\textbf{Our Contributions.} This paper makes three main contributions: (i) It presents Attentive Normalization which harnesses the best of feature normalization and feature attention (channel-wise). To our knowledge, AN is the first work that studies self-attention based conditional and adaptive feature normalization in visual recognition tasks.
(ii) It presents a lightweight integration method for deploying AN in different widely used building blocks of ResNets, DenseNets, MobileNetsV2 and AOGNets.
(iii) It obtains consistently better results than the vanilla feature normalization backbones by a large margin across different neural architectures in two large-scale benchmarks, ImageNet-1000 and MS-COCO.
\section{The Proposed Attentive Normalization}\label{sec:AN}
In this section, we present details of the proposed attentive normalization.
Consider a DNN for 2D images, denote by $\mathbf{x}$ a feature map with axes in the convention order of $(N, C, H, W)$ (i.e., batch, channel, height and width). $\mathbf{x}$ is represented by a 4D tensor. Let $i=(i_N, i_C, i_H, i_W)$ be the address index in the 4D tensor. $\mathbf{x}_i$ represents the feature response at a position $i$.
\subsection{Background on Feature Normalization}\label{sec:featnorm}
Existing feature normalization schema often consist of two components (Fig.~\ref{fig:AN}):
\textit{i) Block-wise Standardization}. Denote by $B_j$ a block (slice) in a given 4-D tensor $\mathbf{x}$. For example, for BN, we have $j = 1, \cdots, C$ and $B_j=\{\mathbf{x}_i | \forall i, i_C=j\}$. We first compute the empirical mean and standard deviation in $B_j$, denoted by $\mu_j$ and $\sigma_j$ respectively: $\mu_j = \frac{1}{M}\sum_{x\in B_j} x,\quad
\sigma_j = \sqrt{\frac{1}{M}\sum_{x\in B_j} (x-\mu_j)^2 + \epsilon}$,
where $M=|B_j|$ and $\epsilon$ is a small positive constant to ensure $\sigma_j > 0$ for the sake of numeric stability. Then, let $j_i$ be the index of the block that the position $i$ belongs to, and we standardize the feature response by,
\begin{equation}
\hat{\mathbf{x}}_i = \frac{1}{\sigma_{j_i}} (\mathbf{x}_i - \mu_{j_i})
\end{equation}
\textit{ii) Channel-wise Affine Transformation}. Denote by $\gamma_c$ and $\beta_c$ the scalar coefficient (re-scaling) and offset (re-shifting) parameter respectively for the $c$-th channel. The re-calibrated feature response at a position $i$ is then computed by,
\begin{equation}
\Tilde{\mathbf{x}}_i = \gamma_{i_C} \cdot \hat{\mathbf{x}}_i + \beta_{i_C}, \label{eq:featnorm}
\end{equation}
where $\gamma_c$'s and $\beta_c$'s are shared by all the instances in a min-batch across the spatial domain. They are usually frozen in testing and fine-tuning.
\subsection{Background on Feature Attention}\label{sec:featattn}
We focus on channel-wise attention and briefly review the Squeeze-Excitation (SE) module~\cite{SENet}. SE usually takes the feature normalization result (Eqn.~\ref{eq:featnorm}) as its input (the bottom-right of Fig.~\ref{fig:AN}), and learns channel-wise attention weights:
\textit{i) The squeeze module} encodes the inter-dependencies between feature channels in a low dimensional latent space with the reduction rate $r$ (e.g., $r=16$),
\begin{equation}
S(\Tilde{\mathbf{x}};\theta_S) = v,\, v\in \mathbb{R}^{N\times \frac{C}{r}\times 1\times 1}, \label{eq:squeeze}
\end{equation}
which is implemented by a sub-network consisting of a global average pooling layer (AvgPool), a fully-connected (FC) layer and rectified linear unit (ReLU)~\cite{AlexNet}. $\theta_S$ collects all the model parameters.
\textit{ii) The excitation module} computes the channel-wise attention weights, denoted by $\lambda$, by decoding the learned latent representations $v$,
\begin{equation}
E(v;\theta_E)=\lambda,\, \lambda\in \mathbb{R}^{N\times C\times 1 \times 1},
\end{equation}
which is implemented by a sub-network consisting of a FC layer and a sigmoid layer. $\theta_E$ collects all model parameters.
Then, the input, $\Tilde{\mathbf{x}}$ is re-calibrated by,
\begin{align}
\Tilde{\mathbf{x}}^{SE}_i = \lambda_{i_N, i_C} \cdot \Tilde{\mathbf{x}}_i = (\lambda_{i_N, i_C} \cdot \gamma_{i_C}) \cdot \hat{\mathbf{x}}_i + \lambda_{i_N, i_C} \cdot \beta_{i_C}, \label{eq:se}
\end{align}
where the second step is obtained by plugging in Eqn.~\ref{eq:featnorm}. \textbf{It is thus straightforward to see the foundation facilitating the integration between feature normalization and channel-wise feature attention.}
However, the SE module often entails a significant number of extra parameters (e.g., $\sim$2.5M extra parameters for ResNet50~\cite{ResidualNet} which originally consists of $\sim$25M parameters, resulting in 10\% increase). We aim to design more parsimonious integration that can further improve performance
\subsection{Attentive Normalization}\label{sec:attnnorm}
Our goal is to generalize Eqn.~\ref{eq:featnorm} in re-calibrating feature responses to enable dynamic and adaptive control in both training and testing. On the other hand, our goal is to simplify Eqn.~\ref{eq:se} into a single light-weight module, rather than, for example, the two-module setup using BN+SE. In general, we have,
\begin{equation}
\Tilde{\mathbf{x}}^{AN}_i = \Gamma(\mathbf{x};\theta_{\Gamma})_i \cdot \hat{\mathbf{x}}_i + \mathbb{B}(\mathbf{x};\theta_{\mathbb{B}})_i, \label{eq:recalibration}
\end{equation}
where both $\Gamma(\mathbf{x};\theta_{\Gamma})$ and $\mathbb{B}(\mathbf{x};\theta_{\mathbb{B}})$ are functions of the entire input feature map (without standardization~\footnote{We tried the variant of learning $\Gamma()$ and $\mathbb{B}()$ from the standardized features and observed it works worse, so we ignore it in our experiments.}) with parameters $\theta_{\Gamma}$ and $\theta_{\mathbb{B}}$ respectively. They both compute 4D tensors of the size same as the input feature map and can be parameterized by some attention guided light-weight DNNs. The subscript in $\Gamma(\mathbf{x};\theta_{\Gamma})_i$ and $\mathbb{B}(\mathbf{x};\theta_{\mathbb{B}})_i$ represents the learned re-calibration weights at a position $i$.
In this paper, we focus on learning instance-specific channel-wise affine transformations. To that end, we have three components as follows.
\textit{i) Learning a Mixture of $K$ Channel-wise Affine Transformations.} Denote by $\gamma_{k,c}$ and $\beta_{k,c}$ the re-scaling and re-shifting (scalar) parameters respectively for the $c$-th channel in the $k$-th mixture component. They are model parameters learned end-to-end via back-propagation.
\textit{ii) Learning Attention Weights for the $K$ Mixture Components.} Denote by $\lambda_{n, k}$ the instance-specific mixture component weight ($n\in [1, N]$ and $k\in [1, K]$), and by $\lambda$ the $N\times K$ weight matrix. $\lambda$ is learned via some attention-guided function from the entire input feature map,
\begin{equation}
\lambda = A(\mathbf{x}; \theta_{\lambda}), \label{eq:attn_wts}
\end{equation}
where $\theta_{\lambda}$ collects all the parameters.
\textit{iii) Computing the Final Affine Transformation.} With the learned $\gamma_{k,c}$, $\beta_{k,c}$ and $\lambda$, the re-calibrated feature response is computed by,
\begin{equation}
\Tilde{\mathbf{x}}^{AN}_i = \sum_{k=1}^K \lambda_{i_N, k} [\gamma_{k, i_C}\cdot \hat{\mathbf{x}}_i + \beta_{k, i_C}], \label{eq:recalibration1}
\end{equation}
where $\lambda_{i_N, k}$ is shared by the re-scaling parameter and the re-shifting parameter for simplicity. Since the attention weights $\lambda$ are adaptive and dynamic in both training and testing, the proposed AN realizes adaptive and dynamic feature re-calibration. Compared to the general form (Eqn.~\ref{eq:recalibration}), we have,
\begin{equation}
\Gamma(\mathbf{x})_i= \sum_{k=1}^K \lambda_{i_N, k}\cdot \gamma_{k,i_C},\,\,
\mathbb{B}(\mathbf{x})_i=\sum_{k=1}^K \lambda_{i_N, k}\cdot \beta_{k,i_C}.
\end{equation}
Based on the formulation, there are \textbf{a few advantages of the proposed AN in training, fine-tuning and testing }a DNN:
\begin{itemize}
\item The channel-wise affine transformation parameters, $\gamma_{k,i_C}$'s and $\beta_{k,i_C}$'s, are shared across spatial dimensions and by data instances, which can learn population-level knowledge in a more fine-grained manner than a single affine transformation in the vanilla feature normalization.
\item $\lambda_{i_N, k}$'s are instance specific and learned from features that are not standardized. Combining them with $\gamma_{k,i_C}$'s and $\beta_{k,i_C}$'s (Eqn.~\ref{eq:recalibration1}) enables AN paying attention to both the population (what the common and useful information are) and the individuals (what the specific yet critical information are). The latter is particularly useful for testing samples slightly ``drifted" from training population, that is to improve generalizability.
Their weighted sum encodes more direct and ``actionable" information for re-calibrating standardized features (Eqn.~\ref{eq:recalibration1}) without being delayed until back-propagation updates as done in the vanilla feature normalization.
\item In fine-tuning, especially between different tasks (\emph{e.g.}, from image classification to object detection), $\gamma_{k,i_C}$'s and $\beta_{k,i_C}$'s are usually frozen as done in the vanilla feature normalization. They carry information from a source task. But, $\theta_{\lambda}$ (Eqn.~\ref{eq:attn_wts}) are allowed to be fine-tuned, thus potentially better realizing transfer learning for a target task.
This is a desirable property since we can decouple training correlation between tasks. For example, when GN~\cite{GroupNorm} is applied in object detection in MS-COCO, it is fine-tuned from a feature backbone with GN trained in ImageNet, instead of the one with BN that usually has better performance in ImageNet.
As we shall show in experiments, the proposed AN facilitates a smoother transition. We can use the proposed AN (with BN) as the normalization backbone in pre-training in ImageNet, and then use AN (with GN) as the normalization backbone for the head classifiers in MS-COCO with significant improvement.
\end{itemize}
\subsubsection{Details of Learning Attention Weights}\label{sec:learn_wts}
We present a simple method for computing the attention weights $A(\mathbf{x}; \theta_{\lambda})$ (Eqn.~\ref{eq:attn_wts}). Our goal is to learn a weight coefficient for each component from each individual instance in a mini-batch (i.e, a $N\times K$ matrix).
The question of interest is how to characterize the underlying importance of a channel $c$ from its realization across the spatial dimensions $(H, W)$ in an instance, such that we will learn a more informative instance-specific weight coefficient for a channel $c$ in re-calibrating the feature map $\mathbf{x}$.
In realizing Eqn.~\ref{eq:attn_wts}, the proposed method is similar in spirit to the squeeze module in SENets~\cite{SENet} to maintain light-weight implementation. To show the difference, let's first rewrite the vanilla squeeze module (Eqn.~\ref{eq:squeeze}),
\begin{equation}
v=S(\mathbf{x};\theta_S) = ReLU(fc(AvgPool(\mathbf{x});\theta_S))\, , \label{eq:squeeze1}
\end{equation}
where the mean of a channel $c$ (via global average pooling, $AvgPool(\cdot)$) is used to characterize its underlying importance. We generalize this assumption by taking into account both mean and standard deviation empirically computed for a channel $c$, denoted by $\mu_c$ and $\sigma_c$ respectively. More specifically, we compare three different designs using:
\begin{enumerate}
\item [i)] The mean $\mu_c$ only as done in SENets.
\item [ii)] The concatenation of the mean and standard deviation, $(\mu_c, \sigma_c)$.
\item [iii)] The coefficient of variation or the relative standard deviation (RSD), $\frac{\sigma_c}{\mu_c}$. RSD measures the dispersion of an underlying distribution (i.e., the extent to which the distribution is stretched or squeezed) which intuitively conveys more information in learning attention weights for re-calibration.
\end{enumerate}
RSD is indeed observed to work better in our experiments\footnote{In implementation, we use the reverse $\frac{\mu_c}{\sigma_c + \epsilon}$ for numeric stability, which is equivalent to the original formulation when combing with the $fc$ layer.}. Eqn.~\ref{eq:attn_wts} is then expanded with two choices,
\begin{align}
\textit{Choice 1: } A_1(\mathbf{x}; \theta_{\lambda}) &= Act(fc(RSD(\mathbf{x});\theta_{\lambda})), \label{eq:attn_wts1} \\
\nonumber \textit{Choice 2: } A_2(\mathbf{x}; \theta_{\lambda}) &= Act(BN(fc(RSD(\mathbf{x});\theta_{fc});\theta_{BN})),
\end{align}
where $Act(\cdot)$ represents a non-linear activation function for which we compare three designs:
\begin{itemize}
\item [i)] The vanilla $ReLU(\cdot)$ as used in the squeeze module of SENets.
\item [ii)] The vanilla $sigmoid(\cdot)$ as used in the excitation module of SENets.
\item [iii)] The channel-wise $softmax(\cdot)$.
\item [iv)] The piece-wise linear hard analog of the sigmoid function, so-called $hsigmoid$ function~\cite{mobilenetv3}, $hsigmoid(a) = \min(\max(a+3.0, 0), 6.0) / 6.0$.
\end{itemize}
The $hsigmoid(\cdot)$ is observed to work better in our experiments. In the Choice 2 (Eqn.~\ref{eq:attn_wts1}), we apply the vanilla BN~\cite{BatchNorm} after the FC layer, which normalizes the learned attention weights across all the instances in a mini-batch with the hope of balancing the instance-specific attention weights better. The Choice 2 improves performance in our experiments in ImageNet.
In AN, we have another hyper-parameter, $K$. For stage-wise building block based neural architectures such as the four neural architectures tested in our experiments, we use different $K$'s for different stages with smaller values for early stages. For example, for the 4-stage setting, we typically use $K=10, 10, 20, 20$ for the four stages respectively based on our ablation study. The underlying assumption is that early stages often learn low-to-middle level features which are considered to be shared more between different categories, while later stages learn more category-specific features which may entail larger mixtures.
\section{Experiments}\label{sec:exp}
In this section, we first show the ablation study verifying the design choices in the proposed AN. Then, we present detailed comparisons and analyses.
\begin{figure} [t]
\centering
\includegraphics[width=0.9\linewidth]{Fig/AN_Integration.pdf}
\caption{Illustration of integrating the proposed AN in different building blocks. The first two show the vanilla residual block and the SE-residual block. The remaining four are: the Basicblock and Bottleneck design of a residual block, the inverted residual block (used in MobileNetV2), and the DenseBlock.
For the residual block and its variants, the proposed AN is used to replace the vanilla BN(s) followed the last $3\times 3$ convolution in different blocks. This potentially enables jointly integrating local spatial attention (conveyed by the $3\times 3$ convolution) in learning the instance-specific attention weights, which is also observed helpful in~\cite{SWhitenning} and is shown beneficial for the SE module itself in our experiments (Table~\ref{table:se-results}). For the dense block, we replace the second vanilla BN (after the $1\times 1$ convolution applied to the concatenated features) with our AN.
\label{fig:ResBlockAN}
\end{figure}
\textbf{Data and Evaluation Metric.} We use two benchmarks, the ImageNet-1000 classification benchmark (ILSVRC2012)~\cite{ImageNet} and the MS-COCO object detection and instance segmentation benchmark~\cite{COCO}. The ImageNet-1000 benchmark consists of about $1.28$ million images for training, and $50,000$ for validation, from $1,000$ classes. We apply a single-crop with size $224\times224$ in evaluation. Following the common protocol, we report the top-1 and top-5 classification error rates tested using a single model on the validation set. For the MS-COCO benchmark, there are 80 categories of objects. We use {\tt train2017} in training and evaluate the trained models using {\tt val2107}. We report the standard COCO metrics of Average Precision (AP) at different intersection-over-union (IoU) thresholds, \emph{e.g.}, AP$_{50}$ and AP$_{75}$, for bounding box detection (AP$^{bb}_{IoU}$) and instance segmentation (AP$^m_{IoU}$), and the mean AP over IoU=$0.5:0.05:0.75$, AP$^{bb}$ and AP$^m$ for bounding box detection and instance segmentation respectively.
\textbf{Neural Architectures and Vanilla Feature Normalization Backbones.} We use four representative neural architectures: (i) \textit{ResNets}~\cite{ResidualNet} (ResNet50 and ResNet101), which are the most widely used architectures in practice, (ii) \textit{DenseNets}~\cite{DenseNet}, which are popular alternatives to ResNets, (iii) \textit{MobileNetV2}~\cite{sandler2018mobilenetv2}. MobileNets are popular architectures under mobile settings and MobileNetV2 uses inverted residuals and linear Bottlenecks, and (iv) \textit{AOGNets}~\cite{AOGNets}, which are grammar-guided networks and represent an interesting direction of network architecture engineering with better performance than ResNets and DenseNets. So, the improvement by our AN will be both broadly useful for existing ResNets, DenseNets and MobileNets based deployment in practice and potentially insightful for on-going and future development of more advanced and more powerful DNNs in the community.
In classification, we use BN~\cite{BatchNorm} as the feature normalization backbone for our proposed AN, denoted by \textbf{AN (w/ BN)}. We compare with the vanilla BN, GN~\cite{GroupNorm} and SN~\cite{SwitchNorm}. In object detection and instance segmentation, we use the Mask-RCNN framework~\cite{maskrcnn} and its cascade variant~\cite{cascadercnn} in the MMDetection code platform~\cite{mmdetection}. We fine-tune feature backbones pretrained on the ImageNet-1000 dataset. We also test the proposed AN using GN as the feature normalization backbone, denoted by \textbf{AN (w/ GN)} in the head classifier of Mask-RCNN.
\textbf{Where to Apply AN?} Fig.~\ref{fig:ResBlockAN} illustrates the integration of our proposed AN in different building blocks. At the first thought, it is straightforward to replace all vanilla feature normalization modules (\emph{e.g.}, BN) in a DNN. It may not be necessary to do so, similar in spirit to the SE-residual block which re-calibrates the residual part once in a building block. As we shall see, our ablation study supports the design choice shown in Fig.~\ref{fig:ResBlockAN}.
\textbf{Initialization of our AN.} The initialization of $\gamma_{k,c}$'s and $\beta_{k,c}$'s (Eqn.~\ref{eq:recalibration1}) is based on, $\gamma_{k, c} = 1.0 + \mathcal{N}(0, 1) \times 0.1$ and $\beta_{k,c}=\mathcal{N}(0, 1)\times 0.1$, where $\mathcal{N}(0, 1)$ represents the standard Gaussian distribution. This type of initialization is also adopted for conditional BN used in the BigGAN~\cite{BigGAN}.
\subsection{Ablation Study}\label{sec:ablation} \begin{wraptable}{r}{0.6\textwidth}
\centering
\small{
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{r|cccc}
\hline
Design Choices in AN (w/ BN) & \#Params & FLOPS & top-1 & top-5 \\ \hline
\textbf{mean} + $A_2(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & 21.85 & 5.92 \\ \hline
\textbf{(mean,std)} + $A_2(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.82M & 4.09G & 21.73 & 5.85 \\ \hline
RSD + ${\textbf{A}_1}(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & 21.76 & 6.05 \\ \hline
RSD + $A_2(\cdot)$ + \textbf{softmax} + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & 21.72 & 5.90 \\ \hline
RSD + $A_2(\cdot)$ + \textbf{relu} + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20} \end{smallmatrix}\bigr)$ & 25.96M & 4.09G & 21.89 & 6.04 \\ \hline
RSD + $A_2(\cdot)$ + \textbf{sigmoid} + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & 21.96 & 5.91 \\ \hline
RSD + $A_2(\cdot)$ + hsigmoid + $\textbf{K}=\bigl(\begin{smallmatrix} {5}\\ {5}\\ {10}\\{10}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & 21.92 & 5.93 \\ \hline
RSD + $A_2(\cdot)$ + hsigmoid + $\textbf{K}=\bigl(\begin{smallmatrix} {20}\\ {20}\\ {40}\\{40} \end{smallmatrix}\bigr)$ & 25.96M & 4.09G & 21.62 & 5.63 \\ \hline
RSD + $A_2(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 25.76M & 4.09G & \textbf{21.59} & \textbf{5.58} \\ \bottomrule
\textbf{*} RSD + $A_2(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$ & 26.96M & 4.10G & 22.15 & 6.24 \\ \bottomrule
\end{tabular} }}
\\ [1ex]
\caption{Ablation study on different design choices in AN with BN as feature normalization backbone using ResNet50+Bottleneck in ImageNet-1000.
* means AN is applied to all the BNs of the network. }\label{table:ablation}
\end{wraptable}
We compare different design choices in our proposed AN using ResNet50 in ImageNet-1000. Table~\ref{table:ablation} summarizes the results. There are four categories of design choices: The first three are related to the realization of learning the attention weights (Eqn.~\ref{eq:attn_wts}): three types of inputs, two architectural choices and four activation function choices. The last one refers to the number $K$ of components in the mixture of affine transformation which is used for each of the four stages in ResNet50 and we empirically select three options for simplicity. All the models are trained using the same settings (the vanilla setup in Section~\ref{sec:results_imagenet}).
\textbf{The best combination} is RSD + $A_2(\cdot)$ + hsigmoid + $K=\bigl(\begin{smallmatrix} {10}\\ {10}\\ {20}\\{20}
\end{smallmatrix}\bigr)$. During our development, we first observed the best combination based on our intuitive reasoning and small experiments (a few epochs) in the process, and then design this ablation study to verify the design choices. Based on the observed best combination, we further verify that \textit{replacing all vanilla BNs is not helpful} (the last row in Table~\ref{table:ablation}). One explanation is that we may not need to re-calibrate the features using our AN (as well as other channel-wise feature attention methods) for both before and after a $1\times 1$ convolution, since channel-wise re-calibration can be tackled by the $1\times 1$ convolution kernel and the vanilla feature normalization themselves in training.
The ablation study is in support of the intuitions and design choices discussed in Section~\ref{sec:learn_wts}.
\subsection{Image Classification in ImageNet-1000}\label{sec:results_imagenet}
\textit{Common Training Settings.}
We use 8 GPUs (NVIDIA V100) to train models \begin{wraptable}{r}{0.5\textwidth}
\centering
\small{
\resizebox{0.49\textwidth}{!}{
\begin{tabular}{lrccc}
\toprule
\multicolumn{5}{c}{\textit{The Vanilla Setup}} \\ \hline
Method & \#Params & FLOPS & top-1 & top-5 \\ \hline
ResNet34+BN & 21.80M & 3.68G & 25.58$_{\downarrow (1.15)}$ & 8.19$_{\downarrow (0.76)}$ \\
\textbf{ResNet34+AN} & 21.92M & 3.68G & \textbf{24.43} & \textbf{7.43} \\\bottomrule
ResNet50-BN & 25.56M & 4.09G & 23.01$_{\downarrow (1.42)}$ & 6.68$_{\downarrow (0.80)}$ \\
$^\dagger$ResNet50-GN~\cite{GroupNorm} & 25.56M & 4.09G & 23.52$_{\downarrow (1.93)}$ & 6.85$_{\downarrow (0.97)}$ \\
$^\dagger$ResNet50-SN~\cite{SwitchNorm} & 25.56M & - & 22.43$_{\downarrow (0.83)}$ & 6.35$_{\downarrow (0.47)}$ \\
$^\dagger$ResNet50-SE~\cite{SENet} & 28.09M & 4.12G & 22.37$_{\downarrow (0.78)}$ & 6.36$_{\downarrow (0.48)}$ \\
ResNet50-SE & 28.09M & 4.12G & 22.35$_{\downarrow (0.76)}$ & 6.09$_{\downarrow (0.21)}$ \\
\textbf{ResNet50-AN} & 25.76M & 4.09G & \textbf{21.59} & \textbf{5.88} \\
\bottomrule
ResNet101-BN & 44.57M & 8.12G & 21.33$_{\downarrow (0.72)}$ & 5.85$_{\downarrow (0.44)}$ \\
\textbf{ResNet101-AN} & 45.00M & 8.12G & \textbf{20.61} & \textbf{5.41} \\\bottomrule
DenseNet121-BN & 7.98M & 2.86G & 25.35$_{\downarrow (2.73)}$ & 7.83$_{\downarrow (1.41)}$ \\
\textbf{DenseNet121-AN} & 8.34M & 2.86G & \textbf{22.62} & \textbf{6.42} \\ \bottomrule
MobileNetV2-BN & 3.50M & 0.34G & 28.69$_{\downarrow (2.02)}$ & 9.33$_{\downarrow (0.77)}$ \\
\textbf{MobileNetV2-AN} & 3.56M & 0.34G & \textbf{26.67} & \textbf{8.56} \\ \bottomrule
AOGNet12M-BN & 12.26M & 2.19G & 22.22$_{\downarrow (0.94)}$ & 6.06$_{\downarrow (0.30)}$ \\
\textbf{AOGNet12M-AN} & 12.37M & 2.19G & \textbf{21.28} & \textbf{5.76} \\ \hline
AOGNet40M-BN & 40.15M & 7.51G & 19.84$_{\downarrow (0.51)}$ & 4.94$_{\downarrow (0.22)}$ \\
\textbf{AOGNet40M-AN} & 40.39M & 7.51G & \textbf{19.33} & \textbf{4.72} \\ \bottomrule \bottomrule
\multicolumn{5}{c}{\textit{The State-of-the-Art Setup}} \\ \hline
Method & \#Params & FLOPS & top-1 & top-5 \\ \hline
ResNet50-BN & 25.56M & 4.09G & 21.08$_{\downarrow (1.16)}$ & 5.56$_{\downarrow (0.52)}$ \\
\textbf{ResNet50-AN} & 25.76M & 4.09G & \textbf{19.92} & \textbf{5.04} \\ \bottomrule
ResNet101-BN & 44.57M & 8.12G & 19.71$_{\downarrow (0.86)}$ & 4.89$_{\downarrow (0.26)}$ \\
\textbf{ResNet101-AN} & 45.00M & 8.12G & \textbf{18.85} & \textbf{4.63} \\\bottomrule
AOGNet12M-BN & 12.26M & 2.19G & 21.63$_{\downarrow (1.06)}$ & 5.60$_{\downarrow (0.22)}$ \\
\textbf{AOGNet12M-AN} & 12.37M & 2.19G & \textbf{20.57} & \textbf{5.38} \\ \bottomrule
AOGNet40M-BN & 40.15M & 7.51G & 18.70$_{\downarrow (0.57)}$ & 4.47$_{\downarrow (0.21)}$ \\
\textbf{AOGNet40M-AN} & 40.39M & 7.51G & \textbf{18.13} & \textbf{4.26} \\ \bottomrule
\end{tabular} }}
\caption{Comparisons between BN and our AN (w/ BN) in terms of the top-1 and top-5 error rates (\%) in the ImageNet-1000 validation set using \textit{the vanilla setup} and \textit{the state-of-the-art setup}.
$^\dagger$ means the model is not trained by us.
All other models are trained from scratch under the same settings.
}\label{table:imagenet-results}
\end{wraptable} using the same settings for apple-to-apple comparisons. The method proposed in~\cite{KaimingNormInit} is used to initialize all convolutions for all models. The batch size is 128 per GPU.
with FP16 optimization used in training to reduce the training time.
The mean and standard deviation for block-wise standardization are computed \textit{within} each GPU. The initial learning rate is $0.4$, and the cosine learning rate scheduler~\cite{cosine_lr} is used with $5$ warm-up epochs and weight decay $1\times10^{-4}$ and momentum $0.9$. For AN, the best practice observed in our ablation study (Table~\ref{table:ablation}) is used. AN is not used in the stem layer in all the models. In addition to the common settings, we have two different setups in experimental comparisons:
\textit{i) The Vanilla Setup.} We adopt the basic data augmentation scheme (random crop and horizontal flip) in training as done in~\cite{ResidualNet}. We train the models for 120 epochs. All ResNets~\cite{ResidualNet} use the vanilla stem layer with $7\times 7$ convolution. The MobileNetsV2 uses $3\times 3$ convolution in the stem layer. The AOGNets use two consecutive $3\times 3$ convolution in the stem layer. All the $\gamma$ and $\beta$ parameters of the feature normalization backbones are initialized to $1$ and $0$ respectively.
\textit{ii) The State-of-the-Art Setup.} There are different aspects in the vanilla setup which have better variants developed with better performance shown~\cite{BagofTricksImgCls}. \textit{We want to address whether the improvement by our proposed AN are truly fundamental or will disappear with more advanced tips and tricks added in training ConvNets.} First, on top of the basic data augmentation, we also use label smoothing~\cite{LabelSmoothing} (with rate 0.1) and the mixup (with rate 0.2)~\cite{mixup}. We increase the total number of epochs to 200. We use the same stem layer with two consecutive $3\times 3$ convolution for all models. For ResNets, we add the zero $\gamma$ initialization trick, which uses 0 to initialize the last normalization layer to make the initial state of a residual block to be identity.
\textbf{Results Summary.} Table~\ref{table:imagenet-results} shows the comparison results for the two setups respectively. \textbf{Our proposed AN consistently obtains the best top-1 and top-5 accuracy results with more than 0.5\% absolute top-1 accuracy increase (up to 2.7\%) in all models without bells and whistles.} \textit{The improvement is often obtained with negligible extra parameters} (e.g., 0.06M parameter increase in MobileNetV2 for 2.02\% absolute top-1 accuracy increase, and 0.2M parameter increase in ResNet50 with 1.42\% absolute top-1 accuracy increase) \textit{at almost no extra computational cost} (up to the precision used in measuring FLOPs). With ResNet50, our AN also outperforms
GN~\cite{GroupNorm} and SN~\cite{SwitchNorm} by 1.93\% and 0.83\% in top-1 accuracy respectively. For GN, it is known that it works (slightly) worse than BN under the normal (big) mini-batch setting~\cite{GroupNorm}. For SN, \begin{wraptable}{r}{0.5\textwidth}
\centering
\small{
\resizebox{0.49\textwidth}{!}{
\begin{tabular}{lrccc}
\hline
Method & \#Params & FLOPS & top-1 & top-5 \\ \hline
ResNet50-SE (BN$_3$) & 28.09M & 4.12G & 22.35$_{\downarrow (0.76)}$ & 6.09$_{\downarrow (0.21)}$ \\
ResNet50-SE (BN$_2$) & 26.19M & 4.12G & 22.10$_{\downarrow (0.55)}$ & 6.02$_{\downarrow (0.14)}$ \\
ResNet50-SE (All) & 29.33M & 4.13G & 22.13$_{\downarrow (0.52)}$ & $5.96_{\downarrow (0.08)}$ \\ \hline
ResNet50-AN (w/BN$_3$) & 26.35M & 4.11G & 21.78$_{\downarrow (0.19)}$ & 5.98$_{\downarrow (0.1)}$ \\
\textbf{ResNet50-AN} (w/BN$_2$) & \textbf{25.76M} & \textbf{4.09G} & \textbf{21.59} & \textbf{5.88} \\
ResNet50-AN (All) & 25.92M & 4.10G & 21.85$_{\downarrow (0.26)}$ & 6.06$_{\downarrow (0.18)}$ \\
\bottomrule
\end{tabular} }}
\caption{Comparisons between SE and our AN (w/ BN) in terms of the top-1 and top-5 error rates (\%) in the ImageNet-1000 validation set using \textit{the vanilla setup}. By ``(All)", it means SE or AN is used for all the three BNs in a bottleneck block.}\label{table:se-results}
\end{wraptable} our result shows that it is more beneficial to improve the re-calibration component than to learn-to-switch between different feature normalization schema.
We observe that the proposed AN is more effective for small ConvNets in terms of performance gain. Intuitively, this makes sense. Small ConvNets usually learn less expressive features. With the mixture of affine transformations and the instance-specific channel-wise feature re-calibration, the proposed AN offers the flexibility of clustering intra-class data better while separating inter-class data better in training.
\textbf{Comparisons with the SE module.} Our proposed AN provides a strong alternative to the widely used SE module. Table~\ref{table:se-results} shows the comparisons. We observe that applying SE after the second BN in the bottleneck in ResNet50 is also beneficial with better performance and smaller number of extra parameters.
\begin{table*} [t]
\centering
\small{
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{l|ll|c|ccc|ccc}
Architecture & Backbone & Head & \#Params & AP$^{bb}$ & AP$^{bb}_{50}$ & AP$^{bb}_{75}$ & AP$^{m}$ & AP$^{m}_{50}$ & AP$^{m}_{75}$ \\ \toprule
\multirow{2}{*}{\rotatebox[origin=c]{0}{MobileNetV2}} & $\mathbb{BN}$ & - & 22.72M & 34.2$_{\downarrow {(1.8)}}$ & 54.6$_{\downarrow {(2.4)}}$ & 37.1$_{\downarrow {(1.8)}}$ & 30.9$_{\downarrow {(1.6)}}$ & 51.1$_{\downarrow {(2.7)}}$ & 32.6$_{\downarrow {(1.9)}}$ \\
& AN (w/ BN) & - & 22.78M & \textbf{36.0} & \textbf{57.0} & \textbf{38.9} & \textbf{32.5} & \textbf{53.8} & \textbf{34.5} \\ \bottomrule
\multirow{5}{*}{\rotatebox[origin=c]{0}{ResNet50}}
& $\mathbb{BN}$ & - & 45.71M & 39.2$_{\downarrow {(1.6)}}$ & 60.0$_{\downarrow {(2.1)}}$ & 43.1$_{\downarrow {(1.4)}}$ & 35.2$_{\downarrow {(1.2)}}$ & 56.7$_{\downarrow {(2.2)}}$ & 37.6$_{\downarrow {(1.1)}}$\\
& $\mathbb{BN}+SE(BN_3)$ & - & 48.23M & 40.1$_{\downarrow {(0.7)}}$ & 61.2$_{\downarrow {(0.9)}}$ & 43.8$_{\downarrow {(0.7)}}$ & 35.9$_{\downarrow {(0.5)}}$ & 57.9$_{\downarrow {(1.0)}}$ & 38.1$_{\downarrow {(0.6)}}$ \\
& $\mathbb{BN}+SE(BN_2)$ & - & 46.34M & 40.1$_{\downarrow {(0.7)}}$ & 61.2$_{\downarrow {(0.9)}}$ & 43.8$_{\downarrow {(0.7)}}$ & 35.9$_{\downarrow {(0.5)}}$ & 57.9$_{\downarrow {(1.0)}}$ & 38.4$_{\downarrow {(0.3)}}$ \\
& AN (w/ BN) & - & 45.91M & \textbf{40.8} & \textbf{62.1} & \textbf{44.5} & \textbf{36.4} & \textbf{58.9} & \textbf{38.7} \\ \cline{2-10}
& $^\dagger$GN & GN~\cite{GroupNorm} & 45.72M & 40.3$_{\downarrow {(1.3)}}$ & 61.0$_{\downarrow {(1.0)}}$ & 44.0$_{\downarrow {(1.7)}}$ & 35.7$_{\downarrow {(1.7)}}$ & 57.9$_{\downarrow {(1.6)}}$ & 37.7$_{\downarrow {(2.2)}}$ \\
& $^\dagger$SN & SN~\cite{SwitchNorm} & - & 41.0$_{\downarrow {(0.6)}}$ & \textbf{62.3}$_{\downarrow {(-0.3)}}$ & 45.1$_{\downarrow {(0.6)}}$ & 36.5$_{\downarrow {(0.9)}}$ & 58.9$_{\downarrow {(0.6)}}$ & 38.7$_{\downarrow {(1.2)}}$ \\
& AN (w/ BN) & AN (w/ GN) & 45.96M & \textbf{41.6} & 62.0 & \textbf{45.7} & \textbf{37.4} & \textbf{59.5} & \textbf{39.9} \\ \bottomrule
\multirow{4}{*}{\rotatebox[origin=c]{0}{ResNet101}}
& $\mathbb{BN}$ & - & 64.70M & 41.4$_{\downarrow {(1.7)}}$ & 62.0$_{\downarrow {(2.1)}}$ & 45.5$_{\downarrow {(1.8)}}$ & 36.8$_{\downarrow {(1.4)}}$ & 59.0$_{\downarrow {(2.0)}}$ & 39.1$_{\downarrow {(1.6)}}$ \\
& AN (w/ BN) & - & 65.15M & \textbf{43.1} & \textbf{64.1} & \textbf{47.3} & \textbf{{38.2}} & \textbf{61.0} & \textbf{40.7 } \\ \cline{2-10}
& $^\dagger$GN & GN~\cite{GroupNorm} & 64.71M & 41.8$_{\downarrow {(1.4)}}$ & 62.5$_{\downarrow {(1.5)}}$ & 45.4$_{\downarrow {(1.9)}}$ & 36.8$_{\downarrow {(2.0)}}$ & 59.2$_{\downarrow {(2.1)}}$ & 39.0$_{\downarrow {(2.6)}}$ \\
& AN (w/ BN) & AN (w/ GN) & 65.20M & \textbf{43.2} & \textbf{64.0} & \textbf{47.3} & \textbf{38.8} & \textbf{61.3} & \textbf{41.6} \\ \bottomrule
\multirow{3}{*}{\rotatebox[origin=c]{0}{AOGNet12M}} & $\mathbb{BN}$ & - & 33.09M & 40.7$_{\downarrow {(1.3)}}$ & 61.4$_{\downarrow {(1.7)}}$ & 44.6$_{\downarrow {(1.5)}}$ & 36.4$_{\downarrow {(1.4)}}$ & 58.4$_{\downarrow {(1.7)}}$ & 38.8$_{\downarrow {(1.6)}}$ \\
& AN (w/ BN) & - & 33.21M & {42.0}$_{\downarrow {(1.0)}}$& 63.1$_{\downarrow {(1.1)}}$ & 46.1$_{\downarrow {(0.7)}}$ & {37.8}$_{\downarrow {(0.9)}}$ & 60.1$_{\downarrow {(1.0)}}$ & 40.4$_{\downarrow {(1.3)}}$ \\
& AN (w/ BN) & AN (w/ GN) & 33.26M & \textbf{43.0} & \textbf{64.2} & \textbf{46.8} & \textbf{38.7} & \textbf{61.1} & \textbf{41.7} \\ \bottomrule
\multirow{3}{*}{\rotatebox[origin=c]{0}{AOGNet40M}} & $\mathbb{BN}$ & - & 60.73M & 43.4$_{\downarrow {(0.7)}}$ & 64.2$_{\downarrow {(0.9)}}$ & 47.5$_{\downarrow {(0.7)}}$ & 38.5$_{\downarrow {(0.5)}}$ & 61.0$_{\downarrow {(1.0)}}$ & 41.4$_{\downarrow {(0.4)}}$ \\
& AN (w/ BN) & - & 60.97M & {44.1}$_{\downarrow {(0.8)}}$ & 65.1$_{\downarrow {(1.1)}}$ & 48.2$_{\downarrow {(0.9)}}$ & {39.0}$_{\downarrow {(1.2)}}$ & 62.0$_{\downarrow {(1.2)}}$ & 41.8$_{\downarrow {(1.5)}}$ \\
& AN (w/ BN) & AN (w/ GN) & 61.02M & \textbf{44.9} & \textbf{66.2} & \textbf{49.1} & \textbf{40.2} & \textbf{63.2} & \textbf{43.3} \\ \bottomrule
\end{tabular}
}
}
\caption{Detection and segmentation results in MS-COCO {\tt val2017}~\cite{COCO}. All models use 2x lr scheduling (180k iterations). $\mathbb{BN}$ means BN is frozen in fine-tuning for object detection. $^\dagger$ means that models are not trained by us. All other models are trained from scratch under the same settings. The numbers show sequential improvement in the two AOGNet models indicating the importance of adding our AN in the backbone and the head respectively.
}\label{table:coco-results}
\end{table*}
\subsection{Object Detection and Segmentation in COCO}
In object detection and segmentation, high-resolution input images are beneficial and often entailed for detecting medium to small objects, but limit the batch-size in training (often 1 or 2 images per GPU). GN~\cite{GroupNorm} and SN~\cite{SwitchNorm} have shown significant progress in handling the applicability discrepancies of feature normalization schema from ImageNet to MS-COCO.
{We test our AN in MS-COCO following the standard protocol, as done in GN~\cite{GroupNorm}. We build on the MMDetection code platform~\cite{mmdetection}. We observe further performance improvement.}
We first summarize the details of implementation. Following the terminologies used in MMDetection~\cite{mmdetection}, there are four modular components in the R-CNN detection framework~\cite{FastRCNN,FasterRCNN,maskrcnn}:
\textit{i) Feature Backbones}. We use the pre-trained networks in Table~\ref{table:imagenet-results} (with the vanilla setup) for fair comparisons in detection, since we compare with some models which are not trained by us from scratch and use the feature backbones pre-trained in a way similar to our vanilla setup and with on par top-1 accuracy. In fine-tuning a network with AN (w/ BN) pre-trained in ImageNet such as ResNet50+AN (w/ BN) in Table~\ref{table:imagenet-results}, we freeze the stem layer and the first stage as commonly done in practice. For the remaining stages, we freeze the standardization component only (the learned mixture of affine transformations and the learned running mean and standard deviation), but allow the attention weight sub-network to be fine-tuned.
\textit{ii) Neck Backbones}: We test the feature pyramid network (FPN)~\cite{FPN} which is widely used in practice.
\textit{iii) Head Classifiers}. We test two setups:
\textit{ (a) The vanilla setup} as done in GN~\cite{GroupNorm} and SN~\cite{SwitchNorm}. In this setup, we further have two settings: with vs without feature normalization in the bounding box head classifier. The former is denoted by ``-" in Table~\ref{table:coco-results}, and the latter is denoted by the corresponding type of feature normalization scheme in Table~\ref{table:coco-results} (\emph{e.g.}, GN, SN and AN (w/ GN)). We experiment on using AN (w/ GN) in the bounding box head classifier and keeping GN in the mask head unchanged for simplicity. Adding AN (w/ GN) in the mask head classifier may further help improve the performance. When adding AN (w/ GN) in the bounding box head, we adopt the same design choices except for ``Choice 1, $A_1(\cdot)$" (Eqn.~\ref{eq:attn_wts1}) used in learning attention weights.
\textit{(b) The state-of-the-art setup } which is based on the cascade generalization of head classifiers~\cite{cascadercnn} and does not include feature normalization scheme, also denoted by ``-" in Table~\ref{table:coco-csc-results}.
\textit{iv) RoI Operations}. We test the RoIAlign operation~\cite{maskrcnn}.
\begin{table*} [t]
\centering
\small{
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{l|ll|r|ccc|ccc}
Architecture & Backbone & Head & \#Params & AP$^{bb}$ & AP$^{bb}_{50}$ & AP$^{bb}_{75}$ & AP$^{m}$ & AP$^{m}_{50}$ & AP$^{m}_{75}$ \\ \hline
\multirow{2}{*}{ResNet101}
& $\mathbb{BN}$ & - & 96.32M & 44.4$_{\downarrow (1.4)}$ & 62.5$_{\downarrow (1.8)}$ & 48.4$_{\downarrow (1.4)}$ & 38.2$_{\downarrow (1.4)}$ & 59.7$_{\downarrow (2.0)}$ & 41.3$_{\downarrow (1.4)}$ \\
& AN (w/ BN)& - & 96.77M & \textbf{45.8} & \textbf{64.3} & \textbf{49.8} & \textbf{39.6} & \textbf{61.7} & \textbf{42.7} \\ \hline
\multirow{2}{*}{AOGNet40M} & $\mathbb{BN}$& - & 92.35M & 45.6$_{\downarrow (0.9)}$ & 63.9$_{\downarrow (1.1)}$ & 49.7$_{\downarrow (1.1)}$ & 39.3$_{\downarrow (0.7)}$ & 61.2$_{\downarrow (1.1)}$ & 42.7$_{\downarrow (0.4)}$ \\
& AN (w/ BN)& - & 92.58M & \textbf{46.5} & \textbf{65.0} & \textbf{50.8} & \textbf{40.0} & \textbf{62.3} & \textbf{43.1} \\ \hline
\end{tabular} }}
\caption{Results in MS-COCO using the cascade variant~\cite{cascadercnn} of Mask R-CNN.
}\label{table:coco-csc-results}
\end{table*}
\textbf{Result Summary.} The results are summarized in Table~\ref{table:coco-results} and Table~\ref{table:coco-csc-results}. Compared with the vanilla BN that are frozen in fine-tuning, our AN (w/ BN) improves performance by a large margin in terms of both bounding box AP and mask AP (\textit{1.8\% $\&$ 1.6\%} for MobileNetV2, \textit{1.6\% $\&$ 1.2\%} for ResNet50, \textit{1.7\% $\&$ 1.4\%} for ResNet101, \textit{1.3\% $\&$ 1.4\%} for AOGNet12M and \textit{0.7\% $\&$ 0.5\%} for AOGNet40M). It shows the advantages of the self-attention based dynamic and adaptive control of the mixture of affine transformations (although they themselves are frozen) in fine-tuning.
With the AN further integrated in the bounding box head classifier of Mask-RCNN and trained from scratch, we also obtain better performance than GN and SN. Compared with the vanilla GN~\cite{GroupNorm}, our AN (w/ GN) improves bounding box and mask AP by 1.3\% and 1.7\% for ResNet50, and 1.4\% and 2.2\% for ResNet101. Compared with SN~\cite{SwitchNorm} which outperforms the vanilla GN in ResNet50, our AN (w/ GN) is also better by 0.6\% bounding box AP and 0.9\% mask AP increase respectively. Slightly less improvements are observed with AOGNets.
Similar in spirit to the ImageNet experiments, we want to verify whether the advantages of our AN will disappear if we use state-of-the-art designs for head classifiers of R-CNN such as the widely used cascade R-CNN~\cite{cascadercnn}. Table~\ref{table:coco-csc-results} shows that similar improvements are obtained with ResNet101 and AOGNet40M.
\section{Conclusion}
This paper presents Attentive Normalization (AN) that aims to harness the best of feature normalization and feature attention in a single lightweight module. AN learns a mixture of affine transformations and uses the weighted sum via a self-attention module for re-calibrating standardized features in a dynamic and adaptive way. AN provides a strong alternative to the Squeeze-and-Excitation (SE) module. In experiments, AN is tested with BN and GN as the feature normalization backbones. AN is tested in both ImageNet-1000 and MS-COCO using four representative networks (ResNets, DenseNets, MobileNetsV2 and AOGNets). It consistently obtains better performance, often by a large margin, than the vanilla feature normalization schema and some state-of-the-art variants.
\section*{Acknowledgement}
This work is supported in part by NSF IIS-1909644, ARO Grant W911NF1810295, NSF IIS-1822477 and NSF IUSE-2013451.
The views presented in this paper are those of the authors and should not be interpreted as representing any funding agencies.
\bibliographystyle{splncs04}
|
2,877,628,090,786 | arxiv | \section*{Introduction}
In \cite{WITTEN} Witten pointed out the mathematical inconsistency of an
$SU(2)$ gauge theory coupled to only an odd number of Weyl fermions.
The inconsistency arises when one consi\-ders the behaviour of the effective
action under topologically non-trivial gauge transformations
($\pi_4[SU(2)] = {\mathbb {Z}}_2$), i.e. those that cannot be continuously
deformed to the identity mapping.
Let $g$ be one of those non-trivial mappings.
The gauge fields $A_{\mu}$ and its gauge transformed
\begin{equation}
A^{\rm g}_{\mu} = {\rm g} A_{\mu} {\rm g}^{-1} + {\rm g} \partial_{\mu}
{\rm g}^{-1} \ ,
\end{equation}
are not connected by some smooth gauge transformations.
However, they are connected
in the space of all gauge fields since it is a vector space. Thus
\begin{eqnarray}
A^t_{\mu} = (1-t)A_{\mu} + tA^{\rm g}_{\mu}, \quad t \in [0,1] \ ,
\label{PATH}
\end{eqnarray}
is a well defined potential which interpolates between $A_{\mu}$ and
$A^{\rm g}_{\mu}$.
Witten showed that along this trajectory an odd number
of eigenvalues of the square root of the Dirac operator crosses zero, leading to
a switch of sign in the fermionic determinant \cite{WITTEN}.
The theory is thus ill-defined because the fermion determinant cannot be
defined in a gauge invariant and smooth way.
Afterwards, the Witten anomaly has been established by other arguments.
In particular, the change in the phase of the effective action under $SU(2)$
gauge transformations can be calculated using the so-called
{\sl embedding technique} \cite{ELITZUR}. Here one uses the fact
that $SU(2)$ can be embedded in another group with a trivial $\pi_4$,
say $SU(3)$. The gauge fields $A_{\mu}$ and $A^g_{\mu}$ are then connected
by some smooth family $\Omega$ of
gauge transformations in $SU(3)$. In this context the
change in the phase of the effective action is given by \cite{WITTEN2}
\begin{equation}
\Delta \Gamma_{\rm eff}(\Omega d \Omega) =
i\frac{-i}{240\pi^2} \int {\rm tr} [(\Omega d \Omega)^5] = i\pi \ ,
\label{CS}
\end{equation}
leading to the switch of sign in the fermionic determinant.
During the last years encouraging progress has been made to formulate
chiral gauge theories on the lattice as has been summarised in
\cite{PLENARY}. Any proper lattice formulation of an $SU(2)$
theory coupled to one doublet of chiral fermions should exhibit
the Witten anomaly. This has been stu\-died in the context of the
overlap formalism \cite{NEU2}. Our purpose is showing how global anomalies
arise in the recently developed action formalism \cite{ABEL,SU2}.
\section*{Weyl fermions on the lattice}
Exact chiral symmetry \cite{SYMMETRY} can be achieved on the lattice
provided the Wilson-Dirac operator $D$ fulfils the Ginsparg-Wilson relation
\cite{GW}.
Having exact chiral symmetry allows us to split the fermion
fields in left- and right-handed, independently transforming
components \cite{SYMMETRY,NIED,NARA}. In particular,
we can restrict ourselves to the left-handed fields imposing the constraints
\begin{eqnarray}
\label{CONS1}
\hat{P}_- \psi &=& \psi \ ,\\
\bar{\psi}P_+ &=& \bar{\psi} \ .
\eea
One of the new features of the present approach
is that $\hat{P}_- = \frac{1}{2} (1 - \gamma_5(1 -aD))$ depends on
the gauge field. Let us discuss some of its geometrical implications.
Consider a path in configuration space
$U_{\rm t} (x,\mu)$, where $t \in [0,1]$ is the path parameter.
We define an unitary operator $Q_{\rm t}$ through the differential equation
\begin{eqnarray}
\partial_t Q_t = [\partial_t P_t,P_t] Q_t \ , \qquad\mbox{$Q_0 = \id$} \ ,
\label{DE}
\eea
where $P_{\rm t} \equiv {\hat P} |_{U=U_{\rm t}}$.
The operator $Q_{\rm t}$ is such that
$P_{\rm t} Q_{\rm t} = Q_{\rm t} P_0$. In this way $Q_{\rm t}$ is the
transporter of $P_{\rm t}$ along the path.
One key point here is that if the path is a closed loop,
the operator $Q_1$ is not necessarily the identity map. Indeed,
$Q_1 \neq \id$ is an indication of a non-trivial bundle structure of
the gauge field. As a measure for this let us define the quantity
\begin{equation}
\mathcal T = \det [1 -P_0 + P_0 Q_1 ] \ ,
\ee
for all closed loops.
Using the unitarity of $Q_{\rm t}$
one easily proves that $\mathcal T$ is a phase.
In particular, for gauge fields in $SU(2)$, using charge conjugation
symmetry and the reality properties of the $SU(2)$ representations
we have $\mathcal T = \pm 1$.
Finally we remark that the composition law
\begin{equation}
\mathcal T_{[\Gamma_1 \circ \Gamma2]} = \mathcal T_{\Gamma1} \, \mathcal T_{\Gamma_2}
\label{COMPO}
\ee
holds for two closed loops that have the same starting point.
\section*{Fermionic Measures}
The action of the classical gauge theory coupled
to a single Weyl fermion reads
\begin{equation}
S_{F,L} = a^4 \sum_x \bar{\psi}(x) [P_+ D \hat{P}_- \psi](x) \ .
\end{equation}
In order to set up the quantum theory we have to define a measure
for the fields in the path integral. Here the difficulty arises
due to the gauge field dependence of the constraint (\ref{CONS1}).
In fact, an infinitesimal deformation $\eta_{\mu}(x)$
of the gauge field $U(x,\mu)$, induces a change in the phase
of the fermionic measure given by the so-called measure term \cite{SU2}
\begin{eqnarray}
{\mathfrak L}_{\eta} = i \sum_j (v_{\rm j}, \delta_{\eta} v_{\rm j})
\equiv a^4 \sum_x \eta^c_{\mu}(x)j^c_{\mu}(x) \ ,
\label{MEASU}
\eea
where $c$ is the colour index and
$\{v_{\rm j}\}$ is a basis of left-handed fields at $U(x,\mu)$.
We are then left with a gauge dependent
phase ambiguity which has to be fixed in order to achieve
the gauge invariance and the locality of the effective action.
In \cite{ABEL,SU2} the problem is solved
by choosing a current $j_{\mu}(x)$, local and gauge invariant
function of the gauge fields, and such that an integrability
condition is fulfilled.
This condition is formulated in terms of the Wilson line
\begin{equation}
{W} = \exp \left\{ i \int_0^1 dt {\mathfrak L}_{\eta} \right\} \ ,
\ee
where $a\eta_{\mu}(x) \;=\;\partial_tU(x,\mu)U(x,\mu)^{-1}$. $W$ measures
the total change of phase along a given path $U_{\rm t} (x,\mu)$ in the set
of gauge fields.
The integrability condition states that for all closed loops in
configuration space the Wilson line must satisfy
\begin{equation}
W=\mathcal T \ .
\label{IC}
\ee
Let us discuss the meaning of this condition.
A local and gauge invariant current defines $W$ only locally.
In (\ref{IC}) it is pointed out that the
current must take into account the global geometry of the bundle
underlying the gauge field.
Global anomalies arise when (\ref{IC}) is not satisfied.
In the classical continuum limit, requiring a local and gauge
invariant effective action implies \cite{SU2}
\begin{equation}
j_{\mu}(x) = 0 + {\mathcal O}(a) \ .
\label{JCC}
\ee
Therefore $W=1 + {\mathcal O}(a)$ in this limit and the integrability
condition can only be satisfied if $\mathcal T~=~1$~for all closed loops.
In the next section we will show that in the $SU(2)$ theory
with a single Weyl fermion there are closed loops in configuration
space on which $\mathcal T = -1$. Therefore the anomaly arises because the proper
classical continuum limit cannot be reproduced.
\section*{SU(2) Global Anomaly}
Let $g$ be a non-trivial $SU(2)$ gauge transformation,
and consider its lattice version acting on the classical vacuum, $U(x,\mu)=\id$.
There are three different paths in configuration space connecting
the classical vacuum with its gauge transformed, $g(x) g(x+a\mu)^{-1}$:
\begin{itemize}
\item{$\Gamma_1 \equiv$} $[g(x)^tg(x+a\mu)^{-t}]$ pure gauge $SU(2)$
\item{$\Gamma_2 \equiv$} $[\Omega(t,x) \Omega(t,x + a \mu)^{-1}]$
pure gauge $SU(3)$
\item{$\Gamma_3 \equiv$} $(g(x) g(x+a\mu)^{-1})^t$
\end{itemize}
$\Gamma_1$ has no continuum limit. $\Gamma_2$ is the lattice analogue
of the $SU(3)$ pure gauge path defined in the introduction and
$\Gamma_3$ is a lattice version of Witten's path (\ref{PATH}).
Our aim is computing $\mathcal T$ on the
closed loop $[\Gamma_3 \circ - \Gamma_1]$.
Using the composition law (\ref{COMPO}) we split $\mathcal T$ in the following way
\begin{equation}
\mathcal T_{[\Gamma_3 \circ - \Gamma_1]} = \mathcal T_{[\Gamma_3 \circ - \Gamma_2]} \,
\mathcal T_{[\Gamma_2 \circ - \Gamma_1]} \ .
\end{equation}
Along pure gauge loops we have,
\begin{eqnarray}
{\mathcal T} = \exp \left\{-i\int_0^1 \: dt \sum_x \omega_t^a(x)
{\cal A}^a(x) \right\} \ ,
\eea
where ${\cal A}^a(x)$is the anomaly on the lattice.
For the vacuum configuration the anomaly vanishes because
it is invariant under translations but
also odd under parity, ${\cal A}^{a}(-x) = - {\cal A}^a(x)$.
That means $\mathcal T_{[\Gamma_2 \circ - \Gamma_1]} = 1$.
This argument holds for the vacuum
in $SU(N)$. However, in $SU(2)$ the anomaly is identically
zero in any configuration.
We are then left with the calculation of $\mathcal T$ for the
loop $[\Gamma_3 \circ - \Gamma_2]$. To deal with it we
convert the line integral into a surface integral. On this
surface the vector potential depends on two parameters ($t,s$).
It can be shown \cite{FUTURE} that
\begin{equation}
\partial_{\rm s} \ln \: {\mathcal T} =
\int_0^1 dt \: {\rm Tr} {\hat P}_- \, [\partial_s {\hat P}_-, \partial_t {\hat P}_-] \ .
\label{EQCON}
\ee
The r.h.s. of equation (\ref{EQCON}) can be expanded in powers of the
lattice spacing $a$ and to leading order one finds
\begin{equation}
\partial_{\rm s}\ln{\mathcal T} \\
= -i c_2 \int_0^1 dt \int d^4 x \, d^{abc}_R \epsilon_{\mu \nu \rho \sigma
}
\eta^a_{\mu} \xi^b_{\nu} F^c_{\rho \sigma}
\label{ASYM}
\ee
Here we take over the notation of \cite{SU2}. The deformations
$\eta$ and $\xi$ corresponds to the $t$ and $s$ directions respectively.
In (\ref{ASYM}) we substitute our parameterisation of the vector
potential on the surface.
Integrating over $s$ we finally end up with the integral
(\ref{CS}) and find $\mathcal T_{[\Gamma_3 \circ - \Gamma_2]} = -1$.
Altogether we get $\mathcal T_{[\Gamma_3 \circ - \Gamma_1]} = -1$.
We want to point out that our proof is not restricted to have
the classical vacuum as starting configuration
(although it would be sufficient for the theory to be inconsistent).
Since $\mathcal T$ is a homotopy invariant, smooth deformations of the loop
cannot change its value.
\section*{Witten Anomaly}
Next we want to show that the global anomaly we discovered
in the previous section
can be brought into a form that its equivalence to Witten's anomaly
\cite{WITTEN} is transparent. It will turn out that if we restrict
ourselves to a
real fermion determinant, $\mathcal T=-1$ implies a change of sign along the
loop $[\Gamma_3 \circ - \Gamma_1]$.
To start with consider the function
\begin{eqnarray}
f(t) = \det \, (1-P_+ + P_+D_tQ_tD_0^{\dagger}),
\eea
that is a smooth function of $t$ for smooth paths. In addition it is
real if the gauge group is $SU(2)$. For closed loops that function
satisfies
\begin{eqnarray}
f(0)\,>\,0\,, & & f(1) = {\cal T}f(0) \ ,
\eea
if $D$ has no zero mode at the starting point. This implies that $f$
passes through zero an odd number of times for $0\leq t\leq 1$ if and
only if ${\cal T}=-1$.
Next one can show
\begin{eqnarray}\label{f_squared}
f^2 (t) = \det D_t \det D_0^{\dagger}.
\eea
If $D^{\dagger}=\gamma_5 D \gamma_5$, the eigenvalues of $D$
come in complex conjugate pairs, i.e.
\begin{eqnarray}
\det D=\prod\lambda_i\lambda_i^* \ .
\eea
According to \pref{f_squared} a
passing through zero of $f(t)$ at $t_0$ implies a passing through zero
of an odd number of
eigenvalues $\lambda_i(t)$. One can prove this by expanding
both $f$ and $\lambda_i$ around $t_0$.
Hence we have found that for any given
loop in field space, starting at a point where $D$ has no zero modes,
an odd number of pairs of eigenvalues of $D$ cross zero if and only if
$\mathcal T = -1$.
Hence we find the same behaviour of the eigenvalues that Witten proved in
\cite{WITTEN} using
the Atiyah--Singer index theorem and that he used to conclude a change
of sign of the fermion determinant.
We can find the same here. In general, the fermion determinant is
given by \cite{SU2}
\begin{equation}\label{formula_det}
\det M_t\det M_0^{\dagger} = f(t) \, W^{-1} \ .
\ee
If we define $j_{\mu}\equiv1$ we end up with a real fermion
determinant all along the path. According to the properties of $f$, we find that
$\det M_1 \det M_0^{\dagger}$ is negative. Therefore $\det M_t$ changes sign
along a loop with $\mathcal T=-1$.
Note however, that a change of sign is closely connected to the
definition of a real fermion determinant. If we allow for
complex values too and define a
current $j_{\mu}$ such that $W$ equals $-1$ at the end of the
loop, the fermion determinant is single valued. In
that case the anomaly shows up in the disguise of a conflict with the
correct continuum limit of $j_{\mu}$ (\ref{JCC}) and $W$.
\section*{Spectral flow of Neuberger's operator}
In the last section we have seen that ${\cal T} =-1$ implies that an
odd number of pairs of eigenvalues of $D$ cross zero. On a finite
lattice it is possible to confirm this crossing by a numerical
computation of the spectral flow of an appropriate lattice Dirac
operator, as we will see now.
Neuberger's operator \cite{Neuberger98a} is an example for a Dirac
operator that satisfies the Ginsparg-Wilson relation.
It is explicitly given by
\begin{eqnarray}\label{def_Neub_op}
aD & = & 1-A(A^{\dagger}A)^{-1/2}\,,\\
A & = & 1 - aD_w\,.
\eea
$D_w$ denotes the usual Wilson--Dirac operator.
The eigenvalues $\lambda_j$ of $aD$ lie on a unit circle around 1 in
the complex plane.
They can be parameterised by an angle $\theta$ according to
\begin{eqnarray}\label{def_ev}
\lambda_j & = & 1 -e^{i\theta_j}\,.
\eea
Furthermore the eigenvalues come in complex conjugate pairs
$\lambda_j,\lambda_j^*$ due to $D^{\dagger}=\gamma_5 D \gamma_5$.
We are interested in the spectral flow of Neuberger's operator along
the loop $\Gamma = [\Gamma_3\circ-\Gamma_1]$. According to the last
section we expect an odd number of eigenvalues that cross zero and go
over into its complex conjugate value.
Along $\Gamma_1$ the eigenvalues are constant as a function of $t$
because it is a path of gauge transformations.
The non--trivial part is along $\Gamma_3$ where we numerically
computed the eigenvalues.
This has been done in two steps.
First of all we computed the low lying eigenvalues of the hermitian operator
$aD^{\dagger}aD$.
They are given by the squared magnitude $|\lambda_j|^2$ of the
eigenvalues \pref{def_ev}.
In a second step we calculated the imaginary part of $\lambda_0$,
i.e. the eigenvalue with the smallest imaginary part at the beginning
of the path.
This is sufficient for our purpose.
Before discussing the results let us make
some remarks concerning the numerical computation itself.
We used a power series expansion into Chebyshev polynomials for the
inverse square root of $A^{\dagger}A$ in \pref{def_Neub_op}.
The Conjugate Gradient algorithm \cite{Bunk94a,Kalkreuther96a} has been
employed for the computation of
the eigenvalues. In that way both the truncation
and the numerical error are theoretically well under control.
For numerical reasons the path we used differs slightly from
$\Gamma_3$.
We did not start at the classical vacuum configuration but at a
constant gauge field instead.
This results in a gap of the spectrum at $t=0$ and $t=1$ and is
advantageous in the numerical computation.
Figure \ref{fig1} shows the first six lowest lying eigenvalues of
$aD^{\dagger}aD$ as a function of the path parameter $t$.
The obvious symmetry of the spectrum is due to our path
parameterisation and the symmetry properties
of the particular $g(x)$ we used.
The magnitude of all but one eigenvalue is unequal to zero for the whole
path.
Only the lowest eigenvalue becomes zero for $t=0.5$.
Hence only the imaginary part of $\lambda_0$ may cross zero and changes sign.
To see if it really changes sign we calculated directly the imaginary
part of $\lambda_0$.
The result is shown in
fig.~\ref{fig2} and indeed we find a crossing.
Hence only one pair of eigenvalues crosses zero and we numerically
confirmed a spectral flow that we have expected.
\begin{figure}[t]
\epsfysize=7.5cm
\epsffile{fig1.ps}
\caption{\small The lowest six eigenvalues of
$aD^{\dagger}aD$ on a $8^4$ lattice. The total error is smaller than the
size of the data points.}
\label{fig1}
\end{figure}
\begin{figure}[t]
\epsfysize=7.5cm
\epsffile{fig2.ps}
\caption{\small The imaginary part of $\lambda_0$. The error bars
incorporate both truncation and numerical error.}
\label{fig2}
\end{figure}
The change of sign of the fermion determinant can now be shown very
explicitly. In terms of the angles $\theta_j$ the fermion determinant
reads as
\begin{eqnarray}\label{def_detM}
\det M_t & = & \prod_j\,2\,\sin\frac{\theta_j(t)}{2}.
\eea
Here we fixed the phase of the fermion determinant such
that $\det\,M_t$ is real.
However, we still have a sign ambiguity.
Suppose we want to fix the sign but also insist on a smooth gauge
field dependence of $\det\,M_t$. This implies that the sign at $t=0$
fixes the sign along the whole path $\Gamma$. In terms of the angle
$\theta_j$ our numerical result runs as follows: Only the angle
$\theta_0$ crosses zero and changes sign. A glance at \pref{def_detM}
immediately tells us that the fermion determinant changes sign along
the closed path $\Gamma$ and is not a single valued function.
\section*{Conclusions}
Let us give a brief summary of our result.
We have shown that there are closed loops in configuration space
on which the phase $\mathcal T = - 1$.
In a finite lattice it would be possible to define a current
$j_{\mu}(x)$ such that the associated measure produces a Wilson line
${W} = -1$. However, the conflict arises because the proper behaviour in the
classical continuum limit cannot be reproduced:
Going to large physical lattices (small lattice spacing) and insisting
in ${W}=-1$, implies giving up the locality or the gauge invariance
of the theory. Our argument is completely analytical, in particular the
behaviour close to the continuum limit is under control.
We have also checked numerically the original argument given by Witten.
We find that along a path connecting two gauge fields that differ
by a topologically non-trivial gauge transformation, the lowest
eigenvalue of Neuberger's operator crosses zero.
Subsequently there exist closed loops in configuration
space where the fermion determinant changes sign.
On the lattice all gauge transformations can be deformed to the identity
mapping. As we have seen it does not mean that global anomalies are
not present in the lattice formulation, but they rather arise in a
different way.
\section*{Acknowledgements}
The authors are grateful to Martin L\"uscher for helpful discussions and advise.
The numerical computations have been performed on the Linux cluster
of the DESY-Hamburg Theory Group and on the RTNN Linux farm in Zaragoza
(Spain).
\newpage
|
2,877,628,090,787 | arxiv | \section{Introduction}
Value-at-Risk (VaR) and Expected Shortfall (ES, also known as TVaR and CVaR) are the most widely used risk measures for regulation in finance and insurance. The former has gained its popularity due to its simplistic approach toward risk as the risk quantile, and the second one is perceived to be useful as a modification of VaR with more appealing properties, such as tail-sensitivity and subadditivity, as studied in the seminal work of \cite{ADEH99}.
In the Fundamental Review of the Trading Book (FRTB), the Basel Committee on Banking Supervision (\citet{BASEL19}) proposed to replace VaR at 99\% confidence with ES with a 97.5\% confidence interval for the internal model-based approach. The main reason, as mentioned in the FRTB, was that $\mathrm{ES}$ can better capture tail risk; see \cite{ELW18} for a concrete risk sharing model where tail risk is captured by ES and ignored by VaR. On the other hand, VaR also has advantages that ES does not have, such as elicitability (e.g., \cite{G11} and \cite{KP16}) or backtesting tractability (e.g., \cite{AS14}), and the two risk measures admit different axiomatic foundations (see \cite{C09} and \cite{WZ21}). We refer to the reviews of \citet{EPRWB14} and \citet{EKT15} for general discussions on VaR and ES, and \citet{MFE15} for a standard treatment on risk management including the use of VaR and ES.
The technical contrasts of the two risk measures and their co-existence in regulatory practice give rise to great interest from both researchers and practitioners to explore the relationship between them.
To understand the balancing point of VaR and ES during the transition in the FRTB, \citet{LW2019} proposed the Probability Equivalent Level of VaR-ES (PELVE). The value of PELVE is the multiplier to the tail probability when replacing VaR with ES such that the capital calculation stays unchanged.
More precisely, the PELVE of $X$ at level $\epsilon$ is the multiplier $c$ such that $\mathrm{ES}_{1-c\epsilon}(X)=\mathrm{VaR}_{1-\epsilon}(X)$; such $c$ uniquely exists under mild conditions.
For instance, if $\mathrm{VaR}_{99\%}(X)=\mathrm{ES}_{97.5\%}(X)$ for a future portfolio loss $X$, then PELVE of $X$ at probability level 0.01 is the multiplier $2.5$. In this case, replacing $\mathrm{VaR}_{99\%}$ with $ \mathrm{ES}_{97.5\%} $ in FRTB does not have much effect on the capital requirement for the bank bearing the loss $X$. Instead, if $\mathrm{ES}_{97.5\%}(X)>\mathrm{VaR}_{99\%}(X)$, then the bank has a larger capital requirement under the new regulatory risk measure; this is often the case for financial assets and portfolios as shown by the empirical studies in \cite{LW2019}. The PELVE enjoys many convenient properties, and it has been extended in a few ways. In particular,
\citet{FG2021} defined generalized PELVE by replacing $\mathrm{VaR}$ and $\mathrm{ES}$ with another pair of monotone risk measures $(\rho,\tilde{\rho})$, and
\cite{B22} extended PELVE by replacing ES with a higher-order ES.
For a given distribution model or a data set,
its PELVE can be computed or estimated in a straightforward manner.
As argued by \cite{LW2019}, the PELVE for a small $\epsilon$ may be seen as a summarizing index measuring tail heaviness in a non-limit sense. As such, one may like to generate models for a given PELVE, in a way similar to constructing models for other given statistical information; see e.g., \cite{EMS02, EHW16} for constructing multivariate models with a given correlation or tail-dependence matrix. Such statistical information may be obtained either from data or from expert opinion, but there is no a priori guarantee that a corresponding model exists.
Since PELVE involves a parameter $\epsilon\in (0,1)$, its information is represented by a curve. The calibration problem, that is, to find a distribution model for given PELVE values or a given PELVE curve, turns out to be highly non-trivial, and it is the main objective of the current paper.
From now on, suppose that we receive some information on the PELVE of a certain random loss from an expert opinion, and we aim to build a distribution model consistent with the supplied information.
Since PELVE is location-scale invariant, such a distribution, if it exists, is not unique.
The calibration problem is trivial if we are supplied with only one point on the PELVE curve.
As the PELVE curve of the generalized Pareto distribution is a constant function, we can use the generalized Pareto distribution to match the given PELVE value, which has a tail index implied from the expert opinion.
The calibration problem becomes more involved if we are supplied with two points on the PELVE curve, because the value of the PELVE at two different probability levels interact with each other.
The situation becomes more complicated as the number of points increase, and we further turn to the problem of calibration from a fully specified PELVE curve. Calibrating distribution from the PELVE curve can be reformulated as solving for a function $f$ via the integral equation $\int_{0}^{y}f\left(s\right)\mathrm{d} s=yf\left(z_{f}\left(y\right)y\right)$, where the curve $z_{f}$ is computed from the PELVE curve. This integral equation can be
further converted to an advanced differential equation (see \cite{BC63}), and
we develop a numerical method to compute $f$.
For the case that $z_{f}$ is a constant curve, we can explicitly obtain all solutions for $f$. We find other distributions that also have constant PELVE curves other than the simple ones with a polynomial or exponential distribution function. As a consequence, a PELVE curve does not characterize a unique location-scale family of distributions; this provides a negative answer to a question posed by \citet[Section 7, Question (iv)]{LW2019}.
Furthermore, we study technical properties of PELVE, such as monotonicity and convergence as the probability level goes to $0$. A decreasing PELVE indicates a relatively larger impact of $\mathrm{ES}$ in risk assessment than $\mathrm{VaR}$ moving towards the tail. As we will see, while for the most known parametric distributions the PELVE is decreasing, there exists some examples at some risk levels it is not decreasing. This means that for those examples $\mathrm{VaR}$ becomes a stricter risk measure when moving towards the tail. To obtain conditions for monotonicity, we define the dual PELVE by moving the multiplier $c$ from the $\mathrm{ES}$ side to the $\mathrm{VaR}$ side.
PELVE can be seen as a functional measure of tail heaviness in the sense that a heavier-tailed distribution has a higher PELVE curve (\citet[Theorem 1]{LW2019}). The hazard rate, on the other hand, is another functional measure of tail heaviness.
We show that the PELVE is decreasing (increasing) if the inverse of hazard rate is convex (concave).
Monotonicity also leads to conditions for the PELVE to have a limit at the tail, which from the risk management perspective, identifies the ultimate relative positions of $\mathrm{ES}$ and $\mathrm{VaR}$ in the tail region. From a mathematical perspective,
the limit of PELVE at $0$ allows us to extend the domain of PELVE to include $0$ as a measure of tail heaviness.
The rest of the paper is organized as follows.
Section \ref{sec:definition} introduce the background and examples of the PELVE.
In Section \ref{sec:points} we calibrate distribution from finite-point in PELVE curve. The proofs of Section \ref{sec:points} are provided in Appendix \ref{app:proof_sec2}.
Section \ref{sec:function} calibrates distribution from a general given PELVE curve numerically.
The constant curve constraint is discussed in Section \ref{sec:constant}.
In Section \ref{sec:property}, we study the monotonicity and convergence of the PELVE.
A conclusion is given in Section \ref{sec:conclusion}.
\section{Definitions and background}\label{sec:definition}
disLet us consider an atom free probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$, where $\mathcal{F}$ is the set of the measurable sets and $\mathbb{P}$ is the probability measure. Let $L^{1}$ be the set of integrable random variables, i.e., $L^{1}= \{ X : \mathbb{E} [ |X | ] <\infty \}$, where $\mathbb{E}$ is the expectation with respect to $\mathbb{P}$.
We first define $\mathrm{VaR}$ and $\mathrm{ES}$ in $L^{1}$, the two most popular risk measures. The $\mathrm{VaR}$ and at probability level $p\in(0,1)$ is defined as
\begin{equation}\label{eq:left_VaR}
\mathrm{VaR}_{p}(X) =\inf\{x\in\mathbb{R}:\mathbb{P}(X\le x)\ge p\}=F_{X}^{-1}(p),~~~~X\in L^{1},
\end{equation}
and the $\mathrm{ES}$ at probability level $p\in[0,1)$ is defined as
$$
\mathrm{ES}_{p}(X) =\frac{1}{1-p}\int_{p}^{1}\mathrm{VaR}_{q}(X)\mathrm{d} q,~~~~X\in L^{1}.
$$
Moreover, let $\mathrm{VaR}_{1}(X)=\mathrm{ES}_{1}(X)=\mathrm{ess\mbox{-}sup}(X)$ and $\mathrm{VaR}_{0}(X)=\mathrm{ess\mbox{-}inf}(X)$.
Note that $\mathrm{ES}_{0}(X)$ is the mean of $X$.
For $\epsilon\in(0,1)$, the PELVE, proposed by \citet{LW2019}, is defined as
$$
\Pi_{X}(\epsilon) =\inf\left\{ c\in[1,1/\epsilon],\mathrm{ES}_{1-c\epsilon}(X)\le\mathrm{VaR}_{1-\epsilon}(X)\right\} ,\quad X\in L^{1},
$$
where $\inf (\varnothing) =\infty$.
\cite{LW2019} used $\Pi_\epsilon (X)$ for our $\Pi_X(\epsilon)$, and our choice of notation is due to the fact that the curve $\epsilon \to \Pi_X(\epsilon)$ is the main quantity of interest in this paper.
The PELVE of $X$ is finite if and only if $\mathrm{VaR}_{1-\epsilon}(X)\ge \mathbb{E}[X]$.
In most cases, the value of the PELVE is the multiplier $c$ such that $\mathrm{ES}_{1-c\epsilon}(X)=\mathrm{VaR}_{1-\epsilon}(X).$
If $\mathrm{VaR}_{1-\epsilon}(X)$ is not a constant for $\epsilon \in (0,\epsilon]$, then the PELVE is the unique solution for the multiplier. By Theorem 1 in \cite{LW2019}, the PELVE is location-scale invariant. The distribution with heavy tail will have a higher PELVE value.
If $X$ is a normal distributed random variable and $\epsilon=1\%$, we have $\Pi_{X}(\epsilon)\approx2.5$. It means that $\mathrm{ES}_{97.5\%}(X)\approx\mathrm{VaR}_{99\%}(X)$. That is, the replacement suggested by BCBS is fair for normal distributed risks. In other words, a higher PELVE will result in a higher capital requirement after the replacement.
In this paper, we are generally interested in the question of which distributions have a specified or partially specified PELVE curve.
We first look at a few simple examples.
\begin{example}[Constant PELVE]
We first list some distributions that have constant PELVE curves.
From the definition of the PELVE, we know that the PELVE should be larger than 1.
As we can see from Table \ref{constant-PELVE}, the PELVE for the generalized Pareto distribution takes value on $(1, \infty)$. For $X \sim \text{GPD}(\xi)$, we have $1<\Pi_{X}(\epsilon)<e$ when $\xi <0$, $\Pi_{X}(\epsilon)=e$ when $\xi =0$ and $\Pi_{X}(\epsilon)>e$ when $\xi >0$. Furthermore, if $X$ follows the point-mass distribution $\delta_{c}$ or the Bernoulli distribution, we have the PELVE equals to 1.
\end{example}
\begin{table}[htbp]
\def1.4{1.4}
\centering{}
\caption{Example of Constant PELVE}\label{constant-PELVE}
\begin{threeparttable}
\begin{tabular}{m{2cm}<{\centering}| m{7cm}<{\centering}|m{4cm}<{\centering}}
\hline\hline
Distribution & Distribution or probability function & $\Pi_{X}(\epsilon)$\\
\hline
$\delta_c$ &$\mathbb{P}(X=c)=1$ &$\Pi_{X}(\epsilon)=1$ for $\epsilon \in (0,1)$\\
\hline
$\mathrm{B}(1,p)$& $\mathbb{P}(X=1)=p$ and $\mathbb{P}(X=0)=1-p$ & $\Pi_{X}(\epsilon)=1$ for $\epsilon \in (0,p)$\\
\hline
$\mathrm{U}(0,1)$ & $F_X(t)=t$ for $t \in (0,1)$ & $\Pi_{X}(\epsilon)=2$ for $0<\epsilon<1/2$\\
\hline
$\text{EXP}(\lambda)$ & $F_X(t)=1-\exp(-\lambda t),~~\lambda>0$ & $\Pi_{X}(\epsilon)=e$ for $0<\epsilon<1/e$\\
\hline
$\text{GPD}(\xi)$\tnote{1}&
$F_X(x)=\left\{\begin{aligned}
&1-\left(1+\xi x\right)^{-\frac{1}{\xi}}~~~& \xi \neq 0\\
&1-\exp(-x)~~~& \xi=0
\end{aligned}\right.$
&
$\Pi_{X}(\epsilon)=(1-\xi)^{-\frac{1}{\xi}}$ for $0<\epsilon<(1-\xi)^{\frac{1}{\xi}}$\\
\hline\hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] The distribution $\text{GPD}(\xi)$ is called the standard generalized Pareto distribution. As $\mathbb{E}[X]<\infty$ when $\xi<1$, the PELVE exists only when $\xi<1$.
The support of $\text{GPD}(\xi)$ is $[0, \infty)$ when $\xi>0$ and $[0,-\frac{1}{\xi}]$ when $\xi<0$. When $\xi=0$, the $\text{GPD}(\xi)$ is exactly exponential distribution with $\lambda=1/\sigma$. There is a three-parameter $\text{GPD}(\mu,\sigma,\xi)$, which is a location-scale transform of standard GPD. Therefore, $\text{GPD}(\mu,\sigma,\xi)$ has the same PELVE as $\text{GPD}(\xi)$.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{example}\label{ex:decreasing-PELVE}
Here we present some non-constant PELVE examples. We write $\mathrm{t}(v)$ for the t-distribution with parameter $(0,1,v)$, and $\mathrm{LN}(\sigma)$ for the log-normal distribution with parameter $(0,\sigma^2)$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.32]{PELVE_N}
\includegraphics[scale=0.32]{PELVE_t}
\includegraphics[scale=0.32]{PELVE_LN}
\caption{PELVE for Normal distribution, t-distribution and Lognormal distribution}
\label{PELVE}
\end{figure}
As we can see, for normal distribution and t-distribution, the PELVE curve is decreasing as $\epsilon$ increasing. The monotonicity of the PELVE of the lognormal distribution depends on the value of $\sigma$. The monotonicity of the PELVE will be further discussed in Section \ref{sec:property}.
For more PELVE examples, see \cite{LW2019}.
\end{example}
\section{Calibration from finite-point constraints}\label{sec:points}
In this section, we discuss the calibration problem when some points of the PELVE are given.
We will focus on the case where one point or two points on the PELVE curve are specified, for which we can explicitly construct the corresponding quantile functions.
We first note that the calibrated distribution is not unique.
For example, if we are given $\Pi_{X}(0.01)=2.5$, we can assume the distribution of $X$ is the Normal distribution or the generalized Pareto distribution with tail parameter $\xi$ satisfying $(1-\xi)^{-1/\xi}=2.5$ from Table \ref{PELVE}. Therefore, the distributions obtained in our results are only some possible choices, which we choose to have a
generalized Pareto tail, as Pareto tails are standard in risk management applications.
\subsection{Calibration from a one-point constraint}
Based on Table \ref{constant-PELVE}, we can calibrate the distribution for $X$ from one given PELVE point $(\epsilon_1, c_1)$ such that $\Pi_{X}({\epsilon_1})=c_1$.
A simple idea is to take the generalized Pareto distribution when $c_1>1$ and $\delta_c$ when $c_1=1$. We summarize the idea in the following Proposition.
\begin{proposition}\label{pro:one-point}
Let $\epsilon_1 \in (0,1)$ and $c_1 \in [1,\infty)$ such that $c_1\epsilon_1\le 1$.
If $c_1>1$, let $\xi \in \mathbb{R}$ such that $(1-\xi)^{-\frac{1}{\xi}}=c_1$.
Then, $X \sim \text{GPD}(\xi)$ has $\Pi_{X}({\epsilon_1})=c_1$.
If $c_1=1$, then $X=k$ for some constant $k \in \mathbb{R}$ has $\Pi_{X}({\epsilon_1})=c_1$.
\end{proposition}
The proof can be direct derived from Table \ref{constant-PELVE} and it is omitted the proof.
By Proposition \ref{pro:one-point}, if we have the value of PELVE at point $\epsilon_1$, we can find a distribution of $X$ which has the same PELVE value at $\epsilon_1$. If we also have the value of $\mathrm{VaR}$ at $1-\epsilon_1$, we can determine the scale parameter ($\sigma$) for the GPD distribution or the value of $k$ to match the value of $\mathrm{VaR}$.
\subsection{Calibration from a two-point constraint}
The calibration problem would be much difficult when we are given two points of PELVE curve. Given two points $(\epsilon_1, c_1)$ and $(\epsilon_2, c_2)$ such that $\epsilon_1<\epsilon_2$, we want to find a distribution for $X \in L^1$ such that $\Pi_{X}({\epsilon_1})=c_1$ and $\Pi_{X}(\epsilon_2)=c_2$. Nevertheless, the choices of $(\epsilon_1, c_1)$ and $(\epsilon_2, c_2)$ are not arbitrary. First, we need $1 \le c_1\le 1/\epsilon_1$ and $1 \le c_2\le 1/\epsilon_2$ by the definition of the PELVE. Then, we will show that the value of $c_2$ will be restricted if $(\epsilon_1, c_1)$ and $\epsilon_2$ are given.
\begin{lemma}\label{lem:bound_1}
For any $X \in L^1$, let $\epsilon_1,\epsilon_2 \in (0,1)$ be such that $\mathbb{E}[X]\le \mathrm{VaR}_{1-\epsilon_2}(X)$ and $\epsilon_1<\epsilon_2$. Then, we have $\epsilon_1 \Pi_{X}(\epsilon_1)\le \epsilon_2 \Pi_{X}(\epsilon_2)$.
\end{lemma}
By Lemma \ref{lem:bound_1}, for
given $\epsilon_1, \epsilon_2$ and $c_1$, the value of $c_2$ is bounded below by both 1 and $ {c_1\epsilon_1}/{\epsilon_2}$.
We also note that if $c_2=1$, then $p\mapsto \mathrm{VaR}_{1-p}(X)$ is constant on $(0,\epsilon_2)$, which implies $c_1=1$.
The following lemma shows that the above lower bound is achieved if and only if $\mathrm{VaR}_{1-\epsilon_1}(X)=\mathrm{VaR}_{1-\epsilon_2}(X)$.
From the definition of the PELVE and Lemma \ref{lem:bound_1}, for $\epsilon_1<\epsilon_2$, the possible choices of $(\epsilon_1,c_1)$ and $(\epsilon_2,c_2)$ should satisfy $1\le c_1\le1/\epsilon_1$, $1\le c_2\le 1/\epsilon_2$ and $c_1\epsilon_1\le c_2\epsilon_2$.
We denote by $\Delta$ the admissible set for $(\epsilon_1,c_1,\epsilon_2 ,c_2) $, that is,
$$\Delta=\{(\epsilon_1,c_1,\epsilon_2 ,c_2) \in ((0,1)\times[1, \infty))^2: \epsilon_1<\epsilon_2,~ c_1\epsilon_1\le 1,~ c_2\epsilon_2\le 1,~c_1\epsilon_1\le c_2\epsilon_2 \}.
$$
We illustrate the possible region of $(c_1,c_2)$ with given $\epsilon_1$ and $\epsilon_2$ in Figure \ref{fig:region}. We divide the region into 5 cases and calibrate distribution for each case.
\begin{figure}[!htb]
\centering
\begin{tikzpicture}
[scale=.4,description/.style=auto]
\draw[-latex] (0,-1) -- (0,16);
\draw[-latex] (-1,0) -- (18,0);
\node[left] at (0,16){{\small $c_2$}};
\node[below] at (18,0){{\small $c_1$}};
\draw [top color=gray!50, bottom color=gray!50, white, opacity =0.5] (6,2)--(2,2)--(2,12)--(6,12)--(6,2);
\fill [white,opacity=0.3,postaction={pattern=north east lines}] (6,2)--(6,12)--(14,12)--(6,2);
\draw[thin] (2,0)--(2,14);
\draw[very thick] (2,2)--(2,12);
\draw[thin] (14,0)--(14,14);
\node[below] at (2,0){{\small 1}};
\node[below] at (14,0){{\small $1/\epsilon_1$}};
\draw[thin] (0,2)--(2,2);
\draw[dash pattern={on 0.84pt off 2.51pt}] (2,2)--(14,2);
\draw[thin] (0,12)--(16,12);
\node[left] at (0,2){{\small 1}};
\node[left] at (0,12){{\small $1/\epsilon_2$}};
\draw[very thick] (6,2)--(14,12);
\node[below] at (6,0){{\small $\epsilon_2/\epsilon_1$}};
\draw[dash pattern={on 0.84pt off 2.51pt}] (6,0)--(6,12);
\node at (2,2){$\bullet$};
\path[<-, draw, thick, dashed] (2,2)
to[out=270, in=180] (-1,1)
node[left] {\small Case 1};
\path[<-, draw, thick, dashed] (2,7)
to[out=200, in=45] (-1,5)
node[left] {\small Case 2};
\path[<-, draw, thick, dashed] (10,7)
to[out=300, in=90] (12,4)
node[below] {\small Case 3};
\node at (4,7){{\small Case 4}};
\node at (8.7,9){{\small Case 5}};
\end{tikzpicture}
\caption{Admissible region of $(c_1,c_2)$}
\label{fig:region}
\end{figure}
The calibration process is to construct a continuous and decreasing quantile function that can satisfy two equivalent conditions between $\mathrm{VaR}$ and $\mathrm{ES}$, which are
\begin{equation}\label{eq:VaR-ES}\mathrm{ES}_{1-c_1\epsilon_1}(X)=\mathrm{VaR}_{1-\epsilon_1}(X) \text{~~~ and ~~~} \mathrm{ES}_{1-c_2\epsilon_2}(X)=\mathrm{VaR}_{1-\epsilon_2}(X).\end{equation}
In addition, we need $\mathrm{VaR}_{1-\epsilon}(X)$ not to be a constant on $(0,\epsilon_1]$ so that $\Pi_{X}({\epsilon_1})=c_1$ and $\Pi_{X}(\epsilon_2)=c_2$.
As we can see, only the values of $\mathrm{VaR}_{1-\epsilon}(X)$ for $\epsilon \in (0,c_2\epsilon_2]$ matters for the equivalent condition \eqref{eq:VaR-ES}. Therefore, we focus on constructing $\mathrm{VaR}_{1-\epsilon}(X)$ for $\epsilon \in (0,c_2\epsilon_2]$.
The case $c_1=1$ or $c_2=1$ is special, which means that $\mathrm{VaR}_{1-\epsilon}(X)$ is a constant on tail part. If $c_1>1$, we can set the tail distribution as the generalized Pareto distribution from Table \ref{PELVE}.
For $\mathbf z=(\epsilon_1,c_1,\epsilon_2,c_2) \in \Delta$,
we will construct a class of functions, denoted by $G_{\mathbf z}$, in five different cases according to Figure \ref{fig:region}.
The function $t\mapsto G_{\mathbf z}(1-t)$ will be our desired quantile function.
Let $\hat k,\tilde k \in \mathbb{R}$ be any two constants satisfying $\tilde k<\hat k$. For $c_1>1$, let $\xi \in (-\infty,1)$ be such that $(1-\xi)^{-{1}/{\xi}}=c_1$,
$$
k(\epsilon)=\left\{\begin{aligned}
&\frac{1}{\xi}(\epsilon^{-\xi}-1), &\xi\neq0,\\
&- \log(\epsilon), &\xi=0,
\end{aligned}\right.$$
and $k=\int_0^{\epsilon_1} k(\epsilon) \mathrm{d}\epsilon$.
First, the function $G_{\mathbf z}$ is an arbitrary continuous and decreasing function on $[c_2\epsilon_2,1)$ since the values of $\mathrm{VaR}_{1-t}(X)$ for $t\in [c_2\epsilon_2,1)$ do not affect its PELVE at $\epsilon_1$ and $\epsilon_2$. The value of $G_{\mathbf z}$ on $(0,c_2\epsilon_2]$ is given by
\begin{enumerate}[(i)]
\item \underline{Case 1}, $c_2=1$ (which implies $c_1=1$):
$G_{\mathbf z}(\epsilon)=\hat k;$
\item \underline{Case 2}, $c_1=1$ and $1<c_2\le 1/\epsilon_2$:
$$
G_{\mathbf z}(\epsilon)=\left\{\begin{aligned}
&\hat k, &\epsilon \in (0,\epsilon_1),\\
&a_1\epsilon+b_1, &\epsilon \in [\epsilon_1,\epsilon_2),\\
&a_2\epsilon+b_2, &\epsilon \in [\epsilon_2,c_2\epsilon_2],
\end{aligned}\right.
~~~\text{where} ~~~
\left\{
\begin{aligned}
&a_1=\frac{\tilde k-\hat k}{\epsilon_2-\epsilon_1},\\
&b_1=\hat k-a_1\epsilon_1,\\
&a_2=\frac{(\tilde k-\hat k)(\epsilon_1+\epsilon_2)}{(c_2\epsilon_2-\epsilon_2)^2},\\
&b_2=\tilde k-a_2\epsilon_2;
\end{aligned}\right.$$
\item \underline{Case 3}, $1<c_1\le 1/\epsilon_1$ and $c_2=\frac{c_1\epsilon_1}{\epsilon_2}$:
$$
G_{\mathbf z}(\epsilon)=\left\{\begin{aligned}
&k(\epsilon), &\epsilon \in (0,\epsilon_1),\\
&k(\epsilon_1), &\epsilon \in [\epsilon_1,\epsilon_2),\\
&a\epsilon+b, &\epsilon \in [\epsilon_2,c_2\epsilon_2],
\end{aligned}\right.
\text{~~~where~~~}
\left\{\begin{aligned}
&a=\frac{2(k(\epsilon_1)\epsilon_1-k)}{(c_2\epsilon_2-\epsilon_2)^2},\\
&b=k(\epsilon_1)-a\epsilon_2;
\end{aligned}\right.$$
\item \underline{Case 4}, $1<c_1\le \epsilon_2/\epsilon_1$ and $1<c_2\le 1/\epsilon_2$:
$$
G_{\mathbf z}(\epsilon)=\left\{\begin{aligned}
&k(\epsilon), &\epsilon \in (0,c_1\epsilon_1),\\
&k(c_1\epsilon_1), &\epsilon \in [c_1\epsilon_1, \epsilon_2),\\
&a \epsilon+b, &\epsilon \in [\epsilon_2,c_2\epsilon_2],
\end{aligned}\right.
~~~\text{where}~~~
\left\{\begin{aligned}
&a=\frac{2c_1\epsilon_1\left(k(c_1\epsilon_1)-k(\epsilon_1)\right)}{\left(c_2\epsilon_2-\epsilon_2\right)^2},\\
&b=k(c_1\epsilon_1)-a\epsilon_2;
\end{aligned}\right.$$
\item \underline{Case 5}, $\epsilon_2/\epsilon_1<c_1\le 1/\epsilon_1$ and $\frac{c_1\epsilon_1}{\epsilon_2}<c_2\le1/\epsilon_2$:
$$
G_{\mathbf z}(\epsilon)=\left\{\begin{aligned}
&k(\epsilon), &\epsilon \in (0,\epsilon_1),\\
&a_1\epsilon+b_1, & \epsilon \in [\epsilon_1, \epsilon_2),\\
&a_1\epsilon_2+b_1, &\epsilon \in [\epsilon_2,c_1\epsilon_1),\\
&a_2\epsilon+b_2, &\epsilon \in [c_1\epsilon_1,c_2\epsilon_2],
\end{aligned}\right.~~~\text{where}~~~
\left\{
\begin{aligned}
&a_1=\frac{k(\epsilon_1)\epsilon_1-k}{(\epsilon_2-\epsilon_1)(c_1\epsilon_1-1/2(\epsilon_1+\epsilon_2))},\\
&b_1=k(\epsilon_1)-a_1\epsilon_1,\\
&a_2=\frac{2c_1\epsilon_1(a_1\epsilon_2+b_1-k(\epsilon_1))}{(c_1\epsilon_1-c_2\epsilon_2)^2},\\
&b_2=a_1\epsilon_2+b_1-a_2c_1\epsilon_1.
\end{aligned}\right.~~~$$
\end{enumerate}
An illustration of the functions $G_{\mathbf z}$ in Case 2 to Case 5 is presented in Figure \ref{fig:calibration},
and we omit Case 1 in which $G_{\mathbf z}$ is a constant function.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=.93,description/.style=auto]
\draw[<->] (0,5) -- (0,0) -- (5,0);
\node[below] at (0,0) {$0$};
\node[below] at (1.5,0) {$\epsilon_1$};
\node[below] at (3,0) {$\epsilon_2$};
\node[below] at (4.5,0) {$c_2\epsilon_2$};
\node[left] at (0,5) {$G_{\mathbf z}(\epsilon)$};
\node[left] at (0,4) {$G_{\mathbf z}(\epsilon_1)$};
\draw (0,4)--(1.5,4);
\draw (1.5,4)--(3,3);
\draw (3,3)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (1.5,0)--(1.5,4);
\draw[dash pattern={on 0.84pt off 2.51pt}] (3,0)--(3,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (4.5,0)--(4.5,0.5);
\end{tikzpicture}
\caption{\footnotesize The function $G_{\mathbf z}$ in Case 2}
\end{subfigure}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=.93,description/.style=auto]
\draw[<->] (0,5) -- (0,0) -- (5,0);
\node[below] at (0,0) {$0$};
\node[below] at (1.5,0) {$\epsilon_1$};
\node[below] at (3,0) {$\epsilon_2$};
\node[below] at (4.5,0) {$c_2\epsilon_2$};
\node[left] at (0,5) {$G_{\mathbf z}(\epsilon)$};
\node[left] at (0,3) {$\begin{aligned} &G_{\mathbf z}(\epsilon_1)\\ &G_{\mathbf z}(\epsilon_2)\end{aligned}$};
\draw (0.2,5) .. controls (0.5,3.7) and (1,3.5).. (1.5,3);
\draw (1.5,3)--(3,3);
\draw (3,3)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (1.5,0)--(1.5,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (3,0)--(3,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (4.5,0)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (0,3)--(1.5,3);
\end{tikzpicture}
\caption{\footnotesize The function $G_{\mathbf z}$ in Case 3}
\end{subfigure}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=.93,description/.style=auto]
\draw[<->] (0,5) -- (0,0) -- (5,0);
\node[below] at (0,0) {$0$};
\node[below] at (1.5,0) {$\epsilon_1$};
\node[below] at (2.5,0) {$c_1\epsilon_1$};
\node[below] at (3.5,0) {$\epsilon_2$};
\node[below] at (4.5,0) {$c_2\epsilon_2$};
\node[left] at (0,5) {$G_{\mathbf z}(\epsilon)$};
\node[left] at (0,3) {$\begin{aligned} &G_{\mathbf z}(c_1\epsilon_1)\\ &G_{\mathbf z}(\epsilon_2)\end{aligned}$};
\draw (0.2,5) .. controls (1,3.5) and (2,3.1).. (2.5,3);
\draw (2.5,3)--(3.5,3);
\draw (3.5,3)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (1.5,0)--(1.5,3.3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (2.5,0)--(2.5,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (3.5,0)--(3.5,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (4.5,0)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (0,3)--(3.5,3);
\end{tikzpicture}
\caption{\footnotesize The function $G_{\mathbf z}$ in Case 4}
\end{subfigure}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\begin{tikzpicture}[scale=.93,description/.style=auto]
\draw[<->] (0,5) -- (0,0) -- (5,0);
\node[below] at (0,0) {$0$};
\node[below] at (1.5,0) {$\epsilon_1$};
\node[below] at (2.5,0) {$\epsilon_2$};
\node[below] at (3.5,0) {$c_1\epsilon_1$};
\node[below] at (4.5,0) {$c_2\epsilon_2$};
\node[left] at (0,5) {$G_{\mathbf z}(\epsilon)$};
\node[left] at (0,2) {$\begin{aligned} &G_{\mathbf z}(c_1\epsilon_1)\\ &G_{\mathbf z}(\epsilon_2)\end{aligned}$};
\draw (0.2,5) .. controls (0.5,3.5) and (1,3.1).. (1.5,3);
\draw (1.5,3)--(2.5,2);
\draw (2.5,2)--(3.5,2);
\draw (3.5,2)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (1.5,0)--(1.5,3);
\draw[dash pattern={on 0.84pt off 2.51pt}] (2.5,0)--(2.5,2);
\draw[dash pattern={on 0.84pt off 2.51pt}] (3.5,0)--(3.5,2);
\draw[dash pattern={on 0.84pt off 2.51pt}] (4.5,0)--(4.5,0.5);
\draw[dash pattern={on 0.84pt off 2.51pt}] (0,2)--(2.5,2);
\end{tikzpicture}
\caption{ \footnotesize The function $G_{\mathbf z}$ in Case 5}
\end{subfigure}
\caption{An illustration of $G_{\mathbf z}$ in Cases 2 to 5}
\label{fig:calibration}
\end{figure}
\begin{theorem}\label{thm:two_points}
For $\mathbf z=(\epsilon_1,c_1,\epsilon_2,c_2) \in \Delta$, the random variable $X$ with a continuous quantile function given by $t\mapsto \mathrm{VaR}_{t}(X) =G_{\mathbf z}(1-t)$ satisfies $\Pi_{X}({\epsilon_1})=c_1$ and $\Pi_{X}(\epsilon_2)=c_2$.
\end{theorem}
\begin{remark}\label{rem:cannot}
As we can see from Figure \ref{fig:calibration}, some parts of the calibrated quantile function are flat, corresponding to the existence of atoms in the distribution.
This may be considered as undesirable from a modeling perspective, and indeed it is forced by the boundary cases of $(\epsilon_1,c_1,\epsilon_2,c_2) \in \Delta$ in Figure \ref{fig:region}.
If $(\epsilon_1,c_1,\epsilon_2,c_2)$ is in the interior of $\Delta$ (which is possible in Cases 4 and 5), then the flat part can be replaced by a strictly monotone linear segment, which can be shown by a similar argument to the proof of Theorem \ref{thm:two_points}. The interested reader is referred to Propositions \ref{lem:low_bound} and \ref{pro:c_2_bound} in Appendix \ref{app:proof_sec2}, where we show that a strictly increasing quantile function cannot attain the boundary cases $(\epsilon_1,c_1,\epsilon_2,c_2) $, and hence the flat parts are necessary to include and unify these cases.
\end{remark}
We can easily get the distribution of $X$ from $\mathrm{VaR}_{1-\epsilon}(X)$. As the PELVE is scale-location invariant, we can scale or move the distribution we get to match more information. For example, if $\mathrm{VaR}_{1-\epsilon_1}(X)$ and $\mathrm{VaR}_{1-\epsilon_2}(X)$ are given, we can choose two constants $\lambda$ and $\mu$ such that $\lambda X+\mu$ matches the specified VaR values.
In a similar spirit, the calibration problem can be extended to calibrate the distributions from some given $\mathrm{ES}$ and $\mathrm{VaR}$ values. The two points calibration problem can be regarded as given two $\mathrm{ES}$ and $\mathrm{VaR}$ values. Calibrating from only $\mathrm{ES}$ or $\mathrm{VaR}$ would be easy. However, the choices of $\mathrm{ES}$ values will also be limited by $\mathrm{VaR}$ values if we consider them at the same time, which is the same as the choice of $c_1, c_2$ as we discussed in this section.
As we see above, the PELVE calibration problem is complicated even when only two points on the PELVE curve are given. If we extend the calibration problem to more points, the problem will be more difficult to solve. In the next section, we will characterize distribution from a given PELVE function.
\section{Calibration from a curve constraint}\label{sec:function}
By the location-scale invariance properties of the PELVE, we know that the solution cannot be unique. Conversely, it would be interesting to ask whether all solutions can be linearly transformed from a particular solution; that is, for a given function $\epsilon \mapsto \Pi(\epsilon)$, whether the set
$\left\{ X\in\mathcal {X}:\Pi_{X} =\Pi \right\}$
is a location-scale class.
This question, as well as identifying $X$ satisfying $\Pi_{X}=\Pi$, is the main objective of this section.
\subsection{PELVE and dual PELVE}\label{var-es}
First, we note that calibrated distributions from the an entire PELVE curve $\epsilon \mapsto \Pi(\epsilon)$ on $ (0,1)$ would be unnatural, because the existence of the PELVE requires $\mathbb{E}[X]\le \mathrm{VaR}_{1-\epsilon}(X)$ which may not hold for $\epsilon$ not very small. Thus, the PELVE curve $\Pi_{X}$ does not behave well on some part of $(0,1)$.
To address this issue, we introduce a new notion called the dual PELVE and an integral equation which can help us to calibrate the distribution by differential equations.
The dual PELVE is defined by moving the multiplier in PELVE from the $\mathrm{ES}$ side to the $\mathrm{VaR}$ side.
\begin{definition}
For $X\in L^{1}$, the dual PELVE function of $X$ at level $\epsilon\in(0,1]$ is defined as
$$
\pi_{X}(\epsilon) =\inf\left\{ d\ge1:\mathrm{ES}_{1-\epsilon}(X)\le\mathrm{VaR}_{1-\epsilon/d}(X)\right\} ,~~\epsilon\in(0,1].
$$
\end{definition}
The existence and uniqueness of $\pi_{X}(\epsilon)$ can be shown in the same way as the existence and uniqueness of the PELVE. There are advantages and disadvantages of working with both notions; see \citet[Remark 2]{LW2019}. In our context, the main advantage of using the dual PELVE is that $\pi_{X}(\epsilon)$ is finite for all $\epsilon\in(0,1]$, while $\Pi_{X}(\epsilon)$ is finite only when $\mathbb{E}[X]\le\mathrm{VaR}_{1-\epsilon}(X)$.
Note that for $X$ with a discontinuous quantile function,
there may not exist $d $ such that $\mathrm{ES}_{1-\epsilon}(X)= \mathrm{VaR}_{1-\epsilon/d}(X)$.
In order to guarantee the above equivalence, we limit our discussion to continuous and strictly increasing distribution functions, which cover the most common models in risk management. Let $\mathcal {X}$ be the set of all random variables in $L^{1}$ with continuous and strictly increasing distribution functions. Let $\mathcal{M}$ be the set of distribution functions for $X\in\mathcal {X}$. For $X\in\mathcal {X}$ and $\epsilon\in(0,1]$, $\pi_{X}(\epsilon)$ is the unique solution $d\ge1$ to the equation
$$
\mathrm{ES}_{1-\epsilon}(X)=\mathrm{VaR}_{1-\epsilon/d}(X).
$$
The PELVE and dual PELVE are closely related.
It is straightforward to check that for any $X\in\mathcal {X}$, we have $\Pi_{X}(\epsilon/\pi_{X}(\epsilon))=\pi_{X}(\epsilon)$ for $\epsilon\in(0,1]$, and if $\mathbb{E}[X]\le\mathrm{VaR}_{1-\epsilon}(X)$, then $\pi_{X}(\Pi_{X}(\epsilon)\epsilon)=\Pi_{X}(\epsilon)$.
\subsection{An integral equation associated with dual PELVE}\label{abstract}
In order to calibrate distributions from the dual PELVE, we can equivalently focus on quantile functions. Let us consider $X\in\mathcal {X}$, and let $f(s)=\mathrm{VaR}_{1-s}(X)$. Then, solving $\pi_{X}(\epsilon)$ is the same as solving $z$ in following equation:
\begin{equation}\label{eq:abstract}
\int_{0}^{y}f\left(s\right)\mathrm{d}s=yf\left(zy\right)
\end{equation}
for $y=\epsilon$. The solution is $z=1/\pi_{X}(y)$.
As $f(s)=\mathrm{VaR}_{1-s}(X)$, $f$ is strictly decreasing and continuous.
Let $\mathcal{C}$ be the set of all continuous and strictly decreasing functions $f:(0,1)\to\mathbb{R}$ satisfying $\int_{0}^{1}\left|f(s)\right|\mathrm{d}s<\infty$. For any $f\in\mathcal{C}$, the existence of the solution $z$ is guaranteed by the mean-value theorem and its uniqueness is obvious.
For $y\in(0,1]$, let $z_{f}(y)$ be the solution to \eqref{eq:abstract}. Clearly, $z_{f}(y)\le1$ and $y\mapsto yz_{f}(y)$ is strictly increasing.
This is similar to Lemma \ref{lem:bound_1} for the two-point case.
Obviously, $z_{f}(y)$ is also location-scale invariant under linear transformation on $\mathcal{C}$. That is, $z_{\lambda f+b}=z_{f}$ for $\lambda>0$ and $b\in\mathbb{R}$.
The next proposition is a simple connection between $z_f$ and $\pi_X$.
\begin{proposition}\label{prop:abstract}
For any $f \in \mathcal{C}$, $X=f(U)$ for some $U \sim \mathrm{U}(0,1)$ has the dual PELVE $\pi_X(y)=1/z(y)$ for all $y \in (0,1)$ where $z_f$ is solution to \eqref{eq:abstract}.
For $X \in \mathcal {X}$, there exists $f \in \mathcal{C}$ such that $X=f(U)$ for some $U \sim \mathrm{U}(0,1)$ and the solution to \eqref{eq:abstract} is $z(y)=1/\pi_X(y)$ for all $y \in (0,1)$.
\end{proposition}
\begin{proof}
For any $f\in\mathcal{C}$, let $F(x)=1-f^{-1}(x)$.
Then, $F$ is a continuous and strictly increasing distribution function and $F^{-1}(s)=f(1-s)$ for $s \in (0,1)$.
Let $U\sim \mathrm{U}(0,1)$ and $X=F^{-1}(U)=f(1-U)$.
Then $X \in \mathcal {C}$ and $X\sim F$.
As $F^{-1}(1-s)=f(s)$, we have $\pi_X(y)=1/z_f(y)$.
Take $U'=1-U$. We have $X=f(U')$ and $U'\sim \mathrm{U}(0,1)$.
For $X\in \mathcal {X}$, let $f(s)=\mathrm{VaR}_{1-s}(X)$.
Then, we have $z_f(y)=1/\pi_X(y)$ for $y \in (0,1]$.
Furthermore, we have $F^{-1}(s)=f(1-s)$.
Therefore, there exists $U \sim \mathrm{U}(0,1)$ such that $X=f(1-U)$.
Let $U'=1-U$. Then, we have $X=f(U')$ and $U'\sim \mathrm{U}(0,1)$.
\end{proof}
Proposition \ref{prop:abstract} allows us to study $z_{f}$ instead of $\pi_X$ for the calibration problem. The integral equation \eqref{eq:abstract} can be very helpful in characterizing the distribution from the dual PELVE.
Some examples of $\pi_X$ and $z_{f}$ are listed in Table \ref{ex:zf}, which is corresponding to the PELVE presented in Table \ref{constant-PELVE}.
\begin{table}[htbp]
\def1.4{1.4}
\centering{}
\caption{Example of $\pi_X$ and $z_{f}$}\label{ex:zf} %
\begin{tabular}{m{1.5cm}<{\centering} | m{3cm}<{\centering}|m{5cm}<{\centering}|m{2.8cm}<{\centering}}
\hline \hline
$X$& $\pi_X(\epsilon)$ & $f$ & $z_f$ \\
\hline
$\mathrm{U}(0,1)$ & $\pi_X(\epsilon)=2$ & $f(x)=1-x$ & $z_{f}(y)=1/2$ \\
\hline
$\mathrm{Exp}(\lambda)$& $\pi_X(\epsilon)=e$ & $f(x)=-\log(x)/\lambda$&$z_{f}(y)=1/e$ \\
\hline
$\mathrm{GPD}(\xi)$ & $\pi_X(\epsilon)=(1-\xi)^{-\frac{1}{\xi}}$ & $f(x)=\left\{\begin{aligned}
&1/\xi\left(x^{-\xi}-1\right) &\xi\neq 0\\
&-\log(x) &\xi=0 \\\end{aligned}\right.$ &$z_{f}=(1-\xi)^{\frac{1}{\xi}}$\\
\hline \hline
\end{tabular}
\end{table}
For a given dual PELVE curve $\pi$, we find the solution to the integral equation by the following steps.
\begin{enumerate}
\item Let $z(y)=\frac{1}{\pi(y)}$ for all $y\in(0,1]$.
\item Find $f\in\mathcal{C}$ that satisfies $\int_{0}^{y}f(s)\mathrm{d} s=yf\left(z(y)y\right)$ for all $y\in(0,1]$.
\item By Proposition \ref{prop:abstract}, $X=f(U)$ for some $U \sim \mathrm{U}(0,1)$ will have the given dual PELVE $\pi$.
\end{enumerate}
Therefore, we will focus on characterizing $f$ from a given $z:(0,1]\to(0,1]$ below. Generally, it is hard to characterize $f$ explicitly.
We first formulate the problem as an advanced differential equation, which helps us to find solutions.
\subsection{Advanced differential equations}
In this section, we show that the main objective \eqref{eq:abstract} can be represented by a differential equation. The idea of using differential equation in computing risk measures has not been actively developed, the only paper we know is \cite{B20} which addresses a different problem.
Let us recall the integral equation from Section \ref{abstract}.
For a function $f\in\mathcal{C}$, we solve function $z_{f}:(0,1)\to\mathbb{R}$ from \eqref{eq:abstract}.
Then, we have $\int_{0}^{y}f(s)\mathrm{d} s=yf\left(z_{f}(y)y\right)$ for all $y\in(0,1]$.
It is easy to see that $z_{f}\left(y\right)\le1$.
Let $\omega_{f}(y)=yz_{f}\left(y\right)$.
We have $\omega_{f}$ is strictly increasing on $(0,1]$ and $\omega_{f}(y)\le y$.
Let $u_{f}$ be the inverse function of $\omega_{f}$.
Then, $u_{f}$ is a function defined from $(0,z_{f}(1)]$ to $(0,1]$.
We get that $u_{f}$ is strictly increasing and $u_{f}\left(w\right)\ge w$.
We assume both $u_f$ and $f$ are differentiable.
The following equality holds
$$
\int_{0}^{y}f\left(s\right)\mathrm{d} s=yf\left(z_{f}\left(y\right)y\right) ~\Longleftrightarrow ~f'\left(w\right)+\frac{u_{f}'\left(w\right)}{u_{f}\left(w\right)}\left(f\left(w\right)-f\left(u_{f}\left(w\right)\right)\right)=0.
$$
Since $u_{f}$ is strictly increasing and $u_{f}\left(w\right)\ge w$, we can rewrite the above equation in the following form
\begin{equation}
f'\left(w\right)+A\left(w\right)f\left(w\right)-B\left(w\right)f\left(h\left(w\right)\right)=0,\label{DDE}
\end{equation}
where $A_{f}\left(w\right)=\frac{u_{f}'\left(w\right)}{u_{f}\left(w\right)}\ge0,B_{f}\left(w\right)=\frac{u_{f}'\left(w\right)}{u_{f}\left(w\right)}\ge0$ and $h\left(w\right)=u_{f}\left(w\right)\ge w$.
For a given $z:(0,1]\to\mathbb{R}$, $u_{f}$ is known. As such, \eqref{DDE} is a linear advanced differential equation which is well studied in literature. In \citet{BB11},
it is shown that
if $0\le B(w)\le A(w)$ and $h(w)\ge w$, then there exists a non-oscillatory solution for \eqref{DDE}.
In general, the solution to \eqref{DDE} is not unique.
We will see in the following sections that how different solutions of \eqref{DDE} can be constructed.
\iffalse
To see that let us first prove the following lemma:
\begin{lemmma}\label{lem:1}
Let us consider two functions $f_{1}$ and $f_{2}$ in $\mathcal{C}$, that are almost surely differentiable. In addition suppose that both of them are solutions to \eqref{DDE} where $u_{f_{1}}=u_{f_{2}}=u$. Then $f=f_{1}\vee f_{2}$ and $y=u_{f}\left(w\right)\Leftrightarrow w=yz_{f}\left(y\right)f_{1}\wedge f_{2}$ are also a solution to \eqref{DDE}.
\end{lemmma}
\begin{proof}
We only prove it for $f=f_{1}\vee f_{2}$, the other one follows similarly.
As both $f_{1}$ and $f_{2}$ are continuous, and almost surely differentiable, they are of finite variation. So one can identify a set $I=\cup_{i}I_{i}$, where $\left\{ I_{i}\right\} _{i}$ are intervals, where $f_{1}\vee f_{2}=f_{1}$ on $I$, and $f_{1}\vee f_{2}=f_{2}$, otherwise. So we have
\begin{multline}
f^{'}\left(w\right)+A\left(w\right)f\left(w\right)-B\left(w\right)f\left(h\left(w\right)\right)\\
=f_{1}^{'}\left(w\right)+A\left(w\right)f_{1}\left(w\right)-B\left(w\right)f_{1}\left(h\left(w\right)\right)=0,a.s,\text{ on }\mathrm{int}\left(I\right),\label{DDE-1}
\end{multline}
and
\begin{multline}
f^{'}\left(w\right)+A\left(w\right)f\left(w\right)-B\left(w\right)f\left(h\left(w\right)\right)\\
=f_{2}^{'}\left(w\right)+A\left(w\right)f_{2}\left(w\right)-B\left(w\right)f_{2}\left(h\left(w\right)\right)=0,a.s,\text{ on }\mathrm{int}\left(\mathbb{R}\setminus I\right).\label{DDE-1-1}
\end{multline}
This implies that \eqref{DDE} holds for $f$.
\end{proof}
\begin{corollary}\label{cor:1}
For a given PELVE, if $f_{1}$ and $f_{2}$ are solutions to \eqref{eq:abstract} then $f=f_{1}\vee f_{2}$ and $f=f_{1}\wedge f_{2}$ are also solutions to \eqref{eq:abstract}. In particular, for any constant number $d$, $f=f_{1}\vee d$ and $f_{1}\wedge d$ satisfy
\eqref{eq:abstract}.
\end{corollary}
\begin{proof}
The first part of the corollary is a direct result of Lemma \ref{lem:1}.
The second part can be derived by sending $\epsilon>0$ to zero when
$f=f_{1}\vee\left(d+\epsilon f_{1}\right)$ is verified in the the integral equation \eqref{eq:abstract}.
\end{proof}
%
\begin{corollary}
For any two co-monotone random variable $X_{1}$ and $X_{2}$ so that $\pi_{X_{1}}=\pi_{X_{2}}=\pi$, then $\pi_{X_{1}\vee X_{2}}=\pi_{X_{1}\wedge X_{2}}=\pi$.
\end{corollary}
\fi
\subsection{A numerical method}
For a given $z:(0,1)\to\mathbb{R}$, let $\omega(y)=yz(y)$ and $u$ be the inverse function of $\omega$. The goal is to solve the following different equation for $f$:
\begin{equation}
f'\left(w\right)+\frac{u'\left(w\right)}{u\left(w\right)}f\left(w\right)=\frac{u'\left(w\right)}{u\left(w\right)}f\left(u\left(w\right)\right),\quad0\le w\le a,\label{eq:DDE}
\end{equation}
where $a=\omega(1)=z(1)$. It is hard to get explicit solution to \eqref{eq:DDE}.
Here we present a numerical method to solve \eqref{eq:DDE}.
Let us introduce the following process.
\begin{enumerate}
\item Let $a_{0}=1$, $a_{1}=a$, ...,$a_{n}=u^{-1}\left(a_{n-1}\right)$.
\item
For $a \in (0,1)$, let $\xi$ be the solution to $(1-\xi)^{\frac{1}{\xi}}=a$.
Let
\begin{equation}\label{eq:GPD}
f_0(x)=\left\{\begin{aligned}
&\frac{1}{\xi}\left(x^{-\xi}-1\right), &\xi\neq 0,\\
&-\log(x), &\xi=0, \\\end{aligned}\right.
\end{equation} on $[a,1]$.
\item We can solve the following ODE on $\left[a_{2},a_{1}\right]$:
$$ f_{1}'\left(w\right)+\frac{u'\left(w\right)}{u\left(w\right)}f_{1}\left(w\right) =\frac{u'\left(w\right)}{u\left(w\right)}f_{0}\left(u\left(w\right)\right),\quad w\in\left[a_{2},a_{1}\right].
$$
\item Now we can repeat step 3 by induction on $\left[a_{n+1},a_{n}\right]$ for $n>1$ by solving
$$
f_{n}'\left(w\right)+\frac{u'\left(w\right)}{u\left(w\right)}f_{n}\left(w\right) =\frac{u'\left(w\right)}{u\left(w\right)}f_{n-1}\left(u\left(w\right)\right),\quad w\in\left[a_{n+1},a_{n}\right].
$$
\item In general, the solution for differential equation $\frac{dy}{dx}+P(x)y=Q(x)$ is
$$
y =e^{-\int^{x}P(\lambda)\,\mathrm{d}\lambda}\left[\int^{x}e^{\int^{\lambda}P(\varepsilon)\mathrm{d}\varepsilon}Q(\lambda)\mathrm{d}\lambda+C\right].
$$
So, we get the following solution for $f_{n}$:
$$
f_{n}\left(w\right) =e^{\int_{w}^{a_{n}}\frac{u'(\lambda)}{u(\lambda)}\mathrm{d}\lambda}\left[f_{n-1}(a_{n})-\int_{w}^{a_{n}}e^{-\int_{\lambda}^{a_{n}}\frac{u'(\varepsilon)}{u(\varepsilon)}\mathrm{d}
\varepsilon}\frac{u'(\lambda)}{u(\lambda)}f_{n-1}\left(u\left(\lambda\right)\right)\mathrm{d}\lambda\right],~w\in[a_{n+1},a_{n}].\label{fn}
$$
\item Finally, let $f=f_{n}$ on $[a_{n+1},a_{n}]$.
\end{enumerate}
Note that since we start with a strictly decreasing function, then from equation (\ref{eq:DDE}) we have
$$
f'(w) =\frac{u'\left(w\right)}{u\left(w\right)}\left(f\left(u(w)\right)-f\left(w\right)\right)<0,
$$
so $f$ remains strictly decreasing.
Especially, when $z(y)=c$ for some constant $c$, we have $u(x)=x/c$.
Therefore, \eqref{fn} gives
$$
f_n(w)=\frac{a_n}{w}\left[f_{n-1}(a_n)-\int_{w}^{a_n}\frac{1}{a_n}f_{n-1}\left(\frac{\lambda}{c}\right)\mathrm{d}\lambda\right].
$$
If we set $f_{0}$ as \eqref{eq:GPD}, we can have $f_{1}$ is also in the form of \eqref{eq:GPD}.
Then, it is obvious that $f_n$ is also in the form of \eqref{eq:GPD}.
Therefore, the numerical method gives the simplest power function or logarithm function when $z(y)$ is a constant on $(0,1]$ as Table \ref{ex:zf}, which leads to the generalized Pareto distribution for $X$.
The solution to the equation (\ref{eq:DDE}) is heavily relying on $f_{0}$.
Thus, the solution to the equation (\ref{eq:DDE}) is not unique but the solution from the above process is unique.
We set $f_{0}$ as \eqref{eq:GPD} because we assume $z$ can be extended from $(0,1]$ to $\mathbb{R}^{+}$ and set $z(y)=a$ for all $y>1$.
\label{sec:44}
\subsection{Numerical examples}
Now let us explore the method in Section \ref{sec:44} with simulation. Here we present the results for a few cases. In Figures \ref{linear} to \ref{exp}, we compare the solution from numerical method with the standard formula in Table \ref{ex:zf} in the left panel, and
compare $\int_{0}^{y}f(s)\mathrm{d} s$ with $yf(z(y)y)$ to validate the equation \eqref{eq:abstract} in the right panel.
We first try some examples where $z$ is constant as shown in Table \ref{ex:zf}, i.e. $z(x)=1/2$ (Figure \ref{linear}), $z(x)=1/e$ (Figure \ref{log}) and $z(x)=0.9^{10}$ (Figure \ref{power}). For Figure \ref{linear} to \ref{power}, we can see that the numerical method provides exactly the same function $f$ as Table \ref{ex:zf}.
In Figure \ref{exp}, we check the case $f(x)=e^{-x}$. When $f(x)=e^{-x}$, we can solve $z(x)=\log\left(x/(1-e^{-x})\right)/x$ from \eqref{eq:abstract}. We can see that the solution from numerical method is close to a function of the form $f(x)=\lambda e^{-x}+b$, which is known to satisfy the integral equation.
\begin{figure}[h]
\centering\includegraphics[scale=0.45]{linear}
\includegraphics[scale=0.45]{check-linear}\caption{Calibrated function and validation for $z(x)=1/2$}
\label{linear}
\end{figure}
\begin{figure}[h]
\centering\includegraphics[scale=0.45]{log}
\includegraphics[scale=0.45]{check-log}\caption{Calibrated function and validation for $z(x)=1/e$}
\label{log}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[scale=0.45]{power}
\includegraphics[scale=0.45]{check-power}
\caption{Calibrated function and validation for $z(x)=0.9^{10}$}
\label{power}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[scale=0.45]{exp1} \includegraphics[scale=0.45]{check-exp}
\caption{Calibrated function and validation for $z(x)=\log\left(x/(1-e^{-x})\right)/x$}
\label{exp}
\end{figure}
\section{The constant PELVE curve}\label{sec:constant}
Although there are no explicit solutions for $f$ in \eqref{abstract} for a general $z$ function, we can find explicit solutions for $z$ being constant. In this section, we assume $z(y)=c$ for all $y\in(0,1]$ and some constant $c \in\left(0,1\right)$. As we can see from Table \ref{ex:zf}, the power function and logarithm function have constant $z_f$. If $f(x)=\lambda x^{\alpha}+b$ for $\alpha>-1$, we can see that $ (\alpha+1)^{-1/\alpha}=c$.
In this section, we can characterize all the other solutions which can not be expressed as a linear transformation of the power function. That is, we will see that the set
$$\left\{ f\in \mathcal{C}:z_{f}(y)=z(y), ~y \in (0,1]\right\}$$
is not a location-scale class.
\begin{theorem}\label{thm:constant}
For $c \in (0,1)$, any $X\in\mathcal {X}$ with $\pi_{X}(\epsilon)=1/c$ for $\epsilon \in (0,1)$ can be written as $X=f(U)$ for some $U \sim \mathrm{U}(0,1)$ and $f\in \mathcal {C}$. Furthermore, such $f$ has the form
$$f\left(y\right) =C_{1}+C_{2}y^{\alpha}+O\left(y^{\zeta}\right),$$
where $\alpha$ is the root of $(\alpha+1)^{-1/\alpha}=c$, $\zeta>\max\{0,\alpha\}$, $C_1, C_2\in \mathbb{R}$, $C_2\alpha<0$ and $O (y^{\zeta} )$ is a function such that $\lim_{y \to 0} O (y^{\zeta} )/y^{\zeta}$ is a constant.
\end{theorem}
The proof of Theorem \ref{thm:constant} is provided in Appendix \ref{app:proof_sec5}. As we can see, Theorem \ref{thm:constant} characterizes all the solutions that represent $f$.
If $c \in (0,1/e)$, $\alpha$ is negative. As $\zeta>0$, we can see that $X=f(U)$ is regularly varying of index $\alpha$.
Hence, one can then consider the Pareto distribution with survival function $S(x)=x^{\alpha}$ as a representative solution for the tail behavior. An open question is that, in the general case that the PELVE is not necessarily constant, whether all the solutions behave similarly regarding their tail behavior.
Another interesting implication of the theorem and its proof is that one can give a non-trivial solution for $z_f$ is a constant.
\begin{example}\label{f}
For $c \in (0,1)$, let $(\theta,\eta)$ be a solution of
$$
\begin{cases}
c\log c=-\frac{\eta\exp\left(-\frac{\eta}{\tan\left(\eta\right)}\right)}{\sin\left(\eta\right)},\\
\theta=-\frac{\eta}{\tan\left(\eta\right)}.
\end{cases}
$$
Then,
\begin{equation}\label{eq:sol_f}
f(y)=C_1+C_2y^{\alpha}+C_3y^{\zeta}\sin(-\sigma\log(y)),\quad 0<y<1,
\end{equation}
satisfies $\int_{0}^{y}f(s)\mathrm{d} s=yf(cy)$ and $f\in \mathcal {C}$,
where $\alpha$ solves $(\alpha+1)^{-1/\alpha}=c$, $\zeta=\theta/\log c-1$, $\sigma=-\eta/\log c$, $C_2$ is a constant such that $C_2\alpha<0$
and $0<C_3<-C_2\alpha/(\zeta+|\sigma|)$.
\iffalse
We can verify the solution directly.
First, we need to check if $f\in\mathcal{C}$.
It is clear that $f$ is obviously.
We need to check if $f$ is strictly decreasing. For monotonicity, we have
$$
\begin{aligned}
f'\left(y\right)
& =\alpha C_2 y^{\alpha-1}+\zeta C_3y^{\zeta-1}\sin\left(-\sigma\log\left(y\right)\right)-C_3\sigma y^{\zeta-1}\cos\left(-\sigma\log\left(y\right)\right)\\
& =\alpha C_2 y^{\alpha-1}+C_3y^{\zeta-1}\left(\zeta\sin\left(-\sigma\log\left(y\right)\right)-\sigma\cos\left(-\sigma\log\left(y\right)\right)\right)\\
& \le \alpha C_2 y^{\alpha-1}+C_3y^{\zeta-1}\left(\zeta+\left|\sigma\right|\right)
=y^{\alpha-1}\left(C_2\alpha+C_3y^{\zeta-\alpha}\left(\zeta+\left|\sigma\right|\right)\right)\\
& <y^{\alpha-1}C_2\alpha\left(1-y^{\theta-\alpha}\right)
<0.
\end{aligned}
$$
Thus, $f$ is strictly decreasing.
Next, we check if $f$ satisfies $\int_{0}^{y}f(s)\mathrm{d} s=yf(cy)$. Using the same change of variable of $x\left(t\right)=e^{-t}f\left(e^{-t}\right)$, we know that $x(t)=e^{\lambda t+\sigma ti}$ satisfies $x'(t)=-e^{-a}x(t-a)$ for $a=-\log\left(c\right)$. So this gives us
$$\left(\lambda+i\sigma\right)e^{\left(\lambda+i\sigma\right)t} =-ce^{\left(\lambda+i\sigma\right)\left(t-a\right)}.$$
As the real and the imaginary part of the both sides are equal, we get
$$
\begin{aligned}
\lambda\cos\left(\sigma t\right)-\sigma\sin\left(\sigma t\right) & =-ce^{-\lambda a}\cos\left(\sigma\left(t-a\right)\right),\\
\sigma\cos\left(\sigma t\right)+\lambda\sin\left(\sigma t\right) & =-ce^{-\lambda a}\sin\left(\sigma\left(t-a\right)\right).
\end{aligned}
$$
In addition, we have
$$
\begin{aligned}
ame^{am}=-ae^{-a}
& \Rightarrow\left|m\right|^{2}e^{2a\lambda}=e^{-2a}\\
& \Rightarrow\left(\lambda^{2}+\sigma^{2}\right)e^{2a\lambda}=e^{-2a}\\
& \Rightarrow\lambda^{2}+\sigma^{2}=e^{-2a\left(1+\lambda\right)}=e^{2a\zeta}=c^{-2\zeta}.
\end{aligned}
$$
In order to verify that $f$ satisfies $\int_{0}^{y}f(s)\mathrm{d} s=yf(cy)$, we can separately check each part in the summation.
As $(\alpha+1)^{-1/\alpha}=c$, we have $(\alpha+1)c^{\alpha}=1$.
For the first part $y^{\alpha}$, we have
$$\int_{0}^{y}s^{\alpha}ds =\frac{y^{\alpha+1}}{(1+\alpha)}=c^{\alpha}y^{\alpha+1}=y(cy)^{\alpha}.$$
Now let us focus on $\lambda\left(y\right)=y^{\zeta}\sin\left(-\sigma\log\left(y\right)\right)$.
For $y\in(0,1)$, taking integral from $l$ we have
$$\begin{aligned}
\int_{0}^{y}\lambda\left(s\right)ds
& =\int_{0}^{y}s^{\zeta}\sin\left(-\sigma\log\left(s\right)\right)ds\\
& =-\int_{-\infty}^{\log\left(y\right)}e^{\left(\zeta+1\right)w}\sin\left(\sigma w\right)dw\\
& =-\int_{-\infty}^{\log\left(y\right)}e^{-\lambda w}\sin\left(\sigma w\right)dw\\
& =\left.\frac{e^{-\lambda w}}{\lambda^{2}+\sigma^{2}}\left(\lambda\sin\left(\sigma w\right)+\sigma\cos\left(\sigma w\right)\right)\right|_{-\infty}^{\log\left(y\right)}\\
& =\frac{y^{-\lambda}}{\lambda^{2}+\sigma^{2}}\left(\lambda\sin\left(\sigma\log\left(y\right)\right)+\sigma\cos\left(\sigma\log\left(y\right)\right)\right)\\
& =\frac{-y^{-\lambda}}{\lambda^{2}+\sigma^{2}}ce^{-\lambda a}\sin\left(\sigma\left(\log\left(y\right)-a\right)\right)\\
& =\frac{-y\left(cy\right)^{-\lambda-1}}{\lambda^{2}+\sigma^{2}}c^{2\lambda+2}\sin\left(\sigma\left(\log\left(y\right)-a\right)\right)\\
& =\frac{y\left(cy\right)^{\zeta}}{\lambda^{2}+\sigma^{2}}c^{-2\zeta}\sin\left(-\sigma\log\left(cy\right)\right)\\
& =y\left(cy\right)^{\zeta}\sin\left(-\sigma\log\left(cy\right)\right)=y\lambda\left(cy\right).
\end{aligned}$$
As $z_{f}$ is location-scale invariant with respect to $f$, any linear transformations of $f$ also gives that $z_{f}$ is constant. Therefore, we can get $z_f(x)=c$.
\fi
\end{example}
If we take $C_3=0$, we get the simplest power function for $z_f(x)=c$.
If $C_3\neq0$, the solution \eqref{eq:sol_f} is not a linear transformation of the power function solution.
Let us look at the uniform distribution example where $\pi_{X}(\epsilon)=2$ for all $\epsilon\in(0,1]$, which means $z_f(y)=1/2$ for $y\in(0,1]$.
As we have seen in Table \ref{ex:zf}, $f(y)=1-y$ can be a solution.
Furthermore, according to Theorem \ref{f}, we can have another solution
$$f(y) =1-y^{\alpha}+Cy^{\zeta}\sin(-\sigma\log(y)),$$
where $\alpha=1$, $C=0.05096$, $\zeta=4.0184$ and $\sigma=-15.4090$.
In the left of Figure \ref{fig:Non-unique-answer}, we have depicted the two solutions for $f$.
We can see they are quite different when $y$ goes to 1.
In the right of Figure \ref{fig:Non-unique-answer}, we numerically calculate $z_{f}$ for $f(y)=1-y^{\alpha}+Cy^{\zeta}\sin(-\sigma\log(y))$. We can see its numerical value is almost 1/2 and the discrepancy is due to limited computational accuracy.
\begin{figure}[htbp]
\centering{}\includegraphics[scale=0.45]{f_constant} \includegraphics[scale=0.45]{z_constant}
\caption{\label{fig:Non-unique-answer}Non-unique calibrated functions for $z(y)=1/2$.}
\end{figure}
By letting $X=f(U)$, we get $\pi_{X}(\epsilon)=2$ for all $\epsilon\in(0,1]$ and such $X$ does not follow the uniform distribution.
\section{Technical properties of the PELVE}\label{sec:property}
We now take a turn to study several additional properties of PELVE. In particular, we will obtain results on monotonicity and convergence of the dual PELVE as well as the PELVE.
\subsection{Basic properties of dual PELVE}
The following proposition that shows the PELVE and dual PELVE share some basic properties such as monotonicity (i), location-scale invariance (ii) and shape relevance (iii)-(iv) below.
\begin{proposition}\label{property}
Suppose $X\in\mathcal {X}$, $\epsilon\in(0,1]$.
\begin{enumerate}[(i)]
\item $\Pi_{X}(\epsilon)$ is increasing (decreasing) in $\epsilon$ if and only if so is $\pi_X(\epsilon)$.
\item For all $\lambda>0$ and $a\in\mathbb{R}$, $\pi_{\lambda X+a}(\epsilon)=\pi_{X}(\epsilon)$.
\item $\pi_{f(X)}(\epsilon)\le\pi_{X}(\epsilon)$ for all strictly increasing concave functions: $f:\mathbb{R}\to\mathbb{R}$ with $f(X)\in\mathcal {X}$.
\item $\pi_{g(X)}(\epsilon)\ge\pi_{X}(\epsilon)$ for all strictly increasing convex functions: $g:\mathbb{R}\to\mathbb{R}$ with $g(X)\in\mathcal {X}$.
\end{enumerate}
\end{proposition}
The statements (ii)-(iv) are parallel to the corresponding statements in Theorem 1 of \cite{LW2019} on PELVE.
The proof of Proposition \ref{property} is put in Appendix \ref{app:proof_sec6}. Proposition \ref{property} allows us to study monotonicity and convergence of the PELVE by analyzing the corresponding properties of the dual PELVE, which is more convenient in many cases. In the following sections, we focus on finding the conditions which make the dual PELVE monotone and convergent at 0. By Proposition \ref{property}, those conditions can also apply to the PELVE.
\subsection{Non-monotone and non-convergent examples}
In this section, we study monotonicity and convergence of dual PELVE. For monotonicity, we have shown some well-known distributions such as normal distribution, t-distribution and lognormal distribution have monotone PELVE curves in Example \ref{PELVE}.
However, the PELVE is not monotone for all $X$. Below we provide an example.
\begin{example}
[Non-monotone PELVE]
Let us consider the following density function $g$ on $[-2,2]$,
$$
g\left(x\right)
= \frac{1}{2}\left((x+2)\mathds{1}_{\{x\in[-2,-1]\}} -x\mathds{1}_{\{x\in(-1,0]\}}+ x\mathds{1}_{\{x\in(0,1]\}} + (2-x) \mathds{1}_{\{x\in (1,2]\}} \right) .
$$
For $X$ with density function $g$,
Figure \ref{nonmontone} presents the value of $\Pi_{X}(\epsilon)$ for $\epsilon\in(0,0.5)$.
As one can see, the PELVE is not necessarily decreasing, and so is the dual PELVE.
\begin{figure}[htpb]
\centering{}\includegraphics[scale=0.6]{notmonotone} \caption{PELVE for $X$ with density $g$}
\label{nonmontone}
\end{figure}
\end{example}
For the convergence, it is clear that $\pi_{X}(\epsilon)$ is continuous in $(0,1)$.
Therefore, $\lim_{\epsilon\to p}\pi_{X}(\epsilon)$ exists for all $p\in(0,1)$. However, both $\Pi_{X}(\epsilon)$ and $\pi_{X}(\epsilon)$ are not well defined at $\epsilon=0$. If $\lim_{\epsilon\to0} \pi_{X}(\epsilon)$ exists, we can define $\pi_{X}(0)$ as the limit, and $\Pi_{X}(0)$ similarly. However, the following example shows that the limit does not exist for some distributions.
\begin{example}[No limit at 0]
We can construct a random variable $X\in\mathcal {X}$ such that $\lim_{\epsilon\to0}\pi_{X}(\epsilon)$ does not exist from the integral equation \eqref{eq:abstract} in Section \ref{abstract}. Equivalently, we will find a continuous and strictly decreasing function $f\in\mathcal{C}$ such that $\lim_{y\to0}z_{f}(y)$ does not exist.
Let $c$ be the Cantor ternary function on $[0,1]$. Note that $x\mapsto c(x)$ is continuous and increasing on $(0,1)$ and $c(x/3)=c(x)/2$. Let $f(x)=-c(x)-x^{\log2/\log3}$.
It is clear that $f\in\mathcal{C}$ and $f(x/3)=f(x)/2$. For each $y\in(0,1]$, we have
$$
\begin{aligned}
yf\left(z_{f}(y)y\right)
& =\int_{0}^{y}f(x)\mathrm{d} x
\\ & =2\int_{0}^{y}f\left(\frac{1}{3}x\right)\mathrm{d} x
=6\int_{0}^{\frac{1}{3}y}f(x)\mathrm{d} x=2yf\left(\frac{1}{3}yz_{f}\left(\frac{1}{3}y\right)\right) =yf\left(yz_{f}\left(\frac{1}{3}y\right)\right).
\end{aligned}
$$
Since $f$ is strictly decreasing, $z_{f}(y)=z_{f}(y/3)$ for $y\in(0,1]$.
It means that $z_{f}(y)$ is a constant on $(0,1]$ if $\lim_{y\to0}z_{f}(y)$ exists.
Now, let us look at two particular points of $z_{f}(y)$. We can show that $z_{f}(1)\neq z_{f}(4/9)$.
Let $z=(\log2/\log3+1)^{-(\log3/\log2)}$. Then, we have $1/3<z\approx 0.46<1/2$.
For $y=1$, we have $\int_{0}^{1}c(s)\mathrm{d} s=c(z)=1/2$ and $\int_{0}^{c}s^{\log2/\log3} \mathrm{d} s=z^{\log2/\log3}$. Therefore, we get $z_{f}(1)=z<1/2$.
For $y=4/9$, we have
$$
\begin{aligned}
f\left(\frac{4}{9}z_{f}\left(\frac{4}{9}\right)\right)
& =\frac{9}{4}\int_{0}^{4/9}f(s)\mathrm{d} s\\
& =-\frac{9}{4}\left(\frac{1}{\frac{\log2}{\log3}+1}\left(\frac{4}{9}\right)^{\frac{\log2}{\log3}+1}+\frac{1}{12}+\frac{1}{2}\left({\frac{4}{9}-\frac{1}{3}}\right)\right)
<-0.68<f\left(\ensuremath{\frac{2}{9}}\right)\approx-0.64.
\end{aligned}
$$
As $f$ is strictly increasing, we have $(4/9)z_{f}(4/9)>2/9$ which implies $z_{f}(4/9)>1/2>z_{f}(1)$.
As a result, $\lim_{y\to0}z_{f}(y)$ does not exist. Therefore, we have a continuous and strictly decreasing $f$ such that $\lim_{y\to0}z_{f}(y)$ does not exist.
\end{example}
\subsection{Sufficient condition for monotonicity and convergence}\label{monotonicity}
In risk management applications,
for a random variable $X$ modeling a random loss, the behavior of its tail is the most important.
Let $F^{[p,1]}$ be the upper $p$-tail distribution of $F$ (see e.g., \cite{LW21}), namely
$$
F^{[p,1]}(x) =\frac{(F(x)-p)_{+}}{1-p},~~~x\in \mathbb{R}.
$$
We will see that the dual PELVE of $F^{[p,1]}$ is a part of the dual PELVE of $F$.
\begin{lemma}\label{lem:tail}
For $F\in\mathcal{M}$ and $p\in(0,1)$, $X\sim F$ and $X'\sim F^{[p,1]}$, it holds
$$\pi_{X'}(\epsilon) =\pi_{X}(\epsilon(1-p)).$$
\end{lemma}
\begin{proof}
It is clear that $\mathrm{VaR}_{1-\epsilon}(X')=\mathrm{VaR}_{1-{\epsilon(1-p)}}(X)$ and $\mathrm{ES}_{1-\epsilon}(X')=\mathrm{ES}_{1-{\epsilon(1-p)}}(X)$. Therefore,
$$
\begin{aligned}
\pi_{X'}(\epsilon)
& =\inf\{d\ge1:\mathrm{ES}_{1-\epsilon}(X')\le\mathrm{VaR}_{1-\epsilon/d}(X')\}\\
& =\inf\{d\ge1:\mathrm{ES}_{1-\epsilon(1-p)}(X')\le\mathrm{VaR}_{1-\epsilon(1-p)/d}(X')\} =\pi_{X}(\epsilon(1-p)).
\end{aligned}
$$
Thus, we have the desired result.
\end{proof}
The tail distribution can provide a condition to check whether the dual PELVE is decreasing.
\begin{proposition}\label{convex}
For $F \in \mathcal{M}$,
if $x\mapsto F^{-1}\big((1-p)F(x)+p\big)$ is convex (concave) for all $p\in(0,1)$, then $\pi_{X}$ is decreasing (increasing).
\end{proposition}
\begin{proof}
For any $p\in(0,1)$, let $X'\sim F^{[p,1]}$. By Lemma \ref{lem:tail},
we have $\pi_{X'}(\epsilon)=\pi_{X}(\epsilon(1-p))$. Furthermore,
we have
\begin{align*}
{\left(F^{[p,1]}\right)}^{-1}(t) & =F^{-1}\left(\ensuremath{(1-p)t+p}\right) =F^{-1}\Big((1-p)F\big(F^{-1}(t)\big)+p\Big),\quad t\in[0,1].
\end{align*}
Let $U\sim\mathrm{U}(0,1)$, $X=F^{-1}(U)$ and $X'=(F^{[p,1]})^{-1}(U)$.
We assume that $x\mapsto F^{-1}\big((1-p)F(x)+p\big)$ is a convex
function on $(\mathrm{ess\mbox{-}inf}(X),\mathrm{ess\mbox{-}sup}(X))$ first. Let $f:\mathbb{R}\to\mathbb{R}$ be
a strictly increasing convex function such that $f(x)=F^{-1}((1-p)F(x)+p)$
for $x\in(\mathrm{ess\mbox{-}inf}(X),\mathrm{ess\mbox{-}sup}(X)).$ Then, we have $X'=f(X)$. By Proposition
\ref{property}, we get $\pi_{X'}(\epsilon)\ge\pi_{X}(\epsilon)$.
As $\pi_{X'}(\epsilon)=\pi_{X}\ensuremath{(\epsilon(1-p))}$, we have
$\pi_{X}\ensuremath{(\epsilon(1-p))}\ge\pi_{X}(\epsilon)$ for all $p\in(0,1)$.
Thus, $\pi_{X}$ is decreasing.
On the other hand, if $x\mapsto F^{-1}\big((1-p)F(x)+p\big)$ is concave,
we have $\pi_{X}\ensuremath{(\epsilon(1-p))}\le\pi_{X}(\epsilon)$ for
all $p\in(0,1)$ and $\pi_{X}$ is increasing.
\end{proof}
The condition $x\mapsto F^{-1}\big((1-p)F(x)+p\big)$ is convex (concave) for all $p\in(0,1)$ is generally hard to check. Intuitively, this condition means that $F^{[p,1]}$ has a less heavy tail compared to $F$.
We can further simplify this condition by using the hazard rate function. For $X\in\mathcal {X}$ with distribution function $F$ and density function $f$, let $S=1-F$ be the survival function and $\eta=f/S$ be the hazard rate function. As $F$ is continuous and strictly increasing, $S$ is continuous and strictly decreasing.
\begin{theorem}\label{hazard_rate}
For $X\in\mathcal {X}$, let $\eta$ be the hazard rate function of $X$. If $1/\eta$ is second-order differentiable and convex (concave), then $\pi_{X}$ is decreasing (increasing).
\end{theorem}
The proof the Theorem \ref{hazard_rate} is provided in Appendix \ref{app:proof_sec6}.
\begin{example}
For the normal distribution, we can give a short proof of the convexity of $1/\eta$. Let $S$ be the survival function of the standard normal distribution and $f$ its density.
Let $I\left(x\right)=1/\eta\left(x\right)= S (x )/f (x )=\exp\left(x^{2}/2\right)\int_{-\infty}^{-x}\exp\left(-s^{2}/2\right)\mathrm{d} s$.
One can easily see that
\begin{equation}
I'\left(x\right)=xI\left(x\right)-1\label{eq:first-derivative}
\end{equation}
which gives $I''\left(x\right)=xI'\left(x\right)+I\left(x\right)$.
This with \eqref{eq:first-derivative} implies that
\begin{equation}
I''\left(x\right)=\left(1+x^{2}\right)I\left(x\right)-x.\label{eq:second-derivative}
\end{equation}
First consider the negative line i.e., $x<0$. In this case \eqref{eq:first-derivative} and \eqref{eq:second-derivative} imply
$
I' (x ) =xI (x )-1<0,
$
and $
I'' (x ) = (1+x^{2} )I (x )+ (-x )>0.
$
The implication of the two relations is that $I$ is a convex and decreasing function on negative line. Now we consider the case $x>0$.
In this case, let $i\left(x\right)=I'\left(-x\right)$. From what we have proved it is clear that $i$ is an increasing function on $x>0$.
On the other hand, we have $I\left(x\right)+I\left(-x\right)=1/f \left(x\right)=\sqrt{2\pi}\exp\left(x^{2}/2\right)$.
This combined with \eqref{eq:first-derivative} gives us
$$
I'\left(x\right) =x\left(I\left(x\right)+I\left(-x\right)\right)+i\left(x\right)=x\sqrt{2\pi}\exp\left(x^{2}/2\right)+i\left(x\right),x>0.
$$
This means $I'$ is an increasing function on $x>0$ as it is a summation of two other increasing functions, so $I$ is convex on positive line as well.
\end{example}
Figure \ref{hazard} presents the curve $1/\eta$ for the generalized Pareto distribution, the Normal distribution, the t-distribution and the Lognormal distribution.
For distributions $\mathrm{GPD}(1/2)$, $\mathrm{N}(0,1)$ and $\mathrm{t}(2)$, we can see that the curves $1/\eta$ are convex, and this coincides with decreasing PELVE shown in Example \ref{ex:decreasing-PELVE}.
For the Lognormal distribution, the shape of $1/\eta$ depends on $\sigma$. As shown in Example \ref{ex:decreasing-PELVE}, the PELVE for $\mathrm{LN}(\sigma)$ is visibly decreasing for $\sigma^{2}=0.04$ and increasing for $\sigma=1$. Corresponding to the above observations, we see that $1/\eta$ is convex for $\sigma^{2}=0.04$ and concave for $\sigma^{2}=1$.
\begin{figure}[htpb]
\centering{}\includegraphics[width=2.5in]{hazard_convex} \includegraphics[width=2.5in]{hazard_concave}
\caption{$1/\eta$ for $\mathrm{GPD}(1/2)$, $\mathrm{N}(0,1)$, $\mathrm{t}(2)$, $\mathrm{LN}(0,2)$ and $\mathrm{LN}(1)$ in blue curves; in the right panel, the red curve is linear}
\label{hazard}
\end{figure}
\begin{corollary}\label{cor:concave_hazard}
If the hazard rate of a random variable $X$ is second-order differentiable and concave, then $\pi_{X}$ is decreasing.
\end{corollary}
\begin{proof}
Just note that if $\eta$ is concave, then $\eta \eta''$ is non-positive. It follows that $$
\left(\frac{1}{\eta}\right)''=\left(-\frac{\eta'}{\eta^{2}}\right)'=\frac{2\left(\eta'\right)^{2}-{{\eta\eta''}}}{\eta^{3}}\ge0.
$$
Thus, ${1}/{\eta}$ is convex, and the desired statement follows from Theorem \ref{hazard_rate}.
\end{proof}
The corollary above is a result of the fact that the concavity of $\eta$ implies convexity of $1/\eta$. Therefore, concave $\eta$ always leads to decreasing PELVE. For example, the Gamma distribution $\mathrm{G}(\alpha,\lambda)$ with density $f(x)=\frac{\lambda^\alpha t^{\alpha-1}e^{-\lambda t}}{\Gamma(\alpha)}$ has concave hazard rate function when $\alpha>1$.
Furthermore, by Theorem \ref{hazard_rate}, we can easily find more well-known distributions that have decreasing $\pi_{X}$.
As the tail distribution determine $\pi_{X}$ around $0$, we can focus on the tail distribution to discuss the convergence of $\pi_{X}$ at $0$.
Note that if the survival distribution function is regularly varying, then
its tail parameter one-to-one corresponds to the limit of $\Pi_X$ at $0$ as shown by Theorem 3 of \cite{LW2019}. Hence, the limit of $\Pi_X$, if it exists, can be useful as a measure of tail heaviness, and it is well defined even for distributions that do not have a heavy tail.
By the monotone convergence theorem, we have $\lim_{\epsilon\to0}\pi_{X}(\epsilon)$ exists if $\pi_X$ is monotone.
The limit may be finite or infinite.
\begin{corollary}\label{cor:limit}
For $X\in\mathcal {X}$, let $\eta$ be the hazard rate of $X$ and $F$ be the distribution function of $X$. If $1/\eta(x)$ is second-order differentiable and convex (concave) in $\left(\ensuremath{F^{-1}(\delta),\mathrm{ess\mbox{-}sup}(X)}\right)$ for some $\delta\in(0,1)$, then $\lim_{\epsilon\to0}\pi_{X}(\epsilon)$ exists.
In particular, this is true if $\eta$ is second-order differentiable and concave on $\left(\ensuremath{F^{-1}(\delta),\mathrm{ess\mbox{-}sup}(X)}\right)$.
\end{corollary}
\begin{proof}
Let $X'\sim F^{[\delta,1]}$. Then, the survival function for $X'$ is $S_{X'}(x)=S(x)/(1-p)$ for $x\ge F^{-1}(\delta)$.
The density function is $f_{X'}(x)=f(x)/(1-p)$ for $x\ge F^{-1}(\delta)$.
Therefore, the hazard rate function is $\eta_{X'}(x)=f(x)/S(x)=\eta(x)$ for $x\ge F^{-1}(\delta)$.
As $1/\eta(x)$ is convex (concave) when $x>F^{-1}(\delta)$, we have $1/\eta_{X'}(x)$ is convex (concave).
By Theorem \ref{hazard_rate}, we have $\pi_{X'}(\epsilon)$ is decreasing (increasing) on $(0,1)$.
As a result, we have $\pi_{X}(\epsilon)$ is decreasing (increasing) on $(0,\delta)$ and $\lim_{\epsilon\to0}\pi_{X}(\epsilon)$ exists.
By Corollary \ref{cor:concave_hazard}, if $\eta$ is concave on $(F^{-1}(\delta),\mathrm{ess\mbox{-}sup}(X))$, $1/\eta$ is convex on $(F^{-1}(\delta),\mathrm{ess\mbox{-}sup}(X))$ and $\lim_{\epsilon \to 0} \pi_X(\epsilon)$ also exists.
\end{proof}
\begin{example}
We give the numerical values of $\Pi_X(\epsilon)$ at very small probability level $\epsilon$ for normal, t, and log-normal distributions. These distributions do not have a constant PELVE curve, and using Corollary \ref{cor:limit} we can check that their PELVE have limits.
As we can see from Table \ref{table:limit}, PELVE can still distinguish the heaviness of tail even when $\epsilon$ is very small. The heavier tailed distributions report a higher PELVE value. For the normal distribution and the log-normal distribution with $\sigma=0.2$, the value of PELVE is closed to $e\approx2.7183$ as $\epsilon \downarrow 0$.
From the numerical values, it is unclear whether $\Pi_X(\epsilon)\to e$ for all log-normal distributions, but there is no practical relevance to compute $\Pi_X(\epsilon)$ for $\epsilon<10^{-11}$ in applications.
\begin{table}[htbp]
\def1.4{1.4}
\centering{}
\caption{ The value of $\Pi_{X}(\epsilon)$}\label{table:limit} %
\begin{tabular}{m{1.6cm}<{\centering} m{1.6cm}<{\centering} m{1.6cm}<{\centering} m{1.6cm}<{\centering} m{1.6cm}<{\centering} m{1.6cm}<{\centering} m{1.6cm}<{\centering}}
\hline \hline
Distribution& N & LN($1$) & LN($0.5$) &LN($0.2$) & t($2$) & t($3$)\\
\hline
$\epsilon={10}^{-10}$ &2.6884 &2.9167&2.7944 & 2.7290 &4.0000&3.3750 \\
\hline
$\epsilon={10}^{-11}$ &2.6909 &2.9077& 2.7920 &2.7287 &4.0000&3.3750 \\
\hline\hline
\end{tabular}
\end{table}
\end{example}
\section{Conclusion}\label{sec:conclusion}
In this paper, we offer several contributions to the calibration problem and properties of the PELVE.
The calibration problem concerns, with some given values from a PELVE curve, how one can build a distribution that has this PELVE.
We solve a few settings of calibration based on a one-point constraint, a two-point constraint, or the entire curve constraint.
In particular, the calibration for a given PELVE curve involves solving an integral equation $\int_{0}^{y}f\left(s\right)\mathrm{d} s=yf\left(z_{f}\left(y\right)y\right)$ for a function given $z_{f}$, and this requires some advanced analysis and a numerical method in different equations.
For the case that $z_{f}$ is a constant curve, we can identify all solutions, which are surprisingly complicated. In addition, we see that if $\pi_X$ is a constant larger than $e$, which is observed from typical values in financial return data (\cite{LW2019}), $X$ share the same tail behavior with the corresponding Pareto solution.
On the technical side, we study whether the PELVE is monotone and whether it converges at 0.
We show that the monotonicity of the PELVE is associated with the shape of the hazard rate.
If the inverse of hazard rate is convex (concave), the PELVE is decreasing (increasing).
The monotonicity at the tail part of the PELVE leads to conditions to check the convergence of the PELVE at $0$.
If the inverse of hazard rate is convex (concave) at tail of distribution, the limit of the PELVE at $0$ exists.
There are several open questions related to PELVE that we still do not fully understand. One particular such question is whether the tail behaviour, e.g., tail index, of a distribution is completely determined by its PELVE. We have seen that this holds true in the case of a constant PELVE (see Theorem \ref{thm:constant}), but we do not have a general conclusion. In case of regularly varying survival functions, \citet[Theorem 3]{LW2019} showed that the limit of PELVE determines its tail parameter, but it is unclear whether this can be generalized to other distributions.
Another challenging task is, for a specified curve $\pi$ on $[0,1]$, to determine whether there exists a model $X$ with $\pi_X=\pi$. The case of $n$-point constraints for large $n$ may require new design of verification algorithms. This question concerns compatibility of given information with statistical models, which has been studied, in other applications of risk management, by \cite{EMS02, EHW16} and \cite{KSSW18}.
\subsection*{Acknowledgements}
Ruodu Wang is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).
|
2,877,628,090,788 | arxiv | \section*{Introduction}
Let $K$ be a complete field with respect to a non-trivial non-archimedean absolute value $|\cdot|:=|\cdot|_K$.
Mumford built in 1972 some algebraic curves associated to certain subgroups of the linear
group $\PGL_2(K)$ analogous to a construction of Schottky over the complex numbers.
He restricted to the case of discrete absolute value and used the geometry given by formal schemes.
This was generalized to every non-archimedean absolute
value by Gerritzen and van der Put in \cite{GvdP80} in 1980. They
named such curves Mumford curves. Shortly after Mumford's paper,
Drinfeld and Manin in \cite{MD73} showed that the Jacobian of a Mumford
curve is isomorphic to an analytic torus and that it can be built with some theta
functions, in the case $K$ is a finite extension of the $p$-adic
numbers. This construction was done also in the general case by
Gerritzen and van der Put in \cite{GvdP80}. Both took advantage of rigid analytic geometry, introduced by Tate some years ago.
More recently, Dasgupta showed in his thesis (\cite{Das04}) an
equivalent construction of the Jacobian to the ones cited above, but restricted to
the local case, by means of multiplicative integrals, defined
previously by Darmon in \cite{Dar01} and generalized by Longhi in
\cite{Lon02}.
Before that, in 1990 Berkovich introduced an alternative analytic
theory to the one of Tate in his seminal book \cite{Ber90}. The
biggest difference over a variety consists in introducing more
points instead of removing Zariski open sets. This does not stop
from getting equivalent categories of ``good'' enough analytic
varieties which can be seen as generic fibres of formal schemes,
thanks to works of Raynaud, Bosch and L\"{u}tkebohmert.
Concurrently, tropical geometry was developed and found in big
relation with Berkovich analytic geometry.
In this paper, we give a new construction of the Jacobian of a
Mumford curve over any complete non-archimedean field, departing
from Berkovich geometry, which we believe we get clarify the
constructions previously done.
It should be also recognized a great parallelism of this work with
part of the paper by van der Put \cite{vdP92}. Some of the results
are directly related to results by Baker and Rabinoff appeared in
\cite{BR15} in slightly different language.
In order to get the asserted goal, we make the basic constructions
given by Berkovich theory in sections~\ref{trees}~and~\ref{ret},
from which later in section~\ref{schottky} we build our Mumford
curve. They are the Berkovich projective line together
$({{\mathbb{P}}^1_K}^*)^{an}$ with its skeleton ${\mathcal{T}}_K$, which coincides with
the Bruhat-Tits building of $\PGL_2(K)$, the locally finite subtree
${\mathcal{T}}_K({\mathcal{L}})$ associated to a compact set ${\mathcal{L}}$ and the retraction
map
$$
\red_{\mathcal{L}}:({{\mathbb{P}}^1_K}^*)^{an}\longrightarrow\overline{{\mathcal{T}}_K({\mathcal{L}})}.
$$
In sections~\ref{MI}~and~\ref{poisson} we develope the theory of multiplicative integrals and analytic functions that we need -completed later through sections~\ref{APS}~and~\ref{automorphic}. Essentially, we define these integrals, we build the ones in which we are interested and we relate them to analytic functions through the Poisson formula and the map
$$
\tilde{\mu}:{\mathcal{O}}(\Omega_{\mathcal{L}})^*:\longrightarrow{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0
$$
Later we study the automorphic forms for a Schottky group $\Gamma\subset\PGL_2(K)$. Meanwhile, we show some essential results on metric graphs, which can be thought as tropical curves, along sections~\ref{graphs}~and~\ref{tropical}. Mainly, we prove the isomorphism between harmonic measures and harmonic cochains and the isomorphism between the $\Gamma$-invariant harmonic measures and the abelianized of $\Gamma$. Finally, the last part of this work gather all previous topics to build the desired Abel-Jacobi map.
\begin{nota} The ring of integers will be denoted ${\mathcal{O}}_K=\{x\in K \ | \ |x|\le 1\}$ and its maximal ideal $\mathfrak{m}_K=\{x\in K \ | \ |x|< 1\}$. We will denote also $k:={\mathcal{O}}_K/\mathfrak{m}_K$ its residue field.
We denote by $\log$ the natural logarithm. If the absolute value $|\cdot|$ is discretely valued, we will assume that $-\log|x|\in {\mathbb{Z}}$ for any $x\in K^*$ and that is the discrete valuation $\mathit{v}_K$ associated to $|\cdot|$. Otherwise, we define the valuation of $x$ by $\mathit{v}_K(x):=-\log|x|$.
By a complete extension $L|K$ we will refer to a field $L$ containing $K$ complete with respect to an absolute value $|\ |_L$ which extends $|\cdot |$.
The dual projective line ${{\mathbb{P}}^1_K}^*$ over $K$ is equal to the projective spectrum of the polynomial ring $K[X_0,X_1]$, after defining ${{\mathbb{P}}^1_K}^*={\mathbb{P}}(V^*)=\Proj(S^\bullet V)$ and identifying $V:=KX_0\oplus KX_1\cong K^2$.
The $K$-rational points ${{\mathbb{P}}^1}^*(K)$ correspond to $(K^2)^*\setminus\{(0,0)\}$ modulo homothety. We denote the class of $(x_0,x_1)$ by $[x_0:x_1]$.
The infinite point in the dual projective line will be $\infty=[0:1]$ and we embed $K$ in ${{\mathbb{P}}^1_K}^*(K)$ by means of $i^*(z)=[1:-z]$.
Therefore, an $f\in K[X_0,X_1]$ defines a function $K\longrightarrow K$ that by abuse of notation we also denote $f$, by $f(z):=f(1,-z)$.
On the other hand we inject $K$ in ${{\mathbb{P}}^1_K}(K)$ by $i(z)=[z:1]$, taking as infinity of the projective line $[1:0]$.
We will consider the usual left action of the projective linear group $\PGL_2(K)$ on ${{\mathbb{P}}^1_K}^*(K)$ by the contragredient representation, that is $\gamma\cdot\omega:=\omega\gamma^{-1}$ for all $\gamma\in \PGL_2(K),\ \omega\in{{\mathbb{P}}^1_K}^*(K)$ (and also the usual left action on ${{\mathbb{P}}^1_K}(K)$).
Given a point $p=[a:b]\in{{\mathbb{P}}^1_K}^*(K)$, we will denote its corresponding point $p^*=[-b:a]\in{{\mathbb{P}}^1_K}(K)$ (or if $p\in{{\mathbb{P}}^1_K}(K)$, then $p^*\in{{\mathbb{P}}^1_K}^*(K)$). Note that this implies $i^*(z)=i(z)^*$ for all $z\in K$ and $(\gamma\cdot p)^*=\gamma\cdot p^*$ for all $p\in{{\mathbb{P}}^1_K}(K)$ (or $p\in{{\mathbb{P}}^1_K}^*(K)$), $\gamma\in \PGL_2(K)$.
For any $x\in K$, $r\in{\mathbb{R}}_{\geq0}$, we consider the ball in the completion ${\mathbb{C}}_K:=\widehat{\overline{K}}$ of the algebraic closure of $K$, $B(x,r):=\{ y\in {\mathbb{C}}_K \ | \ |y-x|\le r\}$.
\end{nota}
\section{Trees and Skeletons}\label{trees}
The main objective of this section is the construction of a metric
tree associated to an arbitrary compact set
${\mathcal{L}}\subset{{\mathbb{P}}^1}^*(K)$, study its structure and define the open
sets associated to its edges. This subtree generalizes to a
non-discrete setting the one defined by Mumford in \cite{Mum72} and
gives an alternative and more complete definition to the one given
in \cite[Ch.~1]{GvdP80}. In order to do it we recall some well known
notions coming from Berkovich analytic geometry and Bruhat-Tits
theory. This first part is mainly extracted from \cite{Bak08}, but
it is also greatly indebted to \cite{Wer04}, where some ideas we
recall here and along the second section are shown.
Consider the Berkovich analytic projective line $({{\mathbb{P}}^1_K}^*)^{an}$
defined over $K$, which is the set of all the
multiplicative seminorms of the polynomial ring $K[X_0,X_1]$
extending $|\ |$ on $K$ modulo an equivalence relation which is specified below; that is,
the maps
$$\alpha:K[X_0,X_1] \to {\mathbb{R}}_{\ge 0}$$
such that
\begin{enumerate}
\item $\alpha_{|K}=|\ |$.
\item $\alpha(X_0K+X_1K)\ne \{0\}$.
\item $\alpha(f\cdot g)=\alpha(f)\cdot \alpha(g)$
\item $\alpha(f+ g)\le \max\{\alpha(f), \alpha(g)\}$
\end{enumerate}
with $\alpha\sim \beta$ if there exists a constant $C\in{\mathbb{R}}_{>0}$
such that $\alpha(f)=C^d\beta(f)$ for all $f\in K[X_0,X_1]$
homogeneous of degree $d$ and for all $d\ge 0$.
We associate to an $x\in {{\mathbb{P}}^1_K}^*(K)$, $x\ne \infty=[0:1]$ and an
$r\in {\mathbb{R}}_{\ge 0}$ an element $\alpha(x,r) \in ({{\mathbb{P}}^1_K}^*)^{an}$ by
defining
$$
\alpha(x,r)(f)=\sup\{|f(y)|\ : \ y \in B(x,r) \}\text{ for }f\in K[X_0,X_1]
$$
and $\alpha(\infty,0)(f):=|f(0,1)|$.
We will call these seminorms the ones associated to the balls (or to
$K$-rational points if $r=0$).
\begin{obs}\label{frem}
Take $q\in K$ and assume that $f=qX_0+1X_1$ which by abuse of
notation we denote by $q$. Then $\alpha(x,r)(q)=\max\{|q-x|,r\}$. In
order to show this we compute
$$
\alpha(x,r)(q)=\sup\{|q\cdot1+1\cdot(-y)|:\ y\in B(x,r)\}=\sup\{|q-y|:\ |x-y|\leq r\}
$$
If $|q-x|<r$, then $B(x,r)=B(q,r)$ and there exists a sequence $(y_n)_n$ inside $B(x,r)$
such that $\displaystyle{\lim_n|q-y_n|=r}$, so $\alpha(x,r)(q)=r$.\\
If $|q-x|>r$, then $|q-y|=\max\{|q-x|,|x-y|\}=|q-x|$, since $|x-y|\leq r$.\\
If $|q-x|=r$, then $\alpha(x,r)(q)=\sup\{|q-y|:\ |x-y|\leq r\}=r$, since $|q-y|\leq r$.
\end{obs}
\begin{defn}
We call maximal skeleton of $({{\mathbb{P}}^1_K}^*)^{an}$ and denote ${\mathcal{T}}_K$
the set of points associated to balls with $r>0$, and the
compactified skeleton $ \overline{{\mathcal{T}}_K}$ is the skeleton together
with the (points associated to) rational points ${{\mathbb{P}}^1}^*(K)$. It
is well known that this set is a topological space, and together
with a natural metric, which we will recall in the following, forms
a metric tree (\cite{BPR13}).
\end{defn}
\begin{obs}\label{types}
If $K$ is algebraically closed, then it is well-known (look at
\cite{Ber90}) that the points in $({{\mathbb{P}}^1_K}^*)^{an}$ can be divided
in four types, the type I being associated to $K$-rational points,
types II and III associated to (closed) balls with center some $x\in
{{\mathbb{P}}^1}^*(K)$, and with radius $r \in |K^*|$ or $r \in
{\mathbb{R}}_{>0}\setminus |K^*|$ respectively, and a fourth type associated
to sequences of nesting balls with empty intersection. Then the
topological space $({{\mathbb{P}}^1_K}^*)^{an}$ has the structure of a metric
tree. The maximal skeleton ${\mathcal{T}}_K$ of $({{\mathbb{P}}^1_K}^*)^{an}$ is the
set of points of type II and III, which is a metric subtree, and $
\overline{{\mathcal{T}}_K}$ is the set of points of type I, II and III.
Recall that in \cite{BPR13} is defined a skeleton in
$({{\mathbb{P}}^1_K}^*)^{an}$ and corollary 5.56. asserts that ${\mathcal{T}}_K$ is the
inductive limit of all their skeleta. Note also that
$({{\mathbb{P}}^1_K}^*)^{an}$ is homeomorphic to the inverse limit of the set
of all skeleta with respect to the natural retraction maps
(\cite[Thm.~5.57.]{BPR13}).
\end{obs}
Given any two distinct points $x_0$ and $x_1
\in{{\mathbb{P}}^1}^*(K)\setminus\{\infty\}$, if $R=|x_0-x_1|$, we will
denote by ${x_0\vee x_1:=\alpha(x_0,R)=\alpha(x_1,R)}$. For any two
points $\alpha_0=\alpha(x_0,r_0)$ and $\alpha_1=\alpha(x_1,r_1)\in
{\mathcal{T}}_K$, either the corresponding balls $B(x_0,r_0)\cap B(x_1,r_1)\ne
\emptyset$, in which case $\alpha(x_i,r_i)=\alpha(y,r_i)$ for all
$y\in B(x_0,r_0)\cap B(x_1,r_1)$ and $i=0,\ 1$, and we denote
$\alpha_0\vee\alpha_1:=\alpha(y,\max(r_0,r_1))$, or $B(x_0,r_0)\cap
B(x_1,r_1)= \emptyset$ and we denote $\alpha_0\vee\alpha_1:=x_0\vee
x_1$.
Let us consider two points $\alpha=\alpha(x,r),\
\alpha'=\alpha(x,r')$ of the tree $\overline{{\mathcal{T}}_K}$, with $0\leq
r\leq r'$ and $x\neq\infty$. We denote the (oriented) path from
$\alpha$ to $\alpha'$ as $P(\alpha,\alpha')$, being as a set of
points ${\{\alpha(x,s)| r\leq s\leq
r'\}\cong[r,r']\subset{\mathbb{R}}_{\geq0}}$. The (oriented) path
$P(\alpha',\alpha)$ from $\alpha'$ to $\alpha$ is the same set of
points oriented with the opposite direction. Finally, the (oriented)
path $P(\alpha(x,r),\alpha(\infty,0))$ from $\alpha(x,r)$ to
$\alpha(\infty,0)$ is the set of points ${\{\alpha(x,s)|s\geq
r\}\bigcup\{\alpha(\infty,0)\}\cong[r,\infty]\subset{\mathbb{R}}_{\geq0}\bigcup\{\infty\}}$
with the orientation given by the isomorphism (as above), and we
define similarly the opposite path $P(\alpha(\infty,0),\alpha(x,r))$
reversing the orientation. Given two arbitrary points $\alpha,\
\alpha'\in\overline{{\mathcal{T}}_K}\setminus\{\alpha(\infty,0)\}$, the
(oriented) path $P(\alpha,\alpha')$ from $\alpha$ to $\alpha'$ is
the path $P(\alpha,\alpha\vee\alpha')$ followed by the path
$P(\alpha\vee\alpha',\alpha')$.
Recall that given any two distinct points $x_0$ and $x_1 \in
{{\mathbb{P}}^1}^*(K)$, there is a unique line in ${\mathcal{T}}_K$ going from $x_0$ to
$x_1$, being the open path $\mathring{P}(\alpha(x_0,0),\alpha(x_1,0))$
-the interior of the path $P(\alpha(x_0,0),\alpha(x_1,0))$.
This line is homeomorphic as a metric tree to ${\mathbb{R}}$, and we denote
it by ${\mathbb{A}}_{\{x_0,x_1\}}$: it is called an apartment of the skeleton
${\mathcal{T}}_K$. Its closure is, by definition,
$\ds{\overline{{\mathbb{A}}_{\{x_0,x_1\}}}={\mathbb{A}}_{\{x_0,x_1\}}\cup\{x_0,x_1\}}$.
Given two points $\alpha_0=\alpha(x_0,r_0)$ and
$\alpha_1=\alpha(x_0,r_1)\in{\mathbb{A}}_{\{x_0,\infty\}}$, we define
$\displaystyle{d(\alpha_0,\alpha_1)=\left|\log \frac{r_1}{r_0}\right|}$; and in general we define
$$d(\alpha_0,\alpha_1):=d(\alpha_0,\alpha_0\vee
\alpha_1)+d(\alpha_0\vee\alpha_1,\alpha_1).$$ Then $d$ determines a
well defined metric on ${\mathcal{T}}_K$.
A seminorm on $V$ is $\alpha:V=X_0K+X_1K\longrightarrow{\mathbb{R}}_{\ge 0}$ satisfying
(2) and (4) as above and $\alpha(\lambda v)=|\lambda|\alpha(v)$ for
$\lambda\in K,\ v\in V$. We say that a seminorm $\alpha$ on $V$ is
diagonalizable if there exists a basis $v_0,v_1$ of $V$ such that
$\alpha(v)=\max\{|\omega_0(v)|\alpha(v_0),|\omega_1(v)|\alpha(v_1)\}$
for all $v\in V$, where $\omega_0,\omega_1$ is the dual basis of
$v_0,v_1$. We denote that seminorm as
$\alpha_{(v_0,v_1),(\rho_0,\rho_1)}$ with $\rho_0:=\alpha(v_0)$ and
$\rho_1:=\alpha(v_1)$.
\begin{obs}[The action of $\PGL_2(K)$ on $ ({{\mathbb{P}}^1_K}^*)^{an}$]\label{ActS}
The left action of $\PGL_2(K)$ on $V$ induces a left action on
${K[X_0,X_1]\cong S^\bullet V}$. Then, it also induces a left
action on $({{\mathbb{P}}^1_K}^*)^{an}$ by defining
${(\gamma\cdot\alpha)(f):=\alpha(\gamma^{-1}\cdot f)}$.
For any $\gamma\in \PGL_2(K)$ we get
$\gamma\cdot\alpha(x,0)=\alpha(\gamma\cdot x,0)$, making the
injection ${{\mathbb{P}}^1}^*(K)\longrightarrow ({{\mathbb{P}}^1_K}^*)^{an}$ defined by
$x\mapsto\alpha(x,0)$ equivariant. We also have
$$\gamma\cdot\alpha_{(v_0,v_1),(\rho_0,\rho_1)}=\alpha_{(\gamma\cdot v_0,\gamma\cdot v_1),(\rho_0,\rho_1)}.$$
\end{obs}
Next we are going to identify ${\mathcal{T}}_K$ with the Bruhat-Tits tree of $\PGL_2(K)$.
\begin{prop}
The seminorm $\alpha(x,r)$ restricted to $V$ is the seminorm $\alpha:=\alpha_{(v_0,v_1),(\rho_0,\rho_1)}$, diagonalizable with respect to the basis $v_0=(1,0),\ v_1=(x,1)$ and such that $\rho_0=1$ and $\rho_1=r$ when $x\neq\infty$, and $\alpha(\infty,0)=\alpha_{((1,0),(0,1)),(0,1)}$.
\end{prop}
\begin{proof}
The identification works restricting any seminorm in
$\overline{{\mathcal{T}}_K}$ to $KX_0+KX_1$, by means of its
identification with $K^2$.
When the seminorm is $\alpha(x,r)$ for
$x\in K\subset{{\mathbb{P}}^1}^*(K)$ and $r\ge 0$, and we apply it to a vector $v=(a,b)=(a-bx)v_0+bv_1$, we have
$$
\alpha(x,r)(v)=\sup\{|a+b(-y)|:y\in B(x,r)\}=\sup\{|a-bx+b(x-y)|: y\in B(x,r)\}
$$
$$
=\sup_{y\in B(x,r)}\{|a-bx|,|b(x-y)|\}=\max\{|a-bx|,|b|r\}=\alpha(v)
$$
Observe that $\omega_0(a,b)=a-bx$ and also that the seminorm on $K^2$ associated to a rational point $x$ has $x^*$ as its kernel, that is to say, the set of vectors $w\in K^2$ with $|\omega_0(w)|=0$ is the subspace generated by $(x,1)$.
In the case of $\alpha(\infty,0)$ we have
$$
\alpha(\infty,0)(v)=|b|=\max\{|a|0,|b|1\}=\alpha_{((1,0),(0,1)),(0,1)}(v)
$$
\end{proof}
In the following result we will specify how the correspondence between classes of seminorms with form $\alpha(x,r)$ and diagonalizable seminorms on $V$ works.
\begin{lem}
Let $v_0,v_1$ be a basis of $V$, $\omega_0,\omega_1\in V^*$ be its dual basis, $y_0=[\omega_0],y_1=[\omega_1]\in{{\mathbb{P}}^1}^*(K)$ and $\rho_0,\rho_1\in{\mathbb{R}}_{\geq0}$. We suppose that $y_0,y_1\neq\infty$ (look at proposition above for the case in which one point is $\infty$), and then we may take $\omega_i=(1,-y_i)$ for $i=1,2$ (by means of $i^*$).\\
With these hypotheses we get:
$$
\mbox{If }\rho_1<\rho_0,\qquad[\alpha_{(v_0,v_1),(\rho_0,\rho_1)}]=[\alpha_{(v_0,v_1),(1,\frac{\rho_1}{\rho_0})}]
$$
and
$$
\alpha_{(v_0,v_1),(1,\frac{\rho_1}{\rho_0})}=\alpha\left(y_0,\frac{\rho_1}{\rho_0}|y_0-y_1|\right)
$$
$$
\mbox{If }\rho_0<\rho_1,\qquad[\alpha_{(v_0,v_1),(\rho_0,\rho_1)}]=[\alpha_{(v_0,v_1),(\frac{\rho_0}{\rho_1},1)}]
$$
and
$$
\alpha_{(v_0,v_1),(\frac{\rho_0}{\rho_1},1)}=\alpha\left(y_1,\frac{\rho_0}{\rho_1}|y_0-y_1|\right)
$$
$$
\mbox{If }\rho_1=\rho_0,\qquad[\alpha_{(v_0,v_1),(\rho_0,\rho_1)}]=[\alpha_{(v_0,v_1),(1,1)}]
$$
and
$$
\alpha_{(v_0,v_1),(1,1)}=\alpha\left(y_0,|y_0-y_1|\right)=\alpha\left(y_1,|y_0-y_1|\right)
$$
Reciprocally, and for $r\leq|y_0-y_1|$
$$
\alpha(y_0,r)=\alpha_{(v_0,v_1),(1,\frac{r}{|y_0-y_1|})}
$$
$$
\alpha(y_1,r)=\alpha_{(v_0,v_1),(\frac{r}{|y_0-y_1|},1)}.
$$
\end{lem}
\begin{proof}
Assume, just for simplicity, that $\rho_0,\rho_1\neq0$, meaning that $\alpha$ is a norm. Define $\alpha:=\alpha_{(v_0,v_1),(\rho_0,\rho_1)}$.
Next, we start at the end. By definition $\alpha\in{\mathbb{A}}_{\{y_0,y_1\}}$, so $\alpha\in P(y_0,y_0\vee y_1)$ or $\alpha\in P(y_0\vee y_1,y_1)$; for some $r\leq|y_0-y_1|$, in the first case we would get $\alpha=\alpha(y_0,r)$ and in the second we would $\alpha=\alpha(y_1,r)$ up to homothety. Without loss of generality we suppose the first case. Let us take an arbitrary vector $v=(a,b)\in V$. We have
$$
\alpha(y_0,r)(v)=\max\{|a-by_0|,|b|r\}
$$
$$
\alpha(a,b)=\max\{|a-by_0|\rho_0,|a-by_1|\rho_1\}\sim\max\{|a-by_0|,|a-by_1|\frac{\rho_1}{\rho_0}\}
$$
We note that if $|a-by_0|<|b|r$, we have $[\alpha](v)=[\alpha(y_0,r)](v)$ if and only if $\displaystyle{|b|r=|a-by_1|\frac{\rho_1}{\rho_0}}$, or also $\displaystyle{\frac{\rho_1}{\rho_0}=\frac{|b|r}{|a-by_1|}}$.
But since we have $|b||y_0-y_1|\geq|b|r>|a-by_0|$, then we get $|a-by_1|=|a-by_0+b(y_0-y_1)|=\max\{|a-by_0|,|b||y_0-y_1|\}=|b||y_0-y_1|$, so
$$
\frac{\rho_1}{\rho_0}=\frac{r}{|y_0-y_1|}
$$
Therefore we obtain
$$
\displaystyle{[\alpha]=\left[\alpha\left(y_0,\frac{\rho_1}{\rho_0}|y_0-y_1|\right)\right]}
$$
after assuming $r\leq|y_0-y_1|$, that is $\rho_1\leq\rho_0$. In the same way, when $\rho_1\geq\rho_0$ we get
$$
\displaystyle{[\alpha]=\left[\alpha\left(y_1,\frac{\rho_0}{\rho_1}|y_0-y_1|\right)\right]}.
$$
Note that we see the extreme cases too, that is, when $\rho_1=0$ then $[\alpha]=[\alpha(y_0,0)]$, and when $\rho_0=0$, $[\alpha]=[\alpha(y_1,0)]$.
\end{proof}
We keep together the last two results in the next:
\begin{cor}\label{IdentificationBT}
The maximal and the compactified skeletons ${\mathcal{T}}_K$ and $\overline{{\mathcal{T}}_K}$ can be canonically identified with the
set of classes
modulo homothety of nontrivial diagonalizable norms and seminorms on
$K^2$ respectively. These are the Bruhat-Tits tree of $\PGL_2(K)$ and its compactification.
\end{cor}
\begin{proof}
The classes of seminorms associated to balls correspond to the classes of diagonalizable norms and seminorms on $K^2$ by the two previous results.
\end{proof}
And now we are going to show that $d$ is invariant with respect to the action of $\PGL_2(K)$.
Consider any apartment ${\mathbb{A}}_{\{x_0,x_1\}}$ for $x_0,x_1\in{{\mathbb{P}}^1}^*(K)$ and choose representatives $\omega_0,\omega_1\in\ V^*$ respectively. Let $v_0,v_1\in V$ be the dual basis of $\omega_0,\omega_1$. For any two elements in this apartment $\alpha:=\alpha_{(v_0,v_1),(\rho_0,\rho_1)},\alpha':=\alpha_{(v_0,v_1),(\rho'_0,\rho'_1)}$ we define a distance in this apartment as:
$$
d_{x_0,x_1}(\alpha,\alpha'):=\left|\log\left(\frac{\rho_1\rho'_0}{\rho_0\rho'_1}\right)\right|=\left|\log\left(\frac{\rho_1}{\rho_0}\right)-\log\left(\frac{\rho'_1}{\rho'_0}\right)\right|
$$
Note that the homeomorphism (up to orientation) ${\mathbb{A}}_{\{x_0,x_1\}}\longrightarrow{\mathbb{R}}$ is given by $\displaystyle{\alpha\mapsto\log\left(\frac{\rho_1}{\rho_0}\right)}$, so $d_{x_0,x_1}$ is the transported distance from the natural one in ${\mathbb{R}}$.
\begin{lem}
The two definitions of distance coincide, that is, for any $x_0,x_1\in{{\mathbb{P}}^1}^*(K)$ we have
$$
d_{|{\mathbb{A}}_{\{x_0,x_1\}}}=d_{x_0,x_1}
$$
\end{lem}
\begin{proof}
For any
$\alpha:=\alpha_{(v_0,v_1),(\rho_0,\rho_1)},\alpha':=\alpha_{(v_0,v_1),(\rho'_0,\rho'_1)}\in{\mathbb{A}}_{\{x_0,x_1\}}$
we want to see $d(\alpha,\alpha')=d_{x_0,x_1}(\alpha,\alpha')$.
First, we may assume that there exists an $x\in{{\mathbb{P}}^1}^*(K)$ such
that $\alpha,\alpha'\in{\mathbb{A}}_{\{x,\infty\}}$. Otherwise
$d(\alpha,\alpha')=d(\alpha,\alpha\vee\alpha')+d(\alpha\vee\alpha',\alpha')$
and by definition $d_{x_0,x_1}$ satisfies the same equality.
Moreover, it is enough to prove that if
$\alpha,\alpha'\in{\mathbb{A}}_{\{x_0,x_1\}}\bigcap{\mathbb{A}}_{\{y_0,y_1\}}$ then
$d_{x_0,x_1}(\alpha,\alpha')=d_{y_0,y_1}(\alpha,\alpha')$, since for
the particular case $y_0=x,\ y_1=\infty$ we have $d_{x,\infty}=d$.
We may reduce to the case $y_0=x_0$ by applying the result in two
steps. Let us denote $x_2:=y_1\in{{\mathbb{P}}^1}^*(K)$ and let it be
represented by $\omega_2=\lambda\omega_0+\mu\omega_1\in V^*,\
\mu\neq0$. Then
$\displaystyle{u_0=v_0-\frac{\lambda}{\mu}v_1,u_2=\frac{v_1}{\mu}\in
V}$ is the dual basis of $\omega_0,\omega_2$. Now we have that
$$
\alpha:=\alpha_{(v_0,v_1),(\rho_0,\rho_1)}=\alpha_{(u_0,u_2),(\eta_0,\eta_2)}\mbox{ with }\eta_0=\max\left\{\rho_0,\left|\frac{\lambda}{\mu}\right|\rho_1\right\},\ \eta_2=\left|\frac{1}{\mu}\right|\rho_1
$$
and
$$
\alpha':=\alpha_{(v_0,v_1),(\rho'_0,\rho'_1)}=\alpha_{(u_0,u_2),(\eta'_0,\eta'_2)}\mbox{ with }\eta'_0=\max\left\{\rho'_0,\left|\frac{\lambda}{\mu}\right|\rho'_1\right\},\ \eta'_2=\left|\frac{1}{\mu}\right|\rho'_1
$$
Note that $\rho_1=|\mu|\eta_2$ implies that $\eta_0=\max\{\rho_0,|\lambda|\eta_2\}$. Furthermore $\rho_0=\alpha(v_0)=\max\{\eta_0,|\lambda|\eta_2\}$, since $v_0=u_0+\lambda u_2$. Then $\eta_0=\rho_0$. Identically we get $\eta'_0=\rho'_0$.
Therefore
$$
d_{x_0,x_2}(\alpha,\alpha')=\left|\log\frac{\eta_2\eta'_0}{\eta_0\eta'_2}\right|=\left|\log\frac{\rho_1\rho'_0}{\rho_0\rho'_1}\right|=d_{x_0,x_1}(\alpha,\alpha')
$$
\end{proof}
\begin{cor}
The distance is $\PGL_2(K)$-invariant, that is to say, $d(\alpha,\alpha')=d(\gamma\cdot\alpha,\gamma\cdot\alpha')$ for any $\gamma\in \PGL_2(K)$.
\end{cor}
\begin{proof}
First we recall that $\gamma\cdot\alpha_{(v_0,v_1),(\rho_0,\rho_1)}=\alpha_{(\gamma\cdot v_0,\gamma\cdot v_1),(\rho_0,\rho_1)}$. Let us to take now any apartment ${\mathbb{A}}_{\{x_0,x_1\}}$ which contains $\alpha,\alpha'$ as above. Then
$d(\alpha,\alpha')=d_{x_0,x_1}(\alpha,\alpha')=d_{\gamma\cdot x_0,\gamma\cdot x_1}(\gamma\cdot\alpha,\gamma\cdot\alpha')=d(\gamma\cdot\alpha,\gamma\cdot\alpha')$, where the second equality is due to the remark~\ref{ActS}.
\end{proof}
Let $x_0$, $x_1$ and $x_2$ be three distinct points in ${{\mathbb{P}}^1}^*(K)$. Then
there exists a unique point $t(x_0,x_1,x_2)\in {\mathcal{T}}_K$ which is
contained in the three lines they form. If $x_2=\infty$, then
$t(x_0,x_1,\infty)=\alpha(x_0,R)=x_0\vee x_1$, where $|x_1-x_0|=R$.
If none of them is equal to $\infty$, it corresponds to the smallest
ball containing all three points.
Observe that the points $t(x_0,x_1,x_2)$ are always of type II, so
they have the form $\alpha(x_0,r)$ with $r\in |K^*|$.
\begin{defn}\label{tree} Let ${\mathcal{L}}$ be a subset of ${{\mathbb{P}}^1}^*(K)$ which contain at least two points. Denote by
$${\mathcal{T}}_K({\mathcal{L}}):=\bigcup_{\{x_0,x_1\}\subset {\mathcal{L}}} {\mathbb{A}}_{\{x_0,x_1\}}$$
the metric tree associated to ${\mathcal{L}}$ (which is the subspace of
${\mathcal{T}}_K$ generated by the lines between two points in ${\mathcal{L}}$). Note
that $\overline{{\mathcal{T}}_K({\mathcal{L}})}:={\mathcal{T}}_K({\mathcal{L}})\cup{\mathcal{L}}$ with the natural
topology.
\end{defn}
It is clear that for any extension of fields $L|K$ the tree
associated to ${\mathcal{L}}$ is always the same: ${\mathcal{T}}_{L}({\mathcal{L}})={\mathcal{T}}_K({\mathcal{L}}),\
\overline{{\mathcal{T}}_{L}({\mathcal{L}})}=\overline{{\mathcal{T}}_K({\mathcal{L}}})$.
We will show in the sequel that ${\mathcal{T}}_K({\mathcal{L}})$ is a locally finite
metric tree if ${\mathcal{L}}$ is compact.
\begin{lem} The points of the form $t(x_0,x_1,x_2)$ for three
distinct points $x_0, x_1, x_2\in
{\mathcal{L}}$ are the points in ${\mathcal{T}}_K({\mathcal{L}})$ with valence greater than $2$.
Suppose that $\infty,x_0\in {\mathcal{L}}$ and consider a point
$\alpha:=\alpha(x_0,r)\in {\mathcal{T}}_K({\mathcal{L}})$ of the form
$t(x_0,x_1,\infty)$ for some $x_1\in {\mathcal{L}}$. Then
$$\{y \in {\mathcal{L}}\setminus\{x_0,\infty\} \ | \
\alpha=t(x_0,y,\infty)\}=\{y \in {\mathcal{L}} \ | \ |y-x_0|=r\}.$$ Moreover,
there is a bijection between the set of directions from $\alpha(x_0,r)$ except the ones which connect with $\infty$ and $x_0$, and the image
of the map
$$\psi:\{y \in {\mathcal{L}} \ | \ |y-x_0|=r\} \to k^*$$ given by sending
$$\psi(y)=\frac{y-x_0}{x_1-x_0} \pmod{\mathfrak{m}_K}.$$
\end{lem}
\begin{proof}
The unique claim that needs a proof is the bijection. From the equality shown, we see that a direction can be identified with a set of points $E_y\subset\{y \in {\mathcal{L}} \ | \ |y-x_0|=r\}$ such that $|y'-y''|<r$ for all $y',y''\in E_y$.
Thus, the only thing we have to prove is that $\psi(y)=\psi(y')$ if and only if $|y-y'|< r$.
To start with this equivalence we note that $\psi(y)=\psi(y')$ means that there exists $z\in\mathfrak{m}_K$, or equivalently $|z|<1$, such that
$$
\frac{y-x_0}{x_1-x_0}=\frac{y'-x_0}{x_1-x_0}+z
$$
We may write this equality as $y-y'=z(x_1-x_0)$ and taking absolute value $|y-y'|=|z|r<r$. Finally, the other option, $|z|=1$, is that for which $\psi(y)\neq\psi(y')$.
\end{proof}
\begin{cor}\label{locft} If ${\mathcal{L}}$ is compact, then ${\mathcal{T}}_K({\mathcal{L}})$ is a locally finite
metric tree, that is to say, any vertex has a finite number of directions arriving
to it and any finite lenght path contains only a finite number of vertices of valence greater than $2$.
\end{cor}
\begin{proof} We suppose ${\mathcal{L}}$ has at least three points and $\infty\in {\mathcal{L}}$ without loss of generality.\\
In order to prove the first claim consider a vertex $\alpha(x_0,r)\in {\mathcal{T}}_K({\mathcal{L}})$ that we may assume of the form
$t(x_0,x_1,\infty)$ for some $x_0$ and $x_1\in {\mathcal{L}}$. Since ${\mathcal{L}}$ is compact and
$\{y\in K \ | \ |y-x_0|=r\}$ is closed, their intersection $\{y
\in {\mathcal{L}} \ | \ |y-x_0|=r\}$ is compact. Now, given any
$t\in k^*$, the set $\psi^{-1}(\{t\})$ is an open subset (the previous proof shows it is an open ball).
Then, if the point had infinite directions arriving to it, the image
of $\psi$ would be infinite so the compact set $\{y \in {\mathcal{L}} \ | \
|y-x_0|=r\}$ would be covered by an infinite number of disjoint open
subsets and we would get a contradiction.\\
To get the second claim we can reduce us to show it for a path $P(\alpha(x,r),\alpha(x,r'))$ with $0<r\leq r'$. We have to show that the set
$$
S_{r,r'}:=\{s\in[r,r']|\ \exists y\in{\mathcal{L}}:|y-x|=s\}
$$
is finite. Consider the set
$$
\{y\in{\mathcal{L}}|\ r\leq|y-x|\leq r'\}={\mathcal{L}}\bigcap\left(B(x,r')\setminus\mathring{B}(x,r)\right)=\bigcup_{s\in S_{r,r'}}\{y\in{\mathcal{L}}|\ |y-x|=s\}
$$
Since it is a closed in ${\mathcal{L}}$, then it is compact. Further, the subsets
$$
{\mathcal{L}}_{x,s}:=\{y\in{\mathcal{L}}|\ |y-x|=s\}=\bigcup_{y\in{\mathcal{L}}_{x,s}}\left({\mathcal{L}}\cap\mathring{B}(y,s)\right)
$$
are open, so we can get a finite covering by them, and this implies necessarily that $S_{r,r'}$ is finite.
\end{proof}
\begin{defn}
With the hypotheses of definition~\ref{tree} we say that
${\mathcal{T}}_K({\mathcal{L}})$ is perfect if for any $\alpha\in{\mathcal{T}}_K({\mathcal{L}})$ and for any
$r\in\mathbb{R}_{>0}$ there exists $\alpha'\in{\mathcal{T}}_K({\mathcal{L}})$ with
valence greater than 2 and such that $d(\alpha,\alpha')>r$.
\end{defn}
One can show that this definition is compatible with the one of
perfect set, so ${\mathcal{T}}_K({\mathcal{L}})$ is perfect if and only if ${\mathcal{L}}$ is
perfect (all the points in ${\mathcal{L}}$ are accumulation points). For
example, if ${\mathcal{L}}$ is a finite set, then ${\mathcal{T}}_K({\mathcal{L}})$ is not perfect,
since it has just a finite number of vertices of valence greater
than $2$.
\begin{defn}
We will call a topological (oriented) edge $\varepsilon:=\varepsilon_{\alpha,\beta}$
(of ${\mathcal{T}}_K({\mathcal{L}})$) a non trivial path
$P(\alpha,\beta)\subset{\mathcal{T}}_K({\mathcal{L}})$, such that all its interior
points have valence two in ${\mathcal{T}}_K({\mathcal{L}})$. We will call the length of
$\varepsilon$ the distance $d(\alpha,\beta)$, and we will denote it by
$l(\varepsilon)$.
\end{defn}
Given any
compact subset ${\mathcal{L}}\subset {{\mathbb{P}}^1}^*(K)$, we consider ${\mathcal{L}}^*:=\{x^*\ |
\ x\in {\mathcal{L}}\}\subset{\mathbb{P}}^1(K)$, which is also a compact subset.
\begin{defn}\label{open}
Let $\varepsilon$ be the topological edge of ${\mathcal{T}}_K({\mathcal{L}})$ induced by the
path $P(\alpha,\beta)$. We may define a compact open set of ${\mathcal{L}}^*$
associated to it as
$$
{\mathcal{B}}(\varepsilon):={\mathcal{B}}(\alpha,\beta):=\{x\in{\mathcal{L}}^*|\ \alpha\not\in
P(x^*,\beta)\}.
$$
\end{defn}
Note that if $\alpha=\alpha(x,r)$, $\alpha'=\alpha(x,s)$ and $x\in
K$, either $r<s$ and so ${\mathcal{B}}(\alpha,\alpha')={\mathcal{L}}^*\setminus
\mathring{B}(x,s)$, or $r>s$ and then
${\mathcal{B}}(\alpha,\alpha')={\mathcal{L}}^*\bigcap B(x,s)$. The following lemma is
elementary.
\begin{lem}\label{oprops} These sets satisfy the next properties:
\begin{itemize}
\item All together are an open basis of the topology of ${\mathcal{L}}^*$ in the strong sense, meaning that any compact open set of ${\mathcal{L}}^*$ is a finite disjoint union
of them.
\item If $\overline{\varepsilon}$ is the opposite of a topological oriented edge $\varepsilon$, then ${\mathcal{B}}(\overline{\varepsilon})={\mathcal{L}}^*\setminus {\mathcal{B}}(\varepsilon)$.
\item If $v$ is a vertex en ${\mathcal{T}}_K({\mathcal{L}})$, the open sets ${\mathcal{B}}(\varepsilon)$ for $\varepsilon$ topological edges with source $v$ covering the different directions from the vertex in the tree are pairwise disjoint and $\bigsqcup {\mathcal{B}}(\varepsilon)={\mathcal{L}}^*$.
\item Let $\varepsilon=P(\alpha,\beta)$, $\varepsilon'=P(\alpha',\beta')$ be two topological edges of ${\mathcal{T}}_K({\mathcal{L}})$. Then
$$
{\mathcal{B}}(\varepsilon)\bigcap{\mathcal{B}}(\varepsilon')\left\{
\begin{array}{l}
={\mathcal{B}}(\varepsilon)={\mathcal{B}}(\varepsilon'),\mbox{ if there is another topological edge containing both}\\
\qquad\qquad\qquad\quad\mbox{ with the same orientation}\\
=\emptyset,\;\mbox{ if }\alpha,\alpha'\in P(\beta,\beta')\\
={\mathcal{B}}(\varepsilon)\subset{\mathcal{B}}(\varepsilon'),\mbox{ if }\alpha\in P(\beta,\alpha')\mbox{ and }\alpha'\not\in P(\beta,\beta')\\
={\mathcal{B}}(\varepsilon')\subset{\mathcal{B}}(\varepsilon),\mbox{ if }\alpha'\in P(\beta',\alpha)\mbox{ and }\alpha\not\in P(\beta',\beta)\\
\left\{\begin{array}{l}
\neq\emptyset\mbox{ and}\\
\subsetneq{\mathcal{B}}(\varepsilon),{\mathcal{B}}(\varepsilon')
\end{array},\right.
\begin{array}{l}\mbox{ if }P(\alpha,\alpha')\mbox{ is not a topological edge}\\
\mbox{ and }\alpha,\alpha'\not\in P(\beta,\beta')\end{array}
\end{array}\right.
$$
\end{itemize}
\end{lem}
\section{The retraction map}\label{ret}
We build the retraction map $\red_{{\mathcal{L}}}:({{\mathbb{P}}^1_{K}}^*)^{an} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}$ generalizing the reduction map constructed by Werner in \cite{Wer04} to the subtrees introduced in the previous section, which, on the other hand, gives the complete description over all the Berkovich analytic points of the reduction map named in \cite[2.3.]{Das05}. Further, we do not restrict to a local field.
Through this section $L|K$ will be an arbitrary extension of valued
complete fields.
Given any compact subset with at least two points ${\mathcal{L}}\subset {{\mathbb{P}}^1}^*(K)$ we define $\Omega_{\mathcal{L}}(L):={{\mathbb{P}}^1}^*(L)\setminus {\mathcal{L}}$. We also define the diameter of ${\mathcal{L}}$ as
$$
d_{\mathcal{L}}=\left\{\begin{array}{l}
\inf\{r\geq0|\ {\mathcal{L}}\subset B(x,r)\mbox{ for some }x\in{\mathcal{L}}\}\mbox{ if }\infty\not\in{\mathcal{L}}\\
+\infty\mbox{ if }\infty\in{\mathcal{L}}
\end{array}\right.
$$
Note that we may fix $x\in{\mathcal{L}}$ and the definition is independent of the chosen point $x$.
\begin{defn} Let ${\mathcal{L}}\subset{{\mathbb{P}}^1}^*(K)$ be as just above. We define the retraction map
$\red_{{\mathcal{L}}}:{{\mathbb{P}}^1}^*(L) \to \overline{{\mathcal{T}}_K({\mathcal{L}})}$ to be
$$
\red_{{\mathcal{L}}}(x)=\left\{\begin{array}{l}
x\mbox{, if }x\in{\mathcal{L}}\\
\alpha(x,\inf\{s\ge 0 \ | \ B(x,s)\cap {\mathcal{L}}\ne \emptyset\})\mbox{, if }B(x,d_{\mathcal{L}})\cap{\mathcal{L}}\neq\emptyset\mbox{ and }x\not\in{\mathcal{L}}\\
\alpha(y,d_{\mathcal{L}})\mbox{ for any }y\in{\mathcal{L}}\mbox{, if }B(x,d_{\mathcal{L}})\cap{\mathcal{L}}=\emptyset\\
\end{array}\right.
$$
for $x\neq\infty$, and
$$
\red_{{\mathcal{L}}}(\infty)=\left\{\begin{array}{l}
\alpha(y,d_{\mathcal{L}})\mbox{ for any }y\in{\mathcal{L}}\mbox{, if }\infty\not\in{\mathcal{L}}\\
\alpha(\infty,0)\mbox{, if }\infty\in{\mathcal{L}}
\end{array}\right.
$$
We also define
$\red_{{\mathcal{L}}}:\Omega_{\mathcal{L}}(L) \to {\mathcal{T}}_K({\mathcal{L}})$ as the restriction.
\end{defn}
\begin{obs}
The retraction map leaves fixed the points of ${\mathcal{L}}$. On the other
hand, if $x\not\in{\mathcal{L}}$, the point $\red_{\mathcal{L}}(x)$ is the only point of
the path $P(\alpha(x,0),\red_{\mathcal{L}}(x))\subset\overline{{\mathcal{T}}_K}$ which
is in ${\mathcal{T}}_K({\mathcal{L}})$.
\end{obs}
Now we want to extend this map to $\red_{{\mathcal{L}}}:\overline{{\mathcal{T}}_{L}}
\longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}$. First, if
$\alpha\in\overline{{\mathcal{T}}_K({\mathcal{L}})}$, then $\red_{{\mathcal{L}}}(\alpha)=\alpha$.
Next, consider $\alpha\in
{\mathcal{T}}_{L}\smallsetminus\overline{{\mathcal{T}}_K({\mathcal{L}})}$. Then
$\alpha=\alpha(x,r)$ for some $x\in L\smallsetminus {\mathcal{L}}$ and some
$r> 0$.
If $B(x,r)\bigcap{\mathcal{L}}=\emptyset$, we define
$\red_{{\mathcal{L}}}(\alpha):=\red_{{\mathcal{L}}}(x)$. We only need to show that
$\red_{{\mathcal{L}}}(\alpha)$ does not depend on the chosen $x$.
When $B(x,d_{\mathcal{L}})\cap{\mathcal{L}}\neq\emptyset$, $\red_{{\mathcal{L}}}(x)=\alpha(x,s)$
and $s>r$ since $\alpha(x,r)\not\in\overline{{\mathcal{T}}_K({\mathcal{L}})}$. Hence, if
$\alpha(x,r)=\alpha(y,r)$, then $\alpha(x,s)=\alpha(y,s)$.
Otherwise, it is clear.
In the other case, $B(x,r)\bigcap{\mathcal{L}}\neq\emptyset$, we have
$\infty\not\in{\mathcal{L}}$ and ${\mathcal{L}}\subset B(x,r)$ (so $r>d_{\mathcal{L}}$). Then we
define $\red_{{\mathcal{L}}}(\alpha):=\red_{{\mathcal{L}}}(\infty)$.
\begin{cor}\label{retraction}
The retraction map is a retraction. As a consequence, if ${\Gamma\subset\PGL_2(K)}$ acts on ${\mathcal{L}}$, it is $\Gamma$-equivariant.
\end{cor}
\begin{proof}
It follows from the previous remark and construction that the map is
a retraction in the strict sense. The consequence is due to the fact
that the projective linear group acts continuously on ${\mathcal{T}}_K$ and
$\Gamma$ leaves ${\mathcal{T}}_K({\mathcal{L}})$ invariant.
\end{proof}
Next, let us recall that ${\mathbb{C}}_K$ embeds isometrically into a
spherically complete nonarchimedean field ${\mathbb{K}}$, since it admits a
maximally complete extension by \cite[Thm.~24]{Kru32}, and this
condition is equivalent to spherical completeness by
\cite[Thm.~4]{Kap42}. We know by \cite[\S1.4]{Ber90} that
$({{\mathbb{P}}^1_{{\mathbb{K}}}}^*)^{an}$ has no type IV points so we get
$$
\red_{{\mathcal{L}}}:({{\mathbb{P}}^1_{{\mathbb{K}}}}^*)^{an}=\overline{{\mathcal{T}}_{{\mathbb{K}}}} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}
$$
Note that from the beginning of the formalization of the retraction
map, each time that we define it taking an infimum
($\red_{\mathcal{L}}(\alpha)=\alpha(x,\inf\{...\})$) we get this element is
inside the tree ${\mathcal{T}}_K({\mathcal{L}})$ since ${\mathcal{L}}$ is compact.
The following lemma is clear from the properties of the retraction
map.
\begin{lem}\label{extend}
If we have two subsets ${\mathcal{L}}'\subset {\mathcal{L}}\subset {{\mathbb{P}}^1}^*(K)$ as above, then
$$
\red_{{\mathcal{L}}'}(\alpha)=\red_{{\mathcal{L}}'}(\red_{{\mathcal{L}}}(\alpha))\text{ for any }\alpha \in \overline{{\mathcal{T}}_K}.
$$
\end{lem}
\begin{lem}\label{red2points}
For any two points $y_0,\ y_1$ in ${{\mathbb{P}}^1}^*(K)$, with respective
representatives in $(K^2)^*$ given by $\omega_0,\ \omega_1$ and
having dual basis $v_0,\ v_1$, and for any $\alpha\in
\overline{{\mathcal{T}}_{L}}$, the point $\red_{\{y_0,y_1\}}(\alpha)$ is the
seminorm $\eta$ diagonalized by $v_0$ and $v_1$ up to equivalence,
with $\eta(v_i)=\alpha(v_i)$ for $i=0$ and $1$, that is
$[\red_{\{y_0,y_1\}}(\alpha)]=[\alpha_{(v_0,v_1),(\alpha(v_0),\alpha(v_1))}]$.
\end{lem}
\begin{proof}
If $\alpha\in{\mathbb{A}}_{\{y_0,y_1\}}$ there is nothing to prove. From now
on we assume this is not the case.
If one of the two points, let us assume $y_1$, is $\infty$, then, writing $\alpha=\alpha(x,r)$,
$$
\red_{\{y_0,\infty\}}(\alpha)=\alpha(x,|y_0-x|)=\alpha(y_0,|y_0-x|)=\alpha_{(v_0,v_1),(1,|x-y_0|)}
$$
Now we compute
$$
\alpha(x,r)(v_0)=\alpha(x,r)(1,0)=\max\{1,0\}=1
$$
$$
\alpha(x,r)(v_1)=\alpha(x,r)(y_0,1)=\max\{|y_0-x|,r\}=|x-y_0|
$$
since $|x-y_0|>r$ due to $\alpha\not\in{\mathbb{A}}_{\{y_0,y_1\}}$.
Next, suppose $y_0,y_1\neq\infty$, and then we can take
$\omega_i=(1,-y_i)$ for $i=0,1$, so
$$\displaystyle{v_0=\left(\frac{y_1}{y_1-y_0},\frac{1}{y_1-y_0}\right) \mbox{ and } v_1=\left(\frac{y_0}{y_0-y_1},\frac{1}{y_0-y_1}\right)}.$$
Furthermore, either $\{y_0,y_1\}\subset B(x,r)$ or
$B(x,r)\bigcap\{y_0,y_1\}=\emptyset$.
In the first case
$$
\red_{\{y_0,y_1\}}(\alpha)=\red_{\{y_0,y_1\}}(\infty)=\alpha(y_0,|y_0-y_1|\})=\alpha_{(v_0,v_1),(1,1)}
$$
We just need to show that $\alpha(v_0)=\alpha(v_1)$. We have
$$
\alpha(x,r)(v_0)=\alpha(x,r)\left(\frac{y_1}{y_1-y_0},\frac{1}{y_1-y_0}\right)=\max\left\{\left|\frac{y_1-x}{y_1-y_0}\right|,\left|\frac{r}{y_1-y_0}\right|\right\}
$$
and, identically,
$\displaystyle{\alpha(x,r)(v_1)=\max\left\{\left|\frac{y_0-x}{y_0-y_1}\right|,\left|\frac{r}{y_0-y_1}\right|\right\}}$.
Since the condition $\{y_0,y_1\}\subset B(x,r)$ tells us that
$r\geq|y_0-x|,|y_1-x|$ we get the required equality
$\alpha(v_0)=\alpha(v_1)$.
In the second case, which is satisfied
$B(x,r)\bigcap\{y_0,y_1\}=\emptyset$, we have
$$
\red_{\{y_0,y_1\}}(\alpha)=\red_{\{y_0,y_1\}}(x)=
$$
$$
=\left\{\begin{array}{l}
\alpha(x,\min\{|x-y_0|,|x-y_1|\})\mbox{, if }B(x,|y_0-y_1|)\cap\{y_0,y_1\}\neq\emptyset\\
\alpha(y_0,|y_0-y_1|)\mbox{, if }B(x,|y_0-y_1|)\cap\{y_0,y_1\}=\emptyset
\end{array}\right.
$$
after noticing that $d_{\{y_0,y_1\}}=|y_0-y_1|$. The equality $B(x,|y_0-y_1|)\cap\{y_0,y_1\}=\emptyset$ is equivalent to say that
$|y_0-x|=|y_1-x|>|y_0-y_1|$ and $\alpha(y_0,|y_0-y_1|)=\alpha_{(v_0,v_1),(1,1)}$.
All the rest of the proof for this situation works exactly equal as above taking into account that the
condition $B(x,r)\bigcap\{y_0,y_1\}=\emptyset$ implies $|y_0-x|(=|y_1-x|)>r$.\\
Finally, when $B(x,|y_0-y_1|)\cap\{y_0,y_1\}\neq\emptyset$ we have $|y_i-x|\leq|y_0-y_1|$ for $i=0,1$ and at least for one $i$, $|y_i-x|=|y_0-y_1|$; assume this equality for $y_1$. Then, on one hand we get
$$
\alpha(x,\min\{|x-y_0|,|x-y_1|\})=\alpha(x,|x-y_0|)=\alpha(y_0,|x-y_0|)=\alpha_{(v_0,v_1),(1,\frac{|x-y_0|}{|y_0-y_1|})}
$$
On the other hand we have
$$
\alpha(x,r)(v_0)=\max\left\{\left|\frac{y_1-x}{y_1-y_0}\right|,\left|\frac{r}{y_1-y_0}\right|\right\}=\left|\frac{y_1-x}{y_1-y_0}\right|
$$
and
$$
\alpha(x,r)(v_1)=\max\left\{\left|\frac{y_0-x}{y_0-y_1}\right|,\left|\frac{r}{y_0-y_1}\right|\right\}=\left|\frac{y_0-x}{y_0-y_1}\right|
$$
since $B(x,r)\bigcap\{y_0,y_1\}=\emptyset$. Therefore, maintaining and employing the assumption $|y_1-x|=|y_0-y_1|\geq|y_0-x|$, we obtain
$$
\alpha_{(v_0,v_1),(\alpha(v_0),\alpha(v_1))}=\alpha_{(v_0,v_1),(\frac{|y_1-x|}{|y_1-y_0|},\frac{|y_0-x|}{|y_0-y_1|}}=\alpha_{(v_0,v_1),(1,\left|\frac{x-y_0}{y_0-y_1}\right|)},
$$
and so the claimed equality. Note that if we had assumed $|y_0-x|=|y_0-y_1|$ we would have got
$$
\alpha(x,\min\{|x-y_0|,|x-y_1|\})=\alpha(y_1,|x-y_1|)=\alpha_{(v_0,v_1),(\frac{|x-y_1|}{|y_0-y_1|},1)}=\alpha_{(v_0,v_1),(\alpha(v_0),\alpha(v_1))}
$$
too.
\end{proof}
\begin{lem}
Let ${\mathcal{L}}\subset{{\mathbb{P}}^1}^*(K)$ be a compact subset with at least two points. For any two seminorms $\alpha,\alpha'\in\overline{{\mathcal{T}}_{L}}$ such that $\displaystyle{\alpha_{|K[X_0,X_1]}=\alpha'_{|K[X_0,X_1]}}$, then $\red_{\mathcal{L}}(\alpha)=\red_{\mathcal{L}}(\alpha')$.
\end{lem}
\begin{proof}
If ${\mathcal{L}}=\{y_0,y_1\}$ the claim is true due to the last lemma. Otherwise, we always can find two points $y_0,y_1\in{\mathcal{L}}$ such that $\red_{\mathcal{L}}(\alpha),\red_{\mathcal{L}}(\alpha')\in{\mathbb{A}}_{\{y_0,y_1\}}$. Then, using this hypothesis for the outer equalities together with lemmas~\ref{extend}~and~\ref{red2points} for the interior equalities, we get
$$
\red_{\mathcal{L}}(\alpha)=\red_{\{y_0,y_1\}}(\red_{\mathcal{L}}(\alpha))=\red_{\{y_0,y_1\}}(\alpha)=\red_{\{y_0,y_1\}}(\alpha')=\red_{\{y_0,y_1\}}(\red_{\mathcal{L}}(\alpha'))=\red_{\mathcal{L}}(\alpha')
$$
\end{proof}
Finally, recall that we have a retraction map
$\red_{{\mathcal{L}}}:\overline{{\mathcal{T}}_{L}} \longrightarrow
\overline{{\mathcal{T}}_K({\mathcal{L}})}$ with two important particular cases:
$$
\red_{{\mathcal{L}}}:\overline{{\mathcal{T}}_{K}} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}
$$
and
$$
\red_{{\mathcal{L}}}:({{\mathbb{P}}^1_{{\mathbb{K}}}}^*)^{an}=\overline{{\mathcal{T}}_{{\mathbb{K}}}} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}
$$
Now we want extend the first retraction map to $\red_{{\mathcal{L}}}:({{\mathbb{P}}^1_{K}}^*)^{an} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}$. Note first that $({{\mathbb{P}}^1_{K}}^*)^{an}\cong({{\mathbb{P}}^1_{{\mathbb{C}}_K}}^*)^{an}/Gal({\mathbb{C}}_K|K)$ by \cite[Cor.~1.3.6]{Ber90} so we may assume for a while $K={\mathbb{C}}_K$ in order to define the extension.
Then, by remark~\ref{types} we only have to do this for the points of type IV. Let us take such a seminorm point $\alpha\in({{\mathbb{P}}^1_{K}}^*)^{an}$. It is a limit of ball seminorms $\{\alpha(x_i,r_i)\}_{i\in{\mathbb{N}}}$ such that
$$
r_{i+1}\leq r_i,\qquad B(x_{i+1},r_{i+1})\subset B(x_i,r_i)
$$
$$
r:=\lim_{i\rightarrow\infty}{r_i}>0\mbox{ and }\bigcap_{i\in{\mathbb{N}}}{B(x_i,r_i)}=\emptyset
$$
We consider the balls of the same center and radio with points in the spherical completion ${\mathbb{K}}$, that is $B_{\mathbb{K}}(x_i,r_i):=\{ y\in {\mathbb{K}} \ | \ |y-x_i|\le r_i\}$. Denote the associated seminorms in $({{\mathbb{P}}^1_{{\mathbb{K}}}}^*)^{an}$ by $\alpha_{\mathbb{K}}(x_i,r_i)$.\\
Therefore, on one hand we have $\alpha_{\mathbb{K}}(x_i,r_i)_{|K[X_0,X_1]}=\alpha(x_i,r_i)$ and on the other hand we obtain $\displaystyle{\bigcap_{i\in{\mathbb{N}}}{B_{\mathbb{K}}(x_i,r_i)}\neq\emptyset}$ so it is a ball $B_{\mathbb{K}}(\hat{x},r)$ which has an associated seminorm $\alpha_{\mathbb{K}}(\hat{x},r)\in({{\mathbb{P}}^1_{{\mathbb{K}}}}^*)^{an}$. Thus we get
$$
\alpha=\lim_{i\rightarrow\infty}{\alpha(x_i,r_i)}=\lim_{i\rightarrow\infty}{\alpha_{\mathbb{K}}(x_i,r_i)_{|K[X_0,X_1]}}=\alpha_{\mathbb{K}}(\hat{x},r)_{|K[X_0,X_1]}
$$
Finally, we may take $\red_{\mathcal{L}}(\alpha):=\red_{\mathcal{L}}(\alpha_{\mathbb{K}}(\hat{x},r))$ which is well defined by last lemma above.
\begin{obs}
This construction of $\red_{{\mathcal{L}}}:({{\mathbb{P}}^1_{K}}^*)^{an} \longrightarrow \overline{{\mathcal{T}}_K({\mathcal{L}})}$ and
the lemma~\ref{red2points} allows us to note that when $\ds{\overline{{\mathcal{T}}_K({\mathcal{L}})}=\overline{{\mathcal{T}}_{K}}}$,
this definition coincides with the given by Werner in \cite{Wer04}.
\end{obs}
\section{Graphs, their models and harmonic cochains}\label{graphs}
We give a general definition of harmonic cochains over any weighted graph and we prove the isomorphism between harmonic measures on any compact subset ${{\mathcal{L}}^*\subset{\mathbb{P}}^1(K)}$ and harmonic cochains on the associated tree ${\mathcal{T}}_K({\mathcal{L}})$, claimed in \cite[Ex.~2.1.1]{vdP92}.
A weighted graph $\mathfrak{G}$ is a non empty set
$V=V(\mathfrak{G})$ called vertex set together with an oriented edge
set $E=E(\mathfrak{G})$, a weight function $\ell:E\longrightarrow\mb{R}_{>0}$,
an edge assignment map $s\times t:E\longrightarrow V\times V$ which makes
correspond to each edge $e$ a pair $(s(e),t(e))$, where $s(e)$ is
called the source of $e$ and $t(e)$ the target of $e$, and a
bijection $\vartheta:E\longrightarrow E$ verifying $\ell(\vartheta(e))=\ell(e)$, $s\times
t(\vartheta(e))=(t(e),s(e))$ and $\vartheta(e)\neq e$. The edge $\vartheta(e)$ is
called the opposite of $e$ and denoted by $\bar e$ (cf. \cite{BF11}
and \cite{Ser80}).
The topological realization of a weighted graph $\mathfrak{G}$ is a metric graph $G:=|\mathfrak{G}|$, for which the lenght of their edges is given by the weight of the edges of $\mathfrak{G}$ (the same definitions and notations that we have for a weighted graph work for a metric graph). Reciprocally, given a metric graph $G$, a model for it is any weighted graph $\mathfrak{G}$ such that $G$ is obtained as its topological realization, that is $G\cong|\mathfrak{G}|$. A minimal model is one in which all the vertices have valence greater than 2.
We will consider the free abelian group ${\mathbb{Z}}[E(\mathfrak{G})]$ generated by the oriented edges of $\mathfrak{G}$.
Given a weighted graph $\mathfrak{G}$, and a vertex $v$, we denote
by $\Star(v)$ the set of edges of $\mathfrak{G}$ with source $v$. Recall that an harmonic cochain is a
morphism $c:{\mathbb{Z}}[E(\mathfrak{G}))]\to {\mathbb{Z}}$ verifying
\begin{itemize}
\item $c(\bar e)=-c(e)$ for any $e \in E(\mathfrak{G})$, and
\item $\displaystyle{c\Big(\sum_{e\in \Star(v)}
e\Big) =0}$ for any vertex $v \in V(\mathfrak{G})$.
\end{itemize}
We denote the set of harmonic cochains of $\mathfrak{G}$ by $\HC(\mathfrak{G},{\mathbb{Z}})$.
Observe that, if we subdivide an oriented edge $e$ in two oriented
edges $e_1$ and $e_2$, then the properties tell that any
harmonic cochain verifies that $c(e_1)=c(e_2)$. Hence, given a (locally finite) metric graph and two arbitrary models for it, there is a
canonical isomorphism between their harmonic cochains, so we can define them for the metric graph $G=|\mathfrak{G}|$, and we can write $\HC(G,{\mathbb{Z}}):=\HC(\mathfrak{G},{\mathbb{Z}})$.
Let $\mathfrak{G}=(V,E)$ be a weighted graph and $\mathfrak{H}$ be a finite weighted subgraph of $\mathfrak{G}$. We define
$$
\Star(\mathfrak{H}):=\{e\in E|\ s(e)\in \mathfrak{H}, e\not\in E(\mathfrak{H})\}
$$
Note that this generalizes the definition of $\Star(v)$ for a vertex $v$.
\begin{lem}\label{harstar} Let $\mathfrak{H}$ be a finite weighted subgraph of $\mathfrak{G}$. Then, any harmonic cochain $c$ satisfies $\displaystyle{c\Big(\sum_{e\in
\Star(\mathfrak{H})} e\Big) =0}$.
\end{lem}
\begin{proof}
First observe the following properties of stars:
$$
\Star(\mathfrak{H})\sqcup E(\mathfrak{H})=\Star(V(\mathfrak{H}))=\bigsqcup_{v\in V(\mathfrak{H})}{\Star(v)}
$$
Next note that an edge belongs to $\mathfrak{H}$ if and only if its opposite also do. Then, taking into consideration the first equality of stars and the previous remark, because of the first property of the harmonic cochains we get
$$
c\Big(\sum_{e\in \Star(\mathfrak{H})} e\Big) = \sum_{e\in \Star(\mathfrak{H})} c(e) = \sum_{e\in \Star(V(\mathfrak{H}))} c(e)
$$
and because of the second equality of stars and the second property of harmonic cochains we finish as follows:
$$
\sum_{e\in \Star(V(\mathfrak{H}))} c(e) = \sum_{v\in V(\mathfrak{H})} \sum_{e\in \Star(v)} c(e) = 0
$$
\end{proof}
Let $\mathfrak{T}=(V,E)$ be a model for ${\mathcal{T}}={\mathcal{T}}_K({\mathcal{L}})$. For any
edge $e$ there exists a topological edge $\varepsilon$ (both oriented) such
that the first (its topological realization) is contained in the
second (as an oriented path in ${\mathcal{T}}$), that is $|e|\subset\varepsilon$. Then
we define the open set ${\mathcal{B}}(e):={\mathcal{B}}(\varepsilon)$. This is well defined due
to the lemma~\ref{oprops}.
\begin{prop}
Let $\mathfrak{T}=(V,E)$ be a model for ${\mathcal{T}}={\mathcal{T}}_K({\mathcal{L}})$ and let $F\subset E$ be a well oriented finite set of edges, meaning that it satisfies the following hypothesis:
\begin{itemize}
\item it cannot exist a topological edge $\varepsilon$ of ${\mathcal{T}}$ and edges $e,e'\in F$ such that both are contained in $\varepsilon$, $e$ is oriented like $\varepsilon$ and $e'$ is oriented like the topological opposite edge $\overline{\varepsilon}$.
\end{itemize}
Take the source vertices of F, $\sigma:=\sigma(F):=\{s(e)|\ e\in F\}$ and denote by $\mathfrak{T}_\sigma$ the subtree generated by $\sigma$. Then
\begin{enumerate}
\item The open sets $\{{\mathcal{B}}(e)\}_{e\in F}$ are pairwise disjoint if and only if $F\bigcap E(\mathfrak{T}_\sigma)=\emptyset$, which means $|F|\subset |\Star(\mathfrak{T}_\sigma)|$.
\item The equality $\displaystyle{\bigcup_{e\in F}{\mathcal{B}}(e)={\mathcal{L}}^*}$ occurs if and only if $\Star(\mathfrak{T}_\sigma)\subset F$.
\end{enumerate}
\end{prop}
\begin{proof}
We will show the claims by induction on the cardinal of vertices $n=\#V(\mathfrak{T}_\sigma)$.
If $n=1$, then $\mathfrak{T}_\sigma=\{v\}=\sigma(F)$, $F\subset \Star(v)$, the sets ${\mathcal{B}}(e)$ with $e\in \Star(v)$ are pairwise disjoint and $\displaystyle{\bigsqcup_{e\in F}{{\mathcal{B}}(e)}={\mathcal{L}}^*}$ if and only if $F=\Star(v)$.
Next, assume $n>1$ and let $v\in \sigma=\sigma(F)$ be a vertex with valence $1$ in $\mathfrak{T}_\sigma$. Consider the non empty set $F_v:=\{e\in F|\ s(e)=v\}$, proper in $F$ since $n>1$, and let $e_v$ be the edge of $\mathfrak{T}_\sigma$ with target $t(e_v)=v$. Then, if $F':=(F\setminus F_v)\cup\{e_v\}$, we get the next remarkable properties:
\begin{itemize}
\item $\sigma':=\left(\sigma\setminus\{v\}\right)\cup\{s(e_v)\}=\sigma(F')$,
\item and $\#V(\mathfrak{T}_{\sigma'})=n-1$, so we may apply the induction hypothesis on $F'$.
\item $E(\mathfrak{T}_{\sigma'})=E(\mathfrak{T}_\sigma)\setminus \{e_v,\overline{e_v}\}$, so $$F'\cap E(\mathfrak{T}_{\sigma'})=(F\setminus F_v)\bigcap \big(E(\mathfrak{T}_\sigma)\setminus\{e_v,\overline{e_v}\}\big).$$
\item $\Star(\mathfrak{T}_{\sigma'})=(\Star(\mathfrak{T}_\sigma)\setminus \Star(v))\cup\{e_v\}$.
\item $\Star(\mathfrak{T}_{\sigma})=(\Star(\mathfrak{T}_{\sigma'})\setminus \{e_v\})\cup (\Star(v)\setminus\{\overline{e_v}\})$.
\end{itemize}
Suppose that $F\bigcap E(\mathfrak{T}_\sigma)=\emptyset$. Therefore
$F'\bigcap E(\mathfrak{T}_{\sigma'})=\emptyset$. Then, by induction
hypothesis, the open sets $\{{\mathcal{B}}(e)\}_{e\in F'}$ are pairwise
disjoint and, in particular, ${\mathcal{B}}(e_v)\cap{\mathcal{B}}(e)=\emptyset$ for all
$e\in F\setminus F_v$. Recall now that
$$\displaystyle{{\mathcal{B}}(e_v)=\bigsqcup_{e\in \Star(v)\setminus\overline{e_v}}{\mathcal{B}}(e)}$$ and that $F_v\subset \Star(v)$.
But, $e_v$ and $\overline{e_v}$ are edges of $\mathfrak{T}_\sigma$,
so $F\bigcap E(\mathfrak{T}_\sigma)=\emptyset$ implies that $e_v,
\overline{e_v}\not\in F$, and therefore we get that the sets
$\{{\mathcal{B}}(e)\}_{e\in F}$ are also pairwise disjoint.
Now assume that $F\bigcap E(\mathfrak{T}_\sigma)\neq\emptyset$.
Then, either $F'\bigcap E(\mathfrak{T}_{\sigma'})\neq\emptyset$, or
$F'\bigcap E(\mathfrak{T}_{\sigma'})=\emptyset$ but $$\emptyset\neq
F\bigcap E(\mathfrak{T}_\sigma)\subset \{e_v,\overline{e_v}\}.$$
In this last case, $F'\bigcap E(\mathfrak{T}_{\sigma'})=\emptyset$
and $F\bigcap E(\mathfrak{T}_\sigma)\subset \{e_v,\overline{e_v}\}$,
when $e_v\in F$, then ${\mathcal{B}}(e_v)\cap{\mathcal{B}}(e)\neq\emptyset$ for any
$e\in F_v\neq\emptyset$. In the case $\overline{e_v}\in F$, the fact
that $e_v\in F'$ and so that ${\mathcal{B}}(e_v)\cap{\mathcal{B}}(e)=\emptyset$ for any
$e\in F\setminus F_v$ (by induction on $F'$), together with
${\mathcal{B}}(\overline{e_v})={\mathcal{L}}^*\setminus{\mathcal{B}}(e_v)$, implies that
${\mathcal{B}}(\overline{e_v})\cap{\mathcal{B}}(e)\neq\emptyset$ for any $e\in
F\setminus F_v$.
If $F'\bigcap E(\mathfrak{T}_{\sigma'})\neq\emptyset$, the sets
$\{{\mathcal{B}}(e)\}_{e\in F'}$ are not pairwise disjoint, and the collection
of sets $\{{\mathcal{B}}(e)\}_{e\in F}$ include the same except maybe
${\mathcal{B}}(e_v)$, besides the $\{{\mathcal{B}}(e)\}_{e\in F_v}$.
Therefore, if there are $e,e'\in F\setminus F_v$ such that
${\mathcal{B}}(e)\cap{\mathcal{B}}(e')\neq\emptyset$ we get the claim. Otherwise there
is an $e_0\in F\setminus F_v$ such that
${\mathcal{B}}(e_0)\cap{\mathcal{B}}(e_v)\neq\emptyset$ and $e_v\not\in F$. By
definition of $e_v$ we have that $s(e_v)\in P(t(e_v),s(e_0))$. Then,
taking in consideration the lemma~\ref{oprops} we get $s(e_0)\not\in
P(t(e_v),t(e_0))$ (and
${\mathcal{B}}(e_0)\cap{\mathcal{B}}(e_v)={\mathcal{B}}(e_v)\subset{\mathcal{B}}(e_0)$), since otherwise we
would have $s(e_0)\in P(t(e_v),t(e_0))$ and, as a consequence,
${\mathcal{B}}(e_0)\cap{\mathcal{B}}(e_v)=\emptyset$.
Take now an edge $e_1\in F_v$. Assume first $e_1\neq\overline{e_v}$.
Then we obtain that ${\mathcal{B}}(e_1)\subset{\mathcal{B}}(e_v)\subset{\mathcal{B}}(e_0)$ and
that the sets $\{{\mathcal{B}}(e)\}_{e\in F}$ are not pairwise disjoint as we
wanted. To finish the proof of the the first equivalence, we just
have to deal with the case $F_v=\{\overline{e_v}\}$. Since $F$ is
well oriented, there is some vertex of valence three in ${\mathcal{T}}_K({\mathcal{L}})$
between $s(e_0)$ and $t(e_v)$ (excluding them). Then
${\mathcal{B}}(e_0)\cap{\mathcal{B}}(\overline{e_v})\neq\emptyset$ by
lemma~\ref{oprops}.
Recalling the properties we have noted above, we get that $\Star(\mathfrak{T}_\sigma)\subset F$ implies $\Star(\mathfrak{T}_{\sigma'})\subset F'$, so, by hypothesis, $\displaystyle{\bigcup_{e\in F'}{\mathcal{B}}(e)={\mathcal{L}}^*}$. By definition, we know that each edge of $F'$ is an edge of $F$ except at most $e_v$, but we have that $\Star(v)\setminus\{\overline{e_v}\}\subset \Star(\mathfrak{T}_\sigma)\subset F$ and $\displaystyle{{\mathcal{B}}(e_v)=\bigsqcup_{e\in \Star(v)\setminus\overline{e_v}}{\mathcal{B}}(e)}$, so
$$
{\mathcal{L}}^*=\bigcup_{e\in F'}{\mathcal{B}}(e)\subset\bigcup_{e\in F}{\mathcal{B}}(e)={\mathcal{L}}^*.
$$
Suppose that $\Star(\mathfrak{T}_\sigma)\not\subset F$. This means that there is an edge $e\in \Star(\mathfrak{T}_\sigma)\setminus F$, in particular with $s(e)\in \sigma=\sigma(F)$. We may assume that the vertex $v$ we chose above in order to apply the induction method is different from $s(e)$. It is clear that $e\not\in F'$, and by the assumption $e\in \Star(\mathfrak{T}_{\sigma'})$, so $\Star(\mathfrak{T}_{\sigma'})\not\subset F'$ and $\displaystyle{\bigcup_{e\in F'}{\mathcal{B}}(e)\neq{\mathcal{L}}^*}$.
Finally, as we have seen before, we have
$$
\bigcup_{e\in F}{\mathcal{B}}(e)=\bigcup_{e\in F\setminus F_v}{\mathcal{B}}(e)\cup\bigcup_{e\in F_v}{\mathcal{B}}(e)\subset\bigcup_{e\in F'}{\mathcal{B}}(e)\cup{\mathcal{B}}(e_v)\subset\bigcup_{e\in F'}{\mathcal{B}}(e)\subsetneq{\mathcal{L}}^*
$$
\end{proof}
\begin{cor} Let $\{\varepsilon_i\}_{i\in I}$ be a finite set
of topological oriented edges in ${\mathcal{T}}_K({\mathcal{L}})$ such that the open
sets ${\mathcal{B}}(\varepsilon_i)$ for $i\in I$ are pairwise disjoint. Then $
\bigsqcup_{i\in I} {\mathcal{B}}(\varepsilon_i) = {\mathcal{L}}^* \Leftrightarrow
\{\varepsilon_i\}_{i\in I}=\Star(\mathfrak{T})$ for the finite subtree
$\mathfrak{T}$ with source vertices $\{s(\varepsilon_i)\}_{i\in I}$, or
$\{\varepsilon_i\}_{i\in I}=\{\varepsilon_1,\varepsilon_2\}$ existing a topological edge
$\varepsilon$ in ${\mathcal{T}}_K({\mathcal{L}})$ such that $\varepsilon_1\subset\varepsilon$ and
$\varepsilon_2\subset\overline{\varepsilon}$.
\end{cor}
In order to get another point of view for the harmonic cochains we have to define the harmonic measures on a suitable compact space.
\begin{defn}
Let $X$ be a compact space such that the compact open subsets form a
basis for the topology. A $\mb{Z}$-valued measure $\mu$ on $X$ is a finitely additive function on the (disjoint) compact subsets
of $X$. The set of $\mb{Z}$-valued measures on $X$ is denoted $\mathscr{M}(X,\mb{Z})$.
\end{defn}
\begin{defn} Let $\mu\in\mathscr{M}(X,\mb{Z})$ be a $\mb{Z}$-valued
measure on $X$. We say that $\mu$ is harmonic if the total
volume $\mu(X)$ is $0$. We denote the set of harmonic measures by $\mathscr{M}(X,{\mathbb{Z}})_0$.
\end{defn}
Note that as much $\mathscr{M}(X,\mb{Z})$ as $\mathscr{M}(X,{\mathbb{Z}})_0$ are abelian groups.
\begin{cor}\label{HMC} Any harmonic cochain $c$ of the metric tree ${\mathcal{T}}_K({\mathcal{L}})$
determines a unique harmonic measure $\mu(c)$ in $\mathscr{M}(
{\mathcal{L}}^*,\mb{Z})_{0}$ by defining $\mu(c)({\mathcal{B}}(\varepsilon))=c(\varepsilon)$ for any
topological oriented edge $\varepsilon$ in ${\mathcal{T}}_K({\mathcal{L}})$. This induces an
isomorphism between $\mathscr{M}( {\mathcal{L}}^*,\mb{Z})_{0}$ and
$\HC({\mathcal{T}}_K({\mathcal{L}}),{\mathbb{Z}})$.
\end{cor}
\begin{proof}
Essentially, all we have to check is that the map
${\HC({\mathcal{T}}_K({\mathcal{L}}),{\mathbb{Z}})\longrightarrow\mathscr{M}( {\mathcal{L}}^*,\mb{Z})_{0}}$
given by the description above is well defined.
First, it is enough to characterize a measure over the sets ${\mathcal{B}}(e)$ because of these form a basis for the topology of ${\mathcal{L}}^*$.
Next, take a model $\mathfrak{T}=(V,E)$ for ${\mathcal{T}}_K({\mathcal{L}})$. We just have to see that for any open compact set $\mathcal{U}\subset{\mathcal{L}}^*$ and for any partition $\mathcal{U}=\bigsqcup_{e\in I}{{\mathcal{B}}(e)}$ with $I\subset E$ finite, the sum $\sum_{e\in I}{c(e)}$ is invariant. Let us take two finite partitions of ${\mathcal{U}}$:
$$
{\mathcal{U}}=\bigsqcup_{e\in I}{{\mathcal{B}}(e)}=\bigsqcup_{e\in I'}{{\mathcal{B}}(e)}
$$
Since $\mathcal{U}$ is open and compact so it is the complement $\mathcal{V}={\mathcal{L}}^*\setminus\mathcal{U}$ and we can consider another finite partition $\displaystyle{\mathcal{V}=\bigsqcup_{e\in\tilde{I}}{{\mathcal{B}}(e)}}$, $\tilde{I}\subset E$. Then we have
$$
{\mathcal{L}}^*={\mathcal{U}}\sqcup\mathcal{V}=\bigsqcup_{e\in I}{{\mathcal{B}}(e)}\sqcup\bigsqcup_{e\in\tilde{I}}{{\mathcal{B}}(e)}=\bigsqcup_{e\in I'}{{\mathcal{B}}(e)}\sqcup\bigsqcup_{e\in\tilde{I}}{{\mathcal{B}}(e)}
$$
Therefore, by the previous corollary, we get $I\sqcup\tilde{I}=\Star(\mathfrak{T})$ and $I'\sqcup\tilde{I}=\Star(\mathfrak{T}')$ for certain finite subtrees of ${\mathcal{T}}$ (or any or both disjoint unions can be the degenerated case, which the reader can do as an easy exercise). Then we have
$$
\sum_{e\in I}{c(e)}+\sum_{e\in\tilde{I}}{c(e)}=\sum_{e\in I\sqcup\tilde{I}}{c(e)}=\sum_{e\in \Star(\mathfrak{T})}{c(e)}=0
$$
and
$$
\sum_{e\in I'}{c(e)}+\sum_{e\in\tilde{I}}{c(e)}=\sum_{e\in I'\sqcup\tilde{I}}{c(e)}=\sum_{e\in \Star(\mathfrak{T}')}{c(e)}=0
$$
after apply lemma~\ref{harstar}, so we get
$$
\sum_{e\in I}{c(e)}=\sum_{e\in I'}{c(e)}
$$
as we wanted to prove.
Once we have the map well defined, it follows immediately from the definition that it is an isomorphism of abelian groups. Indeed, the kernel has to be zero and the same definition provides the exhaustivity.
\end{proof}
\section{Multiplicative Integrals}\label{MI}
The following definition was introduced by Longhi \cite{Lon02} as a
generalization of Darmon \cite{Dar01}.
\begin{defn}
Let $X$ be a compact space such that the compact open subsets form a
basis for the topology. Let $G$ be a complete topological abelian
group (written multiplicatively) such that a basic system of
neighbourhoods of the identity consists of open subgroups. Let
$\ds{f:X\longrightarrow G}$ be a continuous function and let
$\mu\in\mathscr{M}(X,\mb{Z})$ be a $\mb{Z}$-valued measure on $X$. The \textbf{multiplicative integral} of $f$ with respect to
$\mu$ is defined as
$$
\mathop{\mathrlap{\pushMI}}\!\int_{X}{fd\mu}:=\mathop{\mathrlap{\pushMI}}\!\int_{X}{f(t)d\mu(t)}:=
\lim_{\substack{\rightarrow\\\mc{C}_\alpha}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha\\t_n^\alpha\in\mc{U}_n^\alpha}}{f(t_n^\alpha)^{\mu(\mc{U}_n^\alpha)}}}}
$$
where the limit is taken over the direct system of finite covers
$\ds{\mc{C}_\alpha=\mc{C}_\alpha(X)}$ of $X$ by disjoint open compact subsets
$\ds{\mc{U}_n^\alpha}$, and the $\ds{t_n^\alpha}$ are arbitrary points in
them.
\end{defn}
It is well defined since the limit exists and it does not depend on the choice of the $t_n^\alpha$'s. (\cite[Prop.~5]{Lon02})
\begin{prop}\label{propertiesmultiplicativeintegral} For any measure $\mu\in\mathscr{M}(X,\mb{Z})$, we
have
\begin{enumerate}
\item For any compact open subset $U$ of $X$, and for any $\gamma\in G$, denote by $\chi_{U,\gamma}(t)$
the function sending $x\in X$ to $\gamma$ if $x\in U$, and to $1$
otherwise. Then $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{X}{\chi_{U,\gamma} d\mu}=\gamma^{\mu(U)}}$.
\item If $\ds{f,g:X\longrightarrow G}$ are continuous functions on $X$, then
$$\mathop{\mathrlap{\pushMI}}\!\int_{X}{(f\cdot g) d\mu}=\left(\mathop{\mathrlap{\pushMI}}\!\int_{X}{f d\mu}\right)\left(\mathop{\mathrlap{\pushMI}}\!\int_{X}{ g d\mu}\right)$$
\end{enumerate}
\end{prop}
Note that for any harmonic measure $\mu$ and any constant function
$\ds{f:X\longrightarrow G}$ such that $f(x)=\lambda$ for all $x\in X$, we have
$\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{X}{fd\mu}=1}$.
Now, let ${\mathcal{L}}$ be a compact subset of ${{\mathbb{P}}^1}^*(K)$ with at least two points and let $L|K$ be an arbitrary complete extension of fields. We
get from them the set ${\mathcal{L}}^*\subset{\mathbb{P}}^1(K)$, the
space $\Omega_{\mathcal{L}}(L)$ and the tree ${\mathcal{T}}_K({\mathcal{L}})$. With these objects we give the next definitions and lemmas.
\begin{defn}\label{FutS}
Let ${\mathcal{P}}$ be a finite set of points in $\Omega_{\mathcal{L}}(L)$, and consider
$D:=\sum_{p\in {\mathcal{P}}} m_p p$ a divisor of degree zero. We denote by
$f_D$ the element of $\mathscr{M}aps({\mathcal{L}}^*, L^*)/L^*$ which is
defined up to scalars as follows: if we choose representatives
$v_p\in (L^2)^*$ for any $p\in {\mathcal{P}}$ and $v_q\in K^2$ for $q$, then
$f_D(q)=\prod_{p\in {\mathcal{P}}} v_p(v_q)^{m_p}$ does not depend on $v_q$.
Any other election of the vectors $v_p$ change $f_D$ to $\lambda
f_D$ for some $\lambda\in L^*$.
Similarly, if ${\mathcal{A}}$ is a finite set of points in ${\mathcal{T}}_{L}$, and
consider $D:=\sum_{[\alpha]\in {\mathcal{A}}} m_{[\alpha]} [\alpha]$ a degree
zero divisor, then we denote by $|f|_D$ the element of
$\mathscr{M}aps({\mathcal{L}}^*, {\mathbb{R}}_{>0})/{\mathbb{R}}_{>0}^*$ being defined up to
scalars by $|f|_D(q)=\prod_{\alpha\in {\mathcal{A}}} \alpha(q)^{m_{[\alpha]}
}$ (remind that the points $[\alpha]$ are classes modulo homothety
of diagonalizable seminorms $\alpha$).
We note that we will be flexible when using these notations, not making difference between the map and the class of the map.
We note also that any representant of $f_D$ can be seen as a map
which extends to a meromorphic function on ${\mathbb{P}}^1$ with divisor $D$.
\end{defn}
\begin{obs}
We can see the degree zero divisor $0$ as the divisor $0p$ for any ${p\in\Omega_{\mathcal{L}}(L)}$. Therefore, as $m_p=0$, we get $f_0\equiv1$ and $|f|_0\equiv1$.
\end{obs}
As a particular case, if we consider the divisor $D:=\alpha(x,s)-\alpha(x,r)$ in ${\mathcal{T}}_K({\mathcal{L}})$, where $s>r$, then we have
$$
|f|_{D}(q)=\left\{\begin{array}{ll} \frac sr & \mbox{ if }
q\in B(x^*,r) \\
\frac{s}{|q-x^*|} & \mbox{ if } q\in B(x^*,s)\setminus B(x^*,r) \\
1 & \mbox{ if } q\not\in B(x^*,s)
\end{array} \right.
$$
for any $q\in {\mathcal{L}}^*$.
Observe that, if the path from $\alpha(x,r)$
to $\alpha(x,s)$ is a topological edge, then ${\mathcal{L}} \cap (B(x,s-\epsilon)\setminus
B(x,r))$ is empty for any $s-r>\epsilon>0$ (and so the corresponding intersection with ${\mathcal{L}}^*$), and then
$|f|_{D}(q)=1$ or $\frac sr$ for any $q\in {\mathcal{L}}^*$.
\begin{prop}\label{frd} Let ${\mathcal{A}}$ be a finite set of points in ${\mathcal{T}}_K$,
let ${D:=\sum_{\alpha\in {\mathcal{A}}} m_{\alpha} \alpha}$ be a degree zero
divisor and consider its retraction $\red_{{\mathcal{L}}}(D):=\sum_{\alpha\in {\mathcal{A}}} m_{\alpha}
\red_{{\mathcal{L}}}(\alpha)$. Then
$|f|_{D}=|f|_{\red_{{\mathcal{L}}}(D)}$ in $\mathscr{M}aps({\mathcal{L}}^*, {\mathbb{R}}_{>0})/{\mathbb{R}}_{>0}^*$.
\end{prop}
\begin{proof}
First of all, observe that in the case ${\mathcal{L}}=\{y_0,y_1\}$ this is a
consequence of lemma~\ref{red2points}.
Now we do the general case. Fix $x\in {\mathcal{L}}$ and consider any point
$y\in {\mathcal{L}}$, $x\ne y$. Take ${\mathcal{L}}':=\{y,x\}\subset {\mathcal{L}}$. Using the
previous case twice (and taking some representatives) we get that
$${|f|_{D}}_{|{\mathcal{L}}'^*}={|f|_{\red_{{\mathcal{L}}'}(D)}}_{|{\mathcal{L}}'^*}={|f|_{\red_{{\mathcal{L}}'}(\red_{{\mathcal{L}}}(D))}}_{|{\mathcal{L}}'^*}={|f|_{\red_{{\mathcal{L}}}(D)}}_{|{\mathcal{L}}'^*}$$
by applying lemma~\ref{extend}. Since this equality is satisfied for all ${\mathcal{L}}'$ with $x$ fixed, it is satisfied for ${\mathcal{L}}$ too (if we looked to the maps representing these classes modulo homothety, it would appear some scalar at the end of the equality which would not depend on ${\mathcal{L}}'$ or on $y$ due to the fixed $x$).
\end{proof}
\begin{defn}\label{defint} Given any degree 0 divisor
$D=\sum_{i\in I} m_i p_i$ with support in $\Omega_{\mathcal{L}}(L)$ (i.e. $m_i\in {\mathbb{Z}}$, $p_i\in \Omega_{\mathcal{L}}(L)$,
being $I$ a finite set and with $\sum_{i\in I}m_i=0$) we choose
$v_i$ in $(L^2)^*$ representatives of the $p_i\in{{\mathbb{P}}^1}^*(L)$ and consider
the map up to scalars $\ds{f_D\in\mathscr{M}aps({\mathcal{L}}^*, L^*)/K^*}$ given by a representant $\prod_{i\in
I} v_i(x)^{m_i}$ (which depend on the $v_i$'s). Let $\mu\in\mathscr{M}(
{\mathcal{L}}^*,\mb{Z})_{0}$ be a $\mb{Z}$-valued harmonic measure on
${\mathcal{L}}^*$.
We define $$\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*,D}{d\mu}:= \mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_D
d\mu}\in L^*},$$ which is well defined since the integral does not
depend on $f_D$ but only on $D$. Indeed, although the representant
of $f_D$ depend on the elections of the representatives in $(L^2)^*$
of the points in ${{\mathbb{P}}^1}^*(L)$, the multiplicative integral does
not, since the measure is harmonic.
In general, when some ${\mathcal{L}}$ was fixed previously -as along this section-, we will omit its corresponding set, writing $$\mathop{\mathrlap{\pushMI}}\!\int_D{d\mu}:=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*,D}{d\mu},$$ meanwhile we will specify the other sets over which we will integrate.
\end{defn}
Note also that when $D=0$, since $f_0\equiv1$, we have $\ds{\mathop{\mathrlap{\pushMI}}\!\int_0{d\mu}=1}$.
Therefore, this definition gives us a morphism of groups
$$
\xymatrix@R=.1pc{
{\mathbb{Z}}[\Omega_{\mathcal{L}}(L)]_0\ar[rr]^(.42){\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_\bullet {d}}}&&\displaystyle{\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0},{L}^*)}\\
D\ar@{|->}[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_D{d}:\mu\longmapsto \mathop{\mathrlap{\pushMI}}\!\int_{D}{d\mu}}
}
$$
\begin{lem}
Let $\Gamma\subset \PGL_2(K)$ be a subgroup acting on ${\mathcal{L}}$ and so on ${\mathcal{L}}^*$. Then, the map $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_\bullet{d}}$ is $\Gamma$-equivariant.
\end{lem}
\begin{proof}
We want to see that $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma\cdot D}{d}=\gamma\cdot\mathop{\mathrlap{\pushMI}}\!\int_D{d}}$ for any $\gamma\in\Gamma$. That is to say that for any $\gamma\in\Gamma$ and $\mu\in\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}$ we have
$$
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\gamma D}d\mu}=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma\cdot D}{d\mu}=\gamma\cdot\mathop{\mathrlap{\pushMI}}\!\int_D{d\mu}=\mathop{\mathrlap{\pushMI}}\!\int_D{d(\gamma^{-1}\mu)}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_Dd(\gamma^{-1}\mu)}
$$
Let us to compute the first integral:
$$
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\gamma D}d\mu}=
\lim_{\substack{\rightarrow\\\mc{C}_\alpha({\mathcal{L}}^*)}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha({\mathcal{L}}^*)\\t_n^\alpha\in\mc{U}_n^\alpha}}{f_{\gamma D}(t_n^\alpha)^{\mu(\mc{U}_n^\alpha)}}}}=
\lim_{\substack{\rightarrow\\\mc{C}_\alpha({\mathcal{L}}^*)}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha({\mathcal{L}}^*)\\t_n^\alpha\in\mc{U}_n^\alpha}}{(\gamma f_{D})(t_n^\alpha)^{\mu(\mc{U}_n^\alpha)}}}}=
$$
$$
=\lim_{\substack{\rightarrow\\\mc{C}_\alpha({\mathcal{L}}^*)}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha({\mathcal{L}}^*)\\t_n^\alpha\in\mc{U}_n^\alpha}}{f_D(\gamma^{-1} t_n^\alpha)^{\mu(\mc{U}_n^\alpha)}}}}=
\lim_{\substack{\rightarrow\\\mc{C}_\alpha({\mathcal{L}}^*)}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha({\mathcal{L}}^*)\\t_n^\alpha\in\mc{U}_n^\alpha}}{f_D(t_n^\alpha)^{\mu(\gamma\mc{U}_n^\alpha)}}}}=
$$
$$
=\lim_{\substack{\rightarrow\\\mc{C}_\alpha({\mathcal{L}}^*)}}{\ds{\prod_{\substack{\mc{U}_n^\alpha\in\mc{C}_\alpha({\mathcal{L}}^*)\\t_n^\alpha\in\mc{U}_n^\alpha}}{f_D(t_n^\alpha)^{(\gamma^{-1}\mu)(\mc{U}_n^\alpha)}}}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_Dd(\gamma^{-1}\mu)}
$$
Therefore we get the claimed compatibility of the action of $\Gamma$ with the map.
\end{proof}
\begin{defn} Given any degree 0 divisor $D=\sum_{i\in I} m_i \alpha_i$ with support in
${\mathcal{T}}_K({\mathcal{L}})$ (i.e. $m_i\in {\mathbb{Z}}$, $\alpha_i\in {\mathcal{T}}_K({\mathcal{L}})$, with $I$ a finite
set and $\sum_{i\in I}m_i=0$) consider the map up to scalars
$\ds{|f|_D\in\mathscr{M}aps({\mathcal{L}}^*, {\mathbb{R}}_{>0})/{\mathbb{R}}_{>0}^*}$ given by a representant $\prod_{i\in I}
\alpha_i(x)^{m_i}$. Let
$\mu\in\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}$ be a $\mb{Z}$-valued harmonic
measure on ${\mathcal{L}}^*$.\\
We define
$$\displaystyle{\left|\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}\right|_{D}{d\mu}:=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{|f|_D d\mu}\in {\mathbb{R}}_{>0}}$$
since, as above, the value of the integral only depends on $D$, and not on the representant of $|f|_D$, because of the harmonicity of the measure.
We will follow the same rule that above with respect to ${\mathcal{L}}^*$, omiting it when it is a given fixed set and specifying only in case of need:
$$
\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{D}{d\mu}:=\left|\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}\right|_{D}{d\mu}.
$$
\end{defn}
\begin{lem}\label{AVC} Let ${\mathcal{P}}$ be a finite set of points in $\Omega_{\mathcal{L}}(L)$,
and let $D:=\sum_{p\in {\mathcal{P}}} m_p p$ be a degree zero divisor. Denote by
$\alpha_D:=\sum_{p\in {\mathcal{P}}} m_p \alpha_p$, where $\alpha_p$ is the
seminorm associated to $p$. Then $|f_D|=|f|_{\alpha_{D}}$
in $\mathscr{M}aps({\mathcal{L}}^*, {\mathbb{R}}_{>0})/{\mathbb{R}}_{>0}^*$.
\end{lem}
\begin{proof} Take $q\in{\mathcal{L}}^*$ and representatives as in definition~\ref{FutS}. For the sake of simplicity we will assume all the points $p$ and $q$ are non infinite (then we can choose $v_q=(q,1)$ and $v_p=(1,-p)$).
$$
|f_D|(q)=|f_D(q)|=|\prod_{p\in {\mathcal{P}}} v_p(v_q)^{m_p}|=\prod_{p\in {\mathcal{P}}} {|q-p|^{m_p}}=|\prod_{p\in {\mathcal{P}}} {\alpha_p(q)}|=|f|_{\alpha_D}(q)
$$
by having into account for the fourth equality the remark~\ref{frem}.\end{proof}
This has as an immediate consequence the next result.
\begin{cor}\label{absI}
For any degree 0 divisor
$D=\sum_{i\in I} m_i p_i$ with support in $\Omega_{\mathcal{L}}(L)$, consider
the divisor $\red_{{\mathcal{L}}}(D):=\sum_{i\in I}
m_i \red_{{\mathcal{L}}}(p_i)$ on ${\mathcal{T}}_K({\mathcal{L}})$. Then, for any $\mb{Z}$-valued harmonic
measure $\mu\in\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}$ on ${\mathcal{L}}$, we have
$$\left|\mathop{\mathrlap{\pushMI}}\!\int_{D}{d\mu}\right|=\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\red_{\mathcal{L}}(D)}{d\mu}.$$
\end{cor}
\begin{proof}
Applying the previous lemma and proposition~\ref{frd} we obtain
$$
\left|\mathop{\mathrlap{\pushMI}}\!\int_{D}{d\mu}\right|=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{|f_D|d\mu}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{|f|_{\alpha_D}d\mu}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{|f|_{\red_{\mathcal{L}}(D)}d\mu}=\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\red_{\mathcal{L}}(D)}{d\mu}
$$
\end{proof}
\begin{lem} Given $x\in {\mathcal{L}}$, for any
two points $\alpha(x,r),\alpha(x,s)\in {\mathcal{T}}_K({\mathcal{L}})$, with $s>r$,
such that the path $P(\alpha(x,r),\alpha(x,s))$ is a topological edge, then
$$\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\alpha(x,s)-\alpha(x,r)}{d\mu}=\left(\frac sr\right)^{\mu(B(x^*,r)\cap{\mathcal{L}}^*)}$$
\end{lem}
\begin{proof} We have
$$
|f|_D(q)=\left\{\begin{array}{ll} \frac sr & \mbox{ if }
q\in B(x^*,r) \\
1 & \mbox{ if } q\not\in B(x^*,r)
\end{array} \right.
$$
and these are the only two possibilities. Hence
$\displaystyle{|f|_D(q)=\chi_{U,\frac sr}}$ for $U=B(x^*,r)$ in the
notation of Proposition~\ref{propertiesmultiplicativeintegral}.
Now, if we denote by $D=\alpha(x,s)-\alpha(x,r)$, and by applying
Proposition~\ref{propertiesmultiplicativeintegral}, we get
$$
\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{D}{d\mu}= \mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{|f|_D d\mu}=
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{\chi_{U,\frac sr} d\mu}=\left(\frac sr\right)^{\mu(B(x^*,r)\cap{\mathcal{L}}^*)}.
$$
\end{proof}
The following result generalizes \cite[Lem. 4.2]{Das05} to the case
that $K$ is any complete non-archimedean field (and not just a local
field).
\begin{lem}\label{logint}
For any $\alpha,\alpha'\in {\mathcal{T}}_K({\mathcal{L}})$ such that $P(\alpha,\alpha')$ is a topological edge, then
$$
\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{\alpha'-\alpha}{d\mu}\right)=-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\alpha'-\alpha}{d\mu}}=d(\alpha,\alpha')\mu({\mathcal{B}}(\alpha,\alpha'))
$$
where the first equality is by corollary~\ref{absI}.
\end{lem}
\begin{proof}
If $\alpha=\alpha(x,r),\ \alpha'=\alpha(x,s)$ are like in the
previous lemma , then the claim is immediate consequence of that
together with definition~\ref{open}. One only has to observe that
$B(x^*,r)\cap{\mathcal{L}}^*= \mathring{B}(x^*,s)\cap{\mathcal{L}}^*$, so
$$
\mu(B(x^*,r)\cap{\mathcal{L}}^*)=\mu(\mathring{B}(x^*,s)\cap{\mathcal{L}}^*)=-\mu({\mathcal{L}}^*\setminus \mathring{B}(x^*,s))=-\mu({\mathcal{B}}(\alpha,\alpha'))
$$
We get the identity in a similar way when $r>s$.
Otherwise
$$
-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\alpha'-\alpha}{d\mu}}=-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\alpha'\vee\alpha-\alpha}{d\mu}}-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\alpha'-\alpha\vee\alpha'}{d\mu}}=
$$
$$
=d(\alpha,\alpha'\vee\alpha)\mu({\mathcal{B}}(\alpha,\alpha'\vee\alpha))+d(\alpha\vee\alpha',\alpha')\mu({\mathcal{B}}(\alpha\vee\alpha',\alpha'))=
$$
$$
=d(\alpha,\alpha'\vee\alpha)\mu({\mathcal{B}}(\alpha,\alpha'))+d(\alpha\vee\alpha',\alpha')\mu({\mathcal{B}}(\alpha,\alpha'))=d(\alpha,\alpha')\mu({\mathcal{B}}(\alpha,\alpha'))
$$
since ${\mathcal{B}}(\alpha,\alpha')={\mathcal{B}}(\alpha,\alpha'\vee\alpha)={\mathcal{B}}(\alpha\vee\alpha',\alpha')$.
\end{proof}
We may show this result in a more expressive way writing the
topological edge as $\varepsilon$ and defining its boundary $\partial \varepsilon$
as the difference of its target minus its source -as usual in
homology theory (cf. below in section~\ref{tropical}).
Recall that by corollary~\ref{HMC} we have $\ds{\mathscr{M}(
{\mathcal{L}}^*,\mb{Z})_{0}\cong \HC({\mathcal{T}}_K({\mathcal{L}}),{\mathbb{Z}})}$ in such a way that to
each measure $\mu$ corresponds an harmonic cochain $c_\mu$ such that
${c_\mu(\varepsilon)=\mu({\mathcal{B}}(\varepsilon))}$. So, by abuse of notation we may write
$\mu(\varepsilon)=\mu({\mathcal{B}}(\varepsilon))$.
Therefore, we may write the lemma as
$$
\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{\partial\varepsilon}\right)=l(\varepsilon)\mu(\varepsilon).
$$
\section{The Poisson Formula}\label{poisson}
In this section we will show in our context the Poisson formula of
Longhi in \cite[Thm.~6]{Lon02}. To show this, we recall and study in
detail a map introduced by van der Put in \cite[Thm.~2.1]{vdP92}
assigning a mesure to any invertible analytic function.
Let ${\mathcal{L}}$ be a compact set with at least two points and consider the
abelian group of measures $\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}$, as in the
previous section. For any two different points $a,b\in{\mathcal{L}}^*$ we
define the measure $\mu_{a,b}$ by
$$
\mu_{a,b}({\mathcal{U}}):=\left\{\begin{array}{l}
1\mbox{ if }a\in{\mathcal{U}},\ b\not\in{\mathcal{U}}\\
-1\mbox{ if }b\in{\mathcal{U}},\ a\not\in{\mathcal{U}}\\
0\mbox{, otherwise}
\end{array}\right.
$$
In particular, on the open compact subsets ${\mathcal{B}}(e)\subset{\mathcal{L}}^*$, which determine the measure because of being a basis, we note that
$$
\mu_{a,b}(e):=\left\{\begin{array}{l}
1\mbox{ if }e\in P(b^*,a^*)\\
-1\mbox{ if }e\in P(a^*,b^*)\\
0\mbox{, otherwise}
\end{array}\right.
$$
For any $a,b\in{\mathcal{L}}^*$ we take representatives $\tilde{a},\tilde{b}\in K^2$ and for any complete extension $L|K$ we define
the function $\ds{\omega_{\tilde{a}-\tilde{b}}:\Omega_{\mathcal{L}}(L)\longrightarrow L^*}$ as $$\omega_{\tilde{a}-\tilde{b}}(z):=\frac{\tilde{a}(z)}{\tilde{b}(z)}=\frac{z(\tilde{a})}{z(\tilde{b})}.$$
Note that identifying $z$ with $(1,-z)$ or $(0,1)$ if it is $\infty$, this is an analytic function on $\Omega_{\mathcal{L}}(L)$ depending on $a,b$
up to a constant.
Let us write for any $p,q\in{\mathcal{L}}$, $u_{p,q}(z):=\omega_{\tilde{p}^*-\tilde{q}*}$ for suitable representants, so we can put
$$
u_{p,q}(z):=\frac{z-p}{z-q}
$$
where we consider the usual convention when some of the two points are $\infty$ (\cite[Ch.~2(2.2)]{GvdP80}), that is
$$u_{p,q}(z):=\left\{\begin{array}{l}
1\mbox{ if }p=q=\infty\\
z-p\mbox{ if }p\neq\infty=q\\
\ds{\frac{1}{z-q}}\mbox{ if }p=\infty\neq q
\end{array}\right.
$$
On the other hand, let us recall part of the definition~\ref{defint}. For any degree 0 divisor $D=\sum_{i\in I} m_i p_i$ with
support in $\Omega_{\mathcal{L}}(L)$ we could build as above a map up to scalars
$\ds{f_D\in\mathscr{M}aps({\mathcal{L}}^*, L^*)/L^*}$. Let us fix an element $b_0\in{\mathcal{L}}$. Along this section
we will choose a representant of $f_D$ satisfying $f_D(b_0)=1$, so $f_D$ will be a well defined function.
We write the usual notation $\mathcal{O}(\Omega_{\mathcal{L}})$ for the analytic functions on the analytic space $\Omega_{\mathcal{L}}:=({{\mathbb{P}}^1_{K}}^*)^{an}\setminus{\mathcal{L}}$, and we write $\mathcal{O}(\Omega_{\mathcal{L}})^*$ for the ones which vanish nowhere. Then we have $\ds{\omega_{\tilde{a}-\tilde{b}}\in\mathcal{O}(\Omega_{\mathcal{L}})^*}$.
Let $\varepsilon$ be a topological edge of ${\mathcal{T}}_K({\mathcal{L}})$ induced by a path
$P(\alpha(x,r),\alpha(x,s))$ with $x\in{\mathcal{L}}$ and $r\leq s$. Then we
define the (closed) annulus associated to $\varepsilon$ as
$R(\varepsilon):=R_x(r,s):=B(x,s)\setminus\mathring{B}(x,r)$, and the open
annulus associated to $\varepsilon$ as
$\mathring{R}(\varepsilon):=\mathring{R}_x(r,s):=\mathring{B}(x,s)\setminus
B(x,r)$.
The following result is shown by Thuiller in \cite[Lemme
2.2.1]{Thu05}.
\begin{lem}\label{PoiL}
Given $x\in {\mathcal{L}}$, and any two points $\alpha(x,r),\alpha(x,s)\in {\mathcal{T}}_K({\mathcal{L}})$, with $r<s$, such that the path $P(\alpha(x,r),\alpha(x,s))$ is a topological edge (i.e. $\mathring{R}_x(r,s)\cap{\mathcal{L}}=\emptyset$), for any $\omega\in\mathcal{O}(\Omega_{\mathcal{L}})^*$ there exists $k\in{\mathbb{Z}}$ such that for any interior path $P'=P(\alpha(x,r'),\alpha(x,s'))\subset P(\alpha(x,r),\alpha(x,s))$ ($r\leq r'\leq s'\leq s$) satisfying $R_x(r',s')\cap{\mathcal{L}}=\emptyset$, the function $|\omega(z)(z-x)^{-k}|$ is constant on $R_x(r',s')$.
\end{lem}
\begin{proof}
For any $0<r'\leq s'$ let us consider $R_x(r',s')^{an}$, the Berkovich analytic annulus associated to $R_x(r',s')$. Now we can assume without any problem that $x=0$. Then, we have the isomorphism
$$
\mathcal{O}(R_0(r',s')^{an})\cong K\langle r'T^{-1},s'^{-1}T\rangle
$$
where
$$
K\langle r'T^{-1},s'^{-1}T\rangle=
$$
$$
=\left\{\sum_{n=-\infty}^{\infty}{a_nT^n}:\ |a_n|r'^n\rightarrow0\mbox{ as }n\rightarrow-\infty,\ |a_n|s'^n\rightarrow0\mbox{ as }n\rightarrow\infty\right\}
$$
We will prove first the case $r'=s'=1$. We have $\omega\in\mathcal{O}(\Omega_{\mathcal{L}})^*$ and then the restriction of $\omega$ is a unit in $K\langle T^{-1},T\rangle$. Such an element can be expressed as $\omega=c\cdot\omega_1$ for $c\in K^*$ such that $\lVert\omega\rVert_{R_0(1,1)}=|c|$ and $\omega_1\in\mathcal{O}_K\langle T^{-1},T\rangle^*$. Therefore, the reduction of $\omega_1$ to $k[T^{-1},T]^*$ is also invertible so it is of the form $bT^n$ for $b\in k,\ n\in{\mathbb{Z}}$, and so we deduce that we can write $\omega_1=\tilde{b}T^n+\omega_2=\tilde{b}T^n(1+\omega'_2)$ with $\tilde{b}\in\mathcal{O}_K^*,\ \omega_2\in\mathfrak{m}_K\langle T^{-1},T\rangle,\ \omega'_2=\tilde{b}^{-1}T^{-n}\omega_2$ and $\lVert\omega'_2\rVert_{R_0(1,1)}=\lVert\omega_2\rVert_{R_0(1,1)}<1$, so $|\omega(z) z^{-n}|=|c\tilde{b}||1+\omega'_2(z)|=|c\tilde{b}|$.
Observe that writing $\omega=\sum_{n\in{\mathbb{Z}}}{a_nT^n}$ the supremum norm can be expressed by $\lVert\omega\rVert_{R_0(1,1)}:=\max\{|a_m|\}$ and this is reached at just one $m$, which is $n$.
From now on we consider the case $r'<s'$.
Now $\omega$ is a unit $\sum_{n\in{\mathbb{Z}}}{a_nT^n}$ in $K\langle r'T^{-1},s'^{-1}T\rangle$, so for any $r''\in[r',s']$, the image of $\omega$ by the restriction homomorphism $\ds{K\langle r'T^{-1}s'^{-1},T\rangle\longrightarrow K\langle r''T^{-1},r''^{-1}T\rangle}$ is also a unit. Next note that after a non archimedean extension $K'|K$ we have $r''\in|K'^*|$ so there is an isomorphism $\ds{K'\langle r''T^{-1},r''^{-1}T\rangle\cong K'\langle T^{-1},T\rangle}$.
\end{proof}
We say that a sequence of functions $(\omega_n)_n$ in $\mathcal{O}(\Omega_{\mathcal{L}})^*$ converge uniformly to a function $\omega\in\mathcal{O}(\Omega_{\mathcal{L}})^*$ if for each edge $e$ of ${\mathcal{T}}_K({\mathcal{L}})$ and for all $\epsilon>0$ there exists an $n_0=n(e,\epsilon)$ such that for any $N\geq n_0$ we have $\ds{\lVert\omega-\omega_N\rVert_{R(|e|)}<\epsilon}$ (recall that $|e|$ means the topological realization of $e$).
We will write $\displaystyle{\lim_{N\to\infty}{\omega_N}=\omega}$.
\begin{thm}\label{FMeas}
There exists a morphism $\displaystyle{\tilde{\mu}:\mathcal{O}(\Omega_{\mathcal{L}})^*\longrightarrow\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}}$ with kernel $K^*$ and such that commutes with limits in the following sense: if $\displaystyle{\lim_{N\to\infty}{\omega_N}=\omega}$, then $\displaystyle{\tilde{\mu}(\omega)=\lim_{N\rightarrow\infty}{\tilde{\mu}(\omega_N)}}$.
\end{thm}
\begin{proof}
Let us consider $\omega\in\mathcal{O}(\Omega_{\mathcal{L}})^*$. We have to
define $\tilde{\mu}(\omega)$ over each (oriented) edge $e$ of a
model of ${\mathcal{T}}_K({\mathcal{L}})$. By lemma~\ref{oprops} we may assume that
$|e|$ or $|\overline{e}|$ is contained in a topological edge given
by $P(\alpha(x,r),\alpha(x,s))$ with $r<s$ and $x\in{\mathcal{L}}\cap K$.
Depending on if this happens with $e$ or $\overline{e}$, we define
$\tilde{\mu}(\omega)(e):=k$ or $\tilde{\mu}(\omega)(e):=-k$
respectively, where $k$ is the integer obtained in the above lemma.
Henceforth we will work on this edge to prove its properties.
First, $\tilde{\mu}(\omega)$ is a measure because of the definition and the residue theorem (\cite[Thm.~2.3.3 (2)]{FvdP04}).
Form the way we have defined the map $\tilde{\mu}$ it is clear that
it is a morphism and that $K^*$ is inside its kernel. From the
definition of $\tilde{\mu}$, the fact that $\Omega_{\mathcal{L}}$ is connected
implies that if $\tilde{\mu}(\omega)=0$, then the absolute value of
$\omega$ is a constant, and since bounded analytic functions on
$\Omega_{\mathcal{L}}$ are constant (\cite[Ch.~4~Cor.~(2.5)]{GvdP80}), we get
$\Ker(\tilde{\mu})=K^*$.
And now let us see the commutativity with limits in the sense we told. We want to check the equality $\ds{\tilde{\mu}(\omega)(e)=\lim_{N\rightarrow\infty}{\tilde{\mu}(\omega_N)}(e)}$ for any edge $e$ that we can take as above.
We know by hypothesis that for any $\epsilon>0$ there exists an $n_0=n(e,\epsilon)$ such that for any $N\geq n_0$ we have $\lVert\omega-\omega_N\rVert_{R_x(r,s)}<\epsilon$. Note that if we just take $\epsilon=\inf_{z\in R_x(r,s)}{\{|\omega(z)|\}}$, which is strictly positive since $R_x(r,s)$ is compact, then for any $z\in R_x(r,s)$ we get $|\omega(z)-\omega_N(z)|<|\omega(z)|$ and so $\lvert \omega_N(z)\rvert=\lvert \omega(z)\rvert$, therefore $\tilde{\mu}(\omega_N)(e)=\tilde{\mu}(\omega)(e)$.
\end{proof}
\begin{prop}\label{FMeas2}
The morphism $\displaystyle{\tilde{\mu}:\mathcal{O}(\Omega_{\mathcal{L}})^*\longrightarrow\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}}$ satisfies the following properties:
\begin{enumerate}
\item For any two different points $a,b\in{\mathcal{L}}^*$,
$$\displaystyle{\tilde{\mu}\left(\omega_{\tilde{a}-\tilde{b}}\right)=\mu_{b,a}}$$ independently of the chosen representants of $a$ and $b$. In particular, for any $p,q\in{\mathcal{L}}$ we have $\displaystyle{\tilde{\mu}\left(u_{p,q}\right)=\mu_{q^*,p^*}}$.
\item It is natural in the sense that if ${\mathcal{L}}\subset{\mathcal{L}}'$ are both compacts, it commutes with restriction maps:
$$
\xymatrix{
\mathcal{O}(\Omega_{\mathcal{L}})^*\ar[rr]^{\displaystyle{\tilde{\mu}}}\ar[dd]&&\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}\ar[dd]\\
&&\\
\mathcal{O}(\Omega_{{\mathcal{L}}'})^*\ar[rr]^{\displaystyle{\tilde{\mu}}}&&\mathscr{M}({\mathcal{L}}'^*,\mb{Z})_{0}
}
$$
In particular it does not depend on ${\mathcal{L}}$, since given any compacts ${\mathcal{L}}_1,\ {\mathcal{L}}_2$, the definition coincides in ${\mathcal{L}}_1\cap{\mathcal{L}}_2$.
\item It commutes with the action of $\PGL_2(K)$, that is, for each $\displaystyle{\gamma\in \PGL_2(K)}$ the diagram
$$
\xymatrix{
\mathcal{O}(\Omega_{\mathcal{L}})^*\ar[rr]^{\displaystyle{\tilde{\mu}}}\ar[dd]^{\displaystyle{\gamma_*}}&&\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}\ar[dd]^{\displaystyle{\gamma_*}}\\
&&\\
\mathcal{O}(\Omega_{\gamma{\mathcal{L}}})^*\ar[rr]^{\displaystyle{\tilde{\mu}}}&&\mathscr{M}(\gamma{\mathcal{L}}^*,\mb{Z})_{0}
}
$$
is commutative, where $\gamma_*(\omega)=\gamma\cdot\omega$ and $\gamma_*(\mu)=\gamma\cdot\mu$. (Note that $\Omega_{\gamma{\mathcal{L}}}=\gamma\Omega_{{\mathcal{L}}}$ and $\gamma\cdot({\mathcal{L}}^*)=(\gamma\cdot{\mathcal{L}})^*$.)
\end{enumerate}
\end{prop}
\begin{proof}
First, we want to see $\displaystyle{\tilde{\mu}\left(\omega_{\tilde{a}-\tilde{b}}\right)(e)=\mu_{b,a}(e)}$.
If $a,b\in{\mathcal{B}}(e)={\mathcal{L}}^*\setminus \mathring{B}(x^*,s)$, for $z\in R_x(r,s)$ we have
$$|\omega_{\tilde{a}-\tilde{b}}(z)|=\left|\frac{z(\tilde{a})}{z(\tilde{b})}\right|=\frac{|x-a^*|}{|x-b^*|}
$$
(taking into account the above convention if $a$ or $b$ are $\infty$), which is a constant, so ${\tilde{\mu}(\omega_{\tilde{a}-\tilde{b}})(e)=0=\mu_{b,a}(e)}$.
If $a,b\in{\mathcal{B}}(\overline{e})={\mathcal{L}}^*\cap B(x^*,r)$, $z\in R_x(r,s)$ satisfies $|z(\tilde{a})|=|z-a^*|=|z-x|=|z-b^*|=|z(\tilde{b})|$, so we also get a constant ($|\omega_{\tilde{a}-\tilde{b}|R_x(r,s)}|\equiv 1$) and the equality as above.
Finally, assuming $a\in{\mathcal{B}}(e)={\mathcal{L}}^*\setminus \mathring{B}(x^*,s),\ b\in{\mathcal{B}}(\overline{e})={\mathcal{L}}^*\cap B(x^*,r)$, then
$$
|\omega_{\tilde{a}-\tilde{b}}(z)|=\left|\frac{z(\tilde{a})}{z(\tilde{b})}\right|=\frac{|x-a^*|}{|z-b^*|}=\frac{|x-a^*|}{|x-z|}\frac{|x-z|}{|z-b^*|}=|x-a^*|\cdot|z(x^*)|^{-1},
$$
therefore $\ds{\tilde{\mu}(\omega_{\tilde{a}-\tilde{b}})(e)=-1=\mu_{b,a}(e)}$
(once more, one should consider the case in which $a$ is $\infty$,
but we would get a similar result).
Second, the naturality is a direct consequence of the definition of the $\tilde{\mu}$ through the above lemma.
The third property is equivalent to say
$\gamma\cdot\tilde{\mu}(\omega)(e)=\tilde{\mu}(\gamma\cdot\omega)(e)$
for all $\omega\in{\mathcal{O}}(\Omega_{\mathcal{L}})^*$ and
$e\in{\mathcal{T}}_K(\gamma\cdot{\mathcal{L}})$, and the left side of the equality is
$\tilde{\mu}(\omega)(\gamma^{-1}\cdot e)$. Then, this also follows
from the definition by means of the lemma and from the isomorphism
$\displaystyle{\gamma^*:{\mathcal{O}}(R(|e|))\stackrel{\cong}\longrightarrow{\mathcal{O}}(R(|\gamma^{-1}e|))}$,
by which $\gamma^*(\omega)=\gamma^{-1}\cdot\omega$.
\end{proof}
As Longhi remarks (\cite{Lon02}), we may compute a multiplicative
integral on ${\mathcal{L}}^*$ by means of fixing a vertex $v_0\in{\mathcal{T}}_K({\mathcal{L}})$
and defining $l_{v_0}(e)$ as the number of intermediate vertices
between $v_0$ and $e$ in a previously fixed model for ${\mathcal{T}}_K({\mathcal{L}})$.
Then we have
$$
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{fd\mu}=\lim_{n\rightarrow\infty}{\ds{\prod_{\substack{l_{v_0}(e)=n\\t_e\in{\mathcal{B}}(e)}}{f(t_e)^{\mu(e)}}}}
$$
\begin{thm}[Poisson Formula]\label{PF}
Let $u\in\mathcal{O}(\Omega_{\mathcal{L}})^*$ and $z_0\in\Omega_{\mathcal{L}}$. Then, for any $z\in\Omega_{\mathcal{L}}$ the next identity is satisfied:
$$
\frac{u(z)}{u(z_0)}=\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\tilde{\mu}(u)}
$$
\end{thm}
\begin{proof}
We follow the proof of \cite[Thm.~6]{Lon02}.
The partial products
$$
\prod_{\substack{l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)}}{f_{z-z_0}\left(t_e\right)^{\tilde{\mu}(u)(e)}}
$$
converge uniformly on $\Omega_{\mathcal{L}}$ so the integral built with them is a nowhere vanishing analytic function of $z$.
Since by the previous theorem the kernel of $\tilde{\mu}$ is $\ds{K^*}$, in order to prove the identity it is enough to see that $\ds{\tilde{\mu}(u(z))=\tilde{\mu}\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{z-z_0}(t)d\tilde{\mu}(u)(t)}\right)}$. Further, note that
$$
f_{z-z_0}(t_e)=f_{z-z_0}(t_e)/f_{z-z_0}(b_0)=\frac{\tilde{z}(\tilde{t_e})}{\tilde{z_0}(\tilde{t_e})}\frac{\tilde{z_0}(\tilde{b_0})}{\tilde{z}(\tilde{b_0})}=\frac{\tilde{z}(\tilde{t_e})}{\tilde{z}(\tilde{b_0})}\frac{\tilde{z_0}(\tilde{b_0})}{\tilde{z_0}(\tilde{t_e})}=c\cdot\omega_{\tilde{t_e}-\tilde{b_0}}(z),
$$
$$
c\in {K(\tilde{z_0})}^*
$$
Therefore we have $\ds{\tilde{\mu}(f_{z-z_0}(t_e))=\mu_{b_0,t_e}}$ also by the previous theorem. Then, by the commutativity of $\tilde{\mu}$ and limits we obtain
$$
\tilde{\mu}\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{z-z_0}(t)d\tilde{\mu}(u)(t)}\right)=\tilde{\mu}\Big(\lim_{N\rightarrow\infty}{\ds{\prod_{\substack{\l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)}}{f_{z-z_0}\left(t_e\right)^{\tilde{\mu}(u)(e)}}}}\Big)=
$$
$$
=\lim_{N\rightarrow\infty}{\ds{\sum_{\substack{\l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)}}{\tilde{\mu}(u)(e) \tilde{\mu}\left(f_{z-z_0}(t_e)\right)}}}=\lim_{N\rightarrow\infty}{\ds{\sum_{\substack{\l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)}}{\tilde{\mu}(u)(e)\mu_{b_0,t_e}}}}
$$
Let us evaluate on an edge $e'$ of a fixed model of ${\mathcal{T}}_K({\mathcal{L}})$. We
may assume $e'$ points away from $b_0$, so
$b_0\in{\mathcal{B}}(\overline{e'})$. We have $e'\in P(b_0^*,t_e^*)$ if and
only if $t_e\in{\mathcal{B}}(e')$, so we get
$$
\tilde{\mu}\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{z-z_0}(t_e)d\tilde{\mu}(u)(t)}\right)(e')=\lim_{N\rightarrow\infty}{\ds{\sum_{\substack{l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)}}{\tilde{\mu}(u)(e)
\mu_{b_0,t_e}(e')}}}=
$$
$$
=\lim_{N\rightarrow\infty}{\ds{\sum_{\substack{\l_{v_0}(e)=N\\t_e\in{\mathcal{B}}(e)\cap{\mathcal{B}}(e')}}{-\tilde{\mu}(u)(e)}}}=\tilde{\mu}(u)(e')
$$
where the last equality is due to harmonicity applied to the sum
independent of $N\geq l_{v_0}(e')$. \end{proof}
\begin{cor}[Extended Poisson Formula]\label{EPF}
Take $u\in\mathcal{O}(\Omega_{\mathcal{L}})^*$. Then, given any degree 0
divisor $D=\sum{m_p p}$ of $\Omega_{\mathcal{L}}$, we have
$$
\prod_{p\in\Supp(D)}{u(p)^{m_p}}=\mathop{\mathrlap{\pushMI}}\!\int_{D}{d\tilde{\mu}(u)}
$$
\end{cor}
\begin{cor}\label{exh}
The morphism
${\tilde{\mu}:{\mathcal{O}}(\Omega_{\mathcal{L}})^*\longrightarrow{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0}$ is
surjective and for each $z_0\in\Omega_{\mathcal{L}}$ it has a section
${{\mathcal{I}}_{z_0}:{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0\longrightarrow{\mathcal{O}}(\Omega_{\mathcal{L}})^*}$. As a
consequence we get a (non-unique, non-canonical) isomorphism
${{\mathcal{O}}(\Omega_{\mathcal{L}})^*\cong K^*\times{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0}$.
\end{cor}
\begin{proof}
Let us take an harmonic measure $\mu\in{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0$. Let $z_0\in\Omega_{\mathcal{L}}$ be any point. Then, as along the proof of the Poisson formula, we see that the function
$$
{\mathcal{I}}_{\mu,z_0}(z):=\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\mu}
$$
is analytic on $\Omega_{\mathcal{L}}$, and once more, the same steps with $\mu$ instead of $\tilde{\mu}(u)$ prove that ${\tilde{\mu}({\mathcal{I}}_{\mu,z_0})=\mu}$. Then, we define the section by ${\mathcal{I}}_{z_0}(\mu):={\mathcal{I}}_{\mu,z_0}$ and we chech that it is a morphism of groups:
$$
{\mathcal{I}}_{z_0}(\mu+\mu')(z)=\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d(\mu+\mu')}=\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\mu}\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\mu'}=\left({\mathcal{I}}_{z_0}(\mu){\mathcal{I}}_{z_0}(\mu')\right)(z)
$$
Finally, by theorem~\ref{FMeas} we got the short exact sequence
$$
0\longrightarrow K^*\longrightarrow{\mathcal{O}}(\Omega_{\mathcal{L}})^*\longrightarrow{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0\longrightarrow0
$$
which, with the section morphism, gives the asserted isomorphism by elementary homological algebra.
\end{proof}
\section{Schottky Groups and their limit sets}\label{schottky}
Along this section we recall Schottky groups and their main properties, and we build the Mumford curve for which we want to give its Jacobian, and its associated graph. The main novelty is the ``Berkovich analytification'' of some results in \cite{GvdP80}.
Given any $\gamma\in \PGL_2(K)$, we say that $\gamma$ is hyperbolic
if the (two) eigenvalues of $\gamma$ have two distinct absolute
values. Note that in this case, due to the completeness of $K$, the
eigenvalues are in $K$. Hence a $\gamma\in\PGL_2(K)$ is hyperbolic
if and only if it is conjugated to an element of $\PGL_2({\mathcal{O}}_K)$
represented by a matrix
$\left(\begin{array}{cc}q&0\\0&1\end{array}\right)$ with $q\in K$,
$|q|<1$ (look at \cite[Ch.~1~Lem.~I.1.4]{GvdP80}). From this we get
that if $\gamma$ is hyperbolic,
$$
\{x\in{{\mathbb{P}}^1}^*({\mathbb{C}}_K)|\ \gamma x=x\}\subset{{\mathbb{P}}^1}^*(K).
$$
Given any subgroup $\Gamma\subset \PGL_2(K)$, we denote by
${\mathcal{L}}_{\Gamma}$ the set of limit points of $\Gamma$ in the dual projective line, i.e. the set of
points $x\in {{\mathbb{P}}^1}^*({\mathbb{C}}_K)$ such that there exists an infinite set
$\{\gamma_n\}_{n\in {\mathbb{Z}}_{\ge0}}\subset \Gamma$ and $y\in {{\mathbb{P}}^1}^*({\mathbb{C}}_K)$
with $\lim_{n\to \infty} \gamma_n\cdot y=x$. Observe that this set is
closed, and it contains the set $\displaystyle{\Sigma_{\Gamma^{*}}}$ of the points
$x\in {{\mathbb{P}}^1}^*({\mathbb{C}}_K)$ such that there exists $\gamma\in \Gamma$, not of
finite order, satisfying $\gamma\cdot x=x$ (since $x=\lim_{n\to \infty}
\gamma^n\cdot x$). Observe also that $\Gamma$ acts on ${\mathcal{L}}_{\Gamma}$.
Recall that a subgroup $\Gamma\subset \PGL_2(K)$ is discontinuous if
the set of limit points ${\mathcal{L}}_{\Gamma} \ne {{\mathbb{P}}^1}^*({\mathbb{C}}_K)$, and for any $p\in
{{\mathbb{P}}^1}^*({\mathbb{C}}_K)$, the closure of the orbit $\overline{\Gamma p}$ is
compact.
A subgroup $\Gamma\subset \PGL_2(K)$ is a Schottky group if it is
discontinuous, torsion free (so all its elements
$\gamma\neq1_\Gamma$ are hyperbolics) and finitely generated. Then
$\Gamma$ is a free group of finite rank $g(\Gamma)$.
The following lemma is well known, but we didn't find an explicit
reference.
\begin{lem} Let $\Gamma$ be a Schottky group. Then
\begin{enumerate}
\item If $g(\Gamma)=1$, so $\Gamma=<\gamma>\cong {\mathbb{Z}}$, then
${\mathcal{L}}_{\Gamma}=\{y_0,y_1\}=\Sigma_{\Gamma^*}$.
\item If $g(\Gamma)>1$, so $\Gamma$ is not abelian, then
${\mathcal{L}}_{\Gamma}$ is compact, perfect (without isolated points),
${\mathcal{L}}_{\Gamma}=\overline{\Gamma P}$ if $P\in {\mathcal{L}}_{\Gamma}$ and
${\mathcal{L}}_{\Gamma}=\overline{\Gamma P}\setminus \Gamma P$ if $P\not\in
{\mathcal{L}}_{\Gamma}$.
\item The set ${\mathcal{L}}_\Gamma$ always has at least two points.
\item In any case, ${\mathcal{L}}_{\Gamma}=\overline{\Sigma_{\Gamma^*}}$.
\end{enumerate}
\end{lem}
\begin{proof}
The three first claims are proved at paragraphs 1.5 and 1.6 of \cite[Ch.~1]{GvdP80}. Then we have $\overline{\Sigma_{\Gamma^*}}\subset{\mathcal{L}}_\Gamma=\overline{\Gamma P}$ for any $P\in\Sigma_{\Gamma^*}$. But the set of fixed points it is clearly $\Gamma$-invariant, since if we have $\gamma_xx=x$ then $(\gamma\gamma_x\gamma^{-1})\gamma x=\gamma x$ for any $\gamma\in\Gamma$, so ${\mathcal{L}}_\Gamma=\overline{\Gamma P}\subset\overline{\Sigma_{\Gamma^*}}$.
\end{proof}
We will need the two notions we will recall now. A half-line in a tree ${\mathcal{T}}$ is an infinite subtree whose topological realization is homeomorphic to $[0,+\infty)$. Two half-lines are equivalent if they differ in a finite subgraph of the union. An end of ${\mathcal{T}}$ is an equivalence class of half-lines.
\begin{lem}\label{quoGraph}
A Schottky group $\Gamma$ acts freely on ${\mathcal{T}}_K({\mathcal{L}}_{\Gamma})$ (with the induced left action by $\PGL_2(K)$ on ${\mathcal{T}}_K$),
and the quotient $G_{\Gamma}:=\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}}_{\Gamma})$ is a
finite metric graph.
Moreover, if ${\mathcal{L}}'\subset {{\mathbb{P}}^1}^*(K)$ is the union of ${\mathcal{L}}_\Gamma$ and a finite set of orbits of points by the action of $\Gamma$, then there exists a finite connected graph $G_{{\mathcal{L}}'}$ such that
$$
G_\Gamma\subset G_{{\mathcal{L}}'}\subset\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}}')\text{ and }(\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}}'))\setminus G_{{\mathcal{L}}'} = \bigsqcup_{\mathcal{R}_{{\mathcal{L}}'}}{(0,+\infty)}
$$
where $\mathcal{R}_{{\mathcal{L}}'}=\Gamma\backslash({\mathcal{L}}'\setminus{\mathcal{L}}_\Gamma)$ is a finite set.
\end{lem}
\begin{proof}
The fact that $\Gamma$ acts freely on ${\mathcal{T}}_K({\mathcal{L}}_{\Gamma})$ is a
consequence of all its non-neutral elements are hyperbolic with two
fixed points at the ends of the tree.
For the rest of the proof, we
are inspired by the proof given in \cite[Ch.~1~Lem.~(3.2)]{GvdP80}.
Let $B_\Gamma$ be a finite set of generators of $\Gamma$ and their
inverses containing the identity $1_\Gamma$ too. Take $w\in
{\mathcal{T}}_K({\mathcal{L}}_\Gamma)$ and a finite subtree
$\mathfrak{T}_w\subset{\mathcal{T}}_K({\mathcal{L}}_\Gamma)$ containing $B_\Gamma\cdot
w$. Then,
$$
\mathfrak{T}=\bigcup_{\gamma\in\Gamma}{\gamma\cdot\mathfrak{T}_w}
$$
is a subtree of ${\mathcal{T}}_K({\mathcal{L}}_\Gamma)$. The only thing we have to
verify is that it is connected, that is given
$\gamma,\gamma'\in\Gamma$ and $p\in\gamma\cdot\mathfrak{T}_w$,
$p'\in\gamma'\cdot\mathfrak{T}_w$ there exists a path in
$\mathfrak{T}$ between $p$ and $p'$. Through operating by $\gamma'$
on the path, we may suppose $\gamma'=1_\Gamma$. Also, by an
induction process it is enough to show this when $\gamma\in
B_\Gamma$. So, with these hypotheses, we have $p'$ and $\gamma w$
connected by a path in $\mathfrak{T}_w$, and $\gamma w$ and $p$
connected by a path in $\gamma\cdot \mathfrak{T}_w$.
Now we will show $\mathfrak{T}={\mathcal{T}}_K({\mathcal{L}}_\Gamma)$, from what we will
get consequently the finiteness of the quotient.
Let $v$ be any vertex of ${\mathcal{T}}_K({\mathcal{L}}_\Gamma)$ and consider a
half-line through $v$ starting at $w$, whose end corresponds to a
limit point $z\in{\mathcal{L}}_\Gamma$ since ${\mathcal{L}}_\Gamma$ is compact.
Therefore, there exists a sequence
$\{\gamma_n\}_{n\in{\mathbb{N}}}\subset\Gamma$ with $\gamma_0=1_\Gamma$
such that for any $z_0\in{{\mathbb{P}}_K^1}^*\setminus{\mathcal{L}}_\Gamma$,
$\lim_{n\to\infty}{\gamma_nz_0}=z$ (we may assume that the half-line
considered has as end a fixed point for some $\gamma\in\Gamma$ and
take the sequence of powers of $\gamma$ or $\gamma^{-1}$). Then the
fragments $P(\gamma_n w,\gamma_{n+1}w)$ belong to $\mathfrak{T}$,
and they form the unique half-line starting at $w$ in the direction
$z$, so $v\in\mathfrak{T}$.
For the second part, recall that
${\mathcal{T}}_K({\mathcal{L}}_\Gamma)\subset{\mathcal{T}}_K({\mathcal{L}}')$ and that we have the
retraction map
$${\red_{{\mathcal{L}}_\Gamma}:\Omega_{{\mathcal{L}}_\Gamma}\longrightarrow{\mathcal{T}}_K({\mathcal{L}}_\Gamma)}.$$
Choose a $p\in\Omega_{{\mathcal{L}}_\Gamma}$ such that $\Gamma\cdot p$ is one
of the orbits added to ${\mathcal{L}}_\Gamma$ to form ${\mathcal{L}}'$. Take the open
path $L_p:=\mathring{P}(\red_{{\mathcal{L}}_\Gamma}(p),p)$ and then observe
that $L_p\cap{\mathcal{T}}_K({\mathcal{L}}_\Gamma)=\emptyset$. Now it is clear that
$$
\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}}') = G_\Gamma \bigsqcup\left(\bigcup_{\pi_\Gamma(p)\in \mathcal{R}_{{\mathcal{L}}'}}{\pi_\Gamma(L_p)}\right)
$$
but the $\pi_\Gamma(L_p)$ have not to be disjoint. Nevertheless,
note that for any $\gamma\in\Gamma\setminus\{1_\Gamma\}$ the
intersection $L_{\gamma p}\cap L_p$ is empty, since otherwise,
$\red_{{\mathcal{L}}_\Gamma}(p)$ would be a fixed vertex for $\gamma$, which
contradicts the first claim of the result. Take now another
$q\in\Omega_{{\mathcal{L}}_\Gamma}$ such that
$\pi_\Gamma(q)\in\mathcal{R}_{{\mathcal{L}}'}$ and
$\pi_\Gamma(q)\neq\pi_\Gamma(p)$. It may happen that for some
$\gamma\in\Gamma$ (by the previous consideration, for at most one
$\gamma$) we have $L_p\cap L_{\gamma q}\neq\emptyset$. In that case,
in which $\red_{{\mathcal{L}}_\Gamma}(p)=\red_{{\mathcal{L}}_\Gamma}(\gamma q)$, let
$v_{pq}$ be the vertex of valence 3 in the tree $L_p\cup L_{\gamma
q}$. Next, let $v_p$ be one vertex of $L_p$ such that all the
possible $v_{pq}$ with $\pi_\Gamma(q)\in\mathcal{R}_{{\mathcal{L}}'}$ are in
the path $P(\red_{{\mathcal{L}}_\Gamma}(p),v_p)$. Finally take
$$
G_{{\mathcal{L}}'}:=\Gamma\backslash\left({\mathcal{T}}_K({\mathcal{L}}_\Gamma)\bigcup_{\pi_\Gamma(p)\in\mathcal{R}_{{\mathcal{L}}'}}{P(\red_{{\mathcal{L}}_\Gamma}(p),v_p)}\right)
$$
and the claim is immediate.
\end{proof}
\begin{thm} Let $\Gamma$ be a Schottky group and consider ${\mathcal{L}}:={\mathcal{L}}_{\Gamma}$
and $\Omega:=\Omega_{\mathcal{L}}=({{\mathbb{P}}^1}^*)^{an}\setminus {\mathcal{L}}$. Then $\Gamma$
acts on $\Omega$ and $C_{\Gamma}:=\Gamma\backslash \Omega$ is a proper
analytic space and so it is isomorphic to the analytification of a smooth
projective algebraic curve of genus $g(\Gamma)$.
\end{thm}
\begin{proof}
You can see the proof with more detail in \cite[Ch.~2~and~3]{GvdP80}. Here, we will sketch it.
We will suppose that $G_{\Gamma}$ has a model without loops. This is
possible after a finite extension of the base field, if necessary.
The general case can be done by means of Galois descent.
We consider the projection $\pi_\Gamma:{\mathcal{T}}_K({\mathcal{L}})\longrightarrow G_\Gamma$ and
a metric graph model for ${\mathcal{T}}_K({\mathcal{L}})$ given by a pair of sets
($V,E)$. The collection of vertices $V$ is formed by points of the
form $t(x_0,x_1,x_2)$ for $x_0,x_1,x_2 \in {\mathbb{P}}^1(K)$ such that it
includes all the points of valency greater than $2$, it is
$\Gamma$-invariant and the metric graph model for $G_\Gamma$ given
by $\pi_\Gamma(V)$ has no loops. Recall that the set of open edges
for the model of ${\mathcal{T}}_K({\mathcal{L}})$ is the set of connected components of
${\mathcal{T}}_K({\mathcal{L}})\setminus V$, and the edges are obtained from the open
ones adjoining the adherent vertices. We will denote this set by
$E$.
Consider now the restriction to $\Omega_{\mathcal{L}}$ of the retraction map,
that is ${\red_{\mathcal{L}}:\Omega_{\mathcal{L}}\longrightarrow{\mathcal{T}}_K({\mathcal{L}})}$. To each
$e\in E$, we take $U(e):=\red_{{\mathcal{L}}}^{-1}(e)$, and, similarly, to a
vertex $v\in V $ we take $U(v):=\red_{{\mathcal{L}}}^{-1}(v)$. Then, the sets
$U(e)$ and $U(v)$ are strictly affinoid and from them we get back
$\Omega$ by gluing $U(e)$ with $U(e')$ through $U(v)$ when the edges
$e,e'$ have $v$ as a common vertex.
Since the retraction map $\red_{{\mathcal{L}}}$ is $\Gamma$-equivariant, given two
edges $e , e'\in E$ such that $\pi_\Gamma(e)=\pi_\Gamma(e')$ so
there exists $\gamma\in \Gamma$ such that $\gamma\cdot e=e'$, then
$\gamma\cdot U(e)=U(e')$, and similarly for vertices. Therefore, gluing
as before but taking into account these identifications, or what is
the same, gluing according to the graph $G_\Gamma$ we get the
analytic space $C_\Gamma$, which is reduced and separated.
To prove that $C_\Gamma$ is proper we are going to show that it is compact and
its boundary (over $K$) is empty (\cite[Def.~4.2.13.~(ii)]{Tem15}).
The compactness is because we can express $C_\Gamma$ as a finite
union of affinoids: the preimages of the stars of the vertices of
$G_\Gamma$, which is a finite set.
To show that the boundary is empty, take any $x\in C_\Gamma$. We
want to show there exists $x\in U$ affinoid such that
$x\not\in\partial U$. Consider the image of $x$ by the induced
retraction map in the quotients,
$$
\red_{{\mathcal{L}},\Gamma}:C_\Gamma\longrightarrow G_\Gamma.
$$
Now, $\red_{{\mathcal{L}},\Gamma}(x)$ is an interior point of a $\Star(v)$ for
some vertex $v$ in the fixed model of $G_\Gamma$ (if
$\red_{{\mathcal{L}},\Gamma}(x)$ is a vertex we take $v=\red_{{\mathcal{L}},\Gamma}(x)$;
otherwise $v$ is any vertex of the edge to which
$\red_{{\mathcal{L}},\Gamma}(x)$ belongs). Then,
$\red_{{\mathcal{L}},\Gamma}^{-1}(\Star(v))$ is the affinoid we are looking
for.
Consider the following commutative diagram:
$$
\xymatrix{
\Omega\ar^{\displaystyle{\red_{\mathcal{L}}}}[rr]\ar^{\displaystyle{\pi_\Gamma}}[dd]&&{\mathcal{T}}_K({\mathcal{L}})\ar^{\displaystyle{\pi_\Gamma}}[dd]\\
&&\\
C_\Gamma\ar^{\displaystyle{\red_{{\mathcal{L}},\Gamma}}}[rr]&& G_\Gamma
}
$$
Choose a vertex $\tilde{v}$ in ${\mathcal{T}}_K({\mathcal{L}})$ such that $\pi_\Gamma(\tilde{v})=v$. Then $\pi_\Gamma$ gives an isomorphism
$$
\pi_\Gamma:\Star(\tilde{v})\stackrel{\sim}\longrightarrow \Star(v)
$$
since there are no loops in $G_\Gamma$ and the action of $\Gamma$ in ${\mathcal{T}}_K({\mathcal{L}})$ is free. It is clear that
$$
\red_{\mathcal{L}}^{-1}(\Star(\tilde{v}))=\bigcup_{\tilde{v}=s(e)} U(e)
$$
and hence, by construction of $C_\Gamma$, $\pi_\Gamma$ also induces an isomorphism
$$
\pi_\Gamma:\red_{\mathcal{L}}^{-1}(\Star(\tilde{v}))\stackrel{\sim}\longrightarrow \red_{{\mathcal{L}},\Gamma}^{-1}(\Star(v))
$$
Now recall that $\partial U(e)=\{s(e),t(e)\}\subset U(e)$, since
$U(e)$ is an annulus, therefore
$$\partial (\red_{\mathcal{L}}^{-1}(\Star(\tilde{v})))=\{t(e)|\ s(e)=\tilde{v}\}.$$
So we get $\partial
(\red_{{\mathcal{L}},\Gamma}^{-1}(\Star(v)))=\{\pi_\Gamma(t(e))|\
s(e)=\tilde{v}\}\not\ni x$ as we wished.
\end{proof}
\begin{cor}\label{corexh}
If there exists a model of $G_\Gamma$ which is without loops, then
the map $\Omega_{\mathcal{L}}(K)\longrightarrow C_\Gamma(K)$ is surjective.
\end{cor}
\begin{proof}
Choose such a model. By the previous proof we have
$$
C_\Gamma(K)=\bigcup_{e\in E(G_\Gamma)}{U(e)(K)},\qquad\Omega_{\mathcal{L}}(K)=\bigcup_{\tilde{e}\in E}{U(\tilde{e})(K)}
$$
with the same notation. We may assume $\pi_\Gamma(\tilde{e})=e$ so we conclude $U(\tilde{e})=U(e)$.
\end{proof}
\section{The Jacobian of a tropical graph via integration}\label{tropical}
We give a proof of \cite[Thm.~6.4~(2)]{vdP92} from the different
perspective given by multiplicative integrals. This result was
generalized by Baker and Rabinoff in \cite[Thm.~2.9]{BR15}.
Recall the definition of the Jacobian of a finite metric graph (or
more generally, of a tropical curve) (see, for example
\cite[Def.~4.1.4]{CV10}). We only consider metric graphs $G$ with
all vertices of valence greater than or equal to $2$. By the
introduction of section~\ref{graphs}, for any edge $e$ of $G$ we
have a length $\ell(e)\in {\mathbb{R}}_{>0}$.
We choose an orientation for each edge of $G$, and we consider the
free abelian group ${\mathbb{Z}}[E(G)]$ generated by the oriented edges of
some model for $G$ and the map $\partial:{\mathbb{Z}}[E(G)] \to{\mathbb{Z}}[V(G)]$
given by $\partial(e)=t(e)-s(e)$, where $t(e)$ is the target of $e$
and $s(e)$ is the source. Then $H_1(G,{\mathbb{Z}})=\Ker(\partial)$. The
following result is well known.
Let $G$ be a metric graph. Consider the paring $(\ ,\
)_G:{\mathbb{Z}}[E(G))]\times {\mathbb{Z}}[E(G))]\to {\mathbb{R}}$ defined by $(e,e')=0$ if
$e'\ne e$ and $e'\ne \overline{e}$ (the opposite edge of $e$),
$(e,e)=\ell(e)$ and $(e,\overline{e})=-\ell(e)$.
\begin{lem} The pairing $(\ ,\ )_G$
determines a symmetric positive definite bilinear map $(\ ,\
)_G:H_1(G,{\mathbb{Z}})\times H_1(G,{\mathbb{Z}})\to {\mathbb{R}}$.
\end{lem}
The Jacobian of $G$ is the torus given by $H_1(G,{\mathbb{R}})/H_1(G,{\mathbb{Z}})$
together with the metric determined by $(\ ,\ )_G$.
Now, suppose $\Gamma$ is a Schottky group in $\PGL_2(K)$, and
${\mathcal{L}}:={\mathcal{L}}_{\Gamma}$ is its set of limit points. For any $\gamma\in\Gamma$ we denote by $\ds{\lim_{n\rightarrow\infty}{\gamma^n z_0}=:z_\gamma^+\in{\mathcal{L}}}$ and $\ds{\lim_{n\rightarrow-\infty}{\gamma^n z_0}=:z_\gamma^-\in{\mathcal{L}}}$ its attractive and repulsive fixed points respectively.
Further, $\Gamma$ acts
on ${\mathcal{T}}={\mathcal{T}}_K({\mathcal{L}})$, and there is just an apartment fixed by $\gamma$ which is ${\mathbb{A}}_{\{z_\gamma^-,z_\gamma^+\}}$. We will denote it by ${\mathbb{A}}_\gamma$.
The quotient $G_\Gamma:=\Gamma\backslash{\mathcal{T}}$ is a finite metric
graph. Moreover $\pi:{\mathcal{T}}\to\Gamma\backslash{\mathcal{T}}$ is the universal covering,
and the fundamental group of $\Gamma\backslash{\mathcal{T}}$ is canonically isomorphic to
$\Gamma$. Hence the map $\varpi:\Gamma \to H_1(G_\Gamma,{\mathbb{Z}})$
defined sending $\gamma\in \Gamma$ to
$\pi(P(\alpha,\gamma(\alpha))$, where $\alpha$ is any point of ${\mathbb{A}}_\gamma$ (for simplicity, a vertex of any model)
and $P(\alpha,\gamma(\alpha))$ is the oriented path from $\alpha$ to
$\gamma(\alpha)$, determines an isomorphism between the
abelianization $\Gamma^{ab}$ and $H_1(G_\Gamma,{\mathbb{Z}})$.
We denote by $$(\ , \ )_{\Gamma}:\Gamma^{ab}\times \Gamma^{ab}\to
{\mathbb{R}}$$ the bilinear map given by
$$(\gamma, \gamma'
)_{\Gamma}=(\varpi(\gamma),\varpi(\gamma'))_{{\mathcal{T}}/\Gamma}.$$
\begin{lem}
Any finite metric graph $G$ satisfies $H_1(G,{\mathbb{Z}})\cong \HC(G,{\mathbb{Z}})$.
\end{lem}
\begin{proof}
Take a model $\mathfrak{G}$ for $G$. We want to prove
$H_1(G,{\mathbb{Z}})\cong \HC(\mathfrak{G},{\mathbb{Z}})$. Given a cycle
$\displaystyle{z=\sum_{e\in E(G)}{n_e\cdot e\in
H_1(\mathfrak{G},{\mathbb{Z}})\subset{\mathbb{Z}}[E(G)]}}$, we associate to it an
harmonic cochain $c(z)$ defined by $c(z)(e):=n_e$ and $c(z)(\bar
e):=-n_e$ for any $e\in E(G)$. Reciprocally, for each harmonic
cochain $c$ we get a cycle $z_c:=\sum_{e\in E(G)}{c(e)\cdot e}$.
This correspondence defines the bijection.
\end{proof}
\begin{obs}
Note that any Schottky group $\Gamma$ acts on $\HC({\mathcal{T}},{\mathbb{Z}})$ so that $\HC(\Gamma\backslash{\mathcal{T}},{\mathbb{Z}})\cong \HC({\mathcal{T}},{\mathbb{Z}})^\Gamma$.
\end{obs}
\begin{thm}\label{Niso}
The map $\displaystyle{\mu: \Gamma^{ab} \to\mathscr{M}( {\mathcal{L}}^*,\mb{Z})_{0}^\Gamma}$ defined by
$$
\mu_\gamma(\varepsilon):=\mu(\gamma)(\varepsilon):=\frac{(\pi(\varepsilon),\varpi(\gamma))_{{\mathcal{T}}/\Gamma}}{l(\varepsilon)}.
$$
over a (topological) edge $\varepsilon$ is a natural isomorphism such that
for any $\gamma,\gamma'\in \Gamma$, we have
$$
(\gamma, \gamma')_{\Gamma}=-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\gamma\alpha-\alpha}{d\mu_{\gamma'}}}
$$
where $\alpha\in {\mathcal{T}}_K({\mathcal{L}})$ is any point.
\end{thm}
\begin{proof}
The previous results together with section~\ref{graphs} give the composition of isomorphisms
$$
\xymatrix@R=.1pc{
\displaystyle{\Gamma^{ab}}\ar[r]_(.35){\displaystyle{\cong}}&\displaystyle{H_1(G_\Gamma,{\mathbb{Z}})}\ar[r]_(.45){\displaystyle{\cong}}&\displaystyle{\HC(G_\Gamma,{\mathbb{Z}})}\ar[r]_(.42){\displaystyle{\cong}}&\displaystyle{\HC({\mathcal{T}}_K({\mathcal{L}}_\Gamma),{\mathbb{Z}})^\Gamma}\ar[r]_(.55){\displaystyle{\cong}}&\displaystyle{\mathscr{M}( {\mathcal{L}}^*,\mb{Z})_{0}^\Gamma}\\
\displaystyle{\gamma}\ar@{|->}[r]&\displaystyle{\varpi(\gamma)}\ar@{|->}[r]&\displaystyle{c(\varpi(\gamma))}\ar@{|->}[rr]&&\displaystyle{\mu(c(\varpi(\gamma)))}
}
$$
which assigns to $\gamma\in\Gamma^{ab}$ the harmonic cochain defined by
$$
\mu(c(\varpi(\gamma)))(\varepsilon)=c(\varpi(\gamma))(\varepsilon)=\frac{(\pi(\varepsilon),\varpi(\gamma))_{{\mathcal{T}}/\Gamma}}{l(\varepsilon)}
$$
Since the set of points of valence greater than 2 in the path from
$\alpha$ to $\gamma'\alpha$ is finite (by corollary~\ref{locft}),
then we get the equality
$$
(\gamma, \gamma')_{\Gamma}=-\log{\left|\mathop{\mathrlap{\pushMI}}\!\int\right|_{\gamma\alpha-\alpha}{d\mu_{\gamma'}}}
$$
decomposing the path linearly, applying the lemma~\ref{logint} and
the multiplicativity of the integral with respect to the path and
taking into account the definition of the map $\mu$.
\end{proof}
\section{The discrete cross ratio}
In this section we recall some results relating the cross ratio of 4
points in ${\mathbb{P}}^1({\mathbb{C}}_K)$ with the tree they generate. Recall that,
given four points $a_1,a_2,z_1,z_2\in{{\mathbb{P}}_K^1}^*({\mathbb{C}}_K)$, the cross
ratio is defined as
$$
\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)=\frac{(a_1-z_1)(a_2-z_2)}{(a_1-z_2)(a_2-z_1)
$$
Note that formally
$$
\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)=\left(\begin{array}{c}
z_1:a_1\\
z_2:a_2
\end{array}\right)=\left(\begin{array}{c}
a_2:z_2\\
a_1:z_1
\end{array}\right)
$$
and given a fifth point $z_3\in{{\mathbb{P}}_K^1}^*({\mathbb{C}}_K)$,
$$
\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\left(\begin{array}{c}
a_1:z_2\\
a_2:z_3
\end{array}\right)=\left(\begin{array}{c}
a_1:z_1\\
a_2:z_3
\end{array}\right)
$$
The next lemma is known, at least the particular cases and when $K$
is local (\cite{MD73}, \cite{BDG04}), but we prefer to expose a
general and new proof using our results.
\begin{lem}\label{val}
Let $a_1,a_2,z_1,z_2\in{{\mathbb{P}}_K^1}^*({\mathbb{C}}_K)$ be four points such that $a_1\neq a_2$ and $z_1\neq z_2$. Then
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=\left({\mathbb{A}}_{\{a_1,a_2\}},{\mathbb{A}}_{\{z_1,z_2\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}.
$$
\end{lem}
\begin{proof}
To begin, recall the definition of the first term,
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=-\log\left|\frac{(a_1-z_1)(a_2-z_2)}{(a_1-z_2)(a_2-z_1)}\right|.
$$
If $a_i=z_j$ for some $i,j$ it is clear that the valuation of the
cross ratio and the intersection pairing of the apartments are
identically $\pm\infty$ with the sign depending on the combination.
Next we will considerate the case in which the four points are
distinct.
Let us suppose first that one of the four points is $\infty$. By the absolute symmetry among them, we can put $z_2=\infty$. Then, on one hand we have
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=-\log\left|\frac{a_1-z_1}{a_2-z_1}\right|
$$
On the other hand we will compute the intersection of ${\mathbb{A}}_{\{a_1,a_2\}}$ with ${\mathbb{A}}_{\{z_1,\infty\}}$. Note that we may write $\alpha(z_1,r)$ with $r\in{\mathbb{R}}_{>0}$ for the points of the second apartment. Let us assume without loss of generality that $|a_1-z_1|<|a_2-z_1|$, so we see that the intersection between the apartments goes from the point $\alpha(z_1,|z_1-a_1|)$ to the point $\alpha(z_1,|z_1-a_2|)$ and the distance between them, which is the length of the intersection, and it is the product of the pairing (with positive sign because the assumption), is
$$
\left|\log\frac{|a_2-z_1|}{|a_1-z_1|}\right|=-\log\left|\frac{a_1-z_1}{a_2-z_1}\right|=\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)
$$
as we wanted to see.
To finish the proof we have to deal with the case in which none of the four points is $\infty$. Let us define the compact set ${\mathcal{L}}':=\{a_1,a_2\}$ and the radii
$$
\begin{array}{l}
r_1:=d(z_1,{\mathcal{L}}')=\mbox{min}(|z_1-a_1|,|z_1-a_2|)\mbox{, and}\\
r_2:=d(z_2,{\mathcal{L}}')=\mbox{min}(|z_2-a_1|,|z_2-a_2|).
\end{array}
$$
Once more, we can do the assumption $r_1\leq r_2$ without loss of generality. We will consider three cases:
We suppose first $|a_1-a_2|\geq r_2\geq r_1$.
On one hand it can occur that there is an $i\in\{1,2\}$ such that
$r_1=|z_1-a_i|$ and $r_2=|z_2-a_i|$. Then, the starting and ending
points of the intersection between ${\mathbb{A}}_{\{a_1,a_2\}}$ and
${\mathbb{A}}_{\{z_1,z_2\}}$ are $\alpha(a_i,r_1)$ and $\alpha(a_i,r_2)$
respectively (so the intersection pairing is the distance with
positive sign), or the intersection is empty or just a point if
$r_1=r_2$. Anyway,
$$
\left({\mathbb{A}}_{\{a_1,a_2\}},{\mathbb{A}}_{\{z_1,z_2\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}=d(\alpha(a_i,r_1),\alpha(a_i,r_2))=\left|\log\frac{r_2}{r_1}\right|=-\log\frac{r_1}{r_2}
$$
If $i=1$, $r_1\leq|z_1-a_2|\leq\max\{r_1,|a_1-a_2|\}$ so $|z_1-a_2|=|a_1-a_2|$, and $r_2\leq|z_2-a_2|\leq\max\{r_2,|a_1-a_2|\}$ so $|z_2-a_2|=|a_1-a_2|$. If $i=2$, the same computation gives a similar result. In any case we always get
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=-\log\left|\frac{(a_1-z_1)(a_2-z_2)}{(a_1-z_2)(a_2-z_1)}\right|=-\log\frac{r_1 |a_1-a_2|}{r_2 |a_1-a_2|}=-\log\frac{r_1}{r_2}.
$$
On the other hand, writing $\{i,j\}=\{1,2\}$ we have $r_1=|a_i-z_1|$ and $r_2=|a_j-z_2|$. We may assume $i=1$ and $j=2$. The starting and ending points of the intersection are $\alpha(a_1,r_1)$ and $\alpha(a_2,r_2)$. So we have
$$
\left({\mathbb{A}}_{\{a_1,a_2\}},{\mathbb{A}}_{\{z_1,z_2\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}=d(\alpha(a_1,r_1),\alpha(a_2,r_2))=
$$
$$
=d(\alpha(a_1,r_1),\alpha(a_1,|a_1-a_2|))+d(\alpha(a_2,r_2),\alpha(a_2,|a_1-a_2|))=-\log\frac{r_1 r_2}{|a_1-a_2|^2}
$$
(Note that if we assumed $i=2$ and $j=1$, the intersection pairing
would be minus the distance.)
Further,
$r_2\geq|a_1-z_2|\geq\max\{|a_1-a_2|,r_2\}\geq|a_1-a_2|$ so
$|a_1-z_2|=|a_1-a_2|$ and identically $|a_2-z_1|=|a_1-a_2|$.
Therefore
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=-\log\left|\frac{(a_1-z_1)(a_2-z_2)}{(a_1-z_2)(a_2-z_1)}\right|=-\log\frac{r_1 r_2}{|a_1-a_2|^2}.
$$
In second place we suppose $r_2>|a_1-a_2|\geq r_1$. We can assume $r_1=|z_1-a_1|$. Let us observe that $r_2=|z_2-a_1|=|z_2-a_2|$. The starting and ending points of the intersection are $\alpha(a_1,r_1)$ and $\alpha(a_1,|a_1-a_2|)$, so
$$
\left({\mathbb{A}}_{\{a_1,a_2\}},{\mathbb{A}}_{\{z_1,z_2\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}=d(\alpha(a_1,r_1),\alpha(a_1,|a_1-a_2|))=-\log\frac{r_1}{|a_1-a_2|}
$$
(Note that if we assumed $r_1=|z_1-a_2|$, the distance would appear with a minus, and so we would get the inverse value.)
Since we have $|z_1-a_2|=|a_1-a_2|$ by an argument as above, we get
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=-\log\left|\frac{(a_1-z_1)(a_2-z_2)}{(a_1-z_2)(a_2-z_1)}\right|=
$$
$$
=-\log\frac{r_1 r_2}{r_2|a_1-a_2|}=-\log\frac{r_1}{|a_1-a_2|}.
$$
Finally, the third case is $r_1\geq r_2>|z_1-z_2|$. In this case the intersection of the apartments is empty so the intersection pairing of the apartments is zero, and since $|z_1-a_1|=|z_1-a_2|$ and $|z_2-a_1|=|z_2-a_2|$, the valuation of the cross ratio vanishes as well.
\end{proof}
\begin{cor}
Let ${\mathcal{L}}\subset{{\mathbb{P}}_K^1}^*(K)$ be a compact set with at least two points.
If $a_1,a_2,z_1,z_2$ are in ${\mathcal{L}}$ or even in ${{\mathbb{P}}_K^1}^*(K)$, the pairing can be done in ${\mathcal{T}}_K$,
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=\left({\mathbb{A}}_{\{a_1,a_2\}},{\mathbb{A}}_{\{z_1,z_2\}}\right)_{{\mathcal{T}}_{K}}
$$
while if $a_1,a_2\in{\mathcal{L}}$ and $z_1,z_2\in\Omega_{{\mathcal{L}}}({\mathbb{C}}_K)$, we may restrict to ${\mathcal{T}}_K({\mathcal{L}})$:
$$
\mathit{v}_K\left(\left(\begin{array}{c}
a_1:z_1\\
a_2:z_2
\end{array}\right)\right)=\left({\mathbb{A}}_{\{a_1,a_2\}},P\big(\red_{{\mathcal{L}}}(z_1),\red_{{\mathcal{L}}}(z_2)\big)\right)_{{\mathcal{T}}_K({\mathcal{L}})}.
$$
\end{cor}
\section{A peculiar symmetry}\label{APS}
In this section we study some properties of the action of $\Gamma$
on ${\mathcal{T}}_K$, a relation among the measures, and a symmetry among
multiplicative integrals which can be useful to generalize the well
known symmetry between theta functions.
Let $\Gamma\subset \PGL_2(K)$ be a Schottky group, and let
${\mathcal{L}}:={\mathcal{L}}_\Gamma\subset{{\mathbb{P}}^1}^*(K)$ be its set of limit points. We
are going to show a new result which will led to a proof of the
symmetry of bilinear pairing defining the jacobian of the Mumford
curve $C_{\Gamma}$.
We assume that $\Omega_{\mathcal{L}}(K)\neq\emptyset$ and contains at least the
closures of two $\Gamma$-orbits of points. This is possible after a
finite extension of $K$, meanwhile ${\mathcal{L}}$ remains invariant.
Let us define for any $p\in\Omega_{\mathcal{L}}(K)$ the compact set
$\ds{{\mathcal{L}}_p:={\mathcal{L}}\cup\overline{\Gamma\cdot p}\subset {{\mathbb{P}}^1}^*(K)}$
and for any $\gamma,\delta\in\Gamma$ the analytic function
$$
u_{\gamma,\delta,p}(z):=u_{\gamma\delta p,\gamma p}(z)=\frac{z-\gamma\delta p}{z-\gamma p}\in\mathcal{O}(\Omega_{{\mathcal{L}}_p})
$$
Consider now a point $q\in\Omega_{{\mathcal{L}}_p}(K)$, that is $q\in\Omega_{\mathcal{L}}(K)$ such that $\ds{\overline{\Gamma\cdot p}\cap\overline{\Gamma\cdot q}=\emptyset}$. Then, for any $\rho\in\Gamma$, applying the invariance of the cross ratio we obtain
$$
\frac{u_{\gamma,\delta,p}(\rho q)}{u_{\gamma,\delta,p}(q)}=\frac{u_{\gamma^{-1},\rho,q}(\delta p)}{u_{\gamma^{-1},\rho,q}(p)}
$$
Recall from the section on the Poisson formula the equality of measures $$\tilde{\mu}(u_{\gamma,\delta,p})=\tilde{\mu}(u_{\gamma\delta p,\gamma p})=\mu_{\gamma p^*,\gamma\delta p^*},$$ and then
$$
\frac{u_{\gamma,\delta,p}(\rho q)}{u_{\gamma,\delta,p}(q)}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_p^*}{f_{\rho q-q}(t)d\mu_{\gamma p^*,\gamma\delta p^*}}
$$
Therefore, putting together the two last ideas we have
$$
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_p^*}{f_{\rho q-q}(t)d\mu_{\gamma p^*,\gamma\delta p^*}}=\frac{u_{\gamma,\delta,p}(\rho q)}{u_{\gamma,\delta,p}(q)}=\frac{u_{\gamma^{-1},\rho,q}(\delta p)}{u_{\gamma^{-1},\rho,q}(p)}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_q^*}{f_{\delta p-p}(t)d\mu_{\gamma^{-1} q^*,\gamma^{-1}\rho q^*}}
$$
For any $\delta\in\Gamma$, using theorem~\ref{Niso} one defines a
measure $\ds{\mu_\delta\in\mathscr{M}( {\mathcal{L}}^*,\mb{Z})_{0}}$, while we
just defined, for each $\gamma\in\Gamma$, a measure $\ds{\mu_{\gamma
p^*,\gamma\delta p*}\in\mathscr{M}( {\mathcal{L}}_p^*,\mb{Z})_{0}}$. Note that
${\mathcal{L}}^*\subset{\mathcal{L}}_p^*$ and ${\mathcal{T}}_K({\mathcal{L}})\subset{\mathcal{T}}_K({\mathcal{L}}_p)$. We
consider compatible models for these trees, meaning that the model
of ${\mathcal{T}}_K({\mathcal{L}}_p)$ restricts to the model of ${\mathcal{T}}_K({\mathcal{L}})$.
\begin{prop}\label{measum}
With the above notations, for any edge $e$ of ${\mathcal{T}}_K({\mathcal{L}}_p)$ and ${\mathcal{T}}_K({\mathcal{L}})$ we have
$$
\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=-\mu_\delta(e)
$$
and for any edge of ${\mathcal{T}}_K({\mathcal{L}}_p)$ which is not inside ${\mathcal{T}}_K({\mathcal{L}})$, then
$$
\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=0.
$$
\end{prop}
In order to prove the proposition, we observe first that
$$
\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=\sum_{\gamma\in\Gamma}{\mu_{p^*,\delta p^*}}(\gamma^{-1}e)=\sum_{\gamma\in\Gamma}{\mu_{p^*,\delta p^*}}(\gamma e)=\sum_{\{\gamma\in\Gamma|\ \gamma e\in|P(p,\delta p)|\}}{\mu_{p^*,\delta p^*}}(\gamma e)
$$
(where the bars for $|P(p,\delta p)|$ mean that we are considering
just the underlying sets, without orientation) and we proceed by
steps. The first step, which is the main one, lies essentially on
the following lemma.
\begin{lem}
For any $\delta\in\Gamma$ and $p\in\Omega_{\mathcal{L}}(K)$ we have
$|{\mathbb{A}}_{\{p,\delta
p\}}|\cap|{\mathbb{A}}_{\{\delta^2p,\delta^3p\}}|=\emptyset$ and
${\mathbb{A}}_{\{p,\delta
p\}}\cap{\mathbb{A}}_{\{\delta^{-1}p,\delta^2p\}}={\mathbb{A}}_{\{p,\delta
p\}}\cap{\mathbb{A}}_\delta\subset{\mathbb{A}}_{\{p,\delta p\}}\cap{\mathcal{T}}_K({\mathcal{L}})$.
\end{lem}
\begin{proof}
Since $\delta$ is hyperbolic it has the form $\delta=\delta'\left(\begin{array}{cc}q&0\\0&1\end{array}\right)\delta'^{-1}$ with $|q|<1$. Consider $p':=\delta^{-1}p\in\Omega_{\mathcal{L}}$. Then, if we prove the equalities of the lemma for $\left(\begin{array}{cc}q&0\\0&1\end{array}\right)$ and $p'$ instead of $\delta$ and $p$, then allowing $\delta$ act on the apartments we will get the claims. So we may assume $\delta=\left(\begin{array}{cc}q&0\\0&1\end{array}\right)$ with $|q|<1$. In particular, we have ${\mathbb{A}}_\delta={\mathbb{A}}_{\{\infty,0\}}$.
And now, we want to show $|{\mathbb{A}}_{\{p,q
p\}}|\cap|{\mathbb{A}}_{\{q^2p,q^3p\}}|=\emptyset$. Let us observe that
$|q^3p|<|q^2p|<|qp|<p$, so
\begin{eqnarray*}
{\mathbb{A}}_{\{p,q p\}}\cap{\mathbb{A}}_{\{\infty,0\}}=P\left(\alpha(0,|p|),\alpha(0,|qp|)\right)\\
{\mathbb{A}}_{\{q^2p,q^3p\}}\cap{\mathbb{A}}_{\{\infty,0\}}=P\left(\alpha(0,|q^2p|),\alpha(0,|q^3p|)\right)
\end{eqnarray*}
Therefore, if the intersection $|{\mathbb{A}}_{\{p,\delta p\}}|\cap|{\mathbb{A}}_{\{\delta^2p,\delta^3p\}}|$ was non empty it should occur in ${\mathbb{A}}_{\{\infty,0\}}$ since the total space is a tree, but it is clear that $$
P\left(\alpha(0,|p|),\alpha(0,|qp|)\right)\cap P\left(\alpha(0,|q^2p|),\alpha(0,|q^3p|)\right)=\emptyset,
$$
an so we get the first claim.
In order to obtain the second claim we will prove
$\left({\mathbb{A}}_{\{p,qp\}},{\mathbb{A}}_{\{q^{-1}p,q^2p\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}=\left({\mathbb{A}}_{\{p,qp\}},{\mathbb{A}}_{\{\infty,0\}}\right)_{{\mathcal{T}}_{{\mathbb{C}}_K}}$.
Applying the lemma~\ref{val} we see it is enough to check that
$$
\mathit{v}_K\left(\left(\begin{array}{c}
p:q^{-1}p\\
qp:q^2p
\end{array}\right)\right)=
\mathit{v}_K\left(\left(\begin{array}{c}
p:\infty\\
qp:0
\end{array}\right)\right).
$$
So we compute:
$$
\mathit{v}_K\left(\left(\begin{array}{c}
p:q^{-1}p\\
qp:q^2p
\end{array}\right)\right)=-\log\frac{|p-q^{-1}p||qp-q^2p|}{|p-q^2p||qp-q^{-1}p|}=-\log\frac{|q^{-1}p||qp|}{|p||q^{-1}p|}=$$
$$=-\log\frac{|qp|}{|p|}=\mathit{v}_K\left(\left(\begin{array}{c}
p:\infty\\
qp:0
\end{array}\right)\right)
$$
\end{proof}
Next, and under the hypotheses of the previous lemma, it allows us
to subdivide the apartment ${\mathbb{A}}_{p,\delta p}$ in three paths:
${\mathbb{A}}_{\{p,\delta p\}}=S_{p,\delta p}\cup I_{p,\delta p}\cup
T_{p,\delta p}$, where
\begin{eqnarray*}
S_{p,\delta p}=P(p,t(p,\delta p,\delta^{-1}p))\\
I_{p,\delta p}=P(t(p,\delta p,\delta^{-1}p),t(p,\delta p,\delta^2 p))\\
T_{p,\delta p}=P(t(p,\delta p,\delta^2 p),\delta p)
\end{eqnarray*}
Since the first part of the lemma tells that $|{\mathbb{A}}_{\{\delta^{-1}p, p\}}|\cap|{\mathbb{A}}_{\{\delta p,\delta^2p\}}|=\emptyset$, this implies that $|S_{p,\delta p}|\cap|T_{p,\delta p}|=\emptyset$, the intersections of the interior of the paths are empty and the paths are well defined subpaths of ${\mathbb{A}}_{\{p,\delta p\}}$ with the same orientation.\\
The second part of the lemma implies that $I_{p,\delta
p}\subset{\mathcal{T}}_K({\mathcal{L}})$. With this tools, we proceed to get the next
step:
\begin{lem}
Let $e$ be an edge of ${\mathcal{T}}_K({\mathcal{L}}_p)$ and consider the sets
\begin{eqnarray*}
\Gamma_S^e:=\{\gamma\in\Gamma|\ \gamma e\in |S_{p,\delta p}|\}\\
\Gamma_I^e:=\{\gamma\in\Gamma|\ \gamma e\in |I_{p,\delta p}|\}\\
\Gamma_T^e:=\{\gamma\in\Gamma|\ \gamma e\in |T_{p,\delta p}|\}
\end{eqnarray*}
so that we have the decomposition $\{\gamma\in\Gamma|\ \gamma
e\in|P(p,\delta p)|\}=\Gamma_S^e\sqcup\Gamma_I^e\sqcup\Gamma_T^e$.
Then:
\begin{enumerate}
\item There is a bijection $\Gamma_S^e\longleftrightarrow\Gamma_T^e$ which reverses the orientation of the edge in ${\mathbb{A}}_{\{p,\delta p\}}$, that is, if $\gamma'$ corresponds to a $\gamma$ such that $\gamma e$ is in $S_{p,\delta p}$ with the same orientation, the edge $\gamma' e$ is in $T_{p,\delta p}$ with the opposite orientation.
\item If $e$ is not inside ${\mathcal{T}}_K({\mathcal{L}})$, then $\Gamma_I^e=\emptyset$
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item The bijection is defined by
\begin{eqnarray*}
\Gamma_S^e\longrightarrow\Gamma_T^e\\
\gamma\mapsto\ \delta\gamma\
\end{eqnarray*}
Thus, if the oriented edge $\gamma e$ is in
$$S_{p,\delta p}=P(p,t(p,\delta p,\delta^{-1}p))= P(p,\delta^{-1}p)\cap P(p,\delta p),$$
the oriented edge $\delta\gamma e$ is in
$$\delta P(p,\delta^{-1}p)\cap \delta P(p,\delta p)=P(\delta p, p)\cap P(\delta p,\delta^2 p)=T_{p,\delta p}.$$
In general, the orientation of $\gamma e$ with respect to $S_{p,\delta p}$ and $P(p,\delta p)$ is the same as the orientation of $\delta\gamma e$ with respect to $T_{p,\delta p}$ and $P(p,\delta p)$ so the opposite to the orientation of $\gamma e$. Clearly, the inverse map is $\gamma\mapsto\delta^{-1}\gamma$.
\item The result is clear from the remark previous to the lemma. If $e$ is not inside ${\mathcal{T}}_K({\mathcal{L}})$, there is no $\gamma e$ inside ${\mathcal{T}}_K({\mathcal{L}})$ for $\gamma\in\Gamma$, but $\Gamma_I^e=\{\gamma\in\Gamma|\ \gamma e\in |I_{p,\delta p}|\subset|{\mathcal{T}}_K({\mathcal{L}})|\}$ so $\Gamma_I^e=\emptyset$.
\end{enumerate}
\end{proof}
\begin{proof}[Proof of proposition~\ref{measum}]
Let us see first the second claim. If $e$ is not in ${\mathcal{T}}_K({\mathcal{L}})$ we have
$$
\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=\sum_{\{\gamma\in\Gamma|\ \gamma e\in|P(p,\delta p)|\}}{\mu_{p^*,\delta p^*}}(\gamma e)=
$$
$$
=\sum_{\gamma\in\Gamma_S^e}{\mu_{p^*,\delta p^*}}(\gamma e)+\sum_{\gamma\in\Gamma_I^e}{\mu_{p^*,\delta p^*}}(\gamma e)+\sum_{\gamma\in\Gamma_T^e}{\mu_{p^*,\delta p^*}}(\gamma e)
$$
Because of the second part of the previous lemma the second summation is zero and because of the first part and the definition of $\mu_{p^*,\delta p*}$ the sum of the other two summations vanishes, so $\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=0$ as we wanted to see.\\
We assume now that $e$ is in ${\mathcal{T}}_K({\mathcal{L}})$. We have the same equalities that before and also the cancellation of the two extreme summations so
$$
\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=\sum_{\gamma\in\Gamma_I^e}{\mu_{p^*,\delta p^*}}(\gamma e)
$$
and we want to prove this is equal to
$$
-\mu_\delta(e)=-\frac{(\pi(e),\varpi(\delta))_{{\mathcal{T}}/\Gamma}}{\ell(e)}=-\frac{(\pi(e),\pi(P(\alpha,\delta\alpha)))_{{\mathcal{T}}/\Gamma}}{\ell(e)}
$$
where ${\mathcal{T}}={\mathcal{T}}_K({\mathcal{L}})$) and $\alpha$ is any vertex in ${\mathbb{A}}_\delta$.
We take $\alpha=t(p,\delta p,\delta^{-1}p)$, so we have
$\delta\alpha=t(p,\delta p,\delta^2p)$ and
$$
\mu_\delta(e)=\frac{(\pi(e),\pi(P(\alpha,\delta\alpha)))_{{\mathcal{T}}/\Gamma}}{\ell(e)}=\sum_{\substack{|\gamma e|\subset |P(\alpha,\delta\alpha)|\\\gamma\in\Gamma}}{\frac{(\gamma e,P(\alpha,\delta\alpha))_{\mathcal{T}}}{\ell(e)}}=
$$
$$
=\sum_{\gamma\in\Gamma_I^e}{\frac{(\gamma e,P(\alpha,\delta\alpha))_{\mathcal{T}}}{\ell(e)}}=-\sum_{\gamma\in\Gamma_I^e}{\mu_{p^*,\delta p^*}}(\gamma e)=-\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)
$$
where for the third equality we use the definition of $\alpha$ and the fact that the action of $\Gamma$ on ${\mathcal{T}}$ is free, and for the fourth equality we use the definition of $\mu_{p^*,\delta p*}$.
\end{proof}
\begin{cor}
With the above notations we have
$$
\prod_{\gamma\in\Gamma}{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_p^*}{f_{\rho q-q}^{-1}(t)d\mu_{\gamma p^*,\gamma\delta p^*}}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\rho q-q}(t)d\mu_{\delta}}
$$
\end{cor}
\begin{proof}
It is direct from the proposition, taking into account that the inverse of the function $f_{\rho q-q}$ appears due to the negative sign in the equality $\sum_{\gamma\in\Gamma}{\mu_{\gamma p^*,\gamma\delta p^*}}(e)=-\mu_\delta(e)$.
\end{proof}
\begin{cor}\label{APR}
Let $\Gamma\subset \PGL_2(K)$ be a Schottky group, and let ${\mathcal{L}}:={\mathcal{L}}_\Gamma\subset{{\mathbb{P}}^1}^*(K)$ be its set of limit points. For any $\rho,\delta\in\Gamma$ and for any $p,q\in\Omega_{\mathcal{L}}(K)$ such that $\ds{\overline{\Gamma\cdot p}\cap\overline{\Gamma\cdot q}=\emptyset}$ we get
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\rho q-q}{d\mu_{\delta}}=\mathop{\mathrlap{\pushMI}}\!\int_{\delta p-p}{d\mu_{\rho}}
$$
\end{cor}
\begin{proof}
Taking into account the last observation previous to the proposition and the corollary above we get
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\rho q-q}{d\mu_{\delta}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\rho q-q}(t)d\mu_{\delta}}=\prod_{\gamma\in\Gamma}{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_p^*}{f_{\rho q-q}^{-1}(t)d\mu_{\gamma p^*,\gamma\delta p^*}}}=
$$
$$
=\prod_{\gamma\in\Gamma}{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}_q^*}{f_{\delta p-p}^{-1}(t)d\mu_{\gamma^{-1} q^*,\gamma^{-1}\rho q^*}}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\delta p-p}(t)d\mu_{\rho}}=\mathop{\mathrlap{\pushMI}}\!\int_{\delta p-p}{d\mu_{\rho}}
$$
\end{proof}
\section{Automorphic Forms}\label{automorphic}
The main goal of this section is to prove theorem~\ref{AutTh} using
the analytic theory developed along this paper and some results of
\cite{BPR13}, like propositions 2.5, 2.10 and the slope formula
theorem (5.15), instead of using \cite[Ch.~2~(3.2)]{GvdP80}, whose
proof requires more analytic tools.
Let $G$ be a metric graph.
\begin{defn}
We call a tropical function on $G$ a continuous function
$f:G\longrightarrow{\mathbb{R}}$ such that there exists a model
$\mathfrak{G}$ of $G$ that for each edge $e\in E(\mathfrak{G})$ the
restriction
$$
f_{\big|\vert e\vert}:\vert e\vert\longrightarrow{\mathbb{R}}
$$
is linear with integral slope, where by linear we mean that for
every isometric embedding $[a,b]\longrightarrow|e|$, the composition
$[a,b]\longrightarrow|e|\longrightarrow{\mathbb{R}}$ is linear.
Note that this is equivalent to say that for each model of $G$ the function $f$ is piecewise linear (with integral slopes) on each edge.
\end{defn}
Suppose now that $G$ is locally finite. Given a tropical function
$f$ on $G$ and a model $\mathfrak{G}$ of $G$ such that $f$ verifies
the ``edge-linearity'' condition stated on previous definition, we
can associate to it a cochain $D_f$ on the edges of $\mathfrak{G}$
defined by taking $D_f(e)$ to be the slope of $f$ on $e$.
We call $f$ a harmonic function if $D_f$ is an harmonic cochain.
\begin{obs}
If $f$ is harmonic, $f_{\big|\vert e\vert}$ is linear for any edge of any model of $G$.
\end{obs}
Next, let $\Gamma$ be a group with a left action on a metric graph $G$.
\begin{defn}
A tropical function $f$ on $G$ is called an automorphic form for $\Gamma$ if
$$
\forall\ \gamma\in\Gamma\ \exists\ c_f(\gamma)\in{\mathbb{R}}:\ f(z)=c_f(\gamma)+f(\gamma z)\ \forall z\in G
$$
\end{defn}
\begin{lem}
Let $G$ be a locally finite metric graph on which acts a group
$\Gamma$. Let $f$ be an automorphic form for $\Gamma$. Then there
exists a model $\mathfrak{G}$ of $G$, on which acts $\Gamma$, such
that $f$ is linear on its edges, the cochain $D_f$ is
$\Gamma$-invariant and so induces a cochain $\overline{D_f}$ on
$\Gamma\backslash G$.
\end{lem}
\begin{proof}
Since $f$ is tropical there exists a model of $G$ such that $f$ is linear on its edges. Now, the minimal $\Gamma$-invariant model refining the previous satisfies the claims of the lemma immediately, and $D_f$ is $\Gamma$-invariant because $f$ is automorphic for $\Gamma$.
\end{proof}
\begin{lem}\label{BPR210}
Let $G$ be a locally finite metric graph on which acts a group $\Gamma$. Assume there exists a finite connected graph $G'\subset G/\Gamma$ such that
$$
(\Gamma\backslash G)\setminus G'=\bigsqcup_{i\in I}{L_i}\text{ where }I\text{ is finite and }L_i\cong(0,\infty)\ \forall i\in I
$$
such that its closure inside $\Gamma\backslash G$ is $\overline{L_i}\cong[0,\infty)$ (we are choosing an orientation on $L_i$).
Then, any harmonic function on $G$ being an automorphic form for $\Gamma$ verifies:
\begin{enumerate}
\item For any $i\in I$, the restricted cochain is constant: ${\overline{D_f}}_{|L_i}\equiv m_i\in{\mathbb{Z}}$.
\item $\displaystyle{\sum_{i\in I}{m_i}=0}$.
\end{enumerate}
\end{lem}
\begin{proof}
We take a suitable model of $G$ -since $f$ is harmonic, it only has to be $\Gamma$-invariant-. Since $D_f$ is harmonic, so it is $\overline{D_f}$. Now, given two adjacent oriented edges $e,e'$ of $L_i$, due to the hypothesis on $G$ and $G'$ harmonicity implies $\overline{D_f}(e)+\overline{D_f}(\overline{e''})=0$, so $\overline{D_f}(e)=\overline{D_f}(e')$, and this extends obviously to any edge of $L_i$, so the first claim rests proved.
The second claim is a direct consequence of the lemma~\ref{harstar}.
\end{proof}
From now on, let $\Gamma$ be a fixed Schottky group, ${\mathcal{L}}={\mathcal{L}}_\Gamma$ the set of fixed points of $\Gamma$, and $\Omega_{\mathcal{L}}$ as defined above. Let $L|K$ be a field extension.
\begin{defn}
We will say that a $\ds{{\mathbb{C}}_K}$-valued meromorphic function $f\neq0$ on $\ds{\Omega_{{\mathcal{L}}}}$ is an automorphic form for $\Gamma$ with automorphy factor $\ds{c_f:\Gamma\longrightarrow \mb{C}_K^*}$ if
$$f(z)=c_f(\alpha)f(\alpha z)\ \forall z\in\Omega_{\mathcal{L}} \forall \alpha\in\Gamma.$$
We will call it $L$-automorphic if $c_f$ takes values in $\ds{L^*}$.
Let us denote the set of $L$-automorphic forms on $\Omega_{\mathcal{L}}$ by ${\mathcal{A}}_\Gamma(L)$.
\end{defn}
\begin{obs}
By definition, $c_f$ is a group morphism.
\end{obs}
\begin{prop}\label{autMI}
Given a point $z_0\in\Omega_{\mathcal{L}}(K)$ and a $\Gamma$-invariant measure ${\mu\in{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0^\Gamma}$ the function on $\Omega_{\mathcal{L}}$
$$
{\mathcal{I}}_{\mu,z_0}(z):=\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\mu}
$$
is an analytic and automorphic form for $\Gamma$ with automorphy factor independent of $z_0$.
\end{prop}
\begin{proof}
We already know it is analytic, as shown in the proof of theorem~\ref{PF} and remarked in its corollary~\ref{exh}.
In order to see that it is automorphic for $\Gamma$ let us show first that the integral
$$
\mathop{\mathrlap{\pushMI}}\!\int_{ p -\gamma p}{d\mu}
$$
does not depend on $p\in\Omega_{\mathcal{L}}$. Indeed, given $p,q\in\Omega_{\mathcal{L}}$ we have
$$
\frac{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{ p-\gamma p}{d\mu}}}{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{q-\gamma q}{d\mu}}}=\frac{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{p-q}{d\mu}}}{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-\gamma q}{d\mu}}}=1
$$
due to the $\Gamma$-equivariance of the integration and to the $\Gamma$-invariance of $\mu$.
Therefore,
$$
\frac{\ds{{\mathcal{I}}_{\mu,z_0}(z)}}{\ds{{\mathcal{I}}_{\mu,z_0}(\gamma z)}}=\frac{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d\mu}}}{\ds{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma z-z_0}{d\mu}}}=\mathop{\mathrlap{\pushMI}}\!\int_{z-\gamma z}{d\mu}\in K^*
$$
is its automorphy factor.
\end{proof}
\begin{prop}\label{autexh}
For any $c\in \mathrm{Hom}(\Gamma^{ab},L^*)$ there exists an $L$-automorphic form $f$ such that $c=c_f$.
\end{prop}
\begin{proof}
Let us consider the group ${\mathcal{M}}(\Omega_{\mathcal{L}})^*$ of non-zero
meromorphic functions on $\Omega_{\mathcal{L}}$ and its quotient $Q$ by the
constants, so we have the short exact sequence
$$
0\longrightarrow L^*\longrightarrow{\mathcal{M}}(\Omega_{\mathcal{L}})^*\longrightarrow Q\longrightarrow 0
$$
After taking invariants under $\Gamma$ we find the exact sequence
$$
{\mathcal{M}}(C_\Gamma)\longrightarrow Q^\Gamma\longrightarrow \mathrm{Hom}(\Gamma^{ab},L^*)\longrightarrow H^1(\Gamma,{\mathcal{M}}(\Omega_{\mathcal{L}})^*)
$$
We end the proof recalling that $H^1(\Gamma,{\mathcal{M}}(\Omega_{\mathcal{L}})^*)=0$ by \cite[Cor.~5.3]{vdP92} -since $C_\Gamma$ is algebraic-, and noting that $Q^\Gamma$ coincides with the group of $L$-automorphic forms modulo the constants.
\end{proof}
We may express this telling that the morphism
$$
{\mathcal{A}}_\Gamma(L)\longrightarrow \mathrm{Hom}(\Gamma^{ab},L^*)
$$
is surjective.
Let us formalize the notion of infinite divisor as in \cite[\S2]{MD73}.
\begin{defn} We call a function $\textbf{D}:\Omega_{\mathcal{L}}({\mathbb{C}}_K)\longrightarrow{\mathbb{Z}}$ an infinite $L$-divisor on $\Omega_{\mathcal{L}}$ verifying the following properties:
\begin{itemize}
\item $\textbf{D}(z_1)=\textbf{D}(z_2)$ if $z_1=\Gamma z_2$.
\item The set $\Supp(D):=\{z\in\Omega_{\mathcal{L}}|\ \textbf{D}(z)\neq0\}$ has no limit points in $\Omega_{\mathcal{L}}$ and there
is a finite extension $L'|L$ such that $\Supp(D)\subset \Omega_{\mathcal{L}}(L')$.
\end{itemize}
We write such a divisor in the usual form
$$
D=\sum_{n_z=\textbf{D}(z)\neq0}{n_zz}.
$$
\end{defn}
We will say that such an infinite divisor has finite representant $\tilde{D}$ if this is a finite divisor (that is it has finite support) such that
$$
D=\sum_{\gamma\in\Gamma}{\gamma\tilde{D}}=:\Gamma\tilde{D}
$$
We consider now the zeroes and poles of the automorphic forms. Note that if $z$ is a zero (resp. pole) of order $n$ of $f\in{\mathcal{A}}_\Gamma$, for each $\gamma\in\Gamma$, $\gamma z$ is a zero (resp. pole) of order $n$ of $f$ too.
\begin{prop}
Let $f$ be a meromorphic function and $e$ an edge of a model of ${\mathcal{T}}_K({\mathcal{L}})$. Then, the set of zeroes and poles of $f$ restricted to $U(e)$ is finite.
\end{prop}
\begin{proof}
First, a meromorphic function is the quotient of analytic functions so we may assume that $f$ is analytic and we only have to show that it has a finite number of zeroes. But this is proved in \cite[Prop.~3.3.6]{FvdP04} as a consequence of the fact that the affinoid $U(e)$ is a disjoint union of closed discs, the Mittag-Leffler decomposition theorem and the Weierstrass preparation theorem.
\end{proof}
\begin{cor}\label{ftAut}
The set of zeros and poles of an automorphic form $f$ on $\Omega_{\mathcal{L}}$ for $\Gamma$ is a finite union of orbits of points of $\Omega_{\mathcal{L}}$.
\end{cor}
\begin{proof}
Consider a model for ${\mathcal{T}}_K({\mathcal{L}})$ and denote the set of its edges $E$. Consider also a set of
edges $E_\Gamma\subset E$ in bijection by $\pi_\Gamma$ with the edges on the induced model on $G_\Gamma$. Since the quotient graph is
finite so it is the set $E_\Gamma$, and since this is a set of representatives of the graph $G_\Gamma$,
$$
\bigcup_{\gamma\in\Gamma}\gamma\cdot E_\Gamma = E
$$
Therefore, the affinoids $\gamma U(E_\Gamma)$ with $\gamma\in\Gamma$ cover all $\Omega_{\mathcal{L}}$, where
$$
U(E_\Gamma):=\bigcup_{e\in E_\Gamma}{U(e)}.
$$
Now, because of the previous proposition, the set $S_\Gamma(f)$ of zeroes and poles of $f$ on $U(E_\Gamma)$ is finite. And since this set is $\Gamma$-invariant and the orbit of $U(E_\Gamma)$ covers $\Omega_{\mathcal{L}}$, the orbit of $S_\Gamma(f)$ is the set of zeroes and poles of $f$ and it is a finite union of orbits of points.
\end{proof}
Let us denote $S(f)$ the set of zeroes and poles of an automorphic form $f$ on $\Omega_{\mathcal{L}}$, and ${\mathcal{L}}_f:={\mathcal{L}}_\Gamma\cup S(f)$. The set ${\mathcal{L}}_f$ is compact, due to the previous proposition and the fact that $\Gamma$ is a Schottky group.
Note that $f$ has neither zeroes nor poles on $\Omega_{{\mathcal{L}}_f}$, so $f\in{\mathcal{O}}(\Omega_{{\mathcal{L}}_f})^*$.
\begin{thm}
Let $f$ be an automorphic form for $\Gamma$ on $\Omega_{\mathcal{L}}$. Then $$F=-\log|f|_{|{\mathcal{T}}_K({\mathcal{L}}_f)}$$ is a harmonic and automorphic form for $\Gamma$ on ${\mathcal{T}}_K({\mathcal{L}}_f)$.
\end{thm}
\begin{proof}
The first thing we have to check is that $F$ is tropical, that is, given a model of ${\mathcal{T}}_K({\mathcal{L}}_f)$ and an edge $e$ of this model, the restriction of $F$ on $|e|$ is piecewise linear on it.
Since we are going to apply lemma~\ref{PoiL}, we recall the notation used in it. We may suppose that the topological realization of the edge is $|e|=P(\alpha(x,r),\alpha(x,s))$ with $x\in{\mathcal{L}}_f$, $r<s$ and such that its associated annulus satisfies $R_x(r,s)\cap{\mathcal{L}}_f=\emptyset$. We also do not loss generality assuming $x=0$. Now we consider an isometric embedding
$$
\exp:[r_0,s_0]\longrightarrow P(\alpha(0, \exp(r_0)),\alpha(0,\exp(s_0)))\text{ where }r=\exp(r_0),\ s=\exp(s_0)
$$
By the cited lemma, we know that $|f(z)|=r|z^k|$ for some $r\in{\mathbb{R}}_{>0}, k\in{\mathbb{Z}}$ on that path, and $z=\exp(w)$ for $w\in[r_0,s_0]$. Therefore
$$
F(\exp(w))=-\log|f(z)|=-k\log|z|-\log(r)=-kw-\log(r),
$$
so we get the hoped linearity with integral slope $k$, and so $F$ becomes tropical.
In the previous computation we got $D_F(e)=-k$. Recall also the map
$$
\tilde{\mu}:{\mathcal{O}}(\Omega_{{\mathcal{L}}_f})^*\longrightarrow\mathscr{M}({\mathcal{L}}_f^*,\mb{Z})_{0}
$$
by which $\tilde{\mu}(f)(e)=k$. Therefore $D_F=-\tilde{\mu}(f)$, so this is a harmonic cochain and $F$ is harmonic.
Finally we will show that $F$ is automorphic for $\Gamma$ on ${\mathcal{T}}_K({\mathcal{L}}_f)\subset\Omega_{{\mathcal{L}}_f}\subset\Omega_{\mathcal{L}}$. Since $f$ is automorphic on $\Omega_{\mathcal{L}}$ we have that for all $z\in\Omega_{\mathcal{L}}$ and $\gamma\in\Gamma$, $f(z)=c_f(\gamma)f(\gamma z)$ with $c_f(\gamma)\in{\mathbb{C}}_K^*$. Let us restrict to the case when $z\in {\mathcal{T}}_K({\mathcal{L}}_f)$:
$$
F(z)=-\log|f(z)|=-\log|c_f(\gamma)f(\gamma z)|=-\log|c_f(\gamma)|-\log|f(\gamma z)|=
$$
$$
=\mathit{v}_K(c_f(\gamma))+F(\gamma z)\text{ with }\mathit{v}_K(c_f(\gamma))\in{\mathbb{R}}
$$
\end{proof}
We maintain the same hypothesis of the theorem. Consider now the
quotient $\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}}_f)$. By the
lemma~\ref{quoGraph}, its quotient is the disjoint union of a finite
connected graphs with a finite union of ``ends'' which correspond to
the classes of zeroes and poles of $f$ modulo $\Gamma$ -that is
$\Gamma\backslash S(f)$- by the definition of ${\mathcal{L}}_f$. For any $x\in
S(f)$ denote $L_x$ the corresponding end oriented from the interior
to the exterior, like in lemma~\ref{BPR210}. With the previous
theorem, the next completes the slope formula (cf.
\cite[5.15]{BPR13}).
\begin{prop}
With the previous notation we get
$$
\overline{D_F}_{|L_x}\equiv o_x(f)
$$
\end{prop}
\begin{proof}
In order to know the value of $\overline{D_F}_{|L_x}$ we have to
evaluate $D_f$ on any edge $e$ of $L_x$. We can assume its
topological realization is of the form $P(\alpha(x,r),\alpha(x,s))$
with $r<s$. Note that, by the chosen orientation, we have
$\overline{D_F}_{|L_x}=D_f(\overline{e})=-D_F(e)$. Finally, by what
we have seen on the proof of the previous theorem or in
lemma~\ref{PoiL}, we get $D_F(e)=-o_x(f)$, so
$\overline{D_F}_{|L_x}=o_x(f)$.
\end{proof}
Next, we want to build a finite degree zero divisor associated to an
automorphic form on $\Omega_{\mathcal{L}}$. In order to get this, we have to
refine the proof of corollary~\ref{ftAut}.
First, we note that there is a ``semi-open'' (connected) tree (open
at some edges, closed at others) in ${\mathcal{T}}_K({\mathcal{L}})$ in bijection with
$G_\Gamma=\Gamma\backslash{\mathcal{T}}_K({\mathcal{L}})$ by the projection map
$\pi_\Gamma$.
To see this, take a maximal tree $T_\Gamma$ of $G_\Gamma$ and a set $E_\Gamma^c$ of adjacent closed edges of ${\mathcal{T}}_K({\mathcal{L}})$ such that its topological realization $|E_\Gamma^c|$ is a tree in bijection with $T_\Gamma$ by $\pi_\Gamma$. Next take a set of open edges $E_\Gamma^o$ of ${\mathcal{T}}_K({\mathcal{L}})$ corresponding to the open edges which form $G_\Gamma\setminus T_\Gamma$, each one of them adjacent to some edge of $E_\Gamma^c$. Then we have that $|E_\Gamma^c\cup E_\Gamma^o|$ is a subtree of ${\mathcal{T}}_K({\mathcal{L}})$ in bijection with $\pi_\Gamma(E_\Gamma^c\cup E_\Gamma^o)=G_\Gamma$, as the one we claimed the existence.
Now take
$$
U(G_\Gamma):=\left(\bigcup_{e\in E_\Gamma^c}{U(e)}\right)\bigcup\left(\bigcup_{\mathring{e}\in E_\Gamma^o}{U(\mathring{e})}\right)
$$
By construction, for $\gamma'\in\Gamma\setminus\{1_\Gamma\}$, we
have
$$U(G_\Gamma)\bigcap\left(\gamma'\cdot U(G_\Gamma)\right)=\emptyset\text{ and }\bigcup_{\gamma\in\Gamma}{\gamma\cdot U(G_\Gamma)}=\Omega_{\mathcal{L}}.$$
Consider also the set $S_\Gamma(f)=S(f)\bigcap U(G_\Gamma)$ (note that in the proof of corollary~\ref{ftAut} we used the same notation but with a slightly different meaning, since $U(E_\Gamma)\neq U(G_\Gamma)$) and the finite divisor
$$
D_f^\Gamma:=\sum_{p\in S_\Gamma(f)}{o_p(f)p}
$$
By the previous remark on unions and intersections on the orbit of $U(G_\Gamma)$ and the structure of $S(f)$, we get that the divisor of $f$ satisfies
$$
\sum_{p\in S(f)}{o_p(f)p}=\sum_{\gamma\in\Gamma}{\gamma\cdot D_f^\Gamma}
$$
\begin{prop}
An automorphic form has associated an infinite divisor with finite representant of degree zero.
\end{prop}
\begin{proof}
Because of the previous considerations, the only we have to proof is that $D_f^\Gamma$ has degree zero, that is
$$
\sum_{p\in S_\Gamma(f)}{o_p(f)}=0
$$
Next note that there is a bijection between $S_\Gamma(f)$ and $\Gamma\backslash S(f)$. Further, by the previous theorem we have
$$
\sum_{p\in S_\Gamma(f)}{o_p(f)}=\sum_{p\in S_\Gamma(f)}{\overline{D_F}_{|L_p}}=\sum_{\pi_\Gamma(p)\in S(f)/\Gamma}{\overline{D_F}_{|L_p}}
$$
Finally, applying the lemma~\ref{BPR210} to the quotient ${\mathcal{T}}_K({\mathcal{L}}_f)$, which has as ends the sets $L_p$ with $\pi_\Gamma(p)\in\Gamma\backslash S(f)$ by the lemma~\ref{quoGraph}, we get that this sum is zero, as we wanted to prove.
\end{proof}
In order to go in depth, let us take into consideration a special kind of automorphic forms: theta functions.\\
For any $p,p'\in\Omega_{\mathcal{L}}({\mathbb{C}}_K)$, the infinite product
$$
\theta(p-p';z):=\prod_{\gamma\in\Gamma}{\frac{z-\gamma p}{z-\gamma p'}}
$$
defines a meromorphic function on $\Omega_{\mathcal{L}}$, clasically called theta function.
Its construction and the properties we report are done in \cite[Ch.~2]{GvdP80}. It is an $L$-automorphic form for $\Gamma$, where $L|K$ is any complete extension of fields such that $p,p'\in\Omega_{\mathcal{L}}(L)$. If $p$ and $p'$ are in the same $\Gamma$-orbit, the theta function is analytic. If $\Gamma p\neq\Gamma p'$, then $\theta(p-p';z)$ has simple zeroes at the points of $\Gamma p$ and simple poles at the points of $\Gamma p'$ and no other zeroes or poles. The previous considerations show us that $\theta(p-p';z)$ has associated an infinite divisor on $\Omega_{\mathcal{L}}$, which is $\Gamma(p-p')=\Gamma p-\Gamma p'$. Further, for any $\delta\in\Gamma$ and $p\in\Omega_{\mathcal{L}}$, the theta function $\theta(p-\delta p;z)$ does not depend on $p$.
Next we prove a simpler version of \cite[Ch.~2~(3.2)]{GvdP80}.
\begin{thm}\label{AutTh}
Let $f$ be an automorphic form on $\Omega_{\mathcal{L}}$. There is a finite divisor $\sum_{i=1}^r{(p_i-q_i)}$ which represent the infinite divisor associated to $f$ and such that
$$
f(z)=\widetilde{f}(z)\cdot\theta(p_1-q_1;z)\cdots\theta(p_r-q_r;z)
$$
with $\widetilde{f}$ analytic function without zeroes on
$\Omega_{\mathcal{L}}$. Further, if $L$ is a field such that
$p_i,q_i\in\Omega_{\mathcal{L}}(L)$, then $f$ is $L$-automorphic.
\end{thm}
\begin{proof}
First, with the notation of the previous proposition take
$$
D_f^\Gamma=\sum_{i=1}^r{(p_i-q_i)}
$$
Second, consider the automorphic form
$$
\theta_{D_f^\Gamma}(z):=\theta(p_1-q_1;z)\cdots\theta(p_r-q_r;z)
$$
By definition, the zeroes and poles of it are the same as the ones
of $f$, so $\widetilde{f}(z):=f(z)/\theta_{D_f^\Gamma}(z)$ is an
analytic function.
The second claim is immediate.
\end{proof}
Therefore we have an infinite divisor on $\Omega_{\mathcal{L}}$ for any automorphic form. Indeed, the associated infinite divisor to the form of the theorem is
$$
\Gamma\cdot\sum_{i=1}^{r}{(p_i-q_i)}
$$
As a consequence we get a well defined degree zero divisor on $\Gamma\backslash\Omega_{\mathcal{L}}(L)=C_\Gamma(L)$.
Finally let us take into consideration $\delta\in\Gamma$ and the analytic function $\theta(p-\delta p;z)\in{\mathcal{O}}(\Omega_{\mathcal{L}})^*$ for
any $p\in\Omega_{\mathcal{L}}(K)$ (as in the previous section we assume $\Omega_{\mathcal{L}}(K)\neq\emptyset$, if necessary after a finite extension of the base field).
\begin{thm}\label{thMeas}
The image of $\theta(p-\delta p;z)$ by the
morphism
$$
\displaystyle{\tilde{\mu}:\mathcal{O}(\Omega_{\mathcal{L}})^*\longrightarrow\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}}
$$
is $\mu_\delta$. Moreover, it maps any (analytic) automorphic form to a $\Gamma$-invariant measure.
\end{thm}
\begin{proof}
In the same way we did in the previous section, we define ${\mathcal{L}}_{p}:={\mathcal{L}}\cup\overline{\Gamma\cdot p}\subset {{\mathbb{P}}^1}^*(K)$. We recall the analytic functions defined through section~\ref{poisson}.
$$
u_{\gamma p,\gamma\delta p}(z)=\frac{z-\gamma p}{z-\gamma\delta p}\in{\mathcal{O}}(\Omega_{{\mathcal{L}}_p})^*
$$
so
$$
\theta(p-\delta p;z)=\prod_{\gamma\in\Gamma}{u_{\gamma p,\gamma\delta p}(z)}\text{ on }\Omega_{{\mathcal{L}}_p}.
$$
Now, theorem~\ref{FMeas} gives us the map
$$
\tilde{\mu}:{\mathcal{O}}(\Omega_{{\mathcal{L}}_p})^*\longrightarrow\mathscr{M}({\mathcal{L}}_p^*,{\mathbb{Z}})_0
$$
by which we map the previous functions:
$$
\tilde{\mu}(\theta(p-\delta p;z))=\tilde{\mu}\left(\prod_{\gamma\in\Gamma}{u_{\gamma p,\gamma\delta p}(z)}\right)=\sum_{\gamma\in\Gamma}{\tilde{\mu}(u_{\gamma p,\gamma\delta p}(z))}=\sum_{\gamma\in\Gamma}{\mu_{\gamma\delta p^*,\gamma p^*}}
$$
where the second equality is due to the fact that $\tilde{\mu}$ commutes with limits. Thus, applying results of previous section, this
measure coincides with $-\mu_{\delta^{-1}}=\mu_\delta$ when is restricted to ${\mathcal{L}}$, so the image
of $\theta(p-\delta p;z)$ by $\tilde{\mu}$ as an analytic function on ${\mathcal{L}}$ is $\mu_\delta$.
For the second claim, let us take an analytic $K$-automorphic form $f\in{\mathcal{O}}(\Omega_{\mathcal{L}})^*$. To be automorphic means that for any $\gamma\in\Gamma$, $\gamma\cdot f=c_\gamma f$ for some $c_\gamma\in K^*$. Therefore, applying the $\Gamma$-equivariance of $\tilde{\mu}$ -recall the third part of proposition~\ref{FMeas2} and the $\Gamma$-invariance of ${\mathcal{L}}$- we get
$$
\gamma\cdot\tilde{\mu}(f)=\tilde{\mu}(\gamma\cdot f)=\tilde{\mu}(c_\gamma f)=\tilde{\mu}(c_\gamma)+\tilde{\mu}(f)=\tilde{\mu}(f)
$$
Finally, since we can apply this reasoning for any field $K$, this is true for all automorphic forms.
\end{proof}
\begin{cor}
If $f\in{\mathcal{O}}(\Omega_{\mathcal{L}})^*$ is an automorphic form, there exists a $\delta\in\Gamma$ such that $\tilde{\mu}(f)=\mu_\delta$.
\end{cor}
\begin{proof}
By the previous theorem we have $\tilde{\mu}(f)\in \mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma$ and by the isomorphism $\Gamma^{ab}\cong\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma$ (theorem~\ref{Niso}) there exists a $\delta\in\Gamma$ such that $\tilde{\mu}(f)=\mu_\delta$.
\end{proof}
We give a new proof of the complete result cited above \cite[Ch.~2~(3.2)]{GvdP80}.
\begin{cor}
All analytic automorphic forms are products of the theta functions of the form $\theta(p-\delta p;z)$ by constants.
\end{cor}
\begin{proof}
This is due to the first claim of the theorem, to the previous corollary and to the fact that the kernel of $\tilde{\mu}$ are the constants.
\end{proof}
We finish this section extending the corollary~\ref{exh}.
\begin{cor}
We have a commutative rectangle of short exact sequences with sections for each $z_0\in\Omega_{\mathcal{L}}$
$$
\xymatrix@C=1.5pc@R=1.5pc{
0\ar[rr]&&K^*\ar[rr]&&\ds{{\mathcal{O}}(\Omega_{\mathcal{L}})^*}\ar[rr]^(.45){\ds{\tilde{\mu}}}&&\ds{{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0}\ar[rr]\ar@/^1pc/[ll]^{\ds{{\mathcal{I}}_{z_0}}}&&0\\
&&&&&&&&\\
0\ar[rr]&&K^*\ar[rr]\ar@{=}[uu]&&\ds{{\mathcal{A}}_\Gamma\cap{\mathcal{O}}(\Omega_{\mathcal{L}})^*}\ar[rr]^(.45){\ds{\tilde{\mu}}}\ar@{^(->}[uu]&&\ds{{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0^\Gamma}\ar[rr]\ar@/^1pc/[ll]^{\ds{{\mathcal{I}}_{z_0}}}\ar@{^(->}[uu]&&0\\
}
$$
and with (non-canonical) isomorphisms ${{\mathcal{O}}(\Omega_{\mathcal{L}})^*\cong K^*\times{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0}$ and $${{\mathcal{A}}_\Gamma\cap{\mathcal{O}}(\Omega_{\mathcal{L}})^*\cong K^*\times{\mathcal{M}}({\mathcal{L}}^*,{\mathbb{Z}})_0^\Gamma\cong K^*\times\Gamma^{ab}}.$$
\end{cor}
\begin{proof}
We already had the first exact sequence with its section and the corresponding isomorphism by corollary~\ref{exh}. By theorem~\ref{thMeas} the map $\tilde{\mu}$ restricts to analytic automorphic forms and $\Gamma$-invariant measures. The same occurs to the section due to proposition~\ref{autMI}, so we get the exhaustivity and the isomorphism (using for the last part the theorem~\ref{Niso}).
\end{proof}
\section{The Jacobian and the Abel-Jacobi map}
Using the results of the previous sections, we show that the
jacobian and the Abel-Jacobi map of a Mumford curve can be described
in terms of multiplicative integrals. The main theorem generalize
the result of Dasgupta \cite[Thm.~2.5]{Das05} to any field complete
with respect to a non-archimedean absolute value. We give, however,
a distinct and independent proof.
Let $\Gamma\subset \PGL_2(K)$ be a Schottky group, let ${\mathcal{L}}:={\mathcal{L}}_\Gamma\subset{{\mathbb{P}}^1}^*(K)$ be its set of limit points
and let $\Omega_{\mathcal{L}}$ be the functor which associates to any complete extension of fields $L|K$ the set of
points $\Omega_{\mathcal{L}}(L)$.
Now we are going to do the following steps aimed at building an abelian variety associated to $\Gamma$ in a natural way.
Take into consideration the short exact sequence
$$
0\longrightarrow{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0\longrightarrow{\mathbb{Z}}[\Omega_{\mathcal{L}}]\longrightarrow{\mathbb{Z}}\lra0
$$
where the first arrow is the injection of divisors of degree zero
and the second arrow is the degree map. Since $\Gamma$ acts on
$\Omega_{\mathcal{L}}$, we can take the associated long homology sequence, and
in particular, the morphism
$$
\xymatrix@R=.1pc{
\Gamma^{ab}=H_1(\Gamma,{\mathbb{Z}})\ar[r]& H_0(\Gamma,{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0)={{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0}_\Gamma\\
\gamma\ar@{|->}[r]&\gamma p-p
}$$
independent of the chosen $p\in\Omega_{\mathcal{L}}$.
Since the map $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_\bullet{d}:{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0\longrightarrow\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0},{\mathbb{G}}_{m,K})}$ is $\Gamma$-equivariant we may take $\Gamma$-coinvariants so we obtain
$$
\mathop{\mathrlap{\pushMI}}\!\int_\bullet{d}:{{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0}_\Gamma\longrightarrow \mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0},{\mathbb{G}}_{m,K})_\Gamma=\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,{\mathbb{G}}_{m,K})
$$
and after composing with the connecting map above we get
$$
\xymatrix@R=.1pc{
\Gamma^{ab}\ar[r]&{{\mathbb{Z}}[\Omega_{\mathcal{L}}]_0}_\Gamma\ar[r]&\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,{\mathbb{G}}_{m,K})\\
\gamma\ar[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d}:\mu\mapsto\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu}}
}$$
Note that if ${\mathcal{L}}\neq{{\mathbb{P}}^1}^*(K)$, then we may take $p\in\Omega_{\mathcal{L}}(K)$. This occurs unless $K$ is local and $\Gamma$ is cocompact, in which case, since we may take $p$ in any complete extension $L|K$, we also have $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu}\in K^*}$.
Therefore, by theorem~\ref{Niso} we get a pairing
$$
\xymatrix@R=0pc{
\Gamma^{ab}\times\Gamma^{ab}\ar[rr]^{\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{(\ ,\ )}}}&& K^*\\
(\gamma,\gamma')\ar@{|->}[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{(\gamma,\gamma')}:=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu_{\gamma'}}}
}$$
such that, by theorem~\ref{Niso} and corollary~\ref{absI},
$$
\displaystyle{\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma')\right)=(\gamma,\gamma')_\Gamma}
$$
for all $\gamma,\gamma'\in\Gamma$. This equality implies that the pairing is positive definite. Further, using corollary~\ref{APR} we get
$$
\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{(\gamma,\gamma')}=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu_{\gamma'}}=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma' p-p}{d\mu_{\gamma}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{(\gamma',\gamma)}
$$
so the pairing is symmetric too
Summarizing, we have a morphism
$$
\xymatrix@R=0pc{
H_1(\Gamma,{\mathbb{Z}})\ar[rr]^(.35){\ds{\mathop{\mathrlap{\pushMI}}\!\int_\bullet{d}}}&&\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,{\mathbb{G}}_{m,K}):=T\\
\gamma\ar@{|->}[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d}:\mu\mapsto\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu}}
}$$
which is an isomorphism between $H_1(\Gamma,{\mathbb{Z}})\cong\Gamma^{ab}$ and its image $\Lambda$, so that it is a free group of rank $g=\rank(\Gamma)=\mbox{genus}(C_\Gamma)$.\\
Note that, as a consequence of having $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu}\in K^*}$ for any $\gamma\in\Gamma$, we get
$$\Lambda\subset T(K)=\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,K^*)\cong\mathrm{Hom}(\Gamma^{ab},K^*)\cong (K^*)^g$$
Let us consider now the valuation map applied to this:
$$
\xymatrix@R=0pc{
\displaystyle{(K^*)^g}\ar[r]^{\displaystyle{\mathit{v}_K}}&{\mathbb{R}}^g\\
(a_1,\dots,a_g)\ar@{|->}[r]&(\mathit{v}_K(a_1),\dots,\mathit{v}_K(a_g))
}$$
\begin{lem}
The subgroup $\mathit{v}_K(\Lambda)\subset {\mathbb{R}}^g$ is a lattice.
\end{lem}
\begin{proof}
Observe the way in which the isomorphism $T(K)\cong(K^*)^g$ works:
$$
\xymatrix@R=1pc{
T(K)\ar[r]&\displaystyle{(K^*)^g}\\
\ds{\mathop{\mathrlap{\pushMI}}\!\int}\ar@{|->}[r]&\displaystyle{\left(\mathop{\mathrlap{\pushMI}}\!\int(\mu_{\gamma_1}),\dots,\mathop{\mathrlap{\pushMI}}\!\int(\mu_{\gamma_g})\right)}
}
$$
where $\ds{\gamma_1,\dots,\gamma_g}$ is a fixed basis of the free group $\Gamma$.
In particular $\Lambda$ seen inside of $\ds{(K^*)^g}$ is the multiplicative subgroup
$$\ds{\left\{\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma_1),\dots,\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma_g)\right)\right\}_{\gamma\in\Gamma}}.$$
After applying the valuation map to this we get
$$
\left(\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma_1)\right),\dots,\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma_g)\right)\right)=
\left((\gamma,\gamma_1)_\Gamma,\dots,(\gamma,\gamma_g)_\Gamma\right)
$$
that is the image of the map
$$
\xymatrix@R=.1pc{
\Gamma^{ab}\ar[rr]&&\mathrm{Hom}(\Gamma^{ab},{\mathbb{R}})\cong{\mathbb{R}}^g\\
\gamma\ar@{|->}[rr]&&\displaystyle{\mathit{v}_K\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\cdot)\right)}
}
$$
As $\Gamma$ is generated by $\ds{\gamma_1,\dots,\gamma_g}$, $\displaystyle{\mathit{v}_K(\Lambda)\subset{\mathbb{R}}^g}$ is the subgroup generated by
$$
\left((\gamma_1,\gamma_1)_\Gamma,\dots,(\gamma_1,\gamma_g)_\Gamma\right),\dots,\left((\gamma_g,\gamma_1)_\Gamma,\dots,(\gamma_g,\gamma_g)_\Gamma\right)
$$
which, due to the fact that $(\ ,\ )_\Gamma$ is positive definite, is isomorphic to ${\mathbb{Z}}^g$ so it is a discrete subgroup, and has maximal rank. Therefore it is a lattice.
\end{proof}
\begin{thm}
The quotient
$$
T^{an}/\Lambda=\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,{\mathbb{G}}_{m,K})^{an}/\Lambda
$$
is an abelian variety.
\end{thm}
\begin{proof}
By \cite[6.4,~p.~171]{FvdP04} we obtain this quotient
is a proper analytic torus.\\
Note that by means of the identification $\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma\cong\Gamma^{ab}$, the torus is defined by the map $\ds{\gamma\mapsto\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{(\gamma,\cdot)}}$.\\
This torus has principal polarization
$$
\xymatrix@R=.1pc{
\Gamma^{ab}\cong\Lambda\ar[rr]^(.3){\ds{\mu^*}}&&X(T)=\mathrm{Hom}_{K-\mathcal{G}rp}(T,{\mathbb{G}}_{m,K})\cong\Gamma^{ab}\\
\gamma\ar@{|->}[rr]&&\ds{\mu^*(\gamma):\mathop{\mathrlap{\pushMI}}\!\int\longmapsto\mathop{\mathrlap{\pushMI}}\!\int(\mu_\gamma)}
}$$
since
$$\mu^*(\gamma')\left(\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d}\right)=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma p-p}{d\mu_{\gamma'}}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}(\gamma,\gamma')
$$
and this form is symmetric and positive definite. Thus, we conclude that $T^{an}/\Lambda$ is an abelian variety (\cite[Thm.~6.6.1]{FvdP04}).
\end{proof}
Our next goal is to get an isomorphism of abelian varieties
$$
Jac(C_\Gamma)\longrightarrow T/\Lambda
$$
In order to show this we are going to use the well known isomorphism $Jac(C_\Gamma)\cong \Div_0(C_\Gamma)/\Prin(C_\Gamma)$. First we will build for any extension of complete fields $L|K$ a map
$$
\Div_0(C_\Gamma)(L)\longrightarrow(T/\Lambda)(L)
$$
Then, let us fix any extension of complete fields $L|K$. Take a divisor $D\in\Div_0(C_\Gamma)(L)$. It can be written as
$$
D=\sum_{p\in C_\Gamma({\mathbb{C}}_L)}{n_pp}\qquad\text{ verifying }\qquad D^\sigma=D\ \forall\sigma\in Gal({\mathbb{C}}_L|L)
$$
and there exists a finite extension $L'|L$ such that
$\Supp(D)\subset C_\Gamma(L')$ so $D\in\Div_0(C_\Gamma(L'))$. Now,
there is a finite field extension $\tilde{L}|L'$ such that
$G_\Gamma$ has no loops (in fact, this is true for almost any
extension up to a finite number), so by corollary~\ref{corexh}, the
map $\Omega_{\mathcal{L}}(\tilde{L})\longrightarrow C_\Gamma(\tilde{L})$ is
surjective and thus, the maps
$$
\Gamma\backslash\Omega_{\mathcal{L}}(\tilde{L})\longrightarrow C_\Gamma(\tilde{L})\qquad\text{ and }\qquad
\Gamma\backslash{\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0\longrightarrow \Div_0(C_\Gamma(\tilde{L}))
$$
are isomorphisms. Thus, we got a finite extension $\tilde{L}|L$ such that there is a divisor $\tilde{D}\in {\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0$ satisfying $\pi_\Gamma(\tilde{D})=D$, that is
$$
\forall\sigma\in Gal(\tilde{L}|L)\ \exists\ \gamma_\sigma\in\Gamma\text{ such that }\tilde{D}^\sigma=\gamma_\sigma\tilde{D}.
$$
The continuous arrows of the diagram
$$
\xymatrix@R=.1pc{
{{\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0}_\Gamma\ar@{->>}[dddd]\ar[rr]^(.35){\ds{\mathop{\mathrlap{\pushMI}}\!\int_\bullet{d}}}&&\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,\tilde{L}^*)=T(\tilde{L})\ar@{->>}@/^1pc/[rdddd]&\\
\quad \tilde{D}\ar@{|->}[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}:\mu\longmapsto\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu}}\qquad&\\
&&&\\
&&&\\
\Div_0(C_\Gamma(\tilde{L}))\ar@{.>}[rrr]&&& T(\tilde{L})/\Lambda\\
\quad
\pi_\Gamma(\tilde{D})=D\ar@{|.>}[rrr]&&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{D}{d}}
}$$ factorize by the dots arrow, since
$\displaystyle{\Gamma\backslash{{\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0}\cong{{\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0}_\Gamma/H_1(\Gamma,{\mathbb{Z}})}$.
We can finish the construction of the map we told above thanks to the following result.
\begin{lem}\label{Gal} Given a finite extension $\tilde{L}|L$ and any $\tilde{D}\in {\mathbb{Z}}[\Omega_{\mathcal{L}}(\tilde{L})]_0$ satisfying
$$
\forall\sigma\in Gal(\tilde{L}|L)\ \exists\ \gamma_\sigma\in\Gamma\text{ such that }\tilde{D}^\sigma=\gamma_\sigma\tilde{D}.
$$
We have
$$
\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\right)^\sigma\equiv\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\quad (\mbox{mod }\Lambda)
$$
\end{lem}
\begin{proof}
We just have to note how it is defined the integral, as a limit of products of the function $f_{D}$. This is integrated over ${\mathcal{L}}^*$, set of $K$-rational points, so invariant by $\sigma$. Therefore, for any $\mu\in\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}$ we have
$$
\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu}\right)^\sigma\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu}\right)^{-1}=\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\tilde{D}}d\mu}\right)^\sigma\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu}\right)^{-1}=
$$
$$
=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\tilde{D}^\sigma}d\mu}\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu}\right)^{-1}=\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\gamma_\sigma \tilde{D}}d\mu}\left(\mathop{\mathrlap{\pushMI}}\!\int_{{\mathcal{L}}^*}{f_{\tilde{D}}d\mu}\right)^{-1}=
$$
$$
=\mathop{\mathrlap{\pushMI}}\!\int_{\gamma_\sigma \tilde{D}-\tilde{D}}{d\mu}
$$
independent of $\mu$. Finally $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{\gamma_\sigma \tilde{D}-\tilde{D}}{d}\in\Lambda}$.
\end{proof}
\begin{cor}
Under the same hypothesis we get
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\in(T/\Lambda)(L)
$$
\end{cor}
\begin{proof}
It is immediate.
\end{proof}
Therefore, for $D\in\Div_0(C_\Gamma)(L)$ we have build a well defined element
$$
\mathop{\mathrlap{\pushMI}}\!\int_{D}{d}\in(T/\Lambda)(L),
$$
so we get the map
$$
\Div_0(C_\Gamma)(L)\longrightarrow(T/\Lambda)(L)
$$
Next we want to show its exhaustivity and compute its kernel. The next result is crucial to move forward:
\begin{lem}
Let $\tilde{D}$ be a degree zero divisor on $\Omega_{\mathcal{L}}$ which can be represented as $\sum_{i=1}^r{(p_i-q_i)}$ and let us define the automorphic form
$$
\theta_{\tilde{D}}(z):=\theta(p_1-q_1;z)\cdots\theta(p_r-q_r;z)
$$
Then its factor of automorphy is given by
$$
c_{\theta_{\tilde{D}}}(\gamma)=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu_\gamma}\quad\forall\ \gamma\in\Gamma
$$
\end{lem}
\begin{proof}
On one hand we have
$$
c_{\theta_{\tilde{D}}}(\gamma)=\frac{\theta_{\tilde{D}}(z)}{\theta_{\tilde{D}}(\gamma z)}=\frac{\theta(p_1-q_1;z)\cdots\theta(p_r-q_r;z)}{\theta(p_1-q_1;\gamma z)\cdots\theta(p_r-q_r;\gamma z)}=
$$
$$
=\frac{\theta(z-\gamma z;p_1)\cdots\theta(z-\gamma z;p_r)}{\theta(z-\gamma z;q_1)\cdots\theta(z-\gamma z;q_r)}
$$
where the last equality is due to the straightforward symmetry of theta functions. On the other hand, applying the theorem~\ref{thMeas} and the extended Poisson formula (corollary~\ref{EPF}) we have
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu_{\gamma}}=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\tilde{\mu}(\theta(z_0-\gamma z_0;\cdot))}=\prod_{i=1}^r{\frac{\theta(z_0-\gamma z_0;p_i)}{\theta(z_0-\gamma z_0;q_i)}}
$$
Since the right sides of two last chains of equalities are independent of $z$ and $z_0$ respectively, they are equal.
\end{proof}
\begin{lem}
If $h\in{\mathcal{O}}(\Omega_{\mathcal{L}})^*$ is an (analytic) automorphic form, its factor of automorphy $c_{h}$ belongs to $\Lambda$.
\end{lem}
\begin{proof}
First, recall by the final results of the previous section that $\tilde{\mu}(h)=\mu_\delta$ for some $\delta\in\Gamma$. Next, let us compute its automorphic form on a $\gamma\in\Gamma$ by means of applying the Poisson formula:
$$
c_{h}(\gamma)=\frac{h(z)}{h(\gamma z)}=\mathop{\mathrlap{\pushMI}}\!\int_{z-\gamma z}{d\tilde{\mu}(h)}=\mathop{\mathrlap{\pushMI}}\!\int_{z-\gamma z}{d\mu_\delta}=\mathop{\mathrlap{\pushMI}}\!\int_{z-\delta z}{d\mu_\gamma}=\mathop{\mathrlap{\pushMI}}\!\int_{z-\delta z}{d}(\mu_\gamma)
$$
Finally, $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{z-\delta z}{d}}$ belongs to $\Lambda$ by definition.
\end{proof}
\begin{prop} Given an automorphic form $h\in{\mathcal{A}}_\Gamma$ with factor of automorphy $c_h$, there is a finite divisor $\tilde{D}_h$ on $\Omega_{{\mathcal{L}}}$ such that the infinite divisor of $h$ on $\Omega_{\mathcal{L}}$ is $D_h=\Gamma\cdot\tilde{D}_h$ and
$$
c_h\equiv c_{\theta_{\tilde{D}_h}}=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}_h}{d}\ (\mbox{mod }\Lambda)
$$
\end{prop}
\begin{proof}
We take $\tilde{D}_h$ a finite divisor as in theorem~\ref{AutTh}, such that $D_h=\Gamma\cdot\tilde{D}_h$ and $h(z)=h'(z)\theta_{\tilde{D}_h}(z)$ with $h'(z)$ analytic. Then, by the previous lemmas we have
$$
c_h=c_{h'}c_{\theta_{\tilde{D}_h}}\equiv c_{\theta_{\tilde{D}_h}}=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}_h}{d}\ (\mbox{mod }\Lambda)
$$
\end{proof}
\begin{cor}
The map $\displaystyle{\Div_0(C_\Gamma)(L)\longrightarrow (T/\Lambda)(L)}$ factorize by the principal divisors of $C_\Gamma$ and the resulting map
$$
\Div_0(C_\Gamma)(L)/\Prin(C_\Gamma)(L)\longrightarrow (T/\Lambda)(L)
$$
is injective.
\end{cor}
\begin{proof}
First we will show that the map factorize by the principal divisors.\\
A divisor of $Div_0(C_\Gamma)(L)$ is principal when it is the divisor of a meromorphic function on $C_\Gamma$, that is
a $\Gamma$-invariant meromorphic function on $\Omega_{\mathcal{L}}$. Let $D_h$ and $h$ be such a divisor and such a function
respectively. Since $h$ is $\Gamma$-invariant, its factor of automorphy is constant equal to 1. Therefore, by the proposition we get
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D_h}}{d}\equiv1\ (\mbox{mod }\Lambda)
$$
with $D_h=\Gamma\tilde{D}_h$, and so the factorization.
Next we want to prove the injectivity of this factorized map. Take now a $D\in\Div_0(C_\Gamma)(L)$ such that
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\in\Lambda\text{ so there exists a }\delta\in\Gamma \text{ satisfying }\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}=\mathop{\mathrlap{\pushMI}}\!\int_{\delta p-p}{d}
$$
where $D=\Gamma\tilde{D}$ with $\tilde{D}$ divisor on $\Omega_{\mathcal{L}}$ and $p\in\Omega_{\mathcal{L}}$. Now, as above we can build the automorphic form $\theta_{\tilde{D}}$, which has associated infinite divisor $D$.
Further, let us consider the analytic function $\theta(\delta p-p;z)$, and write $c_{\tilde{D}}$ and $c_\delta$ for
the factors of automorphy of the two last automorphic forms. Observe that
$$
c_{\tilde{D}}(\gamma)=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d\mu_{\gamma}}=\mathop{\mathrlap{\pushMI}}\!\int_{\delta p-p}{d\mu_\gamma}=c_\delta(\gamma).
$$
Therefore, $D$ is the divisor associated to the function
$\theta_{\tilde{D}}(z)/\theta(\delta p-p;z)$, which is
$\Gamma$-invariant, so it is principal and thus the injectivity is
done.
\end{proof}
\begin{prop}
There is an isomorphism
$$
\left(\Div_0(C_\Gamma)/\Prin(C_\Gamma)\right)(L)\longrightarrow (T/\Lambda)(L)
$$
\end{prop}
\begin{proof}
Let us check first that this map is well defined.\\
Consider a divisor $D$ in $\left(\Div_0(C_\Gamma)/\Prin(C_\Gamma)\right)(L)$. Then, there is a Galois extension $\tilde{L}|L$ and a divisor $\tilde{D}\in \Div_0(C_\Gamma)(\tilde{L})$ such that $\tilde{D}^\sigma-\tilde{D}\in Prin(C_\Gamma)(\tilde{L})$ for all $\sigma\in Gal(\tilde{L}|L)$. This implies that
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}^\sigma-\tilde{D}}{d}=0_{T/\Lambda}\in(T/\Lambda)(\tilde{L})
$$
and so, as in the proof of the lemma~\ref{Gal} we get the next equalities in $(T/\Lambda)(\tilde{L})$:
$$
\left(\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\right)^\sigma=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}^\sigma}{d}=\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}}{d}\quad\forall\sigma\in Gal(\tilde{L}|L)
$$
Therefore $\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_D{d}\in(T/\Lambda)(L)}$ and we get the morphism
$$
\left(\Div_0(C_\Gamma)/\Prin(C_\Gamma)\right)(L)\longrightarrow (T/\Lambda)(L)
$$
which is injective by the previous corollary.
Next we have to prove its exhaustivity. An element $\Xi\in
(T/\Lambda)(L)$ can be seen in $T(\tilde{L})/\Lambda$, satisfying
$\Xi^\sigma=\Xi$ for each $\sigma\in Gal(\tilde{L}|L)$, where
$\tilde{L}|L$ is a Galois extension. This element is the class of a
$\xi\in T(\tilde{L})\cong \mathrm{Hom}(\Gamma^{ab},\tilde{L}^*)$
such that
$$
\xi^\sigma\equiv\xi\ (\mbox{mod }\Lambda)\quad\text{ for each }\quad\sigma\in Gal(\tilde{L}|L),
$$
which in turn is the factor of automorphy $c_h$ of an automorphic form $h\in{\mathcal{A}}_\Gamma$, by the proposition~\ref{autexh}. Now, by the proposition above we have
$$
\mathop{\mathrlap{\pushMI}}\!\int_{\tilde{D}_h}{d}\equiv c_h=\xi\ (\mbox{mod }\Lambda) \qquad\text{ and so }\qquad\mathop{\mathrlap{\pushMI}}\!\int_{D_h}{d}=\Xi
$$
with $D_h\in \Div_0(C_\Gamma)(\tilde{L})$. By the hypothesis
$$
\left(\mathop{\mathrlap{\pushMI}}\!\int_{D_h}{d}\right)^\sigma=\mathop{\mathrlap{\pushMI}}\!\int_{D_h}{d}\qquad\text{ so }\qquad\mathop{\mathrlap{\pushMI}}\!\int_{D_h^\sigma-D_h}{d}=0_{T/\Lambda}
$$
what, due to the injectivity of the map, gives that $D_h^\sigma-D_h\in\Prin(C_\Gamma)(\tilde{L})$. But this for each $\sigma\in Gal(\tilde{L}|L)$ implies that $D_h\in \left(\Div_0(C_\Gamma)/\Prin(C_\Gamma)\right)(L)$.
\end{proof}
Now we are ready to prove the main theorem, which generalizes to any complete field with respect to a non-trivial non-archimedean valuation \cite[Thm.~2.5]{Das05}:
\begin{thm}\label{mainT}
There is an isomorphism over $K$ of abelian varieties
$$
Jac(C_\Gamma)\longrightarrow T/\Lambda
$$
\end{thm}
\begin{proof}
First, as we told above, we recall the isomorphism
$$
Jac(C_\Gamma)\cong \Div_0(C_\Gamma)/\Prin(C_\Gamma)
$$
Second, we have built an analytic morphism of abelian varieties
$$
\Div_0(C_\Gamma)/\Prin(C_\Gamma)\longrightarrow T/\Lambda
$$
Since they are proper, by GAGA it is an algebraic morphism, and it also respects the group operations, so it is a morphism of abelian varieties. Further, it induces an isomorphism in the corresponding $L$-points for any extension of complete fields $L|K$, and this implies that it is an isomorphism.
\end{proof}
\begin{cor}
The abelian variety $T^{an}/\Lambda$ is the Jacobian of the curve $C_\Gamma$ and the Abel Jacobi map is given, after having fixed some point $z_0\in C_\Gamma$, by
$$
\xymatrix@R=.1pc{
C_\Gamma\ar[rr]^(.25){\displaystyle{i_{z_0}}}&&\displaystyle{\mathrm{Hom}(\mathscr{M}({\mathcal{L}}^*,\mb{Z})_{0}^\Gamma,{\mathbb{G}}_{m,K})/\Lambda}\\
z\ar@{|->}[rr]&&\displaystyle{\mathop{\mathrlap{\pushMI}}\!\int_{z-z_0}{d}}
}
$$
\end{cor}
|
2,877,628,090,789 | arxiv | \section{Introduction}
We consider the following Mac-Laurin development
\begin{equation}\label{x9ed7za}
\frac{(1+xz)^\alpha}{(1-z)^\beta}=\sum_{n=0}^\infty{A_n(\alpha,\beta,x)}z^n,
\end{equation}
where $\alpha>0, \ \beta>0, \ x=e^{i\theta}, \ \theta\in[-\pi,\pi],$
and $z\in{U.}$ It is easily seen, that the radius of convergence of the series (\ref{x9ed7za}) is equal to $1.$
In \cite{5} the author conjectured, that if $\alpha>0, \ \beta>0$ and $|x|=1,$ then $$|A_{2n-1}(\alpha,\beta,x)|\leq{A_{2n-1}(\alpha,\beta,1)},$$
where $n$ is a natural number.
Partial results regarding this question were already proved in \cite{1}, \cite{2}, \cite{5}, \cite{8}.\\
The case $\beta=1,$ $\alpha\in(0,1)$ is still open. Regarding this case partial results were obtained in \ \cite{3}, \cite{4}, \cite{6}, \cite{7}.
We will prove some partial results regarding the case $\beta=1,$ and $\alpha\in(0,1).$ In order to do this we will use an integral representation proved in \ \cite{3}, and the fact that the conjecture was proved for $2n-1\leq51$ in \cite{6}.
\section{Preliminaries}
We inroduce the notation:
$$A_n(\alpha,1,x)=A_n(\alpha,x).$$
It is easily seen that
\begin{eqnarray}\label{n8m9nn}
A_{2n-1}(\alpha,x)=\sum_{k=0}^{2n-1}\frac{(-\alpha)_k(-x)^k}{k!}=1+\frac{\alpha}{1!}x-\frac{\alpha(1-\alpha)}{2!}x^2+\frac{\alpha(1-\alpha)(2-\alpha)}{3!}x^3\\-\frac{\alpha(1-\alpha)(2-\alpha)(3-\alpha)}{4!}x^4+\ldots+
\frac{\alpha(1-\alpha)(2-\alpha)\ldots(2n-2-\alpha)}{(2n-1)!}x^{2n-1}.\nonumber
\end{eqnarray}
We denote $$B(t,\theta)=\frac{\cos\theta+t+t^{2n-1}\cos2n\theta+t^{2n}\cos(2n-1)\theta}{1+t^2+2t\cos\theta},$$
$$C(t,\theta)=\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}.$$
We need the following lemmas in our study.
\begin{lemma
For $\alpha\in(0,1)$ the following equality holds:
\begin{eqnarray}
\Phi(\theta)=-\Gamma(\alpha)\Gamma(-\alpha)A_{2n-1}(\alpha,e^{i\theta})=\int_0^1F(t)\Big[\frac{1}{\alpha}+\sum_{k=1}^{2n-1}(-t)^{k-1}e^{ki\theta}\Big]dt\nonumber\\=\int_0^1F(t)\Big[\frac{1}{\alpha}+B(t,\theta)+iC(t,\theta)\Big]dt,\nonumber
\end{eqnarray}
where $F'(t)=-t^{-1-\alpha}(1-t)^{\alpha-1},$ and $F(1)=0.$
\end{lemma}
\begin{proof}
We use two equalities in order to prove the assertion of the lemma, the first one is the following:
\begin{eqnarray}\label{1745fd55dw}A_n(\alpha,x)=1+\frac{1}{\Gamma(\alpha)\Gamma(-\alpha)}\int_0^1\Big(\sum_{k=1}^n\frac{(-tx)^k}{k}\Big)t^{-\alpha-1}(1-t)^{\alpha-1}dt,
\end{eqnarray}
which has been deduced in \cite{3},
while the second one is the well-known equality $B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$
Replacing $p=\alpha$ and $q=1-\alpha,$ we get
$$\int_0^1t^{\alpha-1}(1-t)^{-\alpha}dt=\frac{\Gamma(\alpha)\Gamma(1-\alpha)}{\Gamma(1)}=-\alpha\Gamma(\alpha)\Gamma(-\alpha)=\int_0^1t^{-\alpha}(1-t)^{\alpha-1}dt.$$
This equality and (\ref{1745fd55dw}) imply that
\begin{eqnarray}\Gamma(\alpha)\Gamma(-\alpha)A_n(\alpha,x)=\frac{1}{\alpha}\int_0^1tF'(t)dt-\int_0^1\Big(\sum_{k=1}^n\frac{(-tx)^k}{k}\Big){F'(t)}dt,\nonumber
\end{eqnarray}
or equivalently
\begin{eqnarray}\label{1dp3k07}-\Gamma(\alpha)\Gamma(-\alpha)A_n(\alpha,x)=\int_0^1{F'(t)}\Big(\frac{-t}{\alpha}+\sum_{k=1}^n\frac{(-tx)^k}{k}\Big)dt.\nonumber
\end{eqnarray}
Now integrating by parts and using that $\lim_{t\rightarrow0}F(t)\Big(\frac{t}{\alpha}+\sum_{k=1}^n\frac{(-tx)^k}{k}\Big)=0$ and $\lim_{t\rightarrow1}F(t)\Big(\frac{t}{\alpha}+\sum_{k=1}^n\frac{(-tx)^k}{k}\Big)=0,$ we infer that
\begin{eqnarray}\label{2ww4dp3k07}-\Gamma(\alpha)\Gamma(-\alpha)A_n(\alpha,x)=\int_0^1{F(t)}\Big(\frac{1}{\alpha}+\sum_{k=1}^n{(-t)^{k-1}}x^k\Big)dt.\nonumber
\end{eqnarray}
We get the desired equality replacing $n$ by $2n-1$.
\end{proof}
\begin{remark}
It is easily seen that the condition $\alpha\in(0,1)$ implies the existence of the integrals in the previous lemma and its proof.
\end{remark}
We denote
\begin{eqnarray}\label{234n234}
\Phi(\theta)=-\Gamma(\alpha)\Gamma(-\alpha)A_{2n-1}(\alpha,e^{i\theta})=\int_0^1F(t)\Big[\frac{1}{\alpha}\nonumber\\+\frac{\cos\theta+t+t^{2n-1}\cos2n\theta+t^{2n}\cos(2n-1)\theta}{1+t^2+2t\cos\theta}\\+i\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}\Big]dt\nonumber.
\end{eqnarray}
Since $|\Phi(\theta)|^2\in\mathbb{R}$ it follows that
\begin{eqnarray}\label{a3x234n2x34}
|\Phi(\theta)|^2=\Phi(\theta)\overline{\Phi(\theta)}=\Big[\int_0^1F(t)\Big(\frac{1}{\alpha}\nonumber\\+\frac{\cos\theta+t+t^{2n-1}\cos2n\theta+t^{2n}\cos(2n-1)\theta}{1+t^2+2t\cos\theta}\nonumber\\+i\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}\Big)dt\Big]\nonumber\\\Big[\int_0^1F(v)\Big(\frac{1}{\alpha}+\frac{\cos\theta+v+v^{2n-1}\cos2n\theta+v^{2n}\cos(2n-1)\theta}{1+v^2+2v\cos\theta}\\-i\frac{\sin\theta+v^{2n-1}\sin2n\theta+v^{2n}\sin(2n-1)\theta}{1+v^2+2v\cos\theta}\Big)dv\Big]\nonumber\\=\int_0^1\int_0^1F(t)F(v)\Big[\Big(\frac{1}{\alpha}+\frac{\cos\theta+t+t^{2n-1}\cos2n\theta+t^{2n}\cos(2n-1)\theta}{1+t^2+2t\cos\theta}\Big)\nonumber\\
\Big(\frac{1}{\alpha}+\frac{\cos\theta+v+v^{2n-1}\cos2n\theta+v^{2n}\cos(2n-1)\theta}{1+v^2+2v\cos\theta}\Big)\nonumber\\
+\Big(\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}\Big)\nonumber\\
\Big(\frac{\sin\theta+v^{2n-1}\sin2n\theta+v^{2n}\sin(2n-1)\theta}{1+v^2+2v\cos\theta}\Big)\Big].\nonumber
\end{eqnarray}
\begin{lemma
(a) \ Let $f,g:[0,1]\rightarrow\mathbb{R}$ be two continuous function. If there is a point $t^*\in(0,1)$ such that $f$ is decreasing on $(t^*,1),$ and the equation $g(t)=0$ has a unique root $t_0\in[t^*, 1), $ such that $g(t)\leq0, \ t\in[t_0,1],$ \ $g(t)\geq0, \ t\in{[0,t_0]},$ and $f(v)\geq{f(t_0)}$ for $v\in[0,t^*],$ then we have $$\int_0^1f(t)g(t)dt\geq{f(t_0)}\int_0^1g(t)dt.$$
(b) \ Let $f,g:[0,1]\rightarrow\mathbb{R}$ two continuous functions. If $f$ is a decreasing function, and if there is a point $t_0\in(0,1)$ such that $g(t)\geq0, \ t\in(0,t_0)$ and $g(t)\leq0, \ t\in(t_0,1),$ then
$$\int_0^1f(t)g(t)dt\geq{f(t_0)}\int_0^1g(t)dt.$$
The statement (b) is a particular case of (a).\\
(c) \ A well-known result is the following statement. (Chebyshev's inequality) If $f$ and $g$ are monotonic functions with different monotony, then $$\int_0^1f(t)g(t)dt\leq\int_0^1f(t)dt\int_0^1g(t)dt$$
and in case of the same monotony we have $$\int_0^1f(t)g(t)dt\geq\int_0^1f(t)dt\int_0^1g(t)dt.$$
\end{lemma}
\begin{proof}
We have
\begin{eqnarray}\int_0^1f(t)g(t)dt=\int_0^{t_0}f(t)g(t)dt+\int_{t_0}^1f(t)g(t)dt\geq\int_0^{t_0}f(t)g(t_0)dt\nonumber\\+\int_{t_0}^1f(t)g(t_0)dt=\int_{0}^1f(t)g(t_0)dt.\nonumber\end{eqnarray}
\end{proof}
\begin{lemma
If $\theta \in[0,\frac{\pi}{2}],$ and $n\geq27,$ then
\begin{eqnarray}\label{1q2} \ \ \ \ \ (a) \ \ \ \ \ \ \ \int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ \ \nonumber\\\geq\int_0^1F(t)\frac{\frac{1}{2}(1-\cos{\theta})+t^{2n-1}(1-\cos2n\theta)+t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt,\end{eqnarray}
\begin{eqnarray}\label{z3x61q2} \ \ \ \ \ \ (b) \ \ \ \ \ \ \ \ \int_0^1F(t)\Big(1+B(t,\theta)\Big)dt \ \ \ \ \ \nonumber\\\geq\int_0^1F(t)\frac{(1+t)(1+\cos{\theta})+t^{2n-1}(1+\cos2n\theta)+t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}dt.\end{eqnarray}
\end{lemma}
\begin{proof} According to Lemma 2 (c), we have
\begin{equation}\label{1234}
\int_0^1\frac{t^{2n}}{1+t}dt\leq\int_0^1{t^{2n}}dt\int_0^1\frac{1}{1+t}dt=\frac{\ln2}{2n+1}.
\end{equation}
We use assertion (b) of Lemma 2 putting $f(t)=\frac{F(t)}{1+t^2+2t\cos\theta}$ and $g(t)=\frac{\frac{1}{2}-t-2t^{2n}}{1+t}.$ If $\theta\in\big[0,\frac{\pi}{2}\big],$ then the mapping $f:[0,1]\rightarrow[0,\infty)$ is strictly decreasing and we get
\begin{eqnarray}\label{1tds0nm}\int_0^1F(t)\frac{\frac{1}{2}-t-2t^{2n}}{(1+t)(1+t^2+2t\cos\theta)}dt\geq\frac{F(t_n)}{1+t_n^2+2t_n\cos\theta}\int_0^1\Big(\frac{\frac{1}{2}-t}{1+t}\nonumber\\-2\frac{t^{2n}}{1+t}\Big)dt\geq\frac{F(t_n)}{1+t_n^2+2t_n\cos\theta}\Big(\frac{3}{2}\ln2-1-\frac{2\ln2}{2n+1}\Big)>0,
\end{eqnarray}
where $t_n$ denotes the unique root of the equation $\frac{1}{2}-t-2t^{2n}=0,$ in the interval $ \ t\in(0,1).$
The following equality holds:
\begin{eqnarray}\label{x7v1q2}\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ =\int_0^1F(t)\frac{\frac{1}{2}(1-\cos{\theta})}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\+\int_0^1F(t)\frac{(t^{2n}+t^{2n-1})(1-\cos2n\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt+\int_0^1F(t)\frac{(t^{2n}+t^{2n+1})(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\
+\int_0^1F(t)\frac{(\frac{1}{2}-t-2t^{2n})(1-\cos\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt. \ \ \ \ \end{eqnarray}
The equality (\ref{x7v1q2}) and the inequality (\ref{1tds0nm}) imply that
\begin{eqnarray}\label{x1mn7v1q2}\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ \geq\int_0^1F(t)\frac{\frac{1}{2}(1-\cos{\theta})}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\+\int_0^1F(t)\frac{t^{2n-1}(1-\cos2n\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt+\int_0^1F(t)\frac{t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt. \ \
\end{eqnarray}
We use assertion (b) of Lemma 2 putting $f(t)=\frac{F(t)}{1+t^2+2t\cos\theta}$ and $g(t)={t^2-2t^{2n}-t^{2n-1}}.$ If $\theta\in[0,\frac{\pi}{2}],$ then $f$ is strictly decreasing and we get
\begin{eqnarray}\int_0^1F(t)\frac{t^{2}-t^{2n}-t^{2n-1}}{1+t^2+2t\cos\theta}dt\geq\frac{{F}(t_n)}{1+t_n^2+2t_n\cos\theta}\int_0^1({t^{2}-t^{2n}-t^{2n-1}})dt\nonumber\\=\frac{{F}(t_n)}{1+t_n^2+2t_n\cos\theta}\Big(\frac{1}{3}-\frac{1}{2n+1}-\frac{1}{2n}\Big)>0.\end{eqnarray}
In order to finish the proof of the second inequality, we take notice of the fact that in case $\theta\in[0,\frac{\pi}{2}]$ each member of the following sum is positive:
\begin{eqnarray}
\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt= \int_0^1F(t)\frac{(1+t)(1+\cos{\theta})}{1+t^2+2t\cos\theta}dt\nonumber\\+\int_0^1F(t)\frac{t^{2n-1}(1+\cos2n\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt+\int_0^1F(t)\frac{t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}dt\nonumber\\+\int_0^1F(t)\frac{t^{2}-t^{2n}-t^{2n-1}}{1+t^2+2t\cos\theta}dt+ \int_0^1F(t)\frac{t\cos\theta}{1+t^2+2t\cos\theta}dt.
\nonumber\end{eqnarray}
Thus we get
\begin{eqnarray}
\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\geq \int_0^1F(t)\frac{(1+t)(1+\cos{\theta})}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\+\int_0^1F(t)\frac{t^{2n-1}(1+\cos2n\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt+\int_0^1F(t)\frac{t^{2n}(1+\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt,
\nonumber\end{eqnarray}
and the proof is done.
\end{proof}
\begin{lemma
If $\theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big],$ then
\begin{eqnarray}\label{jm5nnz4x1q2}\frac{5}{2}+\frac{t+\cos\theta}{1+t^2+2t\cos\theta}\geq\frac{50}{23}\frac{(1+t)(1+\cos\theta)}{1+t^2+2t\cos\theta}, \ \ (\forall) \ t\in[0,1].\end{eqnarray}
\end{lemma}
\begin{proof}
The inequality (\ref{jm5nnz4x1q2}) is equivalent to
\begin{eqnarray}\label{gjm5s4nnz4x1q2}f(t)=\frac{5}{2}t^2+t\Big(\frac{65}{23}\cos\theta-\frac{27}{23}\Big)+\frac{15}{46}-\frac{27}{23}\cos\theta\geq0, \ \ (\forall) \ t\in[0,1].\end{eqnarray}
The function $f$ has a minimum point at $t^*=\frac{65}{115}\cos\theta-\frac{27}{115}\in(0,1),$ for every $\theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].$
Thus we get
$$f(t)\geq{f(t^*)}=\frac{15}{46}-\frac{27}{23}\cos\theta-\frac{1}{10}\Big(\frac{27-65\cos\theta}{23}\Big)^2, \ (\forall) \ t\in[0,1] \ \textrm{and} \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].$$
Consequently in order to prove (\ref{jm5nnz4x1q2}) we have to show that
$$\frac{15}{46}-\frac{27}{23}\cos\theta-\frac{1}{10}\Big(\frac{27-65\cos\theta}{23}\Big)^2\geq0, \ (\forall) \ \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].$$
Using the notation $x=\frac{13-35\cos\theta}{35},$ we get
$$\frac{5}{12}-\frac{13}{12}\cos\theta-\frac{5}{2}\Big(\frac{13-35\cos\theta}{60}\Big)^2=-\frac{483}{2990}+\frac{27}{65}x-\frac{1}{10}x^2.$$
$$ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big]\Rightarrow \ x\in\Big[\frac{37}{23},\frac{119}{46}\Big].$$
The equality implies
\begin{eqnarray}\frac{5}{12}-\frac{13}{12}\cos\theta-\frac{5}{2}\Big(\frac{13-35\cos\theta}{60}\Big)^2\geq\min_{ x\in[\frac{37}{23},\frac{119}{46}\Big]}\Big\{-\frac{483}{2990}+\frac{27}{65}x-\frac{1}{10}x^2\Big\}\nonumber\\
\geq\min_{ x\in\Big[\frac{3}{2},\frac{13}{5}\Big]}\Big\{-\frac{1}{5}+\frac{5}{13}x-\frac{1}{10}x^2\Big\}=\min\Big\{g\Big(\frac{3}{2}\Big),g\Big(\frac{13}{5}\Big)\Big\}\nonumber\\=g\Big(\frac{13}{5}\Big)=\frac{31}{250}, \ \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big],\nonumber
\end{eqnarray}
where $g(x)=-\frac{1}{5}+\frac{5}{13}x-\frac{1}{10}x^2.$
and consequently (\ref{jm5nnz4x1q2}) holds.
\end{proof}
\begin{lemma
If $\theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big],$ and $n\geq27,$ then
\begin{eqnarray}\label{z4x1q2}\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ \ \nonumber\\\geq\int_0^1F(t)\frac{\frac{27}{50}(1-\cos{\theta})+t^{2n-1}(1-\cos2n\theta)+t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt \ \ \ \ \end{eqnarray}
\begin{eqnarray}\label{q3sz3x61q2}\int_0^1F(t)\Big(2+B(t,0)+B(t,\theta)\Big)dt\geq\int_0^1F(t) \ \ \ \ \ \nonumber\\\frac{\frac{50}{27}(1+t)(1+\cos{\theta})+2t^{2n-1}(1+\cos2n\theta)+2t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}dt \ \ \end{eqnarray}
\end{lemma}
\begin{proof}
We use assertion (b) of Lemma 2, putting $f(t)=\frac{F(t)}{(1+t)(1+t^2+2t\cos\theta)}$ and $g(t)={\frac{27}{50}-t-2t^{2n}}.$ If $\theta\in\big[\frac{\pi}{2},\frac{2\pi}{3}\big]$ the mapping $f:[0,1]\rightarrow[0,\infty)$ is strictly decreasing and we get
\begin{eqnarray}\label{1td5fs0mb6nn2m}\int_0^1F(t)\frac{\frac{27}{50}-t-2t^{2n}}{(1+t)(1+t^2+2t\cos\theta)}dt\geq\frac{F(t_n)}{(1+t_n)(1+t_n^2+2t_n\cos\theta)}\int_0^1\Big({\frac{27}{50}-t}\nonumber\\-2{t^{2n}}\Big)dt>\frac{F(t_n)}{(1+t_n)(1+t_n^2+2t_n\cos\theta)}\Big(\frac{27}{50}-\frac{1}{2}-\frac{2}{2n+1}\Big)>0 \ \ \ \
\end{eqnarray}
where $t_n$ denotes the unique root of the equation $\frac{27}{50}-t-2t^{2n}=0,$ in the interval $ \ (0,1).$\\
\begin{eqnarray}\label{2b3mx7v1q2}\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ =\int_0^1F(t)\frac{\frac{23}{50}(1-\cos{\theta})}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\+\int_0^1F(t)\frac{(t^{2n}+t^{2n-1})(1-\cos2n\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt+\int_0^1F(t)\frac{(t^{2n}+t^{2n+1})(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt\nonumber\\
+\int_0^1F(t)\frac{\big(\frac{27}{50}-t-2t^{2n}\big)(1-\cos\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt v\ \ \ \ \ \ \end{eqnarray}
The equality (\ref{2b3mx7v1q2}) and the inequality (\ref{1td5fs0mb6nn2m}) imply (\ref{z4x1q2}).\\
In order to prove the second inequality, we remark that
\begin{eqnarray}\label{xzzxy4xvc44fg} \int_0^1F(t)\Big(2+B(t,0)+B(t,\theta)\Big)dt= \int_0^1F(t)\Big(\frac{5}{2}+\frac{1+t^{2n-1}}{1+t}-\frac{1}{2}+B(t,\theta)\Big)dt\nonumber\\
= \int_0^1F(t)\Big(\frac{5}{2}+\frac{1+2t^{2n-1}-t}{2(1+t)}+\frac{t+\cos\theta+t^{2n}\cos(2n-1)\theta+t^{2n-1}\cos(2n\theta)}{1+t^2+2t\cos\theta}\Big)dt \ \ \ \ \\
= \int_0^1F(t)\Big(\frac{5}{2}+\frac{t+\cos\theta}{1+t^2+2t\cos\theta}+\frac{1+2t^{2n-1}-t}{2(1+t)}+\frac{t^{2n}\cos(2n-1)\theta+t^{2n-1}\cos(2n\theta)}{1+t^2+2t\cos\theta}\Big)dt\nonumber\\
= \int_0^1F(t)\Big(\frac{5}{2}+\frac{t+\cos\theta}{1+t^2+2t\cos\theta}+\frac{2t^{2n}(1+\cos(2n-1)\theta)+2t^{2n-1}(1+\cos(2n\theta))}{1+t^2+2t\cos\theta}\Big)dt\nonumber\\
+ \int_0^1F(t)\frac{2t^{2n-1}}{1+t}dt+ \int_0^1F(t)\Big(\frac{1-t}{2(1+t)}-\frac{t^{2n}(2+\cos(2n-1)\theta)+t^{2n-1}(2+\cos(2n\theta))}{1+t^2+2t\cos\theta}\Big)dt\nonumber
\end{eqnarray}
We put $g_1(t)=1-2t+2t^2-t^3-24t^{2n-1}$ $g_2(t)=2t^3-8t^{2n-1},$ and $f(t)=\frac{F(t)}{2(1+t^3)}$ in the assertion (b) of Lemma 2, and we get
\begin{eqnarray}\label{d2s1xcc4b3fm12nijf}
\int_0^1F(t)\Big(\frac{1-t}{2(1+t)}-\frac{t^{2n}(2+\cos(2n-1)\theta)+t^{2n-1}(2+\cos(2n\theta))}{1+t^2+2t\cos\theta}\Big)dt\nonumber\\
\geq \int_0^1F(t)\Big(\frac{1-t}{2(1+t)}-\frac{3t^{2n}+3t^{2n-1}}{1+t^2+2t\cos\theta}\Big)dt\geq \int_0^1F(t)\Big(\frac{1-t}{2(1+t)}\\-\frac{6t^{2n-1}}{1+t^2-t}\Big)dt
\geq \int_0^1\frac{F(t)}{2(1+t^3)}\Big(1-2t+2t^2-t^3-12t^{2n-1}(1+t)\Big)dt \nonumber\\
\geq \int_0^1\frac{F(t)}{2(1+t^3)}\Big(1-2t+2t^2-t^3-24t^{2n-1}\Big)dt \nonumber\\
\frac{F(t^*)}{2(1+(t^*)^3)} \int_0^1\Big(1-2t+2t^2-t^3-24t^{2n-1}\Big)dt\nonumber\\ = \frac{F(t^*)}{2(1+(t^*)^3)} \Big(\frac{5}{12}-\frac{12}{n} \Big)>0, \ \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].\nonumber
\end{eqnarray}
Finally equality (\ref{xzzxy4xvc44fg}), Lemma 4 and inequality (\ref{d2s1xcc4b3fm12nijf}) imply (\ref{q3sz3x61q2}), and the proof is done.
\end{proof}
\section{The Main Result}
\begin{theorem}
If $n$ is a natural number, $n\geq52$ and $\alpha\in(0,1)$ then the following inequality holds
\begin{equation}\label{us5n6sw}
|A_{2n-1}(\alpha,e^{i\theta})|\leq|A_{2n-1}(\alpha,1)|, \ \ \textrm{for \ all} \ \theta\in[-\frac{\pi}{2},\frac{\pi}{2}].
\end{equation}
\end{theorem}
\begin{proof}
According to (\ref{234n234}) and (\ref{a3x234n2x34}) the inequality (\ref{us5n6sw}) is equivalent to \begin{equation}\label{lfd8sn}|\Phi(0)|^2\geq|\Phi(\theta)|^2, \ \theta\in[-\pi,\pi]\Leftrightarrow |\Phi(0)|^2-|\Phi(\theta)|^2\geq0, \ \theta\in[-\frac{\pi}{2},\frac{\pi}{2}].\end{equation}
We denote $$B(t,\theta)=\frac{\cos\theta+t+t^{2n-1}\cos2n\theta+t^{2n}\cos(2n-1)\theta}{1+t^2+2t\cos\theta},$$
$$C(t,\theta)=\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}.$$
The equality (\ref{a3x234n2x34}) implies that
\begin{eqnarray}\label{b5x3yo04d6z}
\Phi^2(0)-|\Phi(\theta)|^2=\int_0^1\int_0^1F(t)F(v)\Big[\Big(\frac{1}{\alpha}+B(t,0)\Big)\Big(\frac{1}{\alpha}+B(v,0)\Big) \\-\Big(\frac{1}{\alpha}+B(t,\theta)\Big)\Big(\frac{1}{\alpha}+B(v,\theta)\Big)-C(t,\theta)C(v,\theta)\Big]dtdv\nonumber\\
=\int_0^1\int_0^1F(t)F(v)\Big[\frac{1}{\alpha}\Big(B(t,0)-B(t,\theta)\Big)+\frac{1}{\alpha}\Big(B(v,0)-B(v,\theta)\Big)\nonumber\\+B(t,0)B(v,0)-B(t,\theta)B(v,\theta)-C(t,\theta)C(v,\theta)\Big]dtdv.\nonumber
\end{eqnarray}
It is easily seen that $\Phi^2(0)-|\Phi(\theta)|^2$ is an even function with respect to $\theta,$ consequently we have to prove (\ref{lfd8sn}) only for $\theta\in[0,\frac{\pi}{2}].$\\
Lemma 3, (a) implies that in case $\theta\in[0,\frac{\pi}{2}],$ the following inequality holds $\int_0^1F(t)\Big(B(t,0)-{B}(t,\theta)\Big)dt\geq0.$
Thus we infer that
\begin{eqnarray}\label{f5b5xy4d6z}
\Phi^2(0)-|\Phi(\theta)|^2=\int_0^1\int_0^1F(t)F(v)\Big[\Big(B(t,0)-B(t,\theta)\Big)\Big(1+B(v,\theta)\Big)\nonumber\\+\Big(B(v,0)-B(v,\theta)\Big)\Big(1+B(t,0)\Big)\Big]dtdv-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv\nonumber\\
+\int_0^1\int_0^1F(t)F(v)\Big(B(t,0)-B(t,\theta)\Big)\Big(B(v,0)-B(v,\theta)\Big)dtdv\nonumber\\=\int_0^1\int_0^1F(t)F(v)\Big[\Big(B(t,0)-B(t,\theta)\Big)\Big(1+B(v,\theta)\Big)\nonumber\\+\Big(B(v,0)-B(v,\theta)\Big)\Big(1+B(t,0)\Big)\Big]dtdv-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv\nonumber\\
+\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv\nonumber\\\nonumber\\\geq\int_0^1\int_0^1F(t)F(v)\Big[\Big(B(t,0)-B(t,\theta)\Big)\Big(1+B(v,\theta)\Big)\nonumber\\+\Big(B(v,0)-B(v,\theta)\Big)\Big(1+B(t,0)\Big)\Big]dtdv-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv\nonumber
\end{eqnarray}
This inequality is equivalent to
\begin{eqnarray}\label{w1w2c4x2zz9}
\Phi^2(0)-|\Phi(\theta)|^2\geq\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(1+B(v,\theta)\Big)dv\nonumber\\+
\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv \ \ \ \ \ \ \\-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv \ \ \ \ \ \ \nonumber
\end{eqnarray}
The inequality between the arithmetic and geometric means leads to
\begin{eqnarray}\label{111w1w2c4x2zz9}
\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(1+B(v,\theta)\Big)dv\nonumber\\+
\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv \ \ \ \ \ \ \\-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv \ \ \ \ \ \ \nonumber\\
\geq2\Big[\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(1+B(v,\theta)\Big)dv\nonumber\\
\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv\Big]^{\frac{1}{2}} \nonumber \\-\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv\nonumber\\
=2\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \nonumber \\-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\nonumber
\end{eqnarray}
The Cauchy-Schwarz inequality for integrals implies
\begin{eqnarray}\label{wndlkoop}
2\int_0^1F(t)\Big(1+B(t,\theta)\Big)dt\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt \ \ \ \ \ \ \nonumber \\-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2
\geq2\Big\{\int_0^1F(t)\Big[\Big(1+B(t,\theta)\Big)\Big(B(t,0)-B(t,\theta)\Big)\Big]^{\frac{1}{2}}dt\Big\}^2 \ \ \ \ \\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2 \ \ \ \ \ \nonumber
\end{eqnarray}
Lemma 3 implies
\begin{eqnarray}\label{ed55f}
2\Big\{\int_0^1F(t)\Big[\Big(1+B(t,\theta)\Big)\Big(B(t,0)-B(t,\theta)\Big)\Big]^{\frac{1}{2}}dt\Big\}^2 \ \ \ \ \\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\geq\nonumber\\
2\Big(\int_0^1F(t)\sqrt{\frac{\frac{1}{2}(1-\cos{\theta})+t^{2n-1}(1-\cos2n\theta)+t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}}\nonumber\\
\sqrt{\frac{(1+t)(1+\cos{\theta})+t^{2n-1}(1+\cos2n\theta)+t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}}dt\Big)^2\nonumber\\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2.\nonumber
\end{eqnarray}
Putting $a_1^2=\frac{\frac{1}{2}(1-\cos{\theta})}{(1+v)(1+v^2+2v\cos\theta)},$ \ $b_1^2=\frac{(1+t)(1+\cos{\theta})}{1+v^2+2v\cos\theta}$ and so an, in the inequality
$$\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}\geq{|a_1b_1|+|a_2b_2|+|a_3b_3|},$$
we get
\begin{eqnarray}\label{xcs3bnm3xcxc2}
2\Big(\int_0^1F(t)\sqrt{\frac{\frac{1}{2}(1-\cos{\theta})+t^{2n-1}(1-\cos2n\theta)+t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}}\nonumber\\
\sqrt{\frac{(1+t)(1+\cos{\theta})+t^{2n-1}(1+\cos2n\theta)+t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}}dt\Big)^2\\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\geq\nonumber\\
2\Big(\int_0^1F(t)\frac{\sqrt{\frac{1}{2}(1+t)}|\sin\theta|+t^{2n-1}|\sin2n\theta|+t^{2n}|\sin(2n-1)\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}
dt\Big)^2\nonumber\\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2.\nonumber
\end{eqnarray}
On the other hand we have
\begin{eqnarray}\label{r5txcvcckxcxc2}
2\Big(\int_0^1F(t)\frac{\sqrt{\frac{1}{2}(1+t)}|\sin\theta|+t^{2n-1}|\sin2n\theta|+t^{2n}|\sin(2n-1)\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}
dt\Big)^2\\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\geq\nonumber\\
\Big(\int_0^1F(t)\frac{\sqrt{(1+t)}|\sin\theta|+\sqrt{2}t^{2n-1}|\sin2n\theta|+\sqrt{2}t^{2n}|\sin(2n-1)\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}
dt\Big)^2\nonumber\\
-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\geq\nonumber\\
\Big(\int_0^1F(t)\frac{|\sin\theta|+t^{2n-1}|\sin2n\theta|+t^{2n}|\sin(2n-1)\theta|}{1+t^2+2t\cos\theta}
dt\Big)^2\nonumber\\
-\Big(\int_0^1F(t)\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}dt\Big)^2\geq0, \ \ \theta\in[0,\frac{\pi}{2}],\nonumber
\end{eqnarray}
Finally the inequalities (\ref{w1w2c4x2zz9}), (\ref{111w1w2c4x2zz9}), (\ref{wndlkoop}), (\ref{ed55f}), (\ref{xcs3bnm3xcxc2}) and (\ref{r5txcvcckxcxc2}) imply that
$$
\Phi^2(0)-|\Phi(\theta)|^2\geq \ \theta\in[0,\frac{\pi}{2}],\nonumber
$$ and consequently inequalities (\ref{lfd8sn}) and (\ref{us5n6sw}) hold in case $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}].$\\
\end{proof}
\begin{theorem
If $n$ is a natural number, $n\geq51$ and $\alpha\in(0,1),$ then the following inequality holds
\begin{equation}\label{usxzs5n6sw}
|A_{2n-1}(\alpha,e^{i\theta})|\leq|A_{2n-1}(\alpha,1)|, \ \ \textrm{for \ all} \ \theta\in[-\frac{2\pi}{3},-\frac{\pi}{2}]\cup[\frac{\pi}{2},\frac{2\pi}{3}]
\end{equation}
\end{theorem}
\begin{proof}
Equality (\ref{b5x3yo04d6z}) can be rewritten as follows
\begin{eqnarray}\label{d0c0bb}2\Big(\Phi^2(0)-|\Phi(\theta)|^2\Big) \ \ \nonumber\\=\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(\frac{2}{\alpha}+B(v,0) +B(v,\theta)\Big)dv \ \ \ \\+\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv\int_0^1F(t)\Big(\frac{2}{\alpha}+B(t,0)+B(t,\theta)\Big)\Big]dt \ \ \ \nonumber\\-2\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv. \ \ \ \nonumber
\end{eqnarray}
Since $2\Big(\Phi^2(0)-|\Phi(\theta)|^2\Big)$ defines an even function in order to prove (\ref{usxzs5n6sw}) it is enough to show that
\begin{equation}
\Phi^2(0)-|\Phi(\theta)|^2\geq0, \ \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].
\end{equation}
The inequality between the arithmetic and geometric means and the condition $\alpha\in(0,1]$ imply
\begin{eqnarray}\label{kx13eed0sc0bb}2\Big(\Phi^2(0)-|\Phi(\theta)|^2\Big)\geq2\Big[\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(v)\Big(2+B(v,0) \ \ \ \ \ \ \ \ \ \\ +B(v,\theta)\Big)dv\int_0^1F(v)\Big(B(v,0)-B(v,\theta)\Big)dv\int_0^1F(t)\Big(2+B(t,0)+B(t,\theta)\Big)dt\Big]^{\frac{1}{2}} \ \ \ \ \ \ \nonumber\\-2\int_0^1\int_0^1F(t)F(v)C(t,\theta)C(v,\theta)dtdv \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber
\nonumber\\=2\int_0^1F(t)\Big(B(t,0)-B(t,\theta)\Big)dt\int_0^1F(t)\Big(2+B(t,0)+B(t,\theta\Big)dt \ \ \ \ \ \ \ \ \ \ \nonumber\\-2\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2 \ \ \ \ \ \ \ \ \nonumber\end{eqnarray}
Lemma 5 and inequality (\ref{kx13eed0sc0bb}) lead to
\begin{eqnarray}\label{yhnx13eed0sc0bb}\Phi^2(0)-|\Phi(\theta)|^2\nonumber\\\geq\int_0^1F(t)\frac{\frac{12}{25}(1-\cos{\theta})+t^{2n-1}(1-\cos2n\theta)+t^{2n}(1-\cos(2n-1)\theta)}{(1+t)(1+t^2+2t\cos\theta)}dt \ \ \ \\
\int_0^1F(t) \frac{\frac{25}{12}(1+t)(1+\cos{\theta})+2t^{2n-1}(1+\cos2n\theta)+2t^{2n}(1+\cos(2n-1)\theta)}{1+t^2+2t\cos\theta}dt\nonumber\\-\Big(\int_0^1F(t)C(t,\theta)dt\Big)^2\nonumber\end{eqnarray}
We apply twice the Cauchy-Schwarz inequality jest like in the proof of the previous theorem and we get that in order to prove (\ref{yhnx13eed0sc0bb}) it is enough to show that
\begin{eqnarray}\label{yhnx13evcxed0sc0bb}\Phi^2(0)-|\Phi(\theta)|^2\nonumber\\\geq\Big(\int_0^1F(t)\frac{\sqrt{1+t}|\sin{\theta}|+\sqrt{2}t^{2n-1}|\sin{2n}\theta|+\sqrt{2}t^{2n}|\sin({2n-1})\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}dt\Big)^2\nonumber\\ -\Big(\int_0^1F(t)\frac{\sin\theta+t^{2n-1}\sin2n\theta+t^{2n}\sin(2n-1)\theta}{1+t^2+2t\cos\theta}dt\Big)^2\nonumber\end{eqnarray}
This inequality is equivalent to
\begin{eqnarray}\label{xcxcxc2}
\Phi^2(0)-|\Phi(\theta)|^2\geq\nonumber\\
\Big(\int_0^1F(t)\frac{|\sin\theta|}{1+t^2+2t\cos\theta}dt+\int_0^1F(t)\frac{\sqrt{2}t^{2n-1}|\sin2n\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}dt \ \ \\+\int_0^1F(t)\frac{\sqrt{2}t^{2n}|\sin(2n-1)\theta|}{(1+t^2+2t\cos\theta)\sqrt{1+t}}
dt\Big)^2\nonumber\\
-\Big(\int_0^1F(t)\frac{\sin\theta}{1+t^2+2t\cos\theta}dt+\int_0^1F(t)\frac{t^{2n-1}\sin2n\theta}{(1+t^2+2t\cos\theta)}dt\nonumber\\+\int_0^1F(t)\frac{t^{2n}\sin(2n-1)\theta}{(1+t^2+2t\cos\theta)\sqrt{1+t}}
dt\Big)^2\geq0, \ \ \theta\in\Big[\frac{\pi}{2},\frac{2\pi}{3}\Big].\nonumber
\end{eqnarray}
and the proof is done.
\end{proof}
\begin{theorem
If $n$ is a natural number, $n\geq27,$ and $\alpha\in[\frac{1}{3},1),$ then the following inequality holds
\begin{equation}\label{uscsaxzs5n6sw}
{A_{2n-1}(\alpha,1)}\geq|A_{2n-1}(\alpha,e^{i\theta})|, \ \ \textrm{for \ all} \ \theta\in[-\pi,-\frac{2\pi}{3}]\cup[\frac{2\pi}{3},\pi]
\end{equation}
\end{theorem}
\begin{proof}
We will use the Taylor formula with an integral remainder. Let $f:(-a,a)\rightarrow\mathbb{R}$ be a $2n$ times derivabile function, such that $f^{(2n)}$ is continous. If $x\in(-a,a),$ then
\begin{eqnarray}f(x)=f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\ldots+\frac{f^{(2n-1)}(0)}{(2n-1)!}x^{2n-1}\nonumber\\+\frac{x^{2n}}{(2n-1)!}\int_0^1(1-t)^{2n-1}f^{(2n)}(xt)dt.\nonumber\end{eqnarray}
Let $f$ be the function defined by
$f:(-1,1)\rightarrow\mathbb{R}, \ \ f(x)=(1+x)^\alpha, \ \
\alpha\in(0,1)$ and we get
\begin{eqnarray}(1+x)^\alpha=1+\frac{\alpha}{1!}x+\frac{\alpha(\alpha-1)}{2!}x^2+\ldots+\frac{\alpha(\alpha-1)\ldots(\alpha-2n+2)}{(2n-1)!}x^{2n-1}\nonumber\\
+x^{2n}\frac{\alpha(\alpha-1)\ldots(\alpha-2n+1)}{(2n-1)!}\int_0^1(1-t)^{2n-1}(1+xt)^{\alpha-2n}dt.\nonumber\end{eqnarray}
Since the mapping $f:U\rightarrow\mathbb{R}, \ \
f(z)=(1+z)^\alpha$ is well defined(we take the principal branch of
the multi valued function) it follows that the equality
\begin{eqnarray}(1+z)^\alpha=1+\frac{\alpha}{1!}z+\frac{\alpha(\alpha-1)}{2!}z^2+\ldots+\frac{\alpha(\alpha-1)\ldots(\alpha-2n+2)}{(2n-1)!}z^{2n-1}\nonumber\\
+z^{2n}\frac{\alpha(\alpha-1)\ldots(\alpha-2n+1)}{(2n-1)!}\int_0^1(1-t)^{2n-1}(1+zt)^{\alpha-2n}dt.\nonumber\end{eqnarray}
holds for every $z\in{U}.$ The mapping $f:U\rightarrow\mathbb{R},
\ \ f(z)=(1+z)^\alpha$ is radially continuous and so we infer
that
\begin{eqnarray}(1+e^{i\theta})^\alpha=1+\frac{\alpha}{1!}e^{i\theta}+\frac{\alpha(\alpha-1)}{2!}e^{2i\theta}+\ldots+\frac{\alpha(\alpha-1)\ldots(\alpha-2n+2)}{(2n-1)!}e^{(2n-1)i\theta}\nonumber\\
+e^{2ni\theta}\frac{\alpha(\alpha-1)\ldots(\alpha-2n+1)}{(2n-1)!}\int_0^1(1-t)^{2n-1}(1+e^{i\theta}t)^{\alpha-2n}dt,
\ \theta\in(-\pi,\pi).\nonumber\end{eqnarray} Taking the absolute
value, this equality implies that
\begin{eqnarray}\label{fkfk354}|A_{2n-1}(\alpha,e^{i\theta})|\nonumber\\=\big|1+\frac{\alpha}{1!}e^{i\theta}+\frac{\alpha(\alpha-1)}{2!}e^{2i\theta}+\ldots
+\frac{\alpha(\alpha-1)\ldots(\alpha-2n+2)}{(2n-1)!}e^{(2n-1)i\theta}\big|\nonumber\\
\leq\frac{\alpha(1-\alpha)(2-\alpha)\ldots(2n-1-\alpha)}{(2n-1)!}\int_0^1(1-t)^{2n-1}\big|1+e^{i\theta}t\big|^{\alpha-2n}dt\nonumber\\+\big|1+e^{i\theta}
\big|^\alpha\leq\big|1+e^{i\theta}
\big|^\alpha++\alpha(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1(1-t)^{2n-1}\big|1+e^{i\theta}t\big|^{\alpha-2n}dt,
\ \theta\in(-\pi,\pi).\nonumber\end{eqnarray}
On the other hand we have
\begin{eqnarray}\label{ramglp67}A_{2n-1}(\alpha,1)=1+\frac{\alpha}{1!}-\frac{\alpha(1-\alpha)}{2!}+\frac{\alpha(1-\alpha)(2-\alpha)}{3!}\nonumber\\
-\frac{\alpha(1-\alpha)(2-\alpha)(3-\alpha)}{4!}+\ldots
+\frac{\alpha(1-\alpha)\ldots(2n-2-\alpha)}{(2n-1)!}\geq1+\frac{\alpha(1+\alpha)}{2}\nonumber\end{eqnarray}
Thus in order to prove (\ref{uscsaxzs5n6sw}) we have to show
that the following inequality holds
\begin{eqnarray}\label{ttsjjkk}
1+\frac{\alpha(1+\alpha)}{2}\geq\big|1+e^{i\theta}
\big|^\alpha++\alpha(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1(1-t)^{2n-1}\big|1+e^{i\theta}t\big|^{\alpha-2n}dt,
\ \theta\in[\frac{2\pi}{3},\pi].\nonumber
\end{eqnarray}
We denote $x=-\cos\theta,$ and the inequality (\ref{ttsjjkk}) will be
equivalent to
\begin{eqnarray}\label{t3tsjvjkk}
1+\frac{\alpha(1+\alpha)}{2}\geq\big(2-2x\big)^\frac{\alpha}{2}+\alpha(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1\Big(\frac{1-t}{\sqrt{1+t^2-2tx}}\Big)^{2n-1}\big(\sqrt{1+t^2-2tx}\big)^{\alpha-1}dt,
\ x\in[\frac{1}{2},1],
\end{eqnarray}
and this inequality can be rewritten in the following form:
\begin{eqnarray}\label{t3ts1jwv6zpx42jk2k}
1+\alpha\geq\frac{\big(2-2x\big)^\frac{\alpha}{2}-1}{\frac{\alpha}{2}}+2(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1\Big(\frac{1-t}{\sqrt{1+t^2-2tx}}\Big)^{2n-1}\big(\sqrt{1+t^2-2tx}\big)^{\alpha-1}dt, \ \ \
\ \ \ x\in[\frac{1}{2},1].\ \ \
\end{eqnarray}
It is easily seen that $$2(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1\Big(\frac{1-t}{\sqrt{1+t^2-2tx}}\Big)^{2n-1}\big(\sqrt{1+t^2-2tx}\big)^{\alpha-1}dt$$ is decreasing with respect to $n$ and $x,$ and $\frac{\big(2-2x\big)^\frac{\alpha}{2}-1}{\frac{\alpha}{2}}<0,$ for all $\alpha\in(0,1),$ and $ x\in[\frac{1}{2},1].$
Thus in order to prove (\ref{t3ts1jwv6zpx42jk2k}) it is enough to prove the inequality
\begin{eqnarray}\label{h6xnlnp0xjj1jv6zx4jk2k}
1+\frac{1}{3}\geq2(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\frac{\alpha}{53})\int_0^1\big(1-t\big)^{\alpha-1}dt
\end{eqnarray}
in case $\alpha=1,$ that is
$$\frac{4}{3}\geq6(1-\frac{1}{3})(1-\frac{1}{3\cdot2})(1-\frac{1}{3\cdot3})\ldots(1-\frac{1}{3\cdot27}).$$
This inequality holds and the proof is done.
\end{proof}
\section{Concluding Remarcs}
The following two corollaries are the proof of the Brannan conjecture in two different particular cases.\\
Theorem 1, Theorem 2 and the result of \cite{6} imply the following corollary.
\begin{corollary}
If $x\in\mathbb{C}$ with $|\arg{x}|\leq\frac{2\pi}{3},$ and $|x|=1,$ then the inequality
$$|A_{2n-1}(\alpha,x)|\leq{A_{2n-1}}(\alpha,1)$$
holds for every $\alpha\in(0,1).$
\end{corollary}
Theorem 1, Theorem 2, Theorem 3 and the result of \cite{6} imply the following corollary. This corollary is the solution of the Brannan conjecture in case $\alpha \in [\frac{1}{2},1).$
\begin{corollary}
The inequality
$$|A_{2n-1}(\alpha,x)|\leq{A_{2n-1}}(\alpha,1)$$
holds for every $\alpha\in[\frac{1}{3},1),$ and $x\in\mathbb{C},$ with $|x|=1.$
\end{corollary}
\begin{conjecture}
Numerical resuls suggest that the inequality
\begin{eqnarray}\label{s5d6f7t3tsjvjkk}
1+\frac{\alpha(1+\alpha)}{2}\geq\big(2-2x\big)^\frac{\alpha}{2}+\alpha(1-\alpha)(1-\frac{\alpha}{2})(1-\frac{\alpha}{3})\ldots(1-\ \ \nonumber\\\frac{\alpha}{2n-1})\int_0^1\Big(\frac{1-t}{\sqrt{1+t^2-2tx}}\Big)^{2n-1}\big(\sqrt{1+t^2-2tx}\big)^{\alpha-1}dt,
\ x\in[\frac{1}{2},1],
\end{eqnarray}
holds for every $\alpha\in(0,\frac{1}{3}).$
\end{conjecture}
\begin{remark}
If the previous conjecture holds, then the conjecture of Brannan holds in case $\beta=1$ and every $\alpha\in(0,1).$
\end{remark}
|
2,877,628,090,790 | arxiv | \chapter*{Acknowledgments}
\vspace{-1cm}
\noindent\rule[0.5ex]{\linewidth}{2pt}
\vspace{0.5cm}
\noindent
The completion of this thesis would not have been possible without the many people who have supported and encouraged me along the way. I take this opportunity to express my gratitude towards them.
First and foremost, I am greatly indebted to my supervisors Johan van Leeuwaarden and Bert Zwart. Johan, I think I can safely say that you have been my guiding light throughout this academic journey.
Ever since my bachelor project, your endless optimism, patience and sense of perspective have been exactly what I needed to keep me motivated through the ups and downs that come with research.
Thank you for teaching me to be persistent, critical and positive.
Bert, thank you for the continuous flow of ingenious ideas that came during our many discussions.
I truly admire your drive, enthusiasm and passion for mathematics.
Much of work presented in this thesis is the result of fruitful collaborations with some great researchers, to whom I wish to express my appreciation as well.
First of all, Onno Boxma, your inspiring lectures on stochastic processes may well have marked the starting point of the path that led to this PhD, and I am very happy that you have been willing to be a member of my defense committee.
It has been delightful to work with you, together with Shaul Bar-Lev and David Perry, on the subject of Chapter 7.
Guido Janssen, thank you for getting me acquainted with the mathematical theory behind asymptotics.
Your thoroughness, preciseness and willingness to share your expertise are much appreciated.
Galit Yom-Tov, thank you for the nice and intensive collaboration that resulted in Chapter 5, and for serving on my committee.
Also, I hold nice memories on our joint effort to organize the YEQT workshop together with Jan-Pieter.
I am also thankful to Sandjai Bhulai, Ton de Kok and Josh Reed for being part of my defense committee, for their time to read this thesis and for providing me with valuable feedback.
During my visits to the Technion, I have been warmly welcomed by Avi Mandelbaum and the people at the SEELab.
It has been fascinating to get a glimpse of the goldmine of service system data that they have collected and I am grateful to them for showing me the way.
Avi, thank you for kindly introducing me to the Israeli culture, and for your many advices on research and academic life.
I owe many thanks to my office mates, Fabio, Fiona and Thomas, for creating the most pleasant and comforting work environment.
I am pleasantly surprised you have been able to tolerate my ever-fluctuating stress levels, certainly in the last couple of months.
The fact that I have remained (reasonably) sane throughout these years is in large part thanks to you.
Outside my office door, I have been lucky to be part of an amazing research group.
I want to thank my fellow PhDs of the Stochastics section, including the former generations, for creating and sustaining such an amiable atmosphere.
Further, I am grateful to Remco van der Hofstad and Marko Boon for granting me the opportunity to develop my teaching and lecturing skills, which has brought refreshing variation into my job as a PhD student.
Having been a member of the departmental PhD Council, I got the chance to meet many bright young minds, and it has been great fun to collaborate with them to strengthen the PhD community.
I wish all of them the best for the future.
When you stick around at the same university for almost nine years, you run the risk of befriending some mathematicians.
I am nevertheless very happy to have met Christine, Jorg, Jorn, Laura, Mark, Thomas and many others at this place.
Thank you all for making my time at TU/e very enjoyable.
Rik, thank you for holding my hand throughout these years. You have been my rock.
Finally, to my parents Jan and Willemien and my sister Kim, I owe my deepest gratitude.
Your unconditional support and comfort have kept me grounded and realistic.
I could never have done this without you. \\
\begin{flushright}
\textit{
Britt Mathijsen\\
March 2017}
\end{flushright}
\chapter{Novel heavy-traffic regimes}
\begin{chapterstart}
In this chapter, we introduce a family of heavy-traffic scalings for a large-scale service system meant to serve jobs coming from a large pool of customers.
The scaling rules are inspired by the classical QED regime discussed in Chapter 1, but lead to a range of different system behavior s that include the ED, QED and QD regime behavior as special cases.
To determine the scaling limits, we describe the performance measures in terms of Pollaczek integrals and use asymptotic techniques to evaluate these integrals in the large-system limit.
\end{chapterstart}
\begin{flushright}
Based on \\
\textbf{Novel heavy-traffic regimes for large-scale service systems}\\
\textit{Guido Janssen, Johan van Leeuwaarden \& Britt Mathijsen}\\
In \textit{SIAM Journal of Applied Mathematics, 75(2), 787-812 (2015)}
\end{flushright}
\newpage
\section{Introduction and motivation}
We study the workload process of the queue at equidistant time epochs.
Demand is assumed to be generated by $n$ stochastically identical and independent sources.
Let $A_{i,j}$ denote the workload brought into the system by source $i$ in period $j$, for which $\mathbb{E}[A_{i,j}] =\mu$ and ${\rm Var}\, A_{i,j} = \sigma^2$.
Then the total amount of demand arriving to the system in period $j$ is $A^{(n)}_j=\sum_{i=1}^n A_{i,j}$ with $\mathbb{E}[A^{(n)}_j] = n\mu$ and ${\rm Var}\, A^{(n)}_j = n\sigma^2$.
As explained in Chapter 1, a good capacity sizing rule for achieving economies-of-scale is $s_n = n\mu+\beta\sqrt{n}\sigma$ for some $\beta>0$.
If we denote the system utilization by $\rho_n := n\mu/s_n$, then this dimensioning rule in the bulk service queue with many sources is tantamount to the heavy-traffic scaling
\begin{equation}\label{bb1}
\sqrt{n}(1-\rho_n) \to \gamma = \frac{\beta\sigma}{\mu}, \qquad {\rm as }\ n\to\infty.
\end{equation}
Starting from this setting, we introduce a novel family described in terms of a parameter $\eta$ for which we assume that
\begin{equation}\label{bb}
n^{\eta}(1-\rho_n)\rightarrow \gamma, \quad {\rm as} \ n\to \infty, \ \gamma> 0.
\end{equation}
The parameter $\eta\geq 0$ defines a whole range of possible scaling regimes, including the classic case $\eta=1/2$.
In terms of a capacity sizing rule for systems with many customers, the condition \eqref{bb} is tantamount to $s_n=n\mu+\beta \sigma n^{1-\eta}$.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as {\it moderate} heavy traffic: heavy traffic conditions in which the full occupancy is reached more slowly, as a function of $n$, than for classical heavy traffic. For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to {\it extreme} heavy traffic due to a relatively small variability hedge. Note that the case $\eta=0$ does not lead to 100\% system utilization when $n\to \infty$.
In this chapter we show that economies-of-scale can be achieved for a large range of $\eta$, although the nature of the benefits obtained by operating on large scale depends on the precise capacity sizing rule (hence the parameter $\eta$). We quantify performance in terms of stationary measures: The mean and variance of the congestion in the system, and the probability of an empty system. For these performance measures we derive heavy-traffic limits under the scalings \eqref{bb} that
are relatively simple functions of only the first two moments of the demand per period. Such parsimonious expressions are useful for quantifying and improving system behavior. The heavy-traffic limits, however, provide also qualitative insight into the system behavior. Our asymptotic analysis shows that mean congestion is $O(n^\eta)$, which implies
that delays experienced by the customers are negligible for all values of $\eta\in [0,1)$, are roughly constant for $\eta=1$, and grow without bound for $\eta>1$. We expect this qualitative behavior to be universal for a wide range of stochastic models to which the regime \eqref{bb} is applied.
We further show the existence of the following trichotomy as $n\to \infty$ under \eqref{bb}: For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, for $\eta\in (1/2,1)$ it converges to $0$, while only for $\eta=1/2$ there is a limiting value in $(0,1)$. Hence, as expected, the system performance deteriorates with $\eta$, in a rather crude way for the empty-system probability, and in only a mild way for mean congestion levels. The regime \eqref{bb} thus presents a range of possible capacity sizing rules that all lead to economies-of-scale, and depending on what is the desired nature of performance for a particular service system, an appropriate $\eta$ can be selected. From the quantitative perspective, our detailed asymptotic analysis leads to more precise asymptotic estimates for the performance measures in heavy traffic, which reveal the exact manner in which the mean congestion is influenced by $\eta$ and $\gamma$.\\
\\
\noindent\textbf{Motivating examples.}
Our stochastic model is one of the canonical models in queueing theory, having a wide range of applications in fields like digital communication, wireless networks, road traffic, reservation systems, health care and many more (see \cite{Bruneel1993} and \cite[Chap.~5]{johanthesis} for an overview). In road traffic, the basic model for congestion at an intersection, known as the fixed-cycle traffic-light queue \cite{Newell1960,Leeuwaarden2006}, is related to our discrete bulk service queue. Then $s_n$ represents the maximum number of delayed cars in front of a traffic light that can depart during one green period, while $A^{(n)}_j$ is the number of newly arriving cars during a consecutive green and red period. An example from health care is panel sizing \cite{Zacharias2014}. Say a general practitioner has a pool of $n$ clients (typically in the order of \cite{Green2008}), all of which are potential patients, and together require $A^{(n)}_j$ consults per day. Further assume that the practitioner can see a maximum number of $s_n$ patients per day. What is then an appropriate patient panel size $n$, which strikes a reasonable balance between accessing medical care in a timely manner and restricting the time that the practitioner sits idle? The panel size application is one of many examples of an appointment book, referring to some schedule of appointments for a fixed period, with capacity $s_n$ appointments per period and newly arriving appointment $A^{(n)}_j$ per period. See \cite{Dai2014} for another recent example of an appointment book in a health care setting, again in terms of our bulk service queue, with $A^{(n)}_j$ the new patients per day and $s_n$ the number of available beds. For all examples above, and many more, our new class of heavy-traffic scalings \eqref{bb} presents capacity sizing rules for which the expected performance can be quantified using the results in this paper. This will be helpful in dimensioning the systems (How much capacity is needed to achieve a certain target performance?) while exploiting economies-of-scale. For appointment books our model together with the capacity sizing rules \eqref{bb} are particularly relevant for {\it advanced access} \cite{Green2008}, a scheduling approach in health care designed to reduce delays by offering every patient a same-day appointment, regardless of the urgency of the problem. In that way, patients do not have to wait long for appointments, and practices do not waste capacity by holding appointments in anticipation of urgent.
\noindent\textbf{Pollaczek's formula.}
Next to the freedom to model different situations, another advantage of our model is that it is mathematically tractable, in the sense that it can be subjected to powerful mathematical methods from complex and asymptotic analysis. In order to establish the heavy-traffic limits we start from Pollaczek's formula for the transform of the stationary queue length distribution in terms of a contour integral. From this famous transform representation, contour integrals for the empty-system probability and the mean and variance of the congestion immediately follow. Contour integrals are often amenable to asymptotic evaluation (see e.g.~\cite{Cohen1982}), particular for obtaining classical heavy-traffic asymptotics. We also subject the contour integral representations to asymptotic evaluation, but not under classical heavy traffic scaling. This asymptotic analysis requires a {\it non-standard} saddle point method (see \cite{flajolet} for an historical account on the application of the saddle point method in mathematics), tailored to the specific form of the integral expressions that arise under the capacity sizing rule \eqref{bb}. The saddle point method in its standard form is typically suited for large deviations regimes, for instance to characterize rare event probabilities, and cannot be applied to asymptotically characterize other stationary measures like the mean or mass at zero. Indeed, for our model the saddle point converges to one (as $n\to \infty$), which is a singular point of the integrand, and renders the standard saddle point method useless. Our non-standard saddle point method, originally proposed by \cite{debruijn}, is made specifically to overcome this complication. This leads to asymptotic expansions for the performance measures, of which the limiting forms correspond to the heavy-traffic limits, and pre-limit forms present refined approximations for pre-limit systems ($n<\infty$) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as {\em corrected diffusion approximations} \cite{Siegmund1978,Blanchet2006,Asmussen2003}.
\noindent{\bf Further connections to the literature.}
We now discuss two classes of stochastic systems for which the heavy-traffic regime \eqref{bb1} has been studied extensively, and for which our new family of regimes \eqref{bb} is largely unexplored. We discuss these classes because, despite the Pollaczek formula not to hold, we believe the qualitative results that we reveal for our particular model should to a large extent carry over to these settings as well, presenting some interesting avenues for further research (see Section \ref{subsec62}).
The first class concerns so-called {\it nearly-deterministic} systems \cite{Sigman2011a,Sigman2011b}, denoted by $G_n/G_n/1$ system, where $G_n$ stands for {\it cyclic thinning} of order $n$, indicating that some point process is thinned to contain only every $n$th point. As $n\to \infty$, the $G_n/G_n/1$ systems approach the deterministic $D/D/1$ system. For $G_n/G_n/1$ systems, \cite{Sigman2011a} establishes stochastic-process limits, and \cite{Sigman2011b} derives heavy-traffic limits for stationary waiting times. In the framework of \cite{Sigman2011a,Sigman2011b}, our stochastic model corresponds to a $D/G_n/1$ queue, where the sequence of service times $\{A^{(n)}_j\}_{j\geq 1}$ follows from a cyclically thinned sequence of i.i.d.~random variables $A_{i,j}$. It follows from \cite[Theorem 3]{Sigman2011b} that the rescaled stationary waiting time process converges under \eqref{bb1} to a reflected Gaussian random walk. Hence, the performance measures of the nearly deterministic system, under \eqref{lind} and \eqref{bb1}, should be well approximated by the performance measures of the reflected Gaussian random walk, giving rise to heavy-traffic approximations. This connection is discussed in detail in Section \ref{subsec3.2}. It seems likely that results similar as in this paper can be obtained for applying the scaling \eqref{bb} to the nearly-deterministic systems in \cite{Sigman2011a,Sigman2011b}, and because Pollaczek's formula also applies to this setting, the non-standard saddle point method developed in this paper can provide the appropriate methodology.
The second class concerns multi-server systems, and in particular the many-server regime (not to be confused with many-sources regime). When we interpret $s_n$ as the number of servers, instead of capacity per time slot or order of thinning, the scaling \eqref{bb1} is similar to the many-server heavy-traffic regime called QED (Quality and Efficiency Driven) or Halfin-Whitt regime, first developed by Halfin and Whitt \cite{Halfin1981} for the $M/M/s_n$ system. The QED regime \eqref{bb1} is in many situations a highly effective way of scaling, because the probability of delay converges to a non-degenerate limit away from both zero and one, and the mean delay is asymptotically negligible as the number if servers grows large. The QED regime \eqref{bb1} is naturally positioned in between the Quality-Driven (QD) regime and the Efficiency-Driven (ED) regime. In the QD regime, the load remains bounded away from 1, which corresponds to setting $\eta=0$ in \eqref{bb}. Hence, the range $\eta\in(0,1/2)$ bridges the gap between the QED regime and the QD regime. Likewise, the ED regime corresponds to setting $\eta=1$ in \eqref{bb}, so that the range $\eta\in(1/2,1]$ connects the QED regime and ED regime. For the birth-death process describing the $M/M/s_n$ system, Maman \cite{maman} introduced a scaling similar to \eqref{bb}, and called it the QED-$c$ regime, also bridging the ED and QD regimes. \cite[Thm 4.1]{maman} says that the expected waiting time under the scaling $s_n = n\mu+\beta\sigma n^{1-\eta}$ is of order $s_n^{1-\eta}$, which is equivalent to the expected queue length being of order $n^\eta$ by Little's law. We should stress though that we expect the mathematical techniques that are needed to establish heavy-traffic results could be entirely different than in this paper, because Pollaczek's formula does not apply to many-server settings. The specific model assumptions will determine to a large extent the appropriate methodology. Under Markovian assumptions leading to the $M/M/s_n$ system, product-form solutions are available for the stationary distribution. This makes it possible to describe performance measures like the mean congestion directly in terms of real integrals. Where the saddle point method is used for integrals in the complex plane, the Laplace method (see e.g.~\cite{flajolet}) is used for real integrals. Hence, for the asymptotic evaluation of the $M/M/s_n$ system under the scaling \eqref{bb}, the Laplace method seems an appropriate methodology, although again one needs to deal with possible singularities in the integrand. For $G/D/s_n$ systems, which assume deterministic service times, it has been shown in \cite{Jelenkovic2004} that using a decomposition property the dynamics of this multi-server systems can be captured in terms of a single-server system. Hence, for these systems, Pollaczek's formula applies, and our saddle point method can most likely be applied to obtain heavy-traffic results in the regimes \eqref{bb}. Under more general conditions, for instance leading to a $G/G/s_n$ system, it is simply unclear at this stage how to obtain precise heavy-traffic approximations for \eqref{bb}, because a tractable description of the performance measures is not available.\\
\\
\\*
\noindent{\bf Structure of the chapter.}
In Section \ref{sec1} we present in detail the model and the family of heavy-traffic scalings. In Section \ref{spSec} we introduce the saddle point method. In Section \ref{sec3} we apply the saddle point method for the mean congestion level. Theorem \ref{mainthm} gives for all heavy-traffic scalings the limiting behavior in terms of an integral expression. As a consequence, we show in
Proposition \ref{prop1} that there are two types of heavy-traffic behavior, depending on whether $\eta\in(0,1/2)$ or $\eta\geq 1/2$.
In Section \ref{subsec3.2} we discuss for the case $\eta=1/2$ the connection with the Gaussian random walk and the Riemann zeta function.
In fact, we show that for all $\eta\geq 1/2$ there exists a connection between the integral expression in Theorem \ref{mainthm} and the Riemann zeta function.
In Section \ref{more} we apply the saddle point method to obtain several more heavy-traffic results, including refined heavy-traffic approximations for the mean congestion level, and the leading heavy-traffic behaviors for the variance of the stationary congestion level and for the empty-system probability. Numerical examples are given in Section \ref{numm}. Appendix \ref{app} presents a new self-contained derivation of Pollaczek's formula for the transform of the stationary waiting time in the $D/G/1$ system, which forms the point of departure for our analysis. In Section \ref{numm}, however, we do confirm through numerical experiments that under \eqref{bb}, various multi-server systems behave similar to our discrete bulk service queue.
\section{Model description and heavy-traffic regimes}\label{sec1}
We thus consider a discrete stochastic model in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,...$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
We will omit the superscript $(n)$ if no ambiguity is possible.
The system has a service capacity $s_n\in\mathbb{N}$ per period, so that the recursion
\begin{equation}
\label{lind}
Q(j+1) = \max\{Q(j) + A^{(n)}_j - s_n,0\},\qquad j=1,2,...,
\end{equation}
assuming $Q(0)=0$, gives rise to a Markov chain $\{Q(j)\}_{j\geq 1}$ that describes the congestion in the system over time. The probability generation function (pgf)
\begin{equation} \label{e2}
\tilde A(z)=\sum_{k=0}^{\infty} \mathbb{P}\big(A^{(n)}=k\big) z^k
\end{equation}
of $A^{(n)}$ is assumed analytic in a disk $|z|<r$ with $r>1$, which implies that all moments of $A^{(n)}$ exist. We also assume that
\begin{equation} \label{e3}
\tilde A'(1)=\mathbb{E}[A^{(n)}_j]=\mu_A<s_n.
\end{equation}
Under the assumption (\ref{e3}) the function $z^{s_n}-\tilde A(z)$ has exactly $s$ zeros in the closed unit disk, one of these being $z=1$ (see \cite{rouche}).
We further assume that $\mathbb{P}(A^{(n)}=j)>0$ for some $j>s_n$.
Under this assumption the function
$z^{s_n}-\tilde A(z)$ also has zeros outside $|z|\leq1$, and we let $r_0$ be the minimum modulus of these zeros.
The number $r_0$ is the unique zero of $z^{s_n}-\tilde A(z)$ with real $z>1$; see e.g.~\cite{Janssen2005}.
Under the assumption (\ref{e3}) the stationary distribution $\lim_{k\to \infty}\mathbb{P}\left(Q(j)=k\right)=\mathbb{P}(Q=k)$, $j=0,1,\ldots$ exists, with the random variable $Q$ defined as having this stationary distribution.
We let
\begin{equation} \label{e4}
\tilde Q(w)=\sum_{j=0}^{\infty}\mathbb{P}(Q=j)w^j
\end{equation}
be the pgf of the stationary distribution. $\tilde Q(w)$ is analytic in $|w|<r_0$, and given by Pollaczek's formula (see e.g.~\cite{Abate1993, Cohen1982})
\begin{equation} \label{e5}
\tilde Q(w)=\exp\,\Big[\,\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\ln\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,dz\Big] ,
\end{equation}
where $\varepsilon>0$ is such that $|w|<1+\varepsilon<r_0$. In (\ref{e5}), the principal value of $\ln(\frac{w-z}{1-z})$ is chosen, which is analytic in the whole complex $z$-plane, except for a branch cut consisting of the straight line segment from $w$ to 1. In Appendix \ref{app} we present a short proof of Pollaczek's formula in the discrete-queue setting that we have here.
Using $\mathbb{P}(Q=0)=\tilde Q(0)$, $\mu_Q=\tilde Q'(1)$ and $\sigma_Q^2 = \tilde Q''(1)+\tilde Q'(1)-(\tilde Q'(1))^2$, it follows by straightforward manipulations that
\begin{align} \label{e6}
\mathbb{P}(Q=0)&=\exp\,\Big[\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\ln\Big(\frac{z}{z-1}\Big)\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,dz\Big] , \\
\label{e7}
\mu_Q&=\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{1-z}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,dz ,\\
\label{e8}
\sigma_Q^2 &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{-z}{(1-z)^2}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,dz .
\end{align}
Because $s_n$ appears directly in expressions \eqref{e6}-\eqref{e8}, we will be conducting our analysis with respect to $s_n$ rather than $n$. Note that this has no consequences for our results on the convergence speed of the performance metrics, since $s_n = O(n)$. Furthermore, we will omit the index $n$ when describing the capacity $s_n$ in the remainder of the paper for brevity.
We next discuss in more detail the family of heavy-traffic scalings considered in this paper, which combines two features. First, we have assumed that
$A^{(n)}_j$ is in distribution equal to the sum of work generated by all sources, $A_{1,k}+...+A_{n,j}$, where the $A_{i,j}$ are for all $i$ and $k$ i.i.d.~copies of a random variable $X$, of which the pgf $\tilde X(z)=\sum_{k=0}^{\infty}\mathbb{P}(X=k)z^k$ has radius of convergence $r>1$, and
\begin{equation} \label{e9}
0< \mathbb{E}[A^{(n)}] =n\mu = n \tilde X'(1)<s_n .
\end{equation}
Hence
\begin{equation} \label{e10}
\vartheta:=\frac{n}{s_n}\in(0,1/\mu) .
\end{equation}
Second, we scale the system according to \eqref{bb}, for which we assume that
\begin{equation} \label{e11}
\rho_{s_n} =\vartheta\,\mu =1-\frac{\gamma}{s_n^\eta}
\end{equation}
in which $\gamma>0$ is bounded away from 0 and $\infty$ as $s_n\to \infty$.
In the remainder of this chapter, we will omit the subscript in $s_n$.
The condition that $\mathbb{P}(A^{(n)}=k)>0$ for some $k>s$ holds when the degree $d$ of $\tilde X(z)$ (with $d=\infty$ if $\tilde X(z)$ is not a polynomial) is such that $nd>s$.
To avoid certain complications when applying the saddle point method, we further assume that
\begin{equation} \label{e12}
|\tilde X(z)|<\tilde X(r_1) ,~~~~~~|z|=r_1\,,~~z\neq r_1 ,
\end{equation}
for any $r_1\in(0,r)$. This implies that $r_0$ is the unique zero of $z^s-\tilde A(z)$ on $|z|=r_0$.
This condition is related to Cram\'er's condition, see \cite[pp.~189 and 355]{Asmussen2003}, and it has also been used in \cite{relaxation}.
Condition \eqref{e12} holds when the set of all $j=0,1,\ldots$ such that $\mathbb{P}(X=k)>0$ is not contained in an arithmetic progression with a ratio larger than one (see also \cite{rouche}).
\section{Non-standard saddle point method}\label{spSec}\ \
We illustrate our saddle point method for $\mu_Q$.
As a first step, we bring (\ref{e7}) in a form which is amenable to saddle point analysis.
\begin{lemma}
\begin{equation} \label{e18}
\mu_Q = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{g'(z)}{z-1}~\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}\,dz
\end{equation}
with
\begin{equation} \label{e15}
g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}(\tilde X(z)) .
\end{equation}
\end{lemma}
\begin{proof}
With $\tilde A(z)=\tilde X^n(z)$,
\begin{align} \label{e13}
\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)} & = \frac{s\,z^{s-1}-n\,\tilde X'(z)\,\tilde X^{n-1}(z)}{z^s-\tilde X^n(z)} \nonumber \\
& = \frac{s}{z}-\frac{s}{z}\,\Big(\frac{n}{s}~\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big)\,\frac{z^{-s}\,\tilde X^n(z)}{1-z^{-s}\,\tilde X^n(z)} .
\end{align}
Write
$
z^{-s}\,\tilde X^n(z)=\exp(s\,g(z))$.
Noting that
\begin{equation} \label{e16}
\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{s}{z}~\frac{1}{1-z}\,dz=0 ,
\end{equation}
and that
\begin{equation} \label{e17}
g'(z)=\frac1z\,\Big(\vartheta\,\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big) ,
\end{equation}
gives \eqref{e18}. \end{proof}
Let us now explain how the standard saddle point method can be applied to \eqref{e18}.
Since
\begin{equation} \label{e19}
g(1)=g(r_0)=0~;~~~~~~g(z)<0\,,~~1<z<r_0 ,
\end{equation}
and by strict convexity of
\begin{equation} \label{e20}
z^{-s}\,\tilde X^n(z)=z^{-s}\tilde A(z)=\sum_{k=0}^{\infty}\,a_k\,z^{k-s} ,~~~~~~z\in(0,r) ,
\end{equation}
$g(z)$ has a unique minimum on $[1,r_0]$. This minimum is found by solving $z\in[1,r_0]$ from $g'(z)=0$, and this yields the equation
\begin{equation} \label{e21}
\tilde X(z)=\vartheta\,z\,\tilde X'(z) .
\end{equation}
Denote the solution $z\in(1,r_0)$ of (\ref{e21}) by $z_{\rm sp}$, and observe that $z_{\rm sp}$ is a saddle point of $g(z)$, explaining the notation. Thus, the saddle point method can be used for the integral in (\ref{e18}) by taking $1+\varepsilon=z_{\rm sp}$.
In the case that $\vartheta=n/s$ is bounded away from $1/\mu$ as $s\to \infty$, we have that the minimum value of $g(z)$, $1\leq z\leq r_0$, is negative and bounded away from 0. Furthermore, $z_{\rm sp}$ is bounded away from 1, and the saddle point method can be applied in the classical way by replacing
\begin{equation} \label{e22}
\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}~~~~{\rm by}~~~~\exp(s\,g(z)) ,
\end{equation}
at the expense of an exponentially small relative error, and performing an expansion of $g'(z)/(z_{\rm sp}-1)=d_1(z-z_{\rm sp})+O((z-z_{\rm sp})^2)$ with $d_1=g''(z_{\rm sp})/(z_{\rm sp}-1)\neq 0$.
Using that $g(z^{\ast})=(g(z))^{\ast}$, where the $^*$ denotes complex conjugation, it can be shown that
\begin{equation} \label{e23}
\mu_Q=\frac{\exp(s\,g(z_{\rm sp}))}{(z_{\rm sp}-1)^2\,\sqrt{2\pi s\,g''(z_{\rm sp})}}\,(1+O(s^{-1})) .
\end{equation}
We next explain why the standard saddle point method does not work for the heavy-traffic scaling considered in this paper. Since we operate in (\ref{e11}),
$\vartheta\mu\to 1$ as $s\to \infty$, and
\begin{align} \label{e24}
z_{\rm sp}-1&=\frac{\gamma}{a_2\,s^\eta}+O(s^{-2\eta}) ,\\
\label{e25}
g(z_{\rm sp})&=\frac{-\gamma^2}{2a_2s^{2\eta}}+O(s^{-3\eta}) ,\\
\label{e26}
g''(z_{\rm sp})&=a_2+O(s^{-\eta}) ,
\end{align}
where
\begin{equation} \label{e27}
a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) .
\end{equation}
Hence, $\exp(sg(z))$ near $z=z_{\rm sp}$ is (as $s\to \infty$):
vanishingly small when $\eta\in(0,1/2)$,
bounded away from 1, but non-negligible when $\eta=1/2$,
and tending to 1 when $\eta\in(1/2,\infty)$.
Furthermore, $(z-1)^{-1}$ in \eqref{e18} is unbounded near $z=z_{\rm sp}$ as $s\to \infty$. Therefore, an adaptation of the standard saddle point method is required, and the resulting asymptotic form of $\mu_Q$ will deviate significantly from the standard case (\ref{e23}). In particular, since $z_{\rm sp}\to 1$, this asymptotic form will contain information from $X(z)$ at $z=1$, rather than at a point away from 1 as is the case in (\ref{e23}).
The required adaption of the saddle point method is modeled after a device developed in \cite[Section ~5.12]{debruijn}. We use a substitution $z=z(v)$ in (\ref{e18}) with real $v$ and $z(0)=z_{\rm sp}$ such that for sufficiently small $v$,
\begin{equation} \label{e29}
g(z(v))=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) .
\end{equation}
This is feasible, since
\begin{equation} \label{e30}
g(z)=g(z_{\rm sp})+\tfrac12\,g''(z_{\rm sp})(z-z_{\rm sp})^2\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)
\end{equation}
with $g''(z_{\rm sp})$ positive and bounded away from 0 as $s\to \infty$. Hence, $z(v)$ can be found for small $v$ by inverting the equation
\begin{equation} \label{e31}
(z-z_{\rm sp})\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)^{1/2}=iv .
\end{equation}
By Lagrange's inversion theorem \cite{debruijn}, there is a $\delta>0$ (independent of $s$) such that
\begin{equation} \label{e32}
z(v)=z_{\rm sp}+iv+\sum_{k=2}^{\infty}\,c_k(iv)^k ,~~~~~~|v|<\delta ,
\end{equation}
with real coefficients $c_k$ (since $g(z)$ is real for real $z$) and
\begin{equation} \label{e33}
c_2={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})} .
\end{equation}
Thus
\begin{equation} \label{e34}
z(v)=z_{\rm sp}+iv-c_2\,v^2+O(v^3) ,~~~~~~|v|\leq\tfrac12\,\delta ,
\end{equation}
where the order term holds uniformly in $s$. The uniformity statement follows from an inspection of the usual argument
by which Lagrange's theorem is proved, noting that the inversion in \eqref{e29} with $g$ as in \eqref{e15} is considered for $\vartheta\to 1/\mu$, $z_{\rm sp}\to 1$ with radius
of convergence $r$ away from $1$.
By (\ref{e12}) we can restrict the integration in (\ref{e18}) to a fixed but arbitrarily small subset of $|z|=z_{\rm sp}$ near $z=z_{\rm sp}$, at the expense of an exponentially small error. Furthermore, by Cauchy's theorem and again at the expense of an exponentially small error, the integration path can be deformed in accordance with the transformation in (\ref{e29})--(\ref{e34}). Set
\begin{equation} \label{e35}
q(v)=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp})
\end{equation}
and note that from \eqref{e29}
\begin{equation} \label{e36}
g'(z(v))\,z'(v)={-}v\,g''(z_{\rm sp}) .
\end{equation}
Then substituting $z=z(v)$ in (\ref{e18}), $\mu_Q$ is given with exponentially small error by
\begin{eqnarray} \label{e37a}
\frac{s}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{g'(z(v))}{z(v)-1}~\frac{\exp(s\,g(z(v)))}{1-\exp(s\,g(z(v)))}z'(v)\,dv,
\end{eqnarray}
which gives the following result.
\begin{lemma} \label{lemma2} The mean stationary congestion level is given with exponentially small error by
\begin{equation} \label{e37}
\mu_Q =~\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v}{z(v)-1}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,dv .
\end{equation}
\end{lemma}
In a similar fashion we get that $\mathbb{P}(Q=0)$ and $\sigma_Q^2$, see (\ref{e6}) and (\ref{e8}), are given, both with exponentially small error, by
\begin{equation} \label{e39}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,v\,{\rm ln}\Big(\frac{z(v)}{z(v)-1}\Big)\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,dv
\end{equation}
and
\begin{equation} \label{e38}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,dv,
\end{equation}
respectively.
\section{Heavy-traffic limits for the mean congestion level} \label{sec3}
In this section we apply the non-standard saddle point method explained in Section \ref{sec1} to the Pollaczek integral representation for the mean stationary congestion level $\mu_Q$. In Section \ref{subsec3.1} we first derive an integral representation for the leading order behavior of $\mu_Q$ with a relative error of order $O(s^{-1})$, which serves as a heavy-traffic approximation in the regime $\rho_s=1-\gamma/s^\eta$ with $\eta>0$. We also consider separately the cases of moderate heavy traffic ($\eta\in(0,1/2)$) and extreme heavy traffic ($\eta\in(1/2,\infty)$), for which the integral representation leads to vastly different alternative expressions. We find that $\mu_Q\to 0$ more rapidly than any power of $1/s$ when $\eta\in(0,1/2)$. When $\eta\geq 1/2$ the saddle point method yields an integral representation with relative error $O(s^{-\min(1,\eta)})$.
In Section \ref{subsec3.2} we specialize this general result to the CLT case $\eta=1/2$, and make a connection with existing results.
\subsection{Leading order behavior in integral form} \label{subsec3.1}
\begin{theorem}\label{mainthm}
The mean stationary congestion level is given by
\begin{equation} \label{e48a}
\mu_Q=\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\int_0^{\infty}\,\frac{t^2}{d^2(s)+t^2}~\frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,dt\,\left(1+O({s^{{-}\min(1,\eta)}})\right)
\end{equation}
with $
d^2(s) = s^{1-2\eta}\gamma^2\mu/(2\sigma^2)$.
\end{theorem}
\begin{proof}
According to Lemma \ref{lemma2}, $\mu_Q$ is given with exponentially small error by (\ref{e37}) with $q(v)$ given in (\ref{e35}). Since $z({-}v)=z^{\ast}(v)$ for real $v$, we have
\begin{eqnarray} \label{e40}
\frac{v}{z(v)-1}+\frac{-v}{z({-}v)-1} &=& {-}2iv\,\frac{{\rm Im}(z(v))}{|z(v)-1|^2}\nonumber\\
&=&~\frac{-2iv^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} ,
\end{eqnarray}
where (\ref{e34}) and $c_k\in\mathbb{R}$ have been used. Using (\ref{e40}) in (\ref{e37}) and extending the integration range from $[{-}\tfrac12\delta,\tfrac12\,\delta]$ to $({-}\infty,\infty)$ while using symmetry of $q(v)$, we get that $\mu_Q$ is given with exponentially small error by
\begin{eqnarray} \label{e41}
\frac{s\,g''(z_{\rm sp})}{\pi}\,\int_0^{\infty}\,\frac{v^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)}\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}{\rm d} v .
\end{eqnarray}
With
\begin{equation} \label{e42}
B=\exp(s\,g(z_{\rm sp})) ,~~~~~~\alpha =g''(z_{\rm sp}),
\end{equation}
(\ref{e41}) takes the form
\begin{eqnarray} \label{e43}
\frac{s\alpha }{\pi}\,\int_0^{\infty}\,\frac{v^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} \cdot \frac{B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{eqnarray}
In leading order, the integrand in (\ref{e43}) has the form
\begin{equation} \label{e43a}
\frac{B\,v^2\,\exp(-s\,D\,v^2)}{(v^2+C\,s^{-2\eta})(1-\exp({-}s\,D\,v^2))},
\end{equation}
and this is reminiscent of the integrand in \cite[(5.12.3)]{debruijn} for the case $\kappa=2\eta$. Proceeding as in \cite[Section 5.12]{debruijn}, the substitution $v=t\sqrt{{2}/(s\alpha )}$ brings (\ref{e43}) into the form
\begin{eqnarray} \label{e44}
\frac{2}{\pi}\sqrt{\tfrac12 s\,\alpha }\int_0^{\infty}\frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2+O(t^4/s)} \,\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}{\rm d} t .
\end{eqnarray}
From (\ref{e24})--(\ref{e27}) and (\ref{e42}),
\begin{align}
\frac{2}{\pi}\,\sqrt{\frac{s\alpha }{2}} &= \frac{2}{\pi}\,\sigma_X\,\sqrt{\frac{s}{2\,\mu}}\,(1+O(s^{-\eta})),\label{y45}\\
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &= d^2(s) + O(s^{1-3\eta}),\label{y46}\\
2\,c_2(z_{\rm sp}-1) &= O(s^{-\eta}),\label{y47}\\
s\,g(z_{\rm sp}) &= -d^2(s) + O(s^{1-3\eta}),\label{y48}
\end{align}
where
\begin{equation} \label{y49}
d^2(s) = \frac{b_0^2}{s^{2\eta-1}},\quad b_0^2 := \frac{\gamma^2\mu}{2\,\sigma^2}.
\end{equation}
In the case that $2\eta-1<0$, we have that $\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 \to \infty$ and that
\begin{equation} \label{y50}
B = \exp(s\,g(z_{\rm sp})) = O(\exp({-}b^2s^{1-2\eta}))
\end{equation}
for any $b\in(0,b_0)$. From \eqref{e44} it follows then that $\mu_Q = O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$.
In the case that $2\,\eta-1\geq 0$, we have that $d^2(s)$ is bounded, and using that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, we get
\begin{align}
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &+ t^2-2\,c_2\,(z_{\rm sp}-1)\,t^2+O(t^4/s) \nonumber\\
&= d^2(s) + t^2 + O\left(s^{-\eta}\,(d^2(s)+t^2)\right) + O(t^4/s)\nonumber\\
&= \left(d^2(s)+t^2\right)\left(1+O(s^{-\eta})+O(t^2/s)\right).\label{y51}
\end{align}
Hence, in this case,
\begin{align}
& \frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2+O(t^4/s)}\nonumber\\
& \qquad \qquad \qquad \qquad = \frac{t^2}{d^2(s)+t^2}\left(1+O(s^{-\eta})+O(t^2/s)\right),\label{y52}
\end{align}
where we restrict to $t$ in a range $[0,s^{1/4}]$. Furthermore,
\begin{align}
1-B\,\exp(-t^2) &= 1-\exp({-}d^2(s)-t^2)\,\left(1+d^2(s)\,O(s^{-\eta})\right)\nonumber\\
&=(1-\exp({-}d^2(s)-t^2))\,\Big(1+\frac{d^2(s)}{\exp(d^2(s)+t^2)-1}O(s^{-\eta})\Big)\nonumber\\
&= (1-\exp({-}d^2(s)-t^2))\,(1+O(s^{-\eta})),\label{y53}
\end{align}
It follows therefore that
\begin{equation} \label{y56}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)} = \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,(1+O(s^{-\eta})).
\end{equation}
Combining the three items \eqref{y45}, \eqref{y52} and \eqref{y56}, we obtain for \eqref{e44} the result
\begin{equation} \label{y58}
\frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}} \int_0^{\infty}\frac{t^2}{d^2(s)+t^2} \cdot \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}{\rm d} t
\left(1+O(s^{-\eta})+O(s^{-1})\right),
\end{equation}
where the integration range $[0,\infty)$ is, at the expense of relative errors of type\\ $\exp({-}s^{1/4})$, first restricted to the range $[0,s^{1/4}]$, where \eqref{y52} holds, and then restored again to the full range. \end{proof}
Theorem \ref{mainthm} gives the leading-order behavior of $\mu_Q$ as $s\to \infty$ with a relative error of $O(s^{{-}\min(1,\eta)})$. By considering in more detail the integral expressions, we obtain the following result, describing two different heavy-traffic behaviors.
\begin{proposition}\label{prop1}
If $\eta\in(0,1/2)$ the mean congestion level satisfies
\begin{equation} \label{y59}
\mu_Q=O\left(\exp(-b^2s^{1-2\eta})\right),
\end{equation}
for any $b\in (0,b_0)$. If $\eta\in[1/2,\infty)$ the mean congestion level $\mu_Q$ is given by
\begin{equation} \label{y60}
s^\eta\,\frac{\sigma^2}{2\mu\gamma}\,\left(1+O(s^{\max(1/2-\eta,-1)})\right).
\end{equation}
\end{proposition}
The first assertion in Proposition \ref{prop1} follows from the observation in \eqref{y50}, together with \eqref{e44}. The second assertion is based on a connection between the integral in Theorem \ref{mainthm} and the Riemann zeta function, which is explained in the next subsection.
\subsection{Classical heavy traffic and the Gaussian random walk}
\label{subsec3.2}
We now build on Theorem \ref{mainthm} to obtain further results for the classical heavy traffic case $\eta=1/2$,
for which we know from \cite[Theorem 3]{Sigman2011b} that the rescaled congestion process converges under \eqref{bb1} to a reflected Gaussian random walk. The latter is defined as
$(S_\beta(k))_{k\geq 0}$ with $S_\beta(0)=0$ and
\begin{equation}
S_\beta(j)=Y_1+\ldots+Y_j
\end{equation}
with $Y_1,Y_2,\ldots$ i.i.d.~copies of a normal random variable with mean $-\beta$ and variance 1.
Assume $\beta>0$ (negative drift), and denote the all-time maximum of this random walk by ${M}_\beta$.
Denote by $Q^{(s)}_\infty$ the stationary congestion level for a fixed $s$ (that arises from taking
$j\to \infty$ in \eqref{lind}), and remember that we have assumed $\vartheta=n/s$ fixed.
Then, using $\rho_s=1-\gamma/\sqrt{s}$, with
\begin{equation}\label{gammachoice}
\gamma=\frac{\beta\sigma}{\mu\sqrt{\vartheta}},
\end{equation}
the spatially-scaled stationary congestion levels reach the limit
$Q^{(s)}_\infty/(\sigma\sqrt{n}) {\;\buildrel{d}\over\Rightarrow\;} {M}_\beta$ as $s,n\to \infty$ (see \cite{Jelenkovic2004,Sigman2011a,Sigman2011b}). From \cite[Theorem 4]{Sigman2011b} we then know that under the standard heavy-traffic scaling \eqref{bb1}
\begin{equation}
\frac{\mathbb{E}Q^{(s)}_\infty}{\sigma\sqrt{n}}\to \mathbb{E}{M}_\beta, \quad {\rm as} \ s,n\to \infty,
\end{equation}
from which it follows that
\begin{equation} \label{e48}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}M_\beta.
\end{equation}
The random variable ${M}_\beta$ was studied in \cite{Chang1997,Janssen2006}. In particular, \cite[Thm.~2]{Janssen2006} yields, for $\beta<2\sqrt{\pi}$,
\begin{eqnarray}\label{wdfegfw571}
\mathbb{E}{M}_\beta= \frac{1}{2\beta}+\frac{\zeta(1/2)}{\sqrt{2\pi}}+\frac{\beta}{4}+\frac{\beta^2}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta(-1/2-r)}{r!(2r+1)(2r+2)}\left(\frac{-\beta^2}{2 }\right)^r
\end{eqnarray}
and hence, for small values of $\beta$,
\begin{equation} \label{estimate}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}M_\beta \approx \frac{\sigma\sqrt{n}}{2\beta} = \sqrt{s}\,\frac{\sigma^2}{2\mu\gamma}.
\end{equation}
We will now show how the approximation \eqref{estimate} follows from Theorem \ref{mainthm}, and also how similar steps give rise to Proposition \ref{prop1}.
Consider the integral
\begin{equation} \label{e49}
G_0(b)=G_1(b)-G_2(b)=\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t ,
\end{equation}
where $b>0$ and
\begin{equation} \label{e50}
G_1(b)=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,dt\,,~~~~G_2(b)=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}{\rm d} t .
\end{equation}
We have, as in \cite[Section 2]{Janssen2006},
\begin{align} \label{e51}
G_1(b) & = \sum_{k=0}^{\infty}\:\int_0^{\infty}\,\exp({-}(k+1)(b^2+t^2))\,dt \nonumber \\
& = \frac{\sqrt{\pi}}{2}\,\sum_{k=0}^{\infty}\,\frac{{\rm e}^{-(k+1)b^2}}{\sqrt{k+1}} = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},1/2,1) \nonumber \\
& = \frac{\pi}{2b}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta(\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
where $\Phi(z,s,v)$ is Lerch's transcendent and where the last identity holds when $0<b<\sqrt{2\pi}$.
As to $G_2(b)$, we make a connection with the complementary error function
\begin{equation} \label{e52}
{\rm erfc}(z)=\frac{2}{\sqrt{\pi}}\,\int_z^{\infty}\,{\rm e}^{-t^2}\,dt=\frac{2}{\pi}\,{\rm e}^{-z^2}\,\int_0^{\infty}\,\frac{{\rm e}^{-z^2t^2}}{1+t^2}{\rm d} t ,
\end{equation}
see \cite[Secs.~7.2 and 7.7.1]{ref5}. We thus compute
\begin{align} \label{e53}
G_2(b) & = \sum_{k=0}^{\infty}\,{\rm e}^{-(k+1)b^2}\,\int_0^{\infty}\,\frac{b^2}{b^2+t^2}\,{\rm e}^{-(k+1)t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[(4.3) and (4.23)]{Janssen2006},
\begin{equation} \label{e54}
\sum_{n=1}^{\infty}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^{\infty}\,{\rm e}^{-x^2/2}\,dx= \frac{1}{2\beta^2}-\frac14-\frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^{\infty} \frac{\zeta({-}1/2-r)({-}1/2)^r} {r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
in which $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e54}), we get
\begin{equation} \label{e55}
G_2(b)=\frac{\pi}{4b}-\frac{\pi}{4}\,b-\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results in (\ref{e51}) and (\ref{e55}) can be combined, as in \cite[end of Section \ref{sec4}]{Janssen2006}, and this yields
\begin{equation} \label{e56}
G_0(b)=\frac{\pi}{4b}+\frac{\pi}{4}\,b+\frac{\sqrt{\pi}}{2}\,\zeta(1/2)+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Using (\ref{e56}) in (\ref{e48}), we find that the leading order behavior of $\mu_Q$ is given as
\begin{equation} \label{e57}
\sigma_X\,\sqrt{\dfrac{s}{2\mu}}\,\left[\frac{1}{2b_0}+\frac{b_0}{2}+\frac{\zeta(1/2)}{\sqrt{\pi}}+\frac{2}{\sqrt{\pi}}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r b_0^{2r+2}} {r!\,(2r+1)(2r+2)}\right]
\end{equation}
with relative error of $O(s^{-1/2})$ in which $b_0$ is given by \eqref{y49}. The expression (\ref{e57}) is exactly equal to the right-hand side of \cite[(4.25)]{Janssen2006} times $\sqrt{s}$ when we take there $\sigma=\mu=1$ and $\beta=b_0\,\sqrt{2}$.
Notice that, with $\gamma$ as in \eqref{gammachoice},
\begin{equation} \label{e57a}
\sigma\,\sqrt{\dfrac{s}{2\mu}}\frac{1}{2b_0}=\frac{\sigma\sqrt{n}}{2\beta},
\end{equation}
which confirms the approximation \eqref{estimate}.
According to Theorem \ref{mainthm}, we have for $\eta\geq 1/2$,
\begin{equation} \label{y61}
\mu_Q = \frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}}G_0(d(s))\,\left(1+O(s^{{-}\min(1,\eta)})\right).
\end{equation}
When $\eta=1/2$, so that $d(s) = b_0$ is independent of $s$, the series representation for $G_0$ in \eqref{e56} can be used, as long as $b_0\in(0,\sqrt{2\pi})$. When $\eta>1/2$, we have that $d(s) = b_0/s^{\eta-1/2}\to 0$ as $s\to \infty$, and so this series representation can be used when $s$ is large enough. We then have from \eqref{e56} and $b_0^2 = \gamma^2\mu/2\,\sigma^2$, while replacing the whole series at the right-hand side by $O(b^2)$, for $\mu_Q$ the leading order behavior
\begin{equation} \label{y62}
s^\eta\left[\frac{\sigma^2}{2\,\gamma\,\mu}+\frac{\sigma\,\zeta(1/2)}{\sqrt{2\,\pi\,\mu}}\,\frac{1}{s^{\eta-1/2}}+\frac{1}{4}\,\gamma\,\frac{1}{s^{2\eta-1}}+O(s^{3/2-3\eta})\right]
\end{equation}
with relative error $O(s^{{-}\min(1,\eta)})$. Retaining the constant term $\sigma^2/(2\gamma\mu)$ and estimating the other terms between the brackets in \eqref{y62} as $O(s^{1/2-\eta})$, we get Proposition \ref{prop1}.
\section{More heavy-traffic results}\label{more}
In this section we apply the non-standard saddle point method to obtain several more heavy-traffic results. In Section \ref{subsec3.3} we derive refined heavy-traffic approximations for the mean congestion level by considering higher-order correction terms. In Section \ref{sec4} we derive the leading heavy-traffic behavior for the variance of the stationary congestion level, and in Section \ref{sec5} for the empty-system probability. To keep the developments tractable, we restrict Section \ref{subsec3.3} to $\eta=1/2$, and Section \ref{sec4} and Section \ref{sec5} to $\eta\in(0,1]$, although the same technique will work for all values $\eta>0$.
\subsection{Correction term for the mean congestion level in the case $\eta = 1/2$} \label{subsec3.3}
Our saddle point method not only establishes the leading-order heavy-traffic approximations, but also allows to derive refinements to these approximations. In this section we demonstrate how this works for the mean congestion level in the case $\eta=1/2$.
To obtain a refinement or correction term from (\ref{e44}), we must be more precise about the $O(s^{{-}\eta})$ terms that occur in the approximations in Section \ref{subsec3.1} for $\frac12\,s\,\alpha (z_{\rm sp}-1)^2$, $B$ and $\sqrt{s\,\alpha /2}$. When higher-order corrections are required, we should include higher-order terms in the approximations of these quantities, and be more specific about the $O(t^2/s)$ and $O(t^4/s)$ in the integrand in (\ref{e44}).
Denote, see \eqref{e10} and (\ref{e15}) with $\vartheta=(1-\gamma/s^\eta)\,\mu^{-1}$,
\begin{equation} \label{e58}
a_i=g^{(i)}(1);~~~~~~g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}\,\tilde X(z) .
\end{equation}
Dropping the $X$ from $\mu$ and $\sigma^2$ for brevity, we have
\begin{equation} \label{e59}
a_1={-}\,\frac{\gamma}{s^\eta} ,~~~~~~a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) ,
\end{equation}
\begin{equation} \label{e60}
a_3={-}2+\Big(1-\frac{\gamma}{s^\eta}\Big)\Big(\frac{\tilde X'''(1)}{\tilde X'(1)}-3\tilde X''(1)+2(\tilde X'(1))^2\Big) .
\end{equation}
For the purpose of finding a first-order correction term, we note that
\begin{align} \label{e61}
\alpha &=g''(z_{\rm sp})=a_2+(z_{\rm sp}-1)\,a_3+O(s^{-1}) ,\\
\label{e62}
z_{\rm sp}-1&={-}\,\frac{a_1}{a_2}-\frac{a_3}{2a_2}\,\Big(\frac{a_1}{a_2}\Big)^2+O(s^{-3/2}) ,\\
\label{e63}
c_2&={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})}={-}\,\frac{a_3}{6a_2}+O(s^{-1/2}) ,\\
\label{e64}
g(z_{\rm sp})&={-}\,\frac{a_1^2}{2a_2}-\frac{a_3}{6a_2^3}\,a_1^3+O(s^{-2}) .
\end{align}
This gives rise to
\begin{align} \label{e65}
\sqrt{\tfrac12\,s\,\alpha }&=\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}+O(s^{-1})\Big) ,\\
\tfrac12\,s\,\alpha (z_{\rm sp}-1)^2&=\frac{\gamma^2\,\mu}{2\sigma^2}+\frac{C_2}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e67}
2c_2(z_{\rm sp}-1)&=\frac{C_3}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e68}
B=\exp(s\,g(z_{\rm sp}))&=\exp\Big({-}\,\frac{\gamma^2\,\mu}{2\sigma^2}\Big)\Big(1+\frac{C_4}{\sqrt{s}}+O(s^{-1})\Big) ,
\end{align}
with explicitly computable constants $C_1$, $C_2$, $C_3$, $C_4$. Remembering that $b_0^2=\gamma^2\mu/2\sigma^2$, see \eqref{y49}, we then get with errors of order $1/s$
\begin{eqnarray} \label{e69}
& \mbox{} & \frac{t^2(1+O(t^2/s))}{\frac12\,s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)\,t^2+O(t^4/s)} \nonumber \\[3mm]
& & \qquad =~\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+ b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big) ,
\end{eqnarray}
and
\begin{equation} \label{e70}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}=\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2} .
\end{equation}
Using (\ref{e65}), (\ref{e69}) and (\ref{e70}) in (\ref{e44}) we get with an absolute error of order $1/\sqrt{s}$
\begin{align} \label{e71}
\mu_Q & =\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}\Big)\nonumber \\
& \qquad\qquad \cdot \int_0^{\infty}\,\Big(\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big)\Big) \nonumber \\
& \qquad\qquad\qquad \cdot~\Big(\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\Big){\rm d} t \nonumber \\
& =\frac{2\sigma}{\pi}\,\sqrt{\dfrac{s}{2\mu}}\,G_0(b_0)\nonumber\\
& \qquad\qquad + ~\frac{2\sigma}{\pi}\,\sqrt{\dfrac{1}{2\mu}}\,\big((C_1+C_3)\,G_0(b_0)-(C_2+b_0^2\,C_3)\,G_3(b_0)+C_4\,G_4(b_0)\big) ,
\end{align}
where $G_0$ is as in (\ref{e49}), and
\begin{align} \label{e72}
G_3(b_0)&=\int_0^{\infty}\,\frac{t^2}{(b_0^2+t^2)^2}~\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}{\rm d} t ,\\
\label{e73}
G_4(b_0)&=\int_0^{\infty}\,\frac{t^2}{b_0^2+t^2}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}{\rm d} t .
\end{align}
We shall express the integrals in (\ref{e72}) and (\ref{e73}) in terms of $\zeta$-functions. By partial integration
\begin{align} \label{e74}
G_3(b) & = \frac12\,\int_0^{\infty}\,\frac{1}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,dt -~\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}{\rm d} t \nonumber \\
& = \frac{1}{2b^2}\,G_2(b)-G_4(b) ,
\end{align}
see (\ref{e49}) and (\ref{e73}). Since $G_2(b)$ is expressed in terms of $\zeta$-functions in (\ref{e55}), it is sufficient to consider $G_4(b)$.
As to $G_4(b)$,
\begin{equation} \label{e75}
G_4(b)=G_5(b)-G_6(b) ,
\end{equation}
where
\begin{align} \label{e76}
G_5(b)&=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,dt ,\\
\label{e77}
G_6(b)&=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,dt .
\end{align}
We have, compare (\ref{e51}),
\begin{align} \label{e78}
G_5(b) & = \sum_{k=0}^{\infty}\,(k+1)\,\int_0^{\infty}\,{\rm e}^{-(k+1)(b^2+t^2)}\,dt \nonumber \\[3.5mm]
& = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},{-}\tfrac12,1) = \frac{\pi}{4b^3}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta({-}\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
the last identity being valid when $0<b<\sqrt{2\pi}$. Next we have, compare (\ref{e53}),
\begin{align} \label{e79}
G_6(b) & = \sum_{k=0}^{\infty}\,(k+1)\,b^2\,\int_0^{\infty}\,\frac{\exp({-}(k+1)(b^2+t^2))}{b^2+t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,(k+1)\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[(5.4) and (5.21)]{Janssen2006} we have
\begin{eqnarray} \label{e80}
\sum_{n=1}^{\infty}\frac{n}{\sqrt{2\pi}}\int_{\beta\sqrt{n}}^{\infty}{\rm e}^{-x^2/2}\,dx = \frac{3}{4\beta^4}-\frac{1}{24}-\frac{1}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1/2)^r}{r!\,(2r+1)}\,\beta^{2r+1}
\end{eqnarray}
when $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e80}), we get
\begin{equation} \label{e81}
G_6(b)=\frac{3\pi}{16b^2}-\frac{\pi b}{24}-\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r}{r!\,(2r+1)}\,b^{2r+2}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results (\ref{e78}) and (\ref{e81}) can be combined, as in \cite[end of Section 5]{Janssen2006} and this yields
\begin{equation} \label{e82}
G_4(b)=\frac{\pi}{16b^3}+\frac{\pi b}{24}+\tfrac12\,\zeta({-}1/2)\,\sqrt{\pi}+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Finally, we can use (\ref{e82}) in (\ref{e74}), and we obtain with (\ref{e55}), for $0<b<\sqrt{2\pi} $,
\begin{align} \label{e83}
G_3(b) = & \frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} -~2\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)(2r+3)}.
\end{align}
The right-hand side of (\ref{e83}) equals the right-hand side of \cite[(2.3)]{Janssen2006} multiplied by ${\pi}/{(2b)}$ with $\beta=b\,\sqrt{2}$.
\subsection{Heavy-traffic limits for the variance}\label{sec4}
We have from (\ref{e38}) in Section \ref{sec1}, using the same approach and notation as in Section \ref{subsec3.1} for $\mu_Q$, that $\sigma_Q^2$ is given with exponentially small error by
\begin{equation} \label{e84}
\frac{-s\,\alpha }{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v,
\end{equation}
with $B$ and $\alpha $ given in (\ref{e42}). From $z({-}v)=z^{\ast}(v)$ for real $v$ we now compute
\begin{equation} \label{e85}
\frac{z(v)}{(z(v)-1)^2}-\frac{z({-}v)}{(z({-}v)-1)^2}={-}2i\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,{\rm Im}(z(v)) ,
\end{equation}
and so (\ref{e84}) becomes
\begin{equation} \label{e86}
\frac{s\alpha }{\pi}\,\int_0^{\frac12\delta}\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,v\,{\rm Im}(z(v))\,\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
From
\begin{equation} \label{e87}
{\rm Im}(z(v))=v+O(v^3) ,~~~~~~|z(v)|^2-1=z_{\rm sp}^2-1+O(v^2) ,
\end{equation}
we get for the expression in \eqref{e86}
\begin{equation} \label{y70}
\frac{s\alpha }{\pi}\,\int_0^{\frac{1}{2}\delta}\,\frac{v^2\,(z_{\rm sp}^2-1+O(v^2))(1+O(v^2))}{((z_{\rm sp}-1)^2+v^2 + O((z_{\rm sp}-1)\,v^2)+O(v^4))^2}
\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v.
\end{equation}
When $2\eta-1<0$, we have as for the case of $\mu_Q$ in Section \ref{subsec3.1} that the whole expression in \eqref{y70} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. When $2\eta-1\geq 0$, we get as in the case of $\mu_Q$ after substitution $v = t\sqrt{{2}/{(s\,\alpha })}$ for the expression in \eqref{y70}
\begin{equation} \label{y71}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}\,\int_0^\infty\frac{t^2\,(z_{\rm sp}^2-1+O(t^2/s))(1+O(t^2/s))}{(d^2(s)+t^2)^2\,(1+O(1/s^{\eta})+O(t^2/s))}~\frac{B\,{\rm e}^{{-}t^2}}{1-B\,{\rm e}^{{-}t^2}}{\rm d} t.
\end{equation}
When $2\eta-1\geq 0$, the leading order behavior of $\sigma_Q^2$ depends crucially on the factor $z_{\rm sp}^2-1+O(t^2/s)$, where
\begin{equation} \label{y72}
z_{\rm sp}^2-1 = \frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\left(1+O(s^{-\eta})\right)
\end{equation}
is dominant when $\eta<1$, while the $O(t^2/s)$ is dominant when $\eta>1$. In the case that $\eta\in(1/2,1)$, we get for the leading order behavior of $\sigma_Q^2$
\begin{align}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}& \,\frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\int_0^\infty\frac{t^2}{(d^2(s)+t^2)^2}\cdot~\frac{{\rm e}^{{-}d^2(s)-t^2}}{1-{\rm e}^{{-}d^2(s)-t^2}}{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{\gamma\,\sigma}{\pi}\,\Big(\frac{2}{\mu}\Big)^{1/2}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right), \label{y73}
\end{align}
where \eqref{e26}, \eqref{e27} and \eqref{e42} have been used for $\alpha = g''(z_{\rm sp})$ and where $G_3$ is given in \eqref{e72}.
When we insert the expansion \eqref{e83} for $G_3(b)$, with the whole series on the second line being $O(b^2)$, we get the leading order behavior of $\sigma_Q^2$ as
\begin{align}
s^{2\eta}\,\Big( \frac{\sigma^4}{4\,\gamma^2\mu^2}- \frac{\sigma^2}{4\,\mu}&\,\frac{1}{s^{2\eta-1}} - \Big(\frac{2\,\sigma^2}{\pi\,\mu}\Big)^{1/2}\,\frac{\gamma\,\zeta(-1/2)}{s^{3\eta-3/2}}\nonumber\\
& - \frac{\gamma^2}{24\,s^{5\eta-5/2}}+O(s^{1-4\eta})\Big)\,\left(1+O(s^{\eta-1})\right)\nonumber \\
&\ = s^{2\eta}\,\frac{\sigma^4}{4\,\gamma^2\,\mu^2}\,\Big(1+O(s^{\max(1-2\eta,\eta-1)})\Big)\label{y74}
\end{align}
when $\eta\in(1/2,1)$. For the case $\eta=1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{align}
\frac{\sigma^2 s}{\mu}\left[ \frac{1}{8\,b_0^2} - \frac{1}{4}-\frac{1}{12}\,b_0^2 - \frac{2\,\zeta(-1/2)}{\sqrt{\pi}}\,b_0- \frac{4}{\sqrt{\pi}}\,\sum_{r=0}^\infty \frac{\zeta(-3/2-r)\,(-1)^r\,b_0^{2r+3}}{r!\,(2r+1)\,(2r+2)\,(2r+3)} \right]\label{y75}
\end{align}
with relative error $O(s^{-1/2})$. The expression between brackets in \eqref{y75} coincides with the right-hand side of \cite{Janssen2006}, (2.3) with $\beta = b_0\,\sqrt{2}$.
This leads to the following two results.
\begin{theorem} \label{varthm}
For $\eta\in[1/2,1)$,
\begin{equation} \label{y76}
\sigma_Q^2 = \frac{\gamma\,\sigma_X}{\pi}\,\sqrt{\frac{2}{\mu}}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right)
\end{equation}
with $G_3$ given in \eqref{e72}.
\end{theorem}
\begin{proposition}\label{varprop}
For $\eta\in(0,1/2)$, and for all $b<b_0$,
\begin{equation}
\sigma_Q^2 = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation}
For $\eta = 1/2$, $\sigma_Q^2$ equals expression \eqref{y75} with relative error $O(s^{-1/2})$. For $\eta\in(1/2,1)$ and $b_0\in(0,\sqrt{2\pi})$, $\sigma_Q^2$ has the form in \eqref{y74}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level
with $\eta=1/2$, it is possible to give a correction term which involves now integrals and series with $\zeta$-functions as considered in \cite[Secs.~4-5]{cumulants}.
\subsection{Heavy-traffic limits for the empty-system probability} \label{sec5}
We have from (\ref{e6}) by proceeding as in (\ref{e13})--(\ref{e17}) that
\begin{align} \label{e100}
{\rm ln}\,[Q(0)] & = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{z}{z-1}\Big)\,\frac{g'(z)\,\exp(s\,g(z))}{1-\exp(s\,g(z))}\,dz \nonumber \\[3.5mm]
& = \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{z(z-1)}\,{\rm ln}\left(1-\exp(s\,g(z))\right)\,dz ,
\end{align}
where in the last step we used partial integration (noting that ${\rm Re}\,[g(z)]<0$ on $|z|=1+\varepsilon$). Then, as in Section \ref{sec1} for $\mu_Q$, the last integral in (\ref{e100}) is, with exponentially small error, given by
\begin{equation} \label{e101}
\frac{1}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{z'(v)}{z(v)(z(v)-1)}\,{\rm ln}\left(1-B\,{\rm e}^{-\frac12 s\alpha v^2}\right)\,dv .
\end{equation}
Now for $v\geq0$ from $z({-}v)=z^{\ast}(v)$, $z'({-}v)={-}(z'(v))^{\ast}$
\begin{eqnarray} \label{e102}
& \mbox{} & \hspace*{-6mm}\frac{z'(v)}{z(v)(z(v)-1)}+\frac{z'({-}v)}{z({-}v)(z({-}v)-1)}=2i\,{\rm Im}\,\Big[\frac{z'(v)}{z(v)(z(v)-1)}\Big] \nonumber \\[3.5mm]
& & \hspace*{-6mm}=~2i\,{\rm Im}\,\Big[\frac{z'(v)\,z^{\ast}(v)(z^{\ast}(v)-1)}{|z(v)|^2\,|z(v)-1|^2}\Big] \nonumber \\[3.5mm]
& & \hspace*{-6mm}=~2i\,\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}+O(v^2))((z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4))}\,,
\end{eqnarray}
where we used \eqref{e32} and the fact that $z_{\rm sp}$ and $c_k$ are real with $z_{\rm sp}>1$. Therefore, we get for the expression in \eqref{e101}
\begin{align}
\frac{1}{\pi} &\int_0^{\frac{1}{2}\delta}\frac{1}{z_{\rm sp}{\rm +}O(v^2)}\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)v^2)+O(v^4)}\nonumber \\
&{\rm ln}\left(1-B\exp(-\tfrac12 s\alpha v^2)\right){\rm d} v.
\label{y77}
\end{align}
In the case that $2\eta-1<0$, we have as earlier that the whole expression in \eqref{y77} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. In the case that $2\eta-1\geq 0$, we substitute $v=t\sqrt{{s}/{(2\,\alpha )}}$, and we get as earlier for the expression \eqref{y77}, assuming also that $\eta<1$,
\begin{align}
\frac{1}{\pi}&\,\sqrt{s\,\alpha /2}\,\int_0^{\infty}\frac{z_{\rm sp}-1+O(t^2/s)}{(d^2(s)+t^2)\,(1+O(s^{-\eta})+O(t^2/s))}\,{\rm ln}(1-B\,{\rm e}^{-t^2}) {\rm d} t\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{\sqrt{s\,\alpha /2} \ (z_{\rm sp}-1)}{d^2(s)+t^2}{\rm ln}(1-B\,{\rm e}^{-t^2}){\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{d(s)}{d^2(s)+t^2}{\rm ln}(1-{\rm e}^{{-}d^2(s)-t^2}){\rm d} t\,\left(1+O(s^{\eta-1})\right).
\label{y78}
\end{align}
Here we also used \eqref{y46} and that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, so that
\begin{equation} \label{y79}
(\tfrac12\,s\,\alpha )^{1/2}\,(z_{\rm sp}-1) = d(s)\,\left(1+O(s^{-\eta})\right) = d(s)\left(1+O(s^{\eta-1})\right),
\end{equation}
since $\eta\geq 1/2$.
We have for $b>0$
\begin{align}
\frac{1}{\pi}&\int_0^\infty \frac{b}{b^2+t^2}\,{\rm ln}(1-\exp({-}b^2-t^2)){\rm d} t =-\frac12\sum_{k=0}^{\infty}\,\frac{1}{k+1}\,{\rm erfc}(b\,\sqrt{k+1}) = -F(b\,\sqrt{2}),\label{y80}
\end{align}
where according to \cite[(3.3) and (3.12)]{Janssen2006} for $\beta>0$
\begin{align}
F(\beta) &= \sum_{n=1}^\infty\,\frac{1}{n}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^\infty {\rm e}^{-x^2/2}dx\nonumber\\
&= -{\rm ln}\,\beta - \frac12\,{\rm ln}2 - \frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^\infty \frac{\zeta(1/2-r)\,(-1/2)^r\,\beta^{2r+1}}{r!\,(2r+1)},\label{y81}
\end{align}
the last identity being valid for $0<\beta<2\sqrt{\pi}$.
Using \eqref{y81} with $\beta^2 = d^2(s)= b_0^2/s^{2\eta-1}$, with the entire series on the second line being $O(\beta)$, we get the leading order behavior of ${\rm ln}[Q(0)]$ as
\begin{equation} \label{y82}
\Big({-}(\eta-1/2)\,{\rm ln}\,s+{\rm ln}(2\,b_0)+O(s^{1/2-\eta})\Big)\left(1+O(s^{\eta-1})\right)
\end{equation}
when $\eta\in(1/2,1)$. For $\eta = 1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{equation} \label{y83}
{\rm ln}(2\,b_0) + \frac{1}{\sqrt{\pi}}\,\sum_{r=0}^\infty \,\frac{\zeta(1/2-r)\,(-1)^r}{r!\,(2r+1)}\,b_0^{2r+1}
\end{equation}
with relative error $O(s^{-1/2})$. The expression \eqref{y83} coincides with ${\rm ln}[\mathbb{P}(M=0)]$ as given by \cite[(2.1)]{Janssen2006} with $\beta = b_0\,\sqrt{2}$. The next two results summarize the above.
\begin{theorem} \label{emptythm}
For $\eta\in(1/2,1)$,
\begin{equation} \label{y84}
{\rm ln}[\mathbb{P}(Q=0)] = - F\big(d(s)\,\sqrt{2}\big)\left(1+O(s^{\eta-1})\right)
\end{equation}
with $F$ given by \eqref{y81}.
\end{theorem}
\begin{proposition} \label{emptyprop}
For $\eta\in (0,1/2)$, and for all $b<b_0$,
\begin{equation}
{\rm ln}[\mathbb{P}(Q=0)] = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation}
For $\eta=1/2$, ${\rm ln}[\mathbb{P}(Q=0)]$ equals $-F(b_0\,\sqrt{2})$ with a relative error $O(1/\sqrt{s})$. For $\eta\in (1/2,1)$ and $0<b_0<\sqrt{2\pi}$, ${\rm ln}[\mathbb{P}(Q=0)]$ has leading order behavior as in \eqref{y82}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level case
with $\eta=1/2$, it is possible to give a correction term which involves now the integrals in \eqref{y80} and \eqref{e51}.
\section{Numerical examples}\label{numm}
\subsection{Accuracy of the approximations}
In this subsection we present a numerical example that serves to illustrate the accuracy of the derived heavy-traffic approximations. Consider the Poisson case
\begin{equation}
\tilde X(z)={\rm e}^{z-1},\quad \mu = \sigma^2 = 1.
\end{equation}
We fix $\mu$ and vary $n$ with the value of $s$, according to
\begin{equation}
\vartheta = \frac{n}{s} = 1-\frac{\gamma}{s^\eta}
\end{equation}
for some $\gamma>0$ and $\eta\geq 1/2$. To calculate the exact value of the mean congestion level we use the expression, see \cite{Boudreau1962},
\begin{equation}\label{x73}
\mu_Q=\frac{\sigma_A^2}{2(s-\mu_A)}-\frac{s-1+\mu_A}{2}+\sum_{k=1}^{s-1}\frac{1}{1-z_k}.
\end{equation}
Here $z_1,\ldots,z_{s-1}$ are the zeros of $z^s-A(z)$ in $|z|<1$. We apply the method of successive substitution described in \cite{Janssen2005} to obtain accurate numerical approximations for $z_1,...,z_{s-1}$ and consequently $\mu_Q$.
From Theorem \ref{mainthm}, we find that the leading order behavior of $\mu_Q$ is given by
\begin{equation} \label{x18}
\frac{\sqrt{2s}}{\pi}\,G_0\Big(\frac{\gamma}{\sqrt{2}\,s^{\eta-\frac{1}{2}}}\Big).
\end{equation}
In order to find the correction terms, we proceed by setting $\eta = 1/2$. Deriving constants $C_1,C_2,C_3,$ and $C_4$ for our setting and substituting these into \eqref{e71}, we get for $\mu_Q$, with an absolute error of $O(s^{-1/2})$, the approximation
\begin{equation}\label{x19}
\frac{\sqrt{2\,s}}{\pi}\Big(\Big(1-\frac{\gamma}{3\,\sqrt{s}}\Big)\,G_0(b_0)-\frac{\gamma^3}{3\,\sqrt{s}}\,(\,G_3(b_0)+G_4(b_0))\Big),
\end{equation}
which by \eqref{e49} and \eqref{e74} reduces to
\begin{equation}\label{x20}
\frac{\sqrt{2\,s}}{\pi}\,G_0(b_0)-\frac{\sqrt{2}\,\gamma}{3\,\pi}\,G_1(b_0).
\end{equation}
\begin{table}
\centering
\begin{tabular}{r|rrrr}
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20}\\
\hline
10 & 0.683 & 0.244 & 0.399 & 0.247 \\
20 & 0.776 & 0.410 & 0.565 & 0.412 \\
50 & 0.858 & 0.739 & 0.893 & 0.741 \\
100 & 0.900 & 1.110 & 1.263 & 1.111 \\
200 & 0.929 & 1.633 & 1.787 & 1.634 \\
500 & 0.955 & 2.672 & 2.825 & 2.673 \\
1000 & 0.968 & 3.843 & 3.996 & 3.843
\label{tab:poisson1}
\end{tabular}
\caption{Numerical results for $\gamma = 1$.}
\end{table}
\begin{table}
\centering
\begin{tabular}{r|rrrr}
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20}\\
\hline
10 & 0.968 & 13.707 & 14.046 &13.732\\
20 & 0.977 & 19.533 & 19.865 &19.551\\
50 & 0.985 & 31.084 & 31.409 &31.095\\
100 & 0.990 & 44.097 & 44.419 &44.106\\
200 & 0.992 & 62.499 & 62.819 &62.505\\
500 & 0.995 & 99.008 & 99.325 &99.011\\
1000 & 0.996 & 140.152 & 140.468 & 140.154
\label{tab:poisson2}
\end{tabular}
\caption{Numerical results for $\gamma = 0.1$.}
\end{table}
\begin{table}
\centering
\begin{tabular}{r|rr|rr|rr}
& \multicolumn{2}{c|}{$\eta=0.6$} & \multicolumn{2}{c|}{$\eta=0.75$} & \multicolumn{2}{c}{$\eta=0.9$}\\
$s$ & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18}\\
\hline
10 & 17.781 & 18.125 & 25.970 & 26.318 & 37.553 & 37.905 \\
20 & 27.309 & 27.647 & 44.391 & 44.734 & 71.195 & 71.541 \\
50 & 47.948 & 48.281 & 89.623 & 89.961 & 164.637 & 164.978 \\
100 & 73.245 & 73.574 & 152.031 & 152.367 & 309.353 & 309.692 \\
200 & 111.752 & 112.079 & 257.435 & 257.769 & 580.170 & 580.507 \\
500 & 195.082 & 195.409 & 515.443 & 515.776 & 1329.581 & 1329.917 \\
1000 & 297.122 & 297.448 & 870.524 & 870.857 & 2487.227 & 2487.562
\label{tab:poisson3}
\end{tabular}
\caption{Numerical results for $\gamma=0.1$ and several values of $\eta$.}
\end{table}
\noindent Numerical results for $\eta=1/2$ and various values of $s$ are given in Table 1 and 2, for $ \gamma = 1$ and $\gamma = 0.1$, respectively.
We note that for small $s$ the leading order approximation is still off by a significant amount, while the refinement only shows an error in the second decimal for $\gamma = 0.1$. This seems to justify the use of the correction term.
In Table 3 we compare the approximation \eqref{x18} against the exact value of $\mu_Q$ for three values of $\eta\geq 1/2$ to assess the influence of $\eta$. Clearly, the leading order approximation is relatively accurate for all three scenarios. As expected, the mean congestion increases along with $\eta$, since utilization approaches 1 more rapidly in this case.
\subsection{Connection to other queueing models}\label{subsec62}
As argued in the introduction, we believe that the heavy-traffic behavior for the discrete model in this paper will up to leading order be universal for a wide range of other models (when subjected to the same heavy traffic regime \eqref{bb}). We shall now substantiate this for many-server systems, for which under \eqref{bb}, it turns out that the mean congestion is $O(s^\eta)$. We compare the mean congestion level in our discrete queue with that in the multi-server systems $M/M/s$, $M/D/s$ and Gamma/Gamma/$s$, all with unit mean service time and occupation rate $1-\gamma/s^\eta$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 2.5,
xmax = 6.5,
ymin = 0,
ymax = 7.2,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,0.04)}},
y label style={at={(0.11,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.4,0.2)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,mark = o,mark options={scale=1.25}] table[x=log_s,y=mms] {tikz/novel_figure1.txt};
\addplot[thick, mark=triangle,dashed,mark options={scale=1.25,solid}] table[x=log_s,y=mds] {tikz/novel_figure1.txt};
\addplot[thick,mark=square,dotted,mark options={scale=1.25,solid}] table[x=log_s,y=ggs] {tikz/novel_figure1.txt};
\legend{$M/M/s$,$M/D/s$,Gamma/Gamma/$s$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ plotted against $s$ on log scale for 3 queues for $\eta=0.75$.}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 0.2,
xmax = 6.5,
ymin = 0,
ymax = 9,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,0.04)}},
y label style={at={(0.075,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.75,4.2)},anchor = west},
yscale = 0.8,
xscale = 1
]
\addplot[thick,only marks,mark = o,mark options={scale=1.25}] table[x=n01,y=m01] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square,mark options={scale=1.25}] table[x=n025,y=m025] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = *,mark options={scale=1.25}] table[x=n05,y=m05] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square*,mark options={scale=1.25}] table[x=n075,y=m075] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = triangle*,mark options={scale=1.25}] table[x=n1,y=m1] {tikz/novel_figure2.txt};
\addplot[dashed] coordinates{ (0,2.25) (7,9.25) };
\addplot[dashed] coordinates{ (0,2.25) (7,7.5) };
\addplot[dashed] coordinates{ (0,2.25) (7,5.75) };
\addplot[dashed] coordinates{ (0,2.25) (7,4) };
\addplot[dashed] coordinates{ (0,2.25) (7,2.95) };
\legend{\ $\eta=0.1$,\ $\eta=0.25$,\ $\eta=0.5$,\ $\eta=0.75$,\ $\eta=1$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ of $M/M/s$ plotted against $s$ on log scale for different values of $\eta$.}
\label{fig2}
\end{figure}
Figure \ref{fig1} shows on logarithmic scale the mean congestion levels for $\gamma=0.1$ and $\eta=0.75$ under the specified scaling for three systems. We also display three lines with slope 0.75 for comparison, which confirms that mean congestion levels are of the order $s^\eta$, also in these multi-server system. Formally establishing this heavy- traffic behavior for these multi-server system is an important open problem and requires other mathematical approaches than the ones taken in this paper (see the introduction for more details).
Figure \ref{fig2} shows the mean queue length in the $M/M/s$ system for several values of $\eta$, again on logarithmic scale, together with lines with slope $\eta$. For $\eta\geq 1/2$, we see the same $O(s^\eta)$ behavior, similar as for $\mu_Q$ in our discrete model. For $\eta<1/2$ the mean queue length decays, again in agreement with our results for $\mu_Q$. We note that this qualitative behavior of the $M/M/s$ system was also observed by \cite[Thm 4.1]{maman}, by proving that the mean waiting time in the $M/M/s$ queue under \eqref{bb} is of order $1/s^{1-\eta}$, which by Little's law implies the mean queue length to be of order $s^\eta$.
\section*{Appendix}
\begin{subappendices}
\renewcommand*\thesection{\Alph{section}}
\section{Proof of Pollaczek's formula in the discrete setting}
\label{app}
In the setting of Subsection \ref{sec1}, we shall show that for any $\varepsilon>0$ with $1+\varepsilon<r_0$,
\begin{equation} \label{e111}
\tilde Q(w)=\exp\Big(\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)}\,dz\Big)
\end{equation}
holds when $|w|<1+\varepsilon$. We shall establish (\ref{e111}) for any $w\in(1,1+\varepsilon)$, and then the full result follows from analyticity of $Q(w)$ and of
\begin{equation} \label{e112}
{\rm ln}\Big(\frac{w-z}{1-z}\Big)={\rm ln}\Big(\frac{1-w/z}{1-1/z}\Big)={-}\,\sum_{k=1}^{\infty}\,\frac1k\,\Big(\Big(\frac{w}{z}\Big)^k-\Big(\frac1z\Big)^k\Big)
\end{equation}
in $w$, $|w|<1+\varepsilon$ for any $z$ with $|z|=1+\varepsilon$.
Our starting point is the formula, see \cite{Boudreau1962},
\begin{equation} \label{e113}
\tilde Q(w)=\frac{(s-\mu_A)(w-1)}{w^s-\tilde A(w)}\,\to \sum_{k=1}^{s-1}\,\frac{w-z_k}{1-z_k}
\end{equation}
that holds for all $w$, $|w|<r_0$, in which $z_1,\ldots,z_{s-1}$ are the $s-1$ zeros of $z^s-\tilde A(z)$ in $|z|<1$. Fix $w\in(1,1+\varepsilon)$. Then ${\rm ln}\,[(w-z)/(1-z)]$ is analytic in $z\in\mathbb{C}\backslash [1,w]$. It follows that
\begin{align}
I_C &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)}\,dz \nonumber \\
&=~\sum_{k=1}^{s-1}\,{\rm ln}\Big(\frac{w-z_k}{1-z_k}\Big)+\frac{1}{2\pi i}\,\int_C\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)}\,dz ,
\label{e114}
\end{align}
where $C$ is a contour encircling $[1,w]$ in the positive sense with none of the $z_k$'s in its interior. We let $\delta\in(0,\frac{w-1}{2})$ and we take $C$ the union of two line segments, from $1+\delta-i0$ to $w-\delta-i0$ and from $w-\delta+i0$ to $1+\delta-i0$, and two circles, of radius $\delta$ and encircling 1 and $w$ in positive sense.
A careful administration of the various contributions to the integral $I_C$ in \eqref{e114}, taking account of the branch cut $[1,w]$, yields
\begin{equation}\label{e115}
I_C = {\rm ln }\left(\frac{(s-\mu_A)(w-1)}{w^s-\tilde A(w)}\right) + O(\delta\,{\rm ln}\, \delta ).
\end{equation}
Using this in \eqref{e113} and letting $\delta \downarrow 0$, we get \eqref{e111} for $w\in(1,1+\varepsilon)$ and the proof is complete.\newline
\newline
\end{subappendices}
\chapter{Novel heavy-traffic regimes}
\begin{chapterstart}
In this chapter, we introduce a family of heavy-traffic scalings for a large-scale service system meant to serve jobs coming from a large pool of customers.
The scaling rules are inspired by the classical QED regime discussed in Chapter 1, but lead to a range of different system behaviors that includes the ED, QED and QD regime as special cases.
To determine the scaling limits, we describe the performance measures in terms of Pollaczek integrals and use asymptotic techniques to evaluate these integrals in the large-system limit.
\end{chapterstart}
\begin{flushright}
Based on \\
\textbf{Novel heavy-traffic regimes for large-scale service systems}\\
\textit{Guido Janssen, Johan van Leeuwaarden \& Britt Mathijsen}\\
In \textit{SIAM Journal of Applied Mathematics, 75(2), 787-812 (2015)}
\end{flushright}
\newpage
\section{Introduction \& motivation}
We study the workload process of a system, experiencing stochastic demand and deterministic capacity $s_n$ per period, at equidistant time epochs.
Demand is assumed to be generated by $n$ stochastically identical and independent sources.
Let $A_{i,j}$ denote the workload brought into the system by source $i$ in period $j$, for which $\mathbb{E}[A_{i,j}] =\mu$ and ${\rm Var}\, A_{i,j} = \sigma^2$.
Then the total amount of demand arriving to the system in period $j$ is $A^{(n)}_j=\sum_{i=1}^n A_{i,j}$ with $\mathbb{E}[A^{(n)}_j] = n\mu$ and ${\rm Var}\, A^{(n)}_j = n\sigma^2$.
As explained in Chapter 1, a good capacity sizing rule for achieving economies-of-scale is $s_n = n\mu+\beta\sqrt{n}\sigma$ for some $\beta>0$.
If we denote the system utilization by $\rho_n := n\mu/s_n$, then this dimensioning rule in the bulk service queue with many sources is tantamount to the heavy-traffic scaling
\begin{equation}\label{bb1}
\sqrt{n}(1-\rho_n) \to \gamma = \frac{\beta\sigma}{\mu}, \qquad {\rm as }\ n\to\infty.
\end{equation}
Starting from this setting, we introduce a novel family described in terms of a parameter $\eta$ for which we assume that
\begin{equation}\label{bb}
n^{\eta}(1-\rho_n)\rightarrow \gamma, \quad {\rm as} \ n\to \infty, \ \gamma> 0.
\end{equation}
The parameter $\eta\geq 0$ defines a whole range of possible scaling regimes, including the classical case $\eta=1/2$.
In terms of a capacity sizing rule for systems with many customers, the condition \eqref{bb} is tantamount to $s_n=n\mu+\beta \sigma n^{1-\eta}$.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as {\it moderate} heavy traffic: heavy-traffic conditions in which the full occupancy is reached more slowly, as a function of $n$, than for classical heavy traffic. For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to {\it extreme} heavy traffic due to a relatively small variability hedge. Note that the case $\eta=0$ does not lead to 100\% system utilization when $n\to\infty$.
In this chapter we show that economies-of-scale can be achieved for a large range of $\eta$, although the nature of the benefits obtained by operating on large scale depends on the precise capacity sizing rule (hence the parameter $\eta$). We quantify performance in terms of stationary measures: The mean and variance of the congestion in the system, and the probability of an empty system. For these performance measures we derive heavy-traffic limits under the scalings \eqref{bb} that
are relatively simple functions of only the first two moments of the demand per period. Such parsimonious expressions are useful for quantifying and improving system behavior. The heavy-traffic limits, however, provide also qualitative insight into the system behavior. Our asymptotic analysis shows that mean congestion is $O(n^\eta)$, which implies
that delays experienced by the customers are negligible for all values of $\eta\in [0,1)$, are roughly constant for $\eta=1$, and grow without bound for $\eta>1$. We expect this qualitative behavior to be universal for a wide range of stochastic models to which the regime \eqref{bb} is applied.
We further show the existence of the following trichotomy as $n\to \infty$ under \eqref{bb}: For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, for $\eta\in (1/2,1)$ it converges to $0$, while only for $\eta=1/2$ there is a limiting value in $(0,1)$. Hence, as expected, the system performance deteriorates with $\eta$, in a rather crude way for the empty-system probability, and in only a mild way for mean congestion levels. The regime \eqref{bb} thus presents a range of possible capacity sizing rules that all lead to economies-of-scale, and depending on what is the desired nature of performance for a particular service system, an appropriate $\eta$ can be selected. From the quantitative perspective, our detailed asymptotic analysis leads to more precise asymptotic estimates for the performance measures in heavy traffic, which reveal the exact manner in which congestion is influenced by $\eta$ and $\gamma$.\\
\\
\noindent\textbf{Motivating examples.}
The bulk service queueing model is one of the canonical models in queueing theory, having a wide range of applications in fields like digital communication, wireless networks, road traffic, reservation systems, health care and many more (see \cite{Bruneel1993} and \cite[Chap.~5]{johanthesis} for an overview).
In road traffic, the basic model for congestion at an intersection, known as the fixed-cycle traffic-light queue \cite{Newell1960,Leeuwaarden2006}, is related to our discrete bulk service queue.
Then $s_n$ represents the maximum number of delayed cars in front of a traffic light that can depart during one green period, while $A^{(n)}_j$ is the number of newly arriving cars during a consecutive green and red period.
An example from health care is panel sizing \cite{Zacharias2014}.
Say a general practitioner has a pool of $n$ clients (typically in the order of \cite{Green2008}), all of which are potential patients, and together require $A^{(n)}_j$ consults per day.
Further assume that the practitioner can see a maximum number of $s_n$ patients per day.
What is then an appropriate patient panel size $n$, which strikes a reasonable balance between accessing medical care in a timely manner and restricting the time that the practitioner sits idle?
The panel size application is one of many examples of an appointment book, referring to some schedule of appointments for a fixed period, with capacity $s_n$ appointments per period and newly arriving appointment $A^{(n)}_j$ per period.
See \cite{Dai2014} for another recent example of an appointment book in a health care setting, again in terms of our bulk service queue, with $A^{(n)}_j$ the new patients per day and $s_n$ the number of available beds.
For all examples above, and many more, our new class of heavy-traffic scalings \eqref{bb} presents capacity sizing rules for which the expected performance can be quantified using the results in this chapter. This will be helpful in dimensioning the systems (How much capacity is needed to achieve a certain target performance?) while exploiting economies-of-scale. For appointment books our model together with the capacity sizing rules \eqref{bb} are particularly relevant for {\it advanced access} \cite{Green2008}, a scheduling approach in health care designed to reduce delays by offering every patient a same-day appointment, regardless of the urgency of the problem. In that way, patients do not have to wait long for appointments, and practices do not waste capacity by holding appointments in anticipation of urgent cases.\\
\\*
\noindent\textbf{Pollaczek's formula.}
Next to the freedom to model different situations, another advantage of our model is that it is mathematically tractable, in the sense that it can be subjected to powerful mathematical methods from complex and asymptotic analysis. In order to establish the heavy-traffic limits we start from Pollaczek's formula for the transform of the stationary queue length distribution in terms of a contour integral. From this famous transform representation, contour integrals for the empty-system probability and the mean and variance of the congestion immediately follow. Contour integrals are often amenable to asymptotic evaluation (see e.g.~\cite{Cohen1982}), particularly for obtaining classical heavy-traffic asymptotics.
We also subject the contour integral representations to asymptotic evaluation, but not under classical heavy-traffic scaling.
This asymptotic analysis requires a {\it non-standard} saddle point method, tailored to the specific form of the integral expressions that arise under the capacity sizing rule \eqref{bb}. \\
\\*
\noindent
\textbf{Saddle point method.}
In complex analysis, the saddle point method in its standard form is a useful technique to estimate the asymptotic behavior of integrals of the form
\begin{equation}
\label{eq:sp_integral}
I(n) = \int_C h(z)\, {\rm e}^{n f(z)}\, {\rm d} z,
\end{equation}
as $n\to\infty$, where $C$ is a contour in the complex plane, and $f(z)$ and $h(z)$ are functions that are analytic in some neighborhood of $C$.
The main idea behind the saddle point method is that if the integrand in \eqref{eq:sp_integral} exhibits a sharp peak along the contour, then one may naturally expect that a small neighborhood around this peak provides the dominant contribution to the integral.
More specifically, for large values of $n$, the function $f$ and its associated maximum $f(z^*)$ for $z^*\in C$ to a large extent determine the magnitude of the integrand (where $z^*$ is well-defined due to analytically of $f$.
In the setting of this chapter, $C$ is a closed curve, which implies that the value $z^*$ must be a \textit{saddle-point} of $f$, i.e.~$f'(z^*) = 0$.
Subsequently, one can replace $f(z)$ in \eqref{eq:sp_integral} by its Taylor expansion around $z^*$ and deduce through the Laplace method, see e.g.~\cite{debruijn}, that
\begin{equation}
I(n) = \sqrt{2\pi}\,i\frac{h(z^*)\,{\rm e}^{n f(z^*)}}{\sqrt{n\, |f''(z^*)|}}\Bigl( 1+ O(1/n)\Bigr),
\end{equation}
as $n\to\infty$.
In Section \ref{spSec}, we show how the contour integrals describing stationary measures for the queue length, derived through Pollaczek's formula, can be reformulated into the shape of \eqref{eq:sp_integral}.
However, we will show that the saddle-point method in its standard from cannot be applied to asymptotically characterize other stationary measures like the mean or mass at zero.
Indeed, for our model the saddle point (the solution of \eqref{e21}) converges to one (as $n\to\infty$), which is a singular point of the integrand, and renders the standard saddle point method useless.
Non-standard saddle point method discussed in this chapter, originally proposed by \cite{debruijn}, is made specifically to overcome this complication. This leads to asymptotic expansions for the performance measures, of which the limiting forms correspond to the heavy-traffic limits, and pre-limit forms present refined approximations for pre-limit systems ($n<\infty$) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as {\em corrected diffusion approximations} \cite{Siegmund1978,Blanchet2006,Asmussen2003}.
\noindent{\bf Further connections to the literature.}
We now discuss two classes of stochastic systems for which the heavy-traffic regime \eqref{bb1} has been studied extensively, and for which our new family of regimes \eqref{bb} is largely unexplored. We discuss these classes because, despite the Pollaczek formula not to hold, we believe the qualitative results that we reveal for our particular model should to a large extent carry over to these settings as well, presenting some interesting avenues for further research (see Section \ref{subsec62}).
The first class concerns so-called {\it nearly-deterministic} systems \cite{Sigman2011a,Sigman2011b}, denoted by $G_n/G_n/1$ system, where $G_n$ stands for {\it cyclic thinning} of order $n$, indicating that some point process is thinned to contain only every $n$th point. As $n\to \infty$, the $G_n/G_n/1$ systems approach the deterministic $D/D/1$ system. For $G_n/G_n/1$ systems, \cite{Sigman2011a} establishes stochastic-process limits, and \cite{Sigman2011b} derives heavy-traffic limits for stationary waiting times. In the framework of \cite{Sigman2011a,Sigman2011b}, our stochastic model corresponds to a $D/G_n/1$ queue, where the sequence of service times $\{A^{(n)}_j\}_{j\geq 1}$ follows from a cyclically thinned sequence of i.i.d.~random variables $A_{i,j}$. It follows from \cite[Theorem 3]{Sigman2011b} that the rescaled stationary waiting time process converges under \eqref{bb1} to a reflected Gaussian random walk. Hence, the performance measures of the nearly deterministic system, under \eqref{lind} and \eqref{bb1}, should be well approximated by the performance measures of the reflected Gaussian random walk, giving rise to heavy-traffic approximations. This connection is discussed in detail in Section \ref{subsec3.2}. It seems likely that results similar as in this chapter can be obtained for applying the scaling \eqref{bb} to the nearly-deterministic systems in \cite{Sigman2011a,Sigman2011b}, and because Pollaczek's formula also applies to this setting, the non-standard saddle point method developed in this chapter can provide the appropriate methodology.
The second class concerns multi-server systems, and in particular the many-server regime. When we interpret $s_n$ as the number of servers, instead of capacity per time slot or order of thinning, the scaling \eqref{bb1} is similar to the QED or Halfin-Whitt regime for the $M/M/s_n$ system.
As we have reviewed in Chapter 1, the QED regime is characterized by a delay probability that converges to a non-degenerate limit away from both zero and one, and the mean delay is asymptotically negligible as the number if servers grows large. The QED regime \eqref{bb1} is naturally positioned in between the Quality-Driven (QD) regime and the Efficiency-Driven (ED) regime. In the QD regime, the load remains bounded away from 1, which corresponds to setting $\eta=0$ in \eqref{bb}. Hence, the range $\eta\in(0,1/2)$ bridges the gap between the QED regime and the QD regime. Likewise, the ED regime corresponds to setting $\eta=1$ in \eqref{bb}, so that the range $\eta\in(1/2,1]$ connects the QED regime and ED regime. For the birth-death process describing the $M/M/s_n$ system, Maman \cite{maman} introduced a scaling similar to \eqref{bb}, and called it the QED-$c$ regime, also bridging the ED and QD regimes.
Theorem 4.1 of \cite{maman} says that the expected waiting time under the scaling $s_n = n\mu+\beta\sigma n^{1-\eta}$ is of order $s_n^{1-\eta}$, which is equivalent to the expected queue length being of order $n^\eta$ by Little's law. We should stress though that we expect the mathematical techniques that are needed to establish heavy-traffic results could be entirely different than in this chapter, because Pollaczek's formula does not apply to many-server settings.
The specific model assumptions will determine to a large extent the appropriate methodology. Under Markovian assumptions leading to the $M/M/s_n$ system, product-form solutions are available for the stationary distribution. This makes it possible to describe performance measures like the mean congestion directly in terms of real integrals. Where the saddle point method is used for integrals in the complex plane, the Laplace method (see e.g.~\cite{flajolet}) is used for real integrals. Hence, for the asymptotic evaluation of the $M/M/s_n$ system under the scaling \eqref{bb}, the Laplace method seems an appropriate methodology, although again one needs to deal with possible singularities in the integrand. For $G/D/s_n$ systems, which assume deterministic service times, it has been shown in \cite{Jelenkovic2004} that using a decomposition property the dynamics of this multi-server systems can be captured in terms of a single-server system. Hence, for these systems, Pollaczek's formula applies, and our saddle point method can most likely be applied to obtain heavy-traffic results in the regimes \eqref{bb}. Under more general conditions, for instance leading to a $G/G/s_n$ system, it is simply unclear at this stage how to obtain precise heavy-traffic approximations for \eqref{bb}, because a tractable description of the performance measures is not available; see Section 1.2.4 for details.\\
\\
\\*
\noindent{\bf Structure of the chapter.}
In Section \ref{sec1} we present in detail the model and the family of heavy-traffic scalings. In Section \ref{spSec} we introduce the saddle point method. In Section \ref{sec3} we apply the saddle point method for the mean congestion level. Theorem \ref{mainthm} gives for all heavy-traffic scalings the limiting behavior in terms of an integral expression. As a consequence, we show in
Proposition \ref{prop1} that there are two types of heavy-traffic behavior, depending on whether $\eta\in(0,1/2)$ or $\eta\geq 1/2$.
In Section \ref{subsec3.2} we discuss for the case $\eta=1/2$ the connection with the Gaussian random walk and the Riemann zeta function.
In fact, we show that for all $\eta\geq 1/2$ there exists a connection between the integral expression in Theorem \ref{mainthm} and the Riemann zeta function.
In Section \ref{more} we apply the saddle point method to obtain several more heavy-traffic results, including refined heavy-traffic approximations for the mean congestion level, and the leading heavy-traffic behavior for the variance of the stationary congestion level and for the empty-system probability.
Finally, in Section \ref{numm} we confirm through numerical experiments the accuracy of our heavy-traffic approximations, and moreover show that under \eqref{bb}, various multi-server systems behave similar to our discrete bulk service queue.
\section{Model description \& heavy-traffic regimes}\label{sec1}
We thus consider a discrete stochastic model in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,\ldots$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
We will omit the superscript $(n)$ if no ambiguity is possible.
The system has a service capacity $s_n\in\mathbb{N}$ per period, so that the recursion
\begin{equation}
\label{lind}
Q(j+1) = \max\{Q(j) + A^{(n)}_j - s_n,0\},\qquad j=1,2,...,
\end{equation}
assuming $Q(0)=0$, gives rise to a Markov chain $\{Q(j)\}_{j\geq 1}$ that describes the congestion in the system over time. The probability generation function (pgf)
\begin{equation} \label{e2}
\tilde A(z)=\sum_{k=0}^{\infty} \mathbb{P}\big(A^{(n)}=k\big) z^k
\end{equation}
of $A^{(n)}$ is assumed analytic in a disk $|z|<r$ with $r>1$, which implies that all moments of $A^{(n)}$ exist. We also assume that
\begin{equation} \label{e3}
\tilde A'(1)=\mathbb{E}[A^{(n)}_j]=\mu_A<s_n.
\end{equation}
Under the assumption (\ref{e3}) the function $z^{s_n}-\tilde A(z)$ has exactly $s_n$ zeros in the closed unit disk, one of these being $z=1$ (see \cite{rouche}).
We further assume that $\mathbb{P}(A^{(n)}=j)>0$ for some $j>s_n$.
Under this assumption the function
$z^{s_n}-\tilde A(z)$ also has zeros outside $|z|\leq 1$, and we let $r_0$ be the minimum modulus of these zeros.
The number $r_0$ is the unique zero of $z^{s_n}-\tilde A(z)$ with real $z>1$; see e.g.~\cite{Janssen2005}.
Under the assumption (\ref{e3}) the stationary distribution $\lim_{k\to \infty}\mathbb{P}\left(Q(j)=k\right)=\mathbb{P}(Q=k)$, $k=0,1,\ldots$ exists, with the random variable $Q$ defined as having this stationary distribution.
We let
\begin{equation} \label{e4}
\tilde Q(w)=\sum_{j=0}^{\infty}\mathbb{P}(Q=j)w^j
\end{equation}
be the pgf of the stationary distribution. $\tilde Q(w)$ is analytic in $|w|<r_0$, and given by Pollaczek's formula (see e.g.~\cite{Abate1993, Cohen1982}).
In our discrete setting, we shall first derive a useful expression for $\tilde{Q}(w)$.
\begin{lemma}
For any $\varepsilon>0$ with $1+\varepsilon<r_0$,
\begin{equation} \label{e111}
\tilde Q(w)=\exp\Big(\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big)
\end{equation}
holds when $|w|<1+\varepsilon$.
\end{lemma}
\begin{proof}
We shall establish (\ref{e111}) for any $w\in(1,1+\varepsilon)$, and then the full result follows from analyticity of $\tilde{Q}(w)$ and of
\begin{equation} \label{e112}
{\rm ln}\Big(\frac{w-z}{1-z}\Big)={\rm ln}\Big(\frac{1-w/z}{1-1/z}\Big)={-}\,\sum_{k=1}^{\infty}\,\frac1k\,\Big(\Big(\frac{w}{z}\Big)^k-\Big(\frac1z\Big)^k\Big)
\end{equation}
in $w$, $|w|<1+\varepsilon$ for any $z$ with $|z|=1+\varepsilon$.
Our starting point is the formula, see \cite{Boudreau1962},
\begin{equation} \label{e113}
\tilde Q(w)=\frac{(s_n-\mu_A)(w-1)}{w^{s_n}-\tilde A(w)}\,\to \sum_{k=1}^{s_n-1}\,\frac{w-z_k}{1-z_k}
\end{equation}
that holds for all $w$, $|w|<r_0$, in which $z_1,\ldots,z_{s_n-1}$ are the $s_n-1$ zeros of $z^{s_n}-\tilde A(z)$ in $|z|<1$. Fix $w\in(1,1+\varepsilon)$. Then ${\rm ln}\,[(w-z)/(1-z)]$ is analytic in $z\in\mathbb{C}\backslash [1,w]$. It follows that
\begin{align}
I_C &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z \nonumber \\
&=~\sum_{k=1}^{s_n-1}\,{\rm ln}\Big(\frac{w-z_k}{1-z_k}\Big)+\frac{1}{2\pi i}\,\int_C\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,
\label{e114}
\end{align}
where $C$ is a contour encircling $[1,w]$ in the positive sense with none of the $z_k$'s in its interior. We let $\delta\in(0,\frac{w-1}{2})$ and we take $C$ the union of two line segments, from $1+\delta-i0$ to $w-\delta-i0$ and from $w-\delta+i0$ to $1+\delta-i0$, and two circles, of radius $\delta$ and encircling 1 and $w$ in positive sense.
A careful administration of the various contributions to the integral $I_C$ in \eqref{e114}, taking account of the branch cut $[1,w]$, yields
\begin{equation}\label{e115}
I_C = {\rm ln }\left(\frac{(s_n-\mu_A)(w-1)}{w^s-\tilde A(w)}\right) + O(\delta\,{\rm ln}\, \delta ).
\end{equation}
Using this in \eqref{e113} and letting $\delta \downarrow 0$, we get \eqref{e111} for $w\in(1,1+\varepsilon)$ and the proof is complete.
\end{proof}
Using $\mathbb{P}(Q=0)=\tilde Q(0)$, $\mu_Q=\tilde Q'(1)$ and $\sigma_Q^2 = \tilde Q''(1)+\tilde Q'(1)-(\tilde Q'(1))^2$, it follows by straightforward manipulations that
\begin{align} \label{e6}
\mathbb{P}(Q=0)&=\exp\,\Big[\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\ln\Big(\frac{z}{z-1}\Big)\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big] , \\
\label{e7}
\mu_Q&=\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{1-z}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,\\
\label{e8}
\sigma_Q^2 &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{-z}{(1-z)^2}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z .
\end{align}
Because $s_n$ appears directly in expressions \eqref{e6}-\eqref{e8}, we will be conducting our analysis with respect to $s_n$ rather than $n$. Note that this has no consequences for our results on the convergence speed of the performance metrics, since $s_n = O(n)$. Furthermore, we will omit the index $n$ when describing the capacity $s_n$ in the remainder of the chapter for brevity.
We next discuss in more detail the family of heavy-traffic scalings considered in this chapter, which combines two features. First, we have assumed that
$A^{(n)}_j$ is in distribution equal to the sum of work generated by all sources, $A_{1,k}+...+A_{n,j}$, where the $A_{i,j}$ are for all $i$ and $k$ i.i.d.~copies of a random variable $X$, of which the pgf $\tilde X(z)=\sum_{k=0}^{\infty}\mathbb{P}(X=k)z^k$ has radius of convergence $r>1$, and
\begin{equation} \label{e9}
0< \mathbb{E}[A^{(n)}] =n\mu = n \tilde X'(1)<s_n .
\end{equation}
Hence
\begin{equation} \label{e10}
\vartheta:=\frac{n}{s_n}\in(0,1/\mu) .
\end{equation}
Second, we scale the system according to \eqref{bb}, for which we assume that
\begin{equation} \label{e11}
\rho_{s_n} =\vartheta\,\mu =1-\frac{\gamma}{s_n^\eta}
\end{equation}
in which $\gamma>0$ is bounded away from 0 and $\infty$ as $s_n\to \infty$.
In the remainder of this chapter, we will omit the subscript in $s_n$.
The condition that $\mathbb{P}(A^{(n)}=k)>0$ for some $k>s$ holds when the degree $d$ of $\tilde X(z)$ (with $d=\infty$ if $\tilde X(z)$ is not a polynomial) is such that $nd>s$.
To avoid certain complications when applying the saddle point method, we further assume that
\begin{equation} \label{e12}
|\tilde X(z)|<\tilde X(r_1) ,~~~~~~|z|=r_1\,,~~z\neq r_1 ,
\end{equation}
for any $r_1\in(0,r)$. This implies that $r_0$ is the unique zero of $z^s-\tilde A(z)$ on $|z|=r_0$.
This condition is related to Cram\'er's condition, see \cite[pp.~189 and 355]{Asmussen2003}, and it has also been used in \cite{relaxation}.
Condition \eqref{e12} holds when the set of all $j=0,1,\ldots$ such that $\mathbb{P}(X=k)>0$ is not contained in an arithmetic progression with a ratio larger than one (see also \cite{rouche}).
\section{Non-standard saddle point method}\label{spSec}
\noindent
We illustrate our saddle point method for $\mu_Q$.
As a first step, we bring (\ref{e7}) in a form which is amenable to saddle point analysis.
\begin{lemma}
\begin{equation} \label{e18}
\mu_Q = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{g'(z)}{z-1}~\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}\,{\rm d} z
\end{equation}
with
\begin{equation} \label{e15}
g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}(\tilde X(z)) .
\end{equation}
\end{lemma}
\begin{proof}
With $\tilde A(z)=\tilde X^n(z)$,
\begin{align} \label{e13}
\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)} & = \frac{s\,z^{s-1}-n\,\tilde X'(z)\,\tilde X^{n-1}(z)}{z^s-\tilde X^n(z)} \nonumber \\
& = \frac{s}{z}-\frac{s}{z}\,\Big(\frac{n}{s}~\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big)\,\frac{z^{-s}\,\tilde X^n(z)}{1-z^{-s}\,\tilde X^n(z)} .
\end{align}
Write
$
z^{-s}\,\tilde X^n(z)=\exp(s\,g(z))$.
Noting that
\begin{equation} \label{e16}
\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{s}{z}~\frac{1}{1-z}\,{\rm d} z=0 ,
\end{equation}
and that
\begin{equation} \label{e17}
g'(z)=\frac1z\,\Big(\vartheta\,\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big) ,
\end{equation}
gives \eqref{e18}. \end{proof}
Let us now explain how the standard saddle point method can be applied to \eqref{e18}.
Since
\begin{equation} \label{e19}
g(1)=g(r_0)=0~;~~~~~~g(z)<0\,,~~1<z<r_0 ,
\end{equation}
and by strict convexity of
\begin{equation} \label{e20}
z^{-s}\,\tilde X^n(z)=z^{-s}\tilde A(z)=\sum_{k=0}^{\infty}\,a_k\,z^{k-s} ,~~~~~~z\in(0,r) ,
\end{equation}
$g(z)$ has a unique minimum on $[1,r_0]$. This minimum is found by solving $z\in[1,r_0]$ from $g'(z)=0$, and this yields the equation
\begin{equation} \label{e21}
\tilde X(z)=\vartheta\,z\,\tilde X'(z) .
\end{equation}
Denote the solution $z\in(1,r_0)$ of (\ref{e21}) by $z_{\rm sp}$, and observe that $z_{\rm sp}$ is a saddle point of $g(z)$, explaining the notation. Thus, the saddle point method can be used for the integral in (\ref{e18}) by taking $1+\varepsilon=z_{\rm sp}$.
In the case that $\vartheta=n/s$ is bounded away from $1/\mu$ as $s\to \infty$, we have that the minimum value of $g(z)$, $1\leq z\leq r_0$, is negative and bounded away from 0. Furthermore, $z_{\rm sp}$ is bounded away from 1, and the saddle point method can be applied in the classical way by replacing
\begin{equation} \label{e22}
\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}~~~~{\rm by}~~~~\exp(s\,g(z)) ,
\end{equation}
at the expense of an exponentially small relative error, and performing an expansion of $g'(z)/(z_{\rm sp}-1)=d_1(z-z_{\rm sp})+O((z-z_{\rm sp})^2)$ with $d_1=g''(z_{\rm sp})/(z_{\rm sp}-1)\neq 0$.
Using that $g(z^{\ast})=(g(z))^{\ast}$, where the $^*$ denotes complex conjugation, it can be shown that
\begin{equation} \label{e23}
\mu_Q=\frac{\exp(s\,g(z_{\rm sp}))}{(z_{\rm sp}-1)^2\,\sqrt{2\pi s\,g''(z_{\rm sp})}}\,(1+O(s^{-1})) .
\end{equation}
We next explain why the standard saddle point method does not work for the heavy-traffic scaling considered in this chapter. Since we operate in (\ref{e11}),
$\vartheta\mu\to 1$ as $s\to \infty$, and
\begin{align} \label{e24}
z_{\rm sp}-1&=\frac{\gamma}{a_2\,s^\eta}+O(s^{-2\eta}) ,\\
\label{e25}
g(z_{\rm sp})&=\frac{-\gamma^2}{2a_2s^{2\eta}}+O(s^{-3\eta}) ,\\
\label{e26}
g''(z_{\rm sp})&=a_2+O(s^{-\eta}) ,
\end{align}
where
\begin{equation} \label{e27}
a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) .
\end{equation}
Hence, $\exp(sg(z))$ near $z=z_{\rm sp}$ is (as $s\to \infty$):
vanishingly small when $\eta\in(0,1/2)$,
bounded away from 1, but non-negligible when $\eta=1/2$,
and tending to 1 when $\eta\in(1/2,\infty)$.
Furthermore, $(z-1)^{-1}$ in \eqref{e18} is unbounded near $z=z_{\rm sp}$ as $s\to \infty$. Therefore, an adaptation of the standard saddle point method is required, and the resulting asymptotic form of $\mu_Q$ will deviate significantly from the standard case (\ref{e23}). In particular, since $z_{\rm sp}\to 1$, this asymptotic form will contain information from $X(z)$ at $z=1$, rather than at a point away from 1 as is the case in (\ref{e23}).
The required adaption of the saddle point method is modeled after a device developed in \cite[Sec.~5.12]{debruijn}. We use a substitution $z=z(v)$ in (\ref{e18}) with real $v$ and $z(0)=z_{\rm sp}$ such that for sufficiently small $v$,
\begin{equation} \label{e29}
g(z(v))=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) .
\end{equation}
This is feasible, since
\begin{equation} \label{e30}
g(z)=g(z_{\rm sp})+\tfrac12\,g''(z_{\rm sp})(z-z_{\rm sp})^2\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)
\end{equation}
with $g''(z_{\rm sp})$ positive and bounded away from 0 as $s\to \infty$. Hence, $z(v)$ can be found for small $v$ by inverting the equation
\begin{equation} \label{e31}
(z-z_{\rm sp})\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)^{1/2}=iv .
\end{equation}
By Lagrange's inversion theorem \cite{debruijn}, there is a $\delta>0$ (independent of $s$) such that
\begin{equation} \label{e32}
z(v)=z_{\rm sp}+iv+\sum_{k=2}^{\infty}\,c_k(iv)^k ,~~~~~~|v|<\delta ,
\end{equation}
with real coefficients $c_k$ (since $g(z)$ is real for real $z$) and
\begin{equation} \label{e33}
c_2={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})} .
\end{equation}
Thus
\begin{equation} \label{e34}
z(v)=z_{\rm sp}+iv-c_2\,v^2+O(v^3) ,~~~~~~|v|\leq\tfrac12\,\delta ,
\end{equation}
where the order term holds uniformly in $s$. The uniformity statement follows from an inspection of the usual argument
by which Lagrange's theorem is proved, noting that the inversion in \eqref{e29} with $g$ as in \eqref{e15} is considered for $\vartheta\to 1/\mu$, $z_{\rm sp}\to 1$ with radius
of convergence $r$ away from $1$.
By (\ref{e12}) we can restrict the integration in (\ref{e18}) to a fixed but arbitrarily small subset of $|z|=z_{\rm sp}$ near $z=z_{\rm sp}$, at the expense of an exponentially small error. Furthermore, by Cauchy's theorem and again at the expense of an exponentially small error, the integration path can be deformed in accordance with the transformation in (\ref{e29})--(\ref{e34}). Set
\begin{equation} \label{e35}
q(v)=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp})
\end{equation}
and note that from \eqref{e29}
\begin{equation} \label{e36}
g'(z(v))\,z'(v)={-}v\,g''(z_{\rm sp}) .
\end{equation}
Then substituting $z=z(v)$ in (\ref{e18}), $\mu_Q$ is given with exponentially small error by
\begin{equation} \label{e37a}
\frac{s}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{g'(z(v))}{z(v)-1}~\frac{\exp(s\,g(z(v)))}{1-\exp(s\,g(z(v)))}z'(v)\,{\rm d} v,
\end{equation}
which gives the following result.
\begin{lemma} \label{lemma2} The mean stationary congestion level is given with exponentially small error by
\begin{equation} \label{e37}
\mu_Q =~\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v}{z(v)-1}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v .
\end{equation}
\end{lemma}
In a similar fashion we get that $\mathbb{P}(Q=0)$ and $\sigma_Q^2$, see (\ref{e6}) and (\ref{e8}), are given, both with exponentially small error, by
\begin{equation} \label{e39}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,v\,{\rm ln}\Big(\frac{z(v)}{z(v)-1}\Big)\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v
\end{equation}
and
\begin{equation} \label{e38}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v,
\end{equation}
respectively.
\section{Heavy-traffic limits for the mean congestion level} \label{sec3}
In this section we apply the non-standard saddle point method explained in Section \ref{spSec} to the Pollaczek integral representation for the mean stationary congestion level $\mu_Q$. In Section \ref{subsec3.1} we first derive an integral representation for the leading order behavior of $\mu_Q$ with a relative error of order $O(s^{-1})$, which serves as a heavy-traffic approximation in the regime $\rho_s=1-\gamma/s^\eta$ with $\eta>0$. We also consider separately the cases of moderate heavy traffic ($\eta\in(0,1/2)$) and extreme heavy traffic ($\eta\in(1/2,\infty)$), for which the integral representation leads to vastly different alternative expressions. We find that $\mu_Q\to 0$ more rapidly than any power of $1/s$ when $\eta\in(0,1/2)$. When $\eta\geq 1/2$ the saddle point method yields an integral representation with relative error $O(s^{-\min(1,\eta)})$.
In Section \ref{subsec3.2} we specialize this general result to the CLT case $\eta=1/2$, and make a connection with existing results.
\subsection{Leading order behavior in integral form} \label{subsec3.1}
\begin{theorem}\label{mainthm}
The mean stationary congestion level is given by
\begin{equation} \label{e48a}
\mu_Q=\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\int_0^{\infty}\,\frac{t^2}{d^2(s)+t^2}~\frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,{\rm d} t\,\left(1+O({s^{{-}\min(1,\eta)}})\right)
\end{equation}
with $
d^2(s) = s^{1-2\eta}\gamma^2\mu/(2\sigma^2)$.
\end{theorem}
\begin{proof}
According to Lemma \ref{lemma2}, $\mu_Q$ is given with exponentially small error by (\ref{e37}) with $q(v)$ given in (\ref{e35}). Since $z({-}v)=z^{\ast}(v)$ for real $v$, we have
\begin{eqnarray} \label{e40}
\frac{v}{z(v)-1}+\frac{-v}{z({-}v)-1} &=& {-}2iv\,\frac{{\rm Im}(z(v))}{|z(v)-1|^2}\nonumber\\
&=&~\frac{-2iv^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} ,
\end{eqnarray}
where (\ref{e34}) and $c_k\in\mathbb{R}$ have been used. Using (\ref{e40}) in (\ref{e37}) and extending the integration range from $[{-}\tfrac12\delta,\tfrac12\,\delta]$ to $({-}\infty,\infty)$ while using symmetry of $q(v)$, we get that $\mu_Q$ is given with exponentially small error by
\begin{eqnarray} \label{e41}
\frac{s\,g''(z_{\rm sp})}{\pi}\,\int_0^{\infty}\,\frac{v^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)}\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}{\rm d} v .
\end{eqnarray}
With
\begin{equation} \label{e42}
B=\exp(s\,g(z_{\rm sp})) ,~~~~~~\alpha =g''(z_{\rm sp}),
\end{equation}
Equation \eqref{e41} takes the form
\begin{eqnarray} \label{e43}
\frac{s\alpha }{\pi}\,\int_0^{\infty}\,\frac{v^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} \cdot \frac{B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{eqnarray}
In leading order, the integrand in (\ref{e43}) has the form
\begin{equation} \label{e43a}
\frac{B\,v^2\,\exp(-s\,D\,v^2)}{(v^2+C\,s^{-2\eta})(1-\exp({-}s\,D\,v^2))},
\end{equation}
and this is reminiscent of the integrand in \cite[(5.12.3)]{debruijn} for the case $\kappa=2\eta$. Proceeding as in \cite[Sec.~5.12]{debruijn}, the substitution $v=t\sqrt{{2}/(s\alpha )}$ brings (\ref{e43}) into the form
\begin{eqnarray} \label{e44}
\frac{2}{\pi}\sqrt{\tfrac12 s\,\alpha }\int_0^{\infty}\frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2+O(t^4/s)} \,\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}{\rm d} t .
\end{eqnarray}
From (\ref{e24})--(\ref{e27}) and (\ref{e42}),
\begin{align}
\frac{2}{\pi}\,\sqrt{\frac{s\alpha }{2}} &= \frac{2}{\pi}\,\sigma_X\,\sqrt{\frac{s}{2\,\mu}}\,(1+O(s^{-\eta})),\label{y45}\\
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &= d^2(s) + O(s^{1-3\eta}),\label{y46}\\
2\,c_2(z_{\rm sp}-1) &= O(s^{-\eta}),\label{y47}\\
s\,g(z_{\rm sp}) &= -d^2(s) + O(s^{1-3\eta}),\label{y48}
\end{align}
where
\begin{equation} \label{y49}
d^2(s) = \frac{b_0^2}{s^{2\eta-1}},\quad b_0^2 := \frac{\gamma^2\mu}{2\,\sigma^2}.
\end{equation}
In the case that $2\eta-1<0$, we have that $\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 \to \infty$ and that
\begin{equation} \label{y50}
B = \exp(s\,g(z_{\rm sp})) = O(\exp({-}b^2s^{1-2\eta}))
\end{equation}
for any $b\in(0,b_0)$. From \eqref{e44} it then follows that $\mu_Q = O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$.
In the case that $2\,\eta-1\geq 0$, we have that $d^2(s)$ is bounded, and using that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, we get
\begin{align}
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &+ t^2-2\,c_2\,(z_{\rm sp}-1)\,t^2+O(t^4/s) \nonumber\\
&= d^2(s) + t^2 + O\left(s^{-\eta}\,(d^2(s)+t^2)\right) + O(t^4/s)\nonumber\\
&= \left(d^2(s)+t^2\right)\left(1+O(s^{-\eta})+O(t^2/s)\right).\label{y51}
\end{align}
Hence, in this case,
\begin{align}
& \frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2+O(t^4/s)}\nonumber\\
& \qquad \qquad \qquad \qquad = \frac{t^2}{d^2(s)+t^2}\left(1+O(s^{-\eta})+O(t^2/s)\right),\label{y52}
\end{align}
where we restrict to $t$ in a range $[0,s^{1/4}]$. Furthermore,
\begin{align}
1-B\,\exp(-t^2) &= 1-\exp({-}d^2(s)-t^2)\,\left(1+d^2(s)\,O(s^{-\eta})\right)\nonumber\\
&=(1-\exp({-}d^2(s)-t^2))\,\Big(1+\frac{d^2(s)}{\exp(d^2(s)+t^2)-1}O(s^{-\eta})\Big)\nonumber\\
&= (1-\exp({-}d^2(s)-t^2))\,(1+O(s^{-\eta})),\label{y53}
\end{align}
It follows therefore that
\begin{equation} \label{y56}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)} = \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,(1+O(s^{-\eta})).
\end{equation}
Combining the three items \eqref{y45}, \eqref{y52} and \eqref{y56}, we obtain for \eqref{e44} the result
\begin{equation} \label{y58}
\frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}} \int_0^{\infty}\frac{t^2}{d^2(s)+t^2} \cdot \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}{\rm d} t
\left(1+O(s^{-\eta})+O(s^{-1})\right),
\end{equation}
where the integration range $[0,\infty)$ is, at the expense of relative errors of type\\ $\exp({-}s^{1/4})$, first restricted to the range $[0,s^{1/4}]$, where \eqref{y52} holds, and then restored again to the full range. \end{proof}
Theorem \ref{mainthm} gives the leading-order behavior of $\mu_Q$ as $s\to \infty$ with a relative error of $O(s^{{-}\min(1,\eta)})$. By considering in more detail the integral expressions, we obtain the following result, describing two different heavy-traffic behaviors.
\begin{proposition}\label{prop1}
If $\eta\in(0,1/2)$ the mean congestion level satisfies
\begin{equation} \label{y59}
\mu_Q=O\left(\exp(-b^2s^{1-2\eta})\right),
\end{equation}
for any $b\in (0,b_0)$. If $\eta\in[1/2,\infty)$ the mean congestion level $\mu_Q$ is given by
\begin{equation} \label{y60}
s^\eta\,\frac{\sigma^2}{2\mu\gamma}\,\left(1+O(s^{\max(1/2-\eta,-1)})\right).
\end{equation}
\end{proposition}
The first assertion in Proposition \ref{prop1} follows from the observation in \eqref{y50}, together with \eqref{e44}. The second assertion is based on a connection between the integral in Theorem \ref{mainthm} and the Riemann zeta function, which is explained in the next subsection.
\subsection{Classical heavy traffic and the Gaussian random walk}
\label{subsec3.2}
We now build on Theorem \ref{mainthm} to obtain further results for the classical heavy traffic case $\eta=1/2$,
for which we know from \cite[Thm.~3]{Sigman2011b} that the rescaled congestion process converges under \eqref{bb1} to a reflected Gaussian random walk. The latter is defined as
$(S_\beta(k))_{k\geq 0}$ with $S_\beta(0)=0$ and
\begin{equation}
S_\beta(j)=Y_1+\ldots+Y_j
\end{equation}
with $Y_1,Y_2,\ldots$ i.i.d.~copies of a normal random variable with mean $-\beta$ and variance 1.
Assume $\beta>0$ (negative drift), and denote the all-time maximum of this random walk by ${M}_\beta$.
Denote by $Q^{(s)}_\infty$ the stationary congestion level for a fixed $s$ (that arises from taking
$j\to \infty$ in \eqref{lind}), and remember that we have assumed $\vartheta=n/s$ fixed.
Then, using $\rho_s=1-\gamma/\sqrt{s}$, with
\begin{equation}\label{gammachoice}
\gamma=\frac{\beta\sigma}{\mu\sqrt{\vartheta}},
\end{equation}
the spatially-scaled stationary congestion levels reach the limit
$Q^{(s)}_\infty/(\sigma\sqrt{n}) {\;\buildrel{d}\over\Rightarrow\;} {M}_\beta$ as $s,n\to \infty$ (see \cite{Jelenkovic2004,Sigman2011a,Sigman2011b}). From \cite[Thm.~4]{Sigman2011b} we then know that under the standard heavy-traffic scaling \eqref{bb1}
\begin{equation}
\frac{\mathbb{E}Q^{(s)}_\infty}{\sigma\sqrt{n}}\to \mathbb{E}{M}_\beta, \quad {\rm as} \ s,n\to \infty,
\end{equation}
from which it follows that
\begin{equation} \label{e48}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}M_\beta.
\end{equation}
The random variable ${M}_\beta$ was studied in \cite{Chang1997,Janssen2006}. In particular, \cite[Thm.~2]{Janssen2006} yields, for $\beta<2\sqrt{\pi}$,
\begin{equation}\label{wdfegfw571}
\mathbb{E}{M}_\beta= \frac{1}{2\beta}+\frac{\zeta(1/2)}{\sqrt{2\pi}}+\frac{\beta}{4}+\frac{\beta^2}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta(-1/2-r)}{r!(2r+1)(2r+2)}\left(\frac{-\beta^2}{2 }\right)^r,
\end{equation}
where $\zeta$ denotes the Riemann zeta function.
Hence, for small values of $\beta$,
\begin{equation} \label{estimate}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}M_\beta \approx \frac{\sigma\sqrt{n}}{2\beta} = \sqrt{s}\,\frac{\sigma^2}{2\mu\gamma}.
\end{equation}
We will now show how the approximation \eqref{estimate} follows from Theorem \ref{mainthm}, and also how similar steps give rise to Proposition \ref{prop1}.
Consider the integral
\begin{equation} \label{e49}
G_0(b)=G_1(b)-G_2(b)=\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t ,
\end{equation}
where $b>0$ and
\begin{equation} \label{e50}
G_1(b)=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t\,,~~~~G_2(b)=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}{\rm d} t .
\end{equation}
We have, as in \cite[Sec.~2]{Janssen2006},
\begin{align} \label{e51}
G_1(b) & = \sum_{k=0}^{\infty}\:\int_0^{\infty}\,\exp({-}(k+1)(b^2+t^2))\,{\rm d} t \nonumber \\
& = \frac{\sqrt{\pi}}{2}\,\sum_{k=0}^{\infty}\,\frac{{\rm e}^{-(k+1)b^2}}{\sqrt{k+1}} = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},1/2,1) \nonumber \\
& = \frac{\pi}{2b}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta(\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
where $\Phi(z,s,v)$ is Lerch's transcendent and where the last identity holds when $0<b<\sqrt{2\pi}$.
As to $G_2(b)$, we make a connection with the complementary error function
\begin{equation} \label{e52}
{\rm erfc}(z)=\frac{2}{\sqrt{\pi}}\,\int_z^{\infty}\,{\rm e}^{-t^2}\,{\rm d} t=\frac{2}{\pi}\,{\rm e}^{-z^2}\,\int_0^{\infty}\,\frac{{\rm e}^{-z^2t^2}}{1+t^2}{\rm d} t ,
\end{equation}
see \cite[Secs.~7.2 and 7.7.1]{NIST} We thus compute
\begin{align} \label{e53}
G_2(b) & = \sum_{k=0}^{\infty}\,{\rm e}^{-(k+1)b^2}\,\int_0^{\infty}\,\frac{b^2}{b^2+t^2}\,{\rm e}^{-(k+1)t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[(4.3) and (4.23)]{Janssen2006},
\begin{equation} \label{e54}
\sum_{n=1}^{\infty}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^{\infty}\,{\rm e}^{-x^2/2}\,dx= \frac{1}{2\beta^2}-\frac14-\frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^{\infty} \frac{\zeta({-}1/2-r)({-}1/2)^r} {r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
in which $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e54}), we get
\begin{equation} \label{e55}
G_2(b)=\frac{\pi}{4b}-\frac{\pi}{4}\,b-\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results in (\ref{e51}) and (\ref{e55}) can be combined, as in \cite[Sec.~ \ref{sec4}]{Janssen2006}, and this yields
\begin{equation} \label{e56}
G_0(b)=\frac{\pi}{4b}+\frac{\pi}{4}\,b+\frac{\sqrt{\pi}}{2}\,\zeta(1/2)+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Using (\ref{e56}) in (\ref{e48}), we find that the leading order behavior of $\mu_Q$ is given as
\begin{equation} \label{e57}
\sigma_X\,\sqrt{\dfrac{s}{2\mu}}\,\left[\frac{1}{2b_0}+\frac{b_0}{2}+\frac{\zeta(1/2)}{\sqrt{\pi}}+\frac{2}{\sqrt{\pi}}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r b_0^{2r+2}} {r!\,(2r+1)(2r+2)}\right]
\end{equation}
with relative error of $O(s^{-1/2})$ in which $b_0$ is given by \eqref{y49}. The expression (\ref{e57}) is exactly equal to the right-hand side of \cite[(4.25)]{Janssen2006} times $\sqrt{s}$ when we take there $\sigma=\mu=1$ and $\beta=b_0\,\sqrt{2}$.
Notice that, with $\gamma$ as in \eqref{gammachoice},
\begin{equation} \label{e57a}
\sigma\,\sqrt{\dfrac{s}{2\mu}}\frac{1}{2b_0}=\frac{\sigma\sqrt{n}}{2\beta},
\end{equation}
which confirms the approximation \eqref{estimate}.
According to Theorem \ref{mainthm}, we have for $\eta\geq 1/2$,
\begin{equation} \label{y61}
\mu_Q = \frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}}G_0(d(s))\,\left(1+O(s^{{-}\min(1,\eta)})\right).
\end{equation}
When $\eta=1/2$, so that $d(s) = b_0$ is independent of $s$, the series representation for $G_0$ in \eqref{e56} can be used, as long as $b_0\in(0,\sqrt{2\pi})$. When $\eta>1/2$, we have that $d(s) = b_0/s^{\eta-1/2}\to 0$ as $s\to \infty$, and so this series representation can be used when $s$ is large enough. We then have from \eqref{e56} and $b_0^2 = \gamma^2\mu/2\,\sigma^2$, while replacing the whole series at the right-hand side by $O(b^2)$, for $\mu_Q$ the leading order behavior
\begin{equation} \label{y62}
s^\eta\left[\frac{\sigma^2}{2\,\gamma\,\mu}+\frac{\sigma\,\zeta(1/2)}{\sqrt{2\,\pi\,\mu}}\,\frac{1}{s^{\eta-1/2}}+\frac{1}{4}\,\gamma\,\frac{1}{s^{2\eta-1}}+O(s^{3/2-3\eta})\right]
\end{equation}
with relative error $O(s^{{-}\min(1,\eta)})$. Retaining the constant term $\sigma^2/(2\gamma\mu)$ and estimating the other terms between the brackets in \eqref{y62} as $O(s^{1/2-\eta})$, we get Proposition \ref{prop1}.
\section{More heavy-traffic results}\label{more}
In this section we apply the non-standard saddle point method to obtain several more heavy-traffic results. In Section \ref{subsec3.3} we derive refined heavy-traffic approximations for the mean congestion level by considering higher-order correction terms. In Section \ref{sec4} we derive the leading heavy-traffic behavior for the variance of the stationary congestion level, and in Section \ref{sec5} for the empty-system probability. To keep the developments tractable, we restrict Section \ref{subsec3.3} to $\eta=1/2$, and Section \ref{sec4} and Section \ref{sec5} to $\eta\in(0,1]$, although the same technique will work for all values $\eta>0$.
\subsection{Correction term for the mean congestion level for $\eta = 1/2$} \label{subsec3.3}
Our saddle point method not only establishes the leading-order heavy-traffic approximations, but also allows to derive refinements to these approximations. In this section we demonstrate how this works for the mean congestion level in the case $\eta=1/2$.
To obtain a refinement or correction term from (\ref{e44}), we must be more precise about the $O(s^{{-}\eta})$ terms that occur in the approximations in Section \ref{subsec3.1} for $\frac12\,s\,\alpha (z_{\rm sp}-1)^2$, $B$ and $\sqrt{s\,\alpha /2}$. When higher-order corrections are required, we should include higher-order terms in the approximations of these quantities, and be more specific about the $O(t^2/s)$ and $O(t^4/s)$ in the integrand in (\ref{e44}).
Denote, see \eqref{e10} and (\ref{e15}) with $\vartheta=(1-\gamma/s^\eta)\,\mu^{-1}$,
\begin{equation} \label{e58}
a_i=g^{(i)}(1);~~~~~~g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}\,\tilde X(z) .
\end{equation}
Dropping the $X$ from $\mu$ and $\sigma^2$ for brevity, we have
\begin{equation} \label{e59}
a_1={-}\,\frac{\gamma}{s^\eta} ,~~~~~~a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) ,
\end{equation}
\begin{equation} \label{e60}
a_3={-}2+\Big(1-\frac{\gamma}{s^\eta}\Big)\Big(\frac{\tilde X'''(1)}{\tilde X'(1)}-3\tilde X''(1)+2(\tilde X'(1))^2\Big) .
\end{equation}
For the purpose of finding a first-order correction term, we note that
\begin{align} \label{e61}
\alpha &=g''(z_{\rm sp})=a_2+(z_{\rm sp}-1)\,a_3+O(s^{-1}) ,\\
\label{e62}
z_{\rm sp}-1&={-}\,\frac{a_1}{a_2}-\frac{a_3}{2a_2}\,\Big(\frac{a_1}{a_2}\Big)^2+O(s^{-3/2}) ,\\
\label{e63}
c_2&={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})}={-}\,\frac{a_3}{6a_2}+O(s^{-1/2}) ,\\
\label{e64}
g(z_{\rm sp})&={-}\,\frac{a_1^2}{2a_2}-\frac{a_3}{6a_2^3}\,a_1^3+O(s^{-2}) .
\end{align}
This gives rise to
\begin{align} \label{e65}
\sqrt{\tfrac12\,s\,\alpha }&=\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}+O(s^{-1})\Big) ,\\
\tfrac12\,s\,\alpha (z_{\rm sp}-1)^2&=\frac{\gamma^2\,\mu}{2\sigma^2}+\frac{C_2}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e67}
2c_2(z_{\rm sp}-1)&=\frac{C_3}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e68}
B=\exp(s\,g(z_{\rm sp}))&=\exp\Big({-}\,\frac{\gamma^2\,\mu}{2\sigma^2}\Big)\Big(1+\frac{C_4}{\sqrt{s}}+O(s^{-1})\Big) ,
\end{align}
with explicitly computable constants $C_1$, $C_2$, $C_3$, $C_4$. Remembering that $b_0^2=\gamma^2\mu/2\sigma^2$, see \eqref{y49}, we then get with errors of order $1/s$
\begin{eqnarray} \label{e69}
& \mbox{} & \frac{t^2(1+O(t^2/s))}{\frac12\,s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)\,t^2+O(t^4/s)} \nonumber \\[3mm]
& & \qquad =~\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+ b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big) ,
\end{eqnarray}
and
\begin{equation} \label{e70}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}=\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2} .
\end{equation}
Using (\ref{e65}), (\ref{e69}) and (\ref{e70}) in (\ref{e44}) we get with an absolute error of order $1/\sqrt{s}$
\begin{align} \label{e71}
\mu_Q & =\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}\Big)\nonumber \\
& \qquad\qquad \cdot \int_0^{\infty}\,\Big(\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big)\Big) \nonumber \\
& \qquad\qquad\qquad \cdot~\Big(\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\Big){\rm d} t \nonumber \\
& =\frac{2\sigma}{\pi}\,\sqrt{\dfrac{s}{2\mu}}\,G_0(b_0)\nonumber\\
& \qquad\qquad + ~\frac{2\sigma}{\pi}\,\sqrt{\dfrac{1}{2\mu}}\,\big((C_1+C_3)\,G_0(b_0)-(C_2+b_0^2\,C_3)\,G_3(b_0)+C_4\,G_4(b_0)\big) ,
\end{align}
where $G_0$ is as in (\ref{e49}), and
\begin{align} \label{e72}
G_3(b_0)&=\int_0^{\infty}\,\frac{t^2}{(b_0^2+t^2)^2}~\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}{\rm d} t ,\\
\label{e73}
G_4(b_0)&=\int_0^{\infty}\,\frac{t^2}{b_0^2+t^2}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}{\rm d} t .
\end{align}
We shall express the integrals in (\ref{e72}) and (\ref{e73}) in terms of $\zeta$-functions. By partial integration
\begin{align} \label{e74}
G_3(b) & = \frac12\,\int_0^{\infty}\,\frac{1}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,{\rm d} t \nonumber \\
&\qquad\qquad -~\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}{\rm d} t \nonumber \\
& = \frac{1}{2b^2}\,G_2(b)-G_4(b) ,
\end{align}
see (\ref{e49}) and (\ref{e73}). Since $G_2(b)$ is expressed in terms of $\zeta$-functions in (\ref{e55}), it is sufficient to consider $G_4(b)$.
As to $G_4(b)$,
\begin{equation} \label{e75}
G_4(b)=G_5(b)-G_6(b) ,
\end{equation}
where
\begin{align} \label{e76}
G_5(b)&=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t ,\\
\label{e77}
G_6(b)&=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t .
\end{align}
We have, compare (\ref{e51}),
\begin{align} \label{e78}
G_5(b) & = \sum_{k=0}^{\infty}\,(k+1)\,\int_0^{\infty}\,{\rm e}^{-(k+1)(b^2+t^2)}\,{\rm d} t \nonumber \\[3.5mm]
& = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},{-}\tfrac12,1) = \frac{\pi}{4b^3}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta({-}\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
the last identity being valid when $0<b<\sqrt{2\pi}$. Next we have, compare (\ref{e53}),
\begin{align} \label{e79}
G_6(b) & = \sum_{k=0}^{\infty}\,(k+1)\,b^2\,\int_0^{\infty}\,\frac{\exp({-}(k+1)(b^2+t^2))}{b^2+t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,(k+1)\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[(5.4) and (5.21)]{Janssen2006} we have
\begin{eqnarray} \label{e80}
\sum_{n=1}^{\infty}\frac{n}{\sqrt{2\pi}}\int_{\beta\sqrt{n}}^{\infty}{\rm e}^{-x^2/2}\,dx = \frac{3}{4\beta^4}-\frac{1}{24}-\frac{1}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1/2)^r}{r!\,(2r+1)}\,\beta^{2r+1}
\end{eqnarray}
when $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e80}), we get
\begin{equation} \label{e81}
G_6(b)=\frac{3\pi}{16b^2}-\frac{\pi b}{24}-\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r}{r!\,(2r+1)}\,b^{2r+2}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results (\ref{e78}) and (\ref{e81}) can be combined, as in \cite[Sec.~5]{Janssen2006} and this yields
\begin{equation} \label{e82}
G_4(b)=\frac{\pi}{16b^3}+\frac{\pi b}{24}+\tfrac12\,\zeta({-}1/2)\,\sqrt{\pi}+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Finally, we can use (\ref{e82}) in (\ref{e74}), and we obtain with (\ref{e55}), for $0<b<\sqrt{2\pi} $,
\begin{align} \label{e83}
G_3(b) = & \frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} -~2\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)(2r+3)}.
\end{align}
The right-hand side of (\ref{e83}) equals the right-hand side of \cite[(2.3)]{Janssen2006} multiplied by ${\pi}/{(2b)}$ with $\beta=b\,\sqrt{2}$.
\subsection{Variance of the congestion level}\label{sec4}
We have from (\ref{e38}) in Section \ref{sec1}, using the same approach and notation as in Section \ref{subsec3.1} for $\mu_Q$, that $\sigma_Q^2$ is given with exponentially small error by
\begin{equation} \label{e84}
\frac{-s\,\alpha }{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v,
\end{equation}
with $B$ and $\alpha $ given in (\ref{e42}). From $z({-}v)=z^{\ast}(v)$ for real $v$ we now compute
\begin{equation} \label{e85}
\frac{z(v)}{(z(v)-1)^2}-\frac{z({-}v)}{(z({-}v)-1)^2}={-}2i\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,{\rm Im}(z(v)) ,
\end{equation}
and so (\ref{e84}) becomes
\begin{equation} \label{e86}
\frac{s\alpha }{\pi}\,\int_0^{\frac12\delta}\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,v\,{\rm Im}(z(v))\,\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
From
\begin{equation} \label{e87}
{\rm Im}(z(v))=v+O(v^3) ,~~~~~~|z(v)|^2-1=z_{\rm sp}^2-1+O(v^2) ,
\end{equation}
we get for the expression in \eqref{e86}
\begin{equation} \label{y70}
\frac{s\alpha }{\pi}\,\int_0^{\frac{1}{2}\delta}\,\frac{v^2\,(z_{\rm sp}^2-1+O(v^2))(1+O(v^2))}{((z_{\rm sp}-1)^2+v^2 + O((z_{\rm sp}-1)\,v^2)+O(v^4))^2}
\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v.
\end{equation}
When $2\eta-1<0$, we have as for the case of $\mu_Q$ in Section \ref{subsec3.1} that the whole expression in \eqref{y70} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. When $2\eta-1\geq 0$, we get as in the case of $\mu_Q$ after substitution $v = t\sqrt{{2}/{(s\,\alpha })}$ for the expression in \eqref{y70}
\begin{equation} \label{y71}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}\,\int_0^\infty\frac{t^2\,(z_{\rm sp}^2-1+O(t^2/s))(1+O(t^2/s))}{(d^2(s)+t^2)^2\,(1+O(1/s^{\eta})+O(t^2/s))}~\frac{B\,{\rm e}^{{-}t^2}}{1-B\,{\rm e}^{{-}t^2}}{\rm d} t.
\end{equation}
When $2\eta-1\geq 0$, the leading order behavior of $\sigma_Q^2$ depends crucially on the factor $z_{\rm sp}^2-1+O(t^2/s)$, where
\begin{equation} \label{y72}
z_{\rm sp}^2-1 = \frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\left(1+O(s^{-\eta})\right)
\end{equation}
is dominant when $\eta<1$, while the $O(t^2/s)$ is dominant when $\eta>1$. In the case that $\eta\in(1/2,1)$, we get for the leading order behavior of $\sigma_Q^2$
\begin{align}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}& \,\frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\int_0^\infty\frac{t^2}{(d^2(s)+t^2)^2}\cdot~\frac{{\rm e}^{{-}d^2(s)-t^2}}{1-{\rm e}^{{-}d^2(s)-t^2}}{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{\gamma\,\sigma}{\pi}\,\Big(\frac{2}{\mu}\Big)^{1/2}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right), \label{y73}
\end{align}
where \eqref{e26}, \eqref{e27} and \eqref{e42} have been used for $\alpha = g''(z_{\rm sp})$ and where $G_3$ is given in \eqref{e72}.
When we insert the expansion \eqref{e83} for $G_3(b)$, with the whole series on the second line being $O(b^2)$, we get the leading order behavior of $\sigma_Q^2$ as
\begin{align}
s^{2\eta}\,\Big( \frac{\sigma^4}{4\,\gamma^2\mu^2}- \frac{\sigma^2}{4\,\mu}&\,\frac{1}{s^{2\eta-1}} - \Big(\frac{2\,\sigma^2}{\pi\,\mu}\Big)^{1/2}\,\frac{\gamma\,\zeta(-1/2)}{s^{3\eta-3/2}}\nonumber\\
& - \frac{\gamma^2}{24\,s^{5\eta-5/2}}+O(s^{1-4\eta})\Big)\,\left(1+O(s^{\eta-1})\right)\nonumber \\
&\ = s^{2\eta}\,\frac{\sigma^4}{4\,\gamma^2\,\mu^2}\,\Big(1+O(s^{\max(1-2\eta,\eta-1)})\Big)\label{y74}
\end{align}
when $\eta\in(1/2,1)$. For the case $\eta=1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{align}
\frac{\sigma^2 s}{\mu}\left[ \frac{1}{8\,b_0^2} - \frac{1}{4}-\frac{1}{12}\,b_0^2 - \frac{2\,\zeta(-1/2)}{\sqrt{\pi}}\,b_0- \frac{4}{\sqrt{\pi}}\,\sum_{r=0}^\infty \frac{\zeta(-3/2-r)\,(-1)^r\,b_0^{2r+3}}{r!\,(2r+1)\,(2r+2)\,(2r+3)} \right]\label{y75}
\end{align}
with relative error $O(s^{-1/2})$. The expression between brackets in \eqref{y75} coincides with the right-hand side of \cite{Janssen2006}, (2.3) with $\beta = b_0\,\sqrt{2}$.
This leads to the following two results.
\begin{theorem} \label{varthm}
For $\eta\in[1/2,1)$,
\begin{equation} \label{y76}
\sigma_Q^2 = \frac{\gamma\,\sigma_X}{\pi}\,\sqrt{\frac{2}{\mu}}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right)
\end{equation}
with $G_3$ given in \eqref{e72}.
\end{theorem}
\begin{proposition}\label{varprop}
For $\eta\in(0,1/2)$, and for all $b<b_0$,
\begin{equation}
\sigma_Q^2 = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation}
For $\eta = 1/2$, $\sigma_Q^2$ equals expression \eqref{y75} with relative error $O(s^{-1/2})$. For $\eta\in(1/2,1)$ and $b_0\in(0,\sqrt{2\pi})$, $\sigma_Q^2$ has the form in \eqref{y74}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level
with $\eta=1/2$, it is possible to give a correction term which involves now integrals and series with $\zeta$-functions as considered in \cite[Secs.~4-5]{Janssen2007}.
\subsection{The empty-system probability} \label{sec5}
We have from (\ref{e6}) by proceeding as in (\ref{e13})--(\ref{e17}) that
\begin{align} \label{e100}
{\rm ln}\,[Q(0)] & = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{z}{z-1}\Big)\,\frac{g'(z)\,\exp(s\,g(z))}{1-\exp(s\,g(z))}\,dz \nonumber \\[3.5mm]
& = \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{z(z-1)}\,{\rm ln}\left(1-\exp(s\,g(z))\right)\,dz ,
\end{align}
where in the last step we used partial integration (noting that ${\rm Re}\,[g(z)]<0$ on $|z|=1+\varepsilon$). Then, as in Section \ref{sec1} for $\mu_Q$, the last integral in (\ref{e100}) is, with exponentially small error, given by
\begin{equation} \label{e101}
\frac{1}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{z'(v)}{z(v)(z(v)-1)}\,{\rm ln}\left(1-B\,{\rm e}^{-\frac12 s\alpha v^2}\right)\,{\rm d} v .
\end{equation}
Now for $v\geq0$ from $z({-}v)=z^{\ast}(v)$, $z'({-}v)={-}(z'(v))^{\ast}$
\begin{eqnarray} \label{e102}
& \mbox{} & \hspace*{-6mm}\frac{z'(v)}{z(v)(z(v)-1)}+\frac{z'({-}v)}{z({-}v)(z({-}v)-1)}=2i\,{\rm Im}\,\Big[\frac{z'(v)}{z(v)(z(v)-1)}\Big] \nonumber \\[3.5mm]
& & \hspace*{-6mm}=~2i\,{\rm Im}\,\Big[\frac{z'(v)\,z^{\ast}(v)(z^{\ast}(v)-1)}{|z(v)|^2\,|z(v)-1|^2}\Big] \nonumber \\[3.5mm]
& & \hspace*{-6mm}=~2i\,\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}+O(v^2))((z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4))}\,,
\end{eqnarray}
where we used \eqref{e32} and the fact that $z_{\rm sp}$ and $c_k$ are real with $z_{\rm sp}>1$. Therefore, we get for the expression in \eqref{e101}
\begin{align}
\frac{1}{\pi} &\int_0^{\frac{1}{2}\delta}\frac{1}{z_{\rm sp}{\rm +}O(v^2)}\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)v^2)+O(v^4)}\nonumber \\
&{\rm ln}\left(1-B\exp(-\tfrac12 s\alpha v^2)\right){\rm d} v.
\label{y77}
\end{align}
In the case that $2\eta-1<0$, we have as earlier that the whole expression in \eqref{y77} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. In the case that $2\eta-1\geq 0$, we substitute $v=t\sqrt{{s}/{(2\,\alpha )}}$, and we get as earlier for the expression \eqref{y77}, assuming also that $\eta<1$,
\begin{align}
\frac{1}{\pi}&\,\sqrt{s\,\alpha /2}\,\int_0^{\infty}\frac{z_{\rm sp}-1+O(t^2/s)}{(d^2(s)+t^2)\,(1+O(s^{-\eta})+O(t^2/s))}\,{\rm ln}(1-B\,{\rm e}^{-t^2}) {\rm d} t\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{\sqrt{s\,\alpha /2} \ (z_{\rm sp}-1)}{d^2(s)+t^2}{\rm ln}(1-B\,{\rm e}^{-t^2}){\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{d(s)}{d^2(s)+t^2}{\rm ln}(1-{\rm e}^{{-}d^2(s)-t^2}){\rm d} t\,\left(1+O(s^{\eta-1})\right).
\label{y78}
\end{align}
Here we also used \eqref{y46} and that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, so that
\begin{equation} \label{y79}
(\tfrac12\,s\,\alpha )^{1/2}\,(z_{\rm sp}-1) = d(s)\,\left(1+O(s^{-\eta})\right) = d(s)\left(1+O(s^{\eta-1})\right),
\end{equation}
since $\eta\geq 1/2$.
We have for $b>0$
\begin{align}
\frac{1}{\pi}&\int_0^\infty \frac{b}{b^2+t^2}\,{\rm ln}(1-\exp({-}b^2-t^2)){\rm d} t =-\frac12\sum_{k=0}^{\infty}\,\frac{1}{k+1}\,{\rm erfc}(b\,\sqrt{k+1}) = -F(b\,\sqrt{2}),\label{y80}
\end{align}
where according to \cite[(3.3) and (3.12)]{Janssen2006} for $\beta>0$
\begin{align}
F(\beta) &= \sum_{n=1}^\infty\,\frac{1}{n}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^\infty {\rm e}^{-x^2/2}dx\nonumber\\
&= -{\rm ln}\,\beta - \frac12\,{\rm ln}2 - \frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^\infty \frac{\zeta(1/2-r)\,(-1/2)^r\,\beta^{2r+1}}{r!\,(2r+1)},\label{y81}
\end{align}
the last identity being valid for $0<\beta<2\sqrt{\pi}$.
Using \eqref{y81} with $\beta^2 = d^2(s)= b_0^2/s^{2\eta-1}$, with the entire series on the second line being $O(\beta)$, we get the leading order behavior of ${\rm ln}[Q(0)]$ as
\begin{equation} \label{y82}
\Big({-}(\eta-1/2)\,{\rm ln}\,s+{\rm ln}(2\,b_0)+O(s^{1/2-\eta})\Big)\left(1+O(s^{\eta-1})\right)
\end{equation}
when $\eta\in(1/2,1)$. For $\eta = 1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{equation} \label{y83}
{\rm ln}(2\,b_0) + \frac{1}{\sqrt{\pi}}\,\sum_{r=0}^\infty \,\frac{\zeta(1/2-r)\,(-1)^r}{r!\,(2r+1)}\,b_0^{2r+1}
\end{equation}
with relative error $O(s^{-1/2})$. The expression \eqref{y83} coincides with ${\rm ln}[\mathbb{P}(M=0)]$ as given by \cite[(2.1)]{Janssen2006} with $\beta = b_0\,\sqrt{2}$. The next two results summarize the above.
\begin{theorem} \label{emptythm}
For $\eta\in(1/2,1)$,
\begin{equation} \label{y84}
{\rm ln}[\mathbb{P}(Q=0)] = - F\big(d(s)\,\sqrt{2}\big)\left(1+O(s^{\eta-1})\right)
\end{equation}
with $F$ given by \eqref{y81}.
\end{theorem}
\begin{proposition} \label{emptyprop}
For $\eta\in (0,1/2)$, and for all $b<b_0$,
\begin{equation}
{\rm ln}[\mathbb{P}(Q=0)] = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation}
For $\eta=1/2$, ${\rm ln}[\mathbb{P}(Q=0)]$ equals $-F(b_0\,\sqrt{2})$ with a relative error $O(1/\sqrt{s})$. For $\eta\in (1/2,1)$ and $0<b_0<\sqrt{2\pi}$, ${\rm ln}[\mathbb{P}(Q=0)]$ has leading order behavior as in \eqref{y82}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level case
with $\eta=1/2$, it is possible to give a correction term which involves now the integrals in \eqref{y80} and \eqref{e51}.
\section{Numerical examples}\label{numm}
\subsection{Accuracy of the approximations}
In this subsection we present a numerical example that serves to illustrate the accuracy of the derived heavy-traffic approximations. Consider the Poisson case
\begin{equation}
\tilde X(z)={\rm e}^{z-1},\quad \mu = \sigma^2 = 1.
\end{equation}
We fix $\mu$ and vary $n$ with the value of $s$, according to
\begin{equation}
\vartheta = \frac{n}{s} = 1-\frac{\gamma}{s^\eta}
\end{equation}
for some $\gamma>0$ and $\eta\geq 1/2$. To calculate the exact value of the mean congestion level we use the expression, see \cite{Boudreau1962},
\begin{equation}\label{x73}
\mu_Q=\frac{\sigma_A^2}{2(s-\mu_A)}-\frac{s-1+\mu_A}{2}+\sum_{k=1}^{s-1}\frac{1}{1-z_k}.
\end{equation}
Here $z_1,\ldots,z_{s-1}$ are the zeros of $z^s-A(z)$ in $|z|<1$. We apply the method of successive substitution described in \cite{Janssen2005} to obtain accurate numerical approximations for $z_1,...,z_{s-1}$ and consequently $\mu_Q$.
From Theorem \ref{mainthm}, we find that the leading order behavior of $\mu_Q$ is given by
\begin{equation} \label{x18}
\frac{\sqrt{2s}}{\pi}\,G_0\Big(\frac{\gamma}{\sqrt{2}\,s^{\eta-\frac{1}{2}}}\Big).
\end{equation}
In order to find the correction terms, we proceed by setting $\eta = 1/2$. Deriving constants $C_1,C_2,C_3,$ and $C_4$ for our setting and substituting these into \eqref{e71}, we get for $\mu_Q$, with an absolute error of $O(s^{-1/2})$, the approximation
\begin{equation}\label{x19}
\frac{\sqrt{2\,s}}{\pi}\Big(\Big(1-\frac{\gamma}{3\,\sqrt{s}}\Big)\,G_0(b_0)-\frac{\gamma^3}{3\,\sqrt{s}}\,(\,G_3(b_0)+G_4(b_0))\Big),
\end{equation}
which by \eqref{e49} and \eqref{e74} reduces to
\begin{equation}\label{x20}
\frac{\sqrt{2\,s}}{\pi}\,G_0(b_0)-\frac{\sqrt{2}\,\gamma}{3\,\pi}\,G_1(b_0).
\end{equation}
\begin{table}\label{tab:poisson1}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.683 & 0.244 & 0.399 & 0.247 \bigstrut[t] \\
20 & 0.776 & 0.410 & 0.565 & 0.412 \\
50 & 0.858 & 0.739 & 0.893 & 0.741 \\
100 & 0.900 & 1.110 & 1.263 & 1.111 \\
200 & 0.929 & 1.633 & 1.787 & 1.634 \\
500 & 0.955 & 2.672 & 2.825 & 2.673 \\
1000 & 0.968 & 3.843 & 3.996 & 3.843 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 1$.}
\end{table}
\begin{table}\label{tab:poisson2}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.968 & 13.707 & 14.046 &13.732 \bigstrut[t] \\
20 & 0.977 & 19.533 & 19.865 &19.551\\
50 & 0.985 & 31.084 & 31.409 &31.095\\
100 & 0.990 & 44.097 & 44.419 &44.106\\
200 & 0.992 & 62.499 & 62.819 &62.505\\
500 & 0.995 & 99.008 & 99.325 &99.011\\
1000 & 0.996 & 140.152 & 140.468 & 140.154 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 0.1$.}
\end{table}
\begin{table}\label{tab:poisson3}
\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\eta=0.6$} & \multicolumn{2}{c|}{$\eta=0.75$} & \multicolumn{2}{c|}{$\eta=0.9$} \bigstrut\\
\hline
$s$ & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} \bigstrut\\
\hline
10 & 17.781 & 18.125 & 25.970 & 26.318 & 37.553 & 37.905 \bigstrut[t] \\
20 & 27.309 & 27.647 & 44.391 & 44.734 & 71.195 & 71.541 \\
50 & 47.948 & 48.281 & 89.623 & 89.961 & 164.637 & 164.978 \\
100 & 73.245 & 73.574 & 152.031 & 152.367 & 309.353 & 309.692 \\
200 & 111.752 & 112.079 & 257.435 & 257.769 & 580.170 & 580.507 \\
500 & 195.082 & 195.409 & 515.443 & 515.776 & 1329.581 & 1329.917 \\
1000 & 297.122 & 297.448 & 870.524 & 870.857 & 2487.227 & 2487.562 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma=0.1$ and several values of $\eta$.}
\end{table}
\noindent Numerical results for $\eta=1/2$ and various values of $s$ are given in Table 1 and 2, for $ \gamma = 1$ and $\gamma = 0.1$, respectively.
We note that for small $s$ the leading order approximation is still off by a significant amount, while the refinement only shows an error in the second decimal for $\gamma = 0.1$. This seems to justify the use of the correction term.
In Table 3 we compare the approximation \eqref{x18} against the exact value of $\mu_Q$ for three values of $\eta\geq 1/2$ to assess the influence of $\eta$. Clearly, the leading order approximation is relatively accurate for all three scenarios. As expected, the mean congestion increases along with $\eta$, since utilization approaches 1 more rapidly in this case.
\subsection{Connection to other queueing models}\label{subsec62}
As argued in the introduction, we believe that the heavy-traffic behavior for the discrete model in this chapter will up to leading order be universal for a wide range of other models (when subjected to the same heavy-traffic regime \eqref{bb}). We shall now substantiate this for many-server systems, for which under \eqref{bb}, it turns out that the mean congestion is $O(s^\eta)$. We compare the mean congestion level in our discrete queue with that in the multi-server systems $M/M/s$, $M/D/s$ and Gamma/Gamma/$s$, all with unit mean service time and occupation rate $1-\gamma/s^\eta$.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 2.5,
xmax = 6.5,
ymin = 0,
ymax = 7.2,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.4,0.2)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,mark = o,mark options={scale=1.25}] table[x=log_s,y=mms] {novel_figure1.txt};
\addplot[thick, mark=triangle,dashed,mark options={scale=1.25,solid}] table[x=log_s,y=mds] {novel_figure1.txt};
\addplot[thick,mark=square,dotted,mark options={scale=1.25,solid}] table[x=log_s,y=ggs] {novel_figure1.txt};
\legend{$M/M/s$,$M/D/s$,Gamma/Gamma/$s$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ plotted against $s$ on log scale for 3 queues for $\eta=0.75$.}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 0.2,
xmax = 6.5,
ymin = 0,
ymax = 9,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.75,4.2)},anchor = west},
yscale = 0.8,
xscale = 1
]
\addplot[thick,only marks,mark = o,mark options={scale=1.25}] table[x=n01,y=m01] {novel_figure2.txt};
\addplot[thick,only marks,mark = square,mark options={scale=1.25}] table[x=n025,y=m025] {novel_figure2.txt};
\addplot[thick,only marks,mark = *,mark options={scale=1.25}] table[x=n05,y=m05] {novel_figure2.txt};
\addplot[thick,only marks,mark = square*,mark options={scale=1.25}] table[x=n075,y=m075] {novel_figure2.txt};
\addplot[thick,only marks,mark = triangle*,mark options={scale=1.25}] table[x=n1,y=m1] {novel_figure2.txt};
\addplot[dashed] coordinates{ (0,2.25) (7,9.25) };
\addplot[dashed] coordinates{ (0,2.25) (7,7.5) };
\addplot[dashed] coordinates{ (0,2.25) (7,5.75) };
\addplot[dashed] coordinates{ (0,2.25) (7,4) };
\addplot[dashed] coordinates{ (0,2.25) (7,2.95) };
\legend{\ $\eta=0.1$,\ $\eta=0.25$,\ $\eta=0.5$,\ $\eta=0.75$,\ $\eta=1$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ of $M/M/s$ plotted against $s$ on log scale for different values of $\eta$.}
\label{fig2}
\end{figure}
Figure \ref{fig1} shows on logarithmic scale the mean congestion levels for $\gamma=0.1$ and $\eta=0.75$ under the specified scaling for three systems. We also display three lines with slope 0.75 for comparison, which confirms that mean congestion levels are of the order $s^\eta$, also in these multi-server system. Formally establishing this heavy-traffic behavior for these multi-server system is an important open problem and requires other mathematical approaches than the ones taken in this chapter (see the introduction for more details).
Figure \ref{fig2} shows the mean queue length in the $M/M/s$ system for several values of $\eta$, again on logarithmic scale, together with lines with slope $\eta$. For $\eta\geq 1/2$, we see the same $O(s^\eta)$ behavior, similar as for $\mu_Q$ in our discrete model. For $\eta<1/2$ the mean queue length decays, again in agreement with our results for $\mu_Q$. We note that this qualitative behavior of the $M/M/s$ system was also observed by \cite[Thm.~4.1]{maman}, by proving that the mean waiting time in the $M/M/s$ queue under \eqref{bb} is of order $1/s^{1-\eta}$, which by Little's law implies the mean queue length to be of order $s^\eta$.
\chapter{Novel heavy-traffic regimes}
\begin{chapterstart}
In this chapter, we introduce a family of heavy-traffic scalings for a large-scale service system meant to serve jobs coming from a large pool of customers.
The scaling rules are inspired by the classical QED regime discussed in Chapter 1, but lead to a range of different system behaviors that includes the ED, QED and QD regime as special cases.
To determine the scaling limits, we describe the performance measures in terms of Pollaczek integrals and use asymptotic techniques to evaluate these integrals in the large-system limit.
\end{chapterstart}
\begin{flushright}
Based on \\
\textbf{Novel heavy-traffic regimes for large-scale service systems}\\
\textit{Guido Janssen, Johan van Leeuwaarden \& Britt Mathijsen}\\
In \textit{SIAM Journal of Applied Mathematics, 75(2), 787-812 (2015)}
\end{flushright}
\newpage
\section{Introduction \& motivation}
We study the workload process of a system, experiencing stochastic demand and deterministic capacity $s_n$ per period, at equidistant time epochs.
Demand is assumed to be generated by $n$ stochastically identical and independent sources.
Let $A_{i,j}$ denote the workload brought into the system by source $i$ in period $j$, for which $\mathbb{E}[A_{i,j}] =\mu$ and ${\rm Var}\, A_{i,j} = \sigma^2$.
Then the total amount of demand arriving to the system in period $j$ is $A^{(n)}_j=\sum_{i=1}^n A_{i,j}$ with $\mathbb{E}[A^{(n)}_j] = n\mu$ and ${\rm Var}\, A^{(n)}_j = n\sigma^2$.
As explained in Chapter 1, a good capacity sizing rule for achieving economies-of-scale is $s_n = n\mu+\beta\sqrt{n}\sigma$ for some $\beta>0$.
If we denote the system utilization by $\rho_n := n\mu/s_n$, then this dimensioning rule in the bulk service queue with many sources is tantamount to the heavy-traffic scaling
\begin{equation}\label{bb1}
\sqrt{n}(1-\rho_n) \to \gamma = \frac{\beta\sigma}{\mu}, \qquad {\rm as }\ n\to\infty.
\end{equation}
Starting from this setting, we introduce a novel family described in terms of a parameter $\eta$ for which we assume that
\begin{equation}\label{bb}
n^{\eta}(1-\rho_n)\rightarrow \gamma, \quad {\rm as} \ n\to \infty, \ \gamma> 0.
\end{equation}
The parameter $\eta\geq 0$ defines a whole range of possible scaling regimes, including the classical case $\eta=1/2$.
In terms of a capacity sizing rule for systems with many customers, the condition \eqref{bb} is tantamount to $s_n=n\mu+\beta \sigma n^{1-\eta}$.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as {\it moderate} heavy traffic: heavy-traffic conditions in which the full occupancy is reached more slowly, as a function of $n$, than for classical heavy traffic. For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to {\it extreme} heavy traffic due to a relatively small variability hedge. Note that the case $\eta=0$ does not lead to 100\% system utilization when $n\to\infty$.
In this chapter we show that economies-of-scale can be achieved for a large range of $\eta$, although the nature of the benefits obtained by operating on large scale depends on the precise capacity sizing rule (hence the parameter $\eta$). We quantify performance in terms of stationary measures: The mean and variance of the congestion in the system, and the probability of an empty system. For these performance measures we derive heavy-traffic limits under the scalings \eqref{bb} that
are relatively simple functions of only the first two moments of the demand per period. Such parsimonious expressions are useful for quantifying and improving system behavior. The heavy-traffic limits, however, provide also qualitative insight into the system behavior. Our asymptotic analysis shows that mean congestion is $O(n^\eta)$, which implies
that delays experienced by the customers are negligible for all values of $\eta\in [0,1)$, are roughly constant for $\eta=1$, and grow without bound for $\eta>1$. We expect this qualitative behavior to be universal for a wide range of stochastic models to which the regime \eqref{bb} is applied.
We further show the existence of the following trichotomy as $n\to \infty$ under \eqref{bb}: For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, for $\eta\in (1/2,1)$ it converges to $0$, while only for $\eta=1/2$ there is a limiting value in $(0,1)$. Hence, as expected, the system performance deteriorates with $\eta$, in a rather crude way for the empty-system probability, and in only a mild way for mean congestion levels. The regime \eqref{bb} thus presents a range of possible capacity sizing rules that all lead to economies-of-scale, and depending on what is the desired nature of performance for a particular service system, an appropriate $\eta$ can be selected. From the quantitative perspective, our detailed asymptotic analysis leads to more precise asymptotic estimates for the performance measures in heavy traffic, which reveal the exact manner in which congestion is influenced by $\eta$ and $\gamma$.\\
\\
\noindent\textbf{Motivating examples.}
The bulk service queueing model is one of the canonical models in queueing theory, having a wide range of applications in fields like digital communication, wireless networks, road traffic, reservation systems, health care and many more (see \cite{Bruneel1993} and \cite[Chap.~5]{johanthesis} for an overview).
In road traffic, the basic model for congestion at an intersection, known as the fixed-cycle traffic-light queue \cite{Newell1960,Leeuwaarden2006}, is related to our discrete bulk service queue.
Then $s_n$ represents the maximum number of delayed cars in front of a traffic light that can depart during one green period, while $A^{(n)}_j$ is the number of newly arriving cars during a consecutive green and red period.
An example from health care is panel sizing \cite{Zacharias2014}.
Say a general practitioner has a pool of $n$ clients (typically in the order of 2,500~\cite{Green2008}), all of which are potential patients, and together require $A^{(n)}_j$ consults per day.
Further assume that the practitioner can see a maximum number of $s_n$ patients per day.
What is then an appropriate patient panel size $n$, which strikes a reasonable balance between accessing medical care in a timely manner and restricting the time that the practitioner sits idle?
The panel size application is one of many examples of an appointment book, referring to some schedule of appointments for a fixed period, with capacity $s_n$ appointments per period and newly arriving number of appointments $A^{(n)}_j$ per period.
See \cite{Dai2014} for another recent example of an appointment book in a health care setting, again in terms of our bulk service queue, with $A^{(n)}_j$ the new patients per day and $s_n$ the number of available beds.
For all examples above, and many more, our new class of heavy-traffic scalings \eqref{bb} presents capacity sizing rules for which the expected performance can be quantified using the results in this chapter. This will be helpful in dimensioning the systems (How much capacity is needed to achieve a certain target performance?) while exploiting economies-of-scale. For appointment books, our model together with the capacity sizing rules \eqref{bb} is particularly relevant for {\it advanced access} \cite{Green2008}, a scheduling approach in health care designed to reduce delays by offering every patient a same-day appointment, regardless of the urgency of the problem. In that way, patients do not have to wait long for appointments, and practices do not waste capacity by holding appointments in anticipation of urgent cases.\\
\\*
\noindent\textbf{Pollaczek's formula.}
Next to the freedom to model different situations, another advantage of our model is that it is mathematically tractable, in the sense that it can be subjected to powerful mathematical methods from complex and asymptotic analysis. In order to establish the heavy-traffic limits we start from Pollaczek's formula for the transform of the stationary queue length distribution in terms of a contour integral. From this famous transform representation, contour integrals for the empty-system probability and the mean and variance of the congestion immediately follow. Contour integrals are often amenable to asymptotic evaluation (see e.g.~\cite{Cohen1982}), particularly for obtaining classical heavy-traffic asymptotics.
We also subject the contour integral representations to asymptotic evaluation, but not under classical heavy-traffic scaling.
This asymptotic analysis requires a {\it non-standard} saddle point method, tailored to the specific form of the integral expressions that arise under the capacity sizing rule \eqref{bb}. \\
\\*
\noindent
\textbf{Saddle point method.}
In complex analysis, the saddle point method in its standard form is a useful technique to estimate the asymptotic behavior of integrals of the form
\begin{equation}
\label{eq:sp_integral}
I(n) = \int_C h(z)\, {\rm e}^{n f(z)}\, {\rm d} z,
\end{equation}
as $n\to\infty$, where $C$ is a contour in the complex plane, and $f(z)$ and $h(z)$ are functions that are analytic in some neighborhood of $C$.
The main idea behind the saddle point method is that if the integrand in \eqref{eq:sp_integral} exhibits a sharp peak along the contour, then one may naturally expect that a small neighborhood around this peak provides the dominant contribution to the integral.
More specifically, for large values of $n$, the function $f$ and its associated maximum $f(z^*)$ for $z^*\in C$ to a large extent determine the magnitude of the integrand (where $z^*$ is well-defined due to analyticity of $f$.
In the setting of this chapter, $C$ is a closed curve, which implies that the value $z^*$ must be a \textit{saddle point} of $f$, i.e.~$f'(z^*) = 0$.
Subsequently, one can replace $f(z)$ in \eqref{eq:sp_integral} by its Taylor expansion around $z^*$ and deduce through the Laplace method, see e.g.~\cite{debruijn}, that
\begin{equation*}
I(n) = \sqrt{2\pi}\,i\frac{h(z^*)\,{\rm e}^{n f(z^*)}}{\sqrt{n\, |f''(z^*)|}}\Bigl( 1+ O(1/n)\Bigr),
\end{equation*}
as $n\to\infty$.
In Section \ref{spSec}, we show how the contour integrals describing stationary measures for the queue length, derived through Pollaczek's formula, can be reformulated into the shape of \eqref{eq:sp_integral}.
However, we will show that the saddle point method in its standard from cannot be applied to asymptotically characterize other stationary measures like the mean or mass at zero.
Indeed, for our model the saddle point (the solution of \eqref{e21}) converges to one (as $n\to\infty$), which is a singular point of the integrand, and renders the standard saddle point method useless.
The non-standard saddle point method discussed in this chapter, originally proposed by \cite{debruijn}, is made specifically to overcome this complication. This leads to asymptotic expansions for the performance measures, of which the limiting forms correspond to the heavy-traffic limits, and pre-limit forms present refined approximations for pre-limit systems ($n<\infty$) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as {\em corrected diffusion approximations} \cite{Siegmund1978,Blanchet2006,Asmussen2003}.
\noindent{\bf Further connections to the literature.}
We now discuss two classes of stochastic systems for which the heavy-traffic regime \eqref{bb1} has been studied extensively, and for which our new family of regimes \eqref{bb} is largely unexplored. We discuss these classes because, despite the fact that the Pollaczek formula does not hold, we believe the qualitative results that we reveal for our particular model should to a large extent carry over to these settings as well, presenting some interesting avenues for further research (see Section \ref{subsec62}).
The first class concerns so-called {\it nearly-deterministic} systems \cite{Sigman2011a,Sigman2011b}, denoted by $G_n/G_n/1$ system, where $G_n$ stands for {\it cyclic thinning} of order $n$, indicating that some point process is thinned to contain only every $n$th point. As $n\to \infty$, the $G_n/G_n/1$ systems approach the deterministic $D/D/1$ system. For $G_n/G_n/1$ systems, \cite{Sigman2011a} establishes stochastic-process limits, and \cite{Sigman2011b} derives heavy-traffic limits for stationary waiting times. In the framework of \cite{Sigman2011a,Sigman2011b}, our stochastic model corresponds to a $D/G_n/1$ queue, where the sequence of service times $\{A^{(n)}_j\}_{j\geq 1}$ follows from a cyclically thinned sequence of i.i.d.~random variables $A_{i,j}$. It follows from \cite[Theorem 3]{Sigman2011b} that the rescaled stationary waiting time process converges under \eqref{bb1} to a reflected Gaussian random walk. Hence, the performance measures of the nearly deterministic system, under \eqref{lind} and \eqref{bb1}, should be well approximated by the performance measures of the reflected Gaussian random walk, giving rise to heavy-traffic approximations. This connection is discussed in detail in Section \ref{subsec3.2}. It seems likely that results similar to those presented in this chapter can be obtained by applying the scaling \eqref{bb} to the nearly-deterministic systems in \cite{Sigman2011a,Sigman2011b}, and because Pollaczek's formula also applies to this setting, the non-standard saddle point method developed in this chapter can provide the appropriate methodology.
The second class concerns multi-server systems, and in particular the many-server regime. When we interpret $s_n$ as the number of servers, instead of capacity per time slot or order of thinning, the scaling \eqref{bb1} is similar to the QED or Halfin-Whitt regime for the $M/M/s_n$ system.
As we have reviewed in Chapter 1, the QED regime is characterized by a delay probability that converges to a non-degenerate limit away from both zero and one, and the mean delay is asymptotically negligible as the number of servers grows large. The QED regime \eqref{bb1} is naturally positioned in between the Quality-Driven (QD) regime and the Efficiency-Driven (ED) regime. In the QD regime, the load remains bounded away from 1, which corresponds to setting $\eta=0$ in \eqref{bb}. Hence, the range $\eta\in(0,1/2)$ bridges the gap between the QED regime and the QD regime. Likewise, the ED regime corresponds to setting $\eta=1$ in \eqref{bb}, so that the range $\eta\in(1/2,1]$ connects the QED regime and ED regime. For the birth-death process describing the $M/M/s_n$ system, Maman \cite{maman} introduced a scaling similar to \eqref{bb}, and called it the QED-$c$ regime, also bridging the ED and QD regimes.
Theorem 4.1 of \cite{maman} says that the expected waiting time under the scaling $s_n = n\mu+\beta\sigma n^{1-\eta}$ is of order $s_n^{1-\eta}$, which is equivalent to the expected queue length being of order $n^\eta$ by Little's law. We should stress though that we expect the mathematical techniques that are needed to establish heavy-traffic results could be entirely different than in this chapter, because Pollaczek's formula does not apply to many-server settings.
The specific model assumptions will determine to a large extent the appropriate methodology. Under Markovian assumptions leading to the $M/M/s_n$ system, simple exact solutions are available for the stationary distribution. This makes it possible to describe performance measures like the mean congestion directly in terms of real integrals. Where the saddle point method is used for integrals in the complex plane, the Laplace method (see e.g.~\cite{flajolet}) is used for real integrals. Hence, for the asymptotic evaluation of the $M/M/s_n$ system under the scaling \eqref{bb}, the Laplace method seems an appropriate methodology, although again one needs to deal with possible singularities in the integrand. For $G/D/s_n$ systems, which assume deterministic service times, it has been shown in \cite{Jelenkovic2004} that using a decomposition property the dynamics of this multi-server systems can be captured in terms of a single-server system. Hence, for these systems, Pollaczek's formula applies, and our saddle point method can most likely be applied to obtain heavy-traffic results in the regimes \eqref{bb}. Under more general conditions, for instance leading to a $G/G/s_n$ system, it is simply unclear at this stage how to obtain precise heavy-traffic approximations for \eqref{bb}, because a tractable description of the performance measures is not available; see Section 1.2.4 for details.\\
\\
\\*
\noindent{\bf Structure of the chapter.}
In Section \ref{sec1} we present in detail the model and the family of heavy-traffic scalings. In Section \ref{spSec} we introduce the saddle point method. In Section \ref{sec3} we apply the saddle point method for the mean congestion level. Theorem \ref{mainthm} gives for all heavy-traffic scalings the limiting behavior in terms of an integral expression. As a consequence, we show in
Proposition \ref{prop1} that there are two types of heavy-traffic behavior, depending on whether $\eta\in(0,1/2)$ or $\eta\geq 1/2$.
In Section \ref{subsec3.2} we discuss for the case $\eta=1/2$ the connection with the Gaussian random walk and the Riemann zeta function.
In fact, we show that for all $\eta\geq 1/2$ there exists a connection between the integral expression in Theorem \ref{mainthm} and the Riemann zeta function.
In Section \ref{more} we apply the saddle point method to obtain several more heavy-traffic results, including refined heavy-traffic approximations for the mean congestion level, and the leading heavy-traffic behavior for the variance of the stationary congestion level and for the empty-system probability.
Finally, in Section \ref{numm} we confirm through numerical experiments the accuracy of our heavy-traffic approximations, and moreover show that under \eqref{bb}, various multi-server systems behave similar to our discrete bulk service queue.
\section{Model description \& heavy-traffic regimes}\label{sec1}
We thus consider a discrete stochastic model in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,\ldots$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
We will omit the superscript $(n)$ if no ambiguity is possible.
The system has a service capacity $s_n\in\mathbb{N}$ per period, so that the recursion
\begin{equation}
\label{lind}
Q(j+1) = \max\{Q(j) + A^{(n)}_j - s_n,0\},\qquad j=1,2,...,
\end{equation}
assuming $Q(0)=0$, gives rise to a Markov chain $\{Q(j)\}_{j\geq 1}$ that describes the congestion in the system over time. The probability generation function (pgf)
\begin{equation*}
\tilde A(z)=\sum_{k=0}^{\infty} \mathbb{P}\big(A^{(n)}=k\big) z^k
\end{equation*}
of $A^{(n)}$ is assumed analytic in a disk $|z|<r$ with $r>1$, which implies that all moments of $A^{(n)}$ exist. We also assume that
\begin{equation} \label{e3}
\tilde A'(1)=\mathbb{E}[A^{(n)}_j]=\mu_A<s_n.
\end{equation}
Under the assumption (\ref{e3}) the function $z^{s_n}-\tilde A(z)$ has exactly $s_n$ zeros in the closed unit disk, one of these being $z=1$ (see \cite{rouche}).
We further assume that $\mathbb{P}(A^{(n)}=j)>0$ for some $j>s_n$.
Under this assumption the function
$z^{s_n}-\tilde A(z)$ also has zeros outside $|z|\leq 1$, and we let $r_0$ be the minimum modulus of these zeros.
The number $r_0$ is the unique zero of $z^{s_n}-\tilde A(z)$ with real $z>1$; see e.g.~\cite{Janssen2005}.
Moreover, under assumption (\ref{e3}) the stationary distribution $\lim_{j\to \infty}\mathbb{P}\left(Q(j)=k\right)=\mathbb{P}(Q=k)$, $k=0,1,\ldots$ exists, with the random variable $Q$ defined as having this stationary distribution.
We let
\begin{equation*}
\tilde Q(w)=\sum_{j=0}^{\infty}\mathbb{P}(Q=j)w^j
\end{equation*}
be the pgf of the stationary distribution. $\tilde Q(w)$ is analytic in $|w|<r_0$, and given by Pollaczek's formula (see e.g.~\cite{Abate1993, Cohen1982}).
In our discrete setting, we shall first derive a useful expression for $\tilde{Q}(w)$.
\begin{lemma}
For any $\varepsilon>0$ with $1+\varepsilon<r_0$,
\begin{equation} \label{e111}
\tilde Q(w)=\exp\Big(\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big)
\end{equation}
holds when $|w|<1+\varepsilon$.
\end{lemma}
\begin{proof}
We shall establish (\ref{e111}) for any $w\in(1,1+\varepsilon)$, and then the full result follows from analyticity of $\tilde{Q}(w)$ and of
\begin{equation*}
{\rm ln}\Big(\frac{w-z}{1-z}\Big)={\rm ln}\Big(\frac{1-w/z}{1-1/z}\Big)={-}\,\sum_{k=1}^{\infty}\,\frac1k\,\Big(\Big(\frac{w}{z}\Big)^k-\Big(\frac1z\Big)^k\Big)
\end{equation*}
in $w$, $|w|<1+\varepsilon$ for any $z$ with $|z|=1+\varepsilon$.
Our starting point is the formula, see \cite{Boudreau1962},
\begin{equation} \label{e113}
\tilde Q(w)=\frac{(s_n-\mu_A)(w-1)}{w^{s_n}-\tilde A(w)}\,\prod_{k=1}^{s_n-1}\,\frac{w-z_k}{1-z_k}
\end{equation}
that holds for all $w$, $|w|<r_0$, in which $z_1,\ldots,z_{s_n-1}$ are the $s_n-1$ zeros of $z^{s_n}-\tilde A(z)$ in $|z|<1$. Fix $w\in(1,1+\varepsilon)$.
Then ${\rm ln}\,[(w-z)/(1-z)]$ is analytic in $z\in\mathbb{C}\backslash [1,w]$.
It follows that
\begin{align}
I_C &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z \nonumber \\
&=~\sum_{k=1}^{s_n-1}\,{\rm ln}\Big(\frac{w-z_k}{1-z_k}\Big)+\frac{1}{2\pi i}\,\int_C\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,
\label{e114}
\end{align}
where $C$ is a contour encircling $[1,w]$ in the positive sense with none of the $z_k$'s in its interior. We let $\delta\in(0,\frac{w-1}{2})$ and we take $C$ the union of two line segments, from $1+\delta-i0$ to $w-\delta-i0$ and from $w-\delta+i0$ to $1+\delta-i0$, and two circles, of radius $\delta$ and encircling 1 and $w$ in positive sense.
A careful administration of the various contributions to the integral $I_C$ in \eqref{e114}, taking account of the branch cut $[1,w]$, yields
\begin{equation*}
I_C = {\rm ln }\left(\frac{(s_n-\mu_A)(w-1)}{w^s-\tilde A(w)}\right) + O(\delta\,{\rm ln}\, \delta ).
\end{equation*}
Using this in \eqref{e113} and letting $\delta \downarrow 0$, we get \eqref{e111} for $w\in(1,1+\varepsilon)$ and the proof is complete.
\end{proof}
Using $\mathbb{P}(Q=0)=\tilde Q(0)$, $\mu_Q=\tilde Q'(1)$ and $\sigma_Q^2 = \tilde Q''(1)+\tilde Q'(1)-(\tilde Q'(1))^2$, it follows by straightforward manipulations that
\begin{align} \label{e6}
\mathbb{P}(Q=0)&=\exp\,\Big[\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\ln\Big(\frac{z}{z-1}\Big)\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big] , \\
\label{e7}
\mu_Q&=\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{1-z}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,\\
\label{e8}
\sigma_Q^2 &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{-z}{(1-z)^2}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z .
\end{align}
Because $s_n$ appears directly in expressions \eqref{e6}-\eqref{e8}, we will be conducting our analysis with respect to $s_n$ rather than $n$. Note that this has no consequences for our results on the convergence speed of the performance metrics, since $s_n = O(n)$. Furthermore, we will omit the index $n$ when describing the capacity $s_n$ in the remainder of the chapter for brevity.
We next discuss in more detail the family of heavy-traffic scalings considered in this chapter, which combines two features. First, we have assumed that
$A^{(n)}_j$ is in distribution equal to the sum of work generated by all sources, $A_{1,j}+...+A_{n,j}$, where the $A_{i,k}$ are for all $i$ and $k$ i.i.d.~copies of a random variable $X$, of which the pgf $\tilde X(z)=\sum_{k=0}^{\infty}\mathbb{P}(X=k)z^k$ has radius of convergence $r>1$, and
\begin{equation*}
0< \mathbb{E}[A^{(n)}] =n\mu = n \tilde X'(1)<s_n .
\end{equation*}
Hence
\begin{equation} \label{e10}
\vartheta:=\frac{n}{s_n}\in(0,1/\mu) .
\end{equation}
Second, we scale the system according to \eqref{bb}, for which we assume that
\begin{equation} \label{e11}
\rho_{s_n} =\vartheta\,\mu =1-\frac{\gamma}{s_n^\eta}
\end{equation}
in which $\gamma>0$ is bounded away from 0 and $\infty$ as $s_n\to \infty$.
In the remainder of this chapter, we will omit the subscript in $s_n$.
The condition that $\mathbb{P}(A^{(n)}=k)>0$ for some $k>s$ holds when the degree $d$ of $\tilde X(z)$ (with $d=\infty$ if $\tilde X(z)$ is not a polynomial) is such that $nd>s$.
To avoid certain complications when applying the saddle point method, we further assume that
\begin{equation} \label{e12}
|\tilde X(z)|<\tilde X(r_1) ,~~~~~~|z|=r_1\,,~~z\neq r_1 ,
\end{equation}
for any $r_1\in(0,r)$. This implies that $r_0$ is the unique zero of $z^s-\tilde A(z)$ on $|z|=r_0$.
This condition is related to Cram\'er's condition, see \cite[pp.~189 and 355]{Asmussen2003}, and it has also been used in \cite{relaxation}.
Condition \eqref{e12} holds when the set of all $j=0,1,\ldots$ such that $\mathbb{P}(X=k)>0$ is not contained in an arithmetic progression with a ratio larger than one (see also \cite{rouche}).
\section{Non-standard saddle point method}\label{spSec}
\noindent
We illustrate our saddle point method for $\mu_Q$.
As a first step, we bring (\ref{e7}) in a form which is amenable to saddle point analysis.
\begin{lemma}
\begin{equation} \label{e18}
\mu_Q = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{g'(z)}{z-1}~\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}\,{\rm d} z
\end{equation}
with
\begin{equation} \label{e15}
g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}(\tilde X(z)) .
\end{equation}
\end{lemma}
\begin{proof}
With $\tilde A(z)=\tilde X^n(z)$,
\begin{align} \label{e13}
\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)} & = \frac{s\,z^{s-1}-n\,\tilde X'(z)\,\tilde X^{n-1}(z)}{z^s-\tilde X^n(z)} \nonumber \\
& = \frac{s}{z}-\frac{s}{z}\,\Big(\frac{n}{s}~\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big)\,\frac{z^{-s}\,\tilde X^n(z)}{1-z^{-s}\,\tilde X^n(z)} .
\end{align}
Write
$
z^{-s}\,\tilde X^n(z)=\exp(s\,g(z))$.
Noting that
\begin{equation} \label{e16}
\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{s}{z}~\frac{1}{1-z}\,{\rm d} z=0 ,
\end{equation}
and that
\begin{equation} \label{e17}
g'(z)=\frac1z\,\Big(\vartheta\,\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big) ,
\end{equation}
gives \eqref{e18}. \end{proof}
Let us now explain how the standard saddle point method can be applied to \eqref{e18}.
Since
\begin{equation} \label{e19}
g(1)=g(r_0)=0~;~~~~~~g(z)<0\,,~~1<z<r_0 ,
\end{equation}
and by strict convexity of
\begin{equation*}
z^{-s}\,\tilde X^n(z)=z^{-s}\tilde A(z)=\sum_{k=0}^{\infty}\,a_k\,z^{k-s} ,~~~~~~z\in(0,r) ,
\end{equation*}
$g(z)$ has a unique minimum on $[1,r_0]$. This minimum is found by solving $z\in[1,r_0]$ from $g'(z)=0$, and this yields the equation
\begin{equation} \label{e21}
\tilde X(z)=\vartheta\,z\,\tilde X'(z) .
\end{equation}
Denote the solution $z\in(1,r_0)$ of (\ref{e21}) by $z_{\rm sp}$, and observe that $z_{\rm sp}$ is a saddle point of $g(z)$, explaining the notation. Thus, the saddle point method can be used for the integral in (\ref{e18}) by taking $1+\varepsilon=z_{\rm sp}$.
In the case that $\vartheta=n/s$ is bounded away from $1/\mu$ as $s\to \infty$, we have that the minimum value of $g(z)$, $1\leq z\leq r_0$, is negative and bounded away from 0. Furthermore, $z_{\rm sp}$ is bounded away from 1, and the saddle point method can be applied in the classical way by replacing
\begin{equation*}
\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}~~~~{\rm by}~~~~\exp(s\,g(z)) ,
\end{equation*}
at the expense of an exponentially small relative error, and performing an expansion of $g'(z)/(z_{\rm sp}-1)=d_1(z-z_{\rm sp})+O((z-z_{\rm sp})^2)$ with $d_1=g''(z_{\rm sp})/(z_{\rm sp}-1)\neq 0$.
Using that $g(z^{\ast})=(g(z))^{\ast}$, where the $^*$ denotes complex conjugation, it can be shown that
\begin{equation} \label{e23}
\mu_Q=\frac{\exp(s\,g(z_{\rm sp}))}{(z_{\rm sp}-1)^2\,\sqrt{2\pi s\,g''(z_{\rm sp})}}\,(1+O(s^{-1})) .
\end{equation}
We next explain why the standard saddle point method does not work for the heavy-traffic scaling considered in this chapter.
Since we operate in \eqref{e11},
$\vartheta\mu\to 1$ as $s\to \infty$, and
\begin{align} \label{e24}
z_{\rm sp}-1&=\frac{\gamma}{a_2\,s^\eta}+O(s^{-2\eta}) ,\\
\label{e25}
g(z_{\rm sp})&=\frac{-\gamma^2}{2a_2s^{2\eta}}+O(s^{-3\eta}) ,\\
\label{e26}
g''(z_{\rm sp})&=a_2+O(s^{-\eta}) ,
\end{align}
where
\begin{equation} \label{e27}
a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) .
\end{equation}
Hence, $\exp(sg(z))$ near $z=z_{\rm sp}$ is (as $s\to \infty$):
vanishingly small when $\eta\in(0,1/2)$,
bounded away from 1, but non-negligible when $\eta=1/2$,
and tending to 1 when $\eta\in(1/2,\infty)$.
Furthermore, $(z-1)^{-1}$ in \eqref{e18} is unbounded near $z=z_{\rm sp}$ as $s\to \infty$. Therefore, an adaptation of the standard saddle point method is required, and the resulting asymptotic form of $\mu_Q$ will deviate significantly from the standard case (\ref{e23}). In particular, since $z_{\rm sp}\to 1$, this asymptotic form will contain information from $X(z)$ at $z=1$, rather than at a point away from 1 as is the case in (\ref{e23}).
The required adaptation of the saddle point method is modeled after a device developed in \cite[Sec.~5.12]{debruijn}. We use a substitution $z=z(v)$ in (\ref{e18}) with real $v$ and $z(0)=z_{\rm sp}$ such that for sufficiently small $v$,
\begin{equation} \label{e29}
g(z(v))=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) .
\end{equation}
This is feasible, since
\begin{equation} \label{e30}
g(z)=g(z_{\rm sp})+\tfrac12\,g''(z_{\rm sp})(z-z_{\rm sp})^2\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)
\end{equation}
with $g''(z_{\rm sp})$ positive and bounded away from 0 as $s\to \infty$. Hence, $z(v)$ can be found for small $v$ by inverting the equation
\begin{equation} \label{e31}
(z-z_{\rm sp})\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)^{1/2}=iv .
\end{equation}
By Lagrange's inversion theorem \cite{debruijn}, there is a $\delta>0$ (independent of $s$) such that
\begin{equation} \label{e32}
z(v)=z_{\rm sp}+iv+\sum_{k=2}^{\infty}\,c_k(iv)^k ,~~~~~~|v|<\delta ,
\end{equation}
with real coefficients $c_k$ (since $g(z)$ is real for real $z$) and
\begin{equation} \label{e33}
c_2={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})} .
\end{equation}
Thus
\begin{equation} \label{e34}
z(v)=z_{\rm sp}+iv-c_2\,v^2+O(v^3) ,~~~~~~|v|\leq\tfrac12\,\delta ,
\end{equation}
where the order term holds uniformly in $s$. The uniformity statement follows from an inspection of the usual argument
by which Lagrange's theorem is proved, noting that the inversion in \eqref{e29} with $g$ as in \eqref{e15} is considered for $\vartheta\to 1/\mu$, $z_{\rm sp}\to 1$ with radius
of convergence $r$ away from $1$.
By (\ref{e12}) we can restrict the integration in (\ref{e18}) to a fixed but arbitrarily small subset of $|z|=z_{\rm sp}$ near $z=z_{\rm sp}$, at the expense of an exponentially small error. Furthermore, by Cauchy's theorem and again at the expense of an exponentially small error, the integration path can be deformed in accordance with the transformation in (\ref{e29})--(\ref{e34}). Set
\begin{equation} \label{e35}
q(v)=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp})
\end{equation}
and note that from \eqref{e29},
\begin{equation*}
g'(z(v))\,z'(v)={-}v\,g''(z_{\rm sp}) .
\end{equation*}
Then substituting $z=z(v)$ in (\ref{e18}), $\mu_Q$ is given with exponentially small error by
\begin{equation*}
\frac{s}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{g'(z(v))}{z(v)-1}~\frac{\exp(s\,g(z(v)))}{1-\exp(s\,g(z(v)))}z'(v)\,{\rm d} v,
\end{equation*}
which gives the following result.
\begin{lemma} \label{lemma2} The mean stationary congestion level is given with exponentially small error by
\begin{equation} \label{e37}
\mu_Q =~\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v}{z(v)-1}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v .
\end{equation}
\end{lemma}
In a similar fashion we get that $\mathbb{P}(Q=0)$ and $\sigma_Q^2$, see (\ref{e6}) and (\ref{e8}), are given, both with exponentially small error, by
\begin{equation} \label{e39}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,v\,{\rm ln}\Big(\frac{z(v)}{z(v)-1}\Big)\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v
\end{equation}
and
\begin{equation} \label{e38}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v,
\end{equation}
respectively.
\section{Heavy-traffic limits for the mean congestion level} \label{sec3}
In this section we apply the non-standard saddle point method explained in Section \ref{spSec} to the Pollaczek integral representation for the mean stationary congestion level $\mu_Q$. In Section \ref{subsec3.1} we first derive an integral representation for the leading order behavior of $\mu_Q$ with a relative error of order $O(s^{-1})$, which serves as a heavy-traffic approximation in the regime $\rho_s=1-\gamma/s^\eta$ with $\eta>0$. We also consider separately the cases of moderate heavy traffic ($\eta\in(0,1/2)$) and extreme heavy traffic ($\eta\in(1/2,\infty)$), for which the integral representation leads to vastly different alternative expressions. We find that $\mu_Q\to 0$ more rapidly than any power of $1/s$ when $\eta\in(0,1/2)$. When $\eta\geq 1/2$ the saddle point method yields an integral representation with relative error $O(s^{-\min(1,\eta)})$.
In Section \ref{subsec3.2} we specialize this general result to the CLT case $\eta=1/2$, and make a connection with existing results.
\subsection{Leading order behavior in integral form} \label{subsec3.1}
\begin{theorem}\label{mainthm}
The mean stationary congestion level is given by
\begin{equation} \label{e4.1}
\mu_Q=\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\int_0^{\infty}\,\frac{t^2}{d^2(s)+t^2}~\frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,{\rm d} t\,\left(1+O({s^{{-}\min(1,\eta)}})\right)
\end{equation}
with $
d^2(s) = s^{1-2\eta}\gamma^2\mu/(2\sigma^2)$.
\end{theorem}
\begin{proof}
According to Lemma \ref{lemma2}, $\mu_Q$ is given with exponentially small error by (\ref{e37}) with $q(v)$ given in (\ref{e35}). Since $z({-}v)=z^{\ast}(v)$ for real $v$, we have
\begin{align}
\frac{v}{z(v)-1}+\frac{-v}{z({-}v)-1} &= {-}2iv\,\frac{{\rm Im}(z(v))}{|z(v)-1|^2}\nonumber\\
&=\frac{-2iv^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} \nonumber \\
&=\frac{-2iv^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2},
\label{e40}
\end{align}
for ${-}\tfrac{1}{2} \delta \leq v \leq \tfrac{1}{2} \delta$.
where (\ref{e34}) and $c_k\in\mathbb{R}$ have been used. Using (\ref{e40}) in (\ref{e37}) and extending the integration range from $[{-}\tfrac12\delta,\tfrac12\,\delta]$ to $({-}\infty,\infty)$ while using symmetry of $q(v)$, we get that $\mu_Q$ is given with exponentially small error by
\begin{align} \label{e41}
\frac{s\,g''(z_{\rm sp})}{\pi}\,\int_0^{\infty}\,\frac{v^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2}\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}{\rm d} v .
\end{align}
With
\begin{equation} \label{e42}
B=\exp(s\,g(z_{\rm sp})) ,~~~~~~\alpha =g''(z_{\rm sp}),
\end{equation}
Equation \eqref{e41} takes the form
\begin{equation} \label{e43}
\frac{s\alpha }{\pi}\,\int_0^{\infty}\,\frac{v^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2} \cdot \frac{B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
Since $(z_{\rm sp} - 1)^2 = (\gamma/a_2)^2s^{{-}2\eta} + O(s^{-4\eta})$, see \eqref{e24}, the integrand in \eqref{e43} in leading order has the form
\begin{equation*}
\frac{B\,v^2\,\exp(-s\,D\,v^2)}{(v^2+C\,s^{-2\eta})(1-B\exp({-}s\,D\,v^2))},
\end{equation*}
and this is reminiscent of the integrand in \cite[Eq.~(5.12.3)]{debruijn} for the case $\kappa=2\eta$. Proceeding as in \cite[Sec.~5.12]{debruijn}, the substitution $v=t\sqrt{{2}/(s\alpha )}$ brings (\ref{e43}) into the form
\begin{equation} \label{e44}
\frac{2}{\pi}\sqrt{\tfrac12 s\alpha }\int_0^{\infty}\frac{t^2(1+O(t^2/s))}{\tfrac12 s\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2} \,\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}{\rm d} t .
\end{equation}
From (\ref{e24})--(\ref{e27}) and (\ref{e42}),
\begin{align}
\frac{2}{\pi}\,\sqrt{\frac{s\alpha }{2}} &= \frac{2}{\pi}\,\sigma_X\,\sqrt{\frac{s}{2\,\mu}}\,(1+O(s^{-\eta})),\label{y45}\\
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &= d^2(s) + O(s^{1-3\eta}),\label{y46}\\
2\,c_2(z_{\rm sp}-1) &= O(s^{-\eta}),\label{y47}\\
s\,g(z_{\rm sp}) &= -d^2(s) + O(s^{1-3\eta}),\label{y48}
\end{align}
where
\begin{equation} \label{y49}
d^2(s) = \frac{b_0^2}{s^{2\eta-1}},\quad b_0^2 := \frac{\gamma^2\mu}{2\,\sigma^2}.
\end{equation}
In the case that $2\eta-1<0$, we have that $\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 \to \infty$ and that
\begin{equation} \label{y50}
B = \exp(s\,g(z_{\rm sp})) = O(\exp({-}b^2s^{1-2\eta}))
\end{equation}
for any $b\in(0,b_0)$. From \eqref{e44} it then follows that $\mu_Q = O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$.
In the case that $2\,\eta-1\geq 0$, we have that $d^2(s)$ is bounded, and using that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, we get
\begin{align*}
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 + t^2-2\,c_2\,(z_{\rm sp}-1)\,t^2
&= d^2(s) + t^2 + O\left(s^{-\eta}\,(d^2(s)+t^2)\right) \nonumber\\
&= \big(d^2(s)+t^2\big)\left(1+O(s^{-\eta})\right).
\end{align*}
Hence, in this case,
\begin{equation}
\frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2} = \frac{t^2}{d^2(s)+t^2}\left(1+O(s^{-\eta})+O(t^2/s)\right).\label{y52}
\end{equation}
Furthermore,
\begin{align*}
1-B\,\exp(-t^2) &= 1-\exp({-}d^2(s)-t^2)\,\left(1+d^2(s)\,O(s^{-\eta})\right)\nonumber\\
&=(1-\exp({-}d^2(s)-t^2))\,\Big(1+\frac{d^2(s)}{\exp(d^2(s)+t^2)-1}O(s^{-\eta})\Big)\nonumber\\
&= (1-\exp({-}d^2(s)-t^2))\,(1+O(s^{-\eta})),
\end{align*}
It follows therefore that
\begin{equation} \label{y56}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)} = \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,(1+O(s^{-\eta})).
\end{equation}
Combining the three items \eqref{y45}, \eqref{y52} and \eqref{y56}, we obtain for \eqref{e44} the result
\begin{equation*}
\frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}} \int_0^{\infty}\frac{t^2}{d^2(s)+t^2} \cdot \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}{\rm d} t
\left(1+O(s^{-\eta})+O(s^{-1})\right),
\end{equation*}
and this gives \eqref{e4.1}.
\end{proof}
Theorem \ref{mainthm} gives the leading-order behavior of $\mu_Q$ as $s\to \infty$ with a relative error of $O(s^{{-}\min(1,\eta)})$. By considering in more detail the integral expressions, we obtain the following result, describing two different heavy-traffic behaviors.
\begin{proposition}\label{prop1}
If $\eta\in(0,1/2)$ the mean congestion level satisfies
\begin{equation*}
\mu_Q=O\left(\exp(-b^2s^{1-2\eta})\right),
\end{equation*}
for any $b\in (0,b_0)$. If $\eta\in[1/2,\infty)$ the mean congestion level is given by
\begin{equation*}
\mu_Q = s^\eta\,\frac{\sigma^2}{2\mu\gamma}\,\left(1+O(s^{\max(1/2-\eta,-1)})\right).
\end{equation*}
\end{proposition}
The first assertion in Proposition \ref{prop1} follows from the observation in \eqref{y50}, together with \eqref{e44}. The second assertion is based on a connection between the integral in Theorem \ref{mainthm} and the Riemann zeta function, which is explained in the next subsection.
\subsection{Classical heavy traffic and the Gaussian random walk}
\label{subsec3.2}
We now build on Theorem \ref{mainthm} to obtain further results for the classical heavy traffic case $\eta=1/2$,
for which we know from \cite[Thm.~3]{Sigman2011b} that the rescaled congestion process converges under \eqref{bb1} to a reflected Gaussian random walk. The latter is defined as
$(S_\beta(k))_{k\geq 0}$ with $S_\beta(0)=0$ and
\begin{equation*}
S_\beta(j)=Y_1+\ldots+Y_j
\end{equation*}
with $Y_1,Y_2,\ldots$ i.i.d.~copies of a normal random variable with mean $-\beta$ and variance 1.
Assume $\beta>0$ (negative drift), and denote the all-time maximum of this random walk by ${M}_\beta$.
Denote by $Q^{(s)}_\infty$ the stationary congestion level for a fixed $s$ (that arises from taking
$j\to \infty$ in \eqref{lind}), and remember that we have assumed $\vartheta=n/s$ fixed.
Then, using $\rho_s=1-\gamma/\sqrt{s}$, with
\begin{equation}\label{gammachoice}
\gamma=\frac{\beta\sigma}{\mu\sqrt{\vartheta}},
\end{equation}
the spatially-scaled stationary congestion levels reach the limit
$Q^{(s)}_\infty/(\sigma\sqrt{n}) {\;\buildrel{d}\over\Rightarrow\;} {M}_\beta$ as $s,n\to \infty$ (see \cite{Jelenkovic2004,Sigman2011a,Sigman2011b}). From \cite[Thm.~4]{Sigman2011b} we then know that under the standard heavy-traffic scaling \eqref{bb1}
\begin{equation}
\frac{\mathbb{E}[Q^{(s)}_\infty]}{\sigma\sqrt{n}}\to \mathbb{E}[{M}_\beta], \quad {\rm as} \ s,n\to \infty,
\end{equation}
from which it follows that
\begin{equation} \label{e48}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}[M_\beta].
\end{equation}
The random variable ${M}_\beta$ was studied in \cite{Chang1997,Janssen2006}. In particular, \cite[Thm.~2]{Janssen2006} yields, for $\beta<2\sqrt{\pi}$,
\begin{equation*}
\mathbb{E}[{M}_\beta]= \frac{1}{2\beta}+\frac{\zeta(1/2)}{\sqrt{2\pi}}+\frac{\beta}{4}+\frac{\beta^2}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta(-1/2-r)}{r!(2r+1)(2r+2)}\left(\frac{-\beta^2}{2 }\right)^r,
\end{equation*}
where $\zeta$ denotes the Riemann zeta function, which is defined as, see (1.26).
Hence, for small values of $\beta$,
\begin{equation} \label{estimate}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}[M_\beta] \approx \frac{\sigma\sqrt{n}}{2\beta} = \sqrt{s}\,\frac{\sigma^2}{2\mu\gamma}.
\end{equation}
We will now show how the approximation \eqref{estimate} follows from Theorem \ref{mainthm}, and also how similar steps give rise to Proposition \ref{prop1}.
Consider the integral
\begin{equation} \label{e49}
G_0(b)=G_1(b)-G_2(b)=\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t ,
\end{equation}
where $b>0$ and
\begin{equation} \label{e50}
G_1(b)=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t\,,~~~~G_2(b)=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}{\rm d} t .
\end{equation}
We have, as in \cite[Sec.~2]{Janssen2006},
\begin{align}
G_1(b) & = \sum_{k=0}^{\infty}\:\int_0^{\infty}\,\exp({-}(k+1)(b^2+t^2))\,{\rm d} t \nonumber \\
& = \frac{\sqrt{\pi}}{2}\,\sum_{k=0}^{\infty}\,\frac{{\rm e}^{-(k+1)b^2}}{\sqrt{k+1}} = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},1/2,1) \nonumber \\
& = \frac{\pi}{2b}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta(\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} , \label{e51}
\end{align}
where the last identity holds when $0<b<\sqrt{2\pi}$ and $\Phi(z,s,v)$ is Lerch's transcendent, which is defined as, see \cite[Eq.~25.14.1]{NIST},
\begin{equation*}
\Phi(z,s,v) = \sum_{n=0}^\infty \frac{z^n}{(v+n)^s}, \qquad \text{for }v\neq 0,{-}1,{-}2,\ldots, \ |z|<1; \, \Re s > 1,\ |z|=1.
\end{equation*}
As to $G_2(b)$, we make a connection with the complementary error function
\begin{equation*}
{\rm erfc}(z)=\frac{2}{\sqrt{\pi}}\,\int_z^{\infty}\,{\rm e}^{-t^2}\,{\rm d} t=\frac{2}{\pi}\,{\rm e}^{-z^2}\,\int_0^{\infty}\,\frac{{\rm e}^{-z^2t^2}}{1+t^2}{\rm d} t ,
\end{equation*}
see \cite[Secs.~7.2 and 7.7.1]{NIST}. We thus compute
\begin{align} \label{e53}
G_2(b) & = \sum_{k=0}^{\infty}\,{\rm e}^{-(k+1)b^2}\,\int_0^{\infty}\,\frac{b^2}{b^2+t^2}\,{\rm e}^{-(k+1)t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[Eq.~(4.3) \& (4.23)]{Janssen2006},
\begin{equation} \label{e54}
\sum_{n=1}^{\infty}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^{\infty}\,{\rm e}^{-x^2/2}\,dx= \frac{1}{2\beta^2}-\frac14-\frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^{\infty} \frac{\zeta({-}1/2-r)({-}1/2)^r} {r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
in which $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e54}), we get
\begin{equation} \label{e55}
G_2(b)=\frac{\pi}{4b}-\frac{\pi}{4}\,b-\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results in (\ref{e51}) and (\ref{e55}) can be combined, as in \cite[Sec.~\ref{sec4}]{Janssen2006}, and this yields
\begin{equation} \label{e56}
G_0(b)=\frac{\pi}{4b}+\frac{\pi}{4}\,b+\frac{\sqrt{\pi}}{2}\,\zeta(1/2)+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Using (\ref{e56}) in (\ref{e48}), we find that the leading order behavior of $\mu_Q$ is given as
\begin{equation} \label{e57}
\sigma_X\,\sqrt{\dfrac{s}{2\mu}}\,\left[\frac{1}{2b_0}+\frac{b_0}{2}+\frac{\zeta(1/2)}{\sqrt{\pi}}+\frac{2}{\sqrt{\pi}}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r b_0^{2r+2}} {r!\,(2r+1)(2r+2)}\right]
\end{equation}
with relative error of $O(s^{-1/2})$ in which $b_0$ is given by \eqref{y49}. The expression (\ref{e57}) is exactly equal to the right-hand side of \cite[Eq.~(4.25)]{Janssen2006} times $\sqrt{s}$ when we take there $\sigma=\mu=1$ and $\beta=b_0\,\sqrt{2}$.
Notice that, with $\gamma$ as in \eqref{gammachoice},
\begin{equation*}
\sigma\,\sqrt{\dfrac{s}{2\mu}}\frac{1}{2b_0}=\frac{\sigma\sqrt{n}}{2\beta},
\end{equation*}
which confirms the approximation \eqref{estimate}.
According to Theorem \ref{mainthm}, we have for $\eta\geq 1/2$,
\begin{equation*}
\mu_Q = \frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}}G_0(d(s))\,\left(1+O(s^{{-}\min(1,\eta)})\right).
\end{equation*}
When $\eta=1/2$, so that $d(s) = b_0$ is independent of $s$, the series representation for $G_0$ in \eqref{e56} can be used, as long as $b_0\in(0,\sqrt{2\pi})$. When $\eta>1/2$, we have that $d(s) = b_0/s^{\eta-1/2}\to 0$ as $s\to \infty$, and so this series representation can be used when $s$ is large enough. We then have from \eqref{e56} and $b_0^2 = \gamma^2\mu/2\,\sigma^2$, while replacing the whole series at the right-hand side by $O(b^2)$, for $\mu_Q$ the leading order behavior
\begin{equation} \label{y62}
s^\eta\left[\frac{\sigma^2}{2\,\gamma\,\mu}+\frac{\sigma\,\zeta(1/2)}{\sqrt{2\,\pi\,\mu}}\,\frac{1}{s^{\eta-1/2}}+\frac{1}{4}\,\gamma\,\frac{1}{s^{2\eta-1}}+O(s^{3/2-3\eta})\right]
\end{equation}
with relative error $O(s^{{-}\min(1,\eta)})$. Retaining the constant term $\sigma^2/(2\gamma\mu)$ and estimating the other terms between the brackets in \eqref{y62} as $O(s^{1/2-\eta})$, we get Proposition \ref{prop1}.
\section{More heavy-traffic results}\label{more}
In this section we apply the non-standard saddle point method to obtain several more heavy-traffic results. In Section \ref{subsec3.3} we derive refined heavy-traffic approximations for the mean congestion level by considering higher-order correction terms. In Section \ref{sec4} we derive the leading heavy-traffic behavior for the variance of the stationary congestion level, and in Section \ref{sec5} for the empty-system probability. To keep the developments tractable, we restrict Section \ref{subsec3.3} to $\eta=1/2$, and Section \ref{sec4} and Section \ref{sec5} to $\eta\in(0,1]$, although the same technique will work for all values $\eta>0$.
\subsection{Correction term for the mean congestion level for $\eta = 1/2$} \label{subsec3.3}
Our saddle point method not only establishes the leading-order heavy-traffic approximations, but also allows to derive refinements to these approximations. In this section we demonstrate how this works for the mean congestion level in the case $\eta=1/2$.
To obtain a refinement or correction term from (\ref{e44}), we must be more precise about the $O(s^{{-}\eta})$ terms that occur in the approximations in Section \ref{subsec3.1} for $\frac12\,s\,\alpha (z_{\rm sp}-1)^2$, $B$ and $\sqrt{s\,\alpha /2}$. When higher-order corrections are required, we should include higher-order terms in the approximations of these quantities, and be more specific about the $O(t^2/s)$ and $O(t^4/s)$ in the integrand in (\ref{e44}).
Let $g^{(i)}, \ i=1,2,...$~denote the $i^{\rm th}$ derivative of $g$ and define, see \eqref{e10} and (\ref{e15}) with $\vartheta=(1-\gamma/s^\eta)\,\mu^{-1}$,
\begin{equation*}
a_i=g^{(i)}(1);~~~~~~g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}\,\tilde X(z) .
\end{equation*}
Dropping the $X$ from $\mu$ and $\sigma^2$ for brevity, we have
\begin{equation*}
a_1={-}\,\frac{\gamma}{s^\eta} ,~~~~~~a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big),
\end{equation*}
\begin{equation*}
a_3={-}2+\Big(1-\frac{\gamma}{s^\eta}\Big)\Big(\frac{\tilde X'''(1)}{\tilde X'(1)}-3\tilde X''(1)+2(\tilde X'(1))^2\Big) .
\end{equation*}
For the purpose of finding a first-order correction term, we note that
\begin{align*}
\alpha &=g''(z_{\rm sp})=a_2+(z_{\rm sp}-1)\,a_3+O(s^{-1}) ,\\
z_{\rm sp}-1&={-}\,\frac{a_1}{a_2}-\frac{a_3}{2a_2}\,\Big(\frac{a_1}{a_2}\Big)^2+O(s^{-3/2}) ,\\
c_2&={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})}={-}\,\frac{a_3}{6a_2}+O(s^{-1/2}) ,\\
g(z_{\rm sp})&={-}\,\frac{a_1^2}{2a_2}-\frac{a_3}{6a_2^3}\,a_1^3+O(s^{-2}) .
\end{align*}
This gives rise to
\begin{align} \label{e65}
\sqrt{\tfrac12\,s\,\alpha }&=\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}+O(s^{-1})\Big) ,\\
\tfrac12\,s\,\alpha (z_{\rm sp}-1)^2&=\frac{\gamma^2\,\mu}{2\sigma^2}+\frac{C_2}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e67}
2c_2(z_{\rm sp}-1)&=\frac{C_3}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e68}
B=\exp(s\,g(z_{\rm sp}))&=\exp\Big({-}\,\frac{\gamma^2\,\mu}{2\sigma^2}\Big)\Big(1+\frac{C_4}{\sqrt{s}}+O(s^{-1})\Big) ,
\end{align}
with explicitly computable constants $C_1$, $C_2$, $C_3$, $C_4$. Remembering that $b_0^2=\gamma^2\mu/2\sigma^2$, see \eqref{y49}, we then get with errors of order $1/s$
\begin{align}
& \frac{t^2(1+O(t^2/s))}{\frac12\,s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)\,t^2} \nonumber \\
& \qquad \qquad =~\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+ b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big) ,\label{e69}
\end{align}
and
\begin{equation} \label{e70}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}=\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2} .
\end{equation}
Using (\ref{e65}), (\ref{e69}) and (\ref{e70}) in (\ref{e44}) we get with an absolute error of order $1/\sqrt{s}$
\begin{align} \label{e71}
\mu_Q & =\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}\Big)\nonumber \\
& \qquad\qquad \cdot \int_0^{\infty}\,\Big(\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big)\Big) \nonumber \\
& \qquad\qquad\qquad \cdot~\Big(\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\Big){\rm d} t \nonumber \\
& =\frac{2\sigma}{\pi}\,\sqrt{\dfrac{s}{2\mu}}\,G_0(b_0)\nonumber\\
& \qquad\qquad + ~\frac{2\sigma}{\pi}\,\sqrt{\dfrac{1}{2\mu}}\,\big((C_1+C_3)\,G_0(b_0)-(C_2+b_0^2\,C_3)\,G_3(b_0)+C_4\,G_4(b_0)\big) ,
\end{align}
where $G_0$ is as in (\ref{e49}), and
\begin{align} \label{e72}
G_3(b_0)&=\int_0^{\infty}\,\frac{t^2}{(b_0^2+t^2)^2}~\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,{\rm d} t ,\\
\label{e73}
G_4(b_0)&=\int_0^{\infty}\,\frac{t^2}{b_0^2+t^2}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\,{\rm d} t .
\end{align}
We shall express the integrals in (\ref{e72}) and (\ref{e73}) in terms of $\zeta$-functions. By partial integration
\begin{align} \label{e74}
G_3(b) & = \frac12\,\int_0^{\infty}\,\frac{1}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,\,{\rm d} t \nonumber \\
&\qquad\qquad -~\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t \nonumber \\
& = \frac{1}{2b^2}\,G_2(b)-G_4(b) ,
\end{align}
see (\ref{e49}) and (\ref{e73}). Since $G_2(b)$ is expressed in terms of $\zeta$-functions in (\ref{e55}), it is sufficient to consider $G_4(b)$.
As to $G_4(b)$,
\begin{equation*}
G_4(b)=G_5(b)-G_6(b) ,
\end{equation*}
where
\begin{align*}
G_5(b)&=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t ,\\
G_6(b)&=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t .
\end{align*}
We have, compare \eqref{e51},
\begin{align} \label{e78}
G_5(b) & = \sum_{k=0}^{\infty}\,(k+1)\,\int_0^{\infty}\,{\rm e}^{-(k+1)(b^2+t^2)}\,{\rm d} t \nonumber \\[3.5mm]
& = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},{-}\tfrac12,1) = \frac{\pi}{4b^3}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta({-}\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
the last identity being valid when $0<b<\sqrt{2\pi}$. Next we have, compare (\ref{e53}),
\begin{align*}
G_6(b) & = \sum_{k=0}^{\infty}\,(k+1)\,b^2\,\int_0^{\infty}\,\frac{\exp({-}(k+1)(b^2+t^2))}{b^2+t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,(k+1)\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align*}
From \cite[Eq.~(5.4) \& (5.21)]{Janssen2006} we have
\begin{equation} \label{e80}
\sum_{n=1}^{\infty}\frac{n}{\sqrt{2\pi}}\int_{\beta\sqrt{n}}^{\infty}{\rm e}^{-x^2/2}\,dx = \frac{3}{4\beta^4}-\frac{1}{24}-\frac{1}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1/2)^r}{r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
when $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in (\ref{e80}), we get
\begin{equation} \label{e81}
G_6(b)=\frac{3\pi}{16b^2}-\frac{\pi b}{24}-\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r}{r!\,(2r+1)}\,b^{2r+2}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results (\ref{e78}) and (\ref{e81}) can be combined, as in \cite[Sec.~5]{Janssen2006} and this yields
\begin{equation} \label{e82}
G_4(b)=\frac{\pi}{16b^3}+\frac{\pi b}{24}+\tfrac12\,\zeta({-}1/2)\,\sqrt{\pi}+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
{\color{blue}
Finally, we can rewrite
\begin{align}
\frac{1}{2b^2}G_2(b) &= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \frac{\sqrt{\pi}}{2} \sum_{r=0}^\infty \frac{\zeta({-}1/2-r)(1-)^rb^{2r}}{r!(2r+1)} \nonumber\\
&= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \frac{\sqrt{\pi}}{2} \sum_{r=-1}^\infty \frac{\zeta({-}3/2-r)(-1)^{r+1}b^{2r+2}}{(r+1)!(2r+3)} \nonumber \\
&= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \tfrac{1}{2}\zeta(-1/2)\sqrt{\pi}
+ \sqrt{\pi} \sum_{r=0}^\infty \frac{\zeta(-3/2-r)(-1)^r b^{2r+2}}{r!\,(2r+2)(2r+3)}
\label{e82a}
\end{align}
and use \eqref{e82} and \eqref{e82a} in \eqref{e74}, by which we obtain for $0<b<\sqrt{2\pi} $,
\begin{align}
G_3(b) &=
\frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} \nonumber \\
&\qquad +\sqrt{\pi} \sum_{r=0}^\infty \frac{ \zeta(-3/2-r)(-1)^rb^{2r+2}}{r!\,(2r+2)} \Big[ \frac{1}{2r+3}-\frac{1}{2r+1}\Big]\nonumber\\
&= \frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} -~2\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)(2r+3)}.\label{e83}
\end{align} }
The right-hand side of (\ref{e83}) equals the right-hand side of \cite[Eq.~(2.3)]{Janssen2006} multiplied by ${\pi}/{(2b)}$ with $\beta=b\,\sqrt{2}$.
\subsection{Variance of the congestion level}\label{sec4}
We have from (\ref{e38}) in Section \ref{sec1}, using the same approach and notation as in Section \ref{subsec3.1} for $\mu_Q$, that $\sigma_Q^2$ is given with exponentially small error by
\begin{equation} \label{e84}
\frac{-s\,\alpha }{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v,
\end{equation}
with $B$ and $\alpha $ given in (\ref{e42}). From $z({-}v)=z^{\ast}(v)$ for real $v$ we now compute
\begin{equation*}
\frac{z(v)}{(z(v)-1)^2}-\frac{z({-}v)}{(z({-}v)-1)^2}={-}2i\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,{\rm Im}(z(v)) ,
\end{equation*}
and so \eqref{e84} becomes
\begin{equation} \label{e86}
\frac{s\alpha }{\pi}\,\int_0^{\frac12\delta}\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,v\,{\rm Im}(z(v))\,\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
From
\begin{equation*}
{\rm Im}(z(v))=v+O(v^3) ,\qquad |z(v)|^2-1=z_{\rm sp}^2-1+O(v^2) ,
\end{equation*}
we get for the expression in \eqref{e86}
\begin{equation} \label{y70}
\frac{s\alpha }{\pi}\,\int_0^{\frac{1}{2}\delta}\,\frac{v^2\,(z_{\rm sp}^2-1+O(v^2))(1+O(v^2))}{((z_{\rm sp}-1)^2+v^2 + O((z_{\rm sp}-1)\,v^2)+O(v^4))^2}
\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v.
\end{equation}
When $2\eta-1<0$, we have as for the case of $\mu_Q$ in Section \ref{subsec3.1} that the whole expression in \eqref{y70} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. When $2\eta-1\geq 0$, we get as in the case of $\mu_Q$ after substitution $v = t\sqrt{{2}/{(s\,\alpha })}$ for the expression in \eqref{y70}
\begin{equation*}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}\,\int_0^\infty\frac{t^2\,(z_{\rm sp}^2-1+O(t^2/s))(1+O(t^2/s))}{(d^2(s)+t^2)^2\,(1+O(1/s^{\eta})+O(t^2/s))}~\frac{B\,{\rm e}^{{-}t^2}}{1-B\,{\rm e}^{{-}t^2}}{\rm d} t.
\end{equation*}
When $2\eta-1\geq 0$, the leading order behavior of $\sigma_Q^2$ depends crucially on the factor $z_{\rm sp}^2-1+O(t^2/s)$, where
\begin{equation*}
z_{\rm sp}^2-1 = \frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\left(1+O(s^{-\eta})\right)
\end{equation*}
is dominant when $\eta<1$, while the $O(t^2/s)$ is dominant when $\eta>1$. In the case that $\eta\in(1/2,1)$, we get for the leading order behavior of $\sigma_Q^2$
\begin{align*}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}& \,\frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\int_0^\infty\frac{t^2}{(d^2(s)+t^2)^2}\cdot~\frac{{\rm e}^{{-}d^2(s)-t^2}}{1-{\rm e}^{{-}d^2(s)-t^2}}{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{\gamma\,\sigma}{\pi}\,\Big(\frac{2}{\mu}\Big)^{1/2}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right),
\end{align*}
where \eqref{e26}, \eqref{e27} and \eqref{e42} have been used for $\alpha = g''(z_{\rm sp})$ and where $G_3$ is given in \eqref{e72}.
When we insert the expansion \eqref{e83} for $G_3(b)$, with the whole series on the second line being $O(b^2)$, we get the leading order behavior of $\sigma_Q^2$ as
\begin{align}
s^{2\eta}\,\Big( \frac{\sigma^4}{4\,\gamma^2\mu^2}- \frac{\sigma^2}{4\,\mu}&\,\frac{1}{s^{2\eta-1}} - \Big(\frac{2\,\sigma^2}{\pi\,\mu}\Big)^{1/2}\,\frac{\gamma\,\zeta(-1/2)}{s^{3\eta-3/2}}\nonumber\\
& - \frac{\gamma^2}{24\,s^{5\eta-5/2}}+O(s^{1-4\eta})\Big)\,\left(1+O(s^{\eta-1})\right)\nonumber \\
&\ = s^{2\eta}\,\frac{\sigma^4}{4\,\gamma^2\,\mu^2}\,\Big(1+O(s^{\max(1-2\eta,\eta-1)})\Big)\label{y74}
\end{align}
when $\eta\in(1/2,1)$. For the case $\eta=1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{align}
\frac{\sigma^2 s}{\mu}\left[ \frac{1}{8\,b_0^2} - \frac{1}{4}-\frac{1}{12}\,b_0^2 - \frac{2\,\zeta(-1/2)}{\sqrt{\pi}}\,b_0- \frac{4}{\sqrt{\pi}}\,\sum_{r=0}^\infty \frac{\zeta(-3/2-r)\,(-1)^r\,b_0^{2r+3}}{r!\,(2r+1)\,(2r+2)\,(2r+3)} \right]\label{y75}
\end{align}
with relative error $O(s^{-1/2})$. The expression between brackets in \eqref{y75} coincides with the right-hand side of \cite{Janssen2006}, (2.3) with $\beta = b_0\,\sqrt{2}$.
This leads to the following two results.
\begin{theorem} \label{varthm}
For $\eta\in[1/2,1)$,
\begin{equation*}
\sigma_Q^2 = \frac{\gamma\,\sigma_X}{\pi}\,\sqrt{\frac{2}{\mu}}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right)
\end{equation*}
with $G_3$ given in \eqref{e72}.
\end{theorem}
\begin{proposition}\label{varprop}
For $\eta\in(0,1/2)$, and for all $b<b_0$,
\begin{equation*}
\sigma_Q^2 = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation*}
For $\eta = 1/2$, $\sigma_Q^2$ equals expression \eqref{y75} with relative error $O(s^{-1/2})$. For $\eta\in(1/2,1)$ and $b_0\in(0,\sqrt{2\pi})$, $\sigma_Q^2$ has the form in \eqref{y74}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level
with $\eta=1/2$, it is possible to give a correction term which involves now integrals and series with $\zeta$-functions as considered in \cite[Secs.~4-5]{Janssen2007}.
\subsection{The empty-system probability} \label{sec5}
We have from (\ref{e6}) by proceeding as in (\ref{e13})--(\ref{e17}) that
\begin{align} \label{e100}
{\rm ln}\,[\mathbb{P}(Q=0)] & = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{z}{z-1}\Big)\,\frac{g'(z)\,\exp(s\,g(z))}{1-\exp(s\,g(z))}\,{\rm d} z \nonumber \\[3.5mm]
& = \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{z(z-1)}\,{\rm ln}\left(1-\exp(s\,g(z))\right)\,{\rm d} z ,
\end{align}
where in the last step we used partial integration (noting that ${\rm Re}\,[g(z)]<0$ on $|z|=1+\varepsilon$). Then, as in Section \ref{sec1} for $\mu_Q$, the last integral in (\ref{e100}) is, with exponentially small error, given by
\begin{equation} \label{e101}
\frac{1}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{z'(v)}{z(v)(z(v)-1)}\,{\rm ln}\left(1-B\,\exp\big(-\tfrac{1}{2} s\,\alpha v^2\big)\right)\,{\rm d} v .
\end{equation}
Now for $v\geq0$ from $z({-}v)=z^{\ast}(v)$, $z'({-}v)={-}(z'(v))^{\ast}$,
\begin{align*}
& \frac{z'(v)}{z(v)(z(v)-1)}+\frac{z'({-}v)}{z({-}v)(z({-}v)-1)}=2i\,{\rm Im}\,\Big[\frac{z'(v)}{z(v)(z(v)-1)}\Big] \nonumber \\
& \qquad \qquad =~2i\,{\rm Im}\,\Big[\frac{z'(v)\,z^{\ast}(v)(z^{\ast}(v)-1)}{|z(v)|^2\,|z(v)-1|^2}\Big] \nonumber \\
& \qquad \qquad =~2i\,\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}+O(v^2))((z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4))}\,,
\end{align*}
where we used \eqref{e32} and the fact that $z_{\rm sp}$ and $c_2$ are real with $z_{\rm sp}>1$. Therefore, we get for the expression in \eqref{e101}
\begin{align}
\frac{1}{\pi} &\int_0^{\frac{1}{2}\delta}\frac{1}{z_{\rm sp}{\rm +}O(v^2)}\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)v^2)+O(v^4)} \nonumber \\
& \qquad \qquad \qquad \qquad \cdot {\rm ln}\left(1-B\exp(-\tfrac12 s\,\alpha v^2)\right){\rm d} v.
\label{y77}
\end{align}
In the case that $2\eta-1<0$, we have as earlier that the whole expression in \eqref{y77} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. In the case that $2\eta-1\geq 0$, we substitute $v=t\sqrt{{s}/{(2\,\alpha )}}$, and we get as earlier for the expression \eqref{y77}, assuming also that $\eta<1$,
\begin{align*}
\frac{1}{\pi}&\,\sqrt{s\,\alpha /2}\,\int_0^{\infty}\frac{z_{\rm sp}-1+O(t^2/s)}{(d^2(s)+t^2)\,(1+O(s^{-\eta})+O(t^2/s))}\,{\rm ln}(1-B\,{\rm e}^{-t^2}) \,{\rm d} t\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{\sqrt{s\,\alpha /2} \ (z_{\rm sp}-1)}{d^2(s)+t^2}{\rm ln}(1-B\,{\rm e}^{-t^2})\,{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{d(s)}{d^2(s)+t^2}{\rm ln}(1-{\rm e}^{{-}d^2(s)-t^2})\,{\rm d} t\,\left(1+O(s^{\eta-1})\right).
\end{align*}
Here we also used \eqref{y46} and that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, so that
\begin{equation*}
(\tfrac12\,s\,\alpha )^{1/2}\,(z_{\rm sp}-1) = d(s)\,\left(1+O(s^{-\eta})\right) = d(s)\left(1+O(s^{\eta-1})\right),
\end{equation*}
since $\eta\geq 1/2$.
We have for $b>0$
\begin{align}
\frac{1}{\pi}&\int_0^\infty \frac{b}{b^2+t^2}\,{\rm ln}(1-\exp({-}b^2-t^2)){\rm d} t =-\frac12\sum_{k=0}^{\infty}\,\frac{1}{k+1}\,{\rm erfc}(b\,\sqrt{k+1}) = -F(b\,\sqrt{2}),\label{y80}
\end{align}
where according to \cite[Eq.~(3.3) \& (3.12)]{Janssen2006} for $\beta>0$
\begin{align}
F(\beta) &= \sum_{n=1}^\infty\,\frac{1}{n}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^\infty {\rm e}^{-x^2/2}dx\nonumber\\
&= -{\rm ln}\,\beta - \frac12\,{\rm ln}\,2 - \frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^\infty \frac{\zeta(1/2-r)\,(-1/2)^r\,\beta^{2r+1}}{r!\,(2r+1)},\label{y81}
\end{align}
the last identity being valid for $0<\beta<2\sqrt{\pi}$.
Using \eqref{y81} with $\beta^2 = d^2(s)= b_0^2/s^{2\eta-1}$, with the entire series on the second line being $O(\beta)$, we get the leading order behavior of ${\rm ln}[\mathbb{P}(Q=0)]$ as
\begin{equation} \label{y82}
\Big({-}(\eta-1/2)\,{\rm ln}\,s+{\rm ln}(2\,b_0)+O(s^{1/2-\eta})\Big)\left(1+O(s^{\eta-1})\right)
\end{equation}
when $\eta\in(1/2,1)$. For $\eta = 1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{equation} \label{y83}
{\rm ln}(2\,b_0) + \frac{1}{\sqrt{\pi}}\,\sum_{r=0}^\infty \,\frac{\zeta(1/2-r)\,(-1)^r}{r!\,(2r+1)}\,b_0^{2r+1}
\end{equation}
with relative error $O(s^{-1/2})$. The expression \eqref{y83} coincides with ${\rm ln}[\mathbb{P}(M=0)]$ as given by \cite[Eq.~(2.1)]{Janssen2006} with $\beta = b_0\,\sqrt{2}$. The next two results summarize the above.
\begin{theorem} \label{emptythm}
For $\eta\in(1/2,1)$,
\begin{equation*}
{\rm ln}[\mathbb{P}(Q=0)] = - F\big(d(s)\,\sqrt{2}\big)\left(1+O(s^{\eta-1})\right)
\end{equation*}
with $F$ given by \eqref{y81}.
\end{theorem}
\begin{proposition} \label{emptyprop}
For $\eta\in (0,1/2)$, and for all $b<b_0$,
\begin{equation*}
{\rm ln}[\mathbb{P}(Q=0)] = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation*}
For $\eta=1/2$, ${\rm ln}[\mathbb{P}(Q=0)]$ equals $-F(b_0\,\sqrt{2})$ with a relative error $O(1/\sqrt{s})$. For $\eta\in (1/2,1)$ and $0<b_0<\sqrt{2\pi}$, ${\rm ln}[\mathbb{P}(Q=0)]$ has leading order behavior as in \eqref{y82}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level case
with $\eta=1/2$, it is possible to give a correction term which involves now the integrals in \eqref{y80} and \eqref{e51}.
\section{Numerical examples}\label{numm}
\subsection{Accuracy of the approximations}
In this subsection we present a numerical example that serves to illustrate the accuracy of the derived heavy-traffic approximations. Consider the Poisson case
\begin{equation*}
\tilde X(z)={\rm e}^{z-1},\quad \mu = \sigma^2 = 1.
\end{equation*}
We fix $\mu$ and vary $n$ with the value of $s$, according to
\begin{equation*}
\vartheta = \frac{n}{s} = 1-\frac{\gamma}{s^\eta}
\end{equation*}
for some $\gamma>0$ and $\eta\geq 1/2$. To calculate the exact value of the mean congestion level we use the expression, see \cite{Boudreau1962},
\begin{equation*}
\mu_Q=\frac{\sigma_A^2}{2(s-\mu_A)}-\frac{s-1+\mu_A}{2}+\sum_{k=1}^{s-1}\frac{1}{1-z_k}.
\end{equation*}
Here $z_1,\ldots,z_{s-1}$ are the zeros of $z^s-A(z)$ in $|z|<1$. We apply the method of successive substitution described in \cite{Janssen2005} to obtain accurate numerical approximations for $z_1,...,z_{s-1}$ and consequently $\mu_Q$.
From Theorem \ref{mainthm}, we find that the leading order behavior of $\mu_Q$ is given by
\begin{equation} \label{x18}
\frac{\sqrt{2s}}{\pi}\,G_0\Big(\frac{\gamma}{\sqrt{2}\,s^{\eta-\frac{1}{2}}}\Big).
\end{equation}
In order to find the correction terms, we proceed by setting $\eta = 1/2$. Deriving constants $C_1,C_2,C_3,$ and $C_4$ for our setting and substituting these into \eqref{e71}, we get for $\mu_Q$, with an absolute error of $O(s^{-1/2})$, the approximation
\begin{equation*}
\frac{\sqrt{2\,s}}{\pi}\Big(\Big(1-\frac{\gamma}{3\,\sqrt{s}}\Big)\,G_0(b_0)-\frac{\gamma^3}{3\,\sqrt{s}}\,(\,G_3(b_0)+G_4(b_0))\Big),
\end{equation*}
which by \eqref{e49} and \eqref{e74} reduces to
\begin{equation}\label{x20}
\frac{\sqrt{2\,s}}{\pi}\,G_0(b_0)-\frac{\sqrt{2}\,\gamma}{3\,\pi}\,G_1(b_0).
\end{equation}
\begin{table}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.683 & 0.244 & 0.399 & 0.247 \bigstrut[t] \\
20 & 0.776 & 0.410 & 0.565 & 0.412 \\
50 & 0.858 & 0.739 & 0.893 & 0.741 \\
100 & 0.900 & 1.110 & 1.263 & 1.111 \\
200 & 0.929 & 1.633 & 1.787 & 1.634 \\
500 & 0.955 & 2.672 & 2.825 & 2.673 \\
1000 & 0.968 & 3.843 & 3.996 & 3.843 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 1$.}\label{tab:poisson1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.968 & 13.707 & 14.046 &13.732 \bigstrut[t] \\
20 & 0.977 & 19.533 & 19.865 &19.551\\
50 & 0.985 & 31.084 & 31.409 &31.095\\
100 & 0.990 & 44.097 & 44.419 &44.106\\
200 & 0.992 & 62.499 & 62.819 &62.505\\
500 & 0.995 & 99.008 & 99.325 &99.011\\
1000 & 0.996 & 140.152 & 140.468 & 140.154 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 0.1$.}\label{tab:poisson2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\eta=0.6$} & \multicolumn{2}{c|}{$\eta=0.75$} & \multicolumn{2}{c|}{$\eta=0.9$} \bigstrut\\
\hline
$s$ & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} \bigstrut\\
\hline
10 & 17.781 & 18.125 & 25.970 & 26.318 & 37.553 & 37.905 \bigstrut[t] \\
20 & 27.309 & 27.647 & 44.391 & 44.734 & 71.195 & 71.541 \\
50 & 47.948 & 48.281 & 89.623 & 89.961 & 164.637 & 164.978 \\
100 & 73.245 & 73.574 & 152.031 & 152.367 & 309.353 & 309.692 \\
200 & 111.752 & 112.079 & 257.435 & 257.769 & 580.170 & 580.507 \\
500 & 195.082 & 195.409 & 515.443 & 515.776 & 1329.581 & 1329.917 \\
1000 & 297.122 & 297.448 & 870.524 & 870.857 & 2487.227 & 2487.562 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma=0.1$ and several values of $\eta$.}\label{tab:poisson3}
\end{table}
\noindent Numerical results for $\eta=1/2$ and various values of $s$ are given in Table \ref{tab:poisson1} and \ref{tab:poisson2}, for $ \gamma = 1$ and $\gamma = 0.1$, respectively.
We note that for small $s$ the leading order approximation is still off by a significant amount, while the refinement only shows an error in the second decimal for $\gamma = 0.1$. This seems to justify the use of the correction term.
In Table \ref{tab:poisson3} we compare the approximation \eqref{x18} against the exact value of $\mu_Q$ for three values of $\eta\geq 1/2$ to assess the influence of $\eta$. Clearly, the leading order approximation is relatively accurate for all three scenarios. As expected, the mean congestion increases along with $\eta$, since utilization approaches 1 more rapidly in this case.
\subsection{Connection to other queueing models}\label{subsec62}
As argued in the introduction, we believe that the heavy-traffic behavior for the discrete model in this chapter will up to leading order be universal for a wide range of other models (when subjected to the same heavy-traffic regime \eqref{bb}). We shall now substantiate this for many-server systems, for which under \eqref{bb}, it turns out that the mean congestion is $O(s^\eta)$. We compare the mean congestion level in our discrete queue with that in the multi-server systems $M/M/s$, $M/D/s$ and Gamma/Gamma/$s$, all with unit mean service time and occupation rate $1-\gamma/s^\eta$.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 2.5,
xmax = 6.5,
ymin = 0,
ymax = 7.2,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.4,0.2)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,mark = o,mark options={scale=1.25}] table[x=log_s,y=mms] {tikz/novel_figure1.txt};
\addplot[thick, mark=triangle,dashed,mark options={scale=1.25,solid}] table[x=log_s,y=mds] {tikz/novel_figure1.txt};
\addplot[thick,mark=square,dotted,mark options={scale=1.25,solid}] table[x=log_s,y=ggs] {tikz/novel_figure1.txt};
\legend{$M/M/s$,$M/D/s$,Gamma/Gamma/$s$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ plotted against $s$ on log scale for 3 queues for $\eta=0.75$.}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 0.2,
xmax = 6.5,
ymin = 0,
ymax = 10,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 0.4,7.35)},anchor = west},
yscale = 0.8,
xscale = 1
]
\addplot[thick,only marks,mark = o,mark options={scale=1.25}] table[x=n01,y=m01] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square,mark options={scale=1.25}] table[x=n025,y=m025] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = *,mark options={scale=1.25}] table[x=n05,y=m05] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square*,mark options={scale=1.25}] table[x=n075,y=m075] {tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = triangle*,mark options={scale=1.25}] table[x=n1,y=m1] {tikz/novel_figure2.txt};
\addplot[dashed] coordinates{ (0,2.25) (7,9.25) };
\addplot[dashed] coordinates{ (0,2.25) (7,7.5) };
\addplot[dashed] coordinates{ (0,2.25) (7,5.75) };
\addplot[dashed] coordinates{ (0,2.25) (7,4) };
\addplot[dashed] coordinates{ (0,2.25) (7,2.95) };
\legend{\ $\eta=0.1$,\ $\eta=0.25$,\ $\eta=0.5$,\ $\eta=0.75$,\ $\eta=1$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ of $M/M/s$ plotted against $s$ on log scale for different values of $\eta$.}
\label{fig2}
\end{figure}
Figure \ref{fig1} shows on logarithmic scale the mean congestion levels for $\gamma=0.1$ and $\eta=0.75$ under the specified scaling for three systems. We also display three lines with slope 0.75 for comparison, which confirms that mean congestion levels are of the order $s^\eta$, also in these multi-server system. Formally establishing this heavy-traffic behavior for these multi-server system is an important open problem and requires other mathematical approaches than the ones taken in this chapter (see the introduction for more details).
Figure \ref{fig2} shows the mean queue length in the $M/M/s$ system for several values of $\eta$, again on logarithmic scale, together with lines with slope $\eta$. For $\eta\geq 1/2$, we see the same $O(s^\eta)$ behavior, similar as for $\mu_Q$ in our discrete model. For $\eta<1/2$ the mean queue length decays, again in agreement with our results for $\mu_Q$. We note that this qualitative behavior of the $M/M/s$ system was also observed by \cite[Thm.~4.1]{maman}, by proving that the mean waiting time in the $M/M/s$ queue under \eqref{bb} is of order $1/s^{1-\eta}$, which by Little's law implies the mean queue length to be of order $s^\eta$.
\chapter{Novel heavy-traffic regimes}
\begin{chapterstart}
In this chapter, we introduce a family of heavy-traffic scalings for a large-scale service system meant to serve jobs coming from a large pool of customers.
The scaling rules are inspired by the classical QED regime discussed in Chapter 1, but lead to a range of different system behaviors that includes the ED, QED and QD regime as special cases.
To determine the scaling limits, we describe the performance measures in terms of Pollaczek integrals and use asymptotic techniques to evaluate these integrals in the large-system limit.
\end{chapterstart}
\begin{flushright}
Based on \\
\textbf{Novel heavy-traffic regimes for large-scale service systems}\\
\textit{Guido Janssen, Johan van Leeuwaarden \& Britt Mathijsen}\\
In \textit{SIAM Journal of Applied Mathematics, 75(2), 787-812 (2015)}
\end{flushright}
\newpage
\section{Introduction \& motivation}
We study the workload process of a system, experiencing stochastic demand and deterministic capacity $s_n$ per period, at equidistant time epochs.
Demand is assumed to be generated by $n$ stochastically identical and independent sources.
Let $A_{i,j}$ denote the workload brought into the system by source $i$ in period $j$, for which $\mathbb{E}[A_{i,j}] =\mu$ and ${\rm Var}\, A_{i,j} = \sigma^2$.
Then the total amount of demand arriving to the system in period $j$ is $A^{(n)}_j=\sum_{i=1}^n A_{i,j}$ with $\mathbb{E}[A^{(n)}_j] = n\mu$ and ${\rm Var}\, A^{(n)}_j = n\sigma^2$.
As explained in Chapter 1, a good capacity sizing rule for achieving economies-of-scale is $s_n = n\mu+\beta\sqrt{n}\sigma$ for some $\beta>0$.
If we denote the system utilization by $\rho_n := n\mu/s_n$, then this dimensioning rule in the bulk service queue with many sources is tantamount to the heavy-traffic scaling
\begin{equation}\label{bb1}
\sqrt{n}(1-\rho_n) \to \gamma = \frac{\beta\sigma}{\mu}, \qquad {\rm as }\ n\to\infty.
\end{equation}
Starting from this setting, we introduce a novel family described in terms of a parameter $\eta$ for which we assume that
\begin{equation}\label{bb}
n^{\eta}(1-\rho_n)\rightarrow \gamma, \quad {\rm as} \ n\to \infty, \ \gamma> 0.
\end{equation}
The parameter $\eta\geq 0$ defines a whole range of possible scaling regimes, including the classical case $\eta=1/2$.
In terms of a capacity sizing rule for systems with many customers, the condition \eqref{bb} is tantamount to $s_n=n\mu+\beta \sigma n^{1-\eta}$.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as {\it moderate} heavy traffic: heavy-traffic conditions in which the full occupancy is reached more slowly, as a function of $n$, than for classical heavy traffic. For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to {\it extreme} heavy traffic due to a relatively small variability hedge. Note that the case $\eta=0$ does not lead to 100\% system utilization when $n\to\infty$.
In this chapter we show that economies-of-scale can be achieved for a large range of $\eta$, although the nature of the benefits obtained by operating on large scale depends on the precise capacity sizing rule (hence the parameter $\eta$). We quantify performance in terms of stationary measures: The mean and variance of the congestion in the system, and the probability of an empty system. For these performance measures we derive heavy-traffic limits under the scalings \eqref{bb} that
are relatively simple functions of only the first two moments of the demand per period. Such parsimonious expressions are useful for quantifying and improving system behavior. The heavy-traffic limits, however, provide also qualitative insight into the system behavior. Our asymptotic analysis shows that mean congestion is $O(n^\eta)$, which implies
that delays experienced by the customers are negligible for all values of $\eta\in [0,1)$, are roughly constant for $\eta=1$, and grow without bound for $\eta>1$. We expect this qualitative behavior to be universal for a wide range of stochastic models to which the regime \eqref{bb} is applied.
We further show the existence of the following trichotomy as $n\to \infty$ under \eqref{bb}: For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, for $\eta\in (1/2,1)$ it converges to $0$, while only for $\eta=1/2$ there is a limiting value in $(0,1)$. Hence, as expected, the system performance deteriorates with $\eta$, in a rather crude way for the empty-system probability, and in only a mild way for mean congestion levels. The regime \eqref{bb} thus presents a range of possible capacity sizing rules that all lead to economies-of-scale, and depending on what is the desired nature of performance for a particular service system, an appropriate $\eta$ can be selected. From the quantitative perspective, our detailed asymptotic analysis leads to more precise asymptotic estimates for the performance measures in heavy traffic, which reveal the exact manner in which congestion is influenced by $\eta$ and $\gamma$.\\
\\
\noindent\textbf{Motivating examples.}
The bulk service queueing model is one of the canonical models in queueing theory, having a wide range of applications in fields like digital communication, wireless networks, road traffic, reservation systems, health care and many more (see \cite{Bruneel1993} and \cite[Chap.~2]{johanthesis} for an overview).
In road traffic, the basic model for congestion at an intersection, known as the fixed-cycle traffic-light queue \cite{Newell1960,Leeuwaarden2006}, is related to our discrete bulk service queue.
Then $s_n$ represents the maximum number of delayed cars in front of a traffic light that can depart during one green period, while $A^{(n)}_j$ is the number of newly arriving cars during a consecutive green and red period.
An example from health care is panel sizing \cite{Zacharias2014}.
Say a general practitioner has a pool of $n$ clients (typically in the order of 2,500~\cite{Green2008}), all of which are potential patients, and together require $A^{(n)}_j$ consults per day.
Further assume that the practitioner can see a maximum number of $s_n$ patients per day.
What is then an appropriate patient panel size $n$, which strikes a reasonable balance between accessing medical care in a timely manner and restricting the time that the practitioner sits idle?
The panel size application is one of many examples of an appointment book, referring to some schedule of appointments for a fixed period, with capacity $s_n$ appointments per period and newly arriving number of appointments $A^{(n)}_j$ per period.
See \cite{Dai2015} for another recent example of an appointment book in a health care setting, again in terms of our bulk service queue, with $A^{(n)}_j$ the new patients per day and $s_n$ the number of available beds.
For all examples above, and many more, our new class of heavy-traffic scalings \eqref{bb} presents capacity sizing rules for which the expected performance can be quantified using the results in this chapter. This will be helpful in dimensioning the systems (How much capacity is needed to achieve a certain target performance?) while exploiting economies-of-scale. For appointment books, our model together with the capacity sizing rules \eqref{bb} is particularly relevant for {\it advanced access} \cite{Green2008}, a scheduling approach in health care designed to reduce delays by offering every patient a same-day appointment, regardless of the urgency of the problem. In that way, patients do not have to wait long for appointments, and practices do not waste capacity by holding appointments in anticipation of urgent cases.\\
\\*
\noindent\textbf{Pollaczek's formula.}
Next to the freedom to model different situations, another advantage of our model is that it is mathematically tractable, in the sense that it can be subjected to powerful mathematical methods from complex and asymptotic analysis. In order to establish the heavy-traffic limits we start from Pollaczek's formula for the transform of the stationary queue length distribution in terms of a contour integral. From this famous transform representation, contour integrals for the empty-system probability and the mean and variance of the congestion immediately follow. Contour integrals are often amenable to asymptotic evaluation (see e.g.~\cite{Cohen1982}), particularly for obtaining classical heavy-traffic asymptotics.
We also subject the contour integral representations to asymptotic evaluation, but not under classical heavy-traffic scaling.
This asymptotic analysis requires a {\it non-standard} saddle point method, tailored to the specific form of the integral expressions that arise under the capacity sizing rule \eqref{bb}. \\
\\*
\noindent
\textbf{Saddle point method.}
In complex analysis, the saddle point method in its standard form is a useful technique to estimate the asymptotic behavior of integrals of the form
\begin{equation}
\label{eq:sp_integral}
I(n) = \int_C h(z)\, {\rm e}^{n f(z)}\, {\rm d} z,
\end{equation}
as $n\to\infty$, where $C$ is a contour in the complex plane, and $f(z)$ and $h(z)$ are functions that are analytic in some neighborhood of $C$.
The main idea behind the saddle point method is that if the integrand in \eqref{eq:sp_integral} exhibits a sharp peak along the contour, then one may naturally expect that a small neighborhood around this peak provides the dominant contribution to the integral.
More specifically, for large values of $n$, the function $f$ and its associated maximum $f(z^*)$ for $z^*\in C$ to a large extent determine the magnitude of the integrand (where $z^*$ is well-defined due to analyticity of $f$.
In the setting of this chapter, $C$ is a closed curve, which implies that the value $z^*$ must be a \textit{saddle point} of $f$, i.e.~$f'(z^*) = 0$.
Subsequently, one can replace $f(z)$ in \eqref{eq:sp_integral} by its Taylor expansion around $z^*$ and deduce through the Laplace method, see e.g.~\cite{debruijn}, that
\begin{equation*}
I(n) = \sqrt{2\pi}\,i\frac{h(z^*)\,{\rm e}^{n f(z^*)}}{\sqrt{n\, |f''(z^*)|}}\Bigl( 1+ O(1/n)\Bigr),
\end{equation*}
as $n\to\infty$.
In Section \ref{spSec}, we show how the contour integrals describing stationary measures for the queue length, derived through Pollaczek's formula, can be reformulated into the shape of \eqref{eq:sp_integral}.
However, we will show that the saddle point method in its standard from cannot be applied to asymptotically characterize other stationary measures like the mean or mass at zero.
Indeed, for our model the saddle point (the solution of \eqref{e21}) converges to one (as $n\to\infty$), which is a singular point of the integrand, and renders the standard saddle point method useless.
The non-standard saddle point method discussed in this chapter, originally proposed by \cite{debruijn}, is made specifically to overcome this complication. This leads to asymptotic expansions for the performance measures, of which the limiting forms correspond to the heavy-traffic limits, and pre-limit forms present refined approximations for pre-limit systems ($n<\infty$) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as {\em corrected diffusion approximations} \cite{Siegmund1978,Blanchet2006,Asmussen2003}.
\noindent{\bf Further connections to the literature.}
We now discuss two classes of stochastic systems for which the heavy-traffic regime \eqref{bb1} has been studied extensively, and for which our new family of regimes \eqref{bb} is largely unexplored. We discuss these classes because, despite the fact that the Pollaczek formula does not hold, we believe the qualitative results that we reveal for our particular model should to a large extent carry over to these settings as well, presenting some interesting avenues for further research (see Section \ref{subsec62}).
The first class concerns so-called {\it nearly-deterministic} systems \cite{Sigman2011a,Sigman2011b}, denoted by $G_n/G_n/1$ system, where $G_n$ stands for {\it cyclic thinning} of order $n$, indicating that some point process is thinned to contain only every $n$th point. As $n\to \infty$, the $G_n/G_n/1$ systems approach the deterministic $D/D/1$ system. For $G_n/G_n/1$ systems, \cite{Sigman2011a} establishes stochastic-process limits, and \cite{Sigman2011b} derives heavy-traffic limits for stationary waiting times. In the framework of \cite{Sigman2011a,Sigman2011b}, our stochastic model corresponds to a $D/G_n/1$ queue, where the sequence of service times $\{A^{(n)}_j\}_{j\geq 1}$ follows from a cyclically thinned sequence of i.i.d.~random variables $A_{i,j}$. It follows from \cite[Theorem 3]{Sigman2011b} that the rescaled stationary waiting time process converges under \eqref{bb1} to a reflected Gaussian random walk. Hence, the performance measures of the nearly deterministic system, under \eqref{lind} and \eqref{bb1}, should be well approximated by the performance measures of the reflected Gaussian random walk, giving rise to heavy-traffic approximations. This connection is discussed in detail in Section \ref{subsec3.2}. It seems likely that results similar to those presented in this chapter can be obtained by applying the scaling \eqref{bb} to the nearly-deterministic systems in \cite{Sigman2011a,Sigman2011b}, and because Pollaczek's formula also applies to this setting, the non-standard saddle point method developed in this chapter can provide the appropriate methodology.
The second class concerns multi-server systems, and in particular the many-server regime. When we interpret $s_n$ as the number of servers, instead of capacity per time slot or order of thinning, the scaling \eqref{bb1} is similar to the QED or Halfin-Whitt regime for the $M/M/s_n$ system.
As we have reviewed in Chapter 1, the QED regime is characterized by a delay probability that converges to a non-degenerate limit away from both zero and one, and the mean delay is asymptotically negligible as the number of servers grows large. The QED regime \eqref{bb1} is naturally positioned in between the Quality-Driven (QD) regime and the Efficiency-Driven (ED) regime. In the QD regime, the load remains bounded away from 1, which corresponds to setting $\eta=0$ in \eqref{bb}. Hence, the range $\eta\in(0,1/2)$ bridges the gap between the QED regime and the QD regime. Likewise, the ED regime corresponds to setting $\eta=1$ in \eqref{bb}, so that the range $\eta\in(1/2,1]$ connects the QED regime and ED regime. For the birth-death process describing the $M/M/s_n$ system, Maman \cite{maman} introduced a scaling similar to \eqref{bb}, and called it the QED-$c$ regime, also bridging the ED and QD regimes.
Theorem 4.1 of \cite{maman} says that the expected waiting time under the scaling $s_n = n\mu+\beta\sigma n^{1-\eta}$ is of order $s_n^{1-\eta}$, which is equivalent to the expected queue length being of order $n^\eta$ by Little's law. We should stress though that we expect the mathematical techniques that are needed to establish heavy-traffic results could be entirely different than in this chapter, because Pollaczek's formula does not apply to many-server settings.
The specific model assumptions will determine to a large extent the appropriate methodology. Under Markovian assumptions leading to the $M/M/s_n$ system, simple exact solutions are available for the stationary distribution. This makes it possible to describe performance measures like the mean congestion directly in terms of real integrals. Where the saddle point method is used for integrals in the complex plane, the Laplace method (see e.g.~\cite{flajolet}) is used for real integrals. Hence, for the asymptotic evaluation of the $M/M/s_n$ system under the scaling \eqref{bb}, the Laplace method seems an appropriate methodology, although again one needs to deal with possible singularities in the integrand. For $G/D/s_n$ systems, which assume deterministic service times, it has been shown in \cite{Jelenkovic2004} that using a decomposition property the dynamics of this multi-server systems can be captured in terms of a single-server system. Hence, for these systems, Pollaczek's formula applies, and our saddle point method can most likely be applied to obtain heavy-traffic results in the regimes \eqref{bb}. Under more general conditions, for instance leading to a $G/G/s_n$ system, it is simply unclear at this stage how to obtain precise heavy-traffic approximations for \eqref{bb}, because a tractable description of the performance measures is not available; see Section 1.2.4 for details.\\
\\
\\*
\noindent{\bf Structure of the chapter.}
In Section \ref{sec1} we present in detail the model and the family of heavy-traffic scalings. In Section \ref{spSec} we introduce the saddle point method. In Section \ref{sec3} we apply the saddle point method for the mean congestion level. Theorem \ref{mainthm} gives for all heavy-traffic scalings the limiting behavior in terms of an integral expression. As a consequence, we show in
Proposition \ref{prop1} that there are two types of heavy-traffic behavior, depending on whether $\eta\in(0,1/2)$ or $\eta\geq 1/2$.
In Section \ref{subsec3.2} we discuss for the case $\eta=1/2$ the connection with the Gaussian random walk and the Riemann zeta function.
In fact, we show that for all $\eta\geq 1/2$ there exists a connection between the integral expression in Theorem \ref{mainthm} and the Riemann zeta function.
In Section \ref{more} we apply the saddle point method to obtain several more heavy-traffic results, including refined heavy-traffic approximations for the mean congestion level, and the leading heavy-traffic behavior for the variance of the stationary congestion level and for the empty-system probability.
Finally, in Section \ref{numm} we confirm through numerical experiments the accuracy of our heavy-traffic approximations, and moreover show that under \eqref{bb}, various multi-server systems behave similar to our discrete bulk service queue.
\section{Model description \& heavy-traffic regimes}\label{sec1}
We thus consider a discrete stochastic model in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,\ldots$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
We will omit the superscript $(n)$ if no ambiguity is possible.
The system has a service capacity $s_n\in\mathbb{N}$ per period, so that the recursion
\begin{equation}
\label{lind}
Q(j+1) = \max\{Q(j) + A^{(n)}_j - s_n,0\},\qquad j=1,2,...,
\end{equation}
assuming $Q(0)=0$, gives rise to a Markov chain $\{Q(j)\}_{j\geq 1}$ that describes the congestion in the system over time. The probability generation function (pgf)
\begin{equation*}
\tilde A(z)=\sum_{k=0}^{\infty} \mathbb{P}\big(A^{(n)}=k\big) z^k
\end{equation*}
of $A^{(n)}$ is assumed analytic in a disk $|z|<r$ with $r>1$, which implies that all moments of $A^{(n)}$ exist. We also assume that
\begin{equation} \label{e3}
\tilde A'(1)=\mathbb{E}[A^{(n)}_j]=\mu_A<s_n.
\end{equation}
Under the assumption \eqref{e3} the function $z^{s_n}-\tilde A(z)$ has exactly $s_n$ zeros in the closed unit disk, one of these being $z=1$ (see \cite{rouche}).
We further assume that $\mathbb{P}(A^{(n)}=j)>0$ for some $j>s_n$.
Under this assumption the function
$z^{s_n}-\tilde A(z)$ also has zeros outside $|z|\leq 1$, and we let $r_0$ be the minimum modulus of these zeros.
The number $r_0$ is the unique zero of $z^{s_n}-\tilde A(z)$ with real $z>1$; see e.g.~\cite{Janssen2005}.
Moreover, under assumption \eqref{e3} the stationary distribution $\lim_{j\to \infty}\mathbb{P}\left(Q(j)=k\right)=\mathbb{P}(Q=k)$, $k=0,1,\ldots$ exists, with the random variable $Q$ defined as having this stationary distribution.
We let
\begin{equation*}
\tilde Q(w)=\sum_{j=0}^{\infty}\mathbb{P}(Q=j)w^j
\end{equation*}
be the pgf of the stationary distribution. $\tilde Q(w)$ is analytic in $|w|<r_0$, and given by Pollaczek's formula (see e.g.~\cite{Abate1993, Cohen1982}).
In our discrete setting, we shall first derive a useful expression for $\tilde{Q}(w)$.
\begin{lemma}
For any $\varepsilon>0$ with $1+\varepsilon<r_0$,
\begin{equation} \label{e111}
\tilde Q(w)=\exp\Big(\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big)
\end{equation}
holds when $|w|<1+\varepsilon$.
\end{lemma}
\begin{proof}
We shall establish \eqref{e111} for any $w\in(1,1+\varepsilon)$, and then the full result follows from analyticity of $\tilde{Q}(w)$ and of
\begin{equation*}
{\rm ln}\Big(\frac{w-z}{1-z}\Big)={\rm ln}\Big(\frac{1-w/z}{1-1/z}\Big)={-}\,\sum_{k=1}^{\infty}\,\frac1k\,\Big(\Big(\frac{w}{z}\Big)^k-\Big(\frac1z\Big)^k\Big)
\end{equation*}
in $w$, $|w|<1+\varepsilon$ for any $z$ with $|z|=1+\varepsilon$.
Our starting point is the formula, see \cite{Boudreau1962},
\begin{equation} \label{e113}
\tilde Q(w)=\frac{(s_n-\mu_A)(w-1)}{w^{s_n}-\tilde A(w)}\,\prod_{k=1}^{s_n-1}\,\frac{w-z_k}{1-z_k}
\end{equation}
that holds for all $w$, $|w|<r_0$, in which $z_1,\ldots,z_{s_n-1}$ are the $s_n-1$ zeros of $z^{s_n}-\tilde A(z)$ in $|z|<1$. Fix $w\in(1,1+\varepsilon)$.
Then ${\rm ln}\,[(w-z)/(1-z)]$ is analytic in $z\in\mathbb{C}\backslash [1,w]$.
It follows that
\begin{align}
I_C &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z \nonumber \\
&=~\sum_{k=1}^{s_n-1}\,{\rm ln}\Big(\frac{w-z_k}{1-z_k}\Big)+\frac{1}{2\pi i}\,\int_C\,{\rm ln}\Big(\frac{w-z}{1-z}\Big)\,\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,
\label{e114}
\end{align}
where $C$ is a contour encircling $[1,w]$ in the positive sense with none of the $z_k$'s in its interior. We let $\delta\in(0,\frac{w-1}{2})$ and we take $C$ the union of two line segments, from $1+\delta-i0$ to $w-\delta-i0$ and from $w-\delta+i0$ to $1+\delta-i0$, and two circles, of radius $\delta$ and encircling 1 and $w$ in positive sense.
A careful administration of the various contributions to the integral $I_C$ in \eqref{e114}, taking account of the branch cut $[1,w]$, yields
\begin{equation*}
I_C = {\rm ln }\left(\frac{(s_n-\mu_A)(w-1)}{w^s-\tilde A(w)}\right) + O(\delta\,{\rm ln}\, \delta ).
\end{equation*}
Using this in \eqref{e113} and letting $\delta \downarrow 0$, we get \eqref{e111} for $w\in(1,1+\varepsilon)$ and the proof is complete.
\end{proof}
Using $\mathbb{P}(Q=0)=\tilde Q(0)$, $\mu_Q=\tilde Q'(1)$ and $\sigma_Q^2 = \tilde Q''(1)+\tilde Q'(1)-(\tilde Q'(1))^2$, it follows by straightforward manipulations that
\begin{align} \label{e6}
\mathbb{P}(Q=0)&=\exp\,\Big[\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\ln\Big(\frac{z}{z-1}\Big)\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z\Big] , \\
\label{e7}
\mu_Q&=\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{1-z}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z ,\\
\label{e8}
\sigma_Q^2 &= \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{-z}{(1-z)^2}~\frac{(z^{s_n}-\tilde A(z))'}{z^{s_n}-\tilde A(z)}\,{\rm d} z .
\end{align}
Because $s_n$ appears directly in expressions \eqref{e6}-\eqref{e8}, we will be conducting our analysis with respect to $s_n$ rather than $n$. Note that this has no consequences for our results on the convergence speed of the performance metrics, since $s_n = O(n)$. Furthermore, we will omit the index $n$ when describing the capacity $s_n$ in the remainder of the chapter for brevity.
We next discuss in more detail the family of heavy-traffic scalings considered in this chapter, which combines two features. First, we have assumed that
$A^{(n)}_j$ is in distribution equal to the sum of work generated by all sources, $A_{1,j}+...+A_{n,j}$, where the $A_{i,k}$ are for all $i$ and $k$ i.i.d.~copies of a random variable $X$, of which the pgf $\tilde X(z)=\sum_{k=0}^{\infty}\mathbb{P}(X=k)z^k$ has radius of convergence $r>1$, and
\begin{equation*}
0< \mathbb{E}[A^{(n)}] =n\mu = n \tilde X'(1)<s_n .
\end{equation*}
Hence
\begin{equation} \label{e10}
\vartheta:=\frac{n}{s_n}\in(0,1/\mu) .
\end{equation}
Second, we scale the system according to \eqref{bb}, for which we assume that
\begin{equation} \label{e11}
\rho_{s_n} =\vartheta\,\mu =1-\frac{\gamma}{s_n^\eta}
\end{equation}
in which $\gamma>0$ is bounded away from 0 and $\infty$ as $s_n\to \infty$.
In the remainder of this chapter, we will omit the subscript in $s_n$.
The condition that $\mathbb{P}(A^{(n)}=k)>0$ for some $k>s$ holds when the degree $d$ of $\tilde X(z)$ (with $d=\infty$ if $\tilde X(z)$ is not a polynomial) is such that $nd>s$.
To avoid certain complications when applying the saddle point method, we further assume that
\begin{equation} \label{e12}
|\tilde X(z)|<\tilde X(r_1) ,~~~~~~|z|=r_1\,,~~z\neq r_1 ,
\end{equation}
for any $r_1\in(0,r)$. This implies that $r_0$ is the unique zero of $z^s-\tilde A(z)$ on $|z|=r_0$.
This condition is related to Cram\'er's condition, see \cite[pp.~189 and 355]{Asmussen2003}, and it has also been used in \cite{relaxation}.
Condition \eqref{e12} holds when the set of all $j=0,1,\ldots$ such that $\mathbb{P}(X=k)>0$ is not contained in an arithmetic progression with a ratio larger than one (see also \cite{rouche}).
\section{Non-standard saddle point method}\label{spSec}
\noindent
We illustrate our saddle point method for $\mu_Q$.
As a first step, we bring \eqref{e7} in a form which is amenable to saddle point analysis.
\begin{lemma}
\begin{equation} \label{e18}
\mu_Q = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{g'(z)}{z-1}~\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}\,{\rm d} z
\end{equation}
with
\begin{equation} \label{e15}
g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}(\tilde X(z)) .
\end{equation}
\end{lemma}
\begin{proof}
With $\tilde A(z)=\tilde X^n(z)$,
\begin{align} \label{e13}
\frac{(z^s-\tilde A(z))'}{z^s-\tilde A(z)} & = \frac{s\,z^{s-1}-n\,\tilde X'(z)\,\tilde X^{n-1}(z)}{z^s-\tilde X^n(z)} \nonumber \\
& = \frac{s}{z}-\frac{s}{z}\,\Big(\frac{n}{s}~\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big)\,\frac{z^{-s}\,\tilde X^n(z)}{1-z^{-s}\,\tilde X^n(z)} .
\end{align}
Write
$
z^{-s}\,\tilde X^n(z)=\exp(s\,g(z))$.
Noting that
\begin{equation} \label{e16}
\frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{s}{z}~\frac{1}{1-z}\,{\rm d} z=0 ,
\end{equation}
and that
\begin{equation} \label{e17}
g'(z)=\frac1z\,\Big(\vartheta\,\frac{z\,\tilde X'(z)}{\tilde X(z)}-1\Big) ,
\end{equation}
gives \eqref{e18}. \end{proof}
Let us now explain how the standard saddle point method can be applied to \eqref{e18}.
Since
\begin{equation} \label{e19}
g(1)=g(r_0)=0~;~~~~~~g(z)<0\,,~~1<z<r_0 ,
\end{equation}
and by strict convexity of
\begin{equation*}
z^{-s}\,\tilde X^n(z)=z^{-s}\tilde A(z)=\sum_{k=0}^{\infty}\,a_k\,z^{k-s} ,~~~~~~z\in(0,r) ,
\end{equation*}
$g(z)$ has a unique minimum on $[1,r_0]$. This minimum is found by solving $z\in[1,r_0]$ from $g'(z)=0$, and this yields the equation
\begin{equation} \label{e21}
\tilde X(z)=\vartheta\,z\,\tilde X'(z) .
\end{equation}
Denote the solution $z\in(1,r_0)$ of \eqref{e21} by $z_{\rm sp}$, and observe that $z_{\rm sp}$ is a saddle point of $g(z)$, explaining the notation. Thus, the saddle point method can be used for the integral in \eqref{e18} by taking $1+\varepsilon=z_{\rm sp}$.
In the case that $\vartheta=n/s$ is bounded away from $1/\mu$ as $s\to \infty$, we have that the minimum value of $g(z)$, $1\leq z\leq r_0$, is negative and bounded away from 0. Furthermore, $z_{\rm sp}$ is bounded away from 1, and the saddle point method can be applied in the classical way by replacing
\begin{equation*}
\frac{\exp(s\,g(z))}{1-\exp(s\,g(z))}~~~~{\rm by}~~~~\exp(s\,g(z)) ,
\end{equation*}
at the expense of an exponentially small relative error, and performing an expansion of $g'(z)/(z_{\rm sp}-1)=d_1(z-z_{\rm sp})+O((z-z_{\rm sp})^2)$ with $d_1=g''(z_{\rm sp})/(z_{\rm sp}-1)\neq 0$.
Using that $g(z^{\ast})=(g(z))^{\ast}$, where the $^*$ denotes complex conjugation, it can be shown that
\begin{equation} \label{e23}
\mu_Q=\frac{\exp(s\,g(z_{\rm sp}))}{(z_{\rm sp}-1)^2\,\sqrt{2\pi s\,g''(z_{\rm sp})}}\,(1+O(s^{-1})) .
\end{equation}
We next explain why the standard saddle point method does not work for the heavy-traffic scaling considered in this chapter.
Since we operate in \eqref{e11},
$\vartheta\mu\to 1$ as $s\to \infty$, and
\begin{align} \label{e24}
z_{\rm sp}-1&=\frac{\gamma}{a_2\,s^\eta}+O(s^{-2\eta}) ,\\
\label{e25}
g(z_{\rm sp})&=\frac{-\gamma^2}{2a_2s^{2\eta}}+O(s^{-3\eta}) ,\\
\label{e26}
g''(z_{\rm sp})&=a_2+O(s^{-\eta}) ,
\end{align}
where
\begin{equation} \label{e27}
a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big) .
\end{equation}
Hence, $\exp(sg(z))$ near $z=z_{\rm sp}$ is (as $s\to \infty$):
vanishingly small when $\eta\in(0,1/2)$,
bounded away from 1, but non-negligible when $\eta=1/2$,
and tending to 1 when $\eta\in(1/2,\infty)$.
Furthermore, $(z-1)^{-1}$ in \eqref{e18} is unbounded near $z=z_{\rm sp}$ as $s\to \infty$. Therefore, an adaptation of the standard saddle point method is required, and the resulting asymptotic form of $\mu_Q$ will deviate significantly from the standard case \eqref{e23}. In particular, since $z_{\rm sp}\to 1$, this asymptotic form will contain information from $X(z)$ at $z=1$, rather than at a point away from 1 as is the case in \eqref{e23}.
The required adaptation of the saddle point method is modeled after a device developed in \cite[Sec.~5.12]{debruijn}. We use a substitution $z=z(v)$ in \eqref{e18} with real $v$ and $z(0)=z_{\rm sp}$ such that for sufficiently small $v$,
\begin{equation} \label{e29}
g(z(v))=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) .
\end{equation}
This is feasible, since
\begin{equation} \label{e30}
g(z)=g(z_{\rm sp})+\tfrac12\,g''(z_{\rm sp})(z-z_{\rm sp})^2\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)
\end{equation}
with $g''(z_{\rm sp})$ positive and bounded away from 0 as $s\to \infty$. Hence, $z(v)$ can be found for small $v$ by inverting the equation
\begin{equation} \label{e31}
(z-z_{\rm sp})\Big(1+\frac{g'''(z_{\rm sp})}{3g''(z_{\rm sp})}\,(z-z_{\rm sp})+...\Big)^{1/2}=iv .
\end{equation}
By Lagrange's inversion theorem \cite{debruijn}, there is a $\delta>0$ (independent of $s$) such that
\begin{equation} \label{e32}
z(v)=z_{\rm sp}+iv+\sum_{k=2}^{\infty}\,c_k(iv)^k ,~~~~~~|v|<\delta ,
\end{equation}
with real coefficients $c_k$ (since $g(z)$ is real for real $z$) and
\begin{equation} \label{e33}
c_2={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})} .
\end{equation}
Thus
\begin{equation} \label{e34}
z(v)=z_{\rm sp}+iv-c_2\,v^2+O(v^3) ,~~~~~~|v|\leq\tfrac12\,\delta ,
\end{equation}
where the order term holds uniformly in $s$. The uniformity statement follows from an inspection of the usual argument
by which Lagrange's theorem is proved, noting that the inversion in \eqref{e29} with $g$ as in \eqref{e15} is considered for $\vartheta\to 1/\mu$, $z_{\rm sp}\to 1$ with radius
of convergence $r$ away from $1$.
By \eqref{e12} we can restrict the integration in \eqref{e18} to a fixed but arbitrarily small subset of $|z|=z_{\rm sp}$ near $z=z_{\rm sp}$, at the expense of an exponentially small error. Furthermore, by Cauchy's theorem and again at the expense of an exponentially small error, the integration path can be deformed in accordance with the transformation in \eqref{e29}--\eqref{e34}. Set
\begin{equation} \label{e35}
q(v)=g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp})
\end{equation}
and note that from \eqref{e29},
\begin{equation*}
g'(z(v))\,z'(v)={-}v\,g''(z_{\rm sp}) .
\end{equation*}
Then substituting $z=z(v)$ in \eqref{e18}, $\mu_Q$ is given with exponentially small error by
\begin{equation*}
\frac{s}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{g'(z(v))}{z(v)-1}~\frac{\exp(s\,g(z(v)))}{1-\exp(s\,g(z(v)))}z'(v)\,{\rm d} v,
\end{equation*}
which gives the following result.
\begin{lemma} \label{lemma2} The mean stationary congestion level is given with exponentially small error by
\begin{equation} \label{e37}
\mu_Q =~\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v}{z(v)-1}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v .
\end{equation}
\end{lemma}
In a similar fashion we get that $\mathbb{P}(Q=0)$ and $\sigma_Q^2$, see \eqref{e6} and \eqref{e8}, are given, both with exponentially small error, by
\begin{equation} \label{e39}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,v\,{\rm ln}\Big(\frac{z(v)}{z(v)-1}\Big)\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v
\end{equation}
and
\begin{equation} \label{e38}
\frac{-s}{2\pi i}\,g''(z_{\rm sp})\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}\,{\rm d} v,
\end{equation}
respectively.
\section{Heavy-traffic limits for the mean congestion level} \label{sec3}
In this section we apply the non-standard saddle point method explained in Section \ref{spSec} to the Pollaczek integral representation for the mean stationary congestion level $\mu_Q$. In Section \ref{subsec3.1} we first derive an integral representation for the leading order behavior of $\mu_Q$ with a relative error of order $O(s^{-1})$, which serves as a heavy-traffic approximation in the regime $\rho_s=1-\gamma/s^\eta$ with $\eta>0$. We also consider separately the cases of moderate heavy traffic ($\eta\in(0,1/2)$) and extreme heavy traffic ($\eta\in(1/2,\infty)$), for which the integral representation leads to vastly different alternative expressions. We find that $\mu_Q\to 0$ more rapidly than any power of $1/s$ when $\eta\in(0,1/2)$. When $\eta\geq 1/2$ the saddle point method yields an integral representation with relative error $O(s^{-\min(1,\eta)})$.
In Section \ref{subsec3.2} we specialize this general result to the CLT case $\eta=1/2$, and make a connection with existing results.
\subsection{Leading order behavior in integral form} \label{subsec3.1}
\begin{theorem}\label{mainthm}
The mean stationary congestion level is given by
\begin{equation} \label{e4.1}
\mu_Q=\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\int_0^{\infty}\,\frac{t^2}{d^2(s)+t^2}~\frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,{\rm d} t\,\left(1+O({s^{{-}\min(1,\eta)}})\right)
\end{equation}
with $
d^2(s) = s^{1-2\eta}\gamma^2\mu/(2\sigma^2)$.
\end{theorem}
\begin{proof}
According to Lemma \ref{lemma2}, $\mu_Q$ is given with exponentially small error by \eqref{e37} with $q(v)$ given in \eqref{e35}. Since $z({-}v)=z^{\ast}(v)$ for real $v$, we have
\begin{align}
\frac{v}{z(v)-1}+\frac{-v}{z({-}v)-1} &= {-}2iv\,\frac{{\rm Im}(z(v))}{|z(v)-1|^2}\nonumber\\
&=\frac{-2iv^2+O(v^4)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4)} \nonumber \\
&=\frac{-2iv^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2},
\label{e40}
\end{align}
for ${-}\tfrac{1}{2} \delta \leq v \leq \tfrac{1}{2} \delta$.
where \eqref{e34} and $c_k\in\mathbb{R}$ have been used. Using \eqref{e40} in \eqref{e37} and extending the integration range from $[{-}\tfrac12\delta,\tfrac12\,\delta]$ to $({-}\infty,\infty)$ while using symmetry of $q(v)$, we get that $\mu_Q$ is given with exponentially small error by
\begin{align} \label{e41}
\frac{s\,g''(z_{\rm sp})}{\pi}\,\int_0^{\infty}\,\frac{v^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2}\frac{\exp(s\,q(v))}{1-\exp(s\,q(v))}{\rm d} v .
\end{align}
With
\begin{equation} \label{e42}
B=\exp(s\,g(z_{\rm sp})) ,~~~~~~\alpha =g''(z_{\rm sp}),
\end{equation}
Equation \eqref{e41} takes the form
\begin{equation} \label{e43}
\frac{s\alpha }{\pi}\,\int_0^{\infty}\,\frac{v^2\left(1+O(v^2)\right)}{(z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2} \cdot \frac{B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\tfrac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
Since $(z_{\rm sp} - 1)^2 = (\gamma/a_2)^2s^{{-}2\eta} + O(s^{-4\eta})$, see \eqref{e24}, the integrand in \eqref{e43} in leading order has the form
\begin{equation*}
\frac{B\,v^2\,\exp(-s\,D\,v^2)}{(v^2+C\,s^{-2\eta})(1-B\exp({-}s\,D\,v^2))},
\end{equation*}
and this is reminiscent of the integrand in \cite[Eq.~(5.12.3)]{debruijn} for the case $\kappa=2\eta$. Proceeding as in \cite[Sec.~5.12]{debruijn}, the substitution $v=t\sqrt{{2}/(s\alpha )}$ brings \eqref{e43} into the form
\begin{equation} \label{e44}
\frac{2}{\pi}\sqrt{\tfrac12 s\alpha }\int_0^{\infty}\frac{t^2(1+O(t^2/s))}{\tfrac12 s\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2} \,\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}{\rm d} t .
\end{equation}
From \eqref{e24}--\eqref{e27} and \eqref{e42},
\begin{align}
\frac{2}{\pi}\,\sqrt{\frac{s\alpha }{2}} &= \frac{2}{\pi}\,\sigma_X\,\sqrt{\frac{s}{2\,\mu}}\,(1+O(s^{-\eta})),\label{y45}\\
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 &= d^2(s) + O(s^{1-3\eta}),\label{y46}\\
2\,c_2(z_{\rm sp}-1) &= O(s^{-\eta}),\label{y47}\\
s\,g(z_{\rm sp}) &= -d^2(s) + O(s^{1-3\eta}),\label{y48}
\end{align}
where
\begin{equation} \label{y49}
d^2(s) = \frac{b_0^2}{s^{2\eta-1}},\quad b_0^2 := \frac{\gamma^2\mu}{2\,\sigma^2}.
\end{equation}
In the case that $2\eta-1<0$, we have that $\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 \to \infty$ and that
\begin{equation} \label{y50}
B = \exp(s\,g(z_{\rm sp})) = O(\exp({-}b^2s^{1-2\eta}))
\end{equation}
for any $b\in(0,b_0)$. From \eqref{e44} it then follows that $\mu_Q = O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$.
In the case that $2\,\eta-1\geq 0$, we have that $d^2(s)$ is bounded, and using that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, we get
\begin{align*}
\tfrac12\,s\,\alpha \,(z_{\rm sp}-1)^2 + t^2-2\,c_2\,(z_{\rm sp}-1)\,t^2
&= d^2(s) + t^2 + O\left(s^{-\eta}\,(d^2(s)+t^2)\right) \nonumber\\
&= \big(d^2(s)+t^2\big)\left(1+O(s^{-\eta})\right).
\end{align*}
Hence, in this case,
\begin{equation}
\frac{t^2(1+O(t^2/s))}{\tfrac12 s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)t^2} = \frac{t^2}{d^2(s)+t^2}\left(1+O(s^{-\eta})+O(t^2/s)\right).\label{y52}
\end{equation}
Furthermore,
\begin{align*}
1-B\,\exp(-t^2) &= 1-\exp({-}d^2(s)-t^2)\,\left(1+d^2(s)\,O(s^{-\eta})\right)\nonumber\\
&=(1-\exp({-}d^2(s)-t^2))\,\Big(1+\frac{d^2(s)}{\exp(d^2(s)+t^2)-1}O(s^{-\eta})\Big)\nonumber\\
&= (1-\exp({-}d^2(s)-t^2))\,(1+O(s^{-\eta})),
\end{align*}
It follows therefore that
\begin{equation} \label{y56}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)} = \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}\,(1+O(s^{-\eta})).
\end{equation}
Combining the three items \eqref{y45}, \eqref{y52} and \eqref{y56}, we obtain for \eqref{e44} the result
\begin{equation*}
\frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}} \int_0^{\infty}\frac{t^2}{d^2(s)+t^2} \cdot \frac{\exp({-}d^2(s)-t^2)}{1-\exp({-}d^2(s)-t^2)}{\rm d} t
\left(1+O(s^{-\eta})+O(s^{-1})\right),
\end{equation*}
and this gives \eqref{e4.1}.
\end{proof}
Theorem \ref{mainthm} gives the leading-order behavior of $\mu_Q$ as $s\to \infty$ with a relative error of $O(s^{{-}\min(1,\eta)})$. By considering in more detail the integral expressions, we obtain the following result, describing two different heavy-traffic behaviors.
\begin{proposition}\label{prop1}
If $\eta\in(0,1/2)$ the mean congestion level satisfies
\begin{equation*}
\mu_Q=O\left(\exp(-b^2s^{1-2\eta})\right),
\end{equation*}
for any $b\in (0,b_0)$. If $\eta\in[1/2,\infty)$ the mean congestion level is given by
\begin{equation*}
\mu_Q = s^\eta\,\frac{\sigma^2}{2\mu\gamma}\,\left(1+O(s^{\max(1/2-\eta,-1)})\right).
\end{equation*}
\end{proposition}
The first assertion in Proposition \ref{prop1} follows from the observation in \eqref{y50}, together with \eqref{e44}. The second assertion is based on a connection between the integral in Theorem \ref{mainthm} and the Riemann zeta function, which is explained in the next subsection.
\subsection{Classical heavy traffic and the Gaussian random walk}
\label{subsec3.2}
We now build on Theorem \ref{mainthm} to obtain further results for the classical heavy traffic case $\eta=1/2$,
for which we know from \cite[Thm.~3]{Sigman2011b} that the rescaled congestion process converges under \eqref{bb1} to a reflected Gaussian random walk. The latter is defined as
$(S_\beta(k))_{k\geq 0}$ with $S_\beta(0)=0$ and
\begin{equation*}
S_\beta(j)=Y_1+\ldots+Y_j
\end{equation*}
with $Y_1,Y_2,\ldots$ i.i.d.~copies of a normal random variable with mean $-\beta$ and variance 1.
Assume $\beta>0$ (negative drift), and denote the all-time maximum of this random walk by ${M}_\beta$.
Denote by $Q^{(s)}_\infty$ the stationary congestion level for a fixed $s$ (that arises from taking
$j\to \infty$ in \eqref{lind}), and remember that we have assumed $\vartheta=n/s$ fixed.
Then, using $\rho_s=1-\gamma/\sqrt{s}$, with
\begin{equation}\label{gammachoice}
\gamma=\frac{\beta\sigma}{\mu\sqrt{\vartheta}},
\end{equation}
the spatially-scaled stationary congestion levels reach the limit
$Q^{(s)}_\infty/(\sigma\sqrt{n}) {\;\buildrel{d}\over\Rightarrow\;} {M}_\beta$ as $s,n\to \infty$ (see \cite{Jelenkovic2004,Sigman2011a,Sigman2011b}). From \cite[Thm.~4]{Sigman2011b} we then know that under the standard heavy-traffic scaling \eqref{bb1}
\begin{equation}
\frac{\mathbb{E}[Q^{(s)}_\infty]}{\sigma\sqrt{n}}\to \mathbb{E}[{M}_\beta], \quad {\rm as} \ s,n\to \infty,
\end{equation}
from which it follows that
\begin{equation} \label{e48}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}[M_\beta].
\end{equation}
The random variable ${M}_\beta$ was studied in \cite{Chang1997,Janssen2006}. In particular, \cite[Thm.~2]{Janssen2006} yields, for $\beta<2\sqrt{\pi}$,
\begin{equation*}
\mathbb{E}[{M}_\beta]= \frac{1}{2\beta}+\frac{\zeta(1/2)}{\sqrt{2\pi}}+\frac{\beta}{4}+\frac{\beta^2}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta(-1/2-r)}{r!(2r+1)(2r+2)}\left(\frac{-\beta^2}{2 }\right)^r,
\end{equation*}
where $\zeta$ denotes the Riemann zeta function, which is defined as, see (1.26).
Hence, for small values of $\beta$,
\begin{equation} \label{estimate}
\mu_Q\approx \sigma\sqrt{n}\ \mathbb{E}[M_\beta] \approx \frac{\sigma\sqrt{n}}{2\beta} = \sqrt{s}\,\frac{\sigma^2}{2\mu\gamma}.
\end{equation}
We will now show how the approximation \eqref{estimate} follows from Theorem \ref{mainthm}, and also how similar steps give rise to Proposition \ref{prop1}.
Consider the integral
\begin{equation} \label{e49}
G_0(b)=G_1(b)-G_2(b)=\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t ,
\end{equation}
where $b>0$ and
\begin{equation} \label{e50}
G_1(b)=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}\,{\rm d} t\,,~~~~G_2(b)=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b^2-t^2)}{\rm d} t .
\end{equation}
We have, as in \cite[Sec.~2]{Janssen2006},
\begin{align}
G_1(b) & = \sum_{k=0}^{\infty}\:\int_0^{\infty}\,\exp({-}(k+1)(b^2+t^2))\,{\rm d} t \nonumber \\
& = \frac{\sqrt{\pi}}{2}\,\sum_{k=0}^{\infty}\,\frac{{\rm e}^{-(k+1)b^2}}{\sqrt{k+1}} = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},1/2,1) \nonumber \\
& = \frac{\pi}{2b}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta(\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} , \label{e51}
\end{align}
where the last identity holds when $0<b<\sqrt{2\pi}$ and $\Phi(z,s,v)$ is Lerch's transcendent, which is defined as, see \cite[Eq.~25.14.1]{NIST},
\begin{equation*}
\Phi(z,s,v) = \sum_{n=0}^\infty \frac{z^n}{(v+n)^s}, \qquad \text{for }v\neq 0,{-}1,{-}2,\ldots, \ |z|<1; \, \Re s > 1,\ |z|=1.
\end{equation*}
As to $G_2(b)$, we make a connection with the complementary error function
\begin{equation*}
{\rm erfc}(z)=\frac{2}{\sqrt{\pi}}\,\int_z^{\infty}\,{\rm e}^{-t^2}\,{\rm d} t=\frac{2}{\pi}\,{\rm e}^{-z^2}\,\int_0^{\infty}\,\frac{{\rm e}^{-z^2t^2}}{1+t^2}{\rm d} t ,
\end{equation*}
see \cite[Secs.~7.2 and 7.7.1]{NIST}. We thus compute
\begin{align} \label{e53}
G_2(b) & = \sum_{k=0}^{\infty}\,{\rm e}^{-(k+1)b^2}\,\int_0^{\infty}\,\frac{b^2}{b^2+t^2}\,{\rm e}^{-(k+1)t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align}
From \cite[Eq.~(4.3) \& (4.23)]{Janssen2006},
\begin{equation} \label{e54}
\sum_{n=1}^{\infty}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^{\infty}\,{\rm e}^{-x^2/2}\,dx= \frac{1}{2\beta^2}-\frac14-\frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^{\infty} \frac{\zeta({-}1/2-r)({-}1/2)^r} {r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
in which $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in \eqref{e54}, we get
\begin{equation} \label{e55}
G_2(b)=\frac{\pi}{4b}-\frac{\pi}{4}\,b-\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results in \eqref{e51} and \eqref{e55} can be combined, as in \cite[Sec.~\ref{sec4}]{Janssen2006}, and this yields
\begin{equation} \label{e56}
G_0(b)=\frac{\pi}{4b}+\frac{\pi}{4}\,b+\frac{\sqrt{\pi}}{2}\,\zeta(1/2)+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}1/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Using \eqref{e56} in \eqref{e48}, we find that the leading order behavior of $\mu_Q$ is given as
\begin{equation} \label{e57}
\sigma_X\,\sqrt{\dfrac{s}{2\mu}}\,\left[\frac{1}{2b_0}+\frac{b_0}{2}+\frac{\zeta(1/2)}{\sqrt{\pi}}+\frac{2}{\sqrt{\pi}}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}1/2-r)({-}1)^r b_0^{2r+2}} {r!\,(2r+1)(2r+2)}\right]
\end{equation}
with relative error of $O(s^{-1/2})$ in which $b_0$ is given by \eqref{y49}. The expression \eqref{e57} is exactly equal to the right-hand side of \cite[Eq.~(4.25)]{Janssen2006} times $\sqrt{s}$ when we take there $\sigma=\mu=1$ and $\beta=b_0\,\sqrt{2}$.
Notice that, with $\gamma$ as in \eqref{gammachoice},
\begin{equation*}
\sigma\,\sqrt{\dfrac{s}{2\mu}}\frac{1}{2b_0}=\frac{\sigma\sqrt{n}}{2\beta},
\end{equation*}
which confirms the approximation \eqref{estimate}.
According to Theorem \ref{mainthm}, we have for $\eta\geq 1/2$,
\begin{equation*}
\mu_Q = \frac{2}{\pi}\,\sigma\,\sqrt{\frac{s}{2\,\mu}}G_0(d(s))\,\left(1+O(s^{{-}\min(1,\eta)})\right).
\end{equation*}
When $\eta=1/2$, so that $d(s) = b_0$ is independent of $s$, the series representation for $G_0$ in \eqref{e56} can be used, as long as $b_0\in(0,\sqrt{2\pi})$. When $\eta>1/2$, we have that $d(s) = b_0/s^{\eta-1/2}\to 0$ as $s\to \infty$, and so this series representation can be used when $s$ is large enough. We then have from \eqref{e56} and $b_0^2 = \gamma^2\mu/2\,\sigma^2$, while replacing the whole series at the right-hand side by $O(b^2)$, for $\mu_Q$ the leading order behavior
\begin{equation} \label{y62}
s^\eta\left[\frac{\sigma^2}{2\,\gamma\,\mu}+\frac{\sigma\,\zeta(1/2)}{\sqrt{2\,\pi\,\mu}}\,\frac{1}{s^{\eta-1/2}}+\frac{1}{4}\,\gamma\,\frac{1}{s^{2\eta-1}}+O(s^{3/2-3\eta})\right]
\end{equation}
with relative error $O(s^{{-}\min(1,\eta)})$. Retaining the constant term $\sigma^2/(2\gamma\mu)$ and estimating the other terms between the brackets in \eqref{y62} as $O(s^{1/2-\eta})$, we get Proposition \ref{prop1}.
\section{More heavy-traffic results}\label{more}
In this section we apply the non-standard saddle point method to obtain several more heavy-traffic results. In Section \ref{subsec3.3} we derive refined heavy-traffic approximations for the mean congestion level by considering higher-order correction terms. In Section \ref{sec4} we derive the leading heavy-traffic behavior for the variance of the stationary congestion level, and in Section \ref{sec5} for the empty-system probability. To keep the developments tractable, we restrict Section \ref{subsec3.3} to $\eta=1/2$, and Section \ref{sec4} and Section \ref{sec5} to $\eta\in(0,1]$, although the same technique will work for all values $\eta>0$.
\subsection{Correction term for the mean congestion level for $\eta = 1/2$} \label{subsec3.3}
Our saddle point method not only establishes the leading-order heavy-traffic approximations, but also allows to derive refinements to these approximations. In this section we demonstrate how this works for the mean congestion level in the case $\eta=1/2$.
To obtain a refinement or correction term from \eqref{e44}, we must be more precise about the $O(s^{{-}\eta})$ terms that occur in the approximations in Section \ref{subsec3.1} for $\frac12\,s\,\alpha (z_{\rm sp}-1)^2$, $B$ and $\sqrt{s\,\alpha /2}$. When higher-order corrections are required, we should include higher-order terms in the approximations of these quantities, and be more specific about the $O(t^2/s)$ and $O(t^4/s)$ in the integrand in \eqref{e44}.
Let $g^{(i)}, \ i=1,2,...$~denote the $i^{\rm th}$ derivative of $g$ and define, see \eqref{e10} and \eqref{e15} with $\vartheta=(1-\gamma/s^\eta)\,\mu^{-1}$,
\begin{equation*}
a_i=g^{(i)}(1);~~~~~~g(z)={-}{\rm ln}\,z+\vartheta\,{\rm ln}\,\tilde X(z) .
\end{equation*}
Dropping the $X$ from $\mu$ and $\sigma^2$ for brevity, we have
\begin{equation*}
a_1={-}\,\frac{\gamma}{s^\eta} ,~~~~~~a_2=\frac{\sigma^2}{\mu}-\frac{\gamma}{s^\eta}\,\Big(\frac{\sigma^2}{\mu}-1\Big),
\end{equation*}
\begin{equation*}
a_3={-}2+\Big(1-\frac{\gamma}{s^\eta}\Big)\Big(\frac{\tilde X'''(1)}{\tilde X'(1)}-3\tilde X''(1)+2(\tilde X'(1))^2\Big) .
\end{equation*}
For the purpose of finding a first-order correction term, we note that
\begin{align*}
\alpha &=g''(z_{\rm sp})=a_2+(z_{\rm sp}-1)\,a_3+O(s^{-1}) ,\\
z_{\rm sp}-1&={-}\,\frac{a_1}{a_2}-\frac{a_3}{2a_2}\,\Big(\frac{a_1}{a_2}\Big)^2+O(s^{-3/2}) ,\\
c_2&={-}\,\frac{g'''(z_{\rm sp})}{6g''(z_{\rm sp})}={-}\,\frac{a_3}{6a_2}+O(s^{-1/2}) ,\\
g(z_{\rm sp})&={-}\,\frac{a_1^2}{2a_2}-\frac{a_3}{6a_2^3}\,a_1^3+O(s^{-2}) .
\end{align*}
This gives rise to
\begin{align} \label{e65}
\sqrt{\tfrac12\,s\,\alpha }&=\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}+O(s^{-1})\Big) ,\\
\tfrac12\,s\,\alpha (z_{\rm sp}-1)^2&=\frac{\gamma^2\,\mu}{2\sigma^2}+\frac{C_2}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e67}
2c_2(z_{\rm sp}-1)&=\frac{C_3}{\sqrt{s}}+O(s^{-1}) ,\\
\label{e68}
B=\exp(s\,g(z_{\rm sp}))&=\exp\Big({-}\,\frac{\gamma^2\,\mu}{2\sigma^2}\Big)\Big(1+\frac{C_4}{\sqrt{s}}+O(s^{-1})\Big) ,
\end{align}
with explicitly computable constants $C_1$, $C_2$, $C_3$, $C_4$. Remembering that $b_0^2=\gamma^2\mu/2\sigma^2$, see \eqref{y49}, we then get with errors of order $1/s$
\begin{align}
& \frac{t^2(1+O(t^2/s))}{\frac12\,s\,\alpha (z_{\rm sp}-1)^2+t^2-2c_2(z_{\rm sp}-1)\,t^2} \nonumber \\
& \qquad \qquad =~\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+ b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big) ,\label{e69}
\end{align}
and
\begin{equation} \label{e70}
\frac{B\,\exp({-}t^2)}{1-B\,\exp({-}t^2)}=\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2} .
\end{equation}
Using \eqref{e65}, \eqref{e69} and \eqref{e70} in \eqref{e44} we get with an absolute error of order $1/\sqrt{s}$
\begin{align} \label{e71}
\mu_Q & =\frac{2}{\pi}\,\sigma\,\sqrt{\dfrac{s}{2\mu}}\,\Big(1+\frac{C_1}{\sqrt{s}}\Big)\nonumber \\
& \qquad\qquad \cdot \int_0^{\infty}\,\Big(\frac{t^2}{b_0^2+t^2}-\frac{1}{\sqrt{s}}\,\Big((C_2+b_0^2\,C_3)\,\frac{t^2}{(b_0^2+t^2)^2}-C_3\,\frac{t^2}{b_0^2+t^2}\Big)\Big) \nonumber \\
& \qquad\qquad\qquad \cdot~\Big(\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}+\frac{C_4}{\sqrt{s}}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\Big){\rm d} t \nonumber \\
& =\frac{2\sigma}{\pi}\,\sqrt{\dfrac{s}{2\mu}}\,G_0(b_0)\nonumber\\
& \qquad\qquad + ~\frac{2\sigma}{\pi}\,\sqrt{\dfrac{1}{2\mu}}\,\big((C_1+C_3)\,G_0(b_0)-(C_2+b_0^2\,C_3)\,G_3(b_0)+C_4\,G_4(b_0)\big) ,
\end{align}
where $G_0$ is as in \eqref{e49}, and
\begin{align} \label{e72}
G_3(b_0)&=\int_0^{\infty}\,\frac{t^2}{(b_0^2+t^2)^2}~\frac{\exp({-}b_0^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,{\rm d} t ,\\
\label{e73}
G_4(b_0)&=\int_0^{\infty}\,\frac{t^2}{b_0^2+t^2}~\frac{\exp({-}b_0^2-t^2)}{(1-\exp({-}b_0^2-t^2))^2}\,{\rm d} t .
\end{align}
We shall express the integrals in \eqref{e72} and \eqref{e73} in terms of $\zeta$-functions. By partial integration
\begin{align} \label{e74}
G_3(b) & = \frac12\,\int_0^{\infty}\,\frac{1}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{1-\exp({-}b_0^2-t^2)}\,\,{\rm d} t \nonumber \\
&\qquad\qquad -~\int_0^{\infty}\,\frac{t^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t \nonumber \\
& = \frac{1}{2b^2}\,G_2(b)-G_4(b) ,
\end{align}
see \eqref{e49} and \eqref{e73}. Since $G_2(b)$ is expressed in terms of $\zeta$-functions in \eqref{e55}, it is sufficient to consider $G_4(b)$.
As to $G_4(b)$,
\begin{equation*}
G_4(b)=G_5(b)-G_6(b) ,
\end{equation*}
where
\begin{align*}
G_5(b)&=\int_0^{\infty}\,\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t ,\\
G_6(b)&=\int_0^{\infty}\,\frac{b^2}{b^2+t^2}~\frac{\exp({-}b^2-t^2)}{(1-\exp({-}b^2-t^2))^2}\,{\rm d} t .
\end{align*}
We have, compare \eqref{e51},
\begin{align} \label{e78}
G_5(b) & = \sum_{k=0}^{\infty}\,(k+1)\,\int_0^{\infty}\,{\rm e}^{-(k+1)(b^2+t^2)}\,{\rm d} t \nonumber \\[3.5mm]
& = \frac{\sqrt{\pi}}{2}\,{\rm e}^{-b^2}\,\Phi({\rm e}^{-b^2},{-}\tfrac12,1) = \frac{\pi}{4b^3}+\frac{\sqrt{\pi}}{2}\,\sum_{r=0}^{\infty}\,\zeta({-}\tfrac12-r)\,\frac{({-}1)^r\,b^{2r}}{r!} ,
\end{align}
the last identity being valid when $0<b<\sqrt{2\pi}$. Next we have, compare \eqref{e53},
\begin{align*}
G_6(b) & = \sum_{k=0}^{\infty}\,(k+1)\,b^2\,\int_0^{\infty}\,\frac{\exp({-}(k+1)(b^2+t^2))}{b^2+t^2}{\rm d} t \nonumber \\
& = \frac{\pi}{2}\,b\,\sum_{k=0}^{\infty}\,(k+1)\,{\rm erfc}(b\,\sqrt{k+1}) .
\end{align*}
From \cite[Eq.~(5.4) \& (5.21)]{Janssen2006} we have
\begin{equation} \label{e80}
\sum_{n=1}^{\infty}\frac{n}{\sqrt{2\pi}}\int_{\beta\sqrt{n}}^{\infty}{\rm e}^{-x^2/2}\,dx = \frac{3}{4\beta^4}-\frac{1}{24}-\frac{1}{\sqrt{2\pi}}\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1/2)^r}{r!\,(2r+1)}\,\beta^{2r+1}
\end{equation}
when $0<\beta<2\sqrt{\pi}$. Taking $\beta=b\,\sqrt{2}$ in \eqref{e80}, we get
\begin{equation} \label{e81}
G_6(b)=\frac{3\pi}{16b^2}-\frac{\pi b}{24}-\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r}{r!\,(2r+1)}\,b^{2r+2}
\end{equation}
when $0<b<\sqrt{2\pi}$. The two results \eqref{e78} and \eqref{e81} can be combined, as in \cite[Sec.~5]{Janssen2006} and this yields
\begin{equation} \label{e82}
G_4(b)=\frac{\pi}{16b^3}+\frac{\pi b}{24}+\tfrac12\,\zeta({-}1/2)\,\sqrt{\pi}+\sqrt{\pi}\,\sum_{r=0}^{\infty}\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)}
\end{equation}
when $0<b<\sqrt{2\pi}$.
Finally, we can rewrite
\begin{align}
\frac{1}{2b^2}G_2(b) &= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \frac{\sqrt{\pi}}{2} \sum_{r=0}^\infty \frac{\zeta({-}1/2-r)(1-)^rb^{2r}}{r!(2r+1)} \nonumber\\
&= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \frac{\sqrt{\pi}}{2} \sum_{r=-1}^\infty \frac{\zeta({-}3/2-r)(-1)^{r+1}b^{2r+2}}{(r+1)!(2r+3)} \nonumber \\
&= \frac{\pi}{8b^3} - \frac{\pi}{8b} - \tfrac{1}{2}\zeta(-1/2)\sqrt{\pi}
+ \sqrt{\pi} \sum_{r=0}^\infty \frac{\zeta(-3/2-r)(-1)^r b^{2r+2}}{r!\,(2r+2)(2r+3)}
\label{e82a}
\end{align}
and use \eqref{e82} and \eqref{e82a} in \eqref{e74}, by which we obtain for $0<b<\sqrt{2\pi} $,
\begin{align}
G_3(b) &=
\frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} \nonumber \\
&\qquad +\sqrt{\pi} \sum_{r=0}^\infty \frac{ \zeta(-3/2-r)(-1)^rb^{2r+2}}{r!\,(2r+2)} \Big[ \frac{1}{2r+3}-\frac{1}{2r+1}\Big]\nonumber\\
&= \frac{\pi}{16b^3}-\frac{\pi}{8b}-\frac{\pi b}{24}-\zeta({-}1/2)\,\sqrt{\pi} -~2\sqrt{\pi}\,\sum_{r=0}^{\infty}\,\frac{\zeta({-}3/2-r)({-}1)^r\,b^{2r+2}}{r!\,(2r+1)(2r+2)(2r+3)}.\label{e83}
\end{align}
The right-hand side of \eqref{e83} equals the right-hand side of \cite[Eq.~(2.3)]{Janssen2006} multiplied by ${\pi}/{(2b)}$ with $\beta=b\,\sqrt{2}$.
\subsection{Variance of the congestion level}\label{sec4}
We have from \eqref{e38} in Section \ref{sec1}, using the same approach and notation as in Section \ref{subsec3.1} for $\mu_Q$, that $\sigma_Q^2$ is given with exponentially small error by
\begin{equation} \label{e84}
\frac{-s\,\alpha }{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{v\,z(v)}{(z(v)-1)^2}~\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v,
\end{equation}
with $B$ and $\alpha $ given in \eqref{e42}. From $z({-}v)=z^{\ast}(v)$ for real $v$ we now compute
\begin{equation*}
\frac{z(v)}{(z(v)-1)^2}-\frac{z({-}v)}{(z({-}v)-1)^2}={-}2i\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,{\rm Im}(z(v)) ,
\end{equation*}
and so \eqref{e84} becomes
\begin{equation} \label{e86}
\frac{s\alpha }{\pi}\,\int_0^{\frac12\delta}\,\frac{|z(v)|^2-1}{|z(v)-1|^4}\,v\,{\rm Im}(z(v))\,\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v .
\end{equation}
From
\begin{equation*}
{\rm Im}(z(v))=v+O(v^3) ,\qquad |z(v)|^2-1=z_{\rm sp}^2-1+O(v^2) ,
\end{equation*}
we get for the expression in \eqref{e86}
\begin{equation} \label{y70}
\frac{s\alpha }{\pi}\,\int_0^{\frac{1}{2}\delta}\,\frac{v^2\,(z_{\rm sp}^2-1+O(v^2))(1+O(v^2))}{((z_{\rm sp}-1)^2+v^2 + O((z_{\rm sp}-1)\,v^2)+O(v^4))^2}
\frac{B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{1-B\,\exp({-}\frac12\,s\,\alpha \,v^2)}{\rm d} v.
\end{equation}
When $2\eta-1<0$, we have as for the case of $\mu_Q$ in Section \ref{subsec3.1} that the whole expression in \eqref{y70} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. When $2\eta-1\geq 0$, we get as in the case of $\mu_Q$ after substitution $v = t\sqrt{{2}/{(s\,\alpha })}$ for the expression in \eqref{y70}
\begin{equation*}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}\,\int_0^\infty\frac{t^2\,(z_{\rm sp}^2-1+O(t^2/s))(1+O(t^2/s))}{(d^2(s)+t^2)^2\,(1+O(1/s^{\eta})+O(t^2/s))}~\frac{B\,{\rm e}^{{-}t^2}}{1-B\,{\rm e}^{{-}t^2}}{\rm d} t.
\end{equation*}
When $2\eta-1\geq 0$, the leading order behavior of $\sigma_Q^2$ depends crucially on the factor $z_{\rm sp}^2-1+O(t^2/s)$, where
\begin{equation*}
z_{\rm sp}^2-1 = \frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\left(1+O(s^{-\eta})\right)
\end{equation*}
is dominant when $\eta<1$, while the $O(t^2/s)$ is dominant when $\eta>1$. In the case that $\eta\in(1/2,1)$, we get for the leading order behavior of $\sigma_Q^2$
\begin{align*}
\frac{2}{\pi}\,\Big(\frac{s\,\alpha }{2}\Big)^{3/2}& \,\frac{2\,\gamma\,\mu}{\sigma^2\,s^\eta}\,\int_0^\infty\frac{t^2}{(d^2(s)+t^2)^2}\cdot~\frac{{\rm e}^{{-}d^2(s)-t^2}}{1-{\rm e}^{{-}d^2(s)-t^2}}{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{\gamma\,\sigma}{\pi}\,\Big(\frac{2}{\mu}\Big)^{1/2}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right),
\end{align*}
where \eqref{e26}, \eqref{e27} and \eqref{e42} have been used for $\alpha = g''(z_{\rm sp})$ and where $G_3$ is given in \eqref{e72}.
When we insert the expansion \eqref{e83} for $G_3(b)$, with the whole series on the second line being $O(b^2)$, we get the leading order behavior of $\sigma_Q^2$ as
\begin{align}
s^{2\eta}\,\Big( \frac{\sigma^4}{4\,\gamma^2\mu^2}- \frac{\sigma^2}{4\,\mu}&\,\frac{1}{s^{2\eta-1}} - \Big(\frac{2\,\sigma^2}{\pi\,\mu}\Big)^{1/2}\,\frac{\gamma\,\zeta(-1/2)}{s^{3\eta-3/2}}\nonumber\\
& - \frac{\gamma^2}{24\,s^{5\eta-5/2}}+O(s^{1-4\eta})\Big)\,\left(1+O(s^{\eta-1})\right)\nonumber \\
&\ = s^{2\eta}\,\frac{\sigma^4}{4\,\gamma^2\,\mu^2}\,\Big(1+O(s^{\max(1-2\eta,\eta-1)})\Big)\label{y74}
\end{align}
when $\eta\in(1/2,1)$. For the case $\eta=1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{align}
\frac{\sigma^2 s}{\mu}\left[ \frac{1}{8\,b_0^2} - \frac{1}{4}-\frac{1}{12}\,b_0^2 - \frac{2\,\zeta(-1/2)}{\sqrt{\pi}}\,b_0- \frac{4}{\sqrt{\pi}}\,\sum_{r=0}^\infty \frac{\zeta(-3/2-r)\,(-1)^r\,b_0^{2r+3}}{r!\,(2r+1)\,(2r+2)\,(2r+3)} \right]\label{y75}
\end{align}
with relative error $O(s^{-1/2})$. The expression between brackets in \eqref{y75} coincides with the right-hand side of \cite{Janssen2006}, (2.3) with $\beta = b_0\,\sqrt{2}$.
This leads to the following two results.
\begin{theorem} \label{varthm}
For $\eta\in[1/2,1)$,
\begin{equation*}
\sigma_Q^2 = \frac{\gamma\,\sigma_X}{\pi}\,\sqrt{\frac{2}{\mu}}\,s^{3/2-\eta}\,G_3(d(s))\,\left(1+O(s^{\eta-1})\right)
\end{equation*}
with $G_3$ given in \eqref{e72}.
\end{theorem}
\begin{proposition}\label{varprop}
For $\eta\in(0,1/2)$, and for all $b<b_0$,
\begin{equation*}
\sigma_Q^2 = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation*}
For $\eta = 1/2$, $\sigma_Q^2$ equals expression \eqref{y75} with relative error $O(s^{-1/2})$. For $\eta\in(1/2,1)$ and $b_0\in(0,\sqrt{2\pi})$, $\sigma_Q^2$ has the form in \eqref{y74}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level
with $\eta=1/2$, it is possible to give a correction term which involves now integrals and series with $\zeta$-functions as considered in \cite[Secs.~4-5]{Janssen2007}.
\subsection{The empty-system probability} \label{sec5}
We have from \eqref{e6} by proceeding as in \eqref{e13}--\eqref{e17} that
\begin{align} \label{e100}
{\rm ln}\,[\mathbb{P}(Q=0)] & = \frac{s}{2\pi i}\,\int_{|z|=1+\varepsilon}\,{\rm ln}\Big(\frac{z}{z-1}\Big)\,\frac{g'(z)\,\exp(s\,g(z))}{1-\exp(s\,g(z))}\,{\rm d} z \nonumber \\[3.5mm]
& = \frac{1}{2\pi i}\,\int_{|z|=1+\varepsilon}\,\frac{1}{z(z-1)}\,{\rm ln}\left(1-\exp(s\,g(z))\right)\,{\rm d} z ,
\end{align}
where in the last step we used partial integration (noting that ${\rm Re}\,[g(z)]<0$ on $|z|=1+\varepsilon$). Then, as in Section \ref{sec1} for $\mu_Q$, the last integral in \eqref{e100} is, with exponentially small error, given by
\begin{equation} \label{e101}
\frac{1}{2\pi i}\,\int_{-\frac12\delta}^{\frac12\delta}\,\frac{z'(v)}{z(v)(z(v)-1)}\,{\rm ln}\left(1-B\,\exp\big(-\tfrac{1}{2} s\,\alpha v^2\big)\right)\,{\rm d} v .
\end{equation}
Now for $v\geq0$ from $z({-}v)=z^{\ast}(v)$, $z'({-}v)={-}(z'(v))^{\ast}$,
\begin{align*}
& \frac{z'(v)}{z(v)(z(v)-1)}+\frac{z'({-}v)}{z({-}v)(z({-}v)-1)}=2i\,{\rm Im}\,\Big[\frac{z'(v)}{z(v)(z(v)-1)}\Big] \nonumber \\
& \qquad \qquad =~2i\,{\rm Im}\,\Big[\frac{z'(v)\,z^{\ast}(v)(z^{\ast}(v)-1)}{|z(v)|^2\,|z(v)-1|^2}\Big] \nonumber \\
& \qquad \qquad =~2i\,\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}+O(v^2))((z_{\rm sp}-1)^2+v^2-2c_2(z_{\rm sp}-1)\,v^2+O(v^4))}\,,
\end{align*}
where we used \eqref{e32} and the fact that $z_{\rm sp}$ and $c_2$ are real with $z_{\rm sp}>1$. Therefore, we get for the expression in \eqref{e101}
\begin{align}
\frac{1}{\pi} &\int_0^{\frac{1}{2}\delta}\frac{1}{z_{\rm sp}{\rm +}O(v^2)}\frac{z_{\rm sp}-1+O(v^2)}{(z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)v^2)+O(v^4)} \nonumber \\
& \qquad \qquad \qquad \qquad \cdot {\rm ln}\left(1-B\exp(-\tfrac12 s\,\alpha v^2)\right){\rm d} v.
\label{y77}
\end{align}
In the case that $2\eta-1<0$, we have as earlier that the whole expression in \eqref{y77} is $O(\exp({-}b^2\,s^{1-2\eta}))$ for any $b\in(0,b_0)$, as $s\to \infty$. In the case that $2\eta-1\geq 0$, we substitute $v=t\sqrt{{s}/{(2\,\alpha )}}$, and we get as earlier for the expression \eqref{y77}, assuming also that $\eta<1$,
\begin{align*}
\frac{1}{\pi}&\,\sqrt{s\,\alpha /2}\,\int_0^{\infty}\frac{z_{\rm sp}-1+O(t^2/s)}{(d^2(s)+t^2)\,(1+O(s^{-\eta})+O(t^2/s))}\,{\rm ln}(1-B\,{\rm e}^{-t^2}) \,{\rm d} t\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{\sqrt{s\,\alpha /2} \ (z_{\rm sp}-1)}{d^2(s)+t^2}{\rm ln}(1-B\,{\rm e}^{-t^2})\,{\rm d} t\,\left(1+O(s^{\eta-1})\right)\nonumber\\
&= \frac{1}{\pi}\,\int_0^{\infty}\frac{d(s)}{d^2(s)+t^2}{\rm ln}(1-{\rm e}^{{-}d^2(s)-t^2})\,{\rm d} t\,\left(1+O(s^{\eta-1})\right).
\end{align*}
Here we also used \eqref{y46} and that $1/s^{3\eta-1} = O(d^2(s)/s^\eta)$, so that
\begin{equation*}
(\tfrac12\,s\,\alpha )^{1/2}\,(z_{\rm sp}-1) = d(s)\,\left(1+O(s^{-\eta})\right) = d(s)\left(1+O(s^{\eta-1})\right),
\end{equation*}
since $\eta\geq 1/2$.
We have for $b>0$
\begin{align}
\frac{1}{\pi}&\int_0^\infty \frac{b}{b^2+t^2}\,{\rm ln}(1-\exp({-}b^2-t^2)){\rm d} t =-\frac12\sum_{k=0}^{\infty}\,\frac{1}{k+1}\,{\rm erfc}(b\,\sqrt{k+1}) = -F(b\,\sqrt{2}),\label{y80}
\end{align}
where according to \cite[Eq.~(3.3) \& (3.12)]{Janssen2006} for $\beta>0$
\begin{align}
F(\beta) &= \sum_{n=1}^\infty\,\frac{1}{n}\,\frac{1}{\sqrt{2\pi}}\,\int_{\beta\sqrt{n}}^\infty {\rm e}^{-x^2/2}dx\nonumber\\
&= -{\rm ln}\,\beta - \frac12\,{\rm ln}\,2 - \frac{1}{\sqrt{2\pi}}\,\sum_{r=0}^\infty \frac{\zeta(1/2-r)\,(-1/2)^r\,\beta^{2r+1}}{r!\,(2r+1)},\label{y81}
\end{align}
the last identity being valid for $0<\beta<2\sqrt{\pi}$.
Using \eqref{y81} with $\beta^2 = d^2(s)= b_0^2/s^{2\eta-1}$, with the entire series on the second line being $O(\beta)$, we get the leading order behavior of ${\rm ln}[\mathbb{P}(Q=0)]$ as
\begin{equation} \label{y82}
\Big({-}(\eta-1/2)\,{\rm ln}\,s+{\rm ln}(2\,b_0)+O(s^{1/2-\eta})\Big)\left(1+O(s^{\eta-1})\right)
\end{equation}
when $\eta\in(1/2,1)$. For $\eta = 1/2$, we get the leading order behavior, assuming $0<b_0<\sqrt{2\pi}$,
\begin{equation} \label{y83}
{\rm ln}(2\,b_0) + \frac{1}{\sqrt{\pi}}\,\sum_{r=0}^\infty \,\frac{\zeta(1/2-r)\,(-1)^r}{r!\,(2r+1)}\,b_0^{2r+1}
\end{equation}
with relative error $O(s^{-1/2})$. The expression \eqref{y83} coincides with ${\rm ln}[\mathbb{P}(M=0)]$ as given by \cite[Eq.~(2.1)]{Janssen2006} with $\beta = b_0\,\sqrt{2}$. The next two results summarize the above.
\begin{theorem} \label{emptythm}
For $\eta\in(1/2,1)$,
\begin{equation*}
{\rm ln}[\mathbb{P}(Q=0)] = - F\big(d(s)\,\sqrt{2}\big)\left(1+O(s^{\eta-1})\right)
\end{equation*}
with $F$ given by \eqref{y81}.
\end{theorem}
\begin{proposition} \label{emptyprop}
For $\eta\in (0,1/2)$, and for all $b<b_0$,
\begin{equation*}
{\rm ln}[\mathbb{P}(Q=0)] = O(\exp({-}b^2\,s^{1-2\eta})).
\end{equation*}
For $\eta=1/2$, ${\rm ln}[\mathbb{P}(Q=0)]$ equals $-F(b_0\,\sqrt{2})$ with a relative error $O(1/\sqrt{s})$. For $\eta\in (1/2,1)$ and $0<b_0<\sqrt{2\pi}$, ${\rm ln}[\mathbb{P}(Q=0)]$ has leading order behavior as in \eqref{y82}.
\end{proposition}
As in Section \ref{subsec3.3} for the mean congestion level case
with $\eta=1/2$, it is possible to give a correction term which involves now the integrals in \eqref{y80} and \eqref{e51}.
\section{Numerical examples}\label{numm}
\subsection{Accuracy of the approximations}
In this subsection we present a numerical example that serves to illustrate the accuracy of the derived heavy-traffic approximations. Consider the Poisson case
\begin{equation*}
\tilde X(z)={\rm e}^{z-1},\quad \mu = \sigma^2 = 1.
\end{equation*}
We fix $\mu$ and vary $n$ with the value of $s$, according to
\begin{equation*}
\vartheta = \frac{n}{s} = 1-\frac{\gamma}{s^\eta}
\end{equation*}
for some $\gamma>0$ and $\eta\geq 1/2$. To calculate the exact value of the mean congestion level we use the expression, see \cite{Boudreau1962},
\begin{equation*}
\mu_Q=\frac{\sigma_A^2}{2(s-\mu_A)}-\frac{s-1+\mu_A}{2}+\sum_{k=1}^{s-1}\frac{1}{1-z_k}.
\end{equation*}
Here $z_1,\ldots,z_{s-1}$ are the zeros of $z^s-A(z)$ in $|z|<1$. We apply the method of successive substitution described in \cite{Janssen2005} to obtain accurate numerical approximations for $z_1,...,z_{s-1}$ and consequently $\mu_Q$.
From Theorem \ref{mainthm}, we find that the leading order behavior of $\mu_Q$ is given by
\begin{equation} \label{x18}
\frac{\sqrt{2s}}{\pi}\,G_0\Big(\frac{\gamma}{\sqrt{2}\,s^{\eta-\frac{1}{2}}}\Big).
\end{equation}
In order to find the correction terms, we proceed by setting $\eta = 1/2$. Deriving constants $C_1,C_2,C_3,$ and $C_4$ for our setting and substituting these into \eqref{e71}, we get for $\mu_Q$, with an absolute error of $O(s^{-1/2})$, the approximation
\begin{equation*}
\frac{\sqrt{2\,s}}{\pi}\Big(\Big(1-\frac{\gamma}{3\,\sqrt{s}}\Big)\,G_0(b_0)-\frac{\gamma^3}{3\,\sqrt{s}}\,(\,G_3(b_0)+G_4(b_0))\Big),
\end{equation*}
which by \eqref{e49} and \eqref{e74} reduces to
\begin{equation}\label{x20}
\frac{\sqrt{2\,s}}{\pi}\,G_0(b_0)-\frac{\sqrt{2}\,\gamma}{3\,\pi}\,G_1(b_0).
\end{equation}
\begin{table}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.683 & 0.244 & 0.399 & 0.247 \bigstrut[t] \\
20 & 0.776 & 0.410 & 0.565 & 0.412 \\
50 & 0.858 & 0.739 & 0.893 & 0.741 \\
100 & 0.900 & 1.110 & 1.263 & 1.111 \\
200 & 0.929 & 1.633 & 1.787 & 1.634 \\
500 & 0.955 & 2.672 & 2.825 & 2.673 \\
1000 & 0.968 & 3.843 & 3.996 & 3.843 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 1$.}\label{tab:poisson1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rrrr|}
\hline
$s$ & $\rho$ & $\mu_Q$ & \eqref{x18} & \eqref{x20} \bigstrut \\
\hline
10 & 0.968 & 13.707 & 14.046 &13.732 \bigstrut[t] \\
20 & 0.977 & 19.533 & 19.865 &19.551\\
50 & 0.985 & 31.084 & 31.409 &31.095\\
100 & 0.990 & 44.097 & 44.419 &44.106\\
200 & 0.992 & 62.499 & 62.819 &62.505\\
500 & 0.995 & 99.008 & 99.325 &99.011\\
1000 & 0.996 & 140.152 & 140.468 & 140.154 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma = 0.1$.}\label{tab:poisson2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\eta=0.6$} & \multicolumn{2}{c|}{$\eta=0.75$} & \multicolumn{2}{c|}{$\eta=0.9$} \bigstrut\\
\hline
$s$ & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} & $\mu_Q$ & \eqref{x18} \bigstrut\\
\hline
10 & 17.781 & 18.125 & 25.970 & 26.318 & 37.553 & 37.905 \bigstrut[t] \\
20 & 27.309 & 27.647 & 44.391 & 44.734 & 71.195 & 71.541 \\
50 & 47.948 & 48.281 & 89.623 & 89.961 & 164.637 & 164.978 \\
100 & 73.245 & 73.574 & 152.031 & 152.367 & 309.353 & 309.692 \\
200 & 111.752 & 112.079 & 257.435 & 257.769 & 580.170 & 580.507 \\
500 & 195.082 & 195.409 & 515.443 & 515.776 & 1329.581 & 1329.917 \\
1000 & 297.122 & 297.448 & 870.524 & 870.857 & 2487.227 & 2487.562 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for $\gamma=0.1$ and several values of $\eta$.}\label{tab:poisson3}
\end{table}
\noindent Numerical results for $\eta=1/2$ and various values of $s$ are given in Table \ref{tab:poisson1} and \ref{tab:poisson2}, for $ \gamma = 1$ and $\gamma = 0.1$, respectively.
We note that for small $s$ the leading order approximation is still off by a significant amount, while the refinement only shows an error in the second decimal for $\gamma = 0.1$. This seems to justify the use of the correction term.
In Table \ref{tab:poisson3} we compare the approximation \eqref{x18} against the exact value of $\mu_Q$ for three values of $\eta\geq 1/2$ to assess the influence of $\eta$. Clearly, the leading order approximation is relatively accurate for all three scenarios. As expected, the mean congestion increases along with $\eta$, since utilization approaches 1 more rapidly in this case.
\subsection{Connection to other queueing models}\label{subsec62}
As argued in the introduction, we believe that the heavy-traffic behavior for the discrete model in this chapter will up to leading order be universal for a wide range of other models (when subjected to the same heavy-traffic regime \eqref{bb}). We shall now substantiate this for many-server systems, for which under \eqref{bb}, it turns out that the mean congestion is $O(s^\eta)$. We compare the mean congestion level in our discrete queue with that in the multi-server systems $M/M/s$, $M/D/s$ and Gamma/Gamma/$s$, all with unit mean service time and occupation rate $1-\gamma/s^\eta$.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 2.5,
xmax = 6.5,
ymin = 0,
ymax = 7.2,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 6.4,0.2)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,mark = o,mark options={scale=1.25}] table[x=log_s,y=mms] {Chapter_2/tikz/novel_figure1.txt};
\addplot[thick, mark=triangle,dashed,mark options={scale=1.25,solid}] table[x=log_s,y=mds] {Chapter_2/tikz/novel_figure1.txt};
\addplot[thick,mark=square,dotted,mark options={scale=1.25,solid}] table[x=log_s,y=ggs] {Chapter_2/tikz/novel_figure1.txt};
\legend{$M/M/s$,$M/D/s$,Gamma/Gamma/$s$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ plotted against $s$ on log scale for 3 queues for $\eta=0.75$.}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = 0.2,
xmax = 6.5,
ymin = 0,
ymax = 10,
xlabel = {$\log(s)$},
ylabel = {$\log(\mu_Q)$},
x label style={at={(0.6,-0.1)}},
y label style={at={(-0.06,0.7)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 0.4,7.35)},anchor = west},
yscale = 0.8,
xscale = 1
]
\addplot[thick,only marks,mark = o,mark options={scale=1.25}] table[x=n01,y=m01] {Chapter_2/tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square,mark options={scale=1.25}] table[x=n025,y=m025] {Chapter_2/tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = *,mark options={scale=1.25}] table[x=n05,y=m05] {Chapter_2/tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = square*,mark options={scale=1.25}] table[x=n075,y=m075] {Chapter_2/tikz/novel_figure2.txt};
\addplot[thick,only marks,mark = triangle*,mark options={scale=1.25}] table[x=n1,y=m1] {Chapter_2/tikz/novel_figure2.txt};
\addplot[dashed] coordinates{ (0,2.25) (7,9.25) };
\addplot[dashed] coordinates{ (0,2.25) (7,7.5) };
\addplot[dashed] coordinates{ (0,2.25) (7,5.75) };
\addplot[dashed] coordinates{ (0,2.25) (7,4) };
\addplot[dashed] coordinates{ (0,2.25) (7,2.95) };
\legend{\ $\eta=0.1$,\ $\eta=0.25$,\ $\eta=0.5$,\ $\eta=0.75$,\ $\eta=1$}
\end{axis}
\end{tikzpicture}
\caption{$\mu_Q$ of $M/M/s$ plotted against $s$ on log scale for different values of $\eta$.}
\label{fig2}
\end{figure}
Figure \ref{fig1} shows on logarithmic scale the mean congestion levels for $\gamma=0.1$ and $\eta=0.75$ under the specified scaling for three systems. We also display three lines with slope 0.75 for comparison, which confirms that mean congestion levels are of the order $s^\eta$, also in these multi-server system. Formally establishing this heavy-traffic behavior for these multi-server system is an important open problem and requires other mathematical approaches than the ones taken in this chapter (see the introduction for more details).
Figure \ref{fig2} shows the mean queue length in the $M/M/s$ system for several values of $\eta$, again on logarithmic scale, together with lines with slope $\eta$. For $\eta\geq 1/2$, we see the same $O(s^\eta)$ behavior, similar as for $\mu_Q$ in our discrete model. For $\eta<1/2$ the mean queue length decays, again in agreement with our results for $\mu_Q$. We note that this qualitative behavior of the $M/M/s$ system was also observed by \cite[Thm.~4.1]{maman}, by proving that the mean waiting time in the $M/M/s$ queue under \eqref{bb} is of order $1/s^{1-\eta}$, which by Little's law implies the mean queue length to be of order $s^\eta$.
\chapter{Overdispersion}
\begin{chapterstart}
Arrival processes to service systems often display fluctuations that are wilder than anticipated under the Poisson assumption, a phenomenon that is referred to as \textit{overdispersion}.
Motivated by this, we analyze a class of discrete stochastic models for which we derive heavy-traffic approximations that are scalable in the system size.
Subsequently, we show how this leads to novel capacity sizing rules that acknowledge the presence of overdispersion. This, in turn, leads to robust approximations for performance characteristics of systems that are of moderate size and/or may not operate in heavy traffic.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Robust heavy-traffic approximations for\\ service systems facing overdispersed demand}\\
\textit{Britt Mathijsen, Guido Janssen, Johan van Leeuwaarden \& Bert Zwart}\\
arxiv.org/abs/1512.05581
\end{flushright}
\newpage
\section{Introduction}\label{intro}
In the previous chapter, we analyzed the scaling limit of a queueing model in which demand exhibits stochastic fluctuations that are asymptotically proportional to the square-root of the nominal load, while we deliberately chose to deviate from the square-root staffing principle by allocating a variability hedge that does not match the order of these fluctuations.
This chapter in some ways does the opposite.
We assume the demand faced by the queueing system is more volatile than anticipated by the independent many-sources paradigm that leads to Poisson traffic models.
As will become clear in this chapter, this in fact \emph{requires} an adaption of the square-root staffing principle in order to maintain the desirable properties of QED regime.
We start by motivating our research through empirical evidence of the presence of so-called \emph{overdispersion} in arrival processes faced by service systems reported by recent literature. \\
\noindent
\textbf{Motivation.}
The bulk of the queueing literature assumes perfect knowledge about the model primitives, including the mean demand per time period. For large-scale service systems, like health care facilities, communication systems or call centers, the dominant assumption is that demand arrives according to a (non)homogeneous Poisson process, which in practice translates into the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies shows that the variance of demand typically deviates from the mean significantly. Recent work of \cite{Kim2015b,Kim2015a} reports variance being strictly less than the mean in health care settings employing a appointment booking system. This reduced variability, known as underdispersion, can be accredited to the goal of the booking system to create a more predictable arrival pattern. On the other hand, in other scenarios with no control over the arrivals, the variance typically dominates the mean, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2015, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}. The feature that variability is higher than one expects from the Poisson assumption is referred to as overdispersion. The latter concept will be the center of our attention in this chapter.
Stochastic models with the Poisson assumption have been widely applied to optimize capacity levels in service systems. The goal is to minimize operating costs while providing sufficiently high QoS in terms of performance measures such as mean delay or excess delay. When stochastic models, however, do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly under critical loading.
\\
\\*
\noindent
\textbf{Causes of overdispersion.}
The literature discussed above proves that the presence of overdispersion is widespread across applications.
It however does not specify what causes the increased variability in the arrival process.
We name two possible explanations.
First, we revisit the many-sources characterization of demand inflow discussed in Chapter 2.
Recall that in this setting, demand is generated by a large number $n$ of stochastically identical and independent sources, for that workload arriving to the system in period $j$ is given by $A^{(n)}_j = \sum_{i=1}^n A_{i,j}$, where $A_{i,j}$, $i=1,2,\ldots,n$ are i.i.d.~random variables.
This resulted in nominal workload $\mu_n = n\mu$ and $\sigma_n^2 = n\sigma^2$, thus both of order $n$.
If we now relax the assumption on the (pairwise) independence of the sources, but rather consider the scenario in which these are positively correlated, then the nominal load remain to be equal to $n \mu$, while the variance of demand becomes
\begin{equation*}
\sigma_n^2 = {\rm Var}\, A_j^{(n)} = n\, {\rm Var}\, A_{i,1} + n(n-1)\,{\rm Cov}(A_{1,j},A_{2,j}),
\end{equation*}
which is of higher order than $n$ if $n\,{\rm Cov}(A_{1,j},A_{2,j}) \to \infty$ as $n\to\infty$.
A second interpretation of overdispersion in arrival processes relates to \emph{arrival rate uncertainty}.
The canonical process for modeling the arrival process of a service systems is the Poisson process with a given arrival rate $\lambda$.
Since model primitives, in particular the arrival rate, are typically estimated through historical data, these are prone to be subject to forecasting errors.
In the realm of Poisson processes, this inherent uncertainty can be acknowledged by viewing the arrival rate $\Lambda_n$ itself as being stochastic. The resulting doubly stochastic Poisson process, also known as Cox process (first presented in \cite{Cox1955}), implies that demand in a given interval $A_j$ follows a mixed Poisson distribution.
In this case, the expected demand per period equals $\mu_n = \mathbb{E}[\Lambda_n]$, while the variance is $\sigma_n^2 = \mathbb{E}[\Lambda_n]+{\rm Var}\,\Lambda_n$.
By selecting the distribution of the mixing factor $\Lambda_n$, the magnitude of overdispersion can be made arbitrarily large, and only a deterministic $\Lambda_n$ leads to a true Poisson process.
The mixed Poisson model presents a useful way to fit both the mean and variance to real data, particularly in case of overdispersion.
The mixing distribution can be estimated parametrically or non-parametrically, see \cite{koolejongbloed,maman}.
A popular parametric family is the Gamma distribution, which gives rise to an effective data fitting procedure that uses the fact that a Gamma mixed Poisson random variable follows a negative binomial distribution.
We will in this chapter adopt the assumption of a Gamma-Poisson mixture as the demand process.\\
\\*
\textbf{Adapted QED scaling.} To deal with overdispersion
new models are needed, scaling rules must be adapted, and existing capacity sizing rules need to be modified in order to incorporate a correct hedge against (increased) variability.
In this chapter, we consider an extension of the discrete queueing model of Chapter 2 that has a doubly stochastic Poisson process as input, $A_j\sim\,{\rm Pois}(\Lambda_n)$ and we identify the heavy-traffic regime in which it displays QED behavior.
That is, it fits the three asymptotic characteristics in Section 1.2.3 of this thesis.
As we argued in that particular section, a sensible candidate capacity allocation rule is $s_n = \mu_n + \beta \sigma_n$ for some $\beta>0$, which is equivalent to the scaling
\begin{equation*}
\frac{\mu_n}{\sigma_n}\,(1-\rho_n) \to \beta.
\end{equation*}
We will verify mathematically that this is asymptotically the appropriate choice.
Studies that have adressed similar capacity allocation problems with stochastic arrival rates include \cite{Kocaga2015, maman, Whitt1999, Whitt2006}.
Of the aforementioned papers, our work best relates to \cite{maman}, in the sense that we also assess the asymptotic performance of queueing system having a stochastic arrival rate in heavy traffic.
We therefore expand the paradigm of the QED regime, in order to have it accommodate for overdispersed demand that follows from a doubly stochastic Poisson process.
\\
\\*
\textbf{Structure of the chapter}. The remainder of this chapter is structured as follows. Our model is introduced in Section \ref{modelSection} together with some preliminary results.
In Section 3.3 we derive the classical heavy-traffic scaling limits for the queue length process in the presence of overdispersed arrivals both for the moments and the distribution itself.
Section 3.4 presents our main theoretic result, which provides a robust refinement to the heavy-traffic characterization of the queue length measures in pre-limit systems.
In Section 3.5, we describe the numerical results and demonstrate the heavy-traffic approximation for a real data set coming from a health care setting. Section 3.6 provides some concluding remarks.
\section{Model description}\label{modelSection}
We consider the same mathematical model as in Section 2.2, in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,...$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
The system has a service capacity $s_n\in\mathbb{N}$ per period, the steady-state queue length can be characterized as, see (1.27),
\begin{equation}
\label{mm3}
Q^{(n)} {\;\buildrel{d}\over= \;} \max_{k\geq 0}\Bigl\{\sum_{i=1}^k (A^{(n)}_i-s_n)\Bigr\}.
\end{equation}
For brevity, we define $\mu_n:= \mathbb{E} [A^{(n)}_1]$ and $\sigma_n^2 = {\rm Var}\, A^{(n)}_1$.
The behavior of $Q^{(n)}$ predominantly depends on the characteristics of $A^{(n)}$ and $s_n$. As noted before, $\mu_n<s_n$ is a necessary condition for the maximum in \eqref{mm3} to be finite and consequently for the queue to be stable. Before continuing the analysis of $Q^{(n)}$, we impose a set of conditions on the asymptotic properties of $s_n,\mu_n$ and $\sigma_n$.
\begin{assumption}
\label{as1}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item[{\normalfont (a)}] {\rm (Asymptotic growth)}
\begin{equation*}
\mu_n,\sigma_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (b)}] {\rm (Persistence of overdispersion)}
\begin{equation*}
\sigma_n^2/\mu_n \to \infty \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (c)}] {\rm (Heavy-traffic condition)}
The utilization $\rho_n := \mu_n/s_n \to 1$ as $n\to\infty$, while
\begin{equation}\label{mm5}
s_n = \mu_n + \beta\, \sigma_n,
\end{equation}
for some $\beta > 0$. This is equivalent to requiring
\begin{equation}\label{mm4}
(1-\rho_n)\frac{\mu_n}{\sigma_n} \to \beta.
\end{equation}
\end{enumerate}
\end{assumption}
\noindent
Assumption \ref{as1} is assumed to hold throughout the remainder of this chapter.
Since we are mainly interested in the system behavior in heavy traffic, it is appropriate to study the queue length process in a scaled form. Substituting $s_n$ as in Assumption \ref{as1}(c), and dividing both sides of \eqref{mm3} by $\sigma_n$, gives
\begin{equation}
\label{mm6}
\frac{Q^{(n)}}{\sigma_n} = \max_{k\geq 0} \Bigl\{{\textstyle\sum_{i=1}^k} \Bigl(\frac{A^{(n)}_i-\mu_n}{\sigma_n} - \beta\Bigr)\Bigr\}.
\end{equation}
By defining $\hat{Q}^{(n)} := Q^{(n)}/\sigma_n$ and $\hat{A}^{(n)}_i := (A^{(n)}_i-\mu_n)/\sigma_n$, we see that the scaled queue length process is in distribution equal to the maximum of a random walk with i.i.d. increments distributed as $\hat{A}^{(n)}-\beta$. Besides $\mathbb{E}[\hat{A}^{(n)}] = 0$ and ${\rm Var}\, \hat{A}^{(n)}=1$, the scaled and centered arrival counts $\hat{A}^{(n)}$ has a few other nice properties which we turn to later in this section.
The model in \eqref{mm3} is valid for any distribution of $A^{(n)}$, also for the original case where the number of arrivals follows a Poisson distribution with fixed parameter $\lambda_n$, but in that case Assumption \ref{as1}(b) does not hold. Instead, we assume $A^{(n)}$ to be Poisson distributed with uncertain arrival rate rendered by the non-negative random variable $\Lambda_n$. This $\Lambda_n$ is commonly referred to as the \emph{prior} distribution, while $A^{(n)}$ is given the name of a Poisson mixture, see \cite{Grandell1997}. Given that the moment generation function of $\Lambda_n$, denoted by $M^\Lambda_n(\cdot)$, exists, we are able to express the probability generating function (pgf) of $A^{(n)}$ through the former. Namely,
\begin{equation}
\label{mm7}
\tilde{A}^{(n)}(z) = \mathbb{E}[\mathbb{E}[ z^{A^{(n)}} | \Lambda_n ] ] = \mathbb{E}[ \exp(\Lambda_n(z-1))] = M^\Lambda_n(z-1).
\end{equation}
From \eqref{mm7}, we get
\begin{equation}
\label{mm8}
\mu_n = \mathbb{E}[A^{(n)}] = \mathbb{E}[\Lambda_n],\qquad
\sigma_n^2 = {\rm Var}\, A^{(n)} = {\rm Var}\, \Lambda_n + \mathbb{E}[\Lambda_n],
\end{equation}
so that $\mu_n<\sigma_n^2$ if $\Lambda_n$ is non-deterministic. Assumption \ref{as1}(b) hence translates to
\[{\rm Var}\, \Lambda_n/\mathbb{E}[\Lambda_n]\rightarrow \infty, \qquad n\rightarrow\infty.\]
The next result relates the converging behavior of the centered and scaled $\Lambda_n$ to that of $\hat{A}^{(n)}$.
\begin{lemma}\label{gaussStep}
Let $\mu_n,\sigma_n^2\rightarrow\infty$ and $\sigma_n^2/\mu_n\rightarrow\infty$. If
\begin{equation*}
\hat{\Lambda}_n := \frac{\Lambda_n-\mu_n}{\sigma_n}{\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{\normalfont for } n\rightarrow\infty,
\end{equation*}
then $\hat{A}^{(n)}$ converges weakly to a standard normal variable as $n\rightarrow\infty$.
\end{lemma}
\noindent
The proof can be found in Appendix \ref{formalSec}.
The prevalent choice for $\Lambda_n$ is the Gamma distribution. The Gamma-Poisson mixture turns out to provide a very good fit to arrival counts experienced by service systems, as was observed by \cite{koolejongbloed}. Assuming $\Lambda_n$ to be of Gamma type with scale and rate parameters $a_n$ and $1/b_n$, respectively, we get for the pgf of $A^{(n)}$:
\begin{equation}
\label{r0}
\tilde{A}^{(n)}(z) = \Bigl(\frac{1}{1+b_n(1-z)}\Bigr)^{a_n},
\end{equation}
in which we recognize the pgf of a negative binomial distribution with parameters $a_n$ and $1/(b_n+1)$, so that
\begin{equation*}
\label{t21}
\mu_n = a_nb_n,\qquad \sigma_n^2 = a_nb_n(b_n+1).
\end{equation*}
Note that in the context of a Gamma prior, the restrictions in Assumption \ref{as1} reduce to only two rules. For completeness, we include the revised list below.
\begin{assumption}\label{as2}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item {\rm (Asymptotic regime and persistence of overdispersion)}
\begin{equation*}
a_n, b_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item {\rm (Heavy-traffic condition)}
Let
\begin{equation*}
s_n = a_n b_n + \beta \sqrt{a_n b_n(b_n+1)},
\end{equation*}
for some $\beta>0$, or equivalently
\begin{equation*}
(1-\rho_n)\sqrt{a_n} \to \beta, \quad \text{\rm for } n\to\infty.
\end{equation*}
\end{enumerate}
\end{assumption}
The next result follows from the fact that $\Lambda_n$ is a Gamma random variable:
\begin{corollary}\label{scaledLambdaLemma}
Let $\Lambda_n\sim\text{\normalfont Gamma}(a_n,1/b_n)$, $A^{(n)}\sim{\rm Pois }(\Lambda_n)$ and $a_n,b_n\rightarrow \infty$. Then $\hat{A}^{(n)}$ converges weakly to a standard normal random variable as $n\rightarrow \infty$.
\end{corollary}
\begin{proof}
By Lemma \ref{gaussStep}, it is sufficient to prove that $\hat{\Lambda}_n{\;\buildrel{d}\over\Rightarrow\;}\mathcal{N}(0,1)$ for this particular choice of $\Lambda_n$.
We do this by proving the pointwise convergence of the characteristic function (cf) of $\hat{\Lambda}_n$ to $\exp({-} t^2/2)$, the cf of the standard normal distribution.
Let $\phi_{G}(\cdot)$ denote the characteristic function of a random variable $G$. By basic properties of the cf,
\begin{align*}
\phi_{\hat{\Lambda}_n}(t) &= {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n)
= {\rm e}^{-i\mu_nt/\sigma_n} \Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)^{-a_n}\nonumber\\
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n}\, - a_n\,{\rm ln}\Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)\Bigr]\nonumber\\
\label{g13d}
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n} -a_n\Bigl( {-}\frac{i\,b_nt}{\sigma_n} + \frac{b_n^2t^2}{2\sigma_n^2} + O( b_n^3/\sigma_n^3)\Bigr)\Bigr] \nonumber\\
&= \exp\Bigl[ -\frac{b_n\,t^2}{2(b_n+1)} + O\left(1/\sqrt{a_n}\right)\Bigr] \rightarrow \exp\big({-} t^2/2\big),
\end{align*}
for $n\rightarrow\infty$. By L\'evy's continuity theorem this implies $\hat{\Lambda}_n$ is indeed asymptotically standard normal.
\end{proof}
The characterization of the arrival process as a Gamma-Poisson mixture be of vital importance in later sections.
\subsection{Expressions for the stationary distribution\label{expressionsSubsec}}
Our main focus is on the stationary queue length distribution, denoted by
\[\mathbb{P}(Q^{(n)}=i) =\lim_{k\rightarrow\infty} \mathbb{P}(Q^{(n)}(k)=i).\]
Denote the pgf of $Q^{(n)}$ by
\begin{equation*}
\label{t1}
\tilde{Q}^{(n)}(w) := \sum_{i=0}^\infty \mathbb{P}(Q^{(n)}=i) w^i.
\end{equation*}
Furthermore, let $\mu_Q := \mathbb{E}[Q^{(n)}]$ and $\sigma_Q^{2} := {\rm Var}\, Q^{(n)}$ denote the stationary mean and variance of the queue length, respectively.
To avoid notational complexity, we omit the superscript $(n)$ in these definitions.
To continue our analysis of $Q^{(n)}$, we need one more condition on $A^{(n)}$.
\begin{assumption}\label{as3}
The pgf of $A^{(n)}$, denoted by $\tilde{A}^{(n)}(w)$, exists within $|z|<r_0$, for some $r_0>1$, so that all moments of $A^{(n)}$ are finite.
\end{assumption}
We next recall two characterizations of $\tilde{Q}^{(n)}(w)$ that play prominent roles in the remainder of our analysis.
The first characterization of $\tilde{Q}^{(n)}(w)$ originates from a random walk perspective. As we see from \eqref{mm3}, the (scaled) stationary queue length is equal in distribution to the all-time maximum of a random walk with i.i.d. increments distributed as $A^{(n)}-\beta$ (or $\hat{A}^{(n)}-\beta$ in the scaled setting). Spitzer's identity, see e.g. \cite[Theorem VIII4.2]{Asmussen2003} and Section 1.2.2 of this thesis, then gives
\begin{equation*}
\label{t3}
\tilde{Q}^{(n)}(w) = \exp\left\{\sum_{k=1}^\infty \frac{1}{k}\,Big(\mathbb{E}\Big[w^{\left(\sum_{i=1}^k \{A^{(n)}_i-s_n\}\right)^+}\Big]-1\Big)\right\},
\end{equation*}
where $(x)^+ = \max\{x,0\}$. Hence,
\begin{equation*}
\label{t4}
\mu_Q = \tilde{Q}^{(n)\prime}(1) = \sum_{k=1}^\infty \frac{1}{k}\mathbb{E}\Bigl[ {\sum_{i=1}^k} (A^{(n)}_i - s_n) \Bigr]^+,
\end{equation*}
\begin{equation*}
\label{t4a}
\sigma^{2}_Q = \tilde{Q}^{(n)\prime\prime}(1)+Q^{(n)\prime}(1)-\left(\tilde{Q}^{(n)\prime}(1)\right)^2 = \sum_{k=1}^\infty \frac{1}{k}\mathbb{E}\Bigl[ \Big(\sum_{i=1}^k (A^{(n)}_i - s_n) \Big)^+\Bigr]^2,
\end{equation*}
\begin{align*}
\label{t5}
\mathbb{P}(Q^{(n)}=0) = \tilde{Q}_n(0) &= \exp\Bigl\{{-}{\sum_{k=1}^\infty}\frac{1}{k} \mathbb{P}\Bigl({\textstyle\sum_{i=1}^k} (A^{(n)}_i-s_n) > 0\Bigr) \Bigr\}.
\end{align*}
A second characterization follows from Pollaczek's formula, see \cite{Abate1993} and Section 2.2.2 of this thesis:
\begin{equation}
\label{t6}
\tilde{Q}^{(n)}(w) = \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{w-z}{1-z}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\},
\end{equation}
which is analytic for $|w|<r_0$, for some $r_0>1$. Therefore, $\varepsilon>0$ has to be chosen such that $|w|<1+\varepsilon<r_0$. This gives
\begin{align}
\label{t7}
\mu_Q &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{1}{1-z}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)} {\rm d} z,\\
\label{t7a}
\sigma_Q^{2} &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{{-}z}{(1-z)^2}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z,\\
\label{t8}
\mathbb{P}(Q^{(n)}=0) &= \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{z}{z-1}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\}.
\end{align}
\section{Heavy-traffic limits}
In this section we present the result on the convergence of the discrete process $\hat{Q}^{(n)}$ to a non-degenerate limiting process and of the associated stationary moments. The latter requires an interchange of limits. Using this asymptotic result, we derive two sets of approximations for $\mu_Q$, $\sigma^2_Q$ and $\mathbb{P}(Q^{(n)}=0)$, that capture the limiting behavior of $Q^{(n)}$. The first set provides a rather crude estimation for the first cumulants of the queue length process for any arrival process $A^{(n)}$ satisfying Assumption \ref{as1}. The second set, which is the subject of the next section, is derived for the specific case of a Gamma prior and is therefore expected to provide more accurate, robust approximations for the performance metrics.
We start by indicating how the asymptotic properties of the scaled arrival process give rise to a proper limiting random variable describing the stationary queue length. The asymptotic normality of $\hat{A}^{(n)}$ provides a link with the Gaussian random walk and nearly deterministic queues \cite{Sigman2011a,Sigman2011b}.
The main results in \cite{Sigman2011a,Sigman2011b} were obtained under the assumption that $\rho_n\sim 1-\beta/\sqrt{n}$, in which case it follows from \cite[Thm.~3]{Sigman2011b} that the rescaled stationary waiting time process converges to a reflected Gaussian random walk.
We shall also identify the Gaussian random walk as the appropriate scaling limit for our stationary system. However, since the normalized natural fluctuations of our system are given by $\mu_n/\sigma_n$ instead of $\sqrt{n}$, we assume that the load grows like $\rho_n \sim 1 - \frac{\beta}{\mu_n/\sigma_n}$. Hence, in contrast to \cite{Sigman2011a,Sigman2011b}, our systems' characteristics display larger natural fluctuations, due to the mixing factor that renders the arrivals. Yet, by matching this overdispersed demand with the appropriate hedge against variability, we again obtain Gaussian limiting behavior. This is not surprising, since we saw in Lemma \ref{gaussStep} that the increments start resembling Gaussian behavior for $n\rightarrow\infty$. The following result summarizes this.
\begin{theorem}
\label{gaussianThm}
Let $\Lambda_n$ be a non-negative random variable such that $(\Lambda_n-\mu_n)/\sigma_n$ is asymptotically standard normal, with $\mu_n$ and $\sigma_n$ as defined in \eqref{mm8}, and $\mathbb{E}[\Lambda_n^3]<\infty$ for all $n\in\mathbb{N}$. Then under Assumption \ref{as1}, for $n\rightarrow \infty$,
\begin{enumerate}
\item[{\rm (i)}] $\hat{Q}^{(n)} {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[{\rm (ii)}] $\mathbb{P}(Q^{(n)} = 0) \rightarrow \mathbb{P}(M_\beta=0)$,
\item[{\rm (iii)}] $\mathbb{E}[\hat{Q}^{(n)}] \rightarrow \mathbb{E} [M_\beta]$,
\item[{\rm (iv)}] $\sigma^2_Q \rightarrow {\rm Var}\,\, M_\beta$,
\end{enumerate}
where $M_\beta$ is the all-time maximum of a random walk with i.i.d. normal increments with mean $-\beta$ and unit variance.
\end{theorem}
The proof of Theorem \ref{gaussianThm} is given in Appendix \ref{formalSec}. The following result shows that Theorem \ref{gaussianThm} also applies to Gamma mixtures, which is a direct consequence of Corollary \ref{scaledLambdaLemma}.
\begin{corollary}
Let $\Lambda_n\sim$ \normalfont{Gamma}$(a_n,b_n)$. Then under Assumption \ref{as2} the four convergence results of Theorem \ref{gaussianThm} hold true.
\end{corollary}
It follows from Theorem \ref{gaussianThm} that the scaled stationary queueing process converges under \eqref{mm4} to a reflected Gaussian random walk. Hence, the performance measures of the original system should be well approximated by the performance measures of the reflected Gaussian random walk, yielding heavy-traffic approximations.
Like our original system, the Gaussian random walk falls in the classical setting of the reflected one-dimensional random walk, whose behavior is characterized by both Spitzer's identity and Pollaczek's formula. In particular, Pollaczek's formula gives rise to contour integral expressions for performance measures that are easy to evaluate numerically, also in heavy-traffic conditions. The numerical evaluation of such integrals is considered in \cite{Abate1993}. For $\mathbb{E} [M_\beta]$ such an integral is as follows
\begin{equation}
\label{g13e}
\mathbb{E} [M_\beta] = {-}\frac{1}{\pi}\int_0^\infty {\rm Re}\Bigl[\frac{1-\phi(-z)}{z^2}\Bigr]{\rm d} y,
\end{equation}
with $\phi(z) = \exp(-\beta\,z+\tfrac12\,z^2)$, the Laplace transform of a normal random variable with mean $-\beta$ and unit variance, and $z=x+iy$ with an appropriately chosen real part $x$. Note that this integral involves complex-valued numbers. Similar expressions appear for $\mathbb{P}(M_\beta=0)$ and ${\rm Var}\, M_\beta$. The following result simply rewrites these integrals in \eqref{g13e} in terms of a real integral and uses the fact that the scaled queue length process mimics the maximum of the Gaussian random walk for large $n$.
\begin{corollary}\label{abateThm}
Under Assumption \ref{as1}, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$, $\mu_Q$ and $\sigma^2_Q$ as $n\to\infty$ is characterized by
\begin{align}
\label{h1a}
\mathbb{P}(Q^{(n)} = 0) &\approx \exp\Bigl[\frac{1}{\pi} \int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\Bigl(1-e^{-\tfrac12\beta^2-t^2}\Bigr){\rm d} t\Bigr],\\
\label{h1}
\mu_Q &\approx \frac{\sqrt{2}\sigma_n}{\pi}\int_0^\infty \frac{t^2}{\tfrac12\beta^2+t^2}\, \frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t,\\
\label{h1b}
\sigma^2_Q &\approx \frac{\sqrt{2}\beta\sigma_n^2}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t.
\end{align}
\end{corollary}
\begin{proof}
According to \cite[(15)]{Abate1993} we have for the maximum $M_\beta$ of a Gaussian random walk with drift parameter ${-}\beta$ and unit variance
\begin{equation*}
\label{z1}
{-}\,{\rm ln}\,[\mathbb{P}(M_\beta=0)] = c_0,\quad \mathbb{E}[M_\beta]\ = c_1, \quad {\rm Var}\,\, M_\beta = c_2,
\end{equation*}
where
\begin{equation*}
\label{z2}
c_n = \frac{(-1)^nn!}{\pi} \,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp(\beta\,z+\tfrac12 z^2))}{z^{n+1}} {\rm d} y\Bigr],
\end{equation*}
in which $z={-}x+i\,y$, $y\geq 0$, and $x$ is any fixed number between 0 and $2\beta$. We take $x=\beta$, so that
\begin{equation*}
\label{z3}
\beta z+\tfrac12 z^2 = {-}\tfrac12\beta^2 - \tfrac12 y^2\leq 0,\quad y\geq 0.
\end{equation*}
For $n=0$, we then have
\begin{align*}
c_0 &= \frac{1}{\pi}\,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-\tfrac12 y^2))}{{-}\beta+i\,y} {\rm d} y\Bigr] \nonumber\\
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta}{\beta^2+y^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2- \tfrac12 y^2)) {\rm d} y\nonumber\\
\label{z4}
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-t^2)) {\rm d} t,
\end{align*}
where we used that
\begin{equation*}
\label{z5}
{\rm Re }\Bigl[\frac{1}{{-}\beta+i\, y}\Bigr] = \frac{{-}\beta}{\beta^2+y^2},
\end{equation*}
together with the substitution $y=t\sqrt{2}$. For $n=1,2,\ldots,$ we have by partial integration
\begin{align*}
c_n &= \frac{(-1)^n n!}{\pi} \, {\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{({-}\beta+i\,y)^{n+1}} {\rm d} y\nonumber\\
&= \frac{(-1)^{n-1}(n-1)!}{\pi}\,{\rm Im}\Bigl[\int_0^\infty {\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)){\rm d} \Bigl(\frac{1}{(-\beta+i\,y)^n}\Bigr)\Bigr]\nonumber\\
\label{z6}
&= {-}\frac{(-1)^{n-1}(n-1)!}{\pi} {\rm Im}\Bigl[ \int_0^\infty \frac{y}{(-\beta+i\,y)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr],
\end{align*}
where we have used that
\begin{equation*}
\label{z7}
{\rm Im}\Bigl[\frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{(-\beta+i\,y)^n}\Bigr]\Bigl|_0^\infty\Bigr. = 0.
\end{equation*}
Using
\begin{equation*}
\label{z8}
\frac{1}{(-\beta+i\,y)^n} = (-1)^n\,\frac{(\beta+i\,y)^n}{(\beta^2+y^2)^n},
\end{equation*}
we then get
\begin{equation*}
\label{z9}
c_n = \frac{(n-1)!}{\pi}\,{\rm Im}\,\Bigl[\int_0^\infty \frac{y(\beta+i\,y)^n}{(\beta^2+y^2)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr].
\end{equation*}
Hence for $n=1,2,$ we finally get by the substitution $y=t\sqrt{2}$
\begin{align}
c_1&=\frac{1}{\pi}\,\int_0^\infty \frac{y^2}{\beta^2+y^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y \nonumber\\
\label{z10}
&= \frac{\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{\tfrac12 \beta^2+t^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)}{\rm d} t,
\end{align}
\begin{align*}
c_2&=\frac{2\beta}{\pi}\,\int_0^\infty \frac{y^2}{(\beta^2+y^2)^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y\nonumber\\
\label{z11}
&= \frac{\beta\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)} {\rm d} t.
\end{align*}
\end{proof}
\section{Robust heavy-traffic approximations}
To obtain more accurate approximations for $\mu_Q$, $\sigma^2_Q$ and $\mathbb{P}(Q^{(n)}=0)$, using the Pollaczek's formula given in \eqref{t6}, we need to be more specific about the arrival process $A^{(n)}$ and its pgf $\tilde{A}^{(n)}(w)$. The remainder of this chapter deals with the case of the Gamma-Poisson mixture with parameters $a_n$ and $b_n$, so that $\tilde{A}^{(n)}(w)$ has the form in \eqref{r0}. As noted earlier, Gamma mixing yields an arrival process that has a negative binomial distribution, which allows us to establish the detailed asymptotic results in the next theorem.
\begin{theorem}\label{saddlepointThm}
Let $a_n,b_n$ and $s_n$ be as in Assumption \ref{as2}. Then the leading order behavior of $\mu_Q$ is given by
\begin{equation}
\label{r1}
\mu_Q = \frac{\sqrt{2}\,\beta_n}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta^2_n+t^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)} {\rm d} t\,(1+o(1)),
\end{equation}
where
\begin{equation}
\label{r2}
\beta_n^2 = s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Furthermore, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$ and $\sigma^2_Q$ is given by
\begin{equation*}
\label{r3}
\exp\Bigl[\frac{1}{\pi}\,\frac{b_n+\rho_n}{b_n+1}\,\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta^2_n+t^2}\,{\rm ln}\,\Bigl(1-{\rm e}^{{-}\tfrac12\beta^2_n-t^2}\Bigr){\rm d} t\Bigr],
\end{equation*}
and
\begin{equation}
\label{r4}
\frac{\beta_n^3/\sqrt{2}}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)^2\Bigl(\frac{b_n+1}{b_n+\rho_n}+1\Bigr)\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t,
\end{equation}
respectively.
\end{theorem}
To obtain pre-limit approximations to the queue length descriptors, we take an original approach that starts from Pollaczek's formula, see \eqref{t6}.
From this classical transform representation, contour integrals for the zero-queue probability and the mean queue length follow immediately.
Subsequently, we apply the non-standard saddle-point method---originally proposed by \cite{debruijn} and also applied in Chapter 2 of this thesis---to turn these contour integrals into practical approximations.
In contrast to the setting of Chapter 2, in which the saddle-point was bounded away from one as $n\rightarrow\infty$, the relevant saddle point and the analyticity radius tend to one, which is a singular point of the integrand, in the setting with overdispersion.
For the proof of Theorem \ref{saddlepointThm}, we therefore modify the special saddle-point method developed in Chapter 2 to account for this circumstance.
\begin{proof}
Our starting point is the probability generating function of the number of arrivals per time slot, given in \eqref{r0}, which is analytic for $|z|<1+1/b_n=:r$. Under Assumption \ref{as2}, we consider $\mu_Q$ as given in \eqref{t7}. We set
\begin{equation}
\label{a7}
g(z) = -{\rm ln }\,z+\frac{1}{s_n}\,{\rm ln }\Bigl[\tilde{A}^{(n)}(z)\Bigr] = -{\rm ln }\,z - \frac{a_n}{s_n}\,{\rm ln }\left(1+(1-z)b_n\right),
\end{equation}
to be considered in the entire complex plane with branch cuts $(-\infty,0]$ and $[r,\infty)$. The relevant saddle point $z_{\rm sp}$ is the unique zero $z$ of $g'(z)$ with $z\in(1,r_0)$. Since
\begin{equation}
\label{a8}
g'(z) = -\frac{1}{z} + \frac{\rho_n}{1+(1-z)b_n},
\end{equation}
this yields,
\begin{equation}
\label{a9}
1+(1-z_{\rm sp})b_n = \rho_n z_{\rm sp},\quad {\rm i.e., } \quad z_{\rm sp} = 1+\frac{1-\rho_n}{\rho_n+b_n}.
\end{equation}
We then find
\begin{equation}
\label{a10}
\mu_Q = \frac{s_n}{2\pi i} \int_{|z| = 1+\varepsilon} \frac{g'(z)}{z-1}\,\frac{\exp(s_n\,g(z))}{1-\exp(s_n\,g(z))}{\rm d} z,
\end{equation}
and we take here $1+\varepsilon = z_{\rm sp}$. There are no problems with the branch cuts since we consider $\exp(s_ng(z))$ with integer $s_n$. \\
We continue as in Chapter 2, Section 3 and thus we intend to substitute $z=z(v)$ in the integral in \eqref{a10}, where $z(v)$ satisfies
\begin{equation*}
\label{k1}
g(z(v)) = g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) =: q(v)
\end{equation*}
on a range ${-}\tfrac12\delta_n \leq v\leq \tfrac12 \delta_n$. Thus, we consider the approximate representation
\begin{equation}
\label{k2}
\frac{-s_n\,g''(z_{\rm sp})}{2\pi i}\int_{-\tfrac12 \delta_n}^{\tfrac12 \delta_n}\frac{v}{z(v)-1}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))} {\rm d} v
\end{equation}
of $\mu_Q$. We have to operate here with additional care, since in the present case, the analyticity radius $r=1+1/b_n$, the saddle point $z_{\rm sp}$ as well as the outside zero $r_0$ tend to 1 as $n\rightarrow\infty$. Specifically, proceeding under the assumptions that $(1-\rho_n)^2a_n$ is bounded while $a_n\rightarrow\infty$ and $b_n\geq 1$,
we have from \eqref{a9} that
\begin{equation}\label{a19}
z_{\rm sp}-1=\frac{1-\rho_n}{b_n+\rho_n} = \frac{1-\rho_n}{b_n} + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr),
\end{equation}
where the $O$-term is small compared to the first term of the right-hand side of \eqref{a19} when $b_n\rightarrow\infty$. Next, we approximate $r_0$, using that $r_0>1$ satisfies
\begin{equation*}
\label{a20}
{-}{\rm ln}\, r_0 - \frac{\rho_n}{b_n}\, {\rm ln}\,(1+(1-r_0)b_n) = 0.
\end{equation*}
Write $r_0 = 1+u/b_n$, so that we get the equation
\begin{align*}
0 &= {-}{\rm ln}\,\left(1+\frac{u}{b_n}\right) - \frac{\rho_n}{b_n}\,{\rm ln }(1-u)\nonumber \\
\label{a21}
&= {-}\frac{u}{b_n}\Bigl(1-\rho_n-\tfrac12\Bigl(\frac{1}{b_n}+\rho_n\Bigr)u-\tfrac{1}{3}\Bigl(\frac{-1}{b^2_n}+\rho_n\Bigr)u^2+\cdots\Bigr),
\end{align*}
where we have used the Taylor expansion of ${\rm ln}(1+x)$ at $x=0$. Thus we find
\begin{equation*}
\label{a22}
u=\frac{2(1-\rho_n)}{\rho_n+1/b_n}+O(u^2) = 2(1-\rho_n)+O((1-\rho_n)^2)+O\Bigl(\frac{1-\rho_n}{b_n}\Bigr),
\end{equation*}
and so,
\begin{equation*}
\label{a23}
r_0 = 1+2\,\frac{1-\rho_n}{b_n}+O\Bigl(\frac{(1-\rho_n)^2}{b_n}\Bigr) + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr).
\end{equation*}
In \eqref{k2} we choose $\delta_n$ so large that the integral has converged within exponentially small error using $\pm\delta_n$ as integration limits, and, at the same time, so small that there is a convergence power series
\begin{equation}
\label{a26}
z(v) = z_{\rm sp}+iv+ \sum_{k=2}^\infty c_k(iv)^k, \qquad \text{for } |v| \leq \tfrac12 \delta_n.
\end{equation}
To achieve these goals, we supplement the information on $g(z)$, as given by $\eqref{a7}-\eqref{a9}$, by
\begin{equation}
\label{a27}
g''(z)=\frac{1}{z^2}+\frac{\rho_nb_n}{(1+(1-z)b_n)^2},\quad g''(1) = 1+\rho_nb_n,\quad g''(z_{\rm sp}) =\frac{1}{z_{\rm sp}^2}\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Now
\begin{equation*}
\label{a36}
\exp(s_n\,q(v)) = \exp(s_n\,g(z_{\rm sp}))\exp(-\tfrac12\,s_n\,g''(z_{\rm sp})\,v^2),
\end{equation*}
and
\begin{equation*}
\label{a37} s_n\, g''(z_{\rm sp})v^2 = s_n\,b_nv^2(1+o(1)) = a_n(b_n\,v)^2(1+o(1)).
\end{equation*}
Therefore, \eqref{k2} approximates $\mu_Q$ with exponentially small error where we take $\tfrac12 \delta_n$ of the order $1/b_n$.
We next aim at showing that we have a power series for $z(v)$ as in \eqref{a26} that converges for $|v|\leq\tfrac12\delta_n$ with $\tfrac12\delta_n$ of the order $1/b_n$.
\begin{lemma}
Let
\begin{equation*}
\label{a38}
r_n:=\frac{1}{2\,b_n}-(z_{\rm sp} -1 ),\quad m_n:= \tfrac{2}{3}\rho_nr_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}},
\end{equation*}
where we assume $r_n>0$, see \eqref{a19}. Then \eqref{a26} holds with real coefficients $c_k$ satisfying
\begin{equation}
\label{a39}
|c_k|\leq\frac{r_n}{m_n^k},\quad k=2,3,\ldots.
\end{equation}
\end{lemma}
\begin{proof}
We let
\begin{equation}
\label{a40}
G(z):=\frac{2(g(z)-g(z_{\rm sp}))}{g''(z_{\rm sp})(z-z_{\rm sp})^2}.
\end{equation}
Then $G(z_{\rm sp})=1$ and so we can write \eqref{k1} as
\begin{equation}
\label{a41}
F(z):=(z-z_{\rm sp})\sqrt{G(z)} = i v
\end{equation}
when $|z-z_{\rm sp}|$ is sufficiently small. Since $F(z_{\rm sp})=0$, $F'(z_{\rm sp})=1$, the B\"urmann-Lagrange inversion theorem implies validity of a power series as in \eqref{a40}, with real $c_k$ since $G(z)$ is positive and real for real $z$ close to $z_{\rm sp}$. We therefore just need to estimate the convergence radius of this series from below.
To this end, we start by showing that
\begin{equation}
\label{a42}
{\rm Re}[g''(z)] > \frac{4}{9}\,\rho_n^2\frac{b_n+\rho_n^{-1}}{b_n+\rho_n},\quad |z-z_{\rm sp}|\leq r_n.
\end{equation}
For this, we consider the representation
\begin{equation}
\label{a43}
G(z) = 2\int_{0}^1\int_0^1 \frac{g''(z_{\rm sp}+s\,t(z-z_{\rm sp}))}{g''(z_{\rm sp})} \,t{\rm d} s{\rm d} t.
\end{equation}
We have for $\zeta\in\mathbb{C}$ and $|\zeta-1|\leq 1/2b_{n}\leq 1/2$ from \eqref{a27} that
\begin{equation}
\label{a44}
{\rm Re}[g''(\zeta)] = {\rm Re}(1/\zeta^2) + \rho_nb_n\,{\rm Re}\Bigl[\Bigl(\frac{1}{1+(1-\zeta)b_n}\Bigr)^2\Bigr]\geq \tfrac{4}{9}(1+\rho_nb_n).
\end{equation}
To show the inequality in \eqref{a44}, it suffices to show that
\begin{equation}
\label{a45}
\min_{|\xi-1|\leq 1/2} {\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{4}{9}.
\end{equation}
The minimum in \eqref{a45} is assumed at the boundary $|\xi-1|=1/2$, and for a boundary point $\xi$, we write
\begin{equation*}
\label{a46}
\xi= 1+\tfrac12\cos\theta+\tfrac12 i \sin\theta, \quad 0\leq \theta\leq 2\pi,
\end{equation*}
so that
\begin{equation*}
\label{a47}
{\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{1+\cos\theta+\tfrac{1}{4}\cos 2\theta}{(\tfrac{5}{4}+\cos\theta)^2}.
\end{equation*}
Now
\begin{equation*}
\label{a48}
\frac{{\rm d}}{d\theta} \Bigl[\frac{1+\cos\theta+\tfrac{1}{4}\cos2\theta}{(\tfrac{5}{4}+\cos\theta)^2}\Bigr] = \frac{\sin \theta\,(1-\cos \theta)}{4(\tfrac{5}{4}+\cos\theta)^3}
\end{equation*}
vanishes for $\theta=0,\pi,2\pi$, where ${\rm Re}(1/\xi^2)$ assumes the values $4/9$, 4, 4/9, respectively. This shows \eqref{a45}.
We use \eqref{a45} with $\zeta+\xi$ and with $\xi=1+(1-\zeta)b_n$, with
\begin{equation}
\label{a49}
\zeta = \zeta(s,t) = z_{\rm sp} + s\,t\,(z-z_{\rm sp}),\quad 0\leq s,\, t\leq 1,
\end{equation}
where we take $\zeta$ such that $|\zeta-1|\leq 1/2b_n$. It is easy to see from
$1<z_{\rm sp}<1+1/2b_n$ that $|\zeta-1|\leq 1/2b_n$ holds when $|z-z_{\rm sp}|\leq r_n=1/2b_n-(z_{\rm sp}-1)$. We have, furthermore, from \eqref{a9} that $0<g''(z_{\rm sp})\leq 1+b_n/\rho_n$. Using this, together with \eqref{a44} where $\zeta$ is as in \eqref{a49}, yields
\begin{equation*}
\label{a50}
{\rm Re}[G(z)] \leq \frac{4}{9}\,\frac{1+\rho_nb_n}{1+b_n/\rho_n}\,2\,\int_0^1\int_0^1 t\,{\rm d} s{\rm d} t = \tfrac{4}{9}\,\rho_n^2\,\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}
\end{equation*}
when $|z-z_{\rm sp}|\leq r_n$, and this is \eqref{a42}.\\
We therefore have from \eqref{a41} that
\begin{equation*}
\label{a51}
|F(z)|>r_n\cdot\frac{2}{3}\rho_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}} = m_n,\quad |z-z_{\rm sp}|=r_n.
\end{equation*}
Hence, for any $v$ with $|v|\leq m_n$, there is exactly one solution $z=z(v)$ of the equation $F(z)-iv=0$ in $|z-z_{\rm sp}|\leq r_n$ by Rouch\'e's theorem. This $z(v)$ is given by
\begin{equation*}
\label{a52}
z(v) = \frac{1}{2\pi i}\,\int_{|z-z_{\rm sp}|=r_n} \frac{F'(z)\,z}{F(z)-iv}{\rm d} z,
\end{equation*}
and depends analytically on $v$, $|v|\leq m_n$. From $|z(v)-z_{\rm sp}|\leq r_n$, we can finally bound the power series coefficients $c_k$ according to
\begin{equation*}
\label{a53}
|c_k| = \Bigl|\frac{1}{2\pi i}\int_{|iv|=m_n} \frac{z(v)-z_{\rm sp}}{(iv)^{k+1}}{\rm d}(iv)\Bigr| \leq \frac{r_n}{m_n^k},
\end{equation*}
and this completes the proof of the lemma.
\end{proof}
\begin{remark}
We have $z_{\rm sp}-1=o(1/b_n)$, see \eqref{a19}, and so
\begin{equation*}
\label{a54}
r_n = \frac{1}{2b_n}(1+o(1)),\quad m_n = \frac{1}{3b_n}(1+o(1)),
\end{equation*}
implying that the radius of convergence of the series in \eqref{a26} is indeed of order $1/b_n$ (since we have assumed $b_n\geq 1$).
\end{remark}
We let $\delta_n=m_n$, and we write for $0\leq v\leq \tfrac12\delta_n$
\begin{equation*}
\label{a55}
\frac{v}{z(v)-1}+\frac{{-}v}{z({-}v)-1} = \frac{-2iv\,{\rm Im}(z(v))}{|z(v)-1|^2},
\end{equation*}
where we have used that all $c_k$ are real, so that $z(-v)=z(v)^*$. Now from \eqref{a39} and realness of the $c_k$, we have
\begin{equation}
\label{a56}
{\rm Im}(z(v)) = v+\sum_{l=1}^\infty c_{2l+1}(-1)^l\,v^{2l+1} = v+O(v^3),
\end{equation}
and in similar fashion
\begin{equation}
\label{a57}
|z(v)-1|^2 = (z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)^2v^2) + O(v^4)
\end{equation}
when $0\leq v\leq \tfrac12\delta_n$. The order terms in \eqref{a56}-\eqref{a57} are negligible in leading order, and so we get for $\mu_{Q^{(n)}}$ via \eqref{k2} the leading order expression
\begin{equation*}
\label{a58}
\frac{{-}s_n\,g''(z_{\rm sp})}{2\pi i}\,\int_0^{\tfrac12\delta_n}\frac{{-}2iv^2}{(z_{\rm sp}-1)^2+v^2}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))}{\rm d} v.
\end{equation*}
We finally approximate $q(v) = g(z_{\rm sp})-\tfrac12 g''(z_{\rm sp})v^2$.
There is a $z_1$, $1\leq z_1\leq z_{\rm sp}$ such that
\begin{equation*}
\label{a59}
g(z_{\rm sp}) = {-}\tfrac12(z_{\rm sp}-1)^2\,g''(z_1),
\end{equation*}
and, see \eqref{a19} and \eqref{a27},
\begin{equation*}
\label{a60}
g''(z_1) = g''(z_{\rm sp}) + O((1-\rho_n)b_n).
\end{equation*}
Hence
\begin{align}
s_n\,q(v) &= {-}\tfrac12 s_n\,g''(z_{\rm sp})\,[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)b_ns_n(z_{\rm sp}-1)^2),\nonumber\\
&= {-}\tfrac12 s_n\,g''(z_{\rm sp})[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)^2a_n),\label{a61}
\end{align}
where \eqref{a19} has been used and $a_nb_n = s_n(1+o(1))$ Therefore, the O-term in \eqref{a61} tends to 0 by our assumption that $(1-\rho_n)^2a_n$ is bounded. Thus, we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation}\label{a62}
\frac{s_n g''(z_{\rm sp})}{\pi} \int_{0}^{\tfrac12\delta_n}\frac{v^2}{(z_{\rm sp}-1)^2+v^2}\,
\frac{\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))}{1-\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))} {\rm d} v,
\end{equation}
When we substitute $t=v\sqrt{s_n\,g''(z_{\rm sp})/2}$ and extend the integration in \eqref{a62} to all $t\geq 0$ (at the expense of an exponentially small error), we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation*}
\label{a63}
=\frac{1}{\pi}\,\sqrt{2\,s_n\,g''(z_{\rm sp})}\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta_n^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)}{\rm d} t,
\end{equation*}
where
\begin{equation*}
\label{a64}
\beta^2_n = s_n\,g''(z_{\rm sp})(z_{\rm sp}-1)^2.
\end{equation*}
Now using \eqref{a9} and \eqref{a27}, we get the result of Theorem \ref{saddlepointThm}. A separate analysis of $\beta_n$ is provided in Subsection \ref{convRobust}.
A similar analysis, modeled after the one given in Chapter 2 gives under Condition \ref{as1} the leading-order expression
\begin{equation}
\label{a65}
\frac{1}{z_{\rm sp} \pi}\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta_n^2+t^2}\,{\rm ln}(1-e^{-\tfrac12\beta_n^2-t^2}){\rm d} t
\end{equation}
for ${\rm ln}\,\mathbb{P}(Q^{(n)}=0)$. Observe that the quantity in \eqref{a65} is negative, but bounded away from ${-}\infty$ when $\beta_n$ is bounded away from 0.
Furthermore, we find for the variance of $Q^{(n)}$ the approximation
\begin{equation*}
\label{a66}
\frac{\beta_n^3/\sqrt{2}}{\pi}\frac{z_{\rm sp}+1}{(z_{\rm sp}-1)^2}\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t.
\end{equation*}
\end{proof}
\noindent
Note that we can write \eqref{r1} as
\begin{equation*}
\label{ra1}
\mu_Q = \tilde{\sigma}_n\,\mathbb{E}[ M_{\beta_n}]\quad \text{and}\quad \sigma^2_Q \approx \tilde{\sigma}^2_n\, {\rm Var}\, M_{\beta_n}
\end{equation*}
with
\begin{equation}
\label{ra5}
\tilde{\sigma}_n = \beta_n \Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr).
\end{equation}
This robust approximation for $\mu_Q$ is suggestive of the following two properties that extend beyond the mean system behavior, and hold at the level of approximating the queue by $\sigma_n$ times the Gaussian random walk:
\begin{itemize}
\item[\rm (i)] At the process level, the space should be normalized with $\sigma_n$, as in \eqref{mm7}. The approximation \eqref{r1} suggests that it is better to normalize with $\tilde{\sigma}_n$. Although $\tilde \sigma_n\to\sigma_n$ for $n\to\infty$, the $\tilde \sigma_n$ is expected to lead to sharper approximations for finite $n$.
\item[\rm (ii)] Again at the process level, it seems better to replace the original hedge $\beta$ by the robust hedge $\beta_n$. This thus means that the original system for finite $n$ is approximated by a Gaussian random walk with drift $-\beta_n$. Apart from this approximation being asymptotically correct for $n\to \infty$, it is also expected to approximate the behavior better for finite $n$.
\end{itemize}
\section{Numerical \& empirical studies}
\subsection{Convergence of the robust hedge\label{convRobust}}
We next examine the accuracy of the heavy-traffic approximations for $\mu_Q$ and $\sigma^2_Q$, following Corollary \ref{abateThm} and Theorem \ref{saddlepointThm}. We expect the robust approximation to be considerably better than the classical approximation when $\beta_n$ and $\tilde{\sigma}_n$ differ substantially from their limiting counterparts. Before substantiating this claim numerically, we present a result on the convergence rates of $\beta_n$ to $\beta$ and $\tilde{\sigma}_n$ to $\sigma_n$.
\begin{proposition}\label{gammanProp}
Let $a_n,b_n$ and $s_n$ as in Assumption \ref{as2}. Then
\begin{equation}
\label{r3a}
\beta_n^2 = \beta^2\Bigl(1 - \frac{1}{1+b_n+\sigma_n/\beta}\Bigr).
\end{equation}
\end{proposition}
\begin{proof}
From \eqref{r2}, we have
\begin{align*}
\beta_n^2 &= s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr)= \frac{1}{s_n}\Bigl(\frac{s_n-a_nb_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{s_n}{a_n}\Bigr)\nonumber\\
\label{x1}
&= \frac{1}{s_n}\frac{\beta^2\,a_nb_n(b_n+1)}{(b_n+1)^2}\Bigl(1+\frac{s_n}{a_n}\Bigr) = \beta^2\,\frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) =:\beta^2\,\bar{F}_n.
\end{align*}
Now consider the factor $\bar{F}_n$.
\begin{align*}
\bar{F_n} &= \frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) = \frac{b_n}{b_n+1}+\frac{1}{b_n+1}\,\frac{a_nb_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\Bigl(1-\frac{a_nb_n}{s_n}\Bigr) = 1-\frac{1}{b_n+1}\,\frac{\beta\,\sigma_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\frac{1}{1+\frac{\mu_n}{\beta\sigma_n}} = 1-\frac{1}{b_n+1+\frac{1}{\beta}\sqrt{a_nb_n(b_n+1)}},
\end{align*}
which together with $\sigma_n^2=a_nb_n(b_n+1)$ proves the proposition.
\end{proof}
Note that $\beta_n$ always approaches $\beta$ from below. Also, \eqref{r3a} shows that $b_n$ is the dominant factor in determining the rate of convergence of $\beta_n$.
\begin{proposition}\label{sigmanProp}
Let $\tilde{\sigma}_n$ as in \eqref{ra5}. Then
\begin{equation*}
\tilde{\sigma}_n = \sigma_n + b_n\beta_n + O(1).
\end{equation*}
\end{proposition}
\begin{proof}
Straightforward calculations give
\begin{align*}
\tilde{\sigma}_n &= \beta_n\,\Bigl(\frac{s_nb_n+a_nb_n}{s_n-a_nb_n}\Bigr) \nonumber\\
&= \frac{\beta_n}{\beta}\,\frac{b_n}{\sigma_n}\,(s_n+a_n)
= \frac{\beta_n}{\beta}\,\sqrt{\frac{b_n}{a_n(b_n+1)}}\left(a_n(b_n+1)+\beta\sqrt{a_nb_n(b_n+1)}\right)\nonumber\\
&= \frac{\beta_n}{\beta}\left(\sqrt{a_nb_n(b_n+1)}+\beta b_n\right) = \frac{\beta_n}{\beta}\,\sigma_n + \beta_n b_n.
\end{align*}
Applying Proposition \ref{gammanProp} together with the observation
\begin{equation*}
\sigma_n \sqrt{1 - \frac{1}{1+b_n+\sigma_n/\beta}} = \sigma_n(1 + O(1/\sqrt{a_n}b_n)) = \sigma_n + O(1)
\end{equation*}
yields the result.
\end{proof}
In Figure \ref{fig:convHedge}, we visualize the convergence speed of both parameters in case $\mu_n=n$, $\sigma_n = n^\delta$ with $\delta=0.7$ and $\beta=1$. This implies $a_n = n/(n^{2\delta}-1)$ and $b_n = n^{2\delta}-1$.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.05,
xlabel = {$x$},
ylabel = {$\tilde{\beta}_n/\beta_n$},
y label style={at={(0.04,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.1)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {./tikz/gamman.txt};
\addplot[thick,col4] table[x=n,y=d075] {./tikz/gamman.txt};
\addplot[thick,col5] table[x=n,y=d09] {./tikz/gamman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\beta_n$.}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.7,
xlabel = {$x$},
ylabel = {$\tilde{\sigma}_n/\sigma_n$},
y label style={at={(0.06,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.1)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {./tikz/sigman.txt};
\addplot[thick,col4] table[x=n,y=d075] {./tikz/sigman.txt};
\addplot[thick,col5] table[x=n,y=d09] {./tikz/sigman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\tilde{\sigma}_n$.}
\end{subfigure}
\caption{}
\label{fig:convHedge}
\end{figure}
We observe that $\beta_n$ starts resembling $\beta$ fairly quickly, as predicted by Proposition \ref{gammanProp}; $\tilde{\sigma}_n$, on the other hand, converges extremely slowly to its limiting counterpart. Since $\mu_Q$ and $\sigma^2_Q$ are approximated by $\tilde{\beta}_n$ and $\tilde{\sigma}_n$, multiplied by a term that remains almost constant as $n$ grows, the substitution of $\sigma_n$ by $\tilde{\sigma}_n$, is essential for obtaining accurate approximations, as we illustrate further in the next subsection.
\subsection{Comparison between heavy-traffic approximations}
We set, so that $\mu_n=n$ and $\sigma^2_n=n^{2\delta}$ with $\delta>\tfrac{1}{2}$, so that $s_n = n+\beta n^{\delta}$, and $a_n =n/(n^{2\delta-1}-1)$ and $b_n = n^{2\delta-1}-1$.
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sqrt{\sigma^2_Q}$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.609 & 0.343 & 0.246 & 0.363 & 1.002 & 0.835 & 0.978 \bigstrut[t] \\
10 & 0.683 & 0.535 & 0.400 & 0.551 & 1.239 & 1.063 & 1.216 \\
50 & 0.815 & 1.405 & 1.168 & 1.405 & 1.995 & 1.817 & 1.971 \\
100 & 0.855 & 2.113 & 1.824 & 2.105 & 2.445 & 2.270 & 2.420 \\
500 & 0.920 & 5.446 & 5.006 & 5.412 & 3.923 & 3.762 & 3.899 \bigstrut[b] \\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.6$.}
\label{gammaPoisson1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.550 & 0.462 & 0.284 & 0.479 & 1.162 & 0.896 & 1.130 \bigstrut[t]\\
10 & 0.587 & 0.852 & 0.521 & 0.855 & 1.570 & 1.213 & 1.528 \\
50 & 0.668 & 3.197 & 2.093 & 3.106 & 3.025 & 2.433 & 2.947 \\
100 & 0.700 & 5.561 & 3.784& 5.377 & 3.983 & 3.270 & 3.887\\
500 & 0.766 & 19.887 & 14.741 & 19.202 & 7.514 & 6.455 & 7.361 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.8$.}
\label{gammaPoisson2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.949 & 11.532 & 11.306 & 11.495 & 3.634 & 3.559 & 3.602 \bigstrut[t] \\
10 & 0.961 & 17.565 & 17.268 & 17.548 & 4.474& 4.398 & 4.444 \\
50 & 0.979 & 46.368 & 45.869 & 46.418 & 7.241 & 7.168 & 7.218 \\
100 & 0.984 & 70.340 & 69.735 & 70.430 & 8.910 & 8.839 & 8.888 \\
500 & 0.991 & 184.900 & 183.989 & 185.108 & 14.422 & 14.357 & 14.404 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.6$.}
\label{gammaPoisson3}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.931 & 15.730 & 15.209 & 15.909 & 4.276 & 4.127 & 4.233 \bigstrut[t]\\
10 & 0.939 & 27.561 & 26.672 & 27.958 & 5.652 & 5.466 & 5.605 \\
50 & 0.955 & 100.660 & 97.967 & 102.070 & 10.760 & 10.476 & 10.698 \\
100 & 0.961 & 175.591 & 171.360 & 177.818 & 14.189 & 13.855 & 14.117 \\
500 & 0.971 & 638.097 & 626.346 & 644.105 & 26.963 & 26.490 & 26.864 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.8$.}
\label{gammaPoisson4}
\end{table}
Tables \ref{gammaPoisson1} to \ref{gammaPoisson4} present numerical results for various parameter values. The exact values are calculated using the method in Appendix \ref{numprocs}.
Several conclusions are drawn from these tables. First observe that the heavy-traffic approximations based on the Gaussian random walk, \eqref{h1} and \eqref{h1b}, capture the right order of magnitude for both $\mu_Q$ and $\sigma_Q$. However, the values are off, in particular for small $s_n$ and low $\rho_n := \mathbb{E}[A^{(n)}] / s_n$. The inaccuracy also increases with the level of overdispersion. In contrast, the approximations that follow from Theorem \ref{saddlepointThm}, \eqref{r1} and \eqref{r4} are remarkably accurate. Even for small systems with $s_n = 5$ or 10, the approximations for $\mu_Q$ are within 6$\%$ of the exact value for small $\rho_n$ and within $2\%$ for $\rho_n$ close to 1. For $\sigma_Q^2$, these percentages even reduce to $3\%$ and $1\%$, respectively. For larger values of $s_n$ these relative errors naturally reduce further. Overall, we observe that the approximations improve for heavily loaded systems, and the corrected approximations are particularly useful for systems with increased overdispersion.
\subsection{Capacity allocation in health care}
We next apply our model and robust approximations to real-life patient arrivals. We consider emergency patients who require diagnostic tests at the radiology department of a hospital. Green \cite{Green2004} points out that patients at the radiology department can be roughly categorized into three groups: Inpatients, outpatients and emergency patients. Inpatient and outpatient arrivals are relatively predictable as these are usually scheduled by appointment. Emergency patients, on the other hand, are inherently unpredictable: They typically require urgent care and therefore timely access to testing facilities, as well as a quick assessment of the test results. This leads to prioritization of emergency patients over the other two groups in such settings, so that they do not experience any delay caused by the groups of lower priority. However, patients from the same top-priority group can still cause considerable congestion. A careful evaluation of capacity allocation is hence required, bearing in mind that additional sophisticated pieces of medical equipment are very costly.
In the setting we study, capacity is defined by the number of time slots available to perform radiology tests on emergency patients in a given time period, which we set at 24 hours. As radiology tests are commonly performed in appointment slots of fixed length, the number of slots available per day is also indirectly fixed. In terms of our model parameters, see Section \ref{modelSection}, we have $s$ as the number of slots per day allocated to emergency patients, and $A(k)$ the number of test requests received by the department on day $k$. We omit the subscript $n$ in this section due to the absence of limits. Then $\mathbb{E}[Q]$ can be interpreted as the expected number of patients whose test is carried over to the next day. A more natural performance measure in this setting is the expected waiting time, namely the time between the physician's request and the actual start of the test. However, Little's law implies that there is a linear relation between the two, hence we choose to only evaluate $\mathbb{E}[Q]$.
The data set on which our empirical study is based originates from the emergency department of SKHospital, monitored over a period of 76 days from September to November 2012. We extracted information of ED patients referred to the radiology department by the ED physicians, which includes the time the test request was made and the exact test type performed. The two test types, X-ray and CT scans, are performed on different equipment and hence it makes sense to analyze their queueing processes separately.
We refer to test requests as arrivals. The empirical cumulative distribution function of the number of arrivals per day, for each type, are depicted by the black lines in Figure \ref{fig:fittedHospital}. The sample means equal 69.81 and 17.47, for the X-ray and CT scans respectively, whereas the sample variances are 121.8 and 26.12. These values suggest that fitting a Poisson distribution is inappropriate, which is visually backed up by the fitted Poisson cdf, depicted in Figure \ref{fig:fittedHospital} by the red line. To strengthen this claim, we tested both samples for the Poisson assumption using the \emph{dispersion test}, see Appendix \ref{statproc}, and obtained $p$-values equal $7.01\cdot 10^{-3}$ and $3.57\cdot 10^{-3}$ respectively, which allow us to safely reject the Poisson hypothesis in both cases.
In search for a better distributional fit with the arrivals count, we resort to Gamma-Poisson mixtures. Here we employ the procedure in \cite{koolejongbloed}, which is basically a maximum log-likelihood method, to obtain estimates for the parameters $a$ and $b$. This yields
\begin{equation*}
\label{parameterEstimators}
\hat{a}_{\rm X-ray} = 95.68,\quad \hat{b}_{\rm X-ray} = 0.7297,\quad \hat{a}_{\rm CT} = 37.19,\quad \hat{b}_{\rm CT} = 0.4698.
\end{equation*}
Applying the bootstrapping method to the data and the fitted model, also described in the appendix of \cite{koolejongbloed}, returns p-values that equal 0.7354 and 0.2120 for X-ray and CT scans, respectively. Therefore, the null hypothesis, stating that the data originated from a Gamma-Poisson mixture, cannot be rejected. The cdfs of the fitted Gamma-Poisson distributions, plotted in Figure \ref{fig:fittedHospital}, give visual confirmation of this claim as well.
Naturally, we also compared the estimated densities to the empirical pdf of the data. However, these fail to give a convincing visual fit due to the relatively small sample size and are therefore omitted here.
\begin{figure}
\begin{center}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 40,
xmax = 110,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 109,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\addplot[thick,col1] table[x=x,y=poisson] {./tikz/xray.txt};
\addplot[thick,col4] table[x=x,y=fitted] {./tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{X-ray}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 5,
xmax = 32,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 32,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {./tikz/ct.txt};
\addplot[thick,col1] table[x=x,y=poisson] {./tikz/ct.txt};
\addplot[thick,col4] table[x=x,y=fitted] {./tikz/ct.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/ct.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{CT scan}
\end{subfigure}
\end{center}
\caption{Empirical, fitted Poisson and fitted Gamma-Poisson cumulative distribution functions of the number of arrivals.}
\label{fig:fittedHospital}
\end{figure}
We now have clear evidence that both the X-ray and CT scan facilities face an overdispersed arrival stream. In our final step of the empirical study we now evaluate the accuracy of our performance measure of interest $\mathbb{E}[Q]$, and the consequences of assessing system performance while ignoring the presence of overdispersion. We take the following approach: Trivially, we need to choose $s> \mathbb{E}[A]$ in order for the system to be stable. Hence, we vary $s$ from 70 to 80 for X-rays and from 18 to 24 for CT scans and simulate the queue length process by sampling the number of requests per day from the actual arrival counts. The number of iterations performed is $10^8$ for each configuration, excluding a warm-up interval of length $10^7$ (days). Around the mean of $Q$ obtained from this simulation, we create a 95\% confidence interval. Next, we approximate the expected stationary queue length under two scaling rules. Assuming Poisson arrivals, the appropriate capacity allocation rule would be $s=\hat{\mu}+\beta\sqrt{\hat{\mu}}$, for some $\beta>0$. Our novel capacity sizing rule prescribes $s = \hat{\mu} + \beta\hat{\sigma} = \hat{a}\hat{b}+\beta\sqrt{\hat{a}\hat{b}(\hat{b}+1)}$. We compute the first approximation based on square-root safety capacity sizing by deducing $\beta$ for each $s$, which we substitute in $\mathbb{E}[Q^{\rm srs}] \approx \sqrt{\hat{\mu}}\,\mathbb{E}[M_{\beta}]$. Similarly, we obtain $\beta$ from the new rule, and plug this value, together with the fitted parameters $\hat{a}$ and $\hat{b}$, into \eqref{r1}. The results are given in Tables \ref{tab:simXRay} and \ref{tab:simCT}. The last column shows the 95\% relative error bound of the second approximation.
\begin{table}[h]
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q] \ (\pm $ conf. iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
70 & 0.997 & $328.313 \pm\ 6.6\cdot 10^{-2}$ & 186.664 & 324.231 & 325.221 & $9.6\cdot 10^{-3}$ \bigstrut[t]\\
71 & 0.983 & $45.073 \pm\ 1.0\cdot 10^{-2}$ & 24.946 & 45.290 & 45.308 & $5.4\cdot 10^{-3}$ \\
72 & 0.970 & $21.988 \pm\ 5.4\cdot 10^{-3}$ & 11.650 & 21.982 & 22.129 & $6.6\cdot 10^{-3}$ \\
73 & 0.956 & $13.546 \pm\ 3.6\cdot 10^{-3}$ & 6.847 & 13.455 & 13.649 & $7.8\cdot 10^{-3}$ \\
74 & 0.943 & $9.230 \pm\ 2.7\cdot 10^{-3}$ & 4.438 & 9.106 & 9.319 & $1.0\cdot 10^{-2}$ \\
75 & 0.931 & $6.655 \pm\ 2.1\cdot 10^{-3}$ & 3.031 & 6.513 & 6.731 & $1.2\cdot 10^{-2}$ \\
76 & 0.919 & $4.949 \pm\ 1.7\cdot 10^{-3}$ & 2.136 & 4.821 & 5.037 & $1.8\cdot 10^{-2}$ \\
77 & 0.907 & $3.755 \pm\ 1.4\cdot 10^{-3}$ & 1.534 & 3.650 & 3.861 & $2.8\cdot 10^{-2}$ \\
78 & 0.895 & $2.884 \pm\ 1.1\cdot 10^{-3}$ & 1.115 & 2.807 & 3.009 & $4.4\cdot 10^{-2}$ \\
79 & 0.884 & $2.230 \pm\ 1.0\cdot 10^{-3}$ & 0.816 & 2.183 & 2.374 & $6.5\cdot 10^{-2}$ \\
80 & 0.873 & $1.734 \pm\ 8.5\cdot 10^{-4}$ & 0.600 & 1.710 & 1.890 & $9.1\cdot 10^{-2}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for X-ray.}
\label{tab:simXRay}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q]\ (\pm $ conf.iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
18 & 0.970 & 22.116 $\pm\ 4.9\cdot 10^{-3}$ & 14.235 & 21.965 & 21.724 & $1.8\cdot 10^{-2}$ \bigstrut[t] \\
19 & 0.919 & 6.289 $\pm\ 1.7\cdot 10^{-3}$ & 3.640 & 5.941 & 6.040 & 4.0$\cdot 10^{-2}$ \\
20 & 0.873 & 3.101 $\pm\ 1.0\cdot 10^{-3}$ & 1.589 & 2.772 & 2.917 & 6.0$\cdot 10^{-2}$ \\
21 & 0.832 & 1.767 $\pm\ 6.6\cdot 10^{-4}$ & 0.800 & 1.507 & 1.658 & 6.1$\cdot 10^{-2}$ \\
22 & 0.794 & 1.066 $\pm\ 4.6\cdot 10^{-4}$ & 0.425 & 0.874 & 1.016 & 4.7$\cdot 10^{-2}$ \\
23 & 0.760 & 0.653 $\pm\ 3.3\cdot 10^{-4}$ & 0.230 & 0.522 & 0.649 & 7.1$\cdot 10^{-3}$\\
24 & 0.728 & 0.377 $\pm\ 2.3\cdot 10^{-4}$ & 0.124 & 0.315 & 0.424 & 1.2$\cdot 10^{-1}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for CT scan.}
\label{tab:simCT}
\end{table}
Based on these figures, we make several remarks. First, assuming the conventional regime (neglecting overdispersion) the approximation severely overestimates system performance in both queues. Because the square-root safety margin underestimates the stochastic fluctuations in the arrival process, the safety parameter $\beta$ is overestimated, which leads to a smaller magnitude of the approximated queue length process. This clearly illustrates the distorted view estimated performance characteristics can give under the wrong scaling.
Secondly, it is worth noticing the very good proximity of $\eqref{r1}$ to the values obtained through simulation. As we expected, the quality of the approximation deteriorates with increasing values of $s$. This makes sense, because we assumed the system to be in heavy traffic in the derivation of the formulas. What is surprising, on the other hand, is the fact that the approximation performs very well, even though the parameter $b$ is very small for these particular data sets, while the analysis of Theorem \ref{saddlepointThm} assumes that $a$ and $b$ are large. This demonstrates that the approximation scheme is remarkably robust and is able to capture the pre-limit behavior of these types of queues very well.
\section{Conclusion \& future research}
In this chapter, we proposed an adaption to the square-root staffing rule for service systems facing overdispersed demand, using the bulk service queue as a vehicle for our analysis.
Subsequently, we derive two sets of asymptotic approximations for the scaled steady-state queue length moments for large arrival volumes.
The first set being based on the resemblance with the maximum of a Gaussian random walk, the second set being derived through a non-standard saddle point method, assuming arrivals follow a Gamma-Poisson mixture.
Numerical experiments indicated that our approximations capture the pre-limit behavior well under different order of overdispersion, and are robust against any parameter estimation errors.
Although our method provides a robust way to approximate and dimension queues with overdispersed arrival processes, we see have some interesting directions for future research.
First, we accentuate that our model is a discrete time queueing model in which a deterministic amount of workload can we handled within each period.
This approach allowed us to use Pollaczek's formula as a starting point to obtain more refined asymptotic approximations for the performance indicators of the system.
In case we consider queueing models of birth-death-type, such as the $M/M/s$ queue, in the presence of overdispersion demand, different techniques are required to provide derive scaling limits and corresponding capacity allocation rules, see e.g.~\cite{maman}.
Although we expect that, just as in the novel scaling regimes of Chapter 2, the asymptotic behavior between the bulk service queue and the multi-server queueing models to be similar, this needs proper analysis and understanding.
Second, empirical work, see e.g. \cite{Avramidis:2004}, shows that in real-life settings, demand in consecutive time periods is often positively correlated, rather than independently distributed as assumed in this chapter.
This correlation structure obviously alters the queue's dynamics and presumably requires an adaption of the square-root staffing rule as well, making it a worthwhile direction for further analysis.
Last, we have only considered the analysis of the queueing model in steady state.
Typical service systems however do not face a constant expected arrival rate, nor do they run infinitely long.
Henceforth, it would be interesting to study the influence of overdispersion on the transient dynamics of the queue and to investigate the capacity allocation problem in scenarios with time-varying demand.
The theory developed in this chapter may serve as a building block to tackle these more profound questions.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Proofs of convergence results}
\label{formalSec}
This section presents the details of the proof of Lemma \ref{gaussStep} and Theorem \ref{gaussianThm}, using the random walk perspective of the process $\{Q^{(n)}(k)\}_{k=0}^\infty$. This section is structured as follows. The next two lemmata are necessary for proving the first assertion of Theorem \ref{gaussianThm}, concerning the weak convergence of the scaled process to the maximum of the Gaussian random walk, which is summarized in Proposition \ref{prop6}. The two remaining propositions of this section show convergence of $\hat{Q}^{(n)}$ at the process level as well as in terms of the three characteristics.
Let us first fix some notation:
\begin{equation}
\label{b1}
Y^{(n)}_k := \hat{A}^{(n)}_k-\beta,\quad
S^{(n)}_k = \sum_{i=1}^k Y^{(n)}_i,
\end{equation}
with $S_0^{(n)} = 0$ and $k=1,2,...$. Then \eqref{mm6} can be rewritten as
\begin{equation}
\label{g5a}
\hat{Q}^{(n)} {\;\buildrel{d}\over= \;} \max_{0\leq i \leq k} \Bigl\{{\textstyle \sum}_{i=1}^k Y^{(n)}_i\Bigr\} =: M_\beta^{(n)},
\end{equation}
Last, we introduce the sequence of independent normal random variables $Z_1,Z_2,\ldots$ with mean $\-\beta$ and unit variance 1, and
\begin{equation*}
M_\beta {\;\buildrel{d}\over= \;} \max_{k\geq 0} \{{\textstyle \sum}_{i=1}^k Z_i\}
\end{equation*}
\subsection{Proof of Lemma \ref{gaussStep}}
\begin{proof}
We show weak convergence of the random variable $\hat{A}^{(n)}$, as defined in \eqref{b1}, to a standard normal random variable. Since $\hat{\Lambda}_n$ is asymptotically standard normal, its characteristic function converges pointwise to the corresponding limiting characteristic function, i.e.
\begin{equation}
\label{g8}
\lim_{n\rightarrow\infty} \phi_{\hat{\Lambda}_n}(t) = \lim_{n\rightarrow \infty} {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n) = {\rm e}^{{-}t^2/2},\qquad \forall t\in \mathbb{R}.
\end{equation}
Furthermore, by definition of $A^{(n)}$,
\begin{equation*}
\label{g9}
\phi_{A^{(n)}}(t) = \mathbb{E}\left[ \exp(\Lambda_n({\rm e}^{it}-1))\right] = \phi_{\Lambda_n}\left(-i({\rm e}^{it}-1)\right),
\end{equation*}
so that
\begin{equation}
\label{g10}
\phi_{\hat{A}_k^{(n)}}(t) = {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{A_k^{(n)}}(t/\sigma_n) = {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}\left(-i({\rm e}^{it/\sigma_n}-1)\right).
\end{equation}
Now fix $t\in\mathbb{R}$. By using
\begin{equation*}
\label{g11}
-i(\e^{it/\sigma_n} - 1) = \frac{t}{\sigma_n} -\frac{it^2}{2\sigma_n^2} + O\left(t^3/\sigma_n^3\right),
\end{equation*}
we expand the last term in \eqref{g10},
\begin{equation*}
\label{g12}
\phi_{\Lambda_n}(t/\sigma_n) + \Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(t/\sigma_n) + O\Bigl(\Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(\frac{t^3}{\sigma_n^3}\right)\Bigr)^2\phi_{\Lambda_n}''\Big(\frac{t}{\sigma_n}\Big)\Bigr)
\end{equation*}
\begin{equation*}
\label{g13}
= \phi_{\Lambda_n}(t/\sigma_n) - \Bigl(\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(\zeta)
\end{equation*}
for some $\zeta$ such that $|\zeta - t/\sigma_n| < |i(1-{\rm e}^{it/\sigma_n})-t/\sigma_n|$. Also,
\begin{align}
|\phi_{\Lambda_n}'(u)| &= \left|\frac{\delta}{{\rm d} u}\int_{-\infty}^\infty {\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| = \left|\int_{0}^{\infty} ix\,{\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| \nonumber\\
\label{g13a}
&\leq \int_{-\infty}^\infty |ix\,{\rm e}^{iux}|\,{\rm d} F_{\Lambda_n}(x) = \int_0^\infty x{\rm d} F_{\Lambda_n}(x) = \mu_n
\end{align}
for all $u\in\mathbb{R}$. Hence, by substituting \eqref{g10},
\begin{align}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| &= \left|{\rm e}^{-i\mu_nt/\sigma_n}\,\left(\frac{i\,t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right)\,\phi_{\Lambda_n}'(\zeta)\right|\nonumber\\
& \leq \left(\frac{t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right) |\phi_{\Lambda_n}'(\zeta)|\nonumber\\
& = \frac{\mu_n t^2}{\sigma_n^2} + O\left(\frac{\mu_nt^3}{\sigma_n^3}\right),
\label{g13b}
\end{align}
which tends to zero as $n\rightarrow \infty$ by our assumption that $\mu_n/\sigma_n^2\rightarrow 0$.
Finally,
\begin{equation*}
\label{g13c}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-\tfrac12 t^2}\right| \leq \left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| +
\left| {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n) - {\rm e}^{-\tfrac12 t^2}\right|,
\end{equation*}
in which both terms go to zero for $n\rightarrow \infty$, by \eqref{g8} and \eqref{g13b}. Hence $\phi_{\hat{A}^{(n)}_k}(t)$ converges to ${\rm e}^{{-}t^2/2}$ for all $t\in\mathbb{R}$, so that we can conclude by L\'evy's continuity theorem that $\hat{A}_k^{(n)} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1)$.
\end{proof}
\subsection{Proof of Theorem \ref{gaussianThm}}
To secure convergence in distribution of $\hat{Q}^{(n)}$ to $M_\beta$, i.e. the maximum of a Gaussian random walk with negative drift, the first assertion of Theorem \ref{gaussianThm}.
the following property of the sequence $\{Y_k^{(n)}\}_{n\in\mathbb{N}}$ needs to hold.
\begin{lemma}\label{uilemma}
Let $Y^{(n)}_k$ be defined as in \eqref{b1} with $\mu_n,\sigma_n^2 < \infty$ for all $n\in\mathbb{N}$. Then the sequence $\{(Y_k^{(n)})^+\}_{n\in\mathbb{N}}$ is uniform integrable, i.e.
\begin{equation*}
\label{g14}
\lim_{K\rightarrow\infty}\sup_n \mathbb{E}[Y^{(n)\,+}_k|\mathbbm{1}_{\{|Y^{(n)\,+}_k|\geq K\}}] = 0.
\end{equation*}
\end{lemma}
\begin{proof}
Because the sequence $\{Y^{(n)}_k\}_{k\in\mathbb{N}}$ is i.i.d. for all $n$, we omit the index $k$ in this proof. First, fix $K>0$ and note that
\begin{equation*}
\label{g15}
\mathbb{E}[|Y^{(n)+}|\mathbbm{1}{\{|Y^{(n)\,+}|\geq K\}}] = \mathbb{E}[Y^{(n)+}\mathbbm{1}{\{Y^{(n)+}\geq K\}}] = \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}].
\end{equation*}
This last expression can be bounded from above using the Cauchy-Schwarz inequality, so that
\begin{equation*}
\label{g16}
\mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}] \leq \mathbb{E}[ Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K)^{1/2}.
\end{equation*}
By the definition of $Y^{(n)}$, we know $\mathbb{E} [Y^{(n)}] = -\beta$ and ${\rm Var}\, Y^{(n)} = {\rm Var}\, A^{(n)} / \sigma_n^2 = 1$. Using this information, we find
\begin{equation*}
\label{g17}
\mathbb{E}[Y^{(n)\,2}] = {\rm Var}\, Y^{(n)} + (\mathbb{E}[Y^{(n)}])^2 = 1+\beta^2
\end{equation*}
and
\begin{align*}
\mathbb{P}(Y^{(n)}\geq K )&=\mathbb{P}(Y^{(n)}+\beta\geq K+\beta) \leq \mathbb{P}(|Y^{(n)}+\beta|\geq K+\beta)\nonumber\\
&\leq \frac{{\rm Var}\, Y^{(n)}}{(K+\beta)^2} = \frac{1}{(K+\beta)^2},
\end{align*}
where we used Chebyshev's inequality for the last upper bound. Therefore,
\begin{align*}
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[|Y^{(n)\,+}|\mathbbm{1}_{\{|Y^{(n)\,+}|\geq K\}}] &=
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}]\nonumber\\
&\leq \lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K )^{1/2}\nonumber\\
&\leq \lim_{K\rightarrow \infty} \frac{\sqrt{1+\beta^2}}{K+\beta} = 0.
\end{align*}
\end{proof}
By combining the properties proved in Lemma \ref{gaussStep} and \ref{uilemma} with Assumption \ref{as2}, the next result follows directly by \cite[Thm.~X6.1]{Asmussen2003}.
\begin{proposition}\label{maxRWprop}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}. Then
\begin{equation*}
\hat{Q}^{(n)}{\;\buildrel{d}\over\Rightarrow\;} M_\beta,\qquad {\rm as}\ n\rightarrow\infty.
\end{equation*}
\end{proposition}
Although Proposition \ref{maxRWprop} tells us that the properly scaled $Q^{(n)}$ converges to a non-degenerate limiting random variable, it does not cover the convergence of its mean, variance and the empty-queue probability. In order to secure convergence of these performance measures as well, we follow the approach similar \cite{Sigman2011b}, using Assumptions \ref{as2} and \ref{as3}.
\begin{proposition}\label{prop6}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}, $\mu_n,\sigma_n^2 \rightarrow \infty$ such that both $\sigma_n^2/\mu_n\rightarrow \infty$ and $\mathbb{E}[\hat{A}^{(n)3}]$ $<\infty$. Then
\begin{align*}
\label{b16}
\mathbb{P}(\hat{Q}^{(n)}= 0)&\rightarrow \mathbb{P}(M_\beta = 0),\\
\mathbb{E} [\hat{Q}^{(n)}]&\rightarrow \mathbb{E} [M_\beta],\\
{\rm Var}\, \hat{Q}^{(n)}&\rightarrow {\rm Var}\, M_\beta,
\end{align*}
as $n\rightarrow\infty$.
\end{proposition}
\proof
First, we recall that $\hat{Q}^{(n)}{\;\buildrel{d}\over= \;} M_\beta^{(n)}$ for all $n\in\mathbb{N}$, so that $\mathbb{P}(\hat{Q}^{(n)} = 0) = \mathbb{P}(M_\beta^{(n)}=0)$, $\mathbb{E}[\hat{Q}^{(n)}]=\mathbb{E}[M_\beta^{(n)}]$ and ${\rm Var}\,\,\hat{Q}^{(n)}={\rm Var}\,\,M_\beta^{(n)}$ as defined in \eqref{b1}. Our starting point is Spitzer's identity, see \cite[p.~230]{Asmussen2003},
\begin{equation}
\label{b17}
\mathbb{E}[{\rm e}^{it M_\beta^{(n)}}] = \exp\Bigl( \sum_{k=1}^\infty \frac{1}{k} (\mathbb{E}[{\rm e}^{it(S^{(n)}_k)^+}]-1)\Bigr),
\end{equation}
with $S^{(n)}_k$ as in \eqref{b1}, and $M_\beta^{(n)}$ the all-time maximum of the associated random walk. Simple manipulations of \eqref{b17} give
\begin{align}
\label{y1}
{\rm ln}\,\mathbb{P}(M_\beta^{(n)} = 0) &= -\sum_{k=1}^\infty \frac{1}{k}\,\mathbb{P}(S^{(n)}_k > 0),\\
\label{y2}
\mathbb{E}[M_\beta^{(n)}] &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[S^{(n)\,+}_k] = \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x) {\rm d} x,\\
\label{y3}
{\rm Var}\, M_\beta^{(n)} &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[(S^{(n)\,+}_k)^2] =\sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}) {\rm d} x.
\end{align}
By Lemma \ref{gaussStep}, we know
\begin{equation*}
\label{y4}
\mathbb{P}(S^{(n)}_k > y) = \mathbb{P}\left( {\sum_{i=1}^k} Y^{(n)}_i > y \right) \rightarrow \mathbb{P}\left({\textstyle\sum_{i=1}^k} Z(i) > y\right),
\end{equation*}
for $n\rightarrow \infty$, where the $Z(i)$'s are independent and identically normally distributed with mean $-\beta$ and variance 1.
Because equivalent expressions to \eqref{y1}-\eqref{y3} apply to the limiting Gaussian random walk, it is sufficient to show that the sums converge uniformly in $n$, so that we can apply dominated convergence to prove the result.
We start with the empty-queue probability. To justify interchangeability of the infinite sum and limit, note
\begin{equation*}
\label{y5}
\mathbb{P}(S^{(n)}_k > 0) \leq \mathbb{P}(|S^{(n)}_k+k\beta| > k\beta )\leq \frac{k}{\beta^2k^2} = \frac{1}{\beta^2k},
\end{equation*}
where we used that $\mathbb{E}[ S^{(n)}_k] = k\mathbb{E} [Y^{(n)}(1)] = -k\beta$ and ${\rm Var}\, S^{(n)}_k = k$ and apply Chebychev's inequality, so that
\begin{equation*}
\label{y6}
\sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) \leq \sum_{k=1}^\infty \frac{1}{\beta^2 k^2} < \infty, \qquad \forall n\in\mathbb{N}.
\end{equation*}
Hence,
\begin{align*}
\lim_{n\rightarrow\infty} {\rm ln}\,\mathbb{P}(\hat{Q}^{(n)}= 0) &= \lim_{n\rightarrow\infty} - \sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) = -\sum_{k=1}^\infty \frac{1}{k} \lim_{n\rightarrow\infty}\mathbb{P}(S^{(n)}_k > 0)\nonumber\\
&= -\sum_{k=1}^\infty \frac{1}{k} \mathbb{P}({\textstyle\sum_{i=1}^k} Z(i) > 0) = {\rm ln}\,\mathbb{P}(M_\beta = 0),
\end{align*}
Finding a suitable upper bound on $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>x) dx$ and $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>\sqrt{x}) {\rm d} x$ requires a bit more work. We initially focus on the former, the latter follows easily. The following inequality from \cite{Nagaev1979} proves to be very useful:
\begin{equation}
\label{y8}
\mathbb{P}(\bar{S}(k)>y) \leq C_r\,\Bigl(\frac{k\,\sigma^2}{y^2}\Bigr)^2 + k\,\mathbb{P}(X>y/r),
\end{equation}
where $\bar{S}(k)$ is the sum of $k$ i.i.d. random variables distributed as $X$, with $\mathbb{E}[X] = 0$ and ${\rm Var}\,\, X=\sigma^2$, $y > 0$, $r>0$ and $C_r$ a constant only depending on $r$. We take $r=3$ for brevity in the remainder of the proof, although any $r>2$ will suffice. We analyze the integral in two parts, one for the interval $(0,k)$ and one for $[k,\infty)$. For the first part, we have
\begin{align}
\label{y9}
\int_0^k\mathbb{P}(S^{(n)}_k>x) {\rm d} x &=\int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > x+k\beta)dx\, \leq\, \int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > k\beta){\rm d} x \nonumber\\
&= k\,\mathbb{P}({\textstyle \sum_{i=1}^k }\hat{A}^{(n)}_i > k\beta) \,\leq\, \frac{C_3}{k^2\beta^6} + k^2\mathbb{P}(\hat{A}^{(n)}(1)> \tfrac{1}{3}k),
\end{align}
where we used \eqref{y8} in the last inequality.
Hence,
\begin{align}
\label{y10}
\sum_{k=1}^\infty\frac{1}{k}\, \int_0^k \mathbb{P}(S^{(n)}_k>x){\rm d} x &\leq \, \frac{C_3}{\beta^6}\sum_{k=1}^\infty k^{-3} +\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}k) \nonumber \\
&\leq C_1^*+\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}k).
\end{align}
With the help of the inequality (see \cite{Sigman2011b}),
\begin{equation}
\label{y11}
(b-a)a\,\mathbb{P}(X>b) \leq \int_a^b x\,\mathbb{P}(X>x) {\rm d} x \qquad \forall 0<a<b,
\end{equation}
we get by taking $a=(k-1)/3$ and $b=k/3$,
\begin{align}
\label{y12}
k\,\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}k) &\leq \frac{9\,k}{k-1}\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}(1)>x) {\rm d} x \nonumber \\
&\leq 18\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}(1)>x) {\rm d} x,
\end{align}
for $k\geq 2$. Since the tail probability for $k=1$ is obviously bounded by 1, this yields
\begin{align}
\label{y13}
\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}k) &\leq 1+18\sum_{k=2}^\infty\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}(1)>x) {\rm d} x\nonumber\\
&\leq 1+ \int_{0}^{\infty} x\,\mathbb{P}(\hat{A}^{(n)}(1)>x){\rm d} x \leq 1+\mathbb{E}[\hat{A}^{(n)}(1)^2] < \infty,
\end{align}
since $\hat{A}^{(n)}(1)$ has finite variance by assumption. This completes the integral over the first interval. For the second part, we use \eqref{y8} again to find
\begin{align}
\label{y14}
\int_k^\infty \mathbb{P}(S^{(n)}_k>x)dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > x+k\beta)dx \leq \int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > x){\rm d} x\nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{x^6} dx + k\int_k^\infty \mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}x){\rm d} x\nonumber \\
&= \frac{5 C_3}{k^3}+ k\int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) {\rm d} x.
\end{align}
So,
\begin{equation}
\label{y15}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>x)dx \leq C_2^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) {\rm d} x,
\end{equation}
for some constant $C_2^*$. Last, we are able to upper bound the second term in \eqref{y15} by Tonelli's theorem:
\begin{align}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) dx &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}x) {\rm d} x \nonumber\\
\label{y16}
&\leq 9\int_0^\infty y\mathbb{P}(\hat{A}^{(n)}(1)>y) dy = 9\mathbb{E}[\hat{A}^{(n)}(1)^2] < \infty.
\end{align}
Combining the results in \eqref{y10}, \eqref{y13}, \eqref{y15} and \eqref{y16}, we find
\begin{equation*}
\label{y17}
\sum_{k=1}^\infty \frac{1}{k} \int_0^\infty \mathbb{P}(S^{(n)}_k>x){\rm d} x < \infty,
\end{equation*}
and thus
\begin{align*}
\lim_{n\rightarrow\infty} \mathbb{E}[\hat{Q}^{(n)}] &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x){\rm d} x \nonumber\\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z(i) > x){\rm d} x = \mathbb{E} [M_\beta].
\end{align*}
Finally, we show how the proof changes for the convergence of ${\rm Var}\, \hat{Q}^{(n)}$. The expressions for $\mathbb{E} [\hat{Q}^{(n)}]$ and ${\rm Var}\, \hat{Q}^{(n)}$ in \eqref{y1} and \eqref{y2} only differ in the term $\sqrt{x}$. Hence only minor modifications are needed to also prove convergence of the variance. Note that boundedness of the integral over the interval $(0,k)$ in \eqref{y9}-\eqref{y13} remains to hold when substituting $\sqrt{x}$ for $x$. \eqref{y14} changes into
\begin{align*}
\label{y18}
\int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x})dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > \sqrt{x}+k\beta){\rm d} x \nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{(\sqrt{x}+k\beta)^6} dx + k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
&\leq \frac{C_4^*}{k}+ k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}\sqrt{x}) {\rm d} x,
\end{align*}
for some constant $C_4^*$, so that
\begin{equation*}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x}){\rm d} x \leq C_4^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}\sqrt{x}) {\rm d} x.
\end{equation*}
Lastly, we have
\begin{align*}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}\sqrt{x}) {\rm d} x &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}(1)>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
\label{y17a}
&\leq 18\int_0^\infty y^2\mathbb{P}(\hat{A}^{(n)}(1)>y) {\rm d} y = 18\mathbb{E}[\hat{A}^{(n)}(1)^3] < \infty.
\end{align*}
Therefore the sum describing the variance is also uniformly convergent in $n$, so that interchanging of infinite sum and limit is permitted and
\begin{align*}
\lim_{n\rightarrow\infty} {\rm Var}\,\,\hat{Q}^{(n)} &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}){\rm d} x \nonumber \\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z(i) > \sqrt{x}){\rm d} x = {\rm Var}\, M_\beta.
\end{align*}
\section{Numerical procedures}\label{numprocs}
An alternative characterization of the stationary distribution is based on the analysis in \cite{Boudreau1962} and considers a factorization in terms of (complex) roots:
\begin{equation*}
\label{t9}
Q^{(n)}(w) = \frac{(s_n-\mathbb{E} [A^{(n)}])(w-1)}{w^{s_n}-\tilde{A}^{(n)}(w)}\,\prod_{k=1}^{s_n-1} \frac{w-z^n_k}{1-z^n_k},
\end{equation*}
where $z_1^n,z_2^n...,z_{s_n-1}^n$ are the $s_n-1$ zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$, in $|z|<1$, yielding
\begin{equation*}
\label{c2}
\mu_Q = \frac{\sigma_n^2}{2(s_n-\mu_n)}-\frac{s_n-1+\mu_n}{2} + \sum_{k-1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c3}
\mathbb{P}(Q^{(n)}=0) = \frac{s_n-\mu_A}{\tilde{A}^{(n)}(0)}\prod_{k=1}^{s-1}\frac{z^n_k}{z^n_k-1},
\end{equation*}
which for our choice of $\tilde{A}^{(n)}(z)$ becomes
\begin{equation*}
\label{c4}
\mu_Q = \frac{a_nb_n(b_n+1)}{2\beta\sqrt{a_n}b_n}-\frac{2a_nb_n+\beta\sqrt{a_nb_n(b_n+1)}-1}{2}+\sum_{k=1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c5}
\mathbb{P}(Q^{(n)}=0) = \beta \sqrt{a_nb_n(b_n+1)}(1+b_n)^{a_n}\prod_{k=1}^{s_n-1} \frac{z^n_k}{z^n_k-1}.
\end{equation*}
where $z_1,...,z_{s_n-1}$ denote the zeros of $z^{s_n} - \tilde{A}^{(n)}(z)$ in $|z|<1$. A robust numerical procedure to obtain these zeros is essential for a base of comparison. We discuss two methods that fit these requirements. The first follows directly from \cite{Janssen2005}. \\
\begin{lemma}\label{fixedIterLemma}
Define the iteration scheme
\begin{equation}
\label{c6}
z_k^{n,l+1} = w^n_k [\tilde{A}^{(n)}(z_k^{n,l})]^{1/s_n},
\end{equation}
with $w^n_k = {\rm e}^{2\pi ik/s_n}$ and $z_k^{n,0}=0$ for $k=0,1,\ldots,s_{n-1}$. Then $z_k^{n,l} \rightarrow z_k^n$ for all $k=0,1,...,s_n-1$ for $l\rightarrow \infty$.
\end{lemma}
\begin{proof}
The successive substitution scheme given in \eqref{c6} is the fixed point iteration scheme described in \cite{Janssen2005}, applied to the pgf of our arrival process. The authors show that, under the assumption of $\tilde{A}^{(n)}(z)$ being zero-free within $|z|\leq 1$, the zeros can be approximated arbitrarily closely, given that the function $[\tilde{A}^{(n)}(z)]^{1/s_n}$ is a contraction for $|z|\leq 1$, i.e.
\begin{equation*}
\label{c7}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| < 1.
\end{equation*}
In our case,
\begin{align}
\label{c8}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| = \Bigl|\frac{{\rm d}}{{\rm d} z}\left(1+(1-z)b_n\right)^{-a_n/s_n}\Bigr| = \frac{a_nb_n}{s_n}\Bigl|1+(1-z)b_n\Bigr|^{-a_n/s_n-1},
\end{align}
where $a_nb_n/s_n = \rho_n$ is close to, but less than 1 and
\begin{align*}
\label{c9}
|1+(1-z)b_n| \geq |1+b_n|-|z|b_n = 1+(1-|z|)b_n \geq 1,
\end{align*}
when $|z|\leq 1$. Hence the expression in \eqref{c8} is less than 1 for all $|z|\leq 1$. Evidently, $\tilde{A}^{(n)}(z)$ is also zero-free in $|z|\leq 1$. Thus \cite[Lemma~3.8]{Janssen2005} shows that $z_k^{n,l}$ as in \eqref{c6} converges to the desired roots $z^n_k$ for all $k$ as $l$ tends to infinity.
\end{proof}
\begin{remark}
The asymptotic convergence rate of the iteration in \eqref{c6} equals \\
\noindent $\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}$ evaluated at $z=z_k^n$. Hence, convergence is slow for zeros near 1 and fast for zeros away from 1.
\end{remark}
A different approach is based on the B\"urmann-Lagrange inversion formula.
\begin{lemma}\label{BLLemma}
Let $w^n_k = e^{2\pi ik/s_n}$ and $\alpha_n = a_n/s_n$. Then the zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$ are given by
\begin{equation*}
z_k^n = \sum_{l=1}^\infty \frac{1}{l!}\,\frac{\beta[l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{b_n+1}{b_n}\Bigl(\frac{b_n}{(b_n+1)^{\alpha_n+1}}\Bigr)^l (w_k^n)^l,
\end{equation*}
for $k=0,1,...,s_n-1$.
\end{lemma}
\begin{proof}
Note that we are looking for $z$'s that solve
\begin{equation*}
\label{c10}
z\,[\tilde{A}^{(n)}(z)]^{-1/s_n} = z\left(1+(1-z)b_n\right)^{a_n/s_n} = w,
\end{equation*}
where $w = w_k = {\rm e}^{2\pi i k/s_n}$. The B\"urmann-Lagrange formula for $z=z(w)$, as can be found in \cite[Sec.~2.2]{debruijn} for $z=z(w)$ is given by
\begin{align*}
z(w) &= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(\frac{z}{z(1+(1-z)b_n)^{a_n/s_n}}\right)^l\right]_{z=0}\,w^l\nonumber\\
\label{c11}
&= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(1+(1-z)b_n)^{-l\,a_n/s_n}\right)\right]_{z=0}\,w^l.
\end{align*}
Set $\alpha_n = a_n/s_n$. We compute
\begin{equation*}
\label{c1}
\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[ (1+(1-z)b_n)^{-l\alpha_n}\right]_{z=0} = \frac{\beta(l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{1+b_n}{b_n}\,\left(\frac{b_n}{(1+b_n)^{\alpha_n+1}}\right)^l.
\end{equation*}
With $c_n = b_n/(1+b_n)^{\alpha_n+1}$ and $d_n = (1+b_n)/b_n$, we thus have
\begin{equation*}
\label{c13}
z(w) = d_n\,\sum_{l=1}^\infty \frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} c_n^l\,w^l.
\end{equation*}
By Stirling's formula
\begin{equation*}\label{c14}
\frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} = \frac{D}{l\sqrt{l}}\left(\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\right)^l,
\end{equation*}
where $D=\alpha_n^{1/2}(\alpha_n+1)^{-3/2}(2\pi)^{-1/2}$. Now,
\begin{equation*}
\label{c15}
\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\, c_n = \frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\cdot \frac{b_n}{(1+b_n)^{\alpha_n+1}} = \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation*}
This determines the radius of convergence $r_{\rm BL}$ of the above series for $z(w)$:
\begin{equation}
\label{c16}
\frac{1}{r_{\rm BL}} := \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation}
The derivative with respect to $\rho_n$ of the quantity
\begin{equation}
\label{c17}
\left(1+\frac{\rho_n}{b_n}\right) {\rm ln }\left(\frac{b_n+\rho_n}{b_n+1}\right)+\frac{\rho_n}{b_n}\,{\rm ln}\left(\frac{1}{\rho_n}\right)
\end{equation}
is given by
\begin{equation*}
\label{c18}
\frac{1}{b_n}{\rm ln }\Bigl(\frac{b_n+\rho_n}{b_n\rho_n+\rho_n}\Bigr) > 0
\end{equation*}
for $0<\rho_n<1$ and $b_n>0$. Furthermore, the quantity in \eqref{c17} vanishes at $\rho_n=1$ and is therefore negative for $0<\rho_n<1$ and $b_n>0$.
\begin{remark}
The formula for the radius of convergence in \eqref{c16} clearly shows the decremental effect of both having a large $b_n$ and or having $\rho_n$ close to 1. The quantities $\beta(l\alpha+l-1)/(\beta(l+1)\beta(l\alpha))$ in the power series for $z(w)$ are not very convenient for recursive computation, although normally $\alpha = a_n/s_n$ is a rational number.\end{remark}
\end{proof}
\section{Statistical procedures}\label{statproc}
To calibrate our model to real data, we now discuss some statistical procedures to show the presence of overdispersion and to estimate the parameters of the mixed Gamma-Poisson distribution. Let $x_1,...,x_n$ denote the observed number of arrivals in consecutive time slots. These observations can be interpreted as realizations of the random variables $A_1,...,A_N$, and
\begin{equation*}
\bar{a}_N=\frac{1}{N}\sum_{i=1}^N x_i, \qquad \bar{s}_N^2 = \frac{1}{N-1}\sum_{i=1}(x_i-\bar{x}_i)^2,
\end{equation*}
the sample mean and variance with equivalent random variables $\bar{A}_N$ and $S_N^2$, respectively. Several tests with null hypothesis that $x_1,...,x_N$ originate from a (constant rate) Poisson distribution were discussed by \cite{Brown2002}. We mention two of them. The first is frequently referred to as the \emph{dispersion test}, and is based on the test statistic
\begin{equation*}
\label{dispTest}
D_N = \frac{(N-1)S_N^2}{\bar{A}_N},
\end{equation*}
which is approximately chi-squared distributed with $N-1$ degrees of freedom. When using a significance level $\alpha$, the critical value is equal to the $(1-\alpha)$-th quantile of chi-squared distribution $\chi^2_{N-1,1-\alpha}$. The second test, which is also used in \cite{koolejongbloed}, involves the test statistic
\begin{equation*}
\label{NStest}
T_N = \sqrt{N/2}\,\Bigl(\frac{S_N^2}{\bar{A}_N}-1\Bigr),
\end{equation*}
which is known as the Neyman-Scott test statistic. Under the null hypothesis $T_N$ tends to a standard normal random variable for large $N$. Hence both test statistics rely on the ratio of the sample variance and sample mean, which should be 1 if $A_1,...,A_N$ are indeed i.i.d. Poisson distributed. Excessive values of $D_N$ and $T_N$ therefore raise the suspicion of overdispersed arrivals.
Once either (or both) of the Poisson tests rejects the hypothesis of arrivals originating from a unicomponent Poisson process, we fit the data to the Gamma-Poisson mixture. Note that if we assume $A_i$ to be distributed as a Poisson random variable with random rate $\Lambda_i$, which is in turn Gamma distributed with parameters $a$ and $1/b$, then $A_i$ is in fact a negative binomial random variable with parameters $r = a$ and $p=b/(b+1)$. Finding estimators $\hat{a}$ and $\hat{b}$ therefore is equivalent to fitting a negative binomial distribution to the data to obtain $\hat{r}$ and $\hat{p}$, followed by retrieving $\hat{a} = \hat{r}$ and $\hat{b} = \hat{p}/(1-\hat{p})$. We proceed by applying the maximum likelihood estimation method described in \cite{koolejongbloed} to find $\hat{r}$ and $\hat{p}$. This method prescribes to set $\hat{r}$ to be the value of $r$ for which the \emph{profile loglikelihood function} defined by
\begin{equation*}
L(r) = \frac{1}{N}\,\sum_{i=1}^N\sum_{j=1}^{a_i} {\rm ln}(r+j+1)+r\,{\rm ln}\,r -(r+\bar{a}_N)\,{\rm ln}(r+\bar{a}_N),
\end{equation*}
is attained. Subsequently, $\hat{p} = \hat{r}/(\hat{r}+\bar{a}_N)$, so that $\hat{a} = \hat{r}$ and $\hat{b} = \hat{r}/\bar{a}_N$.
Finally, given the estimators $\hat{a}$ and $\hat{b}$, we need statistical evidence that the obtained Poisson mixture indeed fits the data reasonably well. Here we again cite \cite{koolejongbloed}, who give a method to retrieve the $p$-value for the goodness-of-fit based on bootstrap and Monte-Carlo simulation. In our experiments, we work with $10^6$ replications of the Monte-Carlo simulation to obtain the approximated $p$-value. We refer to the appendix of \cite{koolejongbloed} for further details on this method.
\resettocdepth
\end{subappendices}
\chapter{Overdispersion}
\begin{chapterstart}
Arrival processes to service systems often display fluctuations that are larger than anticipated under the Poisson assumption, a phenomenon that is referred to as \textit{overdispersion}.
Motivated by this, we analyze a class of discrete stochastic models for which we derive heavy-traffic approximations that are scalable in the system size.
Subsequently, we show how this leads to novel capacity sizing rules that acknowledge the presence of overdispersion. This, in turn, leads to robust approximations for performance characteristics of systems that are of moderate size and/or may not operate in heavy traffic.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Robust heavy-traffic approximations for\\ service systems facing overdispersed demand}\\
\textit{Britt Mathijsen, Guido Janssen, Johan van Leeuwaarden \& Bert Zwart}\\
arxiv.org/abs/1512.05581
\end{flushright}
\newpage
\section{Introduction}\label{intro}
In the previous chapter, we analyzed the scaling limit of a queueing model in which demand exhibits stochastic fluctuations that are asymptotically proportional to the square-root of the nominal load, while we deliberately chose to deviate from the square-root staffing principle by allocating a variability hedge that does not match the order of these fluctuations.
This chapter in some ways does the opposite.
We assume the demand faced by the queueing system is more volatile than anticipated by the independent many-sources paradigm that leads to Poisson traffic models.
As will become clear in this chapter, this in fact \emph{requires} an adaptation of the square-root staffing principle in order to maintain the desirable properties of the QED regime.
We start by motivating our research through empirical evidence of the presence of so-called \emph{overdispersion} in arrival processes faced by service systems reported by recent literature. \\
\noindent
\textbf{Motivation.}
The bulk of the queueing literature assumes perfect knowledge about the model primitives, including the mean demand per time period. For large-scale service systems, like health care facilities, communication systems or call centers, the dominant assumption is that demand arrives according to a (non)homogeneous Poisson process, which in practice translates into the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies shows that the variance of demand typically deviates from the mean significantly. Recent work \cite{Kim2015b,Kim2015a} reports variance being strictly less than the mean in health care settings employing appointment booking systems. This reduced variability, known as underdispersion, can be accredited to the goal of the booking system to create a more predictable arrival pattern.
On the other hand, in other scenarios with no control over the arrivals, the variance typically dominates the mean, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2015, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}.
The feature that variability is higher than one expects from the Poisson assumption is referred to as overdispersion. The latter concept will be the center of our attention in this chapter.
Stochastic models with the Poisson assumption have been widely applied to optimize capacity levels in service systems. The goal is to minimize operating costs while providing sufficiently high QoS in terms of performance measures such as mean delay or excess delay. When stochastic models, however, do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly under critical loading.
\\
\\*
\noindent
\textbf{Causes of overdispersion.}
The literature discussed above proves that the presence of overdispersion is widespread across applications.
It however does not specify what causes the increased variability in the arrival process.
We name two possible explanations.
First, we revisit the many-sources characterization of demand inflow discussed in Chapter 2.
Recall that in this setting, demand is generated by $n$ stochastically identical and independent sources, with $n$ large, so that workload arriving to the system in period $j$ is given by $A^{(n)}_j = \sum_{i=1}^n A_{i,j}$, where $A_{i,j}$, $i=1,2,\ldots,n$ are i.i.d.~random variables.
This resulted in nominal workload $\mu_n = n\mu$ and $\sigma_n^2 = n\sigma^2$, thus both of order $n$.
If we now relax the assumption on the (pairwise) independence of the sources, but rather consider the scenario in which these are positively correlated, then the nominal load remains to be equal to $n \mu$, while the variance of demand becomes
\begin{equation*}
\sigma_n^2 = {\rm Var}\, A_j^{(n)} = n\, {\rm Var}\, A_{1,j} + n(n-1)\,{\rm Cov}(A_{1,j},A_{2,j}),
\end{equation*}
which is of higher order than $n$ if $n\,{\rm Cov}(A_{1,j},A_{2,j}) \to \infty$ as $n\to\infty$.
A second interpretation of overdispersion in arrival processes relates to \emph{arrival rate uncertainty}.
The canonical process for modeling the arrival process of a service system is the Poisson process with a given arrival rate $\lambda$.
Since model primitives, in particular the arrival rate, are typically estimated through historical data, these are prone to be subject to forecasting errors.
In the realm of Poisson processes, this inherent uncertainty can be acknowledged by viewing the arrival rate $\Lambda_n$ itself as being stochastic. The resulting doubly stochastic Poisson process, also known as Cox process (first presented in \cite{Cox1955}), implies that demand in a given interval $A_j$ follows a mixed Poisson distribution.
In this case, the expected demand per period equals $\mu_n = \mathbb{E}[\Lambda_n]$, while the variance is $\sigma_n^2 = \mathbb{E}[\Lambda_n]+{\rm Var}\,\Lambda_n$.
By selecting the distribution of the mixing factor $\Lambda_n$, the magnitude of overdispersion can be made arbitrarily large, and only a deterministic $\Lambda_n$ leads to a true Poisson process.
The mixed Poisson model presents a useful way to fit both the mean and variance to real data, particularly in case of overdispersion.
The mixing distribution can be estimated parametrically or non-parametrically, see \cite{koolejongbloed,maman}.
A popular parametric family is the Gamma distribution, which gives rise to an effective data fitting procedure that uses the fact that a Gamma mixed Poisson random variable follows a negative binomial distribution.
We will in this chapter adopt the assumption of a Gamma-Poisson mixture as the demand process.\\
\\*
\textbf{Adapted QED scaling.} To deal with overdispersion
new models are needed, scaling rules must be adapted, and existing capacity sizing rules need to be modified in order to incorporate a correct hedge against (increased) variability.
In this chapter, we consider an extension of the discrete queueing model of Chapter 2 that has a doubly stochastic Poisson process as input, $A_j\sim\,{\rm Pois}(\Lambda_n)$ and we identify the heavy-traffic regime in which it displays QED behavior.
That is, it fits the three asymptotic characteristics in Section 1.2.3 of this thesis.
As we argued in that particular section, a sensible candidate capacity allocation rule is $s_n = \mu_n + \beta \sigma_n$ for some $\beta>0$, which is equivalent to the scaling
\begin{equation*}
\frac{\mu_n}{\sigma_n}\,(1-\rho_n) \to \beta, \qquad \text{as }n\to\infty.
\end{equation*}
We will verify mathematically that this is asymptotically the appropriate choice.
Studies that have adressed similar capacity allocation problems with stochastic arrival rates include \cite{Kocaga2015, maman, Whitt1999, Whitt2006}.
Of the aforementioned papers, our work best relates to \cite{maman}, in the sense that we also assess the asymptotic performance of a queueing system having a stochastic arrival rate in heavy traffic.
We therefore expand the paradigm of the QED regime, in order to have it accommodate for overdispersed demand that follows from a doubly stochastic Poisson process.
\\
\\*
\textbf{Structure of the chapter}. The remainder of this chapter is structured as follows. Our model is introduced in Section \ref{modelSection} together with some preliminary results.
In Section 3.3 we derive the classical heavy-traffic scaling limits for the queue length process in the presence of overdispersed arrivals both for the moments and the distribution itself.
Section 3.4 presents our main theoretic result, which provides a robust refinement to the heavy-traffic characterization of the queue length measures in pre-limit systems.
In Section 3.5, we describe the numerical results and demonstrate the heavy-traffic approximation for a real data set coming from a health care setting. Section 3.6 provides some concluding remarks.
\section{Model description}\label{modelSection}
We consider the same mathematical model as in Section 2.2, in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,...$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
The system has a service capacity $s_n\in\mathbb{N}$ per period, the steady-state queue length can be characterized as, see (1.27),
\begin{equation}
\label{mm3}
Q^{(n)} {\;\buildrel{d}\over= \;} \max_{k\geq 0}\Bigl\{\sum_{i=1}^k (A^{(n)}_i-s_n)\Bigr\}.
\end{equation}
For brevity, we define $\mu_n:= \mathbb{E} [A^{(n)}_1]$ and $\sigma_n^2 = {\rm Var}\, A^{(n)}_1$.
The behavior of $Q^{(n)}$ predominantly depends on the characteristics of $A^{(n)}$ and $s_n$. As noted before, $\mu_n<s_n$ is a necessary condition for the maximum in \eqref{mm3} to be finite and consequently for the queue to be stable. Before continuing the analysis of $Q^{(n)}$, we impose a set of conditions on the asymptotic properties of $s_n,\mu_n$ and $\sigma_n$.
\begin{assumption}
\label{as1}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item[{\normalfont (a)}] {\rm (Asymptotic growth)}
\begin{equation*}
\mu_n,\sigma_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (b)}] {\rm (Persistence of overdispersion)}
\begin{equation*}
\sigma_n^2/\mu_n \to \infty \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (c)}] {\rm (Heavy-traffic condition)}
The utilization $\rho_n := \mu_n/s_n \to 1$ as $n\to\infty$, while
\begin{equation}\label{mm5}
s_n = \mu_n + \beta\, \sigma_n,
\end{equation}
for some $\beta > 0$. This is equivalent to requiring
\begin{equation}\label{mm4}
(1-\rho_n)\frac{\mu_n}{\sigma_n} \to \beta, \qquad \text{\rm for }n\to\infty.
\end{equation}
\end{enumerate}
\end{assumption}
\noindent
Assumption \ref{as1} is assumed to hold throughout the remainder of this chapter.
Since we are mainly interested in the system behavior in heavy traffic, it is appropriate to study the queue length process in a scaled form. Substituting $s_n$ as in Assumption \ref{as1}(c), and dividing both sides of \eqref{mm3} by $\sigma_n$, gives
\begin{equation}
\label{mm6}
\frac{Q^{(n)}}{\sigma_n} = \max_{k\geq 0} \Bigl\{{\sum_{i=1}^k} \Bigl(\frac{A^{(n)}_i-\mu_n}{\sigma_n} - \beta\Bigr)\Bigr\}.
\end{equation}
By defining $\hat{Q}^{(n)} := Q^{(n)}/\sigma_n$ and $\hat{A}^{(n)}_i := (A^{(n)}_i-\mu_n)/\sigma_n$, we see that the scaled queue length process is in distribution equal to the maximum of a random walk with i.i.d. increments distributed as $\hat{A}^{(n)}-\beta$. Besides $\mathbb{E}[\hat{A}^{(n)}] = 0$ and ${\rm Var}\, \hat{A}^{(n)}=1$, the scaled and centered arrival count $\hat{A}^{(n)}$ has a few other nice properties which we turn to later in this section.
The model in \eqref{mm3} is valid for any distribution of $A^{(n)}$, also for the original case where the number of arrivals follows a Poisson distribution with fixed parameter $\lambda_n$, but in that case Assumption \ref{as1}(b) does not hold. Instead, we assume $A^{(n)}$ to be Poisson distributed with uncertain arrival rate rendered by the non-negative random variable $\Lambda_n$. This $\Lambda_n$ is commonly referred to as the \emph{prior} distribution, while $A^{(n)}$ is given the name of a Poisson mixture, see \cite{Grandell1997}. Given that the moment generation function of $\Lambda_n$, denoted by $M^\Lambda_n(\cdot)$, exists, we are able to express the probability generating function (pgf) of $A^{(n)}$ through the former. Namely,
\begin{equation}
\label{mm7}
\tilde{A}^{(n)}(z) = \mathbb{E}[\mathbb{E}[ z^{A^{(n)}} | \Lambda_n ] ] = \mathbb{E}[ \exp(\Lambda_n(z-1))] = M^\Lambda_n(z-1).
\end{equation}
From \eqref{mm7}, we get
\begin{equation}
\label{mm8}
\mu_n = \mathbb{E}[A^{(n)}] = \mathbb{E}[\Lambda_n],\qquad
\sigma_n^2 = {\rm Var}\, A^{(n)} = {\rm Var}\, \Lambda_n + \mathbb{E}[\Lambda_n],
\end{equation}
so that $\mu_n<\sigma_n^2$ if $\Lambda_n$ is non-deterministic. Assumption \ref{as1}(b) hence translates to
\[{\rm Var}\, \Lambda_n/\mathbb{E}[\Lambda_n]\rightarrow \infty, \qquad n\rightarrow\infty.\]
The next result relates the converging behavior of the centered and scaled $\Lambda_n$ to that of $\hat{A}^{(n)}$.
\begin{lemma}\label{gaussStep}
Let $\mu_n,\sigma_n^2\rightarrow\infty$ and $\sigma_n^2/\mu_n\rightarrow\infty$. If
\begin{equation*}
\hat{\Lambda}_n := \frac{\Lambda_n-\mu_n}{\sigma_n}{\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{\normalfont for } n\rightarrow\infty,
\end{equation*}
then $\hat{A}^{(n)}$ converges weakly to a standard normal variable as $n\rightarrow\infty$.
\end{lemma}
\noindent
The proof can be found in Appendix \ref{formalSec}.
The prevalent choice for $\Lambda_n$ is the Gamma distribution. The Gamma-Poisson mixture turns out to provide a very good fit to arrival counts experienced by service systems, as was observed by \cite{koolejongbloed}. Assuming $\Lambda_n$ to be of Gamma type with scale and rate parameters $a_n$ and $1/b_n$, respectively, we get for the pgf of $A^{(n)}$:
\begin{equation}
\label{r0}
\tilde{A}^{(n)}(z) = \Bigl(\frac{1}{1+b_n(1-z)}\Bigr)^{a_n},
\end{equation}
in which we recognize the pgf of a negative binomial distribution with parameters $a_n$ and $1/(b_n+1)$, so that
\begin{equation*}
\label{t21}
\mu_n = a_nb_n,\qquad \sigma_n^2 = a_nb_n(b_n+1).
\end{equation*}
Note that in the context of a Gamma prior, the restrictions in Assumption \ref{as1} reduce to only two rules. For completeness, we include the revised list below.
\begin{assumption}\label{as2}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item {\rm (Asymptotic regime and persistence of overdispersion)}
\begin{equation*}
a_n, b_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item {\rm (Heavy-traffic condition)}
Let
\begin{equation*}
s_n = a_n b_n + \beta \sqrt{a_n b_n(b_n+1)},
\end{equation*}
for some $\beta>0$, or equivalently
\begin{equation*}
(1-\rho_n)\sqrt{a_n} \to \beta, \quad \text{\rm for } n\to\infty.
\end{equation*}
\end{enumerate}
\end{assumption}
The next result follows from the fact that $\Lambda_n$ is a Gamma random variable:
\begin{corollary}\label{scaledLambdaLemma}
Let $\Lambda_n\sim\text{\normalfont Gamma}(a_n,1/b_n)$, $A^{(n)}\sim{\rm Pois }(\Lambda_n)$ and $a_n,b_n\rightarrow \infty$. Then $\hat{A}^{(n)}$ converges weakly to a standard normal random variable as $n\rightarrow \infty$.
\end{corollary}
\begin{proof}
By Lemma \ref{gaussStep}, it is sufficient to prove that $\hat{\Lambda}_n{\;\buildrel{d}\over\Rightarrow\;}\mathcal{N}(0,1)$ for this particular choice of $\Lambda_n$.
We do this by proving the pointwise convergence of the characteristic function (cf) of $\hat{\Lambda}_n$ to $\exp({-} t^2/2)$, the cf of the standard normal distribution.
Let $\phi_{G}(\cdot)$ denote the characteristic function of a random variable $G$. By basic properties of the cf,
\begin{align*}
\phi_{\hat{\Lambda}_n}(t) &= {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n)
= {\rm e}^{-i\mu_nt/\sigma_n} \Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)^{-a_n}\nonumber\\
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n}\, - a_n\,{\rm ln}\Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)\Bigr]\nonumber\\
\label{g13d}
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n} -a_n\Bigl( {-}\frac{i\,b_nt}{\sigma_n} + \frac{b_n^2t^2}{2\sigma_n^2} + O( b_n^3/\sigma_n^3)\Bigr)\Bigr] \nonumber\\
&= \exp\Bigl[ -\frac{b_n\,t^2}{2(b_n+1)} + O\left(1/\sqrt{a_n}\right)\Bigr] \rightarrow \exp\big({-} t^2/2\big),
\end{align*}
for $n\rightarrow\infty$. By L\'evy's continuity theorem this implies $\hat{\Lambda}_n$ is indeed asymptotically standard normal.
\end{proof}
The characterization of the arrival process as a Gamma-Poisson mixture is of vital importance in later sections.\\
\\*
\noindent
\textbf{Expressions for the stationary distribution.} \label{expressionsSubsec}
Our main focus is on the stationary queue length distribution, denoted by
\[\mathbb{P}(Q^{(n)}=i) =\lim_{k\rightarrow\infty} \mathbb{P}(Q^{(n)}(k)=i).\]
Denote the pgf of $Q^{(n)}$ by
\begin{equation*}
\label{t1}
\tilde{Q}^{(n)}(w) := \sum_{i=0}^\infty \mathbb{P}(Q^{(n)}=i) w^i.
\end{equation*}
Furthermore, let $\mu_Q := \mathbb{E}[Q^{(n)}]$ and $\sigma_Q^{2} := {\rm Var}\, Q^{(n)}$ denote the stationary mean and variance of the queue length, respectively.
To avoid notational complexity, we omit the superscript $(n)$ in these definitions.
To continue our analysis of $Q^{(n)}$, we need one more condition on $A^{(n)}$.
\begin{assumption}\label{as3}
The pgf of $A^{(n)}$, denoted by $\tilde{A}^{(n)}(w)$, exists for $|z|<r_0$, for some $r_0>1$, so that all moments of $A^{(n)}$ are finite.
\end{assumption}
We next recall two characterizations of $\tilde{Q}^{(n)}(w)$ that play prominent roles in the remainder of our analysis.
The first characterization of $\tilde{Q}^{(n)}(w)$ originates from a random walk perspective. As we see from \eqref{mm3}, the (scaled) stationary queue length is equal in distribution to the all-time maximum of a random walk with i.i.d. increments distributed as $A^{(n)}-\beta$ (or $\hat{A}^{(n)}-\beta$ in the scaled setting). Spitzer's identity, see e.g. \cite[Theorem VIII4.2]{Asmussen2003} and Section 1.2.2 of this thesis, then gives
\begin{equation*}
\label{t3}
\tilde{Q}^{(n)}(w) = \exp\left\{\sum_{k=1}^\infty \frac{1}{k}\,\Big(\mathbb{E}\Big[w^{\left(\sum_{i=1}^k \{A^{(n)}_i-s_n\}\right)^+}\Big]-1\Big)\right\},
\end{equation*}
where $(x)^+ = \max\{x,0\}$. Hence,
\begin{equation*}
\label{t4}
\mu_Q = \mathbb{E}[Q^{(n)}] = \tilde{Q}^{(n)\prime}(1) = \sum_{k=1}^\infty \frac{1}{k}\,\mathbb{E}\Bigl[ {\sum_{i=1}^k} (A^{(n)}_i - s_n) \Bigr]^+,
\end{equation*}
\begin{equation*}
\label{t4a}
\sigma^{2}_Q = {\rm Var}\, Q^{(n)} = \tilde{Q}^{(n)\prime\prime}(1)+Q^{(n)\prime}(1)-\left(\tilde{Q}^{(n)\prime}(1)\right)^2 = \sum_{k=1}^\infty \frac{1}{k}\,\mathbb{E}\Bigl[ \Big(\sum_{i=1}^k (A^{(n)}_i - s_n) \Big)^+\Bigr]^2,
\end{equation*}
\begin{align*}
\label{t5}
\mathbb{P}(Q^{(n)}=0) = \tilde{Q}_n(0) &= \exp\Bigl\{{-}{\sum_{k=1}^\infty}\frac{1}{k}\,\mathbb{P}\Bigl({\textstyle\sum_{i=1}^k} (A^{(n)}_i-s_n) > 0\Bigr) \Bigr\}.
\end{align*}
A second characterization follows from Pollaczek's formula, see \cite{Abate1993} and Section 2.2.2 of this thesis:
\begin{equation}
\label{t6}
\tilde{Q}^{(n)}(w) = \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{w-z}{1-z}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\},
\end{equation}
which is analytic for $|w|<r_0$, for some $r_0>1$. Therefore, $\varepsilon>0$ has to be chosen such that $|w|<1+\varepsilon<r_0$. This gives
\begin{align}
\label{t7}
\mu_Q &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{1}{1-z}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)} {\rm d} z,\\
\label{t7a}
\sigma_Q^{2} &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{{-}z}{(1-z)^2}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z,\\
\label{t8}
\mathbb{P}(Q^{(n)}=0) &= \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{z}{z-1}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\}.
\end{align}
{\color{purple}
Pollaczek-type integrals like \eqref{t6}-\eqref{t8} first occurred in the work of Pollaczek on the classical single-server queue (see \cite{Abate1993,Cohen1982,Janssen2008} for historical accounts). These integrals are fairly straightforward to evaluate numerically and hence give rise to efficient algorithms for performance evaluation \cite{Abate1993,boon2017pollaczek}. The integrals also proved useful in establishing heavy-traffic results by asymptotic evaluation of the integrals in various heavy-traffic regimes \cite{Kingman1962,Cohen1982,Janssen2015,boon2017pollaczek2}, and in this paper we follow that approach for a heavy-traffic regime that is suitable for overdispersion.
}
\section{Heavy-traffic limits}
In this section we present the result on the convergence of the discrete process $\hat{Q}^{(n)}$ to a non-degenerate limiting process and of the associated stationary moments. The latter requires an interchange of limits. Using this asymptotic result, we derive two sets of approximations for $\mu_Q$, $\sigma^2_Q$ and $\mathbb{P}(Q^{(n)}=0)$, that capture the limiting behavior of $Q^{(n)}$. The first set provides a rather crude estimation for the first cumulants of the queue length process for any arrival process $A^{(n)}$ satisfying Assumption \ref{as1}. The second set, which is the subject of the next section, is derived for the specific case of a Gamma prior and is therefore expected to provide more accurate, robust approximations for the performance metrics.
We start by indicating how the asymptotic properties of the scaled arrival process give rise to a proper limiting random variable describing the stationary queue length. The asymptotic normality of $\hat{A}^{(n)}$ provides a link with the Gaussian random walk and nearly deterministic queues \cite{Sigman2011a,Sigman2011b}.
The main results in \cite{Sigman2011a,Sigman2011b} were obtained under the assumption that $\rho_n\sim 1-\beta/\sqrt{n}$, in which case it follows from \cite[Thm.~3]{Sigman2011b} that the rescaled stationary waiting time process converges to a reflected Gaussian random walk.
We shall also identify the Gaussian random walk as the appropriate scaling limit for our stationary system. However, since the normalized natural fluctuations of our system are given by $\mu_n/\sigma_n$ instead of $\sqrt{n}$, we assume that the load grows like $\rho_n \sim 1 - \frac{\beta}{\mu_n/\sigma_n}$. Hence, in contrast to \cite{Sigman2011a,Sigman2011b}, our systems' characteristics display larger natural fluctuations, due to the mixing factor that renders the arrivals. Yet, by matching this overdispersed demand with the appropriate hedge against variability, we again obtain Gaussian limiting behavior. This is not surprising, since we saw in Lemma \ref{gaussStep} that the increments start resembling Gaussian behavior for $n\rightarrow\infty$. The following result summarizes this.
\begin{theorem}
\label{gaussianThm}
Let $\Lambda_n$ be a non-negative random variable such that $(\Lambda_n-\mu_n)/\sigma_n$ is asymptotically standard normal, with $\mu_n$ and $\sigma_n$ as defined in \eqref{mm8}, and $\mathbb{E}[\Lambda_n^3]<\infty$ for all $n\in\mathbb{N}$. Then under Assumption \ref{as1}, for $n\rightarrow \infty$,
\begin{enumerate}
\item[{\rm (i)}] $\hat{Q}^{(n)} {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[{\rm (ii)}] $\mathbb{P}(Q^{(n)} = 0) \rightarrow \mathbb{P}(M_\beta=0)$,
\item[{\rm (iii)}] $\mathbb{E}[\hat{Q}^{(n)}] \rightarrow \mathbb{E} [M_\beta]$,
\item[{\rm (iv)}] ${\rm Var}\, \hat{Q}_n \rightarrow {\rm Var}\,\, M_\beta$,
\end{enumerate}
where $M_\beta$ is the all-time maximum of a random walk with i.i.d. normal increments with mean $-\beta$ and unit variance.
\end{theorem}
The proof of Theorem \ref{gaussianThm} is given in Appendix \ref{formalSec}. The following result shows that Theorem \ref{gaussianThm} also applies to Gamma mixtures, which is a direct consequence of Corollary \ref{scaledLambdaLemma}.
\begin{corollary}
Let $\Lambda_n\sim$ \normalfont{Gamma}$(a_n,b_n)$. Then under Assumption \ref{as2} the four convergence results of Theorem \ref{gaussianThm} hold true.
\end{corollary}
It follows from Theorem \ref{gaussianThm} that the scaled stationary queueing process converges under \eqref{mm4} to a reflected Gaussian random walk. Hence, the performance measures of the original system should be well approximated by the performance measures of the reflected Gaussian random walk, yielding heavy-traffic approximations.
Like our original system, the Gaussian random walk falls in the classical setting of the reflected one-dimensional random walk, whose behavior is characterized by both Spitzer's identity and Pollaczek's formula. In particular, Pollaczek's formula gives rise to contour integral expressions for performance measures that are easy to evaluate numerically, also in heavy-traffic conditions. The numerical evaluation of such integrals is considered in \cite{Abate1993}. For $\mathbb{E} [M_\beta]$ such an integral is as follows
\begin{equation}
\label{g13e}
\mathbb{E} [M_\beta] = {-}\frac{1}{\pi}\int_0^\infty {\rm Re}\Bigl[\frac{1-\phi(-z)}{z^2}\Bigr]{\rm d} y,
\end{equation}
where $z=x+iy$ with an appropriately chosen real part $x$, with $\phi(z) = \exp(-\beta\,z+\tfrac12\,z^2)$, the Laplace transform of a normal random variable with mean $-\beta$ and unit variance.
Note that this integral involves complex-valued functions with complex arguments. Similar Pollaczek-type integrals exist for $\mathbb{P}(M_\beta=0)$ and ${\rm Var}\, M_\beta$, see \cite{Abate1993}. The following result simply rewrites these integrals in terms of a real integral and uses the fact that the scaled queue length process mimics the maximum of the Gaussian random walk for large $n$.
\begin{corollary}\label{abateThm}
Under Assumption \ref{as1}, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$, $\mu_Q$ and $\sigma^2_Q$ as $n\to\infty$ are given by, respectively,
\begin{equation}
\label{h1a}
\exp\Bigl[\frac{1}{\pi} \int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\Bigl(1-e^{-\tfrac12\beta^2-t^2}\Bigr){\rm d} t\Bigr],
\end{equation}
\begin{equation}
\label{h1}
\frac{\sqrt{2}\sigma_n}{\pi}\int_0^\infty \frac{t^2}{\tfrac12\beta^2+t^2}\, \frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t,
\end{equation}
\begin{equation}
\label{h1b}
\frac{\sqrt{2}\beta\sigma_n^2}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t.
\end{equation}
\end{corollary}
\begin{proof}
According to \cite[Eq.~(15)]{Abate1993},
\begin{equation*}
\label{z1}
{-}\,{\rm ln}\,[\mathbb{P}(M_\beta=0)] = c_0,\quad \mathbb{E}[M_\beta]\ = c_1, \quad {\rm Var}\,\, M_\beta = c_2,
\end{equation*}
where
\begin{equation*}
\label{z2}
c_n = \frac{(-1)^nn!}{\pi} \,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp(\beta\,z+\tfrac12 z^2))}{z^{n+1}} {\rm d} y\Bigr],
\end{equation*}
in which $z={-}x+i\,y$, $y\geq 0$, and $x$ is any fixed number between 0 and $2\beta$.
Take $x=\beta$, so that
\begin{equation*}
\label{z3}
\beta z+\tfrac12 z^2 = {-}\tfrac12\beta^2 - \tfrac12 y^2\leq 0,\quad y\geq 0.
\end{equation*}
For $n=0$, this gives
\begin{align*}
c_0 &= \frac{1}{\pi}\,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-\tfrac12 y^2))}{{-}\beta+i\,y} {\rm d} y\Bigr] \nonumber\\
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta}{\beta^2+y^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2- \tfrac12 y^2)) {\rm d} y\nonumber\\
\label{z4}
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-t^2)) {\rm d} t,
\end{align*}
where we used that
\begin{equation*}
\label{z5}
{\rm Re }\Bigl[\frac{1}{{-}\beta+i\, y}\Bigr] = \frac{{-}\beta}{\beta^2+y^2},
\end{equation*}
together with the substitution $y=t\sqrt{2}$. For $n=1,2,\ldots,$ partial integration gives
\begin{align*}
c_n &= \frac{(-1)^n n!}{\pi} \, {\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{({-}\beta+i\,y)^{n+1}} {\rm d} y\nonumber\\
&= \frac{(-1)^{n-1}(n-1)!}{\pi}\,{\rm Im}\Bigl[\int_0^\infty {\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)){\rm d} \Bigl(\frac{1}{(-\beta+i\,y)^n}\Bigr)\Bigr]\nonumber\\
\label{z6}
&= {-}\frac{(-1)^{n-1}(n-1)!}{\pi} {\rm Im}\Bigl[ \int_0^\infty \frac{y}{(-\beta+i\,y)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr],
\end{align*}
where we have used that
\begin{equation*}
\label{z7}
{\rm Im}\Bigl[\frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{(-\beta+i\,y)^n}\Bigr]\Bigl|_0^\infty\Bigr. = 0.
\end{equation*}
Using
\begin{equation*}
\label{z8}
\frac{1}{(-\beta+i\,y)^n} = (-1)^n\,\frac{(\beta+i\,y)^n}{(\beta^2+y^2)^n},
\end{equation*}
we then get
\begin{equation*}
\label{z9}
c_n = \frac{(n-1)!}{\pi}\,{\rm Im}\,\Bigl[\int_0^\infty \frac{y(\beta+i\,y)^n}{(\beta^2+y^2)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr],
\end{equation*}
which after the substitution of $y=t\sqrt{2}$ gives
\begin{align}
c_1&=\frac{1}{\pi}\,\int_0^\infty \frac{y^2}{\beta^2+y^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y \nonumber\\
\label{z10}
&= \frac{\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{\tfrac12 \beta^2+t^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)}{\rm d} t,
\end{align}
\begin{align*}
c_2&=\frac{2\beta}{\pi}\,\int_0^\infty \frac{y^2}{(\beta^2+y^2)^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y\nonumber\\
\label{z11}
&= \frac{\beta\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)} {\rm d} t.
\end{align*}
\end{proof}
\section{Robust heavy-traffic approximations}
We shall now establish robust heavy-traffic approximations for the canonical case of Gamma-POisson mixutres; see \eqref{r0}.
As noted earlier, Gamma mixing yields an arrival process that has a negative binomial distribution, which allows us to establish the detailed asymptotic results in the next theorem.
\begin{theorem}\label{saddlepointThm}
Let $a_n,b_n$ and $s_n$ be as in Assumption \ref{as2}. Then the leading order behavior of $\mu_Q$ is given by
\begin{equation}
\label{r1}
\frac{\sqrt{2}\,\beta_n}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta^2_n+t^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)} {\rm d} t\,(1+o(1)),
\end{equation}
where
\begin{equation}
\label{r2}
\beta_n^2 = s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Furthermore, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$ and $\sigma^2_Q$ is given by
\begin{equation*}
\label{r3}
\exp\Bigl[\frac{1}{\pi}\,\frac{b_n+\rho_n}{b_n+1}\,\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta^2_n+t^2}\,{\rm ln}\,\Bigl(1-{\rm e}^{{-}\tfrac12\beta^2_n-t^2}\Bigr){\rm d} t\Bigr],
\end{equation*}
and
\begin{equation}
\label{r4}
\frac{\beta_n^3/\sqrt{2}}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)^2\Bigl(\frac{b_n+1}{b_n+\rho_n}+1\Bigr)\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t,
\end{equation}
respectively.
\end{theorem}
The proof of Theorem \ref{saddlepointThm} requires asymptotic evaluation of the Pollaczek-type integrals \eqref{t6}-\eqref{t8}, for which shall use the \textit{non-standard} saddle point method---originally proposed by \cite{debruijn} and also applied in Chapter 2 of this thesis---to turn these contour integrals into practical approximations.
In contrast to the setting of Chapter 2, both the relevant saddle point and the analyticity radius tend to one as $n\to\infty$, which is a singular point of the integrand, in the setting with overdispersion.
For the proof of Theorem \ref{saddlepointThm}, we therefore modify the special saddle point method developed in Chapter 2 to account for this circumstance.
\begin{proof}
Our starting point is the probability generating function of the number of arrivals per time slot, given in \eqref{r0}, which is analytic for $|z|<1+1/b_n=:r$. Under Assumption \ref{as2}, we consider $\mu_Q$ as given in \eqref{t7}. We set
\begin{equation}
\label{a7}
g(z) = -{\rm ln }\,z+\frac{1}{s_n}\,{\rm ln }\Bigl[\tilde{A}^{(n)}(z)\Bigr] = -{\rm ln }\,z - \frac{a_n}{s_n}\,{\rm ln }\left(1+(1-z)b_n\right),
\end{equation}
to be considered in the entire complex plane with branch cuts $(-\infty,0]$ and $[r,\infty)$. The relevant saddle point $z_{\rm sp}$ is the unique zero $z$ of $g'(z)$ with $z\in(1,r_0)$. Since
\begin{equation}
\label{a8}
g'(z) = -\frac{1}{z} + \frac{\rho_n}{1+(1-z)b_n},
\end{equation}
this yields,
\begin{equation}
\label{a9}
1+(1-z_{\rm sp})b_n = \rho_n z_{\rm sp},\quad {\rm i.e., } \quad z_{\rm sp} = 1+\frac{1-\rho_n}{\rho_n+b_n}.
\end{equation}
We then find
\begin{equation}
\label{a10}
\mu_Q = \frac{s_n}{2\pi i} \int_{|z| = 1+\varepsilon} \frac{g'(z)}{z-1}\,\frac{\exp(s_n\,g(z))}{1-\exp(s_n\,g(z))}{\rm d} z,
\end{equation}
and we take here $1+\varepsilon = z_{\rm sp}$. There are no problems with the branch cuts since we consider $\exp(s_ng(z))$ with integer $s_n$. \\
We continue as in Chapter 2, Section 3 and thus we intend to substitute $z=z(v)$ in the integral in \eqref{a10}, where $z(v)$ satisfies
\begin{equation*}
\label{k1}
g(z(v)) = g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) =: q(v)
\end{equation*}
on a range ${-}\tfrac12\delta_n \leq v\leq \tfrac12 \delta_n$ with $\delta_n \to 0$ as $n\to\infty$.
Note that, this range depends on $n$, whereas these bounds $\pm \tfrac{1}{2} \delta_n$ remained bounded away from zero in \cite{Janssen2015}.
This severely complicates the present analysis.
We consider the approximate representation
\begin{equation}
\label{k2}
\frac{-s_n\,g''(z_{\rm sp})}{2\pi i}\int_{-\tfrac12 \delta_n}^{\tfrac12 \delta_n}\frac{v}{z(v)-1}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))} {\rm d} v
\end{equation}
of $\mu_Q$. We have to operate here with additional care, since both the analyticity radius $r=1+1/b_n$ and the saddle point $z_{\rm sp}$ outside zero $r_0$ tend to 1 as $n\rightarrow\infty$. Specifically, proceeding under the assumptions that $(1-\rho_n)^2a_n$ is bounded while $a_n\rightarrow\infty$ and $b_n\geq 1$, see Assumption \ref{as2}, we have from \eqref{a9} that
\begin{equation}\label{a19}
z_{\rm sp}-1=\frac{1-\rho_n}{b_n+\rho_n} = \frac{1-\rho_n}{b_n} + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr),
\end{equation}
where the $O$-term is small compared to $(1-\rho_n)/b_n$ when $b_n\rightarrow\infty$. Next, we approximate $r_0$, using that $r_0>1$ satisfies
\begin{equation*}
\label{a20}
{-}{\rm ln}\, r_0 - \frac{\rho_n}{b_n}\, {\rm ln}\,(1+(1-r_0)b_n) = 0.
\end{equation*}
Write $r_0 = 1+u/b_n$, so that we get the equation
\begin{align*}
0 &= {-}{\rm ln}\,\left(1+\frac{u}{b_n}\right) - \frac{\rho_n}{b_n}\,{\rm ln }(1-u)\nonumber \\
\label{a21}
&= {-}\frac{u}{b_n}\Bigl(1-\rho_n-\tfrac12\Bigl(\frac{1}{b_n}+\rho_n\Bigr)u-\tfrac{1}{3}\Bigl(\frac{-1}{b^2_n}+\rho_n\Bigr)u^2+\cdots\Bigr),
\end{align*}
where we have used the Taylor expansion of ${\rm ln}(1+x)$ at $x=0$. Thus we find
\begin{equation*}
\label{a22}
u=\frac{2(1-\rho_n)}{\rho_n+1/b_n}+O(u^2) = 2(1-\rho_n)+O((1-\rho_n)^2)+O\Bigl(\frac{1-\rho_n}{b_n}\Bigr),
\end{equation*}
and so,
\begin{equation*}
\label{a23}
r_0 = 1+2\,\frac{1-\rho_n}{b_n}+O\Bigl(\frac{(1-\rho_n)^2}{b_n}\Bigr) + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr).
\end{equation*}
In \eqref{k2} we choose $\delta_n$ so large that the integral has converged within exponentially small error using $\pm\delta_n$ as integration limits, and, at the same time, so small that there is a convergent power series
\begin{equation}
\label{a26}
z(v) = z_{\rm sp}+iv+ \sum_{k=2}^\infty c_k(iv)^k, \qquad \text{for } |v| \leq \tfrac12 \delta_n.
\end{equation}
To achieve these goals, we supplement the information on $g(z)$, as given by $\eqref{a7}-\eqref{a9}$, by
\begin{equation}
\label{a27}
g''(z)=\frac{1}{z^2}+\frac{\rho_nb_n}{(1+(1-z)b_n)^2},\quad g''(1) = 1+\rho_nb_n,\quad g''(z_{\rm sp}) =\frac{1}{z_{\rm sp}^2}\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Now
\begin{equation*}
\label{a36}
\exp(s_n\,q(v)) = \exp(s_n\,g(z_{\rm sp}))\exp(-\tfrac12\,s_n\,g''(z_{\rm sp})\,v^2),
\end{equation*}
and
\begin{equation*}
\label{a37} s_n\, g''(z_{\rm sp})v^2 = s_n\,b_nv^2(1+o(1)) = a_n(b_n\,v)^2(1+o(1)).
\end{equation*}
Therefore, \eqref{k2} approximates $\mu_Q$ with exponentially small error when we take $\tfrac12 \delta_n$ of the order $1/b_n$.
We next aim at showing that we have a power series for $z(v)$ as in \eqref{a26} that converges for $|v|\leq\tfrac12\delta_n$ with $\tfrac12\delta_n$ of the order $1/b_n$.
\begin{lemma}
Let
\begin{equation*}
\label{a38}
r_n:=\frac{1}{2\,b_n}-(z_{\rm sp} -1 ),\quad m_n:= \tfrac{2}{3}\rho_nr_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}},
\end{equation*}
where we assume $r_n>0$. Then \eqref{a26} holds with real coefficients $c_k$ satisfying
\begin{equation}
\label{a39}
|c_k|\leq\frac{r_n}{m_n^k},\quad k=2,3,\ldots.
\end{equation}
\end{lemma}
\begin{proof}
We let
\begin{equation}
\label{a40}
G(z):=\frac{2(g(z)-g(z_{\rm sp}))}{g''(z_{\rm sp})(z-z_{\rm sp})^2}.
\end{equation}
Then $G(z_{\rm sp})=1$ and so we can write \eqref{k1} as
\begin{equation}
\label{a41}
F(z):=(z-z_{\rm sp})\sqrt{G(z)} = i v
\end{equation}
when $|z-z_{\rm sp}|$ is sufficiently small. Since $F(z_{\rm sp})=0$, $F'(z_{\rm sp})=1$, the B\"urmann-Lagrange inversion theorem implies validity of a power series as in \eqref{a26}, with real $c_k$ since $G(z)$ is positive and real for real $z$ close to $z_{\rm sp}$. We therefore just need to estimate the convergence radius of this series from below.
To this end, we start by showing that
\begin{equation}
\label{a42}
{\rm Re}[g''(z)] > \frac{4}{9}\,\rho_n^2\frac{b_n+\rho_n^{-1}}{b_n+\rho_n},\quad |z-z_{\rm sp}|\leq r_n.
\end{equation}
For this, we consider the representation
\begin{equation}
\label{a43}
G(z) = 2\int_{0}^1\int_0^1 \frac{g''(z_{\rm sp}+s\,t(z-z_{\rm sp}))}{g''(z_{\rm sp})} \,t{\rm d} s{\rm d} t.
\end{equation}
We have for $\zeta\in\mathbb{C}$ and $|\zeta-1|\leq 1/2b_{n}\leq 1/2$ from \eqref{a27} that
\begin{equation}
\label{a44}
{\rm Re}[g''(\zeta)] = {\rm Re}(1/\zeta^2) + \rho_nb_n\,{\rm Re}\Bigl[\Bigl(\frac{1}{1+(1-\zeta)b_n}\Bigr)^2\Bigr]\geq \tfrac{4}{9}(1+\rho_nb_n).
\end{equation}
To show the inequality in \eqref{a44}, it suffices to show that
\begin{equation}
\label{a45}
\min_{|\xi-1|\leq 1/2} {\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{4}{9}.
\end{equation}
The minimum in \eqref{a45} is assumed at the boundary $|\xi-1|=1/2$, and for a boundary point $\xi$, we write
\begin{equation*}
\label{a46}
\xi= 1+\tfrac12\cos\theta+\tfrac12 i \sin\theta, \quad 0\leq \theta\leq 2\pi,
\end{equation*}
so that
\begin{equation*}
\label{a47}
{\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{1+\cos\theta+\tfrac{1}{4}\cos 2\theta}{(\tfrac{5}{4}+\cos\theta)^2}.
\end{equation*}
Now
\begin{equation*}
\label{a48}
\frac{{\rm d}}{d\theta} \Bigl[\frac{1+\cos\theta+\tfrac{1}{4}\cos2\theta}{(\tfrac{5}{4}+\cos\theta)^2}\Bigr] = \frac{\sin \theta\,(1-\cos \theta)}{4(\tfrac{5}{4}+\cos\theta)^3}
\end{equation*}
vanishes for $\theta=0,\pi,2\pi$, where ${\rm Re}(1/\xi^2)$ assumes the values $4/9$, 4, 4/9, respectively. This shows \eqref{a45}.
We use \eqref{a45} with $\xi=\zeta$ and with $\xi=1+(1-\zeta)b_n$, with
\begin{equation}
\label{a49}
\zeta = \zeta(s,t) = z_{\rm sp} + s t\,(z-z_{\rm sp}),\quad 0\leq s,\, t\leq 1,
\end{equation}
where we take $\zeta$ such that $|\zeta-1|\leq 1/2b_n$. It is easy to see from
$1<z_{\rm sp}<1+1/2b_n$ that $|\zeta-1|\leq 1/2b_n$ holds when $|z-z_{\rm sp}|\leq r_n=1/2b_n-(z_{\rm sp}-1)$. We have, furthermore, from \eqref{a9} that $0<g''(z_{\rm sp})\leq 1+b_n/\rho_n$. Using this, together with \eqref{a44} where $\zeta$ is as in \eqref{a49}, yields
\begin{equation*}
\label{a50}
{\rm Re}[G(z)] \leq \frac{4}{9}\,\frac{1+\rho_nb_n}{1+b_n/\rho_n}\,2\,\int_0^1\int_0^1 t\,{\rm d} s{\rm d} t = \tfrac{4}{9}\,\rho_n^2\,\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}
\end{equation*}
when $|z-z_{\rm sp}|\leq r_n$, and this is \eqref{a42}.
We therefore have from \eqref{a41} that
\begin{equation*}
\label{a51}
|F(z)|>r_n\cdot\frac{2}{3}\rho_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}} = m_n,\quad |z-z_{\rm sp}|=r_n.
\end{equation*}
Hence, for any $v$ with $|v|\leq m_n$, there is exactly one solution $z=z(v)$ of the equation $F(z)-iv=0$ in $|z-z_{\rm sp}|\leq r_n$ by Rouch\'e's theorem. This $z(v)$ is given by
\begin{equation*}
\label{a52}
z(v) = \frac{1}{2\pi i}\,\int_{|z-z_{\rm sp}|=r_n} \frac{F'(z)\,z}{F(z)-iv}{\rm d} z,
\end{equation*}
and depends analytically on $v$, $|v|\leq m_n$. From $|z(v)-z_{\rm sp}|\leq r_n$, we can finally bound the power series coefficients $c_k$ according to
\begin{equation*}
\label{a53}
|c_k| = \Bigl|\frac{1}{2\pi i}\int_{|iv|=m_n} \frac{z(v)-z_{\rm sp}}{(iv)^{k+1}}{\rm d}(iv)\Bigr| \leq \frac{r_n}{m_n^k},
\end{equation*}
and this completes the proof of the lemma.
\end{proof}
\begin{remark}
We have $z_{\rm sp}-1=o(1/b_n)$, see \eqref{a19}, and so
\begin{equation*}
\label{a54}
r_n = \frac{1}{2b_n}(1+o(1)),\quad m_n = \frac{1}{3b_n}(1+o(1)),
\end{equation*}
implying that the radius of convergence of the series in \eqref{a26} is indeed of order $1/b_n$ (since we have assumed $b_n\geq 1$).
\end{remark}
We let $\delta_n=m_n$, and we write for $0\leq v\leq \tfrac12\delta_n$
\begin{equation*}
\label{a55}
\frac{v}{z(v)-1}+\frac{{-}v}{z({-}v)-1} = \frac{-2iv\,{\rm Im}(z(v))}{|z(v)-1|^2},
\end{equation*}
where we have used that all $c_k$ are real, so that $z(-v)=z(v)^*$, where $ ^*$ denotes the complex conjugate. Now from \eqref{a39} and realness of the $c_k$, we have
\begin{equation}
\label{a56}
{\rm Im}(z(v)) = v+\sum_{l=1}^\infty c_{2l+1}(-1)^l\,v^{2l+1} = v+O(v^3),
\end{equation}
and in similar fashion
\begin{equation}
\label{a57}
|z(v)-1|^2 = (z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)^2v^2) + O(v^4)
\end{equation}
when $0\leq v\leq \tfrac12\delta_n$. The order terms in \eqref{a56}-\eqref{a57} are negligible in leading order, and so we get for $\mu_{Q^{(n)}}$ via \eqref{k2} the leading order expression
\begin{equation*}
\label{a58}
\frac{{-}s_n\,g''(z_{\rm sp})}{2\pi i}\,\int_0^{\tfrac12\delta_n}\frac{{-}2iv^2}{(z_{\rm sp}-1)^2+v^2}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))}{\rm d} v.
\end{equation*}
We finally approximate $q(v) = g(z_{\rm sp})-\tfrac12 g''(z_{\rm sp})v^2$.
There is a $z_1$, $1\leq z_1\leq z_{\rm sp}$ such that
\begin{equation*}
\label{a59}
g(z_{\rm sp}) = {-}\tfrac12(z_{\rm sp}-1)^2\,g''(z_1),
\end{equation*}
and, see \eqref{a19} and \eqref{a27},
\begin{equation*}
\label{a60}
g''(z_1) = g''(z_{\rm sp}) + O((1-\rho_n)b_n).
\end{equation*}
Hence
\begin{align}
s_n\,q(v) &= {-}\tfrac12 s_n\,g''(z_{\rm sp})\,[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)b_ns_n(z_{\rm sp}-1)^2)\nonumber\\
&= {-}\tfrac12 s_n\,g''(z_{\rm sp})[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)^2a_n),\label{a61}
\end{align}
where \eqref{a19} has been used and $a_nb_n = s_n(1+o(1))$ Therefore, the $O$-term in \eqref{a61} tends to 0 by our assumption that $(1-\rho_n)^2a_n$ is bounded. Thus, we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation}\label{a62}
\frac{s_n g''(z_{\rm sp})}{\pi} \int_{0}^{\tfrac12\delta_n}\frac{v^2}{(z_{\rm sp}-1)^2+v^2}\,
\frac{\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))}{1-\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))} {\rm d} v,
\end{equation}
When we substitute $t=v\sqrt{s_n\,g''(z_{\rm sp})/2}$ and extend the integration in \eqref{a62} to all $t\geq 0$ (at the expense of an exponentially small error), we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation*}
\label{a63}
\frac{1}{\pi}\,\sqrt{2\,s_n\,g''(z_{\rm sp})}\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta_n^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)}{\rm d} t,
\end{equation*}
where
\begin{equation*}
\label{a64}
\beta^2_n = s_n\,g''(z_{\rm sp})(z_{\rm sp}-1)^2.
\end{equation*}
Now using \eqref{a9} and \eqref{a27}, we get the result of Theorem \ref{saddlepointThm}. A separate analysis of $\beta_n$ is provided in Subsection \ref{convRobust}.
\section{Numerical \& empirical studies}
A similar analysis, modeled after the one given in Chapter 2 gives under Assumption \ref{as1} the leading-order expression
\begin{equation}
\label{a65}
\frac{1}{z_{\rm sp} \pi}\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta_n^2+t^2}\,{\rm ln}(1-e^{-\tfrac12\beta_n^2-t^2}){\rm d} t
\end{equation}
for ${\rm ln}\,\mathbb{P}(Q^{(n)}=0)$. Observe that the quantity in \eqref{a65} is negative, but bounded away from ${-}\infty$ when $\beta_n$ is bounded away from 0.
Furthermore, we find for the variance of $Q^{(n)}$ the approximation
\begin{equation*}
\label{a66}
\frac{\beta_n^3/\sqrt{2}}{\pi}\frac{z_{\rm sp}+1}{(z_{\rm sp}-1)^2}\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t.
\end{equation*}
\end{proof}
\noindent
Note that we can write \eqref{r1} as
\begin{equation*}
\label{ra1}
\mu_Q \approx \tilde{\sigma}_n\,\mathbb{E}[ M_{\beta_n}]\quad \text{and}\quad \sigma^2_Q \approx \tilde{\sigma}^2_n\, {\rm Var}\, M_{\beta_n}
\end{equation*}
with
\begin{equation}
\label{ra5}
\tilde{\sigma}_n = \beta_n \Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr).
\end{equation}
This robust approximation for $\mu_Q$ is suggestive of the following two properties that extend beyond the mean system behavior, and hold at the level of approximating the queue by $\sigma_n$ times the Gaussian random walk:
\begin{itemize}
\item[\rm (i)] At the process level, the space should be normalized with $\sigma_n$, as in \eqref{mm7}. The approximation \eqref{r1} suggests that it is better to normalize with $\tilde{\sigma}_n$. Although $\tilde \sigma_n\to\sigma_n$ for $n\to\infty$, the $\tilde \sigma_n$ is expected to lead to sharper approximations for finite $n$.
\item[\rm (ii)] Again at the process level, it seems better to replace the original hedge $\beta$ by the robust hedge $\beta_n$. This thus means that the original system for finite $n$ is approximated by a Gaussian random walk with drift $-\beta_n$. Apart from this approximation being asymptotically correct for $n\to \infty$, it is also expected to approximate the behavior better for finite $n$.
\end{itemize}
\subsection{Convergence of the robust hedge\label{convRobust}}
We next examine the accuracy of the heavy-traffic approximations for $\mu_Q$ and $\sigma^2_Q$, following Corollary \ref{abateThm} and Theorem \ref{saddlepointThm}. We expect the robust approximation to be considerably better than the classical approximation when $\beta_n$ and $\tilde{\sigma}_n$ differ substantially from their limiting counterparts. Before substantiating this claim numerically, we present a result on the convergence rates of $\beta_n$ to $\beta$ and $\tilde{\sigma}_n$ to $\sigma_n$.
\begin{proposition}\label{gammanProp}
Let $a_n,b_n$ and $s_n$ as in Assumption \ref{as2}. Then
\begin{equation}
\label{r3a}
\beta_n^2 = \beta^2\Bigl(1 - \frac{1}{1+b_n+\sigma_n/\beta}\Bigr).
\end{equation}
\end{proposition}
\begin{proof}
From \eqref{r2}, we have
\begin{align*}
\beta_n^2 &= s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr)= \frac{1}{s_n}\Bigl(\frac{s_n-a_nb_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{s_n}{a_n}\Bigr)\nonumber\\
\label{x1}
&= \frac{1}{s_n}\frac{\beta^2\,a_nb_n(b_n+1)}{(b_n+1)^2}\Bigl(1+\frac{s_n}{a_n}\Bigr) = \beta^2\,\frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) =:\beta^2\,\bar{F}_n.
\end{align*}
Now,
\begin{align*}
\bar{F_n} &= \frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) = \frac{b_n}{b_n+1}+\frac{1}{b_n+1}\,\frac{a_nb_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\Bigl(1-\frac{a_nb_n}{s_n}\Bigr) = 1-\frac{1}{b_n+1}\,\frac{\beta\,\sigma_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\frac{1}{1+\frac{\mu_n}{\beta\sigma_n}} = 1-\frac{1}{b_n+1+\frac{1}{\beta}\sqrt{a_nb_n(b_n+1)}},
\end{align*}
which together with $\sigma_n^2=a_nb_n(b_n+1)$ proves the proposition.
\end{proof}
Note that $\beta_n$ always approaches $\beta$ from below. Also, \eqref{r3a} shows that $b_n$ is the dominant factor in determining the rate of convergence of $\beta_n$.
\begin{proposition}\label{sigmanProp}
Let $\tilde{\sigma}_n$ as in \eqref{ra5}. Then
\begin{equation*}
\tilde{\sigma}_n = \sigma_n + b_n\beta_n + O(1).
\end{equation*}
\end{proposition}
\begin{proof}
Straightforward calculations give
\begin{align*}
\tilde{\sigma}_n &= \beta_n\,\Bigl(\frac{s_nb_n+a_nb_n}{s_n-a_nb_n}\Bigr) \nonumber\\
&= \frac{\beta_n}{\beta}\,\frac{b_n}{\sigma_n}\,(s_n+a_n)
= \frac{\beta_n}{\beta}\,\sqrt{\frac{b_n}{a_n(b_n+1)}}\left(a_n(b_n+1)+\beta\sqrt{a_nb_n(b_n+1)}\right)\nonumber\\
&= \frac{\beta_n}{\beta}\left(\sqrt{a_nb_n(b_n+1)}+\beta b_n\right) = \frac{\beta_n}{\beta}\,\sigma_n + \beta_n b_n.
\end{align*}
Applying Proposition \ref{gammanProp} together with the observation
\begin{equation*}
\sigma_n \sqrt{1 - \frac{1}{1+b_n+\sigma_n/\beta}} = \sigma_n(1 + O(1/\sqrt{a_n}b_n)) = \sigma_n + O(1)
\end{equation*}
yields the result.
\end{proof}
In Figure \ref{fig:convHedge}, we visualize the convergence speed of both parameters in case $\mu_n=n$, $\sigma_n = n^\delta$ with $\delta=0.7$ and $\beta=1$. This implies $a_n = n/(n^{2\delta}-1)$ and $b_n = n^{2\delta}-1$.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.05,
xlabel = {$x$},
ylabel = {$\tilde{\beta}_n/\beta_n$},
y label style={at={(-0.09,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.052)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {./tikz/gamman.txt};
\addplot[thick,col4] table[x=n,y=d075] {./tikz/gamman.txt};
\addplot[thick,col5] table[x=n,y=d09] {./tikz/gamman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\beta_n$.}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.7,
xlabel = {$x$},
ylabel = {$\tilde{\sigma}_n/\sigma_n$},
y label style={at={(-0.09,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.1)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {./tikz/sigman.txt};
\addplot[thick,col4] table[x=n,y=d075] {./tikz/sigman.txt};
\addplot[thick,col5] table[x=n,y=d09] {./tikz/sigman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\tilde{\sigma}_n$.}
\end{subfigure}
\caption{}
\label{fig:convHedge}
\end{figure}
We observe that $\beta_n$ starts resembling $\beta$ fairly quickly, as predicted by Proposition \ref{gammanProp}; $\tilde{\sigma}_n$, on the other hand, converges extremely slowly to its limiting counterpart. Since $\mu_Q$ and $\sigma^2_Q$ are approximated by $\tilde{\beta}_n$ and $\tilde{\sigma}_n^2$, multiplied by a term that remains almost constant as $n$ grows, the substitution of $\sigma_n$ by $\tilde{\sigma}_n$, is essential for obtaining accurate approximations, as we illustrate further in the next subsection.
\subsection{Comparison between heavy-traffic approximations}
We set $\mu_n=n$ and $\sigma^2_n=n^{2\delta}$ with $\delta>\tfrac{1}{2}$, so that $s_n = n+\beta n^{\delta}$, and $a_n =n/(n^{2\delta-1}-1)$ and $b_n = n^{2\delta-1}-1$.
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.609 & 0.343 & 0.246 & 0.363 & 1.002 & 0.835 & 0.978 \bigstrut[t] \\
10 & 0.683 & 0.535 & 0.400 & 0.551 & 1.239 & 1.063 & 1.216 \\
50 & 0.815 & 1.405 & 1.168 & 1.405 & 1.995 & 1.817 & 1.971 \\
100 & 0.855 & 2.113 & 1.824 & 2.105 & 2.445 & 2.270 & 2.420 \\
500 & 0.920 & 5.446 & 5.006 & 5.412 & 3.923 & 3.762 & 3.899 \bigstrut[b] \\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.6$.}
\label{gammaPoisson1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.550 & 0.462 & 0.284 & 0.479 & 1.162 & 0.896 & 1.130 \bigstrut[t]\\
10 & 0.587 & 0.852 & 0.521 & 0.855 & 1.570 & 1.213 & 1.528 \\
50 & 0.668 & 3.197 & 2.093 & 3.106 & 3.025 & 2.433 & 2.947 \\
100 & 0.700 & 5.561 & 3.784& 5.377 & 3.983 & 3.270 & 3.887\\
500 & 0.766 & 19.887 & 14.741 & 19.202 & 7.514 & 6.455 & 7.361 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.8$.}
\label{gammaPoisson2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.949 & 11.532 & 11.306 & 11.495 & 3.634 & 3.559 & 3.602 \bigstrut[t] \\
10 & 0.961 & 17.565 & 17.268 & 17.548 & 4.474& 4.398 & 4.444 \\
50 & 0.979 & 46.368 & 45.869 & 46.418 & 7.241 & 7.168 & 7.218 \\
100 & 0.984 & 70.340 & 69.735 & 70.430 & 8.910 & 8.839 & 8.888 \\
500 & 0.991 & 184.900 & 183.989 & 185.108 & 14.422 & 14.357 & 14.404 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.6$.}
\label{gammaPoisson3}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.931 & 15.730 & 15.209 & 15.909 & 4.276 & 4.127 & 4.233 \bigstrut[t]\\
10 & 0.939 & 27.561 & 26.672 & 27.958 & 5.652 & 5.466 & 5.605 \\
50 & 0.955 & 100.660 & 97.967 & 102.070 & 10.760 & 10.476 & 10.698 \\
100 & 0.961 & 175.591 & 171.360 & 177.818 & 14.189 & 13.855 & 14.117 \\
500 & 0.971 & 638.097 & 626.346 & 644.105 & 26.963 & 26.490 & 26.864 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.8$.}
\label{gammaPoisson4}
\end{table}
Tables \ref{gammaPoisson1}-\ref{gammaPoisson4} present numerical results for various parameter values.
In these tables, we fixed $s_n$ to integer values, and use the associated value of $n$ in our calculations.
The exact values of the performance measures are calculated using the method in Appendix \ref{numprocs}.
Several conclusions are drawn from these tables. Observe that the heavy-traffic approximations based on the Gaussian random walk, \eqref{h1} and \eqref{h1b}, capture the right order of magnitude for both $\mu_Q$ and $\sigma_Q$. However, the values are off, in particular for small $s_n$ and relatively low $\rho_n := \mathbb{E}[A^{(n)}] / s_n$. The inaccuracy also increases with the level of overdispersion. In contrast, the approximations that follow from Theorem \ref{saddlepointThm}, \eqref{r1} and \eqref{r4} are remarkably accurate. Even for small systems with $s_n = 5$ or 10, the approximations for $\mu_Q$ are within 6$\%$ of the exact value for small $\rho_n$ and within $2\%$ for $\rho_n$ close to 1. For $\sigma_Q^2$, these percentages even reduce to $3\%$ and $1\%$, respectively. For larger values of $s_n$ these relative errors naturally reduce further. Overall, we observe that the approximations improve for heavily loaded systems, and the corrected approximations are particularly useful for systems with increased overdispersion.
\subsection{Capacity allocation in health care}
We next apply our model and robust approximations to real-life patient arrivals. We consider emergency patients who require diagnostic tests at the radiology department of a hospital. Green \cite{Green2004} points out that patients at the radiology department can be roughly categorized into three groups: Inpatients, outpatients and emergency patients. Inpatient and outpatient arrivals are relatively predictable as these are usually scheduled by appointment. Emergency patients, on the other hand, are inherently unpredictable: They typically require urgent care and therefore timely access to testing facilities, as well as a quick assessment of the test results. This leads to prioritization of emergency patients over the other two groups in such settings, so that they do not experience any delay caused by the groups of lower priority. However, patients from the same top-priority group can still cause considerable congestion. A careful evaluation of capacity allocation is hence required, bearing in mind that additional sophisticated pieces of medical equipment are very costly.
In the setting we study, capacity is defined by the number of time slots available to perform radiology tests on emergency patients in a given time period, which we set at 24 hours. As radiology tests are commonly performed in appointment slots of fixed length, the number of slots available per day is also indirectly fixed. In terms of our model parameters, see Section \ref{modelSection}, we have $s$ as the number of slots per day allocated to emergency patients, and $A(k)$ the number of test requests received by the department on day $k$. We omit the subscript $n$ in this section due to the absence of limits. Then $\mathbb{E}[Q]$ can be interpreted as the expected number of patients whose test is carried over to the next day. A more natural performance measure in this setting is the expected waiting time, namely the time between the physician's request and the actual start of the test. However, Little's law implies that there is a linear relation between the two, hence we choose to only evaluate $\mathbb{E}[Q]$.
The data set on which our empirical study is based originates from the emergency department of SKHospital, monitored over a period of 76 days from September to November 2012. We extracted information of ED patients referred to the radiology department by the ED physicians, which includes the time the test request was made and the exact test type performed. The two test types, X-ray and CT scans, are performed on different equipment and hence it makes sense to analyze their queueing processes separately.
We refer to test requests as arrivals. The empirical cumulative distribution functions of the number of arrivals per day, for each type, are depicted by the black lines in Figure \ref{fig:fittedHospital}. The sample means equal 69.81 and 17.47, for the X-ray and CT scans respectively, whereas the sample variances are 121.8 and 26.12. These values suggest that fitting a Poisson distribution is inappropriate, which is visually backed up by the fitted Poisson cdf, depicted in Figure \ref{fig:fittedHospital} by the red line. To strengthen this claim, we tested both samples for the Poisson assumption using the \emph{dispersion test}, see Appendix \ref{statproc}, and obtained $p$-values equal $7.01\cdot 10^{-3}$ and $3.57\cdot 10^{-3}$ respectively, which allow us to safely reject the Poisson hypothesis in both cases.
In search for a better distributional fit with the arrivals count, we resort to Gamma-Poisson mixtures. Here we employ the procedure in \cite{koolejongbloed}, which is basically a maximum log-likelihood method, to obtain estimates for the parameters $a$ and $b$. This yields
\begin{equation*}
\label{parameterEstimators}
\hat{a}_{\rm X-ray} = 95.68,\quad \hat{b}_{\rm X-ray} = 0.7297,\quad \hat{a}_{\rm CT} = 37.19,\quad \hat{b}_{\rm CT} = 0.4698.
\end{equation*}
Applying the bootstrapping method to the data and the fitted model, also described in the appendix of \cite{koolejongbloed}, returns p-values that equal 0.7354 and 0.2120 for X-ray and CT scans, respectively. Therefore, the null hypothesis, stating that the data originated from a Gamma-Poisson mixture, cannot be rejected. The cdfs of the fitted Gamma-Poisson distributions, plotted in Figure \ref{fig:fittedHospital}, give visual confirmation of this claim as well.
Naturally, we also compared the estimated densities to the empirical pdf of the data. However, these fail to give a convincing visual fit due to the relatively small sample size and are therefore omitted here.
\begin{figure}
\begin{center}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 40,
xmax = 110,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 109,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\addplot[thick,col1] table[x=x,y=poisson] {./tikz/xray.txt};
\addplot[thick,col4] table[x=x,y=fitted] {./tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{X-ray}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 5,
xmax = 32,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 32,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {./tikz/ct.txt};
\addplot[thick,col1] table[x=x,y=poisson] {./tikz/ct.txt};
\addplot[thick,col4] table[x=x,y=fitted] {./tikz/ct.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {./tikz/ct.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{CT scan}
\end{subfigure}
\end{center}
\caption{Empirical, fitted Poisson and fitted Gamma-Poisson cumulative distribution functions of the number of arrivals.}
\label{fig:fittedHospital}
\end{figure}
We now have clear evidence that both the X-ray and CT scan facilities face an overdispersed arrival stream. In our final step of the empirical study we now evaluate the accuracy of our performance measure of interest $\mathbb{E}[Q]$, and the consequences of assessing system performance while ignoring the presence of overdispersion. We take the following approach: Trivially, we need to choose $s> \mathbb{E}[A]$ in order for the system to be stable. Hence, we vary $s$ from 70 to 80 for X-rays and from 18 to 24 for CT scans and simulate the queue length process by sampling the number of requests per day from the actual arrival counts. The number of iterations performed is $10^8$ for each configuration, excluding a warm-up interval of length $10^7$ (days). Around the mean of $Q$ obtained from this simulation, we create a 95\% confidence interval. Next, we approximate the expected stationary queue length under two scaling rules. Assuming Poisson arrivals, the appropriate capacity allocation rule would be $s=\hat{\mu}+\beta\sqrt{\hat{\mu}}$, for some $\beta>0$. Our novel capacity sizing rule prescribes $s = \hat{\mu} + \beta\hat{\sigma} = \hat{a}\hat{b}+\beta\sqrt{\hat{a}\hat{b}(\hat{b}+1)}$. We compute the first approximation based on square-root safety capacity sizing by deducing $\beta$ for each $s$, which we substitute in $\mathbb{E}[Q^{\rm srs}] \approx \sqrt{\hat{\mu}}\,\mathbb{E}[M_{\beta}]$. Similarly, we obtain $\beta$ from the new rule, and plug this value, together with the fitted parameters $\hat{a}$ and $\hat{b}$, into \eqref{r1}. The results are given in Tables \ref{tab:simXRay} and \ref{tab:simCT}. The last column shows the 95\% relative error bound of the second approximation.
\begin{table}[h]
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q] \ (\pm $ conf. iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
70 & 0.997 & $328.313 \pm\ 6.6\cdot 10^{-2}$ & 186.664 & 324.231 & 325.221 & $9.6\cdot 10^{-3}$ \bigstrut[t]\\
71 & 0.983 & $45.073 \pm\ 1.0\cdot 10^{-2}$ & 24.946 & 45.290 & 45.308 & $5.4\cdot 10^{-3}$ \\
72 & 0.970 & $21.988 \pm\ 5.4\cdot 10^{-3}$ & 11.650 & 21.982 & 22.129 & $6.6\cdot 10^{-3}$ \\
73 & 0.956 & $13.546 \pm\ 3.6\cdot 10^{-3}$ & 6.847 & 13.455 & 13.649 & $7.8\cdot 10^{-3}$ \\
74 & 0.943 & $9.230 \pm\ 2.7\cdot 10^{-3}$ & 4.438 & 9.106 & 9.319 & $1.0\cdot 10^{-2}$ \\
75 & 0.931 & $6.655 \pm\ 2.1\cdot 10^{-3}$ & 3.031 & 6.513 & 6.731 & $1.2\cdot 10^{-2}$ \\
76 & 0.919 & $4.949 \pm\ 1.7\cdot 10^{-3}$ & 2.136 & 4.821 & 5.037 & $1.8\cdot 10^{-2}$ \\
77 & 0.907 & $3.755 \pm\ 1.4\cdot 10^{-3}$ & 1.534 & 3.650 & 3.861 & $2.8\cdot 10^{-2}$ \\
78 & 0.895 & $2.884 \pm\ 1.1\cdot 10^{-3}$ & 1.115 & 2.807 & 3.009 & $4.4\cdot 10^{-2}$ \\
79 & 0.884 & $2.230 \pm\ 1.0\cdot 10^{-3}$ & 0.816 & 2.183 & 2.374 & $6.5\cdot 10^{-2}$ \\
80 & 0.873 & $1.734 \pm\ 8.5\cdot 10^{-4}$ & 0.600 & 1.710 & 1.890 & $9.1\cdot 10^{-2}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for X-ray.}
\label{tab:simXRay}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q]\ (\pm $ conf.iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
18 & 0.970 & 22.116 $\pm\ 4.9\cdot 10^{-3}$ & 14.235 & 21.965 & 21.724 & $1.8\cdot 10^{-2}$ \bigstrut[t] \\
19 & 0.919 & 6.289 $\pm\ 1.7\cdot 10^{-3}$ & 3.640 & 5.941 & 6.040 & 4.0$\cdot 10^{-2}$ \\
20 & 0.873 & 3.101 $\pm\ 1.0\cdot 10^{-3}$ & 1.589 & 2.772 & 2.917 & 6.0$\cdot 10^{-2}$ \\
21 & 0.832 & 1.767 $\pm\ 6.6\cdot 10^{-4}$ & 0.800 & 1.507 & 1.658 & 6.1$\cdot 10^{-2}$ \\
22 & 0.794 & 1.066 $\pm\ 4.6\cdot 10^{-4}$ & 0.425 & 0.874 & 1.016 & 4.7$\cdot 10^{-2}$ \\
23 & 0.760 & 0.653 $\pm\ 3.3\cdot 10^{-4}$ & 0.230 & 0.522 & 0.649 & 7.1$\cdot 10^{-3}$\\
24 & 0.728 & 0.377 $\pm\ 2.3\cdot 10^{-4}$ & 0.124 & 0.315 & 0.424 & 1.2$\cdot 10^{-1}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for CT scan.}
\label{tab:simCT}
\end{table}
Based on these figures, we make several remarks. First, assuming the conventional regime (neglecting overdispersion) the approximation severely overestimates system performance in both queues. Because the square-root safety margin underestimates the stochastic fluctuations in the arrival process, the safety parameter $\beta$ is overestimated, which leads to a smaller magnitude of the approximated queue length process. This clearly illustrates the distorted view estimated performance characteristics can give under the wrong scaling.
Secondly, it is worth noticing the very good proximity of $\eqref{r1}$ to the values obtained through simulation. As we expected, the quality of the approximation deteriorates with increasing values of $s$. This makes sense, because we assumed the system to be in heavy traffic in the derivation of the formulas. What is surprising, on the other hand, is the fact that the approximation performs very well, even though the parameter $b$ is very small for these particular data sets, while the analysis of Theorem \ref{saddlepointThm} assumes that $a$ and $b$ are large. This demonstrates that the approximation scheme is remarkably robust and is able to capture the pre-limit behavior of these types of queues very well.
\section{Conclusion \& future research}
In this chapter, we proposed an adaptation to the square-root staffing rule for service systems facing overdispersed demand, using the bulk service queue as a vehicle for our analysis.
Subsequently, we derive two sets of asymptotic approximations for the scaled steady-state queue length moments for large arrival volumes.
The first set being based on the resemblance with the maximum of a Gaussian random walk, the second set being derived through a non-standard saddle point method, assuming arrivals follow a Gamma-Poisson mixture.
Numerical experiments indicated that our approximations capture the pre-limit behavior well under different order of overdispersion, and are robust against any parameter estimation errors.
Although our method provides a robust way to approximate and dimension queues with overdispersed arrival processes, we see some interesting directions for future research.
First, we accentuate that our model is a discrete time queueing model in which a deterministic amount of workload can we handled within each period.
This approach allowed us to use Pollaczek's formula as a starting point to obtain more refined asymptotic approximations for the performance indicators of the system.
In case we consider queueing models of birth-death-type, such as the $M/M/s$ queue, in the presence of overdispersion demand, different techniques are required to provide scaling limits and corresponding capacity allocation rules, see e.g.~\cite{maman}.
Although we expect that, just as in the novel scaling regimes of Chapter 2, the asymptotic behavior of the bulk service queue and the multi-server queueing models to be similar, this needs proper analysis and understanding.
Second, empirical work, see e.g. \cite{Avramidis:2004}, shows that in real-life settings, demand in consecutive time periods is often positively correlated, rather than independently distributed as assumed in this chapter.
This correlation structure obviously alters the queue's dynamics and presumably requires an adaptation of the square-root staffing rule as well, making it a worthwhile direction for further analysis.
Last, we have only considered the analysis of the queueing model in steady state.
Typical service systems however do not face a constant expected arrival rate, nor do they run infinitely long.
Henceforth, it would be interesting to study the influence of overdispersion on the transient dynamics of the queue and to investigate the capacity allocation problem in scenarios with time-varying demand.
The theory developed in this chapter may serve as a building block to tackle these more profound questions.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Proofs of convergence results}
\label{formalSec}
This section presents the details of the proof of Lemma \ref{gaussStep} and Theorem \ref{gaussianThm}, using the random walk perspective of the process $\{Q^{(n)}(k)\}_{k=0}^\infty$. This section is structured as follows. The next two lemmata are necessary for proving the first assertion of Theorem \ref{gaussianThm}, concerning the weak convergence of the scaled process to the maximum of the Gaussian random walk, which is summarized in Proposition \ref{prop6}. The two remaining propositions of this section show convergence of $\hat{Q}^{(n)}$ at the process level as well as in terms of the three characteristics.
Let us first fix some notation:
\begin{equation}
\label{b1}
Y^{(n)}_k := \hat{A}^{(n)}_k-\beta,\quad
S^{(n)}_k = \sum_{i=1}^k Y^{(n)}_i,
\end{equation}
with $S_0^{(n)} = 0$ and $k=1,2,...$. Then \eqref{mm6} can be rewritten as
\begin{equation}
\label{g5a}
\hat{Q}^{(n)} {\;\buildrel{d}\over= \;} \max_{k\geq 0} \Bigl\{{\textstyle \sum}_{i=1}^k Y^{(n)}_i\Bigr\} =: M_\beta^{(n)},
\end{equation}
Last, we introduce the sequence of independent normal random variables $Z_1,Z_2,\ldots$ with mean $\-\beta$ and unit variance 1, and
\begin{equation*}
M_\beta {\;\buildrel{d}\over= \;} \max_{k\geq 0} \{{\textstyle \sum}_{i=1}^k Z_i\}.
\end{equation*}
\subsection{Proof of Lemma \ref{gaussStep}}
\begin{proof}
We show weak convergence of the random variable $\hat{A}^{(n)}$, as defined in Section \ref{modelSection}, to a standard normal random variable. Since $\hat{\Lambda}_n$ is asymptotically standard normal, its characteristic function converges pointwise to the corresponding limiting characteristic function, i.e.
\begin{equation}
\label{g8}
\lim_{n\rightarrow\infty} \phi_{\hat{\Lambda}_n}(t) = \lim_{n\rightarrow \infty} {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n) = {\rm e}^{{-}t^2/2},\qquad \forall t\in \mathbb{R}.
\end{equation}
Furthermore, by definition of $A^{(n)}$,
\begin{equation*}
\label{g9}
\phi_{A^{(n)}}(t) = \mathbb{E}\left[ \exp(\Lambda_n({\rm e}^{it}-1))\right] = \phi_{\Lambda_n}\left(-i({\rm e}^{it}-1)\right),
\end{equation*}
so that
\begin{equation}
\label{g10}
\phi_{\hat{A}_k^{(n)}}(t) = {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{A_k^{(n)}}(t/\sigma_n) = {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}\left(-i({\rm e}^{it/\sigma_n}-1)\right).
\end{equation}
Now fix $t\in\mathbb{R}$. By using
\begin{equation*}
\label{g11}
-i({\rm e}^{it/\sigma_n} - 1) = \frac{t}{\sigma_n} -\frac{it^2}{2\sigma_n^2} + O\left(t^3/\sigma_n^3\right),
\end{equation*}
we expand the last term in \eqref{g10},
\begin{equation*}
\label{g12}
\phi_{\Lambda_n}(t/\sigma_n) + \Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(t/\sigma_n) + O\Bigl(\Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(\frac{t^3}{\sigma_n^3}\right)\Bigr)^2\phi_{\Lambda_n}''\Big(\frac{t}{\sigma_n}\Big)\Bigr)
\end{equation*}
\begin{equation*}
\label{g13}
= \phi_{\Lambda_n}(t/\sigma_n) - \Bigl(\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(\zeta)
\end{equation*}
for some $\zeta$ such that $|\zeta - t/\sigma_n| < |i(1-{\rm e}^{it/\sigma_n})-t/\sigma_n|$. Also,
\begin{align}
|\phi_{\Lambda_n}'(u)| &= \left|\frac{\delta}{{\rm d} u}\int_{-\infty}^\infty {\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| = \left|\int_{0}^{\infty} ix\,{\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| \nonumber\\
\label{g13a}
&\leq \int_{-\infty}^\infty |ix\,{\rm e}^{iux}|\,{\rm d} F_{\Lambda_n}(x) = \int_0^\infty x{\rm d} F_{\Lambda_n}(x) = \mu_n
\end{align}
for all $u\in\mathbb{R}$. Hence, by substituting \eqref{g10},
\begin{align}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| &= \left|{\rm e}^{-i\mu_nt/\sigma_n}\,\left(\frac{i\,t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right)\,\phi_{\Lambda_n}'(\zeta)\right|\nonumber\\
& \leq \left(\frac{t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right) |\phi_{\Lambda_n}'(\zeta)|\nonumber\\
& = \frac{\mu_n t^2}{\sigma_n^2} + O\left(\frac{\mu_nt^3}{\sigma_n^3}\right),
\label{g13b}
\end{align}
which tends to zero as $n\rightarrow \infty$ by our assumption that $\mu_n/\sigma_n^2\rightarrow 0$.
Finally,
\begin{equation*}
\label{g13c}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-\tfrac12 t^2}\right| \leq \left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| +
\left| {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n) - {\rm e}^{-\tfrac12 t^2}\right|,
\end{equation*}
in which both terms go to zero for $n\rightarrow \infty$, by \eqref{g8} and \eqref{g13b}. Hence $\phi_{\hat{A}^{(n)}_k}(t)$ converges to ${\rm e}^{{-}t^2/2}$ for all $t\in\mathbb{R}$, so that we can conclude by L\'evy's continuity theorem that $\hat{A}_k^{(n)} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1)$.
\end{proof}
\subsection{Proof of Theorem \ref{gaussianThm}}
To secure convergence in distribution of $\hat{Q}^{(n)}$ to $M_\beta$, i.e. the maximum of a Gaussian random walk with negative drift, the first assertion of Theorem \ref{gaussianThm},
the following property of the sequence $\{Y_k^{(n)}\}_{n\in\mathbb{N}}$ needs to hold.
\begin{lemma}\label{uilemma}
Let $Y^{(n)}_k$ be defined as in \eqref{b1} with $\mu_n,\sigma_n^2 < \infty$ for all $n\in\mathbb{N}$. Then the sequence $\{(Y_k^{(n)})^+\}_{n\in\mathbb{N}}$ is uniform integrable, i.e.
\begin{equation*}
\label{g14}
\lim_{K\rightarrow\infty}\sup_n \mathbb{E}\Big[Y^{(n)\,+}_k|\mathbbm{1}_{\{|Y^{(n)\,+}_k|\geq K\}}\Big] = 0.
\end{equation*}
\end{lemma}
\begin{proof}
Because the sequence $\{Y^{(n)}_k\}_{k\in\mathbb{N}}$ is i.i.d. for all $n$, we omit the index $k$ in this proof. First, fix $K>0$ and note that
\begin{equation*}
\label{g15}
\mathbb{E}[|Y^{(n)+}|\mathbbm{1}{\{|Y^{(n)\,+}|\geq K\}}] = \mathbb{E}[Y^{(n)+}\mathbbm{1}{\{Y^{(n)+}\geq K\}}] = \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}].
\end{equation*}
This last expression can be bounded from above using the Cauchy-Schwarz inequality, so that
\begin{equation*}
\label{g16}
\mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}] \leq \mathbb{E}[ Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K)^{1/2}.
\end{equation*}
By the definition of $Y^{(n)}$, we know $\mathbb{E} [Y^{(n)}] = -\beta$ and ${\rm Var}\, Y^{(n)} = {\rm Var}\, A^{(n)} / \sigma_n^2 = 1$. Using this information, we find
\begin{equation*}
\label{g17}
\mathbb{E}[Y^{(n)\,2}] = {\rm Var}\, Y^{(n)} + (\mathbb{E}[Y^{(n)}])^2 = 1+\beta^2
\end{equation*}
and
\begin{align*}
\mathbb{P}(Y^{(n)}\geq K )&=\mathbb{P}(Y^{(n)}+\beta\geq K+\beta) \leq \mathbb{P}(|Y^{(n)}+\beta|\geq K+\beta)\nonumber\\
&\leq \frac{{\rm Var}\, Y^{(n)}}{(K+\beta)^2} = \frac{1}{(K+\beta)^2},
\end{align*}
where we used Chebyshev's inequality for the last upper bound. Therefore,
\begin{align*}
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[|Y^{(n)\,+}|\mathbbm{1}_{\{|Y^{(n)\,+}|\geq K\}}] &=
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}]\nonumber\\
&\leq \lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K )^{1/2}\nonumber\\
&\leq \lim_{K\rightarrow \infty} \frac{\sqrt{1+\beta^2}}{K+\beta} = 0.
\end{align*}
\end{proof}
By combining the properties proved in Lemma \ref{gaussStep} and \ref{uilemma} with Assumption \ref{as2}, the next result follows directly by \cite[Thm.~X6.1]{Asmussen2003}.
\begin{proposition}\label{maxRWprop}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}. Then
\begin{equation*}
\hat{Q}^{(n)}{\;\buildrel{d}\over\Rightarrow\;} M_\beta,\qquad {\rm as}\ n\rightarrow\infty.
\end{equation*}
\end{proposition}
Although Proposition \ref{maxRWprop} tells us that the properly scaled $Q^{(n)}$ converges to a non-degenerate limiting random variable, it does not cover the convergence of its mean, variance and the empty-queue probability. In order to secure convergence of these performance measures as well, we follow the approach similar to \cite{Sigman2011b}, using Assumptions \ref{as2} and \ref{as3}.
\begin{proposition}\label{prop6}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}, $\mu_n,\sigma_n^2 \rightarrow \infty$ such that both $\sigma_n^2/\mu_n\rightarrow \infty$ and $\mathbb{E}[\hat{A}^{(n)3}]$ $<\infty$. Then
\begin{align*}
\label{b16}
\mathbb{P}(\hat{Q}^{(n)}= 0)&\rightarrow \mathbb{P}(M_\beta = 0),\\
\mathbb{E} [\hat{Q}^{(n)}]&\rightarrow \mathbb{E} [M_\beta],\\
{\rm Var}\, \hat{Q}^{(n)}&\rightarrow {\rm Var}\, M_\beta,
\end{align*}
as $n\rightarrow\infty$.
\end{proposition}
\proof
First, we recall that $\hat{Q}^{(n)}{\;\buildrel{d}\over= \;} M_\beta^{(n)}$ for all $n\in\mathbb{N}$, so that $\mathbb{P}(\hat{Q}^{(n)} = 0) = \mathbb{P}(M_\beta^{(n)}=0)$, $\mathbb{E}[\hat{Q}^{(n)}]=\mathbb{E}[M_\beta^{(n)}]$ and ${\rm Var}\,\,\hat{Q}^{(n)}={\rm Var}\,\,M_\beta^{(n)}$ as defined in \eqref{b1}. Our starting point is Spitzer's identity, see \cite[p.~230]{Asmussen2003},
\begin{equation}
\label{b17}
\mathbb{E}[{\rm e}^{it M_\beta^{(n)}}] = \exp\Bigl( \sum_{k=1}^\infty \frac{1}{k} (\mathbb{E}[{\rm e}^{it(S^{(n)}_k)^+}]-1)\Bigr),
\end{equation}
with $S^{(n)}_k$ as in \eqref{b1}, and $M_\beta^{(n)}$ the all-time maximum of the associated random walk. Simple manipulations of \eqref{b17} give
\begin{align}
\label{y1}
{\rm ln}\,\mathbb{P}(M_\beta^{(n)} = 0) &= -\sum_{k=1}^\infty \frac{1}{k}\,\mathbb{P}(S^{(n)}_k > 0),\\
\label{y2}
\mathbb{E}[M_\beta^{(n)}] &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[S^{(n)\,+}_k] = \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x) {\rm d} x,\\
\label{y3}
{\rm Var}\, M_\beta^{(n)} &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[(S^{(n)\,+}_k)^2] =\sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}) {\rm d} x.
\end{align}
By Lemma \ref{gaussStep}, we know
\begin{equation*}
\label{y4}
\mathbb{P}(S^{(n)}_k > y) = \mathbb{P}\left( {\sum_{i=1}^k} Y^{(n)}_i > y \right) \rightarrow \mathbb{P}\left({\textstyle\sum_{i=1}^k} Z_i > y\right),
\end{equation*}
for $n\rightarrow \infty$, where the $Z_i$'s are independent and identically normally distributed with mean $-\beta$ and variance 1.
Because equivalent expressions to \eqref{y1}-\eqref{y3} apply to the limiting Gaussian random walk, it is sufficient to show that the sums converge uniformly in $n$, so that we can apply dominated convergence to prove the result.
We start with the empty-queue probability. To justify interchangeability of the infinite sum and limit, note
\begin{equation*}
\label{y5}
\mathbb{P}(S^{(n)}_k > 0) \leq \mathbb{P}(|S^{(n)}_k+k\beta| > k\beta )\leq \frac{k}{\beta^2k^2} = \frac{1}{\beta^2k},
\end{equation*}
where we used that $\mathbb{E}[ S^{(n)}_k] = k\mathbb{E} [Y^{(n)}_1] = -k\beta$ and ${\rm Var}\, S^{(n)}_k = k$ and apply Chebychev's inequality, so that
\begin{equation*}
\label{y6}
\sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) \leq \sum_{k=1}^\infty \frac{1}{\beta^2 k^2} < \infty, \qquad \forall n\in\mathbb{N}.
\end{equation*}
Hence,
\begin{align*}
\lim_{n\rightarrow\infty} {\rm ln}\,\mathbb{P}(\hat{Q}^{(n)}= 0) &= \lim_{n\rightarrow\infty} - \sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) = -\sum_{k=1}^\infty \frac{1}{k} \lim_{n\rightarrow\infty}\mathbb{P}(S^{(n)}_k > 0)\nonumber\\
&= -\sum_{k=1}^\infty \frac{1}{k} \mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > 0) = {\rm ln}\,\mathbb{P}(M_\beta = 0).
\end{align*}
Finding a suitable upper bound on $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>x) {\rm d} x$ and $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>\sqrt{x}) {\rm d} x$ requires a bit more work. We initially focus on the former, the latter follows easily. The following inequality from \cite{Nagaev1979} proves to be very useful:
\begin{equation}
\label{y8}
\mathbb{P}(\bar{S}(k)>y) \leq C_r\,\Bigl(\frac{k\,\sigma^2}{y^2}\Bigr)^2 + k\,\mathbb{P}(X>y/r),
\end{equation}
where $\bar{S}(k)$ is the sum of $k$ i.i.d. random variables distributed as $X$, with $\mathbb{E}[X] = 0$ and ${\rm Var}\,\, X=\sigma^2$, $y > 0$, $r>0$ and $C_r$ a constant only depending on $r$. We take $r=3$ for brevity in the remainder of the proof, although any $r>2$ will suffice. We analyze the integral in two parts, one for the interval $(0,k)$ and one for $[k,\infty)$. For the first part, we have
\begin{align}
\label{y9}
\int_0^k\mathbb{P}(S^{(n)}_k>x) {\rm d} x &=\int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > x+k\beta){\rm d} x\, \leq\, \int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > k\beta){\rm d} x \nonumber\\
&= k\,\mathbb{P}({\textstyle \sum_{i=1}^k }\hat{A}^{(n)}_i > k\beta) \,\leq\, \frac{C_3}{k^2\beta^6} + k^2\mathbb{P}(\hat{A}^{(n)}> \tfrac{1}{3}k),
\end{align}
where we used \eqref{y8} in the last inequality.
Hence,
\begin{align}
\label{y10}
\sum_{k=1}^\infty\frac{1}{k}\, \int_0^k \mathbb{P}(S^{(n)}_k>x){\rm d} x &\leq \, \frac{C_3}{\beta^6}\sum_{k=1}^\infty k^{-3} +\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) \nonumber \\
&\leq C_1^*+\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k).
\end{align}
With the help of the inequality (see \cite{Sigman2011b}),
\begin{equation}
\label{y11}
(b-a)a\,\mathbb{P}(X>b) \leq \int_a^b x\,\mathbb{P}(X>x) {\rm d} x \qquad \text{\rm for } 0<a<b,
\end{equation}
we get by taking $a=(k-1)/3$ and $b=k/3$,
\begin{align}
\label{y12}
k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) &\leq \frac{9\,k}{k-1}\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x \nonumber \\
&\leq 18\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x,
\end{align}
for $k\geq 2$. Since the tail probability for $k=1$ is obviously bounded by 1, this yields
\begin{align}
\label{y13}
\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) &\leq 1+18\sum_{k=2}^\infty\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x\nonumber\\
&\leq 1+ 18\int_{0}^{\infty} x\,\mathbb{P}(\hat{A}^{(n)}>x){\rm d} x \leq 1+18\,\mathbb{E}[\hat{A}^{(n)2}] < \infty,
\end{align}
since $\hat{A}^{(n)}$ has finite variance by assumption. This completes the integral over the first interval. For the second part, we use \eqref{y8} again to find
\begin{align}
\label{y14}
\int_k^\infty \mathbb{P}(S^{(n)}_k>x)dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)} > x+k\beta)dx \leq \int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)} > x){\rm d} x\nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{x^6} {\rm d} x + k\int_k^\infty \mathbb{P}(\hat{A}^{(n)} >\tfrac{1}{3}x){\rm d} x\nonumber \\
&= \frac{5 C_3}{k^3}+ k\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}x) {\rm d} x.
\end{align}
So,
\begin{equation}
\label{y15}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>x){\rm d} x \leq C_2^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) {\rm d} x,
\end{equation}
for some constant $C_2^*$. Last, we are able to upper bound the second term in \eqref{y15} by Tonelli's theorem:
\begin{align}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) dx &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}x) {\rm d} x \nonumber\\
\label{y16}
&\leq 9\int_0^\infty y\mathbb{P}(\hat{A}^{(n)}>y) dy = 9\mathbb{E}[\hat{A}^{(n)2}] < \infty.
\end{align}
Combining the results in \eqref{y10}, \eqref{y13}, \eqref{y15} and \eqref{y16}, we find
\begin{equation*}
\label{y17}
\sum_{k=1}^\infty \frac{1}{k} \int_0^\infty \mathbb{P}(S^{(n)}_k>x){\rm d} x < \infty,
\end{equation*}
and thus
\begin{align*}
\lim_{n\rightarrow\infty} \mathbb{E}[\hat{Q}^{(n)}] &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x){\rm d} x \nonumber\\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > x){\rm d} x = \mathbb{E} [M_\beta].
\end{align*}
Finally, we show how the proof changes for the convergence of ${\rm Var}\, \hat{Q}^{(n)}$. The expressions for $\mathbb{E} [\hat{Q}^{(n)}]$ and ${\rm Var}\, \hat{Q}^{(n)}$ in \eqref{y1} and \eqref{y2} only differ in the term $\sqrt{x}$. Hence only minor modifications are needed to also prove convergence of the variance. Note that boundedness of the integral over the interval $(0,k)$ in \eqref{y9}-\eqref{y13} remains to hold when substituting $\sqrt{x}$ for $x$. \eqref{y14} changes into
\begin{align*}
\label{y18}
\int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x})dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > \sqrt{x}+k\beta){\rm d} x \nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{(\sqrt{x}+k\beta)^6} dx + k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
&\leq \frac{C_4^*}{k}+ k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x,
\end{align*}
for some constant $C_4^*$, so that
\begin{equation*}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x}){\rm d} x \leq C_4^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x.
\end{equation*}
Lastly, we have
\begin{align*}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
\label{y17a}
&\leq 18\int_0^\infty y^2\mathbb{P}(\hat{A}^{(n)}>y) {\rm d} y = 18\,\mathbb{E}[\hat{A}^{(n)3}] < \infty.
\end{align*}
Therefore the sum describing the variance is also uniformly convergent in $n$, so that interchanging of infinite sum and limit is permitted and
\begin{align*}
\lim_{n\rightarrow\infty} {\rm Var}\,\,\hat{Q}^{(n)} &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}){\rm d} x \nonumber \\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > \sqrt{x}){\rm d} x = {\rm Var}\, M_\beta.
\end{align*}
\section{Numerical procedures}\label{numprocs}
An alternative characterization of the stationary distribution is based on the analysis in \cite{Boudreau1962} and considers a factorization in terms of (complex) roots:
\begin{equation*}
\label{t9}
Q^{(n)}(w) = \frac{(s_n-\mathbb{E} [A^{(n)}])(w-1)}{w^{s_n}-\tilde{A}^{(n)}(w)}\,\prod_{k=1}^{s_n-1} \frac{w-z^n_k}{1-z^n_k},
\end{equation*}
where $z_1^n,z_2^n...,z_{s_n-1}^n$ are the $s_n-1$ zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$, in $|z|<1$, yielding
\begin{equation*}
\label{c2}
\mu_Q = \frac{\sigma_n^2}{2(s_n-\mu_n)}-\frac{s_n-1+\mu_n}{2} + \sum_{k-1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c3}
\mathbb{P}(Q^{(n)}=0) = \frac{s_n-\mu_A}{\tilde{A}^{(n)}(0)}\prod_{k=1}^{s_n-1}\frac{z^n_k}{z^n_k-1},
\end{equation*}
which for our choice of $\tilde{A}^{(n)}(z)$ becomes
\begin{equation*}
\label{c4}
\mu_Q = \frac{a_nb_n(b_n+1)}{2\beta\sqrt{a_n}b_n}-\frac{2a_nb_n+\beta\sqrt{a_nb_n(b_n+1)}-1}{2}+\sum_{k=1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c5}
\mathbb{P}(Q^{(n)}=0) = \beta \sqrt{a_nb_n(b_n+1)}(1+b_n)^{a_n}\prod_{k=1}^{s_n-1} \frac{z^n_k}{z^n_k-1},
\end{equation*}
where $z_1,...,z_{s_n-1}$ denote the zeros of $z^{s_n} - \tilde{A}^{(n)}(z)$ in $|z|<1$. A robust numerical procedure to obtain these zeros is essential for a base of comparison. We discuss two methods that fit these requirements. The first follows directly from \cite{Janssen2005}. \\
\begin{lemma}\label{fixedIterLemma}
Define the iteration scheme
\begin{equation}
\label{c6}
z_k^{n,l+1} = w^n_k [\tilde{A}^{(n)}(z_k^{n,l})]^{1/s_n},
\end{equation}
with $w^n_k = {\rm e}^{2\pi ik/s_n}$ and $z_k^{n,0}=0$ for $k=0,1,\ldots,s_{n-1}$. Then $z_k^{n,l} \rightarrow z_k^n$ for all $k=0,1,...,s_n-1$ for $l\rightarrow \infty$.
\end{lemma}
\begin{proof}
The successive substitution scheme given in \eqref{c6} is the fixed point iteration scheme described in \cite{Janssen2005}, applied to the pgf of our arrival process. The authors show that, under the assumption of $\tilde{A}^{(n)}(z)$ being zero-free within $|z|\leq 1$, the zeros can be approximated arbitrarily closely, given that the function $[\tilde{A}^{(n)}(z)]^{1/s_n}$ is a contraction for $|z|\leq 1$, i.e.
\begin{equation*}
\label{c7}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| < 1.
\end{equation*}
In our case,
\begin{align}
\label{c8}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| = \Bigl|\frac{{\rm d}}{{\rm d} z}\left(1+(1-z)b_n\right)^{-a_n/s_n}\Bigr| = \frac{a_nb_n}{s_n}\Bigl|1+(1-z)b_n\Bigr|^{-a_n/s_n-1},
\end{align}
where $a_nb_n/s_n = \rho_n$ is close to, but less than 1 and
\begin{align*}
\label{c9}
|1+(1-z)b_n| \geq |1+b_n|-|z|b_n = 1+(1-|z|)b_n \geq 1,
\end{align*}
when $|z|\leq 1$. Hence the expression in \eqref{c8} is less than 1 for all $|z|\leq 1$. Evidently, $\tilde{A}^{(n)}(z)$ is also zero-free in $|z|\leq 1$. Thus \cite[Lemma~3.8]{Janssen2005} shows that $z_k^{n,l}$ as in \eqref{c6} converges to the desired roots $z^n_k$ for all $k$ as $l$ tends to infinity.
\end{proof}
\begin{remark}
The asymptotic convergence rate of the iteration in \eqref{c6} equals \\
\noindent $\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}$ evaluated at $z=z_k^n$. Hence, convergence is slow for zeros near 1 and fast for zeros away from 1.
\end{remark}
A different approach is based on the B\"urmann-Lagrange inversion formula.
\begin{lemma}\label{BLLemma}
Let $w^n_k = e^{2\pi ik/s_n}$ and $\alpha_n = a_n/s_n$. Then the zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$ are given by
\begin{equation*}
z_k^n = \sum_{l=1}^\infty \frac{1}{l!}\,\frac{\beta[l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{b_n+1}{b_n}\Bigl(\frac{b_n}{(b_n+1)^{\alpha_n+1}}\Bigr)^l (w_k^n)^l,
\end{equation*}
for $k=0,1,...,s_n-1$.
\end{lemma}
\begin{proof}
Note that we are looking for $z$'s that solve
\begin{equation*}
\label{c10}
z\,[\tilde{A}^{(n)}(z)]^{-1/s_n} = z\left(1+(1-z)b_n\right)^{a_n/s_n} = w,
\end{equation*}
where $w = w_k = {\rm e}^{2\pi i k/s_n}$. The B\"urmann-Lagrange formula for $z=z(w)$, as can be found in \cite[Sec.~2.2]{debruijn} for $z=z(w)$ is given by
\begin{align*}
z(w) &= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(\frac{z}{z(1+(1-z)b_n)^{a_n/s_n}}\right)^l\right]_{z=0}\,w^l\nonumber\\
\label{c11}
&= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(1+(1-z)b_n)^{-l\,a_n/s_n}\right)\right]_{z=0}\,w^l.
\end{align*}
Set $\alpha_n = a_n/s_n$. We compute
\begin{equation*}
\label{c1}
\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[ (1+(1-z)b_n)^{-l\alpha_n}\right]_{z=0} = \frac{\beta(l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{1+b_n}{b_n}\,\left(\frac{b_n}{(1+b_n)^{\alpha_n+1}}\right)^l.
\end{equation*}
With $c_n = b_n/(1+b_n)^{\alpha_n+1}$ and $d_n = (1+b_n)/b_n$, we thus have
\begin{equation*}
\label{c13}
z(w) = d_n\,\sum_{l=1}^\infty \frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} c_n^l\,w^l.
\end{equation*}
By Stirling's formula
\begin{equation*}\label{c14}
\frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} = \frac{D}{l\sqrt{l}}\left(\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\right)^l,
\end{equation*}
where $D=\alpha_n^{1/2}(\alpha_n+1)^{-3/2}(2\pi)^{-1/2}$. Now,
\begin{equation*}
\label{c15}
\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\, c_n = \frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\cdot \frac{b_n}{(1+b_n)^{\alpha_n+1}} = \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation*}
This determines the radius of convergence $r_{\rm BL}$ of the above series for $z(w)$:
\begin{equation}
\label{c16}
\frac{1}{r_{\rm BL}} := \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation}
The derivative with respect to $\rho_n$ of the quantity
\begin{equation}
\label{c17}
\left(1+\frac{\rho_n}{b_n}\right) {\rm ln }\left(\frac{b_n+\rho_n}{b_n+1}\right)+\frac{\rho_n}{b_n}\,{\rm ln}\left(\frac{1}{\rho_n}\right)
\end{equation}
is given by
\begin{equation*}
\label{c18}
\frac{1}{b_n}{\rm ln }\Bigl(\frac{b_n+\rho_n}{b_n\rho_n+\rho_n}\Bigr) > 0
\end{equation*}
for $0<\rho_n<1$ and $b_n>0$. Furthermore, the quantity in \eqref{c17} vanishes at $\rho_n=1$ and is therefore negative for $0<\rho_n<1$ and $b_n>0$.
\begin{remark}
The formula for the radius of convergence in \eqref{c16} clearly shows the decremental effect of both having a large $b_n$ and of having $\rho_n$ close to 1. The quantities $\beta(l\alpha+l-1)/(\beta(l+1)\beta(l\alpha))$ in the power series for $z(w)$ are not very convenient for recursive computation, although normally $\alpha_n = a_n/s_n$ is a rational number.\end{remark}
\end{proof}
\section{Statistical procedures}\label{statproc}
To calibrate our model to real data, we now discuss some statistical procedures to show the presence of overdispersion and to estimate the parameters of the mixed Gamma-Poisson distribution. Let $x_1,...,x_n$ denote the observed arrival counts in consecutive time slots. These observations can be interpreted as realizations of the random variables $A_1,...,A_N$, and
\begin{equation*}
\bar{a}_N=\frac{1}{N}\sum_{i=1}^N x_i, \qquad \bar{s}_N^2 = \frac{1}{N-1}\sum_{i=1}(x_i-\bar{x}_i)^2,
\end{equation*}
the sample mean and variance with equivalent random variables $\bar{A}_N$ and $S_N^2$, respectively. Several tests with null hypothesis that $x_1,...,x_N$ originate from a (constant rate) Poisson distribution were discussed by \cite{Brown2002}. We mention two of them. The first is frequently referred to as the \emph{dispersion test}, and is based on the test statistic
\begin{equation*}
\label{dispTest}
D_N = \frac{(N-1)S_N^2}{\bar{A}_N},
\end{equation*}
which is approximately chi-squared distributed with $N-1$ degrees of freedom. When using a significance level $\alpha$, the critical value is equal to the $(1-\alpha)$-th quantile of chi-squared distribution $\chi^2_{N-1,1-\alpha}$. The second test, which is also used in \cite{koolejongbloed}, involves the test statistic
\begin{equation*}
\label{NStest}
T_N = \sqrt{N/2}\,\Bigl(\frac{S_N^2}{\bar{A}_N}-1\Bigr),
\end{equation*}
which is known as the Neyman-Scott test statistic. Under the null hypothesis $T_N$ tends to a standard normal random variable for large $N$. Hence both test statistics rely on the ratio of the sample variance and sample mean, which should be 1 if $A_1,...,A_N$ are indeed i.i.d. Poisson distributed. Excessive values of $D_N$ and $T_N$ therefore raise the suspicion of overdispersed arrivals.
Once either (or both) of the Poisson tests rejects the hypothesis of arrivals originating from a unicomponent Poisson process, we fit the data to the Gamma-Poisson mixture. Note that if we assume $A_i$ to be distributed as a Poisson random variable with random rate $\Lambda_i$, which is in turn Gamma distributed with parameters $a$ and $1/b$, then $A_i$ is in fact a negative binomial random variable with parameters $r = a$ and $p=b/(b+1)$. Finding estimators $\hat{a}$ and $\hat{b}$ therefore is equivalent to fitting a negative binomial distribution to the data to obtain $\hat{r}$ and $\hat{p}$, followed by retrieving $\hat{a} = \hat{r}$ and $\hat{b} = \hat{p}/(1-\hat{p})$. We proceed by applying the maximum likelihood estimation method described in \cite{koolejongbloed} to find $\hat{r}$ and $\hat{p}$. This method prescribes to set $\hat{r}$ to be the value of $r$ for which the \emph{profile loglikelihood function} defined by
\begin{equation*}
L(r) = \frac{1}{N}\,\sum_{i=1}^N\sum_{j=1}^{a_i} {\rm ln}(r+j+1)+r\,{\rm ln}\,r -(r+\bar{a}_N)\,{\rm ln}(r+\bar{a}_N),
\end{equation*}
is attained. Subsequently, $\hat{p} = \hat{r}/(\hat{r}+\bar{a}_N)$, so that $\hat{a} = \hat{r}$ and $\hat{b} = \hat{r}/\bar{a}_N$.
Finally, given the estimators $\hat{a}$ and $\hat{b}$, we need statistical evidence that the obtained Poisson mixture indeed fits the data reasonably well. Here we again cite \cite{koolejongbloed}, who give a method to retrieve the $p$-value for the goodness-of-fit based on bootstrap and Monte-Carlo simulation. In our experiments, we work with $10^6$ replications of the Monte-Carlo simulation to obtain the approximated $p$-value. We refer to the appendix of \cite{koolejongbloed} for further details on this method.
\resettocdepth
\end{subappendices}
\chapter{Overdispersion}
\begin{chapterstart}
Arrival processes to service systems often display fluctuations that are larger than anticipated under the Poisson assumption, a phenomenon that is referred to as \textit{overdispersion}.
Motivated by this, we analyze a class of discrete stochastic models for which we derive heavy-traffic approximations that are scalable in the system size.
Subsequently, we show how this leads to novel capacity sizing rules that acknowledge the presence of overdispersion. This, in turn, leads to robust approximations for performance characteristics of systems that are of moderate size and/or may not operate in heavy traffic.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Robust heavy-traffic approximations for\\ service systems facing overdispersed demand}\\
\textit{Britt Mathijsen, Guido Janssen, Johan van Leeuwaarden \& Bert Zwart}\\
arxiv.org/abs/1512.05581
\end{flushright}
\newpage
\section{Introduction}\label{intro}
In the previous chapter, we analyzed the scaling limit of a queueing model in which demand exhibits stochastic fluctuations that are asymptotically proportional to the square-root of the nominal load, while we deliberately chose to deviate from the square-root staffing principle by allocating a variability hedge that does not match the order of these fluctuations.
This chapter in some ways does the opposite.
We assume the demand faced by the queueing system is more volatile than anticipated by the independent many-sources paradigm that leads to Poisson traffic models.
As will become clear in this chapter, this in fact \emph{requires} an adaptation of the square-root staffing principle in order to maintain the desirable properties of the QED regime.
We start by motivating our research through empirical evidence of the presence of so-called \emph{overdispersion} in arrival processes faced by service systems reported by recent literature. \\
\noindent
\textbf{Motivation.}
The bulk of the queueing literature assumes perfect knowledge about the model primitives, including the mean demand per time period. For large-scale service systems, like health care facilities, communication systems or call centers, the dominant assumption is that demand arrives according to a (non)homogeneous Poisson process, which in practice translates into the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies shows that the variance of demand typically deviates from the mean significantly. Recent work \cite{Kim2015b,Kim2015a} reports variance being strictly less than the mean in health care settings employing appointment booking systems. This reduced variability, known as underdispersion, can be accredited to the goal of the booking system to create a more predictable arrival pattern.
On the other hand, in other scenarios with no control over the arrivals, the variance typically dominates the mean, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2015, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}.
The feature that variability is higher than one expects from the Poisson assumption is referred to as overdispersion. The latter concept will be the center of our attention in this chapter.
Stochastic models with the Poisson assumption have been widely applied to optimize capacity levels in service systems. The goal is to minimize operating costs while providing sufficiently high QoS in terms of performance measures such as mean delay or excess delay. When stochastic models, however, do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly under critical loading.
\\
\\*
\noindent
\textbf{Causes of overdispersion.}
The literature discussed above proves that the presence of overdispersion is widespread across applications.
It however does not specify what causes the increased variability in the arrival process.
We name two possible explanations.
First, we revisit the many-sources characterization of demand inflow discussed in Chapter 2.
Recall that in this setting, demand is generated by $n$ stochastically identical and independent sources, with $n$ large, so that workload arriving to the system in period $j$ is given by $A^{(n)}_j = \sum_{i=1}^n A_{i,j}$, where $A_{i,j}$, $i=1,2,\ldots,n$ are i.i.d.~random variables.
This resulted in nominal workload $\mu_n = n\mu$ and $\sigma_n^2 = n\sigma^2$, thus both of order $n$.
If we now relax the assumption on the (pairwise) independence of the sources, but rather consider the scenario in which these are positively correlated, then the nominal load remains to be equal to $n \mu$, while the variance of demand becomes
\begin{equation*}
\sigma_n^2 = {\rm Var}\, A_j^{(n)} = n\, {\rm Var}\, A_{1,j} + n(n-1)\,{\rm Cov}(A_{1,j},A_{2,j}),
\end{equation*}
which is of higher order than $n$ if $n\,{\rm Cov}(A_{1,j},A_{2,j}) \to \infty$ as $n\to\infty$.
A second interpretation of overdispersion in arrival processes relates to \emph{arrival rate uncertainty}.
The canonical process for modeling the arrival process of a service system is the Poisson process with a given arrival rate $\lambda$.
Since model primitives, in particular the arrival rate, are typically estimated through historical data, these are prone to be subject to forecasting errors.
In the realm of Poisson processes, this inherent uncertainty can be acknowledged by viewing the arrival rate $\Lambda_n$ itself as being stochastic. The resulting doubly stochastic Poisson process, also known as Cox process (first presented in \cite{Cox1955}), implies that demand in a given interval $A_j$ follows a mixed Poisson distribution.
In this case, the expected demand per period equals $\mu_n = \mathbb{E}[\Lambda_n]$, while the variance is $\sigma_n^2 = \mathbb{E}[\Lambda_n]+{\rm Var}\,\Lambda_n$.
By selecting the distribution of the mixing factor $\Lambda_n$, the magnitude of overdispersion can be made arbitrarily large, and only a deterministic $\Lambda_n$ leads to a true Poisson process.
The mixed Poisson model presents a useful way to fit both the mean and variance to real data, particularly in case of overdispersion.
The mixing distribution can be estimated parametrically or non-parametrically, see \cite{koolejongbloed,maman}.
A popular parametric family is the Gamma distribution, which gives rise to an effective data fitting procedure that uses the fact that a Gamma mixed Poisson random variable follows a negative binomial distribution.
We will in this chapter adopt the assumption of a Gamma-Poisson mixture as the demand process.\\
\\*
\textbf{Adapted QED scaling.} To deal with overdispersion
new models are needed, scaling rules must be adapted, and existing capacity sizing rules need to be modified in order to incorporate a correct hedge against (increased) variability.
In this chapter, we consider an extension of the discrete queueing model of Chapter 2 that has a doubly stochastic Poisson process as input, $A_j\sim\,{\rm Pois}(\Lambda_n)$ and we identify the heavy-traffic regime in which it displays QED behavior.
That is, it fits the three asymptotic characteristics in Section 1.2.3 of this thesis.
As we argued in that particular section, a sensible candidate capacity allocation rule is $s_n = \mu_n + \beta \sigma_n$ for some $\beta>0$, which is equivalent to the scaling
\begin{equation*}
\frac{\mu_n}{\sigma_n}\,(1-\rho_n) \to \beta, \qquad \text{as }n\to\infty.
\end{equation*}
We will verify mathematically that this is asymptotically the appropriate choice.
Studies that have adressed similar capacity allocation problems with stochastic arrival rates include \cite{Kocaga2015, maman, Whitt1999, Whitt2006}.
Of the aforementioned papers, our work best relates to \cite{maman}, in the sense that we also assess the asymptotic performance of a queueing system having a stochastic arrival rate in heavy traffic.
We therefore expand the paradigm of the QED regime, in order to have it accommodate for overdispersed demand that follows from a doubly stochastic Poisson process.
\\
\\*
\textbf{Structure of the chapter}. The remainder of this chapter is structured as follows. Our model is introduced in Section \ref{modelSection} together with some preliminary results.
In Section 3.3 we derive the classical heavy-traffic scaling limits for the queue length process in the presence of overdispersed arrivals both for the moments and the distribution itself.
Section 3.4 presents our main theoretic result, which provides a robust refinement to the heavy-traffic characterization of the queue length measures in pre-limit systems.
In Section 3.5, we describe the numerical results and demonstrate the heavy-traffic approximation for a real data set coming from a health care setting. Section 3.6 provides some concluding remarks.
\section{Model description}\label{modelSection}
We consider the same mathematical model as in Section 2.2, in which time is divided into periods of equal length. At the beginning of each period $j=1,2,3,...$ new demand $A^{(n)}_j$ arrives to the system. The demands per period $A^{(n)}_1,A^{(n)}_2,...$ are assumed independent and equal in distribution to some non-negative integer-valued random variable $A^{(n)}$.
The system has a service capacity $s_n\in\mathbb{N}$ per period, the steady-state queue length can be characterized as, see (1.27),
\begin{equation}
\label{mm3}
Q^{(n)} {\;\buildrel{d}\over= \;} \max_{k\geq 0}\Bigl\{\sum_{i=1}^k (A^{(n)}_i-s_n)\Bigr\}.
\end{equation}
For brevity, we define $\mu_n:= \mathbb{E} [A^{(n)}_1]$ and $\sigma_n^2 = {\rm Var}\, A^{(n)}_1$.
The behavior of $Q^{(n)}$ predominantly depends on the characteristics of $A^{(n)}$ and $s_n$. As noted before, $\mu_n<s_n$ is a necessary condition for the maximum in \eqref{mm3} to be finite and consequently for the queue to be stable. Before continuing the analysis of $Q^{(n)}$, we impose a set of conditions on the asymptotic properties of $s_n,\mu_n$ and $\sigma_n$.
\begin{assumption}
\label{as1}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item[{\normalfont (a)}] {\rm (Asymptotic growth)}
\begin{equation*}
\mu_n,\sigma_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (b)}] {\rm (Persistence of overdispersion)}
\begin{equation*}
\sigma_n^2/\mu_n \to \infty \quad \text{\rm for } n\to\infty.
\end{equation*}
\item[{\normalfont (c)}] {\rm (Heavy-traffic condition)}
The utilization $\rho_n := \mu_n/s_n \to 1$ as $n\to\infty$, while
\begin{equation}\label{mm5}
s_n = \mu_n + \beta\, \sigma_n,
\end{equation}
for some $\beta > 0$. This is equivalent to requiring
\begin{equation}\label{mm4}
(1-\rho_n)\frac{\mu_n}{\sigma_n} \to \beta, \qquad \text{\rm for }n\to\infty.
\end{equation}
\end{enumerate}
\end{assumption}
\noindent
Assumption \ref{as1} is assumed to hold throughout the remainder of this chapter.
Since we are mainly interested in the system behavior in heavy traffic, it is appropriate to study the queue length process in a scaled form. Substituting $s_n$ as in Assumption \ref{as1}(c), and dividing both sides of \eqref{mm3} by $\sigma_n$, gives
\begin{equation}
\label{mm6}
\frac{Q^{(n)}}{\sigma_n} = \max_{k\geq 0} \Bigl\{{\sum_{i=1}^k} \Bigl(\frac{A^{(n)}_i-\mu_n}{\sigma_n} - \beta\Bigr)\Bigr\}.
\end{equation}
By defining $\hat{Q}^{(n)} := Q^{(n)}/\sigma_n$ and $\hat{A}^{(n)}_i := (A^{(n)}_i-\mu_n)/\sigma_n$, we see that the scaled queue length process is in distribution equal to the maximum of a random walk with i.i.d. increments distributed as $\hat{A}^{(n)}-\beta$. Besides $\mathbb{E}[\hat{A}^{(n)}] = 0$ and ${\rm Var}\, \hat{A}^{(n)}=1$, the scaled and centered arrival count $\hat{A}^{(n)}$ has a few other nice properties which we turn to later in this section.
The model in \eqref{mm3} is valid for any distribution of $A^{(n)}$, also for the original case where the number of arrivals follows a Poisson distribution with fixed parameter $\lambda_n$, but in that case Assumption \ref{as1}(b) does not hold. Instead, we assume $A^{(n)}$ to be Poisson distributed with uncertain arrival rate rendered by the non-negative random variable $\Lambda_n$. This $\Lambda_n$ is commonly referred to as the \emph{prior} distribution, while $A^{(n)}$ is given the name of a Poisson mixture, see \cite{Grandell1997}. Given that the moment generation function of $\Lambda_n$, denoted by $M^\Lambda_n(\cdot)$, exists, we are able to express the probability generating function (pgf) of $A^{(n)}$ through the former. Namely,
\begin{equation}
\label{mm7}
\tilde{A}^{(n)}(z) = \mathbb{E}[\mathbb{E}[ z^{A^{(n)}} | \Lambda_n ] ] = \mathbb{E}[ \exp(\Lambda_n(z-1))] = M^\Lambda_n(z-1).
\end{equation}
From \eqref{mm7}, we get
\begin{equation}
\label{mm8}
\mu_n = \mathbb{E}[A^{(n)}] = \mathbb{E}[\Lambda_n],\qquad
\sigma_n^2 = {\rm Var}\, A^{(n)} = {\rm Var}\, \Lambda_n + \mathbb{E}[\Lambda_n],
\end{equation}
so that $\mu_n<\sigma_n^2$ if $\Lambda_n$ is non-deterministic. Assumption \ref{as1}(b) hence translates to
\[{\rm Var}\, \Lambda_n/\mathbb{E}[\Lambda_n]\rightarrow \infty, \qquad n\rightarrow\infty.\]
The next result relates the converging behavior of the centered and scaled $\Lambda_n$ to that of $\hat{A}^{(n)}$.
\begin{lemma}\label{gaussStep}
Let $\mu_n,\sigma_n^2\rightarrow\infty$ and $\sigma_n^2/\mu_n\rightarrow\infty$. If
\begin{equation*}
\hat{\Lambda}_n := \frac{\Lambda_n-\mu_n}{\sigma_n}{\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{\normalfont for } n\rightarrow\infty,
\end{equation*}
then $\hat{A}^{(n)}$ converges weakly to a standard normal variable as $n\rightarrow\infty$.
\end{lemma}
\noindent
The proof can be found in Appendix \ref{formalSec}.
The prevalent choice for $\Lambda_n$ is the Gamma distribution. The Gamma-Poisson mixture turns out to provide a very good fit to arrival counts experienced by service systems, as was observed by \cite{koolejongbloed}. Assuming $\Lambda_n$ to be of Gamma type with scale and rate parameters $a_n$ and $1/b_n$, respectively, we get for the pgf of $A^{(n)}$:
\begin{equation}
\label{r0}
\tilde{A}^{(n)}(z) = \Bigl(\frac{1}{1+b_n(1-z)}\Bigr)^{a_n},
\end{equation}
in which we recognize the pgf of a negative binomial distribution with parameters $a_n$ and $1/(b_n+1)$, so that
\begin{equation*}
\label{t21}
\mu_n = a_nb_n,\qquad \sigma_n^2 = a_nb_n(b_n+1).
\end{equation*}
Note that in the context of a Gamma prior, the restrictions in Assumption \ref{as1} reduce to only two rules. For completeness, we include the revised list below.
\begin{assumption}\label{as2}
\ \\*
\vspace{-6mm}
\begin{enumerate}
\item {\rm (Asymptotic regime and persistence of overdispersion)}
\begin{equation*}
a_n, b_n \to \infty, \quad \text{\rm for } n\to\infty.
\end{equation*}
\item {\rm (Heavy-traffic condition)}
Let
\begin{equation*}
s_n = a_n b_n + \beta \sqrt{a_n b_n(b_n+1)},
\end{equation*}
for some $\beta>0$, or equivalently
\begin{equation*}
(1-\rho_n)\sqrt{a_n} \to \beta, \quad \text{\rm for } n\to\infty.
\end{equation*}
\end{enumerate}
\end{assumption}
The next result follows from the fact that $\Lambda_n$ is a Gamma random variable:
\begin{corollary}\label{scaledLambdaLemma}
Let $\Lambda_n\sim\text{\normalfont Gamma}(a_n,1/b_n)$, $A^{(n)}\sim{\rm Pois }(\Lambda_n)$ and $a_n,b_n\rightarrow \infty$. Then $\hat{A}^{(n)}$ converges weakly to a standard normal random variable as $n\rightarrow \infty$.
\end{corollary}
\begin{proof}
By Lemma \ref{gaussStep}, it is sufficient to prove that $\hat{\Lambda}_n{\;\buildrel{d}\over\Rightarrow\;}\mathcal{N}(0,1)$ for this particular choice of $\Lambda_n$.
We do this by proving the pointwise convergence of the characteristic function (cf) of $\hat{\Lambda}_n$ to $\exp({-} t^2/2)$, the cf of the standard normal distribution.
Let $\phi_{G}(\cdot)$ denote the characteristic function of a random variable $G$. By basic properties of the cf,
\begin{align*}
\phi_{\hat{\Lambda}_n}(t) &= {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n)
= {\rm e}^{-i\mu_nt/\sigma_n} \Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)^{-a_n}\nonumber\\
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n}\, - a_n\,{\rm ln}\Bigl(1-\frac{i b_nt}{\sigma_n}\Bigr)\Bigr]\nonumber\\
\label{g13d}
&= \exp\Bigl[ -\frac{i\mu_nt}{\sigma_n} -a_n\Bigl( {-}\frac{i\,b_nt}{\sigma_n} + \frac{b_n^2t^2}{2\sigma_n^2} + O( b_n^3/\sigma_n^3)\Bigr)\Bigr] \nonumber\\
&= \exp\Bigl[ -\frac{b_n\,t^2}{2(b_n+1)} + O\left(1/\sqrt{a_n}\right)\Bigr] \rightarrow \exp\big({-} t^2/2\big),
\end{align*}
for $n\rightarrow\infty$. By L\'evy's continuity theorem this implies $\hat{\Lambda}_n$ is indeed asymptotically standard normal.
\end{proof}
The characterization of the arrival process as a Gamma-Poisson mixture is of vital importance in later sections.\\
\\*
\noindent
\textbf{Expressions for the stationary distribution.} \label{expressionsSubsec}
Our main focus is on the stationary queue length distribution, denoted by
\[\mathbb{P}(Q^{(n)}=i) =\lim_{k\rightarrow\infty} \mathbb{P}(Q^{(n)}(k)=i).\]
Denote the pgf of $Q^{(n)}$ by
\begin{equation*}
\label{t1}
\tilde{Q}^{(n)}(w) := \sum_{i=0}^\infty \mathbb{P}(Q^{(n)}=i) w^i.
\end{equation*}
Furthermore, let $\mu_Q := \mathbb{E}[Q^{(n)}]$ and $\sigma_Q^{2} := {\rm Var}\, Q^{(n)}$ denote the stationary mean and variance of the queue length, respectively.
To avoid notational complexity, we omit the superscript $(n)$ in these definitions.
To continue our analysis of $Q^{(n)}$, we need one more condition on $A^{(n)}$.
\begin{assumption}\label{as3}
The pgf of $A^{(n)}$, denoted by $\tilde{A}^{(n)}(w)$, exists for $|z|<r_0$, for some $r_0>1$, so that all moments of $A^{(n)}$ are finite.
\end{assumption}
We next recall two characterizations of $\tilde{Q}^{(n)}(w)$ that play prominent roles in the remainder of our analysis.
The first characterization of $\tilde{Q}^{(n)}(w)$ originates from a random walk perspective. As we see from \eqref{mm3}, the (scaled) stationary queue length is equal in distribution to the all-time maximum of a random walk with i.i.d. increments distributed as $A^{(n)}-\beta$ (or $\hat{A}^{(n)}-\beta$ in the scaled setting). Spitzer's identity, see e.g. \cite[Theorem VIII4.2]{Asmussen2003} and Section 1.2.2 of this thesis, then gives
\begin{equation*}
\label{t3}
\tilde{Q}^{(n)}(w) = \exp\left\{\sum_{k=1}^\infty \frac{1}{k}\,\Big(\mathbb{E}\Big[w^{\left(\sum_{i=1}^k \{A^{(n)}_i-s_n\}\right)^+}\Big]-1\Big)\right\},
\end{equation*}
where $(x)^+ = \max\{x,0\}$. Hence,
\begin{equation*}
\label{t4}
\mu_Q = \mathbb{E}[Q^{(n)}] = \tilde{Q}^{(n)\prime}(1) = \sum_{k=1}^\infty \frac{1}{k}\,\mathbb{E}\Bigl[ {\sum_{i=1}^k} (A^{(n)}_i - s_n) \Bigr]^+,
\end{equation*}
\begin{equation*}
\label{t4a}
\sigma^{2}_Q = {\rm Var}\, Q^{(n)} = \tilde{Q}^{(n)\prime\prime}(1)+Q^{(n)\prime}(1)-\left(\tilde{Q}^{(n)\prime}(1)\right)^2 = \sum_{k=1}^\infty \frac{1}{k}\,\mathbb{E}\Bigl[ \Big(\sum_{i=1}^k (A^{(n)}_i - s_n) \Big)^+\Bigr]^2,
\end{equation*}
\begin{align*}
\label{t5}
\mathbb{P}(Q^{(n)}=0) = \tilde{Q}_n(0) &= \exp\Bigl\{{-}{\sum_{k=1}^\infty}\frac{1}{k}\,\mathbb{P}\Bigl({\textstyle\sum_{i=1}^k} (A^{(n)}_i-s_n) > 0\Bigr) \Bigr\}.
\end{align*}
A second characterization follows from Pollaczek's formula, see \cite{Abate1993} and Section 2.2.2 of this thesis:
\begin{equation}
\label{t6}
\tilde{Q}^{(n)}(w) = \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{w-z}{1-z}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\},
\end{equation}
which is analytic for $|w|<r_0$, for some $r_0>1$. Therefore, $\varepsilon>0$ has to be chosen such that $|w|<1+\varepsilon<r_0$. This gives
\begin{align}
\label{t7}
\mu_Q &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{1}{1-z}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)} {\rm d} z,\\
\label{t7a}
\sigma_Q^{2} &= \frac{1}{2\pi i} \int_{|z|=1+\varepsilon} \frac{{-}z}{(1-z)^2}\,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z,\\
\label{t8}
\mathbb{P}(Q^{(n)}=0) &= \exp\Bigl\{ \frac{1}{2\pi i}\int_{|z|=1+\varepsilon} {\rm ln}\Bigl(\frac{z}{z-1}\Bigr) \,\frac{(z^{s_n}-\tilde{A}^{(n)}(z))'}{z^{s_n}-\tilde{A}^{(n)}(z)}{\rm d} z\Bigr\}.
\end{align}
Pollaczek-type integrals like \eqref{t6}-\eqref{t8} first occurred in the work of Pollaczek on the classical single-server queue (see \cite{Abate1993,Cohen1982,Janssen2008} for historical accounts). These integrals are fairly straightforward to evaluate numerically and hence give rise to efficient algorithms for performance evaluation \cite{Abate1993,boon2017pollaczek}. The integrals also proved useful in establishing heavy-traffic results by asymptotic evaluation of the integrals in various heavy-traffic regimes \cite{Kingman1962,Cohen1982,Janssen2015,boon2017pollaczek2}, and in this paper we follow that approach for a heavy-traffic regime that is suitable for overdispersion.
\section{Heavy-traffic limits}
In this section we present the result on the convergence of the discrete process $\hat{Q}^{(n)}$ to a non-degenerate limiting process and of the associated stationary moments. The latter requires an interchange of limits. Using this asymptotic result, we derive two sets of approximations for $\mu_Q$, $\sigma^2_Q$ and $\mathbb{P}(Q^{(n)}=0)$, that capture the limiting behavior of $Q^{(n)}$. The first set provides a rather crude estimation for the first cumulants of the queue length process for any arrival process $A^{(n)}$ satisfying Assumption \ref{as1}. The second set, which is the subject of the next section, is derived for the specific case of a Gamma prior and is therefore expected to provide more accurate, robust approximations for the performance metrics.
We start by indicating how the asymptotic properties of the scaled arrival process give rise to a proper limiting random variable describing the stationary queue length. The asymptotic normality of $\hat{A}^{(n)}$ provides a link with the Gaussian random walk and nearly deterministic queues \cite{Sigman2011a,Sigman2011b}.
The main results in \cite{Sigman2011a,Sigman2011b} were obtained under the assumption that $\rho_n\sim 1-\beta/\sqrt{n}$, in which case it follows from \cite[Thm.~3]{Sigman2011b} that the rescaled stationary waiting time process converges to a reflected Gaussian random walk.
We shall also identify the Gaussian random walk as the appropriate scaling limit for our stationary system. However, since the normalized natural fluctuations of our system are given by $\mu_n/\sigma_n$ instead of $\sqrt{n}$, we assume that the load grows like $\rho_n \sim 1 - \frac{\beta}{\mu_n/\sigma_n}$. Hence, in contrast to \cite{Sigman2011a,Sigman2011b}, our systems' characteristics display larger natural fluctuations, due to the mixing factor that renders the arrivals. Yet, by matching this overdispersed demand with the appropriate hedge against variability, we again obtain Gaussian limiting behavior. This is not surprising, since we saw in Lemma \ref{gaussStep} that the increments start resembling Gaussian behavior for $n\rightarrow\infty$. The following result summarizes this.
\begin{theorem}
\label{gaussianThm}
Let $\Lambda_n$ be a non-negative random variable such that $(\Lambda_n-\mu_n)/\sigma_n$ is asymptotically standard normal, with $\mu_n$ and $\sigma_n$ as defined in \eqref{mm8}, and $\mathbb{E}[\Lambda_n^3]<\infty$ for all $n\in\mathbb{N}$. Then under Assumption \ref{as1}, for $n\rightarrow \infty$,
\begin{enumerate}
\item[{\rm (i)}] $\hat{Q}^{(n)} {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[{\rm (ii)}] $\mathbb{P}(Q^{(n)} = 0) \rightarrow \mathbb{P}(M_\beta=0)$,
\item[{\rm (iii)}] $\mathbb{E}[\hat{Q}^{(n)}] \rightarrow \mathbb{E} [M_\beta]$,
\item[{\rm (iv)}] ${\rm Var}\, \hat{Q}_n \rightarrow {\rm Var}\,\, M_\beta$,
\end{enumerate}
where $M_\beta$ is the all-time maximum of a random walk with i.i.d. normal increments with mean $-\beta$ and unit variance.
\end{theorem}
The proof of Theorem \ref{gaussianThm} is given in Appendix \ref{formalSec}. The following result shows that Theorem \ref{gaussianThm} also applies to Gamma mixtures, which is a direct consequence of Corollary \ref{scaledLambdaLemma}.
\begin{corollary}
Let $\Lambda_n\sim$ \normalfont{Gamma}$(a_n,b_n)$. Then under Assumption \ref{as2} the four convergence results of Theorem \ref{gaussianThm} hold true.
\end{corollary}
It follows from Theorem \ref{gaussianThm} that the scaled stationary queueing process converges under \eqref{mm4} to a reflected Gaussian random walk. Hence, the performance measures of the original system should be well approximated by the performance measures of the reflected Gaussian random walk, yielding heavy-traffic approximations.
Like our original system, the Gaussian random walk falls in the classical setting of the reflected one-dimensional random walk, whose behavior is characterized by both Spitzer's identity and Pollaczek's formula. In particular, Pollaczek's formula gives rise to contour integral expressions for performance measures that are easy to evaluate numerically, also in heavy-traffic conditions. The numerical evaluation of such integrals is considered in \cite{Abate1993}. For $\mathbb{E} [M_\beta]$ such an integral is as follows
\begin{equation}
\label{g13e}
\mathbb{E} [M_\beta] = {-}\frac{1}{\pi}\int_0^\infty {\rm Re}\Bigl[\frac{1-\phi(-z)}{z^2}\Bigr]{\rm d} y,
\end{equation}
where $z=x+iy$ with an appropriately chosen real part $x$, with $\phi(z) = \exp(-\beta\,z+\tfrac12\,z^2)$, the Laplace transform of a normal random variable with mean $-\beta$ and unit variance.
Note that this integral involves complex-valued functions with complex arguments. Similar Pollaczek-type integrals exist for $\mathbb{P}(M_\beta=0)$ and ${\rm Var}\, M_\beta$, see \cite{Abate1993}. The following result simply rewrites these integrals in terms of a real integral and uses the fact that the scaled queue length process mimics the maximum of the Gaussian random walk for large $n$.
\begin{corollary}\label{abateThm}
Under Assumption \ref{as1}, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$, $\mu_Q$ and $\sigma^2_Q$ as $n\to\infty$ are given by, respectively,
\begin{equation}
\label{h1a}
\exp\Bigl[\frac{1}{\pi} \int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\Bigl(1-e^{-\tfrac12\beta^2-t^2}\Bigr){\rm d} t\Bigr],
\end{equation}
\begin{equation}
\label{h1}
\frac{\sqrt{2}\sigma_n}{\pi}\int_0^\infty \frac{t^2}{\tfrac12\beta^2+t^2}\, \frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t,
\end{equation}
\begin{equation}
\label{h1b}
\frac{\sqrt{2}\beta\sigma_n^2}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\frac{\exp(-\tfrac12\beta^2- t^2)}{1-\exp(-\tfrac12 \beta^2 - t^2)} {\rm d} t.
\end{equation}
\end{corollary}
\begin{proof}
According to \cite[Eq.~(15)]{Abate1993},
\begin{equation*}
\label{z1}
{-}\,{\rm ln}\,[\mathbb{P}(M_\beta=0)] = c_0,\quad \mathbb{E}[M_\beta]\ = c_1, \quad {\rm Var}\,\, M_\beta = c_2,
\end{equation*}
where
\begin{equation*}
\label{z2}
c_n = \frac{(-1)^nn!}{\pi} \,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp(\beta\,z+\tfrac12 z^2))}{z^{n+1}} {\rm d} y\Bigr],
\end{equation*}
in which $z={-}x+i\,y$, $y\geq 0$, and $x$ is any fixed number between 0 and $2\beta$.
Take $x=\beta$, so that
\begin{equation*}
\label{z3}
\beta z+\tfrac12 z^2 = {-}\tfrac12\beta^2 - \tfrac12 y^2\leq 0,\quad y\geq 0.
\end{equation*}
For $n=0$, this gives
\begin{align*}
c_0 &= \frac{1}{\pi}\,{\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-\tfrac12 y^2))}{{-}\beta+i\,y} {\rm d} y\Bigr] \nonumber\\
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta}{\beta^2+y^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2- \tfrac12 y^2)) {\rm d} y\nonumber\\
\label{z4}
&= {-}\frac{1}{\pi}\,\int_0^\infty \frac{\beta/\sqrt{2}}{\tfrac12\beta^2+t^2}\,{\rm ln}\,(1-\exp({-}\tfrac12 \beta^2-t^2)) {\rm d} t,
\end{align*}
where we used that
\begin{equation*}
\label{z5}
{\rm Re }\Bigl[\frac{1}{{-}\beta+i\, y}\Bigr] = \frac{{-}\beta}{\beta^2+y^2},
\end{equation*}
together with the substitution $y=t\sqrt{2}$. For $n=1,2,\ldots,$ partial integration gives
\begin{align*}
c_n &= \frac{(-1)^n n!}{\pi} \, {\rm Re}\Bigl[\int_0^\infty \frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{({-}\beta+i\,y)^{n+1}} {\rm d} y\nonumber\\
&= \frac{(-1)^{n-1}(n-1)!}{\pi}\,{\rm Im}\Bigl[\int_0^\infty {\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)){\rm d} \Bigl(\frac{1}{(-\beta+i\,y)^n}\Bigr)\Bigr]\nonumber\\
\label{z6}
&= {-}\frac{(-1)^{n-1}(n-1)!}{\pi} {\rm Im}\Bigl[ \int_0^\infty \frac{y}{(-\beta+i\,y)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr],
\end{align*}
where we have used that
\begin{equation*}
\label{z7}
{\rm Im}\Bigl[\frac{{\rm ln}(1-\exp(-\tfrac12\beta^2-\tfrac12 y^2))}{(-\beta+i\,y)^n}\Bigr]\Bigl|_0^\infty\Bigr. = 0.
\end{equation*}
Using
\begin{equation*}
\label{z8}
\frac{1}{(-\beta+i\,y)^n} = (-1)^n\,\frac{(\beta+i\,y)^n}{(\beta^2+y^2)^n},
\end{equation*}
we then get
\begin{equation*}
\label{z9}
c_n = \frac{(n-1)!}{\pi}\,{\rm Im}\,\Bigl[\int_0^\infty \frac{y(\beta+i\,y)^n}{(\beta^2+y^2)^n}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{\rm d} y\Bigr],
\end{equation*}
which after the substitution of $y=t\sqrt{2}$ gives
\begin{align}
c_1&=\frac{1}{\pi}\,\int_0^\infty \frac{y^2}{\beta^2+y^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y \nonumber\\
\label{z10}
&= \frac{\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{\tfrac12 \beta^2+t^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)}{\rm d} t,
\end{align}
\begin{align*}
c_2&=\frac{2\beta}{\pi}\,\int_0^\infty \frac{y^2}{(\beta^2+y^2)^2}\,\frac{\exp(-\tfrac12\beta^2-\tfrac12 y^2)}{1-\exp(-\tfrac12\beta^2-\tfrac12 y^2)} {\rm d} y\nonumber\\
\label{z11}
&= \frac{\beta\sqrt{2}}{\pi}\,\int_0^\infty \frac{t^2}{(\tfrac12 \beta^2+t^2)^2}\,\frac{\exp(-\tfrac12\beta^2-t^2)}{1-\exp(-\tfrac12\beta^2-t^2)} {\rm d} t.
\end{align*}
\end{proof}
\section{Robust heavy-traffic approximations}
We shall now establish robust heavy-traffic approximations for the canonical case of Gamma-POisson mixutres; see \eqref{r0}.
As noted earlier, Gamma mixing yields an arrival process that has a negative binomial distribution, which allows us to establish the detailed asymptotic results in the next theorem.
\begin{theorem}\label{saddlepointThm}
Let $a_n,b_n$ and $s_n$ be as in Assumption \ref{as2}. Then the leading order behavior of $\mu_Q$ is given by
\begin{equation}
\label{r1}
\frac{\sqrt{2}\,\beta_n}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta^2_n+t^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)} {\rm d} t\,(1+o(1)),
\end{equation}
where
\begin{equation}
\label{r2}
\beta_n^2 = s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Furthermore, the leading order behavior of $\mathbb{P}(Q^{(n)}=0)$ and $\sigma^2_Q$ is given by
\begin{equation*}
\label{r3}
\exp\Bigl[\frac{1}{\pi}\,\frac{b_n+\rho_n}{b_n+1}\,\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta^2_n+t^2}\,{\rm ln}\,\Bigl(1-{\rm e}^{{-}\tfrac12\beta^2_n-t^2}\Bigr){\rm d} t\Bigr],
\end{equation*}
and
\begin{equation}
\label{r4}
\frac{\beta_n^3/\sqrt{2}}{\pi}\Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr)^2\Bigl(\frac{b_n+1}{b_n+\rho_n}+1\Bigr)\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t,
\end{equation}
respectively.
\end{theorem}
The proof of Theorem \ref{saddlepointThm} requires asymptotic evaluation of the Pollaczek-type integrals \eqref{t6}-\eqref{t8}, for which shall use the \textit{non-standard} saddle point method---originally proposed by \cite{debruijn} and also applied in Chapter 2 of this thesis---to turn these contour integrals into practical approximations.
In contrast to the setting of Chapter 2, both the relevant saddle point and the analyticity radius tend to one as $n\to\infty$, which is a singular point of the integrand, in the setting with overdispersion.
For the proof of Theorem \ref{saddlepointThm}, we therefore modify the special saddle point method developed in Chapter 2 to account for this circumstance.
\begin{proof}
Our starting point is the probability generating function of the number of arrivals per time slot, given in \eqref{r0}, which is analytic for $|z|<1+1/b_n=:r$. Under Assumption \ref{as2}, we consider $\mu_Q$ as given in \eqref{t7}. We set
\begin{equation}
\label{a7}
g(z) = -{\rm ln }\,z+\frac{1}{s_n}\,{\rm ln }\Bigl[\tilde{A}^{(n)}(z)\Bigr] = -{\rm ln }\,z - \frac{a_n}{s_n}\,{\rm ln }\left(1+(1-z)b_n\right),
\end{equation}
to be considered in the entire complex plane with branch cuts $(-\infty,0]$ and $[r,\infty)$. The relevant saddle point $z_{\rm sp}$ is the unique zero $z$ of $g'(z)$ with $z\in(1,r_0)$. Since
\begin{equation}
\label{a8}
g'(z) = -\frac{1}{z} + \frac{\rho_n}{1+(1-z)b_n},
\end{equation}
this yields,
\begin{equation}
\label{a9}
1+(1-z_{\rm sp})b_n = \rho_n z_{\rm sp},\quad {\rm i.e., } \quad z_{\rm sp} = 1+\frac{1-\rho_n}{\rho_n+b_n}.
\end{equation}
We then find
\begin{equation}
\label{a10}
\mu_Q = \frac{s_n}{2\pi i} \int_{|z| = 1+\varepsilon} \frac{g'(z)}{z-1}\,\frac{\exp(s_n\,g(z))}{1-\exp(s_n\,g(z))}{\rm d} z,
\end{equation}
and we take here $1+\varepsilon = z_{\rm sp}$. There are no problems with the branch cuts since we consider $\exp(s_ng(z))$ with integer $s_n$. \\
We continue as in Chapter 2, Section 3 and thus we intend to substitute $z=z(v)$ in the integral in \eqref{a10}, where $z(v)$ satisfies
\begin{equation*}
\label{k1}
g(z(v)) = g(z_{\rm sp})-\tfrac12\,v^2\,g''(z_{\rm sp}) =: q(v)
\end{equation*}
on a range ${-}\tfrac12\delta_n \leq v\leq \tfrac12 \delta_n$ with $\delta_n \to 0$ as $n\to\infty$.
Note that, this range depends on $n$, whereas these bounds $\pm \tfrac{1}{2} \delta_n$ remained bounded away from zero in \cite{Janssen2015}.
This severely complicates the present analysis.
We consider the approximate representation
\begin{equation}
\label{k2}
\frac{-s_n\,g''(z_{\rm sp})}{2\pi i}\int_{-\tfrac12 \delta_n}^{\tfrac12 \delta_n}\frac{v}{z(v)-1}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))} {\rm d} v
\end{equation}
of $\mu_Q$. We have to operate here with additional care, since both the analyticity radius $r=1+1/b_n$ and the saddle point $z_{\rm sp}$ outside zero $r_0$ tend to 1 as $n\rightarrow\infty$. Specifically, proceeding under the assumptions that $(1-\rho_n)^2a_n$ is bounded while $a_n\rightarrow\infty$ and $b_n\geq 1$, see Assumption \ref{as2}, we have from \eqref{a9} that
\begin{equation}\label{a19}
z_{\rm sp}-1=\frac{1-\rho_n}{b_n+\rho_n} = \frac{1-\rho_n}{b_n} + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr),
\end{equation}
where the $O$-term is small compared to $(1-\rho_n)/b_n$ when $b_n\rightarrow\infty$. Next, we approximate $r_0$, using that $r_0>1$ satisfies
\begin{equation*}
\label{a20}
{-}{\rm ln}\, r_0 - \frac{\rho_n}{b_n}\, {\rm ln}\,(1+(1-r_0)b_n) = 0.
\end{equation*}
Write $r_0 = 1+u/b_n$, so that we get the equation
\begin{align*}
0 &= {-}{\rm ln}\,\left(1+\frac{u}{b_n}\right) - \frac{\rho_n}{b_n}\,{\rm ln }(1-u)\nonumber \\
\label{a21}
&= {-}\frac{u}{b_n}\Bigl(1-\rho_n-\tfrac12\Bigl(\frac{1}{b_n}+\rho_n\Bigr)u-\tfrac{1}{3}\Bigl(\frac{-1}{b^2_n}+\rho_n\Bigr)u^2+\cdots\Bigr),
\end{align*}
where we have used the Taylor expansion of ${\rm ln}(1+x)$ at $x=0$. Thus we find
\begin{equation*}
\label{a22}
u=\frac{2(1-\rho_n)}{\rho_n+1/b_n}+O(u^2) = 2(1-\rho_n)+O((1-\rho_n)^2)+O\Bigl(\frac{1-\rho_n}{b_n}\Bigr),
\end{equation*}
and so,
\begin{equation*}
\label{a23}
r_0 = 1+2\,\frac{1-\rho_n}{b_n}+O\Bigl(\frac{(1-\rho_n)^2}{b_n}\Bigr) + O\Bigl(\frac{1-\rho_n}{b^2_n}\Bigr).
\end{equation*}
In \eqref{k2} we choose $\delta_n$ so large that the integral has converged within exponentially small error using $\pm\delta_n$ as integration limits, and, at the same time, so small that there is a convergent power series
\begin{equation}
\label{a26}
z(v) = z_{\rm sp}+iv+ \sum_{k=2}^\infty c_k(iv)^k, \qquad \text{for } |v| \leq \tfrac12 \delta_n.
\end{equation}
To achieve these goals, we supplement the information on $g(z)$, as given by $\eqref{a7}-\eqref{a9}$, by
\begin{equation}
\label{a27}
g''(z)=\frac{1}{z^2}+\frac{\rho_nb_n}{(1+(1-z)b_n)^2},\quad g''(1) = 1+\rho_nb_n,\quad g''(z_{\rm sp}) =\frac{1}{z_{\rm sp}^2}\Bigl(1+\frac{b_n}{\rho_n}\Bigr).
\end{equation}
Now
\begin{equation*}
\label{a36}
\exp(s_n\,q(v)) = \exp(s_n\,g(z_{\rm sp}))\exp(-\tfrac12\,s_n\,g''(z_{\rm sp})\,v^2),
\end{equation*}
and
\begin{equation*}
\label{a37} s_n\, g''(z_{\rm sp})v^2 = s_n\,b_nv^2(1+o(1)) = a_n(b_n\,v)^2(1+o(1)).
\end{equation*}
Therefore, \eqref{k2} approximates $\mu_Q$ with exponentially small error when we take $\tfrac12 \delta_n$ of the order $1/b_n$.
We next aim at showing that we have a power series for $z(v)$ as in \eqref{a26} that converges for $|v|\leq\tfrac12\delta_n$ with $\tfrac12\delta_n$ of the order $1/b_n$.
\begin{lemma}
Let
\begin{equation*}
\label{a38}
r_n:=\frac{1}{2\,b_n}-(z_{\rm sp} -1 ),\quad m_n:= \tfrac{2}{3}\rho_nr_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}},
\end{equation*}
where we assume $r_n>0$. Then \eqref{a26} holds with real coefficients $c_k$ satisfying
\begin{equation}
\label{a39}
|c_k|\leq\frac{r_n}{m_n^k},\quad k=2,3,\ldots.
\end{equation}
\end{lemma}
\begin{proof}
We let
\begin{equation}
\label{a40}
G(z):=\frac{2(g(z)-g(z_{\rm sp}))}{g''(z_{\rm sp})(z-z_{\rm sp})^2}.
\end{equation}
Then $G(z_{\rm sp})=1$ and so we can write \eqref{k1} as
\begin{equation}
\label{a41}
F(z):=(z-z_{\rm sp})\sqrt{G(z)} = i v
\end{equation}
when $|z-z_{\rm sp}|$ is sufficiently small. Since $F(z_{\rm sp})=0$, $F'(z_{\rm sp})=1$, the B\"urmann-Lagrange inversion theorem implies validity of a power series as in \eqref{a26}, with real $c_k$ since $G(z)$ is positive and real for real $z$ close to $z_{\rm sp}$. We therefore just need to estimate the convergence radius of this series from below.
To this end, we start by showing that
\begin{equation}
\label{a42}
{\rm Re}[g''(z)] > \frac{4}{9}\,\rho_n^2\frac{b_n+\rho_n^{-1}}{b_n+\rho_n},\quad |z-z_{\rm sp}|\leq r_n.
\end{equation}
For this, we consider the representation
\begin{equation}
\label{a43}
G(z) = 2\int_{0}^1\int_0^1 \frac{g''(z_{\rm sp}+s\,t(z-z_{\rm sp}))}{g''(z_{\rm sp})} \,t{\rm d} s{\rm d} t.
\end{equation}
We have for $\zeta\in\mathbb{C}$ and $|\zeta-1|\leq 1/2b_{n}\leq 1/2$ from \eqref{a27} that
\begin{equation}
\label{a44}
{\rm Re}[g''(\zeta)] = {\rm Re}(1/\zeta^2) + \rho_nb_n\,{\rm Re}\Bigl[\Bigl(\frac{1}{1+(1-\zeta)b_n}\Bigr)^2\Bigr]\geq \tfrac{4}{9}(1+\rho_nb_n).
\end{equation}
To show the inequality in \eqref{a44}, it suffices to show that
\begin{equation}
\label{a45}
\min_{|\xi-1|\leq 1/2} {\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{4}{9}.
\end{equation}
The minimum in \eqref{a45} is assumed at the boundary $|\xi-1|=1/2$, and for a boundary point $\xi$, we write
\begin{equation*}
\label{a46}
\xi= 1+\tfrac12\cos\theta+\tfrac12 i \sin\theta, \quad 0\leq \theta\leq 2\pi,
\end{equation*}
so that
\begin{equation*}
\label{a47}
{\rm Re}\Bigl(\frac{1}{\xi^2}\Bigr) = \frac{1+\cos\theta+\tfrac{1}{4}\cos 2\theta}{(\tfrac{5}{4}+\cos\theta)^2}.
\end{equation*}
Now
\begin{equation*}
\label{a48}
\frac{{\rm d}}{d\theta} \Bigl[\frac{1+\cos\theta+\tfrac{1}{4}\cos2\theta}{(\tfrac{5}{4}+\cos\theta)^2}\Bigr] = \frac{\sin \theta\,(1-\cos \theta)}{4(\tfrac{5}{4}+\cos\theta)^3}
\end{equation*}
vanishes for $\theta=0,\pi,2\pi$, where ${\rm Re}(1/\xi^2)$ assumes the values $4/9$, 4, 4/9, respectively. This shows \eqref{a45}.
We use \eqref{a45} with $\xi=\zeta$ and with $\xi=1+(1-\zeta)b_n$, with
\begin{equation}
\label{a49}
\zeta = \zeta(s,t) = z_{\rm sp} + s t\,(z-z_{\rm sp}),\quad 0\leq s,\, t\leq 1,
\end{equation}
where we take $\zeta$ such that $|\zeta-1|\leq 1/2b_n$. It is easy to see from
$1<z_{\rm sp}<1+1/2b_n$ that $|\zeta-1|\leq 1/2b_n$ holds when $|z-z_{\rm sp}|\leq r_n=1/2b_n-(z_{\rm sp}-1)$. We have, furthermore, from \eqref{a9} that $0<g''(z_{\rm sp})\leq 1+b_n/\rho_n$. Using this, together with \eqref{a44} where $\zeta$ is as in \eqref{a49}, yields
\begin{equation*}
\label{a50}
{\rm Re}[G(z)] \leq \frac{4}{9}\,\frac{1+\rho_nb_n}{1+b_n/\rho_n}\,2\,\int_0^1\int_0^1 t\,{\rm d} s{\rm d} t = \tfrac{4}{9}\,\rho_n^2\,\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}
\end{equation*}
when $|z-z_{\rm sp}|\leq r_n$, and this is \eqref{a42}.
We therefore have from \eqref{a41} that
\begin{equation*}
\label{a51}
|F(z)|>r_n\cdot\frac{2}{3}\rho_n\sqrt{\frac{b_n+\rho_n^{-1}}{b_n+\rho_n}} = m_n,\quad |z-z_{\rm sp}|=r_n.
\end{equation*}
Hence, for any $v$ with $|v|\leq m_n$, there is exactly one solution $z=z(v)$ of the equation $F(z)-iv=0$ in $|z-z_{\rm sp}|\leq r_n$ by Rouch\'e's theorem. This $z(v)$ is given by
\begin{equation*}
\label{a52}
z(v) = \frac{1}{2\pi i}\,\int_{|z-z_{\rm sp}|=r_n} \frac{F'(z)\,z}{F(z)-iv}{\rm d} z,
\end{equation*}
and depends analytically on $v$, $|v|\leq m_n$. From $|z(v)-z_{\rm sp}|\leq r_n$, we can finally bound the power series coefficients $c_k$ according to
\begin{equation*}
\label{a53}
|c_k| = \Bigl|\frac{1}{2\pi i}\int_{|iv|=m_n} \frac{z(v)-z_{\rm sp}}{(iv)^{k+1}}{\rm d}(iv)\Bigr| \leq \frac{r_n}{m_n^k},
\end{equation*}
and this completes the proof of the lemma.
\end{proof}
\begin{remark}
We have $z_{\rm sp}-1=o(1/b_n)$, see \eqref{a19}, and so
\begin{equation*}
\label{a54}
r_n = \frac{1}{2b_n}(1+o(1)),\quad m_n = \frac{1}{3b_n}(1+o(1)),
\end{equation*}
implying that the radius of convergence of the series in \eqref{a26} is indeed of order $1/b_n$ (since we have assumed $b_n\geq 1$).
\end{remark}
We let $\delta_n=m_n$, and we write for $0\leq v\leq \tfrac12\delta_n$
\begin{equation*}
\label{a55}
\frac{v}{z(v)-1}+\frac{{-}v}{z({-}v)-1} = \frac{-2iv\,{\rm Im}(z(v))}{|z(v)-1|^2},
\end{equation*}
where we have used that all $c_k$ are real, so that $z(-v)=z(v)^*$, where $ ^*$ denotes the complex conjugate. Now from \eqref{a39} and realness of the $c_k$, we have
\begin{equation}
\label{a56}
{\rm Im}(z(v)) = v+\sum_{l=1}^\infty c_{2l+1}(-1)^l\,v^{2l+1} = v+O(v^3),
\end{equation}
and in similar fashion
\begin{equation}
\label{a57}
|z(v)-1|^2 = (z_{\rm sp}-1)^2+v^2+O((z_{\rm sp}-1)^2v^2) + O(v^4)
\end{equation}
when $0\leq v\leq \tfrac12\delta_n$. The order terms in \eqref{a56}-\eqref{a57} are negligible in leading order, and so we get for $\mu_{Q^{(n)}}$ via \eqref{k2} the leading order expression
\begin{equation*}
\label{a58}
\frac{{-}s_n\,g''(z_{\rm sp})}{2\pi i}\,\int_0^{\tfrac12\delta_n}\frac{{-}2iv^2}{(z_{\rm sp}-1)^2+v^2}\,\frac{\exp(s_n\,q(v))}{1-\exp(s_n\, q(v))}{\rm d} v.
\end{equation*}
We finally approximate $q(v) = g(z_{\rm sp})-\tfrac12 g''(z_{\rm sp})v^2$.
There is a $z_1$, $1\leq z_1\leq z_{\rm sp}$ such that
\begin{equation*}
\label{a59}
g(z_{\rm sp}) = {-}\tfrac12(z_{\rm sp}-1)^2\,g''(z_1),
\end{equation*}
and, see \eqref{a19} and \eqref{a27},
\begin{equation*}
\label{a60}
g''(z_1) = g''(z_{\rm sp}) + O((1-\rho_n)b_n).
\end{equation*}
Hence
\begin{align}
s_n\,q(v) &= {-}\tfrac12 s_n\,g''(z_{\rm sp})\,[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)b_ns_n(z_{\rm sp}-1)^2)\nonumber\\
&= {-}\tfrac12 s_n\,g''(z_{\rm sp})[(z_{\rm sp}-1)^2+v^2]+O((1-\rho_n)^2a_n),\label{a61}
\end{align}
where \eqref{a19} has been used and $a_nb_n = s_n(1+o(1))$ Therefore, the $O$-term in \eqref{a61} tends to 0 by our assumption that $(1-\rho_n)^2a_n$ is bounded. Thus, we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation}\label{a62}
\frac{s_n g''(z_{\rm sp})}{\pi} \int_{0}^{\tfrac12\delta_n}\frac{v^2}{(z_{\rm sp}-1)^2+v^2}\,
\frac{\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))}{1-\exp(-\tfrac12 g''(z_{\rm sp})s_n((z_{\rm sp}-1)^2+v^2))} {\rm d} v,
\end{equation}
When we substitute $t=v\sqrt{s_n\,g''(z_{\rm sp})/2}$ and extend the integration in \eqref{a62} to all $t\geq 0$ (at the expense of an exponentially small error), we get for $\mu_{Q^{(n)}}$ in leading order
\begin{equation*}
\label{a63}
\frac{1}{\pi}\,\sqrt{2\,s_n\,g''(z_{\rm sp})}\,\int_{0}^\infty \frac{t^2}{\tfrac12\beta_n^2}\,\frac{\exp({-}\tfrac12\beta^2_n-t^2)}{1-\exp({-}\tfrac12\beta^2_n-t^2)}{\rm d} t,
\end{equation*}
where
\begin{equation*}
\label{a64}
\beta^2_n = s_n\,g''(z_{\rm sp})(z_{\rm sp}-1)^2.
\end{equation*}
Now using \eqref{a9} and \eqref{a27}, we get the result of Theorem \ref{saddlepointThm}. A separate analysis of $\beta_n$ is provided in Subsection \ref{convRobust}.
\section{Numerical \& empirical studies}
A similar analysis, modeled after the one given in Chapter 2 gives under Assumption \ref{as1} the leading-order expression
\begin{equation}
\label{a65}
\frac{1}{z_{\rm sp} \pi}\int_0^\infty \frac{\beta_n/\sqrt{2}}{\tfrac12\beta_n^2+t^2}\,{\rm ln}(1-e^{-\tfrac12\beta_n^2-t^2}){\rm d} t
\end{equation}
for ${\rm ln}\,\mathbb{P}(Q^{(n)}=0)$. Observe that the quantity in \eqref{a65} is negative, but bounded away from ${-}\infty$ when $\beta_n$ is bounded away from 0.
Furthermore, we find for the variance of $Q^{(n)}$ the approximation
\begin{equation*}
\label{a66}
\frac{\beta_n^3/\sqrt{2}}{\pi}\frac{z_{\rm sp}+1}{(z_{\rm sp}-1)^2}\int_0^\infty \frac{t^2}{(\tfrac12 \beta_n+t^2)^2}\, \frac{\exp({-}\tfrac12\beta_n-t^2)}{1-\exp({-}\tfrac12\beta_n^2-t^2)}{\rm d} t.
\end{equation*}
\end{proof}
\noindent
Note that we can write \eqref{r1} as
\begin{equation*}
\label{ra1}
\mu_Q \approx \tilde{\sigma}_n\,\mathbb{E}[ M_{\beta_n}]\quad \text{and}\quad \sigma^2_Q \approx \tilde{\sigma}^2_n\, {\rm Var}\, M_{\beta_n}
\end{equation*}
with
\begin{equation}
\label{ra5}
\tilde{\sigma}_n = \beta_n \Bigl(\frac{b_n+\rho_n}{1-\rho_n}\Bigr).
\end{equation}
This robust approximation for $\mu_Q$ is suggestive of the following two properties that extend beyond the mean system behavior, and hold at the level of approximating the queue by $\sigma_n$ times the Gaussian random walk:
\begin{itemize}
\item[\rm (i)] At the process level, the space should be normalized with $\sigma_n$, as in \eqref{mm7}. The approximation \eqref{r1} suggests that it is better to normalize with $\tilde{\sigma}_n$. Although $\tilde \sigma_n\to\sigma_n$ for $n\to\infty$, the $\tilde \sigma_n$ is expected to lead to sharper approximations for finite $n$.
\item[\rm (ii)] Again at the process level, it seems better to replace the original hedge $\beta$ by the robust hedge $\beta_n$. This thus means that the original system for finite $n$ is approximated by a Gaussian random walk with drift $-\beta_n$. Apart from this approximation being asymptotically correct for $n\to \infty$, it is also expected to approximate the behavior better for finite $n$.
\end{itemize}
\subsection{Convergence of the robust hedge\label{convRobust}}
We next examine the accuracy of the heavy-traffic approximations for $\mu_Q$ and $\sigma^2_Q$, following Corollary \ref{abateThm} and Theorem \ref{saddlepointThm}. We expect the robust approximation to be considerably better than the classical approximation when $\beta_n$ and $\tilde{\sigma}_n$ differ substantially from their limiting counterparts. Before substantiating this claim numerically, we present a result on the convergence rates of $\beta_n$ to $\beta$ and $\tilde{\sigma}_n$ to $\sigma_n$.
\begin{proposition}\label{gammanProp}
Let $a_n,b_n$ and $s_n$ as in Assumption \ref{as2}. Then
\begin{equation}
\label{r3a}
\beta_n^2 = \beta^2\Bigl(1 - \frac{1}{1+b_n+\sigma_n/\beta}\Bigr).
\end{equation}
\end{proposition}
\begin{proof}
From \eqref{r2}, we have
\begin{align*}
\beta_n^2 &= s_n\Bigl(\frac{1-\rho_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{b_n}{\rho_n}\Bigr)= \frac{1}{s_n}\Bigl(\frac{s_n-a_nb_n}{b_n+1}\Bigr)^2\Bigl(1+\frac{s_n}{a_n}\Bigr)\nonumber\\
\label{x1}
&= \frac{1}{s_n}\frac{\beta^2\,a_nb_n(b_n+1)}{(b_n+1)^2}\Bigl(1+\frac{s_n}{a_n}\Bigr) = \beta^2\,\frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) =:\beta^2\,\bar{F}_n.
\end{align*}
Now,
\begin{align*}
\bar{F_n} &= \frac{b_n}{b_n+1}\,\Bigl(1+\frac{a_n}{s_n}\Bigr) = \frac{b_n}{b_n+1}+\frac{1}{b_n+1}\,\frac{a_nb_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\Bigl(1-\frac{a_nb_n}{s_n}\Bigr) = 1-\frac{1}{b_n+1}\,\frac{\beta\,\sigma_n}{s_n}\nonumber\\
&= 1-\frac{1}{b_n+1}\,\frac{1}{1+\frac{\mu_n}{\beta\sigma_n}} = 1-\frac{1}{b_n+1+\frac{1}{\beta}\sqrt{a_nb_n(b_n+1)}},
\end{align*}
which together with $\sigma_n^2=a_nb_n(b_n+1)$ proves the proposition.
\end{proof}
Note that $\beta_n$ always approaches $\beta$ from below. Also, \eqref{r3a} shows that $b_n$ is the dominant factor in determining the rate of convergence of $\beta_n$.
\begin{proposition}\label{sigmanProp}
Let $\tilde{\sigma}_n$ as in \eqref{ra5}. Then
\begin{equation*}
\tilde{\sigma}_n = \sigma_n + b_n\beta_n + O(1).
\end{equation*}
\end{proposition}
\begin{proof}
Straightforward calculations give
\begin{align*}
\tilde{\sigma}_n &= \beta_n\,\Bigl(\frac{s_nb_n+a_nb_n}{s_n-a_nb_n}\Bigr) \nonumber\\
&= \frac{\beta_n}{\beta}\,\frac{b_n}{\sigma_n}\,(s_n+a_n)
= \frac{\beta_n}{\beta}\,\sqrt{\frac{b_n}{a_n(b_n+1)}}\left(a_n(b_n+1)+\beta\sqrt{a_nb_n(b_n+1)}\right)\nonumber\\
&= \frac{\beta_n}{\beta}\left(\sqrt{a_nb_n(b_n+1)}+\beta b_n\right) = \frac{\beta_n}{\beta}\,\sigma_n + \beta_n b_n.
\end{align*}
Applying Proposition \ref{gammanProp} together with the observation
\begin{equation*}
\sigma_n \sqrt{1 - \frac{1}{1+b_n+\sigma_n/\beta}} = \sigma_n(1 + O(1/\sqrt{a_n}b_n)) = \sigma_n + O(1)
\end{equation*}
yields the result.
\end{proof}
In Figure \ref{fig:convHedge}, we visualize the convergence speed of both parameters in case $\mu_n=n$, $\sigma_n = n^\delta$ with $\delta=0.7$ and $\beta=1$. This implies $a_n = n/(n^{2\delta}-1)$ and $b_n = n^{2\delta}-1$.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.05,
xlabel = {$x$},
ylabel = {$\tilde{\beta}_n/\beta_n$},
y label style={at={(-0.09,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.052)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {Chapter_3/tikz/gamman.txt};
\addplot[thick,col4] table[x=n,y=d075] {Chapter_3/tikz/gamman.txt};
\addplot[thick,col5] table[x=n,y=d09] {Chapter_3/tikz/gamman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\beta_n$.}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0,
ymax = 1.7,
xlabel = {$x$},
ylabel = {$\tilde{\sigma}_n/\sigma_n$},
y label style={at={(-0.09,0.75)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 195,0.1)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick,col1] table[x=n,y=d06] {Chapter_3/tikz/sigman.txt};
\addplot[thick,col4] table[x=n,y=d075] {Chapter_3/tikz/sigman.txt};
\addplot[thick,col5] table[x=n,y=d09] {Chapter_3/tikz/sigman.txt};
\addplot[dashed] coordinates { (0,1) (200,1) };
\legend{$\delta = 0.6$, $\delta=0.75$, $\delta=0.9$};
\end{axis}
\end{tikzpicture}
\caption{Convergence of $\tilde{\sigma}_n$.}
\end{subfigure}
\caption{}
\label{fig:convHedge}
\end{figure}
We observe that $\beta_n$ starts resembling $\beta$ fairly quickly, as predicted by Proposition \ref{gammanProp}; $\tilde{\sigma}_n$, on the other hand, converges extremely slowly to its limiting counterpart. Since $\mu_Q$ and $\sigma^2_Q$ are approximated by $\tilde{\beta}_n$ and $\tilde{\sigma}_n^2$, multiplied by a term that remains almost constant as $n$ grows, the substitution of $\sigma_n$ by $\tilde{\sigma}_n$, is essential for obtaining accurate approximations, as we illustrate further in the next subsection.
\subsection{Comparison between heavy-traffic approximations}
We set $\mu_n=n$ and $\sigma^2_n=n^{2\delta}$ with $\delta>\tfrac{1}{2}$, so that $s_n = n+\beta n^{\delta}$, and $a_n =n/(n^{2\delta-1}-1)$ and $b_n = n^{2\delta-1}-1$.
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.609 & 0.343 & 0.246 & 0.363 & 1.002 & 0.835 & 0.978 \bigstrut[t] \\
10 & 0.683 & 0.535 & 0.400 & 0.551 & 1.239 & 1.063 & 1.216 \\
50 & 0.815 & 1.405 & 1.168 & 1.405 & 1.995 & 1.817 & 1.971 \\
100 & 0.855 & 2.113 & 1.824 & 2.105 & 2.445 & 2.270 & 2.420 \\
500 & 0.920 & 5.446 & 5.006 & 5.412 & 3.923 & 3.762 & 3.899 \bigstrut[b] \\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.6$.}
\label{gammaPoisson1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.550 & 0.462 & 0.284 & 0.479 & 1.162 & 0.896 & 1.130 \bigstrut[t]\\
10 & 0.587 & 0.852 & 0.521 & 0.855 & 1.570 & 1.213 & 1.528 \\
50 & 0.668 & 3.197 & 2.093 & 3.106 & 3.025 & 2.433 & 2.947 \\
100 & 0.700 & 5.561 & 3.784& 5.377 & 3.983 & 3.270 & 3.887\\
500 & 0.766 & 19.887 & 14.741 & 19.202 & 7.514 & 6.455 & 7.361 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=1$ and $\delta=0.8$.}
\label{gammaPoisson2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.949 & 11.532 & 11.306 & 11.495 & 3.634 & 3.559 & 3.602 \bigstrut[t] \\
10 & 0.961 & 17.565 & 17.268 & 17.548 & 4.474& 4.398 & 4.444 \\
50 & 0.979 & 46.368 & 45.869 & 46.418 & 7.241 & 7.168 & 7.218 \\
100 & 0.984 & 70.340 & 69.735 & 70.430 & 8.910 & 8.839 & 8.888 \\
500 & 0.991 & 184.900 & 183.989 & 185.108 & 14.422 & 14.357 & 14.404 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.6$.}
\label{gammaPoisson3}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrr|rrr|}
\hline
$s_n$ & $\rho_n$ & $\mu_Q$ & \eqref{h1} & \eqref{r1} & $\sigma_Q$ & \eqref{h1b} & \eqref{r4} \bigstrut \\
\hline
5 & 0.931 & 15.730 & 15.209 & 15.909 & 4.276 & 4.127 & 4.233 \bigstrut[t]\\
10 & 0.939 & 27.561 & 26.672 & 27.958 & 5.652 & 5.466 & 5.605 \\
50 & 0.955 & 100.660 & 97.967 & 102.070 & 10.760 & 10.476 & 10.698 \\
100 & 0.961 & 175.591 & 171.360 & 177.818 & 14.189 & 13.855 & 14.117 \\
500 & 0.971 & 638.097 & 626.346 & 644.105 & 26.963 & 26.490 & 26.864 \bigstrut[b]\\
\hline
\end{tabular}
\caption{Numerical results for the Gamma-Poisson case with $\beta=0.1$ and $\delta=0.8$.}
\label{gammaPoisson4}
\end{table}
Tables \ref{gammaPoisson1}-\ref{gammaPoisson4} present numerical results for various parameter values.
In these tables, we fixed $s_n$ to integer values, and use the associated value of $n$ in our calculations.
The exact values of the performance measures are calculated using the method in Appendix \ref{numprocs}.
Several conclusions are drawn from these tables. Observe that the heavy-traffic approximations based on the Gaussian random walk, \eqref{h1} and \eqref{h1b}, capture the right order of magnitude for both $\mu_Q$ and $\sigma_Q$. However, the values are off, in particular for small $s_n$ and relatively low $\rho_n := \mathbb{E}[A^{(n)}] / s_n$. The inaccuracy also increases with the level of overdispersion. In contrast, the approximations that follow from Theorem \ref{saddlepointThm}, \eqref{r1} and \eqref{r4} are remarkably accurate. Even for small systems with $s_n = 5$ or 10, the approximations for $\mu_Q$ are within 6$\%$ of the exact value for small $\rho_n$ and within $2\%$ for $\rho_n$ close to 1. For $\sigma_Q^2$, these percentages even reduce to $3\%$ and $1\%$, respectively. For larger values of $s_n$ these relative errors naturally reduce further. Overall, we observe that the approximations improve for heavily loaded systems, and the corrected approximations are particularly useful for systems with increased overdispersion.
\subsection{Capacity allocation in health care}
We next apply our model and robust approximations to real-life patient arrivals. We consider emergency patients who require diagnostic tests at the radiology department of a hospital. Green \cite{Green2004} points out that patients at the radiology department can be roughly categorized into three groups: Inpatients, outpatients and emergency patients. Inpatient and outpatient arrivals are relatively predictable as these are usually scheduled by appointment. Emergency patients, on the other hand, are inherently unpredictable: They typically require urgent care and therefore timely access to testing facilities, as well as a quick assessment of the test results. This leads to prioritization of emergency patients over the other two groups in such settings, so that they do not experience any delay caused by the groups of lower priority. However, patients from the same top-priority group can still cause considerable congestion. A careful evaluation of capacity allocation is hence required, bearing in mind that additional sophisticated pieces of medical equipment are very costly.
In the setting we study, capacity is defined by the number of time slots available to perform radiology tests on emergency patients in a given time period, which we set at 24 hours. As radiology tests are commonly performed in appointment slots of fixed length, the number of slots available per day is also indirectly fixed. In terms of our model parameters, see Section \ref{modelSection}, we have $s$ as the number of slots per day allocated to emergency patients, and $A(k)$ the number of test requests received by the department on day $k$. We omit the subscript $n$ in this section due to the absence of limits. Then $\mathbb{E}[Q]$ can be interpreted as the expected number of patients whose test is carried over to the next day. A more natural performance measure in this setting is the expected waiting time, namely the time between the physician's request and the actual start of the test. However, Little's law implies that there is a linear relation between the two, hence we choose to only evaluate $\mathbb{E}[Q]$.
The data set on which our empirical study is based originates from the emergency department of SKHospital, monitored over a period of 76 days from September to November 2012. We extracted information of ED patients referred to the radiology department by the ED physicians, which includes the time the test request was made and the exact test type performed. The two test types, X-ray and CT scans, are performed on different equipment and hence it makes sense to analyze their queueing processes separately.
We refer to test requests as arrivals. The empirical cumulative distribution functions of the number of arrivals per day, for each type, are depicted by the black lines in Figure \ref{fig:fittedHospital}. The sample means equal 69.81 and 17.47, for the X-ray and CT scans respectively, whereas the sample variances are 121.8 and 26.12. These values suggest that fitting a Poisson distribution is inappropriate, which is visually backed up by the fitted Poisson cdf, depicted in Figure \ref{fig:fittedHospital} by the red line. To strengthen this claim, we tested both samples for the Poisson assumption using the \emph{dispersion test}, see Appendix \ref{statproc}, and obtained $p$-values equal $7.01\cdot 10^{-3}$ and $3.57\cdot 10^{-3}$ respectively, which allow us to safely reject the Poisson hypothesis in both cases.
In search for a better distributional fit with the arrivals count, we resort to Gamma-Poisson mixtures. Here we employ the procedure in \cite{koolejongbloed}, which is basically a maximum log-likelihood method, to obtain estimates for the parameters $a$ and $b$. This yields
\begin{equation*}
\label{parameterEstimators}
\hat{a}_{\rm X-ray} = 95.68,\quad \hat{b}_{\rm X-ray} = 0.7297,\quad \hat{a}_{\rm CT} = 37.19,\quad \hat{b}_{\rm CT} = 0.4698.
\end{equation*}
Applying the bootstrapping method to the data and the fitted model, also described in the appendix of \cite{koolejongbloed}, returns p-values that equal 0.7354 and 0.2120 for X-ray and CT scans, respectively. Therefore, the null hypothesis, stating that the data originated from a Gamma-Poisson mixture, cannot be rejected. The cdfs of the fitted Gamma-Poisson distributions, plotted in Figure \ref{fig:fittedHospital}, give visual confirmation of this claim as well.
Naturally, we also compared the estimated densities to the empirical pdf of the data. However, these fail to give a convincing visual fit due to the relatively small sample size and are therefore omitted here.
\begin{figure}
\begin{center}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 40,
xmax = 110,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 109,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {Chapter_3/tikz/xray.txt};
\addplot[thick,col1] table[x=x,y=poisson] {Chapter_3/tikz/xray.txt};
\addplot[thick,col4] table[x=x,y=fitted] {Chapter_3/tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {Chapter_3/tikz/xray.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{X-ray}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.74]
\begin{axis}[
xmin = 5,
xmax = 32,
ymin = 0,
ymax = 1,
xlabel = {$x$},
ylabel = {$\mathbb{P}(A\leq x)$},
y label style={at={(-0.1,0.6)}},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 32,0.05)},anchor = south east},
yscale = 0.8,
xscale = 1
]
\addplot[thick] table[x=x,y=emp] {Chapter_3/tikz/ct.txt};
\addplot[thick,col1] table[x=x,y=poisson] {Chapter_3/tikz/ct.txt};
\addplot[thick,col4] table[x=x,y=fitted] {Chapter_3/tikz/ct.txt};
\addplot[thick] table[x=x,y=emp] {Chapter_3/tikz/xray.txt};
\addplot[thick] table[x=x,y=emp] {Chapter_3/tikz/ct.txt};
\legend{Empirical,Poisson,Gamma-Poisson};
\end{axis}
\end{tikzpicture}
\caption{CT scan}
\end{subfigure}
\end{center}
\caption{Empirical, fitted Poisson and fitted Gamma-Poisson cumulative distribution functions of the number of arrivals.}
\label{fig:fittedHospital}
\end{figure}
We now have clear evidence that both the X-ray and CT scan facilities face an overdispersed arrival stream. In our final step of the empirical study we now evaluate the accuracy of our performance measure of interest $\mathbb{E}[Q]$, and the consequences of assessing system performance while ignoring the presence of overdispersion. We take the following approach: Trivially, we need to choose $s> \mathbb{E}[A]$ in order for the system to be stable. Hence, we vary $s$ from 70 to 80 for X-rays and from 18 to 24 for CT scans and simulate the queue length process by sampling the number of requests per day from the actual arrival counts. The number of iterations performed is $10^8$ for each configuration, excluding a warm-up interval of length $10^7$ (days). Around the mean of $Q$ obtained from this simulation, we create a 95\% confidence interval. Next, we approximate the expected stationary queue length under two scaling rules. Assuming Poisson arrivals, the appropriate capacity allocation rule would be $s=\hat{\mu}+\beta\sqrt{\hat{\mu}}$, for some $\beta>0$. Our novel capacity sizing rule prescribes $s = \hat{\mu} + \beta\hat{\sigma} = \hat{a}\hat{b}+\beta\sqrt{\hat{a}\hat{b}(\hat{b}+1)}$. We compute the first approximation based on square-root safety capacity sizing by deducing $\beta$ for each $s$, which we substitute in $\mathbb{E}[Q^{\rm srs}] \approx \sqrt{\hat{\mu}}\,\mathbb{E}[M_{\beta}]$. Similarly, we obtain $\beta$ from the new rule, and plug this value, together with the fitted parameters $\hat{a}$ and $\hat{b}$, into \eqref{r1}. The results are given in Tables \ref{tab:simXRay} and \ref{tab:simCT}. The last column shows the 95\% relative error bound of the second approximation.
\begin{table}[h]
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q] \ (\pm $ conf. iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
70 & 0.997 & $328.313 \pm\ 6.6\cdot 10^{-2}$ & 186.664 & 324.231 & 325.221 & $9.6\cdot 10^{-3}$ \bigstrut[t]\\
71 & 0.983 & $45.073 \pm\ 1.0\cdot 10^{-2}$ & 24.946 & 45.290 & 45.308 & $5.4\cdot 10^{-3}$ \\
72 & 0.970 & $21.988 \pm\ 5.4\cdot 10^{-3}$ & 11.650 & 21.982 & 22.129 & $6.6\cdot 10^{-3}$ \\
73 & 0.956 & $13.546 \pm\ 3.6\cdot 10^{-3}$ & 6.847 & 13.455 & 13.649 & $7.8\cdot 10^{-3}$ \\
74 & 0.943 & $9.230 \pm\ 2.7\cdot 10^{-3}$ & 4.438 & 9.106 & 9.319 & $1.0\cdot 10^{-2}$ \\
75 & 0.931 & $6.655 \pm\ 2.1\cdot 10^{-3}$ & 3.031 & 6.513 & 6.731 & $1.2\cdot 10^{-2}$ \\
76 & 0.919 & $4.949 \pm\ 1.7\cdot 10^{-3}$ & 2.136 & 4.821 & 5.037 & $1.8\cdot 10^{-2}$ \\
77 & 0.907 & $3.755 \pm\ 1.4\cdot 10^{-3}$ & 1.534 & 3.650 & 3.861 & $2.8\cdot 10^{-2}$ \\
78 & 0.895 & $2.884 \pm\ 1.1\cdot 10^{-3}$ & 1.115 & 2.807 & 3.009 & $4.4\cdot 10^{-2}$ \\
79 & 0.884 & $2.230 \pm\ 1.0\cdot 10^{-3}$ & 0.816 & 2.183 & 2.374 & $6.5\cdot 10^{-2}$ \\
80 & 0.873 & $1.734 \pm\ 8.5\cdot 10^{-4}$ & 0.600 & 1.710 & 1.890 & $9.1\cdot 10^{-2}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for X-ray.}
\label{tab:simXRay}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|r|rrrr|r|}
\hline
$s$ & $\rho$ & $\mathbb{E}[Q]\ (\pm $ conf.iv.) & $\mathbb{E}[Q^{\rm srs}]$ & $\eqref{h1}$ & $\eqref{r1}$ & rel. error \bigstrut \\
\hline
18 & 0.970 & 22.116 $\pm\ 4.9\cdot 10^{-3}$ & 14.235 & 21.965 & 21.724 & $1.8\cdot 10^{-2}$ \bigstrut[t] \\
19 & 0.919 & 6.289 $\pm\ 1.7\cdot 10^{-3}$ & 3.640 & 5.941 & 6.040 & 4.0$\cdot 10^{-2}$ \\
20 & 0.873 & 3.101 $\pm\ 1.0\cdot 10^{-3}$ & 1.589 & 2.772 & 2.917 & 6.0$\cdot 10^{-2}$ \\
21 & 0.832 & 1.767 $\pm\ 6.6\cdot 10^{-4}$ & 0.800 & 1.507 & 1.658 & 6.1$\cdot 10^{-2}$ \\
22 & 0.794 & 1.066 $\pm\ 4.6\cdot 10^{-4}$ & 0.425 & 0.874 & 1.016 & 4.7$\cdot 10^{-2}$ \\
23 & 0.760 & 0.653 $\pm\ 3.3\cdot 10^{-4}$ & 0.230 & 0.522 & 0.649 & 7.1$\cdot 10^{-3}$\\
24 & 0.728 & 0.377 $\pm\ 2.3\cdot 10^{-4}$ & 0.124 & 0.315 & 0.424 & 1.2$\cdot 10^{-1}$ \bigstrut[b]\\
\hline
\end{tabular}%
\caption{Computational results for CT scan.}
\label{tab:simCT}
\end{table}
Based on these figures, we make several remarks. First, assuming the conventional regime (neglecting overdispersion) the approximation severely overestimates system performance in both queues. Because the square-root safety margin underestimates the stochastic fluctuations in the arrival process, the safety parameter $\beta$ is overestimated, which leads to a smaller magnitude of the approximated queue length process. This clearly illustrates the distorted view estimated performance characteristics can give under the wrong scaling.
Secondly, it is worth noticing the very good proximity of $\eqref{r1}$ to the values obtained through simulation. As we expected, the quality of the approximation deteriorates with increasing values of $s$. This makes sense, because we assumed the system to be in heavy traffic in the derivation of the formulas. What is surprising, on the other hand, is the fact that the approximation performs very well, even though the parameter $b$ is very small for these particular data sets, while the analysis of Theorem \ref{saddlepointThm} assumes that $a$ and $b$ are large. This demonstrates that the approximation scheme is remarkably robust and is able to capture the pre-limit behavior of these types of queues very well.
\section{Conclusion \& future research}
In this chapter, we proposed an adaptation to the square-root staffing rule for service systems facing overdispersed demand, using the bulk service queue as a vehicle for our analysis.
Subsequently, we derive two sets of asymptotic approximations for the scaled steady-state queue length moments for large arrival volumes.
The first set being based on the resemblance with the maximum of a Gaussian random walk, the second set being derived through a non-standard saddle point method, assuming arrivals follow a Gamma-Poisson mixture.
Numerical experiments indicated that our approximations capture the pre-limit behavior well under different order of overdispersion, and are robust against any parameter estimation errors.
Although our method provides a robust way to approximate and dimension queues with overdispersed arrival processes, we see some interesting directions for future research.
First, we accentuate that our model is a discrete time queueing model in which a deterministic amount of workload can we handled within each period.
This approach allowed us to use Pollaczek's formula as a starting point to obtain more refined asymptotic approximations for the performance indicators of the system.
In case we consider queueing models of birth-death-type, such as the $M/M/s$ queue, in the presence of overdispersion demand, different techniques are required to provide scaling limits and corresponding capacity allocation rules, see e.g.~\cite{maman}.
Although we expect that, just as in the novel scaling regimes of Chapter 2, the asymptotic behavior of the bulk service queue and the multi-server queueing models to be similar, this needs proper analysis and understanding.
Second, empirical work, see e.g. \cite{Avramidis:2004}, shows that in real-life settings, demand in consecutive time periods is often positively correlated, rather than independently distributed as assumed in this chapter.
This correlation structure obviously alters the queue's dynamics and presumably requires an adaptation of the square-root staffing rule as well, making it a worthwhile direction for further analysis.
Last, we have only considered the analysis of the queueing model in steady state.
Typical service systems however do not face a constant expected arrival rate, nor do they run infinitely long.
Henceforth, it would be interesting to study the influence of overdispersion on the transient dynamics of the queue and to investigate the capacity allocation problem in scenarios with time-varying demand.
The theory developed in this chapter may serve as a building block to tackle these more profound questions.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Proofs of convergence results}
\label{formalSec}
This section presents the details of the proof of Lemma \ref{gaussStep} and Theorem \ref{gaussianThm}, using the random walk perspective of the process $\{Q^{(n)}(k)\}_{k=0}^\infty$. This section is structured as follows. The next two lemmata are necessary for proving the first assertion of Theorem \ref{gaussianThm}, concerning the weak convergence of the scaled process to the maximum of the Gaussian random walk, which is summarized in Proposition \ref{prop6}. The two remaining propositions of this section show convergence of $\hat{Q}^{(n)}$ at the process level as well as in terms of the three characteristics.
Let us first fix some notation:
\begin{equation}
\label{b1}
Y^{(n)}_k := \hat{A}^{(n)}_k-\beta,\quad
S^{(n)}_k = \sum_{i=1}^k Y^{(n)}_i,
\end{equation}
with $S_0^{(n)} = 0$ and $k=1,2,...$. Then \eqref{mm6} can be rewritten as
\begin{equation}
\label{g5a}
\hat{Q}^{(n)} {\;\buildrel{d}\over= \;} \max_{k\geq 0} \Bigl\{{\textstyle \sum}_{i=1}^k Y^{(n)}_i\Bigr\} =: M_\beta^{(n)},
\end{equation}
Last, we introduce the sequence of independent normal random variables $Z_1,Z_2,\ldots$ with mean $\-\beta$ and unit variance 1, and
\begin{equation*}
M_\beta {\;\buildrel{d}\over= \;} \max_{k\geq 0} \{{\textstyle \sum}_{i=1}^k Z_i\}.
\end{equation*}
\subsection{Proof of Lemma \ref{gaussStep}}
\begin{proof}
We show weak convergence of the random variable $\hat{A}^{(n)}$, as defined in Section \ref{modelSection}, to a standard normal random variable. Since $\hat{\Lambda}_n$ is asymptotically standard normal, its characteristic function converges pointwise to the corresponding limiting characteristic function, i.e.
\begin{equation}
\label{g8}
\lim_{n\rightarrow\infty} \phi_{\hat{\Lambda}_n}(t) = \lim_{n\rightarrow \infty} {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{\Lambda_n}(t/\sigma_n) = {\rm e}^{{-}t^2/2},\qquad \forall t\in \mathbb{R}.
\end{equation}
Furthermore, by definition of $A^{(n)}$,
\begin{equation*}
\label{g9}
\phi_{A^{(n)}}(t) = \mathbb{E}\left[ \exp(\Lambda_n({\rm e}^{it}-1))\right] = \phi_{\Lambda_n}\left(-i({\rm e}^{it}-1)\right),
\end{equation*}
so that
\begin{equation}
\label{g10}
\phi_{\hat{A}_k^{(n)}}(t) = {\rm e}^{-i\mu_nt/\sigma_n}\,\phi_{A_k^{(n)}}(t/\sigma_n) = {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}\left(-i({\rm e}^{it/\sigma_n}-1)\right).
\end{equation}
Now fix $t\in\mathbb{R}$. By using
\begin{equation*}
\label{g11}
-i({\rm e}^{it/\sigma_n} - 1) = \frac{t}{\sigma_n} -\frac{it^2}{2\sigma_n^2} + O\left(t^3/\sigma_n^3\right),
\end{equation*}
we expand the last term in \eqref{g10},
\begin{equation*}
\label{g12}
\phi_{\Lambda_n}(t/\sigma_n) + \Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(t/\sigma_n) + O\Bigl(\Bigl(-\frac{i\,t^2}{2\sigma_n^2}+O\left(\frac{t^3}{\sigma_n^3}\right)\Bigr)^2\phi_{\Lambda_n}''\Big(\frac{t}{\sigma_n}\Big)\Bigr)
\end{equation*}
\begin{equation*}
\label{g13}
= \phi_{\Lambda_n}(t/\sigma_n) - \Bigl(\frac{i\,t^2}{2\sigma_n^2}+O\left(t^3/\sigma_n^3\right)\Bigr)
\phi_{\Lambda_n}'(\zeta)
\end{equation*}
for some $\zeta$ such that $|\zeta - t/\sigma_n| < |i(1-{\rm e}^{it/\sigma_n})-t/\sigma_n|$. Also,
\begin{align}
|\phi_{\Lambda_n}'(u)| &= \left|\frac{\delta}{{\rm d} u}\int_{-\infty}^\infty {\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| = \left|\int_{0}^{\infty} ix\,{\rm e}^{iux}{\rm d} F_{\Lambda_n}(x)\right| \nonumber\\
\label{g13a}
&\leq \int_{-\infty}^\infty |ix\,{\rm e}^{iux}|\,{\rm d} F_{\Lambda_n}(x) = \int_0^\infty x{\rm d} F_{\Lambda_n}(x) = \mu_n
\end{align}
for all $u\in\mathbb{R}$. Hence, by substituting \eqref{g10},
\begin{align}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| &= \left|{\rm e}^{-i\mu_nt/\sigma_n}\,\left(\frac{i\,t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right)\,\phi_{\Lambda_n}'(\zeta)\right|\nonumber\\
& \leq \left(\frac{t^2}{2\sigma_n^2}+O(t^3/\sigma_n^3)\right) |\phi_{\Lambda_n}'(\zeta)|\nonumber\\
& = \frac{\mu_n t^2}{\sigma_n^2} + O\left(\frac{\mu_nt^3}{\sigma_n^3}\right),
\label{g13b}
\end{align}
which tends to zero as $n\rightarrow \infty$ by our assumption that $\mu_n/\sigma_n^2\rightarrow 0$.
Finally,
\begin{equation*}
\label{g13c}
\left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-\tfrac12 t^2}\right| \leq \left| \phi_{\hat{A}_k^{(n)}}(t)-{\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n)\right| +
\left| {\rm e}^{-i\mu_nt/\sigma_n}\phi_{\Lambda_n}(t/\sigma_n) - {\rm e}^{-\tfrac12 t^2}\right|,
\end{equation*}
in which both terms go to zero for $n\rightarrow \infty$, by \eqref{g8} and \eqref{g13b}. Hence $\phi_{\hat{A}^{(n)}_k}(t)$ converges to ${\rm e}^{{-}t^2/2}$ for all $t\in\mathbb{R}$, so that we can conclude by L\'evy's continuity theorem that $\hat{A}_k^{(n)} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1)$.
\end{proof}
\subsection{Proof of Theorem \ref{gaussianThm}}
To secure convergence in distribution of $\hat{Q}^{(n)}$ to $M_\beta$, i.e. the maximum of a Gaussian random walk with negative drift, the first assertion of Theorem \ref{gaussianThm},
the following property of the sequence $\{Y_k^{(n)}\}_{n\in\mathbb{N}}$ needs to hold.
\begin{lemma}\label{uilemma}
Let $Y^{(n)}_k$ be defined as in \eqref{b1} with $\mu_n,\sigma_n^2 < \infty$ for all $n\in\mathbb{N}$. Then the sequence $\{(Y_k^{(n)})^+\}_{n\in\mathbb{N}}$ is uniform integrable, i.e.
\begin{equation*}
\label{g14}
\lim_{K\rightarrow\infty}\sup_n \mathbb{E}\Big[Y^{(n)\,+}_k|\mathbbm{1}_{\{|Y^{(n)\,+}_k|\geq K\}}\Big] = 0.
\end{equation*}
\end{lemma}
\begin{proof}
Because the sequence $\{Y^{(n)}_k\}_{k\in\mathbb{N}}$ is i.i.d. for all $n$, we omit the index $k$ in this proof. First, fix $K>0$ and note that
\begin{equation*}
\label{g15}
\mathbb{E}[|Y^{(n)+}|\mathbbm{1}{\{|Y^{(n)\,+}|\geq K\}}] = \mathbb{E}[Y^{(n)+}\mathbbm{1}{\{Y^{(n)+}\geq K\}}] = \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}].
\end{equation*}
This last expression can be bounded from above using the Cauchy-Schwarz inequality, so that
\begin{equation*}
\label{g16}
\mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}] \leq \mathbb{E}[ Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K)^{1/2}.
\end{equation*}
By the definition of $Y^{(n)}$, we know $\mathbb{E} [Y^{(n)}] = -\beta$ and ${\rm Var}\, Y^{(n)} = {\rm Var}\, A^{(n)} / \sigma_n^2 = 1$. Using this information, we find
\begin{equation*}
\label{g17}
\mathbb{E}[Y^{(n)\,2}] = {\rm Var}\, Y^{(n)} + (\mathbb{E}[Y^{(n)}])^2 = 1+\beta^2
\end{equation*}
and
\begin{align*}
\mathbb{P}(Y^{(n)}\geq K )&=\mathbb{P}(Y^{(n)}+\beta\geq K+\beta) \leq \mathbb{P}(|Y^{(n)}+\beta|\geq K+\beta)\nonumber\\
&\leq \frac{{\rm Var}\, Y^{(n)}}{(K+\beta)^2} = \frac{1}{(K+\beta)^2},
\end{align*}
where we used Chebyshev's inequality for the last upper bound. Therefore,
\begin{align*}
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[|Y^{(n)\,+}|\mathbbm{1}_{\{|Y^{(n)\,+}|\geq K\}}] &=
\lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)}\mathbbm{1}_{\{Y^{(n)}\geq K\}}]\nonumber\\
&\leq \lim_{K\rightarrow \infty} \sup_n \mathbb{E}[Y^{(n)\,2}]^{1/2}\,\mathbb{P}(Y^{(n)}\geq K )^{1/2}\nonumber\\
&\leq \lim_{K\rightarrow \infty} \frac{\sqrt{1+\beta^2}}{K+\beta} = 0.
\end{align*}
\end{proof}
By combining the properties proved in Lemma \ref{gaussStep} and \ref{uilemma} with Assumption \ref{as2}, the next result follows directly by \cite[Thm.~X6.1]{Asmussen2003}.
\begin{proposition}\label{maxRWprop}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}. Then
\begin{equation*}
\hat{Q}^{(n)}{\;\buildrel{d}\over\Rightarrow\;} M_\beta,\qquad {\rm as}\ n\rightarrow\infty.
\end{equation*}
\end{proposition}
Although Proposition \ref{maxRWprop} tells us that the properly scaled $Q^{(n)}$ converges to a non-degenerate limiting random variable, it does not cover the convergence of its mean, variance and the empty-queue probability. In order to secure convergence of these performance measures as well, we follow the approach similar to \cite{Sigman2011b}, using Assumptions \ref{as2} and \ref{as3}.
\begin{proposition}\label{prop6}
Let $\hat{Q}^{(n)}$ as in \eqref{g5a}, $\mu_n,\sigma_n^2 \rightarrow \infty$ such that both $\sigma_n^2/\mu_n\rightarrow \infty$ and $\mathbb{E}[\hat{A}^{(n)3}]$ $<\infty$. Then
\begin{align*}
\label{b16}
\mathbb{P}(\hat{Q}^{(n)}= 0)&\rightarrow \mathbb{P}(M_\beta = 0),\\
\mathbb{E} [\hat{Q}^{(n)}]&\rightarrow \mathbb{E} [M_\beta],\\
{\rm Var}\, \hat{Q}^{(n)}&\rightarrow {\rm Var}\, M_\beta,
\end{align*}
as $n\rightarrow\infty$.
\end{proposition}
\proof
First, we recall that $\hat{Q}^{(n)}{\;\buildrel{d}\over= \;} M_\beta^{(n)}$ for all $n\in\mathbb{N}$, so that $\mathbb{P}(\hat{Q}^{(n)} = 0) = \mathbb{P}(M_\beta^{(n)}=0)$, $\mathbb{E}[\hat{Q}^{(n)}]=\mathbb{E}[M_\beta^{(n)}]$ and ${\rm Var}\,\,\hat{Q}^{(n)}={\rm Var}\,\,M_\beta^{(n)}$ as defined in \eqref{b1}. Our starting point is Spitzer's identity, see \cite[p.~230]{Asmussen2003},
\begin{equation}
\label{b17}
\mathbb{E}[{\rm e}^{it M_\beta^{(n)}}] = \exp\Bigl( \sum_{k=1}^\infty \frac{1}{k} (\mathbb{E}[{\rm e}^{it(S^{(n)}_k)^+}]-1)\Bigr),
\end{equation}
with $S^{(n)}_k$ as in \eqref{b1}, and $M_\beta^{(n)}$ the all-time maximum of the associated random walk. Simple manipulations of \eqref{b17} give
\begin{align}
\label{y1}
{\rm ln}\,\mathbb{P}(M_\beta^{(n)} = 0) &= -\sum_{k=1}^\infty \frac{1}{k}\,\mathbb{P}(S^{(n)}_k > 0),\\
\label{y2}
\mathbb{E}[M_\beta^{(n)}] &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[S^{(n)\,+}_k] = \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x) {\rm d} x,\\
\label{y3}
{\rm Var}\, M_\beta^{(n)} &= \sum_{k=1}^\infty \frac{1}{k} \mathbb{E}[(S^{(n)\,+}_k)^2] =\sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}) {\rm d} x.
\end{align}
By Lemma \ref{gaussStep}, we know
\begin{equation*}
\label{y4}
\mathbb{P}(S^{(n)}_k > y) = \mathbb{P}\left( {\sum_{i=1}^k} Y^{(n)}_i > y \right) \rightarrow \mathbb{P}\left({\textstyle\sum_{i=1}^k} Z_i > y\right),
\end{equation*}
for $n\rightarrow \infty$, where the $Z_i$'s are independent and identically normally distributed with mean $-\beta$ and variance 1.
Because equivalent expressions to \eqref{y1}-\eqref{y3} apply to the limiting Gaussian random walk, it is sufficient to show that the sums converge uniformly in $n$, so that we can apply dominated convergence to prove the result.
We start with the empty-queue probability. To justify interchangeability of the infinite sum and limit, note
\begin{equation*}
\label{y5}
\mathbb{P}(S^{(n)}_k > 0) \leq \mathbb{P}(|S^{(n)}_k+k\beta| > k\beta )\leq \frac{k}{\beta^2k^2} = \frac{1}{\beta^2k},
\end{equation*}
where we used that $\mathbb{E}[ S^{(n)}_k] = k\mathbb{E} [Y^{(n)}_1] = -k\beta$ and ${\rm Var}\, S^{(n)}_k = k$ and apply Chebychev's inequality, so that
\begin{equation*}
\label{y6}
\sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) \leq \sum_{k=1}^\infty \frac{1}{\beta^2 k^2} < \infty, \qquad \forall n\in\mathbb{N}.
\end{equation*}
Hence,
\begin{align*}
\lim_{n\rightarrow\infty} {\rm ln}\,\mathbb{P}(\hat{Q}^{(n)}= 0) &= \lim_{n\rightarrow\infty} - \sum_{k=1}^\infty \frac{1}{k}\mathbb{P}(S^{(n)}_k > 0) = -\sum_{k=1}^\infty \frac{1}{k} \lim_{n\rightarrow\infty}\mathbb{P}(S^{(n)}_k > 0)\nonumber\\
&= -\sum_{k=1}^\infty \frac{1}{k} \mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > 0) = {\rm ln}\,\mathbb{P}(M_\beta = 0).
\end{align*}
Finding a suitable upper bound on $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>x) {\rm d} x$ and $\frac{1}{k}\int_0^\infty \mathbb{P}(\hat{Q}^{(n)}>\sqrt{x}) {\rm d} x$ requires a bit more work. We initially focus on the former, the latter follows easily. The following inequality from \cite{Nagaev1979} proves to be very useful:
\begin{equation}
\label{y8}
\mathbb{P}(\bar{S}(k)>y) \leq C_r\,\Bigl(\frac{k\,\sigma^2}{y^2}\Bigr)^2 + k\,\mathbb{P}(X>y/r),
\end{equation}
where $\bar{S}(k)$ is the sum of $k$ i.i.d. random variables distributed as $X$, with $\mathbb{E}[X] = 0$ and ${\rm Var}\,\, X=\sigma^2$, $y > 0$, $r>0$ and $C_r$ a constant only depending on $r$. We take $r=3$ for brevity in the remainder of the proof, although any $r>2$ will suffice. We analyze the integral in two parts, one for the interval $(0,k)$ and one for $[k,\infty)$. For the first part, we have
\begin{align}
\label{y9}
\int_0^k\mathbb{P}(S^{(n)}_k>x) {\rm d} x &=\int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > x+k\beta){\rm d} x\, \leq\, \int_0^k \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > k\beta){\rm d} x \nonumber\\
&= k\,\mathbb{P}({\textstyle \sum_{i=1}^k }\hat{A}^{(n)}_i > k\beta) \,\leq\, \frac{C_3}{k^2\beta^6} + k^2\mathbb{P}(\hat{A}^{(n)}> \tfrac{1}{3}k),
\end{align}
where we used \eqref{y8} in the last inequality.
Hence,
\begin{align}
\label{y10}
\sum_{k=1}^\infty\frac{1}{k}\, \int_0^k \mathbb{P}(S^{(n)}_k>x){\rm d} x &\leq \, \frac{C_3}{\beta^6}\sum_{k=1}^\infty k^{-3} +\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) \nonumber \\
&\leq C_1^*+\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k).
\end{align}
With the help of the inequality (see \cite{Sigman2011b}),
\begin{equation}
\label{y11}
(b-a)a\,\mathbb{P}(X>b) \leq \int_a^b x\,\mathbb{P}(X>x) {\rm d} x \qquad \text{\rm for } 0<a<b,
\end{equation}
we get by taking $a=(k-1)/3$ and $b=k/3$,
\begin{align}
\label{y12}
k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) &\leq \frac{9\,k}{k-1}\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x \nonumber \\
&\leq 18\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x,
\end{align}
for $k\geq 2$. Since the tail probability for $k=1$ is obviously bounded by 1, this yields
\begin{align}
\label{y13}
\sum_{k=1}^\infty k\,\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}k) &\leq 1+18\sum_{k=2}^\infty\int_{(k-1)/3}^{k/3} x\,\mathbb{P}(\hat{A}^{(n)}>x) {\rm d} x\nonumber\\
&\leq 1+ 18\int_{0}^{\infty} x\,\mathbb{P}(\hat{A}^{(n)}>x){\rm d} x \leq 1+18\,\mathbb{E}[\hat{A}^{(n)2}] < \infty,
\end{align}
since $\hat{A}^{(n)}$ has finite variance by assumption. This completes the integral over the first interval. For the second part, we use \eqref{y8} again to find
\begin{align}
\label{y14}
\int_k^\infty \mathbb{P}(S^{(n)}_k>x)dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)} > x+k\beta)dx \leq \int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)} > x){\rm d} x\nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{x^6} {\rm d} x + k\int_k^\infty \mathbb{P}(\hat{A}^{(n)} >\tfrac{1}{3}x){\rm d} x\nonumber \\
&= \frac{5 C_3}{k^3}+ k\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}x) {\rm d} x.
\end{align}
So,
\begin{equation}
\label{y15}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>x){\rm d} x \leq C_2^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) {\rm d} x,
\end{equation}
for some constant $C_2^*$. Last, we are able to upper bound the second term in \eqref{y15} by Tonelli's theorem:
\begin{align}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}_i>\tfrac{1}{3}x) dx &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}x) {\rm d} x \nonumber\\
\label{y16}
&\leq 9\int_0^\infty y\mathbb{P}(\hat{A}^{(n)}>y) dy = 9\mathbb{E}[\hat{A}^{(n)2}] < \infty.
\end{align}
Combining the results in \eqref{y10}, \eqref{y13}, \eqref{y15} and \eqref{y16}, we find
\begin{equation*}
\label{y17}
\sum_{k=1}^\infty \frac{1}{k} \int_0^\infty \mathbb{P}(S^{(n)}_k>x){\rm d} x < \infty,
\end{equation*}
and thus
\begin{align*}
\lim_{n\rightarrow\infty} \mathbb{E}[\hat{Q}^{(n)}] &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > x){\rm d} x \nonumber\\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > x){\rm d} x = \mathbb{E} [M_\beta].
\end{align*}
Finally, we show how the proof changes for the convergence of ${\rm Var}\, \hat{Q}^{(n)}$. The expressions for $\mathbb{E} [\hat{Q}^{(n)}]$ and ${\rm Var}\, \hat{Q}^{(n)}$ in \eqref{y1} and \eqref{y2} only differ in the term $\sqrt{x}$. Hence only minor modifications are needed to also prove convergence of the variance. Note that boundedness of the integral over the interval $(0,k)$ in \eqref{y9}-\eqref{y13} remains to hold when substituting $\sqrt{x}$ for $x$. \eqref{y14} changes into
\begin{align*}
\label{y18}
\int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x})dx &=\int_k^\infty \mathbb{P}({\textstyle \sum_{i=1}^\infty}\hat{A}^{(n)}_i > \sqrt{x}+k\beta){\rm d} x \nonumber \\
&\leq C_3\int_k^\infty \frac{k^2}{(\sqrt{x}+k\beta)^6} dx + k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
&\leq \frac{C_4^*}{k}+ k\,\int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x,
\end{align*}
for some constant $C_4^*$, so that
\begin{equation*}
\sum_{k=1}^\infty \frac{1}{k} \int_k^\infty \mathbb{P}(S^{(n)}_k>\sqrt{x}){\rm d} x \leq C_4^* + \sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x.
\end{equation*}
Lastly, we have
\begin{align*}
\sum_{k=1}^\infty \int_k^\infty \mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x &\leq \int_1^\infty x\mathbb{P}(\hat{A}^{(n)}>\tfrac{1}{3}\sqrt{x}) {\rm d} x \nonumber\\
\label{y17a}
&\leq 18\int_0^\infty y^2\mathbb{P}(\hat{A}^{(n)}>y) {\rm d} y = 18\,\mathbb{E}[\hat{A}^{(n)3}] < \infty.
\end{align*}
Therefore the sum describing the variance is also uniformly convergent in $n$, so that interchanging of infinite sum and limit is permitted and
\begin{align*}
\lim_{n\rightarrow\infty} {\rm Var}\,\,\hat{Q}^{(n)} &= \lim_{n\rightarrow\infty} \sum_{k=1}^\infty \frac{1}{k}\int_0^\infty \mathbb{P}(S^{(n)}_k > \sqrt{x}){\rm d} x \nonumber \\
&= \sum_{k=1}^\infty \frac{1}{k} \int_0^\infty\mathbb{P}({\textstyle\sum_{i=1}^k} Z_i > \sqrt{x}){\rm d} x = {\rm Var}\, M_\beta.
\end{align*}
\section{Numerical procedures}\label{numprocs}
An alternative characterization of the stationary distribution is based on the analysis in \cite{Boudreau1962} and considers a factorization in terms of (complex) roots:
\begin{equation*}
\label{t9}
Q^{(n)}(w) = \frac{(s_n-\mathbb{E} [A^{(n)}])(w-1)}{w^{s_n}-\tilde{A}^{(n)}(w)}\,\prod_{k=1}^{s_n-1} \frac{w-z^n_k}{1-z^n_k},
\end{equation*}
where $z_1^n,z_2^n...,z_{s_n-1}^n$ are the $s_n-1$ zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$, in $|z|<1$, yielding
\begin{equation*}
\label{c2}
\mu_Q = \frac{\sigma_n^2}{2(s_n-\mu_n)}-\frac{s_n-1+\mu_n}{2} + \sum_{k=1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c3}
\mathbb{P}(Q^{(n)}=0) = \frac{s_n-\mu_A}{\tilde{A}^{(n)}(0)}\prod_{k=1}^{s_n-1}\frac{z^n_k}{z^n_k-1},
\end{equation*}
which for our choice of $\tilde{A}^{(n)}(z)$ becomes
\begin{equation*}
\label{c4}
\mu_Q = \frac{a_nb_n(b_n+1)}{2\beta\sqrt{a_n}b_n}-\frac{2a_nb_n+\beta\sqrt{a_nb_n(b_n+1)}-1}{2}+\sum_{k=1}^{s_n-1} \frac{1}{1-z^n_k},
\end{equation*}
\begin{equation*}
\label{c5}
\mathbb{P}(Q^{(n)}=0) = \beta \sqrt{a_nb_n(b_n+1)}(1+b_n)^{a_n}\prod_{k=1}^{s_n-1} \frac{z^n_k}{z^n_k-1},
\end{equation*}
where $z_1,...,z_{s_n-1}$ denote the zeros of $z^{s_n} - \tilde{A}^{(n)}(z)$ in $|z|<1$. A robust numerical procedure to obtain these zeros is essential for a base of comparison. We discuss two methods that fit these requirements. The first follows directly from \cite{Janssen2005}. \\
\begin{lemma}\label{fixedIterLemma}
Define the iteration scheme
\begin{equation}
\label{c6}
z_k^{n,l+1} = w^n_k [\tilde{A}^{(n)}(z_k^{n,l})]^{1/s_n},
\end{equation}
with $w^n_k = {\rm e}^{2\pi ik/s_n}$ and $z_k^{n,0}=0$ for $k=0,1,\ldots,s_{n-1}$. Then $z_k^{n,l} \rightarrow z_k^n$ for all $k=0,1,...,s_n-1$ for $l\rightarrow \infty$.
\end{lemma}
\begin{proof}
The successive substitution scheme given in \eqref{c6} is the fixed point iteration scheme described in \cite{Janssen2005}, applied to the pgf of our arrival process. The authors show that, under the assumption of $\tilde{A}^{(n)}(z)$ being zero-free within $|z|\leq 1$, the zeros can be approximated arbitrarily closely, given that the function $[\tilde{A}^{(n)}(z)]^{1/s_n}$ is a contraction for $|z|\leq 1$, i.e.
\begin{equation*}
\label{c7}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| < 1.
\end{equation*}
In our case,
\begin{align}
\label{c8}
\Bigl|\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}\Bigr| = \Bigl|\frac{{\rm d}}{{\rm d} z}\left(1+(1-z)b_n\right)^{-a_n/s_n}\Bigr| = \frac{a_nb_n}{s_n}\Bigl|1+(1-z)b_n\Bigr|^{-a_n/s_n-1},
\end{align}
where $a_nb_n/s_n = \rho_n$ is close to, but less than 1 and
\begin{align*}
\label{c9}
|1+(1-z)b_n| \geq |1+b_n|-|z|b_n = 1+(1-|z|)b_n \geq 1,
\end{align*}
when $|z|\leq 1$. Hence the expression in \eqref{c8} is less than 1 for all $|z|\leq 1$. Evidently, $\tilde{A}^{(n)}(z)$ is also zero-free in $|z|\leq 1$. Thus \cite[Lemma~3.8]{Janssen2005} shows that $z_k^{n,l}$ as in \eqref{c6} converges to the desired roots $z^n_k$ for all $k$ as $l$ tends to infinity.
\end{proof}
\begin{remark}
The asymptotic convergence rate of the iteration in \eqref{c6} equals \\
\noindent $\frac{{\rm d}}{{\rm d} z}[\tilde{A}^{(n)}(z)]^{1/s_n}$ evaluated at $z=z_k^n$. Hence, convergence is slow for zeros near 1 and fast for zeros away from 1.
\end{remark}
A different approach is based on the B\"urmann-Lagrange inversion formula.
\begin{lemma}\label{BLLemma}
Let $w^n_k = e^{2\pi ik/s_n}$ and $\alpha_n = a_n/s_n$. Then the zeros of $z^{s_n}-\tilde{A}^{(n)}(z)$ are given by
\begin{equation*}
z_k^n = \sum_{l=1}^\infty \frac{1}{l!}\,\frac{\beta[l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{b_n+1}{b_n}\Bigl(\frac{b_n}{(b_n+1)^{\alpha_n+1}}\Bigr)^l (w_k^n)^l,
\end{equation*}
for $k=0,1,...,s_n-1$.
\end{lemma}
\begin{proof}
Note that we are looking for $z$'s that solve
\begin{equation*}
\label{c10}
z\,[\tilde{A}^{(n)}(z)]^{-1/s_n} = z\left(1+(1-z)b_n\right)^{a_n/s_n} = w,
\end{equation*}
where $w = w_k = {\rm e}^{2\pi i k/s_n}$. The B\"urmann-Lagrange formula for $z=z(w)$, as can be found in \cite[Sec.~2.2]{debruijn} for $z=z(w)$ is given by
\begin{align*}
z(w) &= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(\frac{z}{z(1+(1-z)b_n)^{a_n/s_n}}\right)^l\right]_{z=0}\,w^l\nonumber\\
\label{c11}
&= \sum_{l=1}^\infty \frac{1}{l!}\,\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[\left(1+(1-z)b_n)^{-l\,a_n/s_n}\right)\right]_{z=0}\,w^l.
\end{align*}
Set $\alpha_n = a_n/s_n$. We compute
\begin{equation*}
\label{c1}
\left(\frac{{\rm d}}{{\rm d} z}\right)^{l-1}\left[ (1+(1-z)b_n)^{-l\alpha_n}\right]_{z=0} = \frac{\beta(l\alpha_n+l-1)}{\beta(l\alpha_n)}\,\frac{1+b_n}{b_n}\,\left(\frac{b_n}{(1+b_n)^{\alpha_n+1}}\right)^l.
\end{equation*}
With $c_n = b_n/(1+b_n)^{\alpha_n+1}$ and $d_n = (1+b_n)/b_n$, we thus have
\begin{equation*}
\label{c13}
z(w) = d_n\,\sum_{l=1}^\infty \frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} c_n^l\,w^l.
\end{equation*}
By Stirling's formula
\begin{equation*}\label{c14}
\frac{\beta(l\alpha_n+l-1)}{\beta(l+1)\beta(l\alpha_n)} = \frac{D}{l\sqrt{l}}\left(\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\right)^l,
\end{equation*}
where $D=\alpha_n^{1/2}(\alpha_n+1)^{-3/2}(2\pi)^{-1/2}$. Now,
\begin{equation*}
\label{c15}
\frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\, c_n = \frac{(\alpha_n+1)^{\alpha_n+1}}{\alpha_n^{\alpha_n}}\cdot \frac{b_n}{(1+b_n)^{\alpha_n+1}} = \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation*}
This determines the radius of convergence $r_{\rm BL}$ of the above series for $z(w)$:
\begin{equation}
\label{c16}
\frac{1}{r_{\rm BL}} := \left(\frac{b_n+\rho_n}{b_n+1}\right)^{\rho_n/b_n + 1}\left(\frac{1}{\rho_n}\right)^{\rho_n/b_n}.
\end{equation}
The derivative with respect to $\rho_n$ of the quantity
\begin{equation}
\label{c17}
\left(1+\frac{\rho_n}{b_n}\right) {\rm ln }\left(\frac{b_n+\rho_n}{b_n+1}\right)+\frac{\rho_n}{b_n}\,{\rm ln}\left(\frac{1}{\rho_n}\right)
\end{equation}
is given by
\begin{equation*}
\label{c18}
\frac{1}{b_n}{\rm ln }\Bigl(\frac{b_n+\rho_n}{b_n\rho_n+\rho_n}\Bigr) > 0
\end{equation*}
for $0<\rho_n<1$ and $b_n>0$. Furthermore, the quantity in \eqref{c17} vanishes at $\rho_n=1$ and is therefore negative for $0<\rho_n<1$ and $b_n>0$.
\begin{remark}
The formula for the radius of convergence in \eqref{c16} clearly shows the decremental effect of both having a large $b_n$ and of having $\rho_n$ close to 1. The quantities $\beta(l\alpha+l-1)/(\beta(l+1)\beta(l\alpha))$ in the power series for $z(w)$ are not very convenient for recursive computation, although normally $\alpha_n = a_n/s_n$ is a rational number.\end{remark}
\end{proof}
\section{Statistical procedures}\label{statproc}
To calibrate our model to real data, we now discuss some statistical procedures to show the presence of overdispersion and to estimate the parameters of the mixed Gamma-Poisson distribution. Let $x_1,...,x_n$ denote the observed arrival counts in consecutive time slots. These observations can be interpreted as realizations of the random variables $A_1,...,A_N$, and
\begin{equation*}
\bar{a}_N=\frac{1}{N}\sum_{i=1}^N x_i, \qquad \bar{s}_N^2 = \frac{1}{N-1}\sum_{i=1}(x_i-\bar{x}_i)^2,
\end{equation*}
the sample mean and variance with equivalent random variables $\bar{A}_N$ and $S_N^2$, respectively. Several tests with null hypothesis that $x_1,...,x_N$ originate from a (constant rate) Poisson distribution were discussed by \cite{Brown2002}. We mention two of them. The first is frequently referred to as the \emph{dispersion test}, and is based on the test statistic
\begin{equation*}
\label{dispTest}
D_N = \frac{(N-1)S_N^2}{\bar{A}_N},
\end{equation*}
which is approximately chi-squared distributed with $N-1$ degrees of freedom. When using a significance level $\alpha$, the critical value is equal to the $(1-\alpha)$-th quantile of chi-squared distribution $\chi^2_{N-1,1-\alpha}$. The second test, which is also used in \cite{koolejongbloed}, involves the test statistic
\begin{equation*}
\label{NStest}
T_N = \sqrt{N/2}\,\Bigl(\frac{S_N^2}{\bar{A}_N}-1\Bigr),
\end{equation*}
which is known as the Neyman-Scott test statistic. Under the null hypothesis $T_N$ tends to a standard normal random variable for large $N$. Hence both test statistics rely on the ratio of the sample variance and sample mean, which should be 1 if $A_1,...,A_N$ are indeed i.i.d. Poisson distributed. Excessive values of $D_N$ and $T_N$ therefore raise the suspicion of overdispersed arrivals.
Once either (or both) of the Poisson tests rejects the hypothesis of arrivals originating from a unicomponent Poisson process, we fit the data to the Gamma-Poisson mixture. Note that if we assume $A_i$ to be distributed as a Poisson random variable with random rate $\Lambda_i$, which is in turn Gamma distributed with parameters $a$ and $1/b$, then $A_i$ is in fact a negative binomial random variable with parameters $r = a$ and $p=b/(b+1)$. Finding estimators $\hat{a}$ and $\hat{b}$ therefore is equivalent to fitting a negative binomial distribution to the data to obtain $\hat{r}$ and $\hat{p}$, followed by retrieving $\hat{a} = \hat{r}$ and $\hat{b} = \hat{p}/(1-\hat{p})$. We proceed by applying the maximum likelihood estimation method described in \cite{koolejongbloed} to find $\hat{r}$ and $\hat{p}$. This method prescribes to set $\hat{r}$ to be the value of $r$ for which the \emph{profile loglikelihood function} defined by
\begin{equation*}
L(r) = \frac{1}{N}\,\sum_{i=1}^N\sum_{j=1}^{a_i} {\rm ln}(r+j+1)+r\,{\rm ln}\,r -(r+\bar{a}_N)\,{\rm ln}(r+\bar{a}_N),
\end{equation*}
is attained. Subsequently, $\hat{p} = \hat{r}/(\hat{r}+\bar{a}_N)$, so that $\hat{a} = \hat{r}$ and $\hat{b} = \hat{r}/\bar{a}_N$.
Finally, given the estimators $\hat{a}$ and $\hat{b}$, we need statistical evidence that the obtained Poisson mixture indeed fits the data reasonably well. Here we again cite \cite{koolejongbloed}, who give a method to retrieve the $p$-value for the goodness-of-fit based on bootstrap and Monte-Carlo simulation. In our experiments, we work with $10^6$ replications of the Monte-Carlo simulation to obtain the approximated $p$-value. We refer to the appendix of \cite{koolejongbloed} for further details on this method.
\resettocdepth
\end{subappendices}
\chapter{Retrial queues in the QED regime}
\begin{chapterstart}
Large-scale queueing systems with retrying customers are intrinsically hard to evaluate analytically.
We in this chapter explore and extend the asymptotic approximation technique proposed by Avram et al.~\cite{Avram2013}, that is able to characterize the impact of slow retrials in the QED regime, in three queueing models.
The technique evolves around a fixed-point equation that quantifies the increased inflow due to retrials implicitly.
We translate this fixed-point method into a powerful and elegant dimensioning procedure that is able to deal with both stationary and time-varying demand.
\end{chapterstart}
\begin{flushright}
Based on \\
\textbf{Delayed workload shifting in many-server systems}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen \& Fiona Sloothaak}\\
In \textit{SIGMETRICS Performance Evaluation Review, 43(2), 10--12 (2015)}\\
and \\
\textbf{Cloud provisioning in the QED regime}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen \& Fiona Sloothaak}\\
In \textit{Proceedings of the 9th EAI International Conference on Performance Evaluation Methodologies and Tools, 180--187 (2016)}
\end{flushright}
\newpage
\section{Introduction}
\textbf{Retrial queues.}
In the previous chapters, we analyzed queueing systems in which all arriving customers join the queue and stay until eventually completing service with one of the servers.
From a practical perspective though, these assumptions are questionable.
For instance, in call centers, customer impatience is known to play a crucial role in the queueing dynamics, see e.g.~\cite{Gans2003,Brown2005,Zeltyn2005}.
Similar features are also seen in health care \cite{Dai2012,Armony2015}.
However, impatience may not be the only cause of customers leaving the system without being seen by a server.
Physical constraints may force system managers to apply some sort of admission policy.
The simplest example of such admission control is the busy-signal in call centers, in which arriving customers finding all servers busy are simply discarded.
But more elaborate strategies can be considered.
A straightforward relaxation of the busy-signal policy is to allow a finite amount of waiting space, and block customers who find a full waiting room upon arrival.
Many other options, such as probabilistic and dynamic admission control policies may be considered, see e.g.~\cite{Janssen2013,Armony2004} and references therein.
Since customers arrive to the system for the purpose of getting assistance from one of the servers, it is reasonable to assume that these refused customers retry getting access tot the system at a later point in time.
In fact, retrials are widely observed in telecommunication systems, see e.g.~\cite{Cohen1957,Falin1997,Mandelbaum2000,Aguir2008}, and customers typically repeat their attempt until successful.
Naturally, retrials have a detrimental effect on the performance of the queueing system in terms of QoS, compared to the setting in which blocked customers do not return.
Hence, one needs to account for their impact in both performance analysis and the staffing decisions.
Unfortunately, the modeling of retrials is analytically challenging \cite{Cohen1957,Falin1997}, and numerical approaches become computationally infeasible as the number of servers increases, which is precisely the regime we are interested in.
We therefore aim to tackle the performance analysis of such retrial systems in an asymptotic manner.
We do so through a clever technique that was recently documented by Avram et al.~\cite{Avram2013}.\\
\\*
\textbf{Fixed-point equation.}
In this chapter, we will show how the asymptotic approximation technique of \cite{Avram2013} can be extended to more complex large-scale retrial queueing systems in the QED regime.
In \cite{Avram2013}, the authors study the $M/M/s/s$ queue with slow retrials.
That is to say, customers retry only after a (stochastic) delay period that is relatively long compared to the service time.
Under this assumption, the authors combine QED limits with a fixed-point equation, which characterizes the impact of retrials implicitly.
We summarize and reformulate their main ideas here for completeness.
Consider the standard $M/M/s/s$ queue with arrival rate $\lambda$ and service rate $\mu=1$, so that the offered load is $R=\lambda$.
Customers finding upon arrival all $s$ servers busy retry after a stochastic delay with mean $1/\delta$.
The first ingredient of the method is Cohen's equation.
This result, first reported by Cohen \cite{Cohen1957}, says that the stationary distribution of a $M/M/s/s$ queue with retrials converges as $1/\delta \to\infty$ to that of a $M/M/s/s$ queue with increased arrival rate $R+\Omega$, where $\Omega$ is the unique positive solution to
\begin{equation}
\Omega = \left( R+\Omega \right)\, B(R+\Omega,s),
\label{eq:cohens_equation}
\end{equation}
where $B(R,s)$ denotes the blocking probability in the $M/M/s/s$ queue with offered load $R$, i.e.~the Erlang-B formula:
\begin{equation}
B(R,s) := \frac{ R^s / s! }{\sum_{k=0}^s R^k/k!} = \frac{\mathbb{P}({\rm Pois}(R) = s)}{\mathbb{P}({\rm Pois}(R) \leq s)}.
\end{equation}
Equation \eqref{eq:cohens_equation} essentially equates the arrival volume generated by retrials, given by definition on the left-hand side, to the right-hand side which quantifies this volume as a fraction of customers blocked times the increased arrival volume.
Indeed, the retrial stream can for long retrial times be considered as independent from the primary arrival stream, yielding a thinned Poisson process \cite{Cohen1957}.
The second crucial observation is that under QED scaling, i.e. $s = R+\beta\sqrt{R} + o(\sqrt{R})$, we have
\begin{equation}
\sqrt{R}\cdot B( R, R+\beta\sqrt{R} ) \to \frac{\varphi(\beta)}{\Phi(\beta)} =: f_0(\beta),
\label{eq:erlangb_limit}
\end{equation}
as $R\to\infty$ for all $\beta\in\mathbb{R}$, see e.g.~Lemma 1 of \cite{Avram2013} or the proof of Proposition 1.1 in this thesis.
Hence, we heuristically deduce that $\Omega = \alpha\sqrt{R}$ for some $\alpha > 0$ and denote $R_{\rm tot} = R + \Omega$ -- a rigorous argument can be found in \cite{Avram2013}.
Rewrite $R = R_{\rm tot}-\alpha \sqrt{R_{\rm tot}} + o(\sqrt{R_{\rm tot}})$ and note that $R_{\rm tot} = O(R)$.
Then \eqref{eq:cohens_equation} becomes
\begin{equation}
\alpha \sqrt{R_{\rm tot}} = R_{\rm tot} \cdot B \big( R_{\rm tot}, R_{\rm tot}+(\beta-\alpha) \sqrt{R_{\rm tot}} \big) + o(\sqrt{R_{\rm tot}}).
\label{eq:cohen_equation_filled}
\end{equation}
Dividing both sides of \eqref{eq:cohen_equation_filled} by $\sqrt{R_{\rm tot}}$ and letting $R_{\rm tot} \to \infty$ then together with \eqref{eq:erlangb_limit} yields the fixed-point equation
\begin{equation}
\alpha = f_0(\beta -\alpha).
\end{equation}
It can be shown that this fixed-point equation has a unique positive root for all $\beta > 0$, which can be computed numerically.
Recalling Cohen's retrial queue characterization, Avram et al.~\cite{Avram2013} conclude that the loss system with slow retrials can in the QED regime be characterized in terms of the original loss system without retrials, but with a corrected QoS-parameter $\beta-\alpha$. \\
\\*
\textbf{Structure of the chapter.}
The fixed-point method of Avram et al. provides a quick and elegant way to approximate the behavior of large-scale loss systems that experience retrials.
In the remainder of this chapter we will explore if and how this technique extends to three more complex queueing settings.
These three models have in common that (i) they exhibit QED limiting behavior, which can be quantified explicitly, and (ii) the blocking probability is $O(1/\sqrt{R})$, so that the retrial volume is $O(\sqrt{R})$.
Note that these were the two essential features for the fixed-point method to work.
In Section \ref{sec:basic_model} we describe a direct extension of the $M/M/s/s$ queue, in which some amount of waiting room is present.
That is, we analyze the $M/M/s/n$ queue with retrials, where $n>s$.
Naturally, this requires a scaling for both $s$ and $n$ as $R\to\infty$, which will become clear in this section.
Motivated by a process related to cloud computing, we in Section \ref{sec:cloud_model} study a tandem queueing network, in which total number of concurrent admissions is limited.
Section \ref{sec:abandonments_retrials} analyzes a queueing model in which all customers are admitted upon arrival, but make the deliberate decision to abandon the queue and retry later in case their patience runs out.
In Section \ref{sec:retrial_dimensioning} we show how the fixed-point method together with QED scaling can be used for dimensioning purposes in both stationary and time-varying environments.
We end the chapter in Section \ref{sec:retrial_conclusion} with some final remarks and suggestions for future research.
\section{The $M/M/s/n$ queue}
\label{sec:basic_model}
In this section, we discuss a simple extension of the loss model of \cite{Avram2013}, namely the $M/M/s/n$ queue with retrials with $n>s$, to expose typical behavior of retrial queues and the influence of the retrial rate $\delta$.
Second, we illustrate the fixed-point method for this model and perform numerical experiments to verify its accuracy.
\subsection{Markov process}
We consider the standard $M/M/s/n$ queue with arrival rate $\lambda$ and service rate $\mu$.
Without loss of generality, we set $\mu=1$ throughout this chapter, so that offered load $R$ equals $\lambda$.
A customer that finds upon arrival a free server occupies this server immediately, while customers that meet more than $s$ but fewer than $n>s$ customers in the system are admitted and wait in a queue for a server to become available.
Customers who meet upon arrival $n$ customers are not admitted directly, but will retry after an exponentially distributed time with mean $1/\delta$.
Each initially blocked customer performs retrials until admitted eventually; see Figure \ref{fig:BasicModel}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1]
\draw[dashed] (0,0.25) rectangle (4.75,2.75) node[right,above] {$n$};
\draw[thick] (0.75,1) -- (2.5,1) -- (2.5,2) -- (0.75,2);
\draw[thick] (1.5,1.1) -- (1.5,1.9);
\draw[thick] (1.75,1.1) -- (1.75,1.9);
\draw[thick] (2,1.1) -- (2,1.9);
\draw[thick] (2.25,1.1) -- (2.25,1.9);
\draw[thick] (3.5,1.5) circle (0.5) node {$s$};
\draw[thick,->] (-2.25,1.5) -- (1,1.5);
\draw[thick,->] (4.25,1.5) -- (5.5,1.5);
\node at (-2.25,1.8) {Pois$(\lambda)$};
\node at (3.5,2.35) {$\exp(\mu)$};
\draw[thick] (-0.5,1.5) -- (-0.5,1);
\draw[thick] (-0.5,1) to [in=0,out=270] (-1,0.5);
\draw[thick] (-1,0.5) to [in=270,out=180] (-1.5,1);
\draw[thick,->] (-1.5,1) to [in=180,out=90] (-1,1.45);
\node at (-2,0.4) {$\exp(\delta)$};
\end{tikzpicture}
\caption{An $M/M/s$ queue with space constraints and retrials.}
\label{fig:BasicModel}
\end{figure}
\noindent
\textbf{Quasi-birth-death process.} The system state can be described by a two-dimensional process $\{ (Q(t),N(t))\}_{t\geq 0}$ with $Q(t)$ the number of customers inside the system (either being served or waiting), and $N(t)$ the number of customers in the retrial orbit.
Under the above assumptions, this process is a continuous-time Markov chain on the semi-infinite strip $\{0,1,\ldots,n\}\times \{0,1,\ldots\}$.
Its transition diagram is presented in Figure \ref{fig:transition_diagram}.
From this diagram it is evident that the process is a quasi-birth-death (QBD) process.
Under stability condition $R>s$, the QBD structure of the process allows for numerical computation of the stationary distribution $\pi(i,j)$, where
\begin{equation*}
\pi(i,j) = \lim_{t\to\infty} \mathbb{P}\left( Q(t) = i, N(t) = j\right).
\end{equation*}
The stationary probability that an arriving customer has to wait or is blocked is given by, respectively,
\begin{align}
\mathbb{P}_r({\rm delay}) = \sum_{i=s}^n \sum_{j=0}^\infty \pi(i,j),\qquad
\mathbb{P}_r({\rm block}) = \sum_{j=0}^\infty \pi(n,j).
\label{eq:performance_measures}
\end{align}
Here, the subscript $r$ is meant to indicate that we consider the system with retrials.\\
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/transition_diagram}
\caption{Transition diagram of the Markov process $(Q(t),N(t))$.}
\label{fig:transition_diagram}
\end{figure}
\noindent
\textbf{Influence of retrial rate $\delta$.}
We first compute the stationary distribution of the Markov process numerically, in order to understand the influence of retrials on the queue performance.
In particular, we investigate the effect of varying $\delta$ on the delay and blocking probability as defined in \eqref{eq:performance_measures}.
In Figure \ref{fig:influence_of_delta} we fix $R=10$ and $s=12$, and plot the delay and blocking probability as a function of $\log_{10}(\delta)$ for several values of $n$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = -3,
xmax = 3,
ymin = 0,
ymax = 0.5,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
grid = both,
axis line style={->},
axis lines = middle,
xscale=1,
yscale=0.8,
every axis x label/.style={at={(current axis.right of origin)},anchor=west},
every axis y label/.style={at={(current axis.north west)},above=2mm},
legend pos = south east]
\addplot[col1,thick] table[x=log_delta,y=n12] {Chapter_4/tikz/pdelay_small.txt};
\addplot[col2,thick] table[x=log_delta,y=n13] {Chapter_4/tikz/pdelay_small.txt};
\addplot[col3,thick] table[x=log_delta,y=n14] {Chapter_4/tikz/pdelay_small.txt};
\addplot[col4,thick] table[x=log_delta,y=n16] {Chapter_4/tikz/pdelay_small.txt};
\addplot[col5,thick] table[x=log_delta,y=n20] {Chapter_4/tikz/pdelay_small.txt};
\addplot[dashed,thick] coordinates { (-3,0.449) (3,0.449) };
\legend{{$n=12$},{$n=13$},{$n=14$},{$n=16$},{$n=20$}};
\end{axis}
\node at (3.45,-0.5) {\small $\log_{10}(\delta)$};
\node at (3.45,5) {\small $\mathbb{P}_r({\rm delay})$};
\end{tikzpicture}
\caption{Delay probability}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = -3,
xmax = 3,
ymin = 0,
ymax = 0.5,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
grid = both,
axis line style={->},
axis lines = middle,
xscale=1,
yscale=0.8,
every axis x label/.style={at={(current axis.north east)},anchor=west},
every axis y label/.style={at={(current axis.north west)},above=2mm},
legend pos = south east]
\addplot[col1,thick] table[x=log_delta,y=n12] {Chapter_4/tikz/pblock_small.txt};
\addplot[col2,thick] table[x=log_delta,y=n13] {Chapter_4/tikz/pblock_small.txt};
\addplot[col3,thick] table[x=log_delta,y=n14] {Chapter_4/tikz/pblock_small.txt};
\addplot[col4,thick] table[x=log_delta,y=n16] {Chapter_4/tikz/pblock_small.txt};
\addplot[col5,thick] table[x=log_delta,y=n20] {Chapter_4/tikz/pblock_small.txt};
\end{axis}
\node at (3.45,-0.5) {\small $\log_{10}(\delta)$};
\node at (3.45,5) {\small $\mathbb{P}_r({\rm block})$};
\end{tikzpicture}
\caption{Blocking probability}
\end{subfigure}
\caption{Performance metrics of the basic model with $R=10$ and $s=12$ as a function of $\log_{10}(\delta)$ for several $n$.}
\label{fig:influence_of_delta}
\end{figure}
We see that the value of $\delta$ indeed does influence the performance of the queue, and its effect is particularly pronounced in systems with $n$ close to $s$.
Both the delay probability and the blocking probability increase with $\delta$.
This can be explained as follows.
If a customer finds $n$ customers on arrival (or retrial) and hence gets blocked, she is more likely to find a less congested system in case she retries after a relatively long amount of time than a short retrial time, because the system might not yet have had enough time to recover from the congested period.
Slow retrials hence create an opportunity to smooth out workload over time, resulting in better quality-of-service.
Figure \ref{fig:influence_of_delta} also suggests that performance no longer changes if $\delta$ is decreased below $10^{-1}$ or increased beyond $10^2$.
Also, we note that the delay probability increases with $n$, and the blocking probability decreases with $n$, regardless of the value of $\delta$.
Fewer customers get blocked if the waiting room $(n-s)$ increases. On the other hand, this allows more customers to enter the system, creating higher congestion levels.
Finally, notice that the delay probability approaches a constant as $\delta\to\infty$.
In fact, this constant equals the delay probability in the standard $M/M/s$ queue, see Equation (1.2), which under these parameter settings equals 0.449 and is represented by the dashed horizontal line.
Indeed, when $\delta\to\infty$ blocked customers retry getting access to the system instantaneously and effectively create a queue (in random order) outside the system, which immediately fills up vacant spaces after service completions.
Therefore, the $M/M/s/n$ queue with instant retrials essentially resembles the behavior of the $M/M/s$ queue.
By similar reasoning, the blocking probability in the $M/M/s/n$ queue approaches as $\delta\to\infty$ the probability that the number of customers in the $M/M/s$ queue exceeds $n$.
Figure \ref{fig:influence_of_delta} shows that the influence of retrials on congestion can be significant.
For fast retrials, we are able to characterize the performance metrics through the standard multi-server queue.
However, for slow retrials, say $\delta < 10^{-1}$, the system behavior is not comparable to that of the open $M/M/s$ queue.
\subsection{QED regime}
Following the approach of Avram et al.~\cite{Avram2013}, we choose to take a step back and consider the model in Figure \ref{fig:BasicModel} without the retrials first.
When blocked customers are simply discarded, the process $\{ (Q(t),N(t))\}_{t\geq 0}$ reduces to that of the $M/M/s/n$ queue.
In this case $N(t)=0$ and $Q(t)$ is a birth-death process with stationary distribution
\[
\pi(i) = \lim_{t\to\infty} \mathbb{P}(Q(t)=i) =
\left\{
\begin{array}{ll}
\pi(0)\, \frac{R^i}{i!}, & \text{if }i < s,\\
\pi(0)\, \frac{R^i}{s!s^{i-s}}, & \text{if }s\leq i \leq n,\\
\end{array}
\right.
\]
where
\[\pi(0) = \Big( \sum_{i=0}^{s-1} \frac{R^i}{i!} + \sum_{i=s}^n \frac{R^i}{s!s^{i-s}} \Big)^{-1}.\]
Hence,
\begin{equation}
\mathbb{P}({\rm delay}) = \pi(0) \sum_{i=s}^n \frac{R^i}{s!s^{i-s}},
\qquad
\mathbb{P}({\rm block}) = \pi(0) \frac{R^n}{s!s^{n-s}}.
\end{equation}
The $M/M/s/n$ queue is well understood.
In particular, Massey \& Wallace \cite{masseywallace} identified the asymptotic scaling regime for $s$ and $n$ under which QED-type behavior prevails.
Namely, under the two-fold scaling rule
\begin{align}
s &= R + \beta\sqrt{R} + o(\sqrt{R}), \nonumber\\
n &= s + \gamma\sqrt{R} + o(\sqrt{R}),
\label{eq:twofold_scaling_basic_model}
\end{align}
for $\beta\in\mathbb{R}$ and $\gamma>0$, they show that the delay probability converges to a value strictly between 0 and 1, while the blocking probability vanishes as $R\to\infty$.
Note that this is in line with our reasoning in Section 1.4.
In the next proposition, we cite the asymptotic results of \cite{masseywallace} for completeness.
\begin{proposition}[\cite{masseywallace}]
If $s$ and $n$ scale according to \eqref{eq:twofold_scaling_basic_model}, then in the $M/M/s/n$ queue,
\begin{align}
\mathbb{P}({\rm delay}) &\to \frac{ 1-{\rm e}^{-\beta\gamma}}{1-{\rm e}^{-\beta\gamma}+\beta\,\Phi(\beta)/\varphi(\beta)} =: g(\beta,\gamma),\label{eq:limit_delay}
\\
\sqrt{R} \, \mathbb{P}({\rm block}) &\to \frac{\beta\,{\rm e}^{-\beta\gamma}}{1-{\rm e}^{-\beta\gamma} + \beta\,\Phi(\beta)/\varphi(\beta)} =: f(\beta,\gamma),
\label{eq:limit_block}
\end{align}
as $R\to\infty$.
\end{proposition}
Before turning to the asymptotic analysis of the model with retrials, we check empirically whether the scaling in \eqref{eq:twofold_scaling_basic_model} also achieves the desirable limiting behavior in case blocked customers are not discarded.
In Figure \ref{fig:sample_paths_retrial} we plot sample paths $Q(t)$ and $N(t)$ in the system with retrials with $\beta = 0.5$ and $\gamma=1$ and slow retrials ($\delta = 0.1$) for increasing values of $R$.
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/sample_paths}
\caption{Sample paths of $Q(t)$ (blue) and $N(t)$ (red) for increasing $R$ while $s$ and $n$ are scaled as in \eqref{eq:twofold_scaling_basic_model} with $\beta=0.5$ and $\gamma=1$ and retrial rate $\delta=0.1$.}
\label{fig:sample_paths_retrial}
\end{figure}
From these sample paths, we observe that indeed the server utilization approaches unity as $R$ tends to infinity, indicating efficient usage of resources.
This should not be surprising, since although retrials occur, all customers eventually receive service, so that the server utilization equals $R/s = R/(R+\beta\sqrt{R}) \to 1$ as $R\to\infty$.
Furthermore, we see that the number of customers in the system concentrates around the level $s$, implying a delay probability away from both 0 and 1.
Observe that the order of magnitude of $N(t)$, the number of customers in the retrial orbit, is smaller than $Q(t)$ or $R$.
This implies that as $R$ grows large, only a small fraction of customers ends up retrying.
Naturally, the order of $N(t)$ also depends on the mean retrial time $1/\delta$.
It can be numerically verified that the expected retrial population grows linearly in $1/\delta$.
Last, observe that $N(t)$ is increasing only if $Q(t)=n$, which is visible through the surges in the sample paths of $N(t)$ in Figure \ref{fig:sample_paths_retrial}.
This is illustrative for the dependency between the two coordinates of the process $\{(Q(t),N(t))\}_{t\geq 0}$ and therefore, we cannot expect to find a simple decoupling in the limit either.
Instead, we propose to evaluate the model with retrials through a heuristic approach which builds upon the asymptotic behavior of the model without retrials.
\subsection{Fixed-point method}
\label{sec:fixed_point}
We continue to translate the ideas behind the fixed-point method by noting that due to \eqref{eq:limit_block}, the fraction of blocked customers is of order $1/\sqrt{R}$, which implies that the mean additional load due to retrials must be of order $\sqrt{R}$.
We can thus assume that the total arrival rate $R_{\rm tot}$ takes the form $R_{\rm tot} = R+\alpha\sqrt{R}$ for some $\alpha>0$.
Then, using that $R = O(R_{\rm tot})$, the first scaling rule in \eqref{eq:twofold_scaling_basic_model} is asymptotically equivalent with
\begin{equation}
s = R_{\rm tot} + (\beta-\alpha) \sqrt{R_{\rm tot}} + o(\sqrt{R_{\rm tot}}),
\end{equation}
while the scaling for $n$ remains unchanged.
We thus argue that the retrial system in the QED regime mimics an $M/M/s/n$ queue with parameters $\beta_\alpha = \beta-\alpha$ and $\gamma$.
Note that the volume of blocked users in this setting is $f(\beta-\alpha,\gamma)\sqrt{R_{\rm tot}}$.
This quantity must equal the mean additional load $\alpha\sqrt{R} \sim \alpha\sqrt{R_{\rm tot}}$ and therefore we obtain the \textit{fixed-point equation}
\begin{equation}
\alpha = f(\beta-\alpha,\gamma).
\label{eq:fixed_point_basic}
\end{equation}
Numerically determining $\alpha$ is straightforward, particularly because it is uniquely defined.
\begin{lemma}
Equation \eqref{eq:fixed_point_basic} has a unique solution for all $\beta,\gamma>0$.
\end{lemma}
\begin{proof}
Let $h(\beta):=\varphi(\beta)/\Phi(\beta)$ and $w(\beta):=(1-{\rm e}^{-\beta\gamma})/\beta$. Write
\begin{align}\label{}
f(\beta):=f(\beta,\gamma)=\frac{(1-\beta w(\beta))h(\beta)}{1+w(\beta)h(\beta)},
\end{align}
so that
\begin{align}\label{}
\beta+f(\beta)=\frac{\beta+h(\beta)}{1+w(\beta)h(\beta)}
\end{align}
For $h(\beta)$ it is known that, see \cite{Sampford1953}, for $\beta\in\mathbb{R}$,
\begin{align}\label{}
h(\beta)>-\beta, \quad -1<h'(\beta)<0, \quad h''(\beta)>0,
\end{align}
so that $h(\beta)$ is non-negative and non-increasing in $\beta\in\mathbb{R}$, while $\beta+h(\beta)$ is positive and strictly increasing in $\beta\in\mathbb{R}$.
Because ${\rm e}^x\geq 1+x$,
\begin{align}\label{}
w'(\beta)=\frac{{\rm e}^{-\beta\gamma}}{\beta^2}(1+\beta\gamma-{\rm e}^{\beta\gamma})\leq 0
\end{align}
so $w(\beta)$ is also non-negative and non-increasing in $\beta\in \mathbb{R}$. It thus follows that $\beta+f(\beta)$ is strictly increasing in $\beta\in\mathbb{R}$.
Moreover, $\beta+f(\beta)\to 0$ as $\beta\to-\infty$ and $\beta+f(\beta)\to \infty$ as $\beta\to \infty$.
Let $\Delta=\beta-\alpha$, and rewrite \eqref{eq:fixed_point_basic} as
\begin{align}\label{}
\beta=\Delta+f(\Delta).
\end{align}
Hence, for each fixed $\beta>0$ there is a unique solution $\Delta \in \mathbb{R}$ from which $\alpha=\beta-\Delta$ follows.
\end{proof}
\noindent As a result, the delay probability $\mathbb{P}_r({\rm delay})$ and the blocking probability $P_r({\rm block})$ in the model with retrials can be approximated in the QED regime by
\begin{equation}
\mathbb{P}_r({\rm delay}) \approx g(\beta-\alpha,\gamma),
\qquad
\mathbb{P}_r({\rm block}) \approx \alpha/\sqrt{R},
\label{eq:delay_approx}
\end{equation}
which should become more accurate as $R$ grows large.
We next test the accuracy of the approximated delay probability in \eqref{eq:delay_approx} in the basic model with slow retrials against the true values obtained through simulation.
Given $R,s$, and $n$, we compute $\beta = (s-R)/\sqrt{R}$ and $\gamma = (n-s)/\sqrt{R}$ in order to approximate the delay and blocking probability as in \eqref{eq:delay_approx} with $\alpha$ as in \eqref{eq:fixed_point_basic}.
First, we assess the quality of the fixed-point approximation for a large but finite system with $R=100$.
In Figure \ref{fig:basic_model_accuracy}, we plot the simulated delay probability against the approximation as a function of $s$ (or equivalently $\beta$).
We consider different values of $\gamma$, namely $\gamma=0.5$, $1$ and $2$, which corresponds to waiting room size $\gamma\sqrt{100} = 5,\, 10$ and 20, respectively.
For comparison, we also include $g(\beta,\gamma)$, the asymptotic delay probability in the system with no retrials, in these plots.
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/basic_model_accuracy}
\caption{Accuracy of the delay probability approximation in basic model with $R=100$ and $\delta = 0.01$.}
\label{fig:basic_model_accuracy}
\end{figure}
We observe that the heuristic is remarkably accurate in describing both the delay and blocking probability over all values of $\beta,\gamma>0$ considered here.
The approximation improves as $\gamma$ increases.
Figure \ref{fig:basic_model_accuracy} also clearly illustrates the impact of retrials on the performance measures, which decreases with both $\beta$ and $\gamma$.
\begin{table}
\centering
\small
\begin{tabular}{|r|rr|rr||rr|rr|}
\cline{2-9}\multicolumn{1}{r|}{} & \multicolumn{4}{c||}{$(\beta,\gamma) = (0.5,0.5)$} & \multicolumn{4}{c|}{$(\beta,\gamma) = (1,0.5)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}_r({\rm delay})$ & $\sqrt{R}\mathbb{P}_r({\rm bl.})$ & $s$ & $n$ & $\mathbb{P}_r({\rm delay})$ & $\sqrt{R}\mathbb{P}_r({\rm bl.})$ \bigstrut\\
\hline
5 & 6 & 7 & 0.5019 & 0.5982 & 7 & 8 & 0.2607 & 0.2610 \bigstrut[t]\\
10 & 12 & 14 & 0.3697 & 0.3679 & 13 & 15 & 0.2298 & 0.1966 \\
50 & 54 & 58 & 0.3509 & 0.4931 & 57 & 61 & 0.1765 & 0.2019 \\
100 & 105 & 110 & 0.3640 & 0.6336 & 110 & 115 & 0.1579 & 0.2178 \\
500 & 511 & 522 & 0.3460 & 0.6780 & 522 & 533 & 0.1482 & 0.2297 \\
1000 & 1016 & 1032 & 0.3333 & 0.6481 & 1032 & 1048 & 0.1412 & 0.2141 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \multicolumn{2}{r|}{Approx} & 0.3225 & 0.6734 & \multicolumn{2}{r|}{Approx} & 0.1349 & 0.2206 \bigstrut\\
\cline{2-9}\end{tabular}%
\vspace{5 mm}
\begin{tabular}{|r|rr|rr||rr|rr|}
\cline{2-9}\multicolumn{1}{r|}{} & \multicolumn{4}{c||}{$(\beta,\gamma) = (0.5,1)$} & \multicolumn{4}{c|}{$(\beta,\gamma) = (1,1)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}_r({\rm delay})$ & $\sqrt{R}\mathbb{P}_r({\rm bl.})$ & $s$ & $n$ & $\mathbb{P}_r({\rm delay})$ & $\sqrt{R}\mathbb{P}_r({\rm bl.})$ \bigstrut\\
\hline
5 & 6 & 8 & 0.5337 & 0.4065 & 7 & 9 & 0.2866 & 0.1612 \bigstrut[t]\\
10 & 12 & 15 & 0.3932 & 0.2701 & 13 & 16 & 0.2472 & 0.1374 \\
50 & 54 & 61 & 0.3993 & 0.3171 & 57 & 64 & 0.2063 & 0.1183 \\
100 & 105 & 115 & 0.4333 & 0.3754 & 110 & 120 & 0.1971 & 0.1143 \\
500 & 511 & 533 & 0.4247 & 0.3986 & 522 & 544 & 0.1928 & 0.1202 \\
1000 & 1016 & 1048 & 0.4115 & 0.3689 & 1032 & 1064 & 0.1831 & 0.1088 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \multicolumn{2}{r|}{Approx} & 0.4062 & 0.3828 & \multicolumn{2}{r|}{Approx} & 0.1798 & 0.1106 \bigstrut\\
\cline{2-9}\end{tabular}%
\caption{\normalsize Numerical results of the fixed-point method for the basic model as $R\to\infty$.}
\label{tab:basic_model_accuracy}
\end{table}
Table \ref{tab:basic_model_accuracy} furthermore shows how the accuracy of the approximations increases as $R$ increases.
In this table, we used the simulated delay and blocking probability for systems of increasing size while adhering to the two-fold scaling rule of \eqref{eq:twofold_scaling_basic_model}.
The values of $s$ and $n$ are rounded to the nearest integer.
\section{Cloud model}
\label{sec:cloud_model}
The second model we consider in this chapter is inspired by cloud computing services.
We shall see how our fixed-point heuristic helps cloud providers in their provisioning process.\\
\\*
\subsection{Practical context}
Cloud computing enables network access to a shared pool of configurable computing resources, allowing users (e.g.~companies, service providers) to store and process their data in third-party data centers, without investing in the operating equipment themselves.
At the foundation of cloud computing lies the idea of sharing resources to achieve economies-of-scale in terms of maximizing computing power usage and
reducing the overall cost of resources such as energy and
infrastructure.
Cloud providers, such as Amazon EC2, Windows Azure and Rackspace \cite{Armbrust2010}, offer virtual machine (VM) provisioning, which allows users to request VM instances configured to their preference.
In a service system context, the provider thus serves users by supplying them with a VM that matches their requirements, running on one of the cloud's physical machines.
\begin{figure}
\centering
\includegraphics[scale=0.33]{Chapter_4/images/CloudProvisioning.pdf}
\caption{Cloud provisioning process}
\label{fig:CloudScheme}
\end{figure}
Let us describe the cloud provisioning process in more detail; see Figure \ref{fig:CloudScheme}.
At the highest granularity level there are the \emph{end-users}, devices typically directly operated by humans, using an \emph{application provider} (AP), usually a company that provides software usage over the internet (e.g.~SaaS \cite{Mell2011}). To some extent, the AP will rely on a static set of computing resources, but certainly in case of sudden surges in workload, these might not be sufficient. When the AP recognizes the need for additional capacity, for instance by \emph{auto-scaling} procedures \cite{Amazon}, a VM request is submitted to the cloud provider. The request is handled by a \emph{host server} that starts the set-up of the VM with requested specifications. This includes elementary operations such as copying the VM image and assigning an IP address. Each server is able to host multiple VM instances in parallel, although the VMs in set-up need their dedicated attention, due to concurrency level constraints incurred by large I/O activities. Once the set-up is completed, the VM is ready for use, and the AP may start using the additional computing resources.
Our focus lies on the capacity allocation within the cloud environment, so the right-hand side of Figure \ref{fig:CloudScheme}.
Successful management of cloud systems requires the right scaling of both the number of host servers (denoted by $s$) at the first I/O queue and the maximum number of VMs (denoted by $n$) that can be hosted simultaneously. Moreover, this needs to be done in a dynamic way in order to respond effectively to the time-varying demand. The capacity $n$ defines a hard constraint on whether a new VM request will be accepted immediately or not. Therefore, new requests will be delayed or even dropped if the available host capacity is insufficient, which is more likely to occur during periods in which the $s$ host servers are overloaded.\\
\\*
\subsection{Queueing model}
To describe the cloud system in mathematical terms, we extend the model proposed by Tan et al.~\cite{Tan2012}.
Each host server may host a number of VM instances at the same time, yielding a total number of $n$ parallel VM instances. Requests, arriving to the system according to a Poisson process with rate $\lambda$, are granted only if one of these $n$ positions is available.
Otherwise, the user retries getting access after an exponentially distributed time with mean $1/\delta$.
If granted, the request is assigned to a host server not busy initializing another VM instance, if available, or waits for one to become available. This start-up time is assumed to be exponentially distributed with mean $1/\mu$. On completion of the initialization phase, VM usage is initiated by the client. The VM continues to be occupied for a random amount of time, with mean $1/\kappa$, until release by the user. We note that the model of Tan et al. \cite{Tan2012} has three queues in tandem, one $M/M/s$ queue, followed by two $M/M/\infty$ queues that separately model a second initialization phase and the actual VM usage by the cloud user. We thus replace the two $M/M/\infty$ queues by one $M/G/\infty$ queue with an aggregated service time, which does not alter the system performance analysis.
This yields the queueing model in Figure~\ref{fig:CloudModel}.
\begin{figure}
\centering
\begin{tikzpicture}[scale=1]
\draw[dashed] (0,0.25) rectangle (6.5,2.75) node[right,above] {$n$};
\draw[thick] (0.75,1) -- (2.5,1) -- (2.5,2) -- (0.75,2);
\draw[thick] (1.5,1.1) -- (1.5,1.9);
\draw[thick] (1.75,1.1) -- (1.75,1.9);
\draw[thick] (2,1.1) -- (2,1.9);
\draw[thick] (2.25,1.1) -- (2.25,1.9);
\draw[thick] (3.5,1.5) circle (0.5) node {$s$};
\draw[thick] (5.5,1.5) circle (0.5) node {$\infty$};
\draw[thick,->] (-2.25,1.5) -- (1,1.5);
\draw[thick,->] (4.25,1.5) -- (4.75,1.5);
\draw[thick,->] (6.25,1.5) -- (7.55,1.5);
\node at (-2.25,1.8) {Pois$(\lambda)$};
\node at (3.5,2.35) {$\exp(\mu)$};
\node at (5.5,2.35) {$\exp(\kappa)$};
\draw[thick] (-0.5,1.5) -- (-0.5,1);
\draw[thick] (-0.5,1) to [in=0,out=270] (-1,0.5);
\draw[thick] (-1,0.5) to [in=270,out=180] (-1.5,1);
\draw[thick,->] (-1.5,1) to [in=180,out=90] (-1,1.45);
\node at (-2,0.4) {$\exp(\delta)$};
\end{tikzpicture}
\caption{Abstracted model of VM provisioning process.}
\label{fig:CloudModel}
\end{figure}
\begin{remark}
We mention that a queueing model similar to the one in Figure \ref{fig:CloudModel} without retrials is analyzed by Khudyakov et al.~\cite{Khudyakov2006} in a telecommunication environment.
In their work, $s$ and $n$ represent the number of agents and trunk lines in a call center.
Although the order of the two queues is switched, the stationary analysis of their model and the cloud model is the same, due to the product-form structure of the stationary distribution.
In fact, Tan et al.~\cite{Tan2012} use the results of \cite{Khudyakov2006} in their asymptotic analysis.
\end{remark}
An exact analysis of the cloud model is again obstructed by the absence of a product-form solution in case of retrials.
We therefore turn to the QED paradigm to approximate the system behavior as $R\to\infty$.
Following the approach in \cite{Khudyakov2006,Tan2012}, we argue that the appropriate QED scaling for $s$ and $n$ should be
\begin{align}
s &= R + \beta\sqrt{R} + o(\sqrt{R}), &\beta>0, \nonumber\\
n &= s + R/\kappa + \gamma\sqrt{R/\kappa} + o(\sqrt{R}), &\gamma>0,
\label{eq:twofold_scaling_cloud_model}
\end{align}
where $R = \lambda / \mu = \lambda$.
To understand why this is indeed the correct scaling regime to obtain non-degenerate limiting behavior, we recall the arguments we presented in Section 1.3.
Namely, in order to achieve QED performance, one allocates the nominal workload brought towards the queue plus a variability hedge that is proportional to the square-root of this amount.
For $s$, this results in the standard square-root staffing rule.
For $n$, this is the sum of the capacity needed at the multi-server queue, i.e. $s$, and the consecutive infinite-server queue.
Since the expected workload at the second queue equals $R/\kappa$, the capacity required at this stage equals $R/\kappa+\gamma\sqrt{R/\kappa}$ for some $\gamma$.
In total, this yields the scaling for the number of VMs $n$ as in \eqref{eq:twofold_scaling_cloud_model}.
The limiting behavior of this queueing model without retrials is documented in \cite{Khudyakov2006,Tan2012}.
\begin{proposition}\label{prop:cloud_model_limits}
Let $s$ and $n$ in the cloud model of Figure \ref{fig:CloudModel} without retrials scale as in \eqref{eq:twofold_scaling_cloud_model}. Then, as $R\to\infty$,
\begin{align}\label{pdelay}
\mathbb{P}^c({\rm delay})&\to \frac{\xi_1-\xi_2}{\eta+\xi_1-\xi_2} =:g_c(\beta,\gamma),\\
\label{pblock}
\sqrt{R}\cdot \mathbb{P}^c({\rm block}) &\to \frac{\nu}{\eta+\xi_1-\xi_2} =:f_c(\beta,\gamma),
\end{align}
where
\[
\eta = \int_{-\infty}^\beta \Phi\Big(\gamma+(\beta-t)\sqrt{\kappa}\Big)\,\varphi(t)\, {\rm d} t, \qquad
\xi_1 = \frac{ \varphi(\beta)\Phi(\gamma) }{\beta},
\]
\[
\xi_2 = \frac{1}{\beta}\,\varphi\left(\sqrt{\beta^2+\gamma^2}\right){\rm e}^{\tfrac{1}{2}(\gamma-\beta/\sqrt{\kappa})^2} \Phi(\gamma-\beta/\sqrt{\kappa}),
\]
\[
\nu = \sqrt{\frac{\kappa}{1+\kappa}}\,\varphi\Big(\frac{\gamma+\beta\sqrt{\kappa}}{\sqrt{1+\kappa}}\Big) \Phi\Big(\frac{\beta-\gamma\sqrt{\kappa}}{\sqrt{1+\kappa}}\Big) + \beta\,\xi_2.
\]
\end{proposition}
\subsection{Fixed-point method}
Proposition \ref{prop:cloud_model_limits} shows that also in this model, the blocking probability vanishes at rate $1/\sqrt{R}$, making it amenable to our fixed-point method for retrials.
Let $\alpha\sqrt{R}$ the volume of retrials, so that a total arrival rate is $R_{\rm tot} = R + \alpha\sqrt{R}$, or equivalently $R = R_{\rm tot} - \alpha \sqrt{R_{\rm tot}} + o(\sqrt{R_{\rm tot}})$.
Substituting this into the two-fold scaling rule in \eqref{eq:twofold_scaling_cloud_model} gives
\begin{align*}
s &= R_{\rm tot} + (\beta - \alpha ) \sqrt{R_{\rm tot}} + o(\sqrt{R_{\rm tot}}),\\
n &= s + \frac{R_{\rm tot}}{\kappa} + \Big( \gamma - \frac{\alpha}{\sqrt{\kappa}} \Big) \sqrt{\frac{R_{\rm tot}}{\kappa}} + o(\sqrt{R_{\rm tot}}).
\end{align*}
Accordingly, the constant $\alpha$ is defined as the solution of the fixed-point equation
\begin{equation}
\label{eq:fixed_point_cloud}
f_c\left(\beta - \alpha, \gamma - \alpha/\sqrt{\kappa}\right) = \alpha.
\end{equation}
Approximations for the delay and blocking probability in the cloud model with retrials are hence given by
\begin{equation}
\mathbb{P}_r^c({\rm delay}) \approx g_c\left(\beta-\alpha,\gamma-\alpha/\sqrt{\kappa}\right), \qquad \mathbb{P}_r^c({\rm block}) \approx \alpha/\sqrt{R}.
\label{eq:cloud_approximations}
\end{equation}
Note that in contrast to the fixed-point equation \eqref{eq:fixed_point_basic} for the basic model, the second argument $\gamma-\alpha/\sqrt{R}$ is also corrected.
\\
\\*
We next test the accuracy of the fixed-point equation for several instances.
In Table \ref{tab:cloud_model_accuracy}, we present the simulation results for $\kappa=0.02$, 0.2 and 1, and two pairs of $(\beta,\gamma)$ for increasing $R$.
\begin{table}
\centering
\small
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr||rr|rr|}
\cline{2-9}\multicolumn{1}{r|}{} & \multicolumn{4}{c||}{$(\beta,\gamma) = (0.5,1)$} & \multicolumn{4}{c|}{$(\beta,\gamma) = (1,1)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ \bigstrut\\
\hline
5 & 6 & 13 & 0.5309 & 0.5540 & 7 & 14 & 0.2800 & 0.2426 \bigstrut[t]\\
10 & 12 & 25 & 0.3864 & 0.3810 & 13 & 26 & 0.2393 & 0.2164 \\
50 & 54 & 111 & 0.3904 & 0.4525 & 57 & 114 & 0.1965 & 0.2010 \\
100 & 105 & 215 & 0.4300 & 0.5474 & 110 & 220 & 0.1859 & 0.1952 \\
500 & 511 & 1033 & 0.4139 & 0.5586 & 522 & 1044 & 0.1787 & 0.2052 \\
1000 & 1016 & 2048 & 0.4003 & 0.5479 & 1032 & 2064 & 0.1660 & 0.1803 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \multicolumn{2}{r|}{Approx} & 0.4029 & 0.5638 & \multicolumn{2}{r|}{Approx} & 0.1709 & 0.1992 \bigstrut\\
\cline{2-9}\end{tabular}%
\caption{$\kappa = 1$}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr||rr|rr|}
\cline{2-9}\multicolumn{1}{r|}{} & \multicolumn{4}{c||}{$(\beta,\gamma) = (0.5,1)$} & \multicolumn{4}{c|}{$(\beta,\gamma) = (1,1)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ \bigstrut\\
\hline
5 & 6 & 36 & 0.5664 & 0.3049 & 7 & 37 & 0.3079 & 0.1457 \bigstrut[t]\\
10 & 12 & 69 & 0.4263 & 0.2227 & 13 & 70 & 0.2683 & 0.1410 \\
50 & 54 & 320 & 0.4444 & 0.2555 & 57 & 323 & 0.2293 & 0.1334 \\
100 & 105 & 627 & 0.4826 & 0.3085 & 110 & 632 & 0.2187 & 0.1379 \\
500 & 511 & 3061 & 0.4842 & 0.3235 & 522 & 3072 & 0.2182 & 0.1358 \\
1000 & 1016 & 6087 & 0.4630 & 0.2906 & 1032 & 6103 & 0.2039 & 0.1332 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \multicolumn{2}{r|}{Approx} & 0.4687 & 0.3029 & \multicolumn{2}{r|}{Approx} & 0.2042 & 0.1326 \bigstrut\\
\cline{2-9}\end{tabular}%
\caption{$\kappa=0.2$}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr||rr|rr|}
\cline{2-9}\multicolumn{1}{r|}{} & \multicolumn{4}{c||}{$(\beta,\gamma) = (0.5,1)$} & \multicolumn{4}{c|}{$(\beta,\gamma) = (1,1)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^c({\rm bl})$ \bigstrut\\
\hline
5 & 6 & 272 & 0.5836 & 0.1085 & 7 & 273 & 0.3217 & 0.0738 \bigstrut[t]\\
10 & 12 & 534 & 0.4456 & 0.0945 & 13 & 535 & 0.2822 & 0.0781 \\
50 & 54 & 2604 & 0.4683 & 0.1031 & 57 & 2607 & 0.2429 & 0.0764 \\
100 & 105 & 5176 & 0.5106 & 0.1064 & 110 & 5181 & 0.2345 & 0.0759 \\
500 & 511 & 25669 & 0.5130 & 0.1158 & 522 & 25680 & 0.2353 & 0.0795 \\
1000 & 1016 & 51240 & 0.4946 & 0.0975 & 1032 & 51256 & 0.2223 & 0.0744 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \multicolumn{2}{r|}{Approx} & 0.4999 & 0.0862 & \multicolumn{2}{r|}{Approx} & 0.2207 & 0.0595 \bigstrut\\
\cline{2-9}\end{tabular}%
\caption{$\kappa=0.02$}
\end{subtable}
\caption{\normalsize Numerical results of the fixed-point method for the cloud model with slow retrials as $R\to\infty$.}
\label{tab:cloud_model_accuracy}
\end{table}
First, observe from Table \ref{tab:cloud_model_accuracy} that $n$ now lives on a different scale than $s$.
This is required to facilitate the long sojourn time of customers in the second stage, which is proportional to $1/\kappa$, creating the need for larger system size.
Besides that, the numerical results show that the fixed-point approximation is again remarkably accurate over a wide range of parameter settings.
Even for cloud systems as small as 50 servers, the fixed-point method gives accurate approximations.
\section{Abandonments}
\label{sec:abandonments_retrials}
Whereas in the basic model of Section \ref{sec:basic_model}, retrials were governed by the system architecture (arriving customers are requested to reattempt if $n$ customers are present in the system), we now consider a setting in which departures from the queue are customer-initiated.
That is, customers deliberately decide to leave the queue to return for service at a later time.
Hence we consider a queueing system with abandonments and retrials.
\subsection{The Erlang-A model}
The canonical model for abandonments is the $M/M/s+M$ or Erlang-A model \cite{Palm1957,Garnett2002}.
The queueing dynamics of the Erlang-A model are similar to those in the $M/M/s$ queue, with the additional feature that each customer is assigned an i.i.d.~patience time, which is exponentially distributed with mean $1/\theta$.
If a customer's patience time expires before reaching an available server, she leaves (abandons) the system.
As the number of customers in the Erlang-A queue is a birth-death process, its stationary distribution and associated performance measures are fairly well-understood, also in the QED regime \cite{Garnett2002,Zeltyn2005,Zhang2012}.
Most importantly to us, Garnett et al. \cite{Garnett2002} and Zeltyn \& Mandelbaum \cite{Zeltyn2005} identified the asymptotic delay and abandonment probability in the Erlang-A model under QED scaling.
\begin{proposition}{ \cite[Thm.~4.1]{Zeltyn2005} }
\label{prop:abandonment_prop}
Let $s = R+\beta\sqrt{R} + o(\sqrt{R})$ for some $\beta\in\mathbb{R}$. Then in the $M/M/s+M$ queue
\begin{align}
\mathbb{P}^a({\rm delay}) &\to \left( 1 + \sqrt{\theta}\,\frac{h(\beta/\sqrt{\theta})}{h({-}\beta)}\right)^{-1} =: g_a(\beta) \\
\sqrt{R}\,\mathbb{P}^a({\rm abandon}) &\to
\frac{ \sqrt{\theta}\,h(\beta/\sqrt{\theta})- \beta }
{ 1 + \sqrt{\theta}\,h(\beta/\sqrt{\theta})/h({-}\beta)} =: f_a(\beta),
\end{align}
as $R\to\infty$ where $h(\beta) = \varphi(\beta)/\Phi({-}\beta)$.
\end{proposition}
We remark that in \cite{Zeltyn2005}, the QED limits for generally distributed patience time were derived.
Although our heuristic also works for this more general setting, we focus on the exponentially distributed patience here to convey our main ideas. \\
\begin{remark}
Large-scale Markovian multi-server queues with abandonments and retrials have been thoroughly studied in a series of papers by Mandelbaum et al.~\cite{Mandelbaum1999,Mandelbaum1999a,Mandelbaum2002}.
In these works, the authors consider a system with time-varying arrivals and a retrial rate that remains bounded away from zero, for which they deduce fluid and diffusion limits as the system grows large.
These limits provide approximations for the time-dependent queue length and virtual waiting time processes, including their means and variances.
We in this section take a different approach by assuming $\delta\to 0$, which enables us to characterize the steady-state behavior of queues with abandonments and retrials.
\end{remark}
\subsection{Fixed-point method}
Next, we include (slow) retrials.
More specifically, we assume that customers who abandon the queue rejoin the queue after an exponentially distributed time with mean $1/\delta \gg 1$.
Just as in the $M/M/s/n$ queue, the $M/M/s+M$ queue with retrials is analytically intractable, and therefore we apply our fixed-point method to approximate its performance in the QED regime.
Observe through Proposition \ref{prop:abandonment_prop} that the fraction of customers leaving before receiving service is roughly $\alpha/\sqrt{R}$.
Following the reasoning of Section \ref{sec:fixed_point}, the total arrival volume, consisting of new arrivals and reattempting customers, is $R_{\rm tot} = R+\alpha \sqrt{R}$, with
\begin{equation}
\label{eq:abandon_fixed_point}
\alpha = f_a(\beta - \alpha).
\end{equation}
Accordingly, this yields the following approximations for the delay and abandonment probability in the system with retrials
\begin{equation}
\label{eq:abandon_approximation}
\mathbb{P}_r^a({\rm delay}) \approx g_a(\beta-\alpha),\qquad \mathbb{P}_r^a({\rm abandon}) \approx \alpha / \sqrt{R}.
\end{equation}
We test our heuristic in the model with abandonment with parameters $R=100$, $\delta = 0.01$.
For $\theta = 0.2$, customers are quite patient, as they are willing to wait on average 5 times their expected service time.
Customer abandonment becomes more dominant for the cases in which customers are reasonably patient $\theta=1$ and very impatient $\theta = 10$.
In Figures \ref{fig:abandonment_accuracy_delay} and \ref{fig:abandonment_accuracy_ab}, we plot the simulated delay and (scaled) abandonment probability against approximations obtained through the fixed-point method. \\
\noindent
Again, we see a very good match between approximated and actual values.
As $\theta$ decreases, that is, customers become more patient, accuracy of the approximations improves.
This makes sense, since the volume of retrials decreases, and the system behaves more and more like a standard $M/M/s$ queue.
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/abandonments_delay}
\caption{Simulated (solid) and approximated (dashed) delay probability in the $M/M/s+M$ queue with retrials and $R=100$, $\delta=0.01$.}
\label{fig:abandonment_accuracy_delay}
\end{figure}
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/abandonments_ab}
\caption{Simulated (solid) and approximated (dashed) abandonment probability in the $M/M/s+M$ queue with retrials and $R=100$, $\delta=0.01$.}
\label{fig:abandonment_accuracy_ab}
\end{figure}
In Table \ref{tab:numerical_accuracy_abandonments} we also check the asymptotic accuracy of the model with abandonments and retrials and see that the approximation indeed improves as $R$ increases.
\begin{table}
\centering
\small
\begin{subtable}{0.99\textwidth}
\centering
\begin{tabular}{|r|r|rr||rr||rr|}
\cline{3-8}
\multicolumn{2}{c|}{} & \multicolumn{2}{c||}{$\theta = 0.2$} & \multicolumn{2}{c||}{$\theta = 1$} & \multicolumn{2}{c|}{$\theta = 10$} \bigstrut\\
\hline
$R$ & $s$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ \bigstrut\\
\hline
5 & 6 & 0.5703 & 0.1522 & 0.5423 & 0.4158 & 0.4766 & 1.0980 \bigstrut[t]\\
10 & 12 & 0.6612 & 0.2152 & 0.6277 & 0.5619 & 0.5429 & 1.5099 \\
50 & 54 & 0.5521 & 0.1517 & 0.5089 & 0.4009 & 0.3920 & 1.0251 \\
100 & 105 & 0.4896 & 0.1218 & 0.4456 & 0.3276 & 0.3282 & 0.8321 \\
500 & 511 & 0.4877 & 0.1228 & 0.4442 & 0.3302 & 0.3132 & 0.8135 \\
1000 & 1016 & 0.4992 & 0.1274 & 0.4472 & 0.3359 & 0.3148 & 0.8244 \bigstrut[b]\\
\hline
\multicolumn{2}{|r|}{Approx} & 0.4757 & 0.1182 & 0.4254 & 0.3120 & 0.2933 & 0.7695 \bigstrut\\
\hline
\end{tabular}
\caption{$\beta = 0.5$}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}
\centering
\begin{tabular}{|r|r|rr||rr||rr|}
\cline{3-8}
\multicolumn{2}{c|}{} & \multicolumn{2}{c||}{$\theta = 0.2$} & \multicolumn{2}{c||}{$\theta = 1$} & \multicolumn{2}{c|}{$\theta = 10$} \bigstrut\\
\hline
$R$ & $s$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ & $\mathbb{P}^a_r({\rm del})$ & $\sqrt{R}\,\mathbb{P}_r^a({\rm ab})$ \bigstrut\\
\hline
5 & 7 & 0.3130 & 0.0527 & 0.2908 & 0.1563 & 0.2382 & 0.4033 \bigstrut[t]\\
10 & 13 & 0.2732 & 0.0444 & 0.2503 & 0.1331 & 0.1917 & 0.3504 \\
50 & 57 & 0.2344 & 0.0373 & 0.2090 & 0.1120 & 0.1442 & 0.2974 \\
100 & 110 & 0.2244 & 0.0357 & 0.1999 & 0.1077 & 0.1355 & 0.2877 \\
500 & 522 & 0.2232 & 0.0355 & 0.1973 & 0.1079 & 0.1282 & 0.2842 \\
1000 & 1032 & 0.2210 & 0.0359 & 0.1979 & 0.1092 & 0.1287 & 0.2890 \bigstrut[b]\\
\hline
\multicolumn{2}{|r|}{Approx} & 0.2105 & 0.0335 & 0.1842 & 0.1005 & 0.1162 & 0.2642 \bigstrut\\
\hline
\end{tabular}%
\caption{$\beta=1$}
\end{subtable}
\caption{\normalsize Numerical results of the fixed-point method for the $M/M/s+M$ queue with slow retrials.}
\label{tab:numerical_accuracy_abandonments}
\end{table}
\begin{remark}
Note that even though the fixed-point approximations come close to the simulated values as $R$ increases, a small gap remains, especially notable in $\beta=0.5$.
This can be attributed to both rounding errors and the heuristic assumption that the retrial stream is independent Poisson.
The latter is obviously false, as the retrial process naturally depends on the history of the external arrival and service processes.
Therefore, the fixed-point heuristic slightly underestimates congestion levels in the actual system.
However, this error is relatively small and moreover in small to moderate-size systems negligible compared to the effects of rounding.
\end{remark}
\begin{remark}
Our fixed-point heuristic easily extends to the case in which only a fraction of abandoning customers returns to the system later.
If each customer who abandons decides (independent from others and his own retrial history) to return with probability $q\in[0,1]$, then the arrival stream due to retrial becomes $q\cdot\alpha\sqrt{R}$, so that the fixed-point becomes $f_a(\beta-q\alpha) = \alpha$.
Approximations of the performance measures follow accordingly.
\end{remark}
\section{Dimensioning}
\label{sec:retrial_dimensioning}
The asymptotic QED expressions for the systems we considered in Sections \ref{sec:basic_model}-\ref{sec:abandonments_retrials} without retrials together with the corrections obtained through the fixed-point equation provide a method for dimensioning large-scale systems with retrials.
For sufficiently large arrival volumes, we can tune the QoS-levels offered by the systems through the QoS-parameters $\beta$ and $\gamma$.
In this section we demonstrate how to do so in the cloud model, using the delay probability as a vehicle.
First we explore the procedure under stationary conditions, then in a time-varying environment.
The methods we propose easily translate to the two other model settings considered in this chapter, and the blocking probability.
\subsection{Stationary dimensioning}
We consider the dimensioning problem in the cloud model from a constraint satisfaction perspective.
That is, given the offered load $R$, we search for the pair $(s,n)$ that realizes a target delay probability $\e\in(0,1)$.
Relying on the two-fold scaling in \eqref{eq:twofold_scaling_cloud_model}, this under large offered loads $R$ is tantamount to finding the pair $(\beta,\gamma)$ that achieves asymptotic delay probability $\e$.
In a system without retrials, attaining this target performance boils down to finding a pair $(\beta_\e,\gamma_\e)$ such that $g_c(\beta_\e,\gamma_\e) = \e$.
The fixed-point heuristic however tells us that the model with retrials performs slightly worse, namely as if the QoS-parameters were $(\beta_\e-\alpha,\gamma-\alpha/\sqrt{\kappa})$ for $\alpha>0$.
Henceforth, to attain the target delay probability $\e$ in the limit with retrials, larger QoS parameters are required.
To be precise, $\beta^*_\e = \beta_\e+\alpha$ and $\gamma^*_\e = \gamma_\e+\alpha/\sqrt{\kappa}$, with $\alpha$ satisfying $\alpha = f_c(\beta^*-\alpha, \gamma^*_\e -\alpha/\sqrt{\kappa}) = f_c(\beta_\e,\gamma_\e)$.
Finally, we substitute $\beta^*_\e$ and $\gamma^*_\e$ in the scaling \eqref{eq:twofold_scaling_cloud_model} to obtain capacity levels $s$ and $n$.
Altogether, this yields the QED dimensioning procedure in Algorithm \ref{alg:cloud_stationary}, in which $[\cdot]$ denotes the integer rounding operator.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Offered load $R$\\
\hspace{1.1cm}Expected time spent in seconds stage $1/\kappa$\\
\hspace{1.1cm}Target delay probability $\e\in(0,1)$}
\hspace{1.1cm}\KwOut{Capacity levels $s$ and $n$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}
\item[] \hspace{0.5cm} 1. Compute $(\beta_\e,\gamma_\e)$ such that $g_c(\beta_\e,\gamma_\e)=\e$.
\item[] \hspace{0.5cm} 2. Set $\beta _\e^* = \beta_{\e} + f_c(\beta_{\e},\gamma_\e)$ and $\gamma_\e^* = \gamma_\e + f_c(\beta_{\e},\gamma_\e)/\sqrt{\kappa}$.
\item[] \hspace{0.5cm} 3. Return $s = \lceil R + \beta^*_\e\sqrt{R} \rceil$ and $n = [s +R/\kappa + \gamma^*_\e \sqrt{R/\kappa}]$.
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Stationary dimensioning for cloud model with retrials.}
\label{alg:cloud_stationary}
\end{algorithm}
\begin{table}
\centering
\small
\begin{subtable}{0.99\textwidth}
\centering
\begin{tabular}{|r|rr|r||rr|r||rr|r|}
\cline{2-10}\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$\e = 0.10$} & \multicolumn{3}{c||}{$\e = 0.25$} & \multicolumn{3}{c|}{$\e = 0.40$} \bigstrut[t]\\
\cline{2-10}\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (1.17,0.35)$} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (0.73,0.57)$} & \multicolumn{3}{c|}{$(\beta^*_\e,\gamma^*_\e) = (0.48,0.78)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ \bigstrut\\
\hline
10 & 14 & 25 & 0.1231 & 13 & 25 & 0.2268 & 12 & 24 & 0.3717 \bigstrut[t]\\
50 & 59 & 111 & 0.0969 & 56 & 110 & 0.2289 & 54 & 110 & 0.3844 \\
100 & 112 & 215 & 0.1069 & 108 & 214 & 0.2426 & 105 & 213 & 0.4148 \\
500 & 527 & 1035 & 0.0994 & 517 & 1030 & 0.2486 & 511 & 1028 & 0.3996 \\
1000 & 1038 & 2049 & 0.0977 & 1024 & 2042 & 0.2442 & 1016 & 2041 & 0.3925 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\kappa=1$}
\end{subtable}
\vspace{ 5mm }
\begin{subtable}{0.99\textwidth}
\centering
\begin{tabular}{|r|rr|r||rr|r||rr|r|}
\cline{2-10}\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$\e = 0.10$} & \multicolumn{3}{c||}{$\e = 0.25$} & \multicolumn{3}{c|}{$\e = 0.40$} \bigstrut[t]\\
\cline{2-10}
\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (1.34,0.50)$} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (0.87,0.65)$} & \multicolumn{3}{c|}{$(\beta^*_\e,\gamma^*_\e) = (0.59,0.79)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ \bigstrut\\
\hline
10 & 15 & 69 & 0.0893 & 13 & 68 & 0.2628 & 12 & 68 & 0.4238 \bigstrut[t]\\
50 & 60 & 318 & 0.1007 & 57 & 317 & 0.2210 & 55 & 318 & 0.3560 \\
100 & 114 & 625 & 0.0989 & 109 & 624 & 0.2504 & 106 & 624 & 0.4099 \\
500 & 530 & 3055 & 0.1051 & 520 & 3053 & 0.2449 & 514 & 3054 & 0.3857 \\
1000 & 1043 & 6078 & 0.0986 & 1028 & 6074 & 0.2459 & 1019 & 6075 & 0.3981 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\kappa=0.2$}
\end{subtable}
\vspace{ 5mm }
\begin{subtable}{0.99\textwidth}
\centering
\begin{tabular}{|r|rr|r||rr|r||rr|r|}
\cline{2-10}\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$\e = 0.10$} & \multicolumn{3}{c||}{$\e = 0.25$} & \multicolumn{3}{c|}{$\e = 0.40$} \bigstrut[t]\\
\cline{2-10}
\multicolumn{1}{r|}{} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (1.41,0.68)$} & \multicolumn{3}{c||}{$(\beta^*_\e,\gamma^*_\e) = (0.93,0.74)$} & \multicolumn{3}{c|}{$(\beta^*_\e,\gamma^*_\e) = (0.59,0.79)$} \bigstrut\\
\hline
$R$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ & $s$ & $n$ & $\mathbb{P}^c_r({\rm del})$ \bigstrut\\
\hline
10 & 15 & 530 & 0.0996 & 13 & 530 & 0.2814 & 13 & 531 & 0.2816 \bigstrut[t]\\
50 & 60 & 2594 & 0.1146 & 57 & 2594 & 0.2416 & 55 & 2595 & 0.3805 \\
100 & 115 & 5163 & 0.0933 & 110 & 5163 & 0.2323 & 107 & 5163 & 0.3794 \\
500 & 532 & 25640 & 0.1001 & 521 & 25638 & 0.2516 & 515 & 25641 & 0.3873 \\
1000 & 1045 & 51198 & 0.1018 & 1030 & 51196 & 0.2437 & 1021 & 51199 & 0.3900 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\kappa=0.02$}
\end{subtable}
\caption{\normalfont Results of the stationary dimensioning algorithm for $\varepsilon = 0.1,\, 0.25$ and $0.4$.}
\label{tab:cloud_dimensioning}
\end{table}
In Table \ref{tab:cloud_dimensioning} we performed this stationary dimensioning procedure for $\kappa = 1$, 0.2 and 0.02, and $\varepsilon = 0.1$, 0.25 and 0.4, and increasing offered loads $R$, and used simulation to obtain the actual delay probabilities.
We immediately see that the procedure yields remarkably good results, that are very close to the target delay probabilities.
\subsection{Time-varying dimensioning}
We next discuss how the parameters $s$ and $n$ can be adjusted in time-varying environments where the offered load $R(t)$ is a function of time.
For this we use the mean-offered-load (MOL) method, which was developed in \cite{Jennings1996} to approximate and dimension the $M_t/G/s$ system by establishing a relation with the analytically tractable $M_t/G/\infty$ system.
An underlying assumption of the MOL method is that a well-capacitated multi-server queue delays only a small portion of users and only for short periods.
Therefore, the system can be approximated by an infinite-server system.
The MOL approximation \cite{Jennings1996} combines the desirable QoS properties rendered by the QED regime with the analytic tractability of the $M/G/\infty$ queue, see \cite{Eick1993}, to establish a dynamic algorithm for choosing $s(t)$ that stabilizes the system behavior at some QoS-target.
To understand why the MOL approximation is likely to be accurate for the systems in this chapter, observe that under the QED scalings, the blocking probability vanishes asymptotically and hence the main assertion on which the MOL approximation is built continues to hold.
Following the line of thought in \cite{Jennings1996}, we consider the number of users in a system with $s = n = \infty$ to obtain $R_1(t) = \mathbb{E}[R(t-S)]\mathbb{E} [S]$, where $S$ is the service requirement per customer taken to be unit exponentially distributed. Then,
\begin{equation}
R_1(t) = \int_0^\infty e^{-u}R(t-u) \,{\rm d} u.
\label{eq:R1}
\end{equation}
Note that this transformation typically shifts and levels peaks in workload ahead in time with respect to those in $R(t)$. As the time-varying counterparts of $s$ in \eqref{eq:twofold_scaling_cloud_model}, we then get
\begin{align}
\label{eq:sTimeVarying}
s(t) &= R_1(t) + \beta\sqrt{R_1(t)}.
\end{align}
Secondly, the number of customers present in the system strongly depends on the number of customers in the second phase of the system, especially since $\kappa \ll \mu$.
Therefore, we moreover need an approximation for the workload offered to the second queue as a function of $t$, which we denote by $R_2(t)$.
Continuing the reasoning of MOL, we argue that $R_2(t)$ is equal to the output process of the first queue $R_1(t)$.
Then,
\begin{align}
R_2(t) = \mathbb{E}[R_1(t-S_2)]\mathbb{E}[S_2] = \int_0^\infty\int_0^\infty e^{-u-\kappa v} R(t-u-v) {\rm d} u\, {\rm d} v \label{eq:R2}
\end{align}
and the natural shape of $n(t)$ becomes
\begin{equation}
n(t) = s(t) + R_2(t) + \gamma\sqrt{R_2(t)}.
\end{equation}
Combining the above ingredients then leads to Algorithm \ref{alg:cloud_timevarying}.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Offered load function $R(t)$\\
\hspace{1.1cm}Expected time spent in second stage $1/\kappa$\\
\hspace{1.1cm}Target delay probability $\e\in(0,1)$}
\hspace{1cm}\KwOut{Capacity levels $s(t)$ and $n(t)$ achieving $P_r({\rm delay}) = \e$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}[noitemsep]
\item[] \hspace{0.5cm} 1. Compute $\beta^*_\e$ and $\gamma_\e^*$ according to Algorithm \ref{alg:cloud_stationary}.
\item[] \hspace{0.5cm} 2. Compute $R_1(t)$ and $R_2(t)$ as in \eqref{eq:R1} and \eqref{eq:R2}.
\item[] \hspace{0.5cm} 3. Return
\begin{align*}
\hspace{-0.5cm} s(t) &= \Big\lceil R_1(t)+\beta_\e^*\sqrt{R_1(t)}\Big\rceil , \nonumber \\
\hspace{-0.5cm} n(t) &= \Big[s(t) + R_2(t) + \gamma_\e^*\sqrt{R_2(t)}\Big].
\end{align*}
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Time-varying dimensioning algorithm for cloud model with retrials.}
\label{alg:cloud_timevarying}
\end{algorithm}
Observe that if the service times at the infinite-server queue are relatively short compared with the rate of change of the load function, we have $R_2(t)\approx R_1(t)/\kappa$, so that $n(t)$ as in Algorithm \ref{alg:cloud_timevarying} shows resemblance with \ref{eq:twofold_scaling_cloud_model}.
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/cloud_model_dimensioning}
\caption{Arrival rate function $R(t)$ (dashed) and staffing functions $s(t)$ (solid) and $n(t)$ (o) for different values of $\kappa$. The left vertical axis refers to $R(t)$ and $s(t)$, where the right axis refers to $n(t)$.}
\label{fig:staffingCurves}
\end{figure}
\begin{figure}
\centering
\input{Chapter_4/tikz_tex/cloud_model_results}
\caption{Simulated time-dependent delay probabilities in the cloud model with $\delta = 10^{-2}$, targets $\e = 0.1, \e=0.25$ and $\e=0.4$, and capacity levels determined by Algorithm \ref{alg:cloud_timevarying}.}
\label{fig:timevarying}
\end{figure}
To illustrate Algorithm \ref{alg:cloud_timevarying} we consider the time-varying load
\begin{equation}
R(t) = a + b \,\sin\left(2\pi t / T\right),
\end{equation}
where we set $a=1000$ as the mode, $b=500$ as the amplitude and $T=100$ as the cycle length.
This system experiences large fluctuations in load volume over the course of one cycle.
Since $\mu=1$, this implies that one cycle on average consists of 100 service times at the host server queue.
Due to relatively short service times with respect to the cycle length, the MOL approximation for the number of customers at the first queue is roughly equal to the original load, i.e.~$R_1(t) \approx R(t)$.
These short services at the first queue compared to the cycle length are typical for cloud systems, in which case the cycle is usually one day.
First, we examine the functions $s(t)$ and $n(t)$ as prescribed by Algorithm \ref{alg:cloud_timevarying} for $\kappa=1,0.2,0.02$ and $\e = 0.25$.
The resulting values are depicted in Figure~\ref{fig:staffingCurves} together with the arrival rate function.
Note that $n(t)$ lives on a different scale than $s(t)$, and has its own vertical axis at the right side of the plots.
For small and hence realistic values of $\kappa$, the function $n(t)$ displays a shifted phase compared to the real-time offered load, due to the relatively long service time at the second station.
The lag can be observed in~\eqref{eq:R2}.
Hence, while the number of servers $s(t)$ allocated at time $t$ is almost in phase with the arrival rate $R(t)$, $n(t)$ undergoes a shift of its peak capacity somewhat ahead in time.
Observe also that $n(t)$ shows milder fluctuations when $\kappa$ decreases. This can be attributed to the added hedge which is of order $\sqrt{R/\kappa}$.
Remark that the overcapacity is relatively small.
This illustrates the economies-of-scale that can be achieved in these large-scale systems.
Next, we simulate the time-dependent process, given the staffing functions depicted in Figure~\ref{fig:staffingCurves}, as well as the staffing functions designed for the target delay probabilities $\e =0.1$ and $\e =0.4$ for the three values of $\kappa$.
The results of the simulations are depicted in Figure \ref{fig:timevarying}. In all cases, the time-dependent delay probability only mildly fluctuates around the target.
As we increase $\e$, the stabilizing effect of the method weakens somewhat, which for other systems was also observed in \cite{Jennings1996}.
\section{Conclusion}
\label{sec:retrial_conclusion}
In this chapter, we studied the impact of retrying customers in large-scale systems in the QED regime.
The presence of retrials has a detrimental effect on congestion-related performance, compared to systems in which customers are simply discarded upon blockage/abandonment.
On the other hand, compared to similar systems without physical size restrictions or customer impatience, the performance gain can be substantial, if retrial times are relatively long compared to the service times.
Namely, retrials prompt temporary release of pressure from the system by shifting workload ahead in time.
Through our analysis, we have shown how the performance of large-scale queueing systems facing slow retrials can be approximated by appropriately combining a fixed-point technique with QED scaling.
We showed the remarkable accuracy of this approximation scheme in various retrial settings, that are otherwise intractable to analyze.
As we discussed in Section \ref{sec:retrial_dimensioning}, our novel asymptotic analysis technique is furthermore a powerful and elegant tool for dimensioning large-scale systems with slow retrials, which is moreover amenable to deal with time-varying demand. \\
\\*
We illustrate a few directions for future research.
As we explained before, the fixed-point method relies heavily on the premise that the blocking (or abandonment) probability vanishes at rate $1/\sqrt{R}$ in the QED regime, and on the availability of expressions for its limiting behavior.
Since this description likely fits a wide range of queueing models, we henceforth believe that our fixed-point method and the related dimensioning scheme find application beyond the three models we discussed here.
Secondly, in the dimensioning procedure of Section \ref{sec:retrial_dimensioning} we took a constraint satisfaction perspective in which we aimed to achieve a preset target QoS-level.
As an alternative approach, one could define a cost function to quantify the trade-off between capacity costs and customer dissatisfaction.
Specifically, suppose a cost $c_1$ is associated with each server per unit of time, cost $c_2$ is charged for every waiting customer per time unit, and cost $c_3$ is the penalty for each blocked customer.
Then in the $M/M/s/n$ queue with retrials in the QED regime, we use that $s = R+\beta\sqrt{R}$, the blocking probability is roughly $f(\beta-\alpha,\gamma)/\sqrt{R}$ and the expected waiting time is approximately $h(\beta-\alpha,\gamma)/\sqrt{R}$ for some function $h$, see \cite{masseywallace}, yielding total operational cost
\[
c_1\left( R + \beta\sqrt{R} \right)
+ c_2\, R\, \frac{f(\beta-\alpha,\gamma)}{\sqrt{R}}
+ c_3\, R\, \frac{h(\beta-\alpha,\gamma)}{\sqrt{R}} ,
\]
where $\alpha$ satisfies the fixed-point equation.
Hence, asymptotic dimensioning of the system boils down to finding the parameters $\beta^*$ and $\gamma^*$ that minimize
\[ c_1\beta^* + c_2\,f(\beta^*-\alpha^*,\gamma^*) + c_2\,h(\beta^*-\alpha^*,\gamma^*),\]
with corresponding fixed point $\alpha^*$.
Solving this optimization problem is not straightforward and a detailed study of this and related asymptotic dimensioning problems is an interesting avenue for future research.
Last, we remark that even though the fixed-point method works very well for systems with slow retrials, i.e.~$\delta\to 0$, it may also serve as an approximation to systems with short to moderate retrial times.
In these scenarios, the method is likely to underestimate congestion levels as it ignores dependencies between the primary and retrial stream of arrivals.
In the extreme case that $\delta\to\infty$, that is, blocked customers retry immediately, the customers in the retrial orbit basically form a (random order) queue outside the service facility.
When the inside of the facility consists of more than one queue, our fixed-point may be used as an heuristic approach to account for the increased workload that builds up outside the facility.
We explore this heuristic idea in a health care context in the next chapter.
\chapter{Finite-size effects in critically dimensioned emergency departments}
\begin{chapterstart}
Motivated by health care systems with repeated services that have both personnel (nurse/physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we account for are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them.
We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly.
We show that the system dynamics is captured by the fraction of needy time in the network, and that returning customers should be accounted for both in steady-state and transient environments.
We demonstrate the application of our policies in two case-studies of resource allocation in hospitals.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Finite-size effects in critically dimensioned emergency departments}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen, Fiona Sloothaak \& Galit Yom-Tov}\\
Submitted to \textit{Operations Research}
\end{flushright}
\newpage
\section{Introduction}
In recent years, operations research techniques has received increased interest from the health care community, as it is able to design and improve workflow processes in health care facilities\cite{Armony2015,Green2008,Bekker2009b,Hall2006,Hall2012}.
Because these processes are typically stochastic in nature, it is common practice to use queueing theory for performance analyses and workforce planning.
At a first step to gain understanding of the processes going on in health care environments, systems are commonly modeled after a single station queue, such as the $M/M/s$ (Erlang-C), $M/M/s/s$ (Erlang-B) or $M/M/s+M$ (Erlang-A) models, and fluid and diffusion approximations are used to provide insights into the process dynamics.
However, simple single station models often fail to capture the more intricate dynamics of the settings specific to health care contexts.
Think of flows of patients in a hospital from one medical ward to another \citep{Armony2015}, within the Emergency Department (ED) between different stages of treatment \citep{Junfei2015}, or between medical facilities \citep{zychlinski2016bedblocking}.
Queueing networks can capture the dependency between several service stages and several types of resources.
More specifically, we are interested in the ubiquitous feature, particularly present in health care environments, that patients during their stay in the system might require a specific resource multiple times, e.g.~physicians and nurses who treat patients several times during their stay in the medical wards \citep{Jennings2011} or the ED \citep{YomTov2014}, while multiple resources types are limited (e.g.\ physicians and beds).
In this chapter, we concentrate on the dynamics within EDs.
An often ignored yet essential feature of medical facilities concerns the restriction of the number of patients that can reside in the facility simultaneously.
In Chapter 4, we already observed that finite-size restrictions can have a significant effect on the performance of systems.
In this chapter, we investigate the influence of such multiple restrictions on the network dynamics and the required staffing policies in the context of an ED. \\
\\*
\noindent
\textbf{The restricted Erlang-R model.}
The canonical model for service networks with returns is the Erlang-R model \citep{YomTov2014} in which customers, during their stay in the system receive a random number of services from the same pool of servers.
Yom-Tov \& Mandelbaum \cite{YomTov2014} showed that such a simple network model can be used to determine staffing in an Israeli ED both in stable and time-varying conditions.
Nevertheless, empirical studies report that some countries, such as the US, use a different operational mode that apply strict restrictions on entering the ED \citep{EDexperiment}.
In typical US EDs, a patient will not enter the ED until both a bed and a physician are available to treat her.
Those restrictions can be either physical (beds) restrictions or managerial ones---for instance by imposing a patient-to-physician ratio.
In this work, we extend the Erlang-R model by enforcing a constraint on the maximum number of available places inside the facility.
Our model hence incorporates two kinds of resource constraints: servers that provide the actual service and the maximum available places inside the service system.
Both affect the system in a highly interdependent way.
The model, presented in Figure \ref{fig:Erlang_R_model}, assumes $s$ servers and a maximum capacity of $n$ concurrent places.
We assume that customers arrive according to a Poisson process with rate $\lambda(t)$.
In case a new arrival finds $n$ or more customers already present, we consider two options---either she waits outside the service facility in a holding queue until a vacant space becomes available (Figure \ref{fig:Erlang_R_holding}) or she is blocked (Figure \ref{fig:Erlang_R_blocking}), such as is the case when patients are sent to an alternative facility.
Once customers are admitted, they require assistance from one of the $s$ servers for an exponentially distributed duration with mean $1/\mu$.
Then, with probability $1-p$, customers leave the system or, with probability $p$, return to service again after an exponentially distributed time with mean $1/\delta$.
Following Jenning \& de V\'ericourt \cite{Jennings2011} and Yom-Tov \& Mandelbaum \cite{YomTov2014}, we call patients {\it needy} when they require attention from one of the servers and {\it content} when they are in the delayed return phase.
In addition, we call customers {\it holding} when they are waiting outside the facility for an available space. We assume that the arrival process, the needy times and content times are mutually independent.
In the holding queue and the needy queue, we apply the First-Come-First-Served (FCFS) discipline.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-2.75,4.5) -- (-1.25,4.5);
\draw [thick] (-1.5,5) -- (0,5) -- (0,4) -- (-1.5,4);
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (0,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick,->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,2.9) {\footnotesize Poisson($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\node [above] at (-0.75,5) {\footnotesize holding};
\end{tikzpicture}
\caption{Erlang-R model with holding.}
\label{fig:Erlang_R_holding}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-1.5,4.5) -- (2.5,4.5);
\draw [thick, ->] (0,4.5) -- (0,2.5) node[below left] {blocked};
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick, ->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,3.4) {\footnotesize Poisson($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{Erlang-R model with blocking.}
\label{fig:Erlang_R_blocking}
\end{subfigure}
\caption{Restricted Erlang-R models with maximally $n$ customers in system.}
\label{fig:Erlang_R_model}
\end{figure}
As mentioned, we consider two versions of the finite-capacity constraint.
The first version is called \emph{Erlang-R with holding}, in which patients wait for an available space in the system.
The second version is called \emph{Erlang-R with blocking}, in which patients meeting a full system are blocked.
Naturally, intermediate scenarios can be constructed in which a proportion of the total arrival volume of customers indeed leaves upon finding a full system, while the rest joins the holding room.
While this chapter focuses on the two extreme cases, straightforward adaptions can fit these intermediate scenarios. \\
\\*
\noindent
\textbf{Examples of restricted Erlang-R.}
As noted before, an ED operated in the US can be modeled using a restricted Erlang-R model. Another health care example are medical units (MUs) in a hospital.
Such units specialize in specific types of illnesses (cadriatric, oncology, etc) and have limited resources such as nurses and beds. If the unit is full, new patients are either allocated to an alternative medical unit, i.e.\ blocked, or wait for an available bed.
Both policies are problematic in terms of quality-of-care, because the personnel in the alternative unit (or the ED) may be less knowledgeable and waiting in the ED was shown to increase mortality.
Moreover, ED waiting may reduce available capacity for treating ED patients \citep{Carmen2016,israelit}, hence endangering both the delayed patient as well as others. Both the number of personnel (nurses and physicians) and the number of beds impact service dynamics and quality-of-care. Research so far looked at the capacity allocation of those resources separately. Green \& Yankovic \cite{GY2011} and Jennings \& de V\'ericourt \cite{Jennings2008} looked at nurse staffing in medical units, while de Bruin et al.~\cite{Bekker2009b} looked into bed allocation. The unified model we suggest enables us to capture the dependency between those two decisions, and its impact on other medical units in the hospital.
At the same time, we capture the two most commonly used modes of operation---blocking and holding of new patients. \\
\\*
\noindent
\textbf{Two-fold square-root staffing rule.}
Our main goal is to provide staffing policies for the ED that ensures high resource utilization, while at the same time maintains a good quality-of-care.
This goal relates to the philosophy of the Quality-and-Efficiency-Driven (QED) regime that is the recurring theme of this thesis.
In this chapter, we obtain asymptotic results for the Erlang-R model with blocking in the QED regime (Section \ref{sec:QED_limit_block}).
Following \cite{Jennings2008}, we employ a two-fold QED staffing policy: $s=R_1 +\beta \sqrt{R_1}$ for the number of nurses and $n=R_1/r+\gamma \sqrt{R_1/r}$ for the number of patients in the system (beds), where $\beta$ and $\gamma$ are constants, $R_1$ is the offered load of the servers (nurses) and $r$ is the fraction of time a customer spends in the needy state.
We establish limiting expressions for performance measures, such as the probability of delay and blocking, in the form of explicit functions that depend solely on $\beta$ and $\gamma$.
In deriving these limit results, we use the available product-form solution for the stationary distribution.
Likewise, we pursue QED performance for the Erlang-R model with holding.
However, a direct analytic approach is obstructed by the absence of product-form solutions.
We provide two solutions for establishing QED behavior.
First, we provide stochastic performance bounds that stay meaningful in the QED regime, which demonstrate the non-degenerate behavior of the two-fold scaling in the large-system limit.
Second, we develop a heuristic method that quantifies the difference between the scalable holding model and the blocking model.
This method is to a large extend related to the asymptotic approximation method for retrial queues discussed in Chapter 4, in the sense that we approximate the model with holding through the model with blocking, yet with an increased arrival rate.
The increase in arrival rate turns out to be the solution of a fixed-point equation.
Using our results on the asymptotic behavior of the model with blocking in the QED regime, we then obtain approximative QED performance measures for the model with holding.
These theoretical findings ultimately yield algorithms for dimensioning and time-varying staffing. \\
\\*
\textbf{Structure of the chapter.}
We first review related literature on the subject of staffing in health care environments in Section \ref{sec:ed_literature}.
In Section \ref{sec:modeldescription}, we introduce the mathematical models more formally, and deduce preliminary results on their stability conditions and relative performance.
Section \ref{sec:QED_scaling} describes the scaling regime we use for our asymptotic study of the restricted Erlang-R models, and Sections \ref{sec:QED_limit_block} and Section \ref{sec:QED_limit_holding} present our main theoretical findings.
We turn to dimensioning problems in Section \ref{sec:dimensioning}, and show how our asymptotic QED results can be used to make resource allocation decision in realistic settings.
Section \ref{sec:analysis} is devoted to the numerical and comparative analysis of the restricted Erlang-R models, and also shows how our method can be applied in time-varying environments through a case study.
We summarize our findings and give directions for future research in Section \ref{sec:conclusion}.
\section{Literature review}
\label{sec:ed_literature}
Due to increasing demand and tightening budgets in health care, there is a growing need for efficient workforce management \citep{Green2008}. Personnel (nurse and physician) expenditure is one of the biggest factors in hospital costs \citep{Kazahaya2005}, and inadequate nursing levels have been mentioned as a significant factor in medical errors and ED overcrowding. In order to establish appropriate nursing levels, a staffing policy requires assessment of a wide range of variables, such as differing nurse expertise and patient acuity during the day. Current methods, such as the minimum nurse-to-patient ratios, are often too inflexible to capture those varying conditions. The American Hospital Association (AHA) and others call for dynamic staffing policies that can deal with the complex and evolving nature of health care \citep{AHA2007}.
Workforce management in health care systems has been studied extensively; see \cite{Denton2013,Hall2006,Hall2012} for overviews.
In recent years it has become apparent that queueing models can be helpful in developing staffing and routing recommendations, not just for large-scale service systems, but also for the small and complicated health care systems.
The first to try such an approach through queueing models were Green et al.~\cite{Green2006,Green2008}, who used the single station stationary Erlang-C model to set staffing levels in EDs and panel sizes for clinics. Using a similar approach, Bekker \& de Bruin~\cite{Bekker2009a} used Erlang-B model to determine bed allocation for medical wards.
The first to observe the significant impact of interrupted services in a health care setting were Jennings \& de V\'ericourt \cite{Jennings2008,Jennings2011}. Motivated by the need to set nurse-to-patient ratios for internal wards, they considered a closed queueing system with $s$ nurses and $n$ beds. This is essentially the Erlang-C model with the additional restriction that a finite population of the $n$ patients requires care. In their model, all beds are always occupied, and patients alternate between two phases: the needy phase where patients require service of a nurse and the content phase where they do not; see Figure \ref{fig:Jennings}. The system dynamics of restricted Erlang-R model are equivalent to those of the closed ward model of \cite{Jennings2008} if the holding queue would never be empty.
Campello et al.~\cite{Campello2016} analyzed a similar operational decision, referred to as ED case management, which determines the maximal number of patients a physician should handle in parallel. They also used queueing networks and analyzed the stationary distribution. Note that in practice such decision is not only affected by operational measurements such as waiting times, but also by psychological constraints that limit physician capability to manage multiple tasks (patients) in parallel.
KC \cite{diwas} provided empirical evidence that physicians should not treat more than 6-7 patients at the same time. Therefore, many hospitals in the US restrict entrance to EDs even if beds are available if physicians are overloaded.
We too consider such constraints, and analyze their impact on performance. We take a different approach than \cite{Campello2016}; instead of analyzing numerically steady-state distributions, we develop many-server approximations that can produce insight into the system dynamics, and can be incorporated into time-varying staffing procedures; see Section \ref{sec:case_study}.
\begin{figure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=1]
\draw [dashed, thick] (-0.5,-0.1) rectangle (3.5,2.85) node[right] {\footnotesize $n$};
\draw [thick,->] (1.1,0.5) -- (0,0.5) -- (0,2) -- (0.6,2);
\draw [thick,->] (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw [thick] (0.6,1.7) -- (1.6,1.7) -- (1.6,2.3) -- (0.6,2.3);
\draw [thick] (1,1.7) -- (1,2.3);
\draw [thick] (1.2,1.7) -- (1.2,2.3);
\draw [thick] (1.4,1.7) -- (1.4,2.3);
\draw (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw (1.5,0.5) circle [radius=0.4] node[above=0.3cm] {\footnotesize $p/\delta$} ;
\draw (2,2) circle [radius=0.4] node[above=0.3cm] {\footnotesize exp($\mu$)};
\node at (1.5,0.5) {\footnotesize $\infty$};
\node at (2,2) {\footnotesize $s$};
\end{tikzpicture}
\caption{The closed ward model.}
\label{fig:Jennings}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\draw [thick, ->] (0,4.5) node[above=0.3cm,right] {\footnotesize $\lambda$} -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5];
\draw [thick, ->] (4.75,4.5) -- node[above=0.3cm,right] {\footnotesize $1-p$} (7,4.5);
\draw [thick,->] (5.75,4.5) -- node[right] {\footnotesize $p$} (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5];
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node at (4.25,4.5) {\footnotesize $s$};
\node at (3.5,2) {\footnotesize $\infty$};
\node [above] at (4,5.15) {\footnotesize exp($\mu$)};
\node [above] at (3.5,2.4) {\footnotesize exp($\delta$)};
\end{tikzpicture}
\caption{The Erlang-R model.}
\label{fig:ErlangR}
\end{subfigure}
\caption{Related queueing models.}
\end{figure}
The model in~\cite{Jennings2008,Jennings2011} was developed for modeling internal dynamics within an internal ward. However, in the ED, beds are not constantly occupied and the utilization level depends on the flow of patients that arrive from outside the system.
Yom-Tov \& Mandelbaum \cite{YomTov2014} highlight the interrupted services while accounting for the transient nature of patient's arrival process, and introduced the Erlang-R model as a model for an ED. The Erlang-R model is an open two-station queueing network that has the same layout as the restricted Erlang-R model, except that all patients find a bed available upon arrival, see Figure \ref{fig:ErlangR}. In both models patients experience the interrupted services, but the Erlang-R model has no further restrictions on the bed capacity, hence neglecting the finite-size effects. Yom-Tov \& Mandelbaum \cite{YomTov2014} showed, using a simulator tailored to an Israeli ED, that the complicated small ED dynamics can be captured using the relatively simple Erlang-R model, and hence, its recommendations can be implemented in ED workforce management.
Although the feature of interrupted services is present in many systems, it is particularly important for modeling EDs, because the duration of the interruption is typically much longer than the time patients require care from a nurse. This explains why the Erlang-R model is considered to be the canonical model for EDs. The restricted Erlang-R model with holding/blocking thus extends the Erlang-R model with finite-size constraints which, like interrupted services, are expected to have a decisive impact on performance.
\section{Models and performance measures}
\label{sec:modeldescription}
\subsection{Three-dimensional Markov process}
\label{sec:Markov_process}
Since in the restricted Erlang-R model described the arrival process is taken Poisson, and all service and content times are assumed independent and exponential, the system can be characterized in terms of a Markov process.
Let $Q(t) = (H(t),Q_1(t),Q_2(t))$ represent the number of patients in the \emph{holding}, \emph{needy} and \emph{content} state at time $t$, respectively.
In both variants, $n$ is the maximum number of patients admitted to system, we have $Q_1(t)+ Q_2(t)\leq n$ for all $t\geq 0$.
Due to the absence of holding patients in the Erlang-R model with blocking, $H(t)=0$ is enforced in this case, whereas $H(t)$ has unbounded support in the model with holding.
This distinction requires us to explore the stationary distribution of the two variants separately.
Before doing so, we introduce some additional notation.
We define
\begin{equation}
R_1 := \frac{\lambda}{(1-p)\mu}, \qquad R_2 := \frac{p\lambda}{(1-p)\delta},
\label{eq:R1_R2}
\end{equation}
where $R_1$ and $R_2$ can be interpreted as the offered workload brought towards the needy queue and the content (infinite-server) queue, respectively.
Furthermore, we define
\begin{equation}
r:= \frac{\delta}{\delta+p\mu},
\label{eq:delta}
\end{equation}
which is the fraction of time a patient spends in the needy state (in case she experienced no wait during her sojourn). \\
\\*
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.9]
\draw [thick] (-1.25,5) -- (0,5) -- (0,4) -- (-1.25,4);
\draw [thick] (0.5,4.5) circle [radius = 0.5] node {\footnotesize 1} node[above=0.5cm] {\footnotesize exp$(\lambda)$}
node[below =0.5cm] {\footnotesize \color{col1} Station 0} ;
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (1,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {\footnotesize $s$} node[above=0.5cm] {\footnotesize exp$(\mu)$}
node[below = 0.5cm] {\footnotesize \color{col1} Station 1} ;
\draw [thick, ->] (4.75,4.5) -- node[right=0.8cm,above] {\footnotesize $1-p$} (6.5,4.5) -- (6.5,1.6) -- (-2,1.6) -- (-2,4.5) -- (-1.2,4.5);
\draw [thick,->] (5.75,4.5) -- node[left] {\footnotesize $p$} (5.75,2.5) -- (4,2.5);
\draw [thick] (3.5,2.5) circle [radius=0.5] node {\footnotesize $\infty$} node[above=0.4cm] {\footnotesize exp$(\delta)$}
node[below right = 0.35cm] {\footnotesize \color{col1} Station 2} ;;
\draw [thick,->] (3,2.5) -- (1.5,2.5) -- (1.5,4.5);
\draw [thick, dashed] (-3,1.2) rectangle (7.25,5.75) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{The Erlang-R model with blocking viewed as a closed Jackson network.}
\label{fig:ErlangR_blocking}
\end{figure}
\noindent
\textbf{Erlang-R model with blocking.}
In case of the blocking model, $Q(t)$ reduces to a finite-state Markov process $Q(t) = (Q_1(t),Q_2(t))$, where $Q_1(t)+Q_2(t)\leq n$ for all $t\geq 0$.
In fact, this is equivalent to the closed Jackson network depicted in Figure \ref{fig:ErlangR_blocking} with finite population $n$.
Station 1 in Figure \ref{fig:ErlangR_blocking} is an $M/M/s$ queue with service rate $\mu$, modeling the number of needy patients $Q_1(t)$.
Station 2 models the number of content patients $Q_2(t)$, and can therefore be represented as an infinite-server queue with service rate $\delta$.
A patient can enter the unit only if $Q_1(t)+Q_2(t)<n$.
Station 0---a single-server queue---moderates this as it only produces output at rate $\lambda$ in case its queue length is positive, i.e.\ if $n-Q_1(t)-Q_2(t)>0$.
Observe that because patients finding a full network are blocked, the number of patients in the system cannot grow beyond $n$.
Hence, the system is stable for all parameter settings, and hence a steady-state distribution exists. Moreover, the simplification of the model with blocking allows us to express the steady-state distribution of the system in explicit product-form.
Let $\pi_b(j,k)$ denote the steady-state probabilities of having $j$ needy and $k$ content patients in the system. Then,
\begin{equation}\label{eq:pih(i,j)}
\pi_b(j,k) = \left\{
\begin{array}{ll}
\pi_0\,\frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k, & ~~~\text{if }j+k \leq n,\\
0, & ~~~\text{else,}
\end{array}\right.
\end{equation}
where
\begin{equation*}
\kappa(j) := \left\{
\begin{array}{ll}
j! , & ~~\text{if }j \leq s,\\
s!\, s^{j-s}, &~~ \text{else,}
\end{array}\right.
\end{equation*}
and $
\pi_0^{-1} = \sum_{j+k\leq n} \frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k$.\\
\\*
\textbf{Erlang-R model with holding.}
\label{ref:modelsoft}
The Erlang-R model with holding does not lead to a Jackson network with an elegant product-form solution for the steady-state distribution, because the holding queue cannot be modeled as a station that is independent from the other queues in the system.
%
However, we are able to describe the system as a two-dimensional Markov process without loss of information.
To see this, define $N:= \{N(t),\,t\geq 0\}$ with $N(t) := H(t)+Q_1(t) + Q_2(t)$, the total number of patients in the system (including the holding queue).
Using the restriction $Q_1(t)+Q_2(t) \leq n$ together with the fact that no bed is left vacant if a patient is waiting in the holding queue, this yields
\begin{equation*}
H(t) = \left(N(t) - n\right)^+, \quad t\geq 0,
\end{equation*}
where $(\cdot)^+ := \max\{0,\cdot\}$.
For the same reason, $Q_2(t) = N(t) - Q_1(t)$ if $H(t)=0$, and $Q_2(t) = n-Q_1(t)$ otherwise.
In other words,
\begin{equation*}
Q_2(t) = \min\{N(t),n\} - Q_1(t), \quad t \geq 0.
\end{equation*}
Therefore, we can express the state of all three queues in the Erlang-R model with holding using a two-dimensional Markov process $X:= \{X(t),\,t\geq 0\}$, where
\begin{equation*}
X(t) :=\left( N(t), Q_1(t) \right).
\end{equation*}
The process $X$ lives on the semi-infinite strip
\begin{equation*}
X(t) \in \left\{\,(i,j)\, |\, j \leq \min\{i,n\}, i\in \mathbb{N}_0, j \in \{0,1,\ldots,n\}\, \right\},
\end{equation*}
and belongs to the class of Quasi-Birth-Death (QBD) processes.
The reader is referred to Appendix~\ref{app:QBDdescription} for a detailed description of this process, in terms of its transition diagram and generator matrix.
Contrary to the model with blocking, the system with holding \emph{can} become unstable in case capacity is insufficient to satisfy patient demand.
\begin{proposition}\label{prop:StabilityCondition}
The Erlang-R model with holding is stable if and only if
\begin{equation}
\frac{\lambda}{(1-p)\mu s} < \frac{ \sum_{i=0}^s \frac{i}{s}\binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
{ \sum_{i=0}^s \binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
=: \rho_{\max}(s,n).
\label{eq:StabilityCondition}
\end{equation}
\end{proposition}
The proof is given in Appendix~\ref{app:stability} and follows from the general theory for QBD processes.
Observe that $\rho_{\max}(s,n)$ poses an upper bound on the occupancy level of the servers in the holding model, which is clearly smaller than 1 for all $s$ and $n$.
In addition, this implies that the maximum workload $R_{\max}(s,n) := s\cdot\rho_{\max}(s,n)$ the system is able to handle is strictly less than $s$.
If we compare this to the open Erlang-R model, in which the maximal attainable workload equals $s$, we observe the effect of finite-size constraints on operational performance.
Figure \ref{fig:Rmax} shows the influence of both $s$ and $n$ on the maximum feasible workload in case $r=0.25$.
From these graphs, note that if $s\ll rn$, $R_{\max}$ grows almost linearly with $s$.
Furthermore, $R_{\rm max}(s,n)$ is increasing in $n$ for $s$ fixed.
A logical practical consequence is that a larger number of beds allows for a larger patient volume to enter the ED with the same number of nurses.
Moreover, $R_{\rm max}(s,n)$ is increasing in $s$, but as in Figure \ref{fig:Rmax_a}, adding an extra nurse does not increase the stability region in case $n$ is too tight.
Conversely, adding extra beds does not increase $R_{\rm max}(s,n)$ if the number of nurses does not allow for an increase in offered load, see Figure \ref{fig:Rmax_b}.
Additionally, it is easily verified that $R_{\rm max}(s,n)$ is upper bounded by both $s$ and $R_{\rm max}(n,n) = rn$. Therefore, a careful balance is called for between servers (nurses) and beds, so that resources will be efficiently utilized. We observe that when the ratio $s/n\approx r$, the system is better balanced.
We will propose an appropriate balance between resources by defining a synchronized QED capacity recommendation for both servers and beds in Section \ref{sec:QED_scaling}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $s$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*] table[x=s,y=n20] {tikz/stability/r025_n_fixed.txt};
\addplot[col3,thick,mark=*] table[x=s,y=n40] {tikz/stability/r025_n_fixed.txt};
\addplot[col4,thick,mark=*] table[x=s,y=n60] {tikz/stability/r025_n_fixed.txt};
\addplot[col5,thick,mark=*] table[x=s,y=n80] {tikz/stability/r025_n_fixed.txt};
\legend{$n=20$,$n=40$,$n=60$,$n=80$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $s$.}
\label{fig:Rmax_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 100,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $n$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*,mark repeat = 2] table[x=n,y=s5] {tikz/stability/r025_s_fixed.txt};
\addplot[col3,thick,mark=*,mark repeat = 2] table[x=n,y=s10] {tikz/stability/r025_s_fixed.txt};
\addplot[col4,thick,mark=*,mark repeat = 2] table[x=n,y=s15] {tikz/stability/r025_s_fixed.txt};
\addplot[col5,thick,mark=*,mark repeat = 2] table[x=n,y=s20] {tikz/stability/r025_s_fixed.txt};
\legend{$s=5$,$s=10$,$s=15$,$s=20$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $n$.}
\label{fig:Rmax_b}
\end{subfigure}
\caption{The maximum achievable workload in the restricted Erlang-R model with holding for $r=0.25$.}
\label{fig:Rmax}
\end{figure}
Provided that the system is stable, the stationary distribution of the QBD process $X$ can be obtained numerically by the matrix geometric method \citep{Neuts1981}.
Subsequently, we can derive the stationary distribution of the original $Q(t)$, denoted by $\pi_h(\cdot,\cdot,\cdot)$.
\subsection{Performance measures}
\label{sec:performance_metrics}
In this work, we concentrate on five performance measures that are central to our analysis.
In the definitions that follow, we present expressions for these measures in terms of a general three-dimensional measure $\pi$, which one can replace by either $\pi_b$ or $\pi_h$, depending on the scenario considered.
In the remainder of this work, we will augment the measures related to the Erlang-R model with blocking and holding by the superscript $b$ and $h$, respectively\footnote{In line with $H(t)=0$, we use $\pi_b(i,j,k) = \pi_b(j,k)$ if $i=0$, with $\pi_b(j,k)$ as in \eqref{eq:pih(i,j)}, and $\pi_b(i,j,k) = 0$ otherwise, when considering the model with blocking.}.
As relevant performance measures, we consider the probability of holding (blocking) at entering the system, the probability of delay at the needy queue, expected waiting time for a nurse, utilization of nurses and utilization of beds:
\begin{equation}
\mathbb{P}({\rm hold}) = \sum_{i=0}^\iy \sum_{j=0}^n \pi(i,j,n-j), \qquad
\mathbb{P}({\rm delay}) \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \pi(i,j,k),
\label{eq:delay_probability}
\end{equation}
\begin{equation}
\label{eq:EW_exact}
\mathbb{E} [W] \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \frac{\max\{0,j-s+1\}}{\mu}\,\pi(i,j,k),
\end{equation}
\begin{equation}
\label{eq:utilization}
\rho_s = \frac{1}{s}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{j,s\} \pi(i,j,k), \qquad
\rho_n = \frac{1}{n}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{i,n\} \pi(i,j,k).
\end{equation}
It should be stressed that the above expression for the delay probability and the expected waiting time for a nurse is not exact. For the blocking model one can use the Arrival Theorem, see e.g.~\cite{Chen2001a}, whereby the exact expression uses $n-1$ instead of $n$; for the holding model, the arrival process to the needy queue, which consists of both external arrivals and content patients becoming needy, is not Poisson. Therefore, we cannot use the PASTA argument for the holding model. However, for both models, as we will be studying the system as $s$ and $n$ become large, this approximation error will become negligible.
\subsection{Stochastic bounds}
\label{sec:bounds}
Although the two variants of the Erlang-R model differ with respect to the admission policy, and require different mathematical treatment, we would like to be able to capture their relative performance.
We substantiate the intuition that the holding room leads to more patients in the ED, in the following result.
\begin{proposition}\label{thm:stochasticordering}
Let $Q_1^b$, $Q_2^b$, $Q_1^h$, $Q_2^h$ denote the nurse and content queue length processes in the Erlang-R model with blocking and holding, respectively.
Let $H(0) = 0$, $Q_1^b(0)=Q_1^h(0)$ and $Q_2^b(0)=Q_2^b(0)$. For all $t\geq 0$,
\begin{align}
Q_1^b(t) + Q_2^b(t) &\preceq_{\rm st} Q_1^h(t) + Q_2^h(t) \preceq_{\rm st} n ,\\
Q_2^b(t) &\preceq_{\rm st} Q_2^h(t),\\
Q_1^b(t) &\preceq_{\rm st} Q_1^h(t) + H(t),
\end{align}
where $X\preceq_{\rm st} Y$ implies $\mathbb{P}(X\geq k) \leq \mathbb{P}(Y\geq k)$ for all $k\geq 0$.
\end{proposition}
\noindent
The proof of Proposition \ref{thm:stochasticordering} uses sample path coupling and can be found in Appendix \ref{app:stochastic_ordering}.
Note that as an immediate consequence, we have
\[ \mathbb{P}^b( {\rm block}) = \lim_{t\to\iy} \mathbb{P}\big( Q_1^b(t)+Q_2^b(t) \geq n \big) \leq \lim_{t\to\iy} \mathbb{P}\big( Q_1^h(t) + Q_2^h(t) \geq n \big) = \mathbb{P}^h( {\rm hold }) \]
and by similar reasoning $\rho^b_n \leq \rho_n^h$.
In other words, under similar offered load and capacity constraints, utilization levels for the nurses in the Erlang-R model with blocking are lower than in the Erlang-R model with holding.
Moreover, the total number of waiting patients in the setting with holding is stochastically larger than in the setting with blocking, and in the open Erlang-R model.
We further discuss the differences between both models in Section \ref{sec:dimensioning} and Section \ref{sec:analysis}.
\section{Two-fold QED regime}
\label{sec:QED_scaling}
We do not want to waste capacity of either servers or beds without getting significant advantage in term of performance.
We therefore take an asymptotic approach that lets the external arrival rate $\lambda$ grow to infinity, while scaling $s$ and $n$ accordingly.
In doing so, we intend to establish QED-type system behavior, i.e.\ high occupancy levels of both nurses and beds and good quality-of-service.
\subsection{Two-fold scaling rule}
In order to identify the scaling of $s$ and $n$ as $\lambda\to\infty$, we draw inspiration from the two-fold scaling rule used by Jennings \& de V\'ericourt \cite{Jennings2008} and Khudyakov et al.~\cite{Khudyakov2010}, which follows the celebrated square-root staffing principle.
This principle suggests that, in the most general setting, capacity should be equal to the expected offered load entering the system, let us say $R$, plus an additional variability hedge that is proportional to $\sqrt{R}$.
In the restricted Erlang-R model, we have two capacity sources, namely $s$ and $n$, which experience different relevant amount of works.
The offered load the servers in the needy queue experience is given by $R_1$, as in the regular Erlang-R model;
it does not change due to the finite-size effects, since all patients are served eventually. Hence, we only need to account for the interrupted services. It follows that the appropriate staffing rule for the nurses in the QED regime remains $s=R_1+\beta \sqrt{R_1}$ for some constant $\beta >0$.
To establish the bed capacity level, we need to reflect on the load offered to the beds. Observe that beds remain occupied both in needy and content states. This suggests that $R_n:=R_1+R_2=R_1/r$, with $R_1$ and $R_2$ as in \eqref{eq:R1_R2} and $r$ is the expected fraction of time a patient spends at the nurse station defined in \eqref{eq:delta}.
As a result, the appropriate staffing rule is $n=R_n+\gamma \sqrt{R_n}$ for some constant $\gamma>0$. In conclusion, the two-fold QED scaling rule is given by
\begin{equation}\label{eq:twofoldscaling}
\begin{array}{ll}
s &= R_1 + \beta \sqrt{R_1} + o(\sqrt{R_1}) \\
n &= \frac{R_1}{r}+\gamma \sqrt{\frac{R_1}{r}} + o(\sqrt{R_1})
\end{array}
\end{equation}
with $\beta,\gamma>0$ constants and $R_1:=\lambda/((1-p)\mu)$.
Recall that we saw in Figure \ref{fig:Rmax} that resources seem efficiently utilized if $s/n\approx r$.
Scaling \eqref{eq:twofoldscaling} is in line with this reasoning since
\[
\frac{s}{n} = r\left(1+ \frac{\beta - \gamma\sqrt{r}}{\sqrt{R_1}}+ O(1/R_1) \right) .
\]
\begin{remark}
In \cite{Jennings2008}, a similar scaling regime is considered, which only relates $s$ and $n$ through a square-root scaling, namely the regime $s = r n + \hat\gamma\sqrt{n}$,
which is equivalent to the second relation in \eqref{eq:twofoldscaling} if $\hat\gamma = \beta\sqrt{r} - \gamma r$.
Due to the absence of external arrivals in this closed system, they let the number of beds $n$ approach infinity as opposed to $\lambda$ in our settings.
Nevertheless, this results in the same asymptotic regime.
\end{remark}
Before turning to asymptotic expressions for the performance measures concerning the Erlang-R model with blocking or holding, we conduct a few numerical experiments to confirm that the scaling in \eqref{eq:twofoldscaling} indeed leads to desired QED behavior.
In Figure \ref{fig:sample_paths}, we plotted the sample paths of the three-dimensional queue length process of the holding model in which $\beta$ and $\gamma$ are fixed, and $R_1$ is increased.
Observe that the needy queue length $Q_1(t)$, plotted in orange in Figure \ref{fig:sample_paths}, fluctuates around the values $s$, and stabilizes for larger values of $R_1$.
This naturally implies that the server (nurses) utilization approaches 100\%, while the number of patients waiting is $O(\sqrt{R_1})$.
Furthermore, we see that the percentage of occupied beds also tends to 100\%, while the holding queue length remains small.
The holding queue is of much smaller order than $R_1$, which implies that the holding time of a patient becomes negligible as $R_1\to\iy$.
From these empirical findings we deduce that under scaling \eqref{eq:twofoldscaling} the restricted Erlang-R model exhibits QED behavior on two levels: Outside the facility while waiting for an available bed, and inside the facility while waiting for attention of a nurse.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.54]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 28,
ytick = {0,5,10,15,20,25},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\definecolor{col1}{rgb}{0.368417, 0.506779, 0.709798}
\addplot[very thick,col5] file {tikz/sample_paths/R5_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R5_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R5_total.txt};
\addplot[very thick,dashed] coordinates {
(0,7)
(200,7)
};
\addplot[very thick,dashed] coordinates {
(0,24)
(200,24)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=5$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.54]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 128.333,
ytick = {0,20,40,60,80,100,120},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {tikz/sample_paths/R25_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R25_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R25_total.txt};
\addplot[very thick,dashed] coordinates {
(0,30)
(200,30)
};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=25$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.54]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 490,
ytick = {0,100,200,300,400},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {tikz/sample_paths/R100_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R100_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R100_total.txt};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\addplot[very thick,dashed] coordinates {
(0,420)
(200,420)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=100$}
\end{subfigure}
\caption{Sample paths of $H(t)$ (blue), $Q_1(t)$ (orange) and $Q_1(t)+Q_2(t)$ (green) of the Erlang-R model with holding with parameters $\mu = 1$, $\delta=0.25$, $p=0.75$ and $\beta=\gamma=1$. The staffing levels $s$ and $n$ are depicted by the dashed lines.}
\label{fig:sample_paths}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.7,
ytick = {0,0.1,...,0.7},
xlabel = $\lambda$,
grid = both,
axis line style={->},
axis lines = left,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/delayProbErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/delayProbYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/delayProbJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Delay probability nurse}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.3,
ytick = {0,{0.05},0.1,0.15,0.2,0.25,3},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = north east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/EWErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/EWYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/EWJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Expected wait}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.7,
ymax = 1.02,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/rhoErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/rhoYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/rhoJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Nurse utilization}
\end{subfigure}
\caption{Asymptotic behavior of the restricted Erlang-R model with holding and blocking, and the closed ward model for $\mu=1$, $\delta = 0.2$, $p=0.8$ and $\beta=\gamma=0.5$.}
\label{fig:empiricalAsymptotics}
\end{figure}
We also check how the Erlang-R model with blocking or holding and the closed ward model of \cite{Jennings2008} relate under scaling \eqref{eq:twofoldscaling}.
In Figure~\ref{fig:empiricalAsymptotics}, we plot the performance measures, obtained through simulation, for the three models in which we fix $\beta=\gamma=0.5$ and vary the arrival rate $\lambda$.
First, we see that $\mathbb{P}({\rm delay})$ stabilizes as $\lambda\to\iy$ in all three models under scaling \eqref{eq:twofoldscaling}, and the delay probability of the model with holding lies in between the other two.
Second, note that the expected waiting time for a nurse in all models converges to 0 as $\lambda$ increases. In fact, the rate of decay is similar in all three models.
We observe that $\rho_s$ approaches unity in all models, and the rate of convergence seems again comparable.
Finally, and most importantly, we notice an ordering between the three models.
Namely, in all performance restricted considered in Figure \ref{fig:empiricalAsymptotics}, Erlang-R with holding appears to be upper bounded by the closed ward and lower bounded by the Erlang-R with blocking.
In a multitude of parameter settings of $(\beta,\gamma)$, we have seen the same ordering, leading to the following conjecture:
\begin{conjecture}\label{conj:stochorder}
Let $Q^b_1(\iy)$, $Q_1^h(\iy)$ and $Q_1^J(\iy)$ denote the stationary number of needy patients in the Erlang-R model with blocking, holding and the closed ward, respectively. Then,
\begin{equation}
Q_1^b(\iy) \preceq_{\rm st} Q_1^h(\iy) \preceq_{\rm st} Q_1^J(\iy).
\end{equation}
\end{conjecture}
Observe that Conjecture \ref{conj:stochorder} poses a stronger statement than the third assertion in Proposition \ref{thm:stochasticordering}.
The latter does give an upper bound to $Q_1^h(\iy)$ in terms of $Q_1^b(\iy)$, albeit supplemented with the stationary holding queue length.
\subsection{QED limits for Erlang-R with blocking}
\label{sec:QED_limit_block}
We now continue our analysis by examining its limiting behavior under scaling \eqref{eq:twofoldscaling},and obtain QED limits for some performance measures of the Erlang-R model with blocking.
Using the explicit expressions for the blocking model in \eqref{eq:pih(i,j)}, we derive the limiting values of the relevant performance restricted defined in Section \ref{sec:performance_metrics} in terms of $\beta$ and $\gamma$.
\begin{theorem}\label{thm:limits_YT}
Let $s$ and $n$ scale as in \eqref{eq:twofoldscaling} with ${-}\infty<\beta<\infty,\,\gamma>0$ as $\lambda\to\infty$. Then, if $\beta \neq 0$,
\begin{align}
g^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay})\nonumber \\
\label{eq:yt_limit_delay}
&=
\left(1 +
\frac{ \beta \, \int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t) }
{\varphi(\beta)\Phi(\eta) - \varphi(\sqrt{\beta^2+\eta^2}){\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)}
\right)^{-1},\\
f^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber\\
\label{eq:yt_limit_block}
&=
\frac{
\sqrt{r}\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \varphi(\sqrt{\beta^2+\eta^2})\,{\rm e}^{\frac{1}{2} \omega^2} \Phi(\omega)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},\\
h^b(\beta,\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay}
&=
\frac{
\frac{\varphi(\beta)\Phi(\eta)}{\beta^2} +
\left(\frac{\beta}{r}-\frac{\gamma}{\sqrt{r}}-\frac{1}{\beta}\right)\,\frac{\varphi(\sqrt{\eta^2+\beta^2})}{\beta}\, {\rm e}^{\tfrac{1}{2}\omega^2}\, \Phi(\omega)
- \sqrt{\frac{1-r}{r}}\,\frac{\varphi(\beta)\varphi(\eta)}{\beta}
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},
\end{align}
and if $\beta=0$,
\begin{align}
g^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay}) \nonumber\\
\label{eq:yt_limit_delay_beta0}
&=
\left(1+
\frac{
\int_{-\iy}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t)
}{
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
}
\right)^{-1},\\
f^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber \\
\label{eq:yt_limit_block_beta0}
&=
\frac{
\sqrt{r}\,\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \frac{1}{\sqrt{2\pi}} \Phi(\eta)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t) +
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
},\\
h_0^b(\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay_beta0}
&= \frac{1}{2\mu}\, \frac{ \left( \gamma^2/r+1\right) \Phi(\eta) + \eta \varphi(\eta) }
{ \frac{r}{1-r} \sqrt{2\pi} \int_{-\infty}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, d\Phi(t) + \sqrt{\frac{r}{1-r}} \left(\eta \Phi(\eta)+\varphi(\eta)\right)},
\end{align}
where $\eta = \frac{\gamma - \beta\sqrt{r}}{\sqrt{1-r}}$ and $\omega := \frac{\gamma - \beta/\sqrt{r}}{\sqrt{1-r}}$.
\end{theorem}
The proof of Theorem \ref{thm:limits_YT} is given in Appendix C of \cite{YomTov2010} under a parameter transformation.
Theorem \ref{thm:limits_YT} proves that the scaling \eqref{eq:twofoldscaling} results in QED behavior: the probability of waiting in Equations \eqref{eq:yt_limit_delay} and \eqref{eq:yt_limit_delay_beta0} converges to a limit that is strictly between 0 and 1.
Notice that all limits in Theorem \ref{thm:limits_YT} are functions of three parameters: $\beta$ and $\gamma$, which are decision variables, and the fraction of needy time $r$, which is dictated by the physics of the system. Furthermore, the theorem also shows that the probability of blocking (Equations \eqref{eq:yt_limit_block} and \eqref{eq:yt_limit_block_beta0}) is of order $1/\sqrt{R_1}$.
For example, assume that the fraction of needy time $r$ is $0.5$ and the system is large (100 servers).
Using Figure \ref{fig:pdelay_pblock}, we observe that, by choosing the pair $\gamma = 1$ and $\beta = 0.245$, we actually aim at a probability of getting served immediately to be 40\%. At the same time, the probability of getting immediately a bed is 97\%.
Thus, waiting inside the ED occurs at a reasonable level, while wait outside the facility becomes negligible.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
tick label style={/pgf/number format/fixed},
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -1.8,0.5)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,0.99)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {tikz/limit_probabilities_delay.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:pdelay_pblock_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
tick label style={/pgf/number format/fixed},
ylabel = {$f(\beta,\gamma)$},
y label style = {at = {(axis cs: -1.8,1)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,1.98)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {tikz/limit_probabilities_block.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {tikz/limit_probabilities_block.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {tikz/limit_probabilities_block.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {tikz/limit_probabilities_block.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\label{fig:pdelay_pblock_b}
\end{subfigure}
\caption{Asymptotic delay and scaled blocking probability for $r=0.5$ based on Theorem \ref{thm:limits_YT}. }
\label{fig:pdelay_pblock}
\end{figure}
Theorem \ref{thm:limits_YT} further shows that the expected waiting (Equations \eqref{eq:yt_limit_Edelay} and \eqref{eq:yt_limit_Edelay_beta0})
is of order $1/\sqrt{R_1}$ too and hence vanishes in the large-system limit.
We see from Theorem \ref{thm:limits_YT} that achieving target service levels is always an interplay between $\beta$ and $\gamma$.
Figure \ref{fig:pdelay_pblock_a} shows for instance that in order to keep $\mathbb{P}({\rm delay})\in (0.25,0.75)$, choosing $\gamma=-1$ requires $\beta$ to stay within the range $[-1.4,-0.5]$, while $\gamma=1$ corresponds to values of $\beta$ in $[-0.4,0.5]$.
While the two-fold scaling rule in \eqref{eq:twofoldscaling} automatically captures the right dimensioning ratio as the system scales up, Theorem \ref{thm:limits_YT} shows that the parameters $\beta$ and $\gamma$ provide a means to fine-tune the performance.
Figure \ref{fig:pdelay_pblock_b} confirms how adding nurses, i.e.~increasing $\beta$, does not improve the blocking probability if the number of beds, i.e.~$\gamma$, is too tight.
This is in accordance with our previous observations in Figure \ref{fig:Rmax} for the exact steady-state distribution.
To test the accuracy of the asymptotic results in Theorem \ref{thm:limits_YT} as approximations in a realistic setting, we plot in Figure \ref{fig:accuracy_blocking} the exact probability of delay and blocking for an Erlang-R model with $R=8$ and $r=0.25$, as a function of $s$. The exact probabilities are given by Equation
\eqref{eq:delay_probability}, and their respective asymptotic approximations are based on Theorem \ref{thm:limits_YT}.
Despite the realistic moderate size of the system ($R=8$), we see that the QED approximations are remarkably accurate for many settings $(s,n)$.
This fast relaxation is in line with observations made earlier in the QED literature \cite{Borst2004,Janssen2011}.
\begin{table}[htb]
\centering
\begin{tabular}{|r|rrrr|}
\hline
& $\mu$ & $\delta$ & $p$ & $r$ \\
\hline
Case 1 & 1 & 0.10 & 0.90 & 0.10 \\
Case 2 & 1 & 0.25 & 0.75 & 0.25\\
Case 3 & 1 & 0.50 & 0.50 & 0.50 \\
\hline
\end{tabular}
\caption{Parameter settings for numerical experiments.}
\label{tab:parameter_settings}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1270 & 0.0900 & 0.2283 & 0.1553 & 0.0212 & 0.1085 \bigstrut[t]\\
10 & 0.1340 & 0.0910 & 0.1919 & 0.1628 & 0.0206 & 0.1205 \\
25 & 0.1981 & 0.0945 & 0.1614 & 0.2356 & 0.0216 & 0.2145 \\
50 & 0.1513 & 0.0963 & 0.1588 & 0.1830 & 0.0205 & 0.1496 \\
100 & 0.1880 & 0.0956 & 0.1532 & 0.2231 & 0.0224 & 0.2055 \\
250 & 0.1797 & 0.0971 & 0.1399 & 0.2143 & 0.0219 & 0.2057 \\
\hline
\multicolumn{1}{r|}{} & 0.1767 & 0.0981 & 0.1437 & 0.2108 & 0.0217 & 0.1947 \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0237 & 0.0868 & 0.0282 & 0.0322 & 0.0192 & 0.0391 \bigstrut[t]\\
10 & 0.0206 & 0.0872 & 0.0188 & 0.0278 & 0.0183 & 0.0264 \\
25 & 0.0277 & 0.0876 & 0.0123 & 0.0363 & 0.0174 & 0.0174 \\
50 & 0.0185 & 0.0913 & 0.0116 & 0.0249 & 0.0175 & 0.0166 \\
100 & 0.0232 & 0.0888 & 0.0103 & 0.0303 & 0.0183 & 0.0145 \\
250 & 0.0203 & 0.0905 & 0.0079 & 0.0267 & 0.0179 & 0.0109 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & 0.0188 & 0.0914 & 0.0084 & 0.0247 & 0.0177 & 0.0118 \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 1.}
\label{tab:numerics_case1}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0911 & 0.1538 & 0.0479 & 0.1431 & 0.0345 & 0.0909 \bigstrut[t]\\
10 & 0.1010 & 0.1498 & 0.0560 & 0.1520 & 0.0326 & 0.1025 \\
25 & 0.1594 & 0.1509 & 0.1058 & 0.2192 & 0.0405 & 0.1785 \\
50 & 0.1201 & 0.1506 & 0.0726 & 0.1697 & 0.0381 & 0.1248 \\
100 & 0.1514 & 0.1539 & 0.1001 & 0.2088 & 0.0398 & 0.1704 \\
250 & 0.1459 & 0.1524 & 0.0957 & 0.2003 & 0.0397 & 0.1618 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1429} & \textit{0.1569} & \textit{0.0940} & \textit{0.1976} & \textit{0.0391} & \textit{0.1617} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0130 & 0.1484 & 0.0044 & 0.0277 & 0.0294 & 0.0109 \bigstrut[t]\\
10 & 0.0121 & 0.1432 & 0.0042 & 0.0244 & 0.0267 & 0.0098 \\
25 & 0.0182 & 0.1383 & 0.0070 & 0.0319 & 0.0295 & 0.0141 \\
50 & 0.0119 & 0.1415 & 0.0043 & 0.0216 & 0.0301 & 0.0090 \\
100 & 0.0154 & 0.1413 & 0.0059 & 0.0270 & 0.0290 & 0.0119 \\
250 & 0.0136 & 0.1403 & 0.0051 & 0.0236 & 0.0291 & 0.0103 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0126} & \textit{0.1445} & \textit{0.0048} & 0.0220 & \textit{0.0284} & 0.0097 \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 2.}
\label{tab:numerics_case2}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0547 & 0.1945 & 0.0221 & 0.1181 & 0.0604 & 0.0617 \bigstrut[t]\\
10 & 0.0579 & 0.2158 & 0.0237 & 0.1325 & 0.0526 & 0.0746 \\
25 & 0.1113 & 0.2086 & 0.0544 & 0.1959 & 0.0641 & 0.1311 \\
50 & 0.0813 & 0.2050 & 0.0363 & 0.1523 & 0.0562 & 0.0933 \\
100 & 0.1060 & 0.2146 & 0.0509 & 0.1873 & 0.0632 & 0.1250 \\
250 & 0.1006 & 0.2179 & 0.0475 & 0.1820 & 0.0596 & 0.1214 \\
\hline
\multicolumn{1}{r|}{} & 0.1011 & \textit{0.2185} & 0.0478 & 0.1792 & \textit{0.0605} & 0.1199 \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0034 & 0.1888 & 0.0009 & 0.0175 & 0.0510 & 0.0057 \bigstrut[t]\\
10 & 0.0030 & 0.2093 & 0.0008 & 0.0172 & 0.0416 & 0.0058 \\
25 & 0.0070 & 0.1937 & 0.0020 & 0.0243 & 0.0440 & 0.0089 \\
50 & 0.0043 & 0.1946 & 0.0011 & 0.0163 & 0.0414 & 0.0056 \\
100 & 0.0061 & 0.1999 & 0.0017 & 0.0207 & 0.0431 & 0.0076 \\
250 & 0.0052 & 0.2037 & 0.0014 & 0.0185 & 0.0401 & 0.0067 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & 0.0052 & \textit{0.2039} & 0.0014 & 0.0173 & \textit{0.0404} & 0.0063 \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 3.}
\label{tab:numerics_case3}
\end{table}
We furthermore compare the asymptotic delay and blocking probability in the three scenarios given in Table \ref{tab:parameter_settings}.
In Tables \ref{tab:numerics_case1}--\ref{tab:numerics_case3} we compute the exact probabilities of delay and blocking through the explicit forms in \eqref{eq:delay_probability} for increasing values of the offered load, $R_1$.
The numerical results show that $g^b(\beta,\gamma)$, $f^b(\beta,\gamma)$ and $h^b(\beta,\gamma)$ provide accurate approximations to $\mathbb{P}({\rm delay})$, $\sqrt{R_1}\mathbb{P}({\rm block})$ and $\sqrt{R_1}\,\mathbb{E}[W]$ in pre-limit systems.
The quality of the approximations increases with $R_1$.
Naturally, fluctuations occur for relatively small values of $R_1$, because $s$ and $n$ need to be rounded to an integer.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.9]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {tikz/accuracy/accuracy_pdelay_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.9]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {tikz/accuracy/accuracy_pblock_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\end{subfigure}
\caption{Comparison of exact performance restricted (solid) against asymptotic approximations (dashed) with $\beta=(s-R_1)/\sqrt{R_1}$ and $\gamma=(n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_blocking}
\end{figure}
\subsection{QED limits for Erlang-R with holding}
\label{sec:QED_limit_holding}
As explained in Section \ref{sec:QED_scaling}, the model with holding has no product-form steady-state distribution, which makes it hard (if not impossible) to obtain QED limits.
Instead, we derive QED approximations by exploiting a connection with the blocking model.
We first prove that under scaling \eqref{eq:twofoldscaling}, the upper bound on the utilization level of the nurses needed to achieve stability in the model with holding, as given in Proposition \ref{prop:StabilityCondition}, converges to unity as $R\to\infty$.
This facilitates high utilization levels of both nurses and beds, a key characteristic of the QED regime.
\begin{proposition}\label{prop:stability_convergence}
Let $s$ and $n$ scale with $R_1\to\infty$ as in \eqref{eq:twofoldscaling}. Then for $\lambda\to\infty$,
\[
\rho_{\max}(s,n) \to 1.
\]
\end{proposition}
The proof can be found in Appendix \ref{app:proof_stability_convergence}.
Combining Proposition \ref{prop:stability_convergence} with Proposition \ref{prop:StabilityCondition} shows that indeed the scaling we use results in a highly utilized system.
As observed before, the nature of the two variants of the model is similar up to the fact that a fraction of the patients is deferred on arrive in the setting with blocking, whereas all the arriving patients are eventually admitted into the system in the holding model.
This implies that, given $s$ and $n$, the nurses face an increased workload in case of a holding room.
In fact, Theorem \ref{thm:limits_YT} shows that the blocking probability is of order $1/\sqrt{R_1}$, yielding a volume of blocked patients of order $\sqrt{R_1}$ in setting with blocking.
Accordingly, if $R^b = R_1$ and $R^h$ denote the nominal load arriving to the nurses in the model with blocking and holding, respectively, we can argue that
\[R^h = R^b + \alpha \sqrt{R^b} + o(\sqrt{R^b}),\]
for some $\alpha>0$.
Notice that this additional load is of the same order as the safety staffing in the blocking model staffing rule \eqref{eq:twofoldscaling}.
As $s$ and $n$ remain unchanged, we rewrite \eqref{eq:twofoldscaling} in terms of $R^h$,
\begin{align}
s &= R^h + (\beta-\alpha)\sqrt{R^h} + o(\sqrt{R^h}), \nonumber \\
n &= \frac{R^h}{r} + \left(\gamma-\alpha/\sqrt{r}\right)\sqrt{\frac{R^h}{r}} + o(\sqrt{R^h}),
\label{eq:fixed_point_scaling}
\end{align}
where we have used $R^b = O(R^h)$.
Observe that the square-root principle prevails also after this substitution, albeit with different hedging parameters.
We therefore heuristically argue that the holding model under scaling \eqref{eq:twofoldscaling} with parameters $\beta$ and $\gamma$ mimics the blocking model with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, respectively.
Observe, however, that we have not yet specified the value of $\alpha$.
By definition, $\alpha\sqrt{R^b}$ is the expected volume of patients that would be rejected in the model with blocking, that is, $R^h$ times the probability of not being admitted to the ED directly.
By the construction in \eqref{eq:fixed_point_scaling}, this volume asymptotically equals $R^h \cdot \mathbb{P}^b({\rm block})$, with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, which by Theorem \ref{thm:limits_YT} is approximated by
\[f^b\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right) / \sqrt{R^h}\]
as $R^h$ grows large.
In conclusion, $\alpha$ is characterized as the solution of the fixed-point equation
\begin{equation}
\label{eq:fixedpoint}
\alpha = f^h\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right),
\end{equation}
and as a result, we are able to approximate the nurse delay probability in the Erlang-R model with holding as
\begin{equation}
\mathbb{P}^h({\rm delay}) \approx g^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: g^h(\beta,\gamma).
\label{eq:fixed_point_Pwait}
\end{equation}
Likewise, the scaled the mean waiting time for a nurse can be approximated by
\begin{equation}
\sqrt{R_1} \cdot \mathbb{E}[W] \approx h^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: h^h(\beta,\gamma).
\label{eq:fixed_point_Ewait}
\end{equation}
This also implies that the holding queue is $O(\sqrt{R_1})$.
Subsequently, we argue that the expected holding time (pre-entering wait) under the QED policy is $O(1/\sqrt{R_1})$ and hence asymptotically negligible.
We justify this claim numerically in Section \ref{sec:analysis}.
\begin{remark}
\label{rem:holding_limit}
Notice that in the reasoning leading to \eqref{eq:fixedpoint}, we implicitly assumed that the additional volume $\alpha\sqrt{R^b}$ is an independent Poisson process, which is obviously not the case. Therefore, \eqref{eq:fixed_point_Pwait}-\eqref{eq:fixed_point_Ewait} are approximations for pre-limit systems that are not asymptotically correct as $R_1\to\iy$.
Nevertheless, we heuristic approach seems to performs well as we confirm numerically next.
\end{remark}
In Figure \ref{fig:accuracy_holding}, we repeat the numerical experiments of Figure \ref{fig:accuracy_blocking} for the model with holding.
Since the heuristic does not provide an approximation for the holding probability, Figure \ref{fig:accuracy_holding_b} only plots the simulated holding probabilities.
Those are provided to better understand the implication of operational decisions.
Recall that the holding system is only stable (i.e. $\mathbb{P}({\rm hold})<1$) if both $s>R_1=8$ and $n > R_1/r = 32$.
We nevertheless included the boundary case $n=32$ for illustrative purposes.
The graphs in Figure \ref{fig:accuracy_holding} show that the heuristic captures the congestion levels well, even for this moderate-size system.
To see how this heuristic approach performs under different settings, and particularly if $R_1\to \infty$, we compare the approximated delay probability in the Erlang-R model with holding as solution of the fixed-point procedure to the outcomes of simulation experiments for the three scenarios in Table \ref{tab:parameter_settings} again.
We performed 100 runs of length $10^4$ for each parameter setting and all values of $R$, yielding the results presented in Tables \ref{tab:heuristic_case1}--\ref{tab:heuristic_case3}, which are accurate up to a 95\% confidence interval of width $10^{-3}$.
\begin{table}[h] \centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1532 & 0.1031 & 0.1628 & 0.1216 \bigstrut[t]\\
10 & 0.1622 & 0.1272 & 0.1697 & 0.1331 \\
25 & 0.2340 & 0.2116 & 0.2413 & 0.2342 \\
50 & 0.1817 & 0.1468 & 0.1890 & 0.1678 \\
100 & 0.2199 & 0.1931 & 0.2304 & 0.2269 \\
250 & 0.2110 & 0.1852 & 0.2176 & 0.2230 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.2076} & \textit{0.1777} & \textit{0.2187} & \textit{0.2050} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0310 & 0.0121 & 0.0344 & 0.0148 \bigstrut[t]\\
10 & 0.0267 & 0.0123 & 0.0292 & 0.0128 \\
25 & 0.0348 & 0.0171 & 0.0373 & 0.0184 \\
50 & 0.0240 & 0.0108 & 0.0258 & 0.0125 \\
100 & 0.0293 & 0.0143 & 0.0317 & 0.0163 \\
250 & 0.0256 & 0.0120 & 0.0276 & 0.0145 \\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0229} & \textit{0.0104} & \textit{0.0257} & \textit{0.0124} \bigstrut[b]\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 1.}
\label{tab:heuristic_case1}
\end{table}
\begin{table}[h]\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1327 & 0.0740 & 0.1620 & 0.1096 \bigstrut[t]\\
10 & 0.1446 & 0.0894 & 0.1683 & 0.1207 \\
25 & 0.2204 & 0.1631 & 0.2442 & 0.2203 \\
50 & 0.1694 & 0.1122 & 0.1888 & 0.1507 \\
100 & 0.2098 & 0.1524 & 0.2322 & 0.2111 \\
250 & 0.2033 & 0.1534 & 0.2190 & 0.1979 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1840} & \textit{0.1277} & \textit{0.2109} & \textit{0.1759} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0219 & 0.0079 & 0.0322 & 0.0137 \bigstrut[t]\\
10 & 0.0199 & 0.0073 & 0.0284 & 0.0115 \\
25 & 0.0283 & 0.0128 & 0.0375 & 0.0163 \\
50 & 0.0190 & 0.0078 & 0.0255 & 0.0107 \\
100 & 0.0244 & 0.0097 & 0.0314 & 0.0151 \\
250 & 0.0214 & 0.0083 & 0.0272 & 0.0134 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0169} & \textit{0.0066} & 0.0234 & 0.0104 \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 2.}
\label{tab:heuristic_case2}
\end{table}
\begin{table}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0977 & 0.0413 & 0.1521 & 0.0851 \bigstrut[t]\\
10 & 0.1070 & 0.0469 & 0.1648 & 0.1028 \\
25 & 0.1926 & 0.1076 & 0.2421 & 0.1874 \\
50 & 0.1431 & 0.0727 & 0.1876 & 0.1342 \\
100 & 0.1855 & 0.1012 & 0.2282 & 0.1714 \\
250 & 0.1775 & 0.0963 & 0.2217 & 0.1765 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1442} & \textit{0.0711} & \textit{0.1981} & \textit{0.1354} \bigstrut\\
\cline{2-5}\end{tabular}%
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0072 & 0.0019 & 0.0250 & 0.0081 \bigstrut[t]\\
10 & 0.0067 & 0.0018 & 0.0235 & 0.0082 \\
25 & 0.0148 & 0.0043 & 0.0325 & 0.0133 \\
50 & 0.0092 & 0.0025 & 0.0217 & 0.0081 \\
100 & 0.0132 & 0.0038 & 0.0277 & 0.0105 \\
250 & 0.0114 & 0.0033 & 0.0246 & 0.0099 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0078} & \textit{0.0022} & 0.0188 & 0.0069 \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 3.}
\label{tab:heuristic_case3}
\end{table}
We conclude from these tables that the approximation is good. As $R$ increases, the simulated values move closer to the heuristic approximation. Extensive numerical experiments suggest that load is slightly underestimated in the limit.
The best results in terms of accuracy are attained for small $r$.
This suggests that the quality of the heuristic method improves as $r$ gets smaller.
These are exactly the parameter settings for which this model is relevant.
\begin{remark}
The approximation technique that evolves around the fixed-point method can be adapted to accommodate balking behavior of external arrivals. If we assume that an arriving patient finding all beds occupied joins leaves the system instantly with probability $1-q$, for some $q\in(0,1)$, independently of the rest of the arrivals, with the same argumentation, the volume of arrivals blocked is still $\alpha\sqrt{R_1}$, while the volume that will enter the ED eventually is $q\cdot\alpha\sqrt{R_1}$. Therefore, we may argue that in the QED regime, the system with holding and balking behaves as the system with blocking but with corrected parameters $(\beta-q\alpha,\gamma-q\alpha/\sqrt{r})$, where $\alpha$ satisfies
\begin{equation}
\alpha = f^b(\beta-q\alpha,\gamma-q\alpha/\sqrt{r}).
\end{equation}
Note that the choice of $q$ interpolates between the two system variants with holding ($q=0$) and blocking ($q=1$).
\end{remark}
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx32] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx36] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx40] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:accuracy_holding_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {tikz/accuracy/accuracy_holding_probability.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Holding probability}
\label{fig:accuracy_holding_b}
\end{subfigure}
\caption{Comparison of simulated delay probability (solid) against asymptotic approximations (dashed) with $\beta = (s-R_1)/\sqrt{R_1}$ and $\gamma = (n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_holding}
\end{figure}
\section{Dimensioning}
\label{sec:dimensioning}
We will now use the accurate asymptotic approximations of the previous section to define a procedure that determines resource capacity in the restricted Erlang-R models.
That is, we aim to set the number of nurses $s$ and the number of beds $n$, such that a preset performance level is achieved.
We take the probability of delay at the Needy queue and the probability of blocking/holding at the pre-entrant queue as the target performance objectives.
\subsection{Capacity setting for Erlang-R with blocking}
\label{sec:dimensioning_block}
In the setting with blocking, we can readily use the asymptotic results of Theorem \ref{thm:limits_YT} to (numerically) find a pair of parameters $(\beta^*,\gamma^*)$ to meet the performance requirements.
For instance, given that we want the delay probability to be at most $\varepsilon$, we first solve the equation $g^b(\beta^*,\gamma^*)=\varepsilon$ and then assign $s = \lceil R_1 + \beta^*\sqrt{R_1}\rceil$ and $n = \lceil R_1/r+\gamma^*\sqrt{R_1/r}\rceil$. Note that there could be multiple solutions to that problem, i.e.\ there could be multiple combinations of number of beds and number of nurses that can result in the same value of a single performance level.
The ED manager can ultimately decide which of these feasible solutions fits the environment best, for instance taking into account space and cost constraints.
We illustrate the resource allocation decisions in an MU setting, using data originated from two articles: \cite{LS2001} and \cite{GY2011}. Green \& Yankovic describe an MU that has 42 beds, with average occupancy level 78\%, and Average Length of Stay (ALOS) 4.3 days. Lundgren \& Segesten studied nurses' service times in a medical-surgical ward. They found that the average service time in their unit was 15.3 minutes per service, and that the average demand rate for each patient is 0.38 requests per hour. Therefore, we take an average service time of 15 minutes and assume that there are 0.4 requests per hour from each patient. Fitting this data to our model results in the following parameters values: $\lambda =0.32, \mu =4, \delta =0.4$, $p=0.975$ and the fraction of needy time is then approximately $r=0.09$.
This yields nominal offered load $R_1 = 3.2$ and $R_1/r = 34.4$.
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -1.8,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: -1.9,0.05)},anchor = south west},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\draw[->,col1,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: -0.0552366,0.5) -- (axis cs: -0.0552366,0);
\draw[->,col2,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.179728,0.5) -- (axis cs: 0.179728,0);
\draw[->,col3,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.359034,0.5) -- (axis cs: 0.359034,0);
\draw[->,col4,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.459825,0.5) -- (axis cs: 0.459825,0);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma=1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:ratio01_delay}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$f(\beta,\gamma)/\sqrt{R_1}$},
y label style = {at = {(axis cs: -1.8,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\draw[very thick, col1,dashed,->] (axis cs: -0.0552,0) -- (axis cs: -0.0552,0.292798) -- (axis cs: -2,0.292798);
\draw[very thick,col2,dashed,->] (axis cs: 0.179728,0) -- (axis cs: 0.179728,0.164903) -- (axis cs: -2,0.164903);
\draw[very thick,col3,dashed,->] (axis cs: 0.359034,0) -- (axis cs: 0.359034,0.0705547) -- (axis cs: -2,0.0705547);
\draw[very thick,col4,dashed,->] (axis cs: 0.459825,0) -- (axis cs: 0.459825,0.0207909) -- (axis cs: -2,0.0207909);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma= 1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Blocking probability}
\label{fig:ratio01_block}
\end{subfigure}
\caption{Approximate performance of restricted Erlang-R with blocking for $r \approx 0.09$ and $R_1 = 3.2$, as functions of $\beta$.}
\label{fig:ratio01}
\end{figure}
Figure \ref{fig:ratio01} visualizes the dimensioning procedure for this particular MU.
The hospital management can find a pair of $n$ and $s$ to meet certain criteria, for example to achieve target delay probability $\varepsilon = 0.5$ with reasonable blocking probability.
Figure \ref{fig:ratio01}a indicates that this target can be achieved by a variety of pairs, for instance $(\beta_1,\gamma_1) = (-0.06,-1)$, $(\beta_2,\gamma_2) = (0.16,0)$, $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$, among infinitely many others.
According to Figure \ref{fig:ratio01}b, the pairs named above lead to blocking probabilities 0.293, 0.165, 0.071 and 0.021, respectively.
If the manager decides that probability of blocking of more than 10 percent is not acceptable, this leaves the choices $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$ as candidate parameter pairs.
Using the two-fold square-root staffing rule $s_i = \lceil R_1 + \beta_i \sqrt{R_1}\rceil$ and $n_i = [R_1/r + \gamma\sqrt{R_1/r}]$, this yields feasible staffing levels $(s_3,n_3) = (4,40)$ and $(s_4,n_4)=(5,46)$.
The ultimate decision to apply any of these solutions can be based on external factors, such as operational costs or space limitations of number of beds.
\subsection{Capacity setting for Erlang-R with holding}
For the holding model, we need a more sophisticated approach, exploiting the asymptotic approximation with the fixed-point equation in \eqref{eq:fixedpoint}. We propose the following dimensioning procedure to achieve a preset target delay probability at the needy queue.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Target delay probability $\varepsilon$. Parameters $\lambda,\mu,\delta$ and $p$.}
\hspace{1.1cm}\KwOut{Staffing levels $s$ and $n$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}
\item[] \hspace{0.5cm} 1. Set $R_1:= \frac{\lambda}{(1-p)\mu}$ and $r = \frac{\delta}{\delta+p\mu}$.
\item[] \hspace{0.5cm} 2. Determine parameters $(\beta^*,\gamma^*)$ such that $g^b(\beta^*,\gamma^*) = \varepsilon$.
\item[] \hspace{0.5cm} 3. Set $\beta = \beta^* + f^b(\beta^*,\gamma^*)$ and $\gamma = \gamma^* + f^b(\beta^*,\gamma^*)/\sqrt{r}$.
\item[] \hspace{0.5cm} 4. Return $s = \left\lceil R_1 + \beta\sqrt{R_1}\right\rceil$ and $n = \left\lfloor R_1/r + \gamma \sqrt{R_1/r}\right\rfloor$.
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Stationary dimensioning algorithm for ED with holding.}
\label{alg:stationarydimensioning}
\end{algorithm}
\begin{remark}\label{rem:upperboundHW}
In Step 2 of Algorithm \ref{alg:stationarydimensioning} infinitely many pairs $(\beta^*,\gamma^*)$ satisfy the delay probability equation.
For practical purposes, it is convenient to fix either $\beta^*$ or $\gamma^*$ beforehand, and then solve $g^b(\beta^*,\gamma^*) = \varepsilon$ for the remaining unknown.
The preset value should however be chosen with care, since $g^b(\beta^*,\gamma^*)$ is upper bounded by the Halfin-Whitt delay probability formula
\[g_{\rm HW}(\beta^*) = \left( 1 + \frac{\beta^* \Phi(\beta^*)}{\varphi(\beta^*)}\right)^{-1}.\]
Hence, if $\varepsilon > g_{\rm HW}(\beta^*)$, then no feasible solution to $g^b(\beta^*,\gamma^*)=\varepsilon$ exists.
This should be considered when choosing $\beta^*$.
Furthermore, it is evident from Step 3 that the final values $(\beta,\gamma)$ are always larger than $(\beta^*,\gamma^*)$.
\end{remark}
We now use the same example as in Section \ref{sec:dimensioning_block} to demonstrate capacity allocation decisions for the model with holding. This can be viewed as the additional capacity the medical unit needs in terms of nurses and beds, in order to account for the fact that patients are waiting in the ED to be admitted instead of being blocked and transferred to a less preferred unit.
Observe that the holding model leaves less flexibility for management in choosing system parameters due to stability constraints. For example, the policy with $n=30$ ($\gamma=-0.75$) is infeasible in the holding model.
For similar reasons, only nurse staffing levels with $\beta>0$, or $s > R_1=3.2$ are feasible.
Targeting a delay probability of $0.5$ with $n=40$, Figure \ref{fig:ratio01_hold} shows that operating a MU with holding room requires $\beta = 0.475$ or $s=5$.
Recall that under the blocking policy, only $s=4$ nurses were needed to achieve a delay probability of $0.5$.
This example hence shows how the managerial decision to have a holding room, rather than deferring patients to less preferred medical units, requires additional workforce in that unit (as well as the ED).
This example also shows that the facility with holding room is able to treat fewer patients simultaneously than under blocking constraints, in line with the bounds in Section \ref{sec:bounds} and Conjecture \ref{conj:stochorder}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g^h(\beta,\gamma)$},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col5] table[x=beta,y=delay_n35] {tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col4] table[x=beta,y=delay_n40] {tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col1] table[x=beta,y=delay_n45] {tikz/staffing_example/staffing_example_with_holding1.txt};
\draw[very thick,col4,dashed,->] (axis cs: -2,0.5) -- (axis cs: 0.475,0.5) -- (axis cs: 0.475,0);
\legend{$\gamma = -0.75$, $\gamma =0.102$,$\gamma= 0.955$, $\gamma=1.807$};
\end{axis}
\end{tikzpicture}
\caption{Approximate delay probability of restricted Erlang-R system with holding for $r\approx 0.09$ and $R_1=3.2$ }
\label{fig:ratio01_hold}
\end{figure}
\section{Model analysis and managerial implications}
\label{sec:analysis}
In this section, we use the analysis and algorithms developed in earlier sections to gain insights into the importance of the capacity restrictions and customer returns in a restricted Erlang-R system by drawing a comparison to related models studied in the literature.
\subsection{The influence of customer returns or the role of $r$}
Here we study how the parameter $r$ affects the service level in the restricted Erlang-R model with blocking, on the basis of the asymptotic expressions in Theorem \ref{thm:limits_YT}.
To better understand the connection with the single-station model and the importance of returns we examine the role of $r$.
Recall the interpretation of $r$ as the fraction of time a patient is needy during his stay within the system in the idealized scenario with infinite capacity, i.e. for $r\in(0,1)$.
The case $r=1$ corresponds to the setting in which patients are needy all the time, in this case customers get service in one time.
When $r=1$ the infinite-server queue, describing the number of content patients, disappears from the queueing system and we end up with a standard loss model---$M/M/s/n$ queue---in which capacity is scaled as
\[ s = R_1+\beta\sqrt{R_1}, \qquad n = R_1+\gamma\sqrt{R_1}. \]
This staffing rule only makes sense in case $\beta<\gamma$, since no delay is experienced if $n\leq s$.
If indeed $\gamma>\beta$, then the asymptotic delay probability and scaled blocking probability are given by \cite{masseywallace},
\[
g_B(\beta,\gamma) = \frac{1-{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)},
\qquad f_B(\beta,\gamma) = \frac{\beta{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}.
\]
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.8,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,1.28)},anchor= north east},
yscale = 0.75
]
\addplot[thick,col1] file {tikz/influence_r/PdelayB_g1_b025.txt};
\addplot[thick,col3] file {tikz/influence_r/PdelayB_g1_b05.txt};
\addplot[thick,col4] file {tikz/influence_r/PdelayB_g1_b1.txt};
\addplot[thick,col5] file {tikz/influence_r/PdelayB_g1_b2.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability $g^b(\beta,\gamma)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.4,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,0.05)},anchor= south east},
yscale = 0.75
]
\addplot[thick,col1] file {tikz/influence_r/PblockB_g1_b025.txt};
\addplot[thick,col3] file {tikz/influence_r/PblockB_g1_b05.txt};
\addplot[thick,col4] file {tikz/influence_r/PblockB_g1_b1.txt};
\addplot[thick,col5] file {tikz/influence_r/PblockB_g1_b2.txt};
\addplot[thick,dashed] file {tikz/influence_r/PblockB_g1_inf.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability $f^b(\beta,\gamma)$.}
\label{fig:influence_of_r_b}
\end{subfigure}
\caption{Asymptotic performance measures as a function of $r$ in the restricted Erlang-R model with blocking for $\gamma=1$.}
\label{fig:influence_of_r}
\end{figure}
We can see that $f^b(\beta,\gamma)$ for increasing $\beta$ approaches a lower bound that is a function of $r$.
To understand this, observe that as $\beta$ grows, delays at the nurse queue vanish.
Then the sojourn time of an admitted patient only consists of a geometric number of needy and content periods with mean $(1/\mu+p/\delta)/(1-p) = 1/(r\mu(1-p))$.
The blocking model can in this case be modeled as an $M/G/n/n$ queue, with offered load $\lambda/(r\mu(1-p)) =R_1/r$, in which the scaled blocking probability is known to be, see \cite{Avram2013},
\[\sqrt{R_1} \, \mathbb{P}({\rm block}) = \sqrt{R_1} \, \frac{(R_1/r)^n/n!}{\sum_{k=0}^n (R_1/r)^k / k!} \to \sqrt{r} \, \frac{\varphi(\gamma)}{\Phi(\gamma)},\]
as $R_1\to\infty$.
This function of $r$ is plotted in Figure \ref{fig:influence_of_r_b} as the dashed line.
We observe that in general the probability of blocking increases with $r$, regardless of the capacity constraints on the needy station.
We can explain this by observing that $r$ influences only $n$ in the QED staffing rule. When $n$ reduces, more patients are blocked. Therefore, if customers spend relatively more time in needy state, which usually indicates services that are less interrupted, blocking will increase. Delays, on the other hand, will decrease in such situations---the minimal delay possible can be achieved if service is given in one time ($r=1$). Returns or interruptions increase delays significantly under QED staffing.
\subsection{Comparing restricted and unrestricted Erlang-R models}
Given the expressions for the asymptotic delay probability in the open Erlang-R model, and its restricted versions with blocking and holding, we compare the three policies for various values of $\beta$, $\gamma$ and $r$.
Figure \ref{fig:comparison_delay} plots the delay probability for blocking ($g^b(\beta,\gamma)$), holding ($g^h(\beta,\gamma)$) and Erlang-R ($g_{\rm HW}(\beta)$) models, as functions of $\gamma$, while keeping $\beta$ fixed, for three values of $r$.
We make a couple of observations.
Notice that
\[ g^b(\beta,\gamma) \leq g^h(\beta,\gamma) \leq g_{\rm HW}(\beta) \]
for all $\beta,\gamma>0$ and $r$.
In that sense, the holding model is an interpolation between the blocking and the open model.
As expected, the delay probabilities in the restricted models converge to those of the open Erlang-R model, because increasing $\gamma$ is tantamount to lifting the stringent constraints on the system size. Note that the rate of conversion is fast---one can provide probability of waiting close to that of the open model with small values of $\gamma$. Indeed, the fact that when using QED staffing not much of excessive delay results from the beds restriction is important by itself.
Also, we observe that the difference between delay probabilities increases with $r$.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r01_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r01_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r01_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.1$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r025_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r025_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r025_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.25$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r05_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r05_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r05_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.5$.}
\end{subfigure}
\caption{Asymptotic delay probability in open Erlang-R (dashed), restricted Erlang-R with blocking (marked) and restricted Erlang-R with holding (solid), as function of $\gamma$ with $\beta=0.1$ (blue), $\beta=0.5$ (orange) and $\beta=1$ (green) fixed.}
\label{fig:comparison_delay}
\end{figure}
\subsection{The impact of visit number}
\label{subsec:num_visit}
We next reflect on the impact of operational capacity decisions on different customer populations. We measure patient's complexity by the number of times she needs to see the nurse or the physician during her stay. In the ED context, simple-to-treat patients will need to see the physician once, while complex ones will need multiple visits. Hence, we divide the patients into complexity groups by the number of visits in the Needy station. Since the number of visits is geometrically distributed, we have a higher proportion of simple patients than complex ones; that fits well the health care environment.
Figure \ref{fig:wait_by_visit} shows the waiting time in the needy and pre-entring queues, and the total waiting time, as a function $n$ (number of beds), for each complexity group.
Obviously, the expected waiting time in the pre-entring queue decreases with $n$, while the needy waiting time increases.
For patients who require a relative large number of visits of the physician, in this case more than 6, the total needy wait is the dominant part of the total waiting time. Therefore, as $n$ grows, the total waiting time first decreases and then increases.
In fact, Figure \ref{fig:wait_by_visit_b} suggests that there is an optimal number of beds $n$ that minimizes the total wait for each complexity type.
Thus, size restrictions reduce the length-of-stay of patients with complex health conditions (given that the constraint is not too tight).
On the other hand, this figure also shows that no such $n$ exists for patients who only require little assistance.
Hence, there is no $n$ that improves the sojourn time of all patients in the ED simultaneously.
This leaves the decision to the hospital manager to weigh the importance of patients of different complexity levels.
\begin{remark}
From a different perspective, note that in queueing systems such as communication systems, the partition of a job to sizable quantities and scheduling those jobs in a similar dynamic to the Erlang-R model became a popular way for increasing throughput. This is because this effectively schedule jobs by their size even though the total job requirements are uncertain. This in fact creates a shortest-job-first policy without prior knowledge of job size \citep{Comte2016}. Considering that perspective we note that the Erlang-R model actually prioritize simple jobs over complex ones. But without restrictions, when load is too high, such procedures may lead to very long LOS of long jobs. The capacity restriction we analyze in this paper, in both of its versions, limits such delays. Hence, even in cases in which the returns themselves are created by a managerial decision, imposing the additional managerial restriction on entering the system has benefits.
\end{remark}
\begin{figure}
\centering
\begin{subfigure}{0.38\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {tikz/inner_vs_outer_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,col1, thick] table[x=n,y=hold] {tikz/inner_vs_outer_wait.txt};
\end{axis}
\end{tikzpicture}
\caption{Expected pre-entering waiting (red) and needy waiting times (black)}
\end{subfigure}
\begin{subfigure}{0.6\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8,
legend cell align=left,
legend style = {at = {(1.05,0.58)}, anchor = west}
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {tikz/total_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {tikz/total_wait.txt};
\legend{{\small $N=1$},{\small $N=2$},{\small $N=3$},{\small $N=4$},{\small $N=5$},{\small $N=6$},{\small $N=7$},{\small $N=8$},{\small $N=9$},{\small $N=10$}};
\end{axis}
\end{tikzpicture}
\caption{Total expected waiting times\\
\quad \\
\quad }
\label{fig:wait_by_visit_b}
\end{subfigure}
\caption{Expected waiting times as a function of $n$ given the number of visits $N$ in the Erlang-R model with holding with $\lambda=2$ $\mu=1$, $\delta=0.25$, $p=0.75$ and $s=9$.}
\label{fig:wait_by_visit}
\end{figure}
\subsection{Case study: comparison of operational decisions}
\label{sec:case_study}
We now illustrate how the managerial decision to operate under a specific operational regime affects ED performance in terms of efficiency and quality-of-care, through a case study.
The practical environment we investigate is the ED of a moderately-sized hospital, which faces the arrival pattern $\lambda(t)$ plotted in Figure \ref{fig:Case_study_arrival_pattern_a} on a typical workday.
Other parameters of the model are estimated to be $\mu = 6.67,\ \delta = 2.18$ and $p = 0.76$, so that $r = 0.301$. These parameters were taken from \cite{YomTov2014}. In order to set time-varying staffing levels $s(t)$ and $n(t)$, we adopt the \textit{mean-offered load} (MOL) approximation of the demand process of~\cite{Jennings1996}.
This approach initially presumes infinite capacity to obtain the number of customers $R(t)$ in the queueing system as a function of time.
This offered load function then replaces (constant) value of $R$ in the stationary dimensioning scheme under consideration, to determine the adequate number of servers at each point in time.
Following this idea in our two-dimensional queueing system, we find the offered load function for the nurses $R_1(t)$ and the offered load function for the beds $R_1(t)+R_2(t)$ as the solution of the system of ODEs,
\begin{align} \label{eq:offeredloadODE}
\frac{d}{dt} R_1(t) &= \lambda(t) + \delta R_2(t) - \mu R_1(t),\\
\frac{d}{dt} R_2(t) &= p\mu R_1(t) - \delta R_2(t),
\end{align}
see \cite[Thm.~2]{YomTov2014} for details.
For this case study's parameters, these offered load functions are also plotted in Figure \ref{fig:Case_study_arrival_pattern_a}.
While the number of nurses can be adjusted in a relatively flexible manner, the value of $n$, which echoes a hard restriction on the ED capacity, is naturally less amenable to fluctuations. The reason is that the maximum ED capacity is to a large extent determined by its hardware, such as beds and medical equipment.
However, the ED manager might deliberately consider reducing $n$ during more quiet periods of the day, e.g.\ during the night, by imposing bed-to-physician constraints. This is done, for example, when setting a case management constraint \citep{EDexperiment,Campello2016}.
Therefore, we consider the scenario in which both $s$ and $n$ are time-dependent but we do not force a constant case management quantity, rather let our new methodology to recommend an appropriate one.
\begin{figure}
\centering
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.01,0.95)},anchor = north west}
]
\addplot[very thick,black] file {tikz/lambdaFunction.txt};
\addplot[very thick,col1] file {tikz/R1.txt};
\addplot[very thick,col5] file {tikz/R1R2.txt};
\legend{ $\lambda(t)$, $R_1(t)$, $R_1(t)+R_2(t)$};
\end{axis}
\end{tikzpicture}
\caption{Dynamic arrival rate function offered load functions}
\label{fig:Case_study_arrival_pattern_a}
\end{subfigure}
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {tikz/casestudy_new/sFunction.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/nFunction.txt};
\legend{ $s(t)$, $n(t)$};
\end{axis}
\end{tikzpicture}
\caption{Capacity function for $\beta=\gamma=0.5$}
\label{fig:Case_study_arrival_pattern_b}
\end{subfigure}
\caption{Empirical arrival rate and offered load functions $R_1(t)$ and $R_1(t)+R_2(t)$ in Israeli ED and corresponding capacity functions.}
\label{fig:Case_study_arrival_pattern}
\end{figure}
Extrapolating Algorithm \ref{alg:stationarydimensioning} to the time-varying case, Step 4 is replaced by
\begin{align*}
s(t) &= R_1(t) + \beta\sqrt{R_1(t)},\\
n(t) &= R_1(t)+R_2(t) + \gamma\sqrt{R_1(t)+R_2(t)},
\end{align*}
for some $\beta,\gamma>0$.
Since $R_1(t)$ and $R_2(t)$ are given, the QED staffing problem again reduces to finding the pair $(\beta,\gamma)$.
Figure \ref{fig:Case_study_arrival_pattern_b} plots the capacity functions for $\beta = 0.5$ and $\gamma=0.5$, assuming capacity can only be adjusted every 30 minutes.
In this case study, we consider three pairs of parameters $(\beta,\gamma)$.
For each of these we investigate, using simulation, the differences in the time-varying performance indicators between the policy with blocking and holding.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.5]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 1.0,
ytick = {0,0.2,0.4,0.6,0.8,1.0},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=delay_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=delay_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=delay_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=delay_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=delay_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=delay_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm delay})$}
\label{fig:simulation_results_a}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.5]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 0.52,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1,
legend cell align=left,
legend style = {at = {(0.9,0.95)}, anchor = north east}
]
\addplot[very thick,col1] table[x=t,y=block_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4] table[x=t,y=block_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5] table[x=t,y=block_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=block_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4,dashed] table[x=t,y=block_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5,dashed] table[x=t,y=block_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\legend{{$(\beta,\gamma)=(0.1,2)$},{$(\beta,\gamma)=(1,1.5)$},{$(\beta,\gamma)=(2,1)$}};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm block})$ or $\mathbb{P}({\rm hold})$}
\label{fig:simulation_results_b}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.5]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.5,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=ratio_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=ratio_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=ratio_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=ratio_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=ratio_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=ratio_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{Nurse-to-patient ratio.}
\label{fig:simulation_results_c}
\end{subfigure}
\caption{Simulation results for case study. Solid and dashed lines represent time-varying performance in the blocking and holding model, respectively.}
\label{fig:simulation_results}
\end{figure}
The simulation results are presented in Figure \ref{fig:simulation_results}.
Figure \ref{fig:simulation_results_a} shows that the MOL approach for capacity allocation roughly stabilizes the delay probability.
Figure \ref{fig:simulation_results_b} shows that the fraction of patients not entering the ED on arrival in the blocking model is reasonable for all parameter pairs considered and are ordered according to $\gamma$.
We also see a significant difference with holding.
Observe also that the holding probability drops in the period 8--13, which is exactly the period when the system is experiencing peak offered load.
Hence, this temporary reduction is in line with our asymptotic findings that the probability of blocking/holding is $O(1/\sqrt{R_1})$.
Finally note that the three parameter settings lead to different nurse-to-patient ratios.
Clearly, larger $\beta$ leads to small nurse-to-patient ratios (due do larger staffing).
Figure \ref{fig:simulation_results_c} demonstrates that for $(\beta,\gamma) = (1,1.5)$ and $(\beta,\gamma) = (2,1)$ the difference between the holding policy and the blocking policy is small. However, for $(\beta,\gamma) = (0.1,2)$ we see a significant increase in the ratio during night hours.
This may be due to the tight nurse schedule, that causes the holding queue to build up just before midnight.
This queue then empties latter on, causing an increase in workload per nurse in the period 24--7.
To see the direct effect of the size restriction on the queue lengths, we plotted the mean holding and service queue lengths in the holding model as a function of the parameter $\gamma$ in Figure \ref{fig:simulation_queuelengths}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.2,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {tikz/casestudy_new/holdingQueue_g01.txt};
\addplot[very thick,col3] file {tikz/casestudy_new/holdingQueue_g025.txt};
\addplot[very thick,col4] file {tikz/casestudy_new/holdingQueue_g05.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/holdingQueue_g1.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$}
\end{axis}
\end{tikzpicture}
\caption{Mean holding queue length}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 15,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.82,0.05)},anchor = south east}
]
\addplot[very thick,col1] file {tikz/casestudy_new/serviceQueue_g01.txt};
\addplot[very thick,col3] file {tikz/casestudy_new/serviceQueue_g025.txt};
\addplot[very thick,col4] file {tikz/casestudy_new/serviceQueue_g05.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/serviceQueue_g1.txt};
\addplot[very thick,dashed] file {tikz/casestudy_new/serviceQueue_R.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$,Erlang-R}
\end{axis}
\end{tikzpicture}
\caption{Mean service queue length}
\end{subfigure}
\caption{Simulated queue length of holding model with different values of $\gamma$.}
\label{fig:simulation_queuelengths}
\end{figure}
Note that for all $\gamma$ considered, the holding queue length are, as expected, of a smaller order than the number of patients admitted.
Also, the holding queue length decreases as we increase $\gamma$.
The service queue lengths naturally approach the expected queue lengths in the Erlang-R model as $\gamma$ is increased.
\section{Conclusions and future research}
\label{sec:conclusion}
In this research we developed and analyzed a queueing network tailored to a health care environment with finite-size restrictions.
Using the asymptotic approximations, numerical analysis and simulation, we gained insight into staffing problems that arise in EDs.
We summarize our main findings.
First, we showed that resource allocation of personnel and beds should be synchronized in order to avoid waste, and that the QED scaling provides an efficient, flexible, and easy to implement methodology to do so through a two-fold staffing rule.
The system dynamics are mostly influenced by the ratio $r:= \delta/(\delta+p\mu)$, which represents the fraction of time a patient spends being needy during her stay in the system and therefore has a significant influence on operational decisions.
We furthermore explained that enabling customers to hold in ED requires higher workforce levels within the ED, and moreover compared the pros and cons of imposing strict constraints on entering an ED in a case study.
We find that size restrictions have the ability to improve the quality-of-service of the processes within the facility, at the expense of a slight increase in pre-entrant wait and server efficiency levels.
The dimensioning scheme we developed provides a powerful and elegant way of finding adequate staffing levels in emergency departments.
Nonetheless, we see some avenues for further research.
The asymptotic approximations we developed based enabled us to take the first step towards characterizing the pre-entering queue behavior in the
QED regime.
We observed how the holding queue length vanishes at that $1/\sqrt{R_1}$ as $R\to\infty$.
Yet, our analysis did not yield explicit characteristics on the holding queue and holding times.
These performance indicators are naturally important to study if one wants to consider the trade-off between inner waiting (for a nurse) time and outer waiting (holding) time.
Nog meer tekst toevoegen!
\section*{Appendix}
\begin{subappendices}
\section{Description of the QBD process}
\label{app:QBDdescription}
\subsection{The QBD-process}
\label{app:theQBDprocess}
We consider the QBD-process $X=\{N,Q_1\}$ in stationarity. Let $\nu(i)=\min\{i,s\}\mu$. To determine the (outgoing) transition rates of the process $X$ we distinguish between the following cases:
\begin{itemize}
\item \emph{Transitions from $(0,0)$:} There are no patients in the Emergency Department and thus the only possible occurrence is when a new patient arrives. This results in a transition to $(1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), 1 \leq i < n$:} There are exactly $i$ patients assigned to a bed of which none are seen by a nurse. Then either one of those patients becomes needy, or a new patient arrives at the Emergency Department that can immediately be seen by a nurse. The first results in a transition to $(i,1)$ and occurs at rate $i \delta$, and the second results in a transition to $(i+1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), i \geq n$:} Again, the only possible transitions arises from either a newly arrived patient or a patient assigned to a bed becoming needy. However, a newly arrived patient finds all beds occupied and needs to wait. Thus, with rate $\lambda$ we have a transition to $(i+1,0)$ and with rate $n \delta$ a transition to $(i,1)$.
\item \emph{Transitions from $(i,i), i < n$:} In this case all patients assigned to a bed are in need of service. With rate $\lambda$ a new patient arrives at the Emergency Department. She joins the (possible) queue to be seen by a nurse immediately, so this results in a transition to $(i+1,i+1)$. Moreover, since there are only $s < n$ nurses, a service completion occurs with rate $\nu(i)$. With probability $p$ the patient turns to the holding phase, so in total we still have $i$ patients with one patient less in queue for a nurse. With probability $1-p$ the patient leaves the Emergency Department, decreasing both $N$ and $Q_1$ by one. In other words, with rate $p \nu(i)$ we have a transition to $(i,i-1)$ and with rate $(1-p)\nu(i)$ we have a transition to $(i-1,i-1)$.
\item \emph{Transitions from $(n,n)$:} Similar to the previous case, we have a transition to $(n,n-1)$ with rate $p s \mu$ and with rate $(1-p)s \mu$ we have a transition to $(n-1,n-1)$. In this case however, a newly arrived patient finds all beds occupied, resulting in a transition to $(n+1,n)$ with rate $\lambda$.
\item \emph{Transitions from $(i,n), i > n$:} We have a transition to $(i+1,n)$ with rate $\lambda$ and a transition to $(i,n-1)$ with rate $p s \mu$. In case that a patient leaves the Emergency Department there are $i-n>0$ patients in the holding room waiting for an available bed. Thus, one of the $i-n$ patients in the holding room is assigned to the available bed in need of service. That is, with rate $(1-p) s \mu$ we have a transition to $(i-1,n)$.
\item \emph{Transitions from $(i,j), 1 \leq j < i < n$:} There are four possible transitions. First, with rate $\lambda$ there is a new arrival which results in a transition to $(i+1,j+1)$. Second, with rate $(i-j) \delta$ a patient in one of the beds becomes needy, which results in a transition to $(i,j+1)$. Third, with rate $p \nu(j)$ a patient turns to the content state after service completion, which results in a transition to $(i,j-1)$. Last, with rate $(1-p) \nu(j)$ a patient leaves the Emergency Department after service completion, which results in a transition to $(i-1,j-1)$.
\item \emph{Transitions from $(n,j), 1 \leq j < n$:} This case is similar to the previous one. The only difference arises when a new patient arrives, since all $n$ beds are already occupied. Thus, with rate $\lambda$ we have a transition to $(n+1,j)$.
\item \emph{Transitions from $(i,j), i > n, 1 \leq j \leq n$:} This case is the previous one, except when a patient leaves the Emergency Department after service completion. Then one of the $(i-n)$ patients in the holding room will be assigned to a bed in need of service. This results in a transition to $(i-1,j)$ with rate $(1-p) \nu(j)$.
\end{itemize}
\noindent
The state space and transition rates of the Erlang-R model with holding are illustrated in Figure~\ref{fig:QBDIllustration}.
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[step=1cm,gray!50!,very thin] (0,0) grid (15.5,8.5);
\draw[thick,->] (0,0) -- (15.5,0);
\draw[thick,->] (0,0) -- (0,8.5);
\draw[thick] (0,0) -- (8,8);
\draw[thick,dashed,black!50!] (8,0) -- (8,8);
\draw[thick] (8,8) -- (15.5,8);
\foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
\foreach \y in {0,1,2,3,4,5,6,7,8}
\draw[fill] (\x,\y) circle [radius=0.025];
\node [below left] at (0,0) {$0$};
\node [left] at (0,2) {$j$};
\node [left] at (0,5) {$k$};
\node [left] at (0,8) {$n$};
\node [below] at (6,0) {$i$};
\node [below] at (8,0) {$n$};
\node [above left] at (0,8.5) {$Q_1$};
\node [below right] at (15.5,0) {$N$};
\path [->,thick,-latex] (0,0) edge [bend right] (1,1);
\path [->,thick,-latex] (3,0) edge (3,1);
\path [->,thick,-latex] (3,0) edge (4,1);
\path [->,thick,-latex] (4,4) edge [bend right] (5,5);
\path [->,thick,-latex] (4,4) edge (4,3);
\path [->,thick,-latex] (4,4) edge [bend right] (3,3);
\path [->,thick,-latex] (6,2) edge (6,3);
\path [->,thick,-latex] (6,2) edge (7,3);
\path [->,thick,-latex] (6,2) edge (6,1);
\path [->,thick,-latex] (6,2) edge (5,1);
\path [->,thick,-latex] (8,5) edge (8,6);
\path [->,thick,-latex] (8,5) edge (8,4);
\path [->,thick,-latex] (8,5) edge (9,5);
\path [->,thick,-latex] (8,5) edge (7,4);
\path [->,thick,-latex] (8,8) edge (8,7);
\path [->,thick,-latex] (8,8) edge [bend right] (7,7);
\path [->,thick,-latex] (8,8) edge [bend left] (9,8);
\path [->,thick,-latex] (11,8) edge (11,7);
\path [->,thick,-latex] (11,8) edge [bend left] (12,8);
\path [->,thick,-latex] (11,8) edge [bend left] (10,8);
\path [->,thick,-latex] (11,0) edge (11,1);
\path [->,thick,-latex] (11,0) edge [bend left] (12,0);
\path [->,thick,-latex] (12,5) edge (12,6);
\path [->,thick,-latex] (12,5) edge (12,4);
\path [->,thick,-latex] (12,5) edge (13,5);
\path [->,thick,-latex] (12,5) edge (11,5);
\node [above] at (12.75,5) {\scriptsize $\lambda$};
\node [above] at (11.25,5) {\scriptsize $(1-p)\nu(k)$};
\node [right] at (12,5.75) {\scriptsize $(n-k)\delta$};
\node [right] at (12,4.25) {\scriptsize $p \nu(k)$};
\node [above] at (6.75,2.25) {\scriptsize $\lambda$};
\node [below] at (5,2) {\scriptsize $(1-p)\nu(j)$};
\node [above] at (6,3) {\scriptsize $(i-j)\delta$};
\node [right] at (6,1.25) {\scriptsize $p \nu(j)$};
\end{tikzpicture}
\caption{Illustration of state space and the transitions for the Erlang-R model with holding.}
\label{fig:QBDIllustration}
\end{figure}
The state space can be partitioned according to its levels, where level $i$ corresponds to a total queue length $N=i$ patients. This results in an infinite-sized matrix consisting of blocks, where each block corresponds to the transition flow from one level to another. Since the only transitions allowed are within the same level or between two adjacent levels in a QBD-process, we obtain a tridiagonal block structure. Each block consists of elements representing the transition rate of one state to another, and therefore each block is a matrix of size at most $(n+1) \times (n+1)$.
For the Erlang-R model with holding this gives the following result. Let $P$ denote the transition matrix of the process $\{N(t),Q_1(t)\}$. We have the boundary levels $\{1,2,...,n\}$ and $P$ is of the form
\[
P = \left( \begin{array}{cccccccccc}
B_{00} & B_{01} & & & & & & & & \\
B_{10} & B_{11} & B_{12} & & & & & & & \\
& B_{21} & B_{22} & B_{23} & & & & & & \\
& & \ddots & \ddots &\ddots & & & & & \\
& & & & & B_{n \, n-1} & & & & \\
& & & & B_{n-1 \, n} & B_{nn} & A_0 & & & \\
& & & & & A_2 & A_1 & A_0 & & \\
& & & & & & A_2 & A_1 & A_0 & \\
& & & & & & & \ddots & \ddots & \ddots \\
\end{array} \right),
\]
where $B_{ii} \in \mathbb{R_1}^{(i+1) \times (i+1)}$, $B_{i \, i-1} \in \mathbb{R_1}^{(i+1) \times i}$, $B_{i-1 \, i} \in \mathbb{R_1}^{i \times (i+1)}$, and $A_0,A_1,A_2 \in \mathbb{R_1}^{(n+1)\times(n+1)}$. The matrices of transition rates for the boundary states are given by
\[
B_{00}=(-\lambda),
\qquad
B_{i-1 \, i} = \left( \begin{array}{ccccc}
0 & \lambda & & & \\
& \ddots & \lambda & & \\
& & \ddots & \ddots &\\
& & & 0 & \lambda \\
\end{array} \right),
\]
\[
B_{i \, i-1} = \left( \begin{array}{cccc}
0 & & & \\
(1-p)\mu & 0 & & \\
& (1-p)\nu(2)& \ddots & \\
& & \ddots & 0 \\
& & & (1-p)\nu(i)\\
\end{array} \right),
\]
and
\[
\scriptsize
B_{ii} = \left(
\begin{array}{ccccccccc}
-(\lambda+i \delta) & i \delta & & & &\\
p \mu & -(\lambda+\mu+(i-1)\delta) & (i-1)\delta & & &\\
& \ddots & \ddots & \ddots & & \\
& & p \nu(i-1) & -(\lambda+\nu(i-1)+\delta) & \delta \\
& & & & p \nu(i) & -(\lambda+\nu(i)) \\
\end{array} \right).
\]
Moreover, the transition rates are given by
\[
A_0 = \left( \begin{array}{ccccc}
\lambda & & & & \\
& \lambda & & & \\
& & & \ddots & \\
& & & & \lambda \\
\end{array} \right)
\]
\[ A_2 = \left( \begin{array}{ccccccc}
0 & & & & & & \\
& (1-p)\mu & & & & & \\
& & 2(1-p)\mu & & & & \\
& & & \ddots & & & \\
& & & & s(1-p)\mu & & \\
& & & & & \ddots & \\
& & & & & & s(1-p)\mu \\
\end{array} \right),
\]
and
\[
\scriptsize
A_1 = \left( \arraycolsep=0.55pt
\begin{array}{cccccccc}
-(\lambda+n \delta) & n \delta & & & & & & \\
p \mu & -(\lambda+\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(\lambda+s\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & s p\mu & -(\lambda+s\mu+\delta) & \delta & \\
& & & & & s p\mu & -(\lambda+s\mu)\\
\end{array} \right).
\]
\subsection{Stability condition}
\label{app:stability}
From the general theory of QBD processes \citep{Neuts1981} follows that the Markov process $\{N(t),Q_1(t)\}$ is ergodic (stable) if and only if
\begin{equation}
\pi A_0 e < \pi A_2 e,
\label{eq:QBDstableCondition}
\end{equation}
where $e$ is the all one column vector and $\pi=(\pi_0,...,\pi_n)$ is the equilibrium distribution of the Markov process with generator $A_0+A_1+A_2$. In other words, $\pi$ is such that
\begin{equation}
\begin{array}{ll}
\pi(A_0+A_1+A_2) =0, & \pi e =1,
\end{array}
\label{eq:QBDstableProbabilityVector}
\end{equation}
and
\[
A_0+A_1+A_2 = \qquad\qquad\qquad
\]
\begin{align*}
{\scriptsize
\left(
\begin{array}{cccccccc}
-n \delta & n \delta & & & & & & \\
p \mu & -(p\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(ps\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & p s \mu & -(ps\mu+\delta) & \delta & \\
& & & & & p s \mu & -ps\mu\\
\end{array} \right).
}
\end{align*}
Then $\pi$ must satisfy the balance equations
\begin{align*}
- n \delta \pi_0 + p \mu \pi_1 &= 0, \\
(n-j+1)\delta \pi_{j-1} - (p\nu(j) +(n-j)\delta) \pi_j + p \nu(j+1) \pi_{j+1} &= 0, \\
\delta \pi_{n-1} - p s \mu \pi_n &= 0,
\end{align*}
with $\nu(j)=\min\{j,s\}\mu$, and the normalization condition
\[
\sum_{i=0}^n \pi_i=1.
\]
It is readily verified that
\begin{equation}
\pi_i =
\left\{\begin{array}{ll}
\pi_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\pi_0 \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistr}
\end{equation}
with
\begin{align*}
\pi_0= \left(\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right)^{-1}.
\end{align*}
satisfies the balance equations and the normalization condition.
\begin{proposition}
The distribution of the closed two-node Jackson network illustrated in Figure~\ref{fig:Jennings} is given by
\begin{equation}
\hat{\pi_i} =
\left\{\begin{array}{ll}
\hat{\pi}_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\hat{\pi_0} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistrTwistedJennings}
\end{equation}
with
\begin{align*}
\hat{\pi}_0= \left[\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right]^{-1}.
\end{align*}
\label{prop:CriticalTilburgdistr}
\end{proposition}
\begin{proof}
We have a two-node closed Jackson network, with probability transition matrix
\[
P = \left(
\begin{array}{cc}
1-p & p \\
1 & 0
\end{array} \right).
\]
Let $r_i(m)$ denote the rate of service when there are $m$ patient at queue $i$, so $r_1(m)=\min\{m,s\}$ and $r_2(m)=m$. The throughput vector $\gamma = (\gamma_1,\gamma_2) \in \mathbb{R_1}^2$ must satisfy $\gamma = \gamma P$ and we find that $\gamma=(p,1)$ suffices. From the general theory of Jackson networks, see \cite{Jackson1963}, it follows that the stationary distribution is given by
\begin{align*}
\pi_i = G^{-1} g_1(i) g_2(n-i)
\end{align*}
with
\begin{align*}
\begin{array}{ll}
g_1(i)= \frac{(\gamma_1/\mu)^i}{\prod_{m=1}^i r_1(m)}, & g_2(n-i)= \frac{(\gamma_2/\delta)^{n-i}}{\prod_{m=1}^{n-i} r_2(m)},
\end{array}
\end{align*}
and normalization constant $G= \sum_{i=0}^n g_1(i) g_2(n-i)$. Then,
\begin{align*}
g_1(i) &= \left\{\begin{array}{ll}
\frac{1}{i! \mu^i} & \textrm{\normalfont for } 0 \leq i \leq s, \\
\frac{1}{s! s^{i-s} \mu^i} & \textrm{\normalfont for } s+1 \leq i \leq n, \\
\end{array} \right.\\
g_2(n-i) &=\frac{1}{(n-i)!} \left(\frac{p}{\delta}\right)^n \left(\frac{\delta}{p}\right)^i,
\end{align*}
and rewriting the expressions yields~\eqref{eq:eqdistrTwistedJennings}.
\end{proof}
\subsection{Stationary distribution}
\label{app:StationaryDistributrion}
Assuming that the stability condition is satisfied, we can determine the unique stationary distribution of the Markov process $\{N(t),Q_1(t)\}$. The vector $\pi_i$ can be written as $\pi_{n+i}= \pi_n G^{i}$ for $i=0,1,...$, where $G$ is the minimal nonnegative solution of the non-linear matrix equation
\begin{equation}
A_0+G A_1 + G^2 A_2=0.
\label{eq:MG-G}
\end{equation}
The balance equations can be written as
\[
\begin{array}{ll}
\pi_{i-1} A_0+ \pi_i A_1 + \pi_{i+1} A_2=0, & i=n+1,n+2,...
\end{array}
\]
and using $\pi_{n+i}= \pi_n G^{i-n}$ for $i=0,1,...$, this find
\[
\begin{array}{ll}
\pi_n G^{i-n-1} \left(A_0+ G A_1 + G A_2\right)=0, & i=n+1,n+2,....
\end{array}
\]
\noindent
Moreover, we have the boundary equations
\begin{align*}
\pi_0 B_{00} + \pi_1 B_{10} &= 0 \\
\pi_0 B_{01} + \pi_1 B_{11} + \pi_2 B_{21} &= 0 \\
\pi_1 B_{12} + \pi_1 B_{22} + \pi_2 B_{32} &= 0 \\
&\vdots& \\
\pi_{n-2} B_{n-2 \, n-1} + \pi_{n-1} B_{n-1 \, n-1} + \pi_{n} B_{n \, n-1} &= 0 \\
\pi_{n-1} B_{n-1 \, n} + \pi_{n} B_{nn} + \pi_{n+1} A_2 &= 0,
\end{align*}
along with the normalization equation
\[
1 = \sum_{i=0}^{\infty} \pi_i e = \sum_{i=0}^{n-1} \pi_i e + \pi_n(I-G)^{-1}e,
\]
where we slightly abuse notation by using $e$ as the all ones vector of appropriate size. We note that the matrix $G$ has a spectral radius less than one and therefore $(I-G)$ is invertible.
These equations provide the tools for finding the equilibrium probabilities. Although it is hard to solve $G$ analytically from Equation~\eqref{eq:MG-G}, it is easy to solve numerically by using the following algorithm (matrix-geometric method). Rewriting~\eqref{eq:MG-G} gives
\[
G=-(A_0+G^2 A_2) A_1^{-1},
\]
where $A_1$ is invertible, since it is a transient generator matrix. Let
\[
G_{k+1}=-(A_0+G_k^2 A_2) A_1^{-1},
\]
starting with $G_k=0$. We note that $G_k \uparrow G$ as $k$ grows to infinity \citep{Neuts1981}. Once $||G_{k+1}-G_{k}||_2$ is below a certain preset threshold, we approximate $G$ by $G_{k+1}$.
\section{Proof of Proposition \ref{thm:stochasticordering}}\label{app:stochastic_ordering}
First, note that by definition of the Erlang-R model with holding, in which no more that $n$ patients can be admitted in the ED simultaneously, that $Q_1^h(t)+Q_2^h(t) \leq n = Q_1^J(t) + Q_2^J(t)$ follows directly.
Therefore, we only consider the relation between the states in the blocking and holding variants Erlang-R model.
As noted Section \ref{sec:Markov_process}, the model with holding can be characterized as a three-dimensional Markov chain $X^h(t) = (H(t),Q^h_1(t),Q^h_2(t))$ in which the components denote the number of holding, needy and content patients respectively. The Erlang-R model with blocking similarly admits a Markov process description, but with two dimensions, namely $X^b(t) = (Q^b_1(t),Q^b_1(t))$.
We prove the result by constructing a coupling between the Markov processes $X^h$ and $X^b$. Let $Z(t) := \left(\hat{X}^h(t),\hat X^b(t)\right) = \left(\hat{H}(t),\hat{Q}_1^h(t),\hat{Q}_2^h(t),\hat{Q}^b_1(t),\hat{Q}^b_2(t)\right)$.
We first define the transition rates of this five-dimensional Markov process, which naturally only depend on the current state of the system.
After that we show that the transition rates relevant to $\hat{X}^h(t)$ and $X^{h}(t)$ coincide with those of either $X^h(t)$ or $\hat{X}^b(t)$, respectively. The latter implies that the marginal transitions of $\hat{X}^h(t)$ and $X^b(t)$ (and $\hat{X}^b(t)$ and $X^h(t)$) are equal, and hence so are their probability distribution of the Markov processes.
Let $Z(t) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$. While defining the reachable states from this state and associated transition rates, we distinguish four transition types, and further differentiate the transition rates depending on the current state.\\
\\*
\textbf{Arrival.}
Arrivals to occur in both models simultaneously, but are handled differently according to the current queue lengths.
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr1}
(h,q_1^h+1,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr2}
(h+1,q_1^h,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h < n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr3}
(h,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr4}
(h+1,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\end{enumerate}
\noindent \textbf{Departure.}
Basically, we align service completions in the two models, but allow a completion occurring solely in either of one of the two models, only if the queue length in this model is strictly larger than in the other one.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep1}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(h-1,q_1^h,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.
\end{equation}
\item If $q_1^h < q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep2}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h \geq q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep3}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep4}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(0,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent\textbf{Become content.}
The differentiation between transitions is similar to those in the \textit{departure} transition type.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$,
\begin{equation}
\label{eq:con1}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b \wedge s)p\mu,\\
(h,q_1^h-1,q_2^h+1,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$,
\begin{equation}
\label{eq:con2}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^h \wedge s)p\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b+1) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent
\textbf{Become needy.}
\begin{enumerate}
\item If $q_2^h \geq q_2^b$,
\begin{equation}
\label{eq:ne1}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^b\delta,\\
(h,q_1^h+1,q_2^h-1,q_1^b,q_2^b) & \text{with rate }(q_2^h-q_2^b)\delta,\\
\end{array}
\right.\end{equation}
\item If $q_2^h < q_2^b$,
\begin{equation}
\label{eq:ne2}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^h\delta,\\
(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1) & \text{with rate }(q_2^b-q_2^h)\delta,\\
\end{array}
\right.\end{equation}
\end{enumerate}
This set of transitions defines the dynamics of the Markov process $Z(t) = (\hat{X}^h(t),\hat{X}^b(t))$.
Let us now restrict our attention to the transitions in which (at least one of the) first three coordinates of $Z(t)$ changes, that is, the marginal transitions of the process $\hat{X}^h$.
Let $\hat{X}^h(t) = (h,q_1^h,q_2^h)$, then according to the transition scheme above, $\hat{X}^h$ moves to state
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ (and hence necessarily $h=0$),
\[
\left\{
\begin{array}{ll}
(0,q_1^h+1,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h-1,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $q_1^h+q_2^h = n$ and $h=0$,
\[
\left\{
\begin{array}{ll}
(1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $h>0$ (and hence necessarily $q_1^h+q_2^h = n$),
\[
\left\{
\begin{array}{ll}
(h+1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(h-1,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(h,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(h,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\end{enumerate}
One can check that these transitions indeed coincide with the transitions in the original holding model, hence $\hat{X}^h(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Similarly, when the focusing on transitions of $Z(t)$ that are relevant for $\hat{X}^b(t)$, we deduce the following transition scheme. If $\hat{X}^b(t) = (q_1^b,q_2^b)$, then the next move according to the transitions of $Z(t)$ is
\[
\left\{
\begin{array}{ll}
(q_1^b+1_{\{q_1^b + q_2^b < n\}},q_2^b) & \text{with rate } \lambda,\\
(q_1^b-1,q_2^b) & \text{with rate }(q_1^b\wedge s)(1-p)\mu,\\
(q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b\wedge s)p\mu,\\
(q_1^b+1,q_2^b-1) & \text{with rate }q_2^b \delta.
\end{array}
\right.\]
These transition rates clearly coincide with the original Erlang-R model with blocking, and also hence $\hat{X}^b(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Next, we show that under this coupling scheme we have that if $\hat{H}(0) = 0$, $\hat{Q}_1^h(0)=\hat{Q}_1^b(0)$ and $\hat{Q}_1^h(0)=\hat{Q}^b(0)$ then for all $t\geq 0$, $Z(t)$ satisfies the hypothesis:
\begin{itemize}
\item[(i)] $\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \leq \hat{Q}_1^h(t) + \hat{Q}_2^h(t)$,
\item[(ii)] $\hat{Q}_2^b(t) \leq \hat{Q}_2^h(t)$,
\item[(iii)] $\hat{Q}_1^b(t) \leq \hat{Q}_1^h(t) + H(t)$.
\end{itemize}
We do so by induction on the next state reached after a transition of the joint Markov process $Z=(\hat{X}^h,\hat{X}^b)$.
First of all, $Z(0)$ clearly satisfies (i)-(iii).
Next, assume $Z(t^-) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$ satisfies the hypothesis and a transition occurs at $t$.
We show that under the specified coupling scheme, the state reached after the next transition, $Z(t)$ must satisfy (i)-(iii) as well. To do so, we differentiate between the four types of transitions that could occur: arrival, departure, become content and become needy.\\
\\*
\noindent\textbf{Arrival.}
Recall that under our coupling scheme an arrival always occurs in both the holding and blocking model simultaneously, see \eqref{eq:arr1}--\eqref{eq:arr4}. Furthermore, $q_2^h$ and $q_2^b$ are unchanged during this transition, rendering (ii) trivial.
By hypothesis $q_1^b + q_2^b \leq q_1^h+q_2^b$, hence the event $q_1^h+q_2^h < n$ and $q_1^h+q_2^b =n$, with resulting state $(0,q_1^h+1,q_2^h,q_1^b,q_2^b)$, can be excluded from our analysis
We check the conditions for the remaining three cases.
\begin{enumerate}[noitemsep]
\item If $Z(t)= (0,q_1^h+1,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h <n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \less[i] q_1^h+q_2^h+1 =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+1 = \hat Q_1^h(t) = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h =n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \leq n = q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h +1= \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $q_1^b + q_2^b = q_1^h+q_2^h=n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+h+1 = \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Departure.}
By carefully examining the possible state transitions of $Z(t)$ following a departure, we list six reachable states. However, by (iii), we have that if $h=0$, then $q_1^b \leq q_1^h$, which excludes the state $(0,q_1^h,q_2^h,q_1^b,q_2^b)$ in \eqref{eq:dep4} from the reachability graph.
We check the remaining states for conditions (i)--(iii). Again, during a departure, $q_2^b$ and $q_2^h$ are unchanged, so (ii) is automatically satisfied by the induction hypothesis.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 \less[i] q_1^h+q_2^h-1 < q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h + h-1 = \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $h>0$ and $q_1^h \geq q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 \leq q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$ and $q_1^h < q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 < q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^b \less[*] q_1^h + h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h,q_1^b-1,q_2^b)$, then $h=0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = (q_1^b-1)+q_2^b-1 < \less[i] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h-1 = \hat Q_1^h(t) + \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (0,q_1^h-1,q_2^h,q_1^b,q_2^b)$, then $h=0$ and $q_1^h>q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] (q_1^h-1)+q_2^b \less[ii] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = qy \less[*] q_1^h-1 =\hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Content start.}
On the event of a patient becoming content, it is clear that the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected. This means that (i) is directly satisfied by the induction hypothesis.
According to \eqref{eq:con1}--\eqref{eq:con2}, three states can be reached.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1)$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \less[ii] q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b,q_2^b)$, then $q_1^h > q_1^b$ (*),
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[ii] q_2^h < q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h = \hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b+1)$, then $q_1^b > q_1^h$ and hence by (iii) $h > 0$. The latter is only possible if $q_1^h+q_2^h=n$ (*),
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \leq n-q_1^b+1 = (q_1^h+q_2^h)-q_1^b+1 \less[*] q_2^h = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^h+h-1 \less[*] q_1^h+h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent \textbf{Become needy.}
Just as in the event of content start, the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected, whereby (i) is directly satisfied by the induction hypothesis.
By (ii), we have $q_2^h \geq q_2^b$. This excludes the state $(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1)$ from being reached, see \eqref{eq:ne2}.
We check the remaining two possibilities.
\begin{enumerate}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1)$.
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b-1 \less[ii] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+h+1 = \hat Q_1^h(t)+\hat H(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b,q_2^b)$, then $q_2^h > q_2^b$ (*).
\begin{itemize}
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[*] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h =\hat Q_1^h(t) + \hat H(t)$.
\end{itemize}
\end{enumerate}
Hence, the state reached after any feasible transition under the coupling scheme satisfies the conditions (i)--(iii).
Thus we conclude that the joint process\\ $(\hat{H}(t),\hat Q_1^h(t),\hat Q_2^h(t),\hat Q_1^b(t),\hat Q_2^b(t))$ adheres to (i)--(iii) for all $t$. Consequently, we have that (i) implies
\begin{align*}
\mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right) &= \mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right)\\
&=\sum_{j=0}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&=\sum_{j=k}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&\leq \sum_{j=h}^n \mathbb{P}\left( \hat Q_1^h(t)+\hat Q_2^h(t) = j \right)\\
&= \mathbb{P}\left( Q_1^h(t) + Q_2^h(t) \geq k\right) = \mathbb{P}\left(Q_1^h(t) + Q_2^h(t) \geq k\right).
\end{align*}
The other two orderings follow similarly.
\begin{remark}
Note that under this coupling scheme we cannot get the ordering $\hat Q_1^h(t)(t) \geq \hat Q_1^b(t)(t)$ for all $t\geq 0$. A minimal counter example occurs for $s=n=1$. Let $Z(0) = ((0,0,0),(0,0))$. First, two arrivals occur, such that state $((1,1,0),(1,0))$ is reached, followed by a departure transition, yielding $((0,1,0),(0,0))$. Next, the one patient left in the model with holding system becomes content, so that we obtain $((0,0,1),(0,0))$.
At this stage, if an arrival occurs, the arriving patient will be put in the holding queue in the model with holding, and admitted to nurse queue in the model with blocking. Hence we end up in state $((1,0,1),(1,0))$, in which $\hat Q_1^h(t) < \hat Q_1^b(t)$.
\end{remark}
\section{Proof of Proposition \ref{prop:stability_convergence}}\label{app:proof_stability_convergence}
Define
\[
A(s,n) = \sum_{k=0}^s \frac{k}{s} \, \binom{n}{k} b^k ,\quad
B(s,n) = \sum_{k=s+1}^n \frac{k!}{s!} \, \binom{n}{k} s^{s-k} b^k, \quad
C(s,n) = \sum_{k=0}^s \binom{n}{k} \, b^k,
\]
\[
\]
where $b = \delta/p\mu = r/(1-r)$. Then
\[
\rho_{\rm max}(s,n) = \frac{A(s,n)+B(s,n)}{C(s,n)+B(s,n)}.
\]
Proving that $\rho_{\rm max}(s,n) \to 1$ as $R_1\to\infty$ with $s$ and $n$ as in \eqref{eq:twofoldscaling} is equivalent to showing that
\begin{equation}\label{eq:proof_stab_1}
1-\rho_{\rm max}(s,n) = \frac{C(s,n)-A(s,n)}{C(s,n)+B(s,n)} = \frac{(1+b)^{-n}[C(s,n)-A(s,n)]}{(1+b)^{-n}[C(s,n)+B(s,n)]} \to 0.
\end{equation}
First, we rewrite
\begin{align*}
(1+b)^{-n} A(s,n)
&= (1+b)^{-n} \sum_{k=1}^s \frac{n}{s} \binom{n-1}{k-1} b^k \\
&= \frac{n}{s}\left(\frac{b}{1+b}\right)\sum_{k=0}^{s-1} \binom{n-1}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-1-k}\\
&= \frac{r n}{s}\sum_{k=0}^{s-1} \binom{n-1}{k} r^k (1-r)^{n-1-k}\\
&= \frac{r n}{s} \mathbb{P}( {\rm Bin}(n-1,r) \leq s-1 ) \\
&= \frac{rn}{s} \mathbb{P}\left( \frac{{\rm Bin}(n-1,r) - (n-1)r}{\sqrt{nr(1-r)}} \leq \frac{s-1 - (n-1)r}{\sqrt{nr(1-r)}} \right)\\
&\to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
since $nr/s = 1 + O(1/\sqrt{R_1})$.
Also,
\begin{align*}
(1+b)^{-n} C(s,n)
&= \sum_{k=0}^s \binom{n}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-k}\\
&= \sum_{k=0}^s \binom{n}{k} r^k (1-r)^{n-k}\\
&= \mathbb{P}( {\rm Bin}(n,r) \leq s) \to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right).
\end{align*}
Therefore, we have $(1+b)^{-n}[C(s,n)-A(s,n)] \to 0$ as $\lambda\to\infty$.
For the remaining term,
\begin{align*}
(1+b)^{-n} B(s,n)
&= (1+b)^{-n}\sum_{k=s+1}^n \binom{n}{k}\,\frac{k!}{s!} s^{s-k} b^k \\
&= (1+b)^{-n}\frac{n!}{s!}\, s^s\sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{-k}\\
&= (1+b)^{-n} \frac{n!}{s!}\, s^s\, \left(\frac{b}{s}\right)^n \sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{n-k}\\
&= r^n\, \frac{n!}{s!} s^{s-n} \sum_{m=0}^{n-s-1} \frac{1}{m!} \left(\frac{s}{b}\right)^m\\
&= \left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b} \, \mathbb{P}({\rm Pois}(s/b)\leq n-s-1),
\end{align*}
in which
\begin{align*}
\mathbb{P}({\rm Pois}(s/b)\leq n-s-1)
&= \mathbb{P}\left(\frac{{\rm Pois}(s/b)-s/b}{\sqrt{s/b}} \leq \frac{n-s-1-s/b}{\sqrt{s/b}}\right) \\
&\to \Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
as $\lambda\to\infty$.
By Stirling's approximation,
\begin{align*}
\left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b}
&\sim \left(\frac{r}{s}\right)^n \sqrt{\frac{n}{s}} \,\frac{n^n {\rm e}^{-n}}{s^s {\rm e}^{-s}}\, s^s \,{\rm e}^{s/b} \\
&= \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s+s/b} = \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s/r}.
\end{align*}
Since,
\[
\frac{rn}{s} = 1 + \frac{\gamma\sqrt{r}-\beta}{\sqrt{R_1}} + O(1/R),
\]
we find $\sqrt{n/s} = 1/\sqrt{r} + O(1/\sqrt{R_1}$ and
\begin{align*}
\log\left[ \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s/r} \right]
&= n \log\left[ \frac{rn}{s}\right] - n+\frac{s}{r}\\
&= -n \left[ \left(1-\frac{rn}{s}\right) + \frac{1}{2}\left(1-\frac{rn}{s}\right)^2 + O(R^{-3/2}) \right] + \frac{s}{r}\left(1-\frac{rn}{s}\right)\\
&= \frac{s}{r}\left(1-\frac{rn}{s}\right)^2 - \frac{n}{2}\left(1-\frac{rn}{s}\right)^2 + O(1/\sqrt{R_1})\\
&= \frac{(\gamma\sqrt{r} - \beta)^2}{2r} + O(1/\sqrt{R_1}),
\end{align*}
as $\lambda\to\infty$ and hence,
\[
(1+b)^{-n} B(s,n) \to \varphi\left(\frac{\gamma\sqrt{r}-\beta}{\sqrt{r}}\right)\Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right).
\]
Hence, we conclude that the denominator of \eqref{eq:proof_stab_1} converges to a constant value as $R_1$ grows, and hence the $1-\rho_{\rm max}(s,n)\to 0$ as $\lambda\to\infty$.
\end{subappendices}
\chapter{Finite-size effects in critically dimensioned emergency departments}
\begin{chapterstart}
Motivated by health care systems with repeated services that have both personnel (nurse/physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we account for are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them.
We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly.
We show that the system dynamics is captured by the fraction of needy time in the network, and that returning patients should be accounted for both in steady-state and time-varying conditions.
We demonstrate the application of our policies in two case-studies of resource allocation in hospitals.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Finite-size effects in critically dimensioned emergency departments}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen, Fiona Sloothaak \& Galit Yom-Tov}\\
Submitted to \textit{Operations Research}
\end{flushright}
\newpage
\section{Introduction}
In recent years, operations research techniques have received increased interest from the health care community, as they are able to design and improve workflow processes in health care facilities~\cite{Armony2015,Green2008,Bekker2009b,Hall2006,Hall2012}.
Because these processes are typically stochastic in nature, it is common practice to use queueing theory for performance analyses and workforce planning.
As a first step towards understanding the processes going on in health care environments, systems are commonly modeled after a single station queue, such as the $M/M/s$ (Erlang-C), $M/M/s/s$ (Erlang-B) or $M/M/s+M$ (Erlang-A) models, and fluid and diffusion approximations are used to provide insights into the process dynamics.
However, simple single station models often fail to capture the more intricate dynamics of the settings specific to health care contexts.
Prime examples include the flows of patients in a hospital from one medical ward to another \citep{Armony2015}, within the Emergency Department (ED) between different stages of treatment \citep{Junfei2015}, or between medical facilities \citep{zychlinski2016bedblocking}.
Queueing networks can capture the dependency between several service stages and several types of resources.
More specifically, we are interested in the ubiquitous feature, particularly present in health care environments, that patients during their stay in the system might require a specific resource multiple times, e.g.~physicians and nurses who treat patients several times during their stay in the medical wards \citep{Jennings2011} or the ED \citep{YomTov2014}, while multiple resources types are limited (e.g.\ medical staff and beds).
In this chapter, we concentrate on the dynamics within EDs.
An often ignored yet essential feature of medical facilities concerns the restriction of the number of patients that can reside in the facility simultaneously.
In Chapter 4, we already observed that finite-size restrictions can have a significant effect on the performance of queueing systems.
In this chapter, we investigate the influence of such multiple restrictions on the network dynamics and the required staffing policies in the context of an ED. \\
\\*
\noindent
\textbf{The restricted Erlang-R model.}
The canonical model for service networks with returns is the Erlang-R model \citep{YomTov2014} in which customers, during their stay in the system receive a random number of services from the same pool of servers.
Yom-Tov \& Mandelbaum \cite{YomTov2014} showed that such a simple network model can be used to determine staffing in an ED both in stable and time-varying conditions.
Nevertheless, empirical studies report that some countries, such as the US, use a different operational mode that applies strict restrictions on entering the ED \citep{EDexperiment}.
In typical US EDs, a patient will not enter the ED until both a bed and a physician are available to treat her.
Those restrictions can be either physical (beds) restrictions or managerial ones --- for instance by imposing a patient-to-physician ratio.
In this work, we extend the Erlang-R model by enforcing a constraint on the maximum number of available places inside the facility.
Our model hence incorporates two kinds of resource constraints: servers that provide the actual service and the maximum available places inside the service system.
Both affect the system in a highly interdependent way.
The model, presented in Figure \ref{fig:Erlang_R_model}, assumes $s$ servers and a maximum capacity of $n$ concurrent places.
We assume that patients arrive according to a Poisson process with rate $\lambda$.
In case a new arrival finds $n$ or more patients already present, we consider two options: either she waits outside the service facility in a holding queue until a vacant space becomes available (Figure \ref{fig:Erlang_R_holding}) or she is blocked (Figure \ref{fig:Erlang_R_blocking}), such as is the case when patients are sent to an alternative facility.
Once a patient is admitted, she requires assistance from one of the $s$ servers for an exponentially distributed duration with mean $1/\mu$.
Then, with probability $1-p$, the patient leaves the system or, with probability $p$, returns to service again after an exponentially distributed time with mean $1/\delta$.
Following Jenning \& de V\'ericourt \cite{Jennings2011} and Yom-Tov \& Mandelbaum \cite{YomTov2014}, we call patients {\it needy} when they require attention from one of the servers and {\it content} when they are in the delayed return phase.
In addition, we call patients {\it holding} when they are waiting outside the facility for an available space. We assume that the arrival process, the needy times and content times are mutually independent.
In the holding queue and the needy queue, we apply the First-Come-First-Served (FCFS) discipline.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-2.75,4.5) -- (-1.25,4.5);
\draw [thick] (-1.5,5) -- (0,5) -- (0,4) -- (-1.5,4);
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (0,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick,->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,2.9) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\node [above] at (-0.75,5) {\footnotesize holding};
\end{tikzpicture}
\caption{Erlang-R model with holding.}
\label{fig:Erlang_R_holding}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-1.5,4.5) -- (2.5,4.5);
\draw [thick, ->] (0,4.5) -- (0,2.5) node[below left] {blocked};
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick, ->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,3.4) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{Erlang-R model with blocking.}
\label{fig:Erlang_R_blocking}
\end{subfigure}
\caption{Restricted Erlang-R models with maximally $n$ customers in system.}
\label{fig:Erlang_R_model}
\end{figure}
As mentioned, we consider two versions of the finite-capacity constraint.
The first version is called \emph{Erlang-R with holding}, in which patients wait for an available space in the system.
The second version is called \emph{Erlang-R with blocking}, in which patients meeting a full system are blocked.
Naturally, intermediate scenarios can be constructed in which a proportion of the total arrival volume of patients indeed leaves upon finding a full system, while the rest joins the holding queue.
While this chapter focuses on the two extreme cases, straightforward adaptions can fit these intermediate scenarios. \\
\\*
\noindent
\textbf{Examples of restricted Erlang-R.}
As noted before, an ED operated in the US can be modeled using a restricted Erlang-R model. Another health care example is medical units (MUs) in a hospital.
Such units specialize in specific types of illnesses (cadriatric, oncology, etc.) and have limited resources such as nurses and beds. If the unit is full, new patients are either allocated to an alternative medical unit, i.e.\ blocked, or wait for an available bed.
Both policies are problematic in terms of quality-of-care, because the personnel in the alternative unit (or the ED) may be less knowledgeable about the patient's medical condition and waiting in the ED was shown to increase mortality.
Moreover, ED waiting may reduce available capacity for treating ED patients \citep{Carmen2016,israelit}, hence endangering both the delayed patient as well as others. Both the number of personnel (nurses and physicians) and the number of beds impact service dynamics and quality-of-care. Research so far looked at the capacity allocation of those resources separately. Green \& Yankovic \cite{GY2011} and Jennings \& de V\'ericourt \cite{Jennings2008} looked at nurse staffing in medical units, while de Bruin et al.~\cite{Bekker2009b} looked into bed allocation. The unified model we suggest enables us to capture the dependency between those two decisions, and its impact on other medical units in the hospital.
At the same time, we capture the two most commonly used modes of operation --- blocking and holding of new patients. \\
\\*
\noindent
\textbf{Two-fold square-root staffing rule.}
Our main goal is to provide staffing policies for the ED that ensures high resource utilization, while at the same time maintains a good quality-of-care.
This goal relates to the philosophy of the Quality-and-Efficiency-Driven (QED) regime that is the recurring theme of this thesis.
In this chapter, we obtain asymptotic results for the Erlang-R model with blocking in the QED regime (Section \ref{sec:QED_limit_block}).
Following \cite{Jennings2008}, we employ a two-fold QED staffing policy: $s=R_1 +\beta \sqrt{R_1}$ for the number of nurses and $n=R_1/r+\gamma \sqrt{R_1/r}$ for the number of patients in the system (beds), where $\beta$ and $\gamma$ are constants, $R_1$ is the offered load of the servers (nurses) and $r$ is the fraction of time a patient spends in the needy state.
We establish limiting expressions for performance measures, such as the probability of delay and blocking, in the form of explicit functions that depend solely on $\beta$ and $\gamma$.
In deriving these limit results, we use the available product-form solution for the stationary distribution.
Likewise, we pursue QED performance for the Erlang-R model with holding.
However, a direct analytic approach is obstructed by the absence of product-form solutions.
We provide two solutions for establishing QED behavior.
First, we provide stochastic performance bounds that stay meaningful in the QED regime, which demonstrate the non-degenerate behavior of the two-fold scaling in the large-system limit.
Second, we develop a heuristic method that quantifies the difference between the holding model and the blocking model.
This method is to a large extend related to the asymptotic approximation method for retrial queues discussed in Chapter 4, in the sense that we approximate the model with holding through the model with blocking, yet with an increased arrival rate.
The increase in arrival rate turns out to be the solution of a fixed-point equation.
Using our results on the asymptotic behavior of the model with blocking in the QED regime, we then obtain approximative QED performance measures for the model with holding.
These theoretical findings ultimately yield algorithms for dimensioning and time-varying staffing. \\
\\*
\textbf{Structure of the chapter.}
We first review related literature on the subject of staffing in health care environments in Section \ref{sec:ed_literature}.
In Section \ref{sec:modeldescription}, we introduce the mathematical models more formally, and deduce preliminary results on their stability conditions and relative performance.
Section \ref{sec:QED_scaling} describes the scaling regime we use for our asymptotic study of the restricted Erlang-R models, and Sections \ref{sec:QED_limit_block} and Section \ref{sec:QED_limit_holding} present our main theoretical findings.
We turn to dimensioning problems in Section \ref{sec:dimensioning}, and show how our asymptotic QED results can be used to make resource allocation decisions in realistic settings.
Section \ref{sec:analysis_chapter5} is devoted to the numerical and comparative analysis of the restricted Erlang-R models, and also shows how our method can be applied in time-varying environments through a case study.
We summarize our findings and give directions for future research in Section \ref{sec:conclusion}.
\section{Literature review}
\label{sec:ed_literature}
Due to increasing demand and tightening budgets in health care, there is a growing need for efficient workforce management \citep{Green2008}. Personnel (nurse and physician) expenditure is one of the biggest factors in hospital costs \citep{Kazahaya2005}, and inadequate nursing levels have been mentioned as a significant factor in medical errors and ED overcrowding. In order to establish appropriate nursing levels, a staffing policy requires assessment of a wide range of variables, such as differing nurse expertise and patient acuity during the day. Current methods, such as the minimum nurse-to-patient ratios, are often too inflexible to capture those varying conditions. The American Hospital Association (AHA) and others call for dynamic staffing policies that can deal with the complex and evolving nature of health care \citep{AHA2007}.
Workforce management in health care systems has been studied extensively; see \cite{Denton2013,Hall2006,Hall2012} for overviews.
In recent years it has become apparent that queueing models can be helpful in developing staffing and routing recommendations, not just for large-scale service systems, but also for the small and complicated health care systems.
The first to try such an approach through queueing models were Green et al.~\cite{Green2006,Green2008}, who used the single station stationary Erlang-C model to set staffing levels in EDs and panel sizes for clinics. Using a similar approach, Bekker \& de Bruin~\cite{Bekker2009a} used Erlang-B model to determine bed allocation for medical wards.
The first to observe the significant impact of interrupted services in a health care setting were Jennings \& de V\'ericourt \cite{Jennings2008,Jennings2011}. Motivated by the need to set nurse-to-patient ratios for internal wards, they considered a closed queueing system with $s$ nurses and $n$ beds. This is essentially the Erlang-C model with the additional restriction that a finite population of the $n$ patients requires care. In their model, all beds are always occupied, and patients alternate between two phases: the needy phase where patients require service of a nurse and the content phase where they do not; see Figure \ref{fig:Jennings}. The system dynamics of restricted Erlang-R model are equivalent to those of the closed ward model of \cite{Jennings2008} if the holding queue would never be empty.
Campello et al.~\cite{Campello2016} analyzed a similar operational decision, referred to as ED case management, which determines the maximal number of patients a physician should handle in parallel. They also used queueing networks and analyzed the stationary distribution. Note that in practice such decision is not only affected by operational measurements such as waiting times, but also by psychological constraints that limit physician capability to manage multiple tasks (patients) in parallel.
KC \cite{diwas} provided empirical evidence that physicians should not treat more than 6-7 patients at the same time. Therefore, many hospitals in the US restrict entrance to EDs even if beds are available if physicians are overloaded.
We too consider such constraints, and analyze their impact on performance. We take a different approach than \cite{Campello2016}; instead of analyzing numerically steady-state distributions, we develop many-server approximations that can produce insight into the system dynamics, and can be incorporated into time-varying staffing procedures; see Section \ref{sec:case_study}.
\begin{figure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=1]
\draw [dashed, thick] (-0.5,-0.1) rectangle (3.5,2.85) node[right] {\footnotesize $n$};
\draw [thick,->] (1.1,0.5) -- (0,0.5) -- (0,2) -- (0.6,2);
\draw [thick,->] (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw [thick] (0.6,1.7) -- (1.6,1.7) -- (1.6,2.3) -- (0.6,2.3);
\draw [thick] (1,1.7) -- (1,2.3);
\draw [thick] (1.2,1.7) -- (1.2,2.3);
\draw [thick] (1.4,1.7) -- (1.4,2.3);
\draw (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw (1.5,0.5) circle [radius=0.4] node[above=0.3cm] {\footnotesize $p/\delta$} ;
\draw (2,2) circle [radius=0.4] node[above=0.3cm] {\footnotesize exp($\mu$)};
\node at (1.5,0.5) {\footnotesize $\infty$};
\node at (2,2) {\footnotesize $s$};
\end{tikzpicture}
\caption{The closed ward model.}
\label{fig:Jennings}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\draw [thick, ->] (0,4.5) node[above=0.3cm,right] {\footnotesize Pois$(\lambda)$} -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5];
\draw [thick, ->] (4.75,4.5) -- node[above=0.3cm,right] {\footnotesize $1-p$} (7,4.5);
\draw [thick,->] (5.75,4.5) -- node[right] {\footnotesize $p$} (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5];
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node at (4.25,4.5) {\footnotesize $s$};
\node at (3.5,2) {\footnotesize $\infty$};
\node [above] at (4,5.15) {\footnotesize exp($\mu$)};
\node [above] at (3.5,2.4) {\footnotesize exp($\delta$)};
\end{tikzpicture}
\caption{The Erlang-R model.}
\label{fig:ErlangR}
\end{subfigure}
\caption{Related queueing models.}
\end{figure}
The model in~\cite{Jennings2008,Jennings2011} was developed for modeling internal dynamics within an internal ward. However, in the ED, beds are not constantly occupied and the utilization level depends on the flow of patients that arrive from outside the system.
Yom-Tov \& Mandelbaum \cite{YomTov2014} highlight the interrupted services while accounting for the transient nature of patient's arrival process, and introduced the Erlang-R model as a model for an ED. The Erlang-R model is an open two-station queueing network that has the same layout as the restricted Erlang-R model, except that all patients find a bed available upon arrival, see Figure \ref{fig:ErlangR}. In both models patients experience the interrupted services, but the Erlang-R model has no further restrictions on the bed capacity, hence neglecting the finite-size effects. Yom-Tov \& Mandelbaum \cite{YomTov2014} showed, using a simulator tailored to an Israeli ED, that the complicated small ED dynamics can be captured using the relatively simple Erlang-R model, and hence, its recommendations can be implemented in ED workforce management.
Although the feature of interrupted services is present in many systems, it is particularly important for modeling EDs, because the duration of the interruption is typically much longer than the time patients require care from a nurse. This explains why the Erlang-R model is considered to be the canonical model for EDs. The restricted Erlang-R model with holding/blocking thus extends the Erlang-R model with finite-size constraints which, like interrupted services, are expected to have a decisive impact on performance.
\section{Models and performance measures}
\label{sec:modeldescription}
\subsection{Three-dimensional Markov process}
\label{sec:Markov_process}
Since in the restricted Erlang-R model described the arrival process is taken Poisson, and all service and content times are assumed independent and exponential, the system can be characterized in terms of a Markov process.
Let $Q(t) = (H(t),Q_1(t),Q_2(t))$ represent the number of patients in the \emph{holding}, \emph{needy} and \emph{content} state at time $t$, respectively.
In both variants, $n$ is the maximum number of patients admitted to system, we have $Q_1(t)+ Q_2(t)\leq n$ for all $t\geq 0$.
Due to the absence of holding patients in the Erlang-R model with blocking, $H(t)=0$ is enforced in this case, whereas $H(t)$ has unbounded support in the model with holding.
This distinction requires us to explore the stationary distribution of the two variants separately.
Before doing so, we introduce some additional notation.
We define
\begin{equation}
R_1 := \frac{\lambda}{(1-p)\mu}, \qquad R_2 := \frac{p\lambda}{(1-p)\delta},
\label{eq:R1_R2}
\end{equation}
where $R_1$ and $R_2$ can be interpreted as the offered workload brought towards the needy queue and the content (infinite-server) queue, respectively.
Furthermore, we define
\begin{equation}
r:= \frac{\delta}{\delta+p\mu},
\label{eq:delta}
\end{equation}
which is the fraction of time a patient spends in the needy state (in case she experienced no wait during her sojourn). \\
\\*
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.9]
\draw [thick] (-1.25,5) -- (0,5) -- (0,4) -- (-1.25,4);
\draw [thick] (0.5,4.5) circle [radius = 0.5] node {\footnotesize 1} node[above=0.5cm] {\footnotesize exp$(\lambda)$}
node[below =0.5cm] {\footnotesize \color{col1} Station 0} ;
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (1,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {\footnotesize $s$} node[above=0.5cm] {\footnotesize exp$(\mu)$}
node[below = 0.5cm] {\footnotesize \color{col1} Station 1} ;
\draw [thick, ->] (4.75,4.5) -- node[right=0.8cm,above] {\footnotesize $1-p$} (6.5,4.5) -- (6.5,1.6) -- (-2,1.6) -- (-2,4.5) -- (-1.2,4.5);
\draw [thick,->] (5.75,4.5) -- node[left] {\footnotesize $p$} (5.75,2.5) -- (4,2.5);
\draw [thick] (3.5,2.5) circle [radius=0.5] node {\footnotesize $\infty$} node[above=0.4cm] {\footnotesize exp$(\delta)$}
node[below right = 0.35cm] {\footnotesize \color{col1} Station 2} ;;
\draw [thick,->] (3,2.5) -- (1.5,2.5) -- (1.5,4.5);
\draw [thick, dashed] (-3,1.2) rectangle (7.25,5.75) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{The Erlang-R model with blocking viewed as a closed Jackson network.}
\label{fig:ErlangR_blocking}
\end{figure}
\noindent
\textbf{Erlang-R model with blocking.}
In case of the blocking model, $Q(t)$ reduces to a finite-state Markov process $Q(t) = (Q_1(t),Q_2(t))$, where $Q_1(t)+Q_2(t)\leq n$ for all $t\geq 0$.
In fact, this is equivalent to the closed Jackson network depicted in Figure \ref{fig:ErlangR_blocking} with finite population $n$.
Station 1 in Figure \ref{fig:ErlangR_blocking} is an $M/M/s$ queue with service rate $\mu$, modeling the number of needy patients $Q_1(t)$.
Station 2 models the number of content patients $Q_2(t)$, and can therefore be represented as an infinite-server queue with service rate $\delta$.
A patient can enter the unit only if $Q_1(t)+Q_2(t)<n$.
Station 0---a single-server queue---moderates this as it only produces output at rate $\lambda$ in case its queue length is positive, i.e.\ if $n-Q_1(t)-Q_2(t)>0$.
Observe that because patients finding a full network are blocked, the number of patients in the system cannot grow beyond $n$.
Hence, the system is stable for all parameter settings, and hence a steady-state distribution exists. Moreover, the simplification of the model with blocking allows us to express the steady-state distribution of the system in explicit product-form.
Let $\pi_b(j,k)$ denote the steady-state probabilities of having $j$ needy and $k$ content patients in the system. Then,
\begin{equation}\label{eq:pih(i,j)}
\pi_b(j,k) = \left\{
\begin{array}{ll}
\pi_0\,\frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k, & ~~~\text{if }j+k \leq n,\\
0, & ~~~\text{else,}
\end{array}\right.
\end{equation}
where
\begin{equation*}
\kappa(j) := \left\{
\begin{array}{ll}
j! , & ~~\text{if }j \leq s,\\
s!\, s^{j-s}, &~~ \text{else,}
\end{array}\right.
\end{equation*}
and $
\pi_0^{-1} = \sum_{j+k\leq n} \frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k$.\\
\\*
\textbf{Erlang-R model with holding.}
\label{ref:modelsoft}
The Erlang-R model with holding does not lead to a Jackson network with an elegant product-form solution for the steady-state distribution, because the holding queue cannot be modeled as a station that is independent from the other queues in the system.
%
However, we are able to describe the system as a two-dimensional Markov process without loss of information.
To see this, define $N:= \{N(t)\}_{t\geq 0}$ with $N(t) := H(t)+Q_1(t) + Q_2(t)$, the total number of patients in the system (including the holding queue).
Using the restriction $Q_1(t)+Q_2(t) \leq n$ together with the fact that no bed is left vacant if a patient is waiting in the holding queue, this yields
\begin{equation*}
H(t) = \left(N(t) - n\right)^+, \quad t\geq 0,
\end{equation*}
where $(\cdot)^+ := \max\{0,\cdot\}$.
For the same reason, $Q_2(t) = N(t) - Q_1(t)$ if $H(t)=0$, and $Q_2(t) = n-Q_1(t)$ otherwise.
In other words,
\begin{equation*}
Q_2(t) = \min\{N(t),n\} - Q_1(t), \quad t \geq 0.
\end{equation*}
Therefore, we can express the state of all three queues in the Erlang-R model with holding using a two-dimensional Markov process $X:= \{X(t)\}_{t\geq 0}$, where
\begin{equation*}
X(t) :=\left( N(t), Q_1(t) \right).
\end{equation*}
The process $X$ lives on the semi-infinite strip
\begin{equation*}
X(t) \in \left\{\,(i,j)\, |\, j \leq \min\{i,n\}, i\in \mathbb{N}_0, j \in \{0,1,\ldots,n\}\, \right\},
\end{equation*}
and belongs to the class of Quasi-Birth-Death (QBD) processes.
The reader is referred to Appendix~\ref{app:QBDdescription} for a detailed description of this process, in terms of its transition diagram and generator matrix.
Contrary to the model with blocking, the system with holding \emph{can} become unstable in case capacity is insufficient to satisfy patient demand.
\begin{proposition}\label{prop:StabilityCondition}
The Erlang-R model with holding is stable if and only if
\begin{equation}
\frac{\lambda}{(1-p)\mu s} < \frac{ \sum_{i=0}^s \frac{i}{s}\binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
{ \sum_{i=0}^s \binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
=: \rho_{\max}(s,n).
\label{eq:StabilityCondition}
\end{equation}
\end{proposition}
The proof is given in Appendix~\ref{app:stability} and follows from the general theory for QBD processes.
Observe that $\rho_{\max}(s,n)$ poses an upper bound on the occupancy level of the servers in the holding model, which is clearly smaller than 1 for all $s$ and $n$.
In addition, this implies that the maximum workload $R_{\max}(s,n) := s\cdot\rho_{\max}(s,n)$ the system is able to handle is strictly less than $s$.
If we compare this to the open Erlang-R model, in which the maximal attainable workload equals $s$, we observe the effect of finite-size constraints on operational performance.
Figure \ref{fig:Rmax} shows the influence of both $s$ and $n$ on the maximum feasible workload in case $r=0.25$.
From these graphs, note that if $s\ll rn$, $R_{\max}$ grows almost linearly with $s$.
Furthermore, $R_{\rm max}(s,n)$ is increasing in $n$ for $s$ fixed.
A logical practical consequence is that a larger number of beds allows for a larger patient volume to enter the ED with the same number of nurses.
Moreover, $R_{\rm max}(s,n)$ is increasing in $s$, but as in Figure \ref{fig:Rmax_a}, adding an extra nurse does not increase the stability region in case $n$ is too tight.
Conversely, adding extra beds does not increase $R_{\rm max}(s,n)$ if the number of nurses does not allow for an increase in offered load, see Figure \ref{fig:Rmax_b}.
Additionally, it is easily verified that $R_{\rm max}(s,n)$ is upper bounded by both $s$ and $R_{\rm max}(n,n) = rn$. Therefore, a careful balance is called for between servers (nurses) and beds, so that resources will be efficiently utilized. We observe that when the ratio $s/n\approx r$, the system is better balanced.
We will propose an appropriate balance between resources by defining a synchronized QED capacity recommendation for both servers and beds in Section \ref{sec:QED_scaling}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $s$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*] table[x=s,y=n20] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\addplot[col3,thick,mark=*] table[x=s,y=n40] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\addplot[col4,thick,mark=*] table[x=s,y=n60] {tikz/stability/r025_n_fixed.txt};
\addplot[col5,thick,mark=*] table[x=s,y=n80] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\legend{$n=20$,$n=40$,$n=60$,$n=80$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $s$.}
\label{fig:Rmax_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 100,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $n$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*,mark repeat = 2] table[x=n,y=s5] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col3,thick,mark=*,mark repeat = 2] table[x=n,y=s10] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col4,thick,mark=*,mark repeat = 2] table[x=n,y=s15] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col5,thick,mark=*,mark repeat = 2] table[x=n,y=s20] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\legend{$s=5$,$s=10$,$s=15$,$s=20$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $n$.}
\label{fig:Rmax_b}
\end{subfigure}
\caption{The maximum achievable workload in the restricted Erlang-R model with holding for $r=0.25$.}
\label{fig:Rmax}
\end{figure}
Provided that the system is stable, the stationary distribution of the QBD process $X$ can be obtained numerically by the matrix geometric method \citep{Neuts1981}.
Subsequently, we can derive the stationary distribution of the original $Q(t)$, denoted by $\pi_h(\cdot,\cdot,\cdot)$.
\subsection{Performance measures}
\label{sec:performance_metrics}
In this work, we concentrate on five performance measures that are central to our analysis.
In the definitions that follow, we present expressions for these measures in terms of a general three-dimensional measure $\pi$, which one can replace by either $\pi_b$ or $\pi_h$, depending on the scenario considered.
In the remainder of this work, we will augment the measures related to the Erlang-R model with blocking and holding by the superscript $b$ and $h$, respectively\footnote{In line with $H(t)=0$, we use $\pi_b(i,j,k) = \pi_b(j,k)$ if $i=0$, with $\pi_b(j,k)$ as in \eqref{eq:pih(i,j)}, and $\pi_b(i,j,k) = 0$ otherwise, when considering the model with blocking.}.
As relevant performance measures, we consider the probability of holding (cq.\\ \noindent blocking) at entering the system, the probability of delay at the needy queue, expected waiting time for a nurse, utilization of nurses and utilization of beds:
\begin{equation}
\mathbb{P}({\rm hold}) = \sum_{i=0}^\iy \sum_{j=0}^n \pi(i,j,n-j), \qquad
\mathbb{P}({\rm delay}) \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \pi(i,j,k),
\label{eq:delay_probability}
\end{equation}
\begin{equation}
\label{eq:EW_exact}
\mathbb{E} [W] \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \frac{\max\{0,j-s+1\}}{\mu}\,\pi(i,j,k),
\end{equation}
\begin{equation}
\label{eq:utilization}
\rho_s = \frac{1}{s}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{j,s\} \pi(i,j,k), \qquad
\rho_n = \frac{1}{n}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{i,n\} \pi(i,j,k).
\end{equation}
It should be stressed that the above expression for the delay probability and the expected waiting time for a nurse is not exact. For the blocking model one can use the Arrival Theorem, see e.g.~\cite{Chen2001a}, whereby the exact expression sums up to $n-1$ instead of $n$.
Since we consider the system as $n\to\infty$, this discrepancy becomes negligible.
For the holding model, a similar argument holds.
We will therefore use the expressions in \eqref{eq:delay_probability}-\eqref{eq:utilization} as definitions for the performance measures.
\subsection{Stochastic bounds}
\label{sec:bounds}
Although the two variants of the Erlang-R model differ with respect to the admission policy, and require different mathematical treatment, we would like to be able to capture their relative performance.
We substantiate the intuition that the holding room leads to more patients in the ED, in the following result.
\begin{proposition}\label{thm:stochasticordering}
Let $Q_1^b$, $Q_2^b$, $Q_1^h$, $Q_2^h$ denote the nurse and content queue length processes in the Erlang-R model with blocking and holding, respectively.
Let $H(0) = 0$, $Q_1^b(0)=Q_1^h(0)$ and $Q_2^b(0)=Q_2^b(0)$. For all $t\geq 0$,
\begin{align}
Q_1^b(t) + Q_2^b(t) &\preceq_{\rm st} Q_1^h(t) + Q_2^h(t) \preceq_{\rm st} n ,\\
Q_2^b(t) &\preceq_{\rm st} Q_2^h(t),\\
Q_1^b(t) &\preceq_{\rm st} Q_1^h(t) + H(t),
\end{align}
where $X\preceq_{\rm st} Y$ implies $\mathbb{P}(X\geq k) \leq \mathbb{P}(Y\geq k)$ for all $k\geq 0$.
\end{proposition}
\noindent
The proof of Proposition \ref{thm:stochasticordering} uses sample path coupling and can be found in Appendix \ref{app:stochastic_ordering}.
Note that as an immediate consequence, we have
\[ \mathbb{P}^b( {\rm block}) = \lim_{t\to\iy} \mathbb{P}\big( Q_1^b(t)+Q_2^b(t) \geq n \big) \leq \lim_{t\to\iy} \mathbb{P}\big( Q_1^h(t) + Q_2^h(t) \geq n \big) = \mathbb{P}^h( {\rm hold }) \]
and by similar reasoning $\rho^b_n \leq \rho_n^h$.
In other words, under similar offered load and capacity constraints, utilization levels for the nurses in the Erlang-R model with blocking are lower than in the Erlang-R model with holding.
Moreover, the total number of waiting patients in the setting with holding is stochastically larger than in the setting with blocking, and in the open Erlang-R model.
We further discuss the differences between both models in Section \ref{sec:dimensioning} and Section \ref{sec:analysis_chapter5}.
\section{Two-fold QED regime}
\label{sec:QED_scaling}
We do not want to waste capacity of either servers or beds without getting significant advantage in term of performance.
We therefore take an asymptotic approach that lets the external arrival rate $\lambda$ grow to infinity, while scaling $s$ and $n$ accordingly.
In doing so, we intend to establish QED-type system behavior, i.e.\ high occupancy levels of both nurses and beds and good quality-of-service.
\subsection{Two-fold scaling rule}
In order to identify the scaling of $s$ and $n$ as $\lambda\to\infty$, we draw inspiration from the two-fold scaling rule used by Jennings \& de V\'ericourt \cite{Jennings2008} and Khudyakov et al.~\cite{Khudyakov2010}, which follows the celebrated square-root staffing principle.
This principle suggests that, in the most general setting, capacity should be equal to the expected offered load entering the system, let us say $R$, plus an additional variability hedge that is proportional to $\sqrt{R}$.
In the restricted Erlang-R model, we have two capacity sources, namely $s$ and $n$, which experience different relevant amount of works.
The offered load the servers in the needy queue experience is given by $R_1$, as in the regular Erlang-R model;
it does not change due to the finite-size effects, since all patients are served eventually. Hence, we only need to account for the interrupted services. It follows that the appropriate staffing rule for the nurses in the QED regime remains $s=R_1+\beta \sqrt{R_1}$ for some constant $\beta >0$.
To establish the bed capacity level, we need to reflect on the load offered to the beds. Observe that beds remain occupied both in needy and content states. This suggests that $R_n:=R_1+R_2=R_1/r$, with $R_1$ and $R_2$ as in \eqref{eq:R1_R2} and $r$ is the expected fraction of time a patient spends at the nurse station defined in \eqref{eq:delta}.
As a result, the appropriate staffing rule is $n=R_n+\gamma \sqrt{R_n}$ for some constant $\gamma>0$. In conclusion, the two-fold QED scaling rule is given by
\begin{equation}\label{eq:twofoldscaling}
\begin{array}{ll}
s &= R_1 + \beta \sqrt{R_1} + o(\sqrt{R_1}) \\
n &= \frac{R_1}{r}+\gamma \sqrt{\frac{R_1}{r}} + o(\sqrt{R_1})
\end{array}
\end{equation}
with $\beta,\gamma>0$ constants and $R_1:=\lambda/((1-p)\mu)$.
Recall that we saw in Figure \ref{fig:Rmax} that resources seem efficiently utilized if $s/n\approx r$.
Scaling \eqref{eq:twofoldscaling} is in line with this reasoning since
\[
\frac{s}{n} = r\left(1+ \frac{\beta - \gamma\sqrt{r}}{\sqrt{R_1}}+ O(1/R_1) \right) .
\]
\begin{remark}
In \cite{Jennings2008}, a similar scaling regime is considered, which only relates $s$ and $n$ through a square-root scaling, namely the regime $s = r n + \hat\gamma\sqrt{n}$,
which is equivalent to the second relation in \eqref{eq:twofoldscaling} if $\hat\gamma = \beta\sqrt{r} - \gamma r$.
Due to the absence of external arrivals in this closed system, they let the number of beds $n$ approach infinity as opposed to $\lambda$ in our settings.
Nevertheless, this results in the same asymptotic regime.
\end{remark}
Before turning to asymptotic expressions for the performance measures concerning the Erlang-R model with blocking or holding, we conduct a few numerical experiments to confirm that the scaling in \eqref{eq:twofoldscaling} indeed leads to desired QED behavior.
In Figure \ref{fig:sample_paths}, we plotted the sample paths of the three-dimensional queue length process of the holding model in which $\beta$ and $\gamma$ are fixed, and $R_1$ is increased.
Observe that the needy queue length $Q_1(t)$, plotted in orange in Figure \ref{fig:sample_paths}, fluctuates around the values $s$, and stabilizes for larger values of $R_1$.
This naturally implies that the server (nurses) utilization approaches 100\%, while the number of patients waiting is $O(\sqrt{R_1})$.
Furthermore, we see that the percentage of occupied beds also tends to 100\%, while the holding queue length remains small.
The holding queue is of much smaller order than $R_1$, which implies that the holding time of a patient becomes negligible as $R_1\to\iy$.
From these empirical findings we deduce that under scaling \eqref{eq:twofoldscaling} the restricted Erlang-R model exhibits QED behavior on two levels: Outside the facility while waiting for an available bed, and inside the facility while waiting for attention of a nurse.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 28,
ytick = {0,5,10,15,20,25},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\definecolor{col1}{rgb}{0.368417, 0.506779, 0.709798}
\addplot[very thick,col5] file {Chapter_5/tikz/sample_paths/R5_holding.txt};
\addplot[very thick,col2] file {Chapter_5/tikz/sample_paths/R5_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R5_total.txt};
\addplot[very thick,dashed] coordinates {
(0,7)
(200,7)
};
\addplot[very thick,dashed] coordinates {
(0,24)
(200,24)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=5$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 128.333,
ytick = {0,20,40,60,80,100,120},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {Chapter_5/tikz/sample_paths/R25_holding.txt};
\addplot[very thick,col2] file {Chapter_5/tikz/sample_paths/R25_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R25_total.txt};
\addplot[very thick,dashed] coordinates {
(0,30)
(200,30)
};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=25$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 490,
ytick = {0,100,200,300,400},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {Chapter_5/tikz/sample_paths/R100_holding.txt};
\addplot[very thick,col2] file {Chapter_5/tikz/sample_paths/R100_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R100_total.txt};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\addplot[very thick,dashed] coordinates {
(0,420)
(200,420)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=100$}
\end{subfigure}
\caption{Sample paths of $H(t)$ (blue), $Q_1(t)$ (orange) and $Q_1(t)+Q_2(t)$ (green) of the Erlang-R model with holding with parameters $\mu = 1$, $\delta=0.25$, $p=0.75$ and $\beta=\gamma=1$. The staffing levels $s$ and $n$ are depicted by the dashed lines.}
\label{fig:sample_paths}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.7,
ytick = {0,0.1,...,0.7},
xlabel = $\lambda$,
grid = both,
axis line style={->},
axis lines = left,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/delayProbErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/delayProbYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/delayProbJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Delay probability nurse}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.3,
ytick = {0,{0.05},0.1,0.15,0.2,0.25,3},
grid = both,
axis line style={->},
tick label style={/pgf/number format/fixed},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = north east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/EWErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/EWYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/EWJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Expected wait}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.7,
ymax = 1.02,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/rhoErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/rhoYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/rhoJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Nurse utilization}
\end{subfigure}
\caption{Asymptotic behavior of the restricted Erlang-R model with holding and blocking, and the closed ward model for $\mu=1$, $\delta = 0.2$, $p=0.8$ and $\beta=\gamma=0.5$.}
\label{fig:empiricalAsymptotics}
\end{figure}
We also check how the Erlang-R model with blocking or holding and the closed ward model of \cite{Jennings2008} relate under scaling \eqref{eq:twofoldscaling}.
In Figure~\ref{fig:empiricalAsymptotics}, we plot the performance measures, obtained through simulation, for the three models in which we fix $\beta=\gamma=0.5$ and vary the arrival rate $\lambda$.
First, we see that $\mathbb{P}({\rm delay})$ stabilizes as $\lambda\to\iy$ in all three models under scaling \eqref{eq:twofoldscaling}, and the delay probability of the model with holding lies in between the other two.
Second, note that the expected waiting time for a nurse in all models converges to 0 as $\lambda$ increases. In fact, the rate of decay is similar in all three models.
We observe that $\rho_s$ approaches unity in all models, and the rate of convergence seems again comparable.
Finally, and most importantly, we notice an ordering between the three models.
Namely, in all performance measures considered in Figure \ref{fig:empiricalAsymptotics}, Erlang-R with holding appears to be upper bounded by the closed ward and lower bounded by the Erlang-R with blocking.
In a multitude of parameter settings of $(\beta,\gamma)$, we have seen the same ordering, leading to the following conjecture:
\begin{conjecture}\label{conj:stochorder}
Let $Q^b_1(\iy)$, $Q_1^h(\iy)$ and $Q_1^J(\iy)$ denote the stationary number of needy patients in the Erlang-R model with blocking, holding and the closed ward, respectively. Then,
\begin{equation}
Q_1^b(\iy) \preceq_{\rm st} Q_1^h(\iy) \preceq_{\rm st} Q_1^J(\iy).
\end{equation}
\end{conjecture}
Observe that Conjecture \ref{conj:stochorder} poses a stronger statement than the third assertion in Proposition \ref{thm:stochasticordering}.
The latter does give an upper bound to $Q_1^h(\iy)$ in terms of $Q_1^b(\iy)$, albeit supplemented with the stationary holding queue length.
\subsection{QED limits for Erlang-R with blocking}
\label{sec:QED_limit_block}
We now continue our analysis by examining its limiting behavior under scaling \eqref{eq:twofoldscaling},and obtain QED limits for some performance measures of the Erlang-R model with blocking.
Using the explicit expressions for the blocking model in \eqref{eq:pih(i,j)}, we derive the limiting values of the relevant performance measures defined in Section \ref{sec:performance_metrics} in terms of $\beta$ and $\gamma$.
\begin{theorem}\label{thm:limits_YT}
Let $s$ and $n$ scale as in \eqref{eq:twofoldscaling} with ${-}\infty<\beta<\infty,\,\gamma>0$ as $\lambda\to\infty$. Then, if $\beta \neq 0$,
\begin{align}
g^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay})\nonumber \\
\label{eq:yt_limit_delay}
&=
\left(1 +
\frac{ \beta \, \int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) }
{\varphi(\beta)\Phi(\eta) - \varphi(\sqrt{\beta^2+\eta^2}){\rm e}^{
{1}{2} \omega^2} \Phi(\omega)}
\right)^{-1},\\
f^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber\\
\label{eq:yt_limit_block}
&=
\frac{
\sqrt{r}\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \varphi(\sqrt{\beta^2+\eta^2})\,{\rm e}^{\frac{1}{2} \omega^2} \Phi(\omega)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},\\
h^b(\beta,\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay}
&=
\frac{
\frac{\varphi(\beta)\Phi(\eta)}{\beta^2} +
\left(\frac{\beta}{r}-\frac{\gamma}{\sqrt{r}}-\frac{1}{\beta}\right)\,\frac{\varphi(\sqrt{\eta^2+\beta^2})}{\beta}\, {\rm e}^{\tfrac{1}{2}\omega^2}\, \Phi(\omega)
- \sqrt{\frac{1-r}{r}}\,\frac{\varphi(\beta)\varphi(\eta)}{\beta}
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},
\end{align}
and if $\beta=0$,
\begin{align}
g^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay}) \nonumber\\
\label{eq:yt_limit_delay_beta0}
&=
\left(1+
\frac{
\int_{-\iy}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t)
}{
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
}
\right)^{-1},\\
f^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber \\
\label{eq:yt_limit_block_beta0}
&=
\frac{
\sqrt{r}\,\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \frac{1}{\sqrt{2\pi}} \Phi(\eta)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
},\\
h_0^b(\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay_beta0}
&= \frac{1}{2\mu}\, \frac{ \left( \gamma^2/r+1\right) \Phi(\eta) + \eta \varphi(\eta) }
{ \frac{r}{1-r} \sqrt{2\pi} \int_{-\infty}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) + \sqrt{\frac{r}{1-r}} \left(\eta \Phi(\eta)+\varphi(\eta)\right)},
\end{align}
where $\eta = \frac{\gamma - \beta\sqrt{r}}{\sqrt{1-r}}$ and $\omega := \frac{\gamma - \beta/\sqrt{r}}{\sqrt{1-r}}$.
\end{theorem}
The proof of Theorem \ref{thm:limits_YT} is given in Appendix C of \cite{YomTov2010} under a parameter transformation.
Theorem \ref{thm:limits_YT} proves that the scaling \eqref{eq:twofoldscaling} results in QED behavior: the probability of waiting in Equations \eqref{eq:yt_limit_delay} and \eqref{eq:yt_limit_delay_beta0} converges to a limit that is strictly between 0 and 1.
Notice that all limits in Theorem \ref{thm:limits_YT} are functions of three parameters: $\beta$ and $\gamma$, which are decision variables, and the fraction of needy time $r$, which is dictated by the physics of the system. Furthermore, the theorem also shows that the probability of blocking (Equations \eqref{eq:yt_limit_block} and \eqref{eq:yt_limit_block_beta0}) is of order $1/\sqrt{R_1}$.
For example, assume that the fraction of needy time $r$ is $0.5$ and the system is large (100 servers).
Using Figure \ref{fig:pdelay_pblock}, we observe that, by choosing the pair $\gamma = 1$ and $\beta = 0.245$, we actually aim at a probability of getting served immediately to be 40\%. At the same time, the probability of getting immediately a bed is 97\%.
Thus, waiting inside the ED occurs at a reasonable level, while wait outside the facility becomes negligible.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
tick label style={/pgf/number format/fixed},
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,0.5)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,0.99)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {Chapter_5/tikz/limit_probabilities_delay.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:pdelay_pblock_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
tick label style={/pgf/number format/fixed},
ylabel = {$f(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,1)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,1.98)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {Chapter_5/tikz/limit_probabilities_block.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\label{fig:pdelay_pblock_b}
\end{subfigure}
\caption{Asymptotic delay and scaled blocking probability for $r=0.5$ based on Theorem \ref{thm:limits_YT}. }
\label{fig:pdelay_pblock}
\end{figure}
Theorem \ref{thm:limits_YT} further shows that the expected waiting (Equations \eqref{eq:yt_limit_Edelay} and \eqref{eq:yt_limit_Edelay_beta0})
is of order $1/\sqrt{R_1}$ too and hence vanishes in the large-system limit.
We see from Theorem \ref{thm:limits_YT} that achieving target service levels is always an interplay between $\beta$ and $\gamma$.
Figure \ref{fig:pdelay_pblock_a} shows for instance that in order to keep $\mathbb{P}({\rm delay})\in (0.25,0.75)$, choosing $\gamma=-1$ requires $\beta$ to stay within the range $[-1.4,-0.5]$, while $\gamma=1$ corresponds to values of $\beta$ in $[-0.4,0.5]$.
While the two-fold scaling rule in \eqref{eq:twofoldscaling} automatically captures the right dimensioning ratio as the system scales up, Theorem \ref{thm:limits_YT} shows that the parameters $\beta$ and $\gamma$ provide a means to fine-tune the performance.
Figure \ref{fig:pdelay_pblock_b} confirms how adding nurses, i.e.~increasing $\beta$, does not improve the blocking probability if the number of beds, i.e.~$\gamma$, is too tight.
This is in accordance with our previous observations in Figure \ref{fig:Rmax} for the exact steady-state distribution.
To test the accuracy of the asymptotic results in Theorem \ref{thm:limits_YT} as approximations in a realistic setting, we plot in Figure \ref{fig:accuracy_blocking} the exact probability of delay and blocking for an Erlang-R model with $R=8$ and $r=0.25$, as a function of $s$. The exact probabilities are given by Equation
\eqref{eq:delay_probability}, and their respective asymptotic approximations are based on Theorem \ref{thm:limits_YT}.
Despite the realistic moderate size of the system ($R=8$), we see that the QED approximations are remarkably accurate for many settings $(s,n)$.
This fast relaxation is in line with observations made earlier in the QED literature \cite{Borst2004,Janssen2011}.
\begin{table}[htb]
\centering
\begin{tabular}{|r|rrrr|}
\hline
& $\mu$ & $\delta$ & $p$ & $r$ \\
\hline
Case 1 & 1 & 0.10 & 0.90 & 0.10 \\
Case 2 & 1 & 0.25 & 0.75 & 0.25\\
Case 3 & 1 & 0.50 & 0.50 & 0.50 \\
\hline
\end{tabular}
\caption{Parameter settings for numerical experiments.}
\label{tab:parameter_settings}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1270 & 0.0900 & 0.2283 & 0.1553 & 0.0212 & 0.1085 \bigstrut[t]\\
10 & 0.1340 & 0.0910 & 0.1919 & 0.1628 & 0.0206 & 0.1205 \\
25 & 0.1981 & 0.0945 & 0.1614 & 0.2356 & 0.0216 & 0.2145 \\
50 & 0.1513 & 0.0963 & 0.1588 & 0.1830 & 0.0205 & 0.1496 \\
100 & 0.1880 & 0.0956 & 0.1532 & 0.2231 & 0.0224 & 0.2055 \\
250 & 0.1797 & 0.0971 & 0.1399 & 0.2143 & 0.0219 & 0.2057 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1767} & \textit{0.0981} & \textit{0.1437} & \textit{0.2108} & \textit{0.0217} & \textit{0.1947} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0237 & 0.0868 & 0.0282 & 0.0322 & 0.0192 & 0.0391 \bigstrut[t]\\
10 & 0.0206 & 0.0872 & 0.0188 & 0.0278 & 0.0183 & 0.0264 \\
25 & 0.0277 & 0.0876 & 0.0123 & 0.0363 & 0.0174 & 0.0174 \\
50 & 0.0185 & 0.0913 & 0.0116 & 0.0249 & 0.0175 & 0.0166 \\
100 & 0.0232 & 0.0888 & 0.0103 & 0.0303 & 0.0183 & 0.0145 \\
250 & 0.0203 & 0.0905 & 0.0079 & 0.0267 & 0.0179 & 0.0109 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0188} & \textit{0.0914} & \textit{0.0084} & \textit{0.0247} & \textit{0.0177} & \textit{0.0118} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 1.}
\label{tab:numerics_case1}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0911 & 0.1538 & 0.0479 & 0.1431 & 0.0345 & 0.0909 \bigstrut[t]\\
10 & 0.1010 & 0.1498 & 0.0560 & 0.1520 & 0.0326 & 0.1025 \\
25 & 0.1594 & 0.1509 & 0.1058 & 0.2192 & 0.0405 & 0.1785 \\
50 & 0.1201 & 0.1506 & 0.0726 & 0.1697 & 0.0381 & 0.1248 \\
100 & 0.1514 & 0.1539 & 0.1001 & 0.2088 & 0.0398 & 0.1704 \\
250 & 0.1459 & 0.1524 & 0.0957 & 0.2003 & 0.0397 & 0.1618 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1429} & \textit{0.1569} & \textit{0.0940} & \textit{0.1976} & \textit{0.0391} & \textit{0.1617} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0130 & 0.1484 & 0.0044 & 0.0277 & 0.0294 & 0.0109 \bigstrut[t]\\
10 & 0.0121 & 0.1432 & 0.0042 & 0.0244 & 0.0267 & 0.0098 \\
25 & 0.0182 & 0.1383 & 0.0070 & 0.0319 & 0.0295 & 0.0141 \\
50 & 0.0119 & 0.1415 & 0.0043 & 0.0216 & 0.0301 & 0.0090 \\
100 & 0.0154 & 0.1413 & 0.0059 & 0.0270 & 0.0290 & 0.0119 \\
250 & 0.0136 & 0.1403 & 0.0051 & 0.0236 & 0.0291 & 0.0103 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0126} & \textit{0.1445} & \textit{0.0048} & \textit{0.0220} & \textit{0.0284} & \textit{0.0097} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 2.}
\label{tab:numerics_case2}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0547 & 0.1945 & 0.0221 & 0.1181 & 0.0604 & 0.0617 \bigstrut[t]\\
10 & 0.0579 & 0.2158 & 0.0237 & 0.1325 & 0.0526 & 0.0746 \\
25 & 0.1113 & 0.2086 & 0.0544 & 0.1959 & 0.0641 & 0.1311 \\
50 & 0.0813 & 0.2050 & 0.0363 & 0.1523 & 0.0562 & 0.0933 \\
100 & 0.1060 & 0.2146 & 0.0509 & 0.1873 & 0.0632 & 0.1250 \\
250 & 0.1006 & 0.2179 & 0.0475 & 0.1820 & 0.0596 & 0.1214 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1011} & \textit{0.2185} & \textit{0.0478} & \textit{0.1792}& \textit{0.0605} & \textit{0.1199} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0034 & 0.1888 & 0.0009 & 0.0175 & 0.0510 & 0.0057 \bigstrut[t]\\
10 & 0.0030 & 0.2093 & 0.0008 & 0.0172 & 0.0416 & 0.0058 \\
25 & 0.0070 & 0.1937 & 0.0020 & 0.0243 & 0.0440 & 0.0089 \\
50 & 0.0043 & 0.1946 & 0.0011 & 0.0163 & 0.0414 & 0.0056 \\
100 & 0.0061 & 0.1999 & 0.0017 & 0.0207 & 0.0431 & 0.0076 \\
250 & 0.0052 & 0.2037 & 0.0014 & 0.0185 & 0.0401 & 0.0067 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0052} & \textit{0.2039} & \textit{0.0014} & \textit{0.0173} & \textit{0.0404} & \textit{0.0063} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Numerical results for Erlang-R model with blocking for Case 3.}
\label{tab:numerics_case3}
\end{table}
We furthermore compare the asymptotic delay and blocking probability in the three scenarios given in Table \ref{tab:parameter_settings}.
In Tables \ref{tab:numerics_case1}--\ref{tab:numerics_case3} we compute the exact probabilities of delay and blocking through the explicit forms in \eqref{eq:delay_probability} for increasing values of the offered load, $R_1$.
The numerical results show that $g^b(\beta,\gamma)$, $f^b(\beta,\gamma)$ and $h^b(\beta,\gamma)$ provide accurate approximations to $\mathbb{P}({\rm delay})$, $\sqrt{R_1}\mathbb{P}({\rm block})$ and $\sqrt{R_1}\,\mathbb{E}[W]$ in pre-limit systems.
The quality of the approximations increases with $R_1$.
Naturally, fluctuations occur for relatively small values of $R_1$, because $s$ and $n$ need to be rounded to an integer.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\end{subfigure}
\caption{Comparison of exact performance measures (solid) against asymptotic approximations (dashed) with $\beta=(s-R_1)/\sqrt{R_1}$ and $\gamma=(n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_blocking}
\end{figure}
\subsection{QED limits for Erlang-R with holding}
\label{sec:QED_limit_holding}
As explained in Section \ref{sec:QED_scaling}, the model with holding has no product-form steady-state distribution, which makes it hard (if not impossible) to obtain QED limits.
Instead, we derive QED approximations by exploiting a connection with the blocking model.
We first prove that under scaling \eqref{eq:twofoldscaling}, the upper bound on the utilization level of the nurses needed to achieve stability in the model with holding, as given in Proposition \ref{prop:StabilityCondition}, converges to unity as $R\to\infty$.
This facilitates high utilization levels of both nurses and beds, a key characteristic of the QED regime.
\begin{proposition}\label{prop:stability_convergence}
Let $s$ and $n$ scale with $R_1\to\infty$ as in \eqref{eq:twofoldscaling}. Then for $\lambda\to\infty$,
\[
\rho_{\max}(s,n) \to 1.
\]
\end{proposition}
The proof can be found in Appendix \ref{app:proof_stability_convergence}.
Combining Proposition \ref{prop:stability_convergence} with Proposition \ref{prop:StabilityCondition} shows that indeed the scaling we use results in a highly utilized system.
As observed before, the nature of the two variants of the model is similar up to the fact that a fraction of the patients is deferred on arrive in the setting with blocking, whereas all the arriving patients are eventually admitted into the system in the holding model.
This implies that, given $s$ and $n$, the nurses face an increased workload in case of a holding room.
In fact, Theorem \ref{thm:limits_YT} shows that the blocking probability is of order $1/\sqrt{R_1}$, yielding a volume of blocked patients of order $\sqrt{R_1}$ in setting with blocking.
Accordingly, if $R^b = R_1$ and $R^h$ denote the nominal load arriving to the nurses in the model with blocking and holding, respectively, we can argue that
\[R^h = R^b + \alpha \sqrt{R^b} + o(\sqrt{R^b}),\]
for some $\alpha>0$.
Notice that this additional load is of the same order as the safety staffing in the blocking model staffing rule \eqref{eq:twofoldscaling}.
As $s$ and $n$ remain unchanged, we rewrite \eqref{eq:twofoldscaling} in terms of $R^h$,
\begin{align}
s &= R^h + (\beta-\alpha)\sqrt{R^h} + o(\sqrt{R^h}), \nonumber \\
n &= \frac{R^h}{r} + \left(\gamma-\alpha/\sqrt{r}\right)\sqrt{\frac{R^h}{r}} + o(\sqrt{R^h}),
\label{eq:fixed_point_scaling}
\end{align}
where we have used $R^b = O(R^h)$.
Observe that the square-root principle prevails also after this substitution, albeit with different hedging parameters.
We therefore heuristically argue that the holding model under scaling \eqref{eq:twofoldscaling} with parameters $\beta$ and $\gamma$ mimics the blocking model with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, respectively.
Observe, however, that we have not yet specified the value of $\alpha$.
By definition, $\alpha\sqrt{R^b}$ is the expected volume of patients that would be rejected in the model with blocking, that is, $R^h$ times the probability of not being admitted to the ED directly.
By the construction in \eqref{eq:fixed_point_scaling}, this volume asymptotically equals $R^h \cdot \mathbb{P}^b({\rm block})$, with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, which by Theorem \ref{thm:limits_YT} is approximated by
\[f^b\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right) / \sqrt{R^h}\]
as $R^h$ grows large.
In conclusion, $\alpha$ is characterized as the solution of the fixed-point equation
\begin{equation}
\label{eq:fixedpoint}
\alpha = f^h\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right),
\end{equation}
and as a result, we are able to approximate the nurse delay probability in the Erlang-R model with holding as
\begin{equation}
\mathbb{P}^h({\rm delay}) \approx g^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: g^h(\beta,\gamma).
\label{eq:fixed_point_Pwait}
\end{equation}
Likewise, the scaled the mean waiting time for a nurse can be approximated by
\begin{equation}
\sqrt{R_1} \cdot \mathbb{E}[W] \approx h^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: h^h(\beta,\gamma).
\label{eq:fixed_point_Ewait}
\end{equation}
This also implies that the holding queue is $O(\sqrt{R_1})$.
Subsequently, we argue that the expected holding time (pre-entering wait) under the QED policy is $O(1/\sqrt{R_1})$ and hence asymptotically negligible.
We justify this claim numerically in Section \ref{sec:analysis_chapter5}.
\begin{remark}
\label{rem:holding_limit}
Notice that in the reasoning leading to \eqref{eq:fixedpoint}, we implicitly assumed that the additional volume $\alpha\sqrt{R^b}$ is an independent Poisson process, which is obviously not the case. Therefore, \eqref{eq:fixed_point_Pwait}-\eqref{eq:fixed_point_Ewait} are approximations for pre-limit systems that are not asymptotically correct as $R_1\to\iy$.
Nevertheless, we heuristic approach seems to performs well as we confirm numerically next.
\end{remark}
In Figure \ref{fig:accuracy_holding}, we repeat the numerical experiments of Figure \ref{fig:accuracy_blocking} for the model with holding.
Since the heuristic does not provide an approximation for the holding probability, Figure \ref{fig:accuracy_holding_b} only plots the simulated holding probabilities.
Those are provided to better understand the implication of operational decisions.
Recall that the holding system is only stable (i.e. $\mathbb{P}({\rm hold})<1$) if both $s>R_1=8$ and $n > R_1/r = 32$.
We nevertheless included the boundary case $n=32$ for illustrative purposes.
The graphs in Figure \ref{fig:accuracy_holding} show that the heuristic captures the congestion levels well, even for this moderate-size system.
To see how this heuristic approach performs under different settings, and particularly if $R_1\to \infty$, we compare the approximated delay probability in the Erlang-R model with holding as solution of the fixed-point procedure to the outcomes of simulation experiments for the three scenarios in Table \ref{tab:parameter_settings} again.
We performed 100 runs of length $10^4$ for each parameter setting and all values of $R$, yielding the results presented in Tables \ref{tab:heuristic_case1}--\ref{tab:heuristic_case3}, which are accurate up to a 95\% confidence interval of width $10^{-3}$.
\begin{table}[h] \centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1532 & 0.1031 & 0.1628 & 0.1216 \bigstrut[t]\\
10 & 0.1622 & 0.1272 & 0.1697 & 0.1331 \\
25 & 0.2340 & 0.2116 & 0.2413 & 0.2342 \\
50 & 0.1817 & 0.1468 & 0.1890 & 0.1678 \\
100 & 0.2199 & 0.1931 & 0.2304 & 0.2269 \\
250 & 0.2110 & 0.1852 & 0.2176 & 0.2230 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.2076} & \textit{0.1777} & \textit{0.2187} & \textit{0.2050} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0310 & 0.0121 & 0.0344 & 0.0148 \bigstrut[t]\\
10 & 0.0267 & 0.0123 & 0.0292 & 0.0128 \\
25 & 0.0348 & 0.0171 & 0.0373 & 0.0184 \\
50 & 0.0240 & 0.0108 & 0.0258 & 0.0125 \\
100 & 0.0293 & 0.0143 & 0.0317 & 0.0163 \\
250 & 0.0256 & 0.0120 & 0.0276 & 0.0145 \\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0229} & \textit{0.0104} & \textit{0.0257} & \textit{0.0124} \bigstrut[b]\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 1.}
\label{tab:heuristic_case1}
\end{table}
\begin{table}[h]\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1327 & 0.0740 & 0.1620 & 0.1096 \bigstrut[t]\\
10 & 0.1446 & 0.0894 & 0.1683 & 0.1207 \\
25 & 0.2204 & 0.1631 & 0.2442 & 0.2203 \\
50 & 0.1694 & 0.1122 & 0.1888 & 0.1507 \\
100 & 0.2098 & 0.1524 & 0.2322 & 0.2111 \\
250 & 0.2033 & 0.1534 & 0.2190 & 0.1979 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1840} & \textit{0.1277} & \textit{0.2109} & \textit{0.1759} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0219 & 0.0079 & 0.0322 & 0.0137 \bigstrut[t]\\
10 & 0.0199 & 0.0073 & 0.0284 & 0.0115 \\
25 & 0.0283 & 0.0128 & 0.0375 & 0.0163 \\
50 & 0.0190 & 0.0078 & 0.0255 & 0.0107 \\
100 & 0.0244 & 0.0097 & 0.0314 & 0.0151 \\
250 & 0.0214 & 0.0083 & 0.0272 & 0.0134 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0169} & \textit{0.0066} & \textit{0.0234} & \textit{0.0104} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 2.}
\label{tab:heuristic_case2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0977 & 0.0413 & 0.1521 & 0.0851 \bigstrut[t]\\
10 & 0.1070 & 0.0469 & 0.1648 & 0.1028 \\
25 & 0.1926 & 0.1076 & 0.2421 & 0.1874 \\
50 & 0.1431 & 0.0727 & 0.1876 & 0.1342 \\
100 & 0.1855 & 0.1012 & 0.2282 & 0.1714 \\
250 & 0.1775 & 0.0963 & 0.2217 & 0.1765 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1442} & \textit{0.0711} & \textit{0.1981} & \textit{0.1354} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0072 & 0.0019 & 0.0250 & 0.0081 \bigstrut[t]\\
10 & 0.0067 & 0.0018 & 0.0235 & 0.0082 \\
25 & 0.0148 & 0.0043 & 0.0325 & 0.0133 \\
50 & 0.0092 & 0.0025 & 0.0217 & 0.0081 \\
100 & 0.0132 & 0.0038 & 0.0277 & 0.0105 \\
250 & 0.0114 & 0.0033 & 0.0246 & 0.0099 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0078} & \textit{0.0022} & \textit{0.0188} & \textit{0.0069} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated and approximated probability of delay in Erlang-R model with holding for Case 3.}
\label{tab:heuristic_case3}
\end{table}
We conclude from these tables that the approximation is good. As $R$ increases, the simulated values move closer to the heuristic approximation. Extensive numerical experiments suggest that load is slightly underestimated in the limit.
The best results in terms of accuracy are attained for small $r$.
This suggests that the quality of the heuristic method improves as $r$ gets smaller.
These are exactly the parameter settings for which this model is relevant.
\begin{remark}
The approximation technique that evolves around the fixed-point\\ \noindent method can be adapted to accommodate balking behavior of external arrivals. If we assume that an arriving patient finding all beds occupied joins leaves the system instantly with probability $1-q$, for some $q\in(0,1)$, independently of the rest of the arrivals, with the same argumentation, the volume of arrivals blocked is still $\alpha\sqrt{R_1}$, while the volume that will enter the ED eventually is $q\cdot\alpha\sqrt{R_1}$. Therefore, we may argue that in the QED regime, the system with holding and balking behaves as the system with blocking but with corrected parameters $(\beta-q\alpha,\gamma-q\alpha/\sqrt{r})$, where $\alpha$ satisfies
\begin{equation}
\alpha = f^b(\beta-q\alpha,\gamma-q\alpha/\sqrt{r}).
\end{equation}
Note that the choice of $q$ interpolates between the two system variants with holding ($q=0$) and blocking ($q=1$).
\end{remark}
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx32] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx36] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx40] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:accuracy_holding_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Holding probability}
\label{fig:accuracy_holding_b}
\end{subfigure}
\caption{Comparison of simulated delay probability (solid) against asymptotic approximations (dashed) with $\beta = (s-R_1)/\sqrt{R_1}$ and $\gamma = (n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_holding}
\end{figure}
\section{Dimensioning}
\label{sec:dimensioning}
We will now use the accurate asymptotic approximations of the previous section to define a procedure that determines resource capacity in the restricted Erlang-R models.
That is, we aim to set the number of nurses $s$ and the number of beds $n$, such that a preset performance level is achieved.
We take the probability of delay at the Needy queue and the probability of blocking/holding at the pre-entrant queue as the target performance objectives.
\subsection{Capacity setting for Erlang-R with blocking}
\label{sec:dimensioning_block}
In the setting with blocking, we can readily use the asymptotic results of Theorem \ref{thm:limits_YT} to (numerically) find a pair of parameters $(\beta^*,\gamma^*)$ to meet the performance requirements.
For instance, given that we want the delay probability to be at most $\varepsilon$, we first solve the equation $g^b(\beta^*,\gamma^*)=\varepsilon$ and then assign $s = \lceil R_1 + \beta^*\sqrt{R_1}\rceil$ and $n = \lceil R_1/r+\gamma^*\sqrt{R_1/r}\rceil$. Note that there could be multiple solutions to that problem, i.e.\ there could be multiple combinations of number of beds and number of nurses that can result in the same value of a single performance level.
The ED manager can ultimately decide which of these feasible solutions fits the environment best, for instance taking into account space and cost constraints.
We illustrate the resource allocation decisions in an MU setting, using data originated from two articles: \cite{LS2001} and \cite{GY2011}. Green \& Yankovic describe an MU that has 42 beds, with average occupancy level 78\%, and Average Length of Stay (ALOS) 4.3 days. Lundgren \& Segesten studied nurses' service times in a medical-surgical ward. They found that the average service time in their unit was 15.3 minutes per service, and that the average demand rate for each patient is 0.38 requests per hour. Therefore, we take an average service time of 15 minutes and assume that there are 0.4 requests per hour from each patient. Fitting this data to our model results in the following parameters values: $\lambda =0.32, \mu =4, \delta =0.4$, $p=0.975$ and the fraction of needy time is then approximately $r=0.09$.
This yields nominal offered load $R_1 = 3.2$ and $R_1/r = 34.4$.
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: -1.9,0.05)},anchor = south west},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\draw[->,col1,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: -0.0552366,0.5) -- (axis cs: -0.0552366,0);
\draw[->,col2,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.179728,0.5) -- (axis cs: 0.179728,0);
\draw[->,col3,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.359034,0.5) -- (axis cs: 0.359034,0);
\draw[->,col4,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.459825,0.5) -- (axis cs: 0.459825,0);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma=1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:ratio01_delay}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$f(\beta,\gamma)/\sqrt{R_1}$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\draw[very thick, col1,dashed,->] (axis cs: -0.0552,0) -- (axis cs: -0.0552,0.292798) -- (axis cs: -2,0.292798);
\draw[very thick,col2,dashed,->] (axis cs: 0.179728,0) -- (axis cs: 0.179728,0.164903) -- (axis cs: -2,0.164903);
\draw[very thick,col3,dashed,->] (axis cs: 0.359034,0) -- (axis cs: 0.359034,0.0705547) -- (axis cs: -2,0.0705547);
\draw[very thick,col4,dashed,->] (axis cs: 0.459825,0) -- (axis cs: 0.459825,0.0207909) -- (axis cs: -2,0.0207909);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma= 1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Blocking probability}
\label{fig:ratio01_block}
\end{subfigure}
\caption{Approximate performance of restricted Erlang-R with blocking for $r \approx 0.09$ and $R_1 = 3.2$, as functions of $\beta$.}
\label{fig:ratio01}
\end{figure}
Figure \ref{fig:ratio01} visualizes the dimensioning procedure for this particular MU.
The hospital management can find a pair of $n$ and $s$ to meet certain criteria, for example to achieve target delay probability $\varepsilon = 0.5$ with reasonable blocking probability.
Figure \ref{fig:ratio01}a indicates that this target can be achieved by a variety of pairs, for instance $(\beta_1,\gamma_1) = (-0.06,-1)$, $(\beta_2,\gamma_2) = (0.16,0)$, $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$, among infinitely many others.
According to Figure \ref{fig:ratio01}b, the pairs named above lead to blocking probabilities 0.293, 0.165, 0.071 and 0.021, respectively.
If the manager decides that probability of blocking of more than 10 percent is not acceptable, this leaves the choices $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$ as candidate parameter pairs.
Using the two-fold square-root staffing rule $s_i = \lceil R_1 + \beta_i \sqrt{R_1}\rceil$ and $n_i = [R_1/r + \gamma\sqrt{R_1/r}]$, this yields feasible staffing levels $(s_3,n_3) = (4,40)$ and $(s_4,n_4)=(5,46)$.
The ultimate decision to apply any of these solutions can be based on external factors, such as operational costs or space limitations of number of beds.
\subsection{Capacity setting for Erlang-R with holding}
For the holding model, we need a more sophisticated approach, exploiting the asymptotic approximation with the fixed-point equation in \eqref{eq:fixedpoint}. We propose the following dimensioning procedure to achieve a preset target delay probability at the needy queue.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Target delay probability $\varepsilon$. Parameters $\lambda,\mu,\delta$ and $p$.}
\hspace{1.1cm}\KwOut{Staffing levels $s$ and $n$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}
\item[] \hspace{0.5cm} 1. Set $R_1:= \frac{\lambda}{(1-p)\mu}$ and $r = \frac{\delta}{\delta+p\mu}$.
\item[] \hspace{0.5cm} 2. Determine parameters $(\beta^*,\gamma^*)$ such that $g^b(\beta^*,\gamma^*) = \varepsilon$.
\item[] \hspace{0.5cm} 3. Set $\beta = \beta^* + f^b(\beta^*,\gamma^*)$ and $\gamma = \gamma^* + f^b(\beta^*,\gamma^*)/\sqrt{r}$.
\item[] \hspace{0.5cm} 4. Return $s = \left\lceil R_1 + \beta\sqrt{R_1}\right\rceil$ and $n = \left\lfloor R_1/r + \gamma \sqrt{R_1/r}\right\rfloor$.
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Stationary dimensioning algorithm for ED with holding.}
\label{alg:stationarydimensioning}
\end{algorithm}
\begin{remark}\label{rem:upperboundHW}
In Step 2 of Algorithm \ref{alg:stationarydimensioning} infinitely many pairs $(\beta^*,\gamma^*)$ satisfy the delay probability equation.
For practical purposes, it is convenient to fix either $\beta^*$ or $\gamma^*$ beforehand, and then solve $g^b(\beta^*,\gamma^*) = \varepsilon$ for the remaining unknown.
The preset value should however be chosen with care, since $g^b(\beta^*,\gamma^*)$ is upper bounded by the Halfin-Whitt delay probability formula
\[g_{\rm HW}(\beta^*) = \left( 1 + \frac{\beta^* \Phi(\beta^*)}{\varphi(\beta^*)}\right)^{-1}.\]
Hence, if $\varepsilon > g_{\rm HW}(\beta^*)$, then no feasible solution to $g^b(\beta^*,\gamma^*)=\varepsilon$ exists.
This should be considered when choosing $\beta^*$.
Furthermore, it is evident from Step 3 that the final values $(\beta,\gamma)$ are always larger than $(\beta^*,\gamma^*)$.
\end{remark}
We now use the same example as in Section \ref{sec:dimensioning_block} to demonstrate capacity allocation decisions for the model with holding. This can be viewed as the additional capacity the medical unit needs in terms of nurses and beds, in order to account for the fact that patients are waiting in the ED to be admitted instead of being blocked and transferred to a less preferred unit.
Observe that the holding model leaves less flexibility for management in choosing system parameters due to stability constraints. For example, the policy with $n=30$ ($\gamma=-0.75$) is infeasible in the holding model.
For similar reasons, only nurse staffing levels with $\beta>0$, or $s > R_1=3.2$ are feasible.
Targeting a delay probability of $0.5$ with $n=40$, Figure \ref{fig:ratio01_hold} shows that operating a MU with holding room requires $\beta = 0.475$ or $s=5$.
Recall that under the blocking policy, only $s=4$ nurses were needed to achieve a delay probability of $0.5$.
This example hence shows how the managerial decision to have a holding room, rather than deferring patients to less preferred medical units, requires additional workforce in that unit (as well as the ED).
This example also shows that the facility with holding room is able to treat fewer patients simultaneously than under blocking constraints, in line with the bounds in Section \ref{sec:bounds} and Conjecture \ref{conj:stochorder}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g^h(\beta,\gamma)$},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col5] table[x=beta,y=delay_n35] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col4] table[x=beta,y=delay_n40] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col1] table[x=beta,y=delay_n45] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\draw[very thick,col4,dashed,->] (axis cs: -2,0.5) -- (axis cs: 0.475,0.5) -- (axis cs: 0.475,0);
\legend{$\gamma = -0.75$, $\gamma =0.102$,$\gamma= 0.955$, $\gamma=1.807$};
\end{axis}
\end{tikzpicture}
\caption{Approximate delay probability of restricted Erlang-R system with holding for $r\approx 0.09$ and $R_1=3.2$ }
\label{fig:ratio01_hold}
\end{figure}
\section{Model analysis and managerial implications}
\label{sec:analysis_chapter5}
In this section, we use the analysis and algorithms developed in earlier sections to gain insights into the importance of the capacity restrictions and patient returns in a restricted Erlang-R system by drawing a comparison to related models studied in the literature.
\subsection{The influence of patient returns or the role of $r$}
Here we study how the parameter $r$ affects the service level in the restricted Erlang-R model with blocking, on the basis of the asymptotic expressions in Theorem \ref{thm:limits_YT}.
To better understand the connection with the single-station model and the importance of returns we examine the role of $r$.
Recall the interpretation of $r$ as the fraction of time a patient is needy during his stay within the system in the idealized scenario with infinite capacity, i.e. for $r\in(0,1)$.
The case $r=1$ corresponds to the setting in which patients are needy all the time, in this case patients get service in one time.
When $r=1$ the infinite-server queue, describing the number of content patients, disappears from the queueing system and we end up with a standard loss model---$M/M/s/n$ queue---in which capacity is scaled as
\[ s = R_1+\beta\sqrt{R_1}, \qquad n = R_1+\gamma\sqrt{R_1}. \]
This staffing rule only makes sense in case $\beta<\gamma$, since no delay is experienced if $n\leq s$.
If indeed $\gamma>\beta$, then the asymptotic delay probability and scaled blocking probability are given by \cite{masseywallace},
\begin{align*}
g_B(\beta,\gamma) &= \frac{1-{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}, \\
f_B(\beta,\gamma) &= \frac{\beta{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}.
\end{align*}
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.8,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,1.28)},anchor= north east},
yscale = 0.75
]
\addplot[thick,col1] file {Chapter_5/tikz/influence_r/PdelayB_g1_b025.txt};
\addplot[thick,col3] file {Chapter_5/tikz/influence_r/PdelayB_g1_b05.txt};
\addplot[thick,col4] file {Chapter_5/tikz/influence_r/PdelayB_g1_b1.txt};
\addplot[thick,col5] file {Chapter_5/tikz/influence_r/PdelayB_g1_b2.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability $g^b(\beta,\gamma)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.4,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,0.05)},anchor= south east},
yscale = 0.75
]
\addplot[thick,col1] file {Chapter_5/tikz/influence_r/PblockB_g1_b025.txt};
\addplot[thick,col3] file {Chapter_5/tikz/influence_r/PblockB_g1_b05.txt};
\addplot[thick,col4] file {Chapter_5/tikz/influence_r/PblockB_g1_b1.txt};
\addplot[thick,col5] file {Chapter_5/tikz/influence_r/PblockB_g1_b2.txt};
\addplot[thick,dashed] file {Chapter_5/tikz/influence_r/PblockB_g1_inf.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability $f^b(\beta,\gamma)$.}
\label{fig:influence_of_r_b}
\end{subfigure}
\caption{Asymptotic performance measures as a function of $r$ in the restricted Erlang-R model with blocking for $\gamma=1$.}
\label{fig:influence_of_r}
\end{figure}
We can see that $f^b(\beta,\gamma)$ for increasing $\beta$ approaches a lower bound that is a function of $r$.
To understand this, observe that as $\beta$ grows, delays at the nurse queue vanish.
Then the sojourn time of an admitted patient only consists of a geometric number of needy and content periods with mean $(1/\mu+p/\delta)/(1-p) = 1/(r\mu(1-p))$.
The blocking model can in this case be modeled as an $M/G/n/n$ queue, with offered load $\lambda/(r\mu(1-p)) =R_1/r$, in which the scaled blocking probability is known to be, see \cite{Avram2013},
\[\sqrt{R_1} \, \mathbb{P}({\rm block}) = \sqrt{R_1} \, \frac{(R_1/r)^n/n!}{\sum_{k=0}^n (R_1/r)^k / k!} \to \sqrt{r} \, \frac{\varphi(\gamma)}{\Phi(\gamma)},\]
as $R_1\to\infty$.
This function of $r$ is plotted in Figure \ref{fig:influence_of_r_b} as the dashed line.
We observe that in general the probability of blocking increases with $r$, regardless of the capacity constraints on the needy station.
We can explain this by observing that $r$ influences only $n$ in the QED staffing rule. When $n$ reduces, more patients are blocked. Therefore, if patients spend relatively more time in needy state, which usually indicates services that are less interrupted, blocking will increase. Delays, on the other hand, will decrease in such situations---the minimal delay possible can be achieved if service is given in one time ($r=1$). Returns or interruptions increase delays significantly under QED staffing.
\subsection{Comparing restricted and unrestricted Erlang-R models}
Given the expressions for the asymptotic delay probability in the open Erlang-R model, and its restricted versions with blocking and holding, we compare the three policies for various values of $\beta$, $\gamma$ and $r$.
Figure \ref{fig:comparison_delay} plots the delay probability for blocking ($g^b(\beta,\gamma)$), holding ($g^h(\beta,\gamma)$) and Erlang-R ($g_{\rm HW}(\beta)$) models, as functions of $\gamma$, while keeping $\beta$ fixed, for three values of $r$.
We make a couple of observations.
Notice that
\[ g^b(\beta,\gamma) \leq g^h(\beta,\gamma) \leq g_{\rm HW}(\beta) \]
for all $\beta,\gamma>0$ and $r$.
In that sense, the holding model is an interpolation between the blocking and the open model.
As expected, the delay probabilities in the restricted models converge to those of the open Erlang-R model, because increasing $\gamma$ is tantamount to lifting the stringent constraints on the system size. Note that the rate of conversion is fast---one can provide probability of waiting close to that of the open model with small values of $\gamma$. Indeed, the fact that when using QED staffing not much of excessive delay results from the beds restriction is important by itself.
Also, we observe that the difference between delay probabilities increases with $r$.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r01_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r01_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r01_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.1$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r025_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r025_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r025_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.25$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r05_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r05_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r05_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.5$.}
\end{subfigure}
\caption{Asymptotic delay probability in open Erlang-R (dashed), restricted Erlang-R with blocking (marked) and restricted Erlang-R with holding (solid), as function of $\gamma$ with $\beta=0.1$ (blue), $\beta=0.5$ (orange) and $\beta=1$ (green) fixed.}
\label{fig:comparison_delay}
\end{figure}
\subsection{The impact of visit number}
\label{subsec:num_visit}
We next reflect on the impact of operational capacity decisions on different patient populations. We measure patient's complexity by the number of times she needs to see the nurse or the physician during her stay. In the ED context, simple-to-treat patients will need to see the physician once, while complex ones will need multiple visits. Hence, we divide the patients into complexity groups by the number of visits in the Needy station. Since the number of visits is geometrically distributed, we have a higher proportion of simple patients than complex ones; that fits well the health care environment.
Figure \ref{fig:wait_by_visit} shows the waiting time in the needy and pre-entring queues, and the total waiting time, as a function $n$ (number of beds), for each complexity group.
Obviously, the expected waiting time in the pre-entring queue decreases with $n$, while the needy waiting time increases.
For patients who require a relative large number of visits of the physician, in this case more than 6, the total needy wait is the dominant part of the total waiting time. Therefore, as $n$ grows, the total waiting time first decreases and then increases.
In fact, Figure \ref{fig:wait_by_visit_b} suggests that there is an optimal number of beds $n$ that minimizes the total wait for each complexity type.
Thus, size restrictions reduce the length-of-stay of patients with complex health conditions (given that the constraint is not too tight).
On the other hand, this figure also shows that no such $n$ exists for patients who only require little assistance.
Hence, there is no $n$ that improves the sojourn time of all patients in the ED simultaneously.
This leaves the decision to the hospital manager to weigh the importance of patients of different complexity levels.
\begin{remark}
From a different perspective, note that in queueing systems such as communication systems, the partition of a job to sizable quantities and scheduling those jobs in a similar dynamic to the Erlang-R model became a popular way for increasing throughput. This is because this effectively schedule jobs by their size even though the total job requirements are uncertain. This in fact creates a shortest-job-first policy without prior knowledge of job size \citep{Comte2016}. Considering that perspective we note that the Erlang-R model actually prioritize simple jobs over complex ones. But without restrictions, when load is too high, such procedures may lead to very long LOS of long jobs. The capacity restriction we analyze in this chapter, in both of its versions, limits such delays. Hence, even in cases in which the returns themselves are created by a managerial decision, imposing the additional managerial restriction on entering the system has benefits.
\end{remark}
\begin{figure}
\centering
\begin{subfigure}{0.38\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {Chapter_5/tikz/inner_vs_outer_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,col1, thick] table[x=n,y=hold] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\end{axis}
\end{tikzpicture}
\caption{Expected pre-entering waiting (red) and needy waiting times (black)}
\end{subfigure}
\begin{subfigure}{0.6\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8,
legend cell align=left,
legend style = {at = {(1.05,0.58)}, anchor = west}
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {Chapter_5/tikz/total_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {Chapter_5/tikz/total_wait.txt};
\legend{{\small $N=1$},{\small $N=2$},{\small $N=3$},{\small $N=4$},{\small $N=5$},{\small $N=6$},{\small $N=7$},{\small $N=8$},{\small $N=9$},{\small $N=10$}};
\end{axis}
\end{tikzpicture}
\caption{Total expected waiting times\\
\quad \\
\quad }
\label{fig:wait_by_visit_b}
\end{subfigure}
\caption{Expected waiting times as a function of $n$ given the number of visits $N$ in the Erlang-R model with holding with $\lambda=2$ $\mu=1$, $\delta=0.25$, $p=0.75$ and $s=9$.}
\label{fig:wait_by_visit}
\end{figure}
\subsection{Case study: comparison of operational decisions}
\label{sec:case_study}
We now illustrate how the managerial decision to operate under a specific operational regime affects ED performance in terms of efficiency and quality-of-care, through a case study.
The practical environment we investigate is the ED of a moderately-sized hospital, which faces the arrival pattern $\lambda(t)$ plotted in Figure \ref{fig:Case_study_arrival_pattern_a} on a typical workday.
Other parameters of the model are estimated to be $\mu = 6.67,\ \delta = 2.18$ and $p = 0.76$, so that $r = 0.301$. These parameters were taken from \cite{YomTov2014}. In order to set time-varying staffing levels $s(t)$ and $n(t)$, we adopt the \textit{mean-offered load} (MOL) approximation of the demand process of~\cite{Jennings1996}.
This approach initially presumes infinite capacity to obtain the number of patients $R(t)$ in the queueing system as a function of time.
This offered load function then replaces (constant) value of $R$ in the stationary dimensioning scheme under consideration, to determine the adequate number of servers at each point in time.
Following this idea in our two-dimensional queueing system, we find the offered load function for the nurses $R_1(t)$ and the offered load function for the beds $R_1(t)+R_2(t)$ as the solution of the system of ODEs,
\begin{align} \label{eq:offeredloadODE}
\frac{d}{dt} R_1(t) &= \lambda(t) + \delta R_2(t) - \mu R_1(t),\\
\frac{d}{dt} R_2(t) &= p\mu R_1(t) - \delta R_2(t),
\end{align}
see \cite[Thm.~2]{YomTov2014} for details.
For this case study's parameters, these offered load functions are also plotted in Figure \ref{fig:Case_study_arrival_pattern_a}.
While the number of nurses can be adjusted in a relatively flexible manner, the value of $n$, which echoes a hard restriction on the ED capacity, is naturally less amenable to fluctuations. The reason is that the maximum ED capacity is to a large extent determined by its hardware, such as beds and medical equipment.
However, the ED manager might deliberately consider reducing $n$ during more quiet periods of the day, e.g.\ during the night, by imposing bed-to-physician constraints. This is done, for example, when setting a case management constraint \citep{EDexperiment,Campello2016}.
Therefore, we consider the scenario in which both $s$ and $n$ are time-dependent but we do not force a constant case management quantity, rather let our new methodology to recommend an appropriate one.
\begin{figure}
\centering
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.01,0.95)},anchor = north west}
]
\addplot[very thick,black] file {Chapter_5/tikz/lambdaFunction.txt};
\addplot[very thick,col1] file {Chapter_5/tikz/R1.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/R1R2.txt};
\legend{ $\lambda(t)$, $R_1(t)$, $R_1(t)+R_2(t)$};
\end{axis}
\end{tikzpicture}
\caption{Dynamic arrival rate function offered load functions}
\label{fig:Case_study_arrival_pattern_a}
\end{subfigure}
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/sFunction.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/nFunction.txt};
\legend{ $s(t)$, $n(t)$};
\end{axis}
\end{tikzpicture}
\caption{Capacity function for $\beta=\gamma=0.5$}
\label{fig:Case_study_arrival_pattern_b}
\end{subfigure}
\caption{Empirical arrival rate and offered load functions $R_1(t)$ and $R_1(t)+R_2(t)$ in Israeli ED and corresponding capacity functions.}
\label{fig:Case_study_arrival_pattern}
\end{figure}
Extrapolating Algorithm \ref{alg:stationarydimensioning} to the time-varying case, Step 4 is replaced by
\begin{align*}
s(t) &= R_1(t) + \beta\sqrt{R_1(t)},\\
n(t) &= R_1(t)+R_2(t) + \gamma\sqrt{R_1(t)+R_2(t)},
\end{align*}
for some $\beta,\gamma>0$.
Since $R_1(t)$ and $R_2(t)$ are given, the QED staffing problem again reduces to finding the pair $(\beta,\gamma)$.
Figure \ref{fig:Case_study_arrival_pattern_b} plots the capacity functions for $\beta = 0.5$ and $\gamma=0.5$, assuming capacity can only be adjusted every 30 minutes.
In this case study, we consider three pairs of parameters $(\beta,\gamma)$.
For each of these we investigate, using simulation, the differences in the time-varying performance indicators between the policy with blocking and holding.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 1.0,
ytick = {0,0.2,0.4,0.6,0.8,1.0},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=delay_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=delay_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=delay_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=delay_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=delay_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=delay_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm delay})$}
\label{fig:simulation_results_a}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 0.52,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1,
legend cell align=left,
legend style = {at = {(0.9,0.95)}, anchor = north east}
]
\addplot[very thick,col1] table[x=t,y=block_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4] table[x=t,y=block_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5] table[x=t,y=block_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=block_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4,dashed] table[x=t,y=block_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5,dashed] table[x=t,y=block_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\legend{{$(\beta,\gamma)=(0.1,2)$},{$(\beta,\gamma)=(1,1.5)$},{$(\beta,\gamma)=(2,1)$}};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm block})$ or $\mathbb{P}({\rm hold})$}
\label{fig:simulation_results_b}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.5,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=ratio_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=ratio_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=ratio_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=ratio_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=ratio_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=ratio_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{Nurse-to-patient ratio.}
\label{fig:simulation_results_c}
\end{subfigure}
\caption{Simulation results for case study. Solid and dashed lines represent time-varying performance in the blocking and holding model, respectively.}
\label{fig:simulation_results}
\end{figure}
The simulation results are presented in Figure \ref{fig:simulation_results}.
Figure \ref{fig:simulation_results_a} shows that the MOL approach for capacity allocation roughly stabilizes the delay probability.
Figure \ref{fig:simulation_results_b} shows that the fraction of patients not entering the ED on arrival in the blocking model is reasonable for all parameter pairs considered and are ordered according to $\gamma$.
We also see a significant difference with holding.
Observe also that the holding probability drops in the period 8--13, which is exactly the period when the system is experiencing peak offered load.
Hence, this temporary reduction is in line with our asymptotic findings that the probability of blocking/holding is $O(1/\sqrt{R_1})$.
Finally note that the three parameter settings lead to different nurse-to-patient ratios.
Clearly, larger $\beta$ leads to small nurse-to-patient ratios (due do larger staffing).
Figure \ref{fig:simulation_results_c} demonstrates that for $(\beta,\gamma) = (1,1.5)$ and $(\beta,\gamma) = (2,1)$ the difference between the holding policy and the blocking policy is small. However, for $(\beta,\gamma) = (0.1,2)$ we see a significant increase in the ratio during night hours.
This may be due to the tight nurse schedule, that causes the holding queue to build up just before midnight.
This queue then empties latter on, causing an increase in workload per nurse in the period 24--7.
To see the direct effect of the size restriction on the queue lengths, we plotted the mean holding and service queue lengths in the holding model as a function of the parameter $\gamma$ in Figure \ref{fig:simulation_queuelengths}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.2,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/holdingQueue_g01.txt};
\addplot[very thick,col3] file {Chapter_5/tikz/casestudy_new/holdingQueue_g025.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/casestudy_new/holdingQueue_g05.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/holdingQueue_g1.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$}
\end{axis}
\end{tikzpicture}
\caption{Mean holding queue length}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 15,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.82,0.05)},anchor = south east}
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/serviceQueue_g01.txt};
\addplot[very thick,col3] file {Chapter_5/tikz/casestudy_new/serviceQueue_g025.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/casestudy_new/serviceQueue_g05.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/serviceQueue_g1.txt};
\addplot[very thick,dashed] file {Chapter_5/tikz/casestudy_new/serviceQueue_R.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$,Erlang-R}
\end{axis}
\end{tikzpicture}
\caption{Mean service queue length}
\end{subfigure}
\caption{Simulated queue length of holding model with different values of $\gamma$.}
\label{fig:simulation_queuelengths}
\end{figure}
Note that for all $\gamma$ considered, the holding queue length are, as expected, of a smaller order than the number of patients admitted.
Also, the holding queue length decreases as we increase $\gamma$.
The service queue lengths naturally approach the expected queue lengths in the Erlang-R model as $\gamma$ is increased.
\section{Conclusion \& future research}
\label{sec:conclusion}
In this chapter we developed and analyzed a queueing network tailored to a health care environment with finite-size restrictions.
Using the asymptotic approximations, numerical analysis and simulation, we gained insight into staffing problems that arise in EDs, and proposed an efficient, flexible, and easy to implement methodology to dimension medical facilities through a two-fold staffing rule.
The dimensioning scheme we developed provides a powerful and elegant way of finding adequate staffing levels in emergency departments.
Nonetheless, we see some avenues for further research.
The asymptotic approximations we developed based enabled us to take the first step towards characterizing the pre-entering queue behavior in the
QED regime.
We observed how the holding queue length vanishes at that $1/\sqrt{R_1}$ as $R_1\to\infty$.
Yet, our analysis did not yield explicit characteristics on the holding queue and holding times.
These performance indicators are naturally important to study if one wants to consider the trade-off between waiting time inside the ED and waiting time outside the ED (holding) time.
Furthermore, it is worthwhile to study the robustness of our approximations against the service and content time distributions. Since the content phase of a patient is modeled after an infinite-server queue, we expect our approximations to be useful for content time distributions beyond the exponential distribution as well, due to distributional insensitivity of the service time in infinite-server queues. For the needy phase, modeled after a multi-server queue, this insensitivity result does not hold and hence this needs further research.
Finally, the restricted Erlang-R model obviously gives a highly a simplify view of the complex reality of the ED.
In practice, distinctive features such as a triage system (with patient priorities), patient boarding time and availability of medical equipment may play a decisive role on congestion in EDs.
However, we think the analysis and dimensioning algorithms presented in this chapter can serve as a building block for staffing procedures that do account for these case-specific factors.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Description of the QBD process}
\label{app:QBDdescription}
\subsection{The QBD-process}
\label{app:theQBDprocess}
We consider the QBD-process $X(t)=(N(t),Q_1(t))$ in stationarity. Let $\nu(i)=\min\{i,s\}\mu$. To determine the (outgoing) transition rates of the process $X$ we distinguish between the following cases:
\begin{itemize}
\item \emph{Transitions from $(0,0)$:} There are no patients in the Emergency Department and thus the only possible occurrence is when a new patient arrives. This results in a transition to $(1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), 1 \leq i < n$:} There are exactly $i$ patients assigned to a bed of which none are seen by a nurse. Then either one of those patients becomes needy, or a new patient arrives at the Emergency Department that can immediately be seen by a nurse. The first results in a transition to $(i,1)$ and occurs at rate $i \delta$, and the second results in a transition to $(i+1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), i \geq n$:} Again, the only possible transitions arises from either a newly arrived patient or a patient assigned to a bed becoming needy. However, a newly arrived patient finds all beds occupied and needs to wait. Thus, with rate $\lambda$ we have a transition to $(i+1,0)$ and with rate $n \delta$ a transition to $(i,1)$.
\item \emph{Transitions from $(i,i), i < n$:} In this case all patients assigned to a bed are in need of service. With rate $\lambda$ a new patient arrives at the Emergency Department. She joins the (possible) queue to be seen by a nurse immediately, so this results in a transition to $(i+1,i+1)$. Moreover, since there are only $s < n$ nurses, a service completion occurs with rate $\nu(i)$. With probability $p$ the patient turns to the holding phase, so in total we still have $i$ patients with one patient less in queue for a nurse. With probability $1-p$ the patient leaves the Emergency Department, decreasing both $N$ and $Q_1$ by one. In other words, with rate $p \nu(i)$ we have a transition to $(i,i-1)$ and with rate $(1-p)\nu(i)$ we have a transition to $(i-1,i-1)$.
\item \emph{Transitions from $(n,n)$:} Similar to the previous case, we have a transition to $(n,n-1)$ with rate $p s \mu$ and with rate $(1-p)s \mu$ we have a transition to $(n-1,n-1)$. In this case however, a newly arrived patient finds all beds occupied, resulting in a transition to $(n+1,n)$ with rate $\lambda$.
\item \emph{Transitions from $(i,n), i > n$:} We have a transition to $(i+1,n)$ with rate $\lambda$ and a transition to $(i,n-1)$ with rate $p s \mu$. In case that a patient leaves the Emergency Department there are $i-n>0$ patients in the holding room waiting for an available bed. Thus, one of the $i-n$ patients in the holding room is assigned to the available bed in need of service. That is, with rate $(1-p) s \mu$ we have a transition to $(i-1,n)$.
\item \emph{Transitions from $(i,j), 1 \leq j < i < n$:} There are four possible transitions. First, with rate $\lambda$ there is a new arrival which results in a transition to $(i+1,j+1)$. Second, with rate $(i-j) \delta$ a patient in one of the beds becomes needy, which results in a transition to $(i,j+1)$. Third, with rate $p \nu(j)$ a patient turns to the content state after service completion, which results in a transition to $(i,j-1)$. Last, with rate $(1-p) \nu(j)$ a patient leaves the Emergency Department after service completion, which results in a transition to $(i-1,j-1)$.
\item \emph{Transitions from $(n,j), 1 \leq j < n$:} This case is similar to the previous one. The only difference arises when a new patient arrives, since all $n$ beds are already occupied. Thus, with rate $\lambda$ we have a transition to $(n+1,j)$.
\item \emph{Transitions from $(i,j), i > n, 1 \leq j \leq n$:} This case is the previous one, except when a patient leaves the Emergency Department after service completion. Then one of the $(i-n)$ patients in the holding room will be assigned to a bed in need of service. This results in a transition to $(i-1,j)$ with rate $(1-p) \nu(j)$.
\end{itemize}
\noindent
The state space and transition rates of the Erlang-R model with holding are illustrated in Figure~\ref{fig:QBDIllustration}.
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[step=1cm,gray!50!,very thin] (0,0) grid (15.5,8.5);
\draw[thick,->] (0,0) -- (15.5,0);
\draw[thick,->] (0,0) -- (0,8.5);
\draw[thick] (0,0) -- (8,8);
\draw[thick,dashed,black!50!] (8,0) -- (8,8);
\draw[thick] (8,8) -- (15.5,8);
\foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
\foreach \y in {0,1,2,3,4,5,6,7,8}
\draw[fill] (\x,\y) circle [radius=0.025];
\node [below left] at (0,0) {$0$};
\node [left] at (0,2) {$j$};
\node [left] at (0,5) {$k$};
\node [left] at (0,8) {$n$};
\node [below] at (6,0) {$i$};
\node [below] at (8,0) {$n$};
\node [above left] at (0,8.5) {$Q_1$};
\node [below right] at (15.5,0) {$N$};
\path [->,thick,-latex] (0,0) edge [bend right] (1,1);
\path [->,thick,-latex] (3,0) edge (3,1);
\path [->,thick,-latex] (3,0) edge (4,1);
\path [->,thick,-latex] (4,4) edge [bend right] (5,5);
\path [->,thick,-latex] (4,4) edge (4,3);
\path [->,thick,-latex] (4,4) edge [bend right] (3,3);
\path [->,thick,-latex] (6,2) edge (6,3);
\path [->,thick,-latex] (6,2) edge (7,3);
\path [->,thick,-latex] (6,2) edge (6,1);
\path [->,thick,-latex] (6,2) edge (5,1);
\path [->,thick,-latex] (8,5) edge (8,6);
\path [->,thick,-latex] (8,5) edge (8,4);
\path [->,thick,-latex] (8,5) edge (9,5);
\path [->,thick,-latex] (8,5) edge (7,4);
\path [->,thick,-latex] (8,8) edge (8,7);
\path [->,thick,-latex] (8,8) edge [bend right] (7,7);
\path [->,thick,-latex] (8,8) edge [bend left] (9,8);
\path [->,thick,-latex] (11,8) edge (11,7);
\path [->,thick,-latex] (11,8) edge [bend left] (12,8);
\path [->,thick,-latex] (11,8) edge [bend left] (10,8);
\path [->,thick,-latex] (11,0) edge (11,1);
\path [->,thick,-latex] (11,0) edge [bend left] (12,0);
\path [->,thick,-latex] (12,5) edge (12,6);
\path [->,thick,-latex] (12,5) edge (12,4);
\path [->,thick,-latex] (12,5) edge (13,5);
\path [->,thick,-latex] (12,5) edge (11,5);
\node [above] at (12.75,5) {\scriptsize $\lambda$};
\node [above] at (11.25,5) {\scriptsize $(1-p)\nu(k)$};
\node [right] at (12,5.75) {\scriptsize $(n-k)\delta$};
\node [right] at (12,4.25) {\scriptsize $p \nu(k)$};
\node [above] at (6.75,2.25) {\scriptsize $\lambda$};
\node [below] at (5,2) {\scriptsize $(1-p)\nu(j)$};
\node [above] at (6,3) {\scriptsize $(i-j)\delta$};
\node [right] at (6,1.25) {\scriptsize $p \nu(j)$};
\end{tikzpicture}
\caption{Illustration of state space and the transitions for the Erlang-R model with holding.}
\label{fig:QBDIllustration}
\end{figure}
The state space can be partitioned according to its levels, where level $i$ corresponds to a total queue length $N=i$ patients. This results in an infinite-sized matrix consisting of blocks, where each block corresponds to the transition flow from one level to another. Since the only transitions allowed are within the same level or between two adjacent levels in a QBD-process, we obtain a tridiagonal block structure. Each block consists of elements representing the transition rate of one state to another, and therefore each block is a matrix of size at most $(n+1) \times (n+1)$.
For the Erlang-R model with holding this gives the following result. Let $P$ denote the transition matrix of the process $(N(t),Q_1(t))$. We have the boundary levels $\{1,2,...,n\}$ and $P$ is of the form
\[
P = \left( \begin{array}{cccccccccc}
B_{00} & B_{01} & & & & & & & & \\
B_{10} & B_{11} & B_{12} & & & & & & & \\
& B_{21} & B_{22} & B_{23} & & & & & & \\
& & \ddots & \ddots &\ddots & & & & & \\
& & & & & B_{n \, n-1} & & & & \\
& & & & B_{n-1 \, n} & B_{nn} & A_0 & & & \\
& & & & & A_2 & A_1 & A_0 & & \\
& & & & & & A_2 & A_1 & A_0 & \\
& & & & & & & \ddots & \ddots & \ddots \\
\end{array} \right),
\]
where $B_{ii} \in \mathbb{R_1}^{(i+1) \times (i+1)}$, $B_{i \, i-1} \in \mathbb{R_1}^{(i+1) \times i}$, $B_{i-1 \, i} \in \mathbb{R_1}^{i \times (i+1)}$, and $A_0,A_1,A_2 \in \mathbb{R_1}^{(n+1)\times(n+1)}$. The matrices of transition rates for the boundary states are given by
\[
B_{00}=(-\lambda),
\qquad
B_{i-1 \, i} = \left( \begin{array}{ccccc}
0 & \lambda & & & \\
& \ddots & \lambda & & \\
& & \ddots & \ddots &\\
& & & 0 & \lambda \\
\end{array} \right),
\]
\[
B_{i \, i-1} = \left( \begin{array}{cccc}
0 & & & \\
(1-p)\mu & 0 & & \\
& (1-p)\nu(2)& \ddots & \\
& & \ddots & 0 \\
& & & (1-p)\nu(i)\\
\end{array} \right),
\]
and
\[
\scriptsize
B_{ii} = \left(
\begin{array}{ccccccccc}
-(\lambda+i \delta) & i \delta & & & &\\
p \mu & -(\lambda+\mu+(i-1)\delta) & (i-1)\delta & & &\\
& \ddots & \ddots & \ddots & & \\
& & p \nu(i-1) & -(\lambda+\nu(i-1)+\delta) & \delta \\
& & & & p \nu(i) & -(\lambda+\nu(i)) \\
\end{array} \right).
\]
Moreover, the transition rates are given by
\[
A_0 = \left( \begin{array}{ccccc}
\lambda & & & & \\
& \lambda & & & \\
& & & \ddots & \\
& & & & \lambda \\
\end{array} \right)
\]
\[ A_2 = \left( \begin{array}{ccccccc}
0 & & & & & & \\
& (1-p)\mu & & & & & \\
& & 2(1-p)\mu & & & & \\
& & & \ddots & & & \\
& & & & s(1-p)\mu & & \\
& & & & & \ddots & \\
& & & & & & s(1-p)\mu \\
\end{array} \right),
\]
and
\[
\scriptsize
A_1 = \left( \arraycolsep=0.55pt
\begin{array}{cccccccc}
-(\lambda+n \delta) & n \delta & & & & & & \\
p \mu & -(\lambda+\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(\lambda+s\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & s p\mu & -(\lambda+s\mu+\delta) & \delta & \\
& & & & & s p\mu & -(\lambda+s\mu)\\
\end{array} \right).
\]
\subsection{Stability condition}
\label{app:stability}
From the general theory of QBD processes \citep{Neuts1981} follows that the Markov process $(N(t),Q_1(t))$ is ergodic (stable) if and only if
\begin{equation}
\pi A_0 e < \pi A_2 e,
\label{eq:QBDstableCondition}
\end{equation}
where $e$ is the all one column vector and $\pi=(\pi_0,...,\pi_n)$ is the equilibrium distribution of the Markov process with generator $A_0+A_1+A_2$. In other words, $\pi$ is such that
\begin{equation}
\begin{array}{ll}
\pi(A_0+A_1+A_2) =0, & \pi e =1,
\end{array}
\label{eq:QBDstableProbabilityVector}
\end{equation}
and
\[
A_0+A_1+A_2 = \qquad\qquad\qquad
\]
\begin{align*}
{\scriptsize
\left(
\begin{array}{cccccccc}
-n \delta & n \delta & & & & & & \\
p \mu & -(p\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(ps\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & p s \mu & -(ps\mu+\delta) & \delta & \\
& & & & & p s \mu & -ps\mu\\
\end{array} \right).
}
\end{align*}
Then $\pi$ must satisfy the balance equations
\begin{align*}
- n \delta \pi_0 + p \mu \pi_1 &= 0, \\
(n-j+1)\delta \pi_{j-1} - (p\nu(j) +(n-j)\delta) \pi_j + p \nu(j+1) \pi_{j+1} &= 0, \\
\delta \pi_{n-1} - p s \mu \pi_n &= 0,
\end{align*}
with $\nu(j)=\min\{j,s\}\mu$, and the normalization condition
\[
\sum_{i=0}^n \pi_i=1.
\]
It is readily verified that
\begin{equation}
\pi_i =
\left\{\begin{array}{ll}
\pi_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\pi_0 \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistr}
\end{equation}
with
\begin{align*}
\pi_0= \left(\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right)^{-1}.
\end{align*}
satisfies the balance equations and the normalization condition.
\begin{proposition}
The distribution of the closed two-node Jackson network illustrated in Figure~\ref{fig:Jennings} is given by
\begin{equation}
\hat{\pi_i} =
\left\{\begin{array}{ll}
\hat{\pi}_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\hat{\pi_0} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistrTwistedJennings}
\end{equation}
with
\begin{align*}
\hat{\pi}_0= \left[\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right]^{-1}.
\end{align*}
\label{prop:CriticalTilburgdistr}
\end{proposition}
\begin{proof}
We have a two-node closed Jackson network, with probability transition matrix
\[
P = \left(
\begin{array}{cc}
1-p & p \\
1 & 0
\end{array} \right).
\]
Let $r_i(m)$ denote the rate of service when there are $m$ patient at queue $i$, so $r_1(m)=\min\{m,s\}$ and $r_2(m)=m$. The throughput vector $\gamma = (\gamma_1,\gamma_2) \in \mathbb{R_1}^2$ must satisfy $\gamma = \gamma P$ and we find that $\gamma=(p,1)$ suffices. From the general theory of Jackson networks, see \cite{Jackson1963}, it follows that the stationary distribution is given by
\begin{align*}
\pi_i = G^{-1} g_1(i) g_2(n-i)
\end{align*}
with
\begin{align*}
\begin{array}{ll}
g_1(i)= \frac{(\gamma_1/\mu)^i}{\prod_{m=1}^i r_1(m)}, & g_2(n-i)= \frac{(\gamma_2/\delta)^{n-i}}{\prod_{m=1}^{n-i} r_2(m)},
\end{array}
\end{align*}
and normalization constant $G= \sum_{i=0}^n g_1(i) g_2(n-i)$. Then,
\begin{align*}
g_1(i) &= \left\{\begin{array}{ll}
\frac{1}{i! \mu^i} & \textrm{\normalfont for } 0 \leq i \leq s, \\
\frac{1}{s! s^{i-s} \mu^i} & \textrm{\normalfont for } s+1 \leq i \leq n, \\
\end{array} \right.\\
g_2(n-i) &=\frac{1}{(n-i)!} \left(\frac{p}{\delta}\right)^n \left(\frac{\delta}{p}\right)^i,
\end{align*}
and rewriting the expressions yields~\eqref{eq:eqdistrTwistedJennings}.
\end{proof}
\subsection{Stationary distribution}
\label{app:StationaryDistributrion}
Assuming that the stability condition is satisfied, we can determine the unique stationary distribution of the Markov process $(N(t),Q_1(t))$. The vector $\pi_i$ can be written as $\pi_{n+i}= \pi_n G^{i}$ for $i=0,1,...$, where $G$ is the minimal nonnegative solution of the non-linear matrix equation
\begin{equation}
A_0+G A_1 + G^2 A_2=0.
\label{eq:MG-G}
\end{equation}
The balance equations can be written as
\[
\begin{array}{ll}
\pi_{i-1} A_0+ \pi_i A_1 + \pi_{i+1} A_2=0, & i=n+1,n+2,...
\end{array}
\]
and using $\pi_{n+i}= \pi_n G^{i-n}$ for $i=0,1,...$, this find
\[
\begin{array}{ll}
\pi_n G^{i-n-1} \left(A_0+ G A_1 + G A_2\right)=0, & i=n+1,n+2,....
\end{array}
\]
\noindent
Moreover, we have the boundary equations
\begin{align*}
\pi_0 B_{00} + \pi_1 B_{10} &= 0 \\
\pi_0 B_{01} + \pi_1 B_{11} + \pi_2 B_{21} &= 0 \\
\pi_1 B_{12} + \pi_1 B_{22} + \pi_2 B_{32} &= 0 \\
&\vdots& \\
\pi_{n-2} B_{n-2 \, n-1} + \pi_{n-1} B_{n-1 \, n-1} + \pi_{n} B_{n \, n-1} &= 0 \\
\pi_{n-1} B_{n-1 \, n} + \pi_{n} B_{nn} + \pi_{n+1} A_2 &= 0,
\end{align*}
along with the normalization equation
\[
1 = \sum_{i=0}^{\infty} \pi_i e = \sum_{i=0}^{n-1} \pi_i e + \pi_n(I-G)^{-1}e,
\]
where we slightly abuse notation by using $e$ as the all ones vector of appropriate size. We note that the matrix $G$ has a spectral radius less than one and therefore $(I-G)$ is invertible.
These equations provide the tools for finding the equilibrium probabilities. Although it is hard to solve $G$ analytically from Equation~\eqref{eq:MG-G}, it is easy to solve numerically by using the following algorithm (matrix-geometric method). Rewriting~\eqref{eq:MG-G} gives
\[
G=-(A_0+G^2 A_2) A_1^{-1},
\]
where $A_1$ is invertible, since it is a transient generator matrix. Let
\[
G_{k+1}=-(A_0+G_k^2 A_2) A_1^{-1},
\]
starting with $G_k=0$. We note that $G_k \uparrow G$ as $k$ grows to infinity \citep{Neuts1981}. Once $||G_{k+1}-G_{k}||_2$ is below a certain preset threshold, we approximate $G$ by $G_{k+1}$.
\section{Proof of Proposition \ref{thm:stochasticordering}}\label{app:stochastic_ordering}
First, note that by definition of the Erlang-R model with holding, in which no more that $n$ patients can be admitted in the ED simultaneously, that $Q_1^h(t)+Q_2^h(t) \leq n = Q_1^J(t) + Q_2^J(t)$ follows directly.
Therefore, we only consider the relation between the states in the blocking and holding variants Erlang-R model.
As noted Section \ref{sec:Markov_process}, the model with holding can be characterized as a three-dimensional Markov chain $X^h(t) = (H(t),Q^h_1(t),Q^h_2(t))$ in which the components denote the number of holding, needy and content patients respectively. The Erlang-R model with blocking similarly admits a Markov process description, but with two dimensions, namely $X^b(t) = (Q^b_1(t),Q^b_1(t))$.
We prove the result by constructing a coupling between the Markov processes $X^h$ and $X^b$. Let $Z(t) := \left(\hat{X}^h(t),\hat X^b(t)\right) = \left(\hat{H}(t),\hat{Q}_1^h(t),\hat{Q}_2^h(t),\hat{Q}^b_1(t),\hat{Q}^b_2(t)\right)$.
We first define the transition rates of this five-dimensional Markov process, which naturally only depend on the current state of the system.
After that we show that the transition rates relevant to $\hat{X}^h(t)$ and $X^{h}(t)$ coincide with those of either $X^h(t)$ or $\hat{X}^b(t)$, respectively. The latter implies that the marginal transitions of $\hat{X}^h(t)$ and $X^b(t)$ (and $\hat{X}^b(t)$ and $X^h(t)$) are equal, and hence so are their probability distribution of the Markov processes.
Let $Z(t) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$. While defining the reachable states from this state and associated transition rates, we distinguish four transition types, and further differentiate the transition rates depending on the current state.\\
\\*
\textbf{Arrival.}
Arrivals to occur in both models simultaneously, but are handled differently according to the current queue lengths.
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr1}
(h,q_1^h+1,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr2}
(h+1,q_1^h,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h < n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr3}
(h,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr4}
(h+1,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\end{enumerate}
\noindent \textbf{Departure.}
Basically, we align service completions in the two models, but allow a completion occurring solely in either of one of the two models, only if the queue length in this model is strictly larger than in the other one.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep1}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(h-1,q_1^h,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.
\end{equation}
\item If $q_1^h < q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep2}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h \geq q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep3}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep4}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(0,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent\textbf{Become content.}
The differentiation between transitions is similar to those in the \textit{departure} transition type.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$,
\begin{equation}
\label{eq:con1}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b \wedge s)p\mu,\\
(h,q_1^h-1,q_2^h+1,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$,
\begin{equation}
\label{eq:con2}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^h \wedge s)p\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b+1) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent
\textbf{Become needy.}
\begin{enumerate}
\item If $q_2^h \geq q_2^b$,
\begin{equation}
\label{eq:ne1}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^b\delta,\\
(h,q_1^h+1,q_2^h-1,q_1^b,q_2^b) & \text{with rate }(q_2^h-q_2^b)\delta,\\
\end{array}
\right.\end{equation}
\item If $q_2^h < q_2^b$,
\begin{equation}
\label{eq:ne2}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^h\delta,\\
(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1) & \text{with rate }(q_2^b-q_2^h)\delta,\\
\end{array}
\right.\end{equation}
\end{enumerate}
This set of transitions defines the dynamics of the Markov process $Z(t) = (\hat{X}^h(t),\hat{X}^b(t))$.
Let us now restrict our attention to the transitions in which (at least one of the) first three coordinates of $Z(t)$ changes, that is, the marginal transitions of the process $\hat{X}^h$.
Let $\hat{X}^h(t) = (h,q_1^h,q_2^h)$, then according to the transition scheme above, $\hat{X}^h$ moves to state
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ (and hence necessarily $h=0$),
\[
\left\{
\begin{array}{ll}
(0,q_1^h+1,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h-1,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $q_1^h+q_2^h = n$ and $h=0$,
\[
\left\{
\begin{array}{ll}
(1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $h>0$ (and hence necessarily $q_1^h+q_2^h = n$),
\[
\left\{
\begin{array}{ll}
(h+1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(h-1,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(h,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(h,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\end{enumerate}
One can check that these transitions indeed coincide with the transitions in the original holding model, hence $\hat{X}^h(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Similarly, when the focusing on transitions of $Z(t)$ that are relevant for $\hat{X}^b(t)$, we deduce the following transition scheme. If $\hat{X}^b(t) = (q_1^b,q_2^b)$, then the next move according to the transitions of $Z(t)$ is
\[
\left\{
\begin{array}{ll}
(q_1^b+\mathbbm{1}_{\{q_1^b + q_2^b < n\}},q_2^b) & \text{with rate } \lambda,\\
(q_1^b-1,q_2^b) & \text{with rate }(q_1^b\wedge s)(1-p)\mu,\\
(q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b\wedge s)p\mu,\\
(q_1^b+1,q_2^b-1) & \text{with rate }q_2^b \delta.
\end{array}
\right.\]
These transition rates clearly coincide with the original Erlang-R model with blocking, and also hence $\hat{X}^b(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Next, we show that under this coupling scheme we have that if $\hat{H}(0) = 0$, $\hat{Q}_1^h(0)=\hat{Q}_1^b(0)$ and $\hat{Q}_1^h(0)=\hat{Q}^b(0)$ then for all $t\geq 0$, $Z(t)$ satisfies the hypothesis:
\begin{itemize}
\item[(i)] $\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \leq \hat{Q}_1^h(t) + \hat{Q}_2^h(t)$,
\item[(ii)] $\hat{Q}_2^b(t) \leq \hat{Q}_2^h(t)$,
\item[(iii)] $\hat{Q}_1^b(t) \leq \hat{Q}_1^h(t) + H(t)$.
\end{itemize}
We do so by induction on the next state reached after a transition of the joint Markov process $Z=(\hat{X}^h,\hat{X}^b)$.
First of all, $Z(0)$ clearly satisfies (i)-(iii).
Next, assume $Z(t^-) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$ satisfies the hypothesis and a transition occurs at $t$.
We show that under the specified coupling scheme, the state reached after the next transition, $Z(t)$ must satisfy (i)-(iii) as well. To do so, we differentiate between the four types of transitions that could occur: arrival, departure, become content and become needy.\\
\\*
\noindent\textbf{Arrival.}
Recall that under our coupling scheme an arrival always occurs in both the holding and blocking model simultaneously, see \eqref{eq:arr1}--\eqref{eq:arr4}. Furthermore, $q_2^h$ and $q_2^b$ are unchanged during this transition, rendering (ii) trivial.
By hypothesis $q_1^b + q_2^b \leq q_1^h+q_2^b$, hence the event $q_1^h+q_2^h < n$ and $q_1^h+q_2^b =n$, with resulting state $(0,q_1^h+1,q_2^h,q_1^b,q_2^b)$, can be excluded from our analysis
We check the conditions for the remaining three cases.
\begin{enumerate}[noitemsep]
\item If $Z(t)= (0,q_1^h+1,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h <n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \less[i] q_1^h+q_2^h+1 =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+1 = \hat Q_1^h(t) = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h =n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \leq n = q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h +1= \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $q_1^b + q_2^b = q_1^h+q_2^h=n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+h+1 = \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Departure.}
By carefully examining the possible state transitions of $Z(t)$ following a departure, we list six reachable states. However, by (iii), we have that if $h=0$, then $q_1^b \leq q_1^h$, which excludes the state $(0,q_1^h,q_2^h,q_1^b,q_2^b)$ in \eqref{eq:dep4} from the reachability graph.
We check the remaining states for conditions (i)--(iii). Again, during a departure, $q_2^b$ and $q_2^h$ are unchanged, so (ii) is automatically satisfied by the induction hypothesis.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 \less[i] q_1^h+q_2^h-1 < q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h + h-1 = \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $h>0$ and $q_1^h \geq q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 \leq q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$ and $q_1^h < q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 < q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^b \less[*] q_1^h + h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h,q_1^b-1,q_2^b)$, then $h=0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = (q_1^b-1)+q_2^b-1 < \less[i] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h-1 = \hat Q_1^h(t) + \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (0,q_1^h-1,q_2^h,q_1^b,q_2^b)$, then $h=0$ and $q_1^h>q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] (q_1^h-1)+q_2^b \less[ii] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = qy \less[*] q_1^h-1 =\hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Content start.}
On the event of a patient becoming content, it is clear that the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected. This means that (i) is directly satisfied by the induction hypothesis.
According to \eqref{eq:con1}--\eqref{eq:con2}, three states can be reached.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1)$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \less[ii] q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b,q_2^b)$, then $q_1^h > q_1^b$ (*),
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[ii] q_2^h < q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h = \hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b+1)$, then $q_1^b > q_1^h$ and hence by (iii) $h > 0$. The latter is only possible if $q_1^h+q_2^h=n$ (*),
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \leq n-q_1^b+1 = (q_1^h+q_2^h)-q_1^b+1 \less[*] q_2^h = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^h+h-1 \less[*] q_1^h+h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent \textbf{Become needy.}
Just as in the event of content start, the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected, whereby (i) is directly satisfied by the induction hypothesis.
By (ii), we have $q_2^h \geq q_2^b$. This excludes the state $(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1)$ from being reached, see \eqref{eq:ne2}.
We check the remaining two possibilities.
\begin{enumerate}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1)$.
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b-1 \less[ii] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+h+1 = \hat Q_1^h(t)+\hat H(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b,q_2^b)$, then $q_2^h > q_2^b$ (*).
\begin{itemize}
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[*] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h =\hat Q_1^h(t) + \hat H(t)$.
\end{itemize}
\end{enumerate}
Hence, the state reached after any feasible transition under the coupling scheme satisfies the conditions (i)--(iii).
Thus we conclude that the joint process\\ $(\hat{H}(t),\hat Q_1^h(t),\hat Q_2^h(t),\hat Q_1^b(t),\hat Q_2^b(t))$ adheres to (i)--(iii) for all $t$. Consequently, we have that (i) implies
\begin{align*}
\mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right) &= \mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right)\\
&=\sum_{j=0}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&=\sum_{j=k}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&\leq \sum_{j=h}^n \mathbb{P}\left( \hat Q_1^h(t)+\hat Q_2^h(t) = j \right)\\
&= \mathbb{P}\left( Q_1^h(t) + Q_2^h(t) \geq k\right) = \mathbb{P}\left(Q_1^h(t) + Q_2^h(t) \geq k\right).
\end{align*}
The other two orderings follow similarly.
\begin{remark}
Note that under this coupling scheme we cannot get the ordering $\hat Q_1^h(t)(t) \geq \hat Q_1^b(t)(t)$ for all $t\geq 0$. A minimal counter example occurs for $s=n=1$. Let $Z(0) = ((0,0,0),(0,0))$. First, two arrivals occur, such that state $((1,1,0),(1,0))$ is reached, followed by a departure transition, yielding $((0,1,0),(0,0))$. Next, the one patient left in the model with holding system becomes content, so that we obtain $((0,0,1),(0,0))$.
At this stage, if an arrival occurs, the arriving patient will be put in the holding queue in the model with holding, and admitted to nurse queue in the model with blocking. Hence we end up in state $((1,0,1),(1,0))$, in which $\hat Q_1^h(t) < \hat Q_1^b(t)$.
\end{remark}
\section{Proof of Proposition \ref{prop:stability_convergence}}\label{app:proof_stability_convergence}
Define
\[
A(s,n) = \sum_{k=0}^s \frac{k}{s} \, \binom{n}{k} b^k ,\quad
B(s,n) = \sum_{k=s+1}^n \frac{k!}{s!} \, \binom{n}{k} s^{s-k} b^k, \quad
C(s,n) = \sum_{k=0}^s \binom{n}{k} \, b^k,
\]
\[
\]
where $b = \delta/p\mu = r/(1-r)$. Then
\[
\rho_{\rm max}(s,n) = \frac{A(s,n)+B(s,n)}{C(s,n)+B(s,n)}.
\]
Proving that $\rho_{\rm max}(s,n) \to 1$ as $R_1\to\infty$ with $s$ and $n$ as in \eqref{eq:twofoldscaling} is equivalent to showing that
\begin{equation}\label{eq:proof_stab_1}
1-\rho_{\rm max}(s,n) = \frac{C(s,n)-A(s,n)}{C(s,n)+B(s,n)} = \frac{(1+b)^{-n}[C(s,n)-A(s,n)]}{(1+b)^{-n}[C(s,n)+B(s,n)]} \to 0.
\end{equation}
First, we rewrite
\begin{align*}
(1+b)^{-n} A(s,n)
&= (1+b)^{-n} \sum_{k=1}^s \frac{n}{s} \binom{n-1}{k-1} b^k \\
&= \frac{n}{s}\left(\frac{b}{1+b}\right)\sum_{k=0}^{s-1} \binom{n-1}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-1-k}\\
&= \frac{r n}{s}\sum_{k=0}^{s-1} \binom{n-1}{k} r^k (1-r)^{n-1-k}\\
&= \frac{r n}{s} \mathbb{P}( {\rm Bin}(n-1,r) \leq s-1 ) \\
&= \frac{rn}{s} \mathbb{P}\left( \frac{{\rm Bin}(n-1,r) - (n-1)r}{\sqrt{nr(1-r)}} \leq \frac{s-1 - (n-1)r}{\sqrt{nr(1-r)}} \right)\\
&\to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
since $nr/s = 1 + O(1/\sqrt{R_1})$.
Also,
\begin{align*}
(1+b)^{-n} C(s,n)
&= \sum_{k=0}^s \binom{n}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-k}\\
&= \sum_{k=0}^s \binom{n}{k} r^k (1-r)^{n-k}\\
&= \mathbb{P}( {\rm Bin}(n,r) \leq s) \to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right).
\end{align*}
Therefore, we have $(1+b)^{-n}[C(s,n)-A(s,n)] \to 0$ as $\lambda\to\infty$.
For the remaining term,
\begin{align*}
(1+b)^{-n} B(s,n)
&= (1+b)^{-n}\sum_{k=s+1}^n \binom{n}{k}\,\frac{k!}{s!} s^{s-k} b^k \\
&= (1+b)^{-n}\frac{n!}{s!}\, s^s\sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{-k}\\
&= (1+b)^{-n} \frac{n!}{s!}\, s^s\, \left(\frac{b}{s}\right)^n \sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{n-k}\\
&= r^n\, \frac{n!}{s!} s^{s-n} \sum_{m=0}^{n-s-1} \frac{1}{m!} \left(\frac{s}{b}\right)^m\\
&= \left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b} \, \mathbb{P}({\rm Pois}(s/b)\leq n-s-1),
\end{align*}
in which
\begin{align*}
\mathbb{P}({\rm Pois}(s/b)\leq n-s-1)
&= \mathbb{P}\left(\frac{{\rm Pois}(s/b)-s/b}{\sqrt{s/b}} \leq \frac{n-s-1-s/b}{\sqrt{s/b}}\right) \\
&\to \Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
as $\lambda\to\infty$.
By Stirling's approximation,
\begin{align*}
\left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b}
&\sim \left(\frac{r}{s}\right)^n \sqrt{\frac{n}{s}} \,\frac{n^n {\rm e}^{-n}}{s^s {\rm e}^{-s}}\, s^s \,{\rm e}^{s/b} \\
&= \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s+s/b} = \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s/r}.
\end{align*}
Since,
\[
\frac{rn}{s} = 1 + \frac{\gamma\sqrt{r}-\beta}{\sqrt{R_1}} + O(1/R),
\]
we find $\sqrt{n/s} = 1/\sqrt{r} + O(1/\sqrt{R_1})$ and
\begin{align*}
\log\left[ \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+\tfrac{s}{r}} \right]
&= n \log\left[ \frac{rn}{s}\right] - n+\frac{s}{r}\\
&= -n \left[ \left(1-\frac{rn}{s}\right) + \frac{1}{2}\left(1-\frac{rn}{s}\right)^2 + O(R^{-\tfrac{3}{2}}) \right] + \frac{s}{r}\left(1-\frac{rn}{s}\right)\\
&= \frac{s}{r}\left(1-\frac{rn}{s}\right)^2 - \frac{n}{2}\left(1-\frac{rn}{s}\right)^2 + O(1/\sqrt{R_1})\\
&= \frac{(\gamma\sqrt{r} - \beta)^2}{2r} + O(1/\sqrt{R_1}),
\end{align*}
as $\lambda\to\infty$ and hence,
\[
(1+b)^{-n} B(s,n) \to \varphi\left(\frac{\gamma\sqrt{r}-\beta}{\sqrt{r}}\right)\Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right).
\]
Hence, we conclude that the denominator of \eqref{eq:proof_stab_1} converges to a constant value as $R_1$ grows, and hence the $1-\rho_{\rm max}(s,n)\to 0$ as $\lambda\to\infty$.
\resettocdepth
\end{subappendices}
\chapter{Finite-size effects in critically dimensioned emergency departments}
\begin{chapterstart}
Motivated by health care systems with repeated services that have both personnel (nurse/physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we account for are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them.
We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly.
We show that the system dynamics is captured by the fraction of needy time in the network, and that returning patients should be accounted for both in steady-state and time-varying conditions.
We demonstrate the application of our policies in two case-studies of resource allocation in hospitals.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Finite-size effects in critically dimensioned emergency departments}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen, Fiona Sloothaak \& Galit Yom-Tov}\\
Submitted to \textit{Operations Research}
\end{flushright}
\newpage
\section{Introduction}
In recent years, operations research techniques have received increased interest from the health care community, as they are able to design and improve workflow processes in health care facilities~\cite{Armony2015,Green2008,Bekker2009b,Hall2006,Hall2012}.
Because these processes are typically stochastic in nature, it is common practice to use queueing theory for performance analyses and workforce planning.
As a first step towards understanding the processes going on in health care environments, systems are commonly modeled after a single station queue, such as the $M/M/s$ (Erlang-C), $M/M/s/s$ (Erlang-B) or $M/M/s+M$ (Erlang-A) models, and fluid and diffusion approximations are used to provide insights into the process dynamics.
However, simple single station models often fail to capture the more intricate dynamics of the settings specific to health care contexts.
Prime examples include the flows of patients in a hospital from one medical ward to another \citep{Armony2015}, within the Emergency Department (ED) between different stages of treatment \citep{Junfei2015}, or between medical facilities \citep{zychlinski2016bedblocking}.
Queueing networks can capture the dependency between several service stages and several types of resources.
More specifically, we are interested in the ubiquitous feature, particularly present in health care environments, that patients during their stay in the system might require a specific resource multiple times, e.g.~physicians and nurses who treat patients several times during their stay in the medical wards \citep{Jennings2011} or the ED \citep{YomTov2014}, while multiple resources types are limited (e.g.\ medical staff and beds).
In this chapter, we concentrate on the dynamics within EDs.
An often ignored yet essential feature of medical facilities concerns the restriction of the number of patients that can reside in the facility simultaneously.
In Chapter 4, we already observed that finite-size restrictions can have a significant effect on the performance of queueing systems.
In this chapter, we investigate the influence of such multiple restrictions on the network dynamics and the required staffing policies in the context of an ED. \\
\\*
\noindent
\textbf{The restricted Erlang-R model.}
The canonical model for service networks with returns is the Erlang-R model, introduced by Yom-Tov \& Mandelbaum~\citep{YomTov2014}.
In this open two-station model, customers arrive according to a Poisson process to an $M/M/s$.
After service completion, the customer with probability $1-p$ leaves the system and with probability $p$ returns to the queue after a random delay.
This delay is modeled as an infinite-server queue.
A schematic visualization of the Erlang-R model is depicted in Figure \ref{fig:ErlangR}
in which customers, during their stay in the system receive a random number of services from the same pool of servers.
Yom-Tov \& Mandelbaum \cite{YomTov2014} showed that such a simple network model can be used to determine staffing in an ED both in stable and time-varying conditions.
Nevertheless, empirical studies report that some countries, such as the US, use a different operational mode that applies strict restrictions on entering the ED \citep{EDexperiment}.
In typical US EDs, a patient will not enter the ED until both a bed and a physician are available to treat her.
Those restrictions can be either physical (beds) restrictions or managerial ones --- for instance by imposing a patient-to-physician ratio.
In this work, we extend the Erlang-R model by enforcing a constraint on the maximum number of available places inside the facility.
Our model hence incorporates two kinds of resource constraints: servers that provide the actual service and the maximum available places inside the service system.
Both affect the system in a highly interdependent way.
The model, presented in Figure \ref{fig:Erlang_R_model}, assumes $s$ servers and a maximum capacity of $n$ concurrent places.
We assume that patients arrive according to a Poisson process with rate $\lambda$.
In case a new arrival finds $n$ or more patients already present, we consider two options: either she waits outside the service facility in a holding queue until a vacant space becomes available (Figure \ref{fig:Erlang_R_holding}) or she is blocked (Figure \ref{fig:Erlang_R_blocking}), such as is the case when patients are sent to an alternative facility.
Once a patient is admitted, she requires assistance from one of the $s$ servers for an exponentially distributed duration with mean $1/\mu$.
Then, with probability $1-p$, the patient leaves the system or, with probability $p$, returns to service again after an exponentially distributed time with mean $1/\delta$.
Following Jennings \& de V\'ericourt \cite{Jennings2011} and Yom-Tov \& Mandelbaum \cite{YomTov2014}, we call patients {\it needy} when they require attention from one of the servers and {\it content} when they are in the delayed return phase.
In addition, we call patients {\it holding} when they are waiting outside the facility for an available space. We assume that the arrival process, the needy times and content times are mutually independent.
In the holding queue and the needy queue, we apply the First-Come-First-Served (FCFS) discipline.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-2.75,4.5) -- (-1.25,4.5);
\draw [thick] (-1.5,5) -- (0,5) -- (0,4) -- (-1.5,4);
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (0,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick,->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,2.9) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\node [above] at (-0.75,5) {\footnotesize holding};
\end{tikzpicture}
\caption{Erlang-R model with holding.}
\label{fig:Erlang_R_holding}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-1.5,4.5) -- (2.5,4.5);
\draw [thick, ->] (0,4.5) -- (0,2.5) node[below left] {blocked};
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick, ->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,3.4) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{Erlang-R model with blocking.}
\label{fig:Erlang_R_blocking}
\end{subfigure}
\caption{Restricted Erlang-R models with maximally $n$ customers in system.}
\label{fig:Erlang_R_model}
\end{figure}
As mentioned, we consider two versions of the finite-capacity constraint.
The first version is called \emph{Erlang-R with holding}, in which patients wait for an available space in the system.
The second version is called \emph{Erlang-R with blocking}, in which patients meeting a full system are blocked.
Naturally, intermediate scenarios can be constructed in which a proportion of the total arrival volume of patients indeed leaves upon finding a full system, while the rest joins the holding queue.
While this chapter focuses on the two extreme cases, straightforward adaptations can fit these intermediate scenarios. \\
\\*
\noindent
\textbf{Examples of restricted Erlang-R.}
As noted before, an ED operated in the US can be modeled using a restricted Erlang-R model. Another health care example is medical units (MUs) in a hospital.
Such units specialize in specific types of illnesses (cardiology, oncology, etc.) and have limited resources such as nurses and beds. If the unit is full, new patients are either allocated to an alternative medical unit, i.e.\ blocked, or wait for an available bed.
Both policies are problematic in terms of quality-of-care, because the personnel in the alternative unit (or the ED) may be less knowledgeable about the patient's medical condition and waiting in the ED was shown to increase mortality.
Moreover, ED waiting may reduce available capacity for treating ED patients \citep{Carmen2016,israelit}, hence endangering both the delayed patient as well as others. Both the number of personnel (nurses and physicians) and the number of beds impact service dynamics and quality-of-care. Research so far looked at the capacity allocation of those resources separately. Green \& Yankovic \cite{GY2011} and Jennings \& de V\'ericourt \cite{Jennings2008} looked at nurse staffing in medical units, while de Bruin et al.~\cite{Bekker2009b} looked into bed allocation. The unified model we suggest enables us to capture the dependency between those two decisions, and its impact on other medical units in the hospital.
At the same time, we capture the two most commonly used modes of operation --- blocking and holding of new patients. \\
\\*
\noindent
\textbf{Two-fold square-root staffing rule.}
Our main goal is to provide staffing policies for the ED that high resource utilization, while at the same time maintain good quality-of-care.
This goal relates to the philosophy of the Quality-and-Efficiency-Driven (QED) regime that is the recurring theme of this thesis.
In this chapter, we obtain asymptotic results for the Erlang-R model with blocking in the QED regime (Section \ref{sec:QED_limit_block}).
Following \cite{Jennings2008}, we employ a two-fold QED staffing policy: $s=R_1 +\beta \sqrt{R_1}$ for the number of nurses and $n=R_1/r+\gamma \sqrt{R_1/r}$ for the number of patients in the system (beds), where $\beta$ and $\gamma$ are constants, $R_1$ is the offered load of the servers (nurses) and $r$ is the fraction of time a patient spends in the needy state.
We establish limiting expressions for performance measures, such as the probability of delay and blocking, in the form of explicit functions that depend solely on $\beta$ and $\gamma$.
In deriving these limit results, we use the available product-form solution for the stationary distribution.
Likewise, we pursue QED performance for the Erlang-R model with holding.
However, a direct analytic approach is obstructed by the absence of product-form solutions.
We provide two solutions for establishing QED behavior.
First, we provide stochastic performance bounds that stay meaningful in the QED regime, which demonstrate the non-degenerate behavior of the two-fold scaling in the large-system limit.
Second, we develop a heuristic method that quantifies the difference between the holding model and the blocking model.
This method is to a large extent related to the asymptotic approximation method for retrial queues discussed in Chapter 4, in the sense that we approximate the model with holding through the model with blocking, yet with an increased arrival rate.
The increase in arrival rate turns out to be the solution of a fixed-point equation.
Using our results on the asymptotic behavior of the model with blocking in the QED regime, we then obtain approximative QED performance measures for the model with holding.
These theoretical findings ultimately yield algorithms for dimensioning and time-varying staffing. \\
\\*
\textbf{Structure of the chapter.}
We first review related literature on the subject of staffing in health care environments in Section \ref{sec:ed_literature}.
In Section \ref{sec:modeldescription}, we introduce the mathematical models more formally, and deduce preliminary results on their stability conditions and relative performance.
Section \ref{sec:QED_scaling} describes the scaling regime we use for our asymptotic study of the restricted Erlang-R models, and Sections \ref{sec:QED_limit_block} and \ref{sec:QED_limit_holding} present our main theoretical findings.
We turn to dimensioning problems in Section \ref{sec:dimensioning}, and show how our asymptotic QED results can be used to make resource allocation decisions in realistic settings.
Section \ref{sec:analysis_chapter5} is devoted to the numerical and comparative analysis of the restricted Erlang-R models, and also shows how our method can be applied in time-varying environments through a case study.
We summarize our findings and give directions for future research in Section \ref{sec:conclusion}.
\section{Literature review}
\label{sec:ed_literature}
Due to increasing demand and tightening budgets in health care, there is a growing need for efficient workforce management \citep{Green2008}. Personnel (nurse and physician) expenditure is one of the biggest factors in hospital costs \citep{Kazahaya2005}, and inadequate nursing levels have been mentioned as a significant factor in medical errors and ED overcrowding. In order to establish appropriate nursing levels, a staffing policy requires assessment of a wide range of variables, such as differing nurse expertise and patient acuity during the day. Current methods, such as the minimum nurse-to-patient ratios, are often too inflexible to capture those varying conditions. The American Hospital Association (AHA) and others call for dynamic staffing policies that can deal with the complex and evolving nature of health care \citep{AHA2007}.
Workforce management in health care systems has been studied extensively; see \cite{Denton2013,Hall2006,Hall2012} for overviews.
In recent years it has become apparent that queueing models can be helpful in developing staffing and routing recommendations, not just for large-scale service systems, but also for the small and complicated health care systems.
The first to try such an approach through queueing models were Green et al.~\cite{Green2006,Green2008}, who used the single station stationary Erlang-C model to set staffing levels in EDs and panel sizes for clinics. Using a similar approach, Bekker \& de Bruin~\cite{Bekker2009a} used the Erlang-B model to determine bed allocation for medical wards.
The first to observe the significant impact of interrupted services in a health care setting were Jennings \& de V\'ericourt \cite{Jennings2008,Jennings2011}. Motivated by the need to set nurse-to-patient ratios for internal wards, they considered a closed queueing system with $s$ nurses and $n$ beds. This is essentially the Erlang-C model with the additional restriction that a finite population of the $n$ patients requires care. In their model, all beds are always occupied, and patients alternate between two phases: the needy phase where patients require service of a nurse and the content phase where they do not; see Figure \ref{fig:Jennings}. The system dynamics of the restricted Erlang-R model are equivalent to those of the closed ward model of \cite{Jennings2008} if the holding queue would never be empty.
Campello et al.~\cite{Campello2016} analyzed a similar operational decision, referred to as ED case management, which determines the maximal number of patients a physician should handle in parallel. They also used queueing networks and analyzed the stationary distribution. Note that in practice such a decision is not only affected by operational measurements such as waiting times, but also by psychological constraints that limit physician capability to manage multiple tasks (patients) in parallel.
KC \cite{diwas} provided empirical evidence that physicians should not treat more than 6-7 patients at the same time. Therefore, many hospitals in the US restrict entrance to EDs even if beds are available if physicians are overloaded.
We too consider such constraints, and analyze their impact on performance. We take a different approach than \cite{Campello2016}; instead of analyzing numerically steady-state distributions, we develop many-server approximations that can produce insight into the system dynamics, and can be incorporated into time-varying staffing procedures; see Section \ref{sec:case_study}.
\begin{figure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=1]
\draw [dashed, thick] (-0.5,-0.1) rectangle (3.5,2.85) node[right] {\footnotesize $n$};
\draw [thick,->] (1.1,0.5) -- (0,0.5) -- (0,2) -- (0.6,2);
\draw [thick,->] (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw [thick] (0.6,1.7) -- (1.6,1.7) -- (1.6,2.3) -- (0.6,2.3);
\draw [thick] (1,1.7) -- (1,2.3);
\draw [thick] (1.2,1.7) -- (1.2,2.3);
\draw [thick] (1.4,1.7) -- (1.4,2.3);
\draw (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw (1.5,0.5) circle [radius=0.4] node[above=0.3cm] {\footnotesize $p/\delta$} ;
\draw (2,2) circle [radius=0.4] node[above=0.3cm] {\footnotesize exp($\mu$)};
\node at (1.5,0.5) {\footnotesize $\infty$};
\node at (2,2) {\footnotesize $s$};
\end{tikzpicture}
\caption{The closed ward model.}
\label{fig:Jennings}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\draw [thick, ->] (0,4.5) node[above=0.3cm,right] {\footnotesize Pois$(\lambda)$} -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5];
\draw [thick, ->] (4.75,4.5) -- node[above=0.3cm,right] {\footnotesize $1-p$} (7,4.5);
\draw [thick,->] (5.75,4.5) -- node[right] {\footnotesize $p$} (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5];
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node at (4.25,4.5) {\footnotesize $s$};
\node at (3.5,2) {\footnotesize $\infty$};
\node [above] at (4,5.15) {\footnotesize exp($\mu$)};
\node [above] at (3.5,2.4) {\footnotesize exp($\delta$)};
\end{tikzpicture}
\caption{The Erlang-R model.}
\label{fig:ErlangR}
\end{subfigure}
\caption{Related queueing models.}
\end{figure}
The model in~\cite{Jennings2008,Jennings2011} was developed for modeling internal dynamics within an internal ward. However, in the ED, beds are not constantly occupied and the utilization level depends on the flow of patients that arrive from outside the system.
Yom-Tov \& Mandelbaum \cite{YomTov2014} highlight the interrupted services while accounting for the transient nature of patient's arrival process, and introduced the Erlang-R model as a model for an ED. The Erlang-R model is an open two-station queueing network that has the same layout as the restricted Erlang-R model, except that all patients find a bed available upon arrival, see Figure \ref{fig:ErlangR}. In both models patients experience the interrupted services, but the Erlang-R model has no further restrictions on the bed capacity, hence neglecting the finite-size effects. Yom-Tov \& Mandelbaum \cite{YomTov2014} showed, using a simulator tailored to an Israeli ED, that the complicated small ED dynamics can be captured using the relatively simple Erlang-R model, and hence, its recommendations can be implemented in ED workforce management.
Although the feature of interrupted services is present in many systems, it is particularly important for modeling EDs, because the duration of the interruption is typically much longer than the time patients require care from a nurse. This explains why the Erlang-R model is considered to be the canonical model for EDs. The restricted Erlang-R model with holding/blocking thus extends the Erlang-R model with finite-size constraints which, like interrupted services, are expected to have a decisive impact on performance.
\section{Models and performance measures}
\label{sec:modeldescription}
\subsection{Three-dimensional Markov process}
\label{sec:Markov_process}
Since in the restricted Erlang-R model described above the arrival process is taken Poisson, and all service and content times are assumed independent and exponential, the system can be characterized in terms of a Markov process.
Let $Q(t) = (H(t),Q_1(t),Q_2(t))$ represent the number of patients in the \emph{holding}, \emph{needy} and \emph{content} state at time $t$, respectively.
In both variants, $n$ is the maximum number of patients admitted to system, we have $Q_1(t)+ Q_2(t)\leq n$ for all $t\geq 0$.
Due to the absence of holding patients in the Erlang-R model with blocking, $H(t)=0$ is enforced in this case, whereas $H(t)$ has unbounded support in the model with holding.
This distinction requires us to explore the stationary distribution of the two variants separately.
Before doing so, we introduce some additional notation.
We define
\begin{equation}
R_1 := \frac{\lambda}{(1-p)\mu}, \qquad R_2 := \frac{p\lambda}{(1-p)\delta},
\label{eq:R1_R2}
\end{equation}
where $R_1$ and $R_2$ can be interpreted as the offered workload brought towards the needy queue and the content (infinite-server) queue, respectively.
Furthermore, we define
\begin{equation}
r:= \frac{\delta}{\delta+p\mu},
\label{eq:delta}
\end{equation}
which is the fraction of time a patient spends in the needy state (in case she experienced no wait during her sojourn). \\
\\*
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.9]
\draw [thick] (-1.25,5) -- (0,5) -- (0,4) -- (-1.25,4);
\draw [thick] (0.5,4.5) circle [radius = 0.5] node {\footnotesize 1} node[above=0.5cm] {\footnotesize exp$(\lambda)$}
node[below =0.5cm] {\footnotesize \textit{Station 0}} ;
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (1,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {\footnotesize $s$} node[above=0.5cm] {\footnotesize exp$(\mu)$}
node[below = 0.5cm] {\footnotesize \textit{Station 1}} ;
\draw [thick, ->] (4.75,4.5) -- node[right=0.8cm,above] {\footnotesize $1-p$} (6.5,4.5) -- (6.5,1.6) -- (-2,1.6) -- (-2,4.5) -- (-1.2,4.5);
\draw [thick,->] (5.75,4.5) -- node[left] {\footnotesize $p$} (5.75,2.5) -- (4,2.5);
\draw [thick] (3.5,2.5) circle [radius=0.5] node {\footnotesize $\infty$} node[above=0.4cm] {\footnotesize exp$(\delta)$}
node[below right = 0.35cm] {\footnotesize \textit{Station 2}} ;;
\draw [thick,->] (3,2.5) -- (1.5,2.5) -- (1.5,4.5);
\draw [thick, dashed] (-3,1.2) rectangle (7.25,5.75) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{The Erlang-R model with blocking viewed as a closed Jackson network.}
\label{fig:ErlangR_blocking}
\end{figure}
\noindent
\textbf{Erlang-R model with blocking.}
In case of the blocking model, $Q(t)$ reduces to a finite-state Markov process $Q(t) = (Q_1(t),Q_2(t))$, where $Q_1(t)+Q_2(t)\leq n$ for all $t\geq 0$.
In fact, this is equivalent to the closed Jackson network depicted in Figure \ref{fig:ErlangR_blocking} with finite population $n$.
Station 1 in Figure \ref{fig:ErlangR_blocking} is an $M/M/s$ queue with service rate $\mu$, modeling the number of needy patients $Q_1(t)$.
Station 2 models the number of content patients $Q_2(t)$, and can therefore be represented as an infinite-server queue with service rate $\delta$.
A patient can enter the unit only if $Q_1(t)+Q_2(t)<n$.
Station 0---a single-server queue---moderates this as it only produces output at rate $\lambda$ in case its queue length is positive, i.e.\ if $n-Q_1(t)-Q_2(t)>0$.
Observe that because patients finding a full network are blocked, the number of patients in the system cannot grow beyond $n$.
Hence, the system is stable for all parameter settings, and hence a steady-state distribution exists. Moreover, the simplification of the model with blocking allows us to express the steady-state distribution of the system in explicit product-form.
Let $\pi_b(j,k)$ denote the steady-state probabilities of having $j$ needy and $k$ content patients in the system. Then,
\begin{equation}\label{eq:pih(i,j)}
\pi_b(j,k) = \left\{
\begin{array}{ll}
\pi_0\,\frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k, & ~~~\text{if }j+k \leq n,\\
0, & ~~~\text{else,}
\end{array}\right.
\end{equation}
where
\begin{equation*}
\kappa(j) := \left\{
\begin{array}{ll}
j! , & ~~\text{if }j \leq s,\\
s!\, s^{j-s}, &~~ \text{else,}
\end{array}\right.
\end{equation*}
and $
\pi_0^{-1} = \sum_{j+k\leq n} \frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k$.\\
\\*
\textbf{Erlang-R model with holding.}
\label{ref:modelsoft}
The Erlang-R model with holding does not lead to a Jackson network with an elegant product-form solution for the steady-state distribution, because the holding queue cannot be modeled as a station that is independent from the other queues in the system.
%
However, we are able to describe the system as a two-dimensional Markov process without loss of information.
To see this, define $N:= \{N(t)\}_{t\geq 0}$ with $N(t) := H(t)+Q_1(t) + Q_2(t)$, the total number of patients in the system (including the holding queue).
Using the restriction $Q_1(t)+Q_2(t) \leq n$ together with the fact that no bed is left vacant if a patient is waiting in the holding queue, this yields
\begin{equation*}
H(t) = \left(N(t) - n\right)^+, \quad t\geq 0,
\end{equation*}
where $(\cdot)^+ := \max\{0,\cdot\}$.
For the same reason, $Q_2(t) = N(t) - Q_1(t)$ if $H(t)=0$, and $Q_2(t) = n-Q_1(t)$ otherwise.
In other words,
\begin{equation*}
Q_2(t) = \min\{N(t),n\} - Q_1(t), \quad t \geq 0.
\end{equation*}
Therefore, we can express the state of all three queues in the Erlang-R model with holding using a two-dimensional Markov process $X:= \{X(t)\}_{t\geq 0}$, where
\begin{equation*}
X(t) :=\left( N(t), Q_1(t) \right).
\end{equation*}
The process $X$ lives on the semi-infinite strip
\begin{equation*}
X(t) \in \left\{\,(i,j)\, |\, j \leq \min\{i,n\}, i\in \mathbb{N}_0, j \in \{0,1,\ldots,n\}\, \right\},
\end{equation*}
and belongs to the class of Quasi-Birth-Death (QBD) processes.
The reader is referred to Appendix~\ref{app:QBDdescription} for a detailed description of this process, in terms of its transition diagram and generator matrix.
Contrary to the model with blocking, the system with holding \emph{can} become unstable in case capacity is insufficient to satisfy patient demand.
\begin{proposition}\label{prop:StabilityCondition}
The Erlang-R model with holding is stable if and only if
\begin{equation}
\frac{\lambda}{(1-p)\mu s} < \frac{ \sum_{i=0}^s \frac{i}{s}\binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
{ \sum_{i=0}^s \binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
=: \rho_{\max}(s,n).
\label{eq:StabilityCondition}
\end{equation}
\end{proposition}
The proof is given in Appendix~\ref{app:stability} and follows from the general theory for QBD processes.
Observe that $\rho_{\max}(s,n)$ poses an upper bound on the occupancy level of the servers in the holding model, which is clearly smaller than 1 for all $s$ and $n$.
In addition, this implies that the maximum workload $R_{\max}(s,n) := s\cdot\rho_{\max}(s,n)$ the system is able to handle is strictly less than $s$.
If we compare this to the open Erlang-R model, in which the maximal attainable workload equals $s$, we observe the effect of finite-size constraints on operational performance.
Figure \ref{fig:Rmax} shows the influence of both $s$ and $n$ on the maximum feasible workload in case $r=0.25$.
From these graphs, note that if $s\ll rn$, $R_{\max}$ grows almost linearly with $s$.
Furthermore, $R_{\rm max}(s,n)$ is increasing in $n$ for $s$ fixed.
A logical practical consequence is that a larger number of beds allows for a larger patient volume to enter the ED with the same number of nurses.
Moreover, $R_{\rm max}(s,n)$ is increasing in $s$, but as in Figure \ref{fig:Rmax_a}, adding an extra nurse does not increase the stability region in case $n$ is too tight.
Conversely, adding extra beds does not increase $R_{\rm max}(s,n)$ if the number of nurses does not allow for an increase in offered load, see Figure \ref{fig:Rmax_b}.
Additionally, it is easily verified that $R_{\rm max}(s,n)$ is upper bounded by both $s$ and $R_{\rm max}(n,n) = rn$. Therefore, a careful balance is called for between servers (nurses) and beds, so that resources will be efficiently utilized. We observe that when the ratio $s/n\approx r$, the system is better balanced.
We will propose an appropriate balance between resources by defining a synchronized QED capacity recommendation for both servers and beds in Section \ref{sec:QED_scaling}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $s$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*] table[x=s,y=n20] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\addplot[col3,thick,mark=*] table[x=s,y=n40] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\addplot[col4,thick,mark=*] table[x=s,y=n60] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\addplot[col5,thick,mark=*] table[x=s,y=n80] {Chapter_5/tikz/stability/r025_n_fixed.txt};
\legend{$n=20$,$n=40$,$n=60$,$n=80$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $s$.}
\label{fig:Rmax_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 100,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $n$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*,mark repeat = 2] table[x=n,y=s5] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col3,thick,mark=*,mark repeat = 2] table[x=n,y=s10] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col4,thick,mark=*,mark repeat = 2] table[x=n,y=s15] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\addplot[col5,thick,mark=*,mark repeat = 2] table[x=n,y=s20] {Chapter_5/tikz/stability/r025_s_fixed.txt};
\legend{$s=5$,$s=10$,$s=15$,$s=20$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $n$.}
\label{fig:Rmax_b}
\end{subfigure}
\caption{The maximum achievable workload in the restricted Erlang-R model with holding for $r=0.25$.}
\label{fig:Rmax}
\end{figure}
Provided that the system is stable, the stationary distribution of the QBD process $X$ can be obtained numerically by the matrix geometric method \citep{Neuts1981}.
Subsequently, we can derive the stationary distribution of the original $Q(t)$, denoted by $\pi_h(\cdot,\cdot,\cdot)$.
\subsection{Performance measures}
\label{sec:performance_metrics}
In this work, we concentrate on five performance measures that are central to our analysis.
In the definitions that follow, we present expressions for these measures in terms of a general three-dimensional measure $\pi$, which one can replace by either $\pi_b$ or $\pi_h$, depending on the scenario considered.
In the remainder of this work, we will augment the measures related to the Erlang-R model with blocking and holding by the superscript $b$ and $h$, respectively\footnote{In line with $H(t)=0$, we use $\pi_b(i,j,k) = \pi_b(j,k)$ if $i=0$, with $\pi_b(j,k)$ as in \eqref{eq:pih(i,j)}, and $\pi_b(i,j,k) = 0$ otherwise, when considering the model with blocking.}.
As relevant performance measures, we consider the probability of holding (cq.\\ \noindent blocking) at entering the system, the probability of delay at the needy queue, expected waiting time for a nurse, utilization of nurses and utilization of beds:
\begin{equation}
\mathbb{P}({\rm hold}) = \sum_{i=0}^\iy \sum_{j=0}^n \pi(i,j,n-j), \qquad
\mathbb{P}({\rm delay}) \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \pi(i,j,k),
\label{eq:delay_probability}
\end{equation}
\begin{equation}
\label{eq:EW_exact}
\mathbb{E} [W] \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \frac{\max\{0,j-s+1\}}{\mu}\,\pi(i,j,k),
\end{equation}
\begin{equation}
\label{eq:utilization}
\rho_s = \frac{1}{s}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{j,s\} \pi(i,j,k), \qquad
\rho_n = \frac{1}{n}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{i,n\} \pi(i,j,k).
\end{equation}
It should be stressed that the above expression for the delay probability and the expected waiting time for a nurse are not exact. For the blocking model one can use the Arrival Theorem, see e.g.~\cite{Chen2001a}, whereby the exact expression sums up to $n-1$ instead of $n$.
Since we consider the system as $n\to\infty$, this discrepancy becomes negligible.
For the holding model, a similar argument holds.
We will therefore use the expressions in \eqref{eq:delay_probability}-\eqref{eq:utilization} as definitions for the performance measures.
\subsection{Stochastic bounds}
\label{sec:bounds}
Although the two variants of the Erlang-R model differ with respect to the admission policy, and require different mathematical treatment, we would like to be able to capture their relative performance.
We substantiate the intuition that the holding room leads to more patients in the ED, in the following result.
\begin{proposition}\label{thm:stochasticordering}
Let $Q_1^b$, $Q_2^b$, $Q_1^h$, $Q_2^h$ denote the nurse and content queue length processes in the Erlang-R model with blocking and holding, respectively.
Let $H(0) = 0$, $Q_1^b(0)=Q_1^h(0)$ and $Q_2^b(0)=Q_2^h(0)$. For all $t\geq 0$,
\begin{align}
Q_1^b(t) + Q_2^b(t) &\preceq_{\rm st} Q_1^h(t) + Q_2^h(t) \preceq_{\rm st} n ,\\
Q_2^b(t) &\preceq_{\rm st} Q_2^h(t),\\
Q_1^b(t) &\preceq_{\rm st} Q_1^h(t) + H(t),
\end{align}
where $X\preceq_{\rm st} Y$ implies $\mathbb{P}(X\geq k) \leq \mathbb{P}(Y\geq k)$ for all $k\geq 0$.
\end{proposition}
\noindent
The proof of Proposition \ref{thm:stochasticordering} uses sample path coupling and can be found in Appendix \ref{app:stochastic_ordering}.
Note that as an immediate consequence, we have
\[ \mathbb{P}^b( {\rm block}) = \lim_{t\to\iy} \mathbb{P}\big( Q_1^b(t)+Q_2^b(t) \geq n \big) \leq \lim_{t\to\iy} \mathbb{P}\big( Q_1^h(t) + Q_2^h(t) \geq n \big) = \mathbb{P}^h( {\rm hold }) \]
and by similar reasoning $\rho^b_n \leq \rho_n^h$.
In other words, under similar offered load and capacity constraints, utilization levels for the nurses in the Erlang-R model with blocking are lower than in the Erlang-R model with holding.
Moreover, the total number of waiting patients in the setting with holding is stochastically larger than in the setting with blocking, and in the open Erlang-R model.
We further discuss the differences between both models in Section \ref{sec:dimensioning} and Section \ref{sec:analysis_chapter5}.
\section{Two-fold QED regime}
\label{sec:QED_scaling}
We do not want to waste capacity of either servers or beds without getting significant advantage in terms of performance.
We therefore take an asymptotic approach that lets the external arrival rate $\lambda$ grow to infinity, while scaling $s$ and $n$ accordingly.
In doing so, we intend to establish QED-type system behavior, i.e.\ high occupancy levels of both nurses and beds and good quality-of-service.
\subsection{Two-fold scaling rule}
In order to identify the scaling of $s$ and $n$ as $\lambda\to\infty$, we draw inspiration from the two-fold scaling rule used by Jennings \& de V\'ericourt \cite{Jennings2008} and Khudyakov et al.~\cite{Khudyakov2010}, which follows the celebrated square-root staffing principle.
This principle suggests that, in the most general setting, capacity should be equal to the expected offered load entering the system, let us say $R$, plus an additional variability hedge that is proportional to $\sqrt{R}$.
In the restricted Erlang-R model, we have two capacity sources, namely $s$ and $n$, which experience different relevant amounts of work.
The offered load the servers in the needy queue experience is given by $R_{\rm nurse} = R_1$, as in the regular Erlang-R model;
it does not change due to the finite-size effects, since all patients are served eventually. Hence, we only need to account for the interrupted services. It follows that the appropriate staffing rule for the nurses in the QED regime remains $s=R_1+\beta \sqrt{R_1}$ for some constant $\beta >0$.
To establish the bed capacity level, we need to reflect on the load offered to the beds. Observe that beds remain occupied both in needy and content states. This suggests that $R_{\rm bed} :=R_1+R_2=R_1/r$, with $R_1$ and $R_2$ as in \eqref{eq:R1_R2} and $r$ is the expected fraction of time a patient spends at the nurse station defined in \eqref{eq:delta}.
As a result, the appropriate staffing rule is $n=R_{\rm bed}+\gamma \sqrt{R_{\rm bed}}$ for some constant $\gamma>0$. In conclusion, the two-fold QED scaling rule is given by
\begin{equation}\label{eq:twofoldscaling}
\begin{array}{ll}
s &= R_1 + \beta \sqrt{R_1} + o(\sqrt{R_1}) \\
n &= \frac{R_1}{r}+\gamma \sqrt{\frac{R_1}{r}} + o(\sqrt{R_1})
\end{array}
\end{equation}
with $\beta,\gamma>0$ constants and $R_1:=\lambda/((1-p)\mu)$.
Recall that we saw in Figure \ref{fig:Rmax} that resources seem efficiently utilized if $s/n\approx r$.
Scaling \eqref{eq:twofoldscaling} is in line with this reasoning since
\[
\frac{s}{n} = r\left(1+ \frac{\beta - \gamma\sqrt{r}}{\sqrt{R_1}}+ O(1/R_1) \right) .
\]
\begin{remark}
In \cite{Jennings2008}, a similar scaling regime is considered, which only relates $s$ and $n$ through a square-root scaling, namely the regime $s = r n + \hat\gamma\sqrt{n}$,
which is equivalent to the second relation in \eqref{eq:twofoldscaling} if $\hat\gamma = \beta\sqrt{r} - \gamma r$.
Due to the absence of external arrivals in this closed system, they let the number of beds $n$ approach infinity as opposed to $\lambda$ in our settings.
Nevertheless, this results in the same asymptotic regime.
\end{remark}
Before turning to asymptotic expressions for the performance measures concerning the Erlang-R model with blocking or holding, we conduct a few numerical experiments to confirm that the scaling in \eqref{eq:twofoldscaling} indeed leads to desired QED behavior.
In Figure \ref{fig:sample_paths}, we plotted the sample paths of the three-dimensional queue length process of the holding model in which $\beta$ and $\gamma$ are fixed, and $R_1$ is increased.
Observe that the needy queue length $Q_1(t)$, plotted in orange in Figure \ref{fig:sample_paths}, fluctuates around the values $s$, and stabilizes for larger values of $R_1$.
This naturally implies that the server (nurses) utilization approaches 100\%, while the number of patients waiting is $O(\sqrt{R_1})$.
Furthermore, we see that the percentage of occupied beds also tends to 100\%, while the holding queue length remains small.
The holding queue is of much smaller order than $R_1$, which implies that the holding time of a patient becomes negligible as $R_1\to\iy$.
From these empirical findings we deduce that under scaling \eqref{eq:twofoldscaling} the restricted Erlang-R model exhibits QED behavior on two levels: Outside the facility while waiting for an available bed, and inside the facility while waiting for attention of a nurse.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 28,
ytick = {0,5,10,15,20,25},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\definecolor{col1}{rgb}{0.368417, 0.506779, 0.709798}
\addplot[very thick,col5] file {Chapter_5/tikz/sample_paths/R5_holding.txt};
\addplot[very thick,col2] file {Chapter_5/tikz/sample_paths/R5_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R5_total.txt};
\addplot[very thick,dashed] coordinates {
(0,7)
(200,7)
};
\addplot[very thick,dashed] coordinates {
(0,24)
(200,24)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=5$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 128.333,
ytick = {0,20,40,60,80,100,120},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {Chapter_5//tikz/sample_paths/R25_holding.txt};
\addplot[very thick,col2] file {Chapter_5//tikz/sample_paths/R25_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R25_total.txt};
\addplot[very thick,dashed] coordinates {
(0,30)
(200,30)
};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=25$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 490,
ytick = {0,100,200,300,400},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {Chapter_5/tikz/sample_paths/R100_holding.txt};
\addplot[very thick,col2] file {Chapter_5/tikz/sample_paths/R100_service.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/sample_paths/R100_total.txt};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\addplot[very thick,dashed] coordinates {
(0,420)
(200,420)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=100$}
\end{subfigure}
\caption{Sample paths of $H(t)$ (blue), $Q_1(t)$ (orange) and $Q_1(t)+Q_2(t)$ (green) of the Erlang-R model with holding with parameters $\mu = 1$, $\delta=0.25$, $p=0.75$ and $\beta=\gamma=1$. The staffing levels $s$ and $n$ are depicted by the dashed lines.}
\label{fig:sample_paths}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.7,
ytick = {0,0.1,...,0.7},
xlabel = $\lambda$,
grid = both,
axis line style={->},
axis lines = left,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/delayProbErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/delayProbYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/delayProbJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Delay probability nurse}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.3,
ytick = {0,{0.05},0.1,0.15,0.2,0.25,3},
grid = both,
axis line style={->},
tick label style={/pgf/number format/fixed},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = north east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/EWErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/EWYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/EWJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Expected wait}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.7,
ymax = 1.02,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {Chapter_5/tikz/empirical/rhoErlangH.txt};
\addplot[thick,col4,mark=*] file {Chapter_5/tikz/empirical/rhoYomTov.txt};
\addplot[thick,col5,mark=*] file {Chapter_5/tikz/empirical/rhoJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Nurse utilization}
\end{subfigure}
\caption{Asymptotic behavior of the restricted Erlang-R model with holding and blocking, and the closed ward model for $\mu=1$, $\delta = 0.2$, $p=0.8$ and $\beta=\gamma=0.5$.}
\label{fig:empiricalAsymptotics}
\end{figure}
We also check how the Erlang-R model with blocking or holding and the closed ward model of \cite{Jennings2008} relate under scaling \eqref{eq:twofoldscaling}.
In Figure~\ref{fig:empiricalAsymptotics}, we plot the performance measures, obtained through simulation, for the three models in which we fix $\beta=\gamma=0.5$ and vary the arrival rate $\lambda$.
First, we see that $\mathbb{P}({\rm delay})$ stabilizes as $\lambda\to\iy$ in all three models under scaling \eqref{eq:twofoldscaling}, and the delay probability of the model with holding lies in between the other two.
Second, note that the expected waiting time for a nurse in all models converges to 0 as $\lambda$ increases. In fact, the rate of decay is similar in all three models.
We observe that $\rho_s$ approaches unity in all models, and the rate of convergence seems again comparable.
Finally, and most importantly, we notice an ordering between the three models.
Namely, in all performance measures considered in Figure \ref{fig:empiricalAsymptotics}, Erlang-R with holding appears to be upper bounded by the closed ward and lower bounded by the Erlang-R with blocking.
In a multitude of parameter settings of $(\beta,\gamma)$, we have seen the same ordering, leading to the following conjecture:
\begin{conjecture}\label{conj:stochorder}
Let $Q^b_1(\iy)$, $Q_1^h(\iy)$ and $Q_1^J(\iy)$ denote the stationary number of needy patients in the Erlang-R model with blocking, holding and the closed ward, respectively. Then,
\begin{equation}
Q_1^b(\iy) \preceq_{\rm st} Q_1^h(\iy) \preceq_{\rm st} Q_1^J(\iy).
\end{equation}
\end{conjecture}
Observe that Conjecture \ref{conj:stochorder} poses a stronger statement than the third assertion in Proposition \ref{thm:stochasticordering}.
The latter does give an upper bound to $Q_1^h(\iy)$ in terms of $Q_1^b(\iy)$, albeit supplemented with the stationary holding queue length.
\subsection{QED limits for Erlang-R with blocking}
\label{sec:QED_limit_block}
We now continue our analysis by examining its limiting behavior under scaling \eqref{eq:twofoldscaling}, and obtain QED limits for some performance measures of the Erlang-R model with blocking.
Using the explicit expressions for the blocking model in \eqref{eq:pih(i,j)}, we derive the limiting values of the relevant performance measures defined in Section \ref{sec:performance_metrics} in terms of $\beta$ and $\gamma$.
\begin{theorem}\label{thm:limits_YT}
Let $s$ and $n$ scale as in \eqref{eq:twofoldscaling} with ${-}\infty<\beta<\infty,\,\gamma>0$ as $\lambda\to\infty$. Then, if $\beta \neq 0$,
\begin{align}
g^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay})\nonumber \\
\label{eq:yt_limit_delay}
&=
\left(1 +
\frac{ \beta \, \int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) }
{\varphi(\beta)\Phi(\eta) - \varphi(\sqrt{\beta^2+\eta^2}){\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)}
\right)^{-1},\\
f^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber\\
\label{eq:yt_limit_block}
&=
\frac{
\sqrt{r}\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \varphi(\sqrt{\beta^2+\eta^2})\,{\rm e}^{\frac{1}{2} \omega^2} \Phi(\omega)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},\\
h^b(\beta,\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay}
&=
\frac{
\frac{\varphi(\beta)\Phi(\eta)}{\beta^2} +
\left(\frac{\beta}{r}-\frac{\gamma}{\sqrt{r}}-\frac{1}{\beta}\right)\,\frac{\varphi(\sqrt{\eta^2+\beta^2})}{\beta}\, {\rm e}^{\tfrac{1}{2}\omega^2}\, \Phi(\omega)
- \sqrt{\frac{1-r}{r}}\,\frac{\varphi(\beta)\varphi(\eta)}{\beta}
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},
\end{align}
and if $\beta=0$,
\begin{align}
g^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay}) \nonumber\\
\label{eq:yt_limit_delay_beta0}
&=
\left(1+
\frac{
\int_{-\iy}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t)
}{
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
}
\right)^{-1},\\
f^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber \\
\label{eq:yt_limit_block_beta0}
&=
\frac{
\sqrt{r}\,\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \frac{1}{\sqrt{2\pi}} \Phi(\eta)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
},\\
h_0^b(\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay_beta0}
&= \frac{1}{2\mu}\, \frac{ \left( \gamma^2/r+1\right) \Phi(\eta) + \eta \varphi(\eta) }
{ \frac{r}{1-r} \sqrt{2\pi} \int_{-\infty}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) + \sqrt{\frac{r}{1-r}} \left(\eta \Phi(\eta)+\varphi(\eta)\right)},
\end{align}
where $\eta = \frac{\gamma - \beta\sqrt{r}}{\sqrt{1-r}}$ and $\omega := \frac{\gamma - \beta/\sqrt{r}}{\sqrt{1-r}}$.
\end{theorem}
The proof of Theorem \ref{thm:limits_YT} is given in Appendix C of \cite{YomTov2010} under a parameter transformation.
Theorem \ref{thm:limits_YT} proves that the scaling \eqref{eq:twofoldscaling} results in QED behavior: the probability of waiting in Equations \eqref{eq:yt_limit_delay} and \eqref{eq:yt_limit_delay_beta0} converges to a limit that is strictly between 0 and 1.
Notice that all limits in Theorem \ref{thm:limits_YT} are functions of three parameters: $\beta$ and $\gamma$, which are decision variables, and the fraction of needy time $r$, which is dictated by the physics of the system. Furthermore, the theorem also shows that the probability of blocking (Equations \eqref{eq:yt_limit_block} and \eqref{eq:yt_limit_block_beta0}) is of order $1/\sqrt{R_1}$.
For example, assume that the fraction of needy time $r$ is $0.5$ and the system is large (100 servers).
Using Figure \ref{fig:pdelay_pblock}, we observe that, by choosing the pair $\gamma = 1$ and $\beta = 0.245$, we actually aim at a probability of getting served immediately to be 40\%. At the same time, the probability of getting immediately a bed is 97\%.
Thus, waiting inside the ED occurs at a reasonable level, while wait outside the facility becomes negligible.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
tick label style={/pgf/number format/fixed},
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,0.5)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,0.99)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {Chapter_5/tikz/limit_probabilities_delay.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {Chapter_5/tikz/limit_probabilities_delay.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:pdelay_pblock_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
tick label style={/pgf/number format/fixed},
ylabel = {$f(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,1)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,1.98)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {Chapter_5/tikz/limit_probabilities_block.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {Chapter_5/tikz/limit_probabilities_block.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\label{fig:pdelay_pblock_b}
\end{subfigure}
\caption{Asymptotic delay and scaled blocking probability for $r=0.5$ based on Theorem \ref{thm:limits_YT}. }
\label{fig:pdelay_pblock}
\end{figure}
Theorem \ref{thm:limits_YT} further shows that the expected waiting (Equations \eqref{eq:yt_limit_Edelay} and \eqref{eq:yt_limit_Edelay_beta0})
is of order $1/\sqrt{R_1}$ too and hence vanishes in the large-system limit.
We see from Theorem \ref{thm:limits_YT} that achieving target service levels is always an interplay between $\beta$ and $\gamma$.
Figure \ref{fig:pdelay_pblock_a} shows for instance that in order to keep $\mathbb{P}({\rm delay})\in (0.25,0.75)$, choosing $\gamma=-1$ requires $\beta$ to stay within the range $[-1.4,-0.5]$, while $\gamma=1$ corresponds to values of $\beta$ in $[-0.4,0.5]$.
While the two-fold scaling rule in \eqref{eq:twofoldscaling} automatically captures the right dimensioning ratio as the system scales up, Theorem \ref{thm:limits_YT} shows that the parameters $\beta$ and $\gamma$ provide a means to fine-tune the performance.
Figure \ref{fig:pdelay_pblock_b} confirms how adding nurses, i.e.~increasing $\beta$, does not improve the blocking probability if the number of beds, i.e.~$\gamma$, is too tight.
This is in accordance with our previous observations in Figure \ref{fig:Rmax} for the exact steady-state distribution.
To test the accuracy of the asymptotic results in Theorem \ref{thm:limits_YT} as approximations in a realistic setting, we plot in Figure \ref{fig:accuracy_blocking} the exact probability of delay and blocking for an Erlang-R model with $R=8$ and $r=0.25$, as a function of $s$. The exact probabilities are given by Equation
\eqref{eq:delay_probability}, and their respective asymptotic approximations are based on Theorem \ref{thm:limits_YT}.
Despite the realistic moderate size of the system ($R=8$), we see that the QED approximations are remarkably accurate for many settings $(s,n)$.
This fast relaxation is in line with observations made earlier in the QED literature \cite{Borst2004,Janssen2011}.
\begin{table}[htb]
\centering
\begin{tabular}{|r|rrrr|}
\hline
& $\mu$ & $\delta$ & $p$ & $r$ \\
\hline
Case 1 & 1 & 0.10 & 0.90 & 0.10 \\
Case 2 & 1 & 0.25 & 0.75 & 0.25\\
Case 3 & 1 & 0.50 & 0.50 & 0.50 \\
\hline
\end{tabular}
\caption{Parameter settings for numerical experiments.}
\label{tab:parameter_settings}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1270 & 0.0900 & 0.2283 & 0.1553 & 0.0212 & 0.1085 \bigstrut[t]\\
10 & 0.1340 & 0.0910 & 0.1919 & 0.1628 & 0.0206 & 0.1205 \\
25 & 0.1981 & 0.0945 & 0.1614 & 0.2356 & 0.0216 & 0.2145 \\
50 & 0.1513 & 0.0963 & 0.1588 & 0.1830 & 0.0205 & 0.1496 \\
100 & 0.1880 & 0.0956 & 0.1532 & 0.2231 & 0.0224 & 0.2055 \\
250 & 0.1797 & 0.0971 & 0.1399 & 0.2143 & 0.0219 & 0.2057 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1767} & \textit{0.0981} & \textit{0.1437} & \textit{0.2108} & \textit{0.0217} & \textit{0.1947} \bigstrut\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0237 & 0.0868 & 0.0282 & 0.0322 & 0.0192 & 0.0391 \bigstrut[t]\\
10 & 0.0206 & 0.0872 & 0.0188 & 0.0278 & 0.0183 & 0.0264 \\
25 & 0.0277 & 0.0876 & 0.0123 & 0.0363 & 0.0174 & 0.0174 \\
50 & 0.0185 & 0.0913 & 0.0116 & 0.0249 & 0.0175 & 0.0166 \\
100 & 0.0232 & 0.0888 & 0.0103 & 0.0303 & 0.0183 & 0.0145 \\
250 & 0.0203 & 0.0905 & 0.0079 & 0.0267 & 0.0179 & 0.0109 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0188} & \textit{0.0914} & \textit{0.0084} & \textit{0.0247} & \textit{0.0177} & \textit{0.0118} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 1. The last row presents the asymptotic approximations.}
\label{tab:numerics_case1}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0911 & 0.1538 & 0.0479 & 0.1431 & 0.0345 & 0.0909 \bigstrut[t]\\
10 & 0.1010 & 0.1498 & 0.0560 & 0.1520 & 0.0326 & 0.1025 \\
25 & 0.1594 & 0.1509 & 0.1058 & 0.2192 & 0.0405 & 0.1785 \\
50 & 0.1201 & 0.1506 & 0.0726 & 0.1697 & 0.0381 & 0.1248 \\
100 & 0.1514 & 0.1539 & 0.1001 & 0.2088 & 0.0398 & 0.1704 \\
250 & 0.1459 & 0.1524 & 0.0957 & 0.2003 & 0.0397 & 0.1618 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1429} & \textit{0.1569} & \textit{0.0940} & \textit{0.1976} & \textit{0.0391} & \textit{0.1617} \bigstrut\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0130 & 0.1484 & 0.0044 & 0.0277 & 0.0294 & 0.0109 \bigstrut[t]\\
10 & 0.0121 & 0.1432 & 0.0042 & 0.0244 & 0.0267 & 0.0098 \\
25 & 0.0182 & 0.1383 & 0.0070 & 0.0319 & 0.0295 & 0.0141 \\
50 & 0.0119 & 0.1415 & 0.0043 & 0.0216 & 0.0301 & 0.0090 \\
100 & 0.0154 & 0.1413 & 0.0059 & 0.0270 & 0.0290 & 0.0119 \\
250 & 0.0136 & 0.1403 & 0.0051 & 0.0236 & 0.0291 & 0.0103 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0126} & \textit{0.1445} & \textit{0.0048} & \textit{0.0220} & \textit{0.0284} & \textit{0.0097} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 2. The last row presents the asymptotic approximations.}
\label{tab:numerics_case2}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0547 & 0.1945 & 0.0221 & 0.1181 & 0.0604 & 0.0617 \bigstrut[t]\\
10 & 0.0579 & 0.2158 & 0.0237 & 0.1325 & 0.0526 & 0.0746 \\
25 & 0.1113 & 0.2086 & 0.0544 & 0.1959 & 0.0641 & 0.1311 \\
50 & 0.0813 & 0.2050 & 0.0363 & 0.1523 & 0.0562 & 0.0933 \\
100 & 0.1060 & 0.2146 & 0.0509 & 0.1873 & 0.0632 & 0.1250 \\
250 & 0.1006 & 0.2179 & 0.0475 & 0.1820 & 0.0596 & 0.1214 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1011} & \textit{0.2185} & \textit{0.0478} & \textit{0.1792}& \textit{0.0605} & \textit{0.1199} \bigstrut\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0034 & 0.1888 & 0.0009 & 0.0175 & 0.0510 & 0.0057 \bigstrut[t]\\
10 & 0.0030 & 0.2093 & 0.0008 & 0.0172 & 0.0416 & 0.0058 \\
25 & 0.0070 & 0.1937 & 0.0020 & 0.0243 & 0.0440 & 0.0089 \\
50 & 0.0043 & 0.1946 & 0.0011 & 0.0163 & 0.0414 & 0.0056 \\
100 & 0.0061 & 0.1999 & 0.0017 & 0.0207 & 0.0431 & 0.0076 \\
250 & 0.0052 & 0.2037 & 0.0014 & 0.0185 & 0.0401 & 0.0067 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0052} & \textit{0.2039} & \textit{0.0014} & \textit{0.0173} & \textit{0.0404} & \textit{0.0063} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 3. . The last row presents the asymptotic approximations.}
\label{tab:numerics_case3}
\end{table}
We furthermore compare the asymptotic delay and blocking probability in the three scenarios given in Table \ref{tab:parameter_settings}.
In Tables \ref{tab:numerics_case1}--\ref{tab:numerics_case3} we compute the exact probabilities of delay and blocking through the explicit forms in \eqref{eq:delay_probability} for increasing values of the offered load, $R_1$.
The numerical results show that $g^b(\beta,\gamma)$, $f^b(\beta,\gamma)$ and $h^b(\beta,\gamma)$ provide accurate approximations to $\mathbb{P}({\rm delay})$, $\sqrt{R_1}\mathbb{P}({\rm block})$ and $\sqrt{R_1}\,\mathbb{E}[W]$ in pre-limit systems.
The quality of the approximations increases with $R_1$.
Naturally, fluctuations occur for relatively small values of $R_1$, because $s$ and $n$ need to be rounded to an integer.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {Chapter_5/tikz/accuracy/accuracy_pdelay_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {Chapter_5/tikz/accuracy/accuracy_pblock_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\end{subfigure}
\caption{Comparison of exact performance measures (solid) against asymptotic approximations (dashed) with $\beta=(s-R_1)/\sqrt{R_1}$ and $\gamma=(n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_blocking}
\end{figure}
\subsection{QED limits for Erlang-R with holding}
\label{sec:QED_limit_holding}
As explained in Section \ref{sec:QED_scaling}, the model with holding has no product-form steady-state distribution, which makes it hard (if not impossible) to obtain QED limits.
Instead, we derive QED approximations by exploiting a connection with the blocking model.
We first prove that under scaling \eqref{eq:twofoldscaling}, the upper bound on the utilization level of the nurses needed to achieve stability in the model with holding, as given in Proposition \ref{prop:StabilityCondition}, converges to unity as $R\to\infty$.
This facilitates high utilization levels of both nurses and beds, a key characteristic of the QED regime.
\begin{proposition}\label{prop:stability_convergence}
Let $s$ and $n$ scale with $R_1\to\infty$ as in \eqref{eq:twofoldscaling}. Then for $\lambda\to\infty$,
\[
\rho_{\max}(s,n) \to 1.
\]
\end{proposition}
The proof can be found in Appendix \ref{app:proof_stability_convergence}.
Combining Proposition \ref{prop:stability_convergence} with Proposition \ref{prop:StabilityCondition} shows that indeed the scaling we use results in a highly utilized system.
As observed before, the nature of the two variants of the model is similar up to the fact that a fraction of the patients is deferred on arrival in the setting with blocking, whereas all the arriving patients are eventually admitted into the system in the holding model.
This implies that, given $s$ and $n$, the nurses face an increased workload in case of a holding room.
In fact, Theorem \ref{thm:limits_YT} shows that the blocking probability is of order $1/\sqrt{R_1}$, yielding a volume of blocked patients of order $\sqrt{R_1}$ in setting with blocking.
Accordingly, if $R^b = R_1$ and $R^h$ denote the nominal load arriving to the nurses in the model with blocking and holding, respectively, we can argue that
\[R^h = R^b + \alpha \sqrt{R^b} + o(\sqrt{R^b}),\]
for some $\alpha>0$.
Notice that this additional load is of the same order as the safety staffing in the blocking model staffing rule \eqref{eq:twofoldscaling}.
As $s$ and $n$ remain unchanged, we rewrite \eqref{eq:twofoldscaling} in terms of $R^h$,
\begin{align}
s &= R^h + (\beta-\alpha)\sqrt{R^h} + o(\sqrt{R^h}), \nonumber \\
n &= \frac{R^h}{r} + \left(\gamma-\alpha/\sqrt{r}\right)\sqrt{\frac{R^h}{r}} + o(\sqrt{R^h}),
\label{eq:fixed_point_scaling}
\end{align}
where we have used $R^b = O(R^h)$.
Observe that the square-root principle prevails also after this substitution, albeit with different hedging parameters.
We therefore heuristically argue that the holding model under scaling \eqref{eq:twofoldscaling} with parameters $\beta$ and $\gamma$ mimics the blocking model with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, respectively.
Observe, however, that we have not yet specified the value of $\alpha$.
By definition, $\alpha\sqrt{R^b}$ is the expected volume of patients that would be rejected in the model with blocking, that is, $R^h$ times the probability of not being admitted to the ED directly.
By the construction in \eqref{eq:fixed_point_scaling}, this volume asymptotically equals $R^h \cdot \mathbb{P}^b({\rm block})$, with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, which by Theorem \ref{thm:limits_YT} is approximated by
\[f^b\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right) / \sqrt{R^h}\]
as $R^h$ grows large.
In conclusion, $\alpha$ is characterized as the solution of the fixed-point equation
\begin{equation}
\label{eq:fixedpoint}
\alpha = f^h\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right),
\end{equation}
and as a result, we are able to approximate the nurse delay probability in the Erlang-R model with holding as
\begin{equation}
\mathbb{P}^h({\rm delay}) \approx g^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: g^h(\beta,\gamma).
\label{eq:fixed_point_Pwait}
\end{equation}
Likewise, the scaled mean waiting time for a nurse can be approximated by
\begin{equation}
\sqrt{R_1} \cdot \mathbb{E}[W] \approx h^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: h^h(\beta,\gamma).
\label{eq:fixed_point_Ewait}
\end{equation}
This also implies that the holding queue is $O(\sqrt{R_1})$.
Subsequently, we argue that the expected holding time (pre-entering wait) under the QED policy is $O(1/\sqrt{R_1})$ and hence asymptotically negligible.
We justify this claim numerically in Section \ref{sec:analysis_chapter5}.
\begin{remark}
\label{rem:holding_limit}
Notice that in the reasoning leading to \eqref{eq:fixedpoint}, we implicitly assumed that the additional volume $\alpha\sqrt{R^b}$ is an independent Poisson process, which is obviously not the case. Therefore, \eqref{eq:fixed_point_Pwait}-\eqref{eq:fixed_point_Ewait} are approximations for pre-limit systems that are not asymptotically correct as $R_1\to\iy$.
Nevertheless, the heuristic approach seems to performs well as we confirm numerically next.
\end{remark}
In Figure \ref{fig:accuracy_holding}, we repeat the numerical experiments of Figure \ref{fig:accuracy_blocking} for the model with holding.
Since the heuristic does not provide an approximation for the holding probability, Figure \ref{fig:accuracy_holding_b} only plots the simulated holding probabilities.
Those are provided to better understand the implication of operational decisions.
Recall that the holding system is only stable (i.e. $\mathbb{P}({\rm hold})<1$) if both $s>R_1=8$ and $n > R_1/r = 32$.
We nevertheless included the boundary case $n=32$ for illustrative purposes.
The graphs in Figure \ref{fig:accuracy_holding} show that the heuristic captures the congestion levels well, even for this moderate-size system.
To see how this heuristic approach performs under different settings, and particularly if $R_1\to \infty$, we again compare the approximated delay probability in the Erlang-R model with holding as solution of the fixed-point procedure to the outcomes of simulation experiments for the three scenarios in Table \ref{tab:parameter_settings}.
We performed 100 runs of length $10^4$ for each parameter setting and all values of $R$, yielding the results presented in Tables \ref{tab:heuristic_case1}--\ref{tab:heuristic_case3}, which are accurate up to a 95\% confidence interval of width $10^{-3}$.
\begin{table}[h] \centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1532 & 0.1031 & 0.1628 & 0.1216 \bigstrut[t]\\
10 & 0.1622 & 0.1272 & 0.1697 & 0.1331 \\
25 & 0.2340 & 0.2116 & 0.2413 & 0.2342 \\
50 & 0.1817 & 0.1468 & 0.1890 & 0.1678 \\
100 & 0.2199 & 0.1931 & 0.2304 & 0.2269 \\
250 & 0.2110 & 0.1852 & 0.2176 & 0.2230 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.2076} & \textit{0.1777} & \textit{0.2187} & \textit{0.2050} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0310 & 0.0121 & 0.0344 & 0.0148 \bigstrut[t]\\
10 & 0.0267 & 0.0123 & 0.0292 & 0.0128 \\
25 & 0.0348 & 0.0171 & 0.0373 & 0.0184 \\
50 & 0.0240 & 0.0108 & 0.0258 & 0.0125 \\
100 & 0.0293 & 0.0143 & 0.0317 & 0.0163 \\
250 & 0.0256 & 0.0120 & 0.0276 & 0.0145 \\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0229} & \textit{0.0104} & \textit{0.0257} & \textit{0.0124} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 1. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case1}
\end{table}
\begin{table}[h]\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1327 & 0.0740 & 0.1620 & 0.1096 \bigstrut[t]\\
10 & 0.1446 & 0.0894 & 0.1683 & 0.1207 \\
25 & 0.2204 & 0.1631 & 0.2442 & 0.2203 \\
50 & 0.1694 & 0.1122 & 0.1888 & 0.1507 \\
100 & 0.2098 & 0.1524 & 0.2322 & 0.2111 \\
250 & 0.2033 & 0.1534 & 0.2190 & 0.1979 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1840} & \textit{0.1277} & \textit{0.2109} & \textit{0.1759} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0219 & 0.0079 & 0.0322 & 0.0137 \bigstrut[t]\\
10 & 0.0199 & 0.0073 & 0.0284 & 0.0115 \\
25 & 0.0283 & 0.0128 & 0.0375 & 0.0163 \\
50 & 0.0190 & 0.0078 & 0.0255 & 0.0107 \\
100 & 0.0244 & 0.0097 & 0.0314 & 0.0151 \\
250 & 0.0214 & 0.0083 & 0.0272 & 0.0134 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0169} & \textit{0.0066} & \textit{0.0234} & \textit{0.0104} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 2. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0977 & 0.0413 & 0.1521 & 0.0851 \bigstrut[t]\\
10 & 0.1070 & 0.0469 & 0.1648 & 0.1028 \\
25 & 0.1926 & 0.1076 & 0.2421 & 0.1874 \\
50 & 0.1431 & 0.0727 & 0.1876 & 0.1342 \\
100 & 0.1855 & 0.1012 & 0.2282 & 0.1714 \\
250 & 0.1775 & 0.0963 & 0.2217 & 0.1765 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1442} & \textit{0.0711} & \textit{0.1981} & \textit{0.1354} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0072 & 0.0019 & 0.0250 & 0.0081 \bigstrut[t]\\
10 & 0.0067 & 0.0018 & 0.0235 & 0.0082 \\
25 & 0.0148 & 0.0043 & 0.0325 & 0.0133 \\
50 & 0.0092 & 0.0025 & 0.0217 & 0.0081 \\
100 & 0.0132 & 0.0038 & 0.0277 & 0.0105 \\
250 & 0.0114 & 0.0033 & 0.0246 & 0.0099 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0078} & \textit{0.0022} & \textit{0.0188} & \textit{0.0069} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 3. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case3}
\end{table}
We conclude from these tables that the approximation is good. As $R$ increases, the simulated values move closer to the heuristic approximation. Extensive numerical experiments suggest that load is slightly underestimated in the limit.
The best results in terms of accuracy are attained for small $r$.
This suggests that the quality of the heuristic method improves as $r$ gets smaller.
These are exactly the parameter settings for which this model is relevant.
\begin{remark}
The approximation technique that evolves around the fixed-point\\ \noindent method can be adapted to accommodate balking behavior of external arrivals. If we assume that an arriving patient finding all beds occupied leaves the system instantly with probability $1-q$, for some $q\in(0,1)$, independently of the rest of the arrivals, with the same argumentation, the volume of arrivals blocked is still $\alpha\sqrt{R_1}$, while the volume that will enter the ED eventually is $q\cdot\alpha\sqrt{R_1}$. Therefore, we may argue that in the QED regime, the system with holding and balking behaves as the system with blocking but with corrected parameters $(\beta-q\alpha,\gamma-q\alpha/\sqrt{r})$, where $\alpha$ satisfies
\begin{equation}
\alpha = f^b(\beta-q\alpha,\gamma-q\alpha/\sqrt{r}).
\end{equation}
Note that the choice of $q$ interpolates between the two system variants with holding ($q=0$) and blocking ($q=1$).
\end{remark}
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx32] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx36] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx40] {Chapter_5/tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:accuracy_holding_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {Chapter_5/tikz/accuracy/accuracy_holding_probability.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Holding probability}
\label{fig:accuracy_holding_b}
\end{subfigure}
\caption{Comparison of simulated delay probability (solid) against asymptotic approximations (dashed) with $\beta = (s-R_1)/\sqrt{R_1}$ and $\gamma = (n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_holding}
\end{figure}
\section{Dimensioning}
\label{sec:dimensioning}
We will now use the accurate asymptotic approximations of the previous section to define a procedure that determines resource capacity in the restricted Erlang-R models.
That is, we aim to set the number of nurses $s$ and the number of beds $n$, such that a preset performance level is achieved.
We take the probability of delay at the needy queue and the probability of blocking/holding at the pre-entrant queue as the target performance objectives.
\subsection{Capacity setting for Erlang-R with blocking}
\label{sec:dimensioning_block}
In the setting with blocking, we can readily use the asymptotic results of Theorem \ref{thm:limits_YT} to (numerically) find a pair of parameters $(\beta^*,\gamma^*)$ to meet the performance requirements.
For instance, given that we want the delay probability to be at most $\varepsilon$, we first solve the equation $g^b(\beta^*,\gamma^*)=\varepsilon$ and then assign $s = \lceil R_1 + \beta^*\sqrt{R_1}\rceil$ and $n = \lceil R_1/r+\gamma^*\sqrt{R_1/r}\rceil$. Note that there could be multiple solutions to that problem, i.e.\ there could be multiple combinations of number of beds and number of nurses that can result in the same value of a single performance level.
The ED manager can ultimately decide which of these feasible solutions fits the environment best, for instance taking into account space and cost constraints.
We illustrate the resource allocation decisions in an MU setting, using data originated from two articles: \cite{LS2001} and \cite{GY2011}. Green \& Yankovic describe an MU that has 42 beds, with average occupancy level 78\%, and Average Length of Stay (ALOS) 4.3 days. Lundgren \& Segesten studied nurses' service times in a medical-surgical ward. They found that the average service time in their unit was 15.3 minutes per service, and that the average demand rate for each patient is 0.38 requests per hour. Therefore, we take an average service time of 15 minutes and assume that there are 0.4 requests per hour from each patient. Fitting this data to our model results in the following parameters values: $\lambda =0.32, \mu =4, \delta =0.4$, $p=0.975$ and the fraction of needy time is then approximately $r=0.09$.
This yields nominal offered load $R_1 = 3.2$ and $R_1/r = 34.4$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: -1.9,0.05)},anchor = south west},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking1.txt};
\draw[->,col1,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: -0.0552366,0.5) -- (axis cs: -0.0552366,0);
\draw[->,col2,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.179728,0.5) -- (axis cs: 0.179728,0);
\draw[->,col3,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.359034,0.5) -- (axis cs: 0.359034,0);
\draw[->,col4,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.459825,0.5) -- (axis cs: 0.459825,0);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma=1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:ratio01_delay}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$f(\beta,\gamma)/\sqrt{R_1}$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {Chapter_5/tikz/staffing_example/staffing_example_with_blocking2.txt};
\draw[very thick, col1,dashed,->] (axis cs: -0.0552,0) -- (axis cs: -0.0552,0.292798) -- (axis cs: -2,0.292798);
\draw[very thick,col2,dashed,->] (axis cs: 0.179728,0) -- (axis cs: 0.179728,0.164903) -- (axis cs: -2,0.164903);
\draw[very thick,col3,dashed,->] (axis cs: 0.359034,0) -- (axis cs: 0.359034,0.0705547) -- (axis cs: -2,0.0705547);
\draw[very thick,col4,dashed,->] (axis cs: 0.459825,0) -- (axis cs: 0.459825,0.0207909) -- (axis cs: -2,0.0207909);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma= 1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Blocking probability}
\label{fig:ratio01_block}
\end{subfigure}
\caption{Approximate performance of restricted Erlang-R with blocking for $r \approx 0.09$ and $R_1 = 3.2$, as functions of $\beta$.}
\label{fig:ratio01}
\end{figure}
Figure \ref{fig:ratio01} visualizes the dimensioning procedure for this particular MU.
The hospital management can find a pair of $n$ and $s$ to meet certain criteria, for example to achieve target delay probability $\varepsilon = 0.5$ with reasonable blocking probability.
Figure \ref{fig:ratio01}a indicates that this target can be achieved by a variety of pairs, for instance $(\beta_1,\gamma_1) = (-0.06,-1)$, $(\beta_2,\gamma_2) = (0.16,0)$, $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$, among infinitely many others.
According to Figure \ref{fig:ratio01}b, the pairs named above lead to blocking probabilities 0.293, 0.165, 0.071 and 0.021, respectively.
If the manager decides that probability of blocking of more than 10 percent is not acceptable, this leaves the choices $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$ as candidate parameter pairs.
Using the two-fold square-root staffing rule $s_i = \lceil R_1 + \beta_i \sqrt{R_1}\rceil$ and $n_i = [R_1/r + \gamma_i\sqrt{R_1/r}]$, this yields feasible staffing levels $(s_3,n_3) = (4,40)$ and $(s_4,n_4)=(5,46)$.
The ultimate decision to apply any of these solutions can be based on external factors, such as operational costs or space limitations on the number of beds.
\subsection{Capacity setting for Erlang-R with holding}
For the holding model, we need a more sophisticated approach, exploiting the asymptotic approximation with the fixed-point equation in \eqref{eq:fixedpoint}. We propose the following dimensioning procedure to achieve a preset target delay probability at the needy queue.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Target delay probability $\varepsilon$. Parameters $\lambda,\mu,\delta$ and $p$.}
\hspace{1.1cm}\KwOut{Staffing levels $s$ and $n$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}
\item[] \hspace{0.5cm} 1. Set $R_1:= \frac{\lambda}{(1-p)\mu}$ and $r = \frac{\delta}{\delta+p\mu}$.
\item[] \hspace{0.5cm} 2. Determine parameters $(\beta^*,\gamma^*)$ such that $g^b(\beta^*,\gamma^*) = \varepsilon$.
\item[] \hspace{0.5cm} 3. Set $\beta = \beta^* + f^b(\beta^*,\gamma^*)$ and $\gamma = \gamma^* + f^b(\beta^*,\gamma^*)/\sqrt{r}$.
\item[] \hspace{0.5cm} 4. Return $s = \left\lceil R_1 + \beta\sqrt{R_1}\right\rceil$ and $n = \left\lfloor R_1/r + \gamma \sqrt{R_1/r}\right\rfloor$.
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Stationary dimensioning algorithm for ED with holding.}
\label{alg:stationarydimensioning}
\end{algorithm}
\begin{remark}\label{rem:upperboundHW}
In Step 2 of Algorithm \ref{alg:stationarydimensioning} infinitely many pairs $(\beta^*,\gamma^*)$ satisfy the delay probability equation.
For practical purposes, it is convenient to fix either $\beta^*$ or $\gamma^*$ beforehand, and then solve $g^b(\beta^*,\gamma^*) = \varepsilon$ for the remaining unknown.
The preset value should however be chosen with care, since $g^b(\beta^*,\gamma^*)$ is upper bounded by the Halfin-Whitt delay probability formula
\[g_{\rm HW}(\beta^*) = \left( 1 + \frac{\beta^* \Phi(\beta^*)}{\varphi(\beta^*)}\right)^{-1}.\]
Hence, if $\varepsilon > g_{\rm HW}(\beta^*)$, then no feasible solution to $g^b(\beta^*,\gamma^*)=\varepsilon$ exists.
This should be considered when choosing $\beta^*$.
Furthermore, it is evident from Step 3 that the final values $(\beta,\gamma)$ are always larger than $(\beta^*,\gamma^*)$.
\end{remark}
We now use the same example as in Section \ref{sec:dimensioning_block} to demonstrate capacity allocation decisions for the model with holding. This can be viewed as the additional capacity the medical unit needs in terms of nurses and beds, in order to account for the fact that patients are waiting in the ED to be admitted instead of being blocked and transferred to a less preferred unit.
Observe that the holding model leaves less flexibility for management in choosing system parameters due to stability constraints. For example, the policy with $n=30$ ($\gamma=-0.75$) is infeasible in the holding model.
For similar reasons, only nurse staffing levels with $\beta>0$, or $s > R_1=3.2$ are feasible.
Targeting a delay probability of $0.5$ with $n=40$, Figure \ref{fig:ratio01_hold} shows that operating a MU with holding room requires $\beta = 0.475$ or $s=5$.
Recall that under the blocking policy, only $s=4$ nurses were needed to achieve a delay probability of $0.5$.
This example hence shows how the managerial decision to have a holding room, rather than deferring patients to less preferred medical units, requires additional workforce in that unit (as well as the ED).
This example also shows that the facility with holding room is able to treat fewer patients simultaneously than under blocking constraints, in line with the bounds in Section \ref{sec:bounds} and Conjecture \ref{conj:stochorder}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g^h(\beta,\gamma)$},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col5] table[x=beta,y=delay_n35] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col4] table[x=beta,y=delay_n40] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col1] table[x=beta,y=delay_n45] {Chapter_5/tikz/staffing_example/staffing_example_with_holding1.txt};
\draw[very thick,col4,dashed,->] (axis cs: -2,0.5) -- (axis cs: 0.475,0.5) -- (axis cs: 0.475,0);
\legend{$\gamma = -0.75$, $\gamma =0.102$,$\gamma= 0.955$, $\gamma=1.807$};
\end{axis}
\end{tikzpicture}
\caption{Approximate delay probability of restricted Erlang-R system with holding for $r\approx 0.09$ and $R_1=3.2$ }
\label{fig:ratio01_hold}
\end{figure}
\section{Model analysis and managerial implications}
\label{sec:analysis_chapter5}
In this section, we use the analysis and algorithms developed in earlier sections to gain insights into the importance of the capacity restrictions and patient returns in a restricted Erlang-R system by drawing a comparison to related models studied in the literature.
\subsection{The influence of patient returns or the role of $r$}
Here we study how the parameter $r$ affects the service level in the restricted Erlang-R model with blocking, on the basis of the asymptotic expressions in Theorem \ref{thm:limits_YT}.
To better understand the connection with the single-station model and the importance of returns we examine the role of $r$.
Recall the interpretation of $r$ as the fraction of time a patient is needy during his stay within the system in the idealized scenario with infinite capacity, i.e. for $r\in(0,1)$.
The case $r=1$ corresponds to the setting in which patients are needy all the time, in this case patients get service in one time.
When $r=1$ the infinite-server queue, describing the number of content patients, disappears from the queueing system and we end up with a standard loss model---$M/M/s/n$ queue---in which capacity is scaled as
\[ s = R_1+\beta\sqrt{R_1}, \qquad n = R_1+\gamma\sqrt{R_1}. \]
This staffing rule only makes sense in case $\beta<\gamma$, since no delay is experienced if $n\leq s$.
If indeed $\gamma>\beta$, then the asymptotic delay probability and scaled blocking probability are given by \cite{masseywallace},
\begin{align*}
g_B(\beta,\gamma) &= \frac{1-{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}, \\
f_B(\beta,\gamma) &= \frac{\beta{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}.
\end{align*}
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.8,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,1.28)},anchor= north east},
yscale = 0.75
]
\addplot[thick,col1] file {Chapter_5/tikz/influence_r/PdelayB_g1_b025.txt};
\addplot[thick,col3] file {Chapter_5/tikz/influence_r/PdelayB_g1_b05.txt};
\addplot[thick,col4] file {Chapter_5/tikz/influence_r/PdelayB_g1_b1.txt};
\addplot[thick,col5] file {Chapter_5/tikz/influence_r/PdelayB_g1_b2.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability $g^b(\beta,\gamma)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.4,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,0.05)},anchor= south east},
yscale = 0.75
]
\addplot[thick,col1] file {Chapter_5/tikz/influence_r/PblockB_g1_b025.txt};
\addplot[thick,col3] file {Chapter_5/tikz/influence_r/PblockB_g1_b05.txt};
\addplot[thick,col4] file {Chapter_5/tikz/influence_r/PblockB_g1_b1.txt};
\addplot[thick,col5] file {Chapter_5/tikz/influence_r/PblockB_g1_b2.txt};
\addplot[thick,dashed] file {Chapter_5/tikz/influence_r/PblockB_g1_inf.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability $f^b(\beta,\gamma)$.}
\label{fig:influence_of_r_b}
\end{subfigure}
\caption{Asymptotic performance measures as a function of $r$ in the restricted Erlang-R model with blocking for $\gamma=1$.}
\label{fig:influence_of_r}
\end{figure}
We can see that $f^b(\beta,\gamma)$ for increasing $\beta$ approaches a lower bound that is a function of $r$.
To understand this, observe that as $\beta$ grows, delays at the nurse queue vanish.
Then the sojourn time of an admitted patient only consists of a geometric number of needy and content periods with mean $(1/\mu+p/\delta)/(1-p) = 1/(r\mu(1-p))$.
The blocking model can in this case be modeled as an $M/G/n/n$ queue, with offered load $\lambda/(r\mu(1-p)) =R_1/r$, in which the scaled blocking probability is known to be, see \cite{Avram2013},
\[\sqrt{R_1} \, \mathbb{P}({\rm block}) = \sqrt{R_1} \, \frac{(R_1/r)^n/n!}{\sum_{k=0}^n (R_1/r)^k / k!} \to \sqrt{r} \, \frac{\varphi(\gamma)}{\Phi(\gamma)},\]
as $R_1\to\infty$.
This function of $r$ is plotted in Figure \ref{fig:influence_of_r_b} as the dashed line.
We observe that in general the probability of blocking increases with $r$, regardless of the capacity constraints on the needy station.
We can explain this by observing that $r$ influences only $n$ in the QED staffing rule. When $n$ reduces, more patients are blocked. Therefore, if patients spend relatively more time in needy state, which usually indicates services that are less interrupted, blocking will increase. Delays, on the other hand, will decrease in such situations---the minimal delay possible can be achieved if service is given in one time ($r=1$). Returns or interruptions increase delays significantly under QED staffing.
\subsection{Comparing restricted and unrestricted Erlang-R models}
Given the expressions for the asymptotic delay probability in the open Erlang-R model, and its restricted versions with blocking and holding, we compare the three policies for various values of $\beta$, $\gamma$ and $r$.
Figure \ref{fig:comparison_delay} plots the delay probability for blocking ($g^b(\beta,\gamma)$), holding ($g^h(\beta,\gamma)$) and Erlang-R ($g_{\rm HW}(\beta)$) models, as functions of $\gamma$, while keeping $\beta$ fixed, for three values of $r$.
We make a couple of observations.
Notice that
\[ g^b(\beta,\gamma) \leq g^h(\beta,\gamma) \leq g_{\rm HW}(\beta) \]
for all $\beta,\gamma>0$ and $r$.
In that sense, the holding model is an interpolation between the blocking and the open model.
As expected, the delay probabilities in the restricted models converge to those of the open Erlang-R model, because increasing $\gamma$ is tantamount to lifting the stringent constraints on the system size. Note that the rate of conversion is fast---one can provide probability of waiting close to that of the open model with small values of $\gamma$. Indeed, the fact that when using QED staffing not much of excessive delay results from the beds restriction is important by itself.
Also, we observe that the difference between delay probabilities increases with $r$.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r01_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r01_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r01_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r01_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.1$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r025_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r025_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r025_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r025_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.25$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {Chapter_5/tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b01.txt};
\addplot[thick,col5] file {Chapter_5/tikz/comparison/PdelayH_r05_b01.txt};
\addplot[thick,col2,dashed] file {Chapter_5/tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b05.txt};
\addplot[thick,col2] file {Chapter_5/tikz/comparison/PdelayH_r05_b05.txt};
\addplot[thick,col4,dashed] file {Chapter_5/tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {Chapter_5/tikz/comparison/PdelayB_r05_b1.txt};
\addplot[thick,col4] file {Chapter_5/tikz/comparison/PdelayH_r05_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.5$.}
\end{subfigure}
\caption{Asymptotic delay probability in open Erlang-R (dashed), restricted Erlang-R with blocking (marked) and restricted Erlang-R with holding (solid), as function of $\gamma$ with $\beta=0.1$ (blue), $\beta=0.5$ (orange) and $\beta=1$ (green) fixed.}
\label{fig:comparison_delay}
\vspace{3mm}
\end{figure}
\subsection{The impact of visit number}
\label{subsec:num_visit}
We next reflect on the impact of operational capacity decisions on different patient populations. We measure patient's complexity by the number of times she needs to see the nurse or the physician during her stay. In the ED context, simple-to-treat patients will need to see the physician once, while complex ones will need multiple visits. Hence, we divide the patients into complexity groups by the number of visits in the Needy station. Since the number of visits is geometrically distributed, we have a higher proportion of simple patients than complex ones; that fits well the health care environment.
Figure \ref{fig:wait_by_visit} shows the waiting time in the needy and pre-entering queues, and the total waiting time, as a function of $n$ (number of beds), for each complexity group.
Obviously, the expected waiting time in the pre-entering queue decreases with $n$, while the needy waiting time increases.
For patients who require a relative large number of visits of the physician, in this case more than 6, the total needy wait is the dominant part of the total waiting time. Therefore, as $n$ grows, the total waiting time first decreases and then increases.
In fact, Figure \ref{fig:wait_by_visit_b} suggests that there is an optimal number of beds $n$ that minimizes the total wait for each complexity type.
Thus, size restrictions reduce the length-of-stay of patients with complex health conditions (given that the constraint is not too tight).
On the other hand, this figure also shows that no such $n$ exists for patients who only require little assistance.
Hence, there is no $n$ that improves the sojourn time of all patients in the ED simultaneously.
This leaves the decision to the hospital manager to weigh the importance of patients of different complexity levels.
\begin{remark}
From a different perspective, note that in queueing systems such as communication systems, the partitioning of a job to sizable quantities and scheduling those jobs in a similar dynamic to the Erlang-R model became a popular way for increasing throughput. This is because this effectively schedules jobs by their size even though the total job requirements are uncertain. This in fact creates a shortest-job-first policy without prior knowledge of job size \citep{Comte2016}. Considering that perspective we note that the Erlang-R model actually prioritizes simple jobs over complex ones. But without restrictions, when load is too high, such procedures may lead to very long LOS of long jobs. The capacity restriction we analyze in this chapter, in both of its versions, limits such delays. Hence, even in cases in which the returns themselves are created by a managerial decision, imposing the additional managerial restriction on entering the system has benefits.
\end{remark}
\begin{figure}
\centering
\begin{subfigure}{0.38\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {Chapter_5/tikz/inner_vs_outer_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,col1, thick] table[x=n,y=hold] {Chapter_5/tikz/inner_vs_outer_wait.txt};
\end{axis}
\end{tikzpicture}
\caption{Expected pre-entering waiting (red) and needy waiting times (black)}
\end{subfigure}
\begin{subfigure}{0.6\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8,
legend cell align=left,
legend style = {at = {(1.05,0.58)}, anchor = west}
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {Chapter_5/tikz/total_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {Chapter_5/tikz/total_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {Chapter_5/tikz/total_wait.txt};
\legend{{\small $N=1$},{\small $N=2$},{\small $N=3$},{\small $N=4$},{\small $N=5$},{\small $N=6$},{\small $N=7$},{\small $N=8$},{\small $N=9$},{\small $N=10$}};
\end{axis}
\end{tikzpicture}
\caption{Total expected waiting times\\
\quad \\
\quad }
\label{fig:wait_by_visit_b}
\end{subfigure}
\caption{Expected waiting times as a function of $n$ given the number of visits $N$ in the Erlang-R model with holding with $\lambda=2$ $\mu=1$, $\delta=0.25$, $p=0.75$ and $s=9$.}
\label{fig:wait_by_visit}
\end{figure}
\subsection{Case study: comparison of operational decisions}
\label{sec:case_study}
We now illustrate how the managerial decision to operate under a specific operational regime affects ED performance in terms of efficiency and quality-of-care, through a case study.
The practical environment we investigate is the ED of a moderately-sized hospital, which faces the arrival pattern $\lambda(t)$ plotted in Figure \ref{fig:Case_study_arrival_pattern_a} on a typical workday.
Other parameters of the model are estimated to be $\mu = 6.67,\ \delta = 2.18$ and $p = 0.76$, so that $r = 0.301$. These parameters were taken from \cite{YomTov2014}. In order to set time-varying staffing levels $s(t)$ and $n(t)$, we adopt the \textit{mean-offered load} (MOL) approximation of the demand process of~\cite{Jennings1996}.
This approach initially presumes infinite capacity to obtain the number of patients $R(t)$ in the queueing system as a function of time.
This offered load function then replaces the (constant) value of $R$ in the stationary dimensioning scheme under consideration, to determine the adequate number of servers at each point in time.
Following this idea in our two-dimensional queueing system, we find the offered load function for the nurses $R_1(t)$ and the offered load function for the beds $R_1(t)+R_2(t)$ as the solution of the system of ODEs,
\begin{align} \label{eq:offeredloadODE}
\frac{d}{dt} R_1(t) &= \lambda(t) + \delta R_2(t) - \mu R_1(t),\\
\frac{d}{dt} R_2(t) &= p\mu R_1(t) - \delta R_2(t),
\end{align}
see \cite[Thm.~2]{YomTov2014} for details.
For this case study's parameters, these offered load functions are also plotted in Figure \ref{fig:Case_study_arrival_pattern_a}.
While the number of nurses can be adjusted in a relatively flexible manner, the value of $n$, which echoes a hard restriction on the ED capacity, is naturally less amenable to fluctuations. The reason is that the maximum ED capacity is to a large extent determined by its hardware, such as beds and medical equipment.
However, the ED manager might deliberately consider reducing $n$ during more quiet periods of the day, e.g.\ during the night, by imposing bed-to-physician constraints. This is done, for example, when setting a case management constraint \citep{EDexperiment,Campello2016}.
Therefore, we consider the scenario in which both $s$ and $n$ are time-dependent but we do not force a constant case management quantity, rather let our new methodology recommend an appropriate one.
\begin{figure}
\centering
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.01,0.95)},anchor = north west}
]
\addplot[very thick,black] file {Chapter_5/tikz/lambdaFunction.txt};
\addplot[very thick,col1] file {Chapter_5/tikz/R1.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/R1R2.txt};
\legend{ $\lambda(t)$, $R_1(t)$, $R_1(t)+R_2(t)$};
\end{axis}
\end{tikzpicture}
\caption{Dynamic arrival rate function offered load functions}
\label{fig:Case_study_arrival_pattern_a}
\end{subfigure}
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/sFunction.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/nFunction.txt};
\legend{ $s(t)$, $n(t)$};
\end{axis}
\end{tikzpicture}
\caption{Capacity function for $\beta=\gamma=0.5$}
\label{fig:Case_study_arrival_pattern_b}
\end{subfigure}
\caption{Empirical arrival rate and offered load functions $R_1(t)$ and $R_1(t)+R_2(t)$ in Israeli ED and corresponding capacity functions.}
\label{fig:Case_study_arrival_pattern}
\end{figure}
Extrapolating Algorithm \ref{alg:stationarydimensioning} to the time-varying case, Step 4 is replaced by
\begin{align*}
s(t) &= R_1(t) + \beta\sqrt{R_1(t)},\\
n(t) &= R_1(t)+R_2(t) + \gamma\sqrt{R_1(t)+R_2(t)},
\end{align*}
for some $\beta,\gamma>0$.
Since $R_1(t)$ and $R_2(t)$ are given, the QED staffing problem again reduces to finding the pair $(\beta,\gamma)$.
Figure \ref{fig:Case_study_arrival_pattern_b} plots the capacity functions for $\beta = 0.5$ and $\gamma=0.5$, assuming capacity can only be adjusted every 30 minutes.
In this case study, we consider three pairs of parameters $(\beta,\gamma)$.
For each of these we investigate, using simulation, the differences in the time-varying performance indicators between the policy with blocking and holding.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 1.0,
ytick = {0,0.2,0.4,0.6,0.8,1.0},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=delay_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=delay_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=delay_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=delay_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=delay_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=delay_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm delay})$}
\label{fig:simulation_results_a}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 0.52,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1,
legend cell align=left,
legend style = {at = {(0.9,0.95)}, anchor = north east}
]
\addplot[very thick,col1] table[x=t,y=block_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4] table[x=t,y=block_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5] table[x=t,y=block_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=block_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4,dashed] table[x=t,y=block_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5,dashed] table[x=t,y=block_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\legend{{$(\beta,\gamma)=(0.1,2)$},{$(\beta,\gamma)=(1,1.5)$},{$(\beta,\gamma)=(2,1)$}};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm block})$ or $\mathbb{P}({\rm hold})$}
\label{fig:simulation_results_b}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.5,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=ratio_b01g2] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=ratio_b01g2] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=ratio_b1g15] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=ratio_b1g15] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=ratio_b2g1] {Chapter_5/tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=ratio_b2g1] {Chapter_5/tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{Nurse-to-patient ratio.}
\label{fig:simulation_results_c}
\end{subfigure}
\caption{Simulation results for case study. Solid and dashed lines represent time-varying performance in the blocking and holding model, respectively.}
\label{fig:simulation_results}
\end{figure}
The simulation results are presented in Figure \ref{fig:simulation_results}.
Figure \ref{fig:simulation_results_a} shows that the MOL approach for capacity allocation roughly stabilizes the delay probability.
Figure \ref{fig:simulation_results_b} shows that the fraction of patients not entering the ED on arrival in the blocking model is reasonable for all parameter pairs considered and the graphs are ordered according to $\gamma$.
We also see a significant difference with holding.
Observe also that the holding probability drops in the period 8--13, which is exactly the period when the system is experiencing peak offered load.
Hence, this temporary reduction is in line with our asymptotic findings that the probability of blocking/holding is $O(1/\sqrt{R_1})$.
Finally note that the three parameter settings lead to different nurse-to-patient ratios.
Clearly, larger $\beta$ leads to small nurse-to-patient ratios (due do larger staffing).
Figure \ref{fig:simulation_results_c} demonstrates that for $(\beta,\gamma) = (1,1.5)$ and $(\beta,\gamma) = (2,1)$ the difference between the holding policy and the blocking policy is small. However, for $(\beta,\gamma) = (0.1,2)$ we see a significant increase in the ratio during night hours.
This may be due to the tight nurse schedule, that causes the holding queue to build up just before midnight.
This queue then empties latter on, causing an increase in workload per nurse in the period 24--7.
To see the direct effect of the size restriction on the queue lengths, we plotted the mean holding and service queue lengths in the holding model as a function of the parameter $\gamma$ in Figure \ref{fig:simulation_queuelengths}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.2,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/holdingQueue_g01.txt};
\addplot[very thick,col3] file {Chapter_5/tikz/casestudy_new/holdingQueue_g025.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/casestudy_new/holdingQueue_g05.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/holdingQueue_g1.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$}
\end{axis}
\end{tikzpicture}
\caption{Mean holding queue length}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 15,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.82,0.05)},anchor = south east}
]
\addplot[very thick,col1] file {Chapter_5/tikz/casestudy_new/serviceQueue_g01.txt};
\addplot[very thick,col3] file {Chapter_5/tikz/casestudy_new/serviceQueue_g025.txt};
\addplot[very thick,col4] file {Chapter_5/tikz/casestudy_new/serviceQueue_g05.txt};
\addplot[very thick,col5] file {Chapter_5/tikz/casestudy_new/serviceQueue_g1.txt};
\addplot[very thick,dashed] file {Chapter_5/tikz/casestudy_new/serviceQueue_R.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$,Erlang-R}
\end{axis}
\end{tikzpicture}
\caption{Mean service queue length}
\end{subfigure}
\caption{Simulated queue length of holding model with different values of $\gamma$.}
\label{fig:simulation_queuelengths}
\vspace{-6mm}
\end{figure}
Note that for all $\gamma$ considered, the holding queue lengtsh are, as expected, of a smaller order than the number of patients admitted.
Also, the holding queue length decreases as we increase $\gamma$.
The service queue lengths naturally approach the expected queue lengths in the Erlang-R model as $\gamma$ is increased.
\section{Conclusion \& future research}
\label{sec:conclusion}
In this chapter we developed and analyzed a queueing network tailored to a health care environment with finite-size restrictions.
Using the asymptotic approximations, numerical analysis and simulation, we gained insight into staffing problems that arise in EDs, and proposed an efficient, flexible, and easy to implement methodology to dimension medical facilities through a two-fold staffing rule.
The dimensioning scheme we developed provides a powerful and elegant way of finding adequate staffing levels in emergency departments.
Nonetheless, we see some avenues for further research.
The asymptotic approximations we developed enabled us to take the first step towards characterizing the pre-entering queue behavior in the
QED regime.
We observed how the holding queue length vanishes at rate $1/\sqrt{R_1}$ as $R_1\to\infty$.
Yet, our analysis did not yield explicit characteristics on the holding queue and holding times.
These performance indicators are naturally important to study if one wants to consider the trade-off between waiting time inside the ED and waiting time outside the ED time (pre-entering time).
Furthermore, it is worthwhile to study the robustness of our approximations against the service and content time distributions. Since the content phase of a patient is modeled after an infinite-server queue, we expect our approximations to be useful for content time distributions beyond the exponential distribution as well, due to distributional insensitivity of the service time in infinite-server queues. For the needy phase, modeled after a multi-server queue, this insensitivity result does not hold and hence this needs further research.
Finally, the restricted Erlang-R model obviously gives a highly simplified view of the complex reality of the ED.
In practice, distinctive features such as a triage system (with patient priorities), patient boarding time and availability of medical equipment may play a decisive role on ED dynamics.
However, we think the analysis and dimensioning algorithms presented in this chapter can serve as a building block for staffing procedures that do account for these case-specific factors.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Description of the QBD process}
\label{app:QBDdescription}
\subsection{The QBD-process}
\label{app:theQBDprocess}
We consider the QBD-process $X(t)=(N(t),Q_1(t))$ in stationarity. Let $\nu(i)=\min\{i,s\}\mu$. To determine the (outgoing) transition rates of the process $X$ we distinguish between the following cases:
\begin{itemize}
\item \emph{Transitions from $(0,0)$:} There are no patients in the Emergency Department and thus the only possible occurrence is when a new patient arrives. This results in a transition to $(1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), 1 \leq i < n$:} There are exactly $i$ patients assigned to a bed of which none are seen by a nurse. Then either one of those patients becomes needy, or a new patient arrives at the Emergency Department that can immediately be seen by a nurse. The first results in a transition to $(i,1)$ and occurs at rate $i \delta$, and the second results in a transition to $(i+1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), i \geq n$:} Again, the only possible transitions arise from either a newly arrived patient or a patient assigned to a bed becoming needy. However, a newly arrived patient finds all beds occupied and needs to wait. Thus, with rate $\lambda$ we have a transition to $(i+1,0)$ and with rate $n \delta$ a transition to $(i,1)$.
\item \emph{Transitions from $(i,i), i < n$:} In this case all patients assigned to a bed are in need of service. With rate $\lambda$ a new patient arrives at the Emergency Department. She joins the (possible) queue to be seen by a nurse immediately, so this results in a transition to $(i+1,i+1)$. Moreover, since there are only $s < n$ nurses, a service completion occurs with rate $\nu(i)$. With probability $p$ the patient turns to the holding phase, so in total we still have $i$ patients with one patient less in queue for a nurse. With probability $1-p$ the patient leaves the Emergency Department, decreasing both $N$ and $Q_1$ by one. In other words, with rate $p \nu(i)$ we have a transition to $(i,i-1)$ and with rate $(1-p)\nu(i)$ we have a transition to $(i-1,i-1)$.
\item \emph{Transitions from $(n,n)$:} Similar to the previous case, we have a transition to $(n,n-1)$ with rate $p s \mu$ and with rate $(1-p)s \mu$ we have a transition to $(n-1,n-1)$. In this case however, a newly arrived patient finds all beds occupied, resulting in a transition to $(n+1,n)$ with rate $\lambda$.
\item \emph{Transitions from $(i,n), i > n$:} We have a transition to $(i+1,n)$ with rate $\lambda$ and a transition to $(i,n-1)$ with rate $p s \mu$. In case that a patient leaves the Emergency Department there are $i-n>0$ patients in the holding room waiting for an available bed. Thus, one of the $i-n$ patients in the holding room is assigned to the available bed in need of service. That is, with rate $(1-p) s \mu$ we have a transition to $(i-1,n)$.
\item \emph{Transitions from $(i,j), 1 \leq j < i < n$:} There are four possible transitions. First, with rate $\lambda$ there is a new arrival which results in a transition to $(i+1,j+1)$. Second, with rate $(i-j) \delta$ a patient in one of the beds becomes needy, which results in a transition to $(i,j+1)$. Third, with rate $p \nu(j)$ a patient turns to the content state after service completion, which results in a transition to $(i,j-1)$. Last, with rate $(1-p) \nu(j)$ a patient leaves the Emergency Department after service completion, which results in a transition to $(i-1,j-1)$.
\item \emph{Transitions from $(n,j), 1 \leq j < n$:} This case is similar to the previous one. The only difference arises when a new patient arrives, since all $n$ beds are already occupied. Thus, with rate $\lambda$ we have a transition to $(n+1,j)$.
\item \emph{Transitions from $(i,j), i > n, 1 \leq j \leq n$:} This case is the previous one, except when a patient leaves the Emergency Department after service completion. Then one of the $(i-n)$ patients in the holding room will be assigned to a bed in need of service. This results in a transition to $(i-1,j)$ with rate $(1-p) \nu(j)$.
\end{itemize}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.66]
\draw[step=1cm,gray!50!,very thin] (0,0) grid (15.5,8.5);
\draw[thick,->] (0,0) -- (15.5,0);
\draw[thick,->] (0,0) -- (0,8.5);
\draw[thick] (0,0) -- (8,8);
\draw[thick,dashed,black!50!] (8,0) -- (8,8);
\draw[thick] (8,8) -- (15.5,8);
\foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
\foreach \y in {0,1,2,3,4,5,6,7,8}
\draw[fill] (\x,\y) circle [radius=0.025];
\node [below left] at (0,0) {$0$};
\node [left] at (0,2) {$j$};
\node [left] at (0,5) {$k$};
\node [left] at (0,8) {$n$};
\node [below] at (6,0) {$i$};
\node [below] at (8,0) {$n$};
\node [above left] at (0,8.5) {$Q_1$};
\node [below right] at (15.5,0) {$N$};
\path [->,thick,-latex] (0,0) edge [bend right] (1,1);
\path [->,thick,-latex] (3,0) edge (3,1);
\path [->,thick,-latex] (3,0) edge (4,1);
\path [->,thick,-latex] (4,4) edge [bend right] (5,5);
\path [->,thick,-latex] (4,4) edge (4,3);
\path [->,thick,-latex] (4,4) edge [bend right] (3,3);
\path [->,thick,-latex] (6,2) edge (6,3);
\path [->,thick,-latex] (6,2) edge (7,3);
\path [->,thick,-latex] (6,2) edge (6,1);
\path [->,thick,-latex] (6,2) edge (5,1);
\path [->,thick,-latex] (8,5) edge (8,6);
\path [->,thick,-latex] (8,5) edge (8,4);
\path [->,thick,-latex] (8,5) edge (9,5);
\path [->,thick,-latex] (8,5) edge (7,4);
\path [->,thick,-latex] (8,8) edge (8,7);
\path [->,thick,-latex] (8,8) edge [bend right] (7,7);
\path [->,thick,-latex] (8,8) edge [bend left] (9,8);
\path [->,thick,-latex] (11,8) edge (11,7);
\path [->,thick,-latex] (11,8) edge [bend left] (12,8);
\path [->,thick,-latex] (11,8) edge [bend left] (10,8);
\path [->,thick,-latex] (11,0) edge (11,1);
\path [->,thick,-latex] (11,0) edge [bend left] (12,0);
\path [->,thick,-latex] (12,5) edge (12,6);
\path [->,thick,-latex] (12,5) edge (12,4);
\path [->,thick,-latex] (12,5) edge (13,5);
\path [->,thick,-latex] (12,5) edge (11,5);
\node [above] at (12.75,5) {\scriptsize $\lambda$};
\node [above] at (11.25,5) {\scriptsize $(1-p)\nu(k)$};
\node [right] at (12,5.75) {\scriptsize $(n-k)\delta$};
\node [right] at (12,4.25) {\scriptsize $p \nu(k)$};
\node [above] at (6.75,2.25) {\scriptsize $\lambda$};
\node [below] at (5,2) {\scriptsize $(1-p)\nu(j)$};
\node [above] at (6,3) {\scriptsize $(i-j)\delta$};
\node [right] at (6,1.25) {\scriptsize $p \nu(j)$};
\end{tikzpicture}
\caption{Transition diagram for the Erlang-R model with holding.}
\label{fig:QBDIllustration}
\vspace{-3mm}
\end{figure}
\noindent
The state space and transition rates of the Erlang-R model with holding are illustrated in Figure~\ref{fig:QBDIllustration}.
The state space can be partitioned according to its levels, where level $i$ corresponds to a total queue length $N=i$ patients. This results in an infinite-sized matrix consisting of blocks, where each block corresponds to the transition flow from one level to another. Since the only transitions allowed are within the same level or between two adjacent levels in a QBD-process, we obtain a tridiagonal block structure. Each block consists of elements representing the transition rate of one state to another, and therefore each block is a matrix of size at most $(n+1) \times (n+1)$.
For the Erlang-R model with holding this gives the following result. Let $P$ denote the transition matrix of the process $(N(t),Q_1(t))$. We have the boundary levels $\{1,2,...,n\}$ and $P$ is of the form
\[
P = \left( \begin{array}{cccccccccc}
B_{00} & B_{01} & & & & & & & & \\
B_{10} & B_{11} & B_{12} & & & & & & & \\
& B_{21} & B_{22} & B_{23} & & & & & & \\
& & \ddots & \ddots &\ddots & & & & & \\
& & & & & B_{n \, n-1} & & & & \\
& & & & B_{n-1 \, n} & B_{nn} & A_0 & & & \\
& & & & & A_2 & A_1 & A_0 & & \\
& & & & & & A_2 & A_1 & A_0 & \\
& & & & & & & \ddots & \ddots & \ddots \\
\end{array} \right),
\]
where $B_{ii} \in \mathbb{R_1}^{(i+1) \times (i+1)}$, $B_{i \, i-1} \in \mathbb{R_1}^{(i+1) \times i}$, $B_{i-1 \, i} \in \mathbb{R_1}^{i \times (i+1)}$, and $A_0,A_1,A_2 \in \mathbb{R_1}^{(n+1)\times(n+1)}$. The matrices of transition rates for the boundary states are given by
\[
B_{00}=(-\lambda),
\qquad
B_{i-1 \, i} = \left( \begin{array}{ccccc}
0 & \lambda & & & \\
& \ddots & \lambda & & \\
& & \ddots & \ddots &\\
& & & 0 & \lambda \\
\end{array} \right),
\]
\[
B_{i \, i-1} = \left( \begin{array}{cccc}
0 & & & \\
(1-p)\mu & 0 & & \\
& (1-p)\nu(2)& \ddots & \\
& & \ddots & 0 \\
& & & (1-p)\nu(i)\\
\end{array} \right),
\]
and
\[
\scriptsize
B_{ii} = \left(
\begin{array}{ccccccccc}
-(\lambda+i \delta) & i \delta & & & &\\
p \mu & -(\lambda+\mu+(i-1)\delta) & (i-1)\delta & & &\\
& \ddots & \ddots & \ddots & & \\
& & p \nu(i-1) & -(\lambda+\nu(i-1)+\delta) & \delta \\
& & & & p \nu(i) & -(\lambda+\nu(i)) \\
\end{array} \right).
\]
Moreover, the transition rates are given by
\[
A_0 = \left( \begin{array}{ccccc}
\lambda & & & & \\
& \lambda & & & \\
& & & \ddots & \\
& & & & \lambda \\
\end{array} \right)
\]
\[ A_2 = \left( \begin{array}{ccccccc}
0 & & & & & & \\
& (1-p)\mu & & & & & \\
& & 2(1-p)\mu & & & & \\
& & & \ddots & & & \\
& & & & s(1-p)\mu & & \\
& & & & & \ddots & \\
& & & & & & s(1-p)\mu \\
\end{array} \right),
\]
and
\[
\scriptsize
A_1 = \left( \arraycolsep=0.55pt
\begin{array}{cccccccc}
-(\lambda+n \delta) & n \delta & & & & & & \\
p \mu & -(\lambda+\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(\lambda+s\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & s p\mu & -(\lambda+s\mu+\delta) & \delta & \\
& & & & & s p\mu & -(\lambda+s\mu)\\
\end{array} \right).
\]
\subsection{Stability condition}
\label{app:stability}
From the general theory of QBD processes \citep{Neuts1981} follows that the Markov process $(N(t),Q_1(t))$ is ergodic (stable) if and only if
\begin{equation}
\pi A_0 e < \pi A_2 e,
\label{eq:QBDstableCondition}
\end{equation}
where $e$ is the all one column vector and $\pi=(\pi_0,...,\pi_n)$ is the equilibrium distribution of the Markov process with generator $A_0+A_1+A_2$. In other words, $\pi$ is such that
\begin{equation}
\begin{array}{ll}
\pi(A_0+A_1+A_2) =0, & \pi e =1,
\end{array}
\label{eq:QBDstableProbabilityVector}
\end{equation}
and
\[
A_0+A_1+A_2 = \qquad\qquad\qquad
\]
\begin{align*}
{\scriptsize
\left(
\begin{array}{cccccccc}
-n \delta & n \delta & & & & & & \\
p \mu & -(p\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(ps\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & p s \mu & -(ps\mu+\delta) & \delta & \\
& & & & & p s \mu & -ps\mu\\
\end{array} \right).
}
\end{align*}
Then $\pi$ must satisfy the balance equations
\begin{align*}
- n \delta \pi_0 + p \mu \pi_1 &= 0, \\
(n-j+1)\delta \pi_{j-1} - (p\nu(j) +(n-j)\delta) \pi_j + p \nu(j+1) \pi_{j+1} &= 0, \\
\delta \pi_{n-1} - p s \mu \pi_n &= 0,
\end{align*}
with $\nu(j)=\min\{j,s\}\mu$, and the normalization condition
\[
\sum_{i=0}^n \pi_i=1.
\]
It is readily verified that
\begin{equation}
\pi_i =
\left\{\begin{array}{ll}
\pi_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\pi_0 \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistr}
\end{equation}
with
\begin{align*}
\pi_0= \left(\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right)^{-1},
\end{align*}
satisfies the balance equations and the normalization condition.
\begin{proposition}
The distribution of the closed two-node Jackson network illustrated in Figure~\ref{fig:Jennings} is given by
\begin{equation}
\hat{\pi_i} =
\left\{\begin{array}{ll}
\hat{\pi}_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\hat{\pi_0} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistrTwistedJennings}
\end{equation}
with
\begin{align*}
\hat{\pi}_0= \left[\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right]^{-1}.
\end{align*}
\label{prop:CriticalTilburgdistr}
\end{proposition}
\begin{proof}
We have a two-node closed Jackson network, with probability transition matrix
\[
P = \left(
\begin{array}{cc}
1-p & p \\
1 & 0
\end{array} \right).
\]
Let $r_i(m)$ denote the rate of service when there are $m$ patients at queue $i$, so $r_1(m)=\min\{m,s\}$ and $r_2(m)=m$. The throughput vector $\gamma = (\gamma_1,\gamma_2) \in \mathbb{R_1}^2$ must satisfy $\gamma = \gamma P$ and we find that $\gamma=(p,1)$ suffices. From the general theory of Jackson networks, see \cite{Jackson1963}, it follows that the stationary distribution is given by
\begin{align*}
\pi_i = G^{-1} g_1(i) g_2(n-i)
\end{align*}
with
\begin{align*}
\begin{array}{ll}
g_1(i)= \frac{(\gamma_1/\mu)^i}{\prod_{m=1}^i r_1(m)}, & g_2(n-i)= \frac{(\gamma_2/\delta)^{n-i}}{\prod_{m=1}^{n-i} r_2(m)},
\end{array}
\end{align*}
and normalization constant $G= \sum_{i=0}^n g_1(i) g_2(n-i)$. Then,
\begin{align*}
g_1(i) &= \left\{\begin{array}{ll}
\frac{1}{i! \mu^i} & \textrm{\normalfont for } 0 \leq i \leq s, \\
\frac{1}{s! s^{i-s} \mu^i} & \textrm{\normalfont for } s+1 \leq i \leq n, \\
\end{array} \right.\\
g_2(n-i) &=\frac{1}{(n-i)!} \left(\frac{p}{\delta}\right)^n \left(\frac{\delta}{p}\right)^i,
\end{align*}
and rewriting the expressions yields~\eqref{eq:eqdistrTwistedJennings}.
\end{proof}
\subsection{Stationary distribution}
\label{app:StationaryDistributrion}
Assuming that the stability condition is satisfied, we can determine the unique stationary distribution of the Markov process $(N(t),Q_1(t))$. The vector $\pi_i$ can be written as $\pi_{n+i}= \pi_n G^{i}$ for $i=0,1,...$, where $G$ is the minimal nonnegative solution of the non-linear matrix equation
\begin{equation}
A_0+G A_1 + G^2 A_2=0.
\label{eq:MG-G}
\end{equation}
The balance equations can be written as
\[
\begin{array}{ll}
\pi_{i-1} A_0+ \pi_i A_1 + \pi_{i+1} A_2=0, & i=n+1,n+2,...
\end{array}
\]
and using $\pi_{n+i}= \pi_n G^{i-n}$ for $i=0,1,...$, this find
\[
\begin{array}{ll}
\pi_n G^{i-n-1} \left(A_0+ G A_1 + G A_2\right)=0, & i=n+1,n+2,....
\end{array}
\]
\noindent
Moreover, we have the boundary equations
\begin{align*}
\pi_0 B_{00} + \pi_1 B_{10} &= 0 \\
\pi_0 B_{01} + \pi_1 B_{11} + \pi_2 B_{21} &= 0 \\
\pi_1 B_{12} + \pi_1 B_{22} + \pi_2 B_{32} &= 0 \\
&\vdots& \\
\pi_{n-2} B_{n-2 \, n-1} + \pi_{n-1} B_{n-1 \, n-1} + \pi_{n} B_{n \, n-1} &= 0 \\
\pi_{n-1} B_{n-1 \, n} + \pi_{n} B_{nn} + \pi_{n+1} A_2 &= 0,
\end{align*}
along with the normalization equation
\[
1 = \sum_{i=0}^{\infty} \pi_i e = \sum_{i=0}^{n-1} \pi_i e + \pi_n(I-G)^{-1}e,
\]
where we slightly abuse notation by using $e$ as the all ones vector of appropriate size. We note that the matrix $G$ has a spectral radius less than one and therefore $(I-G)$ is invertible.
These equations provide the tools for finding the equilibrium probabilities. Although it is hard to solve $G$ analytically from Equation~\eqref{eq:MG-G}, it is easy to solve numerically by using the following algorithm (matrix-geometric method). Rewriting~\eqref{eq:MG-G} gives
\[
G=-(A_0+G^2 A_2) A_1^{-1},
\]
where $A_1$ is invertible, since it is a transient generator matrix. Let
\[
G_{k+1}=-(A_0+G_k^2 A_2) A_1^{-1},
\]
starting with $G_0=0$. We note that $G_k \uparrow G$ as $k$ grows to infinity \citep{Neuts1981}. Once $||G_{k+1}-G_{k}||_2$ is below a certain preset threshold, we approximate $G$ by $G_{k+1}$.
\section{Proof of Proposition \ref{thm:stochasticordering}}\label{app:stochastic_ordering}
First, note that by definition of the Erlang-R model with holding, in which no more than $n$ patients can be admitted in the ED simultaneously, that $Q_1^h(t)+Q_2^h(t) \leq n = Q_1^J(t) + Q_2^J(t)$ follows directly.
Therefore, we only consider the relation between the states in the blocking and holding variants Erlang-R model.
As noted Section \ref{sec:Markov_process}, the model with holding can be characterized as a three-dimensional Markov chain $X^h(t) = (H(t),Q^h_1(t),Q^h_2(t))$ in which the components denote the number of holding, needy and content patients respectively. The Erlang-R model with blocking similarly admits a Markov process description, but with two dimensions, namely $X^b(t) = (Q^b_1(t),Q^b_2(t))$.
We prove the result by constructing a coupling between the Markov processes $X^h$ and $X^b$. Let $Z(t) := \big(\hat{X}^h(t),\hat X^b(t)\big) = \big(\hat{H}(t),\hat{Q}_1^h(t),\hat{Q}_2^h(t),\hat{Q}^b_1(t),\hat{Q}^b_2(t)\big)$.
We first define the transition rates of this five-dimensional Markov process, which naturally only depend on the current state of the system.
After that we show that the transition rates relevant to $\hat{X}^h(t)$ and $\hat{X}^{b}(t)$ coincide with those of $X^h(t)$ and $X^b(t)$, respectively. The latter implies that the marginal transitions of $\hat{X}^h(t)$ and $X^h(t)$ (and $\hat{X}^b(t)$ and $X^b(t)$) are equal, and hence so are their probability distribution of the Markov processes.
Let $Z(t) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$. While defining the reachable states from this state and associated transition rates, we distinguish four transition types, and further differentiate the transition rates depending on the current state.\\
\\*
\textbf{Arrival.}
Arrivals occur in both models simultaneously, but are handled differently according to the current queue lengths.
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr1}
(h,q_1^h+1,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr2}
(h+1,q_1^h,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h < n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr3}
(h,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr4}
(h+1,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\end{enumerate}
\noindent \textbf{Departure.}
Basically, we align service completions in the two models, but allow a completion occurring solely in either of one of the two models, only if the queue length in this model is strictly larger than in the other one.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep1}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(h-1,q_1^h,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.
\end{equation}
\item If $q_1^h < q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep2}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h \geq q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep3}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep4}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(0,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent\textbf{Become content.}
The differentiation between transitions is similar to those in the \textit{departure} transition type.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$,
\begin{equation}
\label{eq:con1}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b \wedge s)p\mu,\\
(h,q_1^h-1,q_2^h+1,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$,
\begin{equation}
\label{eq:con2}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^h \wedge s)p\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b+1) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent
\textbf{Become needy.}
\begin{enumerate}
\item If $q_2^h \geq q_2^b$,
\begin{equation}
\label{eq:ne1}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^b\delta,\\
(h,q_1^h+1,q_2^h-1,q_1^b,q_2^b) & \text{with rate }(q_2^h-q_2^b)\delta,\\
\end{array}
\right.\end{equation}
\item If $q_2^h < q_2^b$,
\begin{equation}
\label{eq:ne2}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^h\delta,\\
(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1) & \text{with rate }(q_2^b-q_2^h)\delta,\\
\end{array}
\right.\end{equation}
\end{enumerate}
This set of transitions defines the dynamics of the Markov process $Z(t) = (\hat{X}^h(t),\hat{X}^b(t))$.
Let us now restrict our attention to the transitions in which (at least one of) the first three coordinates of $Z(t)$ changes, that is, the marginal transitions of the process $\hat{X}^h$.
Let $\hat{X}^h(t) = (h,q_1^h,q_2^h)$, then according to the transition scheme above, $\hat{X}^h$ moves to state
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ (and hence necessarily $h=0$),
\[
\left\{
\begin{array}{ll}
(0,q_1^h+1,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h-1,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $q_1^h+q_2^h = n$ and $h=0$,
\[
\left\{
\begin{array}{ll}
(1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $h>0$ (and hence necessarily $q_1^h+q_2^h = n$),
\[
\left\{
\begin{array}{ll}
(h+1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(h-1,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(h,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(h,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\end{enumerate}
One can check that these transitions indeed coincide with the transitions in the original holding model, hence $\hat{X}^h(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Similarly, when focusing on transitions of $Z(t)$ that are relevant for $\hat{X}^b(t)$, we deduce the following transition scheme. If $\hat{X}^b(t) = (q_1^b,q_2^b)$, then the next move according to the transitions of $Z(t)$ is
\[
\left\{
\begin{array}{ll}
(q_1^b+\mathbbm{1}_{\{q_1^b + q_2^b < n\}},q_2^b) & \text{with rate } \lambda,\\
(q_1^b-1,q_2^b) & \text{with rate }(q_1^b\wedge s)(1-p)\mu,\\
(q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b\wedge s)p\mu,\\
(q_1^b+1,q_2^b-1) & \text{with rate }q_2^b \delta.
\end{array}
\right.\]
These transition rates clearly coincide with the original Erlang-R model with blocking, and also hence $\hat{X}^b(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Next, we show that under this coupling scheme we have that if $\hat{H}(0) = 0$, $\hat{Q}_1^h(0)=\hat{Q}_1^b(0)$ and $\hat{Q}_2^h(0)=\hat{Q}_2^b(0)$ then for all $t\geq 0$, $Z(t)$ satisfies the hypothesis:
\begin{itemize}
\item[(i)] $\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \leq \hat{Q}_1^h(t) + \hat{Q}_2^h(t)$,
\item[(ii)] $\hat{Q}_2^b(t) \leq \hat{Q}_2^h(t)$,
\item[(iii)] $\hat{Q}_1^b(t) \leq \hat{Q}_1^h(t) + H(t)$.
\end{itemize}
We do so by induction on the next state reached after a transition of the joint Markov process $Z=(\hat{X}^h,\hat{X}^b)$.
First of all, $Z(0)$ clearly satisfies (i)-(iii).
Next, assume $Z(t^-) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$ satisfies the hypothesis and a transition occurs at $t$.
We show that under the specified coupling scheme, the state reached after the next transition, $Z(t)$ must satisfy (i)-(iii) as well. To do so, we differentiate between the four types of transitions that could occur: arrival, departure, become content and become needy.\\
\\*
\noindent\textbf{Arrival.}
Recall that under our coupling scheme an arrival always occurs in both the holding and blocking model simultaneously, see \eqref{eq:arr1}--\eqref{eq:arr4}. Furthermore, $q_2^h$ and $q_2^b$ are unchanged during this transition, rendering (ii) trivial.
By hypothesis $q_1^b + q_2^b \leq q_1^h+q_2^b$, hence the event $q_1^h+q_2^h < n$ and $q_1^h+q_2^b =n$, with resulting state $(0,q_1^h+1,q_2^h,q_1^b,q_2^b)$, can be excluded from our analysis.
We check the conditions for the remaining three cases.
\begin{enumerate}[noitemsep]
\item If $Z(t)= (0,q_1^h+1,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h <n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \less[i] q_1^h+q_2^h+1 =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+1 = \hat Q_1^h(t) = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h =n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \leq n = q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h +1= \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $q_1^b + q_2^b = q_1^h+q_2^h=n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+h+1 = \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Departure.}
By carefully examining the possible state transitions of $Z(t)$ following a departure, we list six reachable states. However, by (iii), we have that if $h=0$, then $q_1^b \leq q_1^h$, which excludes the state $(0,q_1^h,q_2^h,q_1^b,q_2^b)$ in \eqref{eq:dep4} from the reachability graph.
We check the remaining states for conditions (i)--(iii). Again, during a departure, $q_2^b$ and $q_2^h$ are unchanged, so (ii) is automatically satisfied by the induction hypothesis.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 \less[i] q_1^h+q_2^h-1 < q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h + h-1 = \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $h>0$ and $q_1^h \geq q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 \leq q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$ and $q_1^h < q_1^b$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 < q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^b \less[iii] q_1^h + h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h,q_1^b-1,q_2^b)$, then $h=0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = (q_1^b-1)+q_2^b-1 < \less[i] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h-1 = \hat Q_1^h(t) + \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (0,q_1^h-1,q_2^h,q_1^b,q_2^b)$, then $h=0$ and $q_1^h>q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] (q_1^h-1)+q_2^b \less[ii] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 =\hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Content start.}
On the event of a patient becoming content, it is clear that the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected. This means that (i) is directly satisfied by the induction hypothesis.
According to \eqref{eq:con1}--\eqref{eq:con2}, three states can be reached.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1)$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \less[ii] q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b,q_2^b)$, then $q_1^h > q_1^b$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[ii] q_2^h < q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h = \hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b+1)$, then $q_1^b > q_1^h$ (*) and hence by (iii) $h > 0$. The latter is only possible if $q_1^h+q_2^h=n$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \leq n-q_1^b+1 = (q_1^h+q_2^h)-q_1^b+1 \less[*] q_2^h = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^h+h-1 \less[*] q_1^h+h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent \textbf{Become needy.}
Just as in the event of content start, the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected, whereby (i) is directly satisfied by the induction hypothesis.
By (ii), we have $q_2^h \geq q_2^b$. This excludes the state $(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1)$ from being reached, see \eqref{eq:ne2}.
We check the remaining two possibilities.
\begin{enumerate}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1)$.
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b-1 \less[ii] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+h+1 = \hat Q_1^h(t)+\hat H(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b,q_2^b)$, then $q_2^h > q_2^b$ (*).
\begin{itemize}
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[*] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h =\hat Q_1^h(t) + \hat H(t)$.
\end{itemize}
\end{enumerate}
Hence, the state reached after any feasible transition under the coupling scheme satisfies the conditions (i)--(iii).
Thus we conclude that the joint process\\ $(\hat{H}(t),\hat Q_1^h(t),\hat Q_2^h(t),\hat Q_1^b(t),\hat Q_2^b(t))$ adheres to (i)--(iii) for all $t$. Consequently, we have that (i) implies
\begin{align*}
\mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right) &= \mathbb{P}\left(\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \geq k\right)\\
&=\sum_{j=0}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&=\sum_{j=k}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&\leq \sum_{j=h}^n \mathbb{P}\left( \hat Q_1^h(t)+\hat Q_2^h(t) = j \right)\\
&= \mathbb{P}\left( Q_1^h(t) + Q_2^h(t) \geq k\right) = \mathbb{P}\left(Q_1^h(t) + Q_2^h(t) \geq k\right).
\end{align*}
The other two orderings follow similarly.
\begin{remark}
Note that under this coupling scheme we cannot get the ordering $\hat Q_1^h(t)(t) \geq \hat Q_1^b(t)(t)$ for all $t\geq 0$. A minimal counter example occurs for $s=n=1$. Let $Z(0) = ((0,0,0),(0,0))$. First, two arrivals occur, such that state $((1,1,0),(1,0))$ is reached, followed by a departure transition, yielding $((0,1,0),(0,0))$. Next, the one patient left in the model with holding system becomes content, so that we obtain $((0,0,1),(0,0))$.
At this stage, if an arrival occurs, the arriving patient will be put in the holding queue in the model with holding, and admitted to nurse queue in the model with blocking. Hence we end up in state $((1,0,1),(1,0))$, in which $\hat Q_1^h(t) < \hat Q_1^b(t)$.
\end{remark}
\section{Proof of Proposition \ref{prop:stability_convergence}}\label{app:proof_stability_convergence}
Define
\[
A(s,n) = \sum_{k=0}^s \frac{k}{s} \, \binom{n}{k} b^k ,\quad
B(s,n) = \sum_{k=s+1}^n \frac{k!}{s!} \, \binom{n}{k} s^{s-k} b^k, \quad
C(s,n) = \sum_{k=0}^s \binom{n}{k} \, b^k,
\]
\[
\]
where $b = \delta/p\mu = r/(1-r)$. Then
\[
\rho_{\rm max}(s,n) = \frac{A(s,n)+B(s,n)}{C(s,n)+B(s,n)}.
\]
Proving that $\rho_{\rm max}(s,n) \to 1$ as $R_1\to\infty$ with $s$ and $n$ as in \eqref{eq:twofoldscaling} is equivalent to showing that
\begin{equation}\label{eq:proof_stab_1}
1-\rho_{\rm max}(s,n) = \frac{C(s,n)-A(s,n)}{C(s,n)+B(s,n)} = \frac{(1+b)^{-n}[C(s,n)-A(s,n)]}{(1+b)^{-n}[C(s,n)+B(s,n)]} \to 0.
\end{equation}
First, we rewrite
\begin{align*}
(1+b)^{-n} A(s,n)
&= (1+b)^{-n} \sum_{k=1}^s \frac{n}{s} \binom{n-1}{k-1} b^k \\
&= \frac{n}{s}\left(\frac{b}{1+b}\right)\sum_{k=0}^{s-1} \binom{n-1}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-1-k}\\
&= \frac{r n}{s}\sum_{k=0}^{s-1} \binom{n-1}{k} r^k (1-r)^{n-1-k}\\
&= \frac{r n}{s} \mathbb{P}( {\rm Bin}(n-1,r) \leq s-1 ) \\
&= \frac{rn}{s} \mathbb{P}\left( \frac{{\rm Bin}(n-1,r) - (n-1)r}{\sqrt{nr(1-r)}} \leq \frac{s-1 - (n-1)r}{\sqrt{nr(1-r)}} \right)\\
&\to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
since $nr/s = 1 + O(1/\sqrt{R_1})$.
Also,
\begin{align*}
(1+b)^{-n} C(s,n)
&= \sum_{k=0}^s \binom{n}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-k}\\
&= \sum_{k=0}^s \binom{n}{k} r^k (1-r)^{n-k}\\
&= \mathbb{P}( {\rm Bin}(n,r) \leq s) \to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right).
\end{align*}
Therefore, we have $(1+b)^{-n}[C(s,n)-A(s,n)] \to 0$ as $\lambda\to\infty$.
For the remaining term,
\begin{align*}
(1+b)^{-n} B(s,n)
&= (1+b)^{-n}\sum_{k=s+1}^n \binom{n}{k}\,\frac{k!}{s!} s^{s-k} b^k \\
&= (1+b)^{-n}\frac{n!}{s!}\, s^s\sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{-k}\\
&= (1+b)^{-n} \frac{n!}{s!}\, s^s\, \left(\frac{b}{s}\right)^n \sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{n-k}\\
&= r^n\, \frac{n!}{s!} s^{s-n} \sum_{m=0}^{n-s-1} \frac{1}{m!} \left(\frac{s}{b}\right)^m\\
&= \left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b} \, \mathbb{P}({\rm Pois}(s/b)\leq n-s-1),
\end{align*}
in which
\begin{align*}
\mathbb{P}({\rm Pois}(s/b)\leq n-s-1)
&= \mathbb{P}\left(\frac{{\rm Pois}(s/b)-s/b}{\sqrt{s/b}} \leq \frac{n-s-1-s/b}{\sqrt{s/b}}\right) \\
&\to \Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
as $\lambda\to\infty$.
By Stirling's approximation,
\begin{align*}
\left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b}
&\sim \left(\frac{r}{s}\right)^n \sqrt{\frac{n}{s}} \,\frac{n^n {\rm e}^{-n}}{s^s {\rm e}^{-s}}\, s^s \,{\rm e}^{s/b} \\
&= \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s+s/b} = \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s/r}.
\end{align*}
Since,
\[
\frac{rn}{s} = 1 + \frac{\gamma\sqrt{r}-\beta}{\sqrt{R_1}} + O(1/R_1),
\]
we find $\sqrt{n/s} = 1/\sqrt{r} + O(1/\sqrt{R_1})$ and
\begin{align*}
\log\left[ \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+\tfrac{s}{r}} \right]
&= n \log\left[ \frac{rn}{s}\right] - n+\frac{s}{r}\\
&= -n \left[ \left(1-\frac{rn}{s}\right) + \frac{1}{2}\left(1-\frac{rn}{s}\right)^2 + O(R^{-\tfrac{3}{2}}) \right] + \frac{s}{r}\left(1-\frac{rn}{s}\right)\\
&= \frac{s}{r}\left(1-\frac{rn}{s}\right)^2 - \frac{n}{2}\left(1-\frac{rn}{s}\right)^2 + O(1/\sqrt{R_1})\\
&= \frac{(\gamma\sqrt{r} - \beta)^2}{2r} + O(1/\sqrt{R_1}),
\end{align*}
as $\lambda\to\infty$ and hence,
\[
(1+b)^{-n} B(s,n) \to \varphi\left(\frac{\gamma\sqrt{r}-\beta}{\sqrt{r}}\right)\Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right).
\]
Hence, we conclude that the denominator of \eqref{eq:proof_stab_1} converges to a constant value as $R_1$ grows, and hence the $1-\rho_{\rm max}(s,n)\to 0$ as $\lambda\to\infty$.
\resettocdepth
\end{subappendices}
\chapter{Finite-size effects in critically dimensioned emergency departments}
\begin{chapterstart}
Motivated by health care systems with repeated services that have both personnel (nurse/physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we account for are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them.
We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly.
We show that the system dynamics is captured by the fraction of needy time in the network, and that returning patients should be accounted for both in steady-state and time-varying conditions.
We demonstrate the application of our policies in two case-studies of resource allocation in hospitals.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Finite-size effects in critically dimensioned emergency departments}\\
\textit{Johan van Leeuwaarden, Britt Mathijsen, Fiona Sloothaak \& Galit Yom-Tov}\\
Submitted to \textit{Operations Research}
\end{flushright}
\newpage
\section{Introduction}
In recent years, operations research techniques have received increased interest from the health care community, as they are able to design and improve workflow processes in health care facilities~\cite{Armony2015,Green2008,Bekker2009b,Hall2006,Hall2012}.
Because these processes are typically stochastic in nature, it is common practice to use queueing theory for performance analyses and workforce planning.
As a first step towards understanding the processes going on in health care environments, systems are commonly modeled after a single station queue, such as the $M/M/s$ (Erlang-C), $M/M/s/s$ (Erlang-B) or $M/M/s+M$ (Erlang-A) models, and fluid and diffusion approximations are used to provide insights into the process dynamics.
However, simple single station models often fail to capture the more intricate dynamics of the settings specific to health care contexts.
Prime examples include the flows of patients in a hospital from one medical ward to another \citep{Armony2015}, within the Emergency Department (ED) between different stages of treatment \citep{Junfei2015}, or between medical facilities \citep{zychlinski2016bedblocking}.
Queueing networks can capture the dependency between several service stages and several types of resources.
More specifically, we are interested in the ubiquitous feature, particularly present in health care environments, that patients during their stay in the system might require a specific resource multiple times, e.g.~physicians and nurses who treat patients several times during their stay in the medical wards \citep{Jennings2011} or the ED \citep{YomTov2014}, while multiple resources types are limited (e.g.\ medical staff and beds).
In this chapter, we concentrate on the dynamics within EDs.
An often ignored yet essential feature of medical facilities concerns the restriction of the number of patients that can reside in the facility simultaneously.
In Chapter 4, we already observed that finite-size restrictions can have a significant effect on the performance of queueing systems.
In this chapter, we investigate the influence of such multiple restrictions on the network dynamics and the required staffing policies in the context of an ED. \\
\\*
\noindent
\textbf{The restricted Erlang-R model.}
The canonical model for service networks with returns is the Erlang-R model, introduced by Yom-Tov \& Mandelbaum~\citep{YomTov2014}.
In this open two-station model, customers arrive according to a Poisson process to an $M/M/s$.
After service completion, the customer with probability $1-p$ leaves the system and with probability $p$ returns to the queue after a random delay.
This delay is modeled as an infinite-server queue.
A schematic visualization of the Erlang-R model is depicted in Figure \ref{fig:ErlangR}
in which customers, during their stay in the system receive a random number of services from the same pool of servers.
Yom-Tov \& Mandelbaum \cite{YomTov2014} showed that such a simple network model can be used to determine staffing in an ED both in stable and time-varying conditions.
Nevertheless, empirical studies report that some countries, such as the US, use a different operational mode that applies strict restrictions on entering the ED \citep{EDexperiment}.
In typical US EDs, a patient will not enter the ED until both a bed and a physician are available to treat her.
Those restrictions can be either physical (beds) restrictions or managerial ones --- for instance by imposing a patient-to-physician ratio.
In this work, we extend the Erlang-R model by enforcing a constraint on the maximum number of available places inside the facility.
Our model hence incorporates two kinds of resource constraints: servers that provide the actual service and the maximum available places inside the service system.
Both affect the system in a highly interdependent way.
The model, presented in Figure \ref{fig:Erlang_R_model}, assumes $s$ servers and a maximum capacity of $n$ concurrent places.
We assume that patients arrive according to a Poisson process with rate $\lambda$.
In case a new arrival finds $n$ or more patients already present, we consider two options: either she waits outside the service facility in a holding queue until a vacant space becomes available (Figure \ref{fig:Erlang_R_holding}) or she is blocked (Figure \ref{fig:Erlang_R_blocking}), such as is the case when patients are sent to an alternative facility.
Once a patient is admitted, she requires assistance from one of the $s$ servers for an exponentially distributed duration with mean $1/\mu$.
Then, with probability $1-p$, the patient leaves the system or, with probability $p$, returns to service again after an exponentially distributed time with mean $1/\delta$.
Following Jennings \& de V\'ericourt \cite{Jennings2011} and Yom-Tov \& Mandelbaum \cite{YomTov2014}, we call patients {\it needy} when they require attention from one of the servers and {\it content} when they are in the delayed return phase.
In addition, we call patients {\it holding} when they are waiting outside the facility for an available space. We assume that the arrival process, the needy times and content times are mutually independent.
In the holding queue and the needy queue, we apply the First-Come-First-Served (FCFS) discipline.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-2.75,4.5) -- (-1.25,4.5);
\draw [thick] (-1.5,5) -- (0,5) -- (0,4) -- (-1.5,4);
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (0,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick,->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,2.9) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\node [above] at (-0.75,5) {\footnotesize holding};
\end{tikzpicture}
\caption{Erlang-R model with holding.}
\label{fig:Erlang_R_holding}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.55]
\footnotesize
\draw [thick, ->] (-1.5,4.5) -- (2.5,4.5);
\draw [thick, ->] (0,4.5) -- (0,2.5) node[below left] {blocked};
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {$s$} node[below= 0.3 cm] {needy} node[above=0.25cm] {$\exp(\mu)$};
\draw [thick, ->] (4.75,4.5) -- (7.75,4.5);
\draw [thick,->] (5.75,4.5) -- (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5] node {$\infty$} node[below=0.3cm] {content} node[above=0.25cm] {$\exp(\delta)$};
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node [above] at (-1.5,3.4) {\footnotesize Pois($\lambda$)};
\node [below] at (5.5,2) {\footnotesize $p$};
\node [above] at (6.25,4.5) {\footnotesize $1-p$};
\draw [thick, dashed] (0.5,0.75) rectangle (7,5.9) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{Erlang-R model with blocking.}
\label{fig:Erlang_R_blocking}
\end{subfigure}
\caption{Restricted Erlang-R models with maximally $n$ customers in system.}
\label{fig:Erlang_R_model}
\end{figure}
As mentioned, we consider two versions of the finite-capacity constraint.
The first version is called \emph{Erlang-R with holding}, in which patients wait for an available space in the system.
The second version is called \emph{Erlang-R with blocking}, in which patients meeting a full system are blocked.
Naturally, intermediate scenarios can be constructed in which a proportion of the total arrival volume of patients indeed leaves upon finding a full system, while the rest joins the holding queue.
While this chapter focuses on the two extreme cases, straightforward adaptations can fit these intermediate scenarios. \\
\\*
\noindent
\textbf{Examples of restricted Erlang-R.}
As noted before, an ED operated in the US can be modeled using a restricted Erlang-R model. Another health care example is medical units (MUs) in a hospital.
Such units specialize in specific types of illnesses (cardiology, oncology, etc.) and have limited resources such as nurses and beds. If the unit is full, new patients are either allocated to an alternative medical unit, i.e.\ blocked, or wait for an available bed.
Both policies are problematic in terms of quality-of-care, because the personnel in the alternative unit (or the ED) may be less knowledgeable about the patient's medical condition and waiting in the ED was shown to increase mortality.
Moreover, ED waiting may reduce available capacity for treating ED patients \citep{Carmen2016,israelit}, hence endangering both the delayed patient as well as others. Both the number of personnel (nurses and physicians) and the number of beds impact service dynamics and quality-of-care. Research so far looked at the capacity allocation of those resources separately. Green \& Yankovic \cite{GY2011} and Jennings \& de V\'ericourt \cite{Jennings2008} looked at nurse staffing in medical units, while de Bruin et al.~\cite{Bekker2009b} looked into bed allocation. The unified model we suggest enables us to capture the dependency between those two decisions, and its impact on other medical units in the hospital.
At the same time, we capture the two most commonly used modes of operation --- blocking and holding of new patients. \\
\\*
\noindent
\textbf{Two-fold square-root staffing rule.}
Our main goal is to provide staffing policies for the ED that high resource utilization, while at the same time maintain good quality-of-care.
This goal relates to the philosophy of the Quality-and-Efficiency-Driven (QED) regime that is the recurring theme of this thesis.
In this chapter, we obtain asymptotic results for the Erlang-R model with blocking in the QED regime (Section \ref{sec:QED_limit_block}).
Following \cite{Jennings2008}, we employ a two-fold QED staffing policy: $s=R_1 +\beta \sqrt{R_1}$ for the number of nurses and $n=R_1/r+\gamma \sqrt{R_1/r}$ for the number of patients in the system (beds), where $\beta$ and $\gamma$ are constants, $R_1$ is the offered load of the servers (nurses) and $r$ is the fraction of time a patient spends in the needy state.
We establish limiting expressions for performance measures, such as the probability of delay and blocking, in the form of explicit functions that depend solely on $\beta$ and $\gamma$.
In deriving these limit results, we use the available product-form solution for the stationary distribution.
Likewise, we pursue QED performance for the Erlang-R model with holding.
However, a direct analytic approach is obstructed by the absence of product-form solutions.
We provide two solutions for establishing QED behavior.
First, we provide stochastic performance bounds that stay meaningful in the QED regime, which demonstrate the non-degenerate behavior of the two-fold scaling in the large-system limit.
Second, we develop a heuristic method that quantifies the difference between the holding model and the blocking model.
This method is to a large extent related to the asymptotic approximation method for retrial queues discussed in Chapter 4, in the sense that we approximate the model with holding through the model with blocking, yet with an increased arrival rate.
The increase in arrival rate turns out to be the solution of a fixed-point equation.
Using our results on the asymptotic behavior of the model with blocking in the QED regime, we then obtain approximative QED performance measures for the model with holding.
These theoretical findings ultimately yield algorithms for dimensioning and time-varying staffing. \\
\\*
\textbf{Structure of the chapter.}
We first review related literature on the subject of staffing in health care environments in Section \ref{sec:ed_literature}.
In Section \ref{sec:modeldescription}, we introduce the mathematical models more formally, and deduce preliminary results on their stability conditions and relative performance.
Section \ref{sec:QED_scaling} describes the scaling regime we use for our asymptotic study of the restricted Erlang-R models, and Sections \ref{sec:QED_limit_block} and \ref{sec:QED_limit_holding} present our main theoretical findings.
We turn to dimensioning problems in Section \ref{sec:dimensioning}, and show how our asymptotic QED results can be used to make resource allocation decisions in realistic settings.
Section \ref{sec:analysis_chapter5} is devoted to the numerical and comparative analysis of the restricted Erlang-R models, and also shows how our method can be applied in time-varying environments through a case study.
We summarize our findings and give directions for future research in Section \ref{sec:conclusion}.
\section{Literature review}
\label{sec:ed_literature}
Due to increasing demand and tightening budgets in health care, there is a growing need for efficient workforce management \citep{Green2008}. Personnel (nurse and physician) expenditure is one of the biggest factors in hospital costs \citep{Kazahaya2005}, and inadequate nursing levels have been mentioned as a significant factor in medical errors and ED overcrowding. In order to establish appropriate nursing levels, a staffing policy requires assessment of a wide range of variables, such as differing nurse expertise and patient acuity during the day. Current methods, such as the minimum nurse-to-patient ratios, are often too inflexible to capture those varying conditions. The American Hospital Association (AHA) and others call for dynamic staffing policies that can deal with the complex and evolving nature of health care \citep{AHA2007}.
Workforce management in health care systems has been studied extensively; see \cite{Denton2013,Hall2006,Hall2012} for overviews.
In recent years it has become apparent that queueing models can be helpful in developing staffing and routing recommendations, not just for large-scale service systems, but also for the small and complicated health care systems.
The first to try such an approach through queueing models were Green et al.~\cite{Green2006,Green2008}, who used the single station stationary Erlang-C model to set staffing levels in EDs and panel sizes for clinics. Using a similar approach, Bekker \& de Bruin~\cite{Bekker2009a} used the Erlang-B model to determine bed allocation for medical wards.
The first to observe the significant impact of interrupted services in a health care setting were Jennings \& de V\'ericourt \cite{Jennings2008,Jennings2011}. Motivated by the need to set nurse-to-patient ratios for internal wards, they considered a closed queueing system with $s$ nurses and $n$ beds. This is essentially the Erlang-C model with the additional restriction that a finite population of the $n$ patients requires care. In their model, all beds are always occupied, and patients alternate between two phases: the needy phase where patients require service of a nurse and the content phase where they do not; see Figure \ref{fig:Jennings}. The system dynamics of the restricted Erlang-R model are equivalent to those of the closed ward model of \cite{Jennings2008} if the holding queue would never be empty.
Campello et al.~\cite{Campello2016} analyzed a similar operational decision, referred to as ED case management, which determines the maximal number of patients a physician should handle in parallel. They also used queueing networks and analyzed the stationary distribution. Note that in practice such a decision is not only affected by operational measurements such as waiting times, but also by psychological constraints that limit physician capability to manage multiple tasks (patients) in parallel.
KC \cite{diwas} provided empirical evidence that physicians should not treat more than 6-7 patients at the same time. Therefore, many hospitals in the US restrict entrance to EDs even if beds are available if physicians are overloaded.
We too consider such constraints, and analyze their impact on performance. We take a different approach than \cite{Campello2016}; instead of analyzing numerically steady-state distributions, we develop many-server approximations that can produce insight into the system dynamics, and can be incorporated into time-varying staffing procedures; see Section \ref{sec:case_study}.
\begin{figure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=1]
\draw [dashed, thick] (-0.5,-0.1) rectangle (3.5,2.85) node[right] {\footnotesize $n$};
\draw [thick,->] (1.1,0.5) -- (0,0.5) -- (0,2) -- (0.6,2);
\draw [thick,->] (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw [thick] (0.6,1.7) -- (1.6,1.7) -- (1.6,2.3) -- (0.6,2.3);
\draw [thick] (1,1.7) -- (1,2.3);
\draw [thick] (1.2,1.7) -- (1.2,2.3);
\draw [thick] (1.4,1.7) -- (1.4,2.3);
\draw (2.4,2) -- (3,2) -- (3,0.5) -- (1.9,0.5);
\draw (1.5,0.5) circle [radius=0.4] node[above=0.3cm] {\footnotesize $p/\delta$} ;
\draw (2,2) circle [radius=0.4] node[above=0.3cm] {\footnotesize exp($\mu$)};
\node at (1.5,0.5) {\footnotesize $\infty$};
\node at (2,2) {\footnotesize $s$};
\end{tikzpicture}
\caption{The closed ward model.}
\label{fig:Jennings}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\draw [thick, ->] (0,4.5) node[above=0.3cm,right] {\footnotesize Pois$(\lambda)$} -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5];
\draw [thick, ->] (4.75,4.5) -- node[above=0.3cm,right] {\footnotesize $1-p$} (7,4.5);
\draw [thick,->] (5.75,4.5) -- node[right] {\footnotesize $p$} (5.75,2) -- (4,2);
\draw [thick] (3.5,2) circle [radius=0.5];
\draw [thick,->] (3,2) -- (1.5,2) -- (1.5,4.5);
\node at (4.25,4.5) {\footnotesize $s$};
\node at (3.5,2) {\footnotesize $\infty$};
\node [above] at (4,5.15) {\footnotesize exp($\mu$)};
\node [above] at (3.5,2.4) {\footnotesize exp($\delta$)};
\end{tikzpicture}
\caption{The Erlang-R model.}
\label{fig:ErlangR}
\end{subfigure}
\caption{Related queueing models.}
\end{figure}
The model in~\cite{Jennings2008,Jennings2011} was developed for modeling internal dynamics within an internal ward. However, in the ED, beds are not constantly occupied and the utilization level depends on the flow of patients that arrive from outside the system.
Yom-Tov \& Mandelbaum \cite{YomTov2014} highlight the interrupted services while accounting for the transient nature of patient's arrival process, and introduced the Erlang-R model as a model for an ED. The Erlang-R model is an open two-station queueing network that has the same layout as the restricted Erlang-R model, except that all patients find a bed available upon arrival, see Figure \ref{fig:ErlangR}. In both models patients experience the interrupted services, but the Erlang-R model has no further restrictions on the bed capacity, hence neglecting the finite-size effects. Yom-Tov \& Mandelbaum \cite{YomTov2014} showed, using a simulator tailored to an Israeli ED, that the complicated small ED dynamics can be captured using the relatively simple Erlang-R model, and hence, its recommendations can be implemented in ED workforce management.
Although the feature of interrupted services is present in many systems, it is particularly important for modeling EDs, because the duration of the interruption is typically much longer than the time patients require care from a nurse. This explains why the Erlang-R model is considered to be the canonical model for EDs. The restricted Erlang-R model with holding/blocking thus extends the Erlang-R model with finite-size constraints which, like interrupted services, are expected to have a decisive impact on performance.
\section{Models and performance measures}
\label{sec:modeldescription}
\subsection{Three-dimensional Markov process}
\label{sec:Markov_process}
Since in the restricted Erlang-R model described above the arrival process is taken Poisson, and all service and content times are assumed independent and exponential, the system can be characterized in terms of a Markov process.
Let $Q(t) = (H(t),Q_1(t),Q_2(t))$ represent the number of patients in the \emph{holding}, \emph{needy} and \emph{content} state at time $t$, respectively.
In both variants, $n$ is the maximum number of patients admitted to system, we have $Q_1(t)+ Q_2(t)\leq n$ for all $t\geq 0$.
Due to the absence of holding patients in the Erlang-R model with blocking, $H(t)=0$ is enforced in this case, whereas $H(t)$ has unbounded support in the model with holding.
This distinction requires us to explore the stationary distribution of the two variants separately.
Before doing so, we introduce some additional notation.
We define
\begin{equation}
R_1 := \frac{\lambda}{(1-p)\mu}, \qquad R_2 := \frac{p\lambda}{(1-p)\delta},
\label{eq:R1_R2}
\end{equation}
where $R_1$ and $R_2$ can be interpreted as the offered workload brought towards the needy queue and the content (infinite-server) queue, respectively.
Furthermore, we define
\begin{equation}
r:= \frac{\delta}{\delta+p\mu},
\label{eq:delta}
\end{equation}
which is the fraction of time a patient spends in the needy state (in case she experienced no wait during her sojourn). \\
\\*
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.9]
\draw [thick] (-1.25,5) -- (0,5) -- (0,4) -- (-1.25,4);
\draw [thick] (0.5,4.5) circle [radius = 0.5] node {\footnotesize 1} node[above=0.5cm] {\footnotesize exp$(\lambda)$}
node[below =0.5cm] {\footnotesize \textit{Station 0}} ;
\draw [thick] (-0.25,4) -- (-0.25,5);
\draw [thick] (-0.5,4) -- (-0.5,5);
\draw [thick] (-0.75,4) -- (-0.75,5);
\draw [thick, ->] (1,4.5) -- (2.5,4.5);
\draw [thick] (2.25,5) -- (3.75,5) -- (3.75,4) -- (2.25,4);
\draw [thick] (3,4) -- (3,5);
\draw [thick] (3.25,4) -- (3.25,5);
\draw [thick] (3.5,4) -- (3.5,5);
\draw [thick] (4.25,4.5) circle [radius=0.5] node {\footnotesize $s$} node[above=0.5cm] {\footnotesize exp$(\mu)$}
node[below = 0.5cm] {\footnotesize \textit{Station 1}} ;
\draw [thick, ->] (4.75,4.5) -- node[right=0.8cm,above] {\footnotesize $1-p$} (6.5,4.5) -- (6.5,1.6) -- (-2,1.6) -- (-2,4.5) -- (-1.2,4.5);
\draw [thick,->] (5.75,4.5) -- node[left] {\footnotesize $p$} (5.75,2.5) -- (4,2.5);
\draw [thick] (3.5,2.5) circle [radius=0.5] node {\footnotesize $\infty$} node[above=0.4cm] {\footnotesize exp$(\delta)$}
node[below right = 0.35cm] {\footnotesize \textit{Station 2}} ;;
\draw [thick,->] (3,2.5) -- (1.5,2.5) -- (1.5,4.5);
\draw [thick, dashed] (-3,1.2) rectangle (7.25,5.75) node[right] {\footnotesize $n$};
\end{tikzpicture}
\caption{The Erlang-R model with blocking viewed as a closed Jackson network.}
\label{fig:ErlangR_blocking}
\end{figure}
\noindent
\textbf{Erlang-R model with blocking.}
In case of the blocking model, $Q(t)$ reduces to a finite-state Markov process $Q(t) = (Q_1(t),Q_2(t))$, where $Q_1(t)+Q_2(t)\leq n$ for all $t\geq 0$.
In fact, this is equivalent to the closed Jackson network depicted in Figure \ref{fig:ErlangR_blocking} with finite population $n$.
Station 1 in Figure \ref{fig:ErlangR_blocking} is an $M/M/s$ queue with service rate $\mu$, modeling the number of needy patients $Q_1(t)$.
Station 2 models the number of content patients $Q_2(t)$, and can therefore be represented as an infinite-server queue with service rate $\delta$.
A patient can enter the unit only if $Q_1(t)+Q_2(t)<n$.
Station 0---a single-server queue---moderates this as it only produces output at rate $\lambda$ in case its queue length is positive, i.e.\ if $n-Q_1(t)-Q_2(t)>0$.
Observe that because patients finding a full network are blocked, the number of patients in the system cannot grow beyond $n$.
Hence, the system is stable for all parameter settings, and hence a steady-state distribution exists. Moreover, the simplification of the model with blocking allows us to express the steady-state distribution of the system in explicit product-form.
Let $\pi_b(j,k)$ denote the steady-state probabilities of having $j$ needy and $k$ content patients in the system. Then,
\begin{equation}\label{eq:pih(i,j)}
\pi_b(j,k) = \left\{
\begin{array}{ll}
\pi_0\,\frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k, & ~~~\text{if }j+k \leq n,\\
0, & ~~~\text{else,}
\end{array}\right.
\end{equation}
where
\begin{equation*}
\kappa(j) := \left\{
\begin{array}{ll}
j! , & ~~\text{if }j \leq s,\\
s!\, s^{j-s}, &~~ \text{else,}
\end{array}\right.
\end{equation*}
and $
\pi_0^{-1} = \sum_{j+k\leq n} \frac{1}{\kappa(j)}\,\frac{1}{k!}\cdot R_1^j\cdot R_2^k$.\\
\\*
\textbf{Erlang-R model with holding.}
\label{ref:modelsoft}
The Erlang-R model with holding does not lead to a Jackson network with an elegant product-form solution for the steady-state distribution, because the holding queue cannot be modeled as a station that is independent from the other queues in the system.
%
However, we are able to describe the system as a two-dimensional Markov process without loss of information.
To see this, define $N:= \{N(t)\}_{t\geq 0}$ with $N(t) := H(t)+Q_1(t) + Q_2(t)$, the total number of patients in the system (including the holding queue).
Using the restriction $Q_1(t)+Q_2(t) \leq n$ together with the fact that no bed is left vacant if a patient is waiting in the holding queue, this yields
\begin{equation*}
H(t) = \left(N(t) - n\right)^+, \quad t\geq 0,
\end{equation*}
where $(\cdot)^+ := \max\{0,\cdot\}$.
For the same reason, $Q_2(t) = N(t) - Q_1(t)$ if $H(t)=0$, and $Q_2(t) = n-Q_1(t)$ otherwise.
In other words,
\begin{equation*}
Q_2(t) = \min\{N(t),n\} - Q_1(t), \quad t \geq 0.
\end{equation*}
Therefore, we can express the state of all three queues in the Erlang-R model with holding using a two-dimensional Markov process $X:= \{X(t)\}_{t\geq 0}$, where
\begin{equation*}
X(t) :=\left( N(t), Q_1(t) \right).
\end{equation*}
The process $X$ lives on the semi-infinite strip
\begin{equation*}
X(t) \in \left\{\,(i,j)\, |\, j \leq \min\{i,n\}, i\in \mathbb{N}_0, j \in \{0,1,\ldots,n\}\, \right\},
\end{equation*}
and belongs to the class of Quasi-Birth-Death (QBD) processes.
The reader is referred to Appendix~\ref{app:QBDdescription} for a detailed description of this process, in terms of its transition diagram and generator matrix.
Contrary to the model with blocking, the system with holding \emph{can} become unstable in case capacity is insufficient to satisfy patient demand.
\begin{proposition}\label{prop:StabilityCondition}
The Erlang-R model with holding is stable if and only if
\begin{equation}
\frac{\lambda}{(1-p)\mu s} < \frac{ \sum_{i=0}^s \frac{i}{s}\binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
{ \sum_{i=0}^s \binom{n}{i} \left(\frac{\delta}{p\mu}\right)^i + \sum_{i=s+1}^n \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p\mu}\right)^i}
=: \rho_{\max}(s,n).
\label{eq:StabilityCondition}
\end{equation}
\end{proposition}
The proof is given in Appendix~\ref{app:stability} and follows from the general theory for QBD processes.
Observe that $\rho_{\max}(s,n)$ poses an upper bound on the occupancy level of the servers in the holding model, which is clearly smaller than 1 for all $s$ and $n$.
In addition, this implies that the maximum workload $R_{\max}(s,n) := s\cdot\rho_{\max}(s,n)$ the system is able to handle is strictly less than $s$.
If we compare this to the open Erlang-R model, in which the maximal attainable workload equals $s$, we observe the effect of finite-size constraints on operational performance.
Figure \ref{fig:Rmax} shows the influence of both $s$ and $n$ on the maximum feasible workload in case $r=0.25$.
From these graphs, note that if $s\ll rn$, $R_{\max}$ grows almost linearly with $s$.
Furthermore, $R_{\rm max}(s,n)$ is increasing in $n$ for $s$ fixed.
A logical practical consequence is that a larger number of beds allows for a larger patient volume to enter the ED with the same number of nurses.
Moreover, $R_{\rm max}(s,n)$ is increasing in $s$, but as in Figure \ref{fig:Rmax_a}, adding an extra nurse does not increase the stability region in case $n$ is too tight.
Conversely, adding extra beds does not increase $R_{\rm max}(s,n)$ if the number of nurses does not allow for an increase in offered load, see Figure \ref{fig:Rmax_b}.
Additionally, it is easily verified that $R_{\rm max}(s,n)$ is upper bounded by both $s$ and $R_{\rm max}(n,n) = rn$. Therefore, a careful balance is called for between servers (nurses) and beds, so that resources will be efficiently utilized. We observe that when the ratio $s/n\approx r$, the system is better balanced.
We will propose an appropriate balance between resources by defining a synchronized QED capacity recommendation for both servers and beds in Section \ref{sec:QED_scaling}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $s$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*] table[x=s,y=n20] {tikz/stability/r025_n_fixed.txt};
\addplot[col3,thick,mark=*] table[x=s,y=n40] {tikz/stability/r025_n_fixed.txt};
\addplot[col4,thick,mark=*] table[x=s,y=n60] {tikz/stability/r025_n_fixed.txt};
\addplot[col5,thick,mark=*] table[x=s,y=n80] {tikz/stability/r025_n_fixed.txt};
\legend{$n=20$,$n=40$,$n=60$,$n=80$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $s$.}
\label{fig:Rmax_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 100,
ymin = 0,
ymax = 20,
grid = none,
axis line style={->},
axis lines = left,
xlabel = $n$,
ylabel = {$R_{\rm max}(s,n)$},
legend cell align=left,
legend pos = north west
]
\addplot[col1,thick,mark=*,mark repeat = 2] table[x=n,y=s5] {tikz/stability/r025_s_fixed.txt};
\addplot[col3,thick,mark=*,mark repeat = 2] table[x=n,y=s10] {tikz/stability/r025_s_fixed.txt};
\addplot[col4,thick,mark=*,mark repeat = 2] table[x=n,y=s15] {tikz/stability/r025_s_fixed.txt};
\addplot[col5,thick,mark=*,mark repeat = 2] table[x=n,y=s20] {tikz/stability/r025_s_fixed.txt};
\legend{$s=5$,$s=10$,$s=15$,$s=20$}
\end{axis}
\end{tikzpicture}
\caption{$R_{\rm max}$ as a function of $n$.}
\label{fig:Rmax_b}
\end{subfigure}
\caption{The maximum achievable workload in the restricted Erlang-R model with holding for $r=0.25$.}
\label{fig:Rmax}
\end{figure}
Provided that the system is stable, the stationary distribution of the QBD process $X$ can be obtained numerically by the matrix geometric method \citep{Neuts1981}.
Subsequently, we can derive the stationary distribution of the original $Q(t)$, denoted by $\pi_h(\cdot,\cdot,\cdot)$.
\subsection{Performance measures}
\label{sec:performance_metrics}
In this work, we concentrate on five performance measures that are central to our analysis.
In the definitions that follow, we present expressions for these measures in terms of a general three-dimensional measure $\pi$, which one can replace by either $\pi_b$ or $\pi_h$, depending on the scenario considered.
In the remainder of this work, we will augment the measures related to the Erlang-R model with blocking and holding by the superscript $b$ and $h$, respectively\footnote{In line with $H(t)=0$, we use $\pi_b(i,j,k) = \pi_b(j,k)$ if $i=0$, with $\pi_b(j,k)$ as in \eqref{eq:pih(i,j)}, and $\pi_b(i,j,k) = 0$ otherwise, when considering the model with blocking.}.
As relevant performance measures, we consider the probability of holding (cq.\\ \noindent blocking) at entering the system, the probability of delay at the needy queue, expected waiting time for a nurse, utilization of nurses and utilization of beds:
\begin{equation}
\mathbb{P}({\rm hold}) = \sum_{i=0}^\iy \sum_{j=0}^n \pi(i,j,n-j), \qquad
\mathbb{P}({\rm delay}) \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \pi(i,j,k),
\label{eq:delay_probability}
\end{equation}
\begin{equation}
\label{eq:EW_exact}
\mathbb{E} [W] \approx \sum_{i=0}^\infty\sum_{j=s}^{n}\sum_{k=0}^{n-j} \frac{\max\{0,j-s+1\}}{\mu}\,\pi(i,j,k),
\end{equation}
\begin{equation}
\label{eq:utilization}
\rho_s = \frac{1}{s}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{j,s\} \pi(i,j,k), \qquad
\rho_n = \frac{1}{n}\,\sum_{i=0}^\infty \sum_{j=0}^n \sum_{k=0}^{n-j} \min\{i,n\} \pi(i,j,k).
\end{equation}
It should be stressed that the above expression for the delay probability and the expected waiting time for a nurse are not exact. For the blocking model one can use the Arrival Theorem, see e.g.~\cite{Chen2001a}, whereby the exact expression sums up to $n-1$ instead of $n$.
Since we consider the system as $n\to\infty$, this discrepancy becomes negligible.
For the holding model, a similar argument holds.
We will therefore use the expressions in \eqref{eq:delay_probability}-\eqref{eq:utilization} as definitions for the performance measures.
\subsection{Stochastic bounds}
\label{sec:bounds}
Although the two variants of the Erlang-R model differ with respect to the admission policy, and require different mathematical treatment, we would like to be able to capture their relative performance.
We substantiate the intuition that the holding room leads to more patients in the ED, in the following result.
\begin{proposition}\label{thm:stochasticordering}
Let $Q_1^b$, $Q_2^b$, $Q_1^h$, $Q_2^h$ denote the nurse and content queue length processes in the Erlang-R model with blocking and holding, respectively.
Let $H(0) = 0$, $Q_1^b(0)=Q_1^h(0)$ and $Q_2^b(0)=Q_2^h(0)$. For all $t\geq 0$,
\begin{align}
Q_1^b(t) + Q_2^b(t) &\preceq_{\rm st} Q_1^h(t) + Q_2^h(t) \preceq_{\rm st} n ,\\
Q_2^b(t) &\preceq_{\rm st} Q_2^h(t),\\
Q_1^b(t) &\preceq_{\rm st} Q_1^h(t) + H(t),
\end{align}
where $X\preceq_{\rm st} Y$ implies $\mathbb{P}(X\geq k) \leq \mathbb{P}(Y\geq k)$ for all $k\geq 0$.
\end{proposition}
\noindent
The proof of Proposition \ref{thm:stochasticordering} uses sample path coupling and can be found in Appendix \ref{app:stochastic_ordering}.
Note that as an immediate consequence, we have
\[ \mathbb{P}^b( {\rm block}) = \lim_{t\to\iy} \mathbb{P}\big( Q_1^b(t)+Q_2^b(t) \geq n \big) \leq \lim_{t\to\iy} \mathbb{P}\big( Q_1^h(t) + Q_2^h(t) \geq n \big) = \mathbb{P}^h( {\rm hold }) \]
and by similar reasoning $\rho^b_n \leq \rho_n^h$.
In other words, under similar offered load and capacity constraints, utilization levels for the nurses in the Erlang-R model with blocking are lower than in the Erlang-R model with holding.
Moreover, the total number of waiting patients in the setting with holding is stochastically larger than in the setting with blocking, and in the open Erlang-R model.
We further discuss the differences between both models in Section \ref{sec:dimensioning} and Section \ref{sec:analysis_chapter5}.
\section{Two-fold QED regime}
\label{sec:QED_scaling}
We do not want to waste capacity of either servers or beds without getting significant advantage in terms of performance.
We therefore take an asymptotic approach that lets the external arrival rate $\lambda$ grow to infinity, while scaling $s$ and $n$ accordingly.
In doing so, we intend to establish QED-type system behavior, i.e.\ high occupancy levels of both nurses and beds and good quality-of-service.
\subsection{Two-fold scaling rule}
In order to identify the scaling of $s$ and $n$ as $\lambda\to\infty$, we draw inspiration from the two-fold scaling rule used by Jennings \& de V\'ericourt \cite{Jennings2008} and Khudyakov et al.~\cite{Khudyakov2010}, which follows the celebrated square-root staffing principle.
This principle suggests that, in the most general setting, capacity should be equal to the expected offered load entering the system, let us say $R$, plus an additional variability hedge that is proportional to $\sqrt{R}$.
In the restricted Erlang-R model, we have two capacity sources, namely $s$ and $n$, which experience different relevant amounts of work.
The offered load the servers in the needy queue experience is given by $R_{\rm nurse} = R_1$, as in the regular Erlang-R model;
it does not change due to the finite-size effects, since all patients are served eventually. Hence, we only need to account for the interrupted services. It follows that the appropriate staffing rule for the nurses in the QED regime remains $s=R_1+\beta \sqrt{R_1}$ for some constant $\beta >0$.
To establish the bed capacity level, we need to reflect on the load offered to the beds. Observe that beds remain occupied both in needy and content states. This suggests that $R_{\rm bed} :=R_1+R_2=R_1/r$, with $R_1$ and $R_2$ as in \eqref{eq:R1_R2} and $r$ is the expected fraction of time a patient spends at the nurse station defined in \eqref{eq:delta}.
As a result, the appropriate staffing rule is $n=R_{\rm bed}+\gamma \sqrt{R_{\rm bed}}$ for some constant $\gamma>0$. In conclusion, the two-fold QED scaling rule is given by
\begin{equation}\label{eq:twofoldscaling}
\begin{array}{ll}
s &= R_1 + \beta \sqrt{R_1} + o(\sqrt{R_1}) \\
n &= \frac{R_1}{r}+\gamma \sqrt{\frac{R_1}{r}} + o(\sqrt{R_1})
\end{array}
\end{equation}
with $\beta,\gamma>0$ constants and $R_1:=\lambda/((1-p)\mu)$.
Recall that we saw in Figure \ref{fig:Rmax} that resources seem efficiently utilized if $s/n\approx r$.
Scaling \eqref{eq:twofoldscaling} is in line with this reasoning since
\[
\frac{s}{n} = r\left(1+ \frac{\beta - \gamma\sqrt{r}}{\sqrt{R_1}}+ O(1/R_1) \right) .
\]
\begin{remark}
In \cite{Jennings2008}, a similar scaling regime is considered, which only relates $s$ and $n$ through a square-root scaling, namely the regime $s = r n + \hat\gamma\sqrt{n}$,
which is equivalent to the second relation in \eqref{eq:twofoldscaling} if $\hat\gamma = \beta\sqrt{r} - \gamma r$.
Due to the absence of external arrivals in this closed system, they let the number of beds $n$ approach infinity as opposed to $\lambda$ in our settings.
Nevertheless, this results in the same asymptotic regime.
\end{remark}
Before turning to asymptotic expressions for the performance measures concerning the Erlang-R model with blocking or holding, we conduct a few numerical experiments to confirm that the scaling in \eqref{eq:twofoldscaling} indeed leads to desired QED behavior.
In Figure \ref{fig:sample_paths}, we plotted the sample paths of the three-dimensional queue length process of the holding model in which $\beta$ and $\gamma$ are fixed, and $R_1$ is increased.
Observe that the needy queue length $Q_1(t)$, plotted in orange in Figure \ref{fig:sample_paths}, fluctuates around the values $s$, and stabilizes for larger values of $R_1$.
This naturally implies that the server (nurses) utilization approaches 100\%, while the number of patients waiting is $O(\sqrt{R_1})$.
Furthermore, we see that the percentage of occupied beds also tends to 100\%, while the holding queue length remains small.
The holding queue is of much smaller order than $R_1$, which implies that the holding time of a patient becomes negligible as $R_1\to\iy$.
From these empirical findings we deduce that under scaling \eqref{eq:twofoldscaling} the restricted Erlang-R model exhibits QED behavior on two levels: Outside the facility while waiting for an available bed, and inside the facility while waiting for attention of a nurse.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 28,
ytick = {0,5,10,15,20,25},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\definecolor{col1}{rgb}{0.368417, 0.506779, 0.709798}
\addplot[very thick,col5] file {tikz/sample_paths/R5_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R5_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R5_total.txt};
\addplot[very thick,dashed] coordinates {
(0,7)
(200,7)
};
\addplot[very thick,dashed] coordinates {
(0,24)
(200,24)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=5$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 128.333,
ytick = {0,20,40,60,80,100,120},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {tikz/sample_paths/R25_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R25_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R25_total.txt};
\addplot[very thick,dashed] coordinates {
(0,30)
(200,30)
};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=25$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.51]
\begin{axis}[
xmin = 0,
xmax = 200,
ymin = 0.0,
ymax = 490,
ytick = {0,100,200,300,400},
grid = none,
axis line style={->},
axis lines = left,
xlabel = $t$,
legend cell align=left,
legend pos = north east
]
\addplot[very thick,col5] file {tikz/sample_paths/R100_holding.txt};
\addplot[very thick,col2] file {tikz/sample_paths/R100_service.txt};
\addplot[very thick,col4] file {tikz/sample_paths/R100_total.txt};
\addplot[very thick,dashed] coordinates {
(0,110)
(200,110)
};
\addplot[very thick,dashed] coordinates {
(0,420)
(200,420)
};
\end{axis}
\end{tikzpicture}
\caption{$R_1=100$}
\end{subfigure}
\caption{Sample paths of $H(t)$ (blue), $Q_1(t)$ (orange) and $Q_1(t)+Q_2(t)$ (green) of the Erlang-R model with holding with parameters $\mu = 1$, $\delta=0.25$, $p=0.75$ and $\beta=\gamma=1$. The staffing levels $s$ and $n$ are depicted by the dashed lines.}
\label{fig:sample_paths}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.7,
ytick = {0,0.1,...,0.7},
xlabel = $\lambda$,
grid = both,
axis line style={->},
axis lines = left,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/delayProbErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/delayProbYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/delayProbJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Delay probability nurse}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.0,
ymax = 0.3,
ytick = {0,{0.05},0.1,0.15,0.2,0.25,3},
grid = both,
axis line style={->},
tick label style={/pgf/number format/fixed},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = north east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/EWErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/EWYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/EWJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Expected wait}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.53]
\begin{axis}[
xmin = 0,
xmax = 145,
ymin = 0.7,
ymax = 1.02,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\lambda$,
legend cell align=left,
legend pos = south east
]
\addplot[thick,col1,mark=*] file {tikz/empirical/rhoErlangH.txt};
\addplot[thick,col4,mark=*] file {tikz/empirical/rhoYomTov.txt};
\addplot[thick,col5,mark=*] file {tikz/empirical/rhoJennings.txt};
\small
\legend{Holding,Blocking,Closed ward};
\end{axis}
\end{tikzpicture}
\caption{Nurse utilization}
\end{subfigure}
\caption{Asymptotic behavior of the restricted Erlang-R model with holding and blocking, and the closed ward model for $\mu=1$, $\delta = 0.2$, $p=0.8$ and $\beta=\gamma=0.5$.}
\label{fig:empiricalAsymptotics}
\end{figure}
We also check how the Erlang-R model with blocking or holding and the closed ward model of \cite{Jennings2008} relate under scaling \eqref{eq:twofoldscaling}.
In Figure~\ref{fig:empiricalAsymptotics}, we plot the performance measures, obtained through simulation, for the three models in which we fix $\beta=\gamma=0.5$ and vary the arrival rate $\lambda$.
First, we see that $\mathbb{P}({\rm delay})$ stabilizes as $\lambda\to\iy$ in all three models under scaling \eqref{eq:twofoldscaling}, and the delay probability of the model with holding lies in between the other two.
Second, note that the expected waiting time for a nurse in all models converges to 0 as $\lambda$ increases. In fact, the rate of decay is similar in all three models.
We observe that $\rho_s$ approaches unity in all models, and the rate of convergence seems again comparable.
Finally, and most importantly, we notice an ordering between the three models.
Namely, in all performance measures considered in Figure \ref{fig:empiricalAsymptotics}, Erlang-R with holding appears to be upper bounded by the closed ward and lower bounded by the Erlang-R with blocking.
In a multitude of parameter settings of $(\beta,\gamma)$, we have seen the same ordering, leading to the following conjecture:
\begin{conjecture}\label{conj:stochorder}
Let $Q^b_1(\iy)$, $Q_1^h(\iy)$ and $Q_1^J(\iy)$ denote the stationary number of needy patients in the Erlang-R model with blocking, holding and the closed ward, respectively. Then,
\begin{equation}
Q_1^b(\iy) \preceq_{\rm st} Q_1^h(\iy) \preceq_{\rm st} Q_1^J(\iy).
\end{equation}
\end{conjecture}
Observe that Conjecture \ref{conj:stochorder} poses a stronger statement than the third assertion in Proposition \ref{thm:stochasticordering}.
The latter does give an upper bound to $Q_1^h(\iy)$ in terms of $Q_1^b(\iy)$, albeit supplemented with the stationary holding queue length.
\subsection{QED limits for Erlang-R with blocking}
\label{sec:QED_limit_block}
We now continue our analysis by examining its limiting behavior under scaling \eqref{eq:twofoldscaling}, and obtain QED limits for some performance measures of the Erlang-R model with blocking.
Using the explicit expressions for the blocking model in \eqref{eq:pih(i,j)}, we derive the limiting values of the relevant performance measures defined in Section \ref{sec:performance_metrics} in terms of $\beta$ and $\gamma$.
\begin{theorem}\label{thm:limits_YT}
Let $s$ and $n$ scale as in \eqref{eq:twofoldscaling} with ${-}\infty<\beta<\infty,\,\gamma>0$ as $\lambda\to\infty$. Then, if $\beta \neq 0$,
\begin{align}
g^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay})\nonumber \\
\label{eq:yt_limit_delay}
&=
\left(1 +
\frac{ \beta \, \int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) }
{\varphi(\beta)\Phi(\eta) - \varphi(\sqrt{\beta^2+\eta^2}){\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)}
\right)^{-1},\\
f^b(\beta,\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber\\
\label{eq:yt_limit_block}
&=
\frac{
\sqrt{r}\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \varphi(\sqrt{\beta^2+\eta^2})\,{\rm e}^{\frac{1}{2} \omega^2} \Phi(\omega)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},\\
h^b(\beta,\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay}
&=
\frac{
\frac{\varphi(\beta)\Phi(\eta)}{\beta^2} +
\left(\frac{\beta}{r}-\frac{\gamma}{\sqrt{r}}-\frac{1}{\beta}\right)\,\frac{\varphi(\sqrt{\eta^2+\beta^2})}{\beta}\, {\rm e}^{\tfrac{1}{2}\omega^2}\, \Phi(\omega)
- \sqrt{\frac{1-r}{r}}\,\frac{\varphi(\beta)\varphi(\eta)}{\beta}
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\frac{\varphi(\beta)\Phi(\eta)}{\beta} - \frac{\varphi(\sqrt{\beta^2+\eta^2})}{\beta}{\rm e}^{\tfrac{1}{2} \omega^2} \Phi(\omega)
},
\end{align}
and if $\beta=0$,
\begin{align}
g^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \mathbb{P}^b({\rm delay}) \nonumber\\
\label{eq:yt_limit_delay_beta0}
&=
\left(1+
\frac{
\int_{-\iy}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t)
}{
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
}
\right)^{-1},\\
f^b_0(\gamma)
&:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{P}^b({\rm block}) \nonumber \\
\label{eq:yt_limit_block_beta0}
&=
\frac{
\sqrt{r}\,\varphi(\gamma)\Phi(-\omega\sqrt{r}) + \frac{1}{\sqrt{2\pi}} \Phi(\eta)
}{
\int_{-\iy}^\beta \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) +
\sqrt{\frac{1-r}{r}} \frac{1}{\sqrt{2\pi}}\,\left(\eta \,\Phi(\eta) + \varphi(\eta) \right)
},\\
h_0^b(\gamma) &:= \lim_{\lambda\to\iy} \sqrt{R_1}\cdot\mathbb{E}[W] \nonumber\\
\label{eq:yt_limit_Edelay_beta0}
&= \frac{1}{2\mu}\, \frac{ \left( \gamma^2/r+1\right) \Phi(\eta) + \eta \varphi(\eta) }
{ \frac{r}{1-r} \sqrt{2\pi} \int_{-\infty}^0 \Phi\left(\frac{\gamma-t\sqrt{r}}{\sqrt{1-r}}\right)\, {\rm d}\Phi(t) + \sqrt{\frac{r}{1-r}} \left(\eta \Phi(\eta)+\varphi(\eta)\right)},
\end{align}
where $\eta = \frac{\gamma - \beta\sqrt{r}}{\sqrt{1-r}}$ and $\omega := \frac{\gamma - \beta/\sqrt{r}}{\sqrt{1-r}}$.
\end{theorem}
The proof of Theorem \ref{thm:limits_YT} is given in Appendix C of \cite{YomTov2010} under a parameter transformation.
Theorem \ref{thm:limits_YT} proves that the scaling \eqref{eq:twofoldscaling} results in QED behavior: the probability of waiting in Equations \eqref{eq:yt_limit_delay} and \eqref{eq:yt_limit_delay_beta0} converges to a limit that is strictly between 0 and 1.
Notice that all limits in Theorem \ref{thm:limits_YT} are functions of three parameters: $\beta$ and $\gamma$, which are decision variables, and the fraction of needy time $r$, which is dictated by the physics of the system. Furthermore, the theorem also shows that the probability of blocking (Equations \eqref{eq:yt_limit_block} and \eqref{eq:yt_limit_block_beta0}) is of order $1/\sqrt{R_1}$.
For example, assume that the fraction of needy time $r$ is $0.5$ and the system is large (100 servers).
Using Figure \ref{fig:pdelay_pblock}, we observe that, by choosing the pair $\gamma = 1$ and $\beta = 0.245$, we actually aim at a probability of getting served immediately to be 40\%. At the same time, the probability of getting immediately a bed is 97\%.
Thus, waiting inside the ED occurs at a reasonable level, while wait outside the facility becomes negligible.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
tick label style={/pgf/number format/fixed},
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,0.5)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,0.99)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {tikz/limit_probabilities_delay.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {tikz/limit_probabilities_delay.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:pdelay_pblock_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
tick label style={/pgf/number format/fixed},
ylabel = {$f(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.4,1)}},
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 1.95,1.98)},anchor = north east}
]
\addplot[thick,col1] table[x=beta,y=g_min1] {tikz/limit_probabilities_block.txt};
\addplot[thick,col3] table[x=beta,y=g_0] {tikz/limit_probabilities_block.txt};
\addplot[thick,col4] table[x=beta,y=g_1] {tikz/limit_probabilities_block.txt};
\addplot[thick,col5] table[x=beta,y=g_2] {tikz/limit_probabilities_block.txt};
\legend{{$\gamma = -1$},{$\gamma = 0$},{$\gamma = 1$},{$\gamma = 2$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\label{fig:pdelay_pblock_b}
\end{subfigure}
\caption{Asymptotic delay and scaled blocking probability for $r=0.5$ based on Theorem \ref{thm:limits_YT}. }
\label{fig:pdelay_pblock}
\end{figure}
Theorem \ref{thm:limits_YT} further shows that the expected waiting (Equations \eqref{eq:yt_limit_Edelay} and \eqref{eq:yt_limit_Edelay_beta0})
is of order $1/\sqrt{R_1}$ too and hence vanishes in the large-system limit.
We see from Theorem \ref{thm:limits_YT} that achieving target service levels is always an interplay between $\beta$ and $\gamma$.
Figure \ref{fig:pdelay_pblock_a} shows for instance that in order to keep $\mathbb{P}({\rm delay})\in (0.25,0.75)$, choosing $\gamma=-1$ requires $\beta$ to stay within the range $[-1.4,-0.5]$, while $\gamma=1$ corresponds to values of $\beta$ in $[-0.4,0.5]$.
While the two-fold scaling rule in \eqref{eq:twofoldscaling} automatically captures the right dimensioning ratio as the system scales up, Theorem \ref{thm:limits_YT} shows that the parameters $\beta$ and $\gamma$ provide a means to fine-tune the performance.
Figure \ref{fig:pdelay_pblock_b} confirms how adding nurses, i.e.~increasing $\beta$, does not improve the blocking probability if the number of beds, i.e.~$\gamma$, is too tight.
This is in accordance with our previous observations in Figure \ref{fig:Rmax} for the exact steady-state distribution.
To test the accuracy of the asymptotic results in Theorem \ref{thm:limits_YT} as approximations in a realistic setting, we plot in Figure \ref{fig:accuracy_blocking} the exact probability of delay and blocking for an Erlang-R model with $R=8$ and $r=0.25$, as a function of $s$. The exact probabilities are given by Equation
\eqref{eq:delay_probability}, and their respective asymptotic approximations are based on Theorem \ref{thm:limits_YT}.
Despite the realistic moderate size of the system ($R=8$), we see that the QED approximations are remarkably accurate for many settings $(s,n)$.
This fast relaxation is in line with observations made earlier in the QED literature \cite{Borst2004,Janssen2011}.
\begin{table}[htb]
\centering
\begin{tabular}{|r|rrrr|}
\hline
& $\mu$ & $\delta$ & $p$ & $r$ \\
\hline
Case 1 & 1 & 0.10 & 0.90 & 0.10 \\
Case 2 & 1 & 0.25 & 0.75 & 0.25\\
Case 3 & 1 & 0.50 & 0.50 & 0.50 \\
\hline
\end{tabular}
\caption{Parameter settings for numerical experiments.}
\label{tab:parameter_settings}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1270 & 0.0900 & 0.2283 & 0.1553 & 0.0212 & 0.1085 \bigstrut[t]\\
10 & 0.1340 & 0.0910 & 0.1919 & 0.1628 & 0.0206 & 0.1205 \\
25 & 0.1981 & 0.0945 & 0.1614 & 0.2356 & 0.0216 & 0.2145 \\
50 & 0.1513 & 0.0963 & 0.1588 & 0.1830 & 0.0205 & 0.1496 \\
100 & 0.1880 & 0.0956 & 0.1532 & 0.2231 & 0.0224 & 0.2055 \\
250 & 0.1797 & 0.0971 & 0.1399 & 0.2143 & 0.0219 & 0.2057 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1767} & \textit{0.0981} & \textit{0.1437} & \textit{0.2108} & \textit{0.0217} & \textit{0.1947} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0237 & 0.0868 & 0.0282 & 0.0322 & 0.0192 & 0.0391 \bigstrut[t]\\
10 & 0.0206 & 0.0872 & 0.0188 & 0.0278 & 0.0183 & 0.0264 \\
25 & 0.0277 & 0.0876 & 0.0123 & 0.0363 & 0.0174 & 0.0174 \\
50 & 0.0185 & 0.0913 & 0.0116 & 0.0249 & 0.0175 & 0.0166 \\
100 & 0.0232 & 0.0888 & 0.0103 & 0.0303 & 0.0183 & 0.0145 \\
250 & 0.0203 & 0.0905 & 0.0079 & 0.0267 & 0.0179 & 0.0109 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0188} & \textit{0.0914} & \textit{0.0084} & \textit{0.0247} & \textit{0.0177} & \textit{0.0118} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 1. The last row presents the asymptotic approximations.}
\label{tab:numerics_case1}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0911 & 0.1538 & 0.0479 & 0.1431 & 0.0345 & 0.0909 \bigstrut[t]\\
10 & 0.1010 & 0.1498 & 0.0560 & 0.1520 & 0.0326 & 0.1025 \\
25 & 0.1594 & 0.1509 & 0.1058 & 0.2192 & 0.0405 & 0.1785 \\
50 & 0.1201 & 0.1506 & 0.0726 & 0.1697 & 0.0381 & 0.1248 \\
100 & 0.1514 & 0.1539 & 0.1001 & 0.2088 & 0.0398 & 0.1704 \\
250 & 0.1459 & 0.1524 & 0.0957 & 0.2003 & 0.0397 & 0.1618 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1429} & \textit{0.1569} & \textit{0.0940} & \textit{0.1976} & \textit{0.0391} & \textit{0.1617} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0130 & 0.1484 & 0.0044 & 0.0277 & 0.0294 & 0.0109 \bigstrut[t]\\
10 & 0.0121 & 0.1432 & 0.0042 & 0.0244 & 0.0267 & 0.0098 \\
25 & 0.0182 & 0.1383 & 0.0070 & 0.0319 & 0.0295 & 0.0141 \\
50 & 0.0119 & 0.1415 & 0.0043 & 0.0216 & 0.0301 & 0.0090 \\
100 & 0.0154 & 0.1413 & 0.0059 & 0.0270 & 0.0290 & 0.0119 \\
250 & 0.0136 & 0.1403 & 0.0051 & 0.0236 & 0.0291 & 0.0103 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0126} & \textit{0.1445} & \textit{0.0048} & \textit{0.0220} & \textit{0.0284} & \textit{0.0097} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 2. The last row presents the asymptotic approximations.}
\label{tab:numerics_case2}
\end{table}
\begin{table}[h] \centering
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 1,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0547 & 0.1945 & 0.0221 & 0.1181 & 0.0604 & 0.0617 \bigstrut[t]\\
10 & 0.0579 & 0.2158 & 0.0237 & 0.1325 & 0.0526 & 0.0746 \\
25 & 0.1113 & 0.2086 & 0.0544 & 0.1959 & 0.0641 & 0.1311 \\
50 & 0.0813 & 0.2050 & 0.0363 & 0.1523 & 0.0562 & 0.0933 \\
100 & 0.1060 & 0.2146 & 0.0509 & 0.1873 & 0.0632 & 0.1250 \\
250 & 0.1006 & 0.2179 & 0.0475 & 0.1820 & 0.0596 & 0.1214 \\
\hline
\multicolumn{1}{r|}{} & \textit{0.1011} & \textit{0.2185} & \textit{0.0478} & \textit{0.1792}& \textit{0.0605} & \textit{0.1199} \bigstrut[b]\\
\cline{2-7}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rrr|rrr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 1$} & \multicolumn{3}{c|}{$\beta = 2,\ \gamma = 2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{P}({\rm b})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0034 & 0.1888 & 0.0009 & 0.0175 & 0.0510 & 0.0057 \bigstrut[t]\\
10 & 0.0030 & 0.2093 & 0.0008 & 0.0172 & 0.0416 & 0.0058 \\
25 & 0.0070 & 0.1937 & 0.0020 & 0.0243 & 0.0440 & 0.0089 \\
50 & 0.0043 & 0.1946 & 0.0011 & 0.0163 & 0.0414 & 0.0056 \\
100 & 0.0061 & 0.1999 & 0.0017 & 0.0207 & 0.0431 & 0.0076 \\
250 & 0.0052 & 0.2037 & 0.0014 & 0.0185 & 0.0401 & 0.0067 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.0052} & \textit{0.2039} & \textit{0.0014} & \textit{0.0173} & \textit{0.0404} & \textit{0.0063} \bigstrut\\
\cline{2-7}\end{tabular}%
\caption{Exact numerical results for Erlang-R model with blocking for Case 3. . The last row presents the asymptotic approximations.}
\label{tab:numerics_case3}
\end{table}
We furthermore compare the asymptotic delay and blocking probability in the three scenarios given in Table \ref{tab:parameter_settings}.
In Tables \ref{tab:numerics_case1}--\ref{tab:numerics_case3} we compute the exact probabilities of delay and blocking through the explicit forms in \eqref{eq:delay_probability} for increasing values of the offered load, $R_1$.
The numerical results show that $g^b(\beta,\gamma)$, $f^b(\beta,\gamma)$ and $h^b(\beta,\gamma)$ provide accurate approximations to $\mathbb{P}({\rm delay})$, $\sqrt{R_1}\mathbb{P}({\rm block})$ and $\sqrt{R_1}\,\mathbb{E}[W]$ in pre-limit systems.
The quality of the approximations increases with $R_1$.
Naturally, fluctuations occur for relatively small values of $R_1$, because $s$ and $n$ need to be rounded to an integer.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {tikz/accuracy/accuracy_pdelay_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {tikz/accuracy/accuracy_pdelay_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 2,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col1,mark = *] table[x=s,y=n24] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,mark = *] table[x=s,y=n28] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,mark = *] table[x=s,y=n32] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=n36] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=n40] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col1,dashed] table[x=s,y=approx_n24] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col2,dashed] table[x=s,y=approx_n28] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx_n32] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx_n36] {tikz/accuracy/accuracy_pblock_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx_n40] {tikz/accuracy/accuracy_pblock_case2.txt};
\legend{{$n=24$},{$n=28$},{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability}
\end{subfigure}
\caption{Comparison of exact performance measures (solid) against asymptotic approximations (dashed) with $\beta=(s-R_1)/\sqrt{R_1}$ and $\gamma=(n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_blocking}
\end{figure}
\subsection{QED limits for Erlang-R with holding}
\label{sec:QED_limit_holding}
As explained in Section \ref{sec:QED_scaling}, the model with holding has no product-form steady-state distribution, which makes it hard (if not impossible) to obtain QED limits.
Instead, we derive QED approximations by exploiting a connection with the blocking model.
We first prove that under scaling \eqref{eq:twofoldscaling}, the upper bound on the utilization level of the nurses needed to achieve stability in the model with holding, as given in Proposition \ref{prop:StabilityCondition}, converges to unity as $R\to\infty$.
This facilitates high utilization levels of both nurses and beds, a key characteristic of the QED regime.
\begin{proposition}\label{prop:stability_convergence}
Let $s$ and $n$ scale with $R_1\to\infty$ as in \eqref{eq:twofoldscaling}. Then for $\lambda\to\infty$,
\[
\rho_{\max}(s,n) \to 1.
\]
\end{proposition}
The proof can be found in Appendix \ref{app:proof_stability_convergence}.
Combining Proposition \ref{prop:stability_convergence} with Proposition \ref{prop:StabilityCondition} shows that indeed the scaling we use results in a highly utilized system.
As observed before, the nature of the two variants of the model is similar up to the fact that a fraction of the patients is deferred on arrival in the setting with blocking, whereas all the arriving patients are eventually admitted into the system in the holding model.
This implies that, given $s$ and $n$, the nurses face an increased workload in case of a holding room.
In fact, Theorem \ref{thm:limits_YT} shows that the blocking probability is of order $1/\sqrt{R_1}$, yielding a volume of blocked patients of order $\sqrt{R_1}$ in setting with blocking.
Accordingly, if $R^b = R_1$ and $R^h$ denote the nominal load arriving to the nurses in the model with blocking and holding, respectively, we can argue that
\[R^h = R^b + \alpha \sqrt{R^b} + o(\sqrt{R^b}),\]
for some $\alpha>0$.
Notice that this additional load is of the same order as the safety staffing in the blocking model staffing rule \eqref{eq:twofoldscaling}.
As $s$ and $n$ remain unchanged, we rewrite \eqref{eq:twofoldscaling} in terms of $R^h$,
\begin{align}
s &= R^h + (\beta-\alpha)\sqrt{R^h} + o(\sqrt{R^h}), \nonumber \\
n &= \frac{R^h}{r} + \left(\gamma-\alpha/\sqrt{r}\right)\sqrt{\frac{R^h}{r}} + o(\sqrt{R^h}),
\label{eq:fixed_point_scaling}
\end{align}
where we have used $R^b = O(R^h)$.
Observe that the square-root principle prevails also after this substitution, albeit with different hedging parameters.
We therefore heuristically argue that the holding model under scaling \eqref{eq:twofoldscaling} with parameters $\beta$ and $\gamma$ mimics the blocking model with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, respectively.
Observe, however, that we have not yet specified the value of $\alpha$.
By definition, $\alpha\sqrt{R^b}$ is the expected volume of patients that would be rejected in the model with blocking, that is, $R^h$ times the probability of not being admitted to the ED directly.
By the construction in \eqref{eq:fixed_point_scaling}, this volume asymptotically equals $R^h \cdot \mathbb{P}^b({\rm block})$, with parameters $\beta-\alpha$ and $\gamma-\alpha/\sqrt{r}$, which by Theorem \ref{thm:limits_YT} is approximated by
\[f^b\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right) / \sqrt{R^h}\]
as $R^h$ grows large.
In conclusion, $\alpha$ is characterized as the solution of the fixed-point equation
\begin{equation}
\label{eq:fixedpoint}
\alpha = f^h\left(\beta-\alpha,\gamma-\alpha/\sqrt{r}\right),
\end{equation}
and as a result, we are able to approximate the nurse delay probability in the Erlang-R model with holding as
\begin{equation}
\mathbb{P}^h({\rm delay}) \approx g^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: g^h(\beta,\gamma).
\label{eq:fixed_point_Pwait}
\end{equation}
Likewise, the scaled mean waiting time for a nurse can be approximated by
\begin{equation}
\sqrt{R_1} \cdot \mathbb{E}[W] \approx h^b(\beta-\alpha,\gamma-\alpha/\sqrt{r}) =: h^h(\beta,\gamma).
\label{eq:fixed_point_Ewait}
\end{equation}
This also implies that the holding queue is $O(\sqrt{R_1})$.
Subsequently, we argue that the expected holding time (pre-entering wait) under the QED policy is $O(1/\sqrt{R_1})$ and hence asymptotically negligible.
We justify this claim numerically in Section \ref{sec:analysis_chapter5}.
\begin{remark}
\label{rem:holding_limit}
Notice that in the reasoning leading to \eqref{eq:fixedpoint}, we implicitly assumed that the additional volume $\alpha\sqrt{R^b}$ is an independent Poisson process, which is obviously not the case. Therefore, \eqref{eq:fixed_point_Pwait}-\eqref{eq:fixed_point_Ewait} are approximations for pre-limit systems that are not asymptotically correct as $R_1\to\iy$.
Nevertheless, the heuristic approach seems to performs well as we confirm numerically next.
\end{remark}
In Figure \ref{fig:accuracy_holding}, we repeat the numerical experiments of Figure \ref{fig:accuracy_blocking} for the model with holding.
Since the heuristic does not provide an approximation for the holding probability, Figure \ref{fig:accuracy_holding_b} only plots the simulated holding probabilities.
Those are provided to better understand the implication of operational decisions.
Recall that the holding system is only stable (i.e. $\mathbb{P}({\rm hold})<1$) if both $s>R_1=8$ and $n > R_1/r = 32$.
We nevertheless included the boundary case $n=32$ for illustrative purposes.
The graphs in Figure \ref{fig:accuracy_holding} show that the heuristic captures the congestion levels well, even for this moderate-size system.
To see how this heuristic approach performs under different settings, and particularly if $R_1\to \infty$, we again compare the approximated delay probability in the Erlang-R model with holding as solution of the fixed-point procedure to the outcomes of simulation experiments for the three scenarios in Table \ref{tab:parameter_settings}.
We performed 100 runs of length $10^4$ for each parameter setting and all values of $R$, yielding the results presented in Tables \ref{tab:heuristic_case1}--\ref{tab:heuristic_case3}, which are accurate up to a 95\% confidence interval of width $10^{-3}$.
\begin{table}[h] \centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1532 & 0.1031 & 0.1628 & 0.1216 \bigstrut[t]\\
10 & 0.1622 & 0.1272 & 0.1697 & 0.1331 \\
25 & 0.2340 & 0.2116 & 0.2413 & 0.2342 \\
50 & 0.1817 & 0.1468 & 0.1890 & 0.1678 \\
100 & 0.2199 & 0.1931 & 0.2304 & 0.2269 \\
250 & 0.2110 & 0.1852 & 0.2176 & 0.2230 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.2076} & \textit{0.1777} & \textit{0.2187} & \textit{0.2050} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0310 & 0.0121 & 0.0344 & 0.0148 \bigstrut[t]\\
10 & 0.0267 & 0.0123 & 0.0292 & 0.0128 \\
25 & 0.0348 & 0.0171 & 0.0373 & 0.0184 \\
50 & 0.0240 & 0.0108 & 0.0258 & 0.0125 \\
100 & 0.0293 & 0.0143 & 0.0317 & 0.0163 \\
250 & 0.0256 & 0.0120 & 0.0276 & 0.0145 \\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0229} & \textit{0.0104} & \textit{0.0257} & \textit{0.0124} \bigstrut[b]\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 1. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case1}
\end{table}
\begin{table}[h]\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.1327 & 0.0740 & 0.1620 & 0.1096 \bigstrut[t]\\
10 & 0.1446 & 0.0894 & 0.1683 & 0.1207 \\
25 & 0.2204 & 0.1631 & 0.2442 & 0.2203 \\
50 & 0.1694 & 0.1122 & 0.1888 & 0.1507 \\
100 & 0.2098 & 0.1524 & 0.2322 & 0.2111 \\
250 & 0.2033 & 0.1534 & 0.2190 & 0.1979 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1840} & \textit{0.1277} & \textit{0.2109} & \textit{0.1759} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0219 & 0.0079 & 0.0322 & 0.0137 \bigstrut[t]\\
10 & 0.0199 & 0.0073 & 0.0284 & 0.0115 \\
25 & 0.0283 & 0.0128 & 0.0375 & 0.0163 \\
50 & 0.0190 & 0.0078 & 0.0255 & 0.0107 \\
100 & 0.0244 & 0.0097 & 0.0314 & 0.0151 \\
250 & 0.0214 & 0.0083 & 0.0272 & 0.0134 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0169} & \textit{0.0066} & \textit{0.0234} & \textit{0.0104} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 2. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=1,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0977 & 0.0413 & 0.1521 & 0.0851 \bigstrut[t]\\
10 & 0.1070 & 0.0469 & 0.1648 & 0.1028 \\
25 & 0.1926 & 0.1076 & 0.2421 & 0.1874 \\
50 & 0.1431 & 0.0727 & 0.1876 & 0.1342 \\
100 & 0.1855 & 0.1012 & 0.2282 & 0.1714 \\
250 & 0.1775 & 0.0963 & 0.2217 & 0.1765 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{} & \textit{0.1442} & \textit{0.0711} & \textit{0.1981} & \textit{0.1354} \bigstrut\\
\cline{2-5}\end{tabular}%
\vspace{5mm}
\begin{tabular}{|r|rr|rr|}
\cline{2-5}\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=1$} & \multicolumn{2}{c|}{$\beta=2,\ \gamma=2$} \bigstrut\\
\hline
$R_1$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ & $\mathbb{P}({\rm d})$ & $\sqrt{R_1}\mathbb{E}[W]$ \bigstrut\\
\hline
5 & 0.0072 & 0.0019 & 0.0250 & 0.0081 \bigstrut[t]\\
10 & 0.0067 & 0.0018 & 0.0235 & 0.0082 \\
25 & 0.0148 & 0.0043 & 0.0325 & 0.0133 \\
50 & 0.0092 & 0.0025 & 0.0217 & 0.0081 \\
100 & 0.0132 & 0.0038 & 0.0277 & 0.0105 \\
250 & 0.0114 & 0.0033 & 0.0246 & 0.0099 \bigstrut[b]\\
\hline
\multicolumn{1}{r|}{\textit{}} & \textit{0.0078} & \textit{0.0022} & \textit{0.0188} & \textit{0.0069} \bigstrut\\
\cline{2-5}\end{tabular}%
\caption{Simulated probability of delay and scaled expected waiting time in Erlang-R model with holding for Case 3. The last row gives the asymptotic approximations.}
\label{tab:heuristic_case3}
\end{table}
We conclude from these tables that the approximation is good. As $R$ increases, the simulated values move closer to the heuristic approximation. Extensive numerical experiments suggest that load is slightly underestimated in the limit.
The best results in terms of accuracy are attained for small $r$.
This suggests that the quality of the heuristic method improves as $r$ gets smaller.
These are exactly the parameter settings for which this model is relevant.
\begin{remark}
The approximation technique that evolves around the fixed-point\\ \noindent method can be adapted to accommodate balking behavior of external arrivals. If we assume that an arriving patient finding all beds occupied leaves the system instantly with probability $1-q$, for some $q\in(0,1)$, independently of the rest of the arrivals, with the same argumentation, the volume of arrivals blocked is still $\alpha\sqrt{R_1}$, while the volume that will enter the ED eventually is $q\cdot\alpha\sqrt{R_1}$. Therefore, we may argue that in the QED regime, the system with holding and balking behaves as the system with blocking but with corrected parameters $(\beta-q\alpha,\gamma-q\alpha/\sqrt{r})$, where $\alpha$ satisfies
\begin{equation}
\alpha = f^b(\beta-q\alpha,\gamma-q\alpha/\sqrt{r}).
\end{equation}
Note that the choice of $q$ interpolates between the two system variants with holding ($q=0$) and blocking ($q=1$).
\end{remark}
\begin{figure}[h]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col3,dashed] table[x=s,y=approx32] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col4,dashed] table[x=s,y=approx36] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\addplot[thick,col5,dashed] table[x=s,y=approx40] {tikz/accuracy/accuracy_pdelay_holding_case2.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:accuracy_holding_a}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.87]
\small
\begin{axis}[
xmin = 0,
xmax = 16,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $s$,
xscale = 0.9,
yscale = 0.75,
legend cell align=left,
legend style = {at = {(axis cs: 0.1,0.01)},anchor = south west}
]
\addplot[thick,col3,mark = *] table[x=s,y=sim32] {tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col4,mark = *] table[x=s,y=sim36] {tikz/accuracy/accuracy_holding_probability.txt};
\addplot[thick,col5,mark = *] table[x=s,y=sim40] {tikz/accuracy/accuracy_holding_probability.txt};
\legend{{$n=32$},{$n=36$},{$n=40$}};
\end{axis}
\end{tikzpicture}
\caption{Holding probability}
\label{fig:accuracy_holding_b}
\end{subfigure}
\caption{Comparison of simulated delay probability (solid) against asymptotic approximations (dashed) with $\beta = (s-R_1)/\sqrt{R_1}$ and $\gamma = (n-R_1/r)/\sqrt{R_1/r}$ for $\lambda = 2$, $\mu=1$, $\delta=0.25$ and $p=0.75$.}
\label{fig:accuracy_holding}
\end{figure}
\section{Dimensioning}
\label{sec:dimensioning}
We will now use the accurate asymptotic approximations of the previous section to define a procedure that determines resource capacity in the restricted Erlang-R models.
That is, we aim to set the number of nurses $s$ and the number of beds $n$, such that a preset performance level is achieved.
We take the probability of delay at the needy queue and the probability of blocking/holding at the pre-entrant queue as the target performance objectives.
\subsection{Capacity setting for Erlang-R with blocking}
\label{sec:dimensioning_block}
In the setting with blocking, we can readily use the asymptotic results of Theorem \ref{thm:limits_YT} to (numerically) find a pair of parameters $(\beta^*,\gamma^*)$ to meet the performance requirements.
For instance, given that we want the delay probability to be at most $\varepsilon$, we first solve the equation $g^b(\beta^*,\gamma^*)=\varepsilon$ and then assign $s = \lceil R_1 + \beta^*\sqrt{R_1}\rceil$ and $n = \lceil R_1/r+\gamma^*\sqrt{R_1/r}\rceil$. Note that there could be multiple solutions to that problem, i.e.\ there could be multiple combinations of number of beds and number of nurses that can result in the same value of a single performance level.
The ED manager can ultimately decide which of these feasible solutions fits the environment best, for instance taking into account space and cost constraints.
We illustrate the resource allocation decisions in an MU setting, using data originated from two articles: \cite{LS2001} and \cite{GY2011}. Green \& Yankovic describe an MU that has 42 beds, with average occupancy level 78\%, and Average Length of Stay (ALOS) 4.3 days. Lundgren \& Segesten studied nurses' service times in a medical-surgical ward. They found that the average service time in their unit was 15.3 minutes per service, and that the average demand rate for each patient is 0.38 requests per hour. Therefore, we take an average service time of 15 minutes and assume that there are 0.4 requests per hour from each patient. Fitting this data to our model results in the following parameters values: $\lambda =0.32, \mu =4, \delta =0.4$, $p=0.975$ and the fraction of needy time is then approximately $r=0.09$.
This yields nominal offered load $R_1 = 3.2$ and $R_1/r = 34.4$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g(\beta,\gamma)$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: -1.9,0.05)},anchor = south west},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {tikz/staffing_example/staffing_example_with_blocking1.txt};
\draw[->,col1,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: -0.0552366,0.5) -- (axis cs: -0.0552366,0);
\draw[->,col2,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.179728,0.5) -- (axis cs: 0.179728,0);
\draw[->,col3,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.359034,0.5) -- (axis cs: 0.359034,0);
\draw[->,col4,very thick,dashed] (axis cs: -2,0.5) -- (axis cs: 0.459825,0.5) -- (axis cs: 0.459825,0);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma=1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability}
\label{fig:ratio01_delay}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{tikzpicture}[scale = 0.75]
\begin{axis}[
xmin = -2,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$f(\beta,\gamma)/\sqrt{R_1}$},
y label style = {at = {(axis cs: -2.5,0.5)}},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col1] table[x=beta,y=delay_gmin1] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col3] table[x=beta,y=delay_g0] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col4] table[x=beta,y=delay_g1] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\addplot[thick,col5] table[x=beta,y=delay_g2] {tikz/staffing_example/staffing_example_with_blocking2.txt};
\draw[very thick, col1,dashed,->] (axis cs: -0.0552,0) -- (axis cs: -0.0552,0.292798) -- (axis cs: -2,0.292798);
\draw[very thick,col2,dashed,->] (axis cs: 0.179728,0) -- (axis cs: 0.179728,0.164903) -- (axis cs: -2,0.164903);
\draw[very thick,col3,dashed,->] (axis cs: 0.359034,0) -- (axis cs: 0.359034,0.0705547) -- (axis cs: -2,0.0705547);
\draw[very thick,col4,dashed,->] (axis cs: 0.459825,0) -- (axis cs: 0.459825,0.0207909) -- (axis cs: -2,0.0207909);
\legend{$\gamma = -1$, $\gamma =0$,$\gamma= 1$, $\gamma=2$}
\end{axis}
\end{tikzpicture}
\caption{Blocking probability}
\label{fig:ratio01_block}
\end{subfigure}
\caption{Approximate performance of restricted Erlang-R with blocking for $r \approx 0.09$ and $R_1 = 3.2$, as functions of $\beta$.}
\label{fig:ratio01}
\end{figure}
Figure \ref{fig:ratio01} visualizes the dimensioning procedure for this particular MU.
The hospital management can find a pair of $n$ and $s$ to meet certain criteria, for example to achieve target delay probability $\varepsilon = 0.5$ with reasonable blocking probability.
Figure \ref{fig:ratio01}a indicates that this target can be achieved by a variety of pairs, for instance $(\beta_1,\gamma_1) = (-0.06,-1)$, $(\beta_2,\gamma_2) = (0.16,0)$, $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$, among infinitely many others.
According to Figure \ref{fig:ratio01}b, the pairs named above lead to blocking probabilities 0.293, 0.165, 0.071 and 0.021, respectively.
If the manager decides that probability of blocking of more than 10 percent is not acceptable, this leaves the choices $(\beta_3,\gamma_3) = (0.36,1)$ or $(\beta_4,\gamma_4) = (0.46,2)$ as candidate parameter pairs.
Using the two-fold square-root staffing rule $s_i = \lceil R_1 + \beta_i \sqrt{R_1}\rceil$ and $n_i = [R_1/r + \gamma_i\sqrt{R_1/r}]$, this yields feasible staffing levels $(s_3,n_3) = (4,40)$ and $(s_4,n_4)=(5,46)$.
The ultimate decision to apply any of these solutions can be based on external factors, such as operational costs or space limitations on the number of beds.
\subsection{Capacity setting for Erlang-R with holding}
For the holding model, we need a more sophisticated approach, exploiting the asymptotic approximation with the fixed-point equation in \eqref{eq:fixedpoint}. We propose the following dimensioning procedure to achieve a preset target delay probability at the needy queue.
\begin{algorithm}
\hspace{1cm}\rule{10cm}{1pt}\\
\hspace{1.1cm}\KwIn{Target delay probability $\varepsilon$. Parameters $\lambda,\mu,\delta$ and $p$.}
\hspace{1.1cm}\KwOut{Staffing levels $s$ and $n$.}
\vspace{-1mm}
\hspace{1cm}\rule{10cm}{0.5pt}\\
\vspace{-1mm}
\begin{enumerate}
\item[] \hspace{0.5cm} 1. Set $R_1:= \frac{\lambda}{(1-p)\mu}$ and $r = \frac{\delta}{\delta+p\mu}$.
\item[] \hspace{0.5cm} 2. Determine parameters $(\beta^*,\gamma^*)$ such that $g^b(\beta^*,\gamma^*) = \varepsilon$.
\item[] \hspace{0.5cm} 3. Set $\beta = \beta^* + f^b(\beta^*,\gamma^*)$ and $\gamma = \gamma^* + f^b(\beta^*,\gamma^*)/\sqrt{r}$.
\item[] \hspace{0.5cm} 4. Return $s = \left\lceil R_1 + \beta\sqrt{R_1}\right\rceil$ and $n = \left\lfloor R_1/r + \gamma \sqrt{R_1/r}\right\rfloor$.
\end{enumerate}
\vspace{-3 mm}
\hspace{1cm}\rule{10cm}{1pt}\\
\vspace{2 mm}
\caption{Stationary dimensioning algorithm for ED with holding.}
\label{alg:stationarydimensioning}
\end{algorithm}
\begin{remark}\label{rem:upperboundHW}
In Step 2 of Algorithm \ref{alg:stationarydimensioning} infinitely many pairs $(\beta^*,\gamma^*)$ satisfy the delay probability equation.
For practical purposes, it is convenient to fix either $\beta^*$ or $\gamma^*$ beforehand, and then solve $g^b(\beta^*,\gamma^*) = \varepsilon$ for the remaining unknown.
The preset value should however be chosen with care, since $g^b(\beta^*,\gamma^*)$ is upper bounded by the Halfin-Whitt delay probability formula
\[g_{\rm HW}(\beta^*) = \left( 1 + \frac{\beta^* \Phi(\beta^*)}{\varphi(\beta^*)}\right)^{-1}.\]
Hence, if $\varepsilon > g_{\rm HW}(\beta^*)$, then no feasible solution to $g^b(\beta^*,\gamma^*)=\varepsilon$ exists.
This should be considered when choosing $\beta^*$.
Furthermore, it is evident from Step 3 that the final values $(\beta,\gamma)$ are always larger than $(\beta^*,\gamma^*)$.
\end{remark}
We now use the same example as in Section \ref{sec:dimensioning_block} to demonstrate capacity allocation decisions for the model with holding. This can be viewed as the additional capacity the medical unit needs in terms of nurses and beds, in order to account for the fact that patients are waiting in the ED to be admitted instead of being blocked and transferred to a less preferred unit.
Observe that the holding model leaves less flexibility for management in choosing system parameters due to stability constraints. For example, the policy with $n=30$ ($\gamma=-0.75$) is infeasible in the holding model.
For similar reasons, only nurse staffing levels with $\beta>0$, or $s > R_1=3.2$ are feasible.
Targeting a delay probability of $0.5$ with $n=40$, Figure \ref{fig:ratio01_hold} shows that operating a MU with holding room requires $\beta = 0.475$ or $s=5$.
Recall that under the blocking policy, only $s=4$ nurses were needed to achieve a delay probability of $0.5$.
This example hence shows how the managerial decision to have a holding room, rather than deferring patients to less preferred medical units, requires additional workforce in that unit (as well as the ED).
This example also shows that the facility with holding room is able to treat fewer patients simultaneously than under blocking constraints, in line with the bounds in Section \ref{sec:bounds} and Conjecture \ref{conj:stochorder}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 2,
ymin = 0,
ymax = 1,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\beta$,
ylabel = {$g^h(\beta,\gamma)$},
legend cell align=left,
legend style = {at = {(axis cs: 1.9,0.95)},anchor = north east},
xscale = 0.95,
yscale = 0.9
]
\addplot[thick,col5] table[x=beta,y=delay_n35] {tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col4] table[x=beta,y=delay_n40] {tikz/staffing_example/staffing_example_with_holding1.txt};
\addplot[thick,col1] table[x=beta,y=delay_n45] {tikz/staffing_example/staffing_example_with_holding1.txt};
\draw[very thick,col4,dashed,->] (axis cs: -2,0.5) -- (axis cs: 0.475,0.5) -- (axis cs: 0.475,0);
\legend{$\gamma = -0.75$, $\gamma =0.102$,$\gamma= 0.955$, $\gamma=1.807$};
\end{axis}
\end{tikzpicture}
\caption{Approximate delay probability of restricted Erlang-R system with holding for $r\approx 0.09$ and $R_1=3.2$ }
\label{fig:ratio01_hold}
\end{figure}
\section{Model analysis and managerial implications}
\label{sec:analysis_chapter5}
In this section, we use the analysis and algorithms developed in earlier sections to gain insights into the importance of the capacity restrictions and patient returns in a restricted Erlang-R system by drawing a comparison to related models studied in the literature.
\subsection{The influence of patient returns or the role of $r$}
Here we study how the parameter $r$ affects the service level in the restricted Erlang-R model with blocking, on the basis of the asymptotic expressions in Theorem \ref{thm:limits_YT}.
To better understand the connection with the single-station model and the importance of returns we examine the role of $r$.
Recall the interpretation of $r$ as the fraction of time a patient is needy during his stay within the system in the idealized scenario with infinite capacity, i.e. for $r\in(0,1)$.
The case $r=1$ corresponds to the setting in which patients are needy all the time, in this case patients get service in one time.
When $r=1$ the infinite-server queue, describing the number of content patients, disappears from the queueing system and we end up with a standard loss model---$M/M/s/n$ queue---in which capacity is scaled as
\[ s = R_1+\beta\sqrt{R_1}, \qquad n = R_1+\gamma\sqrt{R_1}. \]
This staffing rule only makes sense in case $\beta<\gamma$, since no delay is experienced if $n\leq s$.
If indeed $\gamma>\beta$, then the asymptotic delay probability and scaled blocking probability are given by \cite{masseywallace},
\begin{align*}
g_B(\beta,\gamma) &= \frac{1-{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}, \\
f_B(\beta,\gamma) &= \frac{\beta{\rm e}^{-\beta(\gamma-\beta)}}{1-{\rm e}^{-\beta(\gamma-\beta)}+\beta\Phi(\beta)/\varphi(\beta)}.
\end{align*}
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.8,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,1.28)},anchor= north east},
yscale = 0.75
]
\addplot[thick,col1] file {tikz/influence_r/PdelayB_g1_b025.txt};
\addplot[thick,col3] file {tikz/influence_r/PdelayB_g1_b05.txt};
\addplot[thick,col4] file {tikz/influence_r/PdelayB_g1_b1.txt};
\addplot[thick,col5] file {tikz/influence_r/PdelayB_g1_b2.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Delay probability $g^b(\beta,\gamma)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 0,
ymax = 0.4,
grid = both,
xlabel = $r$,
axis line style={->},
legend cell align=left,
legend style={at={(0.98,0.05)},anchor= south east},
yscale = 0.75
]
\addplot[thick,col1] file {tikz/influence_r/PblockB_g1_b025.txt};
\addplot[thick,col3] file {tikz/influence_r/PblockB_g1_b05.txt};
\addplot[thick,col4] file {tikz/influence_r/PblockB_g1_b1.txt};
\addplot[thick,col5] file {tikz/influence_r/PblockB_g1_b2.txt};
\addplot[thick,dashed] file {tikz/influence_r/PblockB_g1_inf.txt};
\legend{$\beta = 0.25$,$\beta=0.5$,$\beta=1.0$,$\beta=2.0$}
\end{axis}
\end{tikzpicture}
\caption{Scaled blocking probability $f^b(\beta,\gamma)$.}
\label{fig:influence_of_r_b}
\end{subfigure}
\caption{Asymptotic performance measures as a function of $r$ in the restricted Erlang-R model with blocking for $\gamma=1$.}
\label{fig:influence_of_r}
\end{figure}
We can see that $f^b(\beta,\gamma)$ for increasing $\beta$ approaches a lower bound that is a function of $r$.
To understand this, observe that as $\beta$ grows, delays at the nurse queue vanish.
Then the sojourn time of an admitted patient only consists of a geometric number of needy and content periods with mean $(1/\mu+p/\delta)/(1-p) = 1/(r\mu(1-p))$.
The blocking model can in this case be modeled as an $M/G/n/n$ queue, with offered load $\lambda/(r\mu(1-p)) =R_1/r$, in which the scaled blocking probability is known to be, see \cite{Avram2013},
\[\sqrt{R_1} \, \mathbb{P}({\rm block}) = \sqrt{R_1} \, \frac{(R_1/r)^n/n!}{\sum_{k=0}^n (R_1/r)^k / k!} \to \sqrt{r} \, \frac{\varphi(\gamma)}{\Phi(\gamma)},\]
as $R_1\to\infty$.
This function of $r$ is plotted in Figure \ref{fig:influence_of_r_b} as the dashed line.
We observe that in general the probability of blocking increases with $r$, regardless of the capacity constraints on the needy station.
We can explain this by observing that $r$ influences only $n$ in the QED staffing rule. When $n$ reduces, more patients are blocked. Therefore, if patients spend relatively more time in needy state, which usually indicates services that are less interrupted, blocking will increase. Delays, on the other hand, will decrease in such situations---the minimal delay possible can be achieved if service is given in one time ($r=1$). Returns or interruptions increase delays significantly under QED staffing.
\subsection{Comparing restricted and unrestricted Erlang-R models}
Given the expressions for the asymptotic delay probability in the open Erlang-R model, and its restricted versions with blocking and holding, we compare the three policies for various values of $\beta$, $\gamma$ and $r$.
Figure \ref{fig:comparison_delay} plots the delay probability for blocking ($g^b(\beta,\gamma)$), holding ($g^h(\beta,\gamma)$) and Erlang-R ($g_{\rm HW}(\beta)$) models, as functions of $\gamma$, while keeping $\beta$ fixed, for three values of $r$.
We make a couple of observations.
Notice that
\[ g^b(\beta,\gamma) \leq g^h(\beta,\gamma) \leq g_{\rm HW}(\beta) \]
for all $\beta,\gamma>0$ and $r$.
In that sense, the holding model is an interpolation between the blocking and the open model.
As expected, the delay probabilities in the restricted models converge to those of the open Erlang-R model, because increasing $\gamma$ is tantamount to lifting the stringent constraints on the system size. Note that the rate of conversion is fast---one can provide probability of waiting close to that of the open model with small values of $\gamma$. Indeed, the fact that when using QED staffing not much of excessive delay results from the beds restriction is important by itself.
Also, we observe that the difference between delay probabilities increases with $r$.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r01_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r01_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r01_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r01_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.1$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r025_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r025_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r025_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r025_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.25$.}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale = 0.56]
\begin{axis}[
xmin = 0,
xmax = 3,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = $\to \gamma$,
xscale = 0.9,
yscale = 0.8
]
\addplot[thick,col5,dashed] file {tikz/comparison/PdelayR_b01.txt};
\addplot[thick,col5,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b01.txt};
\addplot[thick,col5] file {tikz/comparison/PdelayH_r05_b01.txt};
\addplot[thick,col2,dashed] file {tikz/comparison/PdelayR_b05.txt};
\addplot[thick,col2,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b05.txt};
\addplot[thick,col2] file {tikz/comparison/PdelayH_r05_b05.txt};
\addplot[thick,col4,dashed] file {tikz/comparison/PdelayR_b1.txt};
\addplot[thick,col4,mark=*, mark repeat = 2] file {tikz/comparison/PdelayB_r05_b1.txt};
\addplot[thick,col4] file {tikz/comparison/PdelayH_r05_b1.txt};
\end{axis}
\end{tikzpicture}
\caption{$r=0.5$.}
\end{subfigure}
\caption{Asymptotic delay probability in open Erlang-R (dashed), restricted Erlang-R with blocking (marked) and restricted Erlang-R with holding (solid), as function of $\gamma$ with $\beta=0.1$ (blue), $\beta=0.5$ (orange) and $\beta=1$ (green) fixed.}
\label{fig:comparison_delay}
\vspace{3mm}
\end{figure}
\subsection{The impact of visit number}
\label{subsec:num_visit}
We next reflect on the impact of operational capacity decisions on different patient populations. We measure patient's complexity by the number of times she needs to see the nurse or the physician during her stay. In the ED context, simple-to-treat patients will need to see the physician once, while complex ones will need multiple visits. Hence, we divide the patients into complexity groups by the number of visits in the Needy station. Since the number of visits is geometrically distributed, we have a higher proportion of simple patients than complex ones; that fits well the health care environment.
Figure \ref{fig:wait_by_visit} shows the waiting time in the needy and pre-entering queues, and the total waiting time, as a function of $n$ (number of beds), for each complexity group.
Obviously, the expected waiting time in the pre-entering queue decreases with $n$, while the needy waiting time increases.
For patients who require a relative large number of visits of the physician, in this case more than 6, the total needy wait is the dominant part of the total waiting time. Therefore, as $n$ grows, the total waiting time first decreases and then increases.
In fact, Figure \ref{fig:wait_by_visit_b} suggests that there is an optimal number of beds $n$ that minimizes the total wait for each complexity type.
Thus, size restrictions reduce the length-of-stay of patients with complex health conditions (given that the constraint is not too tight).
On the other hand, this figure also shows that no such $n$ exists for patients who only require little assistance.
Hence, there is no $n$ that improves the sojourn time of all patients in the ED simultaneously.
This leaves the decision to the hospital manager to weigh the importance of patients of different complexity levels.
\begin{remark}
From a different perspective, note that in queueing systems such as communication systems, the partitioning of a job to sizable quantities and scheduling those jobs in a similar dynamic to the Erlang-R model became a popular way for increasing throughput. This is because this effectively schedules jobs by their size even though the total job requirements are uncertain. This in fact creates a shortest-job-first policy without prior knowledge of job size \citep{Comte2016}. Considering that perspective we note that the Erlang-R model actually prioritizes simple jobs over complex ones. But without restrictions, when load is too high, such procedures may lead to very long LOS of long jobs. The capacity restriction we analyze in this chapter, in both of its versions, limits such delays. Hence, even in cases in which the returns themselves are created by a managerial decision, imposing the additional managerial restriction on entering the system has benefits.
\end{remark}
\begin{figure}
\centering
\begin{subfigure}{0.38\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {tikz/inner_vs_outer_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {tikz/inner_vs_outer_wait.txt};
\addplot[mark=*,col1, thick] table[x=n,y=hold] {tikz/inner_vs_outer_wait.txt};
\end{axis}
\end{tikzpicture}
\caption{Expected pre-entering waiting (red) and needy waiting times (black)}
\end{subfigure}
\begin{subfigure}{0.6\textwidth}
\centering
\begin{tikzpicture}[scale=0.66]
\begin{axis}[
xmin = 33,
xmax = 60,
ymin = 0,
ymax = 10,
grid = both,
axis line style={->},
axis lines = left,
xlabel = $n$,
yscale = 0.8,
legend cell align=left,
legend style = {at = {(1.05,0.58)}, anchor = west}
]
\addplot[mark=*,black,opacity=0.1] table[x=n,y=in1] {tikz/total_wait.txt} node[right,pos=1] {$N=1$};
\addplot[mark=*,black,opacity=0.2] table[x=n,y=in2] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.3] table[x=n,y=in3] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.4] table[x=n,y=in4] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.5] table[x=n,y=in5] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.6] table[x=n,y=in6] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.7] table[x=n,y=in7] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.8] table[x=n,y=in8] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=0.9] table[x=n,y=in9] {tikz/total_wait.txt};
\addplot[mark=*,black,opacity=1] table[x=n,y=in10] {tikz/total_wait.txt};
\legend{{\small $N=1$},{\small $N=2$},{\small $N=3$},{\small $N=4$},{\small $N=5$},{\small $N=6$},{\small $N=7$},{\small $N=8$},{\small $N=9$},{\small $N=10$}};
\end{axis}
\end{tikzpicture}
\caption{Total expected waiting times\\
\quad \\
\quad }
\label{fig:wait_by_visit_b}
\end{subfigure}
\caption{Expected waiting times as a function of $n$ given the number of visits $N$ in the Erlang-R model with holding with $\lambda=2$ $\mu=1$, $\delta=0.25$, $p=0.75$ and $s=9$.}
\label{fig:wait_by_visit}
\end{figure}
\subsection{Case study: comparison of operational decisions}
\label{sec:case_study}
We now illustrate how the managerial decision to operate under a specific operational regime affects ED performance in terms of efficiency and quality-of-care, through a case study.
The practical environment we investigate is the ED of a moderately-sized hospital, which faces the arrival pattern $\lambda(t)$ plotted in Figure \ref{fig:Case_study_arrival_pattern_a} on a typical workday.
Other parameters of the model are estimated to be $\mu = 6.67,\ \delta = 2.18$ and $p = 0.76$, so that $r = 0.301$. These parameters were taken from \cite{YomTov2014}. In order to set time-varying staffing levels $s(t)$ and $n(t)$, we adopt the \textit{mean-offered load} (MOL) approximation of the demand process of~\cite{Jennings1996}.
This approach initially presumes infinite capacity to obtain the number of patients $R(t)$ in the queueing system as a function of time.
This offered load function then replaces the (constant) value of $R$ in the stationary dimensioning scheme under consideration, to determine the adequate number of servers at each point in time.
Following this idea in our two-dimensional queueing system, we find the offered load function for the nurses $R_1(t)$ and the offered load function for the beds $R_1(t)+R_2(t)$ as the solution of the system of ODEs,
\begin{align} \label{eq:offeredloadODE}
\frac{d}{dt} R_1(t) &= \lambda(t) + \delta R_2(t) - \mu R_1(t),\\
\frac{d}{dt} R_2(t) &= p\mu R_1(t) - \delta R_2(t),
\end{align}
see \cite[Thm.~2]{YomTov2014} for details.
For this case study's parameters, these offered load functions are also plotted in Figure \ref{fig:Case_study_arrival_pattern_a}.
While the number of nurses can be adjusted in a relatively flexible manner, the value of $n$, which echoes a hard restriction on the ED capacity, is naturally less amenable to fluctuations. The reason is that the maximum ED capacity is to a large extent determined by its hardware, such as beds and medical equipment.
However, the ED manager might deliberately consider reducing $n$ during more quiet periods of the day, e.g.\ during the night, by imposing bed-to-physician constraints. This is done, for example, when setting a case management constraint \citep{EDexperiment,Campello2016}.
Therefore, we consider the scenario in which both $s$ and $n$ are time-dependent but we do not force a constant case management quantity, rather let our new methodology recommend an appropriate one.
\begin{figure}
\centering
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.01,0.95)},anchor = north west}
]
\addplot[very thick,black] file {tikz/lambdaFunction.txt};
\addplot[very thick,col1] file {tikz/R1.txt};
\addplot[very thick,col5] file {tikz/R1R2.txt};
\legend{ $\lambda(t)$, $R_1(t)$, $R_1(t)+R_2(t)$};
\end{axis}
\end{tikzpicture}
\caption{Dynamic arrival rate function offered load functions}
\label{fig:Case_study_arrival_pattern_a}
\end{subfigure}
\begin{subfigure}{0.46\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 45,
ytick = {0,10,20,30,40},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {tikz/casestudy_new/sFunction.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/nFunction.txt};
\legend{ $s(t)$, $n(t)$};
\end{axis}
\end{tikzpicture}
\caption{Capacity function for $\beta=\gamma=0.5$}
\label{fig:Case_study_arrival_pattern_b}
\end{subfigure}
\caption{Empirical arrival rate and offered load functions $R_1(t)$ and $R_1(t)+R_2(t)$ in Israeli ED and corresponding capacity functions.}
\label{fig:Case_study_arrival_pattern}
\end{figure}
Extrapolating Algorithm \ref{alg:stationarydimensioning} to the time-varying case, Step 4 is replaced by
\begin{align*}
s(t) &= R_1(t) + \beta\sqrt{R_1(t)},\\
n(t) &= R_1(t)+R_2(t) + \gamma\sqrt{R_1(t)+R_2(t)},
\end{align*}
for some $\beta,\gamma>0$.
Since $R_1(t)$ and $R_2(t)$ are given, the QED staffing problem again reduces to finding the pair $(\beta,\gamma)$.
Figure \ref{fig:Case_study_arrival_pattern_b} plots the capacity functions for $\beta = 0.5$ and $\gamma=0.5$, assuming capacity can only be adjusted every 30 minutes.
In this case study, we consider three pairs of parameters $(\beta,\gamma)$.
For each of these we investigate, using simulation, the differences in the time-varying performance indicators between the policy with blocking and holding.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 1.0,
ytick = {0,0.2,0.4,0.6,0.8,1.0},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=delay_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=delay_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=delay_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=delay_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=delay_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=delay_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm delay})$}
\label{fig:simulation_results_a}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 0.52,
ytick = {0,0.1,0.2,0.3,0.4,0.5},
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1,
legend cell align=left,
legend style = {at = {(0.9,0.95)}, anchor = north east}
]
\addplot[very thick,col1] table[x=t,y=block_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4] table[x=t,y=block_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5] table[x=t,y=block_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=block_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4,dashed] table[x=t,y=block_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5,dashed] table[x=t,y=block_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\legend{{$(\beta,\gamma)=(0.1,2)$},{$(\beta,\gamma)=(1,1.5)$},{$(\beta,\gamma)=(2,1)$}};
\end{axis}
\end{tikzpicture}
\caption{$\mathbb{P}({\rm block})$ or $\mathbb{P}({\rm hold})$}
\label{fig:simulation_results_b}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{tikzpicture}[scale=0.47]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.5,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.1,
yscale=1
]
\addplot[very thick,col1] table[x=t,y=ratio_b01g2] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col1,dashed] table[x=t,y=ratio_b01g2] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col4] table[x=t,y=ratio_b1g15] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col4,dashed] table[x=t,y=ratio_b1g15] {tikz/casestudy_new/case_study_hold.txt};
\addplot[very thick,col5] table[x=t,y=ratio_b2g1] {tikz/casestudy_new/case_study_block.txt};
\addplot[very thick,col5,dashed] table[x=t,y=ratio_b2g1] {tikz/casestudy_new/case_study_hold.txt};
\end{axis}
\end{tikzpicture}
\caption{Nurse-to-patient ratio.}
\label{fig:simulation_results_c}
\end{subfigure}
\caption{Simulation results for case study. Solid and dashed lines represent time-varying performance in the blocking and holding model, respectively.}
\label{fig:simulation_results}
\end{figure}
The simulation results are presented in Figure \ref{fig:simulation_results}.
Figure \ref{fig:simulation_results_a} shows that the MOL approach for capacity allocation roughly stabilizes the delay probability.
Figure \ref{fig:simulation_results_b} shows that the fraction of patients not entering the ED on arrival in the blocking model is reasonable for all parameter pairs considered and the graphs are ordered according to $\gamma$.
We also see a significant difference with holding.
Observe also that the holding probability drops in the period 8--13, which is exactly the period when the system is experiencing peak offered load.
Hence, this temporary reduction is in line with our asymptotic findings that the probability of blocking/holding is $O(1/\sqrt{R_1})$.
Finally note that the three parameter settings lead to different nurse-to-patient ratios.
Clearly, larger $\beta$ leads to small nurse-to-patient ratios (due do larger staffing).
Figure \ref{fig:simulation_results_c} demonstrates that for $(\beta,\gamma) = (1,1.5)$ and $(\beta,\gamma) = (2,1)$ the difference between the holding policy and the blocking policy is small. However, for $(\beta,\gamma) = (0.1,2)$ we see a significant increase in the ratio during night hours.
This may be due to the tight nurse schedule, that causes the holding queue to build up just before midnight.
This queue then empties latter on, causing an increase in workload per nurse in the period 24--7.
To see the direct effect of the size restriction on the queue lengths, we plotted the mean holding and service queue lengths in the holding model as a function of the parameter $\gamma$ in Figure \ref{fig:simulation_queuelengths}.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 5.2,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend pos = north west
]
\addplot[very thick,col1] file {tikz/casestudy_new/holdingQueue_g01.txt};
\addplot[very thick,col3] file {tikz/casestudy_new/holdingQueue_g025.txt};
\addplot[very thick,col4] file {tikz/casestudy_new/holdingQueue_g05.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/holdingQueue_g1.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$}
\end{axis}
\end{tikzpicture}
\caption{Mean holding queue length}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.6]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0.0,
ymax = 15,
xtick = {0,3,6,9,12,15,18,21,24},
grid = both,
axis line style={->},
axis lines = left,
xlabel = $\to t$,
xscale=1.25,
yscale=1,
legend cell align=left,
legend style = {at = {(0.82,0.05)},anchor = south east}
]
\addplot[very thick,col1] file {tikz/casestudy_new/serviceQueue_g01.txt};
\addplot[very thick,col3] file {tikz/casestudy_new/serviceQueue_g025.txt};
\addplot[very thick,col4] file {tikz/casestudy_new/serviceQueue_g05.txt};
\addplot[very thick,col5] file {tikz/casestudy_new/serviceQueue_g1.txt};
\addplot[very thick,dashed] file {tikz/casestudy_new/serviceQueue_R.txt};
\legend{$\gamma=0.1$, $\gamma=0.25$,$\gamma=0.5$,$\gamma=1$,Erlang-R}
\end{axis}
\end{tikzpicture}
\caption{Mean service queue length}
\end{subfigure}
\caption{Simulated queue length of holding model with different values of $\gamma$.}
\label{fig:simulation_queuelengths}
\vspace{-6mm}
\end{figure}
Note that for all $\gamma$ considered, the holding queue lengtsh are, as expected, of a smaller order than the number of patients admitted.
Also, the holding queue length decreases as we increase $\gamma$.
The service queue lengths naturally approach the expected queue lengths in the Erlang-R model as $\gamma$ is increased.
\section{Conclusion \& future research}
\label{sec:conclusion}
In this chapter we developed and analyzed a queueing network tailored to a health care environment with finite-size restrictions.
Using the asymptotic approximations, numerical analysis and simulation, we gained insight into staffing problems that arise in EDs, and proposed an efficient, flexible, and easy to implement methodology to dimension medical facilities through a two-fold staffing rule.
The dimensioning scheme we developed provides a powerful and elegant way of finding adequate staffing levels in emergency departments.
Nonetheless, we see some avenues for further research.
The asymptotic approximations we developed enabled us to take the first step towards characterizing the pre-entering queue behavior in the
QED regime.
We observed how the holding queue length vanishes at rate $1/\sqrt{R_1}$ as $R_1\to\infty$.
Yet, our analysis did not yield explicit characteristics on the holding queue and holding times.
These performance indicators are naturally important to study if one wants to consider the trade-off between waiting time inside the ED and waiting time outside the ED time (pre-entering time).
Furthermore, it is worthwhile to study the robustness of our approximations against the service and content time distributions. Since the content phase of a patient is modeled after an infinite-server queue, we expect our approximations to be useful for content time distributions beyond the exponential distribution as well, due to distributional insensitivity of the service time in infinite-server queues. For the needy phase, modeled after a multi-server queue, this insensitivity result does not hold and hence this needs further research.
Finally, the restricted Erlang-R model obviously gives a highly simplified view of the complex reality of the ED.
In practice, distinctive features such as a triage system (with patient priorities), patient boarding time and availability of medical equipment may play a decisive role on ED dynamics.
However, we think the analysis and dimensioning algorithms presented in this chapter can serve as a building block for staffing procedures that do account for these case-specific factors.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Description of the QBD process}
\label{app:QBDdescription}
\subsection{The QBD-process}
\label{app:theQBDprocess}
We consider the QBD-process $X(t)=(N(t),Q_1(t))$ in stationarity. Let $\nu(i)=\min\{i,s\}\mu$. To determine the (outgoing) transition rates of the process $X$ we distinguish between the following cases:
\begin{itemize}
\item \emph{Transitions from $(0,0)$:} There are no patients in the Emergency Department and thus the only possible occurrence is when a new patient arrives. This results in a transition to $(1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), 1 \leq i < n$:} There are exactly $i$ patients assigned to a bed of which none are seen by a nurse. Then either one of those patients becomes needy, or a new patient arrives at the Emergency Department that can immediately be seen by a nurse. The first results in a transition to $(i,1)$ and occurs at rate $i \delta$, and the second results in a transition to $(i+1,1)$ and occurs with rate $\lambda$.
\item \emph{Transitions from $(i,0), i \geq n$:} Again, the only possible transitions arise from either a newly arrived patient or a patient assigned to a bed becoming needy. However, a newly arrived patient finds all beds occupied and needs to wait. Thus, with rate $\lambda$ we have a transition to $(i+1,0)$ and with rate $n \delta$ a transition to $(i,1)$.
\item \emph{Transitions from $(i,i), i < n$:} In this case all patients assigned to a bed are in need of service. With rate $\lambda$ a new patient arrives at the Emergency Department. She joins the (possible) queue to be seen by a nurse immediately, so this results in a transition to $(i+1,i+1)$. Moreover, since there are only $s < n$ nurses, a service completion occurs with rate $\nu(i)$. With probability $p$ the patient turns to the holding phase, so in total we still have $i$ patients with one patient less in queue for a nurse. With probability $1-p$ the patient leaves the Emergency Department, decreasing both $N$ and $Q_1$ by one. In other words, with rate $p \nu(i)$ we have a transition to $(i,i-1)$ and with rate $(1-p)\nu(i)$ we have a transition to $(i-1,i-1)$.
\item \emph{Transitions from $(n,n)$:} Similar to the previous case, we have a transition to $(n,n-1)$ with rate $p s \mu$ and with rate $(1-p)s \mu$ we have a transition to $(n-1,n-1)$. In this case however, a newly arrived patient finds all beds occupied, resulting in a transition to $(n+1,n)$ with rate $\lambda$.
\item \emph{Transitions from $(i,n), i > n$:} We have a transition to $(i+1,n)$ with rate $\lambda$ and a transition to $(i,n-1)$ with rate $p s \mu$. In case that a patient leaves the Emergency Department there are $i-n>0$ patients in the holding room waiting for an available bed. Thus, one of the $i-n$ patients in the holding room is assigned to the available bed in need of service. That is, with rate $(1-p) s \mu$ we have a transition to $(i-1,n)$.
\item \emph{Transitions from $(i,j), 1 \leq j < i < n$:} There are four possible transitions. First, with rate $\lambda$ there is a new arrival which results in a transition to $(i+1,j+1)$. Second, with rate $(i-j) \delta$ a patient in one of the beds becomes needy, which results in a transition to $(i,j+1)$. Third, with rate $p \nu(j)$ a patient turns to the content state after service completion, which results in a transition to $(i,j-1)$. Last, with rate $(1-p) \nu(j)$ a patient leaves the Emergency Department after service completion, which results in a transition to $(i-1,j-1)$.
\item \emph{Transitions from $(n,j), 1 \leq j < n$:} This case is similar to the previous one. The only difference arises when a new patient arrives, since all $n$ beds are already occupied. Thus, with rate $\lambda$ we have a transition to $(n+1,j)$.
\item \emph{Transitions from $(i,j), i > n, 1 \leq j \leq n$:} This case is the previous one, except when a patient leaves the Emergency Department after service completion. Then one of the $(i-n)$ patients in the holding room will be assigned to a bed in need of service. This results in a transition to $(i-1,j)$ with rate $(1-p) \nu(j)$.
\end{itemize}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.66]
\draw[step=1cm,gray!50!,very thin] (0,0) grid (15.5,8.5);
\draw[thick,->] (0,0) -- (15.5,0);
\draw[thick,->] (0,0) -- (0,8.5);
\draw[thick] (0,0) -- (8,8);
\draw[thick,dashed,black!50!] (8,0) -- (8,8);
\draw[thick] (8,8) -- (15.5,8);
\foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
\foreach \y in {0,1,2,3,4,5,6,7,8}
\draw[fill] (\x,\y) circle [radius=0.025];
\node [below left] at (0,0) {$0$};
\node [left] at (0,2) {$j$};
\node [left] at (0,5) {$k$};
\node [left] at (0,8) {$n$};
\node [below] at (6,0) {$i$};
\node [below] at (8,0) {$n$};
\node [above left] at (0,8.5) {$Q_1$};
\node [below right] at (15.5,0) {$N$};
\path [->,thick,-latex] (0,0) edge [bend right] (1,1);
\path [->,thick,-latex] (3,0) edge (3,1);
\path [->,thick,-latex] (3,0) edge (4,1);
\path [->,thick,-latex] (4,4) edge [bend right] (5,5);
\path [->,thick,-latex] (4,4) edge (4,3);
\path [->,thick,-latex] (4,4) edge [bend right] (3,3);
\path [->,thick,-latex] (6,2) edge (6,3);
\path [->,thick,-latex] (6,2) edge (7,3);
\path [->,thick,-latex] (6,2) edge (6,1);
\path [->,thick,-latex] (6,2) edge (5,1);
\path [->,thick,-latex] (8,5) edge (8,6);
\path [->,thick,-latex] (8,5) edge (8,4);
\path [->,thick,-latex] (8,5) edge (9,5);
\path [->,thick,-latex] (8,5) edge (7,4);
\path [->,thick,-latex] (8,8) edge (8,7);
\path [->,thick,-latex] (8,8) edge [bend right] (7,7);
\path [->,thick,-latex] (8,8) edge [bend left] (9,8);
\path [->,thick,-latex] (11,8) edge (11,7);
\path [->,thick,-latex] (11,8) edge [bend left] (12,8);
\path [->,thick,-latex] (11,8) edge [bend left] (10,8);
\path [->,thick,-latex] (11,0) edge (11,1);
\path [->,thick,-latex] (11,0) edge [bend left] (12,0);
\path [->,thick,-latex] (12,5) edge (12,6);
\path [->,thick,-latex] (12,5) edge (12,4);
\path [->,thick,-latex] (12,5) edge (13,5);
\path [->,thick,-latex] (12,5) edge (11,5);
\node [above] at (12.75,5) {\scriptsize $\lambda$};
\node [above] at (11.25,5) {\scriptsize $(1-p)\nu(k)$};
\node [right] at (12,5.75) {\scriptsize $(n-k)\delta$};
\node [right] at (12,4.25) {\scriptsize $p \nu(k)$};
\node [above] at (6.75,2.25) {\scriptsize $\lambda$};
\node [below] at (5,2) {\scriptsize $(1-p)\nu(j)$};
\node [above] at (6,3) {\scriptsize $(i-j)\delta$};
\node [right] at (6,1.25) {\scriptsize $p \nu(j)$};
\end{tikzpicture}
\caption{Transition diagram for the Erlang-R model with holding.}
\label{fig:QBDIllustration}
\vspace{-3mm}
\end{figure}
\noindent
The state space and transition rates of the Erlang-R model with holding are illustrated in Figure~\ref{fig:QBDIllustration}.
The state space can be partitioned according to its levels, where level $i$ corresponds to a total queue length $N=i$ patients. This results in an infinite-sized matrix consisting of blocks, where each block corresponds to the transition flow from one level to another. Since the only transitions allowed are within the same level or between two adjacent levels in a QBD-process, we obtain a tridiagonal block structure. Each block consists of elements representing the transition rate of one state to another, and therefore each block is a matrix of size at most $(n+1) \times (n+1)$.
For the Erlang-R model with holding this gives the following result. Let $P$ denote the transition matrix of the process $(N(t),Q_1(t))$. We have the boundary levels $\{1,2,...,n\}$ and $P$ is of the form
\[
P = \left( \begin{array}{cccccccccc}
B_{00} & B_{01} & & & & & & & & \\
B_{10} & B_{11} & B_{12} & & & & & & & \\
& B_{21} & B_{22} & B_{23} & & & & & & \\
& & \ddots & \ddots &\ddots & & & & & \\
& & & & & B_{n \, n-1} & & & & \\
& & & & B_{n-1 \, n} & B_{nn} & A_0 & & & \\
& & & & & A_2 & A_1 & A_0 & & \\
& & & & & & A_2 & A_1 & A_0 & \\
& & & & & & & \ddots & \ddots & \ddots \\
\end{array} \right),
\]
where $B_{ii} \in \mathbb{R_1}^{(i+1) \times (i+1)}$, $B_{i \, i-1} \in \mathbb{R_1}^{(i+1) \times i}$, $B_{i-1 \, i} \in \mathbb{R_1}^{i \times (i+1)}$, and $A_0,A_1,A_2 \in \mathbb{R_1}^{(n+1)\times(n+1)}$. The matrices of transition rates for the boundary states are given by
\[
B_{00}=(-\lambda),
\qquad
B_{i-1 \, i} = \left( \begin{array}{ccccc}
0 & \lambda & & & \\
& \ddots & \lambda & & \\
& & \ddots & \ddots &\\
& & & 0 & \lambda \\
\end{array} \right),
\]
\[
B_{i \, i-1} = \left( \begin{array}{cccc}
0 & & & \\
(1-p)\mu & 0 & & \\
& (1-p)\nu(2)& \ddots & \\
& & \ddots & 0 \\
& & & (1-p)\nu(i)\\
\end{array} \right),
\]
and
\[
\scriptsize
B_{ii} = \left(
\begin{array}{ccccccccc}
-(\lambda+i \delta) & i \delta & & & &\\
p \mu & -(\lambda+\mu+(i-1)\delta) & (i-1)\delta & & &\\
& \ddots & \ddots & \ddots & & \\
& & p \nu(i-1) & -(\lambda+\nu(i-1)+\delta) & \delta \\
& & & & p \nu(i) & -(\lambda+\nu(i)) \\
\end{array} \right).
\]
Moreover, the transition rates are given by
\[
A_0 = \left( \begin{array}{ccccc}
\lambda & & & & \\
& \lambda & & & \\
& & & \ddots & \\
& & & & \lambda \\
\end{array} \right)
\]
\[ A_2 = \left( \begin{array}{ccccccc}
0 & & & & & & \\
& (1-p)\mu & & & & & \\
& & 2(1-p)\mu & & & & \\
& & & \ddots & & & \\
& & & & s(1-p)\mu & & \\
& & & & & \ddots & \\
& & & & & & s(1-p)\mu \\
\end{array} \right),
\]
and
\[
\scriptsize
A_1 = \left( \arraycolsep=0.55pt
\begin{array}{cccccccc}
-(\lambda+n \delta) & n \delta & & & & & & \\
p \mu & -(\lambda+\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(\lambda+s\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & s p\mu & -(\lambda+s\mu+\delta) & \delta & \\
& & & & & s p\mu & -(\lambda+s\mu)\\
\end{array} \right).
\]
\subsection{Stability condition}
\label{app:stability}
From the general theory of QBD processes \citep{Neuts1981} follows that the Markov process $(N(t),Q_1(t))$ is ergodic (stable) if and only if
\begin{equation}
\pi A_0 e < \pi A_2 e,
\label{eq:QBDstableCondition}
\end{equation}
where $e$ is the all one column vector and $\pi=(\pi_0,...,\pi_n)$ is the equilibrium distribution of the Markov process with generator $A_0+A_1+A_2$. In other words, $\pi$ is such that
\begin{equation}
\begin{array}{ll}
\pi(A_0+A_1+A_2) =0, & \pi e =1,
\end{array}
\label{eq:QBDstableProbabilityVector}
\end{equation}
and
\[
A_0+A_1+A_2 = \qquad\qquad\qquad
\]
\begin{align*}
{\scriptsize
\left(
\begin{array}{cccccccc}
-n \delta & n \delta & & & & & & \\
p \mu & -(p\mu+(n-1)\delta) & (n-1)\delta & & & & & \\
& \ddots & \ddots & \ddots & & & & \\
& & s p\mu & -(ps\mu+(n-s)\delta) & (n-s)\delta & & & \\
& & & \ddots & \ddots & \ddots & & \\
& & & & p s \mu & -(ps\mu+\delta) & \delta & \\
& & & & & p s \mu & -ps\mu\\
\end{array} \right).
}
\end{align*}
Then $\pi$ must satisfy the balance equations
\begin{align*}
- n \delta \pi_0 + p \mu \pi_1 &= 0, \\
(n-j+1)\delta \pi_{j-1} - (p\nu(j) +(n-j)\delta) \pi_j + p \nu(j+1) \pi_{j+1} &= 0, \\
\delta \pi_{n-1} - p s \mu \pi_n &= 0,
\end{align*}
with $\nu(j)=\min\{j,s\}\mu$, and the normalization condition
\[
\sum_{i=0}^n \pi_i=1.
\]
It is readily verified that
\begin{equation}
\pi_i =
\left\{\begin{array}{ll}
\pi_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\pi_0 \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistr}
\end{equation}
with
\begin{align*}
\pi_0= \left(\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right)^{-1},
\end{align*}
satisfies the balance equations and the normalization condition.
\begin{proposition}
The distribution of the closed two-node Jackson network illustrated in Figure~\ref{fig:Jennings} is given by
\begin{equation}
\hat{\pi_i} =
\left\{\begin{array}{ll}
\hat{\pi}_0 \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } 0 \leq i \leq s, \\
\hat{\pi_0} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i & \textrm{\normalfont for } s+1 \leq i \leq n \\
\end{array} \right.
\label{eq:eqdistrTwistedJennings}
\end{equation}
with
\begin{align*}
\hat{\pi}_0= \left[\sum_{i=0}^{s} \binom{n}{i} \left(\frac{\delta}{p \mu}\right)^i + \sum_{i=s+1}^{n} \binom{n}{i} \frac{i!}{s!} s^{s-i} \left(\frac{\delta}{p \mu}\right)^i \right]^{-1}.
\end{align*}
\label{prop:CriticalTilburgdistr}
\end{proposition}
\begin{proof}
We have a two-node closed Jackson network, with probability transition matrix
\[
P = \left(
\begin{array}{cc}
1-p & p \\
1 & 0
\end{array} \right).
\]
Let $r_i(m)$ denote the rate of service when there are $m$ patients at queue $i$, so $r_1(m)=\min\{m,s\}$ and $r_2(m)=m$. The throughput vector $\gamma = (\gamma_1,\gamma_2) \in \mathbb{R_1}^2$ must satisfy $\gamma = \gamma P$ and we find that $\gamma=(p,1)$ suffices. From the general theory of Jackson networks, see \cite{Jackson1963}, it follows that the stationary distribution is given by
\begin{align*}
\pi_i = G^{-1} g_1(i) g_2(n-i)
\end{align*}
with
\begin{align*}
\begin{array}{ll}
g_1(i)= \frac{(\gamma_1/\mu)^i}{\prod_{m=1}^i r_1(m)}, & g_2(n-i)= \frac{(\gamma_2/\delta)^{n-i}}{\prod_{m=1}^{n-i} r_2(m)},
\end{array}
\end{align*}
and normalization constant $G= \sum_{i=0}^n g_1(i) g_2(n-i)$. Then,
\begin{align*}
g_1(i) &= \left\{\begin{array}{ll}
\frac{1}{i! \mu^i} & \textrm{\normalfont for } 0 \leq i \leq s, \\
\frac{1}{s! s^{i-s} \mu^i} & \textrm{\normalfont for } s+1 \leq i \leq n, \\
\end{array} \right.\\
g_2(n-i) &=\frac{1}{(n-i)!} \left(\frac{p}{\delta}\right)^n \left(\frac{\delta}{p}\right)^i,
\end{align*}
and rewriting the expressions yields~\eqref{eq:eqdistrTwistedJennings}.
\end{proof}
\subsection{Stationary distribution}
\label{app:StationaryDistributrion}
Assuming that the stability condition is satisfied, we can determine the unique stationary distribution of the Markov process $(N(t),Q_1(t))$. The vector $\pi_i$ can be written as $\pi_{n+i}= \pi_n G^{i}$ for $i=0,1,...$, where $G$ is the minimal nonnegative solution of the non-linear matrix equation
\begin{equation}
A_0+G A_1 + G^2 A_2=0.
\label{eq:MG-G}
\end{equation}
The balance equations can be written as
\[
\begin{array}{ll}
\pi_{i-1} A_0+ \pi_i A_1 + \pi_{i+1} A_2=0, & i=n+1,n+2,...
\end{array}
\]
and using $\pi_{n+i}= \pi_n G^{i-n}$ for $i=0,1,...$, this find
\[
\begin{array}{ll}
\pi_n G^{i-n-1} \left(A_0+ G A_1 + G A_2\right)=0, & i=n+1,n+2,....
\end{array}
\]
\noindent
Moreover, we have the boundary equations
\begin{align*}
\pi_0 B_{00} + \pi_1 B_{10} &= 0 \\
\pi_0 B_{01} + \pi_1 B_{11} + \pi_2 B_{21} &= 0 \\
\pi_1 B_{12} + \pi_1 B_{22} + \pi_2 B_{32} &= 0 \\
&\vdots& \\
\pi_{n-2} B_{n-2 \, n-1} + \pi_{n-1} B_{n-1 \, n-1} + \pi_{n} B_{n \, n-1} &= 0 \\
\pi_{n-1} B_{n-1 \, n} + \pi_{n} B_{nn} + \pi_{n+1} A_2 &= 0,
\end{align*}
along with the normalization equation
\[
1 = \sum_{i=0}^{\infty} \pi_i e = \sum_{i=0}^{n-1} \pi_i e + \pi_n(I-G)^{-1}e,
\]
where we slightly abuse notation by using $e$ as the all ones vector of appropriate size. We note that the matrix $G$ has a spectral radius less than one and therefore $(I-G)$ is invertible.
These equations provide the tools for finding the equilibrium probabilities. Although it is hard to solve $G$ analytically from Equation~\eqref{eq:MG-G}, it is easy to solve numerically by using the following algorithm (matrix-geometric method). Rewriting~\eqref{eq:MG-G} gives
\[
G=-(A_0+G^2 A_2) A_1^{-1},
\]
where $A_1$ is invertible, since it is a transient generator matrix. Let
\[
G_{k+1}=-(A_0+G_k^2 A_2) A_1^{-1},
\]
starting with $G_0=0$. We note that $G_k \uparrow G$ as $k$ grows to infinity \citep{Neuts1981}. Once $||G_{k+1}-G_{k}||_2$ is below a certain preset threshold, we approximate $G$ by $G_{k+1}$.
\section{Proof of Proposition \ref{thm:stochasticordering}}\label{app:stochastic_ordering}
First, note that by definition of the Erlang-R model with holding, in which no more than $n$ patients can be admitted in the ED simultaneously, that $Q_1^h(t)+Q_2^h(t) \leq n = Q_1^J(t) + Q_2^J(t)$ follows directly.
Therefore, we only consider the relation between the states in the blocking and holding variants Erlang-R model.
As noted Section \ref{sec:Markov_process}, the model with holding can be characterized as a three-dimensional Markov chain $X^h(t) = (H(t),Q^h_1(t),Q^h_2(t))$ in which the components denote the number of holding, needy and content patients respectively. The Erlang-R model with blocking similarly admits a Markov process description, but with two dimensions, namely $X^b(t) = (Q^b_1(t),Q^b_2(t))$.
We prove the result by constructing a coupling between the Markov processes $X^h$ and $X^b$. Let $Z(t) := \big(\hat{X}^h(t),\hat X^b(t)\big) = \big(\hat{H}(t),\hat{Q}_1^h(t),\hat{Q}_2^h(t),\hat{Q}^b_1(t),\hat{Q}^b_2(t)\big)$.
We first define the transition rates of this five-dimensional Markov process, which naturally only depend on the current state of the system.
After that we show that the transition rates relevant to $\hat{X}^h(t)$ and $\hat{X}^{b}(t)$ coincide with those of $X^h(t)$ and $X^b(t)$, respectively. The latter implies that the marginal transitions of $\hat{X}^h(t)$ and $X^h(t)$ (and $\hat{X}^b(t)$ and $X^b(t)$) are equal, and hence so are their probability distribution of the Markov processes.
Let $Z(t) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$. While defining the reachable states from this state and associated transition rates, we distinguish four transition types, and further differentiate the transition rates depending on the current state.\\
\\*
\textbf{Arrival.}
Arrivals occur in both models simultaneously, but are handled differently according to the current queue lengths.
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr1}
(h,q_1^h+1,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b < n$,
\begin{equation}
\label{eq:arr2}
(h+1,q_1^h,q_2^h,q_1^b+1,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h < n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr3}
(h,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\item if $q_1^h+q_2^h = n$ and $q_1^b+q_2^b = n$,
\begin{equation}
\label{eq:arr4}
(h+1,q_1^h+1,q_2^h,q_1^b,q_2^b) \qquad \text{with rate }\lambda,
\end{equation}
\end{enumerate}
\noindent \textbf{Departure.}
Basically, we align service completions in the two models, but allow a completion occurring solely in either of one of the two models, only if the queue length in this model is strictly larger than in the other one.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep1}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(h-1,q_1^h,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.
\end{equation}
\item If $q_1^h < q_1^b$ and $h > 0$
\begin{equation}
\label{eq:dep2}
\left\{
\begin{array}{ll}
(h-1,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h \geq q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep3}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^b \wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$ and $h = 0$
\begin{equation}
\label{eq:dep4}
\left\{
\begin{array}{ll}
(0,q_1^h-1,q_2^h,q_1^b-1,q_2^b) & \text{with rate }(q_1^h \wedge s)(1-p)\mu,\\
(0,q_1^h,q_2^h,q_1^b-1,q_2^b) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)](1-p)\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent\textbf{Become content.}
The differentiation between transitions is similar to those in the \textit{departure} transition type.
\begin{enumerate}
\item If $q_1^h \geq q_1^b$,
\begin{equation}
\label{eq:con1}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b \wedge s)p\mu,\\
(h,q_1^h-1,q_2^h+1,q_1^b,q_2^b) & \text{with rate }[(q_1^h \wedge s)-(q_1^b \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\item If $q_1^h < q_1^b$,
\begin{equation}
\label{eq:con2}
\left\{
\begin{array}{ll}
(h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1) & \text{with rate }(q_1^h \wedge s)p\mu,\\
(h,q_1^h,q_2^h,q_1^b-1,q_2^b+1) & \text{with rate }[(q_1^b \wedge s)-(q_1^h \wedge s)]p\mu.
\end{array}
\right.\end{equation}
\end{enumerate}
\noindent
\textbf{Become needy.}
\begin{enumerate}
\item If $q_2^h \geq q_2^b$,
\begin{equation}
\label{eq:ne1}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^b\delta,\\
(h,q_1^h+1,q_2^h-1,q_1^b,q_2^b) & \text{with rate }(q_2^h-q_2^b)\delta,\\
\end{array}
\right.\end{equation}
\item If $q_2^h < q_2^b$,
\begin{equation}
\label{eq:ne2}
\left\{
\begin{array}{ll}
(h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1) & \text{with rate } q_2^h\delta,\\
(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1) & \text{with rate }(q_2^b-q_2^h)\delta,\\
\end{array}
\right.\end{equation}
\end{enumerate}
This set of transitions defines the dynamics of the Markov process $Z(t) = (\hat{X}^h(t),\hat{X}^b(t))$.
Let us now restrict our attention to the transitions in which (at least one of) the first three coordinates of $Z(t)$ changes, that is, the marginal transitions of the process $\hat{X}^h$.
Let $\hat{X}^h(t) = (h,q_1^h,q_2^h)$, then according to the transition scheme above, $\hat{X}^h$ moves to state
\begin{enumerate}
\item If $q_1^h+q_2^h < n$ (and hence necessarily $h=0$),
\[
\left\{
\begin{array}{ll}
(0,q_1^h+1,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h-1,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $q_1^h+q_2^h = n$ and $h=0$,
\[
\left\{
\begin{array}{ll}
(1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(0,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(0,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(0,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\item if $h>0$ (and hence necessarily $q_1^h+q_2^h = n$),
\[
\left\{
\begin{array}{ll}
(h+1,q_1^h,q_2^h) & \text{with rate } \lambda,\\
(h-1,q_1^h,q_2^h) & \text{with rate }(q_1^h\wedge s)(1-p)\mu,\\
(h,q_1^h-1,q_2^h+1) & \text{with rate }(q_1^h\wedge s)p\mu,\\
(h,q_1^h+1,q_2^h-1) & \text{with rate }q_2^h \delta.
\end{array}
\right.\]
\end{enumerate}
One can check that these transitions indeed coincide with the transitions in the original holding model, hence $\hat{X}^h(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Similarly, when focusing on transitions of $Z(t)$ that are relevant for $\hat{X}^b(t)$, we deduce the following transition scheme. If $\hat{X}^b(t) = (q_1^b,q_2^b)$, then the next move according to the transitions of $Z(t)$ is
\[
\left\{
\begin{array}{ll}
(q_1^b+\mathbbm{1}_{\{q_1^b + q_2^b < n\}},q_2^b) & \text{with rate } \lambda,\\
(q_1^b-1,q_2^b) & \text{with rate }(q_1^b\wedge s)(1-p)\mu,\\
(q_1^b-1,q_2^b+1) & \text{with rate }(q_1^b\wedge s)p\mu,\\
(q_1^b+1,q_2^b-1) & \text{with rate }q_2^b \delta.
\end{array}
\right.\]
These transition rates clearly coincide with the original Erlang-R model with blocking, and also hence $\hat{X}^b(t) {\;\buildrel{d}\over= \;} X^h(t)$.
Next, we show that under this coupling scheme we have that if $\hat{H}(0) = 0$, $\hat{Q}_1^h(0)=\hat{Q}_1^b(0)$ and $\hat{Q}_2^h(0)=\hat{Q}_2^b(0)$ then for all $t\geq 0$, $Z(t)$ satisfies the hypothesis:
\begin{itemize}
\item[(i)] $\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \leq \hat{Q}_1^h(t) + \hat{Q}_2^h(t)$,
\item[(ii)] $\hat{Q}_2^b(t) \leq \hat{Q}_2^h(t)$,
\item[(iii)] $\hat{Q}_1^b(t) \leq \hat{Q}_1^h(t) + H(t)$.
\end{itemize}
We do so by induction on the next state reached after a transition of the joint Markov process $Z=(\hat{X}^h,\hat{X}^b)$.
First of all, $Z(0)$ clearly satisfies (i)-(iii).
Next, assume $Z(t^-) = (h,q_1^h,q_2^h,q_1^b,q_2^b)$ satisfies the hypothesis and a transition occurs at $t$.
We show that under the specified coupling scheme, the state reached after the next transition, $Z(t)$ must satisfy (i)-(iii) as well. To do so, we differentiate between the four types of transitions that could occur: arrival, departure, become content and become needy.\\
\\*
\noindent\textbf{Arrival.}
Recall that under our coupling scheme an arrival always occurs in both the holding and blocking model simultaneously, see \eqref{eq:arr1}--\eqref{eq:arr4}. Furthermore, $q_2^h$ and $q_2^b$ are unchanged during this transition, rendering (ii) trivial.
By hypothesis $q_1^b + q_2^b \leq q_1^h+q_2^b$, hence the event $q_1^h+q_2^h < n$ and $q_1^h+q_2^b =n$, with resulting state $(0,q_1^h+1,q_2^h,q_1^b,q_2^b)$, can be excluded from our analysis.
We check the conditions for the remaining three cases.
\begin{enumerate}[noitemsep]
\item If $Z(t)= (0,q_1^h+1,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h <n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \less[i] q_1^h+q_2^h+1 =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+1 = \hat Q_1^h(t) = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b+1,q_2^b)$, then $q_1^b + q_2^b<n$ and $q_1^h + q_2^h =n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b+1 \leq n = q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h +1= \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t)= (h+1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $q_1^b + q_2^b = q_1^h+q_2^h=n$.
\begin{itemize}[noitemsep]
\item[(i)] $\hat Q_1^b(t) +\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h =\hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+h+1 = \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Departure.}
By carefully examining the possible state transitions of $Z(t)$ following a departure, we list six reachable states. However, by (iii), we have that if $h=0$, then $q_1^b \leq q_1^h$, which excludes the state $(0,q_1^h,q_2^h,q_1^b,q_2^b)$ in \eqref{eq:dep4} from the reachability graph.
We check the remaining states for conditions (i)--(iii). Again, during a departure, $q_2^b$ and $q_2^h$ are unchanged, so (ii) is automatically satisfied by the induction hypothesis.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 \less[i] q_1^h+q_2^h-1 < q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h + h-1 = \hat Q_1^h(t) +\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h-1,q_1^h,q_2^h,q_1^b,q_2^b)$, then $h>0$ and $q_1^h \geq q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 \leq q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b)$, then $h>0$ and $q_1^h < q_1^b$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b-1 < q_1^b+q_2^b \less[i] q_1^h+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^b \less[iii] q_1^h + h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h,q_1^b-1,q_2^b)$, then $h=0$.
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = (q_1^b-1)+q_2^b-1 < \less[i] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h-1 = \hat Q_1^h(t) + \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (0,q_1^h-1,q_2^h,q_1^b,q_2^b)$, then $h=0$ and $q_1^h>q_1^b$ (*).
\begin{itemize}
\item[(i)] $\hat Q_1^b(t)+\hat Q_2^b(t) = q_1^b+q_2^b \less[i] (q_1^h-1)+q_2^b \less[ii] (q_1^h-1)+q_2^h = \hat Q_1^h(t)+\hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[*] q_1^h-1 =\hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent\textbf{Content start.}
On the event of a patient becoming content, it is clear that the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected. This means that (i) is directly satisfied by the induction hypothesis.
According to \eqref{eq:con1}--\eqref{eq:con2}, three states can be reached.
\begin{enumerate}[noitemsep]
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b-1,q_2^b+1)$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \less[ii] q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 \less[iii] q_1^h+h-1 = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h-1,q_2^h+1,q_1^b,q_2^b)$, then $q_1^h > q_1^b$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[ii] q_2^h < q_2^h+1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h = \hat Q_1^h(t)+ \hat{H}(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h,q_2^h,q_1^b-1,q_2^b+1)$, then $q_1^b > q_1^h$ (*) and hence by (iii) $h > 0$. The latter is only possible if $q_1^h+q_2^h=n$,
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b+1 \leq n-q_1^b+1 = (q_1^h+q_2^h)-q_1^b+1 \less[*] q_2^h = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b-1 < q_1^h+h-1 \less[*] q_1^h+h = \hat Q_1^h(t)+\hat{H}(t)$.
\end{itemize}
\end{enumerate}
\noindent \textbf{Become needy.}
Just as in the event of content start, the sums $\hat Q_1^h(t)+\hat Q_2^h(t)$ and $\hat Q_1^b(t)+\hat Q_2^b(t)$ and $H(t)$ are unaffected, whereby (i) is directly satisfied by the induction hypothesis.
By (ii), we have $q_2^h \geq q_2^b$. This excludes the state $(h,q_1^h,q_2^h,q_1^b+1,q_2^b-1)$ from being reached, see \eqref{eq:ne2}.
We check the remaining two possibilities.
\begin{enumerate}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b+1,q_2^b-1)$.
\begin{itemize}[noitemsep]
\item[(ii)] $\hat Q_2^b(t) = q_2^b-1 \less[ii] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b+1 \less[iii] q_1^h+h+1 = \hat Q_1^h(t)+\hat H(t)$.
\end{itemize}
\item If $Z(t) = (h,q_1^h+1,q_2^h-1,q_1^b,q_2^b)$, then $q_2^h > q_2^b$ (*).
\begin{itemize}
\item[(ii)] $\hat Q_2^b(t) = q_2^b \less[*] q_2^h-1 = \hat Q_2^h(t)$.
\item[(iii)] $\hat Q_1^b(t) = q_1^b \less[iii] q_1^h+h < q_1^h+1+h =\hat Q_1^h(t) + \hat H(t)$.
\end{itemize}
\end{enumerate}
Hence, the state reached after any feasible transition under the coupling scheme satisfies the conditions (i)--(iii).
Thus we conclude that the joint process\\ $(\hat{H}(t),\hat Q_1^h(t),\hat Q_2^h(t),\hat Q_1^b(t),\hat Q_2^b(t))$ adheres to (i)--(iii) for all $t$. Consequently, we have that (i) implies
\begin{align*}
\mathbb{P}\left(Q_1^b(t) + Q_2^b(t) \geq k\right) &= \mathbb{P}\left(\hat{Q}_1^b(t) + \hat{Q}_2^b(t) \geq k\right)\\
&=\sum_{j=0}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&=\sum_{j=k}^n \mathbb{P}\left( \hat Q_1^b(t) + \hat Q_2^b(t) \geq k , \hat Q_1^h(t)+\hat Q_2^h(t) = j \right) \\
&\leq \sum_{j=h}^n \mathbb{P}\left( \hat Q_1^h(t)+\hat Q_2^h(t) = j \right)\\
&= \mathbb{P}\left( Q_1^h(t) + Q_2^h(t) \geq k\right) = \mathbb{P}\left(Q_1^h(t) + Q_2^h(t) \geq k\right).
\end{align*}
The other two orderings follow similarly.
\begin{remark}
Note that under this coupling scheme we cannot get the ordering $\hat Q_1^h(t)(t) \geq \hat Q_1^b(t)(t)$ for all $t\geq 0$. A minimal counter example occurs for $s=n=1$. Let $Z(0) = ((0,0,0),(0,0))$. First, two arrivals occur, such that state $((1,1,0),(1,0))$ is reached, followed by a departure transition, yielding $((0,1,0),(0,0))$. Next, the one patient left in the model with holding system becomes content, so that we obtain $((0,0,1),(0,0))$.
At this stage, if an arrival occurs, the arriving patient will be put in the holding queue in the model with holding, and admitted to nurse queue in the model with blocking. Hence we end up in state $((1,0,1),(1,0))$, in which $\hat Q_1^h(t) < \hat Q_1^b(t)$.
\end{remark}
\section{Proof of Proposition \ref{prop:stability_convergence}}\label{app:proof_stability_convergence}
Define
\[
A(s,n) = \sum_{k=0}^s \frac{k}{s} \, \binom{n}{k} b^k ,\quad
B(s,n) = \sum_{k=s+1}^n \frac{k!}{s!} \, \binom{n}{k} s^{s-k} b^k, \quad
C(s,n) = \sum_{k=0}^s \binom{n}{k} \, b^k,
\]
\[
\]
where $b = \delta/p\mu = r/(1-r)$. Then
\[
\rho_{\rm max}(s,n) = \frac{A(s,n)+B(s,n)}{C(s,n)+B(s,n)}.
\]
Proving that $\rho_{\rm max}(s,n) \to 1$ as $R_1\to\infty$ with $s$ and $n$ as in \eqref{eq:twofoldscaling} is equivalent to showing that
\begin{equation}\label{eq:proof_stab_1}
1-\rho_{\rm max}(s,n) = \frac{C(s,n)-A(s,n)}{C(s,n)+B(s,n)} = \frac{(1+b)^{-n}[C(s,n)-A(s,n)]}{(1+b)^{-n}[C(s,n)+B(s,n)]} \to 0.
\end{equation}
First, we rewrite
\begin{align*}
(1+b)^{-n} A(s,n)
&= (1+b)^{-n} \sum_{k=1}^s \frac{n}{s} \binom{n-1}{k-1} b^k \\
&= \frac{n}{s}\left(\frac{b}{1+b}\right)\sum_{k=0}^{s-1} \binom{n-1}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-1-k}\\
&= \frac{r n}{s}\sum_{k=0}^{s-1} \binom{n-1}{k} r^k (1-r)^{n-1-k}\\
&= \frac{r n}{s} \mathbb{P}( {\rm Bin}(n-1,r) \leq s-1 ) \\
&= \frac{rn}{s} \mathbb{P}\left( \frac{{\rm Bin}(n-1,r) - (n-1)r}{\sqrt{nr(1-r)}} \leq \frac{s-1 - (n-1)r}{\sqrt{nr(1-r)}} \right)\\
&\to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
since $nr/s = 1 + O(1/\sqrt{R_1})$.
Also,
\begin{align*}
(1+b)^{-n} C(s,n)
&= \sum_{k=0}^s \binom{n}{k} \left(\frac{b}{1+b}\right)^k \left(\frac{1}{1+b}\right)^{n-k}\\
&= \sum_{k=0}^s \binom{n}{k} r^k (1-r)^{n-k}\\
&= \mathbb{P}( {\rm Bin}(n,r) \leq s) \to \Phi\left(\frac{\beta-\gamma\sqrt{r}}{\sqrt{1-r}}\right).
\end{align*}
Therefore, we have $(1+b)^{-n}[C(s,n)-A(s,n)] \to 0$ as $\lambda\to\infty$.
For the remaining term,
\begin{align*}
(1+b)^{-n} B(s,n)
&= (1+b)^{-n}\sum_{k=s+1}^n \binom{n}{k}\,\frac{k!}{s!} s^{s-k} b^k \\
&= (1+b)^{-n}\frac{n!}{s!}\, s^s\sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{-k}\\
&= (1+b)^{-n} \frac{n!}{s!}\, s^s\, \left(\frac{b}{s}\right)^n \sum_{k=s+1}^n \frac{1}{(n-k)!} \left(\frac{s}{b}\right)^{n-k}\\
&= r^n\, \frac{n!}{s!} s^{s-n} \sum_{m=0}^{n-s-1} \frac{1}{m!} \left(\frac{s}{b}\right)^m\\
&= \left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b} \, \mathbb{P}({\rm Pois}(s/b)\leq n-s-1),
\end{align*}
in which
\begin{align*}
\mathbb{P}({\rm Pois}(s/b)\leq n-s-1)
&= \mathbb{P}\left(\frac{{\rm Pois}(s/b)-s/b}{\sqrt{s/b}} \leq \frac{n-s-1-s/b}{\sqrt{s/b}}\right) \\
&\to \Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right),
\end{align*}
as $\lambda\to\infty$.
By Stirling's approximation,
\begin{align*}
\left(\frac{r}{s}\right)^n \frac{n!}{s!} s^s \,{\rm e}^{s/b}
&\sim \left(\frac{r}{s}\right)^n \sqrt{\frac{n}{s}} \,\frac{n^n {\rm e}^{-n}}{s^s {\rm e}^{-s}}\, s^s \,{\rm e}^{s/b} \\
&= \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s+s/b} = \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+s/r}.
\end{align*}
Since,
\[
\frac{rn}{s} = 1 + \frac{\gamma\sqrt{r}-\beta}{\sqrt{R_1}} + O(1/R_1),
\]
we find $\sqrt{n/s} = 1/\sqrt{r} + O(1/\sqrt{R_1})$ and
\begin{align*}
\log\left[ \left(\frac{rn}{s}\right)^n \sqrt{\frac{n}{s}} {\rm e}^{-n+\tfrac{s}{r}} \right]
&= n \log\left[ \frac{rn}{s}\right] - n+\frac{s}{r}\\
&= -n \left[ \left(1-\frac{rn}{s}\right) + \frac{1}{2}\left(1-\frac{rn}{s}\right)^2 + O(R^{-\tfrac{3}{2}}) \right] + \frac{s}{r}\left(1-\frac{rn}{s}\right)\\
&= \frac{s}{r}\left(1-\frac{rn}{s}\right)^2 - \frac{n}{2}\left(1-\frac{rn}{s}\right)^2 + O(1/\sqrt{R_1})\\
&= \frac{(\gamma\sqrt{r} - \beta)^2}{2r} + O(1/\sqrt{R_1}),
\end{align*}
as $\lambda\to\infty$ and hence,
\[
(1+b)^{-n} B(s,n) \to \varphi\left(\frac{\gamma\sqrt{r}-\beta}{\sqrt{r}}\right)\Phi\left(\frac{\gamma-\beta/\sqrt{r}}{\sqrt{1-r}}\right).
\]
Hence, we conclude that the denominator of \eqref{eq:proof_stab_1} converges to a constant value as $R_1$ grows, and hence the $1-\rho_{\rm max}(s,n)\to 0$ as $\lambda\to\infty$.
\resettocdepth
\end{subappendices}
\chapter{Transient error approximation in a L\'evy queue}
\begin{chapterstart}
Motivated by a capacity allocation problem within a finite planning period, we conduct a transient analysis of a single-server queue with L\'evy input. From a cost minimization perspective, we investigate the error induced by using stationary congestion measures as opposed to time-dependent measures. Invoking recent results from fluctuation theory of L\'evy processes, we derive a refined cost function, that accounts for transient effects. This leads to a corrected capacity allocation rule for the transient single-server queue. Extensive numerical experiments indicate that the cost reductions achieved by this correction can by significant.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Transient error approximation in a L\'evy queue}\\
\textit{Britt Mathijsen and Bert Zwart}\\
To appear in \textit{Queueing Systems}
\end{flushright}
\newpage
\section{Introduction}
The issue of matching a service system's capacity to stochastic demand induced by its clients arises in many practical settings. Typically, the resources available to satisfy demand are scarce and hence expensive. This forces the manager to consider a trade-off between the system efficiency and the quality of service perceived by its clients. In this chapter, we focus on this trade-off in the context of the $M/G/1$ queue, in which the variable amenable for optimization is the server speed $\mu$.
In general, optimizing the server speed $\mu$ in a single-server queue in a time-homogeneous environment, while trading off congestion levels against capacity allocation costs, does not pose any technical challenges. Typically, the objective function to be minimized, the total cost function, has the shape
\begin{equation}\label{eq:intro}
\Pi_\iy(\mu) = \mathbb{E}[Q_\mu(\infty)] + \aaa\mu = \frac{\la\mathbb{E}[B^2] }{2(\mu-\la\mathbb{E}[B])} + \aaa\mu,
\end{equation}
where $\mathbb{E}[Q_\mu(\infty)]$ denotes the expected steady-state amount of work given server speed $\mu$, and $B$ describes the service requirement per arrival. The parameter $\aaa>0$ represents the relative capacity allocation costs incurred by deploying service rate $\mu$. This one-dimensional optimization problem yields the optimizer
\begin{equation*}
\mui = \lambda \mathbb{E}[B] + \sqrt{\frac{\la\mathbb{E}[B^2]}{2\aaa}}.
\end{equation*}
Despite the simplicity and tractability of the problem described above, the presence of the \emph{steady-state} measure in the cost function in \eqref{eq:intro} should be handled carefully. By employing this particular cost structure, one automatically agrees with the underlying assumption of the system being sufficiently close to its steady state.
However, referring the practical applications of the single-server model, system parameters rarely remain constant over time. Moreover, planning periods for the optimization problem are naturally finite. Hence, the \emph{true} expected costs incurred, which we denote by $\Pi_T(\mu)$, in addition depend on the length of the planning period $T$. Consequently, the usage of steady-state models for decision making needs to be justified by a more elaborate time-dependent or \emph{transient} analysis for these type of settings.\\
\\*
\noindent
\textbf{Related literature}.
The time-dependent behavior of the single-server queue received much attention in queueing theory. First efforts to analyze the time-dependent properties of the $M/G/1$ queue date back to the 1950s and 1960s, e.g. \cite{Benes1957,Gaver1959,Kendall1951,Takacs1955,Takacs1962}. The analyses in these papers mostly yield implicit expressions for performance characteristics through Laplace transforms, integro-differential equations and infinite convolutions.
More specifically, there is vast literature on the transient analysis of the $M/M/1$ queue, with the goal to derive explicit expressions for queue length characteristics, see e.g. \cite{Abate1987,Cohen1982,Pegden1982,Prabhu1964}.
These works provide a variety of explicit expressions for the transient dynamics, although the complexity of the resulting expressions, typically involving Bessel functions, expose the intricate intractability of the matter. Consequently, approximation methods for insightful quantification of the dynamics based on numerical \cite{Neuts1966} or asymptotic methods, have become prevalent in more recent literature.
The asymptotic methods either exploit knowledge on the evolution of the queueing process as time $t$ grows large \cite{Abate1987,Newell1982,Odoni1983}, or as the arrival rate $\la$ is increased to infinity \cite{Abate1987a,Abate1987b,Gaver1968}.
It is noteworthy that a substantial contribution to the transient literature is made by Abate and Whitt \cite{Abate1987a,Abate1987b,Abate1987,Abate1994}, who exploit the existence of a decomposition of the mean transient queue length and obtain expressions for the moments of the queue length and virtual waiting through probabilistic arguments in several queueing models.
More recently, asymptotic methods have been used to justify the application of stationary performance measures in Markovian environments or to refine them, see e.g. \cite{Green1991,Whitt1991}.
Other approximative methods known as uniform acceleration expansions \cite{Massey1998} have been developed to reveal the asymptotic behavior of the single-server queue as a function of $t$, which are moreover able to capture time-varying arrival rates.
The majority of the works mentioned above do reflect on the error imposed by usage of steady-state performance metrics instead of the correct time-dependent counterpart. However, no light has been shed on the accumulation of this error over a finite period of time. To the best of our knowledge, the only work that addresses this issue is the paper by Steckley and Henderson \cite{Steckley2007}, who compute an approximation for the error accumulated between the steady-state and transient delay probability. Our analysis on the other hand is centered around the mean workload, which requires a different approach. In addition, the focus in \cite{Steckley2007} is on performance measures only, while the main goal of our work is to investigate the quality of staffing rules. \\
\\*
\noindent\textbf{L\'evy input}.
Although the $M/G/1$ queue serves as the leading example in our analysis, we choose to use a more general framework for the arrival process of the queue. Namely, we let the server face a L\'evy process.
This gives the advantage that once we have obtained the results, we can apply them to broader queue input classes, such as Brownian motion and the Gamma process.
To shed light on the influence of the transience of the queueing process on traditional staffing questions, we will study the capacity allocation problem in the context of cost minimization in which the objective function is $\Pi_T(\mu)$, i.e. a function of both $\mu$ and $T$. We investigate how the invalidity of the stationary assumption is echoed through the operational cost accounting for congestion-related penalties.
Furthermore, we establish a result on the strict convexity of the function $\Pi_T(\mu)$, for almost all values of $T$ (with a few minor exceptions for certain deterministic initial states), which is an essential property for convergence of both cost function and corresponding minimizer to their stationary counterparts.
\\
\\*
\noindent\textbf{Corrected staffing rule}.
As it will appear that an exact analysis of this disparity is intractable, we will present an explicit approximate correction to the conventional stationary objective function given by $\Psi(\mu)/T$ and prove that
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \frac{\Psi(\mu)}{T} + O(1/T^2),
\end{equation*}
with the help of recent results from the fluctuation theory of L\'evy processes.
Based on this refinement we ultimately examine how incorporating transient effects\\ \noindent changes the optimal capacity level and propose a refinement to the steady-state capacity allocation rule,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T).
\end{equation*}
We moreover deduce an explicit expression for $\mu_\bullet$ in terms of the initial state and the first three moments of the service requirement per arrival.
It is noteworthy that similar refined square-root staffing rules have been proposed for multi-server queues in the Halfin-Whitt regime, see e.g. \cite{Janssen2015,Janssen2008,Janssen2011,Randhawa2014,Zhang2012}. In those cases, the relevant decision value is the number of servers and refinements are derived for $\la\to\iy$, whereas we consider the regime $T\to\infty$.
Building upon the insights gained through the analysis of this optimality gap, we reflect on the parameter settings of the underlying queueing process in which our refined capacity sizing rule yields significant improvement and in which cases it has little effect. Special emphasis is put on the relationship between the accuracy of the standard procedure and the length of the planning period.
\\
\\*
\noindent\textbf{Structure of the chapter}.
The remainder of this chapter is structured as follows. Section \ref{sec:model_description} is devoted to the model description and presents some preliminary results. The main result will be given in Section \ref{sec:analysis} and results regarding the optimization problem will be discussed in Section \ref{sec:optimization}, followed by the validation of our novel techniques through numerical experiments in Section \ref{sec:numerics}. We will give some concluding remarks and topics for further research in Section \ref{sec:conclusion_chapter6}. We have deferred all proofs to the appendix.
\section{Model description}
\label{sec:model_description}
\subsection{A queueing model with L\'evy input \label{sec:levymodel}}
The model that inspired our study is the standard $M/G/1$ queue starting out of equilibrium. Customers arrive to the queue according to a Poisson process with rate $\la$ and each arrival has iid service requirement $B_i$, stemming from a common random variable $B$.
Without loss of generality we will assume $\mathbb{E}[B] = 1$ throughout. The server is able to remove $\mu$ amounts of work from the system per time unit; a variable we will refer to as the \emph{server speed}.
E.g. if $\mu = 3$ and two customers are in the system with remaining service times $4$ and $2$, then the queue will be empty 2 time units later, provided that no new arrivals occur in the meantime.
Let $N_\la(t)$ denote the number of arrivals until time $t$.
Accordingly, the total work generated by the customers is given by
\begin{equation*}
Z_\la(t) = \sum_{i=1}^{N_\la(t)} B_i.
\end{equation*}
Furthermore, define $X_{\la,\mu}(t) := Z_\la(t) - \mu t$. We call $X_{\la,\mu}$ the \emph{net-input process}.
More generally, we assume throughout the chapter that $X_{\la,\mu}$ is a L\'evy process.
Specifically, we let $Z_\la$ be of the form $Z_\la(t) = U(\la t)$, where $U$ is a spectrally positive L\'evy process generated by the triplet $(a,\s,\nu)$ and $\mathbb{E}[U(1)] = 1$.
This restriction to spectrally positive processes is equivalent to stating $\nu(-\infty,0)=0$ and is a vital assumption to our analysis.
Subsequently, we assume the net-input process $X_{\la,\mu}$ to be
\begin{equation}
\label{eq:Xlmprocess}
X_{\la,\mu}(t) = U(\la t) - \mu t, \qquad t \geq 0.
\end{equation}
Note that by setting $a=\s=0$ and $\nu = \la\, F_B$, where $F_B$ is the cumulative distribution function of $B$, we retrieve the original $M/G/1$ queue.
The stochastic process central to our analysis is the \emph{workload process} $Q_{\la,\mu}(t)$, $t\geq 0$, which describes the amount of work the server is facing at time $t$.
The net-input process $X_{\la,\mu}$ completely determines the trajectory of $Q_{\la,\mu}$, namely
\begin{equation}\label{eq:Qlm}
Q_{\la,\mu}(t) = \max\left\{ Q(0) + X_{\la,\mu}(t), \sup_{s\in[0,t]} [X_{\la,\mu}(t)-X_{\la,\mu}(s)]\right\}, \qquad t\geq 0,
\end{equation}
where $Q(0)$ is the initial workload in the system.
In fact, $Q_{\la,\mu}$ is the reflected version of $X_{\la,\mu}$ with reflection barrier at zero.
Careful inspection of the structure also reveals that $X_{\la,\mu}(t) \equiv X_{\la/\mu,1}(\mu t) \equiv X_{1,\mu/\la}(\la t)$, so that
\begin{equation}
\label{eq:Qidentity}
Q_{\la,\mu}(t) {\;\buildrel{d}\over= \;} Q_{\la/\mu,1}(\mu t) {\;\buildrel{d}\over= \;} Q_{1,\mu/\la}(\la t)
\end{equation}
for all $\la,\mu,t>0$.
This identity will prove to be convenient for the numerical analysis in Section \ref{sec:numerics}. For reasons of clarity, we omit the subscript $\la$ in our expressions if no ambiguity is possible.
The process $Q_{\mu}$ is a natural indicator of the level of congestion in the system and therefore a good choice for quantifying the Quality of Service (QoS) received by a client.
We remark that alternative processes characterizing congestion in the system can be deduced directly from $Q_{\mu}(t)$. For example, consider the virtual waiting time process $V_{\mu}(t)$, which is the waiting time a customer would experience if he arrives at time $t$. This satisfies the relation $V_{\mu}(t) \equiv Q_{\mu}(t)/\mu$ for all $t\geq 0$.
Likewise, the expected number of the customers in the system $L_{\mu}(t)$ at time $t\geq 0$ is given by Little's law
\begin{equation*}
\mathbb{E}[L_{\mu}(t)] = \la\, \mathbb{E}[V_{\mu}(t)] = \frac{\la}{\mu}\, \mathbb{E}[Q_{\mu}(t)].
\end{equation*}
To facilitate our investigation of the queueing model, we end this subsection by introducing some notation regarding the net-input and workload process and by stating a useful preliminary result concerning the stationary process $\Qlm(\iy)$.
Throughout the chapter we assume $\mu>\la$ to ensure ergodicity of the queue and convergence in distribution to the limit
\begin{equation*}
\Qlm(\iy) := \lim_{t\to\iy} \Qlm(t),
\end{equation*}
for any initial state $Q(0)<\iy$. This random variable necessarily coincides with the stationary distribution of $\Qlm(t)$.
By $\ka_U(\cdot)$ and $\ka_{\mu}(\cdot)$ we denote the L\'evy exponents of the processes $U$ and $\Xlm$, respectively:
\begin{equation*}
\ka_{\mu}(\thh) = \log \mathbb{E}[{\rm e}^{\thh \Xlm(1)}] = \log \mathbb{E}[{\rm e}^{\thh(U(\la) - \mu)}] = \la \ka_U(\thh) - \mu \thh.
\end{equation*}
Furthermore, define $u_k = \mathbb{E}[\{U(1) - \mathbb{E} U(1)\}^k]$ for $k=2,3,...$.
Using this representation we obtain the following preliminary result.
\begin{lemma}\label{lemma:workloadmoments}
Let $\mathbb{E}|U(1)|<\infty$, $u_2, u_3 < \iy$ and $\mu > \la$. If $Q_{\mu}(\infty)$ represents the steady-state distribution of the workload process, then
\begin{equation*}
\mathbb{E}[\Qlm(\infty)] = \frac{\la u_2}{2(\mu-\la)},\qquad \mathbb{E}[Q_{\mu}^2(\iy)]=\frac{\la^2u_2^2}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)}.
\end{equation*}
\end{lemma}
The proof of Lemma \ref{lemma:workloadmoments} follows directly by differentiation of the Laplace transform of $Q_\mu(\iy)$ and is given in Appendix \ref{app:proof_lemma_workload_moments}.
\subsection{Finite horizon}
For the purpose of our research, we are interested in the dynamics of the workload process within a fixed time frame of length $T>0$.
For all $0\leq t \leq T$, we assume that the parameters of the queue, $\la,\mu,u_2,u_3$, remain unchanged.
If at $t=0$ the queue is not in steady-state corresponding to the specified parameters of the starting period, the process $\{\Qlm(t)\}_{t\in[0,T]}$ differs from its stationary counterpart $\Qlm(\infty)$.
To illustrate this, Figure \ref{fig:transientmeans} depicts the expected value $\Qlm$ in a $M/M/1$ queue as a function of time for several initial workloads $Q(0)$ for a particular setting of $\la$ and $\mu$.
Clearly, transient behavior of $\mathbb{E}[\Qlm(t)]$, for $Q(0) \neq \Qlm(\iy)$, differs significantly from the steady-state mean with the same system parameters.
Note that even if $Q(0) \equiv \mathbb{E}[\Qlm(\iy)]$, the time-dependent mean does not coincide with the steady-state mean. Moreover, $\mathbb{E}[\Qlm(t)]$ is not even a strictly increasing nor decreasing function of time. This phenomenon is a consequence of the decomposition of the transient mean into one strictly increasing, and a strictly decreasing term for $Q(0)>0$, as discussed in \cite{Abate1987}.
Nonetheless, $\Qlm(t)$ converges in distribution to $\Qlm(\infty)$ as $t\to\iy$, if $\mu>\la$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.15,yscale=0.225]
\draw (0,0) -- coordinate (x axis mid) (50,0);
\draw (0,0) -- coordinate (y axis mid) (0,21);
\node[right] at (51,0) {$t$};
\node[rotate=90, above=0.7 cm] at (y axis mid) {$\mathbb{E}[\Qlm(t)]$};
\draw[dashed, thick, gray] (0,10) -- coordinate (eq) (51,10);
\draw[->] (24,6.4) -- coordinate (a1) (21.65,8.49574);
\node[right=0.6cm,below=0.3cm] at (a1) {$Q(0)\equiv 0$};
\draw[->] (14,6.2) -- coordinate (a2) (13.,8.49434);
\node[right=0.2cm,below=0.3cm] at (a2) {$Q(0)\equiv10$};
\draw[->] (9,15.5) -- coordinate (a3) (7.5,13.7648);
\node[right=0.6cm,above=0.1cm] at (a3) {$Q(0)\equiv20$};
\draw[->] (40,12.5) -- coordinate (a4) (38,10.9712);
\node[right,above=0.3cm] at (a4) { $Q(0)\sim \exp\left(\tfrac{1}{15}\right)$ };
\foreach \x in {0,10,...,50}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {\x};
\foreach \y in {5,10,15,20}
\draw (1pt,\y) -- (-20pt,\y)
node[anchor=east] {\y};
\draw[thick,color = col1] plot
file {Chapter_6/tikz/means0.txt};
\draw[thick,color = col3] plot
file {Chapter_6/tikz/means10.txt};
\draw[thick,color = col4] plot
file {Chapter_6/tikz/means20.txt};
\draw[thick,color = col5] plot
file {Chapter_6/tikz/meansExp.txt};
\end{tikzpicture}
\caption{Time-dependent mean workload in a $M/M/1$ queue with $\la = 10$ and $\mu=11$ for different initial states $Q(0)$. The dashed line depicts $\mathbb{E}\Qlm(\iy)$.}
\label{fig:transientmeans}
\end{figure}
Since the time horizon of our analysis is limited to $t\leq T$, the process may not approach the steady-state distribution sufficiently close to appropriately use its steady-state properties for capacity allocation.
To overcome this disparity, we propose a way to include the influence of this transient phase in the capacity allocation problem.
\subsection{Cost structure}
As mentioned before, we are interested in balancing the QoS and efficiency of the queue by choosing the optimal server speed $\mu$.
The adjective \emph{optimal} indicates that we intend to choose the speed according to some objective function.
In our case, we conduct our analysis based on a cost function, which consists of a part accounting for the penalty for congestion in the system and a part for staffing cost. The cost value of both parts is governed by the variable $\mu$.
The instantaneous cost incurred at time $t$ equals
\begin{equation*}
\mathbb{E}[\Qlm(t)] + \aaa \mu,
\end{equation*}
where $\aaa$ is a positive constant defining the \emph{relative staffing cost}.
Hence, the cost structure we apply is a combination of the transient mean of the workload process and a linear staffing cost.
Accumulated and normalized over the period $[0,T]$, the cost function on which the rest of this chapter will be based equals
\begin{equation}\label{eq:PiT}
\Pi_{T}(\mu) := \frac{1}{T}\int_0^T\left( \mathbb{E}[\Qlm(t)] + \aaa\mu\, \right) {\rm d} t
= \frac{1}{T} \int_0^T \mathbb{E}[\Qlm(t)] {\rm d} t + \aaa\mu.
\end{equation}
We use shorthand notation for the normalized congestion costs:
\begin{equation}\label{eq:CTmu}
C_{T}(\mu) := \frac{1}{T}\int_0^T \mathbb{E}[Q_{\mu}(t)] {\rm d} t,
\end{equation}
and $C_{\iy}(\mu) := \mathbb{E}[\Qlm(\iy)]$.
In order to compare the actual costs incurred over the interval $[0,T]$ to the cost function of the queue in stationary conditions, we define
\begin{equation}\label{eq:PiInf}
\Pi_{\iy}(\mu) := C_{\iy}(\mu) + \aaa \mu = \mathbb{E}[Q_\mu(\iy)] + \aaa\mu,
\end{equation}
which allows an explicit expression by Lemma \ref{lemma:workloadmoments}.
Under mild conditions on the net-input process and the distribution of the initial state, the cost functions coincide for $T\to\iy$.
\begin{proposition}\label{prop:cost_convergence}
Let $\mu>\la$ and assume $\mathbb{E}[U(1)],\, \mathbb{E}[Q(0)] < \iy$. Then
\begin{equation*}
\lim_{T\to\iy} \Pi_{T}(\mu) = \Pi_{\iy}(\mu).
\end{equation*}
\end{proposition}
\noindent Rewriting \eqref{eq:PiT} gives the relation
\begin{align}
\Pi_{T}(\mu) &= \frac{1}{T}\int_0^{T} \left( \mathbb{E}[\Qlm(t)] - \mathbb{E}[\Qlm(\iy)] \right) {\rm d} t + \mathbb{E}[\Qlm(\iy)] + \aaa\mu = \Omega_{T}(\mu) + \Pi_{\infty}(\mu).
\label{eq:decomp}
\end{align}
Section \ref{sec:analysis} is concerned with the analysis of the correction factor $\Omega_{T}(\mu)$.
Ultimately, we are concerned with the additional costs incurred by choosing the server speed through minimization of $\Pi_{\iy}(\mu)$ instead of $\Pi_{T}(\mu)$.
Therefore, we formulate the exact and approximate optimization problems as follows
\begin{equation}\label{eq:muStar}
\mu_T^\star := \arg\min_{\mu\geq 0} \Pi_{T}(\mu), \qquad \qquad \mu_\infty^\star := \arg\min_{\mu\geq 0} \Pi_{\iy}(\mu),
\end{equation}
\begin{equation}\label{eq:piStar}
\Pi_{T}^\star := \Pi_{T}(\mu_T^\star), \qquad \qquad \Pi_{\iy}^\star := \Pi_{T}(\mu_\iy^\star).
\end{equation}
In Section \ref{sec:optimization} we turn to the comparison of $\mu_T^{\star}$ and $\mu_\iy^\star$ as well as the \emph{optimality gap} $\Pi_{\iy}^\star - \Pi_{T}^\star$.
\section{Analysis of the objective function}
\label{sec:analysis}
From \eqref{eq:decomp} it is evident that, for finding an explicit characterization of $\Pi_{T}(\mu)$, it suffices to study the term $\Omega_T(\mu)$ in more detail. We start by stating the main result of this section, which describes the leading order behavior of $\Omega_T(\mu)$ as $T$ increases.
\begin{theorem}\label{thm:mainresult}
Let $X_\mu(t)$ be of the form \eqref{eq:Xlmprocess}. If $\mathbb{E}[\max(Q(0),Q_\mu(\infty))^3] < \iy$ and $u_2,u_3 < \iy$, then
\begin{align*}
\Omega_T(\mu) &= \frac{\mathbb{E}[Q(0)^2] - \mathbb{E}[Q_\mu(\iy)^2]}{2T(\mu-\la)} + O\left(\frac{1}{T^2}\right) \nonumber\\
&= \frac{1}{2T(\mu-\la)}\left( \mathbb{E}[Q(0)^2] - \frac{\la^2 u_2^2}{2(\mu-\la)^2} - \frac{\la u_3}{3(\mu-\la)}\right) + O\left(\frac{1}{T^2}\right),
\end{align*}
for $\mu>\la$.
\end{theorem}
Note that this expression provides an \emph{approximation} of the actual cost function
$\Pi_T(\mu)$. We elaborate on the implications of this additional information on the optimization problem in Section \ref{sec:optimization}.
In the remainder of this section we provide a detailed description of the steps taken to obtain this outcome.
We assume a fixed service rate $\mu$ throughout the analysis in this section and therefore omit the subscript $\mu$. Proofs of the intermediate results can be found in Appendix \ref{app:proofs_analysis}.
\subsection{Constructing a coupling}
Before starting our analysis of the correction term $\Omega_{T}(\mu)$ we introduce some auxiliary notation.
By $Q^A(t)$ we denote the workload process as described in Subsection \ref{sec:levymodel} with initial state $A$ and $\mathbb{E}_A$ the expectation with respect to any non-negative random variable $A$, which is independent of the net-input process $X$.
To be able to compare $\mathbb{E}[Q(t)]$ and $\mathbb{E}[Q(\iy)]$ as in $\Omega_T(\mu)$, we will use a coupling technique.
Observe that by definition of the stationary distribution $Q(\iy) {\;\buildrel{d}\over= \;} Q^{Q(\iy)}(t)$ for all $t \geq 0$ and therefore $\mathbb{E}[Q(\iy)] = \mathbb{E}_{Q(\iy)}[Q^{Q(\iy)}(t)]$. Furthermore, $\mathbb{E}[Q(t)] = \mathbb{E}_{Q(0)}[Q^{Q(0)}(t)]$.
Hence, quantifying the difference between the transient and stationary mean is equivalent to comparing the workload processes of two queues starting in two different (random) states at $t=0$.
We starting our analysis for two queues starting in two \textit{deterministic} states $x,y\geq 0$, respectively. At the end of our analysis we will obtain the original form by replacing $x$ with $Q(0)$ and $y$ with $Q(\iy)$.
Equation \eqref{eq:Qlm} shows that all randomness in the workload process originates from the process $X(t)$.
With this in mind, we couple the processes $Q^x(t)$ and $Q^{y}(t)$ on a sample path level by feeding both queues the same net-input process $X(t)$ for $t\geq 0$.
This allows us to compare the processes in the same probability space, so that $\mathbb{E}[Q^x(t)] - \mathbb{E}[Q^y(t)] = \mathbb{E}[Q^x(t) - Q^y(t)]$ for all $t\geq 0$.
Define
\begin{equation*}
Y^{x,y}(t) := Q^x(t) - Q^y(t)
\end{equation*}
and
\begin{equation*}
\Omega_{T}^{x,y} := \frac{1}{T}\,\int_0^T \mathbb{E}\left[Y^{x,y}(t)\right] \, {\rm d} t.
\end{equation*}
A possible sample path triple for $Q^x(t)$, $Q^0(t)$ and $Y^{x,0}(t)$ is depicted in Figure \ref{fig:samplePaths}.
As we see from this figure, $Y^{x,0}(t)$ has nice structural properties which we will exploit in the next subsection.
\begin{figure}
\centering
\begin{tikzpicture}[y=0.6cm, x=0.01cm]
\draw (0,0) -- coordinate (x axis mid) (800,0);
\draw (0,0) -- coordinate (y axis mid) (0,6.5);
\node[below=0.2cm] at (x axis mid) {$\to t$};
\node[rotate=90, above=0.2cm] at (y axis mid) {$Q(t)$};
\node[above=1.3cm,left =0.08 cm] at (y axis mid) {$x$};
\draw plot
file {Chapter_6/tikz/samplePathLevy.txt};
\draw[color = gray] plot
file {Chapter_6/tikz/samplePathLevy2.txt};
\draw[thick,color=col1] plot
file {Chapter_6/tikz/runningMinimumLevy.txt};
\end{tikzpicture}
\caption{Sample path visualization of the processes $Q^x(t)$ (solid), $Q^0(t)$ (gray) and $Y^{x,0}(t)$ (red).}
\label{fig:samplePaths}
\end{figure}
\subsection{Difference process and leading order behavior of the correction term}
We further examine the \emph{difference process} $Y^{x,y}(t)$ with $x>y$. Recall from \eqref{eq:Qlm},
\begin{equation}\label{eq:Wz}
Q^z(t) = \max\{ z + X(t),\, \sup_{0<s\leq t} [X(t)-X(s)]\} = X(t) + \max\{ z, -\inf_{0\leq s\leq t} X(s)\},
\end{equation}
for any initial state $z\geq 0$, where $X(t)$ is a L\'evy process with no negative jumps.
Let $\tau^z(w)$, $0\leq w<z$ denote the first passage time of level $w$ by the process starting in $z$, i.e.
\begin{equation*}
\tau^z(w) := \inf \left\{ t \geq 0\, |\, Q^z(t) \leq w \,\right\}.
\end{equation*}
Then it is easily seen that for all $z\geq 0$,
\begin{equation*}
Q^z(t) = \left\{
\begin{array}{ll}
z + X(t), & {\rm if }\ t <\tau^z(0), \\
\sup_{0<s\leq t} [X(t)-X(s)], & {\rm if }\ t \geq \tau^z(0).
\end{array}\right.
\end{equation*}
Consequently,
\begin{equation}\label{eq:Yxy}
Y^{x,y}(t) = \left\{
\begin{array}{ll}
x - y, & \text{if }t < \tau^y(0),\\
\inf_{0<s\leq t} \{ x+X(s)\}, & \text{if }\tau^y(0) \leq t < \tau^x(0),\\
0, & \text{if }t \geq \tau^x(0).
\end{array}\right.
\end{equation}
Using this representation we can identify
\begin{equation*}
\Omega^{x,y}_T = \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^x(0)\wedge T} Y^{x,y}(t) {\rm d} t\right],
\end{equation*}
where $\wedge$ denotes the minimum operator, due to the fact $Y^{x,y}(t) = 0$ for $t\geq \tau^x(0)$.
Subsequently, we decompose $\Omega_T^{x,y}$ into two terms
\begin{equation}
\Psi^{x,y}_T := \frac{1}{T} \int_0^\infty \mathbb{E}[Y^{x,y}(t)]\, {\rm d} t \qquad
\text{and}
\qquad
\Delta_T^{x,y} := \Omega_T^{x,y} - \Psi_T^{x,y}.
\label{eq:Deltaxy}
\end{equation}
Note that $\Psi_T^{x,y}$ is obtained by replacing $T$ by $\infty$ only in the integration bound.
It is customary in the literature, particularly in the area of stochastic simulation, to compare the truncated integral to its natural expansion of the integration range to a semi-infinite interval, see e.g. \cite[Prop.~2.1]{Awad2007}.The truncated integral connects to the long-run average estimator of a certain performance metric, whereas the infinite integral reflects its exact expectation.
The decomposition in \eqref{eq:Deltaxy} is insightful, because $\Psi_T^{x,y}$ prescribes the leading order behavior of $\Omega_T^{x,y}$, while $\Delta_T^{x,y}$ captures the smaller order error term.
In this section, we only consider $\Psi_T^{x,y}$. Subsection \ref{sec:trunc} investigates the magnitude of $\Delta_T^{x,y}$.
The next preliminary result presents a useful property of $\Psi_T^{x,y}$.
\begin{lemma}\label{lemma:psixy}
Let $x>y$. If $\mathbb{E}[\tau^x(0)]<\iy$, then
\begin{equation}\label{eq:H(x,y)}
\Psi^{x,y}_T = \frac{1}{T}\,\mathbb{E}[\tau^{y}(0)](x-y) + \Psi^{x-y,0}_T.
\end{equation}
\end{lemma}
This leaves us with two unknowns $\mathbb{E}[\tau^y(0)]$ and $\Psi_T^{x-y,0}$.
The next lemma gives an equivalent form for the latter.
\begin{lemma}\label{lemma:psiz0}
If $\mathbb{E}[\tau^z(0)] < \iy$, then for all $z\geq 0$
\begin{equation}\label{eq:H(x,0)}
\Psi^{z,0}_T = \int_0^z \mathbb{E}[\tau^w(0)]\, {\rm d} w.
\end{equation}
\end{lemma}
Since the term $\mathbb{E}[\tau^z(0)]$, for several values of $z$, appears in many of the preliminary results, we devote our attention to this in the next subsection.\\
\\*
\noindent
\textbf{First passage time}.
When studying the first passage time of level $0\leq w < z$, $\tau^z(w)$, of the workload process starting in $z$, we first observe that $\{\tau^z(z-w)\}_{w=0}^z$ is a spectrally positive L\'evy process itself, also visible through Figure \ref{fig:samplePaths}.
More precisely, it is a subordinator, i.e. a L\'evy process whose paths are almost surely non-decreasing \cite{Kyprianou2006}.
In order to calculate $\mathbb{E}[\tau^z(z-w)]$ we use theory presented in \cite[Section 46]{Sato1999}, although results presented there are valid for spectrally \emph{negative} L\'evy processes, as opposed to the absence of negative jumps in our case.
Nonetheless, our setting is easily transformed into this framework by observing that $\hat{X} \equiv -X$, that is $\hat{X}(t) = -X(t)$ for all $t\geq 0$, is spectrally negative.
Furthermore, let
\begin{equation}
\label{eq:transformedTau}
\hat{\tau}^0(w) := \inf\{ t \geq 0\,:\, \hat{X}(t) \geq w\} = \inf\{ t \geq 0\,:\, z+X(t) \leq z-w\} = \tau^z(z-w).
\end{equation}
For completeness, we cite \cite[Thm.~46.3]{Sato1999}.
\begin{theorem}
Let $\hat{X}(t)$ be a spectrally negative L\'evy process with generating triplet $(-a,\s,\hat{\nu})$ and $\hat{\tau}^0(y)$ its corresponding hitting time process. Define $\Upsilon(\thh)$ for $\thh\geq 0$ as
\begin{equation}\label{eq:thmCharExp}
\Upsilon(\thh) = -a\thh + \tfrac{1}{2}\s^2\thh^2 + \int_{-\infty}^0 ({\rm e}^{\thh x}-1-\thh x{\bf 1}_{[-1,0)}(x))\, \hat{\nu}({\rm d} x).
\end{equation}
Then $\Upsilon(\thh)$ is strictly increasing and continuous, $\Upsilon(0)=0$, and $\Upsilon(\thh)\to\infty$ as $\thh\to\infty$. For $w\geq 0$ and $0\leq u < \infty$ we have
\begin{equation}\label{eq:invCharExp}
\mathbb{E}[\exp(-u\hat{\tau}^0(w))] = \exp(-w\,\Upsilon^{-1}(u)),
\end{equation}
where $\thh=\Upsilon^{-1}(u)$ is the inverse function of $u=\Upsilon(\thh)$.
\end{theorem}
\noindent This immediately induces an expression for $\mathbb{E}[\tau^w(0)]$ and henceforth $\Psi^{z,0}$.
\begin{corollary}\label{cor:Psixy}
Let $X(t)$ be a spectrally positive L\'evy process defined as in \eqref{eq:Xlmprocess} with $\mu > \la$. Let $\Psi^{z,0}_T$ as in \eqref{eq:H(x,0)}. Then
\begin{equation*}
\Psi^{z,0}_T = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
Furthermore, if $x,y\geq 0$, then
\begin{equation}\label{eq:mainResult}
\Psi^{x,y}_T = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation}
\end{corollary}
\noindent\textbf{Randomization}.
As we stated before, we easily obtain the original $\Omega_T$ from $\Omega_T^{x,y}$ through substitution of $x$ and $y$ by $Q(0)$ and $Q(\iy)$, respectively, and taking the expectation.
In the previous paragraph, we deduced an explicit expression for $\Psi_T^{x,y}$, the leading order term for $\Omega_T^{x,y}$.
Therefore we equivalently get an approximation for $\Omega_T$, given by
\begin{equation*}
\Psi_T := \frac{1}{T} \int_0^\iy \left( \mathbb{E}[Q(t)]-\mathbb{E}[Q(\iy)] \right)\, {\rm d} t,
\end{equation*}
through randomization of $x$ and $y$ in $\Psi_T^{x,y}$.
By combining the results in Corollary \ref{cor:Psixy}, Lemma \ref{lemma:workloadmoments} and Proposition \ref{prop:truncation_error}, which is given at the end of this section, we directly prove the result in Theorem \ref{thm:mainresult}.
\subsection{Truncation error}\label{sec:trunc}
In order to get a better comprehension of the properties of $\Psi_T$, we depict the value in terms of the (infinite) region between the curves $\mathbb{E}[Q(t)]$, $\mathbb{E}[Q(\iy)]$ and the vertical axis for the case $Q(0)\equiv 0$ in Figure \ref{fig:PsiVisualization}.
In this figure, $\Omega_T$ is given by the area enclosed by the two curves, the vertical axis and the line $t=T$.
One can see that the main contribution to the correction term $\Omega_T$ is given for small $t$.
As $t$ increases, the difference between transient and stationary mean decreases.
Hence for moderate values of $T$, the contribution to the integral in \eqref{eq:Deltaxy} is only minor compared to the contribution over the interval $[0,T]$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.13,yscale=0.3]
\node[below=0.4cm,right=0.5cm] at (x axis mid) {$\to t$};
\draw[dashed, thick, fill =gray!30] (0,0) rectangle coordinate (eq) (50,10);
\node[] at (-7,10) {$\mathbb{E}[Q(\infty)]$};
\node[] at (-3,0) {$0$};
\draw[->] (18,6.4) coordinate (a1) -- (21.65,8.49574);
\node[below] at (a1) {$x=0$};
\foreach \x in {30}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {$T$};
\foreach \y in {10}
\draw (1pt,\y) -- (-20pt,\y);
\draw[thick,color = gray,fill=white] plot
file {Chapter_6/tikz/means0_2.txt};
\draw[thick] (0,0) -- coordinate (x axis mid) (50,0);
\draw[thick] (0,0) -- coordinate (y axis mid) (0,12);
\draw[color=white,very thick] (50,0.05) -- (50,9.56);
\draw[very thick, dotted] (30,0) -- (30,10);
\draw[->] (18,6.8) coordinate (delta) -- (17,8.7);
\node[below] at (delta) {$\Psi_{T}$};
\draw[->] (38,8.1) coordinate (delta) -- (36,9.7);
\node[below] at (delta) {$\Delta_{T}$};
\end{tikzpicture}
\caption{Visualization of $\Omega_T$ and $\Psi_T$ as the area between the curves $\mathbb{E}[Q(t)]$, $\mathbb{E}[Q(\iy)]$ for $Q(0) = 0$.}
\label{fig:PsiVisualization}
\end{figure}
Recall the definition of $\Delta^{x,y}_T$ as in \eqref{eq:Deltaxy}. As we eluded to in Subsection 3.2
we claim the contribution of $\Delta^{x,y}_T$ to $\Omega_T^{x,y}$ is negligible compared to $\Psi^{x,y}_T$. Also note that
\begin{equation}
\label{eq:Delta}
\Delta_T := \Omega_T - \Psi_T = {-}\frac{1}{T} \int_T^\iy \mathbb{E}[Q(t)] - \mathbb{E}[Q(\iy)]\,{\rm d} t.
\end{equation}
can be derived through $\Delta^{x,y}_T$ in a similar manner as we did for $\Psi^{x,y}_T$ to obtain $\Psi_T$.
To substantiate our claim, we compute an upper bound for $\Delta^{x,y}_T$ of order $1/T^2$. The existence of such an upper bound poses a limit on the error this tail integral contributed to the cost structure as a whole.
\begin{proposition} \label{prop:truncation_error}
Let $x,y\geq 0$ and $\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3] < \iy$. Then
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right)
\end{equation*}
and
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proposition}
\textit{Remark.} In case the net-input process $X$ is light-tailed, that is there exists $u>0$ such that $\mathbb{E}[{\rm e}^{u X(1)}] < \iy$, it can be shown that the truncation error is of order ${\rm e}^{-\beta T}/T$ for some $\beta>0$. See Appendix \ref{sec:proof_truncation} for details.
\section{Optimization}
\label{sec:optimization}
The result in Theorem \ref{thm:mainresult}, characterizing the leading order behavior of $\Omega_T(\mu)$, also reveals the behavior of $\Pi_T(\mu)$ in leading order. Namely,
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \Psi_T(\mu) + O(1/T^2).
\end{equation*}
In fact, this representation naturally gives rise to an \emph{approximation} of the actual cost function:
\begin{align}\label{eq:decomposition}
\hat{\Pi}_{T}(\mu) := \Pi_{\iy}(\mu) + \Psi_T(\mu)
\end{align}
Denote the corresponding minimizer of $\Pih$ by
\begin{equation}\label{eq:muhat}
\hat{\mu}_T^\star := \arg\min_{\mu\geq 0} \Pih(\mu), \qquad \Pih^\star := \Pih(\hat{\mu}_T^\star)
\end{equation}
in addition to the definitions in \eqref{eq:muStar} and \eqref{eq:piStar}.
This section is devoted to the analysis of the minimizers $\muT$, $\muh$ and $\mui$, and the optimality gap for the two approximations.
Throughout this section, we assume that $u_2, u_3 <\iy$ and $\mathbb{E}[Q(0)^2] <\iy$.
By its definition in \eqref{eq:PiInf} and Lemma \ref{lemma:workloadmoments}, we have an exact expression for the steady-state cost function:
\begin{equation*}
\Pi_{\iy}(\mu) = \frac{\la u_2}{2(\mu-\la)} + \aaa\mu.
\end{equation*}
It is easily verified that $\Pi_{\iy}$ is strictly convex in $\mu$, for instance by observing that $\Pi_{\iy}''(\mu) > 0$ for all $\mu > \la$. Therefore $\Pi_{\iy}$ has a unique global minimizer and
\begin{equation}
\label{eq:muInf}
\mui = \la + \sqrt{\frac{\la u_2}{2\aaa}}, \qquad \Pi_{\iy}^\star = \aaa\la + \sqrt{2\aaa\la u_2}.
\end{equation}
We are interested in the relation between $\mui$ and $\muT$, and between $\muh$ and $\muT$.
Since $\Pi_{T}(\mu) = \Pi_{\iy}(\mu) + O(1/T)$ for all $\mu > \la$, we have pointwise convergence of the sequence $\Pi_{T}$, as well as $\hat{\Pi}_{T}$, to $\Pi_{\iy}$ for $T\to\iy$, we also expect $\muT \to \mui$ and $\muh\to\mui$ for $T\to\iy$.
Before proving that this convergence indeed holds, we present a result on the strict convexity of the function $\Pi_{T}$.
\begin{lemma}\label{lemma:strict_convexity}
Let $\mu\geq 0$. The function $\Pi_{T}(\mu)$ is
\begin{itemize}
\item convex in $\mu$, if $Q(0)\equiv x$, $T<x/\mu$ and $\sigma=0$,
\item strictly convex in $\mu$, otherwise.
\end{itemize}
\end{lemma}
Building upon strict convexity of both $\Pi_T(\mu)$ and $\Pi_\iy(\mu)$ for $\mu>\la$, we derive the following convergence result.
\begin{proposition}\label{prop:min_convergence_mu}
Let $\muT$, $\muh$ and $\mui$ be as defined in \eqref{eq:muStar} and \eqref{eq:muhat}. Then
\begin{equation*}
\muT \to \mui\, \qquad \text{\rm and } \qquad \muh \to \mui,
\end{equation*}
for $T\to\infty$.
\end{proposition}
The next result describes a refinement of $\muT$ in terms of $\mui$.
\begin{proposition}\label{prop:muBullet}
For $T$ sufficiently large,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T),
\end{equation*}
where
\begin{equation}\label{eq:muBullet}
\mu_\bullet = \frac{\mathbb{E}[Q(0)^2]}{\sqrt{8\la u_2\aaa}} - \frac{u_3}{3 u_2} - 3\sqrt{\frac{\aaa\la u_2}{8}}.
\end{equation}
\end{proposition}
Based on Proposition \ref{prop:muBullet} we propose a \emph{corrected staffing rule}, accounting for the finite horizon
\begin{equation}
\label{eq:correctedMu}
\tilde{\mu}_T^\star = \left[\mui + \frac{\mu_\bullet}{T}\right]^+,
\end{equation}
with $\mu_\bullet$ as in \eqref{eq:muBullet}.
Here $[x]^+ := \max\{x,0\}$, which ensures the value of $\tilde{\mu}_T^\star$ is non-negative and thus is a feasible solution of the optimization problem.
This refined capacity allocation rule is expected to reduce the costs incurred in transient settings.
However, the value of particular interest to us is the cost penalty for using either one of the approximations rather than the actual minimum $\muT$, that is, the \emph{optimality gap}.
As it happens, we deduce the order of the optimality gap for $\mui$ with the help of the explicit form of $\mu_\bullet$ given in \eqref{eq:muBullet}, which is stated in the next proposition. The proof is given in Appendix \ref{sec:proofProp4}.
\begin{proposition}\label{prop:optimalitygap_mui}
Let $\mui$ be as in \eqref{eq:muInf}. Then,
\begin{equation*}
\Pi_\iy^\star- \Pi_T^\star = O(1/T^2).
\end{equation*}
\end{proposition}
\section{Numerical experiments}
\label{sec:numerics}
\subsection{Influence of $\Omega_{T}(\mu)$}
\label{sec:influence_omega}
We first assess the contribution of the correction to the cost function provided by Theorem 1. In other words, we investigate whether $\hat{\Pi}_{T}(\mu)$ as in \eqref{eq:PiT} yields a significantly better fit to $\Pi_{T}(\mu)$, than $\Pi_{\iy}(\mu)$ does.
Note that these three functions only differ in the costs describing the congestion.
Therefore, we limit our study in this subsection to the evaluation of $C_T(\mu)$ as in \eqref{eq:CTmu} with stationary equivalent $C_{\iy}(\mu) = \mathbb{E}[Q_{\mu}(\iy)]$.
Our novel approximation hence reads
\begin{equation*}
\hat{C}_{T}(\mu) := C_{\infty}(\mu) + \Omega_{T}(\mu),
\end{equation*}
with $\Omega_{T}(\mu)$ given in Theorem \ref{thm:mainresult}.
We conduct our numerical experiments based on three models, namely:
\begin{enumerate}
\item \underline{$M/M/1$ queue}: $U(t)$ is a unit rate compound Poisson process with exponentially distributed increments. We have $u_2 = 2$, $u_3=3$, so that
\begin{equation}\label{eq:MM1cor}
\hat{C}_{T}(\mu) = \frac{\la} {\mu-\la} + \frac{1}{T(\mu-\la)} \left(\frac{x^2}{2} - \frac{\la^2}{(\mu-\la)^2} - \frac{\la} {\mu-\la} \right).
\end{equation}
\item \underline{$M/{\rm Pareto}/1$ queue}: $U(t)$ is a unit rate compound Poisson process with Pareto increments. The Pareto distribution deserves special attention due to its heavy-tailed nature, having tail probability $\bar{F}(x) = (x/k)^{-\gamma}$, if $x\geq k$ and 1 otherwise.
It is well-known that heavy-tailed service times lead to long relaxation time. For our purposes, we fix shape parameter $\gamma = 16/5$ and scale parameter $k=11/16$, so that $\beta = 1$, $u_2 = 121/96$, $u_3 = 1331/256$ and $u_k=\iy$ for all $k>3$. Hence,
\begin{equation}
\label{eq:MP1cor}
\hat{C}_{T}(\mu) = \frac{121\la} {192(\mu-\la)} + \frac{1}{2T(\mu-\la)}
\left( x^2 - \frac{(121\la/96)^2}{2(\mu-\la)^2} - \frac{ 1331\la/256 }{2(\mu-\la)}\right)
\end{equation}
\item \underline{Reflected Brownian motion}: $U(t)$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$. We have $u_2 = \sigma^2$, $u_3=0$, so that
\begin{equation}\label{eq:RBMcor}
\hat{C}_{T}(\mu) = \frac{\la\sigma^2}{2(\mu-\la)} + \frac{1}{2T(\mu-\la)} \left( x^2 - \frac{\la^2\sigma^4}{2(\mu-\la)^2}\right).
\end{equation}
\end{enumerate}
In light of the equivalence relations in \eqref{eq:Qidentity} we only consider the case $\la=1$. The cost values for general values of $\la$ follow by appropriate rescaling of $\mu$ and $T$.\\
\\*
\noindent
For the $M/M/1$ and $M/{\rm Pareto}/1$ queue, we obtained the function $C_{T}(\mu)$ through simulation and the results are accurate up until a 95\% confidence interval of width $10^{-3}$. For reflected Brownian motion, we used the explicit distribution function given in \cite{Harrison1985} for double numerical integration. The results for several values of $T$ and two different starting states are depicted in Figures 4-6. These plots also include the approximated functions $\hat{C}_{T}(\mu)$.
\begin{figure}%
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/mm1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/mm1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/mm1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/mm1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/mm1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/mm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/mm1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/mm1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/mm1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/mm1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/mm1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/mm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/M/1$ queue with $\la=1$.}
\label{fig:cont}%
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/mp1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/mp1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/mp1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/mp1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/mp1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/mp1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mp1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/mp1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/mp1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/mp1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/mp1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/mp1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/mp1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mp1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/$Pareto$/1$ queue with $\la=1$.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/rbm1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/rbm1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/rbm1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/rbm1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/rbm1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/rbm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/rbm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {tikz/rbm1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {tikz/rbm1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {tikz/rbm1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {tikz/rbm1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {tikz/rbm1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {tikz/rbm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/rbm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for reflected Brownian motion with $\sigma=1$.}
\end{figure}
We name a few observations based on these figures.
First, we indeed note the pointwise convergence of $\hat{C}_{T}(\mu)$ to $\hat{C}_{\iy}(\mu)$ as $T$ grows, for all $\mu$ in all three cases. However, the difference between the stationary costs and those for small values of $T$ can be significant. This is most clear in the plots with $x=2.5$ and when $\mu$ is close to $\la$, i.e. it is in heavy-traffic. In these scenarios, it is evident that refinements to the stationary cost function are needed. $\hat{C}_{T}(\mu)$ does a fairly good job at providing such correction, especially for moderate values of $\mu$.
Furthermore, we note that $C_{T}(\mu)$ approaches $C_{\iy}(\mu)$ from below for $x=0$ for any value of $\mu$, while this is not strictly the case for $x>0$.
$\hat{C}_{T}(\mu)$ correctly captures the sign of this correction.
Finally, observe that $\hat{C}_{T}(\mu)\to -\iy$ as $\mu$ approaches $\la$ from above. This divergence is clear from the expressions in \eqref{eq:MM1cor}-\eqref{eq:RBMcor}.
Our correction term relies on the premise that under the coupling scheme, the sample paths of the two queues starting from different states have hit with high probability.
This is equivalent to stating that the `largest' of the two queues is has emptied at least once before time $T$. However, as $\mu$ approaches $\la$, the system enters heavy traffic, and hence the hitting time of the zero barrier is set to run off to infinity.
Consequently, this causes our approximation to be inaccurate for small values of $\mu$.
\subsection{Validation of corrected staffing rule}
\label{sec:num_opt}
In this section, we examine whether the corrected staffing rule $\tilde{\mu}_T^\star$ as in \eqref{eq:correctedMu} indeed yields a significant cost reduction over the choice of $\mui$ by comparing their true costs $\Pi_{T}(\tilde{\mu}_T^\star)$ and $\Pi_{T}(\mui)$.
We conduct this comparison for different values of the parameters, $\aaa$, $T$ and starting state $x$ through numerical experiments.
The three models on which we do our calculations are the $M/M/1$ queue, the $M/$Pareto$/1$ queue and the reflected Brownian motion, as introduced in the previous subsection.
We again focus on $\la=1$ only.
For each of the three models, we adhere to the following set-up. The quality of both staffing rules is assessed for $\aaa = 0.1, 1$ and 2, resembling three modes of valuation of the QoS in the system.
As a benchmark, observe that the expected workload in steady-state conditions with staffing level $\mui$ equals
\begin{equation*}
C_\iy(\mui) = \sqrt{\frac{\aaa\la u_2}{2}}.
\end{equation*}
For each value of $\aaa$, we consider two scenarios: one in which the system starts empty, i.e. $x=0$, and one in which the initial state is double this benchmark value, thus $x=\sqrt{2\aaa\la u_2}$.
The numerics are presented for each model separately. We discuss general conclusions drawn from these results afterwards.\\
\\*
\noindent\textbf{$M/M/1$ queue}
As we discussed before, if $U$ is a unit rate compound Poisson process with exponentially distributed increments, then $\Qlm$ describes the workload process in an $M/M/1$ queue.
For this setting we get
\begin{equation*}
\mui = \la + \sqrt{\frac{\la} {\aaa}},\qquad \tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la} {\aaa}} + \frac{1}{T}\left( \frac{x^2}{4\sqrt{\la\aaa}} - 1 - \frac{3}{2} \sqrt{\la\aaa}\right)\right]^+.
\end{equation*}
Table \ref{tab:mm1} presents the actual costs corresponding to these two staffing levels for different value of $x$ and $\aaa$.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{\aaa}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 4.162 & 0.620 & 2.688 & 0.536 & 0.136 & 4.162 & 0.682 & 2.688 & 0.536 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 4.162 & 0.669 & 3.425 & 0.641 & 0.041 & 4.162 & 0.700 & 3.425 & 0.641 & 0.085 \\
\multicolumn{1}{|c|}{} & 5 & 4.162 & 0.706 & 3.867 & 0.703 & 0.005 & 4.162 & 0.719 & 3.867 & 0.703 & 0.022 \\
\multicolumn{1}{|c|}{} & 10 & 4.162 & 0.719 & 4.015 & 0.719 & 0.001 & 4.162 & 0.726 & 4.015 & 0.719 & 0.010 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.000 & 2.309 & 0.000 & 0.500 & 0.783 & 2.000 & 3.500 & 0.500 & 2.750 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 2.461 & 0.750 & 1.480 & 0.398 & 2.000 & 3.218 & 1.250 & 3.125 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 2.675 & 1.500 & 2.400 & 0.103 & 2.000 & 3.043 & 1.700 & 2.968 & 0.025 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 2.810 & 1.750 & 2.726 & 0.030 & 2.000 & 3.007 & 1.850 & 2.980 & 0.009 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} &1 & 1.707 & 3.744 & 0.000 & 0.500 & 0.866 & 1.707 & 5.889 & 0.000 & 3.328 & 0.435 \\
\multicolumn{1}{|c|}{} &2 & 1.707 & 3.924 & 0.146 & 1.232 & 0.686 & 1.707 & 5.547 & 0.854 & 4.682 & 0.156 \\
\multicolumn{1}{|c|}{} &5 & 1.707 & 4.209 & 1.083 & 3.343 & 0.206 & 1.707 & 5.114 & 1.366 & 4.910 & 0.040 \\
\multicolumn{1}{|c|}{} &10 & 1.707 & 4.424 & 1.395 & 4.108 & 0.071 & 1.707 & 4.945 & 1.536 & 4.868 & 0.016 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/M/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mm1}
\end{table}
\noindent
\textbf{$M$/Pareto/1 queue}
In case the service requirements follow a Pareto distribution with shape parameter $\gamma = 16/5$, the staffing rule becomes
\begin{equation*}
\mui = \la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}}, \ \tilde{\mu}_T^\star = \left[\la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}} + \frac{1}{T}\left( \frac{2 x^2}{11\sqrt{\la\aaa/3}} - \frac{11}{8} - \frac{11\sqrt{3\la\aaa}}{16}\right)\right]^+.
\end{equation*}
The numerical results are given in Table \ref{tab:mp1}.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 11/4\cdot \sqrt{\aaa/3}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.510 & 0.524 & 1.759 & 0.461 & 0.120 & 3.510 & 0.573 & 2.010 & 0.562 & 0.019 \\
\multicolumn{1}{|c|}{} & 2 & 3.510 & 0.555 & 2.635 & 0.539 & 0.029 & 3.510 & 0.580 & 2.760 & 0.574 & 0.010 \\
\multicolumn{1}{|c|}{} & 5 & 3.510 & 0.580 & 3.160 & 0.578 & 0.003 & 3.510 & 0.591 & 3.210 & 0.589 & 0.002 \\
\multicolumn{1}{|c|}{} & 10 & 3.510 & 0.590 & 3.335 & 0.590 & 0.000 & 3.510 & 0.596 & 3.360 & 0.595 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.794 & 2.076 & 0.000 & 0.500 & 0.759 & 1.794 & 2.989 & 0.000 & 2.088 & 0.302 \\
\multicolumn{1}{|c|}{} & 2 & 1.794 & 2.190 & 0.511 & 1.291 & 0.411 & 1.794 & 2.790 & 0.610 & 2.588 & 0.072 \\
\multicolumn{1}{|c|}{} & 5 & 1.794 & 2.345 & 1.281 & 2.108 & 0.101 & 1.794 & 2.638 & 1.320 & 2.607 & 0.012 \\
\multicolumn{1}{|c|}{} & 10 & 1.794 & 2.441 & 1.537 & 2.371 & 0.029 & 1.794 & 2.597 & 1.557 & 2.585 & 0.005 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.561 & 3.427 & 0.000 & 0.500 & 0.854 & 1.561 & 5.087 & 0.000 & 2.745 & 0.460 \\
\multicolumn{1}{|c|}{} & 2 & 1.561 & 3.567 & 0.032 & 1.050 & 0.706 & 1.561 & 4.832 & 0.172 & 3.417 & 0.293 \\
\multicolumn{1}{|c|}{} & 5 & 1.561 & 3.779 & 0.950 & 3.012 & 0.203 & 1.561 & 4.499 & 1.006 & 4.313 & 0.041 \\
\multicolumn{1}{|c|}{} & 10 & 1.561 & 3.935 & 1.255 & 3.356 & 0.147 & 1.561 & 4.351 & 1.284 & 4.304 & 0.011 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/{\rm Pareto}/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mp1}
\end{table}
Just as in the results for the $M/M/1$ queue, we observe a higher reduction for larger value of $\aaa$ and $T$. Also, again $\tilde{\mu}_T < \mui$. Hence, the conclusions for the $M/{\rm Pareto}/1$ queue are similar to those of the $M/M/1$ queue. \\
\\*
\noindent\textbf{Reflected Brownian motion}.
In case the input process $U$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$, the steady-state staffing rule and its corrected version reduce to
\begin{equation*}
\mui = \la + \sqrt{\frac{\la\s^2}{2\aaa}}, \qquad
\tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la\s^2}{2\aaa}} + \frac{1}{2\sqrt{2}\,T}\left(\frac{x^2}{\sqrt{\la \aaa}\s} - 3\s\sqrt{\aaa\la} \right)\right]^+.
\end{equation*}
In Tables \ref{tab:rbm1} and \ref{tab:rbm2}, the costs obtained through numerical evaluation are presented for several values of $x$, $T$. We also vary $\s$ to examine the influence of the volatility of arrival process on the quality of the staffing rules.
The observations on the influence of $\aaa, x$ and $T$ are similar to those of the $M/M/1$ queue and the $M/{\rm Pareto}/1$ queue.
However, here we see little improvement by the corrected staffing rule for small values of $\aaa$ for both values of $x$.
The results in Tables \ref{tab:rbm1}-\ref{tab:rbm2} also suggest that the reduction is smaller for larger values of $\s$.
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = \sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.236 & 0.525 & 2.901 & 0.518 & 0.013 & 3.236 & 0.565 & 3.124 & 0.564 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 3.236 & 0.536 & 3.068 & 0.534 & 0.003 & 3.236 & 0.556 & 3.180 & 0.556 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 3.236 & 0.543 & 3.169 & 0.542 & 0.000 & 3.236 & 0.551 & 3.214 & 0.551 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 3.236 & 0.545 & 3.203 & 0.545 & 0.000 & 3.236 & 0.549 & 3.225 & 0.549 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 1$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm1}
\end{table}
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 5.472 & 0.950 & 4.801 & 0.936 & 0.015 & 5.472 & 1.030 & 5.249 & 1.029 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 5.472 & 0.972 & 5.137 & 0.968 & 0.003 & 5.472 & 1.012 & 5.360 & 1.012 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 5.472 & 0.985 & 5.338 & 0.985 & 0.000 & 5.472 & 1.002 & 5.427 & 1.002 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 5.472 & 0.990 & 5.405 & 0.990 & 0.000 & 5.472 & 0.998 & 5.450 & 0.998 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.414 & 3.176 & 0.293 & 1.546 & 0.513 & 2.414 & 4.633 & 1.707 & 4.228 & 0.087 \\
\multicolumn{1}{|c|}{} & 2 & 2.414 & 3.356 & 1.354 & 2.690 & 0.199 & 2.414 & 4.375 & 2.061 & 4.247 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.414 & 3.573 & 1.990 & 3.411 & 0.045 & 2.414 & 4.094 & 2.273 & 4.073 & 0.005 \\
\multicolumn{1}{|c|}{} & 10 & 2.414 & 3.689 & 2.202 & 3.646 & 0.012 & 2.414 & 3.966 & 2.344 & 3.962 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 2.000 & 4.839 & 0.000 & 1.339 & 0.723 & 2.000 & 7.481 & 1.000 & 5.967 & 0.202 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 5.078 & 0.500 & 2.773 & 0.454 & 2.000 & 7.158 & 1.500 & 6.585 & 0.080 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 5.414 & 1.400 & 4.726 & 0.127 & 2.000 & 6.670 & 1.800 & 6.549 & 0.018 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 5.639 & 1.700 & 5.409 & 0.041 & 2.000 & 6.380 & 1.900 & 6.349 & 0.005 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 2$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm2}
\end{table}
\subsection{Discussion}
Based upon these numerical results in Tables \ref{tab:mm1}-\ref{tab:rbm2}, we make a few remarks. The three models roughly exhibit similar behavior as $T$, $x$ and $\aaa$ are varied.
Non-surprisingly, we note that $\tilde{\mu}_T$ approaches $\mui$ with increasing $T$, which also implies that the cost reduction achieved by the corrected staffing rule vanishes as $T\to\iy$.
Also, we observe that in all scenarios examined, the cost reduction increases with $\aaa$. This can be explained through investigation of the objective function $\Pi_T$ as function of $\mu$. Namely, for $\aaa$ small, the curve is relatively flat around the true optimum $\muT$. Hence, in this case a moderate deviation from $\muT$ will likely not lead to a significant cost increase. However, as $\aaa$ becomes larger, i.e. server efficiency is valued more than minimization of congestion, the curve becomes more sharp around $\muT$, and hence more accurate approximations of $\muT$ are required to achieve an acceptable cost level. Hence, the corrected staffing rule \eqref{eq:correctedMu} proves particularly useful in these cases.
Another point we highlight is that the relative improvement is higher for $x=0$ than for $x=\sqrt{2\aaa\la u_2}$. Moreover, even though the initial state of the system is above the optimal equilibrium, $\tilde{\mu}_T$ is smaller than $\mui$. This is somewhat counter-intuitive. In fact, from \eqref{eq:muBullet} it follows that $\mu_\bullet$ positively contributes to the corrected staffing function if
\begin{equation*}
\mathbb{E}[Q^2(0)] > 3\aaa\la u_2 + \frac{2 u_2}{3 u_3}\,\sqrt{2\aaa\la u_2}.
\end{equation*}
\section{Conclusion \& further research}
\label{sec:conclusion_chapter6}
Motivated by the time-varying nature of queues in practical applications, we studied the impact that the transient phase has on traditional capacity allocation questions.
By defining a cost minimization problem, in which the objective function contains a correction accounting for the transient period, we identified the leading and second-order behavior of the cost function as a function of the interval length $T$.
As a by-product, this result yields an approximation for the actual cost function, which is a refinement to its stationary counterpart.
Our numerical experiments in Section \ref{sec:influence_omega} demonstrate the improved accuracy achieved by this approximation in a number of settings.
By perturbation analysis of the optimization problem, this furthermore gives rise to a correction to the steady-state optimal capacity allocation of order $1/T$.
The necessity of the refined capacity allocation level is substantiated by the numerics in Section \ref{sec:num_opt}, which show the cost reduction that can be achieved in the number of settings, compared to settings in which stationary metrics are used.
Especially for small values of $T$ and large values of $\alpha$ this reduction is significant.
Additionally, these results also indicate that it is relatively safe to use the stationary cost when $T$ is moderate, or $\alpha$ is small.
The latter reflects the scenario in which QoS is much more valued than service efficiency.
This observation links to the flat nature of the cost function around its optimal value for $\alpha$ small, a statement on the optimality gap that we formally proved in Proposition \ref{prop:optimalitygap_mui}.
Besides the validation of our theoretical results of Sections \ref{sec:analysis} and \ref{sec:optimization}, the numerical results also reveal some phenomena that require more investigation.
As noted, our corrected capacity allocation level $\tilde{\mu}_T^\star$ is in most studied cases less than the steady-state optimal value $\mu_{\iy}^\star$. This implies that congestion levels tends to be higher under our staffing scheme then under stationary staffing.
A possible explanation for this may be the fact that the planning period under consideration is finite.
Clearly, in the setting we analyzed, anything that happens after time $T$ is neglected.
Therefore, it might be beneficial from the cost perspective to end the period with a higher expected congestion level, as it does not need to be canceled out in the future.
Related to this observation, it would be interesting to look at the setting in which staffing decisions need to be made in consecutive periods of equal length, in which the arrival rate changes at the start of each period.
This case requires careful consideration of the correlation among the staffing decisions within the separate periods.
Another question that arises concerns the translation of our (qualitative) findings to more general queues, in particular the $M/G/s$ queue.
Whereas in our analysis, the central decision variable is the server speed $\mu$, the variable of interest in multi-server queues is typically the number of servers.
It may well be that similar explicit corrections to staffing levels can be deduced to account for transience.
Since our analysis heavily relies on the comparibility of the sample paths of two single-server queues, which is due to the equal negative drift for the two processes, another approach must be taken to tackle this extension.
The analysis and findings for the single-server queue with L\'evy input presented in this chapter may serve a stepping stone for investigation of these more elaborate problems.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Proofs of Section \ref{sec:model_description}}
\label{app:proofs_model}
\subsection{Proof of Lemma \ref{lemma:workloadmoments}}
\label{app:proof_lemma_workload_moments}
\begin{proof}
The conditions of \cite[Cor.IX3.4]{Asmussen2003} are satisfied and therefore $Q_{\mu}(t)\Rightarrow \Qlm(\infty)$ in distribution for $t\rightarrow\infty$. Furthermore, its Laplace transform is for ${\rm Re}(s) < 0$
\[\tilde{Q}_{\mu}(s) = \mathbb{E}[s \Qlm(\infty)] = \frac{s \ka_{\mu}'(0)}{\ka_{\mu}(s)} = \frac{s(\la\ka_U'(0) - \mu)}{\la\ka_U(s) - \mu s} = \frac{s(\mu-\la)}{\mu s-\la \ka_U(s)}.\]
It can be checked that $\ka_U'(0) = \mathbb{E}[U(1)] = 1$, $\ka_U''(0) = u_2$ and $\ka_U'''(0) = u_3$, and $\klm'(0) = \la-\mu$, $\klm''(0) = \la u_2$ and $\klm'''(0) = \la u_3$.
Using l'H\^opital's rule we obtain the first moment of $\Qlm(\infty)$:
\begin{align*}
\mathbb{E}[\Qlm(\iy)] &= {-}\lim_{s\to 0} \frac{d}{ds} \tilde{Q}_{\mu}(s) = \lim_{s\to 0} \klm'(0)\, \frac{s\klm'(s)-\klm(s)}{\klm(s)^2}\\
&= \lim_{s\to 0} \klm'(0)\, \frac{{-}s\klm''(s)}{2\klm(s)\klm'(s)}
= \lim_{s\to 0} \klm'(0)\,\frac{ s\klm'''(s)-\klm''(s)}{2\klm'(s)^2 + 2\klm(s)\klm'''(s)} \\
&= {-}\frac{\klm''(0)}{2\klm'(0)} = \frac{\la u_2}{2(\mu-\la)}.
\end{align*}
Similarly we derive the second moment:
\begin{align*}
\mathbb{E}[\Qlm^2(\iy)] &= \lim_{s\to 0} \frac{d^2}{ds^2} \tilde{Q}_{\mu}(s)
= \lim_{s\to 0} \klm'(0)\, \frac{3 \klm''(0)^2 - 2\klm'(0)\klm'''(0)}{6 \klm'(0)^3}\\
&= (\la-\mu)\frac{3\la^2u_2^2 - 2\la u_3(\la-\mu)}{6(\la-\mu)^3}
= \frac{\la^2u_2^2}{2(\mu-\la)^2} +\frac{\la u_3}{3(\mu-\la)}.
\end{align*}
\end{proof}
\subsection{Proof of Proposition \ref{prop:cost_convergence}}
\begin{proof}
We prove the limit by showing that the difference
\[
\Pi_T(\mu) - \Pi_\iy(\mu) = \frac{1}{T} \int_0^T \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t
\]
converges to zero as $T\to\iy$ for $\mu>\la$ fixed. The assumption $\mathbb{E}[U(1)], \mathbb{E}[Q(0)] < \iy$ implies by \cite[Prop.~1]{Abate1994} that $\mathbb{E}[Q_\mu(t)]<\iy$ for all $t\geq 0$.
Following \cite{Abate1994}, we use the decomposition
\[
\mathbb{E}[Q_\mu(t)] = \mathbb{E}[Q^0_\mu(t)] + \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\},
\]
where $Q_\mu^0(t)$ represents the workload process if the system starts empty.
From this decomposition it is revealed that $\mathbb{E}[Q^0_\mu(t)]$ and $\left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\}$ are non-negative monotonically increasing and decreasing functions of $t$, respectively, see \cite[Prop.~2,Thm.~11]{Abate1994}.
Recall $\mathbb{E}[Q_\mu(t)] \to \mathbb{E}[Q_\mu(\iy)]$ for $t\to\iy$ by ergodicity of the workload process for any initial state $\mathbb{E}[Q(0)]< \iy$, if $\mu>\la$.
Henceforth,
\begin{align*}
\mathbb{E}[Q_\mu(t)] &\leq \sup_t \mathbb{E}[Q_\mu^0(t)] + \sup_t \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\} \\
&= \mathbb{E}[Q_\mu(\iy)] + \left\{\mathbb{E}[Q_\mu(0)] - \mathbb{E}[Q_\mu^0(0)]\right\} = \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)],
\end{align*}
for all $t\geq 0$, which proves that the expected workload is bounded.
Fix $\varepsilon>0$. By convergence of $\mathbb{E}[Q_\mu(t)]$ for $t\to\iy$, there exists a value $t^* := t^*(\varepsilon)$ such that for all $t\geq t^*$
\begin{equation}
\left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| < \varepsilon/2.
\end{equation}
Next, set
\[
T^* := T^*(\varepsilon) = \frac{2\,t^*(\varepsilon)}{\varepsilon}\, ( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]).
\]
Then for $T\geq \hat{T}:= \max\{ t^*,T^* \}$, we have
\begin{align*}
\left| \frac{1}{T} \int_0^T \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t \right|
&\leq \frac{1}{T} \int_0^{t^*} \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| {\rm d} t \\
& \qquad+ \frac{1}{T} \int_{t^*}^T \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| \, {\rm d} t \\
&\leq \frac{1}{T} \int_0^{t^*} \mathbb{E}[Q_\mu(t)] + \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t + \frac{1}{T} \int_{t^*}^T \frac{\varepsilon}{2}\, {\rm d} t \\
&< \frac{t^*}{T}\,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{T-t^*}{T} \,\frac{\varepsilon}{2}\\
&< \frac{t^*}{T^*} \,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{\varepsilon}{2} = \varepsilon.
\end{align*}
Hence, for any choice of $\varepsilon>0$ we can find a value $\hat{T}$ such that $\Pi_{\hat{T}}(\mu)$ approaches $\Pi_\iy(\mu)$ within distance $\varepsilon$, which proves the limit.
\end{proof}
\section{Proofs of Section \ref{sec:analysis}}
\label{app:proofs_analysis}
\subsection{Proof of Lemma \ref{lemma:psixy}}
\begin{proof}
Using the representation in \eqref{eq:Yxy} we write
\begin{align*}
\Psi^{x,y}_T &= \frac{1}{T}\int_0^{\infty} \mathbb{E}[Y^{x,y}(t)]{\rm d} t \\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}Y^{x,y}(t)\right] {\rm d} t + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right]
+ \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^\infty Y^{x,y}(t) {\rm d} t\right] ,\\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}(x-y) {\rm d} t\right] + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right] \\
&= \frac{1}{T}\,\mathbb{E}[\tau^y(0)](x-y) + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right].
\end{align*}
By \eqref{eq:Yxy} and the Strong Markov property holding for L\'evy processes \cite{Asmussen2003}, observe that \\* $Y^{x-y,0}(t) {\;\buildrel{d}\over= \;} Y^{x,y}(\tau^y(0)+t)$, whereby
\begin{equation*}
\frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t)\,{\rm d} t\right] = \frac{1}{T}\,\mathbb{E}\left[\int_{0}^{\tau^{x-y}(0)} Y^{x-y,0}(t) {\rm d} t\right] = \Psi^{x-y,0}_T,
\end{equation*}
which completes the proof.
\end{proof}
\subsection{Proof of Lemma \ref{lemma:psiz0}}
\begin{proof}
Note that $Y^{z,0}(t)$ and $\tau^z(w)$ are intimately related. Namely, due to the fact that $X$ has no negative jumps
\begin{equation*}
\{ \tau^z(w) \leq t\} = \{Y^{z,0}(t) \leq w \}.
\end{equation*}
In fact, $Y^{z,0}(\tau^z(w)) = w$, which implies that $\tau^z$ is a right inverse for $Y^{z,0}(t)$. Therefore, the following equality holds
\begin{equation*}
\int_0^{\tau^z(0)} Y^{z,0}(t)\, {\rm d} t = \int_0^z \tau^z(w)\, {\rm d} w,
\end{equation*}
which implies with the help of Fubini's theorem
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^z(w)]\, {\rm d} w = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^{z-w}(0)]\,{\rm d} w =\frac{1}{T}\, \int_0^z \mathbb{E}[\tau^{w}(0)] \,{\rm d} w.
\end{equation*}
\end{proof}
\subsection{Proof of Corollary \ref{cor:Psixy}}
\begin{proof}
From \eqref{eq:invCharExp},
\begin{equation}\label{eq:corEq1}
\mathbb{E}[\hat{\tau}^0(w)] = -\tfrac{{\rm d}}{{\rm d} u} \left. \mathbb{E}[\exp(-u\,\hat{\tau}^0(w))]\right|_{u=0} = w\left.\frac{{\rm d}}{{\rm d} u} \Upsilon^{-1}(u)\right|_{u=0}.
\end{equation}
Since $\Upsilon(\theta)$ is strictly increasing and $\Upsilon(0)=0$, we get $\Upsilon^{-1}(0)=0$ and
\begin{equation*}
\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = \frac{1}{\Upsilon'(\Upsilon^{-1}(0))} = \{ \Upsilon'(0) \}^{-1}.
\end{equation*}
Furthermore,
\begin{align*}
\Upsilon'(\thh) &= -a+ \s^2\thh + \int_{-\infty}^0 (x\, {\rm e}^{\thh x} - x{\bf 1}_{[-1,0)}(x)) \hat{\nu}({\rm d} x) \\
&= -a + \s^2\thh - \int_0^\infty (y\, {\rm e}^{-\thh y} - y{\rm 1}_{(0,1]}(y)) \nu({\rm d} y).
\end{align*}
Thus, $\Upsilon'(0) = -\mathbb{E}[X(1)] = \mu-\la$ and $\mathbb{E}[\hat{\tau}^0(w)] = w/(\mu-\la)$. By \eqref{eq:H(x,0)} and \eqref{eq:transformedTau}, we deduce that
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\, \int_0^z \mathbb{E}[\tau^w(0)] \,{\rm d} w = \frac{1}{T}\, \int_0^z \mathbb{E}[\hat{\tau}^0(w)] {\rm d} w = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
For $x>y$, we use Lemma \ref{lemma:psixy} to conclude
\begin{equation*}
\Psi^{x,y}_T = \frac{y(x-y)}{T(\mu-\la)} + \frac{(x-y)^2}{2 T(\mu-\la)} = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation*}
The result for $x<y$ follows directly by the observation $\Psi^{x,y}_T = -\Psi_T^{x,y}$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:truncation_error}}\label{sec:proof_truncation}
\begin{proof}
To derive the upper bound for $\Delta^{x,y}_T$, we apply the same coupling argument as in described in Section 3. Let us assume without loss of generality $x>y$.
In this case,
\begin{equation*}
|\Delta^{x,y}_T| = \frac{1}{T} \int_T^\iy \mathbb{E}[Q^x(t)-Q^y(t)] {\rm d} t \leq \frac{1}{T}\int_T^\iy \mathbb{E}[Q^x(t)-Q^0(t)]{\rm d} t.
\end{equation*}
By the decomposition in \eqref{eq:Yxy},
\begin{align}
\int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)] {\rm d} t
&= \int_T^\infty \mathbb{E}[(x+\inf_{s\leq t} X(s))\mathbbm{1}_{\{\tau^x(0)>t\}}] {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( x-u + \inf_{s\leq t}X(s) > 0) {\rm d} u {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( \tau^{x-u}(0) > t ) {\rm d} u{\rm d} t \\
&\leq \int_T^\iy \int_0^x \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} u {\rm d} t \nonumber\\
&= \int_0^x \int_T^\iy \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} t{\rm d} u
= \int_0^x \frac{\mathbb{E}[\tau^{w}(0)^2]}{T}\,{\rm d} w. \nonumber
\label{eq:tailprobIntegral}
\end{align}
We obtain $\mathbb{E}[\tau^w(0)^2]$ with the help of its Laplace transform in \eqref{eq:invCharExp}. Namely,
\begin{align*}
\mathbb{E}[\tau^w(0)^2] &= \left.\tfrac{{\rm d}^2}{{\rm d} u^2}\mathbb{E}[\exp(-u \tau^w(0))]\right|_{u=0} \\
&= w^2\,\left(\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0}\right)^2 - w\left. \tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0}.
\end{align*}
As in the previous subsection we have $\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = (\mu-\la)^{-1}$, and
\begin{equation*}
\left.\tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0} = {-}\frac{\Upsilon''(\Upsilon^{-1}(0))}{\Upsilon'(\Upsilon^{-1}(0))^3} = {-}\frac{\Upsilon''(0)}{\Upsilon'(0)^3}.
\end{equation*}
Since $\Upsilon'(0) = \mu-\la$ and
\begin{equation*}
\Upsilon''(0) = \s^2 + \int_0^\infty x^2\,\nu({\rm d} x) = u_2,
\end{equation*}
we conclude
\begin{equation*}
\mathbb{E}[\tau^w(0)^2] = \frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3},
\end{equation*}
so that
\begin{equation}\label{eq:delta_upper}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\int_0^x \frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3} {\rm d} w = \frac{1}{T^2}\left(\frac{x^3}{3(\mu-\la)^2}+\frac{u_2 x^2}{2(\mu-\la)^3}\right).
\end{equation}
For general $x,y\geq 0$,
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right).
\end{equation*}
As a direct consequence,
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proof}
\noindent\textit{Remark.}
Observe that if $X$ is light-tailed, that is $\mathbb{E}[\exp\{ -\theta X(1) \}]$ $= \mathbb{E}[\exp\{\kappa(\theta)\}] < \iy$ for some $\theta<0$, then $\Upsilon(\theta)$ as in \eqref{eq:invCharExp} has an analytic continuation in the negative half-plane, and in this region $\Upsilon(\theta)<0$. Consequently, we can replace the upper bound on the tail probability of $\tau^{x-u}(0)$ by
\begin{equation*}
\mathbb{P}\left( \tau^{x-u}(0) > t\right) = \mathbb{P}\left( {\rm e}^{\beta \tau^{x-u}(0)} > {\rm e}^{\beta t} \right) \leq {\rm e}^{-\beta t} \, {\rm e}^{ (x-u)\Upsilon^{-1}(-\beta)},
\end{equation*}
for some $\beta > 0$, so that
\[ \int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)]\, {\rm d} t \leq {\rm e}^{-\beta T}\, \frac{{\rm e}^{x\Upsilon^{-1}(-\beta)}-1}{\beta\,\Upsilon^{-1}(-\beta)}. \]
Along similar lines we deduce
\[ |\Delta^{x,y}_T| \leq \frac{ {\rm e}^{-\beta T}}{T}\, \frac{{\rm e}^{x\Upsilon^{-1}(-\beta)} + {\rm e}^{y\Upsilon^{-1}(-\beta)} -2}{\beta\,\Upsilon^{-1}(-\beta)}
\]
and
\[ |\Delta_T| \leq \frac{{\rm e}^{-\beta T}}{T}\, \frac{\mathbb{E}[{\rm e}^{Q(0)\Upsilon^{-1}(-\beta)}] + \mathbb{E}[{\rm e}^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}] -2}{\beta\,\Upsilon^{-1}(-\beta)},\]
assuming that $\mathbb{E}[{\rm e}^{-y Q(0)}] < \iy$ for all $y>0$. The condition $\mathbb{E}[{\rm e}^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}]<\iy$ follows from Lemma \ref{lemma:workloadmoments}.
Hence, the error decays exponentially fast for light-tailed input processes.
\section{Proofs of Section \ref{sec:optimization}}
\label{app:proofs_optimization}
\subsection{Proof of Lemma \ref{lemma:strict_convexity}}
\begin{proof}
Since the term $\aaa\mu$ is convex, the strictness should come from the term $C_T(\mu)$.
Furthermore, observe that if a function $f_\mu(t)$ is convex for all $t\geq 0$, and strictly convex for all $t\geq\e$ for some $\e\in[0,T)$, i.e. for any $\mu_1,\mu_2>0$ and $a\in (0,1)$
\begin{equation*}
a\, f_{\mu_1}(t) + (1-a) f_{\mu_2}(t) > f_{a\mu_1+(1-a)\mu_2}(t),
\end{equation*}
then,
\begin{equation*}
a\int_0^T \, f_{\mu_1}(t)\, {\rm d} t + (1-a)\int_0^T f_{\mu_2}(t) {\rm d} t =
\int_0^T \, a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t
\end{equation*}
\begin{align*}
&= \int_0^\e a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t + \int_{\rm e}^T \, a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t \\
&> \int_0^\e f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t + \int_{\rm e}^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t. \\
&= \int_0^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t.
\end{align*}
Hence, it suffices to prove the convexity of $\mathbb{E}[Q_\mu(t)]$ as a function of $\mu$ for all $t\geq 0$, and strict convexity for $t\geq \e$ for some $\e\in[0,T)$.
Let $\tau^x_{\mu}(0)$ denote the first passage time of level 0 is the process $Q_\mu$ with $Q(0)=x$. Then,
\begin{align}
Q_\mu(t) &= U(t)-\mu t + \max\left\{ x , -\inf_{s\leq t} [ U(s)-\mu s] \right\}\\
&=
\left\{
\begin{array}{ll}
x+U(t)-\mu t, & \text{if } t<\tau^x_{\mu}(0),\\
U(t)-\mu t -\inf_{s\leq t} [ U(s)-\mu s], & \text{if } t\geq \tau^x_{\mu}(0) ,
\end{array}\right.\label{eq:Qrep}
\end{align}
where
\begin{equation*}
\tau^x_{\mu}(0) := \inf\{ t \geq 0\,:\, x+U(t)-\mu t \leq 0\}
\end{equation*}
and $U(t)$ is a spectrally positive L\'evy process.
Fix $\mu_1, \mu_2>0$ and $a\in(0,1)$. Define $\mu_3 := a\mu_1+(1-a)\mu_2$, and
\begin{equation*}
D(t) := a Q_{\mu_1}(t) + (1-a) Q_{\mu_2}(t) - Q_{\mu_3}(t).
\end{equation*}
In order to prove strict convexity we have to show that $D(t) \geq 0$ for all $t\geq 0$, thereby implying $\mathbb{E} [D(t)] \geq 0$, i.e. convexity, for all $t\geq 0$, and $D(t)>0$ with positive probability for $t\in[\e,T]$, for some $\e \in[0,T)$.
We distinguish two cases: $x>0$ and $x=0$. \\
\\*
\textbf{The case $x>0$.}
We start by noticing that if $Q_{\mu_1}$, $Q_{\mu_2}$ and $Q_{\mu_3}$ experience the same input process $U(t)$, then by absence of negative jumps in $U(t)$, it holds that
\begin{equation}\label{eq:stochDom}
\tau^x_{\mu_2}(0) < \tau^x_{\mu_3}(0) < \tau^x_{\mu_1}(0).
\end{equation}
We use shorthand notation
\begin{equation*}
I_k(t) := \inf_{0\leq s\leq t}[U(s)-\mu_k s],
\end{equation*}
for $k=1,2,3$.
Using representation \eqref{eq:Qrep} of the workload process, we obtain
\begin{equation*}
D(t) = \left\{
\begin{array}{ll}
0,
& \text{if } t < \tau^x_{\mu_2}(0),\\
-(1-a)\left(x+I_2(t) \right),
& \text{if } \tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0),\\
a x - (1-a)I_2(t) + I_3(t),
& \text{if } \tau^x_{\mu_3}(0) \leq t < \tau^x_{\mu_1}(0),\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tau^x_{\mu_1}(0).
\end{array}
\right.
\end{equation*}
This partition allows us to spot when strict convexity can occur.
Note that by definition $t \geq \tau^x_{\mu_2}(0)$, $I_2(t) = \inf_{0\leq s\leq t}[U(s)-\mu_2s]\leq -x$, so that $D(t)\geq 0$ if $\tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0)$.
Moreover, by subadditivity of the infimum,
\begin{align*}
I_3(t) &= \inf_{0\leq s\leq t}[U(s)-\mu_3s] = \inf_{0\leq s\leq t}[a(U(s)-\mu_1s)+(1-a)(U(s)-\mu_2s)] \\
&\geq a \inf_{0\leq s\leq t}[U(s)-\mu_1s] + (1-a) \inf_{0\leq s\leq t}[U(s)-\mu_2s] = a I_1(t) + (1-a) I_2(t),
\end{align*}
and hence $D(t)\geq 0$ for $t \geq \tau^x_{\mu_1}(0)$.
Using the same argument, we deduce
\begin{equation*}
ax - (1-a)I_2(t) + I_3(t) \geq a x - (1-a) I_2(t) + a I_1(t) + (1-a) I_2(t) = a(x + I_1(t)).
\end{equation*}
In particular for $t < \tau^x_{\mu_1}(0)$, this value is strictly positive.
As a result, $D(t)\geq 0$ for all $t\geq 0$.
On top of that $D(t) > 0$ for $t\in[\tau^x_{\mu_3}(0),\tau^x_{\mu_1}(0))$.
Accordingly, the latter implies strict positivity of $\mathbb{E} D(t)$, and therefore strict convexity of $\mathbb{E} Q_\mu(t)$, if the event $\{\tau^x_{\mu_3}(0)\leq t< \tau^x_{\mu_1}(0)\}$ occurs with positive probability.
That is,
\begin{align}
P(D(t)>0) &\geq P\left( a(x+I_1(t))\mathbbm{1}_{\{\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\}} > 0 \right)\nonumber\\
&= P\left( x+ I_1(t) > 0 , \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left( x+ I_1(t) > 0 | \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right) = P(\tau^x_{\mu_3}(0)\leq t) - P(\tau^x_{\mu_1}(0) \leq t) > 0, \label{eq:strictConv}
\end{align}
by the stochastic dominance in \eqref{eq:stochDom}. To ensure the strict inequality in \eqref{eq:strictConv} we have to enforce the condition
\begin{equation}\label{eq:condition}
P(\tau^x_{\mu_1}(0)<T) > 0.
\end{equation}
\textit{Remark.}
An example illustrating the need for this condition is the case in which $U(t)$ is a compound Poisson process and $T < x/\mu_2 < x/\mu_1$. Then
\[Q_{\mu_k}(t) = x + U(t) - \mu_k t,\]
for all $t\in[0,T]$, since $U(t)\geq 0$ and therefore $\tau^x_{\mu_1}(0) > T$. Consequently, for all $a\in(0,1)$,
\[ a\,Q_{\mu_1} + (1-a)\,Q_{\mu_2}(t) = Q_{\mu_3}(t),\]
proving only convexity of $\mathbb{E} Q_{\mu}(t)$ and subsequently $\int_0^T \mathbb{E}[Q_\mu(t)]\,{\rm d} t$. In case $\sigma>0$, the probability in \eqref{eq:condition} is necessarily positive.
\\*
\noindent \textbf{The case $x=0$.}
By the fact that $\tau_{\mu}(0) = 0$ for all $\mu>0$, proving that $D(t)>0$ for in the case $x=0$ reduces to showing that the probability of
\begin{equation*}
D(t) = a I_1(t) + (1-a) I_2(t) - I_3(t)>0
\end{equation*}
happening is positive for all $t>0$. Define
\begin{equation*}
t_0 := \inf\{ t > 0\, :\, U(t) > 0 \},
\end{equation*}
and
\begin{equation*}
\tilde{\tau}_\mu := \inf\{ t > t_0\,: U(t) - \mu t \leq 0\}.
\end{equation*}
We note that $t_0$ as defined above, also defines the epoch of the start of a new excursion of the reflection $Q_\mu$ for all $\mu>0$. Namely,
\[U(s) \leq 0 \quad \Rightarrow\quad U(s) - \mu s \leq -\mu s \qquad \text{for all }0\leq s< t_0\]
\[\Rightarrow \inf_{0\leq s < t_0} [U(s)-\mu s] \leq -\mu t_0 \quad
\Rightarrow U(t_0) - \mu t_0 - \inf_{0\leq s < t_0} [U(s)-\mu s] \geq U(t_0) > 0\].
Then $Q_\mu(t_0-) = 0$ for all $\mu>0$.
By the virtue of the Strong Markov Property, not that $Q_\mu(t_0+t) {\;\buildrel{d}\over= \;} Q_\mu(t)$.
Hence we assume without loss of generality $t_0=0$.
Again, we have a stochastic dominance relation similar to \eqref{eq:stochDom}:
\begin{equation*}
\tilde{\tau}_{\mu_2} < \tilde{\tau}_{\mu_3} < \tilde{\tau}_{\mu_1},
\end{equation*}
for all $\mu_1<\mu_3<\mu_2$.
Then
\begin{equation*}
D(t) {\;\buildrel{d}\over= \;} \left\{
\begin{array}{ll}
0,
& \text{if } t < \tilde{\tau}_{\mu_2},\\
-(1-a)I_2(t),
& \text{if } \tilde{\tau}_{\mu_2} \leq t < \tilde{\tau}_{\mu_3},\\
(1-a)I_2(t) + I_3(t),
& \text{if } \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1},\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tilde{\tau}_{\mu_1}.
\end{array}
\right.
\end{equation*}
Clearly, $D(t)\geq 0$ for all $t\geq 0$ and
\begin{equation*}
-(1-a)I_2(t) + I_3(t) \geq a I_1(t) > 0,
\end{equation*}
for $\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}$.
Hence, in a similar manner to \eqref{eq:strictConv},
\begin{align}
P(D(t)>0) &\geq P\left( aI_1(t)\mathbbm{1}_{\{\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\}} > 0 \right)\nonumber\\
&= P\left( I_1(t) > 0 , \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left( I_1(t) > 0 | \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right) = P(\tilde{\tau}_{\mu_3}\leq t) - P(\tilde{\tau}_{\mu_1} \leq t) > 0, \label{eq:strictConv2}
\end{align}
The last inequality is satisfied it $P(\tilde{\tau}_{\mu_1} < T) >0$, which is equivalent to $P( U(T) - \mu T \leq 0 ) >0$, a condition that is clearly true for all our choice of $U$.
In conclusion, for $x=0$, $\mathbb{E}[D(t)] >0$ and therefore $\mathbb{E}[Q_\mu(t)]$ is a strictly convex function of $\mu$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:min_convergence_mu}}
The proof of the proposition relies on the following auxiliary lemma, of which we include the proof for completeness.
\begin{lemma}\label{lemma:minimizerConvergence}
Consider the sequence of functions $f_n:\, [x_0,\infty) \to \mathbb{R}$ and let $f: [x_0,\infty) \to\mathbb{R}$ be the pointwise limit for some $x_0\in \mathbb{R}$.
Assume $f$ and $f_n$ are strictly convex for all $n$.
Furthermore, let $f(y) \to \infty$ for both $y\to x_0^+$ and $y\to \infty$.
If $x_n$ and $x$ are the minimizers for $f_n$ and $f$, respectively, then $x_n\to x$ for $n\to\infty$.
\end{lemma}
\begin{proof}
We start by showing that the sequence $x_n$ is bounded. Fix $u_l, u_r$ such that $x_0<u_l < x < u_r$. We claim that there exists a $N\in\mathbb{N}$ such that $x_n\in[u_l,u_r]$ for all $n \geq N$. First, we prove the upper bound on $x_n$. For any strictly convex function $h$ with minimizer $x_h$, the following statement holds true:
\begin{equation}\label{eq100}
x_h < u_r \quad \Leftrightarrow \quad h \text{ is strictly increasing at } u_r.
\end{equation}
The first implication follows from observing that $h(x_h) < h(y)$ for all $y> x^*$ and definition of convexity:
\[ 0<\frac{h(u_r)-h(x_h)}{u_r-x_h} \leq \frac{h(u_r+\de)-h(u_r)}{\de}, \]
for all $\de>0$. So that $h(u_r)<h(u_r+\de)$, i.e. $h$ is increasing at $u_r$. The converse follows immediately by observing that $h(u_r) < h(u_r+\de)$ for all $\de>0$, so that $x_h < u_r$.
Next, we show that $f_n$ must be increasing at $u_r$ for $n$ sufficiently large. By pointwise convergence of $f_n$ we have
\[ \lim_{n\to\infty} [f_n(u_r+\de) - f_n(u_r)] = f(u_r+\de) - f(u_r).\]
Let $w_r:= f(u_r+\de) - f(u_r)>0$. Then
\[ \exists N_r \in \mathbb{N}:\, \forall n\geq N_r:\, |[f_n(u_r+\de) - f_n(u_r)] - [f(u_r+\de)-f(u_r)] | < w_r/2.\]
Hence for $n\geq N_r$,
\[f(u_r+\de)-f(u_r) - w_r/2 < f_n(u_r+\de) - f_n(u_r) < f(u_r+\de)-f(u_r) + w_r/2\]
\[\Rightarrow 0 < w_r/2 < f_n(u_r+\de) - f_n(u_r).
\]
Hence by \eqref{eq100}, $x_n < u_r$ for sufficiently large $n$. Similarly, we argue
\begin{equation*}
x_h > u_l \quad \Leftrightarrow \quad h \text{ is strictly decreasing at } u_l,
\end{equation*}
for any strictly convex function $h$ with minimizer $x_h$. Note that $x_h > u_l$ implies $h(x_h) - h(u_l) < 0$ and for all $\de>0$ we get by strict convexity
\[\frac{h(u_l)-h(u_l-\de)}{\de} < \frac{h(x_h)-h(u_l)}{x_h-u_l} < 0,\]
by which $h(u_l-\de)>h(u_l)$, i.e. $h$ is decreasing in $u_l$. Moreover, if $h$ is decreasing at $u_l$, then it is decreasing for all $y < u_l$, by arguments similar to the above. Therefore, $h(u_l-\de)> h(u_l)$ for all $\de>0$ and it must hold that $x_h>u_l$. Define $f(u_l) - f(u_l-\de) :=w_l < 0$, then again by pointwise convergence, we have that
\[ \exists N_l \in \mathbb{N}:\, \forall n\geq N_l:\, |[f_n(u_l) - f_n(u_l-\de)] - [f(u_l)-f(u_l-\de)] | < w_l,\]
whereupon
\[ f_n(u_l) - f_n(u_l-\de) < f(u_l) - f(u_l-\de) + w_l = 2w_l < 0.\]
Hence, for sufficiently large $n$, we also have $x_n > u_l$. Fix $N = \max\{N_l,N_r\}$, then for $n\geq N$, $x_n\in( u_l,u_r)$. That is, the sequence $x_n$ is bounded. Therefore, by the theorem of Bolzano-Weierstrass, $x_n$ has to have a convergent subsequence. That is, there exists a sequence $n_k$ such that $n_k \to\infty$ and $x_{n_k}\to a$ as $k\to \infty$ for some $a \in [u_l,u_r]$.
We prove that every subsequence must converge to $x$ by contradiction. Suppose there exists a subsequence $n_k$ such that $x_{n_k}\to a\neq x$. Since, $x_n\in [u_l,u_r]$ for $n\geq N$, we may restrict our attention on the sequence of functions $\hat{f}_n:[u_l,u_r] \to \mathbb{R^+}$, consisting of the original function $f_n$ restricted to the domain $[u_l,u_r]$. To be precise $x_n = \arg\min_y f_n(y) = \arg\min_y \hat{f}_n(y)$ for $n\geq N$. Because $\hat{f}_n$ and $\hat{f}$ are bounded, we furthermore $\hat{f}_n \to \hat{f}$ uniformly.
Fix $\e>0$. By uniform convergence, there exists an $K_0 \in\mathbb{N}$ such that
\[ | \hat{f}_{n_k}( y ) - \hat{f}( y)| < \e /2,\quad \forall k\geq K_0,\ y \in[u_l,u_r].\]
Also, because $\hat{f}$ is convex, it is continuous, so that there exists a $\de := \de(\e)$ so that
\[ |z-y| < \de \quad \Rightarrow \quad |\hat{f}(z) - \hat{f}(y)| < \e/2.\]
Let $K_1$ be such that $|x_{n_k}-a| < \de$ for all $k\geq K_1$. Then for $k \geq K= \max\{K_0,K_1\}$ this implies
\begin{align*}
|f_{n_k}(x_{n_k}) - f(a)| &= |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(a)| \\
&\leq |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(x_{n_k})| + | \hat{f}(x_{n_k}) - f(a)| < \e/2 + \e/2 = \e.
\end{align*}
Hence we conclude $\lim_{k\to\infty} \hat{f}_{n_k}(x_{n_k}) = f(a)$.
Therefore,
\[ \limsup_{n\to \infty} f_n(x_n) \geq f(a) > f(x),\]
by minimality of $x$. However, $f_n(x_n) \leq f_n(x)$, which implies $\limsup_{n\to\infty} f_n(x_n) \leq \lim_{n\to\infty} f_n(x) = f(x)$, contradicting the strict inequality above. Hence we deduce $x=a$. Consequently, every subsequence of $x_n$ converges to $x$ and therefore $x_n\to x$ as $n\to \infty$.
Applying Lemma \ref{lemma:minimizerConvergence} to the functions $\Pi_T$ and $\Pi_\iy$ with $x_0=\la$, together with Lemma \ref{lemma:strict_convexity}, we obtain the result immediately.
\end{proof}
\subsection{Proof of Proposition \ref{prop:muBullet}}
\begin{proof}
Note that $\Pi_\infty$ is a smooth function.
By the first optimality condition $\Pi_\infty'(\mui)$ $= 0$.
We first prove that also $\Pi_T(\mu)$ is differentiable with respect to $\mu$ for all $\mu\geq 0$.
Recall \eqref{eq:PiT}, which defines the cost function as a combination of the accumulated expected transient queue length, and linear staffing costs.
The latter term is clearly differentiable, hence it remains to be proved that
\begin{equation*}
C_T(\mu) = \frac{1}{T}\int_0^\iy \mathbb{E}[Q_\mu(t)] \, {\rm d} t,
\end{equation*}
admits a derivative for all $\mu\geq 0$ with $T$ fixed.
This holds if and only if $\mathbb{E}[Q_\mu(t)]$ is differentiable for all $t\geq 0$.
Let $Q(0)= x\geq 0$.
Following \eqref{eq:Qlm},
\begin{align*}
\mathbb{E}[Q_\mu(t)] &= \mathbb{E}[X_\mu(t)] + \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big]\\
&= (\la-\mu)t+ \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big],
\end{align*}
where the first term is differentiable.
Furthermore,
\begin{align*}
\mathbb{E}[ \max\{ x, \sup_{s\in[0,t]} \{ - X_\mu(s) \} \} ]
&= x + \int_x^\iy P(\sup_{s\in[0,t]} \{ - X_\mu(s) \} > u ){\rm d} u \\
&= x+\int_x^\infty P(\hat\tau^0(u) \leq t ) {\rm d} u,
\end{align*}
with $\hat{\tau}^0(u)$ as defined in \eqref{eq:transformedTau}.
Since $-X_\mu$ is a process with no positive jumps, we may apply Corollary VII3 of \cite{Bertoin1996}, which states that the following equivalence between measures holds:
\begin{equation}
s\,P( \hat\tau^0(u) \in ds ) du = u\,P( -X_\mu(s) \in du ) ds,
\end{equation}
so that
\begin{align}
\int_{u=x}^\infty P(\hat{\tau}^0(u) \leq t )\, {\rm d} u
&=
\int_{u=x}^\iy \int_{s=0}^t P( \hat\tau^0(u) \in ds ) {\rm d} u \nonumber \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,\,P( -X_\mu(s) \in {\rm d} u ) {\rm d} s \nonumber \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,P( X_\mu(s) \in {\rm d} u ) {\rm d} s \nonumber \\
&= \int_{s=0}^t s^{-1} \mathbb{E}[ \max\{x, X_\mu(s)\} ] {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( X_\mu(s)/s > v ) {\rm d} v {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( U(\la s)/s > v + \mu ) {\rm d} v {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{w=x/s+\mu}^\iy P( U(\la s)/s > w) {\rm d} w {\rm d} s,
\end{align}
where the interchange of integrals is justified by Fubini's theorem and this last form is differentiable with respect to $\mu$.
Substituting $Q(0)$ for $x$ straightforwardly yields differentiability of the complete cost function $\Pi_T$ for all $T$.
Consequently we invoke the first optimality condition for $\muT$ to find
\begin{align*}
0=\Pi_T'(\muT)
&= \Pi_\infty'(\muT) + \Psi_T'(\muT) + O(1/T^2)\\
&= \Pi_\infty'(\mui) + \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi_T'''(\xi)+\Psi_T'''(\xi) \right] + O(1/T^2)\\
&= \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi'''(\xi)+\Psi_T'''(\xi)\right] + O(1/T^2).
\end{align*}
for some $\xi \in [\muT,\mui]$. Rearranging this gives
\begin{align*}
\muT-\mui &= \frac{-\Psi_T'(\mui)}{\Pi_\infty''(\mui)+\Psi_T''(\mui) + \frac{1}{2}(\muT-\mui)(\Pi_\infty'''(\muT)+\Psi_T'''(\xi))} + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} \left[1 - \frac{\Psi_T''(\mu)}{\Pi_\infty''(\mui)} - \frac{\muT-\mu_\infty}{2}\frac{\Pi_\infty'''(\mui)+\Psi_T'''(\mui)}{\Pi_\infty''(\mui)}\right] + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} [1 + o(1)]
\end{align*}
for $T\to\infty$, since both $\mu_T - \mu_\infty$ and $\Psi_T''(\mui)$ are $o(1)$.
Let
\begin{equation*}
\mu_\bullet := \lim_{T\to\iy} \frac{T \Psi_T'(\mui)}{\Pi_\iy''(\mui)}.
\end{equation*}
By \eqref{eq:mainResult} we have
\begin{equation*}
T \Psi'_T(\mu) = {-} \frac{\mathbb{E}[Q(0)^2]}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)^3} + \frac{3\la^2u_2^2}{4(\mu-\la)^4}.
\end{equation*}
Together with
\begin{equation*}
\Pi_\iy''(\mu) = \frac{\la u_2}{(\mu-\la)^3}
\end{equation*}
and \eqref{eq:muInf} we obtain the expression for $\mu_\bullet$ in \eqref{eq:muBullet}.
\end{proof}
\subsection{Proof of Proposition \ref{prop:optimalitygap_mui}}\label{sec:proofProp4}
\begin{proof}
We upper bound the optimality gap by using the decomposition in \eqref{eq:decomposition}.
\begin{align}
|\Pi_\iy^\star - \Pi_T^\star| &= \left|\hat{\Pi}_T(\mu_\infty) + \Delta_T(\mui) - \hat{\Pi}_T(\muT) - \Delta_T(\muT)\right|\nonumber\\
&\leq |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + |\Delta_T(\mui)| + |\Delta_T(\muT)|\nonumber\\
&= |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + O(1/T^2),
\end{align}
since $\Delta_T(\mu) = O(1/T^2)$ by Proposition \ref{prop:truncation_error}. Next, we find an upper bound for $|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)|$, with $\hat{\Pi}_T(\cdot)$ as in \eqref{eq:decomposition}, in terms of the difference between $\gamma$ and $\beta$.
For simplicity, denote $\hat{\gamma} = \gamma - \la$ and $\hat{\beta} = \beta-\la$, implying $\hat{\gamma}-\hat{\beta}=\gamma-\beta$. Then, using \eqref{eq:mainResult}, we get
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &=
\left| \aaa(\hat{\gamma}-\hat{\beta})
+\left(\frac{\la u_2}{2} + \frac{\mathbb{E}[Q(0)^2]}{2T}\right)\left(\frac{1}{\hx}-\frac{1}{\hy}\right) \right. \\
& \qquad \left. -\frac{\la^2 u_2^2}{4T}\left(\frac{1}{\hx^3}-\frac{1}{\hy^3}\right)
-\frac{\la u_3}{6T}\left(\frac{1}{\hx^2} - \frac{1}{\hy^2}\right)
\right|.
\end{align*}
Furthermore, we have
\begin{align*}
\frac 1 \hx - \frac 1 \hy &= -\frac{\hx-\hy}{\hy^2} + \frac{(\hx-\hy)^2}{\hy^3} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^2} - \frac 1 {\hy^2} &= -\frac{2(\hx-\hy)}{\hy^3} + \frac{3(\hx-\hy)^2}{\hy^4} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^3} - \frac 1 {\hy^3} &= -\frac{3(\hx-\hy)}{\hy^4} + \frac{6(\hx-\hy)^2}{\hy^5} + O\left((\gamma-\beta)^3\right).\\
\end{align*}
Substituting these yields
\begin{align*}
|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)| &= \left|(\gamma-\beta)\left[ \aaa - \frac{\la u_2}{2 \hy^2} + \frac{1}{2T\hy^2}\left(\mathbb{E}[Q(0)^2] + \frac{3\la^2 u_2^2}{2\hy^2} + \frac{2\la u_3}{3 \hy}\right)\right]\right. \\
&\qquad \left. - (\gamma-\beta)^2\left[ \frac{\la u_2}{2 \hy^3} + \frac{1}{2T\hy^3}\left(\mathbb{E}[Q(0)^2] - \frac{3\la^2 u_2^2}{\hy^2} - \frac{\la u_3}{\hy}\right)\right]\right| \\
& \qquad \qquad + O\left((\gamma-\beta)^3\right).
\end{align*}
Given that $\muT = \mui + \mu_\bullet/T + o(1/T)$, we find
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\mui-\la )^2}\right) + O(1/T^2)\\
&= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\sqrt{\la u_2/2\aaa})^2}\right) + O(1/T^2) = O(1/T^2),
\end{align*}
which concludes the proof.
\end{proof}
\resettocdepth
\end{subappendices}
\chapter{Transient error approximation in a L\'evy queue}
\begin{chapterstart}
Motivated by a capacity allocation problem within a finite planning period, we conduct a transient analysis of a single-server queue with L\'evy input. From a cost minimization perspective, we investigate the error induced by using stationary congestion measures as opposed to time-dependent measures. Invoking recent results from fluctuation theory of L\'evy processes, we derive a refined cost function, that accounts for transient effects. This leads to a corrected capacity allocation rule for the transient single-server queue. Extensive numerical experiments indicate that the cost reductions achieved by this correction can be significant.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Transient error approximation in a L\'evy queue}\\
\textit{Britt Mathijsen \& Bert Zwart}\\
\textit{Queueing Systems, 85(3), 269-304 (2017)}
\end{flushright}
\newpage
\section{Introduction}
The issue of matching a service system's capacity to stochastic demand induced by its clients arises in many practical settings. Typically, the resources available to satisfy demand are scarce and hence expensive. This forces the manager to consider a trade-off between the system efficiency and the quality of service perceived by its clients. In this chapter, we focus on this trade-off in the context of the $M/G/1$ queue, in which the variable amenable for optimization is the server speed $\mu$.
In general, optimizing the server speed $\mu$ in a single-server queue in a time-homogeneous environment, while trading off congestion levels against capacity allocation costs, does not pose any technical challenges. Typically, the objective function to be minimized, the total cost function, has the shape
\begin{equation}\label{eq:intro}
\Pi_\iy(\mu) = \mathbb{E}[Q_\mu(\infty)] + \aaa\mu = \frac{\la\mathbb{E}[B^2] }{2(\mu-\la\mathbb{E}[B])} + \aaa\mu,
\end{equation}
where $\mathbb{E}[Q_\mu(\infty)]$ denotes the expected steady-state amount of work given server speed $\mu$, and $B$ describes the service requirement per arrival. The parameter $\aaa>0$ represents the relative capacity allocation costs incurred by deploying service rate $\mu$. This one-dimensional optimization problem yields the optimizer
\begin{equation*}
\mui = \lambda \mathbb{E}[B] + \sqrt{\frac{\la\mathbb{E}[B^2]}{2\aaa}}.
\end{equation*}
Despite the simplicity and tractability of the problem described above, the presence of the \emph{steady-state} measure in the cost function in \eqref{eq:intro} should be handled carefully. By employing this particular cost structure, one automatically agrees with the underlying assumption of the system being sufficiently close to its steady state.
However, referring to practical applications of the single-server model, system parameters rarely remain constant over time. Moreover, planning periods for the optimization problem are naturally finite. Hence, the \emph{true} expected costs incurred, which we denote by $\Pi_T(\mu)$, in addition depend on the length of the planning period $T$. Consequently, the usage of steady-state models for decision making needs to be justified by a more elaborate time-dependent or \emph{transient} analysis for these type of settings.\\
\\*
\noindent
\textbf{Related literature}.
The time-dependent behavior of the single-server queue received much attention in queueing theory. First efforts to analyze the time-dependent properties of the $M/G/1$ queue date back to the 1950s and 1960s, e.g. \cite{Benes1957,Gaver1959,Kendall1951,Takacs1955,Takacs1962}. The analyses in these papers mostly yield implicit expressions for performance characteristics through Laplace transforms, integro-differential equations and infinite convolutions.
More specifically, there is vast literature on the transient analysis of the $M/M/1$ queue, with the goal to derive explicit expressions for queue length characteristics, see e.g. \cite{Abate1987,Cohen1982,Pegden1982,Prabhu1964}.
These works provide a variety of explicit expressions for the transient dynamics, although the complexity of the resulting expressions, typically involving Bessel functions, exposes the intricate intractability of the matter. Consequently, approximation methods for insightful quantification of the dynamics based on numerical \cite{Neuts1966} or asymptotic methods, have become prevalent in more recent literature.
The asymptotic methods either exploit knowledge on the evolution of the queueing process as time $t$ grows large \cite{Abate1987,Newell1982,Odoni1983}, or as the arrival rate $\la$ is increased to infinity \cite{Abate1987a,Abate1987b,Gaver1968}.
It is noteworthy that a substantial contribution to the transient literature is made by Abate and Whitt \cite{Abate1987a,Abate1987b,Abate1987,Abate1994}, who exploit the existence of a decomposition of the mean transient queue length and obtain expressions for the moments of the queue length and virtual waiting through probabilistic arguments in several queueing models.
More recently, asymptotic methods have been used to justify the application of stationary performance measures in Markovian environments or to refine them, see e.g. \cite{Green1991,Whitt1991}.
Other approximative methods known as uniform acceleration expansions \cite{Massey1998} have been developed to reveal the asymptotic behavior of the single-server queue as a function of $t$, which are moreover able to capture time-varying arrival rates.
The majority of the works mentioned above do reflect on the error imposed by usage of steady-state performance metrics instead of the correct time-dependent counterpart. However, no light has been shed on the accumulation of this error over a finite period of time. To the best of our knowledge, the only work that addresses this issue is the paper by Steckley and Henderson \cite{Steckley2007}, who compute an approximation for the error accumulated between the steady-state and transient delay probability. Our analysis on the other hand is centered around the mean workload, which requires a different approach. In addition, the focus in \cite{Steckley2007} is on performance measures only, while the main goal of our work is to investigate the quality of staffing rules. \\
\\*
\noindent\textbf{L\'evy input}.
Although the $M/G/1$ queue serves as the leading example in our analysis, we choose to use a more general framework for the arrival process of the queue. Namely, we let the server face a L\'evy process.
This gives the advantage that once we have obtained the results, we can apply them to broader queue input classes, such as Brownian motion and the Gamma process.
To shed light on the influence of the transience of the queueing process on traditional staffing questions, we will study the capacity allocation problem in the context of cost minimization in which the objective function is $\Pi_T(\mu)$, i.e. a function of both $\mu$ and $T$. We investigate how the invalidity of the stationary assumption is echoed through the operational cost accounting for congestion-related penalties.
Furthermore, we establish a result on the strict convexity of the function $\Pi_T(\mu)$, for almost all values of $T$ (with a few minor exceptions for certain deterministic initial states), which is an essential property for convergence of both cost function and corresponding minimizer to their stationary counterparts.
\\
\\*
\noindent\textbf{Corrected staffing rule}.
As it will appear that an exact analysis of this disparity is intractable, we will present an explicit approximate correction to the conventional stationary objective function given by $\Psi(\mu)/T$ and prove that
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \frac{\Psi(\mu)}{T} + O(1/T^2),
\end{equation*}
with the help of recent results from the fluctuation theory of L\'evy processes.
Based on this refinement we ultimately examine how incorporating transient effects\\ \noindent changes the optimal capacity level and propose a refinement to the steady-state capacity allocation rule,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T).
\end{equation*}
We moreover deduce an explicit expression for $\mu_\bullet$ in terms of the initial state and the first three moments of the service requirement per arrival.
It is noteworthy that similar refined square-root staffing rules have been proposed for multi-server queues in the Halfin-Whitt regime, see e.g. \cite{Janssen2015,Janssen2008,Janssen2011,Randhawa2014,Zhang2012}. In those cases, the relevant decision value is the number of servers and refinements are derived for $\la\to\iy$, whereas we consider the regime $T\to\infty$.
Building upon the insights gained through the analysis of this optimality gap, we reflect on the parameter settings of the underlying queueing process in which our refined capacity sizing rule yields significant improvement and in which cases it has little effect. Special emphasis is put on the relationship between the accuracy of the standard procedure and the length of the planning period.
\\
\\*
\noindent\textbf{Structure of the chapter}.
The remainder of this chapter is structured as follows. Section \ref{sec:model_description} is devoted to the model description and presents some preliminary results. The main result will be given in Section \ref{sec:analysis} and results regarding the optimization problem will be discussed in Section \ref{sec:optimization}, followed by the validation of our novel techniques through numerical experiments in Section \ref{sec:numerics}. We will give some concluding remarks and topics for further research in Section \ref{sec:conclusion_chapter6}. We have deferred all proofs to the appendix.
\section{Model description}
\label{sec:model_description}
\subsection{A queueing model with L\'evy input \label{sec:levymodel}}
The model that inspired our study is the standard $M/G/1$ queue starting out of equilibrium. Customers arrive to the queue according to a Poisson process with rate $\la$.
All arrivals have i.i.d. service requirement $B_i$, stemming from a common random variable $B$.
Without loss of generality we will assume $\mathbb{E}[B] = 1$ throughout. The server is able to remove $\mu$ amounts of work from the system per time unit; a variable we will refer to as the \emph{server speed}.
E.g. if $\mu = 3$ and two customers are in the system with remaining service times $4$ and $2$, then the queue will be empty 2 time units later, provided that no new arrivals occur in the meantime.
Let $N_\la(t)$ denote the number of arrivals until time $t$.
Accordingly, the total work generated by the customers is given by
\begin{equation*}
Z_\la(t) = \sum_{i=1}^{N_\la(t)} B_i.
\end{equation*}
Furthermore, define $X_{\la,\mu}(t) := Z_\la(t) - \mu t$. We call $X_{\la,\mu}$ the \emph{net-input process}.
More generally, we assume throughout the chapter that $X_{\la,\mu}$ is a L\'evy process.
Specifically, we let $Z_\la$ be of the form $Z_\la(t) = U(\la t)$, where $U$ is a spectrally positive L\'evy process generated by the triplet $(a,\s,\nu)$ and $\mathbb{E}[U(1)] = 1$.
This restriction to spectrally positive processes is equivalent to stating $\nu(-\infty,0)=0$ and is a vital assumption to our analysis.
Subsequently, we assume the net-input process $X_{\la,\mu}$ to be
\begin{equation}
\label{eq:Xlmprocess}
X_{\la,\mu}(t) = U(\la t) - \mu t, \qquad t \geq 0.
\end{equation}
Note that by setting $a=\s=0$ and $\nu = \la\, F_B$, where $F_B$ is the cumulative distribution function of $B$, we retrieve the original $M/G/1$ queue.
The stochastic process central to our analysis is the \emph{workload process} $Q_{\la,\mu}(t)$, $t\geq 0$, which describes the amount of work the server is facing at time $t$.
The net-input process $X_{\la,\mu}$ completely determines the trajectory of $Q_{\la,\mu}$, namely
\begin{equation}\label{eq:Qlm}
Q_{\la,\mu}(t) = \max\left\{ Q(0) + X_{\la,\mu}(t), \sup_{s\in[0,t]} [X_{\la,\mu}(t)-X_{\la,\mu}(s)]\right\}, \qquad t\geq 0,
\end{equation}
where $Q(0)$ is the initial workload in the system.
In fact, $Q_{\la,\mu}$ is the reflected version of $X_{\la,\mu}$ with reflection barrier at zero.
Careful inspection of the structure also reveals that $X_{\la,\mu}(t) \equiv X_{\la/\mu,1}(\mu t) \equiv X_{1,\mu/\la}(\la t)$, so that
\begin{equation}
\label{eq:Qidentity}
Q_{\la,\mu}(t) {\;\buildrel{d}\over= \;} Q_{\la/\mu,1}(\mu t) {\;\buildrel{d}\over= \;} Q_{1,\mu/\la}(\la t)
\end{equation}
for all $\la,\mu,t>0$.
This identity will prove to be convenient for the numerical analysis in Section \ref{sec:numerics}. For reasons of clarity, we omit the subscript $\la$ in our expressions if no ambiguity is possible.
The process $Q_{\mu}$ is a natural indicator of the level of congestion in the system and therefore a good choice for quantifying the Quality of Service (QoS) received by a client.
We remark that alternative processes characterizing congestion in the system can be deduced directly from $Q_{\mu}(t)$. For example, consider the virtual waiting time process $V_{\mu}(t)$, which is the waiting time a customer would experience if he arrives at time $t$. This, under the first-come-first-served policy, satisfies the relation $\mathbb{E}[V_{\mu}(t)] = \mathbb{E}[Q_{\mu}(t)]/\mu$ for all $t\geq 0$.
Likewise, the expected number of the customers in the system $L_{\mu}(t)$ at time $t\geq 0$ is given by Little's law
\begin{equation*}
\mathbb{E}[L_{\mu}(t)] = \la\, \mathbb{E}[V_{\mu}(t)] = \frac{\la}{\mu}\, \mathbb{E}[Q_{\mu}(t)].
\end{equation*}
To facilitate our investigation of the queueing model, we end this subsection by introducing some notation regarding the net-input and workload process and by stating a useful preliminary result concerning the stationary process $\Qlm(\iy)$.
Throughout the chapter we assume $\mu>\la$ to ensure ergodicity of the queue and convergence in distribution to the limit
\begin{equation*}
\Qlm(\iy) := \lim_{t\to\iy} \Qlm(t),
\end{equation*}
for any initial state $Q(0)<\iy$.
The distribution of $Q_{\mu}(\infty)$ coincides with the stationary distribution of $\Qlm(t)$.
By $\ka_U(\cdot)$ and $\ka_{\mu}(\cdot)$ we denote the L\'evy exponents of the processes $U$ and $\Xlm$, respectively:
\begin{equation*}
\ka_{\mu}(\thh) = \log \mathbb{E}[{\rm e}^{\thh \Xlm(1)}] = \log \mathbb{E}[{\rm e}^{\thh(U(\la) - \mu)}] = \la \ka_U(\thh) - \mu \thh.
\end{equation*}
Furthermore, define $u_k = \mathbb{E}[\{U(1) - \mathbb{E} U(1)\}^k]$ for $k=2,3,...$.
Using this representation we obtain the following preliminary result.
\begin{lemma}\label{lemma:workloadmoments}
Let $\mathbb{E}|U(1)|<\infty$, $u_2, u_3 < \iy$ and $\mu > \la$. If $Q_{\mu}(\infty)$ represents the steady-state distribution of the workload process, then
\begin{equation*}
\mathbb{E}[\Qlm(\infty)] = \frac{\la u_2}{2(\mu-\la)},\qquad \mathbb{E}[Q_{\mu}^2(\iy)]=\frac{\la^2u_2^2}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)}.
\end{equation*}
\end{lemma}
The proof of Lemma \ref{lemma:workloadmoments} follows directly by differentiation of the Laplace transform of $Q_\mu(\iy)$ and is given in Appendix \ref{app:proof_lemma_workload_moments}.
\subsection{Finite horizon}
For the purpose of our research, we are interested in the dynamics of the workload process within a fixed time frame of length $T>0$.
For all $0\leq t \leq T$, we assume that the parameters of the queue, $\la,\mu,u_2,u_3$, remain unchanged.
If at $t=0$ the queue is not in steady-state corresponding to the specified parameters of the starting period, the process $\{\Qlm(t)\}_{t\in[0,T]}$ differs from its stationary counterpart $\Qlm(\infty)$.
To illustrate this, Figure \ref{fig:transientmeans} depicts the expected value $\Qlm$ in a $M/M/1$ queue as a function of time for several initial workloads $Q(0)$ for a particular setting of $\la$ and $\mu$.
Clearly, transient behavior of $\mathbb{E}[\Qlm(t)]$, for $Q(0) \neq \Qlm(\iy)$, differs significantly from the steady-state mean with the same system parameters.
Note that even if $Q(0) \equiv \mathbb{E}[\Qlm(\iy)]$, the time-dependent mean does not coincide with the steady-state mean. Moreover, $\mathbb{E}[\Qlm(t)]$ is not even a strictly increasing nor decreasing function of time. This phenomenon is a consequence of the decomposition of the transient mean into one strictly increasing, and a strictly decreasing term for $Q(0)>0$, as discussed in \cite{Abate1987}.
Nonetheless, $\Qlm(t)$ converges in distribution to $\Qlm(\infty)$ as $t\to\iy$, if $\mu>\la$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.15,yscale=0.225]
\draw (0,0) -- coordinate (x axis mid) (50,0);
\draw (0,0) -- coordinate (y axis mid) (0,21);
\node[right] at (51,0) {$t$};
\node[rotate=90, above=0.7 cm] at (y axis mid) {$\mathbb{E}[\Qlm(t)]$};
\draw[dashed, thick, gray] (0,10) -- coordinate (eq) (51,10);
\draw[->] (24,6.4) -- coordinate (a1) (21.65,8.49574);
\node[right=0.6cm,below=0.3cm] at (a1) {$Q(0)\equiv 0$};
\draw[->] (14,6.2) -- coordinate (a2) (13.,8.49434);
\node[right=0.2cm,below=0.3cm] at (a2) {$Q(0)\equiv10$};
\draw[->] (9,15.5) -- coordinate (a3) (7.5,13.7648);
\node[right=0.6cm,above=0.1cm] at (a3) {$Q(0)\equiv20$};
\draw[->] (40,12.5) -- coordinate (a4) (38,10.9712);
\node[right,above=0.3cm] at (a4) { $Q(0)\sim \exp\left(\tfrac{1}{15}\right)$ };
\foreach \x in {0,10,...,50}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {\x};
\foreach \y in {5,10,15,20}
\draw (1pt,\y) -- (-20pt,\y)
node[anchor=east] {\y};
\draw[thick,color = col1] plot
file {Chapter_6/tikz/means0.txt};
\draw[thick,color = col3] plot
file {Chapter_6/tikz/means10.txt};
\draw[thick,color = col4] plot
file {Chapter_6/tikz/means20.txt};
\draw[thick,color = col5] plot
file {Chapter_6/tikz/meansExp.txt};
\end{tikzpicture}
\caption{Time-dependent mean workload in a $M/M/1$ queue with $\la = 10$ and $\mu=11$ for different initial states $Q(0)$. The dashed line depicts $\mathbb{E}\Qlm(\iy)$.}
\label{fig:transientmeans}
\end{figure}
Since the time horizon of our analysis is limited to $t\leq T$, the process may not approach the steady-state distribution sufficiently close to appropriately use its steady-state properties for capacity allocation.
To overcome this disparity, we propose a way to include the influence of this transient phase in the capacity allocation problem.
\subsection{Cost structure}
As mentioned before, we are interested in balancing the QoS and efficiency of the queue by choosing the optimal server speed $\mu$.
The adjective \emph{optimal} indicates that we intend to choose the speed according to some objective function.
In our case, we conduct our analysis based on a cost function, which consists of a part accounting for the penalty for congestion in the system and a part for staffing cost. The cost value of both parts is governed by the variable $\mu$.
The instantaneous cost incurred at time $t$ equals
\begin{equation*}
\mathbb{E}[\Qlm(t)] + \aaa \mu,
\end{equation*}
where $\aaa$ is a positive constant defining the \emph{relative staffing cost}.
Hence, the cost structure we apply is a combination of the transient mean of the workload process and a linear staffing cost.
Accumulated and normalized over the period $[0,T]$, the cost function on which the rest of this chapter will be based equals
\begin{equation}\label{eq:PiT}
\Pi_{T}(\mu) := \frac{1}{T}\int_0^T\left( \mathbb{E}[\Qlm(t)] + \aaa\mu \right) {\rm d} t
= \frac{1}{T} \int_0^T \mathbb{E}[\Qlm(t)] {\rm d} t + \aaa\mu.
\end{equation}
We use shorthand notation for the normalized congestion costs:
\begin{equation}\label{eq:CTmu}
C_{T}(\mu) := \frac{1}{T}\int_0^T \mathbb{E}[Q_{\mu}(t)] {\rm d} t,
\end{equation}
and $C_{\iy}(\mu) := \mathbb{E}[\Qlm(\iy)]$.
In order to compare the actual costs incurred over the interval $[0,T]$ to the cost function of the queue in stationary conditions, we define
\begin{equation}\label{eq:PiInf}
\Pi_{\iy}(\mu) := C_{\iy}(\mu) + \aaa \mu = \mathbb{E}[Q_\mu(\iy)] + \aaa\mu,
\end{equation}
which allows an explicit expression by Lemma \ref{lemma:workloadmoments}.
Under mild conditions on the net-input process and the distribution of the initial state, the cost functions coincide for $T\to\iy$.
\begin{proposition}\label{prop:cost_convergence}
Let $\mu>\la$ and assume $\mathbb{E}[U(1)],\, \mathbb{E}[Q(0)] < \iy$. Then
\begin{equation*}
\lim_{T\to\iy} \Pi_{T}(\mu) = \Pi_{\iy}(\mu).
\end{equation*}
\end{proposition}
\noindent
The proof of Proposition \ref{prop:cost_convergence} can be found in Appendix \ref{app:proof_prop1}.
\noindent
Define
\[
\Omega_T := \frac{1}{T}\int_0^{T} \left( \mathbb{E}[\Qlm(t)] - \mathbb{E}[\Qlm(\iy)] \right) {\rm d} t \]
We can then rewriting \eqref{eq:PiT} as
\begin{align}
\Pi_{T}(\mu) &= \frac{1}{T}\int_0^{T} \left( \mathbb{E}[\Qlm(t)] - \mathbb{E}[\Qlm(\iy)] \right) {\rm d} t + \mathbb{E}[\Qlm(\iy)] + \aaa\mu = \Omega_{T}(\mu) + \Pi_{\infty}(\mu).
\label{eq:decomp}
\end{align}
Section \ref{sec:analysis} is concerned with the analysis of the correction factor $\Omega_{T}(\mu)$.
Ultimately, we are concerned with the additional costs incurred by choosing the server speed through minimization of $\Pi_{\iy}(\mu)$ instead of $\Pi_{T}(\mu)$.
Therefore, we formulate the exact and approximate optimization problems as follows
\begin{equation}\label{eq:muStar}
\mu_T^\star := \arg\min_{\mu\geq 0} \Pi_{T}(\mu), \qquad \qquad \mu_\infty^\star := \arg\min_{\mu\geq 0} \Pi_{\iy}(\mu),
\end{equation}
\begin{equation}\label{eq:piStar}
\Pi_{T}^\star := \Pi_{T}(\mu_T^\star), \qquad \qquad \Pi_{\iy}^\star := \Pi_{T}(\mu_\iy^\star).
\end{equation}
In Section \ref{sec:optimization} we turn to the comparison of $\mu_T^{\star}$ and $\mu_\iy^\star$ as well as the \emph{optimality gap} $\Pi_{\iy}^\star - \Pi_{T}^\star$.
\section{Analysis of the objective function}
\label{sec:analysis}
From \eqref{eq:decomp} it is evident that, for finding an explicit characterization of $\Pi_{T}(\mu)$, it suffices to study the term $\Omega_T(\mu)$ in more detail. We start by stating the main result of this section, which describes the leading order behavior of $\Omega_T(\mu)$ as $T$ increases.
\begin{theorem}\label{thm:mainresult}
Let $X_\mu(t)$ be of the form \eqref{eq:Xlmprocess}. If $\mathbb{E}[\max(Q(0),Q_\mu(\infty))^3] < \iy$ and $u_2,u_3 < \iy$, then
\begin{align*}
\Omega_T(\mu) &= \frac{\mathbb{E}[Q(0)^2] - \mathbb{E}[Q_\mu(\iy)^2]}{2T(\mu-\la)} + O\left(\frac{1}{T^2}\right) \nonumber\\
&= \frac{1}{2T(\mu-\la)}\left( \mathbb{E}[Q(0)^2] - \frac{\la^2 u_2^2}{2(\mu-\la)^2} - \frac{\la u_3}{3(\mu-\la)}\right) + O\left(\frac{1}{T^2}\right),
\end{align*}
for $\mu>\la$.
\end{theorem}
Note that this expression provides an \emph{approximation} of the actual cost function
$\Pi_T(\mu)$. We elaborate on the implications of this additional information on the optimization problem in Section \ref{sec:optimization}.
In the remainder of this section we provide a detailed description of the steps taken to obtain this outcome.
We assume a fixed service rate $\mu$ throughout the analysis in this section and therefore omit the subscript $\mu$. Proofs of the intermediate results can be found in Appendix \ref{app:proofs_analysis}.
\subsection{Constructing a coupling}
Before starting our analysis of the correction term $\Omega_{T}(\mu)$ we introduce some auxiliary notation.
By $Q^A(t)$ we denote the workload process as described in Subsection \ref{sec:levymodel} with initial state $A$ and $\mathbb{E}_A$ the expectation with respect to any non-negative random variable $A$, which is independent of the net-input process $X$.
To be able to compare $\mathbb{E}[Q(t)]$ and $\mathbb{E}[Q(\iy)]$ as in $\Omega_T(\mu)$, we will use a coupling technique.
Observe that by definition of the stationary distribution $Q(\iy) {\;\buildrel{d}\over= \;} Q^{Q(\iy)}(t)$ for all $t \geq 0$ and therefore $\mathbb{E}[Q(\iy)] = \mathbb{E}_{Q(\iy)}[Q^{Q(\iy)}(t)]$. Furthermore, $\mathbb{E}[Q(t)] = \mathbb{E}_{Q(0)}[Q^{Q(0)}(t)]$.
Hence, quantifying the difference between the transient and stationary mean is equivalent to comparing the workload processes of two queues starting in two different (random) states at $t=0$.
We begin our analysis for two queues starting in two \textit{deterministic} states $x,y\geq 0$, respectively. At the end of our analysis we will obtain the original form by replacing $x$ with $Q(0)$ and $y$ with $Q(\iy)$.
Equation \eqref{eq:Qlm} shows that all randomness in the workload process originates from the process $X(t)$.
With this in mind, we couple the processes $Q^x(t)$ and $Q^{y}(t)$ on a sample path level by feeding both queues the same net-input process $X(t)$ for $t\geq 0$.
This allows us to compare the processes in the same probability space, so that $\mathbb{E}[Q^x(t)] - \mathbb{E}[Q^y(t)] = \mathbb{E}[Q^x(t) - Q^y(t)]$ for all $t\geq 0$.
Define
\begin{equation*}
Y^{x,y}(t) := Q^x(t) - Q^y(t)
\end{equation*}
and
\begin{equation*}
\Omega_{T}^{x,y} := \frac{1}{T}\,\int_0^T \mathbb{E}\left[Y^{x,y}(t)\right] \, {\rm d} t.
\end{equation*}
A possible sample path triple for $Q^x(t)$, $Q^0(t)$ and $Y^{x,0}(t)$ is depicted in Figure \ref{fig:samplePaths}.
As we see from this figure, $Y^{x,0}(t)$ has nice structural properties which we will exploit in the next subsection.
\begin{figure}
\centering
\begin{tikzpicture}[y=0.6cm, x=0.01cm]
\draw (0,0) -- coordinate (x axis mid) (800,0);
\draw (0,0) -- coordinate (y axis mid) (0,6.5);
\node[below=0.2cm] at (x axis mid) {$\to t$};
\node[rotate=90, above=0.2cm] at (y axis mid) {$Q(t)$};
\node[above=1.3cm,left =0.08 cm] at (y axis mid) {$x$};
\draw plot
file {Chapter_6/tikz/samplePathLevy.txt};
\draw[color = gray] plot
file {Chapter_6/tikz/samplePathLevy2.txt};
\draw[thick,color=col1] plot
file {Chapter_6/tikz/runningMinimumLevy.txt};
\end{tikzpicture}
\caption{Sample path visualization of the processes $Q^x(t)$ (solid), $Q^0(t)$ (gray) and $Y^{x,0}(t)$ (red).}
\label{fig:samplePaths}
\end{figure}
\subsection{Difference process and leading order behavior of the correction term}
\label{sec:difference}
We further examine the \emph{difference process} $Y^{x,y}(t)$ with $x>y$. Recall from \eqref{eq:Qlm},
\begin{equation}\label{eq:Wz}
Q^z(t) = \max\{ z + X(t),\, \sup_{0<s\leq t} [X(t)-X(s)]\} = X(t) + \max\{ z, -\inf_{0\leq s\leq t} X(s)\},
\end{equation}
for any initial state $z\geq 0$, where $X(t)$ is a L\'evy process with no negative jumps.
Let $\tau^z(w)$, $0\leq w<z$ denote the first passage time of level $w$ by the process starting in $z$, i.e.
\begin{equation*}
\tau^z(w) := \inf \left\{ t \geq 0\, |\, Q^z(t) \leq w \,\right\}.
\end{equation*}
Then it is easily seen that for all $z\geq 0$,
\begin{equation*}
Q^z(t) = \left\{
\begin{array}{ll}
z + X(t), & {\rm if }\ t <\tau^z(0), \\
\sup_{0<s\leq t} [X(t)-X(s)], & {\rm if }\ t \geq \tau^z(0).
\end{array}\right.
\end{equation*}
Consequently,
\begin{equation}\label{eq:Yxy}
Y^{x,y}(t) = \left\{
\begin{array}{ll}
x - y, & \text{if }t < \tau^y(0),\\
\inf_{0<s\leq t} \{ x+X(s)\}, & \text{if }\tau^y(0) \leq t < \tau^x(0),\\
0, & \text{if }t \geq \tau^x(0).
\end{array}\right.
\end{equation}
Using this representation we can identify
\begin{equation*}
\Omega^{x,y}_T = \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^x(0)\wedge T} Y^{x,y}(t) {\rm d} t\right],
\end{equation*}
where $\wedge$ denotes the minimum operator, due to the fact $Y^{x,y}(t) = 0$ for $t\geq \tau^x(0)$.
Subsequently, we decompose $\Omega_T^{x,y}$ into two terms
\begin{equation}
\Psi^{x,y}_T := \frac{1}{T} \int_0^\infty \mathbb{E}[Y^{x,y}(t)]\, {\rm d} t \qquad
\text{and}
\qquad
\Delta_T^{x,y} := \Omega_T^{x,y} - \Psi_T^{x,y}.
\label{eq:Deltaxy}
\end{equation}
Note that $\Psi_T^{x,y}$ is obtained by replacing $T$ by $\infty$ only in the integration bound.
It is customary in the literature, particularly in the area of stochastic simulation, to compare the truncated integral to its natural expansion of the integration range to a semi-infinite interval, see e.g. \cite[Prop.~2.1]{Awad2007}.The truncated integral connects to the long-run average estimator of a certain performance metric, whereas the infinite integral reflects its exact expectation.
The decomposition in \eqref{eq:Deltaxy} is insightful, because $\Psi_T^{x,y}$ prescribes the leading order behavior of $\Omega_T^{x,y}$, while $\Delta_T^{x,y}$ captures the smaller order error term.
In this section, we only consider $\Psi_T^{x,y}$. Subsection \ref{sec:trunc} investigates the magnitude of $\Delta_T^{x,y}$.
The next preliminary result presents a useful property of $\Psi_T^{x,y}$.
\begin{lemma}\label{lemma:psixy}
Let $x>y$. If $\mathbb{E}[\tau^x(0)]<\iy$, then
\begin{equation}\label{eq:H(x,y)}
\Psi^{x,y}_T = \frac{1}{T}\,\mathbb{E}[\tau^{y}(0)](x-y) + \Psi^{x-y,0}_T.
\end{equation}
\end{lemma}
The proof can be found in Appendix \ref{app:psixy}.
This leaves us with two unknowns $\mathbb{E}[\tau^y(0)]$ and $\Psi_T^{x-y,0}$.
The next lemma gives an equivalent form for the latter.
\begin{lemma}\label{lemma:psiz0}
If $\mathbb{E}[\tau^z(0)] < \iy$, then for all $z\geq 0$
\begin{equation}\label{eq:H(x,0)}
\Psi^{z,0}_T = \int_0^z \mathbb{E}[\tau^w(0)]\, {\rm d} w.
\end{equation}
\end{lemma}
The proof can be found in Appendix \ref{app:psiz0}.
Since the term $\mathbb{E}[\tau^z(0)]$, for several values of $z$, appears in many of the preliminary results, we devote our attention to this in the next subsection.\\
\\*
\noindent
\textbf{First passage time}.
When studying the first passage time of level $0\leq w < z$, $\tau^z(w)$, of the workload process starting in $z$, we first observe that $\{\tau^z(z-w)\}_{w=0}^z$ is a spectrally positive L\'evy process itself, also visible through Figure \ref{fig:samplePaths}.
More precisely, it is a subordinator, i.e. a L\'evy process whose paths are almost surely non-decreasing \cite{Kyprianou2006}.
In order to calculate $\mathbb{E}[\tau^z(z-w)]$ we use theory presented in \cite[Section 46]{Sato1999}, although results presented there are valid for spectrally \emph{negative} L\'evy processes, as opposed to the absence of negative jumps in our case.
Nonetheless, our setting is easily transformed into this framework by observing that $\hat{X} \equiv -X$, that is $\hat{X}(t) = -X(t)$ for all $t\geq 0$, is spectrally negative.
Furthermore, let
\begin{equation}
\label{eq:transformedTau}
\hat{\tau}^0(w) := \inf\{ t \geq 0\,:\, \hat{X}(t) \geq w\} = \inf\{ t \geq 0\,:\, z+X(t) \leq z-w\} = \tau^z(z-w).
\end{equation}
For completeness, we cite \cite[Thm.~46.3]{Sato1999}.
\begin{theorem}
Let $\hat{X}(t)$ be a spectrally negative L\'evy process with generating triplet $(-a,\s,\hat{\nu})$ and $\hat{\tau}^0(y)$ its corresponding hitting time process. Define $\Upsilon(\thh)$ for $\thh\geq 0$ as
\begin{equation}\label{eq:thmCharExp}
\Upsilon(\thh) = -a\thh + \tfrac{1}{2}\s^2\thh^2 + \int_{-\infty}^0 ({\rm e}^{\thh x}-1-\thh x{\bf 1}_{[-1,0)}(x))\, \hat{\nu}({\rm d} x).
\end{equation}
Then $\Upsilon(\thh)$ is strictly increasing and continuous, $\Upsilon(0)=0$, and $\Upsilon(\thh)\to\infty$ as $\thh\to\infty$. For $w\geq 0$ and $0\leq u < \infty$ we have
\begin{equation}\label{eq:invCharExp}
\mathbb{E}[\exp(-u\hat{\tau}^0(w))] = \exp(-w\,\Upsilon^{-1}(u)),
\end{equation}
where $\thh=\Upsilon^{-1}(u)$ is the inverse function of $u=\Upsilon(\thh)$.
\end{theorem}
\noindent This immediately induces an expression for $\mathbb{E}[\tau^w(0)]$ and henceforth $\Psi^{z,0}$.
\begin{corollary}\label{cor:Psixy}
Let $X(t)$ be a spectrally positive L\'evy process defined as in \eqref{eq:Xlmprocess} with $\mu > \la$. Let $\Psi^{z,0}_T$ as in \eqref{eq:H(x,0)}. Then
\begin{equation*}
\Psi^{z,0}_T = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
Furthermore, if $x,y\geq 0$, then
\begin{equation}\label{eq:mainResult}
\Psi^{x,y}_T = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation}
\end{corollary}
The proof of Corollary \ref{cor:Psixy} can be found in Appendix \ref{app:Psixy}.
\noindent\textbf{Randomization}.
As we stated before, we easily obtain the original $\Omega_T$ from $\Omega_T^{x,y}$ through substitution of $x$ and $y$ by $Q(0)$ and $Q(\iy)$, respectively, and taking the expectation.
In the previous paragraph, we deduced an explicit expression for $\Psi_T^{x,y}$, the leading order term for $\Omega_T^{x,y}$.
Therefore we equivalently get an approximation for $\Omega_T$, given by
\begin{equation*}
\Psi_T := \frac{1}{T} \int_0^\iy \left( \mathbb{E}[Q(t)]-\mathbb{E}[Q(\iy)] \right)\, {\rm d} t,
\end{equation*}
through randomization of $x$ and $y$ in $\Psi_T^{x,y}$.
By combining the results in Corollary \ref{cor:Psixy}, Lemma \ref{lemma:workloadmoments} and Proposition \ref{prop:truncation_error}, which is given at the end of this section, we directly prove the result in Theorem \ref{thm:mainresult}.
\subsection{Truncation error}\label{sec:trunc}
In order to get a better comprehension of the properties of $\Psi_T$, we depict the value in terms of the (infinite) region between the curves $\mathbb{E}[Q(t)]$, $\mathbb{E}[Q(\iy)]$ and the vertical axis for the case $Q(0)\equiv 0$ in Figure \ref{fig:PsiVisualization}.
In this figure, $\Omega_T$ is given by the area enclosed by the two curves, the vertical axis and the line $t=T$.
One can see that the main contribution to the correction term $\Omega_T$ is given for small $t$.
As $t$ increases, the difference between transient and stationary mean decreases.
Hence for moderate values of $T$, the contribution to the integral in \eqref{eq:Deltaxy} is only minor compared to the contribution over the interval $[0,T]$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.13,yscale=0.3]
\node[below=0.4cm,right=0.5cm] at (x axis mid) {$\to t$};
\draw[dashed, thick, fill =gray!30] (0,0) rectangle coordinate (eq) (50,10);
\node[] at (-7,10) {$\mathbb{E}[Q(\infty)]$};
\node[] at (-3,0) {$0$};
\draw[->] (18,6.4) coordinate (a1) -- (21.65,8.49574);
\node[below] at (a1) {$x=0$};
\foreach \x in {30}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {$T$};
\foreach \y in {10}
\draw (1pt,\y) -- (-20pt,\y);
\draw[thick,color = gray,fill=white] plot
file {Chapter_6/tikz/means0_2.txt};
\draw[thick] (0,0) -- coordinate (x axis mid) (50,0);
\draw[thick] (0,0) -- coordinate (y axis mid) (0,12);
\draw[color=white,very thick] (50,0.05) -- (50,9.56);
\draw[very thick, dotted] (30,0) -- (30,10);
\draw[->] (18,6.8) coordinate (delta) -- (17,8.7);
\node[below] at (delta) {$\Psi_{T}$};
\draw[->] (38,8.1) coordinate (delta) -- (36,9.7);
\node[below] at (delta) {$\Delta_{T}$};
\end{tikzpicture}
\caption{Visualization of $\Omega_T$ and $\Psi_T$ as the area between the curves $\mathbb{E}[Q(t)]$, $\mathbb{E}[Q(\iy)]$ for $Q(0) = 0$.}
\label{fig:PsiVisualization}
\end{figure}
Recall the definition of $\Delta^{x,y}_T$ as in \eqref{eq:Deltaxy}. As we alluded to in Subsection \ref{sec:difference} we claim the contribution of $\Delta^{x,y}_T$ to $\Omega_T^{x,y}$ is negligible compared to $\Psi^{x,y}_T$. Also note that
\begin{equation}
\label{eq:Delta}
\Delta_T := \Omega_T - \Psi_T = {-}\frac{1}{T} \int_T^\iy \big( \mathbb{E}[Q(t)] - \mathbb{E}[Q(\iy)] \big)\,{\rm d} t.
\end{equation}
can be derived through $\Delta^{x,y}_T$ in a similar manner as we did for $\Psi^{x,y}_T$ to obtain $\Psi_T$.
To substantiate our claim, we compute an upper bound for $\Delta^{x,y}_T$ of order $1/T^2$. The existence of such an upper bound poses a limit on the error this tail integral contributed to the cost structure as a whole.
\begin{proposition} \label{prop:truncation_error}
Let $x,y\geq 0$ and $\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3] < \iy$. Then
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right)
\end{equation*}
and
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proposition}
The proof of Proposition \ref{prop:truncation_error} is given in Appendix \ref{sec:proof_truncation}.
\begin{remark}
In case the net-input process $X$ is light-tailed, that is there exists $u>0$ such that $\mathbb{E}[{\rm e}^{u X(1)}] < \iy$, it can be shown that the truncation error is of order ${\rm e}^{-\beta T}/T$ for some $\beta>0$. See Appendix \ref{sec:proof_truncation} for details.
\end{remark}
\section{Optimization}
\label{sec:optimization}
The result in Theorem \ref{thm:mainresult}, characterizing the leading order behavior of $\Omega_T(\mu)$, also reveals the behavior of $\Pi_T(\mu)$ in leading order. Namely,
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \Psi_T(\mu) + O(1/T^2).
\end{equation*}
In fact, this representation naturally gives rise to an \emph{approximation} of the actual cost function:
\begin{align}\label{eq:decomposition}
\hat{\Pi}_{T}(\mu) := \Pi_{\iy}(\mu) + \Psi_T(\mu)
\end{align}
Denote the corresponding minimizer of $\Pih$ by
\begin{equation}\label{eq:muhat}
\hat{\mu}_T^\star := \arg\min_{\mu\geq 0} \Pih(\mu), \qquad \Pih^\star := \Pih(\hat{\mu}_T^\star)
\end{equation}
in addition to the definitions in \eqref{eq:muStar} and \eqref{eq:piStar}.
This section is devoted to the analysis of the minimizers $\muT$, $\muh$ and $\mui$, and the optimality gap for the two approximations.
Throughout this section, we assume that $u_2, u_3 <\iy$ and $\mathbb{E}[Q(0)^2] <\iy$.
By its definition in \eqref{eq:PiInf} and Lemma \ref{lemma:workloadmoments}, we have an exact expression for the steady-state cost function:
\begin{equation*}
\Pi_{\iy}(\mu) = \frac{\la u_2}{2(\mu-\la)} + \aaa\mu.
\end{equation*}
It is easily verified that $\Pi_{\iy}$ is strictly convex in $\mu$, for instance by observing that $\Pi_{\iy}''(\mu) > 0$ for all $\mu > \la$. Therefore $\Pi_{\iy}$ has a unique global minimizer and
\begin{equation}
\label{eq:muInf}
\mui = \la + \sqrt{\frac{\la u_2}{2\aaa}}, \qquad \Pi_{\iy}^\star = \aaa\la + \sqrt{2\aaa\la u_2}.
\end{equation}
We are interested in the relation between $\mui$ and $\muT$, and between $\muh$ and $\muT$.
Since $\Pi_{T}(\mu) = \Pi_{\iy}(\mu) + O(1/T)$ for all $\mu > \la$, we have pointwise convergence of the sequence $\Pi_{T}$, as well as $\hat{\Pi}_{T}$, to $\Pi_{\iy}$ for $T\to\iy$, we also expect $\muT \to \mui$ and $\muh\to\mui$ for $T\to\iy$.
Before proving that this convergence indeed holds, we present a result on the strict convexity of the function $\Pi_{T}$.
\begin{lemma}\label{lemma:strict_convexity}
Let $\mu\geq 0$. The function $\Pi_{T}(\mu)$ is
\begin{itemize}
\item convex in $\mu$, if $Q(0)\equiv x$, $T<x/\mu$ and $\sigma=0$,
\item strictly convex in $\mu$, otherwise.
\end{itemize}
\end{lemma}
Building upon strict convexity of both $\Pi_T(\mu)$ and $\Pi_\iy(\mu)$ for $\mu>\la$, we derive the following convergence result.
\begin{proposition}\label{prop:min_convergence_mu}
Let $\muT$, $\muh$ and $\mui$ be as defined in \eqref{eq:muStar} and \eqref{eq:muhat}. Then
\begin{equation*}
\muT \to \mui\, \qquad \text{\rm and } \qquad \muh \to \mui,
\end{equation*}
for $T\to\infty$.
\end{proposition}
The next result describes a refinement of $\muT$ in terms of $\mui$.
\begin{proposition}\label{prop:muBullet}
For $T$ sufficiently large,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T),
\end{equation*}
where
\begin{equation}\label{eq:muBullet}
\mu_\bullet = \frac{\mathbb{E}[Q(0)^2]}{\sqrt{8\la u_2\aaa}} - \frac{u_3}{3 u_2} - 3\sqrt{\frac{\aaa\la u_2}{8}}.
\end{equation}
\end{proposition}
\noindent
The proofs of the three results above can be found in Appendix \ref{app:proofs_optimization}.
Based on Proposition \ref{prop:muBullet} we propose a \emph{corrected staffing rule}, accounting for the finite horizon
\begin{equation}
\label{eq:correctedMu}
\tilde{\mu}_T^\star = \left[\mui + \frac{\mu_\bullet}{T}\right]^+,
\end{equation}
with $\mu_\bullet$ as in \eqref{eq:muBullet}.
Here $[x]^+ := \max\{x,0\}$, which ensures the value of $\tilde{\mu}_T^\star$ is non-negative and thus is a feasible solution of the optimization problem.
This refined capacity allocation rule is expected to reduce the costs incurred in transient settings.
However, the value of particular interest to us is the cost penalty for using either one of the approximations rather than the actual minimum $\muT$, that is, the \emph{optimality gap}.
As it happens, we deduce the order of the optimality gap for $\mui$ with the help of the explicit form of $\mu_\bullet$ given in \eqref{eq:muBullet}, which is stated in the next proposition. The proof is given in Appendix \ref{sec:proofProp4}.
\begin{proposition}\label{prop:optimalitygap_mui}
Let $\mui$ be as in \eqref{eq:muInf}. Then,
\begin{equation*}
\Pi_\iy^\star- \Pi_T^\star = O(1/T^2).
\end{equation*}
\end{proposition}
\section{Numerical experiments}
\label{sec:numerics}
\subsection{Influence of $\Omega_{T}(\mu)$}
\label{sec:influence_omega}
We first assess the contribution of the correction to the cost function provided by Theorem 1. In other words, we investigate whether $\hat{\Pi}_{T}(\mu)$ as in \eqref{eq:PiT} yields a significantly better fit to $\Pi_{T}(\mu)$, than $\Pi_{\iy}(\mu)$ does.
Note that these three functions only differ in the costs describing the congestion.
Therefore, we limit our study in this subsection to the evaluation of $C_T(\mu)$ as in \eqref{eq:CTmu} with stationary equivalent $C_{\iy}(\mu) = \mathbb{E}[Q_{\mu}(\iy)]$.
Our novel approximation hence reads
\begin{equation*}
\hat{C}_{T}(\mu) := C_{\infty}(\mu) + \Omega_{T}(\mu),
\end{equation*}
with $\Omega_{T}(\mu)$ given in Theorem \ref{thm:mainresult}.
We conduct our numerical experiments based on three models, namely:
\begin{enumerate}
\item \underline{$M/M/1$ queue}: $U(t)$ is a unit rate compound Poisson process with exponentially distributed increments. We have $u_2 = 2$, $u_3=3$, so that
\begin{equation}\label{eq:MM1cor}
\hat{C}_{T}(\mu) = \frac{\la} {\mu-\la} + \frac{1}{T(\mu-\la)} \left(\frac{x^2}{2} - \frac{\la^2}{(\mu-\la)^2} - \frac{\la} {\mu-\la} \right).
\end{equation}
\item \underline{$M/{\rm Pareto}/1$ queue}: $U(t)$ is a unit rate compound Poisson process with Pareto increments. The Pareto distribution deserves special attention due to its heavy-tailed nature, having tail probability $\bar{F}(x) = (x/k)^{-\gamma}$, if $x\geq k$ and 1 otherwise.
It is well-known that heavy-tailed service times lead to long relaxation time. For our purposes, we fix shape parameter $\gamma = 16/5$ and scale parameter $k=11/16$, so that $\beta = 1$, $u_2 = 121/96$, $u_3 = 1331/256$ and $u_k=\iy$ for all $k>3$. Hence,
\begin{equation}
\label{eq:MP1cor}
\hat{C}_{T}(\mu) = \frac{121\la} {192(\mu-\la)} + \frac{1}{2T(\mu-\la)}
\left( x^2 - \frac{(121\la/96)^2}{2(\mu-\la)^2} - \frac{ 1331\la/256 }{2(\mu-\la)}\right)
\end{equation}
\item \underline{Reflected Brownian motion}: $U(t)$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$. We have $u_2 = \sigma^2$, $u_3=0$, so that
\begin{equation}\label{eq:RBMcor}
\hat{C}_{T}(\mu) = \frac{\la\sigma^2}{2(\mu-\la)} + \frac{1}{2T(\mu-\la)} \left( x^2 - \frac{\la^2\sigma^4}{2(\mu-\la)^2}\right).
\end{equation}
\end{enumerate}
In light of the equivalence relations in \eqref{eq:Qidentity} we only consider the case $\la=1$. The cost values for general values of $\la$ follow by appropriate rescaling of $\mu$ and $T$.\\
\\*
\noindent
For the $M/M/1$ and $M/{\rm Pareto}/1$ queue, we obtained the function $C_{T}(\mu)$ through simulation and the results are accurate up until a 95\% confidence interval of width $10^{-3}$. For reflected Brownian motion, we used the explicit distribution function given in \cite{Harrison1985} for double numerical integration. The results for several values of $T$ and two different starting states are depicted in Figures 4-6. These plots also include the approximated functions $\hat{C}_{T}(\mu)$.
\begin{figure}%
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/mm1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/mm1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/mm1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/mm1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/mm1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/mm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/mm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/mm1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/mm1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/mm1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/mm1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/mm1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/mm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/mm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/M/1$ queue with $\la=1$.}
\label{fig:cont}%
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/mp1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/mp1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/mp1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/mp1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/mp1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/mp1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/mp1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/mp1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/mp1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/mp1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/mp1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/mp1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/mp1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/mp1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/$Pareto$/1$ queue with $\la=1$.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/rbm1_0.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/rbm1_0.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/rbm1_0.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/rbm1_0.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/rbm1_0.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/rbm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/rbm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[col1, thick] table[x = mu,y=T2] {Chapter_6/tikz/rbm1_25.txt};
\addplot[col1, dashed, thick] table[x = mu,y=ap2] {Chapter_6/tikz/rbm1_25.txt};
\addplot[col4, thick] table[x = mu,y=T5] {Chapter_6/tikz/rbm1_25.txt};
\addplot[col4, dashed, thick] table[x = mu,y=ap5] {Chapter_6/tikz/rbm1_25.txt};
\addplot[col5, thick] table[x = mu,y=T10] {Chapter_6/tikz/rbm1_25.txt};
\addplot[col5, dashed, thick] table[x = mu,y=ap10] {Chapter_6/tikz/rbm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {Chapter_6/tikz/rbm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for reflected Brownian motion with $\sigma=1$.}
\end{figure}
We name a few observations based on these figures.
First, we indeed note the pointwise convergence of $\hat{C}_{T}(\mu)$ to $\hat{C}_{\iy}(\mu)$ as $T$ grows, for all $\mu$ in all three cases. However, the difference between the stationary costs and those for small values of $T$ can be significant. This is most clear in the plots with $x=2.5$ and when $\mu$ is close to $\la$, i.e. it is in heavy-traffic. In these scenarios, it is evident that refinements to the stationary cost function are needed. $\hat{C}_{T}(\mu)$ does a fairly good job at providing such correction, especially for moderate values of $\mu$.
Furthermore, we note that $C_{T}(\mu)$ approaches $C_{\iy}(\mu)$ from below for $x=0$ for any value of $\mu$, while this is not strictly the case for $x>0$.
$\hat{C}_{T}(\mu)$ correctly captures the sign of this correction.
Finally, observe that $\hat{C}_{T}(\mu)\to -\iy$ as $\mu$ approaches $\la$ from above. This divergence is clear from the expressions in \eqref{eq:MM1cor}-\eqref{eq:RBMcor}.
Our correction term relies on the premise that under the coupling scheme, the sample paths of the two queues starting from different states have hit with high probability.
This is equivalent to stating that the `largest' of the two queues has emptied at least once before time $T$. However, as $\mu$ approaches $\la$, the system enters heavy traffic, and hence the hitting time of the zero barrier is set to run off to infinity.
Consequently, this causes our approximation to be inaccurate for small values of $\mu$.
\subsection{Validation of corrected staffing rule}
\label{sec:num_opt}
In this section, we examine whether the corrected staffing rule $\tilde{\mu}_T^\star$ as in \eqref{eq:correctedMu} indeed yields a significant cost reduction over the choice of $\mui$ by comparing their true costs $\Pi_{T}(\tilde{\mu}_T^\star)$ and $\Pi_{T}(\mui)$.
We conduct this comparison for different values of the parameters, $\aaa$, $T$ and starting state $x$ through numerical experiments.
The three models on which we do our calculations are the $M/M/1$ queue, the $M/$Pareto$/1$ queue and the reflected Brownian motion, as introduced in the previous subsection.
We again focus on $\la=1$ only.
For each of the three models, we adhere to the following set-up. The quality of both staffing rules is assessed for $\aaa = 0.1, 1$ and 2, resembling three modes of valuation of the QoS in the system.
As a benchmark, observe that the expected workload in steady-state conditions with staffing level $\mui$ equals
\begin{equation*}
C_\iy(\mui) = \sqrt{\frac{\aaa\la u_2}{2}}.
\end{equation*}
For each value of $\aaa$, we consider two scenarios: one in which the system starts empty, i.e. $x=0$, and one in which the initial state is double this benchmark value, thus $x=\sqrt{2\aaa\la u_2}$.
The numerics are presented for each model separately. We discuss general conclusions drawn from these results afterwards.\\
\\*
\noindent\textbf{$M/M/1$ queue.}
As we discussed before, if $U$ is a unit rate compound Poisson process with exponentially distributed increments, then $\Qlm$ describes the workload process in an $M/M/1$ queue.
For this setting we get
\begin{equation*}
\mui = \la + \sqrt{\frac{\la} {\aaa}},\qquad \tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la} {\aaa}} + \frac{1}{T}\left( \frac{x^2}{4\sqrt{\la\aaa}} - 1 - \frac{3}{2} \sqrt{\la\aaa}\right)\right]^+.
\end{equation*}
Table \ref{tab:mm1} presents the actual costs corresponding to these two staffing levels for different value of $x$ and $\aaa$.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{\aaa}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 4.162 & 0.620 & 2.688 & 0.536 & 0.136 & 4.162 & 0.682 & 2.688 & 0.536 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 4.162 & 0.669 & 3.425 & 0.641 & 0.041 & 4.162 & 0.700 & 3.425 & 0.641 & 0.085 \\
\multicolumn{1}{|c|}{} & 5 & 4.162 & 0.706 & 3.867 & 0.703 & 0.005 & 4.162 & 0.719 & 3.867 & 0.703 & 0.022 \\
\multicolumn{1}{|c|}{} & 10 & 4.162 & 0.719 & 4.015 & 0.719 & 0.001 & 4.162 & 0.726 & 4.015 & 0.719 & 0.010 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.000 & 2.309 & 0.000 & 0.500 & 0.783 & 2.000 & 3.500 & 0.500 & 2.750 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 2.461 & 0.750 & 1.480 & 0.398 & 2.000 & 3.218 & 1.250 & 3.125 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 2.675 & 1.500 & 2.400 & 0.103 & 2.000 & 3.043 & 1.700 & 2.968 & 0.025 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 2.810 & 1.750 & 2.726 & 0.030 & 2.000 & 3.007 & 1.850 & 2.980 & 0.009 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} &1 & 1.707 & 3.744 & 0.000 & 0.500 & 0.866 & 1.707 & 5.889 & 0.000 & 3.328 & 0.435 \\
\multicolumn{1}{|c|}{} &2 & 1.707 & 3.924 & 0.146 & 1.232 & 0.686 & 1.707 & 5.547 & 0.854 & 4.682 & 0.156 \\
\multicolumn{1}{|c|}{} &5 & 1.707 & 4.209 & 1.083 & 3.343 & 0.206 & 1.707 & 5.114 & 1.366 & 4.910 & 0.040 \\
\multicolumn{1}{|c|}{} &10 & 1.707 & 4.424 & 1.395 & 4.108 & 0.071 & 1.707 & 4.945 & 1.536 & 4.868 & 0.016 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/M/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mm1}
\end{table}
\noindent
\textbf{$M$/Pareto/1 queue.}
In case the service requirements follow a Pareto distribution with shape parameter $\gamma = 16/5$, the staffing rule becomes
\begin{equation*}
\mui = \la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}}, \ \tilde{\mu}_T^\star = \left[\la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}} + \frac{1}{T}\left( \frac{2 x^2}{11\sqrt{\la\aaa/3}} - \frac{11}{8} - \frac{11\sqrt{3\la\aaa}}{16}\right)\right]^+.
\end{equation*}
The numerical results are given in Table \ref{tab:mp1}.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 11/4\cdot \sqrt{\aaa/3}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.510 & 0.524 & 1.759 & 0.461 & 0.120 & 3.510 & 0.573 & 2.010 & 0.562 & 0.019 \\
\multicolumn{1}{|c|}{} & 2 & 3.510 & 0.555 & 2.635 & 0.539 & 0.029 & 3.510 & 0.580 & 2.760 & 0.574 & 0.010 \\
\multicolumn{1}{|c|}{} & 5 & 3.510 & 0.580 & 3.160 & 0.578 & 0.003 & 3.510 & 0.591 & 3.210 & 0.589 & 0.002 \\
\multicolumn{1}{|c|}{} & 10 & 3.510 & 0.590 & 3.335 & 0.590 & 0.000 & 3.510 & 0.596 & 3.360 & 0.595 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.794 & 2.076 & 0.000 & 0.500 & 0.759 & 1.794 & 2.989 & 0.000 & 2.088 & 0.302 \\
\multicolumn{1}{|c|}{} & 2 & 1.794 & 2.190 & 0.511 & 1.291 & 0.411 & 1.794 & 2.790 & 0.610 & 2.588 & 0.072 \\
\multicolumn{1}{|c|}{} & 5 & 1.794 & 2.345 & 1.281 & 2.108 & 0.101 & 1.794 & 2.638 & 1.320 & 2.607 & 0.012 \\
\multicolumn{1}{|c|}{} & 10 & 1.794 & 2.441 & 1.537 & 2.371 & 0.029 & 1.794 & 2.597 & 1.557 & 2.585 & 0.005 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.561 & 3.427 & 0.000 & 0.500 & 0.854 & 1.561 & 5.087 & 0.000 & 2.745 & 0.460 \\
\multicolumn{1}{|c|}{} & 2 & 1.561 & 3.567 & 0.032 & 1.050 & 0.706 & 1.561 & 4.832 & 0.172 & 3.417 & 0.293 \\
\multicolumn{1}{|c|}{} & 5 & 1.561 & 3.779 & 0.950 & 3.012 & 0.203 & 1.561 & 4.499 & 1.006 & 4.313 & 0.041 \\
\multicolumn{1}{|c|}{} & 10 & 1.561 & 3.935 & 1.255 & 3.356 & 0.147 & 1.561 & 4.351 & 1.284 & 4.304 & 0.011 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/{\rm Pareto}/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mp1}
\end{table}
Just as in the results for the $M/M/1$ queue, we observe a higher reduction for larger value of $\aaa$ and $T$. Also, again $\tilde{\mu}_T < \mui$. Hence, the conclusions for the $M/{\rm Pareto}/1$ queue are similar to those of the $M/M/1$ queue. \\
\\*
\noindent\textbf{Reflected Brownian motion}.
In case the input process $U$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$, the steady-state staffing rule and its corrected version reduce to
\begin{equation*}
\mui = \la + \sqrt{\frac{\la\s^2}{2\aaa}}, \qquad
\tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la\s^2}{2\aaa}} + \frac{1}{2\sqrt{2}\,T}\left(\frac{x^2}{\sqrt{\la \aaa}\s} - 3\s\sqrt{\aaa\la} \right)\right]^+.
\end{equation*}
In Tables \ref{tab:rbm1} and \ref{tab:rbm2}, the costs obtained through numerical evaluation are presented for several values of $x$, $T$. We also vary $\s$ to examine the influence of the volatility of arrival process on the quality of the staffing rules.
The observations on the influence of $\aaa, x$ and $T$ are similar to those of the $M/M/1$ queue and the $M/{\rm Pareto}/1$ queue.
However, here we see little improvement by the corrected staffing rule for small values of $\aaa$ for both values of $x$.
The results in Tables \ref{tab:rbm1}-\ref{tab:rbm2} also suggest that the reduction is smaller for larger values of $\s$.
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = \sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.236 & 0.525 & 2.901 & 0.518 & 0.013 & 3.236 & 0.565 & 3.124 & 0.564 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 3.236 & 0.536 & 3.068 & 0.534 & 0.003 & 3.236 & 0.556 & 3.180 & 0.556 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 3.236 & 0.543 & 3.169 & 0.542 & 0.000 & 3.236 & 0.551 & 3.214 & 0.551 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 3.236 & 0.545 & 3.203 & 0.545 & 0.000 & 3.236 & 0.549 & 3.225 & 0.549 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 1$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm1}
\end{table}
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 5.472 & 0.950 & 4.801 & 0.936 & 0.015 & 5.472 & 1.030 & 5.249 & 1.029 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 5.472 & 0.972 & 5.137 & 0.968 & 0.003 & 5.472 & 1.012 & 5.360 & 1.012 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 5.472 & 0.985 & 5.338 & 0.985 & 0.000 & 5.472 & 1.002 & 5.427 & 1.002 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 5.472 & 0.990 & 5.405 & 0.990 & 0.000 & 5.472 & 0.998 & 5.450 & 0.998 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.414 & 3.176 & 0.293 & 1.546 & 0.513 & 2.414 & 4.633 & 1.707 & 4.228 & 0.087 \\
\multicolumn{1}{|c|}{} & 2 & 2.414 & 3.356 & 1.354 & 2.690 & 0.199 & 2.414 & 4.375 & 2.061 & 4.247 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.414 & 3.573 & 1.990 & 3.411 & 0.045 & 2.414 & 4.094 & 2.273 & 4.073 & 0.005 \\
\multicolumn{1}{|c|}{} & 10 & 2.414 & 3.689 & 2.202 & 3.646 & 0.012 & 2.414 & 3.966 & 2.344 & 3.962 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 2.000 & 4.839 & 0.000 & 1.339 & 0.723 & 2.000 & 7.481 & 1.000 & 5.967 & 0.202 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 5.078 & 0.500 & 2.773 & 0.454 & 2.000 & 7.158 & 1.500 & 6.585 & 0.080 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 5.414 & 1.400 & 4.726 & 0.127 & 2.000 & 6.670 & 1.800 & 6.549 & 0.018 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 5.639 & 1.700 & 5.409 & 0.041 & 2.000 & 6.380 & 1.900 & 6.349 & 0.005 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 2$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm2}
\end{table}
\subsection{Discussion}
Based upon these numerical results in Tables \ref{tab:mm1}-\ref{tab:rbm2}, we make a few remarks. The three models roughly exhibit similar behavior as $T$, $x$ and $\aaa$ are varied.
Non-surprisingly, we note that $\tilde{\mu}_T$ approaches $\mui$ with increasing $T$, which also implies that the cost reduction achieved by the corrected staffing rule vanishes as $T\to\iy$.
Also, we observe that in all scenarios examined, the cost reduction increases with $\aaa$. This can be explained through investigation of the objective function $\Pi_T$ as function of $\mu$. Namely, for $\aaa$ small, the curve is relatively flat around the true optimum $\muT$. Hence, in this case a moderate deviation from $\muT$ will likely not lead to a significant cost increase. However, as $\aaa$ becomes larger, i.e. server efficiency is valued more than minimization of congestion, the curve becomes more sharp around $\muT$, and hence more accurate approximations of $\muT$ are required to achieve an acceptable cost level. Hence, the corrected staffing rule \eqref{eq:correctedMu} proves particularly useful in these cases.
Another point we highlight is that the relative improvement is higher for $x=0$ than for $x=\sqrt{2\aaa\la u_2}$. Moreover, even though the initial state of the system is above the optimal equilibrium, $\tilde{\mu}_T$ is smaller than $\mui$. This is somewhat counter-intuitive. In fact, from \eqref{eq:muBullet} it follows that $\mu_\bullet$ positively contributes to the corrected staffing function if
\begin{equation*}
\mathbb{E}[Q^2(0)] > 3\aaa\la u_2 + \frac{2 u_2}{3 u_3}\,\sqrt{2\aaa\la u_2}.
\end{equation*}
\section{Conclusion \& further research}
\label{sec:conclusion_chapter6}
Motivated by the time-varying nature of queues in practical applications, we studied the impact that the transient phase has on traditional capacity allocation questions.
By defining a cost minimization problem, in which the objective function contains a correction accounting for the transient period, we identified the leading and second-order behavior of the cost function as a function of the interval length $T$.
As a by-product, this result yields an approximation for the actual cost function, which is a refinement to its stationary counterpart.
Our numerical experiments in Section \ref{sec:influence_omega} demonstrate the improved accuracy achieved by this approximation in a number of settings.
By perturbation analysis of the optimization problem, this furthermore gives rise to a correction to the steady-state optimal capacity allocation of order $1/T$.
The necessity of the refined capacity allocation level is substantiated by the numerics in Section \ref{sec:num_opt}, which show the cost reduction that can be achieved in a number of settings, compared to settings in which stationary metrics are used.
Especially for small values of $T$ and large values of $\alpha$ this reduction is significant.
Additionally, these results also indicate that it is relatively safe to use the stationary cost when $T$ is moderate, or $\alpha$ is small.
The latter reflects the scenario in which QoS is much more valued than service efficiency.
This observation links to the flat nature of the cost function around its optimal value for $\alpha$ small, a statement on the optimality gap that we formally proved in Proposition \ref{prop:optimalitygap_mui}.
Besides the validation of our theoretical results of Sections \ref{sec:analysis} and \ref{sec:optimization}, the numerical results also reveal some phenomena that require more investigation.
As noted, our corrected capacity allocation level $\tilde{\mu}_T^\star$ is in most studied cases less than the steady-state optimal value $\mu_{\iy}^\star$. This implies that congestion levels tends to be higher under our staffing scheme then under stationary staffing.
A possible explanation for this may be the fact that the planning period under consideration is finite.
Clearly, in the setting we analyzed, anything that happens after time $T$ is neglected.
Therefore, it might be beneficial from the cost perspective to end the period with a higher expected congestion level, as it does not need to be canceled out in the future.
Related to this observation, it would be interesting to look at the setting in which staffing decisions need to be made in consecutive periods of equal length, in which the arrival rate changes at the start of each period.
This case requires careful consideration of the correlation among the staffing decisions within the separate periods.
Another question that arises concerns the translation of our (qualitative) findings to more general queues, in particular the $M/G/s$ queue.
Whereas in our analysis, the central decision variable is the server speed $\mu$, the variable of interest in multi-server queues is typically the number of servers.
It may well be that similar explicit corrections to staffing levels can be deduced to account for transience.
Since our analysis heavily relies on the comparibility of the sample paths of two single-server queues, which is due to the equal negative drift for the two processes, another approach must be taken to tackle this extension.
The analysis and findings for the single-server queue with L\'evy input presented in this chapter may serve a stepping stone for investigation of these more elaborate problems.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Proofs of Section \ref{sec:model_description}}
\label{app:proofs_model}
\subsection{Proof of Lemma \ref{lemma:workloadmoments}}
\label{app:proof_lemma_workload_moments}
\begin{proof}
The conditions of \cite[Cor.IX3.4]{Asmussen2003} are satisfied and therefore $Q_{\mu}(t)\Rightarrow \Qlm(\infty)$ in distribution for $t\rightarrow\infty$. Furthermore, its Laplace transform is for ${\rm Re}(s) < 0$
\[\tilde{Q}_{\mu}(s) = \mathbb{E}\Big[{\rm e}^{s \Qlm(\infty)}\Big] = \frac{s \ka_{\mu}'(0)}{\ka_{\mu}(s)} = \frac{s(\la\ka_U'(0) - \mu)}{\la\ka_U(s) - \mu s} = \frac{s(\mu-\la)}{\mu s-\la \ka_U(s)}.\]
It can be checked that $\ka_U'(0) = \mathbb{E}[U(1)] = 1$, $\ka_U''(0) = u_2$ and $\ka_U'''(0) = u_3$, and $\klm'(0) = \la-\mu$, $\klm''(0) = \la u_2$ and $\klm'''(0) = \la u_3$.
Using l'H\^opital's rule we obtain the first moment of $\Qlm(\infty)$:
\begin{align*}
\mathbb{E}[\Qlm(\iy)] &= \lim_{s\to 0} \frac{{\rm d}}{{\rm d} s} \tilde{Q}_{\mu}(s)
= \lim_{s\to 0} \klm'(0)\, \frac{\klm(s)-s\klm'(s)}{\klm(s)^2}\\
&= \klm'(0)\,\lim_{s\to 0} \frac{{-}s\klm''(s)}{2\klm(s)\klm'(s)}
= \klm'(0)\,\lim_{s\to 0} \frac{ -s\klm'''(s)-\klm''(s)}{2\klm'(s)^2 + 2\klm(s)\klm''(s)} \\
&= {-}\frac{\klm''(0)}{2\klm'(0)} = \frac{\la u_2}{2(\mu-\la)}.
\end{align*}
Similarly, we derive the second moment:
\begin{align*}
\mathbb{E}[\Qlm^2(\iy)] &= \lim_{s\to 0} \frac{{\rm d}^2}{{\rm d} s^2} \tilde{Q}_{\mu}(s)
= \lim_{s\to 0} \klm'(0)\, \frac{2 s \klm'(s)^2-2\klm'(s)\klm(s) - s\klm''(s)\klm(s)}{\klm(s)^3},
\end{align*}
We apply l'H\^opital's rule twice, to find
\begin{align*}
\mathbb{E}[\Qlm^2(\iy)] &=
\klm'(0) \lim_{s\to 0} \frac{ 3s\klm''(s)\klm'(s) - 3 \klm''(s)\klm(s) - s \klm'''(s)\klm(s)}{ 3\klm'(s)\klm(s)^2 }\\
&= \klm'(0) \lim_{s\to 0}
\frac{ 2 s \klm'''(s)\klm'(s) + 3 s\klm''(s)^2 - 4\klm'''(s)\klm(s) - s \klm^{(4)}(s)\klm(s)}
{6\klm'(s)^2\klm(s) + 3\klm''(s)\klm(s)^2}\\
&= \klm'(0) \lim_{s\to 0}
\frac{ s\big[ 2\klm'''(s)\klm'(s) + 3 \klm''(s)^2 - \klm^{(4)}(s)\klm(s)\big] - 4\klm'''(s)\klm(s) }
{\klm(s)\big[6\klm'(s)^2 + 3\klm''(s)\klm(s)\big]}\\
&= \klm'(0) \lim_{s\to 0} \frac{s}{\klm(s)}\,
\frac{ 2\klm'''(s)\klm'(s) + 3 \klm''(s)^2 - \klm^{(4)}(s)\klm(s)}{6\klm'(s)^2 + 3\klm''(s)\klm(s)}\\
&\qquad\qquad -\klm'(0) \lim_{s\to 0} \frac{4\klm'''(s)}{6\klm'(s)^2 + 3\klm''(s)\klm(s)}.
\end{align*}
Since $\klm(0) = 0$ and $\lim_{s\to 0} s/\klm(s) = 1/\klm'(0)$, we have
\begin{align}\label{eq:lemma_eq1}
& \klm'(0) \lim_{s\to 0} \frac{s}{\klm(s)}\,
\frac{ 2\klm'''(s)\klm'(s) + 3 \klm''(s)^2 - \klm^{(4)}(s)\klm(s)}{6\klm'(s)^2 + 3\klm''(s)\klm(s)}
\nonumber\\
&\qquad \qquad \qquad = \frac{ 2\klm'''(0)\klm'(0) + 3 \klm''(0)^2}{6\klm'(0)^2}
= \frac{\klm'''(0)}{3\klm'(0)} + \frac{\klm''(0)^2}{2\klm'(0)^2}
\end{align}
and
\begin{equation}\label{eq:lemma_eq2}
\klm'(0) \lim_{s\to 0} \frac{4\klm'''(s)}{6\klm'(s)^2 + 3\klm''(s)\klm(s)} = \frac{ 2\klm'''(0)}{3\klm'(0)}.
\end{equation}
Combining \eqref{eq:lemma_eq1} and \eqref{eq:lemma_eq2} yields
\[
\mathbb{E}[\Qlm^2(\iy)] = \frac{\klm''(0)^2}{2\klm'(0)^2} - \frac{\klm'''(0)}{3\klm'(0)}
= \frac{ \lambda^2u_2^2}{ 2(\mu-\lambda)^2} + \frac{\lambda u_3}{3(\mu-\lambda)}.
\]
\end{proof}
\subsection{Proof of Proposition \ref{prop:cost_convergence}}
\label{app:proof_prop1}
\begin{proof}
We prove the limit by showing that the difference
\[
\Pi_T(\mu) - \Pi_\iy(\mu) = \frac{1}{T} \int_0^T \Big(\mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)]\Big) \, {\rm d} t
\]
converges to zero as $T\to\iy$ for $\mu>\la$ fixed. The assumption $\mathbb{E}[U(1)], \mathbb{E}[Q(0)] < \iy$ implies by \cite[Prop.~1]{Abate1994} that $\mathbb{E}[Q_\mu(t)]<\iy$ for all $t\geq 0$.
Following \cite{Abate1994}, we use the decomposition
\[
\mathbb{E}[Q_\mu(t)] = \mathbb{E}[Q^0_\mu(t)] + \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\},
\]
where $Q_\mu^0(t)$ represents the workload process if the system starts empty.
From this decomposition it is revealed that $\mathbb{E}[Q^0_\mu(t)]$ and $\left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\}$ are non-negative monotonically increasing and decreasing functions of $t$, respectively, see \cite[Prop.~2,Thm.~11]{Abate1994}.
Recall $\mathbb{E}[Q_\mu(t)] \to \mathbb{E}[Q_\mu(\iy)]$ for $t\to\iy$ by ergodicity of the workload process for any initial state $\mathbb{E}[Q(0)]< \iy$, if $\mu>\la$.
Henceforth,
\begin{align*}
\mathbb{E}[Q_\mu(t)] &\leq \sup_t \mathbb{E}[Q_\mu^0(t)] + \sup_t \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\} \\
&= \mathbb{E}[Q_\mu(\iy)] + \left\{\mathbb{E}[Q_\mu(0)] - \mathbb{E}[Q_\mu^0(0)]\right\} = \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)],
\end{align*}
for all $t\geq 0$, which proves that the expected workload is bounded.
Fix $\varepsilon>0$. By convergence of $\mathbb{E}[Q_\mu(t)]$ for $t\to\iy$, there exists a value $t^* := t^*(\varepsilon)$ such that for all $t\geq t^*$
\begin{equation}
\left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| < \varepsilon/2.
\end{equation}
Next, set
\[
T^* := T^*(\varepsilon) = \frac{2\,t^*(\varepsilon)}{\varepsilon}\, ( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]).
\]
Then for $T\geq \hat{T}:= \max\{ t^*,T^* \}$, we have
\begin{align*}
\left| \frac{1}{T} \int_0^T \big(\mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)]\big) {\rm d} t \right|
&\leq \frac{1}{T} \int_0^{t^*} \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| {\rm d} t \\
& \qquad+ \frac{1}{T} \int_{t^*}^T \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| \, {\rm d} t \\
&\leq \frac{1}{T} \int_0^{t^*} \big(\mathbb{E}[Q_\mu(t)] + \mathbb{E}[Q_\mu(\iy)]\big) {\rm d} t + \frac{1}{T} \int_{t^*}^T \frac{\varepsilon}{2}\, {\rm d} t \\
&< \frac{t^*}{T}\,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{T-t^*}{T} \,\frac{\varepsilon}{2}\\
&< \frac{t^*}{T^*} \,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{\varepsilon}{2} = \varepsilon.
\end{align*}
Hence, for any choice of $\varepsilon>0$ we can find a value $\hat{T}$ such that $\Pi_{\hat{T}}(\mu)$ approaches $\Pi_\iy(\mu)$ within distance $\varepsilon$, which proves the limit.
\end{proof}
\section{Proofs of Section \ref{sec:analysis}}
\label{app:proofs_analysis}
\subsection{Proof of Lemma \ref{lemma:psixy}}\label{app:psixy}
\begin{proof}
Using the representation in \eqref{eq:Yxy} we write
\begin{align*}
\Psi^{x,y}_T &= \frac{1}{T}\int_0^{\infty} \mathbb{E}[Y^{x,y}(t)]{\rm d} t \\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}Y^{x,y}(t)\right] {\rm d} t + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right]
+ \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^\infty Y^{x,y}(t) \, {\rm d} t\right]\\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}(x-y) {\rm d} t\right] + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right] \\
&= \frac{1}{T}\,\mathbb{E}[\tau^y(0)](x-y) + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right].
\end{align*}
By \eqref{eq:Yxy} and the Strong Markov property holding for L\'evy processes \cite{Asmussen2003}, observe that \\* $Y^{x-y,0}(t) {\;\buildrel{d}\over= \;} Y^{x,y}(\tau^y(0)+t)$, whereby
\begin{equation*}
\frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t)\,{\rm d} t\right] = \frac{1}{T}\,\mathbb{E}\left[\int_{0}^{\tau^{x-y}(0)} Y^{x-y,0}(t) {\rm d} t\right] = \Psi^{x-y,0}_T,
\end{equation*}
which completes the proof.
\end{proof}
\subsection{Proof of Lemma \ref{lemma:psiz0}}\label{app:psiz0}
\begin{proof}
Note that $Y^{z,0}(t)$ and $\tau^z(w)$ are intimately related. Namely, due to the fact that $X$ has no negative jumps
\begin{equation*}
\{ \tau^z(w) \leq t\} = \{Y^{z,0}(t) \leq w \}.
\end{equation*}
In fact, $Y^{z,0}(\tau^z(w)) = w$, which implies that $\tau^z$ is a right inverse for $Y^{z,0}(t)$. Therefore, the following equality holds
\begin{equation*}
\int_0^{\tau^z(0)} Y^{z,0}(t)\, {\rm d} t = \int_0^z \tau^z(w)\, {\rm d} w,
\end{equation*}
which implies with the help of Fubini's theorem
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^z(w)]\, {\rm d} w = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^{z-w}(0)]\,{\rm d} w =\frac{1}{T}\, \int_0^z \mathbb{E}[\tau^{w}(0)] \,{\rm d} w.
\end{equation*}
\end{proof}
\subsection{Proof of Corollary \ref{cor:Psixy}}\label{app:Psixy}
\begin{proof}
From \eqref{eq:invCharExp},
\begin{equation}\label{eq:corEq1}
\mathbb{E}[\hat{\tau}^0(w)] = -\tfrac{{\rm d}}{{\rm d} u} \left. \mathbb{E}[\exp(-u\,\hat{\tau}^0(w))]\right|_{u=0} = w\left.\frac{{\rm d}}{{\rm d} u} \Upsilon^{-1}(u)\right|_{u=0}.
\end{equation}
Since $\Upsilon(\theta)$ is strictly increasing and $\Upsilon(0)=0$, we get $\Upsilon^{-1}(0)=0$ and
\begin{equation*}
\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = \frac{1}{\Upsilon'(\Upsilon^{-1}(0))} = \{ \Upsilon'(0) \}^{-1}.
\end{equation*}
Furthermore,
\begin{align*}
\Upsilon'(\thh) &= -a+ \s^2\thh + \int_{-\infty}^0 (x\, {\rm e}^{\thh x} - x{\bf 1}_{[-1,0)}(x)) \hat{\nu}({\rm d} x) \\
&= -a + \s^2\thh - \int_0^\infty (y\, {\rm e}^{-\thh y} - y{\rm 1}_{(0,1]}(y)) \nu({\rm d} y).
\end{align*}
Thus, $\Upsilon'(0) = -\mathbb{E}[X(1)] = \mu-\la$ and $\mathbb{E}[\hat{\tau}^0(w)] = w/(\mu-\la)$. By \eqref{eq:H(x,0)} and \eqref{eq:transformedTau}, we deduce that
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\, \int_0^z \mathbb{E}[\tau^w(0)] \,{\rm d} w = \frac{1}{T}\, \int_0^z \mathbb{E}[\hat{\tau}^0(w)] {\rm d} w = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
For $x>y$, we use Lemma \ref{lemma:psixy} to conclude
\begin{equation*}
\Psi^{x,y}_T = \frac{y(x-y)}{T(\mu-\la)} + \frac{(x-y)^2}{2 T(\mu-\la)} = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation*}
The result for $x<y$ follows directly by the observation $\Psi^{y,x}_T = -\Psi_T^{x,y}$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:truncation_error}}\label{sec:proof_truncation}
\begin{proof}
To derive the upper bound for $\Delta^{x,y}_T$, we apply the same coupling argument as described in Section \ref{sec:analysis}. Let us assume without loss of generality $x>y$.
In this case,
\begin{equation*}
|\Delta^{x,y}_T| = \frac{1}{T} \int_T^\iy \mathbb{E}[Q^x(t)-Q^y(t)] {\rm d} t \leq \frac{1}{T}\int_T^\iy \mathbb{E}[Q^x(t)-Q^0(t)]{\rm d} t.
\end{equation*}
By the decomposition in \eqref{eq:Yxy},
\begin{align}
\int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)] {\rm d} t
&= \int_T^\infty \mathbb{E}[(x+\inf_{s\leq t} X(s))\mathbbm{1}_{\{\tau^x(0)>t\}}] {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( x-u + \inf_{s\leq t}X(s) > 0) {\rm d} u {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( \tau^{x-u}(0) > t ) {\rm d} u{\rm d} t \\
&\leq \int_T^\iy \int_0^x \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} u {\rm d} t \nonumber\\
&= \int_0^x \int_T^\iy \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} t{\rm d} u
= \int_0^x \frac{\mathbb{E}[\tau^{w}(0)^2]}{T}\,{\rm d} w. \nonumber
\label{eq:tailprobIntegral}
\end{align}
We obtain $\mathbb{E}[\tau^w(0)^2]$ with the help of its Laplace transform in \eqref{eq:invCharExp}. Namely,
\begin{align*}
\mathbb{E}[\tau^w(0)^2] &= \left.\tfrac{{\rm d}^2}{{\rm d} u^2}\mathbb{E}[\exp(-u \tau^w(0))]\right|_{u=0} \\
&= w^2\,\left(\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0}\right)^2 - w\left. \tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0}.
\end{align*}
As in the previous subsection we have $\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = (\mu-\la)^{-1}$, and
\begin{equation*}
\left.\tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0} = {-}\frac{\Upsilon''(\Upsilon^{-1}(0))}{\Upsilon'(\Upsilon^{-1}(0))^3} = {-}\frac{\Upsilon''(0)}{\Upsilon'(0)^3}.
\end{equation*}
Since $\Upsilon'(0) = \mu-\la$ and
\begin{equation*}
\Upsilon''(0) = \s^2 + \int_0^\infty x^2\,\nu({\rm d} x) = u_2,
\end{equation*}
we conclude
\begin{equation*}
\mathbb{E}[\tau^w(0)^2] = \frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3},
\end{equation*}
so that
\begin{equation}\label{eq:delta_upper}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\int_0^x \left(\frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3} \right){\rm d} w = \frac{1}{T^2}\left(\frac{x^3}{3(\mu-\la)^2}+\frac{u_2 x^2}{2(\mu-\la)^3}\right).
\end{equation}
For general $x,y\geq 0$,
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right).
\end{equation*}
As a direct consequence,
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proof}
\begin{remark}
Observe that if $X$ is light-tailed, that is $\mathbb{E}[\exp\{ -\theta X(1) \}]$ $= \mathbb{E}[\exp\{\kappa(\theta)\}] < \iy$ for some $\theta<0$, then $\Upsilon(\theta)$ as in \eqref{eq:invCharExp} has an analytic continuation in the negative half-plane, and in this region $\Upsilon(\theta)<0$. Consequently, we can replace the upper bound on the tail probability of $\tau^{x-u}(0)$ by
\begin{equation*}
\mathbb{P}\left( \tau^{x-u}(0) > t\right) = \mathbb{P}\left( {\rm e}^{\beta \tau^{x-u}(0)} > {\rm e}^{\beta t} \right) \leq {\rm e}^{-\beta t} \, {\rm e}^{ (x-u)\Upsilon^{-1}(-\beta)},
\end{equation*}
for some $\beta > 0$, so that
\[ \int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)]\, {\rm d} t \leq {\rm e}^{-\beta T}\, \frac{{\rm e}^{x\Upsilon^{-1}(-\beta)}-1}{\beta\,\Upsilon^{-1}(-\beta)}. \]
Along similar lines we deduce
\[ |\Delta^{x,y}_T| \leq \frac{ {\rm e}^{-\beta T}}{T}\, \frac{{\rm e}^{x\Upsilon^{-1}(-\beta)} + {\rm e}^{y\Upsilon^{-1}(-\beta)} -2}{\beta\,\Upsilon^{-1}(-\beta)}
\]
and
\[ |\Delta_T| \leq \frac{{\rm e}^{-\beta T}}{T}\, \frac{\mathbb{E}[{\rm e}^{Q(0)\Upsilon^{-1}(-\beta)}] + \mathbb{E}[{\rm e}^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}] -2}{\beta\,\Upsilon^{-1}(-\beta)},\]
assuming that $\mathbb{E}[{\rm e}^{-y Q(0)}] < \iy$ for all $y>0$. The condition $\mathbb{E}[{\rm e}^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}]<\iy$ follows from Lemma \ref{lemma:workloadmoments}.
Hence, the error decays exponentially fast for light-tailed input processes.
\end{remark}
\section{Proofs of Section \ref{sec:optimization}}
\label{app:proofs_optimization}
\subsection{Proof of Lemma \ref{lemma:strict_convexity}}
\begin{proof}
Since the term $\aaa\mu$ is convex, the strictness should come from the term $C_T(\mu)$.
Furthermore, observe that if a function $f_\mu(t)$ is convex for all $t\geq 0$, and strictly convex for all $t\geq\e$ for some $\e\in[0,T)$, i.e. for any $\mu_1,\mu_2>0$ and $a\in (0,1)$
\begin{equation*}
a\, f_{\mu_1}(t) + (1-a) f_{\mu_2}(t) > f_{a\mu_1+(1-a)\mu_2}(t),
\end{equation*}
then,
\begin{equation*}
a\int_0^T \, f_{\mu_1}(t)\, {\rm d} t + (1-a)\int_0^T f_{\mu_2}(t) {\rm d} t =
\int_0^T \,\big( a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) \big) {\rm d} t
\end{equation*}
\begin{align*}
&= \int_0^\e \left( a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t)\right) {\rm d} t + \int_\e^T \, \left(a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t)\right) {\rm d} t \\
&> \int_0^\e f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t + \int_\e^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t. \\
&= \int_0^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t.
\end{align*}
Hence, it suffices to prove the convexity of $\mathbb{E}[Q_\mu(t)]$ as a function of $\mu$ for all $t\geq 0$, and strict convexity for $t\geq \e$ for some $\e\in[0,T)$.
Let $\tau^x_{\mu}(0)$ denote the first passage time of level 0 in the process $Q_\mu$ with $Q(0)=x$. Then,
\begin{align}
Q_\mu(t) &= U(t)-\mu t + \max\left\{ x , -\inf_{s\leq t} [ U(s)-\mu s] \right\}\\
&=
\left\{
\begin{array}{ll}
x+U(t)-\mu t, & \text{if } t<\tau^x_{\mu}(0),\\
U(t)-\mu t -\inf_{s\leq t} [ U(s)-\mu s], & \text{if } t\geq \tau^x_{\mu}(0) ,
\end{array}\right.\label{eq:Qrep}
\end{align}
where
\begin{equation*}
\tau^x_{\mu}(0) := \inf\{ t \geq 0\,:\, x+U(t)-\mu t \leq 0\}
\end{equation*}
and $U(t)$ is a spectrally positive L\'evy process.
Fix $\mu_1, \mu_2>0$ and $a\in(0,1)$. Define $\mu_3 := a\mu_1+(1-a)\mu_2$, and
\begin{equation*}
D(t) := a Q_{\mu_1}(t) + (1-a) Q_{\mu_2}(t) - Q_{\mu_3}(t).
\end{equation*}
In order to prove strict convexity we have to show that $D(t) \geq 0$ for all $t\geq 0$, thereby implying $\mathbb{E} [D(t)] \geq 0$, i.e. convexity, for all $t\geq 0$, and $D(t)>0$ with positive probability for $t\in[\e,T]$, for some $\e \in[0,T)$.
We distinguish two cases: $x>0$ and $x=0$. \\
\\*
\textbf{The case $x>0$.}
We start by noticing that if $Q_{\mu_1}$, $Q_{\mu_2}$ and $Q_{\mu_3}$ experience the same input process $U(t)$, then by absence of negative jumps in $U(t)$, it holds that
\begin{equation}\label{eq:stochDom}
\tau^x_{\mu_2}(0) < \tau^x_{\mu_3}(0) < \tau^x_{\mu_1}(0).
\end{equation}
We use shorthand notation
\begin{equation*}
I_k(t) := \inf_{0\leq s\leq t}[U(s)-\mu_k s],
\end{equation*}
for $k=1,2,3$.
Using representation \eqref{eq:Qrep} of the workload process, we obtain
\begin{equation*}
D(t) = \left\{
\begin{array}{ll}
0,
& \text{if } t < \tau^x_{\mu_2}(0),\\
-(1-a)\left(x+I_2(t) \right),
& \text{if } \tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0),\\
a x - (1-a)I_2(t) + I_3(t),
& \text{if } \tau^x_{\mu_3}(0) \leq t < \tau^x_{\mu_1}(0),\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tau^x_{\mu_1}(0).
\end{array}
\right.
\end{equation*}
This partition allows us to spot when strict convexity can occur.
Note that by definition $t \geq \tau^x_{\mu_2}(0)$, $I_2(t) = \inf_{0\leq s\leq t}[U(s)-\mu_2s]\leq -x$, so that $D(t)\geq 0$ if $\tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0)$.
Moreover, by subadditivity of the infimum,
\begin{align*}
I_3(t) &= \inf_{0\leq s\leq t}[U(s)-\mu_3s] = \inf_{0\leq s\leq t}[a(U(s)-\mu_1s)+(1-a)(U(s)-\mu_2s)] \\
&\geq a \inf_{0\leq s\leq t}[U(s)-\mu_1s] + (1-a) \inf_{0\leq s\leq t}[U(s)-\mu_2s] = a I_1(t) + (1-a) I_2(t),
\end{align*}
and hence $D(t)\geq 0$ for $t \geq \tau^x_{\mu_1}(0)$.
Using the same argument, we deduce
\begin{equation*}
ax - (1-a)I_2(t) + I_3(t) \geq a x - (1-a) I_2(t) + a I_1(t) + (1-a) I_2(t) = a(x + I_1(t)).
\end{equation*}
In particular for $t < \tau^x_{\mu_1}(0)$, this value is strictly positive.
As a result, $D(t)\geq 0$ for all $t\geq 0$.
On top of that $D(t) > 0$ for $t\in[\tau^x_{\mu_3}(0),\tau^x_{\mu_1}(0))$.
Accordingly, the latter implies strict positivity of $\mathbb{E} D(t)$, and therefore strict convexity of $\mathbb{E} Q_\mu(t)$, if the event $\{\tau^x_{\mu_3}(0)\leq t< \tau^x_{\mu_1}(0)\}$ occurs with positive probability.
That is,
\begin{align}
P(D(t)>0) &\geq P\left( a(x+I_1(t))\mathbbm{1}_{\{\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\}} > 0 \right)\nonumber\\
&= P\left( x+ I_1(t) > 0 , \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left( x+ I_1(t) > 0 | \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right) = P(\tau^x_{\mu_3}(0)\leq t) - P(\tau^x_{\mu_1}(0) \leq t) > 0, \label{eq:strictConv}
\end{align}
by the stochastic dominance in \eqref{eq:stochDom}. To ensure the strict inequality in \eqref{eq:strictConv} we have to enforce the condition
\begin{equation}\label{eq:condition}
P(\tau^x_{\mu_1}(0)<T) > 0.
\end{equation}
\begin{remark}
An example illustrating the need for this condition is the case in which $U(t)$ is a compound Poisson process and $T < x/\mu_2 < x/\mu_1$. Then
\[Q_{\mu_k}(t) = x + U(t) - \mu_k t,\]
for all $t\in[0,T]$, since $U(t)\geq 0$ and therefore $\tau^x_{\mu_1}(0) > T$. Consequently, for all $a\in(0,1)$,
\[ a\,Q_{\mu_1} + (1-a)\,Q_{\mu_2}(t) = Q_{\mu_3}(t),\]
proving only convexity of $\mathbb{E} Q_{\mu}(t)$ and subsequently convexity of $\int_0^T \mathbb{E}[Q_\mu(t)]\,{\rm d} t$. In case $\sigma>0$, the probability in \eqref{eq:condition} is necessarily positive.
\end{remark}
\noindent \textbf{The case $x=0$.}
By the fact that $\tau_{\mu}(0) = 0$ for all $\mu>0$, proving that $D(t)>0$ in the case $x=0$ reduces to showing that the probability of
\begin{equation*}
D(t) = a I_1(t) + (1-a) I_2(t) - I_3(t)>0
\end{equation*}
happening is positive for all $t>0$. Define
\begin{equation*}
t_0 := \inf\{ t > 0\, :\, U(t) > 0 \},
\end{equation*}
and
\begin{equation*}
\tilde{\tau}_\mu := \inf\{ t > t_0\,: U(t) - \mu t \leq 0\}.
\end{equation*}
We note that $t_0$ as defined above, also defines the epoch of the start of a new excursion of the reflection $Q_\mu$ for all $\mu>0$. Namely,
\[U(s) \leq 0 \quad \Rightarrow\quad U(s) - \mu s \leq -\mu s \qquad \text{for all }0\leq s< t_0\]
\[\Rightarrow \inf_{0\leq s < t_0} [U(s)-\mu s] \leq -\mu t_0 \quad
\Rightarrow U(t_0) - \mu t_0 - \inf_{0\leq s < t_0} [U(s)-\mu s] \geq U(t_0) > 0.\]
Then $Q_\mu(t_0-) = 0$ for all $\mu>0$.
By virtue of the Strong Markov Property, note that $Q_\mu(t_0+t) {\;\buildrel{d}\over= \;} Q_\mu(t)$.
Hence we assume without loss of generality $t_0=0$.
Again, we have a stochastic dominance relation similar to \eqref{eq:stochDom}:
\begin{equation*}
\tilde{\tau}_{\mu_2} < \tilde{\tau}_{\mu_3} < \tilde{\tau}_{\mu_1},
\end{equation*}
for all $\mu_1<\mu_3<\mu_2$.
Then
\begin{equation*}
D(t) {\;\buildrel{d}\over= \;} \left\{
\begin{array}{ll}
0,
& \text{if } t < \tilde{\tau}_{\mu_2},\\
-(1-a)I_2(t),
& \text{if } \tilde{\tau}_{\mu_2} \leq t < \tilde{\tau}_{\mu_3},\\
(1-a)I_2(t) + I_3(t),
& \text{if } \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1},\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tilde{\tau}_{\mu_1}.
\end{array}
\right.
\end{equation*}
Clearly, $D(t)\geq 0$ for all $t\geq 0$ and
\begin{equation*}
-(1-a)I_2(t) + I_3(t) \geq a I_1(t) > 0,
\end{equation*}
for $\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}$.
Hence, in a similar manner to \eqref{eq:strictConv},
\begin{align}
P(D(t)>0) &\geq P\left( aI_1(t)\mathbbm{1}_{\{\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\}} > 0 \right)\nonumber\\
&= P\left( I_1(t) > 0 , \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left( I_1(t) > 0 | \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right) = P(\tilde{\tau}_{\mu_3}\leq t) - P(\tilde{\tau}_{\mu_1} \leq t) > 0. \label{eq:strictConv2}
\end{align}
The last inequality is satisfied it $P(\tilde{\tau}_{\mu_1} < T) >0$, which is equivalent to $P( U(T) - \mu T \leq 0 ) >0$, a condition that is clearly true for all our choices of $U$.
In conclusion, for $x=0$, $\mathbb{E}[D(t)] >0$ and therefore $\mathbb{E}[Q_\mu(t)]$ is a strictly convex function of $\mu$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:min_convergence_mu}}
The proof of the proposition relies on the following auxiliary lemma, of which we include the proof for completeness.
\begin{lemma}\label{lemma:minimizerConvergence}
Consider the sequence of functions $f_n:\, [x_0,\infty) \to \mathbb{R}$ and let $f: [x_0,\infty) \to\mathbb{R}$ be the pointwise limit for some $x_0\in \mathbb{R}$.
Assume $f$ and $f_n$ are strictly convex for all $n$.
Furthermore, let $f(y) \to \infty$ for both $y\to x_0^+$ and $y\to \infty$.
If $x_n$ and $x$ are the minimizers for $f_n$ and $f$, respectively, then $x_n\to x$ for $n\to\infty$.
\end{lemma}
\begin{proof}
We start by showing that the sequence $x_n$ is bounded. Fix $u_l, u_r$ such that $x_0<u_l < x < u_r$. We claim that there exists a $N\in\mathbb{N}$ such that $x_n\in[u_l,u_r]$ for all $n \geq N$. First, we prove the upper bound on $x_n$. For any strictly convex function $h$ with minimizer $x_h$, the following statement holds true:
\begin{equation}\label{eq100}
x_h < u_r \quad \Leftrightarrow \quad h \text{ is strictly increasing at } u_r.
\end{equation}
The first implication follows from observing that $h(x_h) < h(y)$ for all $y> x^*$ and definition of convexity:
\[ 0<\frac{h(u_r)-h(x_h)}{u_r-x_h} \leq \frac{h(u_r+\de)-h(u_r)}{\de}, \]
for all $\de>0$. Hence $h(u_r)<h(u_r+\de)$, i.e. $h$ is increasing at $u_r$. The converse follows immediately by observing that $h(u_r) < h(u_r+\de)$ for all $\de>0$, so that $x_h < u_r$.
Next, we show that $f_n$ must be increasing at $u_r$ for $n$ sufficiently large. By pointwise convergence of $f_n$ we have
\[ \lim_{n\to\infty} [f_n(u_r+\de) - f_n(u_r)] = f(u_r+\de) - f(u_r).\]
Let $w_r:= f(u_r+\de) - f(u_r)>0$. Then
\[ \exists N_r \in \mathbb{N}:\, \forall n\geq N_r:\, |[f_n(u_r+\de) - f_n(u_r)] - [f(u_r+\de)-f(u_r)] | < w_r/2.\]
Hence for $n\geq N_r$,
\[f(u_r+\de)-f(u_r) - w_r/2 < f_n(u_r+\de) - f_n(u_r) < f(u_r+\de)-f(u_r) + w_r/2\]
\[\Rightarrow 0 < w_r/2 < f_n(u_r+\de) - f_n(u_r).
\]
Hence by \eqref{eq100}, $x_n < u_r$ for sufficiently large $n$. Similarly, we argue
\begin{equation*}
x_h > u_l \quad \Leftrightarrow \quad h \text{ is strictly decreasing at } u_l,
\end{equation*}
for any strictly convex function $h$ with minimizer $x_h$. Note that $x_h > u_l$ implies $h(x_h) - h(u_l) < 0$ and for all $\de>0$ we get by strict convexity
\[\frac{h(u_l)-h(u_l-\de)}{\de} < \frac{h(x_h)-h(u_l)}{x_h-u_l} < 0,\]
by which $h(u_l-\de)>h(u_l)$, i.e. $h$ is decreasing in $u_l$. Moreover, if $h$ is decreasing at $u_l$, then it is decreasing for all $y < u_l$, by arguments similar to the above. Therefore, $h(u_l-\de)> h(u_l)$ for all $\de>0$ and it must hold that $x_h>u_l$. Define $f(u_l) - f(u_l-\de) :=w_l < 0$, then again by pointwise convergence, we have that
\[ \exists N_l \in \mathbb{N}:\, \forall n\geq N_l:\, |[f_n(u_l) - f_n(u_l-\de)] - [f(u_l)-f(u_l-\de)] | < w_l,\]
whereupon
\[ f_n(u_l) - f_n(u_l-\de) < f(u_l) - f(u_l-\de) + w_l = 2w_l < 0.\]
Hence, for sufficiently large $n$, we also have $x_n > u_l$. Fix $N = \max\{N_l,N_r\}$, then for $n\geq N$, $x_n\in( u_l,u_r)$. That is, the sequence $x_n$ is bounded. Therefore, by the theorem of Bolzano-Weierstrass, $x_n$ has to have a convergent subsequence. That is, there exists a sequence $n_k$ such that $n_k \to\infty$ and $x_{n_k}\to a$ as $k\to \infty$ for some $a \in [u_l,u_r]$.
We prove that every subsequence must converge to $x$ by contradiction. Suppose there exists a subsequence $n_k$ such that $x_{n_k}\to a\neq x$. Since, $x_n\in [u_l,u_r]$ for $n\geq N$, we may restrict our attention to the sequence of functions $\hat{f}_n:[u_l,u_r] \to \mathbb{R^+}$, consisting of the original function $f_n$ restricted to the domain $[u_l,u_r]$. To be precise $x_n = \arg\min_y f_n(y) = \arg\min_y \hat{f}_n(y)$ for $n\geq N$. Because $\hat{f}_n$ and $\hat{f}$ are bounded, we furthermore have that $\hat{f}_n \to \hat{f}$ uniformly.
Fix $\e>0$. By uniform convergence, there exists an $K_0 \in\mathbb{N}$ such that
\[ | \hat{f}_{n_k}( y ) - \hat{f}( y)| < \e /2,\quad \forall k\geq K_0,\ y \in[u_l,u_r].\]
Also, because $\hat{f}$ is convex, it is continuous, so that there exists a $\de := \de(\e)$ so that
\[ |z-y| < \de \quad \Rightarrow \quad |\hat{f}(z) - \hat{f}(y)| < \e/2.\]
Let $K_1$ be such that $|x_{n_k}-a| < \de$ for all $k\geq K_1$. Then for $k \geq K= \max\{K_0,K_1\}$ this implies
\begin{align*}
|f_{n_k}(x_{n_k}) - f(a)| &= |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(a)| \\
&\leq |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(x_{n_k})| + | \hat{f}(x_{n_k}) - f(a)| < \e/2 + \e/2 = \e.
\end{align*}
Hence we conclude $\lim_{k\to\infty} \hat{f}_{n_k}(x_{n_k}) = f(a)$.
Therefore,
\[ \limsup_{n\to \infty} f_n(x_n) \geq f(a) > f(x),\]
by minimality of $x$. However, $f_n(x_n) \leq f_n(x)$, which implies $\limsup_{n\to\infty} f_n(x_n) \leq \lim_{n\to\infty} f_n(x) = f(x)$, contradicting the strict inequality above. Hence we deduce $x=a$. Consequently, every subsequence of $x_n$ converges to $x$ and therefore $x_n\to x$ as $n\to \infty$.
Applying Lemma \ref{lemma:minimizerConvergence} to the functions $\Pi_T$ and $\Pi_\iy$ with $x_0=\la$, together with Lemma \ref{lemma:strict_convexity}, we obtain the result immediately.
\end{proof}
\subsection{Proof of Proposition \ref{prop:muBullet}}
\begin{proof}
Note that $\Pi_\infty$ is a smooth function.
By the first optimality condition $\Pi_\infty'(\mui)$ $= 0$.
We first prove that also $\Pi_T(\mu)$ is differentiable with respect to $\mu$ for all $\mu\geq 0$.
Recall \eqref{eq:PiT}, which defines the cost function as a combination of the accumulated expected transient queue length, and linear staffing costs.
The latter term is clearly differentiable, hence it remains to be proved that
\begin{equation*}
C_T(\mu) = \frac{1}{T}\int_0^\iy \mathbb{E}[Q_\mu(t)] \, {\rm d} t,
\end{equation*}
admits a derivative for all $\mu\geq 0$ with $T$ fixed.
This holds if and only if $\mathbb{E}[Q_\mu(t)]$ is differentiable for all $t\geq 0$.
Let $Q(0)= x\geq 0$.
Following \eqref{eq:Qlm},
\begin{align*}
\mathbb{E}[Q_\mu(t)] &= \mathbb{E}[X_\mu(t)] + \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big]\\
&= (\la-\mu)t+ \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big],
\end{align*}
where the first term is differentiable.
Furthermore,
\begin{align*}
\mathbb{E}[ \max\{ x, \sup_{s\in[0,t]} \{ - X_\mu(s) \} \} ]
&= x + \int_x^\iy P(\sup_{s\in[0,t]} \{ - X_\mu(s) \} > u ){\rm d} u \\
&= x+\int_x^\infty P(\hat\tau^0(u) \leq t ) {\rm d} u,
\end{align*}
with $\hat{\tau}^0(u)$ as defined in \eqref{eq:transformedTau}.
Since $-X_\mu$ is a process with no positive jumps, we may apply \cite[Cor.~VII.3]{Bertoin1996}, which states that the following equivalence between measures holds:
\begin{equation}
s\,P( \hat\tau^0(u) \in ds ) du = u\,P( -X_\mu(s) \in du ) ds,
\end{equation}
so that
\begin{align}
\int_{u=x}^\infty P(\hat{\tau}^0(u) \leq t )\, {\rm d} u
&=
\int_{u=x}^\iy \int_{s=0}^t P( \hat\tau^0(u) \in ds ) {\rm d} u \nonumber \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,\,P( -X_\mu(s) \in {\rm d} u ) {\rm d} s \nonumber \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,P( X_\mu(s) \in {\rm d} u ) {\rm d} s \nonumber \\
&= \int_{s=0}^t s^{-1} \mathbb{E}[ \max\{x, X_\mu(s)\} ] {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( X_\mu(s)/s > v ) {\rm d} v {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( U(\la s)/s > v + \mu ) {\rm d} v {\rm d} s \nonumber \\
&= \int_{s=0}^t \int_{w=x/s+\mu}^\iy P( U(\la s)/s > w) {\rm d} w {\rm d} s,
\end{align}
where the interchange of integrals is justified by Fubini's theorem and this last form is differentiable with respect to $\mu$.
Substituting $Q(0)$ for $x$ straightforwardly yields differentiability of the complete cost function $\Pi_T$ for all $T$.
Consequently we invoke the first optimality condition for $\muT$ to find
\begin{align*}
0=\Pi_T'(\muT)
&= \Pi_\infty'(\muT) + \Psi_T'(\muT) + O(1/T^2)\\
&= \Pi_\infty'(\mui) + \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi_T'''(\xi)+\Psi_T'''(\xi) \right] + O(1/T^2)\\
&= \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi'''(\xi)+\Psi_T'''(\xi)\right] + O(1/T^2),
\end{align*}
for some $\xi \in [\muT,\mui]$. Rearranging this gives
\begin{align*}
\muT-\mui &= \frac{-\Psi_T'(\mui)}{\Pi_\infty''(\mui)+\Psi_T''(\mui) + \frac{1}{2}(\muT-\mui)(\Pi_\infty'''(\muT)+\Psi_T'''(\xi))} + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} \left[1 - \frac{\Psi_T''(\mui)}{\Pi_\infty''(\mui)} - \frac{\muT-\mu_\infty}{2}\frac{\Pi_\infty'''(\mui)+\Psi_T'''(\mui)}{\Pi_\infty''(\mui)}\right] + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} [1 + o(1)]
\end{align*}
for $T\to\infty$, since both $\mu_T - \mu_\infty$ and $\Psi_T''(\mui)$ are $o(1)$.
Let
\begin{equation*}
\mu_\bullet := \lim_{T\to\iy} \frac{T \Psi_T'(\mui)}{\Pi_\iy''(\mui)}.
\end{equation*}
By \eqref{eq:mainResult} we have
\begin{equation*}
T \Psi'_T(\mu) = {-} \frac{\mathbb{E}[Q(0)^2]}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)^3} + \frac{3\la^2u_2^2}{4(\mu-\la)^4}.
\end{equation*}
Together with
\begin{equation*}
\Pi_\iy''(\mu) = \frac{\la u_2}{(\mu-\la)^3}
\end{equation*}
and \eqref{eq:muInf} we obtain the expression for $\mu_\bullet$ in \eqref{eq:muBullet}.
\end{proof}
\subsection{Proof of Proposition \ref{prop:optimalitygap_mui}}\label{sec:proofProp4}
\begin{proof}
We upper bound the optimality gap by using the decomposition in \eqref{eq:decomposition}.
\begin{align}
|\Pi_\iy^\star - \Pi_T^\star| &= \left|\hat{\Pi}_T(\mu_\infty) + \Delta_T(\mui) - \hat{\Pi}_T(\muT) - \Delta_T(\muT)\right|\nonumber\\
&\leq |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + |\Delta_T(\mui)| + |\Delta_T(\muT)|\nonumber\\
&= |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + O(1/T^2),
\end{align}
since $\Delta_T(\mu) = O(1/T^2)$ by Proposition \ref{prop:truncation_error}. Next, we find an upper bound for $|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)|$, with $\hat{\Pi}_T(\cdot)$ as in \eqref{eq:decomposition}, in terms of the difference between $\gamma$ and $\beta$.
For simplicity, denote $\hat{\gamma} = \gamma - \la$ and $\hat{\beta} = \beta-\la$, implying $\hat{\gamma}-\hat{\beta}=\gamma-\beta$. Then, using \eqref{eq:mainResult}, we get
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &=
\left| \aaa(\hat{\gamma}-\hat{\beta})
+\left(\frac{\la u_2}{2} + \frac{\mathbb{E}[Q(0)^2]}{2T}\right)\left(\frac{1}{\hx}-\frac{1}{\hy}\right) \right. \\
& \qquad \left. -\frac{\la^2 u_2^2}{4T}\left(\frac{1}{\hx^3}-\frac{1}{\hy^3}\right)
-\frac{\la u_3}{6T}\left(\frac{1}{\hx^2} - \frac{1}{\hy^2}\right)
\right|.
\end{align*}
Furthermore, we have
\begin{align*}
\frac 1 \hx - \frac 1 \hy &= -\frac{\hx-\hy}{\hy^2} + \frac{(\hx-\hy)^2}{\hy^3} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^2} - \frac 1 {\hy^2} &= -\frac{2(\hx-\hy)}{\hy^3} + \frac{3(\hx-\hy)^2}{\hy^4} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^3} - \frac 1 {\hy^3} &= -\frac{3(\hx-\hy)}{\hy^4} + \frac{6(\hx-\hy)^2}{\hy^5} + O\left((\gamma-\beta)^3\right).\\
\end{align*}
Substituting these yields
\begin{align*}
|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)| &= \left|(\gamma-\beta)\left[ \aaa - \frac{\la u_2}{2 \hy^2} + \frac{1}{2T\hy^2}\left(\mathbb{E}[Q(0)^2] + \frac{3\la^2 u_2^2}{2\hy^2} + \frac{2\la u_3}{3 \hy}\right)\right]\right. \\
&\qquad \left. - (\gamma-\beta)^2\left[ \frac{\la u_2}{2 \hy^3} + \frac{1}{2T\hy^3}\left(\mathbb{E}[Q(0)^2] - \frac{3\la^2 u_2^2}{\hy^2} - \frac{\la u_3}{\hy}\right)\right]\right| \\
& \qquad \qquad + O\left((\gamma-\beta)^3\right).
\end{align*}
Given that $\muT = \mui + \mu_\bullet/T + o(1/T)$, we find
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\mui-\la )^2}\right) + O(1/T^2)\\
&= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\sqrt{\la u_2/2\aaa})^2}\right) + O(1/T^2) = O(1/T^2),
\end{align*}
which concludes the proof.
\end{proof}
\resettocdepth
\end{subappendices}
\chapter{Transient error approximation in a L\'evy queue}
\begin{chapterstart}
Motivated by a capacity allocation problem within a finite planning period, we conduct a transient analysis of a single-server queue with L\'evy input. From a cost minimization perspective, we investigate the error induced by using stationary congestion measures as opposed to time-dependent measures. Invoking recent results from fluctuation theory of L\'evy processes, we derive a refined cost function, that accounts for transient effects. This leads to a corrected capacity allocation rule for the transient single-server queue. Extensive numerical experiments indicate that the cost reductions achieved by this correction can by significant.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{Transient error approximation in a L\'evy queue}\\
\textit{Britt Mathijsen and Bert Zwart}\\
Submitted to \textit{Queueing Systems}
\end{flushright}
\newpage
\section{Introduction}
The issue of matching a service system's capacity to stochastic demand induced by its clients arises in many practical settings. Typically, the resources available to satisfy demand are scarce and hence expensive. This forces the manager to consider a trade-off between the system efficiency and the quality of service perceived by its clients. In this chapter, we focus on this trade-off in the context of the $M/G/1$ queue, in which the variable amenable for optimization is the server speed $\mu$.
In general, optimizing the server speed $\mu$ in a single-server queue in a time-homogeneous environment, while trading off congestion levels against capacity allocation costs, does not pose any technical challenges. Typically, the objective function to be minimized, the total cost function, has the shape
\begin{equation}\label{eq:intro}
\Pi_\iy(\mu) = \mathbb{E}[Q_\mu(\infty)] + \aaa\mu = \frac{\la\mathbb{E}[B^2] }{2(\mu-\la\mathbb{E}[B])} + \aaa\mu,
\end{equation}
where $\mathbb{E}[Q_\mu(\infty)]$ denotes the expected steady-state amount of work given server speed $\mu$, and $B$ describes the service requirement per arrival. The parameter $\aaa>0$ represents the relative capacity allocation costs incurred by deploying service rate $\mu$. This one-dimensional optimization problem yields the optimizer
\begin{equation*}
\mui = \lambda \mathbb{E}[B] + \sqrt{\frac{\la\mathbb{E}[B^2]}{2\aaa}}.
\end{equation*}
Despite the simplicity and tractability of the problem described above, the presence of the \emph{steady-state} measure in the cost function in \eqref{eq:intro} should be handled carefully. By employing this particular cost structure, one automatically agrees with the underlying assumption of the system being sufficiently close to its steady state.
However, referring the practical applications of the single-server model, system parameters rarely remain constant over time. Moreover, planning periods for the optimization problem are naturally finite. Hence, the \emph{true} expected costs incurred, which we denote by $\Pi_T(\mu)$, in addition depend on the length of the planning period $T$. Consequently, the usage of steady-state models for decision making needs to be justified by a more elaborate time-dependent or \emph{transient} analysis for these type of settings.\\
\\*
\noindent
\textbf{Related literature}.
The time-dependent behavior of the single-server queue received much attention in queueing theory. First efforts to analyze the time-dependent properties of the $M/G/1$ queue date back to the 1950s and 1960s, e.g. \cite{Benes1957,Gaver1959,Kendall1951,Takacs1955,Takacs1962}. The analyses in these papers mostly yield implicit expressions for performance characteristics through Laplace transforms, integro-differential equations and infinite convolutions.
More specifically, there is vast literature on the transient analysis of the $M/M/1$ queue, with the goal to derive explicit expressions for queue length characteristics, see e.g. \cite{Abate1987,Cohen1982,Pegden1982,Prabhu1964}.
These works provide a variety of explicit expressions for the transient dynamics, although the complexity of the resulting expressions, typically involving Bessel functions, expose the intricate intractability of the matter. Consequently, approximation methods for insightful quantification of the dynamics based on numerical \cite{Neuts1966} or asymptotic methods, have become prevalent in more recent literature.
The asymptotic methods either exploit knowledge on the evolution of the queueing process as time $t$ grows large \cite{Abate1987,Newell1982,Odoni1983}, or as the arrival rate $\la$ is increased to infinity \cite{Abate1987a,Abate1987b,Gaver1968}.
It is noteworthy that a substantial contribution to the transient literature is made by Abate and Whitt \cite{Abate1987a,Abate1987b,Abate1987,Abate1994}, who exploit the existence of a decomposition of the mean transient queue length and obtain expressions for the moments of the queue length and virtual waiting through probabilistic arguments in several queueing models.
More recently, asymptotic methods have been used to justify the application of stationary performance measures in Markovian environments or to refine them, see e.g. \cite{Green1991,Whitt1991}.
Other approximative methods known as uniform acceleration expansions \cite{Massey1998} have been developed to reveal the asymptotic behavior of the single-server queue as a function of $t$, which are moreover able to capture time-varying arrival rates.
The majority of the works mentioned above do reflect on the error imposed by usage of steady-state performance metrics instead of the correct time-dependent counterpart. However, no light has been shed on the accumulation of this error over a finite period of time. To the best of our knowledge, the only work that addresses this issue is the paper by Steckley and Henderson \cite{Steckley2007}, who compute an approximation for the error accumulated between the steady-state and transient delay probability. Our analysis on the other hand is centered around the mean workload, which requires a different approach. In addition, the focus in \cite{Steckley2007} is on performance measures only, while the main goal of our work is to investigate the quality of staffing rules. \\
\\*
\noindent\textbf{L\'evy input}.
Although the $M/G/1$ queue serves as the leading example in our analysis, we choose to use a more general framework for the arrival process of the queue. Namely, we let the server face a L\'evy process.
This gives the advantage that once we have obtained the results, we can apply them to broader queue input classes, such as Brownian motion and the Gamma process.
To shed light on the influence of the transience of the queueing process on traditional staffing questions, we will study the capacity allocation problem in the context of cost minimization in which the objective function is $\Pi_T(\mu)$, i.e. a function of both $\mu$ and $T$. We investigate how the invalidity of the stationary assumption is echoed through the operational cost accounting for congestion-related penalties.
Furthermore, we establish a result on the strict convexity of the function $\Pi_T(\mu)$, for almost all values of $T$ (with a few minor exceptions for certain deterministic initial states), which is an essential property for convergence of both cost function and corresponding minimizer to their stationary counterparts.
\\
\\*
\noindent\textbf{Corrected staffing rule}.
As it will appear that an exact analysis of this disparity is intractable, we will present an explicit approximate correction to the conventional stationary objective function given by $\Psi(\mu)/T$ and prove that
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \frac{\Psi(\mu)}{T} + O(1/T^2),
\end{equation*}
with the help of recent results from the fluctuation theory of L\'evy processes.
Based on this refinement we ultimately examine how incorporating transient effects changes the optimal capacity level and propose a refinement to the steady-state capacity allocation rule,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T).
\end{equation*}
We moreover deduce an explicit expression for $\mu_\bullet$ in terms of the initial state and the first three moments of the service requirement per arrival.
It is noteworthy that similar refined square-root staffing rules have been proposed for multi-server queues in the Halfin-Whitt regime, see e.g. \cite{Janssen2015,Janssen2008,Janssen2011,Randhawa2014,Zhang2012}. In those cases, the relevant decision value is the number of servers and refinements are derived for $\la\to\iy$, whereas we consider the regime $T\to\infty$.
Building upon the insights gained through the analysis of this optimality gap, we reflect on the parameter settings of the underlying queueing process in which our refined capacity sizing rule yields significant improvement and in which cases it has little effect. Special emphasis is put on the relationship between the accuracy of the standard procedure and the length of the planning period.
\\
\\*
\noindent\textbf{Structure of the chapter}.
The remainder of this chapter is structured as follows. Section \ref{sec:model_description} is devoted to the model description and presents some preliminary results. The main result will be given in Section \ref{sec:analysis} and results regarding the optimization problem will be discussed in Section \ref{sec:optimization}, followed by the validation of our novel techniques through numerical experiments in Section \ref{sec:numerics}. We will give some concluding remarks and topics for further research in Section \ref{sec:conclusion_chapter6}. We have deferred all proofs to the appendix.
\section{Model description}
\label{sec:model_description}
\subsection{A queueing model with L\'evy input \label{sec:levymodel}}
The model that inspired our study is the standard $M/G/1$ queue starting out of equilibrium. Customers arrive to the queue according to a Poisson process with rate $\la$ and each arrival has iid service requirement $B_i$, stemming from a common random variable $B$.
Without loss of generality we will assume $\mathbb{E}[B] = 1$ throughout. The server is able to remove $\mu$ amounts of work from the system per time unit; a variable we will refer to as the \emph{server speed}.
E.g. if $\mu = 3$ and two customers are in the system with remaining service times $4$ and $2$, then the queue will be empty 2 time units later, provided that no new arrivals occur in the meantime.
Let $N_\la(t)$ denote the number of arrivals until time $t$.
Accordingly, the total work generated by the customers is given by
\begin{equation*}
Z_\la(t) = \sum_{i=1}^{N_\la(t)} B_i.
\end{equation*}
Furthermore, define $X_{\la,\mu}(t) := Z_\la(t) - \mu t$. We call $X_{\la,\mu}$ the \emph{net-input process}.
More generally, we assume throughout the chapter that $X_{\la,\mu}$ is a L\'evy process.
Specifically, we let $Z_\la$ be of the form $Z_\la(t) = U(\la t)$, where $U$ is a spectrally positive L\'evy process generated by the triplet $(a,\s,\nu)$ and $\mathbb{E}[U(1)] = 1$.
This restriction to spectrally positive processes is equivalent to stating $\nu(-\infty,0)=0$ and is a vital assumption to our analysis.
Subsequently, we assume the net-input process $X_{\la,\mu}$ to be
\begin{equation}
\label{eq:Xlmprocess}
X_{\la,\mu}(t) = U(\la t) - \mu t, \qquad t \geq 0.
\end{equation}
Note that by setting $a=\s=0$ and $\nu = \la\, F_B$, where $F_B$ is the cumulative distribution function of $B$, we retrieve the original $M/G/1$ queue.
The stochastic process central to our analysis is the \emph{workload process} $Q_{\la,\mu}(t)$, $t\geq 0$, which describes the amount of work the server is facing at time $t$.
The net-input process $X_{\la,\mu}$ completely determines the trajectory of $Q_{\la,\mu}$, namely
\begin{equation}\label{eq:Qlm}
Q_{\la,\mu}(t) = \max\left\{ Q(0) + X_{\la,\mu}(t), \sup_{s\in[0,t]} [X_{\la,\mu}(t)-X_{\la,\mu}(s)]\right\}, \qquad t\geq 0,
\end{equation}
where $Q(0)$ is the initial workload in the system.
In fact, $Q_{\la,\mu}$ is the reflected version of $X_{\la,\mu}$ with reflection barrier at zero.
Careful inspection of the structure also reveals that $X_{\la,\mu}(t) \equiv X_{\la/\mu,1}(\mu t) \equiv X_{1,\mu/\la}(\la t)$, so that
\begin{equation}
\label{eq:Qidentity}
Q_{\la,\mu}(t) {\;\buildrel{d}\over= \;} Q_{\la/\mu,1}(\mu t) {\;\buildrel{d}\over= \;} Q_{1,\mu/\la}(\la t)
\end{equation}
for all $\la,\mu,t>0$.
This identity will prove to be convenient for the numerical analysis in Section \ref{sec:numerics}. For reasons of clarity, we omit the subscript $\la$ in our expressions if no ambiguity is possible.
The process $Q_{\mu}$ is a natural indicator of the level of congestion in the system and therefore a good choice for quantifying the Quality of Service (QoS) received by a client.
We remark that alternative processes characterizing congestion in the system can be deduced directly from $Q_{\mu}(t)$. For example, consider the virtual waiting time process $V_{\mu}(t)$, which is the waiting time a customer would experience if he arrives at time $t$. This satisfies the relation $V_{\mu}(t) \equiv Q_{\mu}(t)/\mu$ for all $t\geq 0$.
Likewise, the expected number of the customers in the system $L_{\mu}(t)$ at time $t\geq 0$ is given by Little's law
\begin{equation*}
\mathbb{E}[L_{\mu}(t)] = \la\, \mathbb{E}[V_{\mu}(t)] = \frac{\la}{\mu}\, \mathbb{E}[Q_{\mu}(t)].
\end{equation*}
To facilitate our investigation of the queueing model, we end this subsection by introducing some notation regarding the net-input and workload process and by stating a useful preliminary result concerning the stationary process $\Qlm(\iy)$.
Throughout the chapter we assume $\mu>\la$ to ensure ergodicity of the queue and convergence in distribution to the limit
\begin{equation*}
\Qlm(\iy) := \lim_{t\to\iy} \Qlm(t),
\end{equation*}
for any initial state $Q(0)<\iy$. This random variable necessarily coincides with the stationary distribution of $\Qlm(t)$.
By $\ka_U(\cdot)$ and $\ka_{\mu}(\cdot)$ we denote the L\'evy exponents of the processes $U$ and $\Xlm$, respectively:
\begin{equation*}
\ka_{\mu}(\thh) = \log \mathbb{E}[e^{\thh \Xlm(1)}] = \log \mathbb{E}[e^{\thh(U(\la) - \mu)}] = \la \ka_U(\thh) - \mu \thh.
\end{equation*}
Furthermore, define $u_k = \mathbb{E}[\{U(1) - \mathbb{E} U(1)\}^k]$ for $k=2,3,...$.
Using this representation we obtain the following preliminary result.
\begin{lemma}\label{lemma:workloadmoments}
Let $\mathbb{E}|U(1)|<\infty$, $u_2, u_3 < \iy$ and $\mu > \la$. If $Q_{\mu}(\infty)$ represents the steady-state distribution of the workload process, then
\begin{equation*}
\mathbb{E}[\Qlm(\infty)] = \frac{\la u_2}{2(\mu-\la)},\qquad \mathbb{E}[Q_{\mu}^2(\iy)]=\frac{\la^2u_2^2}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)}.
\end{equation*}
\end{lemma}
The proof of Lemma \ref{lemma:workloadmoments} follows directly by differentiation of the Laplace transform of $Q_\mu(\iy)$ and is given in Appendix \ref{app:proof_lemma_workload_moments}.
\subsection{Finite horizon}
For the purpose of our research, we are interested in the dynamics of the workload process within a fixed time frame of length $T>0$.
For all $0\leq t \leq T$, we assume that the parameters of the queue, $\la,\mu,u_2,u_3$, remain unchanged.
If at $t=0$ the queue is not in steady-state corresponding to the specified parameters of the starting period, the process $\{ \Qlm(t)\,:t\in[0,T] \}$ differs from its stationary counterpart $\Qlm(\infty)$.
To illustrate this, Figure \ref{fig:transientmeans} depicts the expected value $\Qlm$ in a $M/M/1$ queue as a function of time for several initial workloads $Q(0)$ for a particular setting of $\la$ and $\mu$.
Clearly, transient behavior of $\mathbb{E}[\Qlm(t)]$, for $Q(0) \neq \Qlm(\iy)$, differs significantly from the steady-state mean with the same system parameters.
Note that even if $Q(0) \equiv \mathbb{E}[\Qlm(\iy)]$, the time-dependent mean does not coincide with the steady-state mean. Moreover, $\mathbb{E}[\Qlm(t)]$ is not even a strictly increasing nor decreasing function of time. This phenomenon is a consequence of the decomposition of the transient mean into one strictly increasing, and a strictly decreasing term for $Q(0)>0$, as discussed in \cite{Abate1987}.
Nonetheless, $\Qlm(t)$ converges in distribution to $\Qlm(\infty)$ as $t\to\iy$, if $\mu>\la$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.15,yscale=0.225]
\draw (0,0) -- coordinate (x axis mid) (50,0);
\draw (0,0) -- coordinate (y axis mid) (0,21);
\node[right] at (51,0) {$t$};
\node[rotate=90, above=0.7 cm] at (y axis mid) {$\mathbb{E}[\Qlm(t)]$};
\draw[dashed, thick, gray] (0,10) -- coordinate (eq) (51,10);
\definecolor{col1}{rgb}{0.368417, 0.506779, 0.709798}
\definecolor{col2}{rgb}{0.880722, 0.611041, 0.142051}
\definecolor{col3}{rgb}{0.560181, 0.691569, 0.194885}
\definecolor{col4}{rgb}{0.922526, 0.385626, 0.209179}
\draw[->] (24,6.4) -- coordinate (a1) (21.65,8.49574);
\node[right=0.6cm,below=0.3cm] at (a1) {$Q(0)\equiv 0$};
\draw[->] (14,6.2) -- coordinate (a2) (13.,8.49434);
\node[right=0.2cm,below=0.3cm] at (a2) {$Q(0)\equiv10$};
\draw[->] (9,15.5) -- coordinate (a3) (7.5,13.7648);
\node[right=0.6cm,above=0.1cm] at (a3) {$Q(0)\equiv20$};
\draw[->] (40,12.5) -- coordinate (a4) (38,10.9712);
\node[right,above=0.3cm] at (a4) { $Q(0)\sim \exp\left(\tfrac{1}{15}\right)$ };
\foreach \x in {0,10,...,50}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {\x};
\foreach \y in {5,10,15,20}
\draw (1pt,\y) -- (-20pt,\y)
node[anchor=east] {\y};
\draw[thick,color = blue] plot
file {tikz/means0.txt};
\draw[thick,color = red] plot
file {tikz/means10.txt};
\draw[thick,color = green] plot
file {tikz/means20.txt};
\draw[thick,color = yellow] plot
file {tikz/meansExp.txt};
\end{tikzpicture}
\caption{Time-dependent mean workload in a $M/M/1$ queue with $\la = 10$ and $\mu=11$ for different initial states $Q(0)$. The dashed line depicts $\mathbb{E}\Qlm(\iy)$.}
\label{fig:transientmeans}
\end{figure}
Since the time horizon of our analysis is limited to $t\leq T$, the process may not approach the steady-state distribution sufficiently close to appropriately use its steady-state properties for capacity allocation.
To overcome this disparity, we propose a way to include the influence of this transient phase in the capacity allocation problem.
\subsection{Cost structure}
As mentioned before, we are interested in balancing the QoS and efficiency of the queue by choosing the optimal server speed $\mu$.
The adjective \emph{optimal} indicates that we intend to choose the speed according to some objective function.
In our case, we conduct our analysis based on a cost function, which consists of a part accounting for the penalty for congestion in the system and a part for staffing cost. The cost value of both parts is governed by the variable $\mu$.
The instantaneous cost incurred at time $t$ equals
\begin{equation*}
\mathbb{E}[\Qlm(t)] + \aaa \mu,
\end{equation*}
where $\aaa$ is a positive constant defining the \emph{relative staffing cost}.
Hence, the cost structure we apply is a combination of the transient mean of the workload process and a linear staffing cost.
Accumulated and normalized over the period $[0,T]$, the cost function on which the rest of this chapter will be based equals
\begin{equation}\label{eq:PiT}
\Pi_{T}(\mu) := \frac{1}{T}\int_0^T\left( \mathbb{E}[\Qlm(t)] + \aaa\mu\, \right) {\rm d} t
= \frac{1}{T} \int_0^T \mathbb{E}[\Qlm(t)] {\rm d} t + \aaa\mu.
\end{equation}
We use shorthand notation for the normalized congestion costs:
\begin{equation}\label{eq:CTmu}
C_{T}(\mu) := \frac{1}{T}\int_0^T \mathbb{E}[Q_{\mu}(t)] {\rm d} t,
\end{equation}
and $C_{\iy}(\mu) := \mathbb{E}[\Qlm(\iy)]$.
In order to compare the actual costs incurred over the interval $[0,T]$ to the cost function of the queue in stationary conditions, we define
\begin{equation}\label{eq:PiInf}
\Pi_{\iy}(\mu) := C_{\iy}(\mu) + \aaa \mu = \mathbb{E}[Q_\mu(\iy)] + \aaa\mu,
\end{equation}
which allows an explicit expression by Lemma \ref{lemma:workloadmoments}.
Under mild conditions on the net-input process and the distribution of the initial state, the cost functions coincide for $T\to\iy$.
\begin{proposition}\label{prop:cost_convergence}
Let $\mu>\la$ and assume $\mathbb{E}[U(1)],\, \mathbb{E}[Q(0)] < \iy$. Then
\begin{equation*}
\lim_{T\to\iy} \Pi_{T}(\mu) = \Pi_{\iy}(\mu).
\end{equation*}
\end{proposition}
\noindent Rewriting \eqref{eq:PiT} gives the relation
\begin{align}
\Pi_{T}(\mu) &= \frac{1}{T}\int_0^{T} \left( \mathbb{E}[\Qlm(t)] - \mathbb{E}[\Qlm(\iy)] \right) {\rm d} t + \mathbb{E}[\Qlm(\iy)] + \aaa\mu = \Omega_{T}(\mu) + \Pi_{\infty}(\mu).
\label{eq:decomp}
\end{align}
Section \ref{sec:analysis} is concerned with the analysis of the correction factor $\Omega_{T}(\mu)$.
Ultimately, we are concerned with the additional costs incurred by choosing the server speed through minimization of $\Pi_{\iy}(\mu)$ instead of $\Pi_{T}(\mu)$.
Therefore, we formulate the exact and approximate optimization problems as follows
\begin{equation}\label{eq:muStar}
\mu_T^\star := \arg\min_{\mu\geq 0} \Pi_{T}(\mu), \qquad \qquad \mu_\infty^\star := \arg\min_{\mu\geq 0} \Pi_{\iy}(\mu),
\end{equation}
\begin{equation}\label{eq:piStar}
\Pi_{T}^\star := \Pi_{T}(\mu_T^\star), \qquad \qquad \Pi_{\iy}^\star := \Pi_{T}(\mu_\iy^\star).
\end{equation}
In Section \ref{sec:optimization} we turn to the comparison of $\mu_T^{\star}$ and $\mu_\iy^\star$ as well as the \emph{optimality gap} $\Pi_{\iy}^\star - \Pi_{T}^\star$.
\section{Analysis of the objective function}
\label{sec:analysis}
From \eqref{eq:decomp} it is evident that, for finding an explicit characterization of $\Pi_{T}(\mu)$, it suffices to study the term $\Omega_T(\mu)$ in more detail. We start by stating the main result of this section, which describes the leading order behavior of $\Omega_T(\mu)$ as $T$ increases.
\begin{theorem}\label{thm:mainresult}
Let $X_\mu(t)$ be of the form \eqref{eq:Xlmprocess}. If $\mathbb{E}[\max(Q(0),Q_\mu(\infty))^3] < \iy$ and $u_2,u_3 < \iy$, then
\begin{align*}
\Omega_T(\mu) &= \frac{\mathbb{E}[Q(0)^2] - \mathbb{E}[Q_\mu(\iy)^2]}{2T(\mu-\la)} + O\left(\frac{1}{T^2}\right) \nonumber\\
&= \frac{1}{2T(\mu-\la)}\left( \mathbb{E}[Q(0)^2] - \frac{\la^2 u_2^2}{2(\mu-\la)^2} - \frac{\la u_3}{3(\mu-\la)}\right) + O\left(\frac{1}{T^2}\right),
\end{align*}
for $\mu>\la$.
\end{theorem}
Note that this expression provides an \emph{approximation} of the actual cost function
$\Pi_T(\mu)$. We elaborate on the implications of this additional information on the optimization problem in Section \ref{sec:optimization}.
In the remainder of this section we provide a detailed description of the steps taken to obtain this outcome.
We assume a fixed service rate $\mu$ throughout the analysis in this section and therefore omit the subscript $\mu$. Proofs of the intermediate results can be found in Appendix \ref{app:proofs_analysis}.
\subsection{Constructing a coupling}
Before starting our analysis of the correction term $\Omega_{T}(\mu)$ we introduce some auxiliary notation.
By $Q^A(t)$ we denote the workload process as described in Subsection \ref{sec:levymodel} with initial state $A$ and $\mathbb{E}_A$ the expectation with respect to any non-negative random variable $A$, which is independent of the net-input process $X$.
To be able to compare $\mathbb{E}[Q(t)]$ and $\mathbb{E}[Q(\iy)]$ as in $\Omega_T(\mu)$, we will use a coupling technique.
Observe that by definition of the stationary distribution $Q(\iy) {\;\buildrel{d}\over= \;} Q^{Q(\iy)}(t)$ for all $t \geq 0$ and therefore $\mathbb{E}[Q(\iy)] = \mathbb{E}_{Q(\iy)}[Q^{Q(\iy)}(t)]$. Furthermore, $\mathbb{E}[Q(t)] = \mathbb{E}_{Q(0)}[Q^{Q(0)}(t)]$.
Hence, quantifying the difference between the transient and stationary mean is equivalent to comparing the workload processes of two queues starting in two different (random) states at $t=0$.
We starting our analysis for two queues starting in two \textit{deterministic} states $x,y\geq 0$, respectively. At the end of our analysis we will obtain the original form by replacing $x$ with $Q(0)$ and $y$ with $Q(\iy)$.
Equation \eqref{eq:Qlm} shows that all randomness in the workload process originates from the process $X(t)$.
With this in mind, we couple the processes $Q^x(t)$ and $Q^{y}(t)$ on a sample path level by feeding both queues the same net-input process $X(t)$ for $t\geq 0$.
This allows us to compare the processes in the same probability space, so that $\mathbb{E}[Q^x(t)] - \mathbb{E}[Q^y(t)] = \mathbb{E}[Q^x(t) - Q^y(t)]$ for all $t\geq 0$.
Define
\begin{equation*}
Y^{x,y}(t) := Q^x(t) - Q^y(t)
\end{equation*}
and
\begin{equation*}
\Omega_{T}^{x,y} := \frac{1}{T}\,\int_0^T \mathbb{E}\left[Y^{x,y}(t)\right] \, {\rm d} t.
\end{equation*}
A possible sample path triple for $Q^x(t)$, $Q^0(t)$ and $Y^{x,0}(t)$ is depicted in Figure \ref{fig:samplePaths}.
As we see from this figure, $Y^{x,0}(t)$ has nice structural properties which we will exploit in the next subsection.
\begin{figure}
\centering
\begin{tikzpicture}[y=0.6cm, x=0.01cm]
\draw (0,0) -- coordinate (x axis mid) (800,0);
\draw (0,0) -- coordinate (y axis mid) (0,6.5);
\node[below=0.2cm] at (x axis mid) {$\to t$};
\node[rotate=90, above=0.2cm] at (y axis mid) {$Q(t)$};
\node[above=1.3cm,left =0.08 cm] at (y axis mid) {$x$};
\draw plot
file {tikz/samplePathLevy.txt};
\draw[color = gray] plot
file {tikz/samplePathLevy2.txt};
\draw[thick,color=red] plot
file {tikz/runningMinimumLevy.txt};
\end{tikzpicture}
\caption{Sample path visualization of the processes $Q^x(t)$ (solid), $Q^0(t)$ (gray) and $Y^{x,0}(t)$ (red).}
\label{fig:samplePaths}
\end{figure}
\subsection{Difference process and leading order behavior of the correction term}
We further examine the \emph{difference process} $Y^{x,y}(t)$ with $x>y$. Recall from \eqref{eq:Qlm},
\begin{equation}\label{eq:Wz}
Q^z(t) = \max\{ z + X(t),\, \sup_{0<s\leq t} [X(t)-X(s)]\} = X(t) + \max\{ z, -\inf_{0\leq s\leq t} X(s)\},
\end{equation}
for any initial state $z\geq 0$, where $X(t)$ is a L\'evy process with no negative jumps.
Let $\tau^z(w)$, $0\leq w<z$ denote the first passage time of level $w$ by the process starting in $z$, i.e.
\begin{equation*}
\tau^z(w) := \inf \left\{ t \geq 0\, |\, Q^z(t) \leq w \,\right\}.
\end{equation*}
Then it is easily seen that for all $z\geq 0$,
\begin{equation*}
Q^z(t) = \left\{
\begin{array}{ll}
z + X(t), & {\rm if }\ t <\tau^z(0), \\
\sup_{0<s\leq t} [X(t)-X(s)], & {\rm if }\ t \geq \tau^z(0).
\end{array}\right.
\end{equation*}
Consequently,
\begin{equation}\label{eq:Yxy}
Y^{x,y}(t) = \left\{
\begin{array}{ll}
x - y, & \text{if }t < \tau^y(0),\\
\inf_{0<s\leq t} \{ x+X(s)\}, & \text{if }\tau^y(0) \leq t < \tau^x(0),\\
0, & \text{if }t \geq \tau^x(0).
\end{array}\right.
\end{equation}
Using this representation we can identify
\begin{equation*}
\Omega^{x,y}_T = \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^x(0)\wedge T} Y^{x,y}(t) {\rm d} t\right],
\end{equation*}
where $\wedge$ denotes the minimum operator, due to the fact $Y^{x,y}(t) = 0$ for $t\geq \tau^x(0)$.
Subsequently, we decompose $\Omega_T^{x,y}$ into two terms
\begin{equation}
\Psi^{x,y}_T := \frac{1}{T} \int_0^\infty \mathbb{E}[Y^{x,y}(t)]\, {\rm d} t \qquad
\text{and}
\qquad
\Delta_T^{x,y} := \Omega_T^{x,y} - \Psi_T^{x,y}.
\label{eq:Deltaxy}
\end{equation}
Note that $\Psi_T^{x,y}$ is obtained by replacing $T$ by $\infty$ only in the integration bound.
It is customary in the literature, particularly in the area of stochastic simulation, to compare the truncated integral to its natural expansion of the integration range to a semi-infinite interval, see e.g. \cite[Prop.~2.1]{Awad2007}.The truncated integral connects to the long-run average estimator of a certain performance metric, whereas the infinite integral reflects its exact expectation.
The decomposition in \eqref{eq:Deltaxy} is insightful, because $\Psi_T^{x,y}$ prescribes the leading order behavior of $\Omega_T^{x,y}$, while $\Delta_T^{x,y}$ captures the smaller order error term.
In this section, we only consider $\Psi_T^{x,y}$. Subsection \ref{sec:trunc} investigates the magnitude of $\Delta_T^{x,y}$.
The next preliminary result presents a useful property of $\Psi_T^{x,y}$.
\begin{lemma}\label{lemma:psixy}
Let $x>y$. If $\mathbb{E}[\tau^x(0)]<\iy$, then
\begin{equation}\label{eq:H(x,y)}
\Psi^{x,y}_T = \frac{1}{T}\,\mathbb{E}[\tau^{y}(0)](x-y) + \Psi^{x-y,0}_T.
\end{equation}
\end{lemma}
This leaves us with two unknowns $\mathbb{E}[\tau^y(0)]$ and $\Psi_T^{x-y,0}$.
The next lemma gives an equivalent form for the latter.
\begin{lemma}\label{lemma:psiz0}
If $\mathbb{E}[\tau^z(0)] < \iy$, then for all $z\geq 0$
\begin{equation}\label{eq:H(x,0)}
\Psi^{z,0}_T = \int_0^z \mathbb{E}[\tau^w(0)]\, {\rm d} w.
\end{equation}
\end{lemma}
Since the term $\mathbb{E}[\tau^z(0)]$, for several values of $z$, appears in many of the preliminary results, we devote our attention to this in the next subsection.\\
\\*
\noindent
\textbf{First passage time}.
When studying the first passage time of level $0\leq w < z$, $\tau^z(w)$, of the workload process starting in $z$, we first observe that $\{\tau^z(z-w)\}_{w=0}^z$ is a spectrally positive L\'evy process itself, also visible through Figure \ref{fig:samplePaths}.
More precisely, it is a subordinator, i.e. a L\'evy process whose paths are almost surely non-decreasing \cite{Kyprianou2006}.
In order to calculate $\mathbb{E}[\tau^z(z-w)]$ we use theory presented in \cite[Section 46]{Sato1999}, although results presented there are valid for spectrally \emph{negative} L\'evy processes, as opposed to the absence of negative jumps in our case.
Nonetheless, our setting is easily transformed into this framework by observing that $\hat{X} \equiv -X$, that is $\hat{X}(t) = -X(t)$ for all $t\geq 0$, is spectrally negative.
Furthermore, let
\begin{equation}
\label{eq:transformedTau}
\hat{\tau}^0(w) := \inf\{ t \geq 0\,:\, \hat{X}(t) \geq w\} = \inf\{ t \geq 0\,:\, z+X(t) \leq z-w\} = \tau^z(z-w).
\end{equation}
For completeness, we cite \cite[Thm.~46.3]{Sato1999}.
\begin{theorem}
Let $\hat{X}(t)$ be a spectrally negative L\'evy process with generating triplet $(-a,\s,\hat{\nu})$ and $\hat{\tau}^0(y)$ its corresponding hitting time process. Define $\Upsilon(\thh)$ for $\thh\geq 0$ as
\begin{equation}\label{eq:thmCharExp}
\Upsilon(\thh) = -a\thh + \tfrac{1}{2}\s^2\thh^2 + \int_{-\infty}^0 (e^{\thh x}-1-\thh x{\bf 1}_{[-1,0)}(x))\, \hat{\nu}({\rm d} x).
\end{equation}
Then $\Upsilon(\thh)$ is strictly increasing and continuous, $\Upsilon(0)=0$, and $\Upsilon(\thh)\to\infty$ as $\thh\to\infty$. For $w\geq 0$ and $0\leq u < \infty$ we have
\begin{equation}\label{eq:invCharExp}
\mathbb{E}[\exp(-u\hat{\tau}^0(w))] = \exp(-w\,\Upsilon^{-1}(u)),
\end{equation}
where $\thh=\Upsilon^{-1}(u)$ is the inverse function of $u=\Upsilon(\thh)$.
\end{theorem}
\noindent This immediately induces an expression for $\mathbb{E}[\tau^w(0)]$ and henceforth $\Psi^{z,0}$.
\begin{corollary}\label{cor:Psixy}
Let $X(t)$ be a spectrally positive L\'evy process defined as in \eqref{eq:Xlmprocess} with $\mu > \la$. Let $\Psi^{z,0}_T$ as in \eqref{eq:H(x,0)}. Then
\begin{equation*}
\Psi^{z,0}_T = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
Furthermore, if $x,y\geq 0$, then
\begin{equation}\label{eq:mainResult}
\Psi^{x,y}_T = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation}
\end{corollary}
\noindent\textbf{Randomization}.
As we stated before, we easily obtain the original $\Omega_T$ from $\Omega_T^{x,y}$ through substitution of $x$ and $y$ by $Q(0)$ and $Q(\iy)$, respectively, and taking the expectation.
In the previous paragraph, we deduced an explicit expression for $\Psi_T^{x,y}$, the leading order term for $\Omega_T^{x,y}$.
Therefore we equivalently get an approximation for $\Omega_T$, given by
\begin{equation*}
\Psi_T := \frac{1}{T} \int_0^\iy \left( \mathbb{E}[Q(t)]-\mathbb{E}[Q(\iy)] \right)\, {\rm d} t,
\end{equation*}
through randomization of $x$ and $y$ in $\Psi_T^{x,y}$.
By combining the results in Corollary \ref{cor:Psixy}, Lemma \ref{lemma:workloadmoments} and Proposition \ref{prop:truncation_error}, which is given at the end of this section, we directly prove the result in Theorem \ref{thm:mainresult}.
\subsection{Truncation error}\label{sec:trunc}
In order to get a better comprehension of the properties of $\Psi_T$, we depict the value in terms of the (infinite) region between the curves $\mathbb{E}[Q(t)]$, $\mathbb{E}[Q(\iy)]$ and the vertical axis for the case $Q(0)\equiv 0$ in Figure \ref{fig:PsiVisualization}.
In this figure, $\Omega_T$ is given by the area enclosed by the two curves, the vertical axis and the line $t=T$.
One can see that the main contribution to the correction term $\Omega_T$ is given for small $t$.
As $t$ increases, the difference between transient and stationary mean decreases.
Hence for moderate values of $T$, the contribution to the integral in \eqref{eq:Deltaxy} is only minor compared to the contribution over the interval $[0,T]$.
\begin{figure}
\centering
\begin{tikzpicture}[xscale=0.13,yscale=0.3]
\node[below=0.4cm,right=0.5cm] at (x axis mid) {$\to t$};
\draw[dashed, thick, fill =gray!30] (0,0) rectangle coordinate (eq) (50,10);
\node[] at (-7,10) {$\mathbb{E}[Q(\infty)]$};
\node[] at (-3,0) {$0$};
\draw[->] (18,6.4) coordinate (a1) -- (21.65,8.49574);
\node[below] at (a1) {$x=0$};
\foreach \x in {30}
\draw (\x,1pt) -- (\x,-10pt)
node[anchor=north] {$T$};
\foreach \y in {10}
\draw (1pt,\y) -- (-20pt,\y);
\draw[thick,color = gray,fill=white] plot
file {tikz/means0_2.txt};
\draw[thick] (0,0) -- coordinate (x axis mid) (50,0);
\draw[thick] (0,0) -- coordinate (y axis mid) (0,12);
\draw[color=white,very thick] (50,0.05) -- (50,9.56);
\draw[very thick, dotted] (30,0) -- (30,10);
\draw[->] (18,6.8) coordinate (delta) -- (17,8.7);
\node[below] at (delta) {$\Psi_{T}$};
\draw[->] (38,8.1) coordinate (delta) -- (36,9.7);
\node[below] at (delta) {$\Delta_{T}$};
\end{tikzpicture}
\caption{Visualization of $\Omega_T$ and $\Psi_T$ as the area between the curves $E[Q(t)]$, $\mathbb{E}[Q(\iy)]$ for $Q(0) = 0$.}
\label{fig:PsiVisualization}
\end{figure}
Recall the definition of $\Delta^{x,y}_T$ as in \eqref{eq:Deltaxy}. As we eluded to in Subsection 3.2
we claim the contribution of $\Delta^{x,y}_T$ to $\Omega_T^{x,y}$ is negligible compared to $\Psi^{x,y}_T$. Also note that
\begin{equation}
\label{eq:Delta}
\Delta_T := \Omega_T - \Psi_T = {-}\frac{1}{T} \int_T^\iy \mathbb{E}[Q(t)] - \mathbb{E}[Q(\iy)]\,{\rm d} t.
\end{equation}
can be derived through $\Delta^{x,y}_T$ in a similar manner as we did for $\Psi^{x,y}_T$ to obtain $\Psi_T$.
To substantiate our claim, we compute an upper bound for $\Delta^{x,y}_T$ of order $1/T^2$. The existence of such an upper bound poses a limit on the error this tail integral contributed to the cost structure as a whole.
\begin{proposition} \label{prop:truncation_error}
Let $x,y\geq 0$ and $\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3] < \iy$. Then
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right)
\end{equation*}
and
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proposition}
\textit{Remark.} In case the net-input process $X$ is light-tailed, that is there exists $u>0$ such that $\mathbb{E}[{\rm e}^{u X(1)}] < \iy$, it can be shown that the truncation error is of order ${\rm e}^{-\beta T}/T$ for some $\beta>0$. See Appendix \ref{sec:proof_truncation} for details.
\section{Optimization}
\label{sec:optimization}
The result in Theorem \ref{thm:mainresult}, characterizing the leading order behavior of $\Omega_T(\mu)$, also reveals the behavior of $\Pi_T(\mu)$ in leading order. Namely,
\begin{equation*}
\Pi_T(\mu) = \Pi_\iy(\mu) + \Psi_T(\mu) + O(1/T^2).
\end{equation*}
In fact, this representation naturally gives rise to an \emph{approximation} of the actual cost function:
\begin{align}\label{eq:decomposition}
\hat{\Pi}_{T}(\mu) := \Pi_{\iy}(\mu) + \Psi_T(\mu)
\end{align}
Denote the corresponding minimizer of $\Pih$ by
\begin{equation}\label{eq:muhat}
\hat{\mu}_T^\star := \arg\min_{\mu\geq 0} \Pih(\mu), \qquad \Pih^\star := \Pih(\hat{\mu}_T^\star)
\end{equation}
in addition to the definitions in \eqref{eq:muStar} and \eqref{eq:piStar}.
This section is devoted to the analysis of the minimizers $\muT$, $\muh$ and $\mui$, and the optimality gap for the two approximations.
Throughout this section, we assume that $u_2, u_3 <\iy$ and $\mathbb{E}[Q(0)^2] <\iy$.
By its definition in \eqref{eq:PiInf} and Lemma \ref{lemma:workloadmoments}, we have an exact expression for the steady-state cost function:
\begin{equation*}
\Pi_{\iy}(\mu) = \frac{\la u_2}{2(\mu-\la)} + \aaa\mu.
\end{equation*}
It is easily verified that $\Pi_{\iy}$ is strictly convex in $\mu$, for instance by observing that $\Pi_{\iy}''(\mu) > 0$ for all $\mu > \la$. Therefore $\Pi_{\iy}$ has a unique global minimizer and
\begin{equation}
\label{eq:muInf}
\mui = \la + \sqrt{\frac{\la u_2}{2\aaa}}, \qquad \Pi_{\iy}^\star = \aaa\la + \sqrt{2\aaa\la u_2}.
\end{equation}
We are interested in the relation between $\mui$ and $\muT$, and between $\muh$ and $\muT$.
Since $\Pi_{T}(\mu) = \Pi_{\iy}(\mu) + O(1/T)$ for all $\mu > \la$, we have pointwise convergence of the sequence $\Pi_{T}$, as well as $\hat{\Pi}_{T}$, to $\Pi_{\iy}$ for $T\to\iy$, we also expect $\muT \to \mui$ and $\muh\to\mui$ for $T\to\iy$.
Before proving that this convergence indeed holds, we present a result on the strict convexity of the function $\Pi_{T}$.
\begin{lemma}\label{lemma:strict_convexity}
Let $\mu\geq 0$. The function $\Pi_{T}(\mu)$ is
\begin{itemize}
\item convex in $\mu$, if $Q(0)\equiv x$, $T<x/\mu$ and $\sigma=0$,
\item strictly convex in $\mu$, otherwise.
\end{itemize}
\end{lemma}
Building upon strict convexity of both $\Pi_T(\mu)$ and $\Pi_\iy(\mu)$ for $\mu>\la$, we derive the following convergence result.
\begin{proposition}\label{prop:min_convergence_mu}
Let $\muT$, $\muh$ and $\mui$ be as defined in \eqref{eq:muStar} and \eqref{eq:muhat}. Then
\begin{equation*}
\muT \to \mui\, \qquad \text{\rm and } \qquad \muh \to \mui,
\end{equation*}
for $T\to\infty$.
\end{proposition}
The next result describes a refinement of $\muT$ in terms of $\mui$.
\begin{proposition}\label{prop:muBullet}
For $T$ sufficiently large,
\begin{equation*}
\muT = \mui + \frac{\mu_\bullet}{T} + o(1/T),
\end{equation*}
where
\begin{equation}\label{eq:muBullet}
\mu_\bullet = \frac{\mathbb{E}[Q(0)^2]}{\sqrt{8\la u_2\aaa}} - \frac{u_3}{3 u_2} - 3\sqrt{\frac{\aaa\la u_2}{8}}.
\end{equation}
\end{proposition}
Based on Proposition \ref{prop:muBullet} we propose a \emph{corrected staffing rule}, accounting for the finite horizon
\begin{equation}
\label{eq:correctedMu}
\tilde{\mu}_T^\star = \left[\mui + \frac{\mu_\bullet}{T}\right]^+,
\end{equation}
with $\mu_\bullet$ as in \eqref{eq:muBullet}.
Here $[x]^+ := \max\{x,0\}$, which ensures the value of $\tilde{\mu}_T^\star$ is non-negative and thus is a feasible solution of the optimization problem.
This refined capacity allocation rule is expected to reduce the costs incurred in transient settings.
However, the value of particular interest to us is the cost penalty for using either one of the approximations rather than the actual minimum $\muT$, that is, the \emph{optimality gap}.
As it happens, we deduce the order of the optimality gap for $\mui$ with the help of the explicit form of $\mu_\bullet$ given in \eqref{eq:muBullet}, which is stated in the next proposition. The proof is given in Appendix \ref{sec:proofProp4}.
\begin{proposition}\label{prop:optimalitygap_mui}
Let $\mui$ be as in \eqref{eq:muInf}. Then,
\begin{equation*}
\Pi_\iy^\star- \Pi_T^\star = O(1/T^2).
\end{equation*}
\end{proposition}
\section{Numerical experiments}
\label{sec:numerics}
\subsection{Influence of $\Omega_{T}(\mu)$}
\label{sec:influence_omega}
We first assess the contribution of the correction to the cost function provided by Theorem 1. In other words, we investigate whether $\hat{\Pi}_{T}(\mu)$ as in \eqref{eq:PiT} yields a significantly better fit to $\Pi_{T}(\mu)$, than $\Pi_{\iy}(\mu)$ does.
Note that these three functions only differ in the costs describing the congestion.
Therefore, we limit our study in this subsection to the evaluation of $C_T(\mu)$ as in \eqref{eq:CTmu} with stationary equivalent $C_{\iy}(\mu) = \mathbb{E}[Q_{\mu}(\iy)]$.
Our novel approximation hence reads
\begin{equation*}
\hat{C}_{T}(\mu) := C_{\infty}(\mu) + \Omega_{T}(\mu),
\end{equation*}
with $\Omega_{T}(\mu)$ given in Theorem \ref{thm:mainresult}.
We conduct our numerical experiments based on three models, namely:
\begin{enumerate}
\item \underline{$M/M/1$ queue}: $U(t)$ is a unit rate compound Poisson process with exponentially distributed increments. We have $u_2 = 2$, $u_3=3$, so that
\begin{equation}\label{eq:MM1cor}
\hat{C}_{T}(\mu) = \frac{\la} {\mu-\la} + \frac{1}{T(\mu-\la)} \left(\frac{x^2}{2} - \frac{\la^2}{(\mu-\la)^2} - \frac{\la} {\mu-\la} \right).
\end{equation}
\item \underline{$M/{\rm Pareto}/1$ queue}: $U(t)$ is a unit rate compound Poisson process with Pareto increments. The Pareto distribution deserves special attention due to its heavy-tailed nature, having tail probability $\bar{F}(x) = (x/k)^{-\gamma}$, if $x\geq k$ and 1 otherwise.
It is well-known that heavy-tailed service times lead to long relaxation time. For our purposes, we fix shape parameter $\gamma = 16/5$ and scale parameter $k=11/16$, so that $\beta = 1$, $u_2 = 121/96$, $u_3 = 1331/256$ and $u_k=\iy$ for all $k>3$. Hence,
\begin{equation}
\label{eq:MP1cor}
\hat{C}_{T}(\mu) = \frac{121\la} {192(\mu-\la)} + \frac{1}{2T(\mu-\la)}
\left( x^2 - \frac{(121\la/96)^2}{2(\mu-\la)^2} - \frac{ 1331\la/256 }{2(\mu-\la)}\right)
\end{equation}
\item \underline{Reflected Brownian motion}: $U(t)$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$. We have $u_2 = \sigma^2$, $u_3=0$, so that
\begin{equation}\label{eq:RBMcor}
\hat{C}_{T}(\mu) = \frac{\la\sigma^2}{2(\mu-\la)} + \frac{1}{2T(\mu-\la)} \left( x^2 - \frac{\la^2\sigma^4}{2(\mu-\la)^2}\right).
\end{equation}
\end{enumerate}
In light of the equivalence relations in \eqref{eq:Qidentity} we only consider the case $\la=1$. The cost values for general values of $\la$ follow by appropriate rescaling of $\mu$ and $T$.
For the $M/M/1$ and $M/{\rm Pareto}/1$ queue, we obtained the function $C_{T}(\mu)$ through simulation and the results are accurate up until a 95\% confidence interval of width $10^{-3}$. For reflected Brownian motion, we used the explicit distribution function given in \cite{Harrison1985} for double numerical integration. The results for several values of $T$ and two different starting states are depicted in Figures 4-6. These plots also include the approximated functions $\hat{C}_{T}(\mu)$.
\begin{figure}%
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/mm1_0.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/mm1_0.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/mm1_0.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/mm1_0.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/mm1_0.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/mm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/mm1_25.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/mm1_25.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/mm1_25.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/mm1_25.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/mm1_25.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/mm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/M/1$ queue with $\la=1$.}
\label{fig:cont}%
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/mp1_0.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/mp1_0.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/mp1_0.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/mp1_0.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/mp1_0.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/mp1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mp1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/mp1_25.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/mp1_25.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/mp1_25.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/mp1_25.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/mp1_25.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/mp1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/mp1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for the $M/$Pareto$/1$ queue with $\la=1$.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 2,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,2)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/rbm1_0.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/rbm1_0.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/rbm1_0.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/rbm1_0.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/rbm1_0.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/rbm1_0.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/rbm1_0.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=0$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = 1,
xmax = 5,
ymin = 0,
ymax = 3,
xlabel = {$\to \mu$},
axis line style={->},
axis lines = left,
legend cell align=left,
legend style = {at = {(axis cs: 5,3)},anchor = north east},
legend columns=2,
yscale = 0.8,
xscale = 1
]
\addplot[red, thick] table[x = mu,y=T2] {tikz/rbm1_25.txt};
\addplot[red, dashed, thick] table[x = mu,y=ap2] {tikz/rbm1_25.txt};
\addplot[green, thick] table[x = mu,y=T5] {tikz/rbm1_25.txt};
\addplot[green, dashed, thick] table[x = mu,y=ap5] {tikz/rbm1_25.txt};
\addplot[blue, thick] table[x = mu,y=T10] {tikz/rbm1_25.txt};
\addplot[blue, dashed, thick] table[x = mu,y=ap10] {tikz/rbm1_25.txt};
\addplot[thick] table[x = mu,y=PSA] {tikz/rbm1_25.txt};
\legend{{$C_2(\mu)$},{$\hat C_2(\mu)$},{$C_5(\mu)$},{$\hat C_5(\mu)$},{$C_{10}(\mu)$},{$\hat C_{10}(\mu)$},{$C_\infty(\mu)$}}
\end{axis}
\end{tikzpicture}
\caption{$x=2.5$}
\end{subfigure}
\caption{Comparison of exact waiting cost function $C_T(\mu)$ against corrected cost function $\hat{C}_T(\mu)$ and PSA cost function $C_\infty(\mu)$ for $T=2,5$ and 10 for reflected Brownian motion with $\sigma=1$.}
\end{figure}
We name a few observations based on these figures.
First, we indeed note the pointwise convergence of $\hat{C}_{T}(\mu)$ to $\hat{C}_{\iy}(\mu)$ as $T$ grows, for all $\mu$ in all three cases. However, the difference between the stationary costs and those for small values of $T$ can be significant. This is most clear in the plots with $x=2.5$ and when $\mu$ is close to $\la$, i.e. it is in heavy-traffic. In these scenarios, it is evident that refinements to the stationary cost function are needed. $\hat{C}_{T}(\mu)$ does a fairly good job at providing such correction, especially for moderate values of $\mu$.
Furthermore, we note that $C_{T}(\mu)$ approaches $C_{\iy}(\mu)$ from below for $x=0$ for any value of $\mu$, while this is not strictly the case for $x>0$.
$\hat{C}_{T}(\mu)$ correctly captures the sign of this correction.
Finally, observe that $\hat{C}_{T}(\mu)\to -\iy$ as $\mu$ approaches $\la$ from above. This divergence is clear from the expressions in \eqref{eq:MM1cor}-\eqref{eq:RBMcor}.
Our correction term relies on the premise that under the coupling scheme, the sample paths of the two queues starting from different states have hit with high probability.
This is equivalent to stating that the `largest' of the two queues is has emptied at least once before time $T$. However, as $\mu$ approaches $\la$, the system enters heavy traffic, and hence the hitting time of the zero barrier is set to run off to infinity.
Consequently, this causes our approximation to be inaccurate for small values of $\mu$.
\subsection{Validation of corrected staffing rule}
\label{sec:num_opt}
In this section, we examine whether the corrected staffing rule $\tilde{\mu}_T^\star$ as in \eqref{eq:correctedMu} indeed yields a significant cost reduction over the choice of $\mui$ by comparing their true costs $\Pi_{T}(\tilde{\mu}_T^\star)$ and $\Pi_{T}(\mui)$.
We conduct this comparison for different values of the parameters, $\aaa$, $T$ and starting state $x$ through numerical experiments.
The three models on which we do our calculations are the $M/M/1$ queue, the $M/$Pareto$/1$ queue and the reflected Brownian motion, as introduced in the previous subsection.
We again focus on $\la=1$ only.
For each of the three models, we adhere to the following set-up. The quality of both staffing rules is assessed for $\aaa = 0.1, 1$ and 2, resembling three modes of valuation of the QoS in the system.
As a benchmark, observe that the expected workload in steady-state conditions with staffing level $\mui$ equals
\begin{equation*}
C_\iy(\mui) = \sqrt{\frac{\aaa\la u_2}{2}}.
\end{equation*}
For each value of $\aaa$, we consider two scenarios: one in which the system starts empty, i.e. $x=0$, and one in which the initial state is double this benchmark value, thus $x=\sqrt{2\aaa\la u_2}$.
The numerics are presented for each model separately. We discuss general conclusions drawn from these results afterwards.\\
\\*
\noindent\textbf{$M/M/1$ queue}
As we discussed before, if $U$ is a unit rate compound Poisson process with exponentially distributed increments, then $\Qlm$ describes the workload process in an $M/M/1$ queue.
For this setting we get
\begin{equation*}
\mui = \la + \sqrt{\frac{\la} {\aaa}},\qquad \tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la} {\aaa}} + \frac{1}{T}\left( \frac{x^2}{4\sqrt{\la\aaa}} - 1 - \frac{3}{2} \sqrt{\la\aaa}\right)\right]^+.
\end{equation*}
Table \ref{tab:mm1} presents the actual costs corresponding to these two staffing levels for different value of $x$ and $\aaa$.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{\aaa}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 4.162 & 0.620 & 2.688 & 0.536 & 0.136 & 4.162 & 0.682 & 2.688 & 0.536 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 4.162 & 0.669 & 3.425 & 0.641 & 0.041 & 4.162 & 0.700 & 3.425 & 0.641 & 0.085 \\
\multicolumn{1}{|c|}{} & 5 & 4.162 & 0.706 & 3.867 & 0.703 & 0.005 & 4.162 & 0.719 & 3.867 & 0.703 & 0.022 \\
\multicolumn{1}{|c|}{} & 10 & 4.162 & 0.719 & 4.015 & 0.719 & 0.001 & 4.162 & 0.726 & 4.015 & 0.719 & 0.010 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.000 & 2.309 & 0.000 & 0.500 & 0.783 & 2.000 & 3.500 & 0.500 & 2.750 & 0.214 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 2.461 & 0.750 & 1.480 & 0.398 & 2.000 & 3.218 & 1.250 & 3.125 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 2.675 & 1.500 & 2.400 & 0.103 & 2.000 & 3.043 & 1.700 & 2.968 & 0.025 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 2.810 & 1.750 & 2.726 & 0.030 & 2.000 & 3.007 & 1.850 & 2.980 & 0.009 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} &1 & 1.707 & 3.744 & 0.000 & 0.500 & 0.866 & 1.707 & 5.889 & 0.000 & 3.328 & 0.435 \\
\multicolumn{1}{|c|}{} &2 & 1.707 & 3.924 & 0.146 & 1.232 & 0.686 & 1.707 & 5.547 & 0.854 & 4.682 & 0.156 \\
\multicolumn{1}{|c|}{} &5 & 1.707 & 4.209 & 1.083 & 3.343 & 0.206 & 1.707 & 5.114 & 1.366 & 4.910 & 0.040 \\
\multicolumn{1}{|c|}{} &10 & 1.707 & 4.424 & 1.395 & 4.108 & 0.071 & 1.707 & 4.945 & 1.536 & 4.868 & 0.016 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/M/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mm1}
\end{table}
\noindent
\textbf{$M$/Pareto/1 queue}
In case the service requirements follow a Pareto distribution with shape parameter $\gamma = 16/5$, the staffing rule becomes
\begin{equation*}
\mui = \la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}}, \ \tilde{\mu}_T^\star = \left[\la + \frac{11}{8}\sqrt{\frac{ \la }{3 \aaa}} + \frac{1}{T}\left( \frac{2 x^2}{11\sqrt{\la\aaa/3}} - \frac{11}{8} - \frac{11\sqrt{3\la\aaa}}{16}\right)\right]^+.
\end{equation*}
The numerical results are given in Table \ref{tab:mp1}.
\begin{table}[h!]
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 11/4\cdot \sqrt{\aaa/3}$} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.510 & 0.524 & 1.759 & 0.461 & 0.120 & 3.510 & 0.573 & 2.010 & 0.562 & 0.019 \\
\multicolumn{1}{|c|}{} & 2 & 3.510 & 0.555 & 2.635 & 0.539 & 0.029 & 3.510 & 0.580 & 2.760 & 0.574 & 0.010 \\
\multicolumn{1}{|c|}{} & 5 & 3.510 & 0.580 & 3.160 & 0.578 & 0.003 & 3.510 & 0.591 & 3.210 & 0.589 & 0.002 \\
\multicolumn{1}{|c|}{} & 10 & 3.510 & 0.590 & 3.335 & 0.590 & 0.000 & 3.510 & 0.596 & 3.360 & 0.595 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.794 & 2.076 & 0.000 & 0.500 & 0.759 & 1.794 & 2.989 & 0.000 & 2.088 & 0.302 \\
\multicolumn{1}{|c|}{} & 2 & 1.794 & 2.190 & 0.511 & 1.291 & 0.411 & 1.794 & 2.790 & 0.610 & 2.588 & 0.072 \\
\multicolumn{1}{|c|}{} & 5 & 1.794 & 2.345 & 1.281 & 2.108 & 0.101 & 1.794 & 2.638 & 1.320 & 2.607 & 0.012 \\
\multicolumn{1}{|c|}{} & 10 & 1.794 & 2.441 & 1.537 & 2.371 & 0.029 & 1.794 & 2.597 & 1.557 & 2.585 & 0.005 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.561 & 3.427 & 0.000 & 0.500 & 0.854 & 1.561 & 5.087 & 0.000 & 2.745 & 0.460 \\
\multicolumn{1}{|c|}{} & 2 & 1.561 & 3.567 & 0.032 & 1.050 & 0.706 & 1.561 & 4.832 & 0.172 & 3.417 & 0.293 \\
\multicolumn{1}{|c|}{} & 5 & 1.561 & 3.779 & 0.950 & 3.012 & 0.203 & 1.561 & 4.499 & 1.006 & 4.313 & 0.041 \\
\multicolumn{1}{|c|}{} & 10 & 1.561 & 3.935 & 1.255 & 3.356 & 0.147 & 1.561 & 4.351 & 1.284 & 4.304 & 0.011 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for the $M/{\rm Pareto}/1$ queue for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:mp1}
\end{table}
Just as in the results for the $M/M/1$ queue, we observe a higher reduction for larger value of $\aaa$ and $T$. Also, again $\tilde{\mu}_T < \mui$. Hence, the conclusions for the $M/{\rm Pareto}/1$ queue are similar to those of the $M/M/1$ queue.
\noindent\textbf{Reflected Brownian motion}.
In case the input process $U$ is Brownian motion with drift 1 and infinitesimal variance $\s^2$, the steady-state staffing rule and its corrected version reduce to
\begin{equation*}
\mui = \la + \sqrt{\frac{\la\s^2}{2\aaa}}, \qquad
\tilde{\mu}_T^\star = \left[\la + \sqrt{\frac{\la\s^2}{2\aaa}} + \frac{1}{2\sqrt{2}\,T}\left(\frac{x^2}{\sqrt{\la \aaa}\s} - 3\s\sqrt{\aaa\la} \right)\right]^+.
\end{equation*}
In Tables \ref{tab:rbm1} and \ref{tab:rbm2}, the costs obtained through numerical evaluation are presented for several values of $x$, $T$. We also vary $\s$ to examine the influence of the volatility of arrival process on the quality of the staffing rules.
The observations on the influence of $\aaa, x$ and $T$ are similar to those of the $M/M/1$ queue and the $M/{\rm Pareto}/1$ queue.
However, here we see little improvement by the corrected staffing rule for small values of $\aaa$ for both values of $x$.
The results in Tables \ref{tab:rbm1}-\ref{tab:rbm2} also suggest that the reduction is smaller for larger values of $\s$.
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = \sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 3.236 & 0.525 & 2.901 & 0.518 & 0.013 & 3.236 & 0.565 & 3.124 & 0.564 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 3.236 & 0.536 & 3.068 & 0.534 & 0.003 & 3.236 & 0.556 & 3.180 & 0.556 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 3.236 & 0.543 & 3.169 & 0.542 & 0.000 & 3.236 & 0.551 & 3.214 & 0.551 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 3.236 & 0.545 & 3.203 & 0.545 & 0.000 & 3.236 & 0.549 & 3.225 & 0.549 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 1.500 & 3.420 & 0.000 & 0.833 & 0.756 & 1.500 & 4.741 & 1.000 & 3.984 & 0.160 \\
\multicolumn{1}{|c|}{} & 2 & 1.500 & 3.539 & 0.750 & 2.386 & 0.326 & 1.500 & 4.579 & 1.250 & 4.293 & 0.063 \\
\multicolumn{1}{|c|}{} & 5 & 1.500 & 3.707 & 1.200 & 3.363 & 0.093 & 1.500 & 4.335 & 1.400 & 4.274 & 0.014 \\
\multicolumn{1}{|c|}{} & 10 & 1.500 & 3.820 & 1.350 & 3.705 & 0.030 & 1.500 & 4.190 & 1.450 & 4.175 & 0.004 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 1$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm1}
\end{table}
\begin{table}
\centering
\resizebox{13cm}{!} {
\begin{tabular}{|c|r|r @{}r|r@{}r|r||r@{}r|r@{}r|r|}
\cline{3-12}
\multicolumn{1}{c}{} & \multicolumn{1}{r|}{} & \multicolumn{5}{c||}{$x = 0$} & \multicolumn{5}{c|}{$x = 2\sqrt{2\aaa} $} \\
\hline
$\aaa$ & $T$ & $\mui$ & \hspace{1pt} $\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt} $\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. & $\mui$ & \hspace{1pt}$\Pi_T(\mui)$ & $\tilde{\mu}_T^\star$ & \hspace{1pt}$\Pi_T(\tilde{\mu}_T^\star)$ & r.c.i. \\
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{0.1}} & 1 & 5.472 & 0.950 & 4.801 & 0.936 & 0.015 & 5.472 & 1.030 & 5.249 & 1.029 & 0.001 \\
\multicolumn{1}{|c|}{} & 2 & 5.472 & 0.972 & 5.137 & 0.968 & 0.003 & 5.472 & 1.012 & 5.360 & 1.012 & 0.000 \\
\multicolumn{1}{|c|}{} & 5 & 5.472 & 0.985 & 5.338 & 0.985 & 0.000 & 5.472 & 1.002 & 5.427 & 1.002 & 0.000 \\
\multicolumn{1}{|c|}{} & 10 & 5.472 & 0.990 & 5.405 & 0.990 & 0.000 & 5.472 & 0.998 & 5.450 & 0.998 & 0.000 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{1}} & 1 & 2.414 & 3.176 & 0.293 & 1.546 & 0.513 & 2.414 & 4.633 & 1.707 & 4.228 & 0.087 \\
\multicolumn{1}{|c|}{} & 2 & 2.414 & 3.356 & 1.354 & 2.690 & 0.199 & 2.414 & 4.375 & 2.061 & 4.247 & 0.029 \\
\multicolumn{1}{|c|}{} & 5 & 2.414 & 3.573 & 1.990 & 3.411 & 0.045 & 2.414 & 4.094 & 2.273 & 4.073 & 0.005 \\
\multicolumn{1}{|c|}{} & 10 & 2.414 & 3.689 & 2.202 & 3.646 & 0.012 & 2.414 & 3.966 & 2.344 & 3.962 & 0.001 \\
\hline
\hline
\multicolumn{1}{|c|}{\multirow{4}[2]{*}{2}} & 1 & 2.000 & 4.839 & 0.000 & 1.339 & 0.723 & 2.000 & 7.481 & 1.000 & 5.967 & 0.202 \\
\multicolumn{1}{|c|}{} & 2 & 2.000 & 5.078 & 0.500 & 2.773 & 0.454 & 2.000 & 7.158 & 1.500 & 6.585 & 0.080 \\
\multicolumn{1}{|c|}{} & 5 & 2.000 & 5.414 & 1.400 & 4.726 & 0.127 & 2.000 & 6.670 & 1.800 & 6.549 & 0.018 \\
\multicolumn{1}{|c|}{} & 10 & 2.000 & 5.639 & 1.700 & 5.409 & 0.041 & 2.000 & 6.380 & 1.900 & 6.349 & 0.005 \\
\hline
\end{tabular}
}
\caption{Comparison of costs for RBM with $\sigma = 2$ for steady-state and corrected staffing rules and relative cost improvement (r.c.i.).}
\label{tab:rbm2}
\end{table}
\subsection{Discussion}
Based upon these numerical results in Tables \ref{tab:mm1}-\ref{tab:rbm2}, we make a few remarks. The three models roughly exhibit similar behavior as $T$, $x$ and $\aaa$ are varied.
Non-surprisingly, we note that $\tilde{\mu}_T$ approaches $\mui$ with increasing $T$, which also implies that the cost reduction achieved by the corrected staffing rule vanishes as $T\to\iy$.
Also, we observe that in all scenarios examined, the cost reduction increases with $\aaa$. This can be explained through investigation of the objective function $\Pi_T$ as function of $\mu$. Namely, for $\aaa$ small, the curve is relatively flat around the true optimum $\muT$. Hence, in this case a moderate deviation from $\muT$ will likely not lead to a significant cost increase. However, as $\aaa$ becomes larger, i.e. server efficiency is valued more than minimization of congestion, the curve becomes more sharp around $\muT$, and hence more accurate approximations of $\muT$ are required to achieve an acceptable cost level. Hence, the corrected staffing rule \eqref{eq:correctedMu} proves particularly useful in these cases.
Another point we highlight is that the relative improvement is higher for $x=0$ than for $x=\sqrt{2\aaa\la u_2}$. Moreover, even though the initial state of the system is above the optimal equilibrium, $\tilde{\mu}_T$ is smaller than $\mui$. This is somewhat counter-intuitive. In fact, from \eqref{eq:muBullet} it follows that $\mu_\bullet$ positively contributes to the corrected staffing function if
\begin{equation*}
\mathbb{E}[Q^2(0)] > 3\aaa\la u_2 + \frac{2 u_2}{3 u_3}\,\sqrt{2\aaa\la u_2}.
\end{equation*}
\section{Conclusion \& further research}
\label{sec:conclusion_chapter6}
Motivated by the time-varying nature of queues in practical applications, we studied the impact that the transient phase has on traditional capacity allocation questions.
By defining a cost minimization problem, in which the objective function contains a correction accounting for the transient period, we identified the leading and second-order behavior of the cost function as a function of the interval length $T$.
As a by-product, this result yields an approximation for the actual cost function, which is a refinement to its stationary counterpart.
Our numerical experiments in Section \ref{sec:influence_omega} demonstrate the improved accuracy achieved by this approximation in a number of settings.
By perturbation analysis of the optimization problem, this furthermore gives rise to a correction to the steady-state optimal capacity allocation of order $1/T$.
The necessity of the refined capacity allocation level is substantiated by the numerics in Section \ref{sec:num_opt}, which show the cost reduction that can be achieved in the number of settings, compared to settings in which stationary metrics are used.
Especially for small values of $T$ and large values of $\alpha$ this reduction is significant.
Additionally, these results also indicate that it is relatively safe to use the stationary cost when $T$ is moderate, or $\alpha$ is small.
The latter reflects the scenario in which QoS is much more valued than service efficiency.
This observation links to the flat nature of the cost function around its optimal value for $\alpha$ small, a statement on the optimality gap that we formally proved in Proposition \ref{prop:optimalitygap_mui}.
Besides the validation of our theoretical results of Sections \ref{sec:analysis} and \ref{sec:optimization}, the numerical results also reveal some phenomena that require more investigation.
As noted, our corrected capacity allocation level $\tilde{\mu}_T^\star$ is in most studied cases less than the steady-state optimal value $\mu_{\iy}^\star$. This implies that congestion levels tends to be higher under our staffing scheme then under stationary staffing.
A possible explanation for this may be the fact that the planning period under consideration is finite.
Clearly, in the setting we analyzed, anything that happens after time $T$ is neglected.
Therefore, it might be beneficial from the cost perspective to end the period with a higher expected congestion level, as it does not need to be canceled out in the future.
Related to this observation, it would be interesting to look at the setting in which staffing decisions need to be made in consecutive periods of equal length, in which the arrival rate changes at the start of each period.
This case requires careful consideration of the correlation among the staffing decisions within the separate periods.
Another question that arises concerns the translation of our (qualitative) findings to more general queues, in particular the $M/G/s$ queue.
Whereas in our analysis, the central decision variable is the server speed $\mu$, the variable of interest in multi-server queues is typically the number of servers.
It may well be that similar explicit corrections to staffing levels can be deduced to account for transience.
Since our analysis heavily relies on the comparibility of the sample paths of two single-server queues, which is due to the equal negative drift for the two processes, another approach must be taken to tackle this extension.
The analysis and findings for the single-server queue with L\'evy input presented in this chapter may serve a stepping stone for investigation of these more elaborate problems.
\section*{Appendix}
\renewcommand*\thesection{\arabic{section}}
\begin{subappendices}
\renewcommand \thesection {\Alph{section}}
\section{Proofs of Section \ref{sec:model_description}}
\label{app:proofs_model}
\subsection{Proof of Lemma \ref{lemma:workloadmoments}}
\label{app:proof_lemma_workload_moments}
\begin{proof}
The conditions of \cite[Cor.IX3.4]{Asmussen2003} are satisfied and therefore $Q_{\mu}(t)\Rightarrow \Qlm(\infty)$ in distribution for $t\rightarrow\infty$. Furthermore, its Laplace transform is for ${\rm Re}(s) < 0$
\[\tilde{Q}_{\mu}(s) = \mathbb{E}[s \Qlm(\infty)] = \frac{s \ka_{\mu}'(0)}{\ka_{\mu}(s)} = \frac{s(\la\ka_U'(0) - \mu)}{\la\ka_U(s) - \mu s} = \frac{s(\mu-\la)}{\mu s-\la \ka_U(s)}.\]
It can be checked that $\ka_U'(0) = \mathbb{E}[U(1)] = 1$, $\ka_U''(0) = u_2$ and $\ka_U'''(0) = u_3$, and $\klm'(0) = \la-\mu$, $\klm''(0) = \la u_2$ and $\klm'''(0) = \la u_3$.
Using l'H\^opital's rule we obtain the first moment of $\Qlm(\infty)$:
\begin{align*}
\mathbb{E}[\Qlm(\iy)] &= {-}\lim_{s\to 0} \frac{d}{ds} \tilde{Q}_{\mu}(s) = \lim_{s\to 0} \klm'(0)\, \frac{s\klm'(s)-\klm(s)}{\klm(s)^2}\\
&= \lim_{s\to 0} \klm'(0)\, \frac{{-}s\klm''(s)}{2\klm(s)\klm'(s)}
= \lim_{s\to 0} \klm'(0)\,\frac{ s\klm'''(s)-\klm''(s)}{2\klm'(s)^2 + 2\klm(s)\klm'''(s)} \\
&= {-}\frac{\klm''(0)}{2\klm'(0)} = \frac{\la u_2}{2(\mu-\la)}.
\end{align*}
Similarly we derive the second moment:
\begin{align*}
\mathbb{E}[\Qlm^2(\iy)] &= \lim_{s\to 0} \frac{d^2}{ds^2} \tilde{Q}_{\mu}(s)
= \lim_{s\to 0} \klm'(0)\, \frac{3 \klm''(0)^2 - 2\klm'(0)\klm'''(0)}{6 \klm'(0)^3}\\
&= (\la-\mu)\frac{3\la^2u_2^2 - 2\la u_3(\la-\mu)}{6(\la-\mu)^3}
= \frac{\la^2u_2^2}{2(\mu-\la)^2} +\frac{\la u_3}{3(\mu-\la)}.
\end{align*}
\end{proof}
\subsection{Proof of Proposition \ref{prop:cost_convergence}}
\begin{proof}
We prove the limit by showing that the difference
\[
\Pi_T(\mu) - \Pi_\iy(\mu) = \frac{1}{T} \int_0^T \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t
\]
converges to zero as $T\to\iy$ for $\mu>\la$ fixed. The assumption $\mathbb{E}[U(1)], \mathbb{E}[Q(0)] < \iy$ implies by \cite[Prop.~1]{Abate1994} that $\mathbb{E}[Q_\mu(t)]<\iy$ for all $t\geq 0$.
Following \cite{Abate1994}, we use the decomposition
\[
\mathbb{E}[Q_\mu(t)] = \mathbb{E}[Q^0_\mu(t)] + \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\},
\]
where $Q_\mu^0(t)$ represents the workload process if the system starts empty.
From this decomposition it is revealed that $\mathbb{E}[Q^0_\mu(t)]$ and $\left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\}$ are non-negative monotonically increasing and decreasing functions of $t$, respectively, see \cite[Prop.~2,Thm.~11]{Abate1994}.
Recall $\mathbb{E}[Q_\mu(t)] \to \mathbb{E}[Q_\mu(\iy)]$ for $t\to\iy$ by ergodicity of the workload process for any initial state $\mathbb{E}[Q(0)]< \iy$, if $\mu>\la$.
Henceforth,
\begin{align*}
\mathbb{E}[Q_\mu(t)] &\leq \sup_t \mathbb{E}[Q_\mu^0(t)] + \sup_t \left\{ \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu^0(t)]\right\} \\
&= \mathbb{E}[Q_\mu(\iy)] + \left\{\mathbb{E}[Q_\mu(0)] - \mathbb{E}[Q_\mu^0(0)]\right\} = \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)],
\end{align*}
for all $t\geq 0$, which proves that the expected workload is bounded.
Fix $\varepsilon>0$. By convergence of $\mathbb{E}[Q_\mu(t)]$ for $t\to\iy$, there exists a value $t^* := t^*(\varepsilon)$ such that for all $t\geq t^*$
\begin{equation}
\left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| < \varepsilon/2.
\end{equation}
Next, set
\[
T^* := T^*(\varepsilon) = \frac{2\,t^*(\varepsilon)}{\varepsilon}\, ( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]).
\]
Then for $T\geq \hat{T}:= \max\{ t^*,T^* \}$, we have
\begin{align*}
\left| \frac{1}{T} \int_0^T \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t \right|
&\leq \frac{1}{T} \int_0^{t^*} \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| {\rm d} t \\
& \qquad+ \frac{1}{T} \int_{t^*}^T \left| \mathbb{E}[Q_\mu(t)] - \mathbb{E}[Q_\mu(\iy)] \right| \, {\rm d} t \\
&\leq \frac{1}{T} \int_0^{t^*} \mathbb{E}[Q_\mu(t)] + \mathbb{E}[Q_\mu(\iy)] \, {\rm d} t + \frac{1}{T} \int_{t^*}^T \frac{\varepsilon}{2}\, {\rm d} t \\
&< \frac{t^*}{T}\,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{T-t^*}{T} \,\frac{\varepsilon}{2}\\
&< \frac{t^*}{T^*} \,( 2 \mathbb{E}[Q_\mu(\iy)] + \mathbb{E}[Q(0)]) + \frac{\varepsilon}{2} = \varepsilon.
\end{align*}
Hence, for any choice of $\varepsilon>0$ we can find a value $\hat{T}$ such that $\Pi_{\hat{T}}(\mu)$ approaches $\Pi_\iy(\mu)$ within distance $\varepsilon$, which proves the limit.
\end{proof}
\section{Proofs of Section \ref{sec:analysis}}
\label{app:proofs_analysis}
\subsection{Proof of Lemma \ref{lemma:psixy}}
\begin{proof}
Using the representation in \eqref{eq:Yxy} we write
\begin{align*}
\Psi^{x,y}_T &= \frac{1}{T}\int_0^{\infty} \mathbb{E}[Y^{x,y}(t)]{\rm d} t \\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}Y^{x,y}(t)\right] {\rm d} t + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right]
+ \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^\infty Y^{x,y}(t) {\rm d} t\right] ,\\
&= \frac{1}{T}\,\mathbb{E}\left[\int_0^{\tau^y(0)}(x-y) {\rm d} t\right] + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right] \\
&= \frac{1}{T}\,\mathbb{E}[\tau^y(0)](x-y) + \frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t) {\rm d} t\right].
\end{align*}
By \eqref{eq:Yxy} and the Strong Markov property holding for L\'evy processes \cite{Asmussen2003}, observe that \\* $Y^{x-y,0}(t) {\;\buildrel{d}\over= \;} Y^{x,y}(\tau^y(0)+t)$, whereby
\begin{equation*}
\frac{1}{T}\,\mathbb{E}\left[\int_{\tau^y(0)}^{\tau^x(0)} Y^{x,y}(t)\,{\rm d} t\right] = \frac{1}{T}\,\mathbb{E}\left[\int_{0}^{\tau^{x-y}(0)} Y^{x-y,0}(t) {\rm d} t\right] = \Psi^{x-y,0}_T,
\end{equation*}
which completes the proof.
\end{proof}
\subsection{Proof of Lemma \ref{lemma:psiz0}}
\begin{proof}
Note that $Y^{z,0}(t)$ and $\tau^z(w)$ are intimately related. Namely, due to the fact that $X$ has no negative jumps
\begin{equation*}
\{ \tau^z(w) \leq t\} = \{Y^{z,0}(t) \leq w \}.
\end{equation*}
In fact, $Y^{z,0}(\tau^z(w)) = w$, which implies that $\tau^z$ is a right inverse for $Y^{z,0}(t)$. Therefore, the following equality holds
\begin{equation*}
\int_0^{\tau^z(0)} Y^{z,0}(t)\, {\rm d} t = \int_0^z \tau^z(w)\, {\rm d} w,
\end{equation*}
which implies with the help of Fubini's theorem
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^z(w)]\, {\rm d} w = \frac{1}{T}\,\int_0^z \mathbb{E}[\tau^{z-w}(0)]\,{\rm d} w =\frac{1}{T}\, \int_0^z \mathbb{E}[\tau^{w}(0)] \,{\rm d} w.
\end{equation*}
\end{proof}
\subsection{Proof of Corollary \ref{cor:Psixy}}
\begin{proof}
From \eqref{eq:invCharExp},
\begin{equation}\label{eq:corEq1}
\mathbb{E}[\hat{\tau}^0(w)] = -\tfrac{{\rm d}}{{\rm d} u} \left. \mathbb{E}[\exp(-u\,\hat{\tau}^0(w))]\right|_{u=0} = w\left.\frac{{\rm d}}{{\rm d} u} \Upsilon^{-1}(u)\right|_{u=0}.
\end{equation}
Since $\Upsilon(\theta)$ is strictly increasing and $\Upsilon(0)=0$, we get $\Upsilon^{-1}(0)=0$ and
\begin{equation*}
\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = \frac{1}{\Upsilon'(\Upsilon^{-1}(0))} = \{ \Upsilon'(0) \}^{-1}.
\end{equation*}
Furthermore,
\begin{align*}
\Upsilon'(\thh) &= -a+ \s^2\thh + \int_{-\infty}^0 (x\, e^{\thh x} - x{\bf 1}_{[-1,0)}(x)) \hat{\nu}({\rm d} x) \\
&= -a + \s^2\thh - \int_0^\infty (y\, e^{-\thh y} - y{\rm 1}_{(0,1]}(y)) \nu({\rm d} y).
\end{align*}
Thus, $\Upsilon'(0) = -\mathbb{E}[X(1)] = \mu-\la$ and $\mathbb{E}[\hat{\tau}^0(w)] = w/(\mu-\la)$. By \eqref{eq:H(x,0)} and \eqref{eq:transformedTau}, we deduce that
\begin{equation*}
\Psi^{z,0}_T = \frac{1}{T}\, \int_0^z \mathbb{E}[\tau^w(0)] \,{\rm d} w = \frac{1}{T}\, \int_0^z \mathbb{E}[\hat{\tau}^0(w)] {\rm d} w = \frac{z^2}{2T(\mu-\la)}.
\end{equation*}
For $x>y$, we use Lemma \ref{lemma:psixy} to conclude
\begin{equation*}
\Psi^{x,y}_T = \frac{y(x-y)}{T(\mu-\la)} + \frac{(x-y)^2}{2 T(\mu-\la)} = \frac{x^2-y^2}{2T(\mu-\la)}.
\end{equation*}
The result for $x<y$ follows directly by the observation $\Psi^{x,y}_T = -\Psi_T^{x,y}$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:truncation_error}}\label{sec:proof_truncation}
\begin{proof}
To derive the upper bound for $\Delta^{x,y}_T$, we apply the same coupling argument as in described in Section 3. Let us assume without loss of generality $x>y$.
In this case,
\begin{equation*}
|\Delta^{x,y}_T| = \frac{1}{T} \int_T^\iy \mathbb{E}[Q^x(t)-Q^y(t)] {\rm d} t \leq \frac{1}{T}\int_T^\iy \mathbb{E}[Q^x(t)-Q^0(t)]{\rm d} t.
\end{equation*}
By the decomposition in \eqref{eq:Yxy},
\begin{align}
\int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)] {\rm d} t
&= \int_T^\infty \mathbb{E}[(x+\inf_{s\leq t} X(s))\textbf{1}_{\{\tau^x(0)>t\}}] {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( x-u + \inf_{s\leq t}X(s) > 0) {\rm d} u {\rm d} t \nonumber\\
&= \int_T^\infty \int_0^x P( \tau^{x-u}(0) > t ) {\rm d} u{\rm d} t \\
&\leq \int_T^\iy \int_0^x \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} u {\rm d} t \nonumber\\
&= \int_0^x \int_T^\iy \frac{\mathbb{E}[\tau^{x-u}(0)^2]}{t^2}{\rm d} t{\rm d} u
= \int_0^x \frac{\mathbb{E}[\tau^{w}(0)^2]}{T}\,{\rm d} w. \nonumber
\label{eq:tailprobIntegral}
\end{align}
We obtain $\mathbb{E}[\tau^w(0)^2]$ with the help of its Laplace transform in \eqref{eq:invCharExp}. Namely,
\begin{align*}
\mathbb{E}[\tau^w(0)^2] &= \left.\tfrac{{\rm d}^2}{{\rm d} u^2}\mathbb{E}[\exp(-u \tau^w(0))]\right|_{u=0} \\
&= w^2\,\left(\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0}\right)^2 - w\left. \tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0}.
\end{align*}
As in the previous subsection we have $\left.\tfrac{{\rm d}}{{\rm d} u}\Upsilon^{-1}(u)\right|_{u=0} = (\mu-\la)^{-1}$, and
\begin{equation*}
\left.\tfrac{{\rm d}^2}{{\rm d} u^2}\Upsilon^{-1}(u)\right|_{u=0} = {-}\frac{\Upsilon''(\Upsilon^{-1}(0))}{\Upsilon'(\Upsilon^{-1}(0))^3} = {-}\frac{\Upsilon''(0)}{\Upsilon'(0)^3}.
\end{equation*}
Since $\Upsilon'(0) = \mu-\la$ and
\begin{equation*}
\Upsilon''(0) = \s^2 + \int_0^\infty x^2\,\nu({\rm d} x) = u_2,
\end{equation*}
we conclude
\begin{equation*}
\mathbb{E}[\tau^w(0)^2] = \frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3},
\end{equation*}
so that
\begin{equation}\label{eq:delta_upper}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\int_0^x \frac{w^2}{(\mu-\la)^2} + \frac{u_2w}{(\mu-\la)^3} {\rm d} w = \frac{1}{T^2}\left(\frac{x^3}{3(\mu-\la)^2}+\frac{u_2 x^2}{2(\mu-\la)^3}\right).
\end{equation}
For general $x,y\geq 0$,
\begin{equation*}
|\Delta^{x,y}_T| \leq \frac{1}{T^2}\left(\frac{\max(y,x)^3}{3(\mu-\la)^2}+\frac{u_2 \max(y,x)^2}{2(\mu-\la)^3}\right).
\end{equation*}
As a direct consequence,
\begin{equation*}
|\Delta_T| \leq \frac{1}{T^2}\left(\frac{\mathbb{E}[\max(Q(0),Q_\mu(\iy))^3]}{3(\mu-\la)^2}+\frac{u_2 \mathbb{E}[\max(Q(0),Q_\mu(\iy))^2]}{2(\mu-\la)^3}\right).
\end{equation*}
\end{proof}
\noindent\textit{Remark.}
Observe that if $X$ is light-tailed, that is $\mathbb{E}[\exp\{ -\theta X(1) \}]$ $= \mathbb{E}[\exp\{\kappa(\theta)\}] < \iy$ for some $\theta<0$, then $\Upsilon(\theta)$ as in \eqref{eq:invCharExp} has an analytic continuation in the negative half-plane, and in this region $\Upsilon(\theta)<0$. Consequently, we can replace the upper bound on the tail probability of $\tau^{x-u}(0)$ by
\begin{equation*}
\mathbb{P}\left( \tau^{x-u}(0) > t\right) = \mathbb{P}\left( e^{\beta \tau^{x-u}(0)} > e^{\beta t} \right) \leq e^{-\beta t} \, e^{ (x-u)\Upsilon^{-1}(-\beta)},
\end{equation*}
for some $\beta > 0$, so that
\[ \int_T^\infty \mathbb{E}[Q^x(t) - Q^0(t)]\, {\rm d} t \leq e^{-\beta T}\, \frac{e^{x\Upsilon^{-1}(-\beta)}-1}{\beta\,\Upsilon^{-1}(-\beta)}. \]
Along similar lines we deduce
\[ |\Delta^{x,y}_T| \leq \frac{ e^{-\beta T}}{T}\, \frac{e^{x\Upsilon^{-1}(-\beta)} + e^{y\Upsilon^{-1}(-\beta)} -2}{\beta\,\Upsilon^{-1}(-\beta)}
\]
and
\[ |\Delta_T| \leq \frac{e^{-\beta T}}{T}\, \frac{\mathbb{E}[e^{Q(0)\Upsilon^{-1}(-\beta)}] + \mathbb{E}[e^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}] -2}{\beta\,\Upsilon^{-1}(-\beta)},\]
assuming that $\mathbb{E}[{\rm e}^{-y Q(0)}] < \iy$ for all $y>0$. The condition $\mathbb{E}[{\rm e}^{Q_\mu(\iy)\Upsilon^{-1}(-\beta)}]<\iy$ follows from Lemma \ref{lemma:workloadmoments}.
Hence, the error decays exponentially fast for light-tailed input processes.
\section{Proofs of Section \ref{sec:optimization}}
\label{app:proofs_optimization}
\subsection{Proof of Lemma \ref{lemma:strict_convexity}}
\begin{proof}
Since the term $\aaa\mu$ is convex, the strictness should come from the term $C_T(\mu)$.
Furthermore, observe that if a function $f_\mu(t)$ is convex for all $t\geq 0$, and strictly convex for all $t\geq\e$ for some $\e\in[0,T)$, i.e. for any $\mu_1,\mu_2>0$ and $a\in (0,1)$
\begin{equation*}
a\, f_{\mu_1}(t) + (1-a) f_{\mu_2}(t) > f_{a\mu_1+(1-a)\mu_2}(t),
\end{equation*}
then,
\begin{equation*}
a\int_0^T \, f_{\mu_1}(t)\, {\rm d} t + (1-a)\int_0^T f_{\mu_2}(t) {\rm d} t =
\int_0^T \, a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t
\end{equation*}
\begin{align*}
&= \int_0^\e a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t + \int_\e^T \, a f_{\mu_1}(t) + (1-a)f_{\mu_2}(t) {\rm d} t \\
&> \int_0^\e f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t + \int_\e^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t. \\
&= \int_0^T f_{a\mu_1+(1-a)\mu_2}(t) {\rm d} t.
\end{align*}
Hence, it suffices to prove the convexity of $\mathbb{E}[Q_\mu(t)]$ as a function of $\mu$ for all $t\geq 0$, and strict convexity for $t\geq \e$ for some $\e\in[0,T)$.
Let $\tau^x_{\mu}(0)$ denote the first passage time of level 0 is the process $Q_\mu$ with $Q(0)=x$. Then,
\begin{align}
Q_\mu(t) &= U(t)-\mu t + \max\left\{ x , -\inf_{s\leq t} [ U(s)-\mu s] \right\}\\
&=
\left\{
\begin{array}{ll}
x+U(t)-\mu t, & \text{if } t<\tau^x_{\mu}(0),\\
U(t)-\mu t -\inf_{s\leq t} [ U(s)-\mu s], & \text{if } t\geq \tau^x_{\mu}(0) ,
\end{array}\right.\label{eq:Qrep}
\end{align}
where
\begin{equation*}
\tau^x_{\mu}(0) := \inf\{ t \geq 0\,:\, x+U(t)-\mu t \leq 0\}
\end{equation*}
and $U(t)$ is a spectrally positive L\'evy process.
Fix $\mu_1, \mu_2>0$ and $a\in(0,1)$. Define $\mu_3 := a\mu_1+(1-a)\mu_2$, and
\begin{equation*}
D(t) := a Q_{\mu_1}(t) + (1-a) Q_{\mu_2}(t) - Q_{\mu_3}(t).
\end{equation*}
In order to prove strict convexity we have to show that $D(t) \geq 0$ for all $t\geq 0$, thereby implying $\mathbb{E} [D(t)] \geq 0$, i.e. convexity, for all $t\geq 0$, and $D(t)>0$ with positive probability for $t\in[\e,T]$, for some $\e \in[0,T)$.
We distinguish two cases: $x>0$ and $x=0$. \\
\\*
\textbf{The case $x>0$.}
We start by noticing that if $Q_{\mu_1}$, $Q_{\mu_2}$ and $Q_{\mu_3}$ experience the same input process $U(t)$, then by absence of negative jumps in $U(t)$, it holds that
\begin{equation}\label{eq:stochDom}
\tau^x_{\mu_2}(0) < \tau^x_{\mu_3}(0) < \tau^x_{\mu_1}(0).
\end{equation}
We use shorthand notation
\begin{equation*}
I_k(t) := \inf_{0\leq s\leq t}[U(s)-\mu_k s],
\end{equation*}
for $k=1,2,3$.
Using representation \eqref{eq:Qrep} of the workload process, we obtain
\begin{equation*}
D(t) = \left\{
\begin{array}{ll}
0,
& \text{if } t < \tau^x_{\mu_2}(0),\\
-(1-a)\left(x+I_2(t) \right),
& \text{if } \tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0),\\
a x - (1-a)I_2(t) + I_3(t),
& \text{if } \tau^x_{\mu_3}(0) \leq t < \tau^x_{\mu_1}(0),\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tau^x_{\mu_1}(0).
\end{array}
\right.
\end{equation*}
This partition allows us to spot when strict convexity can occur.
Note that by definition $t \geq \tau^x_{\mu_2}(0)$, $\inf_{0\leq s\leq t}[U(s)-\mu_2s] \leq x)=I_2(t)\leq x$, so that $D(t)\geq 0$ if $\tau^x_{\mu_2}(0) \leq t < \tau^x_{\mu_3}(0)$.
Moreover, by subadditivity of the infimum,
\[
I_3=\inf_{0\leq s\leq t}[U(s)-\mu_3s] = \inf_{0\leq s\leq t}[a(U(s)-\mu_1s)+(1-a)(U(s)-\mu_2s)]
\]
\begin{equation*}
\geq a \inf_{0\leq s\leq t}[U(s)-\mu_2s] \leq x) + (1-a) \inf_{0\leq s\leq t}[U(s)-\mu_2s] \leq x) = a I_1(t) + (1-a) I_2(t),
\end{equation*}
and hence $D(t)\geq 0$ for $t \geq \tau^x_{\mu_1}(0)$.
Using the same argument, we deduce
\begin{equation*}
ax - (1-a)I_2(t) + I_3(t) \geq a x - (1-a) I_2(t) + a I_1(t) + (1-a) I_2(t) = a(x + I_1(t)).
\end{equation*}
In particular for $t < \tau^x_{\mu_1}(0)$, this value is strictly positive.
As a result, $D(t)\geq 0$ for all $t\geq 0$.
On top of that $D(t) > 0$ for $t\in[\tau^x_{\mu_3}(0),\tau^x_{\mu_1}(0))$.
Accordingly, the latter implies strict positivity of $\mathbb{E} D(t)$, and therefore strict convexity of $\mathbb{E} Q_\mu(t)$, if the event $\{\tau^x_{\mu_3}(0)\leq t< \tau^x_{\mu_1}(0)\}$ occurs with positive probability.
That is,
\begin{align}
P(D(t)>0) &\geq P\left( a(x+I_1(t))\textbf{1}_{\{\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\}} > 0 \right)\nonumber\\
&= P\left( x+ I_1(t) > 0 , \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left( x+ I_1(t) > 0 | \tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right)\nonumber\\
&= P\left(\tau^x_{\mu_3}(0)\leq t < \tau^x_{\mu_1}(0)\right) = P(\tau^x_{\mu_3}(0)\leq t) - P(\tau^x_{\mu_1}(0) \leq t) > 0, \label{eq:strictConv}
\end{align}
by the stochastic dominance in \eqref{eq:stochDom}. To ensure the strict inequality in \eqref{eq:strictConv} we have to enforce the condition
\begin{equation}\label{eq:condition}
P(\tau^x_{\mu_1}(0)<T) > 0.
\end{equation}
\textit{Remark.}
An example illustrating the need for this condition is the case in which $U(t)$ is a compound Poisson process and $T < x/\mu_2 < x/\mu_1$. Then
\[Q_{\mu_k}(t) = x + U(t) - \mu_k t,\]
for all $t\in[0,T]$, since $U(t)\geq 0$ and therefore $\tau^x_{\mu_1}(0) > T$. Consequently, for all $a\in(0,1)$,
\[ a\,Q_{\mu_1} + (1-a)\,Q_{\mu_2}(t) = Q_{\mu_3}(t),\]
proving only convexity of $\mathbb{E} Q_{\mu}(t)$ and subsequently $\int_0^T \mathbb{E}[Q_\mu(t)]\,{\rm d} t$. In case $\sigma>0$, the probability in \eqref{eq:condition} is necessarily positive.
\\*
\noindent \textbf{The case $x=0$.}
By the fact that $\tau_{\mu}(0) = 0$ for all $\mu>0$, proving that $D(t)>0$ for in the case $x=0$ reduces to showing that the probability of
\begin{equation*}
D(t) = a I_1(t) + (1-a) I_2(t) - I_3(t)>0
\end{equation*}
happening is positive for all $t>0$. Define
\begin{equation*}
t_0 := \inf\{ t > 0\, :\, U(t) > 0 \},
\end{equation*}
and
\begin{equation*}
\tilde{\tau}_\mu := \inf\{ t > t_0\,: U(t) - \mu t \leq 0\}.
\end{equation*}
We note that $t_0$ as defined above, also defines the epoch of the start of a new excursion of the reflection $Q_\mu$ for all $\mu>0$. Namely,
\[U(s) \leq 0 \quad \Rightarrow\quad U(s) - \mu s \leq -\mu s \qquad \text{for all }0\leq s< t_0\]
\[\Rightarrow \inf_{0\leq s < t_0} [U(s)-\mu s] \leq -\mu t_0 \quad
\Rightarrow U(t_0) - \mu t_0 - \inf_{0\leq s < t_0} [U(s)-\mu s] \geq U(t_0) > 0\].
Then $Q_\mu(t_0-) = 0$ for all $\mu>0$.
By the virtue of the Strong Markov Property, not that $Q_\mu(t_0+t) {\;\buildrel{d}\over= \;} Q_\mu(t)$.
Hence we assume without loss of generality $t_0=0$.
Again, we have a stochastic dominance relation similar to \eqref{eq:stochDom}:
\begin{equation*}
\tilde{\tau}_{\mu_2} < \tilde{\tau}_{\mu_3} < \tilde{\tau}_{\mu_1},
\end{equation*}
for all $\mu_1<\mu_3<\mu_2$.
Then
\begin{equation*}
D(t) {\;\buildrel{d}\over= \;} \left\{
\begin{array}{ll}
0,
& \text{if } t < \tilde{\tau}_{\mu_2},\\
-(1-a)I_2(t),
& \text{if } \tilde{\tau}_{\mu_2} \leq t < \tilde{\tau}_{\mu_3},\\
(1-a)I_2(t) + I_3(t),
& \text{if } \tilde{\tau}\mu_3) \leq t < \tilde{\tau}\mu_1),\\
- a I_1(t) - (1-a) I_2(t)
+ I_3(t),
& \text{if } t \geq \tilde{\tau}\mu_1).
\end{array}
\right.
\end{equation*}
Clearly, $D(t)\geq 0$ for all $t\geq 0$ and
\begin{equation*}
-(1-a)I_2(t) + I_3(t) \geq a I_1(t) > 0,
\end{equation*}
for $\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}$.
Hence, in a similar manner to \eqref{eq:strictConv},
\begin{align}
P(D(t)>0) &\geq P\left( aI_1(t)\textbf{1}_{\{\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\}} > 0 \right)\nonumber\\
&= P\left( I_1(t) > 0 , \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left( I_1(t) > 0 | \tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right)\nonumber\\
&= P\left(\tilde{\tau}_{\mu_3} \leq t < \tilde{\tau}_{\mu_1}\right) = P(\tilde{\tau}_{\mu_3}\leq t) - P(\tilde{\tau}_{\mu_1} \leq t) > 0, \label{eq:strictConv2}
\end{align}
The last inequality is satisfied it $P(\tilde{\tau}_{\mu_1} < T) >0$, which is equivalent to $P( U(T) - \mu T \leq 0 ) >0$, a condition that is clearly true for all our choice of $U$.
In conclusion, for $x=0$, $\mathbb{E}[D(t)] >0$ and therefore $\mathbb{E}[Q_\mu(t)]$ is a strictly convex function of $\mu$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:min_convergence_mu}}
The proof of the proposition relies on the following auxiliary lemma, of which we include the proof for completeness.
\begin{lemma}\label{lemma:minimizerConvergence}
Consider the sequence of functions $f_n:\, [x_0,\infty) \to \mathbb{R}$ and let $f: [x_0,\infty) \to\mathbb{R}$ be the pointwise limit for some $x_0\in \mathbb{R}$.
Assume $f$ and $f_n$ are strictly convex for all $n$.
Furthermore, let $f(y) \to \infty$ for both $y\to x_0^+$ and $y\to \infty$.
If $x_n$ and $x$ are the minimizers for $f_n$ and $f$, respectively, then $x_n\to x$ for $n\to\infty$.
\end{lemma}
\begin{proof}
We start by showing that the sequence $x_n$ is bounded. Fix $u_l, u_r$ such that $x_0<u_l < x < u_r$. We claim that there exists a $N\in\mathbb{N}$ such that $x_n\in[u_l,u_r]$ for all $n \geq N$. First, we prove the upper bound on $x_n$. For any strictly convex function $h$ with minimizer $x_h$, the following statement holds true:
\begin{equation}\label{eq100}
x_h < u_r \quad \Leftrightarrow \quad h \text{ is strictly increasing at } u_r.
\end{equation}
The first implication follows from observing that $h(x_h) < h(y)$ for all $y> x^*$ and definition of convexity:
\[ 0<\frac{h(u_r)-h(x_h)}{u_r-x_h} \leq \frac{h(u_r+\de)-h(u_r)}{\de}, \]
for all $\de>0$. So that $h(u_r)<h(u_r+\de)$, i.e. $h$ is increasing at $u_r$. The converse follows immediately by observing that $h(u_r) < h(u_r+\de)$ for all $\de>0$, so that $x_h < u_r$.
Next, we show that $f_n$ must be increasing at $u_r$ for $n$ sufficiently large. By pointwise convergence of $f_n$ we have
\[ \lim_{n\to\infty} [f_n(u_r+\de) - f_n(u_r)] = f(u_r+\de) - f(u_r).\]
Let $w_r:= f(u_r+\de) - f(u_r)>0$. Then
\[ \exists N_r \in \mathbb{N}:\, \forall n\geq N_r:\, |[f_n(u_r+\de) - f_n(u_r)] - [f(u_r+\de)-f(u_r)] | < w_r/2.\]
Hence for $n\geq N_r$,
\[f(u_r+\de)-f(u_r) - w_r/2 < f_n(u_r+\de) - f_n(u_r) < f(u_r+\de)-f(u_r) + w_r/2\]
\[\Rightarrow 0 < w_r/2 < f_n(u_r+\de) - f_n(u_r).
\]
Hence by \eqref{eq100}, $x_n < u_r$ for sufficiently large $n$. Similarly, we argue
\begin{equation*}
x_h > u_l \quad \Leftrightarrow \quad h \text{ is strictly decreasing at } u_l,
\end{equation*}
for any strictly convex function $h$ with minimizer $x_h$. Note that $x_h > u_l$ implies $h(x_h) - h(u_l) < 0$ and for all $\de>0$ we get by strict convexity
\[\frac{h(u_l)-h(u_l-\de)}{\de} < \frac{h(x_h)-h(u_l)}{x_h-u_l} < 0,\]
by which $h(u_l-\de)>h(u_l)$, i.e. $h$ is decreasing in $u_l$. Moreover, if $h$ is decreasing at $u_l$, then it is decreasing for all $y < u_l$, by arguments similar to the above. Therefore, $h(u_l-\de)> h(u_l)$ for all $\de>0$ and it must hold that $x_h>u_l$. Define $f(u_l) - f(u_l-\de) :=w_l < 0$, then again by pointwise convergence, we have that
\[ \exists N_l \in \mathbb{N}:\, \forall n\geq N_l:\, |[f_n(u_l) - f_n(u_l-\de)] - [f(u_l)-f(u_l-\de)] | < w_l,\]
whereupon
\[ f_n(u_l) - f_n(u_l-\de) < f(u_l) - f(u_l-\de) + w_l = 2w_l < 0.\]
Hence, for sufficiently large $n$, we also have $x_n > u_l$. Fix $N = \max\{N_l,N_r\}$, then for $n\geq N$, $x_n\in( u_l,u_r)$. That is, the sequence $x_n$ is bounded. Therefore, by the theorem of Bolzano-Weierstrass, $x_n$ has to have a convergent subsequence. That is, there exists a sequence $n_k$ such that $n_k \to\infty$ and $x_{n_k}\to a$ as $k\to \infty$ for some $a \in [u_l,u_r]$.
We prove that every subsequence must converge to $x$ by contradiction. Suppose there exists a subsequence $n_k$ such that $x_{n_k}\to a\neq x$. Since, $x_n\in [u_l,u_r]$ for $n\geq N$, we may restrict our attention on the sequence of functions $\hat{f}_n:[u_l,u_r] \to \mathbb{R^+}$, consisting of the original function $f_n$ restricted to the domain $[u_l,u_r]$. To be precise $x_n = \arg\min_y f_n(y) = \arg\min_y \hat{f}_n(y)$ for $n\geq N$. Because $\hat{f}_n$ and $\hat{f}$ are bounded, we furthermore $\hat{f}_n \to \hat{f}$ uniformly.
Fix $\e>0$. By uniform convergence there exists an $K \in\mathbb{N}$ such that
\[ | \hat{f}_{n_k}( y ) - \hat{f}( y)| < \e /2,\quad \forall k\geq K_0,\ y \in[u_l,u_r].\]
Also, because $\hat{f}$ is convex, it is continuous, so that there exists a $\de := \de(\e)$ so that
\[ |z-y| < \de \quad \Rightarrow \quad |\hat{f}(z) - \hat{f}(y)| < \e/2.\]
Let $K_1$ be such that $|x_{n_k}-a| < \de$ for all $k\geq K_1$. Then for $k \geq K= \max\{K_0,k_1\}$ this implies.
\begin{align*}
|f_{n_k}(x_{n_k}) - f(a)| &= |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(a)| \\
&\leq |\hat{f}_{n_k}(x_{n_k}) - \hat{f}(x_{n_k}) + | \hat{f}(x_{n_k}) - f(a)| < \e/2 + \e/2 = \e.
\end{align*}
Hence we conclude $\lim_{k\to\infty} \hat{f}_{n_k}(x_{n_k}) = f(a)$.
Therefore,
\[ \limsup_{n\to \infty} f_n(x_n) \geq f(a) > f(x),\]
by minimality of $x$. However, $f_n(x_n) \leq f_n(x)$, which implies $\limsup_{n\to\infty} f_n(x_n) \leq \lim_{n\to\infty} f_n(x) = f(x)$, contradicting the strict inequality above. Hence we deduce $x=a$. Consequently, every subsequence of $x_n$ converges to $x$ and therefore $x_n\to x$ as $n\to \infty$.
Applying Lemma \ref{lemma:minimizerConvergence} to the functions $\Pi_T$ and $\Pi_\iy$ with $x_0=\la$, together with Lemma \ref{lemma:strict_convexity}, we obtain the result immediately.
\end{proof}
\subsection{Proof of Proposition \ref{prop:muBullet}}
\begin{proof}
Note that $\Pi_\infty$ is a smooth function.
By the first optimality condition $\Pi_\infty'(\mui)$ $= 0$.
We first prove that also $\Pi_T(\mu)$ is differentiable with respect to $\mu$ for all $\mu\geq 0$.
Recall \eqref{eq:PiT}, which defines the cost function as a combination of the accumulated expected transient queue length, and linear staffing costs.
The latter term is clearly differentiable, hence it remains to be proved that
\begin{equation*}
C_T(\mu) = \frac{1}{T}\int_0^\iy \mathbb{E}[Q_\mu(t)] \, {\rm d} t,
\end{equation*}
admits a derivative for all $\mu\geq 0$ with $T$ fixed.
This holds if and only if $\mathbb{E}[Q_\mu(t)]$ is differentiable for all $t\geq 0$.
Let $Q(0)= x\geq 0$.
Following \eqref{eq:Qlm},
\begin{align*}
\mathbb{E}[Q_\mu(t)] &= \mathbb{E}[X_\mu(t)] + \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big]\\
&= (\la-\mu)t+ \mathbb{E}\Big[ \max\{ x, \sup_{s\in[0,t]}\{- X_\mu(s)\} \}\Big],
\end{align*}
where the first term is differentiable.
Furthermore,
\begin{align*}
\mathbb{E}[ \max\{ x, \sup_{s\in[0,t]} \{ - X_\mu(s) \} \} ]
&= x + \int_x^\iy P(\sup_{s\in[0,t]} \{ - X_\mu(s) \} > u ){\rm d} u \\
&= x+\int_x^\infty P(\hat\tau^0(u) \leq t ) {\rm d} u,
\end{align*}
with $\hat{\tau}^0(u)$ as defined in \eqref{eq:transformedTau}.
Since $-X_\mu$ is a process with no positive jumps, we may apply Corollary VII3 of \cite{Bertoin1996}, which states that the following equivalence between measures holds:
\begin{equation}
s\,P( \hat\tau^0(u) \in ds ) du = u\,P( -X_\mu(s) \in du ) ds,
\end{equation}
so that
\begin{align}
\int_{u=x}^\infty P(\hat{\tau}^0(u) \leq t )\, {\rm d} u
&=
\int_{u=x}^\iy \int_{s=0}^t P( \hat\tau^0(u) \in ds ) {\rm d} u \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,\,P( -X_\mu(s) \in {\rm d} u ) {\rm d} s \\
&= \int_{u=x}^\iy \int_{s=0}^t\,s^{-1} u\,P( X_\mu(s) \in {\rm d} u ) {\rm d} s \\
&= \int_{s=0}^t s^{-1} \mathbb{E}[ \max\{x, X_\mu(s)\} ] {\rm d} s\\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( X_\mu(s)/s > v ) {\rm d} v {\rm d} s\\
&= \int_{s=0}^t \int_{v=x/s}^\iy P( U(\la s)/s > v + \mu ) {\rm d} v {\rm d} s\\
&= \int_{s=0}^t \int_{w=x/s+\mu}^\iy P( U(\la s)/s > w) {\rm d} w {\rm d} s,
\end{align}
where the interchange of integrals is justified by Fubini's theorem and this last form is differentiable with respect to $\mu$.
Substituting $Q(0)$ for $x$ straightforwardly yields differentiability of the complete cost function $\Pi_T$ for all $T$.
Consequently we invoke the first optimality condition for $\muT$ to find
\begin{align*}
0=\Pi_T'(\muT)
&= \Pi_\infty'(\muT) + \Psi_T'(\muT) + O(1/T^2)\\
&= \Pi_\infty'(\mui) + \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi_T'''(\xi)+\Psi_T'''(\xi) \right] + O(1/T^2)\\
&= \Psi_T'(\mui) + (\muT-\mui)\left[ \Pi_\infty''(\mui) + \Psi_T''(\mui) \right] \\
&\qquad + \frac{1}{2}(\mu_T-\mui)^2\left[\Pi'''(\xi)+\Psi_T'''(\xi)\right] + O(1/T^2).
\end{align*}
for some $\xi \in [\muT,\mui]$. Rearranging this gives
\begin{align*}
\muT-\mui &= \frac{-\Psi_T'(\mui)}{\Pi_\infty''(\mui)+\Psi_T''(\mui) + \frac{1}{2}(\muT-\mui)(\Pi_\infty'''(\muT)+\Psi_T'''(\xi))} + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} \left[1 - \frac{\Psi_T''(\mu)}{\Pi_\infty''(\mui)} - \frac{\muT-\mu_\infty}{2}\frac{\Pi_\infty'''(\mui)+\Psi_T'''(\mui)}{\Pi_\infty''(\mui)}\right] + O(1/T)\\
&= {-}\frac{\Psi_T'(\mui)}{\Pi_\iy''(\mui)} [1 + o(1)]
\end{align*}
for $T\to\infty$, since both $\mu_T - \mu_\infty$ and $\Psi_T''(\mui)$ are $o(1)$.
Let
\begin{equation*}
\mu_\bullet := \lim_{T\to\iy} \frac{T \Psi_T'(\mui)}{\Pi_\iy''(\mui)}.
\end{equation*}
By \eqref{eq:mainResult} we have
\begin{equation*}
T \Psi'_T(\mu) = {-} \frac{\mathbb{E}[Q(0)^2]}{2(\mu-\la)^2} + \frac{\la u_3}{3(\mu-\la)^3} + \frac{3\la^2u_2^2}{4(\mu-\la)^4}.
\end{equation*}
Together with
\begin{equation*}
\Pi_\iy''(\mu) = \frac{\la u_2}{(\mu-\la)^3}
\end{equation*}
and \eqref{eq:muInf} we obtain the expression for $\mu_\bullet$ in \eqref{eq:muBullet}.
\end{proof}
\subsection{Proof of Proposition \ref{prop:optimalitygap_mui}}\label{sec:proofProp4}
\begin{proof}
We upper bound the optimality gap by using the decomposition in \eqref{eq:decomposition}.
\begin{align}
|\Pi_\iy^\star - \Pi_T^\star| &= \left|\hat{\Pi}_T(\mu_\infty) + \Delta_T(\mui) - \hat{\Pi}_T(\muT) - \Delta_T(\muT)\right|\nonumber\\
&\leq |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + |\Delta_T(\mui)| + |\Delta_T(\muT)|\nonumber\\
&= |\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| + O(1/T^2),
\end{align}
since $\Delta_T(\mu) = O(1/T^2)$ by Proposition \ref{prop:truncation_error}. Next, we find an upper bound for $|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)|$, with $\hat{\Pi}_T(\cdot)$ as in \eqref{eq:decomposition}, in terms of the difference between $\gamma$ and $\beta$.
For simplicity, denote $\hat{\gamma} = \gamma - \la$ and $\hat{\beta} = \beta-\la$, implying $\hat{\gamma}-\hat{\beta}=\gamma-\beta$. Then using the expression of in \eqref{eq:mainResult} we get
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &=
\left| \aaa(\hat{\gamma}-\hat{\beta})
+\left(\frac{\la u_2}{2} + \frac{\mathbb{E}[Q(0)^2]}{2T}\right)\left(\frac{1}{\hx}-\frac{1}{\hy}\right) \right. \\
& \qquad \left. -\frac{\la^2 u_2^2}{4T}\left(\frac{1}{\hx^3}-\frac{1}{\hy^3}\right)
-\frac{\la u_3}{6T}\left(\frac{1}{\hx^2} - \frac{1}{\hy^2}\right)
\right|.
\end{align*}
Furthermore, we have
\begin{align*}
\frac 1 \hx - \frac 1 \hy &= -\frac{\hx-\hy}{\hy^2} + \frac{(\hx-\hy)^2}{\hy^3} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^2} - \frac 1 {\hy^2} &= -\frac{2(\hx-\hy)}{\hy^3} + \frac{3(\hx-\hy)^2}{\hy^4} + O\left((\gamma-\beta)^3\right),\\
\frac 1 {\hx^3} - \frac 1 {\hy^3} &= -\frac{3(\hx-\hy)}{\hy^4} + \frac{6(\hx-\hy)^2}{\hy^5} + O\left((\gamma-\beta)^3\right).\\
\end{align*}
Substituting these yields
\begin{align*}
|\hat{\Pi}_T(\gamma) - \hat{\Pi}_T(\beta)| &= \left|(\gamma-\beta)\left[ \aaa - \frac{\la u_2}{2 \hy^2} + \frac{1}{2T\hy^2}\left(\mathbb{E}[Q(0)^2] + \frac{3\la^2 u_2^2}{2\hy^2} + \frac{2\la u_3}{3 \hy}\right)\right]\right. \\
&\qquad \left. - (\gamma-\beta)^2\left[ \frac{\la u_2}{2 \hy^3} + \frac{1}{2T\hy^3}\left(\mathbb{E}[Q(0)^2] - \frac{3\la^2 u_2^2}{\hy^2} - \frac{\la u_3}{\hy}\right)\right]\right| \\
& \qquad \qquad + O\left((\gamma-\beta)^3\right).
\end{align*}
Given that $\muT = \mui + \mu_\bullet/T + o(1/T)$, we find
\begin{align*}
|\hat{\Pi}_T(\mui) - \hat{\Pi}_T(\muT)| &= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\mui-\la )^2}\right) + O(1/T^2)\\
&= \frac{|\mu_\bullet|}{T}\left(\aaa - \frac{\la u_2}{2(\sqrt{\la u_2/2\aaa})^2}\right) + O(1/T^2) = O(1/T^2),
\end{align*}
which concludes the proof.
\end{proof}
\end{subappendices}
\chapter{A blood bank model}
\begin{chapterstart}
We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to
independent compound Poisson processes.
Blood is perishable, that is, blood can only be kept in storage for a limited amount of time.
Furthermore, demand for blood is impatient, that is, a demand for blood may be canceled if it cannot be satisfied soon enough.
For a range of perishability functions and demand impatience functions,
we derive the steady-state distribution of the blood inventory level.
Moreover, we deduce fluid and diffusion limits for the inventory process as the arrival rates of of the compound Poisson processes grow indefinitely.
These scaling limits in turn provide normal approximations for the performance of large-scale systems.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{A blood bank model with perishable blood and demand impatience}\\
\textit{Shaul Bar-Lev, Onno Boxma, Britt Mathijsen \& David Perry}\\
Submitted to \textit{Stochastic Systems}
\end{flushright}
\newpage
\section{Introduction}
This chapter is devoted to the study of a stochastic blood bank model in which amounts of blood are offered and demanded according to stochastic processes, and in which blood is perishable (that is, blood can only be kept for a limited amount of time) and demand for blood is impatient (that is, a demand request for blood may be canceled if it cannot be satisfied soon enough).
Let us first provide some background, and subsequently sketch the blood bank model in some more detail.\\
\\*
\textbf{Practical background.}
One of the major issues in securing blood supply to patients worldwide is to
provide blood of the best achievable quality, in the needed quantities.
In most countries, blood, which is collected as whole blood units from human
donors, is separated into different components which are subsequently stored under different storage conditions according to their biological
characteristics, functions and respective expiration dates. Blood units and components are ordered by local hospital blood banks (LBB) from the Central Blood Bank (CBB) according to their operational needs. The CBB
has to run its inventory and supply according to these requests and to the
need to keep sufficient stock for immediate release in emergency situations.
It also has to perform tests to determine the unit's blood type
and to detect the presence of various pathogens which are able to cause
transfusion-transmitted diseases, such as Hepatitis B, Hepatitis C, Human Immunodeficiency Virus (HIV) and Syphilis, see e.g.~Steiner et al.~\cite{Steiner2010}.
Blood consists of several components: red blood cells, plasma and
plate-lets.
In addition, there are $8$ blood groups (types):
$O^{+},O^{-},A^{+},A^{-},B^{+}$ ,$B^{-},AB^{+}$, $AB^{-}$ ($-$ means Rh
negative) where the interrelationship between the transfusion issuing
policies among the $8$ types is quite intricate.
It turns out that each of
the negative types can satisfy the corresponding $+$ type, but not vice
versa.
Blood components are perishable as red blood cells can be used for only $35$ to $42$
days and platelets for only $5$ days (plasma, however, can be frozen and
kept for one year).
Accordingly, if red blood cells and particularly platelets are not
used for blood transfusion within their expiration dates, then they perish.
In most developed countries demand requirements of about $50.000$
blood donations are needed per one million persons per year. About 95\% of
these donations are aggregated by CBBs and the remaining 5\% by LBBs.
Blood units stored at the CBB are usually ordered by LBBs for planned elective
surgeries. However, as it happens rather frequently, elective surgeries
turn out to become emergency ones due to various conditions of the
patient involved. In such cases, hospitals use their own local blood banks
to supply the demand, and they cancel the required demand
from the CBB; this is what we refer to as demand impatience.
A good review on supply chain management in blood products appears in Beli\"{e}n \& Forc\'{e} \cite{Belieen2012} and the
references cited therein. Other relevant studies are
Ghandforoush and Sen \cite{Ghandforoush2010} \&
Stanger et al.\ \cite{Stanger2012}.
\newpage
\noindent
\textbf{Inventory model.}
In this chapter we consider the analysis of blood perishability and demand impatience, concentrating on only one
blood type. We do this by considering the stochastic inventory processes $\{X_b(t)\}_{t \geq 0}$, with $X_b(t)$ the amount of blood
kept in storage at time $t$, and $\{X_d(t)\}_{t \geq 0}$, with $X_d(t)$ the amount of demand for blood (the shortage) at time $t$.
If $X_b(t)>0$ then $X_d(t)=0$, and if $X_d(t)>0$ then $X_b(t)=0$.
We assume that amounts of blood arrive according to a Poisson process,
and that requests for blood arrive according to another, independent, Poisson process.
The delivered and requested amounts of blood are assumed to be random variables.
We represent the perishability of blood by letting the amount of blood, when positive, decrease in a state-dependent way:
if the amount is $v$, then the decrement rate is $\xi_b v + \alpha_b$.
The $\xi_b$ factor is motivated by the fact that a large amount of blood
suggests that some of the blood has been present for quite a while -- and hence there is a relatively high perishability rate when much blood is in inventory.
The $\alpha_b$ factor provides additional modeling flexibility.
One can in this way represent the blood perishability more accurately;
but the $\alpha_b$ term could also, e.g., represent a fluid demand rate of individuals or organizations,
which contact the CBB directly, and that is only satisfied when there is inventory.
Similarly, we represent the demand impatience by a decrement rate $\xi_d v + \alpha_d$.
The $\xi_d$ factor is motivated by the following fact. When there is a large shortage (demand) of blood,
there are probably many patients waiting for blood, so many patients that might become impatient
(that is, they could recover, or die, or become in need of emergency surgery)
leading to a cancellation of the required demand from the CBB.
Again, the $\alpha_d$ factor provides additional modeling flexibility;
it not only allows us to represent demand impatience more accurately, but it could also, e.g.,
represent additional donations of individuals in times of blood shortage.
The inclusion of both the perishability factor $\xi_b v + \alpha_b $ and the demand impatience factor
$\xi_d v + \alpha_d$ makes the analysis of the ensuing model mathematically
quite challenging, but
leads to a very general model that contains many well-known models as special cases.
Our two-sided stochastic process, with both upward and downward jumps,
and with the rather general slope factors $\xi v + \alpha$, could represent a quite large class of stochastic phenomena.
It should for example be noted that this model is a two-sided generalization of the well-known shot-noise model that describes certain physical phenomena, see \cite{Keilson1959}).
In some of our calculations we remove either the $\xi$ factors or the
$\alpha$ factors, and this results in easier calculations and more explicit results.
Our main results are: (i) Determination of the steady-state distributions
of the amounts of blood and of demand in inventory; in particular, we present a detailed analysis of the case in which
the delivered and requested amounts of blood are both exponentially distributed. (ii)
Expressions for mean amounts of blood and demand in storage, and for the probability of not being able to satisfy demand.
(iii) We obtain the fluid and diffusion limits of the blood inventory process, providing in particular sufficient conditions for the limit process to be an Ornstein-Uhlenbeck process.
\\*
\\
\noindent
\textbf{Structure of the chapter.}
The chapter is organized as follows:
Section~\ref{modeldesc} presents a detailed model description.
A steady-state analysis of the densities of demand and of blood amount in storage is
contained in Section~\ref{analysis}, including the special case of exponentially distributed delivered and requested blood amounts when $\alpha_b=\alpha_d=0$ (i.e., pure proportionality).
The fluid and diffusion scalings are discussed in Section~\ref{sectionscaling},
and in Section~\ref{numericals} we present numerical results for certain performance measures like mean net amount of blood and the probability that there is a shortage of blood.
These results indicate, among other things, that the probability that there is a shortage of blood can be accurately approximated via a normal approximation, based on the Ornstein-Uhlenbeck process appearing in the diffusion scaling.
Section~\ref{conclus} contains some conclusions and suggestions for further research.
\section{Model description}
\label{modeldesc}
We consider the following highly simplified model of a blood bank, restricting ourselves to only one type of blood.
Blood amounts arrive according to a Poisson process with rate $\lambda_b$. The amounts which successively arrive are independent, identically
distributed random variables $B_1,B_2,\dots$ with distribution $F_b(\cdot)$; $\bar{F}_b(x) = 1 - F_b(x)$.
Demands for blood arrive according to a Poisson process with rate $\lambda_d$. The successive demand amounts are independent, identically
distributed random variables $D_1,D_2,\dots$ with distribution $F_d(\cdot)$; $\bar{F}_d(x) = 1 - F_d(x)$.
We view these amounts as continuous quantities, measured in, for instance liters.
If there is enough blood for a demand, then that demand is immediately satisfied.
If there is some blood, but not enough to fully satisfy a demand, then that demand is partially satisfied, using all the available blood.
The remainder of the demand may be satisfied later.
Blood has a finite expiration date. We make the assumption
that if the total amount of blood present is $v>0$,
then blood is discarded -- because of its finite expiration date -- at a rate $\xi_b v + \alpha_b$, so linear in $v$.
Blood demands have a finite patience.
We make the assumption
that if the total amount of demand present is $v>0$,
then demand disappears -- because of its finite patience -- at a rate $\xi_d v + \alpha_d$, so linear in $v$.
Notice that {\em either} the total amount of blood present, {\em or} the total amount of demands,
is zero, {\em or} both are zero; they cannot be both positive. Hence we can easily in one figure depict
the two-sided process $\{X(t)\}_{t \geq 0}$ $=\{(X_b(t),X_d(t))\}_{t \geq 0}$
of total blood and total demand amounts present at any time $t$, as we have done in Figure \ref{fig:samplePath}.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = -0.2,
xmax = 4,
ymin = -5,
ymax = 11,
ticks = none,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1
]
\addplot[black,thick] file {tikz/sample_path2.txt};
\node at (axis cs: 0.33,-4.4) {$X_d(t)$};
\node at (axis cs: 0.33,10.2) {$X_b(t)$};
\draw[-stealth] (axis cs: 0,0) -- (axis cs:0,-5);
\end{axis}
\node at (7.05,1.47) {$t$};
\end{tikzpicture}
\caption{Sample path of net amount of blood available as a function of time.}
\label{fig:samplePath}
\end{figure}
For our purposes, we are mainly interested in the characteristics of the process described above in stationarity.
Let us denote by $X_d$ the steady-state total amount of demand and by $X_b$ the steady-state total amount of blood present, with corresponding density functions $f(\cdot)$ and $g(\cdot)$, respectively. Notice that these are defective densities;
we have $\int_{0^+}^{\infty} f(v) {\rm d} v = \pi_d = \mathbb{P}({\rm demand} ~ > ~ 0)$
and
$\int_{0^+}^{\infty} g(v) {\rm d} v = \pi_b = \mathbb{P}({\rm blood} ~ > ~ 0)$.
If $\alpha_b=\alpha_d=0$, then neither $X_b$ nor $X_d$ has probability mass at zero, and $\pi_b+\pi_d=1$
(when there is only a very small amount $v$ present, the "decay" rate $\xi_b v$ or $\xi_d v$ is very small).
However, if $\alpha_b$ and/or $\alpha_d$ is positive, then there is a positive probability $\pi_0$ of being in $0$.
When $\xi_d$ and $\xi_b$ are positive, existence of these steady-state densities is obvious;
otherwise, the conditions for the existence of the steady-state distributions require some discussion, see Section~\ref{sectionvariant}.
\section{Steady-state analysis}
\label{analysis}
In this section we present a global approach towards determining $f(\cdot)$ and $g(\cdot)$ in the most general form of our model.
Using the Level Crossing Technique (LCT),
we derive two integral equations in $f(\cdot)$ and $g(\cdot)$.
Before attempting to solve these equations,
we consider a few important performance measures which can be expressed in $f(\cdot)$
and $g(\cdot)$, $\pi_0$
and the mean length of time during which, uninterruptedly, there is a positive amount of blood (respectively demand).
The latter could be viewed as the busy period of the $X_b$ process (respectively of the $X_d$ process).
First we consider the density $g(\cdot)$ of the amount of blood.
We equate the rate at which some positive blood level $v$ is upcrossed and downcrossed, respectively.
LCT leads to the following integral equation: for $v>0$,
\begin{align}
&\lambda_b \int_0^v g(y) \bar{F}_b(v-y) {\rm d}y
+
\lambda_b \int_0^{\infty} f(y) \bar{F}_b(v+y) {\rm d}y
+ \pi_0 \lambda_b \bar{F}_b(v)
\nonumber
\\
&\qquad =
\lambda_d \int_v^{\infty} g(y) \bar{F}_d(y-v) {\rm d}y
+
(\xi_b v + \alpha_b) g(v).
\label{eq:blood}
\end{align}
Here the three terms in the left-hand side represent the rate of crossing level $v$ from below;
the first term corresponds to a jump from a blood inventory level between $0$ and $v$,
whereas the second term corresponds to a jump from a shortage level, and the third term
corresponds to a jump from level $0$.
The two terms in the right-hand side represent the rate of crossing level $v$ from above;
the first term corresponds to a jump from above $v$, and the second term to a smooth crossing.
Next, we consider the density $f(\cdot)$ of the amount of demand (shortage).
We equate the rate at which some positive demand level $v$ is upcrossed and downcrossed, respectively.
LCT leads to the following integral equation: for $v>0$,
\begin{align}
& \lambda_d \int_0^v f(y) \bar{F}_d(v-y) {\rm d}y
+
\lambda_d \int_0^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y
+ \pi_0 \lambda_d \bar{F}_d(v)
\nonumber
\\
&\qquad =
\lambda_b \int_v^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y
+
(\xi_d v + \alpha_d) f(v).
\label{eq:demand}
\end{align}
It should be noted that these two, coupled, equations are symmetric (swap $f$ and $g$, and the $b$ and $d$ parameters).
In general, it appears to be very difficult to solve these integral equations.
In Section~\ref{sectionexp} we assume that
both $F_b(\cdot)$ and $F_d(\cdot)$ are exponential.
In that case, we are able to obtain explicit expressions of $f(\cdot)$ and $g(\cdot)$, in terms of hypergeometric functions.
In Section~\ref{gener} we consider the case that $F_b(\cdot)$ and $F_d(\cdot)$ are Coxian distributions,
a class of distributions that lies dense in the class of all distributions of non-negative random variables,
and that is suitable for handling the above coupled integral equations via Laplace transforms (LT).
We are able to transform (\ref{eq:blood}) and (\ref{eq:demand})
into inhomogeneous first-order differential equations in the LTs of $f(\cdot)$ and $g(\cdot)$,
and thus to obtain those LTs.\\
\\*
\noindent
\textbf{A few simple performance measures.}
Without solving \eqref{eq:blood}-\eqref{eq:demand} explicitly, we are able to deduce some characteristics of the steady-state inventory level.
First, we can relate $\pi_0$ to the densities $f(\cdot)$ and $g(\cdot)$; see Proposition~\ref{prop. emptiness} below.
Subsequently we express the mean length of time during which there is, uninterruptedly, a positive amount of blood present (we call this the non-emptiness period of the inventory system), into $f(\cdot)$, $g(\cdot)$ and $\pi_0$.
We do the same for the mean length of time during which there is, uninterruptedly, a positive demand, i.e., the non-emptiness period of the demand process, see Proposition~\ref{prop: empty}.
\begin{proposition}
\label{prop. emptiness}Let $\pi _{0}\ $be\ the steady-state atom probability
of the zero period. Then%
\[
\pi _{0}=\frac{\alpha _{d}f(0)+\alpha _{b}g(0)}{\lambda
_{d}+\lambda _{b}}.
\]
\end{proposition}
\begin{proof}
Substitute $v=0$\ in \eqref{eq:blood} and \eqref{eq:demand} and take the sum. The result is
obtained after several steps of elementary algebra.
\end{proof}
The result introduced in the proposition above is very intuitive. By LCT, $%
\alpha _{d}f(0)+\alpha _{b}g(0)$\ is the rate at which
level $0$ is reached (i.e., the process will now really stay at $0$ for a while),
so that $[\alpha _{d}f(0)+\alpha
_{b}g(0)]^{-1}$\ is the expected length of time between two successive times level $0$ is reached by the fluid.
More precisely, the \textit{zero periods} and \textit{non-zero periods}
generate an alternating renewal process whose expected cycle length is $%
[\alpha _{d}f(0)+\alpha _{b}g(0)]^{-1}$. The expected length of the zero
period is $[\lambda _{d}+\lambda _{b}]^{-1}$, since the end of the zero
period is terminated at the moment of the next jump. But the jump process is
a Poisson process with rate $\lambda _{d}+\lambda _{b}$. Now the renewal
reward theorem simply says that
\[
\pi _{0}=\frac{\mathbb{E}[\text{zero period}]}{\mathbb{E}[\text{cycle}]}.
\]%
In preparation of the next proposition,
for the process $\{X(t)\}_{t\geq 0}$ we
define a modified process $\{X_{m}(t)\}_{t\geq 0}$, where $X_{m}$ is constructed by deleting the zero-periods (only the zero periods, not the emptiness periods) from $X$ and gluing together the \textit{non-zero periods}. The modified
process is $X_m$ such that $X_{m}(t)=X_{d}(t)\mathbbm{1}_{\{X_{d}(t)>0\}}+X_{b}(t)\mathbbm{1}_{\{X_{b}(t)>0\}}$ where by definition
of the model $\{X_{d}(t)>0\}\Rightarrow \{X_{b}(t)=0\}$ and $%
\{X_{b}(t)>0\}\Rightarrow \{X_{d}(t)=0\}$.
\begin{proposition}
\label{prop: empty}Let $B_{b}$\ and $I_{b}$ be the generic non-emptiness
period and the emptiness period, respectively, of the inventory system.
Similarly, let $B_{d}$\ and $I_{d}$ be the generic non-emptiness period and the
emptiness period, respectively, of the demand process. Then%
\[
{\rm (i)}\left\{
\begin{array}{l}
\ \mathbb{E} [B_{b}]=\frac{1-\pi _{0}}{\alpha _{b}g(0)+\lambda _{d}\int_{0}^{\infty }%
\bar{F}_{d}(y)g(y){\rm d} y}, \\
\ \mathbb{E} [B_{d}]=\frac{1-\pi _{0}}{\alpha _{d}f(0)+\lambda _{b}\int_{0}^{\infty }%
\bar{F}_{b}(y)f(y){\rm d} y}
\end{array}%
\right.
\]%
and
\\
\[
{\rm (ii)}\left\{
\begin{array}{l}
\ \mathbb{E}[I_{b}]=\frac{1}{\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y)dy+\lambda
_{b}\pi _{0}}
- \mathbb{E}[B_b] ,\\
\ \mathbb{E}[I_{d}]=\frac{1}{\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g(y)dy+\lambda
_{d}\pi _{0}}
-\mathbb{E}[B_d] .
\end{array}%
\right.
\]
\end{proposition}
\begin{proof}
(i) Consider the non-emptiness period of the inventory system.
The steady-state densities of the inventory system and the demand process of $X_m$\ are given by%
\[
g_{m}(x)=\frac{g(x)}{1-\pi _{0}},\ \ \ \ \ f_{m}(x)=\frac{f(x)}{1-\pi _{0}},
\]%
respectively. At the end of the non-emptiness period of the inventory system there are two disjoint
ways (disjoint events) to downcross level $0+$. Either level $0$ is
downcrossed by a negative jump or level $0+$ is reached by the fluid
reduction (both in $X_m$). The rate of the first
event\ is $\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y$\ and by
LCT the rate of the second event is $\alpha _{b}g_{m}(0)$. Since the
events are disjoint, the rate of downcrossings of level $0+$ is $\lambda
_{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y+\alpha _{b}g_{m}(0)$. That
means that the expected length of the non-emptiness period is given by $%
[\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y+\alpha
_{b}g_{m}(0)]^{-1}$. Thus%
\[
\mathbb{E}[B_{b}]=\frac{1-\pi _{0}}{\alpha _{b}g(0)+\lambda _{d}\int_{0}^{\infty }\bar{%
F}_{d}(y)g(y){\rm d} y}.
\]%
The expression for $\mathbb{E}[B_{d}]$ is obtained by symmetry.
\\
(ii) Define\ a \textit{cycle} in the real process $X$ (not the
modified process $X_m$) as the time between two upcrossings of
level $0+$. By definition, the emptiness period plus the non-emptiness
period is a cycle in $X$. That means that the expected length of
the emptiness period is the expected length of the cycle\ minus the expected
length of the non-emptiness period. The non-emptiness period in $X$
and in $X_m$ are identical and the length of the expected cycle
is $[\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y){\rm d} y+\lambda _{b}\pi
_{0}]^{-1}$, since $\lambda _{b}\int_{0}^{\infty }\bar{F}%
_{b}(y)f(y){\rm d} y+\lambda _{b}\pi _{0}$\ \ is the rate of the upcrossings of
level $0+$.\ \ We obtain%
\[
\mathbb{E}[I_{b}] + \mathbb{E}[B_b] = \frac{1}{\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y)dy+\lambda
_{b}\pi _{0}},
\]
yielding $\mathbb{E}[I_b]$.
$\mathbb{E}[I_{d}]$\ is obtained by symmetry.
\end{proof}
For the special case in which $\xi_b=\xi_d=\xi$ and $\alpha_b=\alpha_d = 0$, we are able to deduce that the expected steady-state inventory level $\mathbb{E}[X]$ has a simple form.
\begin{proposition}\label{prop:mean_inventory}
If $\xi_b=\xi_d=\xi$ and $\alpha_b=\alpha_d=0$, then
\begin{equation}
\mathbb{E}[X] = m/\xi,
\end{equation}
where $m=\lambda_b\mathbb{E}[B] - \lambda_d\mathbb{E}[D]$.
\end{proposition}
\begin{proof}
We study the discrete-time embedding of the blood inventory process \\ \noindent $\{X_k\}_{k\geq 1}$, where $X_k$ denotes the blood inventory level \textit{just before} the $k^{th}$ arrival (either blood or demand).
Suppose the process is in steady state.
By the PASTA property, we have that $X_k {\;\buildrel{d}\over= \;} X$ for all $k\geq 1$.
Also, the process $\{X_k\}_{k\geq 1}$ constitutes a Markov chain, of which the evolution is characterized by the recursion
\begin{equation}
X_{k+1} = \left( X_k + \mathbbm{1}_{k,b}B_k - \mathbbm{1}_{k,d} D_k \right)\cdot {\rm e}^{-\xi A_k},
\label{eq:X_recursion}
\end{equation}
where $\mathbbm{1}_{k,b}$ and $\mathbbm{1}_{k,d}$ denote the indicator function of the event that the $k^{th}$ arrival is a blood or demand arrival, respectively.
Remark that the relation holds for both $X_k \geq 0$ and $X_k <0$.
Furthermore, $B_k$ and $D_k$ denote the amount of blood or demand in the $k^{th}$ jump, respectively, and $A_k$ denotes the interarrival time between the $k^{th}$ and $(k+1)^{th}$ arrival.
Note that $A_k$ is the minimum of two exponentially distributed random variables with rate $\lambda_b$ and $\lambda_d$, so that $A_k$ is exponentially distributed with rate $\lambda_b+\lambda_d$.
Next, we take the expectation on both sides of \eqref{eq:X_recursion}, which gives
\begin{equation}
\mathbb{E}[X_{k+1}] = \big( \mathbb{E}[X_k] + p_{k,b}\mathbb{E}[B] - p_{k,d}\mathbb{E}[D]\big)\,\mathbb{E}\big[{\rm e}^{-\xi A_k}\big].
\label{eq:X_recursion_mean}
\end{equation}
Here, we used independence between Poisson processes and their jump sizes, and their memoriless property, and $p_{k,b} = \lambda_b/(\lambda_b+\lambda_d)$ and $p_{k,d} = \lambda_d/(\lambda_b+\lambda_d)$ denote probability of the $k^{th}$ jump being either a blood delivery or demand, respectively.
Since $X_{k} {\;\buildrel{d}\over= \;} X$, we have $\mathbb{E}[X_{k+1}] = \mathbb{E}[X_k] = \mathbb{E}[X]$, and thus we may rewrite \eqref{eq:X_recursion_mean} as
\begin{equation}
\mathbb{E}[X] = \left( \mathbb{E}[X] + \frac{\lambda_b\mathbb{E}[B] - \lambda_d\mathbb{E}[D]}{\lambda_b+\lambda_d} \right)\cdot\frac{\lambda_b+\lambda_d}{\lambda_b+\lambda_d+\xi},
\end{equation}
from which we easily deduce $\mathbb{E}[X] = (\lambda_b\mathbb{E}[B] - \lambda_d \mathbb{E}[D])/\xi = m/\xi$.
\end{proof}
\subsection{The exponential case}
\label{sectionexp}
\textbf{Density functions.}
We assume in this section that $\bar{F}_b(x) = {\rm e}^{-\mu_b x}$
and
$\bar{F}_d(x) = {\rm e}^{-\mu_d x}$.
Let $\rho_d:= \lambda_d/\mu_d$ and $\rho_b:= \lambda_b/\mu_b$ denote the expected amount of demand requested, and amount of blood delivered into the system, per time unit.
Moreover, we take $\alpha_b = \alpha_d = 0$.
Under these assumptions, we can
solve (\ref{eq:demand}) and (\ref{eq:blood}) explicitly.
Equations (\ref{eq:blood}) and (\ref{eq:demand}) reduce to:
\begin{align}
&\lambda_d \int_0^v f(y) {\rm e}^{-\mu_d(v-y)} {\rm d}y
+
\lambda_d {\rm e}^{-\mu_d v} \int_0^{\infty} g(y) {\rm e}^{-\mu_d y} {\rm d}y \nonumber \\
&\qquad =
\lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y
+
\xi_d v f(v),
\label{eq:demand2}
\end{align}
\begin{align}
&\lambda_b \int_0^v g(y) {\rm e}^{-\mu_b(v-y)} {\rm d}y
+
\lambda_b {\rm e}^{-\mu_b v} \int_0^{\infty} f(y) {\rm e}^{-\mu_b y} {\rm d}y \nonumber \\
& \qquad =
\lambda_d \int_v^{\infty} g(y) {\rm e}^{-\mu_d(y-v)} {\rm d}y
+
\xi_b v g(v),
\label{eq:blood2}
\end{align}
for $v>0$.
In our analysis, we concentrate on the derivation of $f(v)$. Notice that, once $f(\cdot)$ has been determined, $g(\cdot)$ follows by swapping parameters (symmetry).
In Appendix \ref{app:transformation_int} we show how the integral equations \eqref{eq:demand2}-\eqref{eq:blood2} can be translated into the following decoupled second order differential equations:
\begin{align}
\xi_d v f''(v) &+ \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber \\
& \qquad + \left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0
\label{eq:demand6}
\end{align}
and
\begin{align}
\xi_b v g''(v) &+ \left(2\xi_b -\lambda_d -\lambda_b + \mu_b\xi_dv -\mu_d \xi_b v\right)g'(v) \nonumber \\
& \qquad + \left(\mu_b\xi_b -\mu_d\xi_b -\mu_b\lambda_d + \mu_d \lambda_b -\mu_d \mu_b \xi_b v\right)g(v) =0,
\label{eq:blood6_1}
\end{align}
with the additional constraint (obtained by applying the level crossing identity for level $v=0$ in either (\ref{eq:demand2}) or (\ref{eq:blood2})):
\begin{equation}
\lambda_b \int_0^{\infty} f(y) {\rm e}^{-\mu_b y} {\rm d}y
=
\lambda_d \int_0^{\infty} g(y) {\rm e}^{-\mu_dy} {\rm d}y .
\label{eq:blood2a}
\end{equation}
Equation (\ref{eq:demand6}) describes a known type of second order differential equation, namely the \textit{extended confluent hypergeometric equation} \cite{Slater1960}, which allows an explicit solution.
A detailed deduction of the solution to \eqref{eq:demand6} is given in Appendix \ref{app:proof_prop_density}, and yields the following result.
\\
\begin{proposition}\label{densityProp}
The probability density functions of the amount of demand $X_d$ and the amount of blood present $X_b$ are given by
\begin{align}
f(v) &= \pi_d\, \frac{\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\,\frac{{\rm e}^{-\mu_d v}U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right)}{ _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},-\tfrac{\mu_b}{\mu_d}\right)}\label{eq:fullf},\\
g(v) &= \pi_b\, \frac{\Gamma\left(1+\frac{\lambda_d}{\xi_b}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_b}\right)}\,\frac{{\rm e}^{-\mu_b v}U\left( 1-\tfrac{\lambda_b}{\xi_b}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_b},(\mu_b+\mu_d)v\right)}{ _2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},-\tfrac{\mu_d}{\mu_b}\right)}\label{eq:fullg},
\end{align}
for $v>0$, respectively.
\end{proposition}
\noindent
Here, $\Gamma(\cdot)$ denotes the gamma function, $ _2F_1(a,b,c,z)$ is the Gaussian hypergeometric function, defined as
\begin{equation}
_2F_1(a,b,c,z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n\, n!}\, z^n
\end{equation}
and $U(a,b,z)$ is Tricomi's confluent hypergeometric function, see \cite{Slater1960},
\begin{align}
U(a,b,x) &= \frac{\Gamma(b-1)}{\Gamma(1+a-b)}\,\sum_{n=0}^\infty \frac{(a)_n}{(b)_n n!} x^n + \frac{\Gamma(b-1)}{\Gamma(a)}\,x^{1-b}\,\sum_{n=0}^\infty \frac{(1+a-b)_n}{(2-b)_n n!} x^n ,
\end{align}
in which $(a)_n$ is the Pochhammer symbol, defined as $(a)_n = a\cdot(a+1)\cdots(a+n-1)$.
As a direct consequence of Proposition \ref{densityProp}, we obtain expressions for the LTs $\phi(s) = \int_0^{\infty} {\rm e}^{-sv} f(v) {\rm d}v$ and $\gamma(s) = \int_0^{\infty} {\rm e}^{-sv} g(v) {\rm d}v$ for ${\rm Re}\,s \geq 0$ through \cite[Eq.~(3.2.51)]{Slater1960}, which we state here for future use.
\begin{corollary}\label{cor:lsts}
The Laplace transforms for $X_d$ and $X_b$, for ${\rm Re}\,s \geq 0$, are given by
\begin{align}
\label{eq:diffLST}
\phi(s) &= \pi_d\, \frac{\mu_d}{\mu_d+s}\frac{ _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},\frac{s-\mu_b}{s+\mu_d}\right)}{ _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)},\\
\gamma(s) &= \pi_b\, \frac{\mu_b}{\mu_b+s}\frac{ _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},\frac{s-\mu_d}{s+\mu_b}\right)}{ _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},
\label{eq:diffLST_2}
\end{align}
respectively.
\end{corollary}
Last, we obtain expressions for $\pi_d$ and $\pi_b$. These follow immediately by
using the normalization equation $\pi_b + \pi_d=1$
and \eqref{eq:blood2a}, or equivalently,
$\lambda_b\phi(\mu_b) = \lambda_d\gamma(\mu_d)$. By filling in $s=\mu_b$ in \eqref{eq:diffLST},
\begin{align}\label{eq:LSTequal2}
&\pi_d\,\,\frac{\lambda_b\mu_d}{\mu_b+\mu_d}\, _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)^{-1} \nonumber\\
&\qquad = \pi_b\,\,\frac{\lambda_d\mu_b}{\mu_b+\mu_d}\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)^{-1},
\end{align}
where we used that $_2F_1(a,b,c,0) = 1$. Using the normalization equation, we obtain
\begin{equation}
\label{eq:piD}
\pi_d = \frac{\rho_b,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)}
{\rho_d,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)}.
\end{equation}
By substituting this result into both \eqref{eq:fullf} and \eqref{eq:diffLST}, we obtain the full pdf for the blood inventory process in steady-state.
\begin{theorem}\label{thm:full_pdf}
The steady-state pdf of the net inventory level $X$ is given by
\begin{equation}
h(v) =
\left\{
\begin{array}{ll}
f(-v), & \text{if }v<0,\\
g(v), & \text{if }v\geq 0,
\end{array}
\right.
\end{equation}
where
\begin{align}
\label{eq:ftotal}
f(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \rho_d \, {\rm e}^{-\mu_d v}\, U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right),\\
g(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_d}{\xi_b}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_b}\right)}\, \rho_b \, {\rm e}^{-\mu_b v}\, U\left( 1-\tfrac{\lambda_b}{\xi_b}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_b},(\mu_b+\mu_d)v\right),
\end{align}
with
\begin{equation}
\bar{C} = \rho_d \,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right).
\end{equation}
\end{theorem}
\begin{remark}
By applying the Pfaff transformation $_2F_1(a,b,c,z)=$ \\
$(1-z)^{-b}\,_2F_1\left(c-a,b,c,\frac{z}{1-z}\right)$, we may reformulate
\begin{equation}
\label{eq:pfaffTransform}
_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) = \frac{\mu_d}{\mu_b+\mu_d}\, _2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b}{\mu_b+\mu_d}\right),
\end{equation}
so that
\begin{equation}
\label{eq:piDalternative}
\pi_d = \frac{\lambda_d\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right)}
{\lambda_d\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right) +
\lambda_b\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_b},1,\tfrac{\lambda_d}{\xi_b},\tfrac{\mu_d}{\mu_b+\mu_d}\right)}.
\end{equation}
By also transforming the hypergeometric term in the numerator of \eqref{eq:fullf}, we get an equivalent form of \eqref{eq:ftotal}, namely
\begin{equation}
\label{eq:ftotalAlternative}
f(v) = \bar{C}^{-1}_{\rm alt}\frac{\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \rho_b\mu_b(\mu_b+\mu_d)\, {\rm e}^{-\mu_d v}\,U\left( 1-\frac{\lambda_d}{\xi_d}, 2-\frac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right),
\end{equation}
with
\begin{equation}
\bar{C}_{\rm alt} = \lambda_d\,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b}{\mu_b+\mu_d}\right) +
\lambda_b\,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_b},1,\frac{\lambda_d}{\xi_b},\frac{\mu_d}{\mu_b+\mu_d}\right).
\end{equation}
As a consequence, \eqref{eq:diffLST} is given by
\begin{equation}
\phi(s) = \pi_d \,\frac{ _2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b-s}{\mu_b+\mu_d}\right)}{ _2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right) } = \bar{C}^{-1}_{\rm alt}\,\lambda_d \,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b-s}{\mu_b+\mu_d}\right).
\end{equation}
\end{remark}
Based on the density functions in Theorem \ref{thm:full_pdf}, we make some comments on its properties, and discuss parameter settings that leads to special cases.
By close inspection of these derived density functions, we can observe the following on the distribution shape around $z=0$.
The confluent hypergeometric function $U(a,b,z)$ has limiting form as $z\rightarrow 0$,
\begin{equation}\label{eq:limit0}
U(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a-b+1)} + \frac{\Gamma(b-1)}{\Gamma(a)}\,z^{1-b} + O(z^{2-b}), \qquad b\leq 2,
\end{equation}
see \cite[Sub.~13.2]{NIST}.
Note that in our model, $b = 2-(\lambda_b+\lambda_d)/\xi_d<2$ for all parameter settings.
Equation \eqref{eq:limit0} shows that $U(a,b,z)$ has a singularity at $z=0$ if Re$(b)>1$, which in our case translates to $f(v)$ and $g(v)$ being analytic at $v=0$ if $\lambda_b+\lambda_d > \xi_d$ and $\lambda_b+\lambda_d > \xi_b$, respectively. Assuming $\lambda_b+\lambda_d > \max\{\xi_b,\xi_d\}$, \eqref{eq:limit0} also implies that
\begin{align}\label{eq:contLimit1}
\lim_{v\rightarrow 0} f(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \lambda_d\mu_b\cdot \frac{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}-1\right)}{\Gamma\left(\frac{\lambda_b}{\xi_d}\right)}\\
&= \bar{C}^{-1}\,\frac{\frac{\lambda_b}{\xi_d}}{\frac{\lambda_b+\lambda_d}{\xi_d}-1}\, \lambda_d\mu_b= \bar{C}^{-1}\,\frac{\lambda_b\lambda_d\mu_b\mu_d}{\lambda_b+\lambda_d-\xi_d}.
\nonumber
\end{align}
Similarly,
\begin{equation}
\lim_{v\rightarrow 0} g(v) = \bar{C}^{-1}\,\frac{\lambda_b\lambda_d\mu_b\mu_d}{\lambda_b+\lambda_d-\xi_b}.
\label{eq:contLimit2}
\end{equation}
By equating these two expressions, we conclude that $\lim_{v\rightarrow 0} f(v) = \lim_{v\rightarrow 0} g(v)< \infty$, i.e. the overall density function $h(v)$ is continuous at $v=0$, if and only if $\xi_b = \xi_d$. \\
\noindent
The asymptotic behavior of $U$ as $z\to\infty$ is given by \cite[p.~60]{Slater1960},
\begin{equation}
U(a,b,z) \sim z^{-a}, ~~~~~~~~~~~~~~~~ z \rightarrow \infty,
\end{equation}
which implies that the density function tail decays as
\begin{equation}
\label{eq:asympt}
f(v) \sim C^*\, e^{-\mu_d v}\, v^{\lambda_d/\xi_d-1}, ~~~~~~~~~~~~~~~~~~~
v\rightarrow\infty ,
\end{equation}
for some constant $C^*$.
\noindent\textbf{Special cases.}
Equation \eqref{eq:asympt} suggests that the case $\lambda_d = \xi_d$ is special.
Indeed, then \eqref{eq:diffLST} reduces to
\begin{equation}
\label{eq:lambdaisxi}
\phi(s) = \bar{C}^{-1}\,\lambda_d\mu_b\, \frac{\mu_d}{\mu_d+s} = \pi_d\, \frac{\mu_d}{\mu_d+s},
\end{equation}
where we used that $_2F_1(0,a,b,z) = 1$ for all $a,b,z$. Hence, conditioned on being positive, the amount of demand present is exponentially distributed with parameter $\mu_d$, regardless of the values of $\lambda_d = \xi_d$, as well as $\lambda_b,\, \xi_b,$ and $\mu_b$.
If we moreover let $\lambda_b = \xi_b$, then
\[
\pi_d = \frac{\lambda_d/\mu_d}{\lambda_b/\mu_b + \lambda_d/\mu_d} = \frac{\rho_d}{\rho_b+\rho_d},
\]
and $X$ has exponential distribution both above and below 0, with parameters $\mu_b$ and $\mu_d$, respectively. \\
A second special case arises when the process is symmetric, that is, $\lambda_b=\lambda_d=\lambda$, $\mu_b=\mu_d=\mu$ and $\xi_b=\xi_d=\xi$. Obviously, we get $\pi_b= \pi_d = \tfrac{1}{2}$ due to the symmetry. If we define $\eta := \lambda/\xi$,
\begin{align}
f(v) &= \frac{\Gamma(1+\eta)\, \mu e^{-\mu v}\, U\left(1-\eta,2(1-\eta),2\mu v\right)}{2\,\Gamma(2\eta) _2F_1\left(2\eta,1,1+\eta,\tfrac{1}{2}\right)}\\
&= \frac{\Gamma(1+\eta)}{2\,\Gamma(2\eta) _2F_1\left(2\eta,1,1+\eta,\tfrac{1}{2}\right)}\, \frac{\mu}{2\sqrt{\pi}}\, \left(2\mu v\right)^{\eta-\tfrac{1}{2}}\, K_{\tfrac{1}{2}-\eta}\left(\mu v\right),
\nonumber
\end{align}
where $K_\alpha(\cdot)$ is the modified Bessel function of the second kind, see \cite[Eq.~(13.6.10)]{NIST}.\\
\\*
\noindent
\textbf{Performance measures.}
Based on Theorem \ref{thm:full_pdf}, we can directly derive a couple of characteristics of the process.
First, we consider the mean inventory level
\begin{corollary}\label{cor:means}
The expected amount of demand (blood) present, given that it is positive equals
\begin{align}
\mathbb{E}[X_d|X_d>0] &= \frac{1}{\xi_d}\left[ \rho_d - \rho_b + \rho_b\, _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},{-}\frac{\mu_b}{\mu_d}\right)^{-1}\right],\label{EXd>0}\\
\mathbb{E}[X_b|X_b>0] &= \frac{1}{\xi_b}\left[ \rho_b - \rho_d + \rho_d \, _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},{-}\frac{\mu_d}{\mu_b}\right)^{-1}\right]\label{EXb>0}.
\end{align}
Accordingly, the expected net amount of blood present equals
\begin{equation}
\mathbb{E}[X] = \left(\rho_b-\rho_d \right)\left(\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right)+\frac{\lambda_b\lambda_d}{\bar{C}}\left(\frac{1}{\xi_b}-\frac{1}{\xi_d}\right).
\end{equation}
\end{corollary}
\begin{proof}
Let us use shorthand notation
\[
F(s) = \Big(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},\frac{s-\mu_b}{s+\mu_d}\Big)
,\]
so that
\[\phi(s)=\pi_d\, \frac{\mu_b}{\mu_b+s}\, \frac{F(s)}{F(0)}.\]
Through \cite[Eq.~(15.5.20)]{NIST},
\begin{equation}\label{eq:proof_a}
\frac{{\rm d}}{{\rm d} z}\, _2F_1(a,1,c,z) = \frac{c-1}{z(1-z)} + \frac{1-c+az}{z(1-z)} \, _2F_1(a,1,c,z),
\end{equation}
where we also used that $ _2F_1(a,1,c,z) = 1$.
Then,
\begin{align*}
\frac{\phi'(0)}{\pi_d} &= \left[ \frac{-\mu_d}{(\mu_d+s)^2} \, \frac{F(s)}{F(0)}
+ \frac{\mu_d}{\mu_d+s}\, \frac{F'(s)}{F(0)} \right]_{s=0} = {-}\frac{1}{\mu_d} + \frac{F'(0)}{F(0)}.
\end{align*}
By \eqref{eq:proof_a}, we find
\begin{align*}
F'(s) &= \Big( \frac{\lambda_b/\xi_d}{\frac{s-\mu_b}{s+\mu_d}\cdot \frac{\mu_b+\mu_d}{s+\mu_d}} + \frac{-\lambda_b/\xi_d + (1-\lambda_d/\xi_d)\frac{s-\mu_b}{s+\mu_d}}{\frac{s-\mu_b}{s+\mu_d}\cdot \frac{\mu_b+\mu_d}{s+\mu_d}}\, F(s)\Big)
\,\frac{{\rm d}}{{\rm d} s} \Big[ \frac{s-\mu_b}{s+\mu_d} \Big]
\\
&= \Big( \frac{ \lambda_b}{\xi_d} + \left[\frac{{-}\lambda_b}{\xi_d} + \Big(1-\frac{\lambda_d}{\xi_d}\Big)\frac{s-\mu_b}{s+\mu_d}\right] F(s) \Big) \frac{ (s+\mu_d)^2} {(s-\mu_b)(\mu_b+\mu_d)}\cdot \frac{\mu_b+\mu_d}{(s+\mu_d)^2}\\
&= \Big( \frac{ \lambda_b}{\xi_d} + \left[{-}\frac{\lambda_b}{\xi_d} + \Big(1-\frac{\lambda_d}{\xi_d}\Big)\frac{s-\mu_b}{s+\mu_d}\right] F(s) \Big)
\frac{1}{s-\mu_b},
\end{align*}
so that
\begin{align*}
F'(0) &= {-} \frac{\lambda_d/\mu_b}{\xi_d} + \left( \frac{\lambda_d/\mu_b}{\xi_d}
+ \frac{1}{\mu_d} - \frac{\lambda_d/\mu_d}{\xi_d}\right)F(0)\\
&= {-}\frac{\rho_b}{\xi_d} + \left( \frac{\rho_b-\rho_d}{\xi_d}
+ \frac{1}{\mu_d}\right)F(0).
\end{align*}
Hence, we find
\begin{align*}
\mathbb{E}[X_d|X_d>0] &= {-}\frac{\phi'(0)}{\pi_d} = \frac{1}{\mu_d}- \frac{1}{F(0)}\left[{-}\frac{\rho_b}{\xi_d} + \left( \frac{\rho_b-\rho_d}{\xi_b}
+ \frac{1}{\mu_d}\right)F(0)\right]\\
&= \frac{1}{\xi_d}\left( \rho_d-\rho_b + \rho_b/F(0)\right) = \frac{1}{\xi_d}\left( -m + \rho_b/F(0)\right),
\end{align*}
which equals \eqref{EXd>0}.
The expression for \eqref{EXb>0} follows by symmetry.
Furthermore,
\begin{align*}
\mathbb{E}[X] &= \pi_b \mathbb{E}[X_b|X_b>0] + \pi_d \mathbb{E}[-X_d|X_d>0]\\
&= m\left[\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right] + \frac{\lambda_d}{\mu_d\,\xi_b}\frac{\pi_b}{_2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},{-}\tfrac{\mu_d}{\mu_b}\right)}\\
&\qquad - \frac{\lambda_b}{\mu_b\,\xi_d}\,\frac{\pi_d}{_2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right)}.
\end{align*}
Note that $\pi_d\,_2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right)^{-1} = \lambda_d\mu_b\bar{C}^{-1}$.
Hence,
\begin{align*}
\mathbb{E}[X] &= m\left[\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right] + \frac{\lambda_b\lambda_d}{\bar{C}}\left(\frac{1}{\xi_b}-\frac{1}{\xi_d}\right),
\end{align*}
which completes the proof.
\end{proof}
\begin{remark}
Note that if $\xi_b = \xi_d = \xi$, we get $\mathbb{E}[X] = m(\pi_b+\pi_d)/\xi = m/\xi$, which is consistent with Proposition \ref{prop:mean_inventory}.
The expression in (\ref{EXd>0}) contains no $\xi_b$. Indeed, while the value of $\xi_b$ influences the probability that $X_d>0$,
it does not influence the mean of $X_d$ given that $X_d >0$.
\end{remark}
In Figure \ref{fig:means}, we plot the behavior of the three performance metrics in Corollary \ref{cor:means} while keeping $m$ fixed. In Figure \ref{fig:means}(a) we set $\lambda_b = 1.2$, $\lambda_d = 1$, $\mu_b=1$, $\mu_d=1.2$, so that $m = 11/30$ and vary $\xi_b=\xi_d=\xi$ between 0 and 1. In Figure \ref{fig:means}b, we fix $\xi_b=\xi_d=0.5$ and take $\lambda_b = 1.2\theta$, $\lambda_d = \theta$, $\mu_b=\theta$, $\mu_d=1.2\theta$, so that still $m=11/30$, and vary $\theta$.
Observe that in Figure \ref{fig:means}b, $\mathbb{E}[X]$ is constant, since the value of $m/\xi$ if unaffected by the parameter $\theta$.
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.78]
\begin{axis}[
xmin = -0.02,
xmax = 1,
ymin = -0.02,
ymax = 5,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1,
xlabel = {$\xi$},
xlabel near ticks,
legend cell align = left,
legend style = {at = {(axis cs: 1,5)},anchor = north east}
]
\addplot[black,dashed,thick] table[x=x,y=Xd] {tikz/means1.txt};
\addplot[black,dotted,thick] table[x=x,y=Xb] {tikz/means1.txt};
\addplot[black,thick] table[x=x,y=Q] {tikz/means1.txt};
\legend{{$\mathbb{E}[X_d|X_d>0]$},{$\mathbb{E}[X_b|X_b>0]$},{$\mathbb{E}[X]$}};
\end{axis}
\end{tikzpicture}
\caption{As a function of $\xi$}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.78]
\begin{axis}[
xmin = -0.02,
xmax = 2,
ymin = -0.02,
ymax = 5,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1,
xlabel = {$\theta$},
xlabel near ticks,
legend cell align = left,
legend style = {at = {(axis cs: 2,5)},anchor = north east}
]
\addplot[black,dashed,thick] table[x=x,y=Xd] {tikz/means2.txt};
\addplot[black,dotted,thick] table[x=x,y=Xb] {tikz/means2.txt};
\addplot[black,thick] table[x=x,y=Q] {tikz/means2.txt};
\legend{{$\mathbb{E}[X_d|X_d>0]$},{$\mathbb{E}[X_b|X_b>0]$},{$\mathbb{E}[X]$}};
\end{axis}
\end{tikzpicture}
\caption{As a function of $\theta$}
\end{subfigure}
\caption{Expected mean amount of blood, demand, and net blood present.}
\label{fig:means}
\end{figure}
Secondly, we present the probability of positive (cq. negative) inventory.
\begin{corollary}\label{cor:pid}
The probability of positive (cq.~negative) inventory is given by,
\begin{align}
\pi_b &= \frac{\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)}
{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},\\
\pi_d &= \frac{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)}
{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},
\end{align}
respectively.
\end{corollary}
\begin{proof}
The expressions follow directly from \eqref{eq:piD} and $\pi_b = 1-\pi_d$.
\end{proof}
The last relevant performance indicator we consider is the fraction of demand that is immediately satisfied from stock.
\begin{corollary}
The probability that a demand request can be fully satisfied from stock is given by
\begin{align}
\mathbb{P}({\rm demand\ satisfied}) = \bar{C}^{-1}\rho_b \left( _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right) - \frac{\mu_b}{\mu_b+\mu_d}\right).
\end{align}
\end{corollary}
\begin{proof}
Using the PASTA property of the Poisson process, we get
\begin{align*}
\mathbb{P}({\rm demand\ satisfied}) &= \mathbb{P}( X > D ) = \mathbb{P}(X_b > D) \\
&= \int_0^\infty g(u) (1-{\rm e}^{-\mu_d u})\, {\rm d} u = \pi_b - \gamma(\mu_d).
\end{align*}
Substituting the expressions for $\pi_b$ as in Corollary \ref{cor:pid} and $\gamma(\mu_b)$ as in \eqref{eq:diffLST_2} yields the result.
\end{proof}
\subsection{The general case}
\label{gener}
In this section we outline how the integral equations (\ref{eq:blood}) and (\ref{eq:demand})
can be solved using Laplace transforms, when we make the restriction that $F_b(\cdot)$ and $F_d(\cdot)$
are Coxian distributions.
This is not a major restriction, because the class of Coxian distributions lies dense in the class of all distributions
of non-negative random variables, see e.g.~\cite[Sec.~III.4]{Asmussen2003}.
Hence, one can approximate $F_b(\cdot)$ arbitrarily closely by a Coxian distribution.
If $X_i$, $i=1,2,\dots,K$ are independent, exponentially distributed random variables,
and $\mathbb{E}[X_i] = \frac{1}{\beta_i}$, $i=1,2,\dots,K$, then a Coxian amount of blood $B$ can be represented
as:
\begin{equation}
B = \sum_{j=1}^i X_j\quad {\rm with ~ probability } \quad p_i \prod_{j=1}^{i-1} (1-p_j), \quad i=1,2,\dots,K.
\end{equation}
In the above case, it is easily verified that one can represent $\bar{F}_b(x)$ as follows:
\begin{equation}
\bar{F}_b(x) = \mathbb{P}(B>x) = \sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
{\rm e}^{-\beta_j x}
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
,
\label{Fbarb}
\end{equation}
if all $\beta_j$ are different. If two $\beta_j$ coincide, then a term with $x {\rm e}^{-\beta_j x}$ (Erlang-$2$) must be added.
We leave this to the reader, but in Remark~\ref{RmErlang} below we outline how Erlang terms can be handled in solving the integral equations
(\ref{eq:demand}) and (\ref{eq:blood}).
The counterpart of (\ref{Fbarb}) for the case that $F_d(\cdot)$ is Coxian, is
\begin{equation}
\bar{F}_d(x) = \mathbb{P}(D>x) = \sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
{\rm e}^{-\delta_j x}
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
.
\label{Fdarb}
\end{equation}
Taking Laplace transforms $\phi(s) = \int_0^{\infty} {\rm e}^{-sy} f(y) {\rm d}y$ and
$\gamma(s) = \int_0^{\infty} {\rm e}^{-sy} g(y) {\rm d}y$
in (\ref{eq:blood}) and (\ref{eq:demand})
results in first-order inhomogeneous differential equations in $\phi(s)$ and $\gamma(s)$, respectively, which can be solved in a straightforward way.
\begin{equation}
\phi'(s) = A_H(s) \phi(s) + A_I(s),
\label{diffeq}
\end{equation}
with the homogeneous term $A_H(s)$ being given by
\begin{align}
A_H(s) &:= -\frac{1}{\xi_d}
\left[
\lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
\frac{1}{\delta_j+s}
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\right.
\nonumber
\\
&\qquad - \lambda_b
\left.\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
\frac{1}{\beta_j-s}
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
- \alpha_d \right],
\end{align}
and the inhomogeneous term $A_I(s)$ being given by
\begin{align}
A_I(s) &:=
- \frac{1}{\xi_d}
\left[
\lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
\frac{1}{\delta_j+s} [\gamma(\delta_j) + \pi_0]
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\right.
\nonumber
\\
&\qquad + \left.\lambda_b
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
\frac{1}{\beta_j-s} \phi(\beta_j)
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\right].
\end{align}
\noindent
The solution of (\ref{diffeq}) is given by the following expression:
\begin{equation}
\phi(s) = \phi(0) {\rm e}^{\int_0^s A_H(z) {\rm d}z} + \int_0^s A_I(u)
{\rm e}^{\int_u^s A_H(z) {\rm d}z} {\rm d}u , ~~~~ s \geq 0.
\label{diffeqsoln}
\end{equation}
$\gamma(s)$ is given by a mirror expression, where $\phi(0)$ is replaced by $\gamma(0)$
and where $A_H(s)$ and $A_I(s)$ are replaced by expressions in which $K$ and $L$ are swapped, and $p$ and $q$, and $\beta_i$ and $\delta_i$.
It should be noticed that $\phi(0)$, $\gamma(0)$ and $\pi_0$ still have to be determined.
Furthermore, it should be noticed that $A_H(s)$ and $A_I(s)$ have singularities at $s=\beta_1,\dots,\beta_K$.
These singularities are removable, but handling Equation \eqref{diffeqsoln} clearly requires some care.
Instead of working out the details, we shall below return to the case
of exponentially distributed amounts of blood and demand -- so $K=L=1$.
For that case, we shall not only work out the solution of the differential equation for $\phi(s)$ in detail,
including the determination of the missing constants, but
we also relate the results to those obtained in Section~\ref{sectionexp}
without resorting to Laplace transforms.
Taking $K=1, p_1=1, \delta_1 = \mu_d$, and $L=1, q_1=1, \beta_1 = \mu_b$, we obtain
the following two inhomogeneous first order differential equations in the LTs $\phi(s)$ and $\gamma(s)$:
\begin{equation}\label{eq:firstPhis}
\phi'(s) = \phi(s)\left[\frac{\lambda_b}{\xi_d} \frac{1}{\mu_b-s} - \frac{\lambda_d}{\xi_d} \frac{1}{\mu_d+s}\right]
-\frac{\lambda_b}{\xi_d} \frac{\phi(\mu_b)}{\mu_b-s} -\frac{\lambda_d}{\xi_d} \frac{\gamma(\mu_d)}{\mu_d+s} ,
\end{equation}
\begin{equation}
\gamma'(s) = \gamma(s)\left[\frac{\lambda_d}{\xi_b} \frac{1}{\mu_d-s} - \frac{\lambda_b}{\xi_b} \frac{1}{\mu_b+s}\right]
-\frac{\lambda_d}{\xi_b} \frac{\gamma(\mu_d)}{\mu_d-s} -\frac{\lambda_b}{\xi_b} \frac{\phi(\mu_b)}{\mu_b+s} .
\end{equation}
They are routinely solved:
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d}{\mu_d+s}\right)^{\frac{\lambda_d}{\xi_d}}
\left[\phi(0) \frac{}{}\right.
\nonumber \\
&\qquad -
\frac{\lambda_d}{\xi_d} \gamma(\mu_d) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}-1} \frac{{\rm d}z}{\mu_d}
\nonumber
\\
&\qquad \qquad -
\frac{\lambda_b}{\xi_d} \phi(\mu_b) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}-1}
\left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}} \frac{{\rm d}z}{\mu_b}\left.\frac{}{}\right].
\label{phis1}
\end{align}
Similarly,
\begin{align}
\gamma(s) &= \left(\frac{\mu_d}{\mu_d-s}\right)^{\frac{\lambda_d}{\xi_b}}
\left(\frac{\mu_b}{\mu_b+s}\right)^{\frac{\lambda_b}{\xi_b}}
\left[\gamma(0) \frac{}{}\right.
\nonumber
\\
&\qquad -
\frac{\lambda_b}{\xi_b} \phi(\mu_b) \int_0^s \left(\frac{\mu_d-z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_b}}
\left(\frac{\mu_b+z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_b}-1} \frac{{\rm d}z}{\mu_b}
\nonumber
\\
&\qquad \qquad -
\frac{\lambda_d}{\xi_b} \gamma(\mu_d) \int_0^s \left(\frac{\mu_d-z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_b}-1}
\left(\frac{\mu_b+z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_b}} \frac{{\rm d}z}{\mu_d}\left.\frac{}{}\right] .
\label{gammas1}
\end{align}
Notice that the exponents in the above integrals have powers which are larger than $-1$ (e.g., $\frac{\lambda_d}{\xi_d}-1$),
so that these integrals do not lead to singularities.
We still need to determine the two constants $\phi(0)=\pi_d$ and $\gamma(0)=\pi_b$.
Together with $\phi(\mu_b)$ and $\gamma(\mu_d)$, we have four unknowns.
We determine these unknowns using the following four equations:
(i) From (\ref{eq:blood2a}), we get
$\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$, while (ii) $\pi_d + \pi_b =1$.
Finally, we take (iii) $s=\mu_b$ in (\ref{phis1}) and (iv) $s=\mu_d$ in (\ref{gammas1}).
Notice that the identity
$\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$ allows us to
reduce the two integrals in (\ref{phis1}) to one integral (and similarly in (\ref{gammas1})):
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d}{\mu_d+s}\right)^{\frac{\lambda_d}{\xi_d}}
\left[\phi(0) \frac{}{}\right.
\nonumber
\\
&\quad -
\frac{\lambda_d}{\xi_d} \gamma(\mu_d)\, \frac{\mu_b+\mu_d}{\mu_b\mu_d} \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}-1} \left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}-1} {\rm d}z \left.\frac{}{}\right].
\label{phis11}
\end{align}
\begin{remark}
We have numerically verified that
the expressions in (\ref{phis1}) and (\ref{eq:diffLST}) coincide.
\end{remark}
\begin{remark}
If $\lambda_b=0$ then we have a known queueing model or shot-noise model
with state-dependent service rate, see Keilson \& Mermin \cite{Keilson1959}
and Bekker et al.~\cite{Bekker2004} for the so-called shot noise model.
\end{remark}
\begin{remark}
\label{R7}
The case $\lambda_d = \xi_d$ is special. Equation \eqref{phis1} now reduces to
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{\mu_d}{\mu_d+s}
\left[\phi(0) \frac{}{}\right.
\label{phis1A}
-
\gamma(\mu_d) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{{\rm d}z}{\mu_d}
\nonumber
\\
&\qquad -
\frac{\lambda_b}{\lambda_d} \phi(\mu_b) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\lambda_d}-1}
\frac{\mu_d+z}{\mu_d} \frac{{\rm d}z}{\mu_b}\left.\frac{}{}\right].
\nonumber
\end{align}
Both integrals are easily evaluated (rewrite,
in the last integral, $\mu_d + z = \mu_d + \mu_b -(\mu_b - z)$).
We find
\begin{align}
\phi(s) &=
\left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{\mu_d}{\mu_d+s}\nonumber\\
&\ \cdot
\left[\phi(0)
+ \frac{\gamma(\mu_d)}{\mu_d} \frac{\lambda_d}{\lambda_b + \lambda_d} \mu_b - \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} - \frac{\phi(\mu_b)}{\mu_d} \frac{\lambda_b}{\lambda_b + \lambda_d} \mu_b \right]
\nonumber
\\
&\ + \frac{\mu_d}{\mu_d+s} \left[
\frac{\gamma(\mu_d)}{\mu_d} \frac{\lambda_d}{\lambda_b + \lambda_d} (\mu_b -s) + \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} - \frac{\phi(\mu_b)}{\mu_d} \frac{\lambda_b}{\lambda_b + \lambda_d} (\mu_b -s)\right] .
\end{align}
Now observe through \eqref{eq:blood2a}, that $\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$.
Hence, in both lines of the above formula, two terms cancel.
Moreover, $\phi(s)$ should be analytic for $s=\mu_b$, yielding
\begin{equation}
\phi(0) = \phi(\mu_b) \frac{\mu_d + \mu_b}{\mu_d}.
\end{equation}
Finally we obtain, see also \eqref{eq:lambdaisxi},
\begin{equation}
\phi(s) = \frac{\mu_d}{\mu_d+s} \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} = \phi(0) \frac{\mu_d}{\mu_d+s} = \pi_d \frac{\mu_d}{\mu_d + s},
\end{equation}
and hence
\begin{equation}\label{eq:exp}
f(x) = \pi_d \mu_d {\rm e}^{-\mu_d x}, ~~~ x > 0;
\end{equation}
the shortage (amount of demand present) is exponentially distributed
when $\lambda_d = \xi_d$.
\\
It should be noticed that, if $\lambda_d = \xi_d$, then the first and last term of (\ref{eq:demand2})
are equal when (\ref{eq:exp}) holds; and using (\ref{eq:blood2a})
it is also readily verified that the second and third term of (\ref{eq:demand2}) are equal.
The constant $\pi_d$ will in general still depend on the parameters
$\lambda_d = \xi_d$, $\lambda_b$, $\mu_b$ and $\xi_b$.
\\
We end this remark with the observation that in the one-sided shot-noise process
(so $\lambda_b=0$), Bekker et al.\ \cite{Bekker2004} also observe that $\lambda_d = \xi_d$
results in an exponential density.
\end{remark}
\subsection{A variant}
\label{sectionvariant}
In this section, we assume that the expiration rate of blood and the patience rate of demand are constant.
So, we take $\xi_b = \xi_d = 0$.
A visualization of a possible sample path is depicted in Figure \ref{FIG2}.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = -0.2,
xmax = 3.25,
ymin = -0.5,
ymax = 1.6,
ticks = none,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1
]
\addplot[black,thick] file {tikz/sample_path_xi0.txt};
\node at (axis cs: 0.25,-0.4) {$X_d(t)$};
\node at (axis cs: 0.25,1.5) {$X_b(t)$};
\draw[-stealth] (axis cs: 0,0) -- (axis cs:0,-0.5);
\end{axis}
\node at (7.05,1.2) {$t$};
\end{tikzpicture}
\caption{Sample path of the net amount of blood present if $\xi_b = \xi_d = 0$.}
\label{FIG2}
\end{figure}
We again restrict ourselves to the case of exponentially distributed amounts of demand and of blood deliveries.
We now need to impose stability conditions.
In the case of positive demand, the drift is towards zero if $\lambda_d \mathbb{E}[D] < \alpha_d + \lambda_b \mathbb{E} [B]$,
while in the case of a positive amount of blood, the drift is towards zero if
$\lambda_b \mathbb{E} [B] < \alpha_b + \lambda_d \mathbb{E} [D]$.
If these two conditions are violated, either the amount of demand or the amount of blood present increases without bound
(see also
below).
In this case, \eqref{eq:demand6} reduces to
\begin{equation}
\alpha_d f''(v) + (-\lambda_d -\lambda_b + \mu_d \alpha_d -\mu_b \alpha_d)f'(v)
+
(-\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \alpha_d)f(v) =0.
\label{eq:demand6b}
\end{equation}
Hence $f(\cdot)$ is a mixture of two exponential terms: $f(v) = R_+ {\rm e}^{- x_+ v} + R_- {\rm e}^{- x_- v}$,
where $x_+$ and $x_-$ are the positive and negative root of the equation
\begin{equation}
\alpha_d x^2 -(\mu_d \alpha_d - \mu_b \alpha_d -\lambda_d -\lambda_b)x +(-\mu_d \lambda_b + \mu_b \lambda_d -\mu_b \mu_d \alpha_d) = 0.
\label{zeros}
\end{equation}
Notice that the last term in the left-hand side of (\ref{zeros}) is negative if the stability condition
$\lambda_d \mathbb{E} [D] < \alpha_d + \lambda_b \mathbb{E}[B]$ holds,
that is, if $\mu_b \lambda_d < \mu_d \lambda_b + \mu_b \mu_d \alpha_d$,
thus guaranteeing that the product of the two roots $x_+$ and $x_-$ is negative,
and hence that there is a positive and a negative root.
One should subsequently observe that
$R_-$ must be zero to have a probability density.
Hence $f(v)$ is simply (a constant times) an exponential;
similarly for $g(v)$.
In addition, the steady-state amounts of demand and of blood have an atom at $0$ (since $\xi_d$ and $\xi_b$ are no longer zero, the demand and blood processes
can reach $0$).
Interestingly, the model of this section is closely related to the model with workload removal
that is considered in \cite{Boucherie1996}. There an $M/G/1$ queue is studied with the extra feature
that, at Poisson epochs, a stochastic amount of work is removed.
In the $M/M/1$ case with removal of exponential amounts of work, see \cite[Sec.~5.1]{Boucherie1996}, one has the model of the present section
when we concentrate on the amount of demand present.
One difference with the model in \cite{Boucherie1996}
is that, when the workload in that model has become zero, the work becomes positive at rate $\lambda_d$,
whereas in the present model the amount of blood can become positive (so zero demand is present)
and the amount of demand does not have to become positive when demands arrive (because they are immediately satisfied, see Figure~\ref{FIG2}). So the atom at zero is in the present model larger than in the model of \cite{Boucherie1996}.
In our model a positive demand level may be reached from below zero (by a jump, i.e., a demand arriving at an epoch
that there is some, but not enough, blood present). The memoryless property of the exponential
demand requirement distribution implies that this jump results in a demand level that is exp($\mu_d$),
just as if the initial demand level had been zero.
In the case of non-exponential demand requirements, our model becomes equivalent with an $M/G/1$ queue with
exponential amounts of work removed, and with the special feature that the first service requirement of a busy period has a different distribution.
Lemmas 4.1 and 4.2 of \cite{Boucherie1996} present the stability condition
of that $M/G/1$ queue with work removal; it amounts to
$\lambda_d \mathbb{E} [D] < \alpha_d + \lambda_b \mathbb{E} [B]$, which indeed is one of the two stability conditions
of the present demand/blood model.
Finally we observe that Equation (5.1) of \cite{Boucherie1996} coincides with \eqref{zeros}
(take $\alpha_d=1$, $\lambda_d = \lambda_+$, $\lambda_b = \lambda_-$, $\mu_d=1/\beta$ and $\mu_b = 1/\gamma$).
\section{Asymptotic analysis}
\label{sectionscaling}
We finally study the model with $\alpha_b = \alpha_d = 0$ from an asymptotic perspective, by obtaining the fluid and diffusion limits of the blood inventory process. That is, we will create a sequence of processes, indexed by $n=1,2,...$, in which we let the rates of blood and demand arrivals grow large. If we then scale the process in a proper manner, we are able to deduce a non-degenerate limiting process, that provides insight in the overall behavior of the arrival volume when the system grows large, which only relies on the first two moments of the blood and demand distributions.
\subsection{Identification of the limiting process}
First, we introduce some additional notation. Let $X_b(t)$ and $X_d(t)$ denote the amount of blood and demand, respectively, at time $t>0$. Let
\begin{equation}
X(t) := X_b(t) - X_d(t),
\end{equation}
be the net amount of blood available at time $t$. Remember that $X_b(t), X_d(t)\geq 0$, and $X_b(t)>0$ \emph{or} $X_d(t)>0$ for all $t$,
since $\alpha_d=\alpha_b=0$. Let $N_b(t)$, $N_d(t)$ be the two independent Poisson processes counting the number of arrivals of blood and demand, respectively. Then the following integral representation holds for $X(t)$,
\begin{equation}\label{eq:integralRep}
X(t) = X(0) - \xi_b\int_0^t X_b(s)\, ds + \xi_d \int_0^t X_d(s)\, ds + \sum_{i=1}^{N_b(t)} B_i - \sum_{i=1}^{N_d(t)} D_i.
\end{equation}
For the sake of exhibition, we will concentrate on the case $\xi_b=\xi_d =: \xi$.
Our analysis may be extended to the general case.
A sketch of this generalization is given at the end of this section without going into the technical difficulties that arise when rigorously proving these limits.
Define
\begin{equation}\label{eq:defX}
Z(t) = \sum_{i=1}^{N_b(t)} B_i - \sum_{i=1}^{N_d(t)} D_i,
\end{equation}
that is, the difference between two compound Poisson processes, so that \eqref{eq:integralRep} reduces to
\begin{equation}\label{eq:simpleRep}
X(t) = X(0) - \xi\int_0^t X(s)\, {\rm d} s + Z(t).
\end{equation}
The first step in the definition of the sequence of processes under investigation is defining the asymptotic scheme we are interested in. As mentioned above, we intend to let the arrival rates grow to infinity. Therefore, in the $n^{th}$ process $X_n(t)$, we replace the rates of the arrival processes by $n\lambda_b$ and $n\lambda_d$. This induces Poisson processes $N^{(n)}_b(t)$ and $N^{(n)}_d(t)$ with arrival rates $n\lambda_b$ and $n\lambda_d$, respectively.
However, we have
\begin{equation}
N_b^{(n)}(t) {\;\buildrel{d}\over= \;} N_b(n t)\qquad \text{and}\qquad N_d^{(n)}(t) {\;\buildrel{d}\over= \;} N_d(n t),
\end{equation}
so that the term $Z(t)$ in \eqref{eq:simpleRep} in this asymptotic scheme can be replaced by
\begin{equation}
Z_n(t) = \sum_{i=1}^{N_b(nt)} B_i - \sum_{i=1}^{N_d(nt)} D_i.
\end{equation}
The first step in our analysis is obtaining the fluid limit of the process. Bearing in mind application of the Functional Law of Large Numbers (FLLN), we scale the process as $\bar{X}_n(t) = X_n(t)/n$,
so that with \eqref{eq:simpleRep}
\begin{equation}\label{eq:fluidRep}
\bar{X}_n(t) = \bar{X}_n(0) -\xi\int_0^t \bar{X}_n(s)\, ds + \bar{Z}_n(t),
\end{equation}
where $\bar{Z}_n(t) = Z_n(t)/n$.
The essential step in establishing a result on the convergence of $\bar{X}_n$ is the application of \cite[Thm.~4.1]{Pang2007}, which we cite here for completeness, slightly rewritten to fit our setting.
\begin{theorem}[{\cite[Thm.~4.1]{Pang2007}}]
\label{thm:pang}Let $D[0,\infty)$ be the space of all one-dimensional real-valued c\`adl\`ag functions defined on $[0,\infty)$, endowed with the usual $J_1$-Skorohod topology.
Consider the integral representation
\begin{equation}\label{eq:pangInt}
x(t) = y(t) + \int_0^t u(x(s))\, ds, \qquad t \geq 0,
\end{equation}
where $u:\mathbb{R}\to\mathbb{R}$ satisfies $u(0)=0$ and is Lipschitz continuous.
The integral representation in \eqref{eq:pangInt} has a unique solution $x$, so that the integral representation constitutes a function $H_u: D[0,\infty) \to D[0,\infty)$ mapping $y$ into $x\equiv H_u(y)$.
In addition, the function $H_u$ is continuous, and if $y$ is continuous, then so is $x$.
\end{theorem}
In our case, we set $u(x) = -\xi x$, to be able to write $\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Z}_n\right)$. Since $u$ is clearly Lipschitz continuous, the mapping $H_u$ is indeed continuous.
Let us rewrite \eqref{eq:fluidRep}, by observing
\begin{equation}
\mathbb{E} \bar{Z}_n(t) = \frac{1}{n}\Big(\mathbb{E} [N_b(nt)]\mathbb{E} [B] - \mathbb{E} [N_d(n t)]\mathbb{E}[D]\Big) = \lambda_b\mathbb{E} [B] t - \lambda_d \mathbb{E} [D] t,
\end{equation}
where the expectation is taken with respect to the compound Poisson processes.
Since $m=\lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]$,
\begin{equation}\label{eq:fluidRep2}
\bar{X}_n(t) = \bar{X}_n(0) -\xi\int_0^t\left(\bar{X}_n(s)-\frac{m}{\xi}\right)\, {\rm d} s + \bar{Y}_n(t),
\end{equation}
where $\bar{Y}_n(t) := \bar{Z}_n(t) - mt$ is now a centered process.
This allows us to state the next result.
\begin{proposition}[Fluid limit]\label{fluidProp}
Let $\mathbb{E}[B],\, \mathbb{E}[D] <\infty$ and $\bar{X}_n(0) = X_n(0)/n \rightarrow q_0 \in \mathbb{R}$, as $n\rightarrow \infty$. Then for $n\rightarrow\infty$,
\begin{equation}\label{eq:fluidLimit}
\bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q,
\end{equation}
where
\begin{equation}
q(t) = \frac{m}{\xi} + \left(q_0 - \frac{m}{\xi}\right) {\rm e}^{-\xi t}.
\label{eq:fluid_function}
\end{equation}
\end{proposition}
\begin{proof}
First, we concentrate on the process $\bar{Y}_n$. Observe that, by the FLLN for renewal-reward processes, which follows from \cite[Thm.~7.4.1]{Whitt2002}, we have
\begin{equation}
\frac{1}{nt}\sum_{i=1}^{N_b(nt)} B_i {\;\buildrel{d}\over\Rightarrow\;} \lambda_b\mathbb{E}[B],\qquad \frac{1}{nt}\sum_{i=1}^{N_d(nt)} D_i {\;\buildrel{d}\over\Rightarrow\;} \lambda_d\mathbb{E}[D],
\end{equation}
for $n\rightarrow\infty$ and for all $t>0$. Hence, $\bar{Z}_n(t) {\;\buildrel{d}\over\Rightarrow\;} \lambda_b\mathbb{E}[B]t-\lambda_d\mathbb{E}[D]t = mt$. By definition of $\bar{Y}_n$ and the assumption of convergence of $\bar{X}_n(0)$, this implies
\begin{equation}
\bar{Y}_n + \bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q_0
\end{equation}
as $n\rightarrow\infty$.
Next, note $\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Z}_n \right) = H_u\left(\bar{X}_n(0) + \bar{Y}_n +I t \right)$, where $I$ denotes the identity map, i.e.~ $I(t) = t$ for all $t \geq 0$. Due to Lipschitz continuity of $u$, $H_u$ constitutes a continuous mapping, and hence we can apply the Continuous Mapping Theorem (CMT), to find
\begin{equation}
\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Y}_n +m I\right) \Rightarrow H_u\left(q_0+m I\right)\equiv q,
\end{equation}
for all $t\geq 0$, where $q(\cdot)$ is the solution of
\begin{align*}
q(t) &= q_0 + \int_0^t u(q(s))\, {\rm d} s = q_0 + mt - \xi \int_0^t q(s) \, {\rm d} s\\
&= q_0 - \xi \int_0^t \left(q(s) - \frac{m}{\xi}\right) \, {\rm d} s.
\end{align*}
The unique solution of this integral equation is given in \eqref{eq:fluid_function}.
\end{proof}
According to Proposition~\ref{fluidProp}, the fluid limit approaches $\mathbb{E}[X] = m/\xi$ exponentially fast.
To obtain an expression for the {\em diffusion limit} of the process, we analyze the fluctuations of the process around the fluid limit in (\ref{eq:fluidLimit}), again by scaling the process in a proper manner. First, we subtract $q(t)$ on both sides of \eqref{eq:fluidRep2}, and multiply by $\sqrt{n}$:
\begin{equation}\label{eq:hatEquation}
\sqrt{n}\left( \bar{X}_n(t) - q(t) \right) = \sqrt{n}\left( \bar{X}_n(0) - q_0 \right) -\xi \int_ 0^t \sqrt{n}\left( \bar{X}_n(s) - q(s) \right)\, ds + \sqrt{n}\,\bar{Y}_n(t).
\end{equation}
Let $\hat{X}_n \equiv \sqrt{n}\left( \bar{X}_n - q \right)$ and $\hat{Y}_n \equiv \sqrt{n}\,\bar{Y}_n$, then this reduces to
\begin{equation}
\hat{X}_n(t) = \hat{X}_n(0) - \xi \int_0^t \hat{X}_n(s) \,ds + \hat{Y}_n(t).
\end{equation}
Again the process $\hat{Y}_n$ needs special attention.
\begin{lemma}\label{diffLemma}
Let $\mathbb{E}[B^2], \mathbb{E}[D^2] < \infty$. Then $\hat{Y}_n {\;\buildrel{d}\over\Rightarrow\;} \sigma W$ as $n\rightarrow \infty$, where $\sigma^2 := \lambda_b \mathbb{E}[B^2]+\lambda_d\mathbb{E}[D^2]$ and $W$ is a standard Brownian motion.
\end{lemma}
\begin{proof}
Recall that
\begin{equation}
\hat{Y}_n(t) {\;\buildrel{d}\over= \;} \sqrt{n}\left[ \Big(\frac{1}{n}\sum_{i=1}^{N_b(n t)} B_i -\lambda_b\mathbb{E} [B]t \Big) -
\Big(\frac{1}{n}\sum_{i=1}^{N_d(nt)} D_i - \lambda_d \mathbb{E}[D] t \Big)
\right] .
\end{equation}
By the Functional Central Limit Theorem (FCLT) for renewal-reward processes given in \cite[Thm.~7.4.1]{Whitt2002}, the process
\begin{equation}
\hat{Y}_n^b(t) = \sqrt{n} \Big(\frac{1}{n}\sum_{i=1}^{N_b(n t)} B_i -\lambda_b\mathbb{E}[ B] t\Big),
\end{equation}
converges weakly to $\sigma_b W_b$, where $W_b$ is a standard Brownian motion, and
\begin{equation}
\sigma_b^2 = \lambda_b\,\text{Var}\,B + \lambda_b(E[B])^2 = \lambda_b\mathbb{E}[B^2].
\end{equation}
Similarly, $\hat{Y}_n^d \Rightarrow \sigma_d W_d$ as $n\to\infty$, with the obvious parameter switches and $W_d$ is standard Brownian motion. Since the processes $\hat{Y}_n^b$ and $\hat{Y}_n^d$ are independent, so are their limits, and
\begin{equation}
\hat{Y}_n \Rightarrow \sqrt{\lambda_b \mathbb{E}[B^2]}\, W_b + \sqrt{\lambda_d \mathbb{E}[D^2]}\, W_d {\;\buildrel{d}\over= \;} \sqrt{\lambda_b\mathbb{E}[B^2]+\lambda_d\mathbb{E}[D^2]}\,W,
\end{equation}
for $n\rightarrow\infty$ and $W$ a standard Brownian motion.
\end{proof}
Now, we are ready to prove the diffusion counterpart of Proposition \ref{fluidProp}.
\begin{proposition}[Diffusion limit]\label{diffProp}
Let $\mathbb{E}[B^2], \mathbb{E}[D^2] < \infty$. If $\hat{X}_n(0) \rightarrow \hat{X}(0)$, then $\hat{X}_n \Rightarrow \hat{X}$ as $n\rightarrow \infty$, where $\hat{X}$ satisfies the integral equation
\begin{equation}\label{diffLimit}
\hat{X}(t) = \hat{X}(0) - \xi \int_0^t \hat{X}(s) \, {\rm d} s + \sigma W(t).
\end{equation}
In other words, $\hat{X}$ is an Ornstein-Uhlenbeck diffusion process with infinitesimal mean $\xi$ and infinitesimal variance $\sigma^2 := \lambda_b\mathbb{E}[B^2] + \lambda_d\mathbb{E}[D^2]$.
\end{proposition}
\begin{proof}
We again rely on the result that the mapping $H_u$ as in the proof of Proposition \ref{fluidProp} is continuous if $u$ is Lipschitz continuous. Here, we set $u(x) = -\xi x$ which again clearly satisfies this condition. We have $\hat{X}_n \equiv H_u(\hat{X}_n(0)+ \hat{Y}_n)$. From Lemma \ref{diffLemma}, we know
\begin{equation}
\hat{X}_n(0) + \hat{Y}_n \Rightarrow \hat{X}(0) + \sigma W,
\end{equation}
for $n\rightarrow\infty$. As a consequence of the CMT, we conclude
\begin{equation}
\hat{X}_n = H_u\left( \hat{X}_n(0) + \hat{Y}_n\right) \Rightarrow H_u\left(\hat{X}(0) + \sigma W\right) \equiv \hat{X},
\end{equation}
where $\hat{X}$ solves \eqref{diffLimit}.
\end{proof}
\subsection{Generalization for $\xi_b\neq \xi_d$}
We now sketch the scaling approach towards fluid and diffusion limits for the general case in which
$\xi_b$ may differ from $\xi_d$.
In case $\xi_b \neq \xi_d$, the integral equation for $\bar{X}_n$ as in \eqref{eq:fluidRep} becomes
\begin{align}
\label{eq:fluidRepNEQ}
\bar{X}_n(t) &= \bar{X}_n(0) + \int_0^t ( -\xi_b \bar{X}_n^+(s) + \xi_d \bar{X}_n^-(s) - m)\, {\rm d} s + \bar{Y}_n(t)\\
&= \bar{X}_n(0) - \int_0^t (\left[\xi_b \mathbbm{1}_{\{ \bar{X}_n(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{ \bar{X}_n(s)<0\}}\right] \bar{X}_n(s) + m) \, {\rm d} s + \bar{Y}_n(t),
\nonumber
\end{align}
where $\bar{Y}_n(t)$ is defined as before. Note that $\hat{X}_n \equiv H_u(\bar{X}_n(0)+\bar{Y}_n)$, where we now have
\begin{equation}
u(x) = - \left[\xi_b \mathbbm{1}_{\{ x\geq 0\}}+\xi_d \mathbbm{1}_{\{x<0\}}\right]x + m,
\end{equation}
which is still Lipschitz continuous. Following the same reasoning of the proof of Proposition \ref{fluidProp}, we obtain the fluid limit $\bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q$, where $q$ is the solution of
\begin{equation}
q(t) = q_0 - \int_0^t (\left[\xi_b \mathbbm{1}_{\{ q(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{q(s)<0\}}\right]q(s) - m)\, {\rm d} s .
\end{equation}
The solution to this integral equation is more elaborate than \eqref{eq:fluidLimit} and depends on the sign of $m$ and $q_0$. Assuming $m\geq 0$, one can check that,
\begin{align}
q(t) &= \frac{m}{\xi_b} + \left(q_0- \frac{m}{\xi_b}\right) e^{-\xi_b t}, & \text{if }q_0\geq 0,\\
q(t) &= \left\{
\begin{array}{ll}
\frac{m}{\xi_d} + \left(q_0- \frac{m}{\xi_d}\right) e^{-\xi_d t}, & \text{if } 0\leq t < t_d^*,\\
\frac{m}{\xi_b}\left(1-e^{-\xi_b (t-t^*_d)}\right), & \text{if } t \geq t_d^*,
\end{array}\right. & \text{if } q_0 <0,
\label{eq:fluid_1}
\end{align}
where \begin{equation}
t_d^* = - \frac{1}{\xi_d}\,\log\left(\frac{m/\xi_d}{m/\xi_d-q_0}\right).
\end{equation}
If $m < 0$,
\begin{align}
q(t) &= \frac{m}{\xi_d} + \left(q_0- \frac{m}{\xi_d}\right) e^{-\xi_d t}, & \text{if }q_0\leq 0,\\
q(t) &= \left\{
\begin{array}{ll}
\frac{m}{\xi_b} + \left(q_0- \frac{m}{\xi_b}\right) e^{-\xi_b t}, & \text{if } 0\leq t < t_b^*,\\
\frac{m}{\xi_d}\left(1-e^{-\xi_d (t-t^*_b)}\right), & \text{if } t \geq t_b^*,
\end{array}\right. & \text{if } q_0>0,
\label{eq:fluid_2}
\end{align}
where \begin{equation}
t_b^* = - \frac{1}{\xi_b}\,\log\left(\frac{m/\xi_b}{m/\xi_b-q_0}\right).
\end{equation}
Note that the equilibrium of the fluid limit also depends on the sign of $m$:
\begin{equation}
\lim_{t\rightarrow\infty} q(t) = \left\{\begin{array}{ll}
m/\xi_b, & \text{if }m\geq 0,\\
m/\xi_d, & \text{if }m <0.
\end{array}\right.
\end{equation}
In the remainder, without loss of generality $m\geq 0$. Furthermore, set $q_0= m/\xi_b$ so that $q \equiv m/\xi_b$. Subtracting $q(t)$ on both sides of \eqref{eq:fluidRepNEQ} yields,
\begin{align}
\left(\bar{X}_n(t)-q(t)\right) &= \left(\bar{X}_n(0)-q_0\right) -
\int_0^t \Big\{ \left[\xi_b \mathbbm{1}_{\{\bar{X}_n(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{\bar{X}_n(s)<0\}}\right] \bar{X}_n(s) \nonumber \\
& \qquad \qquad - \xi_b\,q(s)\Big\} \, {\rm d} s + \bar{Y}_n(t)\\
&= \left(\bar{X}_n(0)-q_0\right) - \int_0^t \xi_b \left(\bar{X}_n(s)-q(s)\right)\, {\rm d} s \nonumber \\
&\qquad \qquad +
\int_0^t \mathbbm{1}_{\{\bar{X}_n(s) < 0\}}(\xi_b-\xi_d)\bar{X}_n(s)\, {\rm d} s.
\end{align}
Let $\hat{X}_n(t) = \sqrt{n}\left(\bar{X}_n(t)-q(t)\right)$. Then
\begin{equation}
\hat{X}_n(t) = \hat{X}_n(0) - \xi_b \int_0^t \hat{X}_n(s)\,{\rm d} s +\int_0^t \mathbbm{1}_{\{\bar{X}_n(s) < 0\}}(\xi_b-\xi_d)\bar{X}_n(s)\, {\rm d} s + \hat{Y}_n(t)
\end{equation}
Now, we argue non-rigorously that the one-but-last term vanishes as $n\rightarrow\infty$. Namely, by defining the function $G: D[0,\infty)\rightarrow D[0,\infty)$ by the integration operator:
\begin{equation}
G(u) = \int_0^t \mathbbm{1}_{\{u(s)<0\}} (\xi_b-\xi_d) u(s)\, {\rm d} s,
\end{equation}
this term can be expressed as $G(\bar{X}_n)$. Hence by the fact that $\hat{X}_n{\;\buildrel{d}\over\Rightarrow\;} m/\xi_b$ and the CMT we see $G(\hat{X}_n)\Rightarrow 0$.
Under this claim, we deduce by the approach of Proposition \ref{diffProp}, that if $\hat{X}_n \Rightarrow \hat{X}$ for $n\rightarrow \infty$, then $\hat{X}$ satisfies the stochastic integral equation
\begin{equation}
\hat{X}(t) = \hat{X}(0) - \xi_b \int_0^t \hat{X}(s)\, {\rm d} s + \sigma W(t),
\end{equation}
which implies that $\hat{X}$ is an Ornstein-Uhlenbeck process with infinitesimal mean $\xi_b$ and variance $\sigma^2 := \lambda_b\mathbb{E}[B^2] + \lambda_d\mathbb{E}[D^2]$.
\\
The result that the scaled process converges to an Ornstein-Uhlenbeck process can be intuitively justified by the so-called \textit{mean-reverting} behavior of the original process. That is, the further the process is away from its mean, the greater the drift towards that equilibrium. This is the defining feature of the OU diffusion process. The decay rates $\xi_b$ and $\xi_d$ are responsible for the original process being `forced' towards 0 and therefore the similarities should not be surprising. However, note that in the diffusion limit $X_n$ has drift $\xi_b$ (cq. $\xi_d$) towards $nm/\xi_b$ (cq. $nm/\xi_d$), if $m>0$ (cq. $<0$) at \emph{any} position of the process. This implies that if $X_n \in(0,nm/\xi_b)$, it has an upward drift equal to $\xi_b$, which is at first sight counter-intuitive.
{\color{blue}
However, we can argue that in case $X_n(t) = v \in (0, nm/\xi_b)$, the mean upward drift of the process $X_n$ equals $n\lambda_b\mathbb{E}[B]$, and the mean downward drift equals $n\lambda_d\mathbb{E}[D] + \xi_b v$, since $v>0$.
Rewrite $v = nm/\xi_b - w\sqrt{n}$ for some $w \in (0,\sqrt{n} m/\xi_b)$.
Then, the mean net drift equals
\[
n\lambda_b\mathbb{E}[B] - n\lambda_d\mathbb{E}[D] - \xi_b \left( \frac{n m}{\xi_b} - w\sqrt{n} \right) = \xi_b w \sqrt{n} >0,
\]
which explains both the sign and magnitude of the drift factor in the scaled process.
\subsection{Related literature}
The Ornstein-Uhlenbeck process is a diffusion process that often arises as the limit of a sequence of stochastic systems, in which the system size tends to infinity.
Particularly in queueing settings with mean reverting behavior, the OU process appears in so-called heavy traffic, i.e.~the arrival rate grows without bound.
We mention a couple of models that exhibit limiting behavior that is similar to ours.
First, it is well-known that the properly normalized $M/M/\infty$ queue length process converges weakly to a OU process as the arrival rate tends to infinity, see e.g.~\cite[Sec.~10.3]{Whitt2002}.
This limiting behavior continues to hold in case the queueing process is modulated by a Markovian background process, see \cite{Anderson2016}.
Another well-known queueing model in which a (piecewise) OU process appears in the limit is the multi-server queue with abandonments.
For the $M/M/s+M$ queue, where $+M$ denotes the exponentially distributed patience of customers, Garnett et al.~\cite{Garnett2002} showed that in the Halfin-Whitt regime, the queue length process, centered and scaled around the number of servers $s$, approaches a hybrid OU process, of which the drift parameter depends on the current state: If the queue length is larger (cq.~smaller) than zero, then the drift is governed by the abandonment rate (cq.~service rate).
Dai et al.~\cite{Dai2010} find a similar piecewise diffusion process under more general assumptions on the model primitives.
For the single-server queue with abandoning customers, Ward \& Glynn \cite{Ward2003,Ward2005} showed that in conventional heavy traffic, the queue length process converges to a OU process with reflecting barrier 0.
Since we in our setting assumed both demand impatience and perishability of inventory (which can be seen as a kind of impatience as well), it should not come as a surprise that we also find our limiting process to be a OU process.
Observe however that in our model, unless $m=0$, we find a OU process with constant, rather than piecewise, parameters, and no reflection barrier, since our (scaled) inventory process can go both positive and negative.
Last, we mention that there is a connection between our blood inventory process and the work of Reed \& Zwart \cite{Reed2011}.
Rather than looking at the OU process as the limit of a sequence of stochastic processes, Reed and Zwart \cite{Reed2011} study a stochastic differential equation that is closely related to Equation \eqref{eq:integralRep}, in the sense that the process has a different (constant) drift term in the upper and lower half plane.
Under the assumption that the input process is a L\'evy process with only one-sided jumps, they develop a methodology to derive the invariant distribution of the solution of the SDE.
Unfortunately, the input in our scenario exhibits both positive and negative jumps, which prevents us from applying their results directly to \eqref{eq:integralRep}.
}
\section{Numerical evaluation}
\label{numericals}
\subsection{Approximation scheme}
The asymptotic results of the previous section regarding the fluid and diffusion limits allude to the fact that for large arrival rates, the normalized inventory process $\{\hat X_n(t)\, |\, t\geq 0\}$, resembles that of the Ornstein-Uhlenbeck process.
Indeed, the sample paths of the scaled process $\bar X_n$ for increasing values of $n$ in Figures \ref{fig:sample_paths_fluid1} and \ref{fig:sample_paths_fluid2} show that the mean-reverting behavior around $m/\xi^*$, that is typical of OU processes, kicks in rather quickly.
Moreover, the fluid limits $q(t)$ as presented by Proposition \ref{fluidProp} and \eqref{eq:fluid_1}-\eqref{eq:fluid_2} predict the mean well for both $\xi_b=\xi_d$ and $\xi_b \neq \xi_d$.
\begin{figure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp1] {tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp10] {tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=10$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp100] {tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=100$}
\end{subfigure}
\caption{Sample paths of the process $\bar{X}_n(t) = X_n(t)/n$ with $\bar{X}_n(0) = 5$, $\lambda_b = 1.2$, $\lambda_d = 1$, $\xi_b = \xi_d = 0.5$ and $\mu_b=0.5$ and $\mu_d=1$. The fluid limit is depicted by the dashed line.}
\label{fig:sample_paths_fluid1}
\end{figure}
\begin{figure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp1] {tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp10] {tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=10$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp100] {tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=100$}
\end{subfigure}
\caption{Sample paths of the process $\bar{X}_n(t) = X_n(t)/n$ with $\bar{X}_n(0) = -2$, $\lambda_b = 2$, $\lambda_d = 1$, $\xi_b = 0.5, \xi_d = 0.1$ and $\mu_b=1$ and $\mu_d=1$. The fluid limit is depicted by the dashed line.}
\label{fig:sample_paths_fluid2}
\end{figure}
Furthermore, we observe that steady state is attained fairly quickly.
This is suggestive of the claim that the steady-state distribution of the normalized process $\hat{X}_n$ is well-described by the steady-state distribution of the OU process $\hat{X}$.
Since the OU process with mean 0, infinitesimal variance $\sigma^2$ and drift $\xi^*$ is known to be normally distributed with mean 0 and variance $\sigma^2/2\xi^*$ in steady-state, this leads to a simpler approximation scheme based on the first two moments of $B$ and $D$ only.
In non-rigorous mathematical terms, we use the approximation that
\begin{equation}
\hat{X}_n = \frac{ X_n - nm/\xi^*}{\sqrt{n}} {\;\buildrel{d}\over\approx \;} Z^*,
\label{eq:normal_approximation}
\end{equation}
where $Z^*$ is a normally distributed random variable with mean 0 and variance $\sigma^2/2\xi^*$.
Note that justification of the conjecture that the normal approximation is indeed an asymptotically correct approximation for systems with large arrival rates requires proof that the interchange-of-limits between $t\to\infty$ and $n\to\infty$ is indeed valid.
Rather than going into the technical details, we provide in the remainder of this section numerical evidence that this interchange indeed holds, and that the normal approximation is able to capture characteristics of processes with exponential jumps as well as generally distributed jumps.
\subsection{Distribution functions}
Since we obtained an explicit expression for the steady-state density function of the net inventory process $X$ in case $B$ and $D$ are exponential, see Theorem \ref{thm:full_pdf}, we will exploit this formula for numerical comparison to the normal approximation arising from the OU process.
Let $h(\cdot)$ as in Theorem \ref{thm:full_pdf} be the pdf of $X$ with parameters $\lambda_b$, $\lambda_d$, $\mu_b$, $\mu_d$, $\xi_b$ and $\xi_d$, and the corresponding cdf $H$, defined as $H(v) = \int_{-\infty}^v h(x) {\rm d} x$.
We denote by $h_n$ and $H_n$ the pdf and cdf, respectively, of the inventory process $X_n$ with arrival rates $n\lambda_b$ and $n\lambda_d$, and the remaining parameters unchanged.
Then, the pdf and cdf of the normalized process are given by $\hat{h}_n(v) = \sqrt{n}\,h_n(v_n)$ and $\hat{H}_n(v) = H_n(v_n)$, respectively, with $v_n = nm/\xi^* + v\sqrt{n}$ for all $v\in\mathbb{R}$.
By the normal approximation scheme, we expect
\begin{equation}
\hat{h}_n(v) \approx \frac{\sqrt{2\xi^*}}{\sigma}\,\varphi\left(\frac{\sqrt{2\xi^*}}{\sigma}v\right),\quad \text{ and } \quad
\hat{H}_n(v) \approx \Phi\left(\frac{\sqrt{2\xi^*}}{\sigma}v\right).
\end{equation}
We perform this numerical comparison of probability functions in Figure \ref{fig:distributions} for three cases: $\xi_b=\xi_d$, $\xi_b>\xi_d$ and $\xi_b<\xi_d$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {tikz/convergence_1_1.txt};
\addplot[thick,col4] table[x=v,y=h5] {tikz/convergence_1_1.txt};
\addplot[thick,col5] table[x=v,y=h10] {tikz/convergence_1_1.txt};
\addplot[thick,dashed] table[x=v,y=ou] {tikz/convergence_1_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=\xi_d=1$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {tikz/convergence_1_1.txt};
\addplot[thick,col4] table[x=v,y=H5] {tikz/convergence_1_1.txt};
\addplot[thick,col5] table[x=v,y=H10] {tikz/convergence_1_1.txt};
\addplot[thick,dashed] table[x=v,y=OU] {tikz/convergence_1_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=\xi_d=1$ (cdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {tikz/convergence_1_2.txt};
\addplot[thick,col4] table[x=v,y=h5] {tikz/convergence_1_2.txt};
\addplot[thick,col5] table[x=v,y=h10] {tikz/convergence_1_2.txt};
\addplot[thick,dashed] table[x=v,y=ou] {tikz/convergence_1_2.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=1,\ \xi_d=2$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {tikz/convergence_1_2.txt};
\addplot[thick,col4] table[x=v,y=H5] {tikz/convergence_1_2.txt};
\addplot[thick,col5] table[x=v,y=H10] {tikz/convergence_1_2.txt};
\addplot[thick,dashed] table[x=v,y=OU] {tikz/convergence_1_2.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=1,\ \xi_d=2$ (cdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {tikz/convergence_2_1.txt};
\addplot[thick,col4] table[x=v,y=h5] {tikz/convergence_2_1.txt};
\addplot[thick,col5] table[x=v,y=h10] {tikz/convergence_2_1.txt};
\addplot[thick,dashed] table[x=v,y=ou] {tikz/convergence_2_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=2,\ \xi_d=1$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {tikz/convergence_2_1.txt};
\addplot[thick,col4] table[x=v,y=H5] {tikz/convergence_2_1.txt};
\addplot[thick,col5] table[x=v,y=H10] {tikz/convergence_2_1.txt};
\addplot[thick,dashed] table[x=v,y=OU] {tikz/convergence_2_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=2,\ \xi_d=1$ (cdf)}
\end{subfigure}
\caption{Probability functions of $\hat{X}_n$ for $n=1,5$ and $10$ with $\lambda_b=1,\lambda_d=0.5,\mu_b=\mu_d=1$, and the probability function of the OU process.}
\label{fig:distributions}
\end{figure}
From Figure \ref{fig:distributions}, in which $m = 1$, so that $\xi^* = \xi_b$, the convergence of the pdf and cdf is evident.
For $n=10$, the distribution functions of the scaled processes are almost aligned with the normal distribution already.
For $\xi_b=\xi_d$, the convergence is fastest.
This can be explained by observing that in cases where $\xi_b\neq \xi_d$, the parameter $\xi_d$ still plays a role in pre-limit systems, whereas it does not appear in the normal limit.
In the cases where $\xi_b\neq \xi_d$ we furthermore see that the functions are not smooth around $v_n=0$ or $v^* = -\sqrt{n}m/\xi^*$, which is the zero-inventory level in the original (unscaled) process.
As $n$ increases, this point of irregularity goes to $-\infty$ and therefore disappears.
\subsection{Approximations to performance metrics}
The plots in the previous section indicate that the normal approximation gives simple yet accurate approximations to the stationary distribution of the inventory process.
We now assess if this also translates to the performance measures.
Again, we choose to fix the parameters $\lambda_b$ and $\lambda_d$, and evaluate the system with arrival rates $n\lambda_b$ and $n\lambda_d$ for increasing $n$.
First, the normal approximation in \eqref{eq:normal_approximation} yields the following approximation for the expected inventory level:
\begin{equation}
\mathbb{E}[X_n] \approx \frac{ n m }{\xi^*} = \frac{n (\lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]) }{\xi^*}.
\label{eq:approx1}
\end{equation}
For the probability of negative inventory, we have
\begin{equation}
\pi_d = \mathbb{P}(X_n < 0) \approx \mathbb{P}\left( Z^* < -\sqrt{n}\,m/\xi^* \right) = \Phi\left(-\sqrt{n/2\xi^*}\,m/\sigma\right).
\label{eq:approx2}
\end{equation}
Last, the probability of demand being satisfied immediately is approximately
\begin{equation}
\mathbb{P}({\rm demand\ satisfied }) = \mathbb{P}( X_n > D ) \approx 1 - \int_0^\infty \Phi\left({-}\frac{\sqrt{2\xi^*}}{\sigma}\, \frac{x-nm/\xi^*}{\sqrt{n}}\right) \,{\rm d} F_d(x).
\label{eq:approx3}
\end{equation}
\begin{remark}
Note that if $\lambda_b$ and $\lambda_d$ are large themselves, the parameter $n$ can be eliminated from \eqref{eq:approx1}-\eqref{eq:approx3}, so that
\begin{equation*}
\mathbb{E}[X] \approx \frac{m}{\xi^*}, \qquad
\pi_d \approx \Phi\left({-}m/(\sigma\sqrt{2\xi^*})\right),
\end{equation*}
\begin{equation*}
\mathbb{P}({\rm demand\ satisfied}) \approx 1-\int_0^\infty \Phi\left({-} \sqrt{2\xi^*}\,\frac{x-m/\xi^*}{\sigma}\right)\, {\rm d} F_d(x),
\end{equation*}
where $m = \lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]$ and $\sigma^2 = \lambda_b \mathbb{E}[B^2] + \lambda_d \mathbb{E}[D^2]$.
\end{remark}
We will now test these approximations under various assumptions on the distribution of $B$ and $D$.
In Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma} we compare the values obtained through the normal approximation against the true values obtained through numerical evaluation (for exponential jump sizes only) and simulation.
All simulation results are accurate up to a 95\% confidence interval of width $10^{-4}$.
We set $\lambda_b=1$ and $\lambda_d=0.5$ and let the mean jump sizes be equal to 1, i.e.~ $\mathbb{E}[B] = 1$ and $\mathbb{E}[B] = 1$ in all numerical experiments.
In Table \ref{tab:accuracy_deterministic}, we let the jump sizes be deterministic, so that ${\rm Var}\, B = {\rm Var}\, D = 0$.
Table \ref{tab:accuracy_exponential} shows the results in case of exponential jump sizes, so that ${\rm Var}\, B = {\rm Var}\, D = 1$.
Last, in Table \ref{tab:accuracy_gamma} we investigate the quality of the approximation for jump sizes that follow a Gamma$(0.25,0.25)$ distribution, yielding ${\rm Var}\, B = {\rm Var}\, D = 4$.
With this set-up we cover jump distributions of increasing variance, so that we are able to study the impact of increased variability on the accuracy of the approximations.
Moreover, we investigate the influence of the decay parameters $\xi_b$ and $\xi_d$ by considering the scenarios $\xi_b=\xi_d$, $\xi_b<\xi_d$ and $\xi_b>\xi_d$.
We make a couple of observations based on the numbers in Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma}.
First, we see that the approximation for the mean blood inventory level $\mathbb{E}[X_n]$ is exact if $\xi_b=\xi_d$, see Proposition \ref{prop:mean_inventory}.
This obviously does not extend to $\pi_d$ and $\mathbb{P}({\rm demand\ satisfied})$, since these performance measures are based on the entire distribution of $X_n$ rather than the mean.
Nonetheless, the normal approximation appears to be very accurate in the case $\xi_b=\xi_d$.
We may explain this by observing that in the approximations \eqref{eq:approx1}-\eqref{eq:approx3}, only $\xi^*$ appears.
In our setting, we have $m = \lambda_b - \lambda_d = 0.5$, so that $\xi^* = \xi_b$.
If $\xi_b\neq \xi_d$, then the value of $\xi_d$ plays a role in pre-limit systems, which induces inaccuracies in the approximation of performance measures.
In case $\xi_b = \xi_d$, we have $\xi^* = \xi_b = \xi_d$, so that this discrepancy is overcome.
Moreover, since $m>0$, we see that $\pi_d\to 0$ and $\mathbb{P}({\rm demand\ satisfied}) \to 1$ as $n$ increases.
This is due to the observation that as $n$ grows large, the inventory process concentrates around the level $nm$ with fluctuations of order $\sqrt{n}$, so that the process stays away from level zero, see Figure \ref{fig:sample_paths_fluid1}.
The approximations \eqref{eq:approx2}-\eqref{eq:approx3} adequately capture this convergence.
As expected, the accuracy of the approximations increases with $n$.
Moreover, increased variability in the jump distributions appears to cause a decrease in accuracy.
However, for all cases considered in Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma}, the normal approximations \eqref{eq:approx1}-\eqref{eq:approx3} seem to yield relatively sharp estimates for the relevant performance measures under various assumptions on the distributions of the jump sizes.
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.2702 & 0.2819 & 0.2598 & 0.2819 \bigstrut[t]\\
2 & 1.000 & 1.000 & 0.2014 & 0.2071 & 0.4859 & 0.5000 \\
5 & 2.500 & 2.500 & 0.0943 & 0.0984 & 0.7814 & 0.7807 \\
10 & 5.000 & 5.000 & 0.0316 & 0.0339 & 0.9306 & 0.9279 \\
20 & 10.000 & 10.000 & 0.0043 & 0.0049 & 0.9908 & 0.9899 \\
50 & 25.000 & 25.000 & 0.0000 & 0.0000 & 1.0000 & 1.0000 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.584 & 0.500 & 0.2522 & 0.2819 & 0.2712 & 0.2819 \bigstrut[t]\\
2 & 1.086 & 1.000 & 0.1809 & 0.2071 & 0.5020 & 0.5000 \\
5 & 2.558 & 2.500 & 0.0837 & 0.0984 & 0.7911 & 0.7807 \\
10 & 5.024 & 5.000 & 0.0286 & 0.0339 & 0.9335 & 0.9279 \\
20 & 10.006 & 10.000 & 0.0040 & 0.0049 & 0.9912 & 0.9899 \\
50 & 25.000 & 25.000 & 0.0000 & 0.0000 & 1.0000 & 1.0000 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.158 & 0.250 & 0.3308 & 0.3415 & 0.1006 & 0.1103 \bigstrut[t]\\
2 & 0.397 & 0.500 & 0.2973 & 0.2819 & 0.2465 & 0.2819 \\
5 & 1.164 & 1.250 & 0.1952 & 0.1807 & 0.5482 & 0.5724 \\
10 & 2.447 & 2.500 & 0.1036 & 0.0984 & 0.7729 & 0.7807 \\
20 & 4.980 & 5.000 & 0.0340 & 0.0339 & 0.9283 & 0.9279 \\
50 & 12.497 & 12.500 & 0.0017 & 0.0019 & 0.9964 & 0.9960 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and deterministic jump sizes, $B\equiv 1$ and $D\equiv 1$.}
\label{tab:accuracy_deterministic}
\end{table}
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.2929 & 0.3415 & 0.3536 & 0.3925 \\
2 & 1.000 & 1.000 & 0.2500 & 0.2819 & 0.5000 & 0.5135 \\
5 & 2.500 & 2.500 & 0.1642 & 0.1807 & 0.7062 & 0.7009 \\
10 & 5.000 & 5.000 & 0.0898 & 0.0984 & 0.8491 & 0.8418 \\
20 & 10.000 & 10.000 & 0.0307 & 0.0339 & 0.9506 & 0.9467 \\
50 & 25.000 & 25.000 & 0.0017 & 0.0019 & 0.9974 & 0.9970 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.621 & 0.500 & 0.2589 & 0.3415 & 0.3705 & 0.3925 \bigstrut[t]\\
2 & 1.153 & 1.000 & 0.2164 & 0.2819 & 0.5224 & 0.5135 \\
5 & 2.656 & 2.500 & 0.1414 & 0.1807 & 0.7254 & 0.7009 \\
10 & 5.113 & 5.000 & 0.0784 & 0.0984 & 0.8598 & 0.8418 \\
20 & 10.050 & 10.000 & 0.0275 & 0.0339 & 0.9538 & 0.9467 \\
50 & 25.004 & 25.000 & 0.0016 & 0.0019 & 0.9975 & 0.9970 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.125 & 0.250 & 0.3548 & 0.3864 & 0.2168 & 0.2942 \bigstrut[t]\\
2 & 0.333 & 0.500 & 0.3333 & 0.3415 & 0.3333 & 0.3925 \\
5 & 1.059 & 1.250 & 0.2647 & 0.2593 & 0.5264 & 0.5570 \\
10 & 2.333 & 2.500 & 0.1856 & 0.1807 & 0.6881 & 0.7009 \\
20 & 4.893 & 5.000 & 0.0995 & 0.0984 & 0.8400 & 0.8418 \\
50 & 12.475 & 12.500 & 0.0198 & 0.0206 & 0.9692 & 0.9678 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and exponentially distributed jump sizes, $B\sim \exp(1)$ and $D\sim\exp(1)$.}
\label{tab:accuracy_exponential}
\end{table}
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.3118 & 0.3981 & 0.4412 & 0.4636 \\
2 & 1.000 & 1.000 & 0.2894 & 0.3575 & 0.5343 & 0.5288 \\
5 & 2.500 & 2.500 & 0.2375 & 0.2819 & 0.6590 & 0.6381 \\
10 & 5.000 & 5.000 & 0.1785 & 0.2071 & 0.7592 & 0.7385 \\
20 & 10.000 & 10.000 & 0.1090 & 0.1241 & 0.8593 & 0.8454 \\
50 & 25.000 & 25.000 & 0.0303 & 0.0339 & 0.9624 & 0.9583 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.667 & 0.500 & 0.2695 & 0.3981 & 0.4636 & 0.4636 \bigstrut[t]\\
2 & 1.253 & 1.000 & 0.2469 & 0.3575 & 0.5632 & 0.5288 \\
5 & 2.863 & 2.500 & 0.2009 & 0.2819 & 0.6895 & 0.6381 \\
10 & 5.385 & 5.000 & 0.1518 & 0.2071 & 0.7834 & 0.7385 \\
20 & 10.328 & 10.000 & 0.0938 & 0.1241 & 0.8739 & 0.8454 \\
50 & 25.124 & 25.000 & 0.0269 & 0.0339 & 0.9658 & 0.9583 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.081 & 0.250 & 0.3694 & 0.4276 & 0.3270 & 0.4104 \bigstrut[t]\\
2 & 0.238 & 0.500 & 0.3593 & 0.3981 & 0.4137 & 0.4636 \\
5 & 0.857 & 1.250 & 0.3237 & 0.3415 & 0.5311 & 0.5528 \\
10 & 2.045 & 2.500 & 0.2739 & 0.2819 & 0.6282 & 0.6381 \\
20 & 4.568 & 5.000 & 0.2039 & 0.2071 & 0.7361 & 0.7385 \\
50 & 12.231 & 12.500 & 0.0966 & 0.0984 & 0.8797 & 0.8779 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and Gamma distributed jump sizes, $B\sim\text{Gamma}(0.25,0.25)$ and $D \sim\text{Gamma}(0.25,0.25)$.}
\label{tab:accuracy_gamma}
\end{table}
\section{Conclusions \& suggestions for further research}
\label{conclus}
In this chapter, we studied a stochastic model for a blood bank.
We have presented a global approach to the model in its full generality,
and obtained very detailed exact expressions for the densities of amount of inventory and amount of demand (shortage)
in special cases (exponential amounts of donated and requested blood; and either $\xi_b=\xi_d=0$ or $\alpha_b=\alpha_d=0$).
Moreover, we have shown how an appropriate scaling, for the model in full generality, leads to an Ornstein-Uhlenbeck diffusion process,
which can be used as a tool to obtain simple yet accurate approximations for some key performance measures.
Our model is a two-sided model, in the sense that we simultaneously consider
the amount of blood in inventory and the amount of demand (shortage), one of the two at any time being zero.
Such two-sided processes arise in many different settings, and thus are of considerable interest.
The present setting is reminiscent of an organ transplantation problem, where there is
either a queue of persons waiting to receive an organ,
or a queue of donor organs. The perishability/impatience aspect features there too \cite{Boxma2011}.
A quite different setting is that of insurance risk. We refer to Albrecher \& Lautscham \cite{Albrecher2013}
who extend the classical Cram\'er-Lundberg insurance risk model by allowing the capital of an insurance company
to become negative -- a situation that is usually indicated by ``ruin" in the insurance literature.
Their process thus becomes two-sided. The capital might become positive again; however,
at a rate $\omega(x)$ when the capital has a negative value $-x$, bankruptcy is declared and the process ends.
Interestingly, similar special functions (like hypergeometric functions) play a role in \cite{Albrecher2013} and in the present study.
The analyses performed in this chapter, which evolved around a simplified version of the inventory process of a blood bank, revealed some interesting avenues for further research.
We name a couple of them.
First, we remark that our results are restricted to one type of blood.
Naturally, it would be very interesting to extend the analysis to multiple types of blood.
Another important extension would be to use our results to facilitate the decision process that is faced by the CBB on a daily basis:
Which amounts of blood, and of which types, should today be sent to the local blood banks (hospitals)?
Knowing that, e.g., blood types $O^-,A^-,B^-,AB^-$ can satisfy the corresponding $+$ type (but not vice versa),
one may try to optimize the blood allocation process on the basis of actual amounts of blood present.
Finally, we mention a significant open research question regarding the process limits that we derived in Section \ref{sectionscaling}, of which the steady-state distributions were used to approximate steady-state performance measures in pre-limit systems.
As we pointed out earlier, the justification that the steady-state distribution of the scaled inventory process indeed converges to the steady-state distribution of the fluid (cq.~diffusion) limit requires a rigorous argument why the order of limits $n\to\infty$ and $t\to \infty$ may be interchanged.
Proving interchange-of-limits statements typically raises many technical challenges, see e.g.~\cite{Dai2014a,Gamarnik2013a,Gurvich2013,Gamarnik2006} for works tackling this issue in the context of queues in heavy traffic.
The usual approach is to prove tightness of the sequence of steady-state distributions of pre-limit, followed by applying Prokhorov's theorem, see e.g.~\cite[Sec.~1.5]{Billingsley1995}.
For our model, such an approach seems to be straightforward for the fluid scaling, since our inventory process can be upper (cq.~lower) bounded by a shot-noise process with only positive (cq.~negative) jumps.
Of the latter, the steady-state behavior is known.
This allows us to derive a uniform bound on the absolute mean of the stationary fluid-scaled process, which gives tightness.
The final step uses the deterministic nature of the differential equation governing the dynamics of the fluid limit, by which the steady-state distribution must be unique.
For the diffusion-scaled process, the steps towards proving the interchange-of-limits are not obvious and hence this needs further investigation.
Our numerical results for various jump size distributions, however, support the conjecture that this interchange is indeed valid.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Transformation integral equation}
\label{app:transformation_int}
In this appendix we show how integral equation (\ref{eq:demand2}) can be transformed into a second-order differential equation,
in the case of exponential $F_b(\cdot)$ and $F_d(\cdot)$.
Differentiate (\ref{eq:demand2}) w.r.t.\ $v$:
\begin{align}
& \lambda_d f(v) - \mu_d \left[\lambda_d \int_0^v f(y) {\rm e}^{-\mu_d(v-y)} {\rm d}y +
\lambda_d \int_0^{\infty} g(y) {\rm e}^{-\mu_d (v+y)} {\rm d}y\right]
\nonumber
\\
&\qquad =
-\lambda_b f(v) + \lambda_b \mu_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y + \xi_d f(v) +\xi_d
v f'(v) .
\label{eq:demand3}
\end{align}
Using (\ref{eq:demand2}) once more, now to replace the term between square brackets in (\ref{eq:demand3}),
we get:
\begin{align}
\xi_d v f'(v) &= (\lambda_d +\lambda_b - \xi_d) f(v)
\nonumber
\\
&\qquad - \mu_d \left(\lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y + \xi_d v f(v)\right)
\nonumber
\\
&\qquad \qquad - \mu_b \lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y,
\label{eq:demand4}
\end{align}
and once more differentiating w.r.t.\ $v$ then gives:
\begin{align}
&\xi_d v f''(v) + \xi_d f'(v) - (\lambda_d +\lambda_b -\xi_d -\mu_d \xi_d v) f'(v)
\nonumber
\\
& \qquad = -\mu_d \xi_d f(v) +(\mu_b+\mu_d) \lambda_b f(v) -\mu_b(\mu_b+\mu_d) \lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y .
\label{eq:demand5}
\end{align}
The integral that appears in \eqref{eq:demand4} can be eliminated by using (\ref{eq:demand5}),
and we thus finally obtain the following second order homogeneous differential equation:
\begin{align}
&\xi_d v f''(v) + \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber\\
& +\left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0.
\end{align}
\section{Proof of Proposition \ref{densityProp}}
\label{app:proof_prop_density}
In the proof, we concentrate on the derivation of $f(v)$, which is the solution to
\begin{align}
\xi_d v f''(v) &+ \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber \\
& \qquad + \left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0
\label{eq:demand6_1}
\end{align}
The expression for $g(v)$ follows directly from exchanging $\lambda_b$ with $\lambda_d$, $\mu_b$ with $\mu_d$, $\xi_b$ with $\xi_d$, and $\pi_b$ with $\pi_d$ in $f(v)$.
We rewrite \eqref{eq:demand6_1} as follows:
\begin{equation}\label{diffvgl}
vf''(v) + (A+Bv)f'(v)+(C+Dv)f(v)=0,
\end{equation}
where
\begin{equation*}
A = 2-\frac{\lambda_b+\lambda_d}{\xi_d}, ~~
B = \mu_d-\mu_b, ~~
C = \mu_d-\mu_b + \frac{\lambda_d\mu_b-\lambda_b\mu_d}{\xi_d}, ~~
D = -\mu_b\mu_d.
\end{equation*}
Note that we divided both sides of equation \eqref{eq:demand6_1} by $\xi_d$ here.
We will try to transform the differential equation into one for which the solution is easily derived. In order to do so, we first guess $f$ to be of the form $f(v) = {\rm e}^{\beta v}h(v)$, where $\beta$ is a constant and $h$ another real-valued function. Substituting this into \eqref{diffvgl} gives
\begin{equation}\label{eq:diff2}
v h''(v) + \left[ (2\beta +B)v + A\right]h'(v) + \left[(\beta^2+B\beta+D)v+ A\beta+C\right] h(v)=0.
\end{equation}
Next, we would like to choose $\beta$ such that $\beta^2+B\beta+D=0$, that is
\begin{equation*}
\beta = \frac{-B \pm \sqrt{B^2-4D}}{2},
\end{equation*}
which equals either $-\mu_d$ or $\mu_b$. Since the solution of \eqref{diffvgl} we are looking for is a density, and necessarily $f(v) = {\rm e}^{\beta v}h(v) \rightarrow 0 $ as $v\rightarrow \infty$, we set $\beta$ equal to the negative root $-\mu_d$. Lastly, we apply a change of variable, $x = \delta v$, and $h(v) = w(x)$, so that \eqref{eq:diff2} is transformed into
\begin{equation*}
x w''(x) + \left[ (2\beta+B)\delta^{-1} x+A\right] w'(x) + \delta^{-1}\left[A\beta+C\right] w(x) = 0.
\end{equation*}
By choosing $(2\beta+B)\delta^{-1} = -1$, i.e.
\begin{equation*}
\delta = {-}(2\beta+B) = \mu_b+\mu_d,
\end{equation*}
we obtain
\begin{equation*}
x w''(x) +[ A - x ] w'(x) + \delta^{-1}\left[A\beta+C\right] w(x) = 0,
\end{equation*}
which is known as Kummer's equation, $x w''(x) + (b-x) w'(x) - aw(x) = 0$, see \cite{Slater1960}, with parameters
\begin{align*}
a &= -\delta^{-1}\left[A\beta+C\right] = 1-\frac{\lambda_d}{\xi_d},\\
b &= A = 2-\frac{\lambda_b+\lambda_d}{\xi_d}.
\end{align*}
Kummer's equation has two linearly independent solutions, namely $w(x) =$\\ \noindent $M(a,b,x)$, where $M$ is Kummer's hypergeometric function, also denoted by \\ \noindent $ _1F_1(a,b,x)$, and $U(a,b,x)$, Tricomi's hypergeometric function. These are defined as, see \cite[Eq.~(1.3.1)]{Slater1960},
\begin{align*}
M(a,b,x) &= \sum_{n=0}^\infty \frac{(a)_n}{(b)_n n!} x^n,\\
U(a,b,x) &= \frac{\Gamma(b-1)}{\Gamma(1+a-b)}\,M(a,b,x) + \frac{\Gamma(b-1)}{\Gamma(a)}\,x^{1-b}\,M(1+a-b,2-b,x),
\end{align*}
where $(.)_n$ is the Pochhammer symbol, which is used to represent $(y)_n = y\cdot(y+1)\cdot...\cdot (y+n-1)$.
We can therefore deduce that $f(v)$ is of the form
\begin{equation*}
{\rm e}^{\beta v}\left[ c_1\, M(a,b,\delta v) + c_2\, U(a,b,\delta v)\right],
\end{equation*}
or
\begin{equation*}
{\rm e}^{-\mu_d v}\left[ c_1 M\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d) v\right) +
c_2 U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d) v\right)\right],
\end{equation*}
where $c_1$ and $c_2$ are constants. From \cite[p.~60]{Slater1960}, we have
\begin{equation*}
M(a,b,x) \sim \frac{\Gamma(b)}{\Gamma(a)}{\rm e}^x x^{a-b}, \qquad \text{as } x\rightarrow \infty.
\end{equation*}
Hence,
\begin{align*}
&{\rm e}^{-\mu_d v} M\left( 1-\frac{\lambda_d}{\xi_d}, 2-\frac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right) \\
&\qquad \qquad \sim
\frac{\Gamma(2-\frac{\lambda_b+\lambda_d}{\xi_d})}{\Gamma(1-\frac{\lambda_d}{\xi_d})}{\rm e}^{\mu_b v}\left((\mu_b+\mu_d)v\right)^{\lambda_b/\xi_d-1}
\to \infty
\end{align*}
for all $\mu_b>0$, which leads us to conclude $c_1 = 0$. We deduce $c_2$ by exploiting the restriction that
\begin{equation*}
\int_0^\infty f(v)\, {\rm d} v = \pi_d,
\end{equation*}
where $\pi_d$ is the probability of positive demand. Hence
\begin{equation*}
\pi_d c_2^{-1} = \int_0^\infty {\rm e}^{-\mu_d v} U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right)\, dv.
\end{equation*}
By slightly transforming \cite[Eq.~(3.2.51)]{Slater1960}, we find
\begin{equation*}
c_2^{-1} = \frac{1}{\pi_d}\,\frac{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}{\Gamma\left(1+\tfrac{\lambda_b}{\xi_d}\right) }\, _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},-\tfrac{\mu_b}{\mu_d}\right)
,
\end{equation*}
where $_2F_1(a_1,a_2,a_3,x) := \sum_{n=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(a_3)_n n!} x^n$
is the hypergeometric function of Gauss.
\section{Laplace Transforms for Coxian jumps}
\label{app:coxian}
We outline how the differential equation (\ref{diffeq}) is obtained.
We take Laplace transforms in (\ref{eq:demand}), considering its five terms and calling them $T_1, T_2, T_3, T_4$ and $T_5$, successively.
Equation (\ref{eq:demand}) then translates into
\begin{equation*}
T_1 + T_2 +T_3 = T_4 + T_5,
\label{Tequ}
\end{equation*}
where
\begin{align}
T_1
&= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^v f(y) \bar{F}_d(v-y) {\rm d}y {\rm d}v \nonumber \\
&=
\lambda_d \phi(s) \frac{1 - \mathbb{E}[{\rm e}^{-sD}]}{s},
\label{T-1}\\
T_2
&= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y {\rm d}v \nonumber\\
&= \lambda_d \int_{y=0}^{\infty} {\rm e}^{sy} g(y) \int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_d(z) {\rm d}z {\rm d}y ,
\label{T-2}\\
T_3
&= \pi_0 \lambda_d \int_0^{\infty} {\rm e}^{-sy} \bar{F}_d(y) {\rm d}y ,\\
T_4
&= \lambda_b \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=v}^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y {\rm d}v \nonumber\\
&= \lambda_b \int_{y=0}^{\infty} {\rm e}^{-sy} f(y) \int_{z=0}^{y} {\rm e}^{sz} \bar{F}_b(z) {\rm d}z {\rm d}y ,
\label{T-4}\\
T_5
&= \xi_d \int_{v=0}^{\infty} v {\rm e}^{-sv} f(v) {\rm d}v + \alpha_d \phi(s) \nonumber\\
&= - \xi_d \phi'(s) + \alpha_d \phi(s) .
\label{T-5}
\end{align}
We now evaluate the terms appearing in the righthand sides of (\ref{T-1})-(\ref{T-4}) for the Coxian case
of (\ref{Fbarb}) and (\ref{Fdarb}):
\begin{align}
\int_{z=0}^y {\rm e}^{sz} \bar{F}_b(z) {\rm d}z &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j-s} (1 - {\rm e}^{(s-\beta_j)y}),
\label{hulp1}\\
\int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_b(z) {\rm d}z &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j+s} {\rm e}^{-(s+\beta_j)y} ,
\label{hulp2}\\
\mathbb{E}[{\rm e}^{-sB}] &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{\beta_j}{\beta_j+s} ,
\label{hulp3}
\end{align}
and hence
\begin{equation}
\frac{1-\mathbb{E}[{\rm e}^{-sB}]}{s} =
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j+s} .
\label{hulp4}
\end{equation}
Combining \eqref{Tequ} with \eqref{T-1}-\eqref{T-5}, and using \eqref{hulp1} and the counterparts of \eqref{hulp2} and \eqref{hulp4}
for $\bar{F}_d(\cdot)$, we find:
\begin{align}
& \lambda_d \phi(s)
\sum_{i=1}^K q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\frac{1}{\delta_j+s}
\nonumber
\\
&\qquad + \lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\frac{1}{\delta_j+s} [\gamma(\delta_j) + \pi_0]
\nonumber
\\
&= \lambda_b
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j-s} (\phi(s) - \phi(\beta_j))
\nonumber
\\
&\qquad - \xi_d \phi'(s) + \alpha_d \phi(s),
\label{eq:3star}
\end{align}
which is readily rewritten into \eqref{diffeq}.
\begin{remark}
If $\xi_d=0$, then $\phi(s)$ is obtained from \eqref{eq:3star}
in a standard manner, see also Section~\ref{sectionvariant}.
\end{remark}
\begin{remark}
We now outline how \eqref{hulp2} and \eqref{hulp3} change when the $B_i$ have an Erlang-($l+1,\beta$) distribution,
and when the $D_i$ have an Erlang-($k+1,\delta$) distribution (see also \eqref{Fbarb} and the line below it);
\eqref{hulp1} and \eqref{hulp4} do not change (but of course $\mathbb{E}[{\rm e}^{-sD}]$ changes).
Firstly,
\begin{equation*}
\int_{z=0}^y {\rm e}^{sz} \bar{F}_b(z) {\rm d}z =
\sum_{j=0}^l \frac{\beta^j}{(\beta-s)^{j+1}}
\left[1 - \sum_{i=0}^j {\rm e}^{-(\beta-s)y} \frac{((\beta-s)y)^i}{i!}\right] .
\end{equation*}
Term $T_4$ now becomes:
\begin{align*}
T_4 &= \lambda_b \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=v}^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y {\rm d}v\nonumber\\
&= \lambda_b \sum_{j=0}^l \frac{\beta^j}{(\beta-s)^{j+1}}
\left[\phi(s) - \sum_{i=0}^j \frac{(\beta-s)^i}{i!} \int_{y=0}^{\infty} y^i {\rm e}^{-\beta y} f(y) {\rm d}y \right].
\end{align*}
It should be noted that $s = \beta$ is a removable singularity. E.g., for $l=0$ one has
$T_4 = \lambda_b \frac{\phi(s) - \phi(\beta)}{\beta - s}$.
\\
Secondly,
\begin{equation*}
\int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_b(z) {\rm d}z =
\sum_{j=0}^k \frac{\delta^j}{(s+\delta)^{j+1}} \sum_{i=0}^j {\rm e}^{-(s+\delta)y} \frac{((s+\delta)y)^i}{i!} .
\end{equation*}
Term $T_2$ now becomes:
\begin{align*}
T_2 &= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y\, {\rm d}v \nonumber\\
&=
\lambda_d
\sum_{j=0}^k \frac{\delta^j}{(s+\delta)^{j+1}} \sum_{i=0}^j \frac{(s+\delta)^i}{i!}
\int_{y=0}^{\infty} y^i {\rm e}^{-\delta y} g(y)\, {\rm d}y .
\end{align*}
It is readily seen that the resulting counterpart of \eqref{eq:3star} can again be written in the form \eqref{diffeq},
and hence the solution is formally still given by \eqref{diffeqsoln}.
\label{RmErlang}
\end{remark}
\resettocdepth
\end{subappendices}
\chapter{A blood bank model}
\begin{chapterstart}
We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to
independent compound Poisson processes.
Blood is perishable, that is, blood can only be kept in storage for a limited amount of time.
Furthermore, demand for blood is impatient, that is, a demand for blood may be canceled if it cannot be satisfied soon enough.
For a range of perishability functions and demand impatience functions,
we derive the steady-state distribution of the blood inventory level.
Moreover, we deduce fluid and diffusion limits for the inventory process as the arrival rates of of the compound Poisson processes grow indefinitely.
These scaling limits in turn provide normal approximations for the performance of large-scale systems.
\end{chapterstart}
\begin{flushright}
Based on\\
\textbf{A blood bank model with perishable blood and demand impatience}\\
\textit{Shaul Bar-Lev, Onno Boxma, Britt Mathijsen \& David Perry}\\
Submitted to \textit{Stochastic Systems}
\end{flushright}
\newpage
\section{Introduction}
This chapter is devoted to the study of a stochastic blood bank model in which amounts of blood are offered and demanded according to stochastic processes, and in which blood is perishable (that is, blood can only be kept for a limited amount of time) and demand for blood is impatient (that is, a demand request for blood may be canceled if it cannot be satisfied soon enough).
Let us first provide some background, and subsequently sketch the blood bank model in some more detail.\\
\\*
\textbf{Practical background.}
One of the major issues in securing blood supply to patients worldwide is to
provide blood of the best achievable quality, in the needed quantities.
In most countries, blood, which is collected as whole blood units from human
donors, is separated into different components which are subsequently stored under different storage conditions according to their biological
characteristics, functions and respective expiration dates. Blood units and components are ordered by local hospital blood banks (LBB) from the Central Blood Bank (CBB) according to their operational needs. The CBB
has to run its inventory and supply according to these requests and to the
need to keep sufficient stock for immediate release in emergency situations.
It also has to perform tests to determine the unit's blood type
and to detect the presence of various pathogens which are able to cause
transfusion-transmitted diseases, such as Hepatitis B, Hepatitis C, Human Immunodeficiency Virus (HIV) and Syphilis, see e.g.~Steiner et al.~\cite{Steiner2010}.
Blood consists of several components: red blood cells, plasma and
plate-lets.
In addition, there are $8$ blood groups (types):
$O^{+},O^{-},A^{+},A^{-},B^{+}$ ,$B^{-},AB^{+}$, $AB^{-}$ ($-$ means Rh
negative) where the interrelationship between the transfusion issuing
policies among the $8$ types is quite intricate.
It turns out that each of
the negative types can satisfy the corresponding $+$ type, but not vice
versa.
Blood components are perishable as red blood cells can be used for only $35$ to $42$
days and platelets for only $5$ days (plasma, however, can be frozen and
kept for one year).
Accordingly, if red blood cells and particularly platelets are not
used for blood transfusion within their expiration dates, then they perish.
In most developed countries demand requirements of about $50.000$
blood donations are needed per one million persons per year. About 95\% of
these donations are aggregated by CBBs and the remaining 5\% by LBBs.
Blood units stored at the CBB are usually ordered by LBBs for planned elective
surgeries. However, as it happens rather frequently, elective surgeries
turn out to become emergency ones due to various conditions of the
patient involved. In such cases, hospitals use their own local blood banks
to supply the demand, and they cancel the required demand
from the CBB; this is what we refer to as demand impatience.
A good review on supply chain management in blood products appears in Beli\"{e}n \& Forc\'{e} \cite{Belieen2012} and the
references cited therein. Other relevant studies are
Ghandforoush and Sen \cite{Ghandforoush2010} \&
Stanger et al.\ \cite{Stanger2012}.
\newpage
\noindent
\textbf{Inventory model.}
In this chapter we consider the analysis of blood perishability and demand impatience, concentrating on only one
blood type. We do this by considering the stochastic inventory processes $\{X_b(t)\}_{t \geq 0}$, with $X_b(t)$ the amount of blood
kept in storage at time $t$, and $\{X_d(t)\}_{t \geq 0}$, with $X_d(t)$ the amount of demand for blood (the shortage) at time $t$.
If $X_b(t)>0$ then $X_d(t)=0$, and if $X_d(t)>0$ then $X_b(t)=0$.
We assume that amounts of blood arrive according to a Poisson process,
and that requests for blood arrive according to another, independent, Poisson process.
The delivered and requested amounts of blood are assumed to be random variables.
We represent the perishability of blood by letting the amount of blood, when positive, decrease in a state-dependent way:
if the amount is $v$, then the decrement rate is $\xi_b v + \alpha_b$.
The $\xi_b$ factor is motivated by the fact that a large amount of blood
suggests that some of the blood has been present for quite a while -- and hence there is a relatively high perishability rate when much blood is in inventory.
The $\alpha_b$ factor provides additional modeling flexibility.
One can in this way represent the blood perishability more accurately;
but the $\alpha_b$ term could also, e.g., represent a fluid demand rate of individuals or organizations,
which contact the CBB directly, and that is only satisfied when there is inventory.
Similarly, we represent the demand impatience by a decrement rate $\xi_d v + \alpha_d$.
The $\xi_d$ factor is motivated by the following fact. When there is a large shortage (demand) of blood,
there are probably many patients waiting for blood, so many patients that might become impatient
(that is, they could recover, or die, or become in need of emergency surgery)
leading to a cancellation of the required demand from the CBB.
Again, the $\alpha_d$ factor provides additional modeling flexibility;
it not only allows us to represent demand impatience more accurately, but it could also, e.g.,
represent additional donations of individuals in times of blood shortage.
The inclusion of both the perishability factor $\xi_b v + \alpha_b $ and the demand impatience factor
$\xi_d v + \alpha_d$ makes the analysis of the ensuing model mathematically
quite challenging, but
leads to a very general model that contains many well-known models as special cases.
Our two-sided stochastic process, with both upward and downward jumps,
and with the rather general slope factors $\xi v + \alpha$, could represent a quite large class of stochastic phenomena.
It should for example be noted that this model is a two-sided generalization of the well-known shot-noise model that describes certain physical phenomena, see \cite{Keilson1959}).
In some of our calculations we remove either the $\xi$ factors or the
$\alpha$ factors, and this results in easier calculations and more explicit results.
Our main results are: (i) Determination of the steady-state distributions
of the amounts of blood and of demand in inventory; in particular, we present a detailed analysis of the case in which
the delivered and requested amounts of blood are both exponentially distributed. (ii)
Expressions for mean amounts of blood and demand in storage, and for the probability of not being able to satisfy demand.
(iii) We obtain the fluid and diffusion limits of the blood inventory process, providing in particular sufficient conditions for the limit process to be an Ornstein-Uhlenbeck process.
\\*
\\
\noindent
\textbf{Structure of the chapter.}
The chapter is organized as follows:
Section~\ref{modeldesc} presents a detailed model description.
A steady-state analysis of the densities of demand and of blood amount in storage is
contained in Section~\ref{analysis}, including the special case of exponentially distributed delivered and requested blood amounts when $\alpha_b=\alpha_d=0$ (i.e., pure proportionality).
The fluid and diffusion scalings are discussed in Section~\ref{sectionscaling},
and in Section~\ref{numericals} we present numerical results for certain performance measures like mean net amount of blood and the probability that there is a shortage of blood.
These results indicate, among other things, that the probability that there is a shortage of blood can be accurately approximated via a normal approximation, based on the Ornstein-Uhlenbeck process appearing in the diffusion scaling.
Section~\ref{conclus} contains some conclusions and suggestions for further research.
\section{Model description}
\label{modeldesc}
We consider the following highly simplified model of a blood bank, restricting ourselves to only one type of blood.
Blood amounts arrive according to a Poisson process with rate $\lambda_b$. The amounts which successively arrive are independent, identically
distributed random variables $B_1,B_2,\dots$ with distribution $F_b(\cdot)$; $\bar{F}_b(x) = 1 - F_b(x)$.
Demands for blood arrive according to a Poisson process with rate $\lambda_d$. The successive demand amounts are independent, identically
distributed random variables $D_1,D_2,\dots$ with distribution $F_d(\cdot)$; $\bar{F}_d(x) = 1 - F_d(x)$.
We view these amounts as continuous quantities, measured in, for instance liters.
If there is enough blood for a demand, then that demand is immediately satisfied.
If there is some blood, but not enough to fully satisfy a demand, then that demand is partially satisfied, using all the available blood.
The remainder of the demand may be satisfied later.
Blood has a finite expiration date. We make the assumption
that if the total amount of blood present is $v>0$,
then blood is discarded -- because of its finite expiration date -- at a rate $\xi_b v + \alpha_b$, so linear in $v$.
Blood demands have a finite patience.
We make the assumption
that if the total amount of demand present is $v>0$,
then demand disappears -- because of its finite patience -- at a rate $\xi_d v + \alpha_d$, so linear in $v$.
Notice that {\em either} the total amount of blood present, {\em or} the total amount of demands,
is zero, {\em or} both are zero; they cannot be both positive. Hence we can easily in one figure depict
the two-sided process $\{X(t)\}_{t \geq 0}$ $=\{(X_b(t),X_d(t))\}_{t \geq 0}$
of total blood and total demand amounts present at any time $t$, as we have done in Figure \ref{fig:samplePath}.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = -0.2,
xmax = 4,
ymin = -5,
ymax = 11,
ticks = none,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1
]
\addplot[black,thick] file {Chapter_7/tikz/sample_path2.txt};
\node at (axis cs: 0.33,-4.4) {$X_d(t)$};
\node at (axis cs: 0.33,10.2) {$X_b(t)$};
\draw[-stealth] (axis cs: 0,0) -- (axis cs:0,-5);
\end{axis}
\node at (7.05,1.47) {$t$};
\end{tikzpicture}
\caption{Sample path of net amount of blood available as a function of time.}
\label{fig:samplePath}
\end{figure}
For our purposes, we are mainly interested in the characteristics of the process described above in stationarity.
Let us denote by $X_d$ the steady-state total amount of demand and by $X_b$ the steady-state total amount of blood present, with corresponding density functions $f(\cdot)$ and $g(\cdot)$, respectively. Notice that these are defective densities;
we have $\int_{0^+}^{\infty} f(v) {\rm d} v = \pi_d = \mathbb{P}({\rm demand} ~ > ~ 0)$
and
$\int_{0^+}^{\infty} g(v) {\rm d} v = \pi_b = \mathbb{P}({\rm blood} ~ > ~ 0)$.
If $\alpha_b=\alpha_d=0$, then neither $X_b$ nor $X_d$ has probability mass at zero, and $\pi_b+\pi_d=1$
(when there is only a very small amount $v$ present, the "decay" rate $\xi_b v$ or $\xi_d v$ is very small).
However, if $\alpha_b$ and/or $\alpha_d$ is positive, then there is a positive probability $\pi_0$ of being in $0$.
When $\xi_d$ and $\xi_b$ are positive, existence of these steady-state densities is obvious;
otherwise, the conditions for the existence of the steady-state distributions require some discussion, see Section~\ref{sectionvariant}.
\section{Steady-state analysis}
\label{analysis}
In this section we present a global approach towards determining $f(\cdot)$ and $g(\cdot)$ in the most general form of our model.
Using the Level Crossing Technique (LCT),
we derive two integral equations in $f(\cdot)$ and $g(\cdot)$.
Before attempting to solve these equations,
we consider a few important performance measures which can be expressed in $f(\cdot)$
and $g(\cdot)$, $\pi_0$
and the mean length of time during which, uninterruptedly, there is a positive amount of blood (respectively demand).
The latter could be viewed as the busy period of the $X_b$ process (respectively of the $X_d$ process).
First we consider the density $g(\cdot)$ of the amount of blood.
We equate the rate at which some positive blood level $v$ is upcrossed and downcrossed, respectively.
LCT leads to the following integral equation: for $v>0$,
\begin{align}
&\lambda_b \int_0^v g(y) \bar{F}_b(v-y) {\rm d}y
+
\lambda_b \int_0^{\infty} f(y) \bar{F}_b(v+y) {\rm d}y
+ \pi_0 \lambda_b \bar{F}_b(v)
\nonumber
\\
&\qquad =
\lambda_d \int_v^{\infty} g(y) \bar{F}_d(y-v) {\rm d}y
+
(\xi_b v + \alpha_b) g(v).
\label{eq:blood}
\end{align}
Here the three terms in the left-hand side represent the rate of crossing level $v$ from below;
the first term corresponds to a jump from a blood inventory level between $0$ and $v$,
whereas the second term corresponds to a jump from a shortage level, and the third term
corresponds to a jump from level $0$.
The two terms in the right-hand side represent the rate of crossing level $v$ from above;
the first term corresponds to a jump from above $v$, and the second term to a smooth crossing.
Next, we consider the density $f(\cdot)$ of the amount of demand (shortage).
We equate the rate at which some positive demand level $v$ is upcrossed and downcrossed, respectively.
LCT leads to the following integral equation: for $v>0$,
\begin{align}
& \lambda_d \int_0^v f(y) \bar{F}_d(v-y) {\rm d}y
+
\lambda_d \int_0^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y
+ \pi_0 \lambda_d \bar{F}_d(v)
\nonumber
\\
&\qquad =
\lambda_b \int_v^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y
+
(\xi_d v + \alpha_d) f(v).
\label{eq:demand}
\end{align}
It should be noted that these two, coupled, equations are symmetric (swap $f$ and $g$, and the $b$ and $d$ parameters).
In general, it appears to be very difficult to solve these integral equations.
In Section~\ref{sectionexp} we assume that
both $F_b(\cdot)$ and $F_d(\cdot)$ are exponential.
In that case, we are able to obtain explicit expressions of $f(\cdot)$ and $g(\cdot)$, in terms of hypergeometric functions.
In Section~\ref{gener} we consider the case that $F_b(\cdot)$ and $F_d(\cdot)$ are Coxian distributions,
a class of distributions that lies dense in the class of all distributions of non-negative random variables,
and that is suitable for handling the above coupled integral equations via Laplace transforms (LT).
We are able to transform (\ref{eq:blood}) and (\ref{eq:demand})
into inhomogeneous first-order differential equations in the LTs of $f(\cdot)$ and $g(\cdot)$,
and thus to obtain those LTs.\\
\\*
\noindent
\textbf{A few simple performance measures.}
Without solving \eqref{eq:blood}-\eqref{eq:demand} explicitly, we are able to deduce some characteristics of the steady-state inventory level.
First, we can relate $\pi_0$ to the densities $f(\cdot)$ and $g(\cdot)$; see Proposition~\ref{prop. emptiness} below.
Subsequently we express the mean length of time during which there is, uninterruptedly, a positive amount of blood present (we call this the non-emptiness period of the inventory system), into $f(\cdot)$, $g(\cdot)$ and $\pi_0$.
We do the same for the mean length of time during which there is, uninterruptedly, a positive demand, i.e., the non-emptiness period of the demand process, see Proposition~\ref{prop: empty}.
\begin{proposition}
\label{prop. emptiness}Let $\pi _{0}\ $be\ the steady-state atom probability
of the zero period. Then%
\[
\pi _{0}=\frac{\alpha _{d}f(0)+\alpha _{b}g(0)}{\lambda
_{d}+\lambda _{b}}.
\]
\end{proposition}
\begin{proof}
Substitute $v=0$\ in \eqref{eq:blood} and \eqref{eq:demand} and take the sum. The result is
obtained after several steps of elementary algebra.
\end{proof}
The result introduced in the proposition above is very intuitive. By LCT, $%
\alpha _{d}f(0)+\alpha _{b}g(0)$\ is the rate at which
level $0$ is reached (i.e., the process will now really stay at $0$ for a while),
so that $[\alpha _{d}f(0)+\alpha
_{b}g(0)]^{-1}$\ is the expected length of time between two successive times level $0$ is reached by the fluid.
More precisely, the \textit{zero periods} and \textit{non-zero periods}
generate an alternating renewal process whose expected cycle length is $%
[\alpha _{d}f(0)+\alpha _{b}g(0)]^{-1}$. The expected length of the zero
period is $[\lambda _{d}+\lambda _{b}]^{-1}$, since the end of the zero
period is terminated at the moment of the next jump. But the jump process is
a Poisson process with rate $\lambda _{d}+\lambda _{b}$. Now the renewal
reward theorem simply says that
\[
\pi _{0}=\frac{\mathbb{E}[\text{zero period}]}{\mathbb{E}[\text{cycle}]}.
\]%
In preparation of the next proposition,
for the process $\{X(t)\}_{t\geq 0}$ we
define a modified process $\{X_{m}(t)\}_{t\geq 0}$, where $X_{m}$ is constructed by deleting the zero-periods (only the zero periods, not the emptiness periods) from $X$ and gluing together the \textit{non-zero periods}. The modified
process is $X_m$ such that $X_{m}(t)=X_{d}(t)\mathbbm{1}_{\{X_{d}(t)>0\}}+X_{b}(t)\mathbbm{1}_{\{X_{b}(t)>0\}}$ where by definition
of the model $\{X_{d}(t)>0\}\Rightarrow \{X_{b}(t)=0\}$ and $%
\{X_{b}(t)>0\}\Rightarrow \{X_{d}(t)=0\}$.
\begin{proposition}
\label{prop: empty}Let $B_{b}$\ and $I_{b}$ be the generic non-emptiness
period and the emptiness period, respectively, of the inventory system.
Similarly, let $B_{d}$\ and $I_{d}$ be the generic non-emptiness period and the
emptiness period, respectively, of the demand process. Then%
\[
{\rm (i)}\left\{
\begin{array}{l}
\ \mathbb{E} [B_{b}]=\frac{1-\pi _{0}}{\alpha _{b}g(0)+\lambda _{d}\int_{0}^{\infty }%
\bar{F}_{d}(y)g(y){\rm d} y}, \\
\ \mathbb{E} [B_{d}]=\frac{1-\pi _{0}}{\alpha _{d}f(0)+\lambda _{b}\int_{0}^{\infty }%
\bar{F}_{b}(y)f(y){\rm d} y}
\end{array}%
\right.
\]%
and
\\
\[
{\rm (ii)}\left\{
\begin{array}{l}
\ \mathbb{E}[I_{b}]=\frac{1}{\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y)dy+\lambda
_{b}\pi _{0}}
- \mathbb{E}[B_b] ,\\
\ \mathbb{E}[I_{d}]=\frac{1}{\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g(y)dy+\lambda
_{d}\pi _{0}}
-\mathbb{E}[B_d] .
\end{array}%
\right.
\]
\end{proposition}
\begin{proof}
(i) Consider the non-emptiness period of the inventory system.
The steady-state densities of the inventory system and the demand process of $X_m$\ are given by%
\[
g_{m}(x)=\frac{g(x)}{1-\pi _{0}},\ \ \ \ \ f_{m}(x)=\frac{f(x)}{1-\pi _{0}},
\]%
respectively. At the end of the non-emptiness period of the inventory system there are two disjoint
ways (disjoint events) to downcross level $0+$. Either level $0$ is
downcrossed by a negative jump or level $0+$ is reached by the fluid
reduction (both in $X_m$). The rate of the first
event\ is $\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y$\ and by
LCT the rate of the second event is $\alpha _{b}g_{m}(0)$. Since the
events are disjoint, the rate of downcrossings of level $0+$ is $\lambda
_{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y+\alpha _{b}g_{m}(0)$. That
means that the expected length of the non-emptiness period is given by $%
[\lambda _{d}\int_{0}^{\infty }\bar{F}_{d}(y)g_{m}(y){\rm d} y+\alpha
_{b}g_{m}(0)]^{-1}$. Thus%
\[
\mathbb{E}[B_{b}]=\frac{1-\pi _{0}}{\alpha _{b}g(0)+\lambda _{d}\int_{0}^{\infty }\bar{%
F}_{d}(y)g(y){\rm d} y}.
\]%
The expression for $\mathbb{E}[B_{d}]$ is obtained by symmetry.
\\
(ii) Define\ a \textit{cycle} in the real process $X$ (not the
modified process $X_m$) as the time between two upcrossings of
level $0+$. By definition, the emptiness period plus the non-emptiness
period is a cycle in $X$. That means that the expected length of
the emptiness period is the expected length of the cycle\ minus the expected
length of the non-emptiness period. The non-emptiness period in $X$
and in $X_m$ are identical and the length of the expected cycle
is $[\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y){\rm d} y+\lambda _{b}\pi
_{0}]^{-1}$, since $\lambda _{b}\int_{0}^{\infty }\bar{F}%
_{b}(y)f(y){\rm d} y+\lambda _{b}\pi _{0}$\ \ is the rate of the upcrossings of
level $0+$.\ \ We obtain%
\[
\mathbb{E}[I_{b}] + \mathbb{E}[B_b] = \frac{1}{\lambda _{b}\int_{0}^{\infty }\bar{F}_{b}(y)f(y)dy+\lambda
_{b}\pi _{0}},
\]
yielding $\mathbb{E}[I_b]$.
$\mathbb{E}[I_{d}]$\ is obtained by symmetry.
\end{proof}
For the special case in which $\xi_b=\xi_d=\xi$ and $\alpha_b=\alpha_d = 0$, we are able to deduce that the expected steady-state inventory level $\mathbb{E}[X]$ has a simple form.
\begin{proposition}\label{prop:mean_inventory}
If $\xi_b=\xi_d=\xi$ and $\alpha_b=\alpha_d=0$, then
\begin{equation}
\mathbb{E}[X] = m/\xi,
\end{equation}
where $m=\lambda_b\mathbb{E}[B] - \lambda_d\mathbb{E}[D]$.
\end{proposition}
\begin{proof}
We study the discrete-time embedding of the blood inventory process \\ \noindent $\{X_k\}_{k\geq 1}$, where $X_k$ denotes the blood inventory level \textit{just before} the $k^{th}$ arrival (either blood or demand).
Suppose the process is in steady state.
By the PASTA property, we have that $X_k {\;\buildrel{d}\over= \;} X$ for all $k\geq 1$.
Also, the process $\{X_k\}_{k\geq 1}$ constitutes a Markov chain, of which the evolution is characterized by the recursion
\begin{equation}
X_{k+1} = \left( X_k + \mathbbm{1}_{k,b}B_k - \mathbbm{1}_{k,d} D_k \right)\cdot {\rm e}^{-\xi A_k},
\label{eq:X_recursion}
\end{equation}
where $\mathbbm{1}_{k,b}$ and $\mathbbm{1}_{k,d}$ denote the indicator function of the event that the $k^{th}$ arrival is a blood or demand arrival, respectively.
Remark that the relation holds for both $X_k \geq 0$ and $X_k <0$.
Furthermore, $B_k$ and $D_k$ denote the amount of blood or demand in the $k^{th}$ jump, respectively, and $A_k$ denotes the interarrival time between the $k^{th}$ and $(k+1)^{th}$ arrival.
Note that $A_k$ is the minimum of two exponentially distributed random variables with rate $\lambda_b$ and $\lambda_d$, so that $A_k$ is exponentially distributed with rate $\lambda_b+\lambda_d$.
Next, we take the expectation on both sides of \eqref{eq:X_recursion}, which gives
\begin{equation}
\mathbb{E}[X_{k+1}] = \big( \mathbb{E}[X_k] + p_{k,b}\mathbb{E}[B] - p_{k,d}\mathbb{E}[D]\big)\,\mathbb{E}\big[{\rm e}^{-\xi A_k}\big].
\label{eq:X_recursion_mean}
\end{equation}
Here, we used independence between Poisson processes and their jump sizes, and their memoriless property, and $p_{k,b} = \lambda_b/(\lambda_b+\lambda_d)$ and $p_{k,d} = \lambda_d/(\lambda_b+\lambda_d)$ denote probability of the $k^{th}$ jump being either a blood delivery or demand, respectively.
Since $X_{k} {\;\buildrel{d}\over= \;} X$, we have $\mathbb{E}[X_{k+1}] = \mathbb{E}[X_k] = \mathbb{E}[X]$, and thus we may rewrite \eqref{eq:X_recursion_mean} as
\begin{equation}
\mathbb{E}[X] = \left( \mathbb{E}[X] + \frac{\lambda_b\mathbb{E}[B] - \lambda_d\mathbb{E}[D]}{\lambda_b+\lambda_d} \right)\cdot\frac{\lambda_b+\lambda_d}{\lambda_b+\lambda_d+\xi},
\end{equation}
from which we easily deduce $\mathbb{E}[X] = (\lambda_b\mathbb{E}[B] - \lambda_d \mathbb{E}[D])/\xi = m/\xi$.
\end{proof}
\subsection{The exponential case}
\label{sectionexp}
\textbf{Density functions.}
We assume in this section that $\bar{F}_b(x) = {\rm e}^{-\mu_b x}$
and
$\bar{F}_d(x) = {\rm e}^{-\mu_d x}$.
Let $\rho_d:= \lambda_d/\mu_d$ and $\rho_b:= \lambda_b/\mu_b$ denote the expected amount of demand requested, and amount of blood delivered into the system, per time unit.
Moreover, we take $\alpha_b = \alpha_d = 0$.
Under these assumptions, we can
solve (\ref{eq:demand}) and (\ref{eq:blood}) explicitly.
Equations (\ref{eq:blood}) and (\ref{eq:demand}) reduce to:
\begin{align}
&\lambda_d \int_0^v f(y) {\rm e}^{-\mu_d(v-y)} {\rm d}y
+
\lambda_d {\rm e}^{-\mu_d v} \int_0^{\infty} g(y) {\rm e}^{-\mu_d y} {\rm d}y \nonumber \\
&\qquad =
\lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y
+
\xi_d v f(v),
\label{eq:demand2}
\end{align}
\begin{align}
&\lambda_b \int_0^v g(y) {\rm e}^{-\mu_b(v-y)} {\rm d}y
+
\lambda_b {\rm e}^{-\mu_b v} \int_0^{\infty} f(y) {\rm e}^{-\mu_b y} {\rm d}y \nonumber \\
& \qquad =
\lambda_d \int_v^{\infty} g(y) {\rm e}^{-\mu_d(y-v)} {\rm d}y
+
\xi_b v g(v),
\label{eq:blood2}
\end{align}
for $v>0$.
In our analysis, we concentrate on the derivation of $f(v)$. Notice that, once $f(\cdot)$ has been determined, $g(\cdot)$ follows by swapping parameters (symmetry).
In Appendix \ref{app:transformation_int} we show how the integral equations \eqref{eq:demand2}-\eqref{eq:blood2} can be translated into the following decoupled second order differential equations:
\begin{align}
\xi_d v f''(v) &+ \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber \\
& \qquad + \left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0
\label{eq:demand6}
\end{align}
and
\begin{align}
\xi_b v g''(v) &+ \left(2\xi_b -\lambda_d -\lambda_b + \mu_b\xi_dv -\mu_d \xi_b v\right)g'(v) \nonumber \\
& \qquad + \left(\mu_b\xi_b -\mu_d\xi_b -\mu_b\lambda_d + \mu_d \lambda_b -\mu_d \mu_b \xi_b v\right)g(v) =0,
\label{eq:blood6_1}
\end{align}
with the additional constraint (obtained by applying the level crossing identity for level $v=0$ in either (\ref{eq:demand2}) or (\ref{eq:blood2})):
\begin{equation}
\lambda_b \int_0^{\infty} f(y) {\rm e}^{-\mu_b y} {\rm d}y
=
\lambda_d \int_0^{\infty} g(y) {\rm e}^{-\mu_dy} {\rm d}y .
\label{eq:blood2a}
\end{equation}
Equation (\ref{eq:demand6}) describes a known type of second order differential equation, namely the \textit{extended confluent hypergeometric equation} \cite{Slater1960}, which allows an explicit solution.
A detailed deduction of the solution to \eqref{eq:demand6} is given in Appendix \ref{app:proof_prop_density}, and yields the following result.
\\
\begin{proposition}\label{densityProp}
The probability density functions of the amount of demand $X_d$ and the amount of blood present $X_b$ are given by
\begin{align}
f(v) &= \pi_d\, \frac{\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\,\frac{{\rm e}^{-\mu_d v}U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right)}{ _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},-\tfrac{\mu_b}{\mu_d}\right)}\label{eq:fullf},\\
g(v) &= \pi_b\, \frac{\Gamma\left(1+\frac{\lambda_d}{\xi_b}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_b}\right)}\,\frac{{\rm e}^{-\mu_b v}U\left( 1-\tfrac{\lambda_b}{\xi_b}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_b},(\mu_b+\mu_d)v\right)}{ _2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},-\tfrac{\mu_d}{\mu_b}\right)}\label{eq:fullg},
\end{align}
for $v>0$, respectively.
\end{proposition}
\noindent
Here, $\Gamma(\cdot)$ denotes the gamma function, $ _2F_1(a,b,c,z)$ is the Gaussian hypergeometric function, defined as
\begin{equation}
_2F_1(a,b,c,z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n\, n!}\, z^n
\end{equation}
and $U(a,b,z)$ is Tricomi's confluent hypergeometric function, see \cite{Slater1960},
\begin{align}
U(a,b,x) &= \frac{\Gamma(b-1)}{\Gamma(1+a-b)}\,\sum_{n=0}^\infty \frac{(a)_n}{(b)_n n!} x^n + \frac{\Gamma(b-1)}{\Gamma(a)}\,x^{1-b}\,\sum_{n=0}^\infty \frac{(1+a-b)_n}{(2-b)_n n!} x^n ,
\end{align}
in which $(a)_n$ is the Pochhammer symbol, defined as $(a)_n = a\cdot(a+1)\cdots(a+n-1)$.
As a direct consequence of Proposition \ref{densityProp}, we obtain expressions for the LTs $\phi(s) = \int_0^{\infty} {\rm e}^{-sv} f(v) {\rm d}v$ and $\gamma(s) = \int_0^{\infty} {\rm e}^{-sv} g(v) {\rm d}v$ for ${\rm Re}\,s \geq 0$ through \cite[Eq.~(3.2.51)]{Slater1960}, which we state here for future use.
\begin{corollary}\label{cor:lsts}
The Laplace transforms for $X_d$ and $X_b$, for ${\rm Re}\,s \geq 0$, are given by
\begin{align}
\label{eq:diffLST}
\phi(s) &= \pi_d\, \frac{\mu_d}{\mu_d+s}\frac{ _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},\frac{s-\mu_b}{s+\mu_d}\right)}{ _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)},\\
\gamma(s) &= \pi_b\, \frac{\mu_b}{\mu_b+s}\frac{ _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},\frac{s-\mu_d}{s+\mu_b}\right)}{ _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},
\label{eq:diffLST_2}
\end{align}
respectively.
\end{corollary}
Last, we obtain expressions for $\pi_d$ and $\pi_b$. These follow immediately by
using the normalization equation $\pi_b + \pi_d=1$
and \eqref{eq:blood2a}, or equivalently,
$\lambda_b\phi(\mu_b) = \lambda_d\gamma(\mu_d)$. By filling in $s=\mu_b$ in \eqref{eq:diffLST},
\begin{align}\label{eq:LSTequal2}
&\pi_d\,\,\frac{\lambda_b\mu_d}{\mu_b+\mu_d}\, _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)^{-1} \nonumber\\
&\qquad = \pi_b\,\,\frac{\lambda_d\mu_b}{\mu_b+\mu_d}\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)^{-1},
\end{align}
where we used that $_2F_1(a,b,c,0) = 1$. Using the normalization equation, we obtain
\begin{equation}
\label{eq:piD}
\pi_d = \frac{\rho_b,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)}
{\rho_d,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)}.
\end{equation}
By substituting this result into both \eqref{eq:fullf} and \eqref{eq:diffLST}, we obtain the full pdf for the blood inventory process in steady-state.
\begin{theorem}\label{thm:full_pdf}
The steady-state pdf of the net inventory level $X$ is given by
\begin{equation}
h(v) =
\left\{
\begin{array}{ll}
f(-v), & \text{if }v<0,\\
g(v), & \text{if }v\geq 0,
\end{array}
\right.
\end{equation}
where
\begin{align}
\label{eq:ftotal}
f(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \rho_d \, {\rm e}^{-\mu_d v}\, U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right),\\
g(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_d}{\xi_b}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_b}\right)}\, \rho_b \, {\rm e}^{-\mu_b v}\, U\left( 1-\tfrac{\lambda_b}{\xi_b}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_b},(\mu_b+\mu_d)v\right),
\end{align}
with
\begin{equation}
\bar{C} = \rho_d \,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right).
\end{equation}
\end{theorem}
\begin{remark}
By applying the Pfaff transformation $_2F_1(a,b,c,z)=$ \\
$(1-z)^{-b}\,_2F_1\left(c-a,b,c,\frac{z}{1-z}\right)$, we may reformulate
\begin{equation}
\label{eq:pfaffTransform}
_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) = \frac{\mu_d}{\mu_b+\mu_d}\, _2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b}{\mu_b+\mu_d}\right),
\end{equation}
so that
\begin{equation}
\label{eq:piDalternative}
\pi_d = \frac{\lambda_d\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right)}
{\lambda_d\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right) +
\lambda_b\,_2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_b},1,\tfrac{\lambda_d}{\xi_b},\tfrac{\mu_d}{\mu_b+\mu_d}\right)}.
\end{equation}
By also transforming the hypergeometric term in the numerator of \eqref{eq:fullf}, we get an equivalent form of \eqref{eq:ftotal}, namely
\begin{equation}
\label{eq:ftotalAlternative}
f(v) = \bar{C}^{-1}_{\rm alt}\frac{\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \rho_b\mu_b(\mu_b+\mu_d)\, {\rm e}^{-\mu_d v}\,U\left( 1-\frac{\lambda_d}{\xi_d}, 2-\frac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right),
\end{equation}
with
\begin{equation}
\bar{C}_{\rm alt} = \lambda_d\,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b}{\mu_b+\mu_d}\right) +
\lambda_b\,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_b},1,\frac{\lambda_d}{\xi_b},\frac{\mu_d}{\mu_b+\mu_d}\right).
\end{equation}
As a consequence, \eqref{eq:diffLST} is given by
\begin{equation}
\phi(s) = \pi_d \,\frac{ _2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b-s}{\mu_b+\mu_d}\right)}{ _2F_1\left( \tfrac{\lambda_b+\lambda_d}{\xi_d},1,\tfrac{\lambda_b}{\xi_d},\tfrac{\mu_b}{\mu_b+\mu_d}\right) } = \bar{C}^{-1}_{\rm alt}\,\lambda_d \,_2F_1\left( \frac{\lambda_b+\lambda_d}{\xi_d},1,\frac{\lambda_b}{\xi_d},\frac{\mu_b-s}{\mu_b+\mu_d}\right).
\end{equation}
\end{remark}
Based on the density functions in Theorem \ref{thm:full_pdf}, we make some comments on its properties, and discuss parameter settings that leads to special cases.
By close inspection of these derived density functions, we can observe the following on the distribution shape around $z=0$.
The confluent hypergeometric function $U(a,b,z)$ has limiting form as $z\rightarrow 0$,
\begin{equation}\label{eq:limit0}
U(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a-b+1)} + \frac{\Gamma(b-1)}{\Gamma(a)}\,z^{1-b} + O(z^{2-b}), \qquad b\leq 2,
\end{equation}
see \cite[Sub.~13.2]{NIST}.
Note that in our model, $b = 2-(\lambda_b+\lambda_d)/\xi_d<2$ for all parameter settings.
Equation \eqref{eq:limit0} shows that $U(a,b,z)$ has a singularity at $z=0$ if Re$(b)>1$, which in our case translates to $f(v)$ and $g(v)$ being analytic at $v=0$ if $\lambda_b+\lambda_d > \xi_d$ and $\lambda_b+\lambda_d > \xi_b$, respectively. Assuming $\lambda_b+\lambda_d > \max\{\xi_b,\xi_d\}$, \eqref{eq:limit0} also implies that
\begin{align}\label{eq:contLimit1}
\lim_{v\rightarrow 0} f(v) &= \bar{C}^{-1}\,\frac{\,\Gamma\left(1+\frac{\lambda_b}{\xi_d}\right)}{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}\, \lambda_d\mu_b\cdot \frac{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}-1\right)}{\Gamma\left(\frac{\lambda_b}{\xi_d}\right)}\\
&= \bar{C}^{-1}\,\frac{\frac{\lambda_b}{\xi_d}}{\frac{\lambda_b+\lambda_d}{\xi_d}-1}\, \lambda_d\mu_b= \bar{C}^{-1}\,\frac{\lambda_b\lambda_d\mu_b\mu_d}{\lambda_b+\lambda_d-\xi_d}.
\nonumber
\end{align}
Similarly,
\begin{equation}
\lim_{v\rightarrow 0} g(v) = \bar{C}^{-1}\,\frac{\lambda_b\lambda_d\mu_b\mu_d}{\lambda_b+\lambda_d-\xi_b}.
\label{eq:contLimit2}
\end{equation}
By equating these two expressions, we conclude that $\lim_{v\rightarrow 0} f(v) = \lim_{v\rightarrow 0} g(v)< \infty$, i.e. the overall density function $h(v)$ is continuous at $v=0$, if and only if $\xi_b = \xi_d$. \\
\noindent
The asymptotic behavior of $U$ as $z\to\infty$ is given by \cite[p.~60]{Slater1960},
\begin{equation}
U(a,b,z) \sim z^{-a}, ~~~~~~~~~~~~~~~~ z \rightarrow \infty,
\end{equation}
which implies that the density function tail decays as
\begin{equation}
\label{eq:asympt}
f(v) \sim C^*\, e^{-\mu_d v}\, v^{\lambda_d/\xi_d-1}, ~~~~~~~~~~~~~~~~~~~
v\rightarrow\infty ,
\end{equation}
for some constant $C^*$.
\noindent\textbf{Special cases.}
Equation \eqref{eq:asympt} suggests that the case $\lambda_d = \xi_d$ is special.
Indeed, then \eqref{eq:diffLST} reduces to
\begin{equation}
\label{eq:lambdaisxi}
\phi(s) = \bar{C}^{-1}\,\lambda_d\mu_b\, \frac{\mu_d}{\mu_d+s} = \pi_d\, \frac{\mu_d}{\mu_d+s},
\end{equation}
where we used that $_2F_1(0,a,b,z) = 1$ for all $a,b,z$. Hence, conditioned on being positive, the amount of demand present is exponentially distributed with parameter $\mu_d$, regardless of the values of $\lambda_d = \xi_d$, as well as $\lambda_b,\, \xi_b,$ and $\mu_b$.
If we moreover let $\lambda_b = \xi_b$, then
\[
\pi_d = \frac{\lambda_d/\mu_d}{\lambda_b/\mu_b + \lambda_d/\mu_d} = \frac{\rho_d}{\rho_b+\rho_d},
\]
and $X$ has exponential distribution both above and below 0, with parameters $\mu_b$ and $\mu_d$, respectively. \\
A second special case arises when the process is symmetric, that is, $\lambda_b=\lambda_d=\lambda$, $\mu_b=\mu_d=\mu$ and $\xi_b=\xi_d=\xi$. Obviously, we get $\pi_b= \pi_d = \tfrac{1}{2}$ due to the symmetry. If we define $\eta := \lambda/\xi$,
\begin{align}
f(v) &= \frac{\Gamma(1+\eta)\, \mu e^{-\mu v}\, U\left(1-\eta,2(1-\eta),2\mu v\right)}{2\,\Gamma(2\eta) _2F_1\left(2\eta,1,1+\eta,\tfrac{1}{2}\right)}\\
&= \frac{\Gamma(1+\eta)}{2\,\Gamma(2\eta) _2F_1\left(2\eta,1,1+\eta,\tfrac{1}{2}\right)}\, \frac{\mu}{2\sqrt{\pi}}\, \left(2\mu v\right)^{\eta-\tfrac{1}{2}}\, K_{\tfrac{1}{2}-\eta}\left(\mu v\right),
\nonumber
\end{align}
where $K_\alpha(\cdot)$ is the modified Bessel function of the second kind, see \cite[Eq.~(13.6.10)]{NIST}.\\
\\*
\noindent
\textbf{Performance measures.}
Based on Theorem \ref{thm:full_pdf}, we can directly derive a couple of characteristics of the process.
First, we consider the mean inventory level
\begin{corollary}\label{cor:means}
The expected amount of demand (blood) present, given that it is positive equals
\begin{align}
\mathbb{E}[X_d|X_d>0] &= \frac{1}{\xi_d}\left[ \rho_d - \rho_b + \rho_b\, _2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},{-}\frac{\mu_b}{\mu_d}\right)^{-1}\right],\label{EXd>0}\\
\mathbb{E}[X_b|X_b>0] &= \frac{1}{\xi_b}\left[ \rho_b - \rho_d + \rho_d \, _2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},{-}\frac{\mu_d}{\mu_b}\right)^{-1}\right]\label{EXb>0}.
\end{align}
Accordingly, the expected net amount of blood present equals
\begin{equation}
\mathbb{E}[X] = \left(\rho_b-\rho_d \right)\left(\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right)+\frac{\lambda_b\lambda_d}{\bar{C}}\left(\frac{1}{\xi_b}-\frac{1}{\xi_d}\right).
\end{equation}
\end{corollary}
\begin{proof}
Let us use shorthand notation
\[
F(s) = \Big(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},\frac{s-\mu_b}{s+\mu_d}\Big)
,\]
so that
\[\phi(s)=\pi_d\, \frac{\mu_b}{\mu_b+s}\, \frac{F(s)}{F(0)}.\]
Through \cite[Eq.~(15.5.20)]{NIST},
\begin{equation}\label{eq:proof_a}
\frac{{\rm d}}{{\rm d} z}\, _2F_1(a,1,c,z) = \frac{c-1}{z(1-z)} + \frac{1-c+az}{z(1-z)} \, _2F_1(a,1,c,z),
\end{equation}
where we also used that $ _2F_1(a,1,c,z) = 1$.
Then,
\begin{align*}
\frac{\phi'(0)}{\pi_d} &= \left[ \frac{-\mu_d}{(\mu_d+s)^2} \, \frac{F(s)}{F(0)}
+ \frac{\mu_d}{\mu_d+s}\, \frac{F'(s)}{F(0)} \right]_{s=0} = {-}\frac{1}{\mu_d} + \frac{F'(0)}{F(0)}.
\end{align*}
By \eqref{eq:proof_a}, we find
\begin{align*}
F'(s) &= \Big( \frac{\lambda_b/\xi_d}{\frac{s-\mu_b}{s+\mu_d}\cdot \frac{\mu_b+\mu_d}{s+\mu_d}} + \frac{-\lambda_b/\xi_d + (1-\lambda_d/\xi_d)\frac{s-\mu_b}{s+\mu_d}}{\frac{s-\mu_b}{s+\mu_d}\cdot \frac{\mu_b+\mu_d}{s+\mu_d}}\, F(s)\Big)
\,\frac{{\rm d}}{{\rm d} s} \Big[ \frac{s-\mu_b}{s+\mu_d} \Big]
\\
&= \Big( \frac{ \lambda_b}{\xi_d} + \left[\frac{{-}\lambda_b}{\xi_d} + \Big(1-\frac{\lambda_d}{\xi_d}\Big)\frac{s-\mu_b}{s+\mu_d}\right] F(s) \Big) \frac{ (s+\mu_d)^2} {(s-\mu_b)(\mu_b+\mu_d)}\cdot \frac{\mu_b+\mu_d}{(s+\mu_d)^2}\\
&= \Big( \frac{ \lambda_b}{\xi_d} + \left[{-}\frac{\lambda_b}{\xi_d} + \Big(1-\frac{\lambda_d}{\xi_d}\Big)\frac{s-\mu_b}{s+\mu_d}\right] F(s) \Big)
\frac{1}{s-\mu_b},
\end{align*}
so that
\begin{align*}
F'(0) &= {-} \frac{\lambda_d/\mu_b}{\xi_d} + \left( \frac{\lambda_d/\mu_b}{\xi_d}
+ \frac{1}{\mu_d} - \frac{\lambda_d/\mu_d}{\xi_d}\right)F(0)\\
&= {-}\frac{\rho_b}{\xi_d} + \left( \frac{\rho_b-\rho_d}{\xi_d}
+ \frac{1}{\mu_d}\right)F(0).
\end{align*}
Hence, we find
\begin{align*}
\mathbb{E}[X_d|X_d>0] &= {-}\frac{\phi'(0)}{\pi_d} = \frac{1}{\mu_d}- \frac{1}{F(0)}\left[{-}\frac{\rho_b}{\xi_d} + \left( \frac{\rho_b-\rho_d}{\xi_b}
+ \frac{1}{\mu_d}\right)F(0)\right]\\
&= \frac{1}{\xi_d}\left( \rho_d-\rho_b + \rho_b/F(0)\right) = \frac{1}{\xi_d}\left( -m + \rho_b/F(0)\right),
\end{align*}
which equals \eqref{EXd>0}.
The expression for \eqref{EXb>0} follows by symmetry.
Furthermore,
\begin{align*}
\mathbb{E}[X] &= \pi_b \mathbb{E}[X_b|X_b>0] + \pi_d \mathbb{E}[-X_d|X_d>0]\\
&= m\left[\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right] + \frac{\lambda_d}{\mu_d\,\xi_b}\frac{\pi_b}{_2F_1\left(1-\tfrac{\lambda_b}{\xi_b},1,1+\tfrac{\lambda_d}{\xi_b},{-}\tfrac{\mu_d}{\mu_b}\right)}\\
&\qquad - \frac{\lambda_b}{\mu_b\,\xi_d}\,\frac{\pi_d}{_2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right)}.
\end{align*}
Note that $\pi_d\,_2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right)^{-1} = \lambda_d\mu_b\bar{C}^{-1}$.
Hence,
\begin{align*}
\mathbb{E}[X] &= m\left[\frac{\pi_b}{\xi_b}+\frac{\pi_d}{\xi_d}\right] + \frac{\lambda_b\lambda_d}{\bar{C}}\left(\frac{1}{\xi_b}-\frac{1}{\xi_d}\right),
\end{align*}
which completes the proof.
\end{proof}
\begin{remark}
Note that if $\xi_b = \xi_d = \xi$, we get $\mathbb{E}[X] = m(\pi_b+\pi_d)/\xi = m/\xi$, which is consistent with Proposition \ref{prop:mean_inventory}.
The expression in (\ref{EXd>0}) contains no $\xi_b$. Indeed, while the value of $\xi_b$ influences the probability that $X_d>0$,
it does not influence the mean of $X_d$ given that $X_d >0$.
\end{remark}
In Figure \ref{fig:means}, we plot the behavior of the three performance metrics in Corollary \ref{cor:means} while keeping $m$ fixed. In Figure \ref{fig:means}(a) we set $\lambda_b = 1.2$, $\lambda_d = 1$, $\mu_b=1$, $\mu_d=1.2$, so that $m = 11/30$ and vary $\xi_b=\xi_d=\xi$ between 0 and 1. In Figure \ref{fig:means}b, we fix $\xi_b=\xi_d=0.5$ and take $\lambda_b = 1.2\theta$, $\lambda_d = \theta$, $\mu_b=\theta$, $\mu_d=1.2\theta$, so that still $m=11/30$, and vary $\theta$.
Observe that in Figure \ref{fig:means}b, $\mathbb{E}[X]$ is constant, since the value of $m/\xi$ if unaffected by the parameter $\theta$.
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.78]
\begin{axis}[
xmin = -0.02,
xmax = 1,
ymin = -0.02,
ymax = 5,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1,
xlabel = {$\xi$},
xlabel near ticks,
legend cell align = left,
legend style = {at = {(axis cs: 1,5)},anchor = north east}
]
\addplot[black,dashed,thick] table[x=x,y=Xd] {Chapter_7/tikz/means1.txt};
\addplot[black,dotted,thick] table[x=x,y=Xb] {Chapter_7/tikz/means1.txt};
\addplot[black,thick] table[x=x,y=Q] {Chapter_7/tikz/means1.txt};
\legend{{$\mathbb{E}[X_d|X_d>0]$},{$\mathbb{E}[X_b|X_b>0]$},{$\mathbb{E}[X]$}};
\end{axis}
\end{tikzpicture}
\caption{As a function of $\xi$}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.78]
\begin{axis}[
xmin = -0.02,
xmax = 2,
ymin = -0.02,
ymax = 5,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1,
xlabel = {$\theta$},
xlabel near ticks,
legend cell align = left,
legend style = {at = {(axis cs: 2,5)},anchor = north east}
]
\addplot[black,dashed,thick] table[x=x,y=Xd] {Chapter_7/tikz/means2.txt};
\addplot[black,dotted,thick] table[x=x,y=Xb] {Chapter_7/tikz/means2.txt};
\addplot[black,thick] table[x=x,y=Q] {Chapter_7/tikz/means2.txt};
\legend{{$\mathbb{E}[X_d|X_d>0]$},{$\mathbb{E}[X_b|X_b>0]$},{$\mathbb{E}[X]$}};
\end{axis}
\end{tikzpicture}
\caption{As a function of $\theta$}
\end{subfigure}
\caption{Expected mean amount of blood, demand, and net blood present.}
\label{fig:means}
\end{figure}
Secondly, we present the probability of positive (cq. negative) inventory.
\begin{corollary}\label{cor:pid}
The probability of positive (cq.~negative) inventory is given by,
\begin{align}
\pi_b &= \frac{\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)}
{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},\\
\pi_d &= \frac{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right)}
{\rho_d\,_2F_1\left(1-\frac{\lambda_d}{\xi_d},1,1+\frac{\lambda_b}{\xi_d},-\frac{\mu_b}{\mu_d}\right) +
\rho_b\,_2F_1\left(1-\frac{\lambda_b}{\xi_b},1,1+\frac{\lambda_d}{\xi_b},-\frac{\mu_d}{\mu_b}\right)},
\end{align}
respectively.
\end{corollary}
\begin{proof}
The expressions follow directly from \eqref{eq:piD} and $\pi_b = 1-\pi_d$.
\end{proof}
The last relevant performance indicator we consider is the fraction of demand that is immediately satisfied from stock.
\begin{corollary}
The probability that a demand request can be fully satisfied from stock is given by
\begin{align}
\mathbb{P}({\rm demand\ satisfied}) = \bar{C}^{-1}\rho_b \left( _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},{-}\tfrac{\mu_b}{\mu_d}\right) - \frac{\mu_b}{\mu_b+\mu_d}\right).
\end{align}
\end{corollary}
\begin{proof}
Using the PASTA property of the Poisson process, we get
\begin{align*}
\mathbb{P}({\rm demand\ satisfied}) &= \mathbb{P}( X > D ) = \mathbb{P}(X_b > D) \\
&= \int_0^\infty g(u) (1-{\rm e}^{-\mu_d u})\, {\rm d} u = \pi_b - \gamma(\mu_d).
\end{align*}
Substituting the expressions for $\pi_b$ as in Corollary \ref{cor:pid} and $\gamma(\mu_b)$ as in \eqref{eq:diffLST_2} yields the result.
\end{proof}
\subsection{The general case}
\label{gener}
In this section we outline how the integral equations (\ref{eq:blood}) and (\ref{eq:demand})
can be solved using Laplace transforms, when we make the restriction that $F_b(\cdot)$ and $F_d(\cdot)$
are Coxian distributions.
This is not a major restriction, because the class of Coxian distributions lies dense in the class of all distributions
of non-negative random variables, see e.g.~\cite[Sec.~III.4]{Asmussen2003}.
Hence, one can approximate $F_b(\cdot)$ arbitrarily closely by a Coxian distribution.
If $X_i$, $i=1,2,\dots,K$ are independent, exponentially distributed random variables,
and $\mathbb{E}[X_i] = \frac{1}{\beta_i}$, $i=1,2,\dots,K$, then a Coxian amount of blood $B$ can be represented
as:
\begin{equation}
B = \sum_{j=1}^i X_j\quad {\rm with ~ probability } \quad p_i \prod_{j=1}^{i-1} (1-p_j), \quad i=1,2,\dots,K.
\end{equation}
In the above case, it is easily verified that one can represent $\bar{F}_b(x)$ as follows:
\begin{equation}
\bar{F}_b(x) = \mathbb{P}(B>x) = \sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
{\rm e}^{-\beta_j x}
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
,
\label{Fbarb}
\end{equation}
if all $\beta_j$ are different. If two $\beta_j$ coincide, then a term with $x {\rm e}^{-\beta_j x}$ (Erlang-$2$) must be added.
We leave this to the reader, but in Remark~\ref{RmErlang} below we outline how Erlang terms can be handled in solving the integral equations
(\ref{eq:demand}) and (\ref{eq:blood}).
The counterpart of (\ref{Fbarb}) for the case that $F_d(\cdot)$ is Coxian, is
\begin{equation}
\bar{F}_d(x) = \mathbb{P}(D>x) = \sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
{\rm e}^{-\delta_j x}
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
.
\label{Fdarb}
\end{equation}
Taking Laplace transforms $\phi(s) = \int_0^{\infty} {\rm e}^{-sy} f(y) {\rm d}y$ and
$\gamma(s) = \int_0^{\infty} {\rm e}^{-sy} g(y) {\rm d}y$
in (\ref{eq:blood}) and (\ref{eq:demand})
results in first-order inhomogeneous differential equations in $\phi(s)$ and $\gamma(s)$, respectively, which can be solved in a straightforward way.
\begin{equation}
\phi'(s) = A_H(s) \phi(s) + A_I(s),
\label{diffeq}
\end{equation}
with the homogeneous term $A_H(s)$ being given by
\begin{align}
A_H(s) &:= -\frac{1}{\xi_d}
\left[
\lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
\frac{1}{\delta_j+s}
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\right.
\nonumber
\\
&\qquad - \lambda_b
\left.\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
\frac{1}{\beta_j-s}
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
- \alpha_d \right],
\end{align}
and the inhomogeneous term $A_I(s)$ being given by
\begin{align}
A_I(s) &:=
- \frac{1}{\xi_d}
\left[
\lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i
\frac{1}{\delta_j+s} [\gamma(\delta_j) + \pi_0]
\prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\right.
\nonumber
\\
&\qquad + \left.\lambda_b
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i
\frac{1}{\beta_j-s} \phi(\beta_j)
\prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\right].
\end{align}
\noindent
The solution of (\ref{diffeq}) is given by the following expression:
\begin{equation}
\phi(s) = \phi(0) {\rm e}^{\int_0^s A_H(z) {\rm d}z} + \int_0^s A_I(u)
{\rm e}^{\int_u^s A_H(z) {\rm d}z} {\rm d}u , ~~~~ s \geq 0.
\label{diffeqsoln}
\end{equation}
$\gamma(s)$ is given by a mirror expression, where $\phi(0)$ is replaced by $\gamma(0)$
and where $A_H(s)$ and $A_I(s)$ are replaced by expressions in which $K$ and $L$ are swapped, and $p$ and $q$, and $\beta_i$ and $\delta_i$.
It should be noticed that $\phi(0)$, $\gamma(0)$ and $\pi_0$ still have to be determined.
Furthermore, it should be noticed that $A_H(s)$ and $A_I(s)$ have singularities at $s=\beta_1,\dots,\beta_K$.
These singularities are removable, but handling Equation \eqref{diffeqsoln} clearly requires some care.
Instead of working out the details, we shall below return to the case
of exponentially distributed amounts of blood and demand -- so $K=L=1$.
For that case, we shall not only work out the solution of the differential equation for $\phi(s)$ in detail,
including the determination of the missing constants, but
we also relate the results to those obtained in Section~\ref{sectionexp}
without resorting to Laplace transforms.
Taking $K=1, p_1=1, \delta_1 = \mu_d$, and $L=1, q_1=1, \beta_1 = \mu_b$, we obtain
the following two inhomogeneous first order differential equations in the LTs $\phi(s)$ and $\gamma(s)$:
\begin{equation}\label{eq:firstPhis}
\phi'(s) = \phi(s)\left[\frac{\lambda_b}{\xi_d} \frac{1}{\mu_b-s} - \frac{\lambda_d}{\xi_d} \frac{1}{\mu_d+s}\right]
-\frac{\lambda_b}{\xi_d} \frac{\phi(\mu_b)}{\mu_b-s} -\frac{\lambda_d}{\xi_d} \frac{\gamma(\mu_d)}{\mu_d+s} ,
\end{equation}
\begin{equation}
\gamma'(s) = \gamma(s)\left[\frac{\lambda_d}{\xi_b} \frac{1}{\mu_d-s} - \frac{\lambda_b}{\xi_b} \frac{1}{\mu_b+s}\right]
-\frac{\lambda_d}{\xi_b} \frac{\gamma(\mu_d)}{\mu_d-s} -\frac{\lambda_b}{\xi_b} \frac{\phi(\mu_b)}{\mu_b+s} .
\end{equation}
They are routinely solved:
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d}{\mu_d+s}\right)^{\frac{\lambda_d}{\xi_d}}
\left[\phi(0) \frac{}{}\right.
\nonumber \\
&\qquad -
\frac{\lambda_d}{\xi_d} \gamma(\mu_d) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}-1} \frac{{\rm d}z}{\mu_d}
\nonumber
\\
&\qquad \qquad -
\frac{\lambda_b}{\xi_d} \phi(\mu_b) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}-1}
\left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}} \frac{{\rm d}z}{\mu_b}\left.\frac{}{}\right].
\label{phis1}
\end{align}
Similarly,
\begin{align}
\gamma(s) &= \left(\frac{\mu_d}{\mu_d-s}\right)^{\frac{\lambda_d}{\xi_b}}
\left(\frac{\mu_b}{\mu_b+s}\right)^{\frac{\lambda_b}{\xi_b}}
\left[\gamma(0) \frac{}{}\right.
\nonumber
\\
&\qquad -
\frac{\lambda_b}{\xi_b} \phi(\mu_b) \int_0^s \left(\frac{\mu_d-z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_b}}
\left(\frac{\mu_b+z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_b}-1} \frac{{\rm d}z}{\mu_b}
\nonumber
\\
&\qquad \qquad -
\frac{\lambda_d}{\xi_b} \gamma(\mu_d) \int_0^s \left(\frac{\mu_d-z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_b}-1}
\left(\frac{\mu_b+z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_b}} \frac{{\rm d}z}{\mu_d}\left.\frac{}{}\right] .
\label{gammas1}
\end{align}
Notice that the exponents in the above integrals have powers which are larger than $-1$ (e.g., $\frac{\lambda_d}{\xi_d}-1$),
so that these integrals do not lead to singularities.
We still need to determine the two constants $\phi(0)=\pi_d$ and $\gamma(0)=\pi_b$.
Together with $\phi(\mu_b)$ and $\gamma(\mu_d)$, we have four unknowns.
We determine these unknowns using the following four equations:
(i) From (\ref{eq:blood2a}), we get
$\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$, while (ii) $\pi_d + \pi_b =1$.
Finally, we take (iii) $s=\mu_b$ in (\ref{phis1}) and (iv) $s=\mu_d$ in (\ref{gammas1}).
Notice that the identity
$\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$ allows us to
reduce the two integrals in (\ref{phis1}) to one integral (and similarly in (\ref{gammas1})):
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\xi_d}}
\left(\frac{\mu_d}{\mu_d+s}\right)^{\frac{\lambda_d}{\xi_d}}
\left[\phi(0) \frac{}{}\right.
\nonumber
\\
&\quad -
\frac{\lambda_d}{\xi_d} \gamma(\mu_d)\, \frac{\mu_b+\mu_d}{\mu_b\mu_d} \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\xi_d}-1} \left(\frac{\mu_d+z}{\mu_d}\right)^{\frac{\lambda_d}{\xi_d}-1} {\rm d}z \left.\frac{}{}\right].
\label{phis11}
\end{align}
\begin{remark}
We have numerically verified that
the expressions in (\ref{phis1}) and (\ref{eq:diffLST}) coincide.
\end{remark}
\begin{remark}
If $\lambda_b=0$ then we have a known queueing model or shot-noise model
with state-dependent service rate, see Keilson \& Mermin \cite{Keilson1959}
and Bekker et al.~\cite{Bekker2004} for the so-called shot noise model.
\end{remark}
\begin{remark}
\label{R7}
The case $\lambda_d = \xi_d$ is special. Equation \eqref{phis1} now reduces to
\begin{align}
\phi(s) &= \left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{\mu_d}{\mu_d+s}
\left[\phi(0) \frac{}{}\right.
\label{phis1A}
-
\gamma(\mu_d) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{{\rm d}z}{\mu_d}
\nonumber
\\
&\qquad -
\frac{\lambda_b}{\lambda_d} \phi(\mu_b) \int_0^s \left(\frac{\mu_b-z}{\mu_b}\right)^{\frac{\lambda_b}{\lambda_d}-1}
\frac{\mu_d+z}{\mu_d} \frac{{\rm d}z}{\mu_b}\left.\frac{}{}\right].
\nonumber
\end{align}
Both integrals are easily evaluated (rewrite,
in the last integral, $\mu_d + z = \mu_d + \mu_b -(\mu_b - z)$).
We find
\begin{align}
\phi(s) &=
\left(\frac{\mu_b}{\mu_b-s}\right)^{\frac{\lambda_b}{\lambda_d}}
\frac{\mu_d}{\mu_d+s}\nonumber\\
&\ \cdot
\left[\phi(0)
+ \frac{\gamma(\mu_d)}{\mu_d} \frac{\lambda_d}{\lambda_b + \lambda_d} \mu_b - \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} - \frac{\phi(\mu_b)}{\mu_d} \frac{\lambda_b}{\lambda_b + \lambda_d} \mu_b \right]
\nonumber
\\
&\ + \frac{\mu_d}{\mu_d+s} \left[
\frac{\gamma(\mu_d)}{\mu_d} \frac{\lambda_d}{\lambda_b + \lambda_d} (\mu_b -s) + \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} - \frac{\phi(\mu_b)}{\mu_d} \frac{\lambda_b}{\lambda_b + \lambda_d} (\mu_b -s)\right] .
\end{align}
Now observe through \eqref{eq:blood2a}, that $\lambda_b \phi(\mu_b) = \lambda_d \gamma(\mu_d)$.
Hence, in both lines of the above formula, two terms cancel.
Moreover, $\phi(s)$ should be analytic for $s=\mu_b$, yielding
\begin{equation}
\phi(0) = \phi(\mu_b) \frac{\mu_d + \mu_b}{\mu_d}.
\end{equation}
Finally we obtain, see also \eqref{eq:lambdaisxi},
\begin{equation}
\phi(s) = \frac{\mu_d}{\mu_d+s} \phi(\mu_b) \frac{\mu_d+\mu_b}{\mu_d} = \phi(0) \frac{\mu_d}{\mu_d+s} = \pi_d \frac{\mu_d}{\mu_d + s},
\end{equation}
and hence
\begin{equation}\label{eq:exp}
f(x) = \pi_d \mu_d {\rm e}^{-\mu_d x}, ~~~ x > 0;
\end{equation}
the shortage (amount of demand present) is exponentially distributed
when $\lambda_d = \xi_d$.
\\
It should be noticed that, if $\lambda_d = \xi_d$, then the first and last term of (\ref{eq:demand2})
are equal when (\ref{eq:exp}) holds; and using (\ref{eq:blood2a})
it is also readily verified that the second and third term of (\ref{eq:demand2}) are equal.
The constant $\pi_d$ will in general still depend on the parameters
$\lambda_d = \xi_d$, $\lambda_b$, $\mu_b$ and $\xi_b$.
\\
We end this remark with the observation that in the one-sided shot-noise process
(so $\lambda_b=0$), Bekker et al.\ \cite{Bekker2004} also observe that $\lambda_d = \xi_d$
results in an exponential density.
\end{remark}
\subsection{A variant}
\label{sectionvariant}
In this section, we assume that the expiration rate of blood and the patience rate of demand are constant.
So, we take $\xi_b = \xi_d = 0$.
A visualization of a possible sample path is depicted in Figure \ref{FIG2}.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmin = -0.2,
xmax = 3.25,
ymin = -0.5,
ymax = 1.6,
ticks = none,
axis line style={->},
axis lines = middle,
yscale = 0.8,
xscale = 1
]
\addplot[black,thick] file {Chapter_7/tikz/sample_path_xi0.txt};
\node at (axis cs: 0.25,-0.4) {$X_d(t)$};
\node at (axis cs: 0.25,1.5) {$X_b(t)$};
\draw[-stealth] (axis cs: 0,0) -- (axis cs:0,-0.5);
\end{axis}
\node at (7.05,1.2) {$t$};
\end{tikzpicture}
\caption{Sample path of the net amount of blood present if $\xi_b = \xi_d = 0$.}
\label{FIG2}
\end{figure}
We again restrict ourselves to the case of exponentially distributed amounts of demand and of blood deliveries.
We now need to impose stability conditions.
In the case of positive demand, the drift is towards zero if $\lambda_d \mathbb{E}[D] < \alpha_d + \lambda_b \mathbb{E} [B]$,
while in the case of a positive amount of blood, the drift is towards zero if
$\lambda_b \mathbb{E} [B] < \alpha_b + \lambda_d \mathbb{E} [D]$.
If these two conditions are violated, either the amount of demand or the amount of blood present increases without bound
(see also
below).
In this case, \eqref{eq:demand6} reduces to
\begin{equation}
\alpha_d f''(v) + (-\lambda_d -\lambda_b + \mu_d \alpha_d -\mu_b \alpha_d)f'(v)
+
(-\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \alpha_d)f(v) =0.
\label{eq:demand6b}
\end{equation}
Hence $f(\cdot)$ is a mixture of two exponential terms: $f(v) = R_+ {\rm e}^{- x_+ v} + R_- {\rm e}^{- x_- v}$,
where $x_+$ and $x_-$ are the positive and negative root of the equation
\begin{equation}
\alpha_d x^2 -(\mu_d \alpha_d - \mu_b \alpha_d -\lambda_d -\lambda_b)x +(-\mu_d \lambda_b + \mu_b \lambda_d -\mu_b \mu_d \alpha_d) = 0.
\label{zeros}
\end{equation}
Notice that the last term in the left-hand side of (\ref{zeros}) is negative if the stability condition
$\lambda_d \mathbb{E} [D] < \alpha_d + \lambda_b \mathbb{E}[B]$ holds,
that is, if $\mu_b \lambda_d < \mu_d \lambda_b + \mu_b \mu_d \alpha_d$,
thus guaranteeing that the product of the two roots $x_+$ and $x_-$ is negative,
and hence that there is a positive and a negative root.
One should subsequently observe that
$R_-$ must be zero to have a probability density.
Hence $f(v)$ is simply (a constant times) an exponential;
similarly for $g(v)$.
In addition, the steady-state amounts of demand and of blood have an atom at $0$ (since $\xi_d$ and $\xi_b$ are no longer zero, the demand and blood processes
can reach $0$).
Interestingly, the model of this section is closely related to the model with workload removal
that is considered in \cite{Boucherie1996}. There an $M/G/1$ queue is studied with the extra feature
that, at Poisson epochs, a stochastic amount of work is removed.
In the $M/M/1$ case with removal of exponential amounts of work, see \cite[Sec.~5.1]{Boucherie1996}, one has the model of the present section
when we concentrate on the amount of demand present.
One difference with the model in \cite{Boucherie1996}
is that, when the workload in that model has become zero, the work becomes positive at rate $\lambda_d$,
whereas in the present model the amount of blood can become positive (so zero demand is present)
and the amount of demand does not have to become positive when demands arrive (because they are immediately satisfied, see Figure~\ref{FIG2}). So the atom at zero is in the present model larger than in the model of \cite{Boucherie1996}.
In our model a positive demand level may be reached from below zero (by a jump, i.e., a demand arriving at an epoch
that there is some, but not enough, blood present). The memoryless property of the exponential
demand requirement distribution implies that this jump results in a demand level that is exp($\mu_d$),
just as if the initial demand level had been zero.
In the case of non-exponential demand requirements, our model becomes equivalent with an $M/G/1$ queue with
exponential amounts of work removed, and with the special feature that the first service requirement of a busy period has a different distribution.
Lemmas 4.1 and 4.2 of \cite{Boucherie1996} present the stability condition
of that $M/G/1$ queue with work removal; it amounts to
$\lambda_d \mathbb{E} [D] < \alpha_d + \lambda_b \mathbb{E} [B]$, which indeed is one of the two stability conditions
of the present demand/blood model.
Finally we observe that Equation (5.1) of \cite{Boucherie1996} coincides with \eqref{zeros}
(take $\alpha_d=1$, $\lambda_d = \lambda_+$, $\lambda_b = \lambda_-$, $\mu_d=1/\beta$ and $\mu_b = 1/\gamma$).
\section{Asymptotic analysis}
\label{sectionscaling}
We finally study the model with $\alpha_b = \alpha_d = 0$ from an asymptotic perspective, by obtaining the fluid and diffusion limits of the blood inventory process. That is, we will create a sequence of processes, indexed by $n=1,2,...$, in which we let the rates of blood and demand arrivals grow large. If we then scale the process in a proper manner, we are able to deduce a non-degenerate limiting process, that provides insight in the overall behavior of the arrival volume when the system grows large, which only relies on the first two moments of the blood and demand distributions.
\subsection{Identification of the limiting process}
First, we introduce some additional notation. Let $X_b(t)$ and $X_d(t)$ denote the amount of blood and demand, respectively, at time $t>0$. Let
\begin{equation}
X(t) := X_b(t) - X_d(t),
\end{equation}
be the net amount of blood available at time $t$. Remember that $X_b(t), X_d(t)\geq 0$, and $X_b(t)>0$ \emph{or} $X_d(t)>0$ for all $t$,
since $\alpha_d=\alpha_b=0$. Let $N_b(t)$, $N_d(t)$ be the two independent Poisson processes counting the number of arrivals of blood and demand, respectively. Then the following integral representation holds for $X(t)$,
\begin{equation}\label{eq:integralRep}
X(t) = X(0) - \xi_b\int_0^t X_b(s)\, ds + \xi_d \int_0^t X_d(s)\, ds + \sum_{i=1}^{N_b(t)} B_i - \sum_{i=1}^{N_d(t)} D_i.
\end{equation}
For the sake of exhibition, we will concentrate on the case $\xi_b=\xi_d =: \xi$.
Our analysis may be extended to the general case.
A sketch of this generalization is given at the end of this section without going into the technical difficulties that arise when rigorously proving these limits.
Define
\begin{equation}\label{eq:defX}
Z(t) = \sum_{i=1}^{N_b(t)} B_i - \sum_{i=1}^{N_d(t)} D_i,
\end{equation}
that is, the difference between two compound Poisson processes, so that \eqref{eq:integralRep} reduces to
\begin{equation}\label{eq:simpleRep}
X(t) = X(0) - \xi\int_0^t X(s)\, {\rm d} s + Z(t).
\end{equation}
The first step in the definition of the sequence of processes under investigation is defining the asymptotic scheme we are interested in. As mentioned above, we intend to let the arrival rates grow to infinity. Therefore, in the $n^{th}$ process $X_n(t)$, we replace the rates of the arrival processes by $n\lambda_b$ and $n\lambda_d$. This induces Poisson processes $N^{(n)}_b(t)$ and $N^{(n)}_d(t)$ with arrival rates $n\lambda_b$ and $n\lambda_d$, respectively.
However, we have
\begin{equation}
N_b^{(n)}(t) {\;\buildrel{d}\over= \;} N_b(n t)\qquad \text{and}\qquad N_d^{(n)}(t) {\;\buildrel{d}\over= \;} N_d(n t),
\end{equation}
so that the term $Z(t)$ in \eqref{eq:simpleRep} in this asymptotic scheme can be replaced by
\begin{equation}
Z_n(t) = \sum_{i=1}^{N_b(nt)} B_i - \sum_{i=1}^{N_d(nt)} D_i.
\end{equation}
The first step in our analysis is obtaining the fluid limit of the process. Bearing in mind application of the Functional Law of Large Numbers (FLLN), we scale the process as $\bar{X}_n(t) = X_n(t)/n$,
so that with \eqref{eq:simpleRep}
\begin{equation}\label{eq:fluidRep}
\bar{X}_n(t) = \bar{X}_n(0) -\xi\int_0^t \bar{X}_n(s)\, ds + \bar{Z}_n(t),
\end{equation}
where $\bar{Z}_n(t) = Z_n(t)/n$.
The essential step in establishing a result on the convergence of $\bar{X}_n$ is the application of \cite[Thm.~4.1]{Pang2007}, which we cite here for completeness, slightly rewritten to fit our setting.
\begin{theorem}[{\cite[Thm.~4.1]{Pang2007}}]
\label{thm:pang}Let $D[0,\infty)$ be the space of all one-dimensional real-valued c\`adl\`ag functions defined on $[0,\infty)$, endowed with the usual $J_1$-Skorohod topology.
Consider the integral representation
\begin{equation}\label{eq:pangInt}
x(t) = y(t) + \int_0^t u(x(s))\, ds, \qquad t \geq 0,
\end{equation}
where $u:\mathbb{R}\to\mathbb{R}$ satisfies $u(0)=0$ and is Lipschitz continuous.
The integral representation in \eqref{eq:pangInt} has a unique solution $x$, so that the integral representation constitutes a function $H_u: D[0,\infty) \to D[0,\infty)$ mapping $y$ into $x\equiv H_u(y)$.
In addition, the function $H_u$ is continuous, and if $y$ is continuous, then so is $x$.
\end{theorem}
In our case, we set $u(x) = -\xi x$, to be able to write $\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Z}_n\right)$. Since $u$ is clearly Lipschitz continuous, the mapping $H_u$ is indeed continuous.
Let us rewrite \eqref{eq:fluidRep}, by observing
\begin{equation}
\mathbb{E} \bar{Z}_n(t) = \frac{1}{n}\Big(\mathbb{E} [N_b(nt)]\mathbb{E} [B] - \mathbb{E} [N_d(n t)]\mathbb{E}[D]\Big) = \lambda_b\mathbb{E} [B] t - \lambda_d \mathbb{E} [D] t,
\end{equation}
where the expectation is taken with respect to the compound Poisson processes.
Since $m=\lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]$,
\begin{equation}\label{eq:fluidRep2}
\bar{X}_n(t) = \bar{X}_n(0) -\xi\int_0^t\left(\bar{X}_n(s)-\frac{m}{\xi}\right)\, {\rm d} s + \bar{Y}_n(t),
\end{equation}
where $\bar{Y}_n(t) := \bar{Z}_n(t) - mt$ is now a centered process.
This allows us to state the next result.
\begin{proposition}[Fluid limit]\label{fluidProp}
Let $\mathbb{E}[B],\, \mathbb{E}[D] <\infty$ and $\bar{X}_n(0) = X_n(0)/n \rightarrow q_0 \in \mathbb{R}$, as $n\rightarrow \infty$. Then for $n\rightarrow\infty$,
\begin{equation}\label{eq:fluidLimit}
\bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q,
\end{equation}
where
\begin{equation}
q(t) = \frac{m}{\xi} + \left(q_0 - \frac{m}{\xi}\right) {\rm e}^{-\xi t}.
\label{eq:fluid_function}
\end{equation}
\end{proposition}
\begin{proof}
First, we concentrate on the process $\bar{Y}_n$. Observe that, by the FLLN for renewal-reward processes, which follows from \cite[Thm.~7.4.1]{Whitt2002}, we have
\begin{equation}
\frac{1}{nt}\sum_{i=1}^{N_b(nt)} B_i {\;\buildrel{d}\over\Rightarrow\;} \lambda_b\mathbb{E}[B],\qquad \frac{1}{nt}\sum_{i=1}^{N_d(nt)} D_i {\;\buildrel{d}\over\Rightarrow\;} \lambda_d\mathbb{E}[D],
\end{equation}
for $n\rightarrow\infty$ and for all $t>0$. Hence, $\bar{Z}_n(t) {\;\buildrel{d}\over\Rightarrow\;} \lambda_b\mathbb{E}[B]t-\lambda_d\mathbb{E}[D]t = mt$. By definition of $\bar{Y}_n$ and the assumption of convergence of $\bar{X}_n(0)$, this implies
\begin{equation}
\bar{Y}_n + \bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q_0
\end{equation}
as $n\rightarrow\infty$.
Next, note $\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Z}_n \right) = H_u\left(\bar{X}_n(0) + \bar{Y}_n +I t \right)$, where $I$ denotes the identity map, i.e.~ $I(t) = t$ for all $t \geq 0$. Due to Lipschitz continuity of $u$, $H_u$ constitutes a continuous mapping, and hence we can apply the Continuous Mapping Theorem (CMT), to find
\begin{equation}
\bar{X}_n = H_u\left(\bar{X}_n(0) + \bar{Y}_n +m I\right) \Rightarrow H_u\left(q_0+m I\right)\equiv q,
\end{equation}
for all $t\geq 0$, where $q(\cdot)$ is the solution of
\begin{align*}
q(t) &= q_0 + \int_0^t u(q(s))\, {\rm d} s = q_0 + mt - \xi \int_0^t q(s) \, {\rm d} s\\
&= q_0 - \xi \int_0^t \left(q(s) - \frac{m}{\xi}\right) \, {\rm d} s.
\end{align*}
The unique solution of this integral equation is given in \eqref{eq:fluid_function}.
\end{proof}
According to Proposition~\ref{fluidProp}, the fluid limit approaches $\mathbb{E}[X] = m/\xi$ exponentially fast.
To obtain an expression for the {\em diffusion limit} of the process, we analyze the fluctuations of the process around the fluid limit in (\ref{eq:fluidLimit}), again by scaling the process in a proper manner. First, we subtract $q(t)$ on both sides of \eqref{eq:fluidRep2}, and multiply by $\sqrt{n}$:
\begin{equation}\label{eq:hatEquation}
\sqrt{n}\left( \bar{X}_n(t) - q(t) \right) = \sqrt{n}\left( \bar{X}_n(0) - q_0 \right) -\xi \int_ 0^t \sqrt{n}\left( \bar{X}_n(s) - q(s) \right)\, ds + \sqrt{n}\,\bar{Y}_n(t).
\end{equation}
Let $\hat{X}_n \equiv \sqrt{n}\left( \bar{X}_n - q \right)$ and $\hat{Y}_n \equiv \sqrt{n}\,\bar{Y}_n$, then this reduces to
\begin{equation}
\hat{X}_n(t) = \hat{X}_n(0) - \xi \int_0^t \hat{X}_n(s) \,ds + \hat{Y}_n(t).
\end{equation}
Again the process $\hat{Y}_n$ needs special attention.
\begin{lemma}\label{diffLemma}
Let $\mathbb{E}[B^2], \mathbb{E}[D^2] < \infty$. Then $\hat{Y}_n {\;\buildrel{d}\over\Rightarrow\;} \sigma W$ as $n\rightarrow \infty$, where $\sigma^2 := \lambda_b \mathbb{E}[B^2]+\lambda_d\mathbb{E}[D^2]$ and $W$ is a standard Brownian motion.
\end{lemma}
\begin{proof}
Recall that
\begin{equation}
\hat{Y}_n(t) {\;\buildrel{d}\over= \;} \sqrt{n}\left[ \Big(\frac{1}{n}\sum_{i=1}^{N_b(n t)} B_i -\lambda_b\mathbb{E} [B]t \Big) -
\Big(\frac{1}{n}\sum_{i=1}^{N_d(nt)} D_i - \lambda_d \mathbb{E}[D] t \Big)
\right] .
\end{equation}
By the Functional Central Limit Theorem (FCLT) for renewal-reward processes given in \cite[Thm.~7.4.1]{Whitt2002}, the process
\begin{equation}
\hat{Y}_n^b(t) = \sqrt{n} \Big(\frac{1}{n}\sum_{i=1}^{N_b(n t)} B_i -\lambda_b\mathbb{E}[ B] t\Big),
\end{equation}
converges weakly to $\sigma_b W_b$, where $W_b$ is a standard Brownian motion, and
\begin{equation}
\sigma_b^2 = \lambda_b\,\text{Var}\,B + \lambda_b(E[B])^2 = \lambda_b\mathbb{E}[B^2].
\end{equation}
Similarly, $\hat{Y}_n^d \Rightarrow \sigma_d W_d$ as $n\to\infty$, with the obvious parameter switches and $W_d$ is standard Brownian motion. Since the processes $\hat{Y}_n^b$ and $\hat{Y}_n^d$ are independent, so are their limits, and
\begin{equation}
\hat{Y}_n \Rightarrow \sqrt{\lambda_b \mathbb{E}[B^2]}\, W_b + \sqrt{\lambda_d \mathbb{E}[D^2]}\, W_d {\;\buildrel{d}\over= \;} \sqrt{\lambda_b\mathbb{E}[B^2]+\lambda_d\mathbb{E}[D^2]}\,W,
\end{equation}
for $n\rightarrow\infty$ and $W$ a standard Brownian motion.
\end{proof}
Now, we are ready to prove the diffusion counterpart of Proposition \ref{fluidProp}.
\begin{proposition}[Diffusion limit]\label{diffProp}
Let $\mathbb{E}[B^2], \mathbb{E}[D^2] < \infty$. If $\hat{X}_n(0) \rightarrow \hat{X}(0)$, then $\hat{X}_n \Rightarrow \hat{X}$ as $n\rightarrow \infty$, where $\hat{X}$ satisfies the integral equation
\begin{equation}\label{diffLimit}
\hat{X}(t) = \hat{X}(0) - \xi \int_0^t \hat{X}(s) \, {\rm d} s + \sigma W(t).
\end{equation}
In other words, $\hat{X}$ is an Ornstein-Uhlenbeck diffusion process with infinitesimal mean $\xi$ and infinitesimal variance $\sigma^2 := \lambda_b\mathbb{E}[B^2] + \lambda_d\mathbb{E}[D^2]$.
\end{proposition}
\begin{proof}
We again rely on the result that the mapping $H_u$ as in the proof of Proposition \ref{fluidProp} is continuous if $u$ is Lipschitz continuous. Here, we set $u(x) = -\xi x$ which again clearly satisfies this condition. We have $\hat{X}_n \equiv H_u(\hat{X}_n(0)+ \hat{Y}_n)$. From Lemma \ref{diffLemma}, we know
\begin{equation}
\hat{X}_n(0) + \hat{Y}_n \Rightarrow \hat{X}(0) + \sigma W,
\end{equation}
for $n\rightarrow\infty$. As a consequence of the CMT, we conclude
\begin{equation}
\hat{X}_n = H_u\left( \hat{X}_n(0) + \hat{Y}_n\right) \Rightarrow H_u\left(\hat{X}(0) + \sigma W\right) \equiv \hat{X},
\end{equation}
where $\hat{X}$ solves \eqref{diffLimit}.
\end{proof}
\subsection{Generalization for $\xi_b\neq \xi_d$}
We now sketch the scaling approach towards fluid and diffusion limits for the general case in which
$\xi_b$ may differ from $\xi_d$.
In case $\xi_b \neq \xi_d$, the integral equation for $\bar{X}_n$ as in \eqref{eq:fluidRep} becomes
\begin{align}
\label{eq:fluidRepNEQ}
\bar{X}_n(t) &= \bar{X}_n(0) + \int_0^t ( -\xi_b \bar{X}_n^+(s) + \xi_d \bar{X}_n^-(s) - m)\, {\rm d} s + \bar{Y}_n(t)\\
&= \bar{X}_n(0) - \int_0^t (\left[\xi_b \mathbbm{1}_{\{ \bar{X}_n(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{ \bar{X}_n(s)<0\}}\right] \bar{X}_n(s) + m) \, {\rm d} s + \bar{Y}_n(t),
\nonumber
\end{align}
where $\bar{Y}_n(t)$ is defined as before. Note that $\hat{X}_n \equiv H_u(\bar{X}_n(0)+\bar{Y}_n)$, where we now have
\begin{equation}
u(x) = - \left[\xi_b \mathbbm{1}_{\{ x\geq 0\}}+\xi_d \mathbbm{1}_{\{x<0\}}\right]x + m,
\end{equation}
which is still Lipschitz continuous. Following the same reasoning of the proof of Proposition \ref{fluidProp}, we obtain the fluid limit $\bar{X}_n {\;\buildrel{d}\over\Rightarrow\;} q$, where $q$ is the solution of
\begin{equation}
q(t) = q_0 - \int_0^t (\left[\xi_b \mathbbm{1}_{\{ q(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{q(s)<0\}}\right]q(s) - m)\, {\rm d} s .
\end{equation}
The solution to this integral equation is more elaborate than \eqref{eq:fluidLimit} and depends on the sign of $m$ and $q_0$. Assuming $m\geq 0$, one can check that,
\begin{align}
q(t) &= \frac{m}{\xi_b} + \left(q_0- \frac{m}{\xi_b}\right) e^{-\xi_b t}, & \text{if }q_0\geq 0,\\
q(t) &= \left\{
\begin{array}{ll}
\frac{m}{\xi_d} + \left(q_0- \frac{m}{\xi_d}\right) e^{-\xi_d t}, & \text{if } 0\leq t < t_d^*,\\
\frac{m}{\xi_b}\left(1-e^{-\xi_b (t-t^*_d)}\right), & \text{if } t \geq t_d^*,
\end{array}\right. & \text{if } q_0 <0,
\label{eq:fluid_1}
\end{align}
where \begin{equation}
t_d^* = - \frac{1}{\xi_d}\,\log\left(\frac{m/\xi_d}{m/\xi_d-q_0}\right).
\end{equation}
If $m < 0$,
\begin{align}
q(t) &= \frac{m}{\xi_d} + \left(q_0- \frac{m}{\xi_d}\right) e^{-\xi_d t}, & \text{if }q_0\leq 0,\\
q(t) &= \left\{
\begin{array}{ll}
\frac{m}{\xi_b} + \left(q_0- \frac{m}{\xi_b}\right) e^{-\xi_b t}, & \text{if } 0\leq t < t_b^*,\\
\frac{m}{\xi_d}\left(1-e^{-\xi_d (t-t^*_b)}\right), & \text{if } t \geq t_b^*,
\end{array}\right. & \text{if } q_0>0,
\label{eq:fluid_2}
\end{align}
where \begin{equation}
t_b^* = - \frac{1}{\xi_b}\,\log\left(\frac{m/\xi_b}{m/\xi_b-q_0}\right).
\end{equation}
Note that the equilibrium of the fluid limit also depends on the sign of $m$:
\begin{equation}
\lim_{t\rightarrow\infty} q(t) = \left\{\begin{array}{ll}
m/\xi_b, & \text{if }m\geq 0,\\
m/\xi_d, & \text{if }m <0.
\end{array}\right.
\end{equation}
In the remainder, without loss of generality $m\geq 0$. Furthermore, set $q_0= m/\xi_b$ so that $q \equiv m/\xi_b$. Subtracting $q(t)$ on both sides of \eqref{eq:fluidRepNEQ} yields,
\begin{align}
\left(\bar{X}_n(t)-q(t)\right) &= \left(\bar{X}_n(0)-q_0\right) -
\int_0^t \Big\{ \left[\xi_b \mathbbm{1}_{\{\bar{X}_n(s)\geq 0\}}+\xi_d \mathbbm{1}_{\{\bar{X}_n(s)<0\}}\right] \bar{X}_n(s) \nonumber \\
& \qquad \qquad - \xi_b\,q(s)\Big\} \, {\rm d} s + \bar{Y}_n(t)\\
&= \left(\bar{X}_n(0)-q_0\right) - \int_0^t \xi_b \left(\bar{X}_n(s)-q(s)\right)\, {\rm d} s \nonumber \\
&\qquad \qquad +
\int_0^t \mathbbm{1}_{\{\bar{X}_n(s) < 0\}}(\xi_b-\xi_d)\bar{X}_n(s)\, {\rm d} s.
\end{align}
Let $\hat{X}_n(t) = \sqrt{n}\left(\bar{X}_n(t)-q(t)\right)$. Then
\begin{equation}
\hat{X}_n(t) = \hat{X}_n(0) - \xi_b \int_0^t \hat{X}_n(s)\,{\rm d} s +\int_0^t \mathbbm{1}_{\{\bar{X}_n(s) < 0\}}(\xi_b-\xi_d)\bar{X}_n(s)\, {\rm d} s + \hat{Y}_n(t)
\end{equation}
Now, we argue non-rigorously that the one-but-last term vanishes as $n\rightarrow\infty$. Namely, by defining the function $G: D[0,\infty)\rightarrow D[0,\infty)$ by the integration operator:
\begin{equation}
G(u) = \int_0^t \mathbbm{1}_{\{u(s)<0\}} (\xi_b-\xi_d) u(s)\, {\rm d} s,
\end{equation}
this term can be expressed as $G(\bar{X}_n)$. Hence by the fact that $\hat{X}_n{\;\buildrel{d}\over\Rightarrow\;} m/\xi_b$ and the CMT we see $G(\hat{X}_n)\Rightarrow 0$.
Under this claim, we deduce by the approach of Proposition \ref{diffProp}, that if $\hat{X}_n \Rightarrow \hat{X}$ for $n\rightarrow \infty$, then $\hat{X}$ satisfies the stochastic integral equation
\begin{equation}
\hat{X}(t) = \hat{X}(0) - \xi_b \int_0^t \hat{X}(s)\, {\rm d} s + \sigma W(t),
\end{equation}
which implies that $\hat{X}$ is an Ornstein-Uhlenbeck process with infinitesimal mean $\xi_b$ and variance $\sigma^2 := \lambda_b\mathbb{E}[B^2] + \lambda_d\mathbb{E}[D^2]$.
\\
The result that the scaled process converges to an Ornstein-Uhlenbeck process can be intuitively justified by the so-called \textit{mean-reverting} behavior of the original process. That is, the further the process is away from its mean, the greater the drift towards that equilibrium. This is the defining feature of the OU diffusion process. The decay rates $\xi_b$ and $\xi_d$ are responsible for the original process being `forced' towards 0 and therefore the similarities should not be surprising. However, note that in the diffusion limit $X_n$ has drift $\xi_b$ (cq. $\xi_d$) towards $nm/\xi_b$ (cq. $nm/\xi_d$), if $m>0$ (cq. $<0$) at \emph{any} position of the process. This implies that if $X_n \in(0,nm/\xi_b)$, it has an upward drift equal to $\xi_b$, which is at first sight counter-intuitive.
However, we can argue that in case $X_n(t) = v \in (0, nm/\xi_b)$, the mean upward drift of the process $X_n$ equals $n\lambda_b\mathbb{E}[B]$, and the mean downward drift equals $n\lambda_d\mathbb{E}[D] + \xi_b v$, since $v>0$.
Rewrite $v = nm/\xi_b - w\sqrt{n}$ for some $w \in (0,\sqrt{n} m/\xi_b)$.
Then, the mean net drift equals
\[
n\lambda_b\mathbb{E}[B] - n\lambda_d\mathbb{E}[D] - \xi_b \left( \frac{n m}{\xi_b} - w\sqrt{n} \right) = \xi_b w \sqrt{n} >0,
\]
which explains both the sign and magnitude of the drift factor in the scaled process.
\subsection{Related literature}
The Ornstein-Uhlenbeck process is a diffusion process that often arises as the limit of a sequence of stochastic systems, in which the system size tends to infinity.
Particularly in queueing settings with mean reverting behavior, the OU process appears in so-called heavy traffic, i.e.~the arrival rate grows without bound.
We mention a couple of models that exhibit limiting behavior that is similar to ours.
First, it is well-known that the properly normalized $M/M/\infty$ queue length process converges weakly to a OU process as the arrival rate tends to infinity, see e.g.~\cite[Sec.~10.3]{Whitt2002}.
This limiting behavior continues to hold in case the queueing process is modulated by a Markovian background process, see \cite{Anderson2016}.
Another well-known queueing model in which a (piecewise) OU process appears in the limit is the multi-server queue with abandonments.
For the $M/M/s+M$ queue, where $+M$ denotes the exponentially distributed patience of customers, Garnett et al.~\cite{Garnett2002} showed that in the Halfin-Whitt regime, the queue length process, centered and scaled around the number of servers $s$, approaches a hybrid OU process, of which the drift parameter depends on the current state: If the queue length is larger (cq.~smaller) than zero, then the drift is governed by the abandonment rate (cq.~service rate).
Dai et al.~\cite{Dai2010} find a similar piecewise diffusion process under more general assumptions on the model primitives.
For the single-server queue with abandoning customers, Ward \& Glynn \cite{Ward2003,Ward2005} showed that in conventional heavy traffic, the queue length process converges to a OU process with reflecting barrier 0.
Since we in our setting assumed both demand impatience and perishability of inventory (which can be seen as a kind of impatience as well), it should not come as a surprise that we also find our limiting process to be a OU process.
Observe however that in our model, unless $m=0$, we find a OU process with constant, rather than piecewise, parameters, and no reflection barrier, since our (scaled) inventory process can go both positive and negative.
Last, we mention that there is a connection between our blood inventory process and the work of Reed \& Zwart \cite{Reed2011}.
Rather than looking at the OU process as the limit of a sequence of stochastic processes, Reed and Zwart \cite{Reed2011} study a stochastic differential equation that is closely related to Equation \eqref{eq:integralRep}, in the sense that the process has a different (constant) drift term in the upper and lower half plane.
Under the assumption that the input process is a L\'evy process with only one-sided jumps, they develop a methodology to derive the invariant distribution of the solution of the SDE.
Unfortunately, the input in our scenario exhibits both positive and negative jumps, which prevents us from applying their results directly to \eqref{eq:integralRep}.
\section{Numerical evaluation}
\label{numericals}
\subsection{Approximation scheme}
The asymptotic results of the previous section regarding the fluid and diffusion limits allude to the fact that for large arrival rates, the normalized inventory process $\{\hat X_n(t)\, |\, t\geq 0\}$, resembles that of the Ornstein-Uhlenbeck process.
Indeed, the sample paths of the scaled process $\bar X_n$ for increasing values of $n$ in Figures \ref{fig:sample_paths_fluid1} and \ref{fig:sample_paths_fluid2} show that the mean-reverting behavior around $m/\xi^*$, that is typical of OU processes, kicks in rather quickly.
Moreover, the fluid limits $q(t)$ as presented by Proposition \ref{fluidProp} and \eqref{eq:fluid_1}-\eqref{eq:fluid_2} predict the mean well for both $\xi_b=\xi_d$ and $\xi_b \neq \xi_d$.
\begin{figure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp1] {Chapter_7/tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp10] {Chapter_7/tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=10$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -0.5,
ymax = 7,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp100] {Chapter_7/tikz/sample_paths_fluid.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=100$}
\end{subfigure}
\caption{Sample paths of the process $\bar{X}_n(t) = X_n(t)/n$ with $\bar{X}_n(0) = 5$, $\lambda_b = 1.2$, $\lambda_d = 1$, $\xi_b = \xi_d = 0.5$ and $\mu_b=0.5$ and $\mu_d=1$. The fluid limit is depicted by the dashed line.}
\label{fig:sample_paths_fluid1}
\end{figure}
\begin{figure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp1] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp10] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=10$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.52]
\begin{axis}[
xmin = 0,
xmax = 25,
ymin = -2.5,
ymax = 4.5,
axis line style={->},
axis lines = middle]
\addplot[] table[x=t,y=sp100] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\addplot[dashed,thick] table[x=t,y=fluid] {Chapter_7/tikz/sample_paths_fluid_min.txt};
\end{axis}
\end{tikzpicture}
\caption{$n=100$}
\end{subfigure}
\caption{Sample paths of the process $\bar{X}_n(t) = X_n(t)/n$ with $\bar{X}_n(0) = -2$, $\lambda_b = 2$, $\lambda_d = 1$, $\xi_b = 0.5, \xi_d = 0.1$ and $\mu_b=1$ and $\mu_d=1$. The fluid limit is depicted by the dashed line.}
\label{fig:sample_paths_fluid2}
\end{figure}
Furthermore, we observe that steady state is attained fairly quickly.
This is suggestive of the claim that the steady-state distribution of the normalized process $\hat{X}_n$ is well-described by the steady-state distribution of the OU process $\hat{X}$.
Since the OU process with mean 0, infinitesimal variance $\sigma^2$ and drift $\xi^*$ is known to be normally distributed with mean 0 and variance $\sigma^2/2\xi^*$ in steady-state, this leads to a simpler approximation scheme based on the first two moments of $B$ and $D$ only.
In non-rigorous mathematical terms, we use the approximation that
\begin{equation}
\hat{X}_n = \frac{ X_n - nm/\xi^*}{\sqrt{n}} {\;\buildrel{d}\over\approx \;} Z^*,
\label{eq:normal_approximation}
\end{equation}
where $Z^*$ is a normally distributed random variable with mean 0 and variance $\sigma^2/2\xi^*$.
Note that justification of the conjecture that the normal approximation is indeed an asymptotically correct approximation for systems with large arrival rates requires proof that the interchange-of-limits between $t\to\infty$ and $n\to\infty$ is indeed valid.
Rather than going into the technical details, we provide in the remainder of this section numerical evidence that this interchange indeed holds, and that the normal approximation is able to capture characteristics of processes with exponential jumps as well as generally distributed jumps.
\subsection{Distribution functions}
Since we obtained an explicit expression for the steady-state density function of the net inventory process $X$ in case $B$ and $D$ are exponential, see Theorem \ref{thm:full_pdf}, we will exploit this formula for numerical comparison to the normal approximation arising from the OU process.
Let $h(\cdot)$ as in Theorem \ref{thm:full_pdf} be the pdf of $X$ with parameters $\lambda_b$, $\lambda_d$, $\mu_b$, $\mu_d$, $\xi_b$ and $\xi_d$, and the corresponding cdf $H$, defined as $H(v) = \int_{-\infty}^v h(x) {\rm d} x$.
We denote by $h_n$ and $H_n$ the pdf and cdf, respectively, of the inventory process $X_n$ with arrival rates $n\lambda_b$ and $n\lambda_d$, and the remaining parameters unchanged.
Then, the pdf and cdf of the normalized process are given by $\hat{h}_n(v) = \sqrt{n}\,h_n(v_n)$ and $\hat{H}_n(v) = H_n(v_n)$, respectively, with $v_n = nm/\xi^* + v\sqrt{n}$ for all $v\in\mathbb{R}$.
By the normal approximation scheme, we expect
\begin{equation}
\hat{h}_n(v) \approx \frac{\sqrt{2\xi^*}}{\sigma}\,\varphi\left(\frac{\sqrt{2\xi^*}}{\sigma}v\right),\quad \text{ and } \quad
\hat{H}_n(v) \approx \Phi\left(\frac{\sqrt{2\xi^*}}{\sigma}v\right).
\end{equation}
We perform this numerical comparison of probability functions in Figure \ref{fig:distributions} for three cases: $\xi_b=\xi_d$, $\xi_b>\xi_d$ and $\xi_b<\xi_d$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,col4] table[x=v,y=h5] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,col5] table[x=v,y=h10] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,dashed] table[x=v,y=ou] {Chapter_7/tikz/convergence_1_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=\xi_d=1$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,col4] table[x=v,y=H5] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,col5] table[x=v,y=H10] {Chapter_7/tikz/convergence_1_1.txt};
\addplot[thick,dashed] table[x=v,y=OU] {Chapter_7/tikz/convergence_1_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=\xi_d=1$ (cdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,col4] table[x=v,y=h5] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,col5] table[x=v,y=h10] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,dashed] table[x=v,y=ou] {Chapter_7/tikz/convergence_1_2.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=1,\ \xi_d=2$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,col4] table[x=v,y=H5] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,col5] table[x=v,y=H10] {Chapter_7/tikz/convergence_1_2.txt};
\addplot[thick,dashed] table[x=v,y=OU] {Chapter_7/tikz/convergence_1_2.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=1,\ \xi_d=2$ (cdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\small
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 0.6,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.48)},anchor=north east},
legend cell align = left,
yscale = 0.8
]
\addplot[thick,col1] table[x=v,y=h1] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,col4] table[x=v,y=h5] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,col5] table[x=v,y=h10] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,dashed] table[x=v,y=ou] {Chapter_7/tikz/convergence_2_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=2,\ \xi_d=1$ (pdf)}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}[scale=0.75]
\begin{axis}[
xmin = -4,
xmax = 4,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = middle,
legend style = {at = {(axis cs: 3.8,0.04)},anchor=south east},
legend cell align = left,
legend pos = north east,
yscale = 0.8
]
\small
\addplot[thick,col1] table[x=v,y=H1] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,col4] table[x=v,y=H5] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,col5] table[x=v,y=H10] {Chapter_7/tikz/convergence_2_1.txt};
\addplot[thick,dashed] table[x=v,y=OU] {Chapter_7/tikz/convergence_2_1.txt};
\legend{$n=1$,$n=5$,$n=10$,OU};
\end{axis}
\end{tikzpicture}
\caption{$\xi_b=2,\ \xi_d=1$ (cdf)}
\end{subfigure}
\caption{Probability functions of $\hat{X}_n$ for $n=1,5$ and $10$ with $\lambda_b=1,\lambda_d=0.5,\mu_b=\mu_d=1$, and the probability function of the OU process.}
\label{fig:distributions}
\end{figure}
From Figure \ref{fig:distributions}, in which $m = 1$, so that $\xi^* = \xi_b$, the convergence of the pdf and cdf is evident.
For $n=10$, the distribution functions of the scaled processes are almost aligned with the normal distribution already.
For $\xi_b=\xi_d$, the convergence is fastest.
This can be explained by observing that in cases where $\xi_b\neq \xi_d$, the parameter $\xi_d$ still plays a role in pre-limit systems, whereas it does not appear in the normal limit.
In the cases where $\xi_b\neq \xi_d$ we furthermore see that the functions are not smooth around $v_n=0$ or $v^* = -\sqrt{n}m/\xi^*$, which is the zero-inventory level in the original (unscaled) process.
As $n$ increases, this point of irregularity goes to $-\infty$ and therefore disappears.
\subsection{Approximations to performance metrics}
The plots in the previous section indicate that the normal approximation gives simple yet accurate approximations to the stationary distribution of the inventory process.
We now assess if this also translates to the performance measures.
Again, we choose to fix the parameters $\lambda_b$ and $\lambda_d$, and evaluate the system with arrival rates $n\lambda_b$ and $n\lambda_d$ for increasing $n$.
First, the normal approximation in \eqref{eq:normal_approximation} yields the following approximation for the expected inventory level:
\begin{equation}
\mathbb{E}[X_n] \approx \frac{ n m }{\xi^*} = \frac{n (\lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]) }{\xi^*}.
\label{eq:approx1}
\end{equation}
For the probability of negative inventory, we have
\begin{equation}
\pi_d = \mathbb{P}(X_n < 0) \approx \mathbb{P}\left( Z^* < -\sqrt{n}\,m/\xi^* \right) = \Phi\left(-\sqrt{n/2\xi^*}\,m/\sigma\right).
\label{eq:approx2}
\end{equation}
Last, the probability of demand being satisfied immediately is approximately
\begin{equation}
\mathbb{P}({\rm demand\ satisfied }) = \mathbb{P}( X_n > D ) \approx 1 - \int_0^\infty \Phi\left({-}\frac{\sqrt{2\xi^*}}{\sigma}\, \frac{x-nm/\xi^*}{\sqrt{n}}\right) \,{\rm d} F_d(x).
\label{eq:approx3}
\end{equation}
\begin{remark}
Note that if $\lambda_b$ and $\lambda_d$ are large themselves, the parameter $n$ can be eliminated from \eqref{eq:approx1}-\eqref{eq:approx3}, so that
\begin{equation*}
\mathbb{E}[X] \approx \frac{m}{\xi^*}, \qquad
\pi_d \approx \Phi\left({-}m/(\sigma\sqrt{2\xi^*})\right),
\end{equation*}
\begin{equation*}
\mathbb{P}({\rm demand\ satisfied}) \approx 1-\int_0^\infty \Phi\left({-} \sqrt{2\xi^*}\,\frac{x-m/\xi^*}{\sigma}\right)\, {\rm d} F_d(x),
\end{equation*}
where $m = \lambda_b \mathbb{E}[B] - \lambda_d \mathbb{E}[D]$ and $\sigma^2 = \lambda_b \mathbb{E}[B^2] + \lambda_d \mathbb{E}[D^2]$.
\end{remark}
We will now test these approximations under various assumptions on the distribution of $B$ and $D$.
In Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma} we compare the values obtained through the normal approximation against the true values obtained through numerical evaluation (for exponential jump sizes only) and simulation.
All simulation results are accurate up to a 95\% confidence interval of width $10^{-4}$.
We set $\lambda_b=1$ and $\lambda_d=0.5$ and let the mean jump sizes be equal to 1, i.e.~ $\mathbb{E}[B] = 1$ and $\mathbb{E}[B] = 1$ in all numerical experiments.
In Table \ref{tab:accuracy_deterministic}, we let the jump sizes be deterministic, so that ${\rm Var}\, B = {\rm Var}\, D = 0$.
Table \ref{tab:accuracy_exponential} shows the results in case of exponential jump sizes, so that ${\rm Var}\, B = {\rm Var}\, D = 1$.
Last, in Table \ref{tab:accuracy_gamma} we investigate the quality of the approximation for jump sizes that follow a Gamma$(0.25,0.25)$ distribution, yielding ${\rm Var}\, B = {\rm Var}\, D = 4$.
With this set-up we cover jump distributions of increasing variance, so that we are able to study the impact of increased variability on the accuracy of the approximations.
Moreover, we investigate the influence of the decay parameters $\xi_b$ and $\xi_d$ by considering the scenarios $\xi_b=\xi_d$, $\xi_b<\xi_d$ and $\xi_b>\xi_d$.
We make a couple of observations based on the numbers in Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma}.
First, we see that the approximation for the mean blood inventory level $\mathbb{E}[X_n]$ is exact if $\xi_b=\xi_d$, see Proposition \ref{prop:mean_inventory}.
This obviously does not extend to $\pi_d$ and $\mathbb{P}({\rm demand\ satisfied})$, since these performance measures are based on the entire distribution of $X_n$ rather than the mean.
Nonetheless, the normal approximation appears to be very accurate in the case $\xi_b=\xi_d$.
We may explain this by observing that in the approximations \eqref{eq:approx1}-\eqref{eq:approx3}, only $\xi^*$ appears.
In our setting, we have $m = \lambda_b - \lambda_d = 0.5$, so that $\xi^* = \xi_b$.
If $\xi_b\neq \xi_d$, then the value of $\xi_d$ plays a role in pre-limit systems, which induces inaccuracies in the approximation of performance measures.
In case $\xi_b = \xi_d$, we have $\xi^* = \xi_b = \xi_d$, so that this discrepancy is overcome.
Moreover, since $m>0$, we see that $\pi_d\to 0$ and $\mathbb{P}({\rm demand\ satisfied}) \to 1$ as $n$ increases.
This is due to the observation that as $n$ grows large, the inventory process concentrates around the level $nm$ with fluctuations of order $\sqrt{n}$, so that the process stays away from level zero, see Figure \ref{fig:sample_paths_fluid1}.
The approximations \eqref{eq:approx2}-\eqref{eq:approx3} adequately capture this convergence.
As expected, the accuracy of the approximations increases with $n$.
Moreover, increased variability in the jump distributions appears to cause a decrease in accuracy.
However, for all cases considered in Tables \ref{tab:accuracy_deterministic}-\ref{tab:accuracy_gamma}, the normal approximations \eqref{eq:approx1}-\eqref{eq:approx3} seem to yield relatively sharp estimates for the relevant performance measures under various assumptions on the distributions of the jump sizes.
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.2702 & 0.2819 & 0.2598 & 0.2819 \bigstrut[t]\\
2 & 1.000 & 1.000 & 0.2014 & 0.2071 & 0.4859 & 0.5000 \\
5 & 2.500 & 2.500 & 0.0943 & 0.0984 & 0.7814 & 0.7807 \\
10 & 5.000 & 5.000 & 0.0316 & 0.0339 & 0.9306 & 0.9279 \\
20 & 10.000 & 10.000 & 0.0043 & 0.0049 & 0.9908 & 0.9899 \\
50 & 25.000 & 25.000 & 0.0000 & 0.0000 & 1.0000 & 1.0000 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.584 & 0.500 & 0.2522 & 0.2819 & 0.2712 & 0.2819 \bigstrut[t]\\
2 & 1.086 & 1.000 & 0.1809 & 0.2071 & 0.5020 & 0.5000 \\
5 & 2.558 & 2.500 & 0.0837 & 0.0984 & 0.7911 & 0.7807 \\
10 & 5.024 & 5.000 & 0.0286 & 0.0339 & 0.9335 & 0.9279 \\
20 & 10.006 & 10.000 & 0.0040 & 0.0049 & 0.9912 & 0.9899 \\
50 & 25.000 & 25.000 & 0.0000 & 0.0000 & 1.0000 & 1.0000 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.158 & 0.250 & 0.3308 & 0.3415 & 0.1006 & 0.1103 \bigstrut[t]\\
2 & 0.397 & 0.500 & 0.2973 & 0.2819 & 0.2465 & 0.2819 \\
5 & 1.164 & 1.250 & 0.1952 & 0.1807 & 0.5482 & 0.5724 \\
10 & 2.447 & 2.500 & 0.1036 & 0.0984 & 0.7729 & 0.7807 \\
20 & 4.980 & 5.000 & 0.0340 & 0.0339 & 0.9283 & 0.9279 \\
50 & 12.497 & 12.500 & 0.0017 & 0.0019 & 0.9964 & 0.9960 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and deterministic jump sizes, $B\equiv 1$ and $D\equiv 1$.}
\label{tab:accuracy_deterministic}
\end{table}
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.2929 & 0.3415 & 0.3536 & 0.3925 \\
2 & 1.000 & 1.000 & 0.2500 & 0.2819 & 0.5000 & 0.5135 \\
5 & 2.500 & 2.500 & 0.1642 & 0.1807 & 0.7062 & 0.7009 \\
10 & 5.000 & 5.000 & 0.0898 & 0.0984 & 0.8491 & 0.8418 \\
20 & 10.000 & 10.000 & 0.0307 & 0.0339 & 0.9506 & 0.9467 \\
50 & 25.000 & 25.000 & 0.0017 & 0.0019 & 0.9974 & 0.9970 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.621 & 0.500 & 0.2589 & 0.3415 & 0.3705 & 0.3925 \bigstrut[t]\\
2 & 1.153 & 1.000 & 0.2164 & 0.2819 & 0.5224 & 0.5135 \\
5 & 2.656 & 2.500 & 0.1414 & 0.1807 & 0.7254 & 0.7009 \\
10 & 5.113 & 5.000 & 0.0784 & 0.0984 & 0.8598 & 0.8418 \\
20 & 10.050 & 10.000 & 0.0275 & 0.0339 & 0.9538 & 0.9467 \\
50 & 25.004 & 25.000 & 0.0016 & 0.0019 & 0.9975 & 0.9970 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Exact & \eqref{eq:approx1} & Exact & \eqref{eq:approx2} & Exact & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.125 & 0.250 & 0.3548 & 0.3864 & 0.2168 & 0.2942 \bigstrut[t]\\
2 & 0.333 & 0.500 & 0.3333 & 0.3415 & 0.3333 & 0.3925 \\
5 & 1.059 & 1.250 & 0.2647 & 0.2593 & 0.5264 & 0.5570 \\
10 & 2.333 & 2.500 & 0.1856 & 0.1807 & 0.6881 & 0.7009 \\
20 & 4.893 & 5.000 & 0.0995 & 0.0984 & 0.8400 & 0.8418 \\
50 & 12.475 & 12.500 & 0.0198 & 0.0206 & 0.9692 & 0.9678 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and exponentially distributed jump sizes, $B\sim \exp(1)$ and $D\sim\exp(1)$.}
\label{tab:accuracy_exponential}
\end{table}
\begin{table}
\centering
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.500 & 0.500 & 0.3118 & 0.3981 & 0.4412 & 0.4636 \\
2 & 1.000 & 1.000 & 0.2894 & 0.3575 & 0.5343 & 0.5288 \\
5 & 2.500 & 2.500 & 0.2375 & 0.2819 & 0.6590 & 0.6381 \\
10 & 5.000 & 5.000 & 0.1785 & 0.2071 & 0.7592 & 0.7385 \\
20 & 10.000 & 10.000 & 0.1090 & 0.1241 & 0.8593 & 0.8454 \\
50 & 25.000 & 25.000 & 0.0303 & 0.0339 & 0.9624 & 0.9583 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 1$, $\xi_d=1$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.667 & 0.500 & 0.2695 & 0.3981 & 0.4636 & 0.4636 \bigstrut[t]\\
2 & 1.253 & 1.000 & 0.2469 & 0.3575 & 0.5632 & 0.5288 \\
5 & 2.863 & 2.500 & 0.2009 & 0.2819 & 0.6895 & 0.6381 \\
10 & 5.385 & 5.000 & 0.1518 & 0.2071 & 0.7834 & 0.7385 \\
20 & 10.328 & 10.000 & 0.0938 & 0.1241 & 0.8739 & 0.8454 \\
50 & 25.124 & 25.000 & 0.0269 & 0.0339 & 0.9658 & 0.9583 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b =1$, $\xi_d=2$.}
\end{subtable}
\vspace{5mm}
\begin{subtable}{0.99\textwidth}\centering
\begin{tabular}{|r|rr|rr|rr|}
\cline{2-7}\multicolumn{1}{r|}{} & \multicolumn{2}{c|}{$\mathbb{E}[X_n]$} & \multicolumn{2}{c|}{$\pi_d$} & \multicolumn{2}{c|}{$\mathbb{P}({\rm dem.sat.})$} \bigstrut\\
\hline
$n$ & Sim. & \eqref{eq:approx1} & Sim. & \eqref{eq:approx2} & Sim. & \eqref{eq:approx3} \bigstrut\\
\hline
1 & 0.081 & 0.250 & 0.3694 & 0.4276 & 0.3270 & 0.4104 \bigstrut[t]\\
2 & 0.238 & 0.500 & 0.3593 & 0.3981 & 0.4137 & 0.4636 \\
5 & 0.857 & 1.250 & 0.3237 & 0.3415 & 0.5311 & 0.5528 \\
10 & 2.045 & 2.500 & 0.2739 & 0.2819 & 0.6282 & 0.6381 \\
20 & 4.568 & 5.000 & 0.2039 & 0.2071 & 0.7361 & 0.7385 \\
50 & 12.231 & 12.500 & 0.0966 & 0.0984 & 0.8797 & 0.8779 \bigstrut[b]\\
\hline
\end{tabular}%
\caption{$\xi_b = 2$, $\xi_d=1$.}
\end{subtable}
\caption{Accuracy of diffusion approximation for the blood inventory process $\mathbb{E}[X_n]$, the probability of negative inventory $\pi_d$ and the probability of demand being fully satisfied $\mathbb{P}(dem.sat)$, with arrival rates $n\lambda_b = n$ and $n\lambda_d = 0.5n$ and Gamma distributed jump sizes, $B\sim\text{Gamma}(0.25,0.25)$ and $D \sim\text{Gamma}(0.25,0.25)$.}
\label{tab:accuracy_gamma}
\end{table}
\section{Conclusions \& suggestions for further research}
\label{conclus}
In this chapter, we studied a stochastic model for a blood bank.
We have presented a global approach to the model in its full generality,
and obtained very detailed exact expressions for the densities of amount of inventory and amount of demand (shortage)
in special cases (exponential amounts of donated and requested blood; and either $\xi_b=\xi_d=0$ or $\alpha_b=\alpha_d=0$).
Moreover, we have shown how an appropriate scaling, for the model in full generality, leads to an Ornstein-Uhlenbeck diffusion process,
which can be used as a tool to obtain simple yet accurate approximations for some key performance measures.
Our model is a two-sided model, in the sense that we simultaneously consider
the amount of blood in inventory and the amount of demand (shortage), one of the two at any time being zero.
Such two-sided processes arise in many different settings, and thus are of considerable interest.
The present setting is reminiscent of an organ transplantation problem, where there is
either a queue of persons waiting to receive an organ,
or a queue of donor organs. The perishability/impatience aspect features there too \cite{Boxma2011}.
A quite different setting is that of insurance risk. We refer to Albrecher \& Lautscham \cite{Albrecher2013}
who extend the classical Cram\'er-Lundberg insurance risk model by allowing the capital of an insurance company
to become negative -- a situation that is usually indicated by ``ruin" in the insurance literature.
Their process thus becomes two-sided. The capital might become positive again; however,
at a rate $\omega(x)$ when the capital has a negative value $-x$, bankruptcy is declared and the process ends.
Interestingly, similar special functions (like hypergeometric functions) play a role in \cite{Albrecher2013} and in the present study.
The analyses performed in this chapter, which evolved around a simplified version of the inventory process of a blood bank, revealed some interesting avenues for further research.
We name a couple of them.
First, we remark that our results are restricted to one type of blood.
Naturally, it would be very interesting to extend the analysis to multiple types of blood.
Another important extension would be to use our results to facilitate the decision process that is faced by the CBB on a daily basis:
Which amounts of blood, and of which types, should today be sent to the local blood banks (hospitals)?
Knowing that, e.g., blood types $O^-,A^-,B^-,AB^-$ can satisfy the corresponding $+$ type (but not vice versa),
one may try to optimize the blood allocation process on the basis of actual amounts of blood present.
Finally, we mention a significant open research question regarding the process limits that we derived in Section \ref{sectionscaling}, of which the steady-state distributions were used to approximate steady-state performance measures in pre-limit systems.
As we pointed out earlier, the justification that the steady-state distribution of the scaled inventory process indeed converges to the steady-state distribution of the fluid (cq.~diffusion) limit requires a rigorous argument why the order of limits $n\to\infty$ and $t\to \infty$ may be interchanged.
Proving interchange-of-limits statements typically raises many technical challenges, see e.g.~\cite{Dai2014a,Gamarnik2013a,Gurvich2013,Gamarnik2006} for works tackling this issue in the context of queues in heavy traffic.
The usual approach is to prove tightness of the sequence of steady-state distributions of pre-limit, followed by applying Prokhorov's theorem, see e.g.~\cite[Sec.~1.5]{Billingsley1995}.
For our model, such an approach seems to be straightforward for the fluid scaling, since our inventory process can be upper (cq.~lower) bounded by a shot-noise process with only positive (cq.~negative) jumps.
Of the latter, the steady-state behavior is known.
This allows us to derive a uniform bound on the absolute mean of the stationary fluid-scaled process, which gives tightness.
The final step uses the deterministic nature of the differential equation governing the dynamics of the fluid limit, by which the steady-state distribution must be unique.
For the diffusion-scaled process, the steps towards proving the interchange-of-limits are not obvious and hence this needs further investigation.
Our numerical results for various jump size distributions, however, support the conjecture that this interchange is indeed valid.
\section*{Appendix}
\addcontentsline{toc}{section}{\hspace{7.1mm} Appendix}
\begin{subappendices}
\settocdepth{chapter}
\section{Transformation integral equation}
\label{app:transformation_int}
In this appendix we show how integral equation (\ref{eq:demand2}) can be transformed into a second-order differential equation,
in the case of exponential $F_b(\cdot)$ and $F_d(\cdot)$.
Differentiate (\ref{eq:demand2}) w.r.t.\ $v$:
\begin{align}
& \lambda_d f(v) - \mu_d \left[\lambda_d \int_0^v f(y) {\rm e}^{-\mu_d(v-y)} {\rm d}y +
\lambda_d \int_0^{\infty} g(y) {\rm e}^{-\mu_d (v+y)} {\rm d}y\right]
\nonumber
\\
&\qquad =
-\lambda_b f(v) + \lambda_b \mu_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y + \xi_d f(v) +\xi_d
v f'(v) .
\label{eq:demand3}
\end{align}
Using (\ref{eq:demand2}) once more, now to replace the term between square brackets in (\ref{eq:demand3}),
we get:
\begin{align}
\xi_d v f'(v) &= (\lambda_d +\lambda_b - \xi_d) f(v)
\nonumber
\\
&\qquad - \mu_d \left(\lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y + \xi_d v f(v)\right)
\nonumber
\\
&\qquad \qquad - \mu_b \lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y,
\label{eq:demand4}
\end{align}
and once more differentiating w.r.t.\ $v$ then gives:
\begin{align}
&\xi_d v f''(v) + \xi_d f'(v) - (\lambda_d +\lambda_b -\xi_d -\mu_d \xi_d v) f'(v)
\nonumber
\\
& \qquad = -\mu_d \xi_d f(v) +(\mu_b+\mu_d) \lambda_b f(v) -\mu_b(\mu_b+\mu_d) \lambda_b \int_v^{\infty} f(y) {\rm e}^{-\mu_b(y-v)} {\rm d}y .
\label{eq:demand5}
\end{align}
The integral that appears in \eqref{eq:demand4} can be eliminated by using (\ref{eq:demand5}),
and we thus finally obtain the following second order homogeneous differential equation:
\begin{align}
&\xi_d v f''(v) + \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber\\
& +\left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0.
\end{align}
\section{Proof of Proposition \ref{densityProp}}
\label{app:proof_prop_density}
In the proof, we concentrate on the derivation of $f(v)$, which is the solution to
\begin{align}
\xi_d v f''(v) &+ \left(2\xi_d -\lambda_d -\lambda_b + \mu_d\xi_dv -\mu_b \xi_d v\right)f'(v) \nonumber \\
& \qquad + \left(\mu_d\xi_d -\mu_b\xi_d -\mu_d\lambda_b + \mu_b \lambda_d -\mu_b \mu_d \xi_d v\right)f(v) =0
\label{eq:demand6_1}
\end{align}
The expression for $g(v)$ follows directly from exchanging $\lambda_b$ with $\lambda_d$, $\mu_b$ with $\mu_d$, $\xi_b$ with $\xi_d$, and $\pi_b$ with $\pi_d$ in $f(v)$.
We rewrite \eqref{eq:demand6_1} as follows:
\begin{equation}\label{diffvgl}
vf''(v) + (A+Bv)f'(v)+(C+Dv)f(v)=0,
\end{equation}
where
\begin{equation*}
A = 2-\frac{\lambda_b+\lambda_d}{\xi_d}, ~~
B = \mu_d-\mu_b, ~~
C = \mu_d-\mu_b + \frac{\lambda_d\mu_b-\lambda_b\mu_d}{\xi_d}, ~~
D = -\mu_b\mu_d.
\end{equation*}
Note that we divided both sides of equation \eqref{eq:demand6_1} by $\xi_d$ here.
We will try to transform the differential equation into one for which the solution is easily derived. In order to do so, we first guess $f$ to be of the form $f(v) = {\rm e}^{\beta v}h(v)$, where $\beta$ is a constant and $h$ another real-valued function. Substituting this into \eqref{diffvgl} gives
\begin{equation}\label{eq:diff2}
v h''(v) + \left[ (2\beta +B)v + A\right]h'(v) + \left[(\beta^2+B\beta+D)v+ A\beta+C\right] h(v)=0.
\end{equation}
Next, we would like to choose $\beta$ such that $\beta^2+B\beta+D=0$, that is
\begin{equation*}
\beta = \frac{-B \pm \sqrt{B^2-4D}}{2},
\end{equation*}
which equals either $-\mu_d$ or $\mu_b$. Since the solution of \eqref{diffvgl} we are looking for is a density, and necessarily $f(v) = {\rm e}^{\beta v}h(v) \rightarrow 0 $ as $v\rightarrow \infty$, we set $\beta$ equal to the negative root $-\mu_d$. Lastly, we apply a change of variable, $x = \delta v$, and $h(v) = w(x)$, so that \eqref{eq:diff2} is transformed into
\begin{equation*}
x w''(x) + \left[ (2\beta+B)\delta^{-1} x+A\right] w'(x) + \delta^{-1}\left[A\beta+C\right] w(x) = 0.
\end{equation*}
By choosing $(2\beta+B)\delta^{-1} = -1$, i.e.
\begin{equation*}
\delta = {-}(2\beta+B) = \mu_b+\mu_d,
\end{equation*}
we obtain
\begin{equation*}
x w''(x) +[ A - x ] w'(x) + \delta^{-1}\left[A\beta+C\right] w(x) = 0,
\end{equation*}
which is known as Kummer's equation, $x w''(x) + (b-x) w'(x) - aw(x) = 0$, see \cite{Slater1960}, with parameters
\begin{align*}
a &= -\delta^{-1}\left[A\beta+C\right] = 1-\frac{\lambda_d}{\xi_d},\\
b &= A = 2-\frac{\lambda_b+\lambda_d}{\xi_d}.
\end{align*}
Kummer's equation has two linearly independent solutions, namely $w(x) =$\\ \noindent $M(a,b,x)$, where $M$ is Kummer's hypergeometric function, also denoted by \\ \noindent $ _1F_1(a,b,x)$, and $U(a,b,x)$, Tricomi's hypergeometric function. These are defined as, see \cite[Eq.~(1.3.1)]{Slater1960},
\begin{align*}
M(a,b,x) &= \sum_{n=0}^\infty \frac{(a)_n}{(b)_n n!} x^n,\\
U(a,b,x) &= \frac{\Gamma(b-1)}{\Gamma(1+a-b)}\,M(a,b,x) + \frac{\Gamma(b-1)}{\Gamma(a)}\,x^{1-b}\,M(1+a-b,2-b,x),
\end{align*}
where $(.)_n$ is the Pochhammer symbol, which is used to represent $(y)_n = y\cdot(y+1)\cdot...\cdot (y+n-1)$.
We can therefore deduce that $f(v)$ is of the form
\begin{equation*}
{\rm e}^{\beta v}\left[ c_1\, M(a,b,\delta v) + c_2\, U(a,b,\delta v)\right],
\end{equation*}
or
\begin{equation*}
{\rm e}^{-\mu_d v}\left[ c_1 M\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d) v\right) +
c_2 U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d) v\right)\right],
\end{equation*}
where $c_1$ and $c_2$ are constants. From \cite[p.~60]{Slater1960}, we have
\begin{equation*}
M(a,b,x) \sim \frac{\Gamma(b)}{\Gamma(a)}{\rm e}^x x^{a-b}, \qquad \text{as } x\rightarrow \infty.
\end{equation*}
Hence,
\begin{align*}
&{\rm e}^{-\mu_d v} M\left( 1-\frac{\lambda_d}{\xi_d}, 2-\frac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right) \\
&\qquad \qquad \sim
\frac{\Gamma(2-\frac{\lambda_b+\lambda_d}{\xi_d})}{\Gamma(1-\frac{\lambda_d}{\xi_d})}{\rm e}^{\mu_b v}\left((\mu_b+\mu_d)v\right)^{\lambda_b/\xi_d-1}
\to \infty
\end{align*}
for all $\mu_b>0$, which leads us to conclude $c_1 = 0$. We deduce $c_2$ by exploiting the restriction that
\begin{equation*}
\int_0^\infty f(v)\, {\rm d} v = \pi_d,
\end{equation*}
where $\pi_d$ is the probability of positive demand. Hence
\begin{equation*}
\pi_d c_2^{-1} = \int_0^\infty {\rm e}^{-\mu_d v} U\left( 1-\tfrac{\lambda_d}{\xi_d}, 2-\tfrac{\lambda_b+\lambda_d}{\xi_d},(\mu_b+\mu_d)v\right)\, dv.
\end{equation*}
By slightly transforming \cite[Eq.~(3.2.51)]{Slater1960}, we find
\begin{equation*}
c_2^{-1} = \frac{1}{\pi_d}\,\frac{\Gamma\left(\frac{\lambda_b+\lambda_d}{\xi_d}\right)}{\Gamma\left(1+\tfrac{\lambda_b}{\xi_d}\right) }\, _2F_1\left(1-\tfrac{\lambda_d}{\xi_d},1,1+\tfrac{\lambda_b}{\xi_d},-\tfrac{\mu_b}{\mu_d}\right)
,
\end{equation*}
where $_2F_1(a_1,a_2,a_3,x) := \sum_{n=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(a_3)_n n!} x^n$
is the hypergeometric function of Gauss.
\section{Laplace Transforms for Coxian jumps}
\label{app:coxian}
We outline how the differential equation (\ref{diffeq}) is obtained.
We take Laplace transforms in (\ref{eq:demand}), considering its five terms and calling them $T_1, T_2, T_3, T_4$ and $T_5$, successively.
Equation (\ref{eq:demand}) then translates into
\begin{equation*}
T_1 + T_2 +T_3 = T_4 + T_5,
\label{Tequ}
\end{equation*}
where
\begin{align}
T_1
&= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^v f(y) \bar{F}_d(v-y) {\rm d}y {\rm d}v \nonumber \\
&=
\lambda_d \phi(s) \frac{1 - \mathbb{E}[{\rm e}^{-sD}]}{s},
\label{T-1}\\
T_2
&= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y {\rm d}v \nonumber\\
&= \lambda_d \int_{y=0}^{\infty} {\rm e}^{sy} g(y) \int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_d(z) {\rm d}z {\rm d}y ,
\label{T-2}\\
T_3
&= \pi_0 \lambda_d \int_0^{\infty} {\rm e}^{-sy} \bar{F}_d(y) {\rm d}y ,\\
T_4
&= \lambda_b \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=v}^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y {\rm d}v \nonumber\\
&= \lambda_b \int_{y=0}^{\infty} {\rm e}^{-sy} f(y) \int_{z=0}^{y} {\rm e}^{sz} \bar{F}_b(z) {\rm d}z {\rm d}y ,
\label{T-4}\\
T_5
&= \xi_d \int_{v=0}^{\infty} v {\rm e}^{-sv} f(v) {\rm d}v + \alpha_d \phi(s) \nonumber\\
&= - \xi_d \phi'(s) + \alpha_d \phi(s) .
\label{T-5}
\end{align}
We now evaluate the terms appearing in the righthand sides of (\ref{T-1})-(\ref{T-4}) for the Coxian case
of (\ref{Fbarb}) and (\ref{Fdarb}):
\begin{align}
\int_{z=0}^y {\rm e}^{sz} \bar{F}_b(z) {\rm d}z &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j-s} (1 - {\rm e}^{(s-\beta_j)y}),
\label{hulp1}\\
\int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_b(z) {\rm d}z &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j+s} {\rm e}^{-(s+\beta_j)y} ,
\label{hulp2}\\
\mathbb{E}[{\rm e}^{-sB}] &=
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{\beta_j}{\beta_j+s} ,
\label{hulp3}
\end{align}
and hence
\begin{equation}
\frac{1-\mathbb{E}[{\rm e}^{-sB}]}{s} =
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j+s} .
\label{hulp4}
\end{equation}
Combining \eqref{Tequ} with \eqref{T-1}-\eqref{T-5}, and using \eqref{hulp1} and the counterparts of \eqref{hulp2} and \eqref{hulp4}
for $\bar{F}_d(\cdot)$, we find:
\begin{align}
& \lambda_d \phi(s)
\sum_{i=1}^K q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\frac{1}{\delta_j+s}
\nonumber
\\
&\qquad + \lambda_d
\sum_{i=1}^L q_i \prod_{h=1}^{i-1} (1-q_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\delta_l}{\delta_l - \delta_j}
\frac{1}{\delta_j+s} [\gamma(\delta_j) + \pi_0]
\nonumber
\\
&= \lambda_b
\sum_{i=1}^K p_i \prod_{h=1}^{i-1} (1-p_h) \sum_{j=1}^i \prod_{l=1; l \neq j}^i \frac{\beta_l}{\beta_l - \beta_j}
\frac{1}{\beta_j-s} (\phi(s) - \phi(\beta_j))
\nonumber
\\
&\qquad - \xi_d \phi'(s) + \alpha_d \phi(s),
\label{eq:3star}
\end{align}
which is readily rewritten into \eqref{diffeq}.
\begin{remark}
If $\xi_d=0$, then $\phi(s)$ is obtained from \eqref{eq:3star}
in a standard manner, see also Section~\ref{sectionvariant}.
\end{remark}
\begin{remark}
We now outline how \eqref{hulp2} and \eqref{hulp3} change when the $B_i$ have an Erlang-($l+1,\beta$) distribution,
and when the $D_i$ have an Erlang-($k+1,\delta$) distribution (see also \eqref{Fbarb} and the line below it);
\eqref{hulp1} and \eqref{hulp4} do not change (but of course $\mathbb{E}[{\rm e}^{-sD}]$ changes).
Firstly,
\begin{equation*}
\int_{z=0}^y {\rm e}^{sz} \bar{F}_b(z) {\rm d}z =
\sum_{j=0}^l \frac{\beta^j}{(\beta-s)^{j+1}}
\left[1 - \sum_{i=0}^j {\rm e}^{-(\beta-s)y} \frac{((\beta-s)y)^i}{i!}\right] .
\end{equation*}
Term $T_4$ now becomes:
\begin{align*}
T_4 &= \lambda_b \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=v}^{\infty} f(y) \bar{F}_b(y-v) {\rm d}y {\rm d}v\nonumber\\
&= \lambda_b \sum_{j=0}^l \frac{\beta^j}{(\beta-s)^{j+1}}
\left[\phi(s) - \sum_{i=0}^j \frac{(\beta-s)^i}{i!} \int_{y=0}^{\infty} y^i {\rm e}^{-\beta y} f(y) {\rm d}y \right].
\end{align*}
It should be noted that $s = \beta$ is a removable singularity. E.g., for $l=0$ one has
$T_4 = \lambda_b \frac{\phi(s) - \phi(\beta)}{\beta - s}$.
\\
Secondly,
\begin{equation*}
\int_{z=y}^{\infty} {\rm e}^{-sz} \bar{F}_b(z) {\rm d}z =
\sum_{j=0}^k \frac{\delta^j}{(s+\delta)^{j+1}} \sum_{i=0}^j {\rm e}^{-(s+\delta)y} \frac{((s+\delta)y)^i}{i!} .
\end{equation*}
Term $T_2$ now becomes:
\begin{align*}
T_2 &= \lambda_d \int_{v=0}^{\infty} {\rm e}^{-sv} \int_{y=0}^{\infty} g(y) \bar{F}_d(v+y) {\rm d}y\, {\rm d}v \nonumber\\
&=
\lambda_d
\sum_{j=0}^k \frac{\delta^j}{(s+\delta)^{j+1}} \sum_{i=0}^j \frac{(s+\delta)^i}{i!}
\int_{y=0}^{\infty} y^i {\rm e}^{-\delta y} g(y)\, {\rm d}y .
\end{align*}
It is readily seen that the resulting counterpart of \eqref{eq:3star} can again be written in the form \eqref{diffeq},
and hence the solution is formally still given by \eqref{diffeqsoln}.
\label{RmErlang}
\end{remark}
\resettocdepth
\end{subappendices}
\chapter{Introduction}
\begin{chapterstart}
Stochastic service systems describe settings in which users compete for service from scarce resources. Think of check-in lines at airports, waiting rooms in hospitals or queues in supermarkets, where the scarce resources are human manpower.
Next to these traditional settings, our increasingly digitalized society creates quite different types of resource sharing systems, such as the internet, wireless networks and cloud computing facilities.
In these virtual environments, geographical location does not play a restricting role on the system size, paving the way for the emergence of large-scale resource sharing networks.
This thesis serves to explain how to analyze and dimension large-scale systems in order to achieve economies-of-scale, by which we mean that the system is highly occupied and hence utilizes efficiently the expensive resources, while at the same time, the offered service levels remain high.
In this chapter, we give an overview of the available machinery that supports such principles and explain how this thesis contributes to the existing study of large-scale service systems. The fundamental law behind these mathematical techniques is the Central Limit Theorem (CLT) -- arguably one of the most important theorems in mathematics and science.
\end{chapterstart}
\newpage
\section{Service systems and queueing theory}
\subsection{Quality vs. Efficiency}
Large-scale service systems take many shapes and forms.
Classical examples of large-scale service systems include call centers \cite{Erlang1917,Palm1957,Whitt1999,Gans2003,Borst2004,Brown2005,Zeltyn2005,Bassamboo2009,Khudyakov2006} and communication systems \cite{Kleinrock1976,Anick1982,Kelly1985,Kleinrock2007,johanthesis}.
More recently, congestion-related issues in health care facilities and cloud-computing facilities have received much attention \cite{Armony2015,Green2007,YomTov2010,Gupta2007,Tan2012}.
In all settings, one can think of service systems as being composed of \textit{customers} and \textit{servers}.
In call centers, customers typically call to request help from one of the center's agents (servers).
In communication networks, the data packets are the customers and the communication channels are the servers.
In health care facilities, patients are the customers, and nurses/physicians are the servers.
The system scale may refer to either the size of the client base it caters to, or the magnitude of its capacity, or both, as is frequently the case.
Next tot the central notions of customers and servers, we view service systems are inherently stochastic, that is, subject to uncertainty.
Although arrival volumes can be anticipated to some extend over a certain planning horizon, for instance through historical data and forecasting methods, one cannot predict with certainty future arrival patterns.
Moreover, service requirements are typically random as well, adding more uncertainty.
This intrinsic stochastic variability is a predominant cause of delay experienced by customers in the system.
Due to the inherent randomness in both their arrival and service processes, stochastic models have proved instrumental in both in quantifying and improving the operational performance of service systems.
Queueing theory and stochastics provide the tools and machinery to describe and evaluate these service systems.
Queueing models are often able to capture and explain fundamental phenomena that are common across applications.
When evaluating the performance of service systems, an important model is the $M/GI/s$ queue, which we will refer to as the \textit{many-server} queue.
This model assumes that customers arrive to the queue according to a Poisson process of rate $\lambda$, and customer service times are mutually independent and identically distributed (i.i.d.) samples from the distribution of a non-negative random variable $B$.
The parameter $s$ represents the number of servers in the system, and hence restricts the number of simultaneous services.
In this thesis we restrict attention to the policy in which customers are handled \textit{first-come-first-served}.
In case $s=1$, we speak of a single-server queue.
First principles say that the queueing process is stable, that is, the number of customers does not explode as time evolves, if and only if the expected workload $R := \lambda\mathbb{E}[B]$ brought into the system per time unit is strictly less than the system capacity.
In other words, the \textit{utilization} of the queue, defined as $\rho := \lambda\mathbb{E}[B] / s$ should remain strictly below 1.
Naturally, a system manager prefers to operate at a utilization level close to 1, so that resources are used efficiently.
However, it is known that pushing the occupation levels to 100\% leads to an explosive increase in congestion, thereby reducing the quality of service (QoS) and also customer satisfaction.
These seemingly conflicting objectives give rise to a classical trade-off between customer satisfaction and costs of resources.
\subsection{Economies-of-scales}
Under the assumption that service times are exponentially distributed with mean $1/\mu$, the many-server queue reduces to the well-studied $M/M/s$ queue.
Despite its simplicity, the analysis of the $M/M/s$ queue explains mathematically the distinctive traits of queues in general, such as the non-linear effect of utilization on the queue size, and pooling effects.
Let $W^{(s)}$ denote the waiting time of a customer and $Q^{(s)}$ the queue length (including the customers in service) in the steady-state $M/M/s$ queue. Without loss of generality, we fix $\mu=1$.
A straightforward balance argument gives the stationary distribution:
\begin{equation}
\label{eq:MMs_stationary_distribution}
\pi_k := \mathbb{P}( Q^{(s)} = k )
= \left\{
\begin{array}{ll}
\pi_0\frac{R^k}{k!}, & \text{if } k\leq s, \\
\pi_0\frac{R^s}{s!}\,\rho^{k-s} & \text{if } k > s,
\end{array}
\right.
\end{equation}
where
\begin{equation*}
\pi_0 := \left( \sum_{k=0}^s \frac{R^k}{k!} + \frac{\rho}{1-\rho} \frac{R^s}{s!}\right)^{-1}.
\end{equation*}
Natural QoS indicators include the expected waiting time $\mathbb{E}[W^{(s)}]$ and the delay probability $\mathbb{P}(W^{(s)}>0)$.
Invoking Little's law and the PASTA (Poisson arrivals see time averages) property \cite{Wolff1982}, it follows that
\begin{equation}
\label{eq:MMs_wait}
\mathbb{P}(W^{(s)} > 0) = \frac{R^s}{s!} \left( (1-\rho) \sum_{k=0}^{s-1} \frac{R^k}{k!} + \frac{R^s}{s!} \right)^{-1},
\quad
\mathbb{E}[W^{(s)}] = \mathbb{P}(W^{(s)} > 0)\,\frac{1/s}{1-\rho}.
\end{equation}
From these formulae, it is readily seen that $\mathbb{P}(W^{(s)} > 0) \to 1$ and $\mathbb{E}[W^{(s)}] \to \infty$ as $\rho \uparrow 1$ . That is, increasing $\lambda$ to $s$, while keeping the latter fixed, leads to a system in which all customers are delayed before service, and the expected delay before reaching a server increases to infinity.
The $M/M/s$ queue also reveals the effect of \textit{resource pooling}.
To illustrate the operational benefits of sharing resources, we compare a system of $s$ separate $M/M/1$ queues, each serving a Poisson arrival stream with rate $\lambda<1$, against one $M/M/s$ queue facing arrival rate $\lambda s$.
The two systems thus experience the same workload and utilization, namely $\rho = \lambda$.
We fix the value of $\lambda$ and vary $s$.
Obviously, the waiting time and queue length distribution in the first scenario are unaffected by the parameter $s$, since there is no interaction between the single-server queues.
This lack of coordination allows for the possibility of having an idle server, while the total number of customers in the system exceeds $s$, therefore wasting resource capacity.
Such an event cannot happen in the many-server scenario, due to the central queue.
This central coordination improves QoS. Indeed Figure \ref{fig:waiting_time_pooling} shows that the reduction in expected waiting time can be substantial.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\small \centering
\input{./tikz_tex/Ewait_pooling.tex}
\caption{Expected waiting time}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\input{./tikz_tex/Pwait_pooling.tex}
\caption{Probability of delay}
\end{subfigure}
\caption{Effects of resource pooling in the $M/M/s$.}
\label{fig:waiting_time_pooling}
\end{figure}
So pooling kills two birds with one stone: QoS for customers improves and the system efficiency increases.
\subsection{Many-server scaling regimes}
\label{sec:intro_many_server_regimes}
Now that we know that economies-of-scale can be achieved, it is relevant to ask how to match capacity $s$ to a demand $\lambda$ in the setting where both $s$ and $\lambda$ become large.
The expressions in \eqref{eq:MMs_wait} provide a starting point for finding such demand-capacity relations, particularly when we apply asymptotic analysis for $s\to\infty$, \cite{Halfin1981,Borst2004,Reed2009}.
Asymptotic theory of many-server systems relies on the prerequisite that the limiting behavior of the service system is determined by the way in which capacity $s$ is adjusted to demand, assuming demand grows large.
We illustrate this idea by investigating typical sample paths of the queue length process $Q = \{Q(t),t\geq 0\}$ of an $M/M/s$ queue for increasing values of $\lambda$.
Sample paths of queueing processes are insightful, because congestion and server utilization are both visualized.
As an example, Figure \ref{fig:sample_path_small} depicts a sample path for $\lambda = 3$ and $s = 4$.
The number of customers queueing at time $t$ is given by $(Q(t)-s)^+$ with $(\cdot)^+ := \max\{0,\cdot\}$.
The number of idle servers is given by $(s-Q(t))^+$.
In Figure \ref{fig:sample_path_small}, the red and green area hence represent the cumulative queue length and cumulative number of idle server, respectively, over the given time period.
Bearing in mind the dual goals of QoS and efficiency, we want to minimize both of these areas simultaneously.
\begin{figure}[b!]
\centering
\input{./tikz_tex/sample_path_small.tex}
\caption{Sample path of the $M/M/s$ queue with $\lambda = 3$ and $s=4$.}
\label{fig:sample_path_small}
\end{figure}
Next, we conduct a similar sample path experiment for increasing values of $\lambda$.
Since $s > \lambda$ is required for stability, the value of $s$ needs to be adjusted accordingly.
We propose three scaling rules:
\begin{equation}
\label{eq:intro_three_scaling_rules}
s^{(1)}_\lambda = \left[ \lambda + \beta \right ], \qquad
s^{(2)}_\lambda = \left[ \lambda + \beta\sqrt{\lambda} \right], \qquad
s^{(3)}_\lambda = \left[ \lambda + \beta\,\lambda \right],
\end{equation}
for some $\beta>0$, where $[\cdot]$ denotes the rounding operator.
Note that these three rules differ in terms of increasing overcapacity $s-\lambda$.
Figure \ref{fig:sample_paths_lambda100} depicts typical sample paths of the queue length process for increasing values of $\lambda$ for the three scaling rules with $\beta = 0.5$.
\begin{figure}
\centering
\input{./tikz_tex/sample_paths_lambda10.tex}
\caption{Sample paths of the $M/M/s$ queue with $\lambda = 10,50$ and $100$ and $s$ set according to the three scaling rules in \eqref{eq:intro_three_scaling_rules}.}
\label{fig:sample_paths_lambda100}
\end{figure}
Observe that for all scaling rules, the stochastic fluctuations of the queue length processes relative to $\lambda$ decrease with the size of the system.
Moreover, the paths in Figure \ref{fig:sample_paths_lambda100} appear to become increasingly continuous in nature with increasing $\lambda$.
Of course, the actual sample path always consists of upwards and downward jumps of size 1, but we will show how proper centering and scaling of the queue length process indeed gives rise to a \textit{diffusion process} in the limit as $\lambda\to\infty$.
Although the difference in performance of the three regimes is not yet evident for relatively small $\lambda$, clear distinctive behavior occurs for large $\lambda$.
Under ${s_\lambda}^{(1)}$, the majority of customers is delayed and server idle time is low, since $\rho = (1+\beta/\lambda)^{-1} \to 1$ as $\lambda \to \infty$.
Systems dimensioned according to this rule value server efficiency over customer satisfaction and therefore this regime is in the literature also known as the \textit{efficiency-driven} (ED) regime \cite{Zeltyn2005}.
In contrast, the third scaling rule $s^{(3)}$ yields a constant utilization level $\rho = 1/(1+\beta)$, which stays away from 1, even for large $\lambda$.
Queues operating in this regime exhibit significant server idle times.
Moreover, for the particular realization of the queueing processes for $\lambda = 50$ and $\lambda=100$ none of the customers waits.
This customer-centered regime is known as the \textit{quality-driven} (QD) regime \cite{Zeltyn2005}.
The scaling rule $s^{(2)}_\lambda$ is in some ways a combination of the other two regimes.
First, we have $\rho = (1 +\beta/\sqrt{\lambda})^{-1} \to 1$ as $\lambda \to \infty$, which indicates efficient usage of resources as the system grows.
Nonetheless, the sample paths indicate that only a fraction of the customers is delayed, and if a queue is present, it seems to be of moderate size, which suggest good QoS.
This regime is therefore called \textit{quality-and-efficiency driven} (QED) regime.
Since this scaling regime and the related \textit{square-root staffing rule}
\begin{equation}
\label{eq:square_root_staffing rule}
s = \lambda + \beta\sqrt{\lambda}
\end{equation}
strikes the right balance between the two profound objectives of capacity allocation in service systems, we discuss in the next section the mathematical foundations of the QED regime and quantify the favorable properties revealed by Figure \ref{fig:sample_paths_lambda100}.
\section{The QED regime: two canonical examples}
\label{sec:intro_QED_regime}
We saw in Figure \ref{fig:waiting_time_pooling} the advantageous effect of resource pooling and economies-of-scale in many-server systems.
The driving force behind this fundamental mathematical insight is the Central Limit Theorem (CLT), arguably one of the most important theorems in mathematics and science in general.
\begin{theorem}[Central Limit Theorem, e.g. {\cite[Thm.~27.1]{Billingsley1995}}]
Suppose $X_1,X_2,\ldots,X_n$ is an independent sequence of random variables having mean $\mu$ and positive variance $\sigma^2$.
Then,
\[
\frac{\sum_{i=1}^n X_i - n\mu }{\sqrt{n}\sigma} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{for }n\to\infty.
\]
where ${\;\buildrel{d}\over= \;}$ denote convergence in distribution and $\mathcal{N}(0,1)$ is the standard normal distribution.
\end{theorem}
Notice that the CLT does not pose any restrictions on the distribution of the samples, apart from its finite mean and variance, its statement is extremely powerful, and its consequences appear in many areas of science.
In this thesis provides the basis of the asymptotic study of many-server systems, as will become clear in this section.
Striking the proper balance between queueing delay and server efficiency asymptotically, i.e.~balancing the green and red areas in Figure \ref{fig:sample_paths_lambda100}, in mathematical terms boils down to choosing a service level $s_\lambda$ such that both $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ and $\mathbb{P}(Q^{(s_\lambda)} < s_\lambda)$ remain strictly smaller than 1 as $\lambda\to\infty$.
In other words, one would like to see that the delay probability $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ converges to a non-degenerate limit $\alpha \in (0,1)$ as $\lambda\to\infty$.
To get a feel for the natural scale of the queue, we first examine the situation with unlimited capacity.
More precisely, let $Q^{(\infty)}$ be the number of customers in a steady-state $M/G/\infty$ queue with mean service requirement $\mathbb{E}[B]=1$.
Notice that in this infinite-server setting, $Q^{(\infty)}$ also represents the steady-state number of busy servers.
It is commonly known that $Q^{(\infty)}$ follows a Poisson distribution with mean $R$, the expected workload.
Moreover, if we assume that $\lambda$ is integer, then a Poisson random variable with rate $\lambda$ can be viewed as the sum of $\lambda$ i.i.d. Poisson random variables with rate 1.
In other words, $Q^{(\infty)} = \sum_{i=1}^\lambda P_i$, where the $P_i$, $i=1,2,\ldots,n$, has Poisson distribution with mean and variance 1.
The CLT thus gives
\begin{equation}
\label{eq:infinite_server_tail}
\mathbb{P}(Q^{(\infty)} \geq x_\lambda )
= \mathbb{P}\left(\frac{Q^{(\infty)} -\lambda }{\sqrt{\lambda}} \geq \frac{ x_\lambda - \lambda}{\sqrt{\lambda}} \right)
\approx 1-\Phi\left( \frac{x_\lambda-\lambda}{\sqrt{\lambda}} \right),
\end{equation}
where $\Phi$ denotes the cumulative distribution function of the standard normal distribution.
which converges to a constant value away from both 0 and 1 if and only if $(x_\lambda - \lambda)/\sqrt{\lambda} \to x \in \mathbb{R}$, or $x_\lambda = \lambda + x \sqrt{\lambda} + o(\sqrt{\lambda})$, as $\lambda\to\infty$.
It also shows that the leading order of the random variable describing the queue length is $\lambda$, while the stochastic fluctuations are of order $\sqrt{\lambda}$.
If we now pretend, for a moment, that the infinite-server queue serves as a good approximation for the many-server queue with $s_\lambda$ servers, then we derive through \eqref{eq:infinite_server_tail} that the steady-state probability of wait for ${s_\lambda} = \lambda +\beta\sqrt\lambda$ obeys the Gaussian approximation
\begin{equation}
\label{eq:infinite_server_approx_delay}
\mathbb{P}(W^{(s_\lambda)}>0) = \mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda ) \approx 1-\Phi(\beta).
\end{equation}
Of course, the infinite-server system ignores the one thing that makes a queueing system unique, namely that a queue is formed when all servers are busy.
During these periods of congestion, customers will depart from a system with a finite number of servers at a slower pace than in its infinite-server counterpart.
So the approximation in \eqref{eq:infinite_server_approx_delay} is likely to overestimate the actual delay probability, and a more careful investigation of the queue length process in many-server settings is needed. Nevertheless, the infinite-server heuristic reveals that in a well-managed system, i.e. queues are of acceptable length, the size at which the system operates is of the order $\lambda$, with fluctuations of order $\sqrt{\lambda}$.
We shall now demonstrate through two canonical examples how these guessed natural scalings can be turned into mathematically rigorous statements.
Both examples which will play a key role in this thesis.
\subsection{The $M/M/s$ queue}
\label{sec:intro_MMsqueue}
\textbf{Converging delay probability}.
Let $Q^{(s)}$ denote the steady-state number of customers in an $M/M/s$ queue with arrival rate $\lambda$ and mean service requirement 1, of which the probability distribution is given in \eqref{eq:MMs_stationary_distribution}.
Halfin and Whitt \cite{Halfin1981} showed that, just as in the infinite-server setting, the delay probability in the $M/M/s$ queue converges under scaling \eqref{eq:square_root_staffing rule} to a value between 0 and 1.
Moreover, they showed that this is in fact the only scaling regime in which such a non-degenerate limit exists and identified its value.
Because this result serves as a key prerequisite, we include the result from \cite{Halfin1981} here and present a slightly modified proof.
\begin{proposition}[{\cite[Prop.~2.1]{Halfin1981}}]
\label{prop:HalfinWhitt_delay_probability}
The probability of delay in the $M/M/s_\lambda$ queue has the non-degenerate limit
\begin{equation}
\lim_{\lambda\to\infty} \mathbb{P}( W^{(s_\lambda)} > 0 ) = \left( 1+ \frac{\beta\,\Phi(\beta)}{\varphi(\beta)} \right)^{-1} =: g(\beta) \in (0,1),
\end{equation}
if and only if
\begin{equation}
\label{eq:HalfinWhitt_scaling}
\lim_{\lambda\to\infty} (1-\rho_{s_\lambda}) \sqrt{s_\lambda} \to \beta, \quad \beta > 0,
\end{equation}
where $\Phi$ and $\varphi$ denote the cumulative distribution function and the probability density function of the standard normal distribution, respectively.
\end{proposition}
\begin{proof}
Rewrite \eqref{eq:MMs_wait} as
\begin{equation}
\label{eq:proof_HW_0}
\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )
= \left( 1 + (1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}\right) ^{-1}.
\end{equation}
Similar to \eqref{eq:infinite_server_tail} we find
\begin{align}
\mathbb{P}({\rm Pois}(\lambda) < {s_\lambda})
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < \frac{{s_\lambda}-\lambda}{\sqrt{\lambda}}\right) \nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\frac{{s_\lambda}}{\sqrt\lambda}\right)\nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\sqrt{{s_\lambda}}\left(1+o(1)\right) \right) \to \Phi(\beta),
\label{eq:proof_HW_1}
\end{align}
for $\lambda\to\infty$.
Using Stirling's approximation gives
\begin{align*}
\mathbb{P}({\rm Pois}(\lambda)=s) &= {\rm e}^{-\lambda}\frac{\lambda^{{s_\lambda}}}{{s_\lambda}!}
\sim {\rm e}^{-\lambda} \lambda^{{s_\lambda}}\cdot \frac{1}{\sqrt{2\pi\,{s_\lambda}}} \left(\frac{\rm e}{{s_\lambda}}\right)^{{s_\lambda}} = \frac{1}{\sqrt{2\pi{s_\lambda}}}\,{\rm e}^{{s_\lambda}-\lambda - {s_\lambda}\log(\rho_{{s_\lambda}})}.
\end{align*}
Since $\log(\rho_{{s_\lambda}}) = -(1-\rho_{{s_\lambda}}) - \tfrac{1}{2}(1-\rho_{{s_\lambda}})^2 + o((1-\rho_{{s_\lambda}})^2)$ we find that
\begin{equation}
\label{eq:proof_HW_2}
\frac{ \mathbb{P}({\rm Pois}(\lambda) = s) }{ 1-\rho_{{s_\lambda}} }
= \frac{1}{(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}} \, \frac{{\rm e}^{ -\tfrac{1}{2}(1-\rho_{{s_\lambda}})^2{s_\lambda} + o\left((1-\rho_{{s_\lambda}})^2{s_\lambda}\right)}}{\sqrt{2\pi}} \to \frac{1}{\beta}\, \frac{{\rm e}^{{-}\tfrac{1}{2} \beta^2}}{\sqrt{2\pi}} = \frac{\varphi(\beta)}{\beta}.
\end{equation}
Substituting \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2} into \eqref{eq:proof_HW_0} proves the result.
\end{proof}
Observe that $g(\beta)$ is a strictly decreasing function on $(0,\infty)$ with $g(\beta) \to 1$ as $\beta\to 0$ and $g(\beta)\to 0$ for $\beta\to\infty$.
Thus the entire range of delay probabilities is achievable in the QED regime, which will prove useful for the dimensioning of systems (see Subsection \ref{sec:intro_dimensioning}).
\begin{figure}
\centering
\input{./tikz_tex/halfin_whitt_accuracy.tex}
\caption{The delay probability $\mathbb{P}(Q^{({s_\lambda})} \geq {s_\lambda})$ with ${s_\lambda} = [ \lambda + \beta \sqrt{\lambda} ]$ for $\beta = 0.1,\ 0.5,\ 1$ as a function of $\lambda$.}
\label{fig:delay_probs_HW_MMs}
\end{figure}
Although Proposition \ref{prop:HalfinWhitt_delay_probability} is an asymptotic result for $\lambda\to\infty$, Figure \ref{fig:delay_probs_HW_MMs} shows that for various values of $\beta$, $g(\beta)$ serves as an accuracy approximation for the delay probability for relatively small $\lambda$.
Moreover, D'Auria \cite{DAuria2012} has proven that $\mathbb{P}(W^{({s_\lambda})}>0) \geq g(\beta)$ for all finite $\lambda$ under scaling \eqref{eq:HalfinWhitt_scaling}.
From Proposition \ref{prop:HalfinWhitt_delay_probability}, it also follows that under \eqref{eq:HalfinWhitt_scaling},
\begin{equation}
\label{eq:halfinwhitt_wait}
\sqrt{{s_\lambda}}\,\mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{P}(W^{({s_\lambda})}>0)}{(1-\rho_{s_\lambda})\sqrt{{s_\lambda}}} \to \frac{g(\beta)}{\beta} =: h(\beta), \qquad \text{ for }\lambda\to\infty,
\end{equation}
where we have used the characterization of $\mathbb{E}[W^{({s_\lambda})}]$ in \eqref{eq:MMs_wait}.
This implies that in the QED regime, the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$ as $\lambda\to\infty$.
By Little's law this implies that the expected queue length is $O(\sqrt{{s_\lambda}})$.
The theoretical results of the QED regime we presented here are based on steady-state queueing analysis.
But at the heart of the QED theory lies a much deeper result in which the entire queue-length process, over all points in time, converges to some other limiting process.
\\*
\textbf{Process-level convergence.}
Obtaining rigorous statements about stochastic-process limits poses considerable mathematical challenges.
Rather than presenting the deep technical details of the convergence results, we give a heuristic explanation of how the limiting process arises and what it should look like.
Having another look at the sample paths of the queue-length process $Q^{(s)}$ in Figure \ref{fig:sample_paths_lambda100} with scaling rule ${s_\lambda} = [\lambda + \beta \sqrt{\lambda}]$, the process appears to concentrate around the level ${s_\lambda}$.
As argued before, the stochastic fluctuations are of order $\sqrt{\lambda}$, or equivalently $\sqrt{{s_\lambda}}$.
For that reason, we consider the centered and scaled process
\begin{equation}
\label{eq:intro_scaled_queue_length_process}
X^{({s_\lambda})}(t) := \frac{ Q^{({s_\lambda})}(t) - {s_\lambda}}{\sqrt{{s_\lambda}}}, \qquad \text{ for\ all } t\geq 0,
\end{equation}
and ask what happens to this process as $\lambda\to\infty$.
First, we consider the expected drift conditioned on $X^{({s_\lambda})}(t) = x$.
When $x> 0$, this corresponds to a state in which $Q^{({s_\lambda})}>{s_\lambda}$ and hence all servers are occupied.
Therefore, the expected rate at which customers leave the system is ${s_\lambda}$, while the arrival rate remains $\lambda$, so that the expected drift of $X^{({s_\lambda})}(t)$ in $x>0$ satisfies
\[
\frac{\lambda - {s_\lambda}}{\sqrt{{s_\lambda}}} \to -\beta, \qquad \text{as }\lambda\to\infty,
\]
under scaling $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to \beta$ in \eqref{eq:HalfinWhitt_scaling}.
When $x\leq 0$, only ${s_\lambda} + x\sqrt{{s_\lambda}}$ servers are working, so that the net drift is
\[
\frac{\lambda - ({s_\lambda} + x\sqrt{{s_\lambda}} )}{\sqrt{{s_\lambda}}} \to -\beta-x, \qquad \text{as }\lambda\to\infty,
\]
Now, imagine what happens to the sample paths of $\{X^{({s_\lambda})}\}_{t\geq 0}$ as we increase $\lambda$.
Within a fixed time interval, larger $\lambda$ and ${s_\lambda}$ will trigger more and more events, both arrivals and departures.
Also, the jump size at each event epoch decreases as $1/\sqrt{{s_\lambda}}$ as a consequence of the scaling in \eqref{eq:intro_scaled_queue_length_process}.
As a result, within each time interval, there will be more events, each with a smaller impact, and in the limit as $\lambda\to\infty$, there will be infinitely many events of infinitesimally small impact.
This heuristic explanation suggests that the process $X^{({s_\lambda})}(t)$ converges to a stochastic-process limit, which is continuous and has infinitesimal drift ${-}\beta$ above zero and ${-}\beta-x$ below zero.
Figure \ref{fig:sample_paths_diffusion} visualizes more clearly how the suggested scaling limit arises with increasing $\lambda$ and ${s_\lambda}$.
\begin{figure}
\centering
\input{./tikz_tex/sample_path_diffusion.tex}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda})}(t)$ with $\lambda = 5$, $\lambda=5$ and $\lambda=500$ and ${s_\lambda} = [\lambda+0.5\sqrt{\lambda}]$.}
\label{fig:sample_paths_diffusion}
\end{figure}
The following theorem by Halfin and Whitt \cite{Halfin1981} characterizes this scaling limit more formally.
\begin{theorem}
\label{thm:Halfin_Whitt_diffusion}
Let $X^{({s_\lambda})}(0)\, {\;\buildrel{d}\over\Rightarrow\;} X(0) \in \mathbb{R}$ and $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to\beta$. Then for all $t\geq 0$,
\[
X^{({s_\lambda})}(t) {\;\buildrel{d}\over\Rightarrow\;} X(t),
\]
as $\lambda\to\infty$, where $X(t)$ is the diffusion process with infinitesimal drift $m(x)$ given by
\[
m(x) = \left\{
\begin{array}{ll}
-\beta, & \text{if }x> 0,\\
-\beta-x, & \text{if } x \leq 0
\end{array}\right.
\]
and infinitesimal variance $\sigma^2(x) = 2$.
\end{theorem}
The limiting diffusion process $\{X(t)\}_{t\geq 0}$ in Theorem \ref{thm:Halfin_Whitt_diffusion} is a combination of a negative-drift Brownian motion in the upper half plane and an Ornstein-Uhlenbeck (O-U) process in the lower half plane.
We refer to this hybrid diffusion process as the Halfin-Whitt diffusion.
Much is known for such diffusion processes with piecewise linear drift coefficient, see \cite{Leeuwaarden2012,Fralix2014}.
Its stationary distribution can for instance be derived, see e.g. \cite{BrowneWhitt1995}.
\begin{theorem}
\label{thm:intro_HW_stationary_distribution}
Let $X(t) {\;\buildrel{d}\over\Rightarrow\;} X(\infty)$ for a random variable $X(\infty)$ and $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}\to \beta$ for $\lambda\to\infty$.
Then
\begin{align}
\mathbb{P}(X(\infty) > 0 ) &= g(\beta),
\mathbb{P}(X(\infty) \geq x | X(\infty) > 0) &= {\rm e}^{-\beta x} ,\quad \text{for }x>0,\\
\mathbb{P}(X(\infty) \leq x | X(\infty) \leq 0 ) &= \frac{\Phi(\beta+x)}{\Phi(\beta)},\quad \text{for }x\leq 0.
\end{align}
\end{theorem}
\noindent
This result shows that as the system grows large, the $Q^{({s_\lambda})}(t)$ concentrates around ${s_\lambda}$, and the fluctuations are of order $\sqrt{{s_\lambda}}$.
Moreover, Theorem \ref{thm:intro_HW_stationary_distribution} iterates the limiting values for the delay probability and scaled expected delay. Namely,
\[ \mathbb{P}\big(W^{({{s_\lambda}})} > 0 \big) \rightarrow \mathbb{P}( X(\infty) > 0 ) = g(\beta)\]
and
\[ \sqrt{{s_\lambda}}\mathbb{E}[W^{({s_\lambda})}] \approx \frac{\mathbb{E}[ Q^{({s_\lambda})}]}{\sqrt{{s_\lambda}}} \rightarrow \mathbb{E}[X(\infty)] = \int_0^\infty g(\beta){\rm e}^{-\beta x} {\rm d} x = \frac{g(\beta)}{\beta},/\]
For obvious reasons, the QED regime is also referred to as the Halfin-Whitt regime, and both these names are used interchangeably in this thesis.
\subsection{The $M/D/s$ queue}
\label{sec:intro_discrete_model}
We next consider a many-server queue with deterministic service requirements equal to 1, a Poisson arrival process of rate $\lambda$ and ${s_\lambda}$ servers.
We let $Q^{({s_\lambda})}(t)$ be process describing the number of customers in the system and only examine the process at discrete time epochs $t=0,1,2,...$.
In our analysis we focus on the queue length process $Z^{({s_\lambda})}(t) := (Q^{({s_\lambda})}(t) - {s_\lambda})^+$.
Since we discretize time, the number of new arrivals per time period is given by the sequence of i.i.d. random variables $\{A_k\}_{k\geq 1}$, which has a Poisson distribution with rate $\lambda$.
At the start of the $k^{\rm th}$ period, $Z^{({s_\lambda})}(k)$ customers are waiting.
Because the service time of a customer is equal to the period length, all $\min\{Q^{({s_\lambda})},{s_\lambda}\}$ customers in service at the beginning of the period will have left the system by time $t=k+1$.
This implies that $\min\{Z^{({s_\lambda})},{s_\lambda}\}$ of the waiting customers are taken into service during period $k$, but could not possibly have departed before its end, due to the deterministic service times.
If $Z^{({s_\lambda})}<{s_\lambda}$, then additionally $\min\{ A_k , s-Z^{({s_\lambda})}(k) \}$ of the new arrivals are taken into service.
This yields a total of $A_k$ arrivals, and $\min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\}$ departures from the queue during period $k$, which gives the Lindley type recursion \cite{Lindley1952}, with $Z^{({s_\lambda})}(0) = 0$,
\begin{equation}
\label{eq:discrete_recursion}
Z^{({s_\lambda})}(k+1) = Z^{({s_\lambda})}(k) + A_k - \min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\} = \max\{ 0,Z^{({s_\lambda})}(k) + A_k - {s_\lambda} \}.
\end{equation}
The queue length process thus gives rise to a random walk with i.i.d. steps of size
$(A^{({s_\lambda})}-{s_\lambda})$, with a reflection barrier at zero. We can iterate the recursion in \eqref{eq:discrete_recursion} to find
\begin{align}
Z^{({s_\lambda})}(k+1) &= \max\left\{ 0 , Z^{({s_\lambda})}(k) + A_k-{s_\lambda} \right\} \nonumber\\
&= \max\left\{ 0 , \max\{ 0 , Z^{({s_\lambda})}(k-1) + (A_{k-1}-{s_\lambda})\} + (A_k-{s_\lambda})\} \right\}\nonumber \\
&= \max\left\{ 0 , (A_k-{s_\lambda}) , Z^{({s_\lambda})}(k-1) + (A_k-{s_\lambda}) + (A_{k-1}-{s_\lambda})\right\}\nonumber \\
&= \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_{k-i}-{s_\lambda})\Big\}
{\;\buildrel{d}\over= \;} \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_i-{s_\lambda}) \Big\},
\label{eq:max_randomwalk}
\end{align}
where the last equality in distribution holds due to the duality principle for random walks, see e.g. \cite[Sec.~7.1]{Ross1996}.
For stability the expected step size satisfies $\mathbb{E}[A_k - {s_\lambda}] = \lambda-{s_\lambda} < 0$.
We use the shorthand notation for the partial sum $S_k := \sum_{i=1}^k (A_i-{s_\lambda})$.
Let $Z^{({s_\lambda})}(\infty):= \lim_{k\to\infty} Z^{({s_\lambda})}(k)$ denote the stationary queue length in this $M/D/s$ queue, which can be shown to exist under our assumptions.
The generating function (pgf) of $Z^{({s_\lambda})}(\infty)$ can then be expressed in terms of the pgf of the positive parts of the partial sum:
\begin{equation}
\label{eq:Spitzers_identity}
\mathbb{E}[ w^{Z^{({s_\lambda})}(\infty)} ]
= \exp\Big\{ - \sum_{k=1}^\infty \tfrac{1}{k}\, (1- \mathbb{E}[w^{S_k^+}]) \Big\},\qquad |w|\leq 1,
\end{equation}
From \eqref{eq:Spitzers_identity} we obtain for the mean queue length and empty-queue probability the expressions
\begin{align}
\mathbb{E}[Z^{({s_\lambda})}(\infty)] &= \sum_{k=1}^\infty \tfrac{1}{k}\, \mathbb{E}[ S_k^+ ],\nonumber\\
\mathbb{P}(Z^{({s_\lambda})}(\infty) = 0 ) &= \exp\Big\{ -\sum_{k=1}^\infty \tfrac{1}{k}\, \mathbb{P}( S_k^+ > 0 ) \Big\}.
\label{eq:spitzer_expressions}
\end{align}
Although explicit, the expressions in \eqref{eq:spitzer_expressions} reveal little of the structure of the queue length process.
Hence, we again turn to asymptotics. \\
\noindent\textbf{Gaussian random walk}.
\label{sec:intro_gaussian_random_walk}
We take another look at the identity in \eqref{eq:max_randomwalk}, and ask ourselves what happens if $\lambda$ grows large.
Since $\mathbb{E}[A_k-{s_\lambda}] = \lambda-{s_\lambda} = -\beta\sqrt{\lambda} + o(\sqrt{\lambda})$ under the QED scaling \eqref{eq:square_root_staffing rule}, it makes sense to consider the scaled queue length process $X^{({s_\lambda})}(k) := Z^{({s_\lambda})}(k)/\sqrt{\lambda}$ for all $k\geq 0$, with scaled steps $Y_k^{({s_\lambda})} := (A_k-{s_\lambda})/\sqrt{\lambda}$.
Dividing both sides of \eqref{eq:max_randomwalk} by $\sqrt{\lambda}$ then gives
\begin{equation}
X^{({s_\lambda})}(k+1) = \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j Y^{({s_\lambda})}_k \Big\}.
\end{equation}
Observe that $A_k \sim {\rm Pois}(\lambda)$.
Hence by the CLT
\begin{equation*}
Y^{({s_\lambda})}_k = \frac{ A_k - {s_\lambda} }{\sqrt\lambda} = \frac{A_k-\lambda}{\sqrt\lambda} - \beta \ {\;\buildrel{d}\over\Rightarrow\;} \ Y_k {\;\buildrel{d}\over= \;} \mathcal{N}(-\beta,1),
\end{equation*}
for $\lambda\to\infty$.
So by intuition, we expect the scaled queue length process to converge in distribution to a reflected random walk with normally distributed increments, i.e. a reflected \textit{Gaussian random walk}.
Indeed, it is easily verified that \cite{Janssen2008a},
\begin{equation}
X^{({s_\lambda})}(k)\ {\;\buildrel{d}\over\Rightarrow\;} \ M_\beta(k) := \max_{0\leq j\leq k} \Big\{\sum_{i=1}^j Y_j \Big\}, \qquad \lambda\to\infty.
\end{equation}
Let $M_\beta:= \lim_{k\to\infty} M_\beta(k)$ denote the all-time maximum of a Gaussian random walk.
It can be shown that $M_\beta$ almost surely exists and that $X^{({s_\lambda})}(\infty) := \lim_{k\to\infty} X^{({s_\lambda})}(k)$ ${\;\buildrel{d}\over\Rightarrow\;} M_\beta$ for instance by \cite[Prop.~19.2]{Spitzer1964} and \cite[Thm.~X6.1]{Asmussen2003}.
The following theorem can be proved using a similar approach as in \cite{Jelenkovic2004}.
(We prove this result in a more general setting in Chapter 3)
\begin{theorem}
Let $X^{({s_\lambda})}(\infty)$ be the scaled queue length in steady-state. If $(1-\rho_{{s_\lambda}})\sqrt{\lambda}\to\beta$, then as $\lambda\to\infty$,
\begin{enumerate}
\item[\normalfont (i)] $X^{({s_\lambda})}(\infty) {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[\normalfont (ii)] $\mathbb{P}(X^{({s_\lambda})}(\infty) = 0) \to \mathbb{P}(M_\beta = 0)$,
\item[\normalfont (iii)] $\mathbb{E}[X^{({s_\lambda})}(\infty)^k] \to \mathbb{E}[M_\beta^k]$, for any $k>0$.
\end{enumerate}
\end{theorem}
The Gaussian random walk is well studied in \cite{Siegmund1978,Chang1997,Janssen2006,Blanchet2006,Janssen2006} and there is an intimate connection with Brownian motion.
The only difference, one could say is that Brownian motion is a continuous-time process, whereas the Gaussian random walk only changes at discrete points in time.
If $\{B(t)\}_{t\geq 0}$ is Brownian motion with drift $-\mu <0$ and infinitesimal variance $\sigma^2$ and $\{W(t)\}_{t = 0}^\infty$ is a random walk with $\mathcal{N}(-\mu,\sigma^2)$ steps and $B(0) = W(0)$, then $W$ can be regarded as the process $B$ embedded at equidistant time epochs.
That is, $W(t) {\;\buildrel{d}\over= \;} B(t)$ for all $t\in\mathbb{N}^+$.
For the maximum of both processes this coupling implies
\begin{equation}
\max_{k\in \mathbb{N}^+} W(k) = \max_{k\in \mathbb{N}^+} B(k) \leq_{\rm st}
\max_{t\in \mathbb{R}^+} B(t),
\label{eq:max_inequality}
\end{equation}
where $\leq_{\rm st}$ denotes stochastic dominance.
This difference in maximum is visualized in Figure \ref{fig:BrownianMotion_vs_GaussianRW}.
It is known that the all-time maximum of Brownian motion with negative drift $-\mu$ and infinitesimal variable $\sigma^2$ has an exponential distribution with mean $\sigma/2\mu$ \cite{Harrison1985}.
Hence, \eqref{eq:max_inequality} implies that $M_\beta$ is stochastically upper bounded by an exponential random variable with mean $1/2\beta$.
\begin{figure}
\centering
\begin{tikzpicture}[scale = 1.1 ]
\begin{axis}[
xmin = 0,
xmax = 10,
ymin = -2.2,
ymax = 5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel={$t$},
ylabel={},
xscale=1,
yscale=1]
\addplot[gray] file {./tikz/Brownian_Motion_SamplePath/BM.txt};
\addplot[only marks, red] file {./tikz/Brownian_Motion_SamplePath/GW.txt};
\addplot[dashed] file {./tikz/Brownian_Motion_SamplePath/maxBM.txt};
\addplot[dotted,very thick, red] file {./tikz/Brownian_Motion_SamplePath/maxGW.txt};
\end{axis}
\end{tikzpicture}
\caption{Brownian motion and embedded Gaussian random walk with their respective running maxima.}
\label{fig:BrownianMotion_vs_GaussianRW}
\end{figure}
Despite this easy bound, precise results for $M_\beta$ are more involved. Let $\zeta$ denote the Rieman zeta function.
\begin{theorem}[{\cite[Thm.~1]{Chang1997}}]
For $0<\beta<2\sqrt{\pi}$,
\begin{equation}
\mathbb{P}(M_\beta = 0) = \sqrt{2}\beta\, \exp \left\{ \frac{\beta}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(1/2-l)}{l!(2l+1)} \left(\frac{-\beta^2}{2}\right)^l \right\}.
\end{equation}
\end{theorem}
In \cite{Janssen2006,Janssen2007,Janssen2009}, similar series expansions are derived for e.g. the mean and variance of the maximum of the Gaussian random walk.
\begin{theorem}[Thm.~2\&3, \cite{Janssen2006}]
For $0<\beta<2\sqrt{\pi}$,
\begin{equation}
\mathbb{E}[M_\beta] = \frac{1}{2\beta} + \frac{\zeta(1/2)}{\sqrt{2\pi}} + \frac{\beta}{4}
+ \frac{\beta^2}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(-1/2-l)}{l!(2l+1)(2l+2)} \left(\frac{-\beta^2}{2}\right)^l,
\end{equation}
\begin{equation}
{\rm Var}\, M_\beta =
\frac{1}{4\beta^2} - \frac{1}{4} - \frac{2\,\zeta(-1/2)}{\sqrt{2\pi}}\beta - \frac{\beta^2}{24} -
\frac{2\beta^3}{\sqrt{2\pi} } \sum_{l=0}^\infty
\frac{\zeta(-3/2-l)}{l!(2l+1)(2l+2)(2l+3)} \Big(\frac{-\beta^2}{2}\Big)^l.
\end{equation}
\end{theorem}
\subsection{Characteristics of the QED regime}
\label{sec:intro_characteristics}
Now that we have seen how the square-root staffing principle \eqref{eq:square_root_staffing rule} yields non-degenerate limiting behavior in two classical queueing models, we can elaborate on how the QED regime fosters three desirable properties.
The first property relates to the efficient usage of resources, expressed as:
\begin{equation}
\rho_{{s_\lambda}} = \frac{\lambda}{{s_\lambda}} = 1 - \frac{\beta}{\sqrt{{s_\lambda}}} + O\big(1/\lambda\big), \tag{Efficiency}
\end{equation}
where we used that ${s_\lambda} = O(\lambda)$.
The second property relates to good QoS:
\begin{equation}
\mathbb{E}[W^{({s_\lambda})}] = \frac{h(\beta)}{\sqrt{{s_\lambda}}} + o(1/\sqrt{{s_\lambda}}) \qquad \text{and} \qquad \mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{E}[M_\beta]}{\sqrt{{s_\lambda}}} + O(1/\sqrt{{s_\lambda}}), \tag{QoS}
\end{equation}
in the $M/M/s$ and $M/D/s$ models, respectively.
Hence the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$.
The third distinctive property is the balance between QoS and efficiency:
\begin{equation}
\mathbb{P}(W^{({s_\lambda})}>0) \to g(\beta), \qquad \text{and} \qquad \mathbb{P}(W^{({s_\lambda})}>0) \to 1-\mathbb{P}(M_\beta=0), \tag{Balance}
\end{equation}
as ${s_\lambda} \to \infty$, both values laying within the interval (0,1), in the $M/M/s$ and $M/D/s$ models, respectively.
The three key properties, reformulated in a model-independent way are, with $s = \lambda +\beta\sqrt\lambda$ and $\rho = \lambda/s$,
\begin{align}
\rho &= 1-\beta/\sqrt\lambda + O(\lambda^{-1}), \tag{*} \\
\mathbb{E}[W] &= O(1/\sqrt\lambda), \tag{**}\\
\mathbb{P}(W>0) &\to \alpha \in(0,1). \tag{***}
\end{align}
Since the mathematical foundation for these properties comes from the CLT, we can expect the properties to hold for a much larger class of models.
These models should then be members of the same universality class (to which the CLT applies).
Let us again show this by example.
Consider a stochastic system in which demand per period is given by some random variable $A$, with mean $\mu_A$ and variance $\sigma_A^2<\infty$.
For systems facing large demand we propose to set the capacity according to the more general rule $s = \mu_A + \beta\sigma_A$, which consists of a minimally required part $\mu_A$ and a variability hedge $\beta\sigma$.
Assume that the workload brought into the system is generated by $n$ stochastically identical and independent sources.
Each source $i$ generates $A_{i,j}$ work in the $j$th period, with $\mathbb{E}[A_{i,j}] = \mu$ and ${\rm Var}\,\,A_{i,j} = \sigma^2$.
Then the total amount of work arriving to the system during one period is $A_j^{(n)} = \sum_{i=1}^n A_{i,j}$ with mean $n\mu$ and variance $n\sigma^2$.
Assume that the system is able to process a deterministic amount of work $s_n$ per period and denote by $U^{(n)}(j)$ the amount of work left over at the end of period $j$.
Then,
\begin{equation}
U(j+1) = \left( U^{(n)}(j) + A^{(n)}_j - s_n \right)^+.
\end{equation}
Given that $s_n > \mathbb{E}[A^{(n)}_1] = n\mu$, the stationary limit $U^{(n)} := \lim_{t\to\infty} U^{(n)}(t)$ exists and satisfies
\begin{equation}
U^{(n)} {\;\buildrel{d}\over= \;} \left( U^{(n)} + A^{(n)}_j - s_n \right)^+.
\label{eq:bulk_service_stationary_recursion}
\end{equation}
This framework is also known as the bulk service queue or the Anick-Sondhi-Mitra model \cite{Anick1982,Janssen2005,Janssen2008}.
In this scenario, increasing the system size is done by increasing $n$, the number of input flows.
As we have seen before, it requires a rescaling of the process $U^{(n)}$ by an increasing function $c(n)$, in order to obtain a non-degenerate scaling limit $U := \lim_{n\to\infty} U^{(n)}/c(n)$.
(We omit the technical details needed to justify the interchange of limits.)
From \eqref{eq:bulk_service_stationary_recursion} it becomes clear that the scaled increment
\begin{equation}
\frac{A^{(n)}_j - s_n}{c(n)} = \frac{\sum_{i=1}^n A_{i,j} - n\mu}{c(n)} + \frac{n\mu - s_n}{c(n)}
\end{equation}
only admits a proper limit if $c(n)$ is of the form $c(n) = O(\sqrt{n})$, by the virtue of the CLT, and $(s_n-n\mu)/c(n) \to \beta >0$ as $n\to\infty$.
Especially for $c(n) = \sigma\sqrt{n}$, this reveals that $U$ has a non-degenerate limit, which is equal in distribution to the maximum of a Gaussian random walk with drift -1 and variance 1, if
\[
s_n = n\mu+\beta \sqrt{n}\sigma + o(\sqrt{n}).
\]
Moreover, the results on the Gaussian random walk presented in Subsection \ref{sec:intro_gaussian_random_walk} are applicable to this model and the key features of the QED scaling carry over to this more general setting as well.
In conclusion, the many-sources framework shows that the QED scaling finds much wider applications than queueing models with Poisson input only.
\subsection{Related literature}
We now provide a partial overview on the literature on heavy-traffic analysis in queueing theory and the QED regime in particular.\\
\\*
\noindent\textbf{Conventional heavy-traffic}.
Before the formal introduction of the Halfin-Whitt scaling regime in 1981, see \cite{Halfin1981}, the existing literature on the asymptotic analysis of many-server queues mostly evolved around two types of scaling regimes.
The idea of studying a sequence of queues in which the utilization approaches 100\%, i.e.~heavy-traffic, was first laid out by Kingman in the 1960s.
In \cite{Kingman1961,Kingman1962} he showed how in the $GI/G/1$ queue, under mild conditional on the arrival and service processes, the scaled steady-state waiting time
distribution $(1-\rho)W^{(1)}$ to an exponentially distributed random variable.
The notion that heavily loaded systems admit a scaling limit that is remarkably simple compared to the otherwise intractable pre-limit queueing systems triggered a surge of research within the field of queueing theory in the 1960s and 1970s, see \cite{Borovkov1965,Iglehart1970,Brumelle1971,Newell1973,Kollerstrom1974,Kollerstrom1979,Whitt1974} among others.
These works conduct their asymptotic analysis in what we now call conventional heavy-traffic.
That is, the service times and number of servers are held fixed, while the arrival rate approaches the critical value from below.
A noteworthy result of these efforts is the extension of Kingman's findings to the $GI/G/s$, which finds that the scaled queue length $(1-\rho)Q^{(s)}$ converges in distribution to an exponential random variable with mean $(c_a^2+c_s^2)/2$, where $c_a$ and $c_s$ denote the coefficient of variation of the interarrival and service time distribution, respectively.
We remark that this limiting result is the key ingredient to the famous Kingman's formula:
\[
\mathbb{E}[W^{(1)}] \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2+c_s^2}{2} \cdot \mathbb{E}[B],
\]
which serves as an approximation to the expected waiting time in the single-server queue and of which the usage is now widespread.
The scaling limits reveal that in the conventional heavy-traffic regime, the expected waiting time explodes as $\rho\to 1$.
Hence, efficient usage of resources is achieved, at the expense of poor QoS.
An alternative regime that received much attention, see e.g. \cite{Iglehart1965,Borovkov1965,Iglehart1973,Iglehart1973a,Whitt1982}, fixed the service time distribution while increasing both the arrival rate $\lambda$ and the number of servers to infinity simultaneously, such that the ratio $\lambda/s$ is constant.
It has been shown that the sequence of queues under this scaling start resembling the behavior of infinite-server queues as $\lambda$ and $s$ grow.
That is, the probability of a customer finding a queue on arrival is negligible.
The sample paths in Figure \ref{fig:sample_paths_lambda100} are illustrative for this regime.
Since the utilization level $\rho$ remains strictly below 1 in the limit, this setting is typically not recognized as heavy-traffic.
Accordingly, server efficiency is not achieved in this case.
However, this regime offers excellent service levels, as customers experience virtually no wait.
As Halfin and Whitt spell out themselves, their novel regime in which service times are held fixed, and $\lambda$ and $s$ tend to infinite while satisfying $(1-\rho)\sqrt{s} \to \beta$, is a hybrid between the two aforementioned regimes.
Namely, it adopts the efficiency property of the conventional heavy-traffic scaling, and the good QoS levels from the resemblance with infinite-server queues.\\
\\*
\noindent
\textbf{The $G/G/s$ queue in the QED regime}.
Since the literature on queues in the QED regime is vast, we choose to give an overview of only the most relevant advances in the performance analysis of the many-server queues in the QED regime under the most general conditions, together with the extension that are rooted within practical service systems.
Whereas \cite{Halfin1981} was able to exploit the Markovian of the exponentially distributed service times, the heavy-traffic analysis of the $G/G/s$ queue requires fundamentally different approaches than Halfin and Whitt's.
For various service distribution classes, the convergence sequence of diffusion-scaled processes has been studied within finite time intervals.
Puhalskii \& Reiman \cite{Puhalskii2000} analyze the multi-class queue with phase-type service times in the Halfin-Whitt regime.
Heavy-traffic limits for queues in which service time distributions are lattice-based and/or have finite support are studied by Mandelbaum \& Momcilovic \cite{Mandelbaum2008} and Gamarnik \& Momcilovic \cite{Gamarnik2008}.
Other approaches by Kang, Kaspi \& Ramanan \cite{Kaspi2011,Kang2012,Kaspi2013}.
The most general class of distributions is considered by Reed \cite{Reed2009} and Puhalskii \& Reed \cite{Puhalskii2010}, who impose no assumption on the service time distribution except for the existence of the first moment.
Considerably less is known for the corresponding steady-state distribution of the $G/G/s$ queue in the QED regime.
Namely, under the assumption of general service time distributions, truly infinite-dimensional limits arise, since the Markovian nature of the service time and `age' process can no longer be exploited.
Works that have been able to characterize limiting behavior for the specific service time distribution classes include Jelenkovic et al.~\cite{Jelenkovic2004}, who assume deterministic service times, and Whitt \cite{Whitt2005}, who identifies the heavy-traffic limit in the case of hyperexponentially distributed service times.
Progress in the understanding of steady-state behavior of $G/G/s$ queues in the Halfin-Whitt regime has been facilitated by Gamarnik \& Goldberg \cite{Goldberg,Gamarnik2013a}, who perform their analysis under the mild assumption that the service distribution has finite $(2+\e)$ moment.
A significant advance has been made by Aghanjani \& Ramanan \cite{Aghajani2016}, who under the most general conditions identify the limit as the steady-state distirbution of infinite-dimensional Markov process, drawing upon previous results by Kang, Kaspi \& Ramanan \cite{Kaspi2011, Kang2012,Kaspi2013}
For an elaborate survey on the techniques required for analysis of $G/G/s$ queues, we refer the reader to \cite{Pang2007} and references therein.\\
\\*
\textbf{Model extensions}
A variety of extensions to the standard many-server queue can be considered.
A feature ubiquitous to service systems involving humans is customer abandonment \cite{Gans2003,Brown2005,Zeltyn2005,Mandelbaum2013}.
The $M/M/s+M$ queue introduced by Palm \cite{Palm1957}, also known as the Erlang-A model \cite{Garnett2002,Leeuwaarden2012}, acknowledges this feature by assigning every customer an exponentially distributed \textit{patience time} upon his arrival (denoted by $+M$ in the model definition).
If a customer has not yet started receiving service by the expiration of his patience, he leaves the system.
Note that abandonments render queues stable under any load.
Under QED scaling, the more general $G/G/s+G$ queue has received much attention under various modeling assumptions, see e.g. \cite{Garnett2002,Gans2003,Whitt2006,Mandelbaum2009,Zeltyn2005,Mandelbaum2012a,Kang2012,Dai2010,Reed2012,Jennings2012,Zhang2013}.
Noteworthy findings include the vanishing abandonment probability \cite{Garnett2002} and insensitivity of the patience time distribution as long as its density at 0, i.e.~the balking probability, is fixed, as the system grows large under QED scaling.
Neat overviews of queues with abandonment and their asymptotic counterpart are given by Zeltyn \& Mandelbaum \cite{Zeltyn2005} and Dai \& He \cite{Dai2012} and Ward \cite{Ward2012}.
Other features model extensions that have been studied in the QED regime include multiple customer classes, see e.g. \cite{Harrison2004,Atar2014,Gurvich2008,Gurvich2009,Tezcan2010}, or heterogeneous servers \cite{Armony2005,Armony2010,Mandelbaum2012b,Stolyar2010}.
These models are all interesting in their own respect and are fairly well-understood.
Therefore, we choose to focus in this thesis on a different set of extensions, which will be discussed in Section \ref{sec:intro_beyond}.
\section{Dimensioning}
\label{sec:intro_dimensioning}
We adopt the term \textit{dimensioning} used by Borst, Mandelbaum \& Reiman~\cite{Borst2004} to say that the capacity of a service system is adapted to the load in order to reach certain performance levels.
In \cite{Borst2004} dimensioning refers to the staffing problem in a large-scale call center and key ingredients are the square-root staffing rule in \eqref{eq:square_root_staffing rule} and the QED regime.
We now revisit the results in \cite{Borst2004} and its follow-up works to explain this connection to the QED regime.
\subsection{Constraint satisfaction}
\label{sec:intro_constraint}
Consider the $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$.
A classical dimensioning problem is to determine the minimum number of servers $s$ necessary to achieve a certain target level of service, say in terms of waiting time.
Suppose we want to determine the minimum number of servers such that the fraction of customer who are delayed in the queue is at most $\varepsilon\in(0,1)$.
Hence we should find
\begin{equation}
s^{*}_\lambda(\varepsilon) := \min \left\{s \geq \lambda\, |\, \mathbb{P}(W^{(s)}>0) \leq \varepsilon \right\}. \tag{A}
\end{equation}
But alternatively, we can use the QED framework, which says that under \eqref{eq:HalfinWhitt_scaling},\ \ $\lim_{s\to\infty} \mathbb{P}(W^{({s_\lambda})} > 0) = g(\beta)$ (see Proposition \ref{prop:HalfinWhitt_delay_probability}).
Then by (A) can be replaced by
\begin{equation}
s^{\rm srs}_\lambda(\varepsilon) = \lceil \lambda + \beta^*(\varepsilon) \sqrt{\lambda}\rceil, \tag{B}
\end{equation}
where $\beta^*(\varepsilon)$ solves
\begin{equation}
g(\beta^*) = \varepsilon. \tag{C}
\end{equation}
In Figure \ref{fig:MMs_staffing_levels} we plot the exact staffing level $s^*_\lambda(\varepsilon)$ and the heuristically obtained staffing level $s^{\rm srs}_\lambda(\varepsilon)$ as a functions of $\varepsilon$ for several loads $\lambda$.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda5_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda10_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda100_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda500_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500.$}
\end{subfigure}
\caption{Staffing levels corresponding to delay probability targets $\varepsilon$.}
\label{fig:MMs_staffing_levels}
\end{figure}
Observe that even for very small values of $\lambda$, the staffing function $s^{\rm srs}(\varepsilon)$ coincides with the exact solution for almost all $\varepsilon\in(0,1)$ and differs no more than by one server for all $\varepsilon$.
Borst et al.~\cite{Borst2004} recognized this in their numerical experiments too, and Janssen, van Leeuwaarden and Zwart \cite{Janssen2011} later confirmed this theoretically.
As the scale of the queue increases, these differences naturally cancel out.
Henceforth, we may argue that the staffing method via the square-root staffing principle performs close to optimal in systems that are `large enough'.
One could easily adapt this asymptotic procedure to fit the staffing problem with a constraint on the mean waiting time or the tail probability of the waiting time, e.g. $\mathbb{P}(W^{(s)}>T)$, which are asymptotically approximated by $h(\beta)/\sqrt{\lambda}$ and $g(\beta){\rm e}^{-\beta \sqrt{\lambda} T}$, respectively, but we do not go into details here.
\subsection{Optimization}
\label{sec:intro_optimization}
A similar line of reasoning holds when one turn form a constraint satisfaction problem to an optimization problem, for instance to strike the right balance between the costs for servers and costs incurred by customer dissatisfaction.
More specifically, assume a salary cost of $a$ per server per unit time, and a penalty cost of $q$ per waiting customer per unit time, yielding the total cost function
\[
\bar{C}_\lambda(s) := a\,s + q\,\lambda\mathbb{E}[W^{(s)}]
\]
and then ask for the staffing level $s$ that minimizes $\bar{C}_\lambda(s)$
in which we target the minimizing number of servers $s$.
Since $s>\lambda$, we have $\bar{C}_\lambda(s) > a\,\lambda$ for all feasible solutions $s$.
Moreover, the minimizing value of $\bar{C}_\lambda$ is invariant with respect to scalar multiplication of the objective function.
Hence we have to optimize
\[
C_\lambda(s) = r\,(s-\lambda) + \lambda\mathbb{E}[W^{(s)}], \qquad r = a/q.
\]
The exact solution to the staffing problem is denoted by $s^*_\lambda(r) := \arg\min_{s > \lambda} C_\lambda(s)$.
With ${s_\lambda} = \lambda + \beta\sqrt{\lambda}$ and the QED limiting results in \eqref{eq:halfinwhitt_wait}, we can replace this optimization problem by its asymptotic counterpart
\begin{align*}
\frac{C_\lambda({s_\lambda})}{\sqrt{\lambda}} = r\,\beta + \sqrt{\lambda} \mathbb{E}[W^{(s)}] \to r\,\beta + \frac{g(\beta)}{\beta} =: \hat{C}(\beta), \qquad \lambda\to\infty.
\end{align*}
Once again we obtain a limiting objective function that is easier to work with than its exact pre-limit counterpart.
Hence, in the spirit of the asymptotic staffing procedure in the previous subsection, we propose the following method to determine the staffing level that minimizes overall costs.
First, (numerically) compute the value $\beta^*(r) = \arg\min_{\beta>0} \hat{C}(\beta)$, which is well-defined, because the function $\hat{C}(\beta)$ is strictly convex for $\beta>0$.
Afterwards, set $s^{\rm srs}_\lambda(r) = [ \lambda + \beta^*(r) \sqrt{\lambda} ]$.
In Figure \ref{fig:MMs_staffing_levels_optimization} we compare the outcomes of this asymptotic staffing procedure against the true optima as a function of $r\in(0,\infty)$, for several values of $\lambda$.
The staffing levels $s^{\rm srs}_\lambda(r)$ and $s^*_\lambda(r)$ are aligned for almost all $r$, and differ no more than one server for all instances.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda5_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda10_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda100_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend style = {at = {(axis cs:4.8,540)},anchor = north east}]
\addplot[very thick] file {./tikz/Optimization/lambda500_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500.$}
\end{subfigure}
\caption{Optimal staffing levels corresponding as a function of $r = a/q$.}
\label{fig:MMs_staffing_levels_optimization}
\end{figure}
These numerical experiments show the power and value of insights gained from asymptotic analysis of queues in the QED regimes, even in small to moderate service systems.
\subsection{Time-varying dimensioning}
So far we have only considered queues in which the model primitives are constant over time.
In practice, though, the arrival rate can fluctuate and depends on the time of day, the day of the week, season or even larger time scales.
It is therefore more realistic to describe these mostly predictable fluctuations through $\lambda(t)$, which represents the instantaneous arrival rate of the arrival process at time $t\in \mathbb{R}$.
The existence of time-varying demand requires a re-evaluation of staffing levels throughout the planning horizon as well.
That is, the number of servers $s(t)$ becomes a function of time, rather then a constant and this clearly asks for an adaption of the dimensioning procedures in Subsections \ref{sec:intro_constraint} and \ref{sec:intro_optimization}.
We explain the concept of time-varying staffing and connection with the QED regime through the time-varying extension of the $M/M/s$ queue denoted by $M_t/M/s_t$ queue, where the subscript $t$ refers to the time-varying nature of both the arrival process and the staffing level.
In this setting, customers arrive according to a non-homogeneous Poisson process with rate function $\lambda(t)$ and customers have exponentially distributed service times with mean $1/\mu$.
Under a constraint satisfaction strategy, we aim to find the staffing function $s(t)$ such that the delay probability is at most $\varepsilon\in(0,1)$ for all $t$.
The analysis and optimization of time-varying many-server queueing systems is known to be intrinsically hard, but many approximation techniques and heuristic methods have been proposed throughout the years \cite{Green1991,Jennings1996}. (nog wat meer toevoegen)
A natural but naive approach is the \textit{pointwise-stationary approximation} (PSA) \cite{Green1991}, which evaluates the system at time $t$ as if it were in steady-state with instantaneous parameters $\lambda=\lambda(t)$, $\mu$ and $s = s(t)$.
Consequently, the analysis and optimization of queues is performed on steady-state performance metrics.
Variants of the PSA method include the \textit{simple-stationary approximation} (SSA) \cite{Green2001}, which uses the long-term (moving) average arrival rate instead of the instantaneous arrival rate, and the \textit{stationary-independent-period-by-period approximation} (SIPP) \cite{Green2001}, which splits the time-horizon into multiple intervals and performs steady-state analysis with the averaged parameters in each of these intervals, among others.
PSA performs well in slowly varying environments with relatively short service times \cite{Green1991,Whitt1991}.
However, when the the model parameters fluctuate significantly, as is often the case in real-life systems, the accuracy of PSA can be poor, as we will see in the numerical experiment at the end of this section.
The main reason why PSA, SSA and SIPP can fail to be accurate is that they neglect that customers actually reside in the system (being in service or waiting in the queue).
In contrast, staffing decisions should be based on the number of customers present in the system rather than the arrival rate at that particular time.
Jennings et al. \cite{Jennings1996} introduced a more sophisticated method that exploits the relation with infinite-server queues.
We explain their idea in the context of the $M_t/M/s_t$ queue.
Let $B_e$ denote the random variable describing the excess service time, which is equal to the original random variable in case $B$ has an exponential distribution.
Then by Eick et al. \cite{Eick1993}, the number of customers in the $M_t/M/\infty$ queue at time $t$ is Poisson distributed with mean
\begin{equation}
\label{eq:offered_load_eick}
R(t) = \mathbb{E}\left[ \lambda(t-B_e)\right] \mathbb{E}[B] = \int_0^\infty \lambda(t-u)\,\mathbb{P}(B>u)\, {\rm d}u = \int_0^\infty \lambda(t-u)\, {\rm e}^{-\mu u} \,{\rm d}u.
\end{equation}
We remark that this result holds for more general service time distributions.
Now, recall that in large systems in the QED regime, the expected delay is negligible (***).
Therefore, under these conditions, the many-server system may be approximated by the infinite-server approximation with offered load as in \eqref{eq:offered_load_eick}.
Accordingly, we can determine the staffing levels $s(t)$ for each $t$ based on steady-state $M/M/s$ measures with offered load $R=R(t)$.
Jennings et al. \cite{Jennings1996} proceed by exploiting the heavy-traffic results of Halfin-Whitt \eqref{eq:halfinwhitt_wait}.
In conjunction with the dimensioning scheme in Subsection \ref{sec:intro_constraint}, the authors propose to set
\begin{equation}
s(t) = \bigg\lceil R(t) + \beta^*(\varepsilon) \sqrt{R(t)} \bigg\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves $g(\beta^*(\varepsilon)) = \varepsilon$.
Remark that the number of servers is rounded up to ensure that the achieved delay probability is indeed below $\varepsilon$.
This method was called in \cite{Jennings1996,Massey1994} to \textit{modified-offered-load} (MOL) approximation, and we adopt this term in this thesis.
Let us show that this approximation scheme works.
Figure \ref{fig:intro_example_arrival}(a) shows an arrival rate pattern $\lambda(t)$ and corresponding offered load function $R(t)$ for $\mu=1/2$.
This arrival rate stems for a real-world emergency department.
The resulting staffing level function based on the PSA and MOL approximations with $\varepsilon = 0.3$ are plotted in
Figure \ref{fig:intro_example_arrival}(b).
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 45,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {$\mathbb{P}(W>0)$},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/TimeVarying/arrival_rate.txt};
\addplot[very thick, red] file {./tikz/TimeVarying/offered_load.txt};
\legend{{$\lambda(t)$},$R(t)$}
\end{axis}
\end{tikzpicture}
\caption{Arrival rate and offered load functions.}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 60,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {$\mathbb{P}(W>0)$},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/TimeVarying/s_PSA.txt};
\addplot[very thick, red] file {./tikz/TimeVarying/s_Jennings.txt};
\legend{PSA,MOL}
\end{axis}
\end{tikzpicture}
\caption{Staffing functions.}
\end{subfigure}
\caption{Time-varying parameters of real-world emergency department.}
\label{fig:intro_example_arrival}
\end{figure}
Through simulation, we evaluate the delay probability as a function of time for $\varepsilon = 0.1,\, 0.3$ and 0.5.
In Figure \ref{fig:intr_timevarying_simulation_results} we see how the PSA approach fails to stabilize the performance of the queue, while the MOL method does stabilize around the target performance.
The erratic nature of the delay probability as a function of time can be explained by rounding effects of the staffing level.
Since this rather simple but elegant technique to address time-varying dimensioning is provably effective, we will adopt the underlying idea of the MOL method in various different settings in this thesis.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, red] file {./tikz/TimeVarying/pdelay_e01_psa.txt};
\addplot[thick, green] file {./tikz/TimeVarying/pdelay_e03_psa.txt};
\addplot[thick, blue] file {./tikz/TimeVarying/pdelay_e05_psa.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{PSA}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, red] file {./tikz/TimeVarying/pdelay_e01_mol.txt};
\addplot[thick, green] file {./tikz/TimeVarying/pdelay_e03_mol.txt};
\addplot[thick, blue] file {./tikz/TimeVarying/pdelay_e05_mol.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{MOL}
\end{subfigure}
\caption{Probability of delay under staffing functions obtained through PSA and MOL approximations.}
\label{fig:intr_timevarying_simulation_results}
\end{figure}
\section{Contributions}
\label{sec:intro_beyond}
We have explained how the QED regime can be used to dimension and staff large-scale service systems.
The basic concepts, however, where explained for the relatively simple $M/M/s$ and $M_t/M/s_t$ queue.
Many real-world service systems have essential features that are not captured by these elementary models.
We will now discuss some of these features and address the need to consider more involved models and extend the existing QED theory.
\subsection{Non-classical scaling regimes and pre-limit behavior}
\label{sec:intro_novel_scalings}
The QED theory is centered around the scaling relation $\sqrt{\lambda}(1-\rho_\lambda) \to \beta$, or equivalently $s_\lambda = \lambda + \beta \sqrt{\lambda} + o(\sqrt{\lambda})$, for $\lambda\to\infty$.
It is worthwhile to study how pre-limit behavior of many-server queues is affected when is deviated from this scaling regime.
We introduce a novel family of heavy-traffic scaling regimes, described in terms of the parameter $\eta$ for which we assume that
\begin{equation}
\label{eq:novel_scaling_rule}
\lambda^\eta (1-\rho_\lambda) \to \beta, \qquad \text{as } \lambda\to\infty,\ \beta > 0.
\end{equation}
The parameter $\eta \geq 0$ defines a whole range of possible scaling regimes, including the classic case $\eta = 1/2$, as well as the cases $\eta=0$ and $\eta=1$ investigated in Subsection \ref{sec:intro_many_server_regimes}.
In terms of a capacity sizing rule, the condition \eqref{eq:novel_scaling_rule} is tantamount to $s_\lambda = \lambda +\beta\,\lambda^{1-\eta}$.
This framework thus bridges the gap between the QD and QED regime if $\eta\in(0,1/2)$ and the QED and ED regime if $\eta\in(1/2,1)$, in the $M/M/s$ model.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\alpha\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\alpha\in(0,1/2)$ can be seen as \textit{moderate} heavy traffic: heavy traffic conditions in which the full occupancy is reached more slowly, as a function of $\lambda$, than for classical heavy traffic. See \cite{Chang1996,Puhalskii1998,Puhalskii1999,Atar2012,Atar2014,Atar2015,Atar2016} for more details.
For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to \textit{extreme} heavy traffic due to a relatively small variability hedge.
We use the insights of Section \ref{sec:intro_QED_regime} and the connection of the QED scaling to the CLT to argue by intuition that the following trichotomy in the qualitative system behavior as $\lambda\to\infty$ holding under scaling \eqref{eq:novel_scaling_rule}.
For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, because the order of the variability hedge $\beta \lambda^{1-\eta}$ is greater than strictly necessary to accommodate the stochastic fluctuations in demand.
Scalings in which $\eta\in(1/2,\infty)$, have adverse behavior, since stochastic fluctuations are not accounted for sufficiently, so that the probability of delay probability converges to 1.
The value $\eta=1/2$ is therefore the tipping point, at which the delay probability converges to a limit between 0 and 1.
Above and below this critical value, the asymptotic performance of the queue overturns to either one of the extremes.
In Chapter 2, we formalize this heuristic argument and conduct an asymptotic analysis to furthermore reveal the rate at which the limit of performance metrics is attained, depending on the parameters $\eta$ and $\beta$ and the system size $\lambda,{s_\lambda}$.
\subsection{Overdispersed arrivals}
\label{sec:intro_overdispersion}
Until now we have assumed queueing systems with perfect knowledge on the model primitives, including the mean demand per time period. For large-scale service systems, the dominant assumption in the literature is that demand arrives according to a non-homogeneous Poisson process, which in practice translates into the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies of service sysyems shows that the variance of demand typically exceeds the mean significantly, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2003, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}. The feature that variability is higher than one expects from the Poisson assumption is referred to as \textit{overdispersion}.
Due to its inherent connection with the CLT, the dimensioning rule in \eqref{eq:square_root_staffing rule} relies heavily on the premise that the variance of the number of customers entering the system over a period of time is of the same order as the mean.
Subsequently, when stochastic models do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly in critical loading.
To deal with overdispersion new models are needed, scaling rules must be adapted, and existing capacity sizing rules need to be modified in order to incorporate a correct hedge against (increased) variability.
Following our findings in Subsection \ref{sec:intro_characteristics}, we propose a capacity allocation rule similar to \eqref{eq:square_root_staffing rule} in which the original variability hedge is replaced by an amount that is proportional to the square-root of the variance of the arrival process.
In Chapter 3, we elaborate on this idea and show how to adapt the scaling of the queueing process appropriately to achieve QED-type behavior in the presence of overdispersion.
\subsection{Finite-size constraints}
The canonical examples in Section \ref{sec:intro_QED_regime} assume an infinite amount of waiting space.
Physical service systems, however, are typically limited in the number of customers that can be held in the system simultaneously.
For instance in a call center, the maximum number of clients in service or queueing is restricted by the number of available trunk lines \cite{Khudyakov2006}, while in the emergency department of a hospital, the number of beds constrains the number of patients that can be admitted \cite{YomTov2010}.
Depending on the practical setting and admission policy, if the maximum capacity, say $n$, is reached, newly arriving customers can either leave the system immediately (blocking), reattempt getting access later (retrials) or queue outside the facility (holding).
In any case, expectations are that the queueing dynamics within the service facility are affected considerably in the presence of such additional capacity constraints.
We illustrate these implications through the $M/M/s/n$ queue, that is, the standard $M/M/s$ queue with additional property that a customer who finds upon arrival $n$ customers already present in the system, is deferred and considered lost.
To avoid trivialities, let $n\geq s$.
Since the expected workload reaching the servers is less than in unconstrained scenario, one expects less congestion and resource utilization.
Consider the $M/M/{s_\lambda}/n_\lambda$ in the QED regime.
That is, we increase $\lambda$ indefinitely and ${s_\lambda}$ scales as ${s_\lambda}=\lambda+\beta\sqrt{\lambda} + o(\sqrt{\lambda})$.
We then ask how $n_\lambda$ should scale along with $\lambda$ and ${s_\lambda}$ to maintain the non-degenerate behavior as seen in Section \ref{sec:intro_QED_regime}.
We provide a heuristic answer.
Let $Q^{({s_\lambda},n_\lambda)}$ and $W^{({s_\lambda},n_\lambda)}$ denote the number of customers in the system and the waiting time in the $M/M/{s_\lambda}/n_\lambda$ queue in steady state.
Note through Theorem \ref{thm:intro_HW_stationary_distribution} that if there were no finite-size constraints, we would have, for $\lambda$ large,
\begin{align}
\mathbb{P}(Q^{({s_\lambda})}\geq n_\lambda)
&= \mathbb{P}\left(\frac{Q^{({s_\lambda})}-{s_\lambda}}{\sqrt{{s_\lambda}}} \geq \frac{n_\lambda-{s_\lambda}}{\sqrt{{s_\lambda}}}\right) \nonumber \\
&\to
\left\{
\begin{array}{ll}
g(\beta), & \text{if }n_\lambda = {s_\lambda} + o({s_\lambda}),\\
g(\beta)\,{\rm e}^{-\beta \gamma}, & \text{if } n_\lambda = {s_\lambda}+\gamma\sqrt{{s_\lambda}} + o(\sqrt{s_\lambda}),\\
0, & \text{if } n_\lambda = {s_\lambda}+\Omega(\sqrt{{s_\lambda}}),
\end{array}
\right.
\end{align}
as $\lambda\to\infty$ for some $\gamma>0$.
Here, the relation $u(\lambda) = o(v(\lambda))$ implies that $u(\lambda)/v(\lambda) \to 0$ as $\lambda\to\infty$ and $u(\lambda) = \Omega(v(\lambda))$ implies $u(\lambda)/v(\lambda) >1$ for $\lambda\to\infty$.
Hence, asymptotically the finite-size effects only play a role if the extra variability hedge of $n_\lambda$ is of order $\sqrt{{s_\lambda}}$ (or equivalently $o(\sqrt{\lambda})$).
Furthermore, if the variability hedge is $o(\sqrt{\lambda})$, then we argue that asymptotically, all customers who do enter the system have zero probability of delay.
That is, asymptotically, we obtain a loss model.
More formally, under the \textit{two-fold scaling rule}
\begin{equation}
\label{eq:intro_twofold_scaling_rule}
\left\{
\begin{array}{ll}
{s_\lambda} = \lambda + \beta\sqrt{\lambda} + o(\sqrt{\lambda}),\\
n_\lambda = {s_\lambda} + \gamma \sqrt{{s_\lambda}} + o(\sqrt{\lambda}),
\end{array}
\right.
\end{equation}
it is not difficult to deduce that, see e.g. \cite{masseywallace},
\begin{equation}
\mathbb{P}(W^{({s_\lambda},n_\lambda)} > 0) \to \left( 1 + \frac{\beta\,\Phi(\beta)}{(1-{\rm e}^{-\beta\gamma})\varphi(\beta)}\right)^{-1}, \quad \text{as } \lambda\to\infty,
\end{equation}
which is strictly smaller than $g(\beta)$ as in \eqref{fig:delay_probs_HW_MMs}, but still bounded away from both 0 and 1.
Furthermore, the buffer size of the queue is $n_\lambda-{s_\lambda} = \gamma\sqrt{{s_\lambda}}$, so that by Little's law, the expected waiting time of an admitted customer is $O(1/\sqrt{{s_\lambda}})$.
Even though resource utilization in the $M/M/{s_\lambda}/n_\lambda$ is less than in the queue with unlimited waiting space, it can easily be shown that $\rho\to 1$ as $\lambda\to\infty$.
Hence, all three key characteristics of the QED regime are carried over to the finite-size setting if adhered to scaling \eqref{eq:intro_twofold_scaling_rule}.
On a process level, adding a capacity constraint translates to adding a reflection barrier to the normalized queue length process $X^{({s_\lambda},n_\lambda)} = (Q^{({s_\lambda},n_\lambda)} -{s_\lambda} ) /\sqrt{{s_\lambda}}$, at $\gamma$, as is illustrated by the sample paths of $X^{{s_\lambda},n_\lambda}$ for three values of $\lambda$ in Figure \ref{fig:sample_paths_MMsn}.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.66]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda5.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.66]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda50.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda = 50$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.66]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xlabel style={right},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda500.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda},n_\lambda)}(t)$ with $\lambda = 5$, $\lambda=50$ and $\lambda=500$ under scaling \eqref{eq:intro_twofold_scaling_rule} with $\beta=0.5$ and $\gamma = 0.5$.}
\label{fig:sample_paths_MMsn}
\end{figure}
It has been shown by \cite{masseywallace} that under \eqref{eq:intro_twofold_scaling_rule}
\begin{equation}
\label{eq:asymptotic_blocking_prob}
\sqrt{{s_\lambda}}\,\mathbb{P}({\rm block}) = \sqrt{{s_\lambda}} \mathbb{P}(Q^{({s_\lambda},n_\lambda}) = n_\lambda) \to f(\beta,\gamma), \quad \text{as } \lambda \to\infty,
\end{equation}
for a non-negative function $f$.
The idea of the two-fold scaling in \eqref{eq:intro_twofold_scaling_rule} can be extended to settings in which the interior is in fact a network of queues, rather that the single-station setting discussed here, see \cite{Khudyakov2006,YomTov2010,Tan2012} for examples of such \textit{semi-open} queueing networks.
In case customers retry of hold upon being initially refused access to the system, the QED analysis becomes much more tricky, and no explicit limiting result are known.
Nevertheless, observe that the volume of blocked arrivals is by \eqref{eq:asymptotic_blocking_prob} of order $\sqrt{\lambda}$, the exact same magnitude as the variability hedge of both ${s_\lambda}$ and $n_\lambda$.
Therefore, retrials and holding customers have a non-negligible effect on the service levels within the facility in the QED regime.
This will be the topic of Chapters 4 and 5.
\subsection{Pre-limit behavior}
The results on queues in the QED regime discussed in Section \ref{sec:intro_QED_regime} are in two ways of an asymptotic nature.
First, the heavy-traffic limits prescribe the the queueing dynamics for $\lambda,{s_\lambda}\to\infty$.
Real-world systems obviously do not experience infinite demand nor have infinite capacity, and hence the heavy-traffic limits only form an approximation for such finite-sized systems.
Although these approximations are qualitatively insightful, these asymptotic analyses do not reveal much about their accuracy with respect to actual performance.
For instance, we would like to know how fast the convergence takes place, and how inaccuracies in asymptotic performance analyses relate to the staffing schemes in pre-limit systems.
To answer such questions, it would be helpful to have an asymptotic estimate for the difference between the (scaled) queueing process and its limiting counterpart, to be able to judge the error made by relying on asymptotic as opposed to actual performance evaluation.
Characterization of the error term gives rise to so-called \textit{corrected diffusion approximations}, \cite{Siegmund1978,Blanchet2006,Janssen2008}.
Correction diffusion approximations are refinements to the heavy-traffic limits and therefore give a more concise description of large but finite load, which makes them a useful tool in the study of large-scale service systems.
We will derive cprrected diffusion approximation in the context of the novel scaling regimes mentioned in Subsection \ref{sec:intro_novel_scalings} in Chapter 2.
Secondly, the bulk of literature on queues is concerned with the performance analysis and optimization of queueing systems, assuming they are in steady-state, which is tantamount to requiring $t\to\infty$.
However, in practice, service systems certainty do not run infinitely long, which renders this assumption questionable.
Validation of the steady-state assumption is related to the \textit{relaxation time} of a queueing process \cite{Abate1987,Abate1988,relaxation,Leeuwaarden2011,Leeuwaarden2012,Gamarnik2013}, which prescribes the time it takes a system starting out of equilibrium to approximate its stationary distribution sufficiently close.
In case the relaxation time is small, stationary performance evaluation is likely to be accurate.
On the contrary, if the relaxation time is the large, transient analysis of the queueing system is required in order to capture realistic behavior.
Subsequently, we can ask ourselves what are the implications of applying staffing principles that are based upon steady-state performance metrics in settings which are inherently transient over the planning period.
We will touch upon this topic in Chapter 6.
\section{Outline of the thesis}
The remainder of this thesis builds upon the ideas behind the QED scaling regime exhibited in this introductory chapter, and is organized as follows.
Chapter 2 is concerned with the analysis of the limiting behavior of queues in case one deviates from the square-root staffing principle as demand grows large.
Using the bulk-service queue together with the many-sources paradigm as a vehicle, we derive corrected diffusion approximations for the performance metrics of pre-limit systems in these alternative scaling regimes.
The work presented in Chapter 2 is based on \cite{Janssen2015}.
In Chapter 3, we also analyze the bulk-service queueing model, but with many correlated sources, so that demand appears to be overdispersed.
As we eluded to in Subsection \ref{sec:intro_overdispersion}, this requires an alternative scaling of the queue length process and associated staffing rule.
This chapter exhibits the ideas of \cite{Mathijsen2016}.
In Chapter 4 we discuss how QED-type behavior prevails in simple settings in which the system size is finite, given appropriate scaling of capacity levels.
More specifically, we show how customer retrials can be incorporated heuristically into the performance analysis of finite-size systems in the QED regime.
The content of this chapter is based on \cite{Leeuwaarden2015} and \cite{Leeuwaarden2016}.
Building upon the insights gained in Chapter 4, we show in Chapter 5 how the heuristic approximation methods carry over to a more complex finite-size queueing system, inspired by delay analysis in a health care facility.
We show how the QED scaling limits for this model offer surprisingly accurate approximations for realistic model parameter in systems of small to moderate size, which moreover gives rise to a means to dimension service systems of such type.
Chapter 5 is based on the ideas of \cite{Leeuwaarden2016a}.
Chapter 6 investigates the validity of a capacity allocation rule based on steady-state performance metrics in practical settings.
Namely, in realistic scenarios, the parameters of a queueing models are typically subject to change over the planning period.
This asks for a more elaborate transient analysis of the queue dynamics, and an adaption of the staffing level.
In this chapter, we present how to do so appropriately in a single-server queueing model facing a L\'evy input process by prescribing a correction to the steady-state optimum, which has square-root form.
This chapter is based on \cite{Mathijsen2016a}.
Chapter 7 presents the analysis of an inventory model with backlogs, perishable goods and consumer impatience.
This model resembles the inventory level of a blood bank, and can be regarded as a shot-noise model with both positive and negative jumps and exponential decay rates above and below zero.
Besides the derivation of the stationary distribution of the inventory level, we show how under appropriate scaling the process converges to an Ornstein-Uhlenbeck process.
The latter allows for a more tractable approximate analysis of the model in case the number of blood deliveries and demand is large.
Chapter 7 is based on \cite{Bar-Lev2015}.
\chapter{Introduction}
\begin{chapterstart}
Stochastic service systems describe situations in which users compete for service from scarce resources. Think of check-in lines at airports, waiting rooms in hospitals or queues in supermarkets, where the scarce resource is human manpower.
Next to these traditional settings, resource sharing is also important in large-scale service systems such as the internet, wireless networks and cloud computing facilities.
In these virtual environments, geographical location does not play a restricting role on the system size, paving the way for the emergence of large-scale resource sharing networks.
This thesis investigates how to design large-scale systems in order to achieve economies-of-scale, by which we mean that the system is highly occupied and hence utilizes efficiently the expensive resources, while at the same time, the offered service levels remain high.
In this introductory chapter, we give an overview of the available machinery that supports such principles and explain how this thesis contributes to the existing study of large-scale service systems.
A crucial concept behind most of the results discussed in the chapter is the Central Limit Theorem (CLT) -- arguably one of the most important theorems in mathematics and science.
\end{chapterstart}
\newpage
\section{Service systems \& queueing theory}
\subsection{Quality vs. Efficiency}
Large-scale service systems take many shapes and forms.
Classical examples of large-scale service systems include call centers \cite{Erlang1917,Palm1957,Whitt1999,Gans2003,Borst2004,Brown2005,Zeltyn2005,Bassamboo2009,Khudyakov2006} and communication systems \cite{Kleinrock1976,Anick1982,Kelly1985,Kleinrock2007,johanthesis}.
More recently, congestion-related issues in health care facilities and cloud-computing facilities have received much attention \cite{Armony2015,Green2007,YomTov2010,Gupta2007,Tan2012}.
In all settings, one can think of service systems as being composed of \textit{customers} and \textit{servers}.
In call centers, customers typically call to request help from one of the agents (servers).
In communication networks, the data packets are the customers and the communication channels are the servers.
In health care facilities, patients are the customers, and nurses/physicians are the servers.
The system scale may refer to the size of the client base it caters to, or the magnitude of its capacity, or both.
Next to the central notions of customers and servers, we emphasize that service systems are inherently stochastic, that is, subject to uncertainty.
Although arrival volumes can be anticipated to some extent over a certain planning horizon, for instance through historical data and forecasting methods, one cannot predict with certainty future arrival patterns.
Moreover, service requirements are typically random as well, adding more uncertainty.
This intrinsic stochastic variability is a predominant cause of delay experienced by customers in the system.
Due to the inherent randomness in both their arrival and service processes, stochastic models have proved instrumental in both quantifying and improving the operational performance of service systems.
Queueing theory and {\color{col1}probability theory} provide the mathematical tools to describe and evaluate these service systems.
Queueing models are often able to capture and explain fundamental phenomena that are common across applications.
A standard model for service systems is the $M/GI/s$ queue, which we will refer to as the \textit{many-server} queue.
This model assumes that customers arrive to the queue according to a Poisson process with rate $\lambda$, and customer service times are mutually independent and identically distributed (i.i.d.) samples from the distribution of a non-negative random variable $B$.
The parameter $s$ denotes the number of servers in the system, and hence restricts the number of simultaneous services.
The case $s=1$ corresponds to a single-server queue.
First principles say that the queueing process is stable, that is, the number of customers does not explode as time evolves, if and only if the expected workload $R := \lambda\mathbb{E}[B]$ brought into the system per time unit is strictly less than the system capacity.
In other words, the \textit{utilization} of the queue, defined as $\rho := \lambda\mathbb{E}[B] / s$ should remain strictly below one.
Naturally, a system manager prefers to operate at a utilization level close to one, so that resources are used efficiently.
However, it is known that pushing the occupation levels to 100\% leads to an explosive increase in congestion.
{\color{col1}
That is, the expected queue length and customer waiting time increase indefinitely,} thereby reducing the quality-of-service (QoS) and also customer satisfaction.
These seemingly conflicting objectives give rise to a classical trade-off between customer satisfaction and costs of resources.
\subsection{Economies-of-scale}
Under the assumption that service times are exponentially distributed with mean $1/\mu$, the many-server queue reduces to the well-studied $M/M/s$ queue.
Despite its simplicity, the analysis of the $M/M/s$ queue explains mathematically the distinctive traits of queues in general, such as the non-linear effect of utilization on the queue size, and pooling effects.
Let $W^{(s)}$ denote the waiting time of a customer and $Q^{(s)}$ the queue length (including the customers in service) in the steady-state $M/M/s$ queue. Without loss of generality, we fix $\mu=1$, so that $\rho = \lambda/s$.
A straightforward balance argument gives the stationary distribution:
\begin{equation}
\label{eq:MMs_stationary_distribution}
\pi_k := \mathbb{P}( Q^{(s)} = k )
= \left\{
\begin{array}{ll}
\pi_0\frac{\lambda^k}{k!}, & \text{if } k < s, \\
\pi_0\frac{\lambda^s}{s!}\,\rho^{k-s} & \text{if } k \geq s,
\end{array}
\right.
\end{equation}
where
\begin{equation*}
\pi_0 := \Big( \sum_{k=0}^{s-1} \frac{\lambda^k}{k!} + \frac{1}{1-\rho} \frac{\lambda^s}{s!}\Big)^{-1}.
\end{equation*}
Natural QoS indicators include the expected waiting time $\mathbb{E}[W^{(s)}]$ and the delay probability $\mathbb{P}(W^{(s)}>0)$.
{\color{blue}
Invoking the PASTA (Poisson arrivals see time averages) property \cite{Wolff1982}, we know that the delay probability equals the probability of the queue length being greater or equal to the number of servers $s$.
Thus,
\begin{equation}
\label{eq:MMs_wait}
\mathbb{P}(W^{(s)} > 0) = \mathbb{P}(Q^{(s)}\geq s) = \frac{\lambda^s}{s!} \Big( (1-\rho) \sum_{k=0}^{s-1} \frac{\lambda^k}{k!} + \frac{\lambda^s}{s!} \Big)^{-1}.
\end{equation}
By Little's law, which says that $\mathbb{E}[(Q^{(s)}-s)^+] =\lambda\mathbb{E}[W^{(s)}]$, we furthermore have
\begin{equation}
\mathbb{E}[W^{(s)}] = \mathbb{P}(W^{(s)} > 0)\,\frac{1/s}{1-\rho}.
\label{eq:MMs_wait2}
\end{equation}
}
From these formulae, it is readily seen that $\mathbb{P}(W^{(s)} > 0) \to 1$ and $\mathbb{E}[W^{(s)}] \to \infty$ as $\rho \uparrow 1$ . That is, increasing $\lambda$ to $s$, while keeping the latter fixed, leads to a system in which all customers are delayed before service, and the expected delay before reaching a server increases to infinity.
The $M/M/s$ queue also reveals the effect of \textit{resource pooling}.
To illustrate the operational benefits of sharing resources, we compare a system of $s$ separate $M/M/1$ queues, each serving a Poisson arrival stream with rate $\lambda<1$, against one $M/M/s$ queue facing arrival rate $\lambda s$.
The two systems thus experience the same workload and utilization, namely $\rho = \lambda$.
We fix the value of $\lambda$ and vary $s$.
Obviously, the waiting time and queue length distribution in the first scenario are unaffected by the parameter $s$, since there is no interaction between the single-server queues.
This lack of coordination tolerates a scenario of having an idle server, while the total number of customers in the system exceeds $s$, therefore wasting resource capacity.
Such an event cannot happen in the many-server scenario, due to the central queue.
This central coordination improves QoS. Indeed Figure \ref{fig:waiting_time_pooling} shows that the reduction in expected waiting time can be substantial.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\input{tikz_tex/Ewait_pooling.tex}
\caption{Expected waiting time}
\end{subfigure}
\hspace{2mm}
\begin{subfigure}{0.48\textwidth}
\input{tikz_tex/Pwait_pooling.tex}
\caption{Probability of delay}
\end{subfigure}
\caption{Effects of resource pooling in the $M/M/s$.}
\label{fig:waiting_time_pooling}
\end{figure}
\noindent
So pooling kills two birds with one stone: QoS for customers improves and the system efficiency increases.
\subsection{Many-server scaling regimes}
\label{sec:intro_many_server_regimes}
Now that we know that economies-of-scale can be achieved, it is relevant to ask how to match capacity $s$ to a demand $\lambda$ in the setting where both $s$ and $\lambda$ become large.
The expressions in \eqref{eq:MMs_wait} and \eqref{eq:MMs_wait2} provide a starting point for finding such demand-capacity relations, particularly when we apply asymptotic analysis for $s\to\infty$, \cite{Halfin1981,Borst2004,Reed2009}.
Asymptotic theory of many-server systems relies on the prerequisite that the limiting behavior of the service system is determined by the way in which capacity $s$ is adjusted to demand, assuming demand grows large.
We illustrate this idea by investigating typical sample paths of the queue length process $Q = \{Q(t)\}_{t\geq 0}$ of an $M/M/s$ queue for increasing values of $\lambda$.
Figure \ref{fig:sample_path_small} depicts a sample path for $\lambda = 3$ and $s = 4$.
The number of customers queueing at time $t$ is given by $(Q(t)-s)^+$ with $(\cdot)^+ := \max\{0,\cdot\}$.
The number of idle servers is given by $(s-Q(t))^+$.
In Figure \ref{fig:sample_path_small}, the red and green area hence represent the cumulative queue length and cumulative number of idle server, respectively, over the given time period.
Bearing in mind the dual goal of QoS and efficiency, we want to minimize both of these areas simultaneously.
\begin{figure}[b!]
\centering
\input{tikz_tex/sample_path_small.tex}
\caption{Sample path of the $M/M/s$ queue with $\lambda = 3$ and $s=4$.}
\label{fig:sample_path_small}
\end{figure}
Next, we conduct a similar sample path experiment for increasing values of $\lambda$.
Since $s > \lambda$ is required for stability, the value of $s$ needs to be adjusted accordingly.
We propose three scaling rules:
\begin{equation}
\label{eq:intro_three_scaling_rules}
s^{(1)}_\lambda = \left[ \lambda + \beta \right ], \qquad
s^{(2)}_\lambda = \left[ \lambda + \beta\sqrt{\lambda} \right], \qquad
s^{(3)}_\lambda = \left[ \lambda + \beta\,\lambda \right],
\end{equation}
for some $\beta>0$, where $[\cdot]$ denotes the rounding operator.
Note that these three rules differ in terms of overcapacity $s-\lambda$.
Figure \ref{fig:sample_paths_lambda100} depicts typical sample paths of the queue length process for increasing values of $\lambda$ for the three scaling rules with $\beta = 0.5$.
\begin{figure}
\centering
\input{tikz_tex/sample_paths_lambda10.tex}
\caption{Sample paths of the $M/M/s$ queue with $\lambda = 10,\,50$ and $100$ and $s$ set according to the three scaling rules in \eqref{eq:intro_three_scaling_rules}.}
\label{fig:sample_paths_lambda100}
\end{figure}
Observe that for all scaling rules, the stochastic fluctuations of the queue length processes relative to $\lambda$ decrease with the system size.
Moreover, the paths in Figure \ref{fig:sample_paths_lambda100} appear to become smoother with increasing $\lambda$.
Of course, the actual sample path always consists of upward and downward jumps of size 1, but we will show how proper centering and scaling of the queue length process indeed gives rise to a \textit{diffusion process} in the limit as $\lambda\to\infty$.
Although the difference in performance of the three regimes is not yet evident for relatively small $\lambda$, clear distinctive behavior occurs for large $\lambda$.
Under ${s_\lambda}^{(1)}$, the majority of customers is delayed and server idle time is low, since $\rho = (1+\beta/\lambda)^{-1} \to 1$ as $\lambda \to \infty$.
Systems dimensioned according to this rule value server efficiency over customer satisfaction and therefore this regime is in the literature also known as the \textit{efficiency-driven} (ED) regime \cite{Zeltyn2005}.
In contrast, the third scaling rule $s^{(3)}_\lambda$ yields a constant utilization level $\rho = 1/(1+\beta)$, which stays away from 1, even for large $\lambda$.
Queues operating in this regime exhibit significant server idle times.
Moreover, for the particular realization of the queueing processes for $\lambda = 50$ and $\lambda=100$ none of the customers waits.
This customer-centered regime is known as the \textit{quality-driven} (QD) regime \cite{Zeltyn2005}.
The scaling rule $s^{(2)}_\lambda$ is in some ways a combination of the other two regimes.
First, we have $\rho = (1 +\beta/\sqrt{\lambda})^{-1} \to 1$ as $\lambda \to \infty$, which indicates efficient usage of resources as the system grows.
The sample paths, however, indicate that only a fraction of the customers is delayed, and only small queues arise, which suggest good QoS.
This regime is therefore called \textit{quality-and-efficiency driven} (QED) regime.
Since this scaling regime and the related \textit{square-root staffing rule}
\begin{equation}
\label{eq:square_root_staffing rule}
s_\lambda = \lambda + \beta\sqrt{\lambda}
\end{equation}
strikes the right balance between the two profound objectives of capacity allocation in service systems, we discuss in the next section the mathematical foundations of the QED regime and quantify the favorable properties revealed by Figure \ref{fig:sample_paths_lambda100}.
\section{The QED regime: two canonical examples}
\label{sec:intro_QED_regime}
We saw in Figure \ref{fig:waiting_time_pooling} the advantageous effect of resource pooling and economies-of-scale in many-server systems.
In this section, we will explain how this is related to the Central Limit Theorem (CLT).
\begin{theorem}[Central Limit Theorem, e.g. {\cite[Thm.~27.1]{Billingsley1995}}]
Consider a sequence $X_1,X_2,\ldots,X_n$ of independent {\color{blue} and identically distributed} random variables having mean $\mu$ and positive variance $\sigma^2$.
Then,
\[
\frac{\sum_{i=1}^n X_i - n\mu }{\sqrt{n}\sigma} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{for }n\to\infty.
\]
where ${\;\buildrel{d}\over\Rightarrow\;}$ denotes convergence in distribution and $\mathcal{N}(0,1)$ is a random variable with standard normal distribution.
\end{theorem}
We shall now apply the CLT to the delay probability in the $M/M/s$ queue.
Striking the proper balance between queueing delay and server efficiency asymptotically, i.e.~balancing the green and red areas in Figure \ref{fig:sample_paths_lambda100}, in mathematical terms boils down to choosing a service level $s_\lambda$ such that both the delay probability $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ and $\mathbb{P}(Q^{(s_\lambda)} < s_\lambda)$ remain strictly smaller than 1 as $\lambda\to\infty$.
In other words, one would like to see that $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ converges to a non-degenerate limit $\alpha \in (0,1)$ as $\lambda\to\infty$.
To get a feel for the natural scale of the queue, we first examine the situation with unlimited capacity.
More precisely, let $Q^{(\infty)}$ be the number of customers in a steady-state $M/GI/\infty$ queue with mean service requirement $\mathbb{E}[B]=1$.
Notice that in this infinite-server setting, $Q^{(\infty)}$ also represents the steady-state number of busy servers.
It is well known that $Q^{(\infty)}$ follows a Poisson distribution with mean equal to the expected workload, in our case $R = \lambda$.
Moreover, if we assume that $\lambda$ is integer, then a Poisson random variable with rate $\lambda$ can be viewed as the sum of $\lambda$ i.i.d. Poisson random variables with rate 1.
In other words, $Q^{(\infty)} = \sum_{i=1}^\lambda P_i$, where the $P_i$'s, $i=1,2,\ldots,n$, have Poisson distribution with unit mean and variance, and are mutually independent.
\noindent
The CLT thus gives
\begin{equation}
\label{eq:infinite_server_tail}
\mathbb{P}(Q^{(\infty)} \geq x_\lambda )
= \mathbb{P}\left(\frac{Q^{(\infty)} -\lambda }{\sqrt{\lambda}} \geq \frac{ x_\lambda - \lambda}{\sqrt{\lambda}} \right)
\approx 1-\Phi\left( \frac{x_\lambda-\lambda}{\sqrt{\lambda}} \right),
\end{equation}
where $\Phi$ denotes the cumulative distribution function of the standard normal distribution for large $\lambda$.
Hence, the probability in \eqref{eq:infinite_server_tail} converges to a constant value away from both 0 and 1 if and only if $(x_\lambda - \lambda)/\sqrt{\lambda} \to x \in \mathbb{R}$, or equivalently $x_\lambda = \lambda + x \sqrt{\lambda} + o(\sqrt{\lambda})$, as $\lambda\to\infty$.
Here, the relation $u(\lambda) = o(v(l))$ implies that $u(\lambda)/v(\lambda)\to 0$ as $\lambda\to\infty$.
Equation \eqref{eq:infinite_server_tail} also shows that the leading order of the random variable describing the queue length is $\lambda$, while the stochastic fluctuations are of order $\sqrt{\lambda}$.
If we now pretend, for a moment, that the infinite-server queue serves as a good approximation for the many-server queue with $s_\lambda$ servers, then \eqref{eq:infinite_server_tail} says that the steady-state probability of delay for ${s_\lambda} = \lambda +\beta\sqrt\lambda$ obeys the Gaussian approximation
\begin{equation}
\label{eq:infinite_server_approx_delay}
\mathbb{P}(W^{(s_\lambda)}>0) = \mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda ) \approx 1-\Phi(\beta),
\end{equation}
where $\Phi$ denotes the cumulative distribution function (cdf) of the standard normal distribution.
Of course, the infinite-server system ignores the one thing that makes a queueing system unique, namely that a queue is formed when all servers are busy.
During these periods of congestion, customers will depart from a system with a finite number of servers at a slower pace than in its infinite-server counterpart.
So the approximation in \eqref{eq:infinite_server_approx_delay} is likely to overestimate the actual delay probability, and a more careful investigation of the queue length process in many-server settings is needed. Nevertheless, the infinite-server heuristic reveals that in a well-managed system, i.e.~queues are of acceptable length, the size at which the system operates is of the order $\lambda$, with fluctuations of order $\sqrt{\lambda}$.
We shall now demonstrate through two canonical examples how these guessed natural scalings can be turned into mathematically rigorous statements.
Both examples which will play a key role in this thesis.
\subsection{The $M/M/s$ queue}
\label{sec:intro_MMsqueue}
\textbf{Converging delay probability.}
Let $Q^{(s)}$ denote the steady-state number of customers in an $M/M/s$ queue with arrival rate $\lambda$ and mean service requirement 1, of which the probability distribution is given in \eqref{eq:MMs_stationary_distribution}.
Halfin \& Whitt \cite{Halfin1981} showed that, just as the tail probability in the infinite-server setting, the delay probability in the $M/M/s$ queue converges under scaling \eqref{eq:square_root_staffing rule} to a value between 0 and 1.
Moreover, they showed that this is in fact the only scaling regime in which such a non-degenerate limit exists and identified its value.
Let $\varphi$ denote the probability density function (pdf) of the standard normal distribution.
\begin{proposition}[{\cite[Prop.~2.1]{Halfin1981}}]
\label{prop:HalfinWhitt_delay_probability}
The probability of delay in the $M/M/s_\lambda$ queue has the non-degenerate limit
\begin{equation}
\lim_{\lambda\to\infty} \mathbb{P}( W^{(s_\lambda)} > 0 ) = \alpha \in (0,1),
\end{equation}
if and only if
\begin{equation}
\label{eq:HalfinWhitt_scaling}
\lim_{\lambda\to\infty} (1-\rho_{s_\lambda}) \sqrt{s_\lambda} \to \beta, \quad \beta > 0,
\end{equation}
where $\alpha$ is given by
\begin{equation}\label{eq:HW_delay_prob}
\alpha = \left( 1+ \frac{\beta\,\Phi(\beta)}{\varphi(\beta)} \right)^{-1} =: g(\beta).
\end{equation}
\end{proposition}
\begin{proof}
We first prove the sufficiency condition.
Rewrite \eqref{eq:MMs_wait} as
\begin{equation}
\label{eq:proof_HW_0}
\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )
= \left( 1 + (1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}\right) ^{-1}.
\end{equation}
Similar to \eqref{eq:infinite_server_tail} we find
\begin{align}
\mathbb{P}({\rm Pois}(\lambda) < {s_\lambda})
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < \frac{{s_\lambda}-\lambda}{\sqrt{\lambda}}\right) \nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\frac{{s_\lambda}}{\sqrt\lambda}\right)\nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\sqrt{{s_\lambda}}\left(1+o(1)\right) \right) \to \Phi(\beta),
\label{eq:proof_HW_1}
\end{align}
for $\lambda\to\infty$.
Using Stirling's approximation, we get
\begin{align*}
\mathbb{P}({\rm Pois}(\lambda)=s) &= {\rm e}^{-\lambda}\frac{\lambda^{{s_\lambda}}}{{s_\lambda}!}
\sim {\rm e}^{-\lambda} \lambda^{{s_\lambda}}\cdot \frac{1}{\sqrt{2\pi\,{s_\lambda}}} \left(\frac{\rm e}{{s_\lambda}}\right)^{{s_\lambda}} = \frac{1}{\sqrt{2\pi{s_\lambda}}}\,{\rm e}^{{s_\lambda}-\lambda - {s_\lambda}{\rm ln}(\rho_{{s_\lambda}})}.
\end{align*}
Since ${\rm ln}(\rho_{{s_\lambda}}) = -(1-\rho_{{s_\lambda}}) - \tfrac{1}{2}(1-\rho_{{s_\lambda}})^2 + o((1-\rho_{{s_\lambda}})^2)$ we find that
\begin{equation}
\label{eq:proof_HW_2}
\frac{ \mathbb{P}({\rm Pois}(\lambda) = s) }{ 1-\rho_{{s_\lambda}} }
= \frac{1}{(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}} \, \frac{{\rm e}^{ -\tfrac{1}{2}(1-\rho_{{s_\lambda}})^2{s_\lambda} + o\left((1-\rho_{{s_\lambda}})^2{s_\lambda}\right)}}{\sqrt{2\pi}} \to \frac{1}{\beta}\, \frac{{\rm e}^{{-}\tfrac{1}{2} \beta^2}}{\sqrt{2\pi}} = \frac{\varphi(\beta)}{\beta}.
\end{equation}
Substituting \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2} into \eqref{eq:proof_HW_0} gives \eqref{eq:HW_delay_prob}.
The necessary condition follows directly by the characterization of $\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )$ as in \eqref{eq:proof_HW_0} by observing, through \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2}, that the term
\begin{equation*}
(1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}
\end{equation*}
has a limiting value in $(0,\infty)$ only if $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}} \to \beta$ for some $\beta>0$.
\end{proof}
Observe that $g(\beta)$ is a strictly decreasing function on $(0,\infty)$ with $g(\beta) \to 1$ as $\beta\to 0$ and $g(\beta)\to 0$ for $\beta\to\infty$.
Thus all possible delay probabilities are achievable in the QED regime, which will prove useful for the dimensioning of systems (see Section \ref{sec:intro_dimensioning}).
\begin{figure}
\centering
\input{tikz_tex/halfin_whitt_accuracy.tex}
\caption{The delay probability $\mathbb{P}(Q^{({s_\lambda})} \geq {s_\lambda})$ with ${s_\lambda} = [ \lambda + \beta \sqrt{\lambda} ]$ for $\beta = 0.1,\ 0.5,$ and 1 as a function of $\lambda$.}
\label{fig:delay_probs_HW_MMs}
\end{figure}
Although Proposition \ref{prop:HalfinWhitt_delay_probability} is an asymptotic result for $\lambda\to\infty$, Figure \ref{fig:delay_probs_HW_MMs} shows that $g(\beta)$ can serve as an accurate approximation for the delay probability for relatively small $\lambda$.
From Proposition \ref{prop:HalfinWhitt_delay_probability}, it also follows that under \eqref{eq:HalfinWhitt_scaling},
\begin{equation}
\label{eq:halfinwhitt_wait}
\sqrt{{s_\lambda}}\,\mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{P}(W^{({s_\lambda})}>0)}{(1-\rho_{s_\lambda})\sqrt{{s_\lambda}}} \to \frac{g(\beta)}{\beta} =: h(\beta), \qquad \text{ as }\lambda\to\infty,
\end{equation}
where we have used the characterization of $\mathbb{E}[W^{({s_\lambda})}]$ in \eqref{eq:MMs_wait2}.
This implies that in the QED regime, the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$ as $\lambda\to\infty$.
By Little's law this implies that the expected queue length is $O(\sqrt{{s_\lambda}})$.
By the relation $u(\lambda) = O(v(\lambda))$ we mean that $\limsup_{\lambda\to\infty} u(\lambda)/v(\lambda)< \infty$.
While these are all steady-state results, similar statements can be made for the entire queue-length process, as shown next.
The theoretical results of the QED regime we presented here are based on steady-state queueing analysis.
But at the heart of the QED theory lies a much deeper result in which the entire queue-length process, over all points in time, converges to some other limiting process.
\\*
\noindent\textbf{Process-level convergence.}
Obtaining rigorous statements about stochastic-process limits poses considerable mathematical challenges.
Rather than presenting the deep technical details of the convergence results, we give a heuristic explanation of how the limiting process arises and what it should look like.
The queue-length process $Q^{({s_\lambda})}$ in Figure \ref{fig:sample_paths_lambda100} with scaling rule ${s_\lambda} = [\lambda + \beta \sqrt{\lambda}]$ appears to concentrate around the level ${s_\lambda}$.
As argued before, the stochastic fluctuations are of order $\sqrt{\lambda}$, or equivalently $\sqrt{{s_\lambda}}$.
For that reason, we consider the centered and scaled process
\begin{equation}
\label{eq:intro_scaled_queue_length_process}
X^{({s_\lambda})}(t) := \frac{ Q^{({s_\lambda})}(t) - {s_\lambda}}{\sqrt{{s_\lambda}}}, \qquad \text{ for\ all } t\geq 0,
\end{equation}
and ask what happens to this process as $\lambda\to\infty$.
First, we consider the expected drift conditioned on $X^{({s_\lambda})}(t) = x$.
When $x> 0$, this corresponds to a state in which $Q^{({s_\lambda})}(t)>{s_\lambda}$ and hence all servers are occupied.
Therefore, the expected rate at which customers leave the system is ${s_\lambda}$, while the arrival rate remains $\lambda$, so that the expected drift of $X^{({s_\lambda})}(t)$ in $x>0$ satisfies
\[
\frac{\lambda - {s_\lambda}}{\sqrt{{s_\lambda}}} \to -\beta, \qquad \text{as }\lambda\to\infty,
\]
under scaling $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to \beta$ in \eqref{eq:HalfinWhitt_scaling}.
When $x\leq 0$, only ${s_\lambda} + x\sqrt{{s_\lambda}}$ servers are working, so that the net drift is
\[
\frac{\lambda - ({s_\lambda} + x\sqrt{{s_\lambda}} )}{\sqrt{{s_\lambda}}} \to -\beta-x, \qquad \text{as }\lambda\to\infty.
\]
Now, imagine what happens to the sample paths of $\{X^{({s_\lambda})}(t)\}_{t\geq 0}$ as we increase $\lambda$.
Within a fixed time interval, larger $\lambda$ and ${s_\lambda}$ will trigger more and more events, both arrivals and departures.
Also, the jump size at each event epoch decreases as $1/\sqrt{{s_\lambda}}$ as a consequence of the scaling in \eqref{eq:intro_scaled_queue_length_process}.
Hence, there will be more events, each with a smaller impact, and in the limit as $\lambda\to\infty$, there will be infinitely many events of infinitesimally small impact.
This heuristic explanation suggests that the process $X^{({s_\lambda})}(t)$ converges to a stochastic-process limit, which is continuous, and has infinitesimal drift ${-}\beta$ above zero and ${-}\beta-x$ below zero.
Figure \ref{fig:sample_paths_diffusion} visualizes the appearance of the suggested scaling limit as $\lambda$ and ${s_\lambda}$ increase.
\begin{figure}
\centering
\input{tikz_tex/sample_path_diffusion.tex}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda})}(t)$ with $\lambda = 5$, $\lambda=5$ and $\lambda=500$ and ${s_\lambda} = [\lambda+0.5\sqrt{\lambda}]$.}
\label{fig:sample_paths_diffusion}
\end{figure}
The following theorem by Halfin \& Whitt \cite{Halfin1981} characterizes this scaling limit more formally.
\begin{theorem}
\label{thm:Halfin_Whitt_diffusion}
Let $X^{({s_\lambda})}(0)\, {\;\buildrel{d}\over\Rightarrow\;} X(0) \in \mathbb{R}$ and $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to\beta$. Then for all $t\geq 0$,
\[
X^{({s_\lambda})}(t) {\;\buildrel{d}\over\Rightarrow\;} X(t),\qquad \text{ as }\lambda\to\infty,
\]
where $X(t)$ is the diffusion process with infinitesimal drift $m(x)$ given by
\[
m(x) = \left\{
\begin{array}{ll}
-\beta, & \text{if }x> 0,\\
-\beta-x, & \text{if } x \leq 0
\end{array}\right.
\]
and infinitesimal variance $\sigma^2(x) = 2$.
\end{theorem}
The limiting diffusion process $\{X(t)\}_{t\geq 0}$ in Theorem \ref{thm:Halfin_Whitt_diffusion} is a combination of a negative-drift Brownian motion in the upper half plane and an Ornstein-Uhlenbeck (OU) process in the lower half plane.
We refer to this hybrid diffusion process as the Halfin-Whitt diffusion.
Much is known for such diffusion processes with piecewise linear drift coefficient, see \cite{Leeuwaarden2012,Fralix2014}.
Its stationary distribution can for instance be derived, see e.g. \cite{Browne1995}.
\begin{proposition}
\label{thm:intro_HW_stationary_distribution}
Let $X(t) {\;\buildrel{d}\over\Rightarrow\;} X(\infty)$ as $t\to\infty$ for a random variable $X(\infty)$ and $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}\to \beta$ for $\lambda\to\infty$.
Then
\begin{align}
\mathbb{P}(X(\infty) > 0 ) &= g(\beta),\\
\mathbb{P}(X(\infty) \geq x | X(\infty) > 0) &= {\rm e}^{-\beta x} ,\quad \text{for }x>0,\\
\mathbb{P}(X(\infty) \leq x | X(\infty) \leq 0 ) &= \frac{\Phi(\beta+x)}{\Phi(\beta)},\quad \text{for }x\leq 0.
\end{align}
\end{proposition}
\noindent
This result shows that as the system grows large, the $Q^{({s_\lambda})}(t)$ concentrates around ${s_\lambda}$, and the fluctuations are of order $\sqrt{{s_\lambda}}$.
Moreover, Proposition \ref{thm:intro_HW_stationary_distribution} iterates the limiting values for the delay probability and scaled expected delay. Namely,
\[ \mathbb{P}\big(W^{({{s_\lambda}})} > 0 \big) \rightarrow \mathbb{P}( X(\infty) > 0 ) = g(\beta)\]
and
\[ \sqrt{{s_\lambda}}\mathbb{E}[W^{({s_\lambda})}] \approx \frac{\mathbb{E}[ Q^{({s_\lambda})}]}{\sqrt{{s_\lambda}}} \rightarrow \mathbb{E}[X(\infty)] = \int_0^\infty g(\beta){\rm e}^{-\beta x} {\rm d} x = \frac{g(\beta)}{\beta},\]
For obvious reasons, the QED regime is also referred to as the Halfin-Whitt regime, and both these names are used interchangeably in this thesis.
\subsection{The $M/D/s$ queue}
\label{sec:intro_discrete_model}
We next consider a many-server queue with deterministic service requirements equal to one, a Poisson arrival process of rate $\lambda$ and ${s_\lambda}$ servers.
We let $Q^{({s_\lambda})}(t)$ be the process describing the number of customers in the system and only examine the process at discrete time epochs $t=0,1,2,\ldots$.
In our analysis, we focus on the queue length process $Z^{({s_\lambda})}(t) := (Q^{({s_\lambda})}(t) - {s_\lambda})^+$.
Since we discretize time, the number of new arrivals per time period is given by the sequence of i.i.d.~random variables $\{A_k\}_{k\geq 1}$, which has a Poisson distribution with mean $\lambda$.
At the start of the $k^{\rm th}$ period, $Z^{({s_\lambda})}(k)$ customers are waiting.
Because the service time of a customer is equal to the period length, all $\min\{Q^{({s_\lambda})}(k),{s_\lambda}\}$ customers in service at the beginning of the period will have left the system by time $t=k+1$.
This implies that $\min\{Z^{({s_\lambda})}(k),{s_\lambda}\}$ of the waiting customers are taken into service during period $k$, but could not possibly have departed before its end, due to the deterministic service times.
If $Z^{({s_\lambda})}<{s_\lambda}$, then additionally $\min\{ A_k , {s_\lambda}-Z^{({s_\lambda})}(k) \}$ of the new arrivals are taken into service.
This yields a total of $A_k$ arrivals, and $\min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\}$ departures from the queue during period $k$, which gives the Lindley type recursion \cite{Lindley1952}, with $Z^{({s_\lambda})}(0) = 0$,
\begin{equation}
\label{eq:discrete_recursion}
Z^{({s_\lambda})}(k+1) = Z^{({s_\lambda})}(k) + A_k - \min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\} = \max\{ 0,Z^{({s_\lambda})}(k) + A_k - {s_\lambda} \}.
\end{equation}
The queue length process thus gives rise to a random walk with i.i.d.~steps of size
$(A_1-{s_\lambda})$, with a reflecting barrier at zero. We can iterate the recursion in \eqref{eq:discrete_recursion} to find
\begin{align}
Z^{({s_\lambda})}(k+1) &= \max\left\{ 0 , Z^{({s_\lambda})}(k) + A_k-{s_\lambda} \right\} \nonumber\\
&= \max\left\{ 0 , \max\{ 0 , Z^{({s_\lambda})}(k-1) + (A_{k-1}-{s_\lambda})\} + (A_k-{s_\lambda})\} \right\}\nonumber \\
&= \max\left\{ 0 , (A_k-{s_\lambda}) , Z^{({s_\lambda})}(k-1) + (A_k-{s_\lambda}) + (A_{k-1}-{s_\lambda})\right\}\nonumber \\
&= \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_{k-i}-{s_\lambda})\Big\}
{\;\buildrel{d}\over= \;} \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_i-{s_\lambda}) \Big\},
\label{eq:max_randomwalk}
\end{align}
where the last equality in distribution holds due to the duality principle for random walks, see e.g.~\cite[Sec.~7.1]{Ross1996}.
For stability, the expected step size satisfies $\mathbb{E}[A_k - {s_\lambda}] = \lambda-{s_\lambda} < 0$.
We use the shorthand notation for the partial sum $S_k := \sum_{i=1}^k (A_i-{s_\lambda})$.
Let $Z^{({s_\lambda})}(\infty):= \lim_{k\to\infty} Z^{({s_\lambda})}(k)$ denote the stationary queue length in this $M/D/s$ queue, which can be shown to exist under our assumptions.
The probability generating function (pgf) of $Z^{({s_\lambda})}(\infty)$ can then be expressed in terms of the pgf of the positive parts of the partial sum:
\begin{equation}
\label{eq:Spitzers_identity}
\mathbb{E}[ w^{Z^{({s_\lambda})}(\infty)} ]
= \exp\Big\{ - \sum_{k=1}^\infty \frac{1}{k}\, (1- \mathbb{E}[w^{S_k^+}]) \Big\},\qquad |w|\leq 1.
\end{equation}
From \eqref{eq:Spitzers_identity}, {\color{blue}which is a special case of Spitzer's identity~\cite{Spitzer1964},} we obtain for the mean queue length and empty-queue probability the expressions
\begin{align}
\mathbb{E}[Z^{({s_\lambda})}(\infty)] &= \sum_{k=1}^\infty \frac{1}{k}\, \mathbb{E}[ S_k^+ ],\nonumber\\
\mathbb{P}(Z^{({s_\lambda})}(\infty) = 0 ) &= \exp\Big\{ -\sum_{k=1}^\infty \frac{1}{k}\, \mathbb{P}( S_k^+ > 0 ) \Big\}.
\label{eq:spitzer_expressions}
\end{align}
Although explicit, the expressions in \eqref{eq:spitzer_expressions} reveal little of the structure of the queue length process.
Hence, we again turn to asymptotics. \\
\noindent\textbf{Gaussian random walk.}
\label{sec:intro_gaussian_random_walk}
We take another look at the identity in \eqref{eq:max_randomwalk}, and ask ourselves what happens if $\lambda$ grows large.
Since $\mathbb{E}[A_k-{s_\lambda}] = \lambda-{s_\lambda} = -\beta\sqrt{\lambda} + o(\sqrt{\lambda})$ under the QED scaling \eqref{eq:square_root_staffing rule}, it makes sense to consider the scaled queue length process $X^{({s_\lambda})}(k) := Z^{({s_\lambda})}(k)/\sqrt{\lambda}$ for all $k\geq 0$, with scaled steps $Y_k^{({s_\lambda})} := (A_k-{s_\lambda})/\sqrt{\lambda}$.
Dividing both sides of \eqref{eq:max_randomwalk} by $\sqrt{\lambda}$ then gives
\begin{equation}
X^{({s_\lambda})}(k+1) = \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j Y^{({s_\lambda})}_i \Big\}.
\end{equation}
Observe that $A_k {\;\buildrel{d}\over= \;} {\rm Pois}(\lambda)$ with ${\rm Pois}(\lambda)$ a random variable with mean $\lambda$.
Hence by the CLT
\begin{equation*}
Y^{({s_\lambda})}_k = \frac{ A_k - {s_\lambda} }{\sqrt\lambda} = \frac{A_k-\lambda}{\sqrt\lambda} - \beta \ {\;\buildrel{d}\over\Rightarrow\;} \ Y_k {\;\buildrel{d}\over= \;} \mathcal{N}(-\beta,1),
\end{equation*}
for $\lambda\to\infty$, where $\mathcal{N}(-\beta,1)$ denotes a normally distributed random variable with mean $-\beta$ and standard deviation 1.
So we expect the scaled queue length process to converge in distribution to a reflected random walk with normally distributed increments, i.e. a reflected \textit{Gaussian random walk}.
Indeed, it is easily verified that \cite{Janssen2008a},
\begin{equation}
X^{({s_\lambda})}(k)\ {\;\buildrel{d}\over\Rightarrow\;} \ M_\beta(k) := \max_{0\leq j\leq k} \Big\{\sum_{i=1}^j Y_i \Big\}, \qquad \lambda\to\infty.
\end{equation}
Let $M_\beta:= \lim_{k\to\infty} M_\beta(k)$ denote the all-time maximum of a Gaussian random walk.
It can be shown that $M_\beta$ almost surely exists and that
\[
X^{({s_\lambda})}(\infty) := \lim_{k\to\infty} X^{({s_\lambda})}(k) {\;\buildrel{d}\over\Rightarrow\;} M_\beta,
\]
for instance by \cite[Prop.~19.2]{Spitzer1964} and \cite[Thm.~X6.1]{Asmussen2003}.
The following theorem can be proved using a similar approach as in \cite{Jelenkovic2004}.
(We prove this result in a more general setting in Chapter 3.)
\begin{theorem}
Let $X^{({s_\lambda})}(\infty)$ be the scaled queue length in steady-state. If $(1-\rho_{{s_\lambda}})\sqrt{\lambda}\to\beta$, then as $\lambda\to\infty$,
\begin{enumerate}
\item[\normalfont (i)] $X^{({s_\lambda})}(\infty) {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[\normalfont (ii)] $\mathbb{P}(X^{({s_\lambda})}(\infty) = 0) \to \mathbb{P}(M_\beta = 0)$,
\item[\normalfont (iii)] $\mathbb{E}[X^{({s_\lambda})}(\infty)^k] \to \mathbb{E}[M_\beta^k]$, for any $k>0$.
\end{enumerate}
\end{theorem}
The Gaussian random walk is well studied \cite{Siegmund1978,Chang1997,Janssen2006,Blanchet2006,Janssen2006} and there is an intimate connection with Brownian motion.
The only difference, one could say, is that Brownian motion is a continuous-time process, whereas the Gaussian random walk only changes at discrete points in time.
If $\{B(t)\}_{t\geq 0}$ is a Brownian motion with drift $-\mu <0$ and infinitesimal variance $\sigma^2$ and $\{W(t)\}_{t \geq 0}$ is a random walk with $\mathcal{N}(-\mu,\sigma^2)$ steps and $B(0) = W(0)$, then $W$ can be regarded as the process $B$ embedded at equidistant time epochs.
That is, $W(t) {\;\buildrel{d}\over= \;} B(t)$ for all $t\in\mathbb{N}^+$.
For the maximum of both processes this coupling implies
\begin{equation}
\max_{k\in \mathbb{N}^+} W(k) = \max_{k\in \mathbb{N}^+} B(k) \leq_{\rm st}
\max_{t\in \mathbb{R}^+} B(t),
\label{eq:max_inequality}
\end{equation}
where $\leq_{\rm st}$ denotes stochastic dominance.
This difference in maximum is visualized in Figure \ref{fig:BrownianMotion_vs_GaussianRW}.
It is known that the all-time maximum of Brownian motion with negative drift $-\mu$ and infinitesimal variable $\sigma^2$ has an exponential distribution with mean $\sigma/2\mu$ \cite{Harrison1985}.
Hence, \eqref{eq:max_inequality} implies that $M_\beta$ is stochastically upper bounded by an exponential random variable with mean $1/2\beta$.
\begin{figure}
\centering
\begin{tikzpicture}[scale = 1]
\begin{axis}[
xmin = 0,
xmax = 10,
ymin = -2.2,
ymax = 5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel={$t$},
ylabel={},
xscale=1,
yscale=0.8]
\addplot[gray] file {tikz/Brownian_Motion_SamplePath/BM.txt};
\addplot[only marks,mark size = 2] file {tikz/Brownian_Motion_SamplePath/GW.txt};
\addplot[dashed] file {tikz/Brownian_Motion_SamplePath/maxBM.txt};
\addplot[dotted] file {tikz/Brownian_Motion_SamplePath/maxGW.txt};
\end{axis}
\end{tikzpicture}
\caption{Brownian motion (gray) and embedded Gaussian random walk (marked) with their respective running maxima (dashed and dotted, respectively).}
\label{fig:BrownianMotion_vs_GaussianRW}
\end{figure}
Despite this easy bound, precise results for $M_\beta$ are more involved. Let $\zeta$ denote the Riemann zeta function, which is defined as, see e.g.~\cite[Eq.~25.2.1]{NIST},
\begin{equation}
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
\end{equation}
\begin{theorem}[{\cite[Thm.~1]{Chang1997} \& \cite[Thm.~2 \& 3]{Janssen2006}}]
For $0<\beta<2\sqrt{\pi}$,
\begin{align}
\mathbb{P}(M_\beta = 0) &= \sqrt{2}\beta\, \exp \left\{ \frac{\beta}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(1/2-l)}{l!(2l+1)} \left(\frac{-\beta^2}{2}\right)^l \right\},\\
\mathbb{E}[M_\beta] &= \frac{1}{2\beta} + \frac{\zeta(1/2)}{\sqrt{2\pi}} + \frac{\beta}{4}
+ \frac{\beta^2}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(-1/2-l)}{l!(2l+1)(2l+2)} \left(\frac{-\beta^2}{2}\right)^l,\\
{\rm Var}\, M_\beta &=
\frac{1}{4\beta^2} - \frac{1}{4} - \frac{2\,\zeta(-1/2)}{\sqrt{2\pi}}\beta - \frac{\beta^2}{24}\nonumber\\
&\qquad\qquad -
\frac{2\beta^3}{\sqrt{2\pi} } \sum_{l=0}^\infty
\frac{\zeta(-3/2-l)}{l!(2l+1)(2l+2)(2l+3)} \Big(\frac{-\beta^2}{2}\Big)^l.
\end{align}
\end{theorem}
\subsection{Characteristics of the QED regime}
\label{sec:intro_characteristics}
Now that we have seen how the square-root staffing rule \eqref{eq:square_root_staffing rule} yields non-degenerate limiting behavior in two classical queueing models, we shall elaborate on how the QED regime gives rise to (at least) three desirable properties.
The first property relates to the efficient usage of resources, expressed as
\begin{equation}
\rho_{{s_\lambda}} = \frac{\lambda}{{s_\lambda}} = 1 - \frac{\beta}{\sqrt{{s_\lambda}}} + O\big(1/\lambda\big), \tag{Efficiency}
\end{equation}
where we have used that ${s_\lambda} = O(\lambda)$.
The second distinctive property is the balance between QoS and efficiency:
\begin{equation}
\mathbb{P}(W^{({s_\lambda})}>0) \to g(\beta), \qquad \text{and} \qquad \mathbb{P}(W^{({s_\lambda})}>0) \to 1-\mathbb{P}(M_\beta=0), \tag{Balance}
\end{equation}
as ${s_\lambda} \to \infty$, in the $M/M/s$ queue and $M/D/s$ queue, respectively.
The third property relates to good QoS:
\begin{equation}
\mathbb{E}[W^{({s_\lambda})}] = \frac{h(\beta)}{\sqrt{{s_\lambda}}} + o(1/\sqrt{{s_\lambda}}) \qquad \text{and} \qquad \mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{E}[M_\beta]}{\sqrt{{s_\lambda}}} + O(1/\sqrt{{s_\lambda}}), \tag{QoS}
\end{equation}
in the $M/M/s$ queue and $M/D/s$ queue, respectively.
Hence the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$.
Both limiting functions $g(\beta)$ and $1-\mathbb{P}(M_\beta=0)$ can take all values in $(0,1)$ by tuning the parameter $\beta$.
Since the mathematical underpinning of these properties comes from the CLT, we can expect the properties to hold for a much larger class of models.
These models should then be members of the same universality class (to which the CLT applies).
Let us again show this by example.
Consider a stochastic system in which demand per period is given by some random variable $A$, with mean $\mu_A$ and variance $\sigma_A^2<\infty$.
For systems facing large demand we propose to set the capacity according to the more general rule
\[s = \mu_A + \beta\sigma_A,\]
which consists of a minimally required part $\mu_A$ and a variability hedge $\beta\sigma_A$.
Assume that the workload brought into the system is generated by $n$ stochastically identical and independent sources.
Each source $i$ generates $A_{i,j}$ work in the $j^{th}$ period, with $\mathbb{E}[A_{i,j}] = \mu$ and ${\rm Var}\,\,A_{i,j} = \sigma^2$.
Then the total amount of work arriving to the system during one period is $A_j^{(n)} = \sum_{i=1}^n A_{i,j}$ with mean $n\mu$ and variance $n\sigma^2$.
Assume that the system is able to process a deterministic amount of work $s_n$ per period and denote by $U^{(n)}(j)$ the amount of work left over at the end of period $j$.
Then,
\begin{equation}
U^{(n)}(j+1) = \left( U^{(n)}(j) + A^{(n)}_j - s_n \right)^+.
\end{equation}
Given that $s_n > \mathbb{E}[A^{(n)}_1] = n\mu$, the steady-state limit $U^{(n)} := \lim_{j\to\infty} U^{(n)}(j)$ exists and satisfies
\begin{equation}
U^{(n)} {\;\buildrel{d}\over= \;} \left( U^{(n)} + A^{(n)}_1 - s_n \right)^+.
\label{eq:bulk_service_stationary_recursion}
\end{equation}
This framework is also known as the bulk service queue or the {\color{blue}Anick-Mitra-Sondhi} model \cite{Anick1982,Janssen2005,Janssen2008}.
In this scenario, increasing the system size is done by increasing $n$, the number of input flows.
As we have seen before, it requires a rescaling of the process $U^{(n)}$ by an increasing function $c(n)$, in order to obtain a non-degenerate scaling limit $U := \lim_{n\to\infty} U^{(n)}/c(n)$.
(We omit the technical details needed to justify the interchange of limits.)
From \eqref{eq:bulk_service_stationary_recursion} it becomes clear that the scaled increment
\begin{equation}
\frac{A^{(n)}_j - s_n}{c(n)} = \frac{\sum_{i=1}^n A_{i,j} - n\mu}{c(n)} + \frac{n\mu - s_n}{c(n)}
\end{equation}
only admits a proper limit if $c(n)$ is of the form $c(n) = O(\sqrt{n})$, by the virtue of the CLT, and $(s_n-n\mu)/c(n) \to \beta >0$ as $n\to\infty$.
Especially for $c(n) = \sigma\sqrt{n}$, this reveals that $U$ has a non-degenerate limit, which is equal in distribution to the maximum of a Gaussian random walk with drift ${-}\beta$ and variance 1, if
\[
s_n = n\mu+\beta \sqrt{n}\sigma + o(\sqrt{n}).
\]
Moreover, the results on the Gaussian random walk presented in Section \ref{sec:intro_gaussian_random_walk} are applicable to this model and the key features of the QED scaling carry over to this more general setting as well.
In conclusion, the many-sources framework shows that the QED scaling finds much wider applications than queueing models with Poisson input only.
\subsection{Related literature}
We now provide a partial overview on the literature on heavy-traffic analysis in queueing theory and the QED regime in particular.\\
\\*
\noindent\textbf{Conventional heavy-traffic.}
Before the formal introduction of the Halfin-Whitt scaling regime in 1981, see \cite{Halfin1981}, the existing literature on the asymptotic analysis of queues mostly evolved around two types of scaling regimes: single-server and infinite-server regimes.
The idea of studying a sequence of queues in which the utilization approaches 100\%, i.e.~heavy-traffic, was first laid out by Kingman in the 1960s.
In \cite{Kingman1961,Kingman1962} he showed how in the $GI/G/1$ queue, under mild conditions on the arrival and service processes, the scaled steady-state waiting time $(1-\rho)W^{(1)}$ converges to an exponentially distributed random variable.
The notion that heavily loaded systems admit a scaling limit that is remarkably simple compared to the otherwise intractable pre-limit queueing systems triggered a surge of research within the field of queueing theory in the 1960s and 1970s, see \cite{Borovkov1965,Iglehart1970,Brumelle1971,Newell1973,Kollerstrom1974,Kollerstrom1979,Whitt1974} among others.
These works conduct their asymptotic analysis in what we now call conventional heavy-traffic.
That is, the service times and number of servers are held fixed, while the arrival rate approaches the critical value from below.
A noteworthy result of these efforts is the extension of Kingman's findings to the $GI/G/s$, which finds that the scaled queue length $(1-\rho)Q^{(s)}$ converges in distribution to an exponential random variable with mean $(c_a^2+c_s^2)/2$, where $c_a$ and $c_s$ denote the coefficient of variation of the interarrival and service time distribution, respectively.
We remark that this limiting result is the key ingredient to the widely applied Kingman formula
\begin{equation}\label{eq:kingman}
\mathbb{E}[W^{(1)}] \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2+c_s^2}{2} \cdot \mathbb{E}[B],
\end{equation}
which serves as an approximation to the expected waiting time in the single-server queue.
The limit \eqref{eq:kingman} reveals that in the conventional heavy-traffic regime, the expected waiting time explodes as $\rho\to 1$.
Hence, efficient usage of resources is achieved, at the expense of poor QoS.
An alternative regime that received much attention, see e.g.~\cite{Iglehart1965,Borovkov1965,Iglehart1973,Iglehart1973a,Whitt1982}, fixes the service time distribution while increasing both the arrival rate $\lambda$ and the number of servers to infinity simultaneously, such that the ratio $\lambda/s$ remains constant.
It has been shown that the sequence of queues under this scaling start resembling the behavior of infinite-server queues as $\lambda$ and $s$ grow.
That is, the probability of a customer finding a queue on arrival is negligible.
The sample paths in Figure \ref{fig:sample_paths_lambda100} are illustrative for this regime.
Since the utilization level $\rho$ remains strictly away from one in the limit, this setting is {\color{col1}in the literature} typically not recognized as heavy-traffic.
As Halfin \& Whitt indicate themselves, the QED regime in which service times are held fixed, and $\lambda$ and $s$ tend to infinite while satisfying $(1-\rho)\sqrt{s} \to \beta$, is a hybrid between the two aforementioned regimes.
Namely, it considers the efficiency property of the conventional heavy-traffic scaling, and the good QoS levels from infinite-server queues.\\
\\*
\noindent
\textbf{The $G/G/s$ queue in the QED regime.}
We have demonstrated in Section \ref{sec:intro_QED_regime} how to obtain QED limits for the $M/M/s$ queue and the $M/D/s$ queue.
When one moves beyond the exponential and deterministic assumptions, establishing QED behavior becomes mathematically more challenging.
The heavy-traffic analysis of the $G/G/s$ queue requires fundamentally different approaches than for Markovian queues.
Most of the research conducted on the $G/G/s$ in the Halfin-Whitt regime evolves around the characterization of the stochastic process limit of the appropriately centered and scaled queueing process in terms of diffusion processes, under various assumptions on the model primitives.
Puhalskii \& Reiman \cite{Puhalskii2000} analyzed the multi-class queue with phase-type service times in the Halfin-Whitt regime.
Heavy-traffic limits for queues in which service time distributions are lattice-based and/or have finite support are studied by Mandelbaum \& Momcilovic \cite{Mandelbaum2008} and Gamarnik \& Momcilovic \cite{Gamarnik2008}.
Approaches through measure-valued processes are taken by Kang, Kaspi \& Ramanan \cite{Kaspi2011,Kang2012,Kaspi2013}.
The most general class of distributions is considered by Reed \cite{Reed2009} and Puhalskii \& Reed \cite{Puhalskii2010}, who impose no assumption on the service time distribution except for the existence of the first moment.
For a survey on the techniques required for the analysis of process limits of $G/G/s$ queues, we refer the reader to \cite{Pang2007} and references therein.
Considerably less is known for the corresponding steady-state distribution of the $G/G/s$ queue in the QED regime.
Namely, under the assumption of general service time distributions, truly infinite-dimensional limits arise, since the Markovian nature of the service time and `age' process can no longer be exploited.
Works that have been able to characterize limiting behavior for the specific service time distribution classes include Jelenkovic et al.~\cite{Jelenkovic2004}, who assume deterministic service times, and Whitt \cite{Whitt2005}, who identifies the heavy-traffic limit in the case of hyperexponentially distributed service times.
Progress in the understanding of steady-state behavior of $G/G/s$ queues in the Halfin-Whitt regime has been facilitated by Gamarnik \& Goldberg \cite{Goldberg,Gamarnik2013a}, who perform their analysis under the mild assumption that the service time distribution has finite $(2+\e)$ moment.
A significant advance has been made by Aghanjani \& Ramanan \cite{Aghajani2016}, who identify the limit as the steady-state distribution of infinite-dimensional Markov process, given that the service time distribution has finite $(3+\varepsilon)$ moment, while drawing upon previous results by Kang, Kaspi \& Ramanan \cite{Kaspi2011, Kang2012,Kaspi2013}.\\
\\*
\textbf{Model extensions.}
Many extensions to the standard many-server queue can be considered.
A feature ubiquitous to service systems involving humans is customer abandonment \cite{Gans2003,Brown2005,Zeltyn2005,Mandelbaum2013}.
The $M/M/s+M$ queue introduced by Palm \cite{Palm1957}, also known as the Erlang-A model \cite{Garnett2002,Leeuwaarden2012}, acknowledges this feature by assigning every customer an exponentially distributed \textit{patience time} upon his arrival (denoted by $+M$ in the model definition).
If a customer has not yet started receiving service by the expiration of his patience, he leaves the system.
Note that abandonments render queues stable under any load.
Under QED scaling, the more general $G/G/s+G$ queue has received much attention under various modeling assumptions, see e.g.~\cite{Garnett2002,Gans2003,Whitt2006,Mandelbaum2009,Zeltyn2005,Mandelbaum2012a,Kang2012,Dai2010,Reed2012,Jennings2012,Zhang2013}.
Noteworthy findings include the vanishing abandonment probability \cite{Garnett2002} and insensitivity of the patience time distribution as long as its density at 0, {\color{blue} i.e.~the probability of abandoning immediately upon arrival}, is fixed, as the system grows large under QED scaling.
Overviews of queues with abandonment and their asymptotic counterpart are given by Zeltyn \& Mandelbaum \cite{Zeltyn2005} and Dai \& He \cite{Dai2012} and Ward \cite{Ward2012}.
Other features that have been studied in the QED regime include multiple customer classes, see e.g.~\cite{Harrison2004,Atar2014,Gurvich2008,Gurvich2009,Tezcan2010}, or heterogeneous servers \cite{Armony2005,Armony2010,Mandelbaum2012b,Stolyar2010}.
These models are all interesting in their own respect and are fairly well-understood.
Therefore, we choose to focus in this thesis on a different set of extensions, which will be discussed in Section \ref{sec:intro_beyond}.
\section{Dimensioning}
\label{sec:intro_dimensioning}
We adopt the term \textit{dimensioning} used by Borst, Mandelbaum \& Reiman~\cite{Borst2004} to say that the capacity of a service system is adapted to the load in order to reach certain performance levels.
In \cite{Borst2004} dimensioning refers to the staffing problem in a large-scale call center and key ingredients are the square-root staffing rule in \eqref{eq:square_root_staffing rule} and the QED regime.
We now revisit the results in \cite{Borst2004} and its follow-up works to explain this connection to the QED regime.
\subsection{Constraint satisfaction}
\label{sec:intro_constraint}
Consider the $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$.
A classical dimensioning problem is to determine the minimum number of servers $s$ necessary to achieve a certain target level of service, say in terms of waiting time.
Suppose we want to determine the minimum number of servers such that the fraction of customers who are delayed in the queue is at most $\varepsilon\in(0,1)$.
Hence we should find
\begin{equation}\label{eq:tagA}
s^{*}_\lambda(\varepsilon) := \min \left\{s \geq \lambda\, |\, \mathbb{P}(W^{(s)}>0) \leq \varepsilon \right\}.
\end{equation}
But alternatively, we can use the QED framework, which says that under \eqref{eq:HalfinWhitt_scaling},\ \ $\lim_{s\to\infty} \mathbb{P}(W^{({s_\lambda})} > 0) = g(\beta)$ (see Proposition \ref{prop:HalfinWhitt_delay_probability}).
Then by \eqref{eq:tagA}, $s^*_\lambda(\varepsilon)$ can be replaced by
\begin{equation}
s^{\rm srs}_\lambda(\varepsilon) = \lceil \lambda + \beta^*(\varepsilon) \sqrt{\lambda}\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves
\begin{equation}
g(\beta^*) = \varepsilon.
\end{equation}
In Figure \ref{fig:MMs_staffing_levels} we plot the exact staffing level $s^*_\lambda(\varepsilon)$ and the heuristically obtained staffing level $s^{\rm srs}_\lambda(\varepsilon)$ as a functions of $\varepsilon$ for several loads $\lambda$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Constraint_Satisfaction/lambda5_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Constraint_Satisfaction/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Constraint_Satisfaction/lambda10_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Constraint_Satisfaction/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Constraint_Satisfaction/lambda100_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Constraint_Satisfaction/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Constraint_Satisfaction/lambda500_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Constraint_Satisfaction/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Staffing levels as a function of the delay probability targets $\varepsilon$.}
\label{fig:MMs_staffing_levels}
\end{figure}
Observe that even for very small values of $\lambda$, the staffing function $s^{\rm srs}(\varepsilon)$ coincides with the exact solution for almost all $\varepsilon\in(0,1)$ and differs no more than by one server for all $\varepsilon$.
Borst et al.~\cite{Borst2004} recognized this in their numerical experiments too, and Janssen, van Leeuwaarden \& Zwart \cite{Janssen2011} later confirmed this theoretically.
One can easily formulate other constraint satisfaction problems and reformulate them in the QED regime.
For instance, constraints on the mean waiting time or the tail probability of the waiting time, e.g.~$\mathbb{P}(W^{(s)}>T)$, which are asymptotically approximated by $h(\beta)/\sqrt{\lambda}$ and $g(\beta){\rm e}^{-\beta \sqrt{\lambda} T}$, respectively.
See \cite{Borst2004} for more examples.
\subsection{Optimization}
\label{sec:intro_optimization}
One can also consider optimization problems, for instance to strike the right balance between the costs for servers and costs incurred by customer dissatisfaction.
More specifically, assume a salary cost of $a$ per server per unit time, and a penalty cost of $q$ per waiting customer per unit time, yielding the total cost function
\[
\bar{C}_\lambda(s) := a\,s + q\,\lambda\mathbb{E}[W^{(s)}]
\]
and then ask for the staffing level $s$ that minimizes $\bar{C}_\lambda(s)$.
Since $s>\lambda$, we have $\bar{C}_\lambda(s) > a\,\lambda$ for all feasible solutions $s$.
Moreover, the minimizing value of $\bar{C}_\lambda$ is invariant with respect to scalar multiplication of the objective function.
Hence we have to optimize
\begin{equation}
\label{eq:optimization_objective}
C_\lambda(s) = r\,(s-\lambda) + \lambda\mathbb{E}[W^{(s)}], \qquad r = a/q.
\end{equation}
Denote by $s^*_\lambda(r) := \arg\min_{s > \lambda} C_\lambda(s)$ the true optimal staffing level.
With ${s_\lambda} = \lambda + \beta\sqrt{\lambda}$ and the QED limit in \eqref{eq:halfinwhitt_wait}, we can replace \eqref{eq:optimization_objective} by its asymptotic counterpart:
\begin{align*}
\frac{C_\lambda({s_\lambda})}{\sqrt{\lambda}} = r\,\beta + \sqrt{\lambda} \mathbb{E}[W^{(s)}] \to r\,\beta + \frac{g(\beta)}{\beta} =: \hat{C}(\beta), \qquad \lambda\to\infty.
\end{align*}
Once again we obtain a limiting objective function that is easier to work with than its exact pre-limit counterpart.
Hence, in the spirit of the asymptotic staffing procedure in the previous subsection, we propose the following method to determine the staffing level that minimizes overall costs.
First, (numerically) compute the value $\beta^*(r) = \arg\min_{\beta>0} \hat{C}(\beta)$, which is well-defined, because the function $\hat{C}(\beta)$ is strictly convex for $\beta>0$.
Then, set $s^{\rm srs}_\lambda(r) = [ \lambda + \beta^*(r) \sqrt{\lambda} ]$.
In Figure \ref{fig:MMs_staffing_levels_optimization} we compare the outcomes of this asymptotic staffing procedure against the true optima as a function of $r\in(0,\infty)$, for several values of $\lambda$.
The staffing levels $s^{\rm srs}_\lambda(r)$ and $s^*_\lambda(r)$ are aligned for almost all $r$, and differ no more than one server for all instances.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
axis y discontinuity = crunch,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
legend style = {at = {(1,1.2)}, anchor = north east}
]
\addplot[very thick] file {tikz/Optimization/lambda5_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Optimization/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Optimization/lambda10_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Optimization/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Optimization/lambda100_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Optimization/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {tikz/Optimization/lambda500_exact.txt};
\addplot[very thick, dashed, col1] file {tikz/Optimization/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Optimal staffing levels as a function of $r = a/q$.}
\label{fig:MMs_staffing_levels_optimization}
\end{figure}
\subsection{Time-varying dimensioning}
So far we have only considered queues in which the model primitives are constant over time.
In practice, though, the arrival rate can fluctuate and depends on the time of day, the day of the week, season or even larger time scales.
It is therefore more realistic to describe these mostly predictable fluctuations through $\lambda(t)$, which represents the instantaneous arrival rate of the arrival process at time $t\in \mathbb{R}$.
The existence of time-varying demand requires a re-evaluation of staffing levels throughout the planning horizon as well.
That is, the number of servers $s(t)$ becomes a function of time, rather than a constant and this clearly asks for an adaptation of the dimensioning procedures in Sections \ref{sec:intro_constraint} and \ref{sec:intro_optimization}.
We explain the concept of time-varying staffing and the connection with the QED regime through the time-varying extension of the $M/M/s$ queue known as the $M_t/M/s_t$ queue, where the subscript $t$ refers to the time-varying nature of both the arrival process and the staffing level.
In this setting, customers arrive according to a non-homogeneous Poisson process with rate function $\lambda(t)$ and customers have exponentially distributed service times with mean $1/\mu$.
Under a constraint satisfaction strategy, we aim to find the staffing function $s(t)$ such that the delay probability is at most $\varepsilon\in(0,1)$ for all $t$.
The analysis and optimization of time-varying many-server queueing systems is known to be intrinsically hard, but many approximation techniques and heuristic methods have been proposed throughout the years \cite{Green1991,Jennings1996}.
A natural but naive approach is the \textit{pointwise-stationary approximation} (PSA) \cite{Green1991}, which evaluates the system at time $t$ as if it were in steady-state with instantaneous parameters $\lambda=\lambda(t)$, $\mu$ and $s = s(t)$.
Consequently, the analysis and optimization of queues is performed on steady-state performance metrics.
Variants of the PSA method include the \textit{simple-stationary approximation} (SSA) \cite{Green2001}, which uses the long-term (moving) average arrival rate instead of the instantaneous arrival rate, and the \textit{stationary-independent-period-by-period approximation} (SIPP) \cite{Green2001}, which splits the time-horizon into multiple intervals and performs steady-state analysis with the averaged parameters in each of these intervals, among others.
PSA performs well in slowly varying environments with relatively short service times \cite{Green1991,Whitt1991}.
However, when the model parameters fluctuate significantly, as is often the case in real-life systems, the accuracy of PSA can be poor, as we will see in the numerical experiment at the end of this section.
The main reason why PSA, SSA and SIPP can fail is that these methods neglect that customers are actually residing in the system (being in service or waiting in the queue) for some time.
In contrast, staffing decisions should be based on the number of customers present in the system rather than the arrival rate at that particular time.
Jennings et al.~\cite{Jennings1996} introduced a more sophisticated method that exploits the relation with infinite-server queues.
We explain their idea in the context of the $M_t/M/s_t$ queue.
By Eick et al. \cite{Eick1993}, the number of customers in the $M_t/M/\infty$ queue at time $t$ is Poisson distributed with mean
\begin{equation}
\label{eq:offered_load_eick}
R(t) = \mathbb{E}\left[ \lambda(t-B_e)\right] \mathbb{E}[B] = \int_0^\infty \lambda(t-u)\,\mathbb{P}(B>u)\, {\rm d}u = \int_0^\infty \lambda(t-u)\, {\rm e}^{-\mu u} \,{\rm d}u.
\end{equation}
We remark that this result holds for more general service time distributions.
Now, recall that in large systems in the QED regime, the expected delay is negligible.
Therefore, under these conditions, the many-server system may be approximated by the infinite-server approximation with offered load as in \eqref{eq:offered_load_eick}.
Accordingly, we can determine the staffing levels $s(t)$ for each $t$ based on steady-state $M/M/s$ measures with offered load $R=R(t)$.
Jennings et al. \cite{Jennings1996} proceed by exploiting the heavy-traffic results of Halfin-Whitt \eqref{eq:halfinwhitt_wait}.
In conjunction with the dimensioning scheme in Section \ref{sec:intro_constraint}, the authors propose to set
\begin{equation}
s(t) = \bigg\lceil R(t) + \beta^*(\varepsilon) \sqrt{R(t)} \bigg\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves $g(\beta^*(\varepsilon)) = \varepsilon$.
Remark that the number of servers is rounded up to ensure that the achieved delay probability is indeed below $\varepsilon$.
This method was called in \cite{Jennings1996,Massey1994} the \textit{modified-offered-load} (MOL) approximation, and we adopt this term in this thesis.
Let us demonstrate that this approximation scheme works.
Figure \ref{fig:intro_example_arrival}(a) shows an arrival rate pattern $\lambda(t)$ and corresponding offered load function $R(t)$ for $\mu=1/2$.
This arrival rate stems from a real-world emergency department~\cite{Sinreich2005}.
The resulting staffing level functions based on the PSA and MOL approximations with $\varepsilon = 0.3$ are plotted in
Figure \ref{fig:intro_example_arrival}(b).
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 45,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,43)},anchor = north west}
]
\addplot[very thick] file {tikz/TimeVarying/arrival_rate.txt};
\addplot[very thick, col1] file {tikz/TimeVarying/offered_load.txt};
\legend{{$\lambda(t)$},$R(t)$}
\end{axis}
\end{tikzpicture}
\caption{Arrival rate and offered load functions}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 60,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,59)},anchor = north west}
]
\addplot[very thick] file {tikz/TimeVarying/s_PSA.txt};
\addplot[very thick, col1] file {tikz/TimeVarying/s_Jennings.txt};
\legend{PSA,MOL}
\end{axis}
\end{tikzpicture}
\caption{Staffing functions.}
\end{subfigure}
\caption{Time-varying parameters of a real-world emergency department.}
\label{fig:intro_example_arrival}
\end{figure}
Through simulation, we evaluate the delay probability as a function of time for $\varepsilon = 0.1,\, 0.3$ and 0.5.
In Figure \ref{fig:intr_timevarying_simulation_results} we see how the PSA approach fails to stabilize the performance of the queue, whereas the MOL method does stabilize around the target performance.
The erratic nature of the delay probability as a function of time can be explained by rounding effects of the staffing level.
Since this rather simple but elegant technique to address time-varying dimensioning is provably effective, we will adopt the underlying idea of the MOL method in various different settings in this thesis.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,0.02)},anchor = south west}]
\addplot[thick, col5] file {tikz/TimeVarying/pdelay_e01_psa.txt};
\addplot[thick, col2] file {tikz/TimeVarying/pdelay_e03_psa.txt};
\addplot[thick, col4] file {tikz/TimeVarying/pdelay_e05_psa.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{PSA}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, col5] file {tikz/TimeVarying/pdelay_e01_mol.txt};
\addplot[thick, col2] file {tikz/TimeVarying/pdelay_e03_mol.txt};
\addplot[thick, col4] file {tikz/TimeVarying/pdelay_e05_mol.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{MOL}
\end{subfigure}
\caption{Probability of delay under staffing functions obtained through PSA and MOL approximations.}
\label{fig:intr_timevarying_simulation_results}
\end{figure}
\section{Contributions}
\label{sec:intro_beyond}
We have explained how the QED regime can be used to dimension and staff large-scale service systems.
The basic concepts, however, were explained for the relatively simple $M/M/s$ and $M_t/M/s_t$ queue.
Many real-world service systems have essential features that are not captured by these elementary models.
We will now discuss some of these features and address the need to consider more involved models and extend the existing QED theory.
\subsection{Non-classical scaling regimes and pre-limit behavior}
\label{sec:intro_novel_scalings}
The QED theory is centered around the scaling relation $\sqrt{\lambda}(1-\rho_\lambda) \to \beta$, or equivalently $s_\lambda = \lambda + \beta \sqrt{\lambda} + o(\sqrt{\lambda})$, for $\lambda\to\infty$.
It is worthwhile to study how pre-limit behavior of many-server queues is affected when one deviates from the square-root staffing rule.
Consider a novel family of heavy-traffic scaling regimes, described in terms of the parameter $\eta$ for which we assume that
\begin{equation}
\label{eq:novel_scaling_rule}
\lambda^\eta (1-\rho_\lambda) \to \beta, \qquad \text{as } \lambda\to\infty,\ \beta > 0.
\end{equation}
The parameter $\eta \geq 0$ defines a whole range of possible scaling regimes, including the classic case $\eta = 1/2$, as well as the cases $\eta=0$ and $\eta=1$ investigated in Subsection \ref{sec:intro_many_server_regimes}.
In terms of a capacity sizing rule, the condition \eqref{eq:novel_scaling_rule} is tantamount to $s_\lambda = \lambda +\beta\,\lambda^{1-\eta}$.
This framework thus bridges the gap between the QD and QED regime if $\eta\in(0,1/2)$ and the QED and ED regime if $\eta\in(1/2,1)$, in the $M/M/s$ model.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as \textit{moderate} heavy traffic: heavy-traffic conditions in which the full occupancy is reached more slowly, as a function of $\lambda$, than for classical heavy traffic. See \cite{Chang1996,Puhalskii1998,Puhalskii1999,Atar2012,Atar2014,Atar2015,Atar2016} for more details.
For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to \textit{extreme} heavy traffic due to a relatively small variability hedge.
We use the insights of Section \ref{sec:intro_QED_regime} and the connection of the QED scaling to the CLT to argue intuitively that the following trichotomy in the qualitative system behavior as $\lambda\to\infty$ holds under scaling \eqref{eq:novel_scaling_rule}.
For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, because the order of the variability hedge $\beta \lambda^{1-\eta}$ is greater than strictly necessary to accommodate the stochastic fluctuations in demand.
Scalings in which $\eta\in(1/2,\infty)$, have adverse behavior, since stochastic fluctuations are not accounted for sufficiently, so that the probability of delay converges to 1.
The value $\eta=1/2$ is therefore the tipping point, at which the delay probability converges to a limit between 0 and 1.
Above and below this critical value, the asymptotic performance of the queue flips to either one of the extremes.
In Chapter 2, we formalize this heuristic argument and conduct an asymptotic analysis to reveal the rate at which the limit of performance metrics is attained, depending on the parameters $\eta$ and $\beta$ and the system size $\lambda,{s_\lambda}$.
\subsection{Overdispersed arrivals}
\label{sec:intro_overdispersion}
Until now we have considered queueing systems with perfect knowledge on the model primitives, including the mean demand per time period. For large-scale service systems, the dominant assumption in the literature is that demand arrives according to a non-homogeneous Poisson process, which in practice translates to the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies of service systems shows that the variance of demand typically exceeds the mean significantly, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2003, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}. The feature that variability is higher than one expects from the Poisson assumption is referred to as \textit{overdispersion}.
Due to its inherent connection with the CLT, the dimensioning rule in \eqref{eq:square_root_staffing rule} relies heavily on the premise that the variance of the number of customers entering the system over a period of time is of the same order as the mean.
Subsequently, when stochastic models do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly in critical loading.
To deal with overdispersion, existing capacity sizing rules like the square-root staffing rule need to be modified in order to incorporate a correct hedge against (increased) variability.
Following our findings in Section \ref{sec:intro_characteristics}, we propose a capacity allocation rule similar to \eqref{eq:square_root_staffing rule} in which the original variability hedge is replaced by an amount that is proportional to the square-root of the variance of the arrival process.
In Chapter 3, we elaborate on this idea and show how to adapt the scaling of the queueing process appropriately to achieve QED-type behavior in the presence of overdispersion.
\subsection{Finite-size constraints}
The canonical examples in Section \ref{sec:intro_QED_regime} assume an infinite amount of waiting space.
Physical service systems, however, are sometimes limited in the number of customers that can be held in the system simultaneously.
For instance in a call center, the maximum number of clients in service or queueing is restricted by the number of available trunk lines \cite{Khudyakov2006}, while in the emergency department of a hospital, the number of beds constrains the number of patients that can be admitted \cite{YomTov2010}.
Depending on the practical setting and admission policy, if the maximum capacity, say $n$, is reached, newly arriving customers either leave the system immediately (blocking), reattempt getting access later (retrials) or queue outside the facility (holding).
In any case, expectations are that the queueing dynamics within the service facility are affected considerably in the presence of such additional capacity constraints.
We illustrate these implications through the $M/M/s/n$ queue, that is, the standard $M/M/s$ queue with additional property that a customer who finds upon arrival $n$ customers already present in the system, is deferred and considered lost.
To avoid trivialities, let $n\geq s$.
Since the expected workload reaching the servers is less than in the unconstrained scenario, one expects less congestion and resource utilization.
Consider the $M/M/{s_\lambda}/n_\lambda$ in the QED regime.
So, let $\lambda$ increase while ${s_\lambda}$ scales as ${s_\lambda}=\lambda+\beta\sqrt{\lambda} + o(\sqrt{\lambda})$.
We then ask how $n_\lambda$ should scale along with $\lambda$ and ${s_\lambda}$ to maintain the non-degenerate behavior as seen in Section \ref{sec:intro_QED_regime}.
We provide a heuristic answer.
Let $Q^{({s_\lambda},n_\lambda)}$ and $W^{({s_\lambda},n_\lambda)}$ denote the number of customers in the system and the waiting time in the $M/M/{s_\lambda}/n_\lambda$ queue in steady state.
Note through Proposition \ref{thm:intro_HW_stationary_distribution} that if there were no finite-size constraints, we would have, for $\lambda$ large,
\begin{align}
\mathbb{P}(Q^{({s_\lambda})}\geq n_\lambda)
&= \mathbb{P}\left(\frac{Q^{({s_\lambda})}-{s_\lambda}}{\sqrt{{s_\lambda}}} \geq \frac{n_\lambda-{s_\lambda}}{\sqrt{{s_\lambda}}}\right) \nonumber \\
&\to
\left\{
\begin{array}{ll}
g(\beta), & \text{if }n_\lambda = {s_\lambda} + o({s_\lambda}),\\
g(\beta)\,{\rm e}^{-\beta \gamma}, & \text{if } n_\lambda = {s_\lambda}+\gamma\sqrt{{s_\lambda}} + o(\sqrt{s_\lambda}),\\
0, & \text{if } n_\lambda = {s_\lambda}+\Omega(\sqrt{{s_\lambda}}),
\end{array}
\right.
\end{align}
as $\lambda\to\infty$ for some $\gamma>0$.
Here, the relation $u(\lambda) = \Omega(v(\lambda))$ implies $u(\lambda)/v(\lambda) >1$ for $\lambda\to\infty$.
Hence, asymptotically the finite-size effects only play a role if the extra variability hedge of $n_\lambda$ is of order $\sqrt{{s_\lambda}}$ (or equivalently $o(\sqrt{\lambda})$).
Furthermore, if the variability hedge is $o(\sqrt{\lambda})$, then we argue that asymptotically, all customers who do enter the system have probability of delay equal to zero.
More formally, under the \textit{two-fold scaling rule}
\begin{equation}
\label{eq:intro_twofold_scaling_rule}
\left\{
\begin{array}{ll}
{s_\lambda} = \lambda + \beta\sqrt{\lambda} + o(\sqrt{\lambda}),\\
n_\lambda = {s_\lambda} + \gamma \sqrt{{s_\lambda}} + o(\sqrt{\lambda}),
\end{array}
\right.
\end{equation}
it is not difficult to deduce that, see e.g. \cite{masseywallace},
\begin{equation}
\mathbb{P}(W^{({s_\lambda},n_\lambda)} > 0) \to \left( 1 + \frac{\beta\,\Phi(\beta)}{(1-{\rm e}^{-\beta\gamma})\varphi(\beta)}\right)^{-1}, \quad \text{as } \lambda\to\infty,
\end{equation}
which is strictly smaller than $g(\beta)$ in \eqref{fig:delay_probs_HW_MMs}, but still bounded away from both 0 and 1.
Furthermore, the buffer size of the queue is $n_\lambda-{s_\lambda} = \gamma\sqrt{{s_\lambda}}$, so that by Little's law, the expected waiting time of an admitted customer is $O(1/\sqrt{{s_\lambda}})$.
Even though resource utilization in the $M/M/{s_\lambda}/n_\lambda$ is less efficient than in the queue with unlimited waiting space, it can easily be shown that $\rho\to 1$ as $\lambda\to\infty$.
Hence, all three key characteristics of the QED regime are carried over to the finite-size setting if adhered to scaling \eqref{eq:intro_twofold_scaling_rule}.
On a process level, adding a capacity constraint translates to adding a reflection barrier to the normalized queue length process $X^{({s_\lambda},n_\lambda)} = (Q^{({s_\lambda},n_\lambda)} -{s_\lambda} ) /\sqrt{{s_\lambda}}$, at $\gamma$, as is illustrated by the sample paths of $X^{{s_\lambda},n_\lambda}$ for three values of $\lambda$ in Figure \ref{fig:sample_paths_MMsn}.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {tikz/SamplePaths_MMsn/lambda5.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {tikz/SamplePaths_MMsn/lambda50.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda = 50$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}\centering
\begin{tikzpicture}[scale = 0.7]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel style={right},
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[thick] file {tikz/SamplePaths_MMsn/lambda100.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}\centering
\begin{tikzpicture}[scale = 0.7]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel style={right},
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {tikz/SamplePaths_MMsn/lambda500.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda},n_\lambda)}(t)$ with $\lambda = 5,\, 50,\, 100$ and $500$ under scaling \eqref{eq:intro_twofold_scaling_rule} with $\beta=0.5$ and $\gamma = 0.5$.}
\label{fig:sample_paths_MMsn}
\end{figure}
\
It has been shown by \cite{masseywallace} that under \eqref{eq:intro_twofold_scaling_rule}
\begin{equation}
\label{eq:asymptotic_blocking_prob}
\sqrt{{s_\lambda}}\,\mathbb{P}({\rm block}) = \sqrt{{s_\lambda}} \mathbb{P}(Q^{({s_\lambda},n_\lambda)} = n_\lambda) \to f(\beta,\gamma), \quad \text{as } \lambda \to\infty,
\end{equation}
for a non-negative function $f$.
The idea of the two-fold scaling in \eqref{eq:intro_twofold_scaling_rule} can be extended to settings in which the interior is in fact a network of queues, rather than the single-station setting discussed here, see \cite{Khudyakov2006,YomTov2010,Tan2012} for examples of such \textit{semi-open} queueing networks.
When customers retry getting access after being blocked initially, the QED analysis becomes much more difficult, and no explicit limiting results are known.
Nevertheless, observe that the volume of blocked arrivals is by \eqref{eq:asymptotic_blocking_prob} of order $\sqrt{\lambda}$, the exact same magnitude as the variability hedge of both ${s_\lambda}$ and $n_\lambda$.
Therefore, retrials and holding customers have a non-negligible effect on the service levels within the facility in the QED regime.
This will be the topic of Chapters 4 and 5.
\subsection{Pre-limit behavior}
The results on queues in the QED regime discussed in Section \ref{sec:intro_QED_regime} are in two ways of an asymptotic nature.
First, the heavy-traffic limits prescribe the queueing dynamics for $\lambda,{s_\lambda}\to\infty$.
Real-world systems obviously do not experience infinite demand nor have infinite capacity, and hence the heavy-traffic limits only form an approximation for such finite-sized systems.
Although these approximations are qualitatively insightful, the asymptotic analyses do not reveal much about their accuracy with respect to actual performance.
For instance, we would like to know how fast the convergence takes place, and how inaccuracies in asymptotic approximations percolate into inaccuracies in pre-limit systems.
To answer such questions, it would be helpful to have an asymptotic estimate for the difference between the (scaled) queueing process and its limiting counterpart, to be able to judge the error made by relying on asymptotic as opposed to actual performance evaluation.
Characterization of the error term gives rise to so-called \textit{corrected diffusion approximations}~ \cite{Siegmund1978,Blanchet2006,Janssen2008}, which are refinements to heavy-traffic limits for finite systems, and are also related to \textit{universal approximations} \cite{Gurvich2014,Huang2016,Braverman2015,Braverman2015a}.
We will derive such corrected diffusion approximations in the context of the novel scaling regimes mentioned in Section \ref{sec:intro_novel_scalings} in Chapter 2.
Second, the bulk of queueing literature is concerned with the performance analysis and optimization of steady-state systems.
However, in practice, service systems certainty do not run infinitely long, which renders this assumption questionable.
Validation of the steady-state assumption is related to the \textit{relaxation time} of a queueing process \cite{Abate1987,Abate1988,relaxation,Leeuwaarden2011,Leeuwaarden2012,Gamarnik2013}, which prescribes the time it takes a system starting out of equilibrium to approximate its stationary distribution.
In case the relaxation time is small, stationary performance evaluation is likely to be accurate.
On the contrary, if the relaxation time is large, a time-dependent analysis of the queueing system is required in order to capture realistic behavior.
Subsequently, we can investigate the implications of applying staffing principles that are based on steady-state performance metrics in settings which are inherently transient over the planning period.
We will touch upon this topic in Chapter 6.
\section{Outline of the thesis}
The remainder of this thesis builds upon the ideas behind the QED scaling regime exhibited in this introductory chapter, and is organized as follows.
Chapter 2 is concerned with the analysis of the limiting behavior of queues in case one deviates from the square-root staffing principle as demand grows large.
Using the bulk-service queue together with the many-sources paradigm as a vehicle, we derive corrected diffusion approximations for the performance metrics of pre-limit systems in these alternative scaling regimes.
The work presented in Chapter 2 is based on \cite{Janssen2015}.
In Chapter 3, we also analyze the bulk-service queueing model, but with many correlated sources, so that demand becomes overdispersed.
As we alluded to in Section \ref{sec:intro_overdispersion}, this requires an alternative scaling of the queue length process and associated staffing rule.
This chapter exhibits the ideas of \cite{Mathijsen2016}.
In Chapter 4, we discuss how QED-type behavior prevails in simple settings in which the system size is finite, given appropriate capacity-sizing rules.
More specifically, we show how customer retrials can be incorporated heuristically into the performance analysis of finite-size systems in the QED regime.
The content of this chapter is based on \cite{Leeuwaarden2015} and \cite{Leeuwaarden2016}.
Building upon the insights gained in Chapter 4, we show in Chapter 5 how the approximation methods carry over to a more complex finite-size queueing system, inspired by delay analysis in a health care facility.
We show how the QED scaling limits for this model offer surprisingly accurate approximations for realistic model parameters in systems of small to moderate size, and develop a staffing algorithm for dimensioning such systems.
Chapter 5 is based on the ideas of \cite{Leeuwaarden2016a}.
Chapter 6 investigates the validity of a capacity allocation rule based on steady-state performance metrics in practical settings.
Namely, in realistic scenarios, the parameters of a queueing model are typically subject to change over the planning period.
This asks for a more elaborate transient analysis of the queue dynamics, and an adaptation of the staffing level.
In this chapter, we present how to do so appropriately in a single-server queueing model facing a L\'evy input process by prescribing a correction to the steady-state optimum, which has a square-root form.
This chapter is based on \cite{Mathijsen2016a}.
Chapter 7 presents the analysis of an inventory model with backlogs, perishable goods and consumer impatience.
This model resembles the inventory level of a blood bank, and can be regarded as a shot-noise model with both positive and negative jumps and exponential decay rates above and below zero.
Besides the derivation of the stationary distribution of the inventory level, we show how under appropriate scaling the process converges to an Ornstein-Uhlenbeck process.
The latter allows for a more tractable approximate analysis of the model in case the number of blood deliveries and demand is large.
Chapter 7 is based on \cite{Bar-Lev2015}.
\chapter{Introduction}
\begin{chapterstart}
Stochastic service systems describe situations in which customers compete for service from scarce resources. Think of check-in lines at airports, waiting rooms in hospitals or queues in supermarkets, where the scarce resource is human manpower.
Next to these traditional settings, resource sharing is also important in large-scale service systems such as the internet, wireless networks and cloud computing facilities.
In these virtual environments, geographical location does not play a restricting role on the system size, paving the way for the emergence of large-scale resource sharing networks.
This thesis investigates how to design large-scale systems in order to achieve economies-of-scale, by which we mean that the system is highly occupied and hence utilizes efficiently the expensive resources, while at the same time, the offered service levels remain high.
In this introductory chapter, we give an overview of the available machinery that supports such principles and explain how this thesis contributes to the existing study of large-scale service systems.
A crucial concept behind most of the results discussed in the chapter is the Central Limit Theorem (CLT) -- arguably one of the most important theorems in mathematics and science.
\end{chapterstart}
\newpage
\section{Service systems \& queueing theory}
\subsection{Quality vs. Efficiency}
Large-scale service systems take many shapes and forms.
Classical examples of large-scale service systems include call centers \cite{Erlang1917,Palm1957,Whitt1999,Gans2003,Borst2004,Brown2005,Zeltyn2005,Bassamboo2009,Khudyakov2006} and communication systems \cite{Kleinrock1976,Anick1982,Kelly1985,Kleinrock2007,johanthesis}.
More recently, congestion-related issues in health care facilities and cloud-computing facilities have received much attention \cite{Armony2015,Green2007,YomTov2010,Gupta2007,Tan2012}.
In all settings, one can think of service systems as being composed of \textit{customers} and \textit{servers}.
In call centers, customers typically call to request help from one of the agents (servers).
In communication networks, the data packets are the customers and the communication channels are the servers.
In health care facilities, patients are the customers, and nurses/physicians are the servers.
The system scale may refer to the size of the client base it caters to, or the magnitude of its capacity, or both.
Next to the central notions of customers and servers, we emphasize that service systems are inherently stochastic, that is, subject to uncertainty.
Although arrival volumes can be anticipated to some extent over a certain planning horizon, for instance through historical data and forecasting methods, one cannot predict with certainty future arrival patterns.
Moreover, service requirements are typically random as well, adding more uncertainty.
This intrinsic stochastic variability is a predominant cause of delay experienced by customers in the system.
Due to the inherent randomness in both their arrival and service processes, stochastic models have proved instrumental in both quantifying and improving the operational performance of service systems.
Queueing theory and stochastics provide the mathematical tools to describe and evaluate these service systems.
Queueing models are often able to capture and explain fundamental phenomena that are common across applications.
A standard model for service systems is the $M/GI/s$ queue, which we will refer to as the \textit{many-server} queue.
This model assumes that customers arrive to the queue according to a Poisson process with rate $\lambda$, and customer service times are mutually independent and identically distributed (i.i.d.) samples from the distribution of a non-negative random variable $B$.
The parameter $s$ denotes the number of servers in the system, and hence restricts the number of simultaneous services.
The case $s=1$ corresponds to a single-server queue.
First principles say that the queueing process is stable, that is, the number of customers does not explode as time evolves, if and only if the expected workload $R := \lambda\mathbb{E}[B]$ brought into the system per time unit is strictly less than the system capacity.
In other words, the \textit{utilization} of the queue, defined as $\rho := \lambda\mathbb{E}[B] / s$ should remain strictly below one.
Naturally, a system manager prefers to operate at a utilization level close to one, so that resources are used efficiently.
However, it is known that pushing the occupation levels to 100\% leads to an explosive increase in congestion.
That is, the expected queue length and customer waiting time increase indefinitely, thereby reducing the quality-of-service (QoS) and also customer satisfaction.
These seemingly conflicting objectives give rise to a classical trade-off between customer satisfaction and costs of resources.
\subsection{Economies-of-scale}
Under the assumption that service times are exponentially distributed with mean $1/\mu$, the many-server queue reduces to the well-studied $M/M/s$ queue.
Despite its simplicity, the analysis of the $M/M/s$ queue explains mathematically the distinctive traits of queues in general, such as the non-linear effect of utilization on the queue size, and pooling effects.
Let $W^{(s)}$ denote the waiting time of a customer and $Q^{(s)}$ the queue length (including the customers in service) in the steady-state $M/M/s$ queue. Without loss of generality, we fix $\mu=1$, so that $\rho = \lambda/s$.
A straightforward balance argument gives the stationary distribution:
\begin{equation}
\label{eq:MMs_stationary_distribution}
\pi_k := \mathbb{P}( Q^{(s)} = k )
= \left\{
\begin{array}{ll}
\pi_0\frac{\lambda^k}{k!}, & \text{if } k < s, \\
\pi_0\frac{\lambda^s}{s!}\,\rho^{k-s} & \text{if } k \geq s,
\end{array}
\right.
\end{equation}
where
\begin{equation*}
\pi_0 := \Big( \sum_{k=0}^{s-1} \frac{\lambda^k}{k!} + \frac{1}{1-\rho} \frac{\lambda^s}{s!}\Big)^{-1}.
\end{equation*}
Natural QoS indicators include the expected waiting time $\mathbb{E}[W^{(s)}]$ and the delay probability $\mathbb{P}(W^{(s)}>0)$.
Invoking the PASTA (Poisson arrivals see time averages) property \cite{Wolff1982}, we know that the delay probability equals the probability of the queue length being greater or equal to the number of servers $s$.
Thus,
\begin{equation}
\label{eq:MMs_wait}
\mathbb{P}(W^{(s)} > 0) = \mathbb{P}(Q^{(s)}\geq s) = \frac{\lambda^s}{s!} \Big( (1-\rho) \sum_{k=0}^{s-1} \frac{\lambda^k}{k!} + \frac{\lambda^s}{s!} \Big)^{-1}.
\end{equation}
By Little's law, which says that $\mathbb{E}[(Q^{(s)}-s)^+] =\lambda\mathbb{E}[W^{(s)}]$, we furthermore have
\begin{equation}
\mathbb{E}[W^{(s)}] = \mathbb{P}(W^{(s)} > 0)\,\frac{1/s}{1-\rho}.
\label{eq:MMs_wait2}
\end{equation}
From these formulae, it is readily seen that $\mathbb{P}(W^{(s)} > 0) \to 1$ and $\mathbb{E}[W^{(s)}] \to \infty$ as $\rho \uparrow 1$ . That is, increasing $\lambda$ to $s$, while keeping the latter fixed, leads to a system in which all customers are delayed before service, and the expected delay before reaching a server increases to infinity.
The $M/M/s$ queue also reveals the effect of \textit{resource pooling}.
To illustrate the operational benefits of sharing resources, we compare a system of $s$ separate $M/M/1$ queues, each serving a Poisson arrival stream with rate $\lambda<1$, against one $M/M/s$ queue facing arrival rate $\lambda s$.
The two systems thus experience the same total workload and utilization, namely $\rho = \lambda$.
We fix the value of $\lambda$ and vary $s$.
Obviously, the waiting time and queue length distribution in the first scenario are unaffected by the parameter $s$, since there is no interaction between the single-server queues.
This lack of coordination tolerates a scenario of having an idle server, while the total number of customers in the system exceeds $s$, therefore wasting resource capacity.
Such an event cannot happen in the many-server scenario, due to the central queue.
This central coordination improves QoS. Indeed Figure \ref{fig:waiting_time_pooling} shows that the reduction in expected waiting time can be substantial.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\input{Introduction/Version_R1/tikz_tex/Ewait_pooling.tex}
\caption{Expected waiting time}
\end{subfigure}
\hspace{2mm}
\begin{subfigure}{0.48\textwidth}
\input{Introduction/Version_R1/tikz_tex/Pwait_pooling.tex}
\caption{Probability of delay}
\end{subfigure}
\caption{Effects of resource pooling in the $M/M/s$ queue.}
\label{fig:waiting_time_pooling}
\end{figure}
\noindent
So pooling kills two birds with one stone: QoS for customers improves and the system efficiency increases.
\subsection{Many-server scaling regimes}
\label{sec:intro_many_server_regimes}
Now that we know that economies-of-scale can be achieved, it is relevant to ask how to match capacity $s$ to a demand $\lambda$ in the setting where both $s$ and $\lambda$ become large.
The expressions in \eqref{eq:MMs_wait} and \eqref{eq:MMs_wait2} provide a starting point for finding such demand-capacity relations, particularly when we apply asymptotic analysis for $s\to\infty$, \cite{Halfin1981,Borst2004,Reed2009}.
Asymptotic theory of many-server systems relies on the prerequisite that the limiting behavior of the service system is determined by the way in which capacity $s$ is adjusted to demand, assuming demand grows large.
We illustrate this idea by investigating typical sample paths of the queue length process $Q = \{Q(t)\}_{t\geq 0}$ of an $M/M/s$ queue for increasing values of $\lambda$.
Figure \ref{fig:sample_path_small} depicts a sample path for $\lambda = 3$ and $s = 4$.
The number of customers queueing at time $t$ is given by $(Q(t)-s)^+$ with $(\cdot)^+ := \max\{0,\cdot\}$.
The number of idle servers is given by $(s-Q(t))^+$.
In Figure \ref{fig:sample_path_small}, the red and green area hence represent the cumulative queue length and cumulative number of idle servers, respectively, over the given time period.
Bearing in mind the dual goal of QoS and efficiency, we want to minimize both of these areas simultaneously.
\begin{figure}[b!]
\centering
\input{Introduction/Version_R1/tikz_tex/sample_path_small.tex}
\caption{Sample path of the $M/M/s$ queue with $\lambda = 3$ and $s=4$.}
\label{fig:sample_path_small}
\end{figure}
Next, we conduct a similar sample path experiment for increasing values of $\lambda$.
Since $s > \lambda$ is required for stability, the value of $s$ needs to be adjusted accordingly.
We propose three scaling rules:
\begin{equation}
\label{eq:intro_three_scaling_rules}
s^{(1)}_\lambda = \left[ \lambda + \beta \right ], \qquad
s^{(2)}_\lambda = \left[ \lambda + \beta\sqrt{\lambda} \right], \qquad
s^{(3)}_\lambda = \left[ \lambda + \beta\,\lambda \right],
\end{equation}
for some $\beta>0$, where $[\cdot]$ denotes the rounding operator.
Note that these three rules differ in terms of overcapacity $s-\lambda$.
Figure \ref{fig:sample_paths_lambda100} depicts typical sample paths of the queue length process for increasing values of $\lambda$ for the three scaling rules with $\beta = 0.5$.
\begin{figure}
\centering
\input{Introduction/Version_R1/tikz_tex/sample_paths_lambda10.tex}
\caption{Sample paths of the $M/M/s$ queue with $\lambda = 10,\,50$ and $100$ and $s$ set according to the three scaling rules in \eqref{eq:intro_three_scaling_rules}.}
\label{fig:sample_paths_lambda100}
\end{figure}
Observe that for all scaling rules, the stochastic fluctuations of the queue length processes relative to $\lambda$ decrease with the system size.
Moreover, the paths in Figure \ref{fig:sample_paths_lambda100} appear to become smoother with increasing $\lambda$.
Of course, the actual sample path always consists of upward and downward jumps of size 1, but we will show how proper centering and scaling of the queue length process indeed gives rise to a \textit{diffusion process} in the limit as $\lambda\to\infty$.
Although the difference in performance of the three regimes is not yet evident for relatively small $\lambda$, clear distinctive behavior occurs for large $\lambda$.
Under ${s_\lambda}^{(1)}$, the majority of customers is delayed and server idle time is low, since $\rho = (1+\beta/\lambda)^{-1} \to 1$ as $\lambda \to \infty$.
Systems dimensioned according to this rule value server efficiency over customer satisfaction and therefore this regime is in the literature also known as the \textit{efficiency-driven} (ED) regime \cite{Zeltyn2005}.
In contrast, the third scaling rule $s^{(3)}_\lambda$ yields a constant utilization level $\rho = 1/(1+\beta)$, which stays away from 1, even for large $\lambda$.
Queues operating in this regime exhibit significant server idle times.
Moreover, for the particular realization of the queueing processes for $\lambda = 50$ and $\lambda=100$ none of the customers waits.
This customer-centered regime is known as the \textit{quality-driven} (QD) regime \cite{Zeltyn2005}.
The scaling rule $s^{(2)}_\lambda$ is in some ways a combination of the other two regimes.
First, we have $\rho = (1 +\beta/\sqrt{\lambda})^{-1} \to 1$ as $\lambda \to \infty$, which indicates efficient usage of resources as the system grows.
The sample paths, however, indicate that only a fraction of the customers is delayed, and only small queues arise, which suggest good QoS.
This regime is therefore called \textit{quality-and-efficiency driven} (QED) regime.
Since this scaling regime and the related \textit{square-root staffing rule}
\begin{equation}
\label{eq:square_root_staffing rule}
s_\lambda = \lambda + \beta\sqrt{\lambda}
\end{equation}
strikes the right balance between the two profound objectives of capacity allocation in service systems, we discuss in the next section the mathematical foundations of the QED regime and quantify the favorable properties revealed by Figure \ref{fig:sample_paths_lambda100}.
\section{The QED regime: two canonical examples}
\label{sec:intro_QED_regime}
We saw in Figure \ref{fig:waiting_time_pooling} the advantageous effect of resource pooling and economies-of-scale in many-server systems.
In this section, we will explain how this is related to the Central Limit Theorem (CLT).
\begin{theorem}[Central Limit Theorem, e.g. {\cite[Thm.~27.1]{Billingsley1995}}]
Consider a sequence $X_1,X_2,\ldots,X_n$ of independent and identically distributed random variables having mean $\mu$ and positive variance $\sigma^2$.
Then,
\[
\frac{\sum_{i=1}^n X_i - n\mu }{\sqrt{n}\sigma} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{for }n\to\infty.
\]
where ${\;\buildrel{d}\over\Rightarrow\;}$ denotes convergence in distribution and $\mathcal{N}(0,1)$ is a random variable with standard normal distribution.
\end{theorem}
We shall now apply the CLT to the delay probability in the $M/M/s$ queue.
Striking the proper balance between queueing delay and server efficiency asymptotically, i.e.~balancing the green and red areas in Figure \ref{fig:sample_paths_lambda100}, in mathematical terms boils down to choosing a service level $s_\lambda$ such that both the delay probability $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ and $\mathbb{P}(Q^{(s_\lambda)} < s_\lambda)$ remain strictly smaller than 1 as $\lambda\to\infty$.
In other words, one would like to see that $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ converges to a non-degenerate limit $\alpha \in (0,1)$ as $\lambda\to\infty$.
To get a feel for the natural scale of the queue, we first examine the situation with unlimited capacity.
More precisely, let $Q^{(\infty)}$ be the number of customers in a steady-state $M/GI/\infty$ queue with mean service requirement $\mathbb{E}[B]=1$.
Notice that in this infinite-server setting, $Q^{(\infty)}$ also represents the steady-state number of busy servers.
It is well known that $Q^{(\infty)}$ follows a Poisson distribution with mean equal to the expected workload, in our case $R = \lambda$.
Moreover, if we assume that $\lambda$ is integer, then a Poisson random variable with rate $\lambda$ can be viewed as the sum of $\lambda$ i.i.d. Poisson random variables with rate 1.
In other words, $Q^{(\infty)} = \sum_{i=1}^\lambda P_i$, where the $P_i$'s, $i=1,2,\ldots,n$, have Poisson distribution with unit mean and variance, and are mutually independent.
\noindent
The CLT thus gives
\begin{equation}
\label{eq:infinite_server_tail}
\mathbb{P}(Q^{(\infty)} \geq x_\lambda )
= \mathbb{P}\left(\frac{Q^{(\infty)} -\lambda }{\sqrt{\lambda}} \geq \frac{ x_\lambda - \lambda}{\sqrt{\lambda}} \right)
\approx 1-\Phi\left( \frac{x_\lambda-\lambda}{\sqrt{\lambda}} \right),
\end{equation}
where $\Phi$ denotes the cumulative distribution function of the standard normal distribution for large $\lambda$.
Hence, the probability in \eqref{eq:infinite_server_tail} converges to a constant value away from both 0 and 1 if and only if $(x_\lambda - \lambda)/\sqrt{\lambda} \to x \in \mathbb{R}$, or equivalently $x_\lambda = \lambda + x \sqrt{\lambda} + o(\sqrt{\lambda})$, as $\lambda\to\infty$.
Here, the relation $u(\lambda) = o(v(l))$ implies that $u(\lambda)/v(\lambda)\to 0$ as $\lambda\to\infty$.
Equation \eqref{eq:infinite_server_tail} also shows that the leading order of the random variable describing the queue length is $\lambda$, while the stochastic fluctuations are of order $\sqrt{\lambda}$.
If we now pretend, for a moment, that the infinite-server queue serves as a good approximation for the many-server queue with $s_\lambda$ servers, then \eqref{eq:infinite_server_tail} says that the steady-state probability of delay for ${s_\lambda} = \lambda +\beta\sqrt\lambda$ obeys the Gaussian approximation
\begin{equation}
\label{eq:infinite_server_approx_delay}
\mathbb{P}(W^{(s_\lambda)}>0) = \mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda ) \approx 1-\Phi(\beta),
\end{equation}
where $\Phi$ denotes the cumulative distribution function (cdf) of the standard normal distribution.
Of course, the infinite-server system ignores the one thing that makes a queueing system unique, namely that a queue is formed when all servers are busy.
During these periods of congestion, customers will depart from a system with a finite number of servers at a slower pace than in its infinite-server counterpart.
So the approximation in \eqref{eq:infinite_server_approx_delay} is likely to overestimate the actual delay probability, and a more careful investigation of the queue length process in many-server settings is needed. Nevertheless, the infinite-server heuristic reveals that in a well-managed system, i.e.~queues are of acceptable length, the size at which the system operates is of the order $\lambda$, with fluctuations of order $\sqrt{\lambda}$.
We shall now demonstrate through two canonical examples how these guessed natural scalings can be turned into mathematically rigorous statements.
Both examples which will play a key role in this thesis.
\subsection{The $M/M/s$ queue}
\label{sec:intro_MMsqueue}
\textbf{Converging delay probability.}
Let $Q^{(s)}$ denote the steady-state number of customers in an $M/M/s$ queue with arrival rate $\lambda$ and mean service requirement 1, of which the probability distribution is given in \eqref{eq:MMs_stationary_distribution}.
Halfin \& Whitt \cite{Halfin1981} showed that, just as the tail probability in the infinite-server setting, the delay probability in the $M/M/s$ queue converges under scaling \eqref{eq:square_root_staffing rule} to a value between 0 and 1.
Moreover, they showed that this is in fact the only scaling regime in which such a non-degenerate limit exists and identified its value.
Let $\varphi$ denote the probability density function (pdf) of the standard normal distribution.
\begin{proposition}[{\cite[Prop.~2.1]{Halfin1981}}]
\label{prop:HalfinWhitt_delay_probability}
The probability of delay in the $M/M/s_\lambda$ queue has the non-degenerate limit
\begin{equation}
\lim_{\lambda\to\infty} \mathbb{P}( W^{(s_\lambda)} > 0 ) = \alpha \in (0,1)
\end{equation}
if and only if
\begin{equation}
\label{eq:HalfinWhitt_scaling}
\lim_{\lambda\to\infty} (1-\rho_{s_\lambda}) \sqrt{s_\lambda} \to \beta, \quad \beta > 0,
\end{equation}
where $\alpha$ is given by
\begin{equation}\label{eq:HW_delay_prob}
\alpha = \left( 1+ \frac{\beta\,\Phi(\beta)}{\varphi(\beta)} \right)^{-1} =: g(\beta).
\end{equation}
\end{proposition}
\begin{proof}
We first prove the sufficiency condition.
Rewrite \eqref{eq:MMs_wait} as
\begin{equation}
\label{eq:proof_HW_0}
\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )
= \left( 1 + (1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}\right) ^{-1}.
\end{equation}
Similar to \eqref{eq:infinite_server_tail} we find
\begin{align}
\mathbb{P}({\rm Pois}(\lambda) < {s_\lambda})
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < \frac{{s_\lambda}-\lambda}{\sqrt{\lambda}}\right) \nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\frac{{s_\lambda}}{\sqrt\lambda}\right)\nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\sqrt{{s_\lambda}}\left(1+o(1)\right) \right) \to \Phi(\beta),
\label{eq:proof_HW_1}
\end{align}
for $\lambda\to\infty$.
Using Stirling's approximation, we get
\begin{align*}
\mathbb{P}({\rm Pois}(\lambda)=s) &= {\rm e}^{-\lambda}\frac{\lambda^{{s_\lambda}}}{{s_\lambda}!}
\sim {\rm e}^{-\lambda} \lambda^{{s_\lambda}}\cdot \frac{1}{\sqrt{2\pi\,{s_\lambda}}} \left(\frac{\rm e}{{s_\lambda}}\right)^{{s_\lambda}} = \frac{1}{\sqrt{2\pi{s_\lambda}}}\,{\rm e}^{{s_\lambda}-\lambda - {s_\lambda}{\rm ln}(\rho_{{s_\lambda}})}.
\end{align*}
Since ${\rm ln}(\rho_{{s_\lambda}}) = -(1-\rho_{{s_\lambda}}) - \tfrac{1}{2}(1-\rho_{{s_\lambda}})^2 + o((1-\rho_{{s_\lambda}})^2)$ we find that
\begin{equation}
\label{eq:proof_HW_2}
\frac{ \mathbb{P}({\rm Pois}(\lambda) = s) }{ 1-\rho_{{s_\lambda}} }
= \frac{1}{(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}} \, \frac{{\rm e}^{ -\tfrac{1}{2}(1-\rho_{{s_\lambda}})^2{s_\lambda} + o\left((1-\rho_{{s_\lambda}})^2{s_\lambda}\right)}}{\sqrt{2\pi}} \to \frac{1}{\beta}\, \frac{{\rm e}^{{-}\tfrac{1}{2} \beta^2}}{\sqrt{2\pi}} = \frac{\varphi(\beta)}{\beta}.
\end{equation}
Substituting \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2} into \eqref{eq:proof_HW_0} gives \eqref{eq:HW_delay_prob}.
The necessary condition follows directly by the characterization of $\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )$ as in \eqref{eq:proof_HW_0} by observing, through \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2}, that the term
\begin{equation*}
(1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}
\end{equation*}
has a limiting value in $(0,\infty)$ only if $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}} \to \beta$ for some $\beta>0$.
\end{proof}
Observe that $g(\beta)$ is a strictly decreasing function on $(0,\infty)$ with $g(\beta) \to 1$ as $\beta\to 0$ and $g(\beta)\to 0$ for $\beta\to\infty$.
Thus all possible delay probabilities are achievable in the QED regime, which will prove useful for the dimensioning of systems (see Section \ref{sec:intro_dimensioning}).
\begin{figure}
\centering
\input{Introduction/Version_R1/tikz_tex/halfin_whitt_accuracy.tex}
\caption{The delay probability $\mathbb{P}(Q^{({s_\lambda})} \geq {s_\lambda})$ with ${s_\lambda} = [ \lambda + \beta \sqrt{\lambda} ]$ for $\beta = 0.1,\ 0.5,$ and 1 as a function of $\lambda$.}
\label{fig:delay_probs_HW_MMs}
\end{figure}
Although Proposition \ref{prop:HalfinWhitt_delay_probability} is an asymptotic result for $\lambda\to\infty$, Figure \ref{fig:delay_probs_HW_MMs} shows that $g(\beta)$ can serve as an accurate approximation for the delay probability for relatively small $\lambda$.
From Proposition \ref{prop:HalfinWhitt_delay_probability}, it also follows that under \eqref{eq:HalfinWhitt_scaling},
\begin{equation}
\label{eq:halfinwhitt_wait}
\sqrt{{s_\lambda}}\,\mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{P}(W^{({s_\lambda})}>0)}{(1-\rho_{s_\lambda})\sqrt{{s_\lambda}}} \to \frac{g(\beta)}{\beta} =: h(\beta), \qquad \text{ as }\lambda\to\infty,
\end{equation}
where we have used the characterization of $\mathbb{E}[W^{({s_\lambda})}]$ in \eqref{eq:MMs_wait2}.
This implies that in the QED regime, the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$ as $\lambda\to\infty$.
By Little's law this implies that the expected queue length is $O(\sqrt{{s_\lambda}})$.
By the relation $u(\lambda) = O(v(\lambda))$ we mean that $\limsup_{\lambda\to\infty} u(\lambda)/v(\lambda)< \infty$.
While these are all steady-state results, similar statements can be made for the entire queue-length process, as shown next.
The theoretical results of the QED regime we presented here are based on steady-state queueing analysis.
But at the heart of the QED theory lies a much deeper result in which the entire queue-length process, over all points in time, converges to some other limiting process.
\\*
\noindent\textbf{Process-level convergence.}
Obtaining rigorous statements about stochastic-process limits poses considerable mathematical challenges.
Rather than presenting the deep technical details of the convergence results, we give a heuristic explanation of how the limiting process arises and what it should look like.
The queue-length process $Q^{({s_\lambda})}$ in Figure \ref{fig:sample_paths_lambda100} with scaling rule ${s_\lambda} = [\lambda + \beta \sqrt{\lambda}]$ appears to concentrate around the level ${s_\lambda}$.
As argued before, the stochastic fluctuations are of order $\sqrt{\lambda}$, or equivalently $\sqrt{{s_\lambda}}$.
For that reason, we consider the centered and scaled process
\begin{equation}
\label{eq:intro_scaled_queue_length_process}
X^{({s_\lambda})}(t) := \frac{ Q^{({s_\lambda})}(t) - {s_\lambda}}{\sqrt{{s_\lambda}}}, \qquad \text{ for\ all } t\geq 0,
\end{equation}
and ask what happens to this process as $\lambda\to\infty$.
First, we consider the expected drift conditioned on $X^{({s_\lambda})}(t) = x$.
When $x> 0$, this corresponds to a state in which $Q^{({s_\lambda})}>{s_\lambda}$ and hence all servers are occupied.
Therefore, the expected rate at which customers leave the system is ${s_\lambda}$, while the arrival rate remains $\lambda$, so that the expected drift of $X^{({s_\lambda})}(t)$ in $x>0$ satisfies
\[
\frac{\lambda - {s_\lambda}}{\sqrt{{s_\lambda}}} \to -\beta, \qquad \text{as }\lambda\to\infty,
\]
under scaling $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to \beta$ in \eqref{eq:HalfinWhitt_scaling}.
When $x\leq 0$, only ${s_\lambda} + x\sqrt{{s_\lambda}}$ servers are working, so that the net drift is
\[
\frac{\lambda - ({s_\lambda} + x\sqrt{{s_\lambda}} )}{\sqrt{{s_\lambda}}} \to -\beta-x, \qquad \text{as }\lambda\to\infty.
\]
Now, imagine what happens to the sample paths of $\{X^{({s_\lambda})}(t)\}_{t\geq 0}$ as we increase $\lambda$.
Within a fixed time interval, larger $\lambda$ and ${s_\lambda}$ will trigger more and more events, both arrivals and departures.
Also, the jump size at each event epoch decreases as $1/\sqrt{{s_\lambda}}$ as a consequence of the scaling in \eqref{eq:intro_scaled_queue_length_process}.
Hence, there will be more events, each with a smaller impact, and in the limit as $\lambda\to\infty$, there will be infinitely many events of infinitesimally small impact.
This heuristic explanation suggests that the process $X^{({s_\lambda})}(t)$ converges to a stochastic-process limit, which is continuous, and has infinitesimal drift ${-}\beta$ above zero and ${-}\beta-x$ below zero.
Figure \ref{fig:sample_paths_diffusion} visualizes the appearance of the suggested scaling limit as $\lambda$ and ${s_\lambda}$ increase.
\begin{figure}
\centering
\input{Introduction/Version_R1/tikz_tex/sample_path_diffusion.tex}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda})}(t)$ with $\lambda = 5$, $\lambda=5$ and $\lambda=500$ and ${s_\lambda} = [\lambda+0.5\sqrt{\lambda}]$.}
\label{fig:sample_paths_diffusion}
\end{figure}
The following theorem by Halfin \& Whitt \cite{Halfin1981} characterizes this scaling limit more formally.
\begin{theorem}
\label{thm:Halfin_Whitt_diffusion}
Let $X^{({s_\lambda})}(0)\, {\;\buildrel{d}\over\Rightarrow\;} X(0) \in \mathbb{R}$ and $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to\beta$. Then for all $t\geq 0$,
\[
X^{({s_\lambda})}(t) {\;\buildrel{d}\over\Rightarrow\;} X(t),\qquad \text{ as }\lambda\to\infty,
\]
where $X(t)$ is the diffusion process with infinitesimal drift $m(x)$ given by
\[
m(x) = \left\{
\begin{array}{ll}
-\beta, & \text{if }x> 0,\\
-\beta-x, & \text{if } x \leq 0
\end{array}\right.
\]
and infinitesimal variance $\sigma^2(x) = 2$.
\end{theorem}
The limiting diffusion process $\{X(t)\}_{t\geq 0}$ in Theorem \ref{thm:Halfin_Whitt_diffusion} is a combination of a negative-drift Brownian motion in the upper half plane and an Ornstein-Uhlenbeck (OU) process in the lower half plane.
We refer to this hybrid diffusion process as the Halfin-Whitt diffusion.
Much is known for such diffusion processes with piecewise linear drift coefficient, see \cite{Leeuwaarden2012,Fralix2014}.
Its stationary distribution can for instance be derived, see e.g. \cite{Browne1995}.
\begin{proposition}
\label{thm:intro_HW_stationary_distribution}
Let $X(t) {\;\buildrel{d}\over\Rightarrow\;} X(\infty)$ as $t\to\infty$ for a random variable $X(\infty)$ and $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}\to \beta$ for $\lambda\to\infty$.
Then
\begin{align}
\mathbb{P}(X(\infty) > 0 ) &= g(\beta),\\
\mathbb{P}(X(\infty) \geq x | X(\infty) > 0) &= {\rm e}^{-\beta x} ,\quad \text{for }x>0,\\
\mathbb{P}(X(\infty) \leq x | X(\infty) \leq 0 ) &= \frac{\Phi(\beta+x)}{\Phi(\beta)},\quad \text{for }x\leq 0.
\end{align}
\end{proposition}
\noindent
This result shows that as the system grows large, the $Q^{({s_\lambda})}(t)$ concentrates around ${s_\lambda}$, and the fluctuations are of order $\sqrt{{s_\lambda}}$.
Moreover, Proposition \ref{thm:intro_HW_stationary_distribution} iterates the limiting values for the delay probability and scaled expected delay. Namely,
\[ \mathbb{P}\big(W^{({{s_\lambda}})} > 0 \big) \rightarrow \mathbb{P}( X(\infty) > 0 ) = g(\beta)\]
and
\[ \sqrt{{s_\lambda}}\mathbb{E}[W^{({s_\lambda})}] \approx \frac{\mathbb{E}[ Q^{({s_\lambda})}]}{\sqrt{{s_\lambda}}} \rightarrow \mathbb{E}[X(\infty)] = \int_0^\infty g(\beta){\rm e}^{-\beta x} {\rm d} x = \frac{g(\beta)}{\beta},\]
For obvious reasons, the QED regime is also referred to as the Halfin-Whitt regime, and both these names are used interchangeably in this thesis.
\subsection{The $M/D/s$ queue}
\label{sec:intro_discrete_model}
We next consider a many-server queue with deterministic service requirements equal to one, a Poisson arrival process of rate $\lambda$ and ${s_\lambda}$ servers.
We let $Q^{({s_\lambda})}(t)$ be the process describing the number of customers in the system and only examine the process at discrete time epochs $t=0,1,2,\ldots$.
In our analysis, we focus on the queue length process $Z^{({s_\lambda})}(t) := (Q^{({s_\lambda})}(t) - {s_\lambda})^+$.
Since we discretize time, the number of new arrivals per time period is given by the sequence of i.i.d.~random variables $\{A_k\}_{k\geq 1}$, which has a Poisson distribution with mean $\lambda$.
At the start of the $k^{\rm th}$ period, $Z^{({s_\lambda})}(k)$ customers are waiting.
Because the service time of a customer is equal to the period length, all $\min\{Q^{({s_\lambda})}(k),{s_\lambda}\}$ customers in service at the beginning of the period will have left the system by time $t=k+1$.
This implies that $\min\{Z^{({s_\lambda})}(k),{s_\lambda}\}$ of the waiting customers are taken into service during period $k$, but could not possibly have departed before its end, due to the deterministic service times.
If $Z^{({s_\lambda})}(k)<{s_\lambda}$, then additionally $\min\{ A_k , {s_\lambda}-Z^{({s_\lambda})}(k) \}$ of the new arrivals are taken into service.
This yields a total of $A_k$ arrivals, and $\min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\}$ departures from the queue during period $k$, which gives the Lindley type recursion \cite{Lindley1952}, with $Z^{({s_\lambda})}(0) = 0$,
\begin{equation}
\label{eq:discrete_recursion}
Z^{({s_\lambda})}(k+1) = Z^{({s_\lambda})}(k) + A_k - \min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\} = \max\{ 0,Z^{({s_\lambda})}(k) + A_k - {s_\lambda} \}.
\end{equation}
The queue length process thus gives rise to a random walk with i.i.d.~steps of size
$(A^{({s_\lambda})}-{s_\lambda})$, with a reflecting barrier at zero. We can iterate the recursion in \eqref{eq:discrete_recursion} to find
\begin{align}
Z^{({s_\lambda})}(k+1) &= \max\left\{ 0 , Z^{({s_\lambda})}(k) + A_k-{s_\lambda} \right\} \nonumber\\
&= \max\left\{ 0 , \max\{ 0 , Z^{({s_\lambda})}(k-1) + (A_{k-1}-{s_\lambda})\} + (A_k-{s_\lambda})\} \right\}\nonumber \\
&= \max\left\{ 0 , (A_k-{s_\lambda}) , Z^{({s_\lambda})}(k-1) + (A_k-{s_\lambda}) + (A_{k-1}-{s_\lambda})\right\}\nonumber \\
&= \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_{k-i}-{s_\lambda})\Big\}
{\;\buildrel{d}\over= \;} \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_i-{s_\lambda}) \Big\},
\label{eq:max_randomwalk}
\end{align}
where the last equality in distribution holds due to the duality principle for random walks, see e.g.~\cite[Sec.~7.1]{Ross1996}.
For stability, the expected step size satisfies $\mathbb{E}[A_k - {s_\lambda}] = \lambda-{s_\lambda} < 0$.
We use the shorthand notation for the partial sum $S_k := \sum_{i=1}^k (A_i-{s_\lambda})$.
Let $Z^{({s_\lambda})}(\infty):= \lim_{k\to\infty} Z^{({s_\lambda})}(k)$ denote the stationary queue length in this $M/D/s$ queue, which can be shown to exist under our assumptions.
The probability generating function (pgf) of $Z^{({s_\lambda})}(\infty)$ can then be expressed in terms of the pgf of the positive parts of the partial sum:
\begin{equation}
\label{eq:Spitzers_identity}
\mathbb{E}[ w^{Z^{({s_\lambda})}(\infty)} ]
= \exp\Big\{ - \sum_{k=1}^\infty \frac{1}{k}\, (1- \mathbb{E}[w^{S_k^+}]) \Big\},\qquad |w|\leq 1.
\end{equation}
From \eqref{eq:Spitzers_identity}, which is a special case of Spitzer's identity~\cite{Spitzer1964}, we obtain for the mean queue length and empty-queue probability the expressions
\begin{align}
\mathbb{E}[Z^{({s_\lambda})}(\infty)] &= \sum_{k=1}^\infty \frac{1}{k}\, \mathbb{E}[ S_k^+ ],\nonumber\\
\mathbb{P}(Z^{({s_\lambda})}(\infty) = 0 ) &= \exp\Big\{ -\sum_{k=1}^\infty \frac{1}{k}\, \mathbb{P}( S_k^+ > 0 ) \Big\}.
\label{eq:spitzer_expressions}
\end{align}
Although explicit, the expressions in \eqref{eq:spitzer_expressions} reveal little of the structure of the queue length process.
Hence, we again turn to asymptotics. \\
\noindent\textbf{Gaussian random walk.}
\label{sec:intro_gaussian_random_walk}
We take another look at the identity in \eqref{eq:max_randomwalk}, and ask ourselves what happens if $\lambda$ grows large.
Since $\mathbb{E}[A_k-{s_\lambda}] = \lambda-{s_\lambda} = -\beta\sqrt{\lambda} + o(\sqrt{\lambda})$ under the QED scaling \eqref{eq:square_root_staffing rule}, it makes sense to consider the scaled queue length process $X^{({s_\lambda})}(k) := Z^{({s_\lambda})}(k)/\sqrt{\lambda}$ for all $k\geq 0$, with scaled steps $Y_k^{({s_\lambda})} := (A_k-{s_\lambda})/\sqrt{\lambda}$.
Dividing both sides of \eqref{eq:max_randomwalk} by $\sqrt{\lambda}$ then gives
\begin{equation}
X^{({s_\lambda})}(k+1) = \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j Y^{({s_\lambda})}_i \Big\}.
\end{equation}
Observe that $A_k {\;\buildrel{d}\over= \;} {\rm Pois}(\lambda)$ with ${\rm Pois}(\lambda)$ a random variable with mean $\lambda$.
Hence by the CLT
\begin{equation*}
Y^{({s_\lambda})}_k = \frac{ A_k - {s_\lambda} }{\sqrt\lambda} = \frac{A_k-\lambda}{\sqrt\lambda} - \beta \ {\;\buildrel{d}\over\Rightarrow\;} \ Y_k {\;\buildrel{d}\over= \;} \mathcal{N}(-\beta,1),
\end{equation*}
for $\lambda\to\infty$, where $\mathcal{N}(-\beta,1)$ denotes a normally distributed random variable with mean $-\beta$ and standard deviation 1.
So we expect the scaled queue length process to converge in distribution to a reflected random walk with normally distributed increments, i.e. a reflected \textit{Gaussian random walk}.
Indeed, it is easily verified that \cite{Janssen2008a},
\begin{equation}
X^{({s_\lambda})}(k)\ {\;\buildrel{d}\over\Rightarrow\;} \ M_\beta(k) := \max_{0\leq j\leq k} \Big\{\sum_{i=1}^j Y_j \Big\}, \qquad \lambda\to\infty.
\end{equation}
Let $M_\beta:= \lim_{k\to\infty} M_\beta(k)$ denote the all-time maximum of a Gaussian random walk.
It can be shown that $M_\beta$ almost surely exists and that
\[
X^{({s_\lambda})}(\infty) := \lim_{k\to\infty} X^{({s_\lambda})}(k) {\;\buildrel{d}\over\Rightarrow\;} M_\beta,
\]
for instance by \cite[Prop.~19.2]{Spitzer1964} and \cite[Thm.~X6.1]{Asmussen2003}.
The following theorem can be proved using a similar approach as in \cite{Jelenkovic2004}.
(We prove this result in a more general setting in Chapter 3.)
\begin{theorem}
Let $X^{({s_\lambda})}(\infty)$ be the scaled queue length in steady-state. If $(1-\rho_{{s_\lambda}})\sqrt{\lambda}\to\beta$, then as $\lambda\to\infty$,
\begin{enumerate}
\item[\normalfont (i)] $X^{({s_\lambda})}(\infty) {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[\normalfont (ii)] $\mathbb{P}(X^{({s_\lambda})}(\infty) = 0) \to \mathbb{P}(M_\beta = 0)$,
\item[\normalfont (iii)] $\mathbb{E}[X^{({s_\lambda})}(\infty)^k] \to \mathbb{E}[M_\beta^k]$, for any $k>0$.
\end{enumerate}
\end{theorem}
The Gaussian random walk is well studied \cite{Siegmund1978,Chang1997,Janssen2006,Blanchet2006,Janssen2006} and there is an intimate connection with Brownian motion.
The only difference, one could say, is that Brownian motion is a continuous-time process, whereas the Gaussian random walk only changes at discrete points in time.
If $\{B(t)\}_{t\geq 0}$ is a Brownian motion with drift $-\mu <0$ and infinitesimal variance $\sigma^2$ and $\{W(t)\}_{t \geq 0}$ is a random walk with $\mathcal{N}(-\mu,\sigma^2)$ steps and $B(0) = W(0)$, then $W$ can be regarded as the process $B$ embedded at equidistant time epochs.
That is, $W(t) {\;\buildrel{d}\over= \;} B(t)$ for all $t\in\mathbb{N}^+$.
For the maximum of both processes this coupling implies
\begin{equation}
\max_{k\in \mathbb{N}^+} W(k) = \max_{k\in \mathbb{N}^+} B(k) \leq_{\rm st}
\max_{t\in \mathbb{R}^+} B(t),
\label{eq:max_inequality}
\end{equation}
where $\leq_{\rm st}$ denotes stochastic dominance.
This difference in maximum is visualized in Figure \ref{fig:BrownianMotion_vs_GaussianRW}.
It is known that the all-time maximum of Brownian motion with negative drift $-\mu$ and infinitesimal variable $\sigma^2$ has an exponential distribution with mean $\sigma/2\mu$ \cite{Harrison1985}.
Hence, \eqref{eq:max_inequality} implies that $M_\beta$ is stochastically upper bounded by an exponential random variable with mean $1/2\beta$.
\begin{figure}
\centering
\begin{tikzpicture}[scale = 1]
\begin{axis}[
xmin = 0,
xmax = 10,
ymin = -2.2,
ymax = 5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel={$t$},
ylabel={},
xscale=1,
yscale=0.8]
\addplot[gray] file {Introduction/Version_R1/tikz/Brownian_Motion_SamplePath/BM.txt};
\addplot[only marks,mark size = 2] file {Introduction/Version_R1/tikz/Brownian_Motion_SamplePath/GW.txt};
\addplot[dashed] file {Introduction/Version_R1/tikz/Brownian_Motion_SamplePath/maxBM.txt};
\addplot[dotted] file {Introduction/Version_R1/tikz/Brownian_Motion_SamplePath/maxGW.txt};
\end{axis}
\end{tikzpicture}
\caption{Brownian motion (gray) and embedded Gaussian random walk (marked) with their respective running maxima (dashed and dotted, respectively).}
\label{fig:BrownianMotion_vs_GaussianRW}
\end{figure}
Despite this easy bound, precise results for $M_\beta$ are more involved. Let $\zeta$ denote the Riemann zeta function, which is defined as, see e.g.~\cite[Eq.~25.2.1]{NIST},
\begin{equation}
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
\end{equation}
\begin{theorem}[{\cite[Thm.~1]{Chang1997} \& \cite[Thm.~2 \& 3]{Janssen2006}}]
For $0<\beta<2\sqrt{\pi}$,
\begin{align}
\mathbb{P}(M_\beta = 0) &= \sqrt{2}\beta\, \exp \left\{ \frac{\beta}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(1/2-l)}{l!(2l+1)} \left(\frac{-\beta^2}{2}\right)^l \right\},\\
\mathbb{E}[M_\beta] &= \frac{1}{2\beta} + \frac{\zeta(1/2)}{\sqrt{2\pi}} + \frac{\beta}{4}
+ \frac{\beta^2}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(-1/2-l)}{l!(2l+1)(2l+2)} \left(\frac{-\beta^2}{2}\right)^l,\\
{\rm Var}\, M_\beta &=
\frac{1}{4\beta^2} - \frac{1}{4} - \frac{2\,\zeta(-1/2)}{\sqrt{2\pi}}\beta - \frac{\beta^2}{24}\nonumber\\
&\qquad\qquad -
\frac{2\beta^3}{\sqrt{2\pi} } \sum_{l=0}^\infty
\frac{\zeta(-3/2-l)}{l!(2l+1)(2l+2)(2l+3)} \Big(\frac{-\beta^2}{2}\Big)^l.
\end{align}
\end{theorem}
\subsection{Characteristics of the QED regime}
\label{sec:intro_characteristics}
Now that we have seen how the square-root staffing rule \eqref{eq:square_root_staffing rule} yields non-degenerate limiting behavior in two classical queueing models, we shall elaborate on how the QED regime gives rise to (at least) three desirable properties.
The first property relates to the efficient usage of resources, expressed as
\begin{equation}
\rho_{{s_\lambda}} = \frac{\lambda}{{s_\lambda}} = 1 - \frac{\beta}{\sqrt{{s_\lambda}}} + O\big(1/\lambda\big), \tag{Efficiency}
\end{equation}
where we have used that ${s_\lambda} = O(\lambda)$.
The second distinctive property is the balance between QoS and efficiency:
\begin{equation}
\mathbb{P}(W^{({s_\lambda})}>0) \to g(\beta), \qquad \text{and} \qquad \mathbb{P}(W^{({s_\lambda})}>0) \to 1-\mathbb{P}(M_\beta=0), \tag{Balance}
\end{equation}
as ${s_\lambda} \to \infty$, in the $M/M/s$ queue and $M/D/s$ queue, respectively.
The third property relates to good QoS:
\begin{equation}
\mathbb{E}[W^{({s_\lambda})}] = \frac{h(\beta)}{\sqrt{{s_\lambda}}} + o(1/\sqrt{{s_\lambda}}) \qquad \text{and} \qquad \mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{E}[M_\beta]}{\sqrt{{s_\lambda}}} + O(1/\sqrt{{s_\lambda}}), \tag{QoS}
\end{equation}
in the $M/M/s$ queue and $M/D/s$ queue, respectively.
Hence the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$.
Both limiting functions $g(\beta)$ and $1-\mathbb{P}(M_\beta=0)$ can take all values in $(0,1)$ by tuning the parameter $\beta$.
Since the mathematical underpinning of these properties comes from the CLT, we can expect the properties to hold for a much larger class of models.
These models should then be members of the same universality class (to which the CLT applies).
Let us again show this by example.
Consider a stochastic system in which demand per period is given by some random variable $A$, with mean $\mu_A$ and variance $\sigma_A^2<\infty$.
For systems facing large demand we propose to set the capacity according to the more general rule
\[s = \mu_A + \beta\sigma_A,\]
which consists of a minimally required part $\mu_A$ and a variability hedge $\beta\sigma_A$.
Assume that the workload brought into the system is generated by $n$ stochastically identical and independent sources.
Each source $i$ generates $A_{i,j}$ work in the $j^{th}$ period, with $\mathbb{E}[A_{i,j}] = \mu$ and ${\rm Var}\,\,A_{i,j} = \sigma^2$.
Then the total amount of work arriving to the system during one period is $A_j^{(n)} = \sum_{i=1}^n A_{i,j}$ with mean $n\mu$ and variance $n\sigma^2$.
Assume that the system is able to process a deterministic amount of work $s_n$ per period and denote by $U^{(n)}(j)$ the amount of work left over at the end of period $j$.
Then,
\begin{equation}
U^{(n)}(j+1) = \left( U^{(n)}(j) + A^{(n)}_j - s_n \right)^+.
\end{equation}
Given that $s_n > \mathbb{E}[A^{(n)}_1] = n\mu$, the steady-state limit $U^{(n)} := \lim_{j\to\infty} U^{(n)}(j)$ exists and satisfies
\begin{equation}
U^{(n)} {\;\buildrel{d}\over= \;} \left( U^{(n)} + A^{(n)}_1 - s_n \right)^+.
\label{eq:bulk_service_stationary_recursion}
\end{equation}
This framework is also known as the bulk service queue or the Anick-Mitra-Sondhi model \cite{Anick1982,Janssen2005,Janssen2008}.
In this scenario, increasing the system size is done by increasing $n$, the number of input flows.
As we have seen before, it requires a rescaling of the process $U^{(n)}$ by an increasing function $c(n)$, in order to obtain a non-degenerate scaling limit $U := \lim_{n\to\infty} U^{(n)}/c(n)$.
(We omit the technical details needed to justify the interchange of limits.)
From \eqref{eq:bulk_service_stationary_recursion} it becomes clear that the scaled increment
\begin{equation}
\frac{A^{(n)}_j - s_n}{c(n)} = \frac{\sum_{i=1}^n A_{i,j} - n\mu}{c(n)} + \frac{n\mu - s_n}{c(n)}
\end{equation}
only admits a proper limit if $c(n)$ is of the form $c(n) = O(\sqrt{n})$, by the virtue of the CLT, and $(s_n-n\mu)/c(n) \to \beta >0$ as $n\to\infty$.
Especially for $c(n) = \sigma\sqrt{n}$, this reveals that $U$ has a non-degenerate limit, which is equal in distribution to the maximum of a Gaussian random walk with drift ${-}\beta$ and variance 1, if
\[
s_n = n\mu+\beta \sqrt{n}\sigma + o(\sqrt{n}).
\]
Moreover, the results on the Gaussian random walk presented in Section \ref{sec:intro_gaussian_random_walk} are applicable to this model and the key features of the QED scaling carry over to this more general setting as well.
In conclusion, the many-sources framework shows that the QED scaling finds much wider applications than queueing models with Poisson input only.
\subsection{Related literature}
We now provide a partial overview on the literature on heavy-traffic analysis in queueing theory and the QED regime in particular.\\
\\*
\noindent\textbf{Conventional heavy-traffic.}
Before the formal introduction of the Halfin-Whitt scaling regime in 1981, see \cite{Halfin1981}, the existing literature on the asymptotic analysis of queues mostly evolved around two types of scaling regimes: single-server and infinite-server regimes.
The idea of studying a sequence of queues in which the utilization approaches 100\%, i.e.~heavy-traffic, was first laid out by Kingman in the 1960s.
In \cite{Kingman1961,Kingman1962} he showed how in the $GI/G/1$ queue, under mild conditions on the arrival and service processes, the scaled steady-state waiting time $(1-\rho)W^{(1)}$ converges to an exponentially distributed random variable.
The notion that heavily loaded systems admit a scaling limit that is remarkably simple compared to the otherwise intractable pre-limit queueing systems triggered a surge of research within the field of queueing theory in the 1960s and 1970s, see \cite{Borovkov1965,Iglehart1970,Brumelle1971,Newell1973,Kollerstrom1974,Kollerstrom1979,Whitt1974} among others.
These works conduct their asymptotic analysis in what we now call conventional heavy-traffic.
That is, the service times and number of servers are held fixed, while the arrival rate approaches the critical value from below.
A noteworthy result of these efforts is the extension of Kingman's findings to the $GI/G/s$, which finds that the scaled queue length $(1-\rho)Q^{(s)}$ converges in distribution to an exponential random variable with mean $(c_a^2+c_s^2)/2$, where $c_a$ and $c_s$ denote the coefficient of variation of the interarrival and service time distribution, respectively.
We remark that this limiting result is the key ingredient to the widely applied Kingman formula
\begin{equation}\label{eq:kingman}
\mathbb{E}[W^{(1)}] \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2+c_s^2}{2} \cdot \mathbb{E}[B],
\end{equation}
which serves as an approximation to the expected waiting time in the single-server queue.
The limit \eqref{eq:kingman} reveals that in the conventional heavy-traffic regime, the expected waiting time explodes as $\rho\to 1$.
Hence, efficient usage of resources is achieved, at the expense of poor QoS.
An alternative regime that received much attention, see e.g.~\cite{Iglehart1965,Borovkov1965,Iglehart1973,Iglehart1973a,Whitt1982}, fixes the service time distribution while increasing both the arrival rate $\lambda$ and the number of servers to infinity simultaneously, such that the ratio $\lambda/s$ remains constant.
It has been shown that the sequence of queues under this scaling start resembling the behavior of infinite-server queues as $\lambda$ and $s$ grow.
That is, the probability of a customer finding a queue on arrival is negligible.
The sample paths in Figure \ref{fig:sample_paths_lambda100} are illustrative for this regime.
Since the utilization level $\rho$ remains strictly away from one in the limit, this setting is in the literature typically not recognized as heavy-traffic.
As Halfin \& Whitt indicate themselves, the QED regime in which service times are held fixed, and $\lambda$ and $s$ tend to infinite while satisfying $(1-\rho)\sqrt{s} \to \beta$, is a hybrid between the two aforementioned regimes.
Namely, it considers the efficiency property of the conventional heavy-traffic scaling, and the good QoS levels from infinite-server queues.\\
\\*
\noindent
\textbf{The $G/G/s$ queue in the QED regime.}
We have demonstrated in Section \ref{sec:intro_QED_regime} how to obtain QED limits for the $M/M/s$ queue and the $M/D/s$ queue.
When one moves beyond the exponential and deterministic assumptions, establishing QED behavior becomes mathematically more challenging.
The heavy-traffic analysis of the $G/G/s$ queue requires fundamentally different approaches than for Markovian queues.
Most of the research conducted on the $G/G/s$ in the Halfin-Whitt regime evolves around the characterization of the stochastic process limit of the appropriately centered and scaled queueing process in terms of diffusion processes, under various assumptions on the model primitives.
Puhalskii \& Reiman \cite{Puhalskii2000} analyzed the multi-class queue with phase-type service times in the Halfin-Whitt regime.
Heavy-traffic limits for queues in which service time distributions are lattice-based and/or have finite support are studied by Mandelbaum \& Mom\v{c}ilovi\'c \cite{Mandelbaum2008} and Gamarnik \& Mom\v{c}ilovi\'c \cite{Gamarnik2008}.
Approaches through measure-valued processes are taken by Kang, Kaspi \& Ramanan \cite{Kaspi2011,Kang2012,Kaspi2013}.
The most general class of distributions is considered by Reed \cite{Reed2009} and Puhalskii \& Reed \cite{Puhalskii2010}, who impose no assumption on the service time distribution except for the existence of the first moment.
For a survey on the techniques required for the analysis of process limits of $G/G/s$ queues, we refer the reader to \cite{Pang2007} and references therein.
Considerably less is known for the corresponding steady-state distribution of the $G/G/s$ queue in the QED regime.
Namely, under the assumption of general service time distributions, truly infinite-dimensional limits arise, since the Markovian nature of the service time and `age' process can no longer be exploited.
Works that have been able to characterize limiting behavior for the specific service time distribution classes include Jelenkovic et al.~\cite{Jelenkovic2004}, who assume deterministic service times, and Whitt \cite{Whitt2005}, who identifies the heavy-traffic limit in the case of hyperexponentially distributed service times.
Progress in the understanding of steady-state behavior of $G/G/s$ queues in the Halfin-Whitt regime has been facilitated by Gamarnik \& Goldberg \cite{Goldberg,Gamarnik2013a}, who perform their analysis under the mild assumption that the service time distribution has finite $(2+\e)$ moment.
A significant advance has been made by Aghanjani \& Ramanan \cite{Aghajani2016}, who identify the limit as the steady-state distribution of infinite-dimensional Markov process, given that the service time distribution has finite $(3+\varepsilon)$ moment, while drawing upon previous results by Kang, Kaspi \& Ramanan \cite{Kaspi2011, Kang2012,Kaspi2013}.\\
\\*
\textbf{Model extensions.}
Many extensions to the standard many-server queue can be considered.
A feature ubiquitous to service systems involving humans is customer abandonment \cite{Gans2003,Brown2005,Zeltyn2005,Mandelbaum2013}.
The $M/M/s+M$ queue introduced by Palm \cite{Palm1957}, also known as the Erlang-A model \cite{Garnett2002,Leeuwaarden2012}, acknowledges this feature by assigning every customer an exponentially distributed \textit{patience time} upon his arrival (denoted by $+M$ in the model definition).
If a customer has not yet started receiving service by the expiration of his patience, he leaves the system.
Note that abandonments render queues stable under any load.
Under QED scaling, the more general $G/G/s+G$ queue has received much attention under various modeling assumptions, see e.g.~\cite{Garnett2002,Gans2003,Whitt2006,Mandelbaum2009,Zeltyn2005,Mandelbaum2012a,Kang2012,Dai2010,Reed2012,Jennings2012,Zhang2013}.
Noteworthy findings include the vanishing abandonment probability \cite{Garnett2002} and insensitivity of the patience time distribution as long as its density at 0, i.e.~the probability of abandoning immediately upon arrival, is fixed, as the system grows large under QED scaling.
Overviews of queues with abandonment and their asymptotic counterpart are given by Zeltyn \& Mandelbaum \cite{Zeltyn2005} and Dai \& He \cite{Dai2012} and Ward \cite{Ward2012}.
Other features that have been studied in the QED regime include multiple customer classes, see e.g.~\cite{Harrison2004,Atar2014,Gurvich2008,Gurvich2009,Tezcan2010}, or heterogeneous servers \cite{Armony2005,Armony2010,Mandelbaum2012b,Stolyar2010}.
These models are all interesting in their own respect and are fairly well-understood.
Therefore, we choose to focus in this thesis on a different set of extensions, which will be discussed in Section \ref{sec:intro_beyond}.
\section{Dimensioning}
\label{sec:intro_dimensioning}
We adopt the term \textit{dimensioning} used by Borst, Mandelbaum \& Reiman~\cite{Borst2004} to say that the capacity of a service system is adapted to the load in order to reach certain performance levels.
In \cite{Borst2004} dimensioning refers to the staffing problem in a large-scale call center and key ingredients are the square-root staffing rule in \eqref{eq:square_root_staffing rule} and the QED regime.
We now revisit the results in \cite{Borst2004} and its follow-up works to explain this connection to the QED regime.
\subsection{Constraint satisfaction}
\label{sec:intro_constraint}
Consider the $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$.
A classical dimensioning problem is to determine the minimum number of servers $s$ necessary to achieve a certain target level of service, say in terms of waiting time.
Suppose we want to determine the minimum number of servers such that the fraction of customers who are delayed in the queue is at most $\varepsilon\in(0,1)$.
Hence we should find
\begin{equation}\label{eq:tagA}
s^{*}_\lambda(\varepsilon) := \min \left\{s \geq \lambda\, |\, \mathbb{P}(W^{(s)}>0) \leq \varepsilon \right\}.
\end{equation}
But alternatively, we can use the QED framework, which says that under \eqref{eq:HalfinWhitt_scaling},\ \ $\lim_{s\to\infty} \mathbb{P}(W^{({s_\lambda})} > 0) = g(\beta)$ (see Proposition \ref{prop:HalfinWhitt_delay_probability}).
Then by \eqref{eq:tagA}, $s^*_\lambda(\varepsilon)$ can be replaced by
\begin{equation}
s^{\rm srs}_\lambda(\varepsilon) = \lceil \lambda + \beta^*(\varepsilon) \sqrt{\lambda}\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves
\begin{equation}
g(\beta^*) = \varepsilon.
\end{equation}
In Figure \ref{fig:MMs_staffing_levels} we plot the exact staffing level $s^*_\lambda(\varepsilon)$ and the heuristically obtained staffing level $s^{\rm srs}_\lambda(\varepsilon)$ as functions of $\varepsilon$ for several loads $\lambda$.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda5_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda10_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda100_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
legend cell align = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda500_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Constraint_Satisfaction/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Staffing levels as a function of the delay probability targets $\varepsilon$.}
\label{fig:MMs_staffing_levels}
\end{figure}
Observe that even for very small values of $\lambda$, the staffing function $s^{\rm srs}(\varepsilon)$ coincides with the exact solution for almost all $\varepsilon\in(0,1)$ and differs no more than by one server for all $\varepsilon$.
Borst et al.~\cite{Borst2004} recognized this in their numerical experiments too, and Janssen, van Leeuwaarden \& Zwart \cite{Janssen2011} later confirmed this theoretically.
One can easily formulate other constraint satisfaction problems and reformulate them in the QED regime.
For instance, constraints on the mean waiting time or the tail probability of the waiting time, e.g.~$\mathbb{P}(W^{(s)}>T)$, which are asymptotically approximated by $h(\beta)/\sqrt{\lambda}$ and $g(\beta){\rm e}^{-\beta \sqrt{\lambda} T}$, respectively.
See \cite{Borst2004} for more examples.
\subsection{Optimization}
\label{sec:intro_optimization}
One can also consider optimization problems, for instance to strike the right balance between the costs for servers and costs incurred by customer dissatisfaction.
More specifically, assume a salary cost of $a$ per server per unit time, and a penalty cost of $q$ per waiting customer per unit time, yielding the total cost function
\[
\bar{C}_\lambda(s) := a\,s + q\,\lambda\mathbb{E}[W^{(s)}]
\]
and then ask for the staffing level $s$ that minimizes $\bar{C}_\lambda(s)$.
Since $s>\lambda$, we have $\bar{C}_\lambda(s) > a\,\lambda$ for all feasible solutions $s$.
Moreover, the minimizing value of $\bar{C}_\lambda$ is invariant with respect to scalar multiplication of the objective function.
Hence we have to optimize
\begin{equation}
\label{eq:optimization_objective}
C_\lambda(s) = r\,(s-\lambda) + \lambda\mathbb{E}[W^{(s)}], \qquad r = a/q.
\end{equation}
Denote by $s^*_\lambda(r) := \arg\min_{s > \lambda} C_\lambda(s)$ the true optimal staffing level.
With ${s_\lambda} = \lambda + \beta\sqrt{\lambda}$ and the QED limit in \eqref{eq:halfinwhitt_wait}, we can replace \eqref{eq:optimization_objective} by its asymptotic counterpart:
\begin{align*}
\frac{C_\lambda({s_\lambda})}{\sqrt{\lambda}} = r\,\beta + \sqrt{\lambda} \mathbb{E}[W^{(s)}] \to r\,\beta + \frac{g(\beta)}{\beta} =: \hat{C}(\beta), \qquad \lambda\to\infty.
\end{align*}
Once again we obtain a limiting objective function that is easier to work with than its exact pre-limit counterpart.
Hence, in the spirit of the asymptotic staffing procedure in the previous subsection, we propose the following method to determine the staffing level that minimizes overall costs.
First, (numerically) compute the value $\beta^*(r) = \arg\min_{\beta>0} \hat{C}(\beta)$, which is well-defined, because the function $\hat{C}(\beta)$ is strictly convex for $\beta>0$.
Then, set $s^{\rm srs}_\lambda(r) = [ \lambda + \beta^*(r) \sqrt{\lambda} ]$.
In Figure \ref{fig:MMs_staffing_levels_optimization} we compare the outcomes of this asymptotic staffing procedure against the true optima as a function of $r\in(0,\infty)$, for several values of $\lambda$.
The staffing levels $s^{\rm srs}_\lambda(r)$ and $s^*_\lambda(r)$ are aligned for almost all $r$, and differ no more than one server for all instances.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
axis y discontinuity = crunch,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
legend style = {at = {(1,1.2)}, anchor = north east}
]
\addplot[very thick] file {Introduction/Version_R1/tikz/Optimization/lambda5_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Optimization/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Optimization/lambda10_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Optimization/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Optimization/lambda100_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Optimization/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}\centering
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
legend cell align = left,
axis y discontinuity=crunch,
legend style = {at = {(1,1.2)}, anchor = north east}]
\addplot[very thick] file {Introduction/Version_R1/tikz/Optimization/lambda500_exact.txt};
\addplot[very thick, dashed, col1] file {Introduction/Version_R1/tikz/Optimization/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Optimal staffing levels as a function of $r = a/q$.}
\label{fig:MMs_staffing_levels_optimization}
\end{figure}
\subsection{Time-varying dimensioning}
So far we have only considered queues in which the model primitives are constant over time.
In practice, though, the arrival rate can fluctuate and depends on the time of day, the day of the week, season or even larger time scales.
It is therefore more realistic to describe these mostly predictable fluctuations through $\lambda(t)$, which represents the instantaneous arrival rate of the arrival process at time $t\in \mathbb{R}$.
The existence of time-varying demand requires a re-evaluation of staffing levels throughout the planning horizon as well.
That is, the number of servers $s(t)$ becomes a function of time, rather than a constant and this clearly asks for an adaptation of the dimensioning procedures in Sections \ref{sec:intro_constraint} and \ref{sec:intro_optimization}.
We explain the concept of time-varying staffing and the connection with the QED regime through the time-varying extension of the $M/M/s$ queue known as the $M_t/M/s_t$ queue, where the subscript $t$ refers to the time-varying nature of both the arrival process and the staffing level.
In this setting, customers arrive according to a non-homogeneous Poisson process with rate function $\lambda(t)$ and customers have exponentially distributed service times with mean $1/\mu$.
Under a constraint satisfaction strategy, we aim to find the staffing function $s(t)$ such that the delay probability is at most $\varepsilon\in(0,1)$ for all $t$.
The analysis and optimization of time-varying many-server queueing systems is known to be intrinsically hard, but many approximation techniques and heuristic methods have been proposed throughout the years \cite{Green1991,Jennings1996}.
A natural but naive approach is the \textit{pointwise-stationary approximation} (PSA) \cite{Green1991}, which evaluates the system at time $t$ as if it were in steady-state with instantaneous parameters $\lambda=\lambda(t)$, $\mu$ and $s = s(t)$.
Consequently, the analysis and optimization of queues is performed on steady-state performance metrics.
Variants of the PSA method include the \textit{simple-stationary approximation} (SSA) \cite{Green2001}, which uses the long-term (moving) average arrival rate instead of the instantaneous arrival rate, and the \textit{stationary-independent-period-by-period approximation} (SIPP) \cite{Green2001}, which splits the time-horizon into multiple intervals and performs steady-state analysis with the averaged parameters in each of these intervals, among others.
PSA performs well in slowly varying environments with relatively short service times \cite{Green1991,Whitt1991}.
However, when the model parameters fluctuate significantly, as is often the case in real-life systems, the accuracy of PSA can be poor, as we will see in the numerical experiment at the end of this section.
The main reason why PSA, SSA and SIPP can fail is that these methods neglect that customers are actually residing in the system (being in service or waiting in the queue) for some time.
In contrast, staffing decisions should be based on the number of customers present in the system rather than the arrival rate at that particular time.
Jennings et al.~\cite{Jennings1996} introduced a more sophisticated method that exploits the relation with infinite-server queues.
We explain their idea in the context of the $M_t/M/s_t$ queue.
By Eick et al. \cite{Eick1993}, the number of customers in the $M_t/M/\infty$ queue at time $t$ is Poisson distributed with mean
\begin{equation}
\label{eq:offered_load_eick}
R(t) = \mathbb{E}\left[ \lambda(t-B_e)\right] \mathbb{E}[B] = \int_0^\infty \lambda(t-u)\,\mathbb{P}(B>u)\, {\rm d}u = \int_0^\infty \lambda(t-u)\, {\rm e}^{-\mu u} \,{\rm d}u.
\end{equation}
We remark that this result holds for more general service time distributions.
Now, recall that in large systems in the QED regime, the expected delay is negligible.
Therefore, under these conditions, the many-server system may be approximated by the infinite-server approximation with offered load as in \eqref{eq:offered_load_eick}.
Accordingly, we can determine the staffing levels $s(t)$ for each $t$ based on steady-state $M/M/s$ measures with offered load $R=R(t)$.
Jennings et al. \cite{Jennings1996} proceed by exploiting the heavy-traffic results of Halfin-Whitt \eqref{eq:halfinwhitt_wait}.
In conjunction with the dimensioning scheme in Section \ref{sec:intro_constraint}, the authors propose to set
\begin{equation}
s(t) = \bigg\lceil R(t) + \beta^*(\varepsilon) \sqrt{R(t)} \bigg\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves $g(\beta^*(\varepsilon)) = \varepsilon$.
Remark that the number of servers is rounded up to ensure that the achieved delay probability is indeed below $\varepsilon$.
This method was called in \cite{Jennings1996,Massey1994} the \textit{modified-offered-load} (MOL) approximation, and we adopt this term in this thesis.
Let us demonstrate that this approximation scheme works.
Figure \ref{fig:intro_example_arrival}(a) shows an arrival rate pattern $\lambda(t)$ and corresponding offered load function $R(t)$ for $\mu=1/2$.
This arrival rate stems from a real-world emergency department~\cite{Sinreich2005}.
The resulting staffing level functions based on the PSA and MOL approximations with $\varepsilon = 0.3$ are plotted in
Figure \ref{fig:intro_example_arrival}(b).
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 45,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,43)},anchor = north west}
]
\addplot[very thick] file {Introduction/Version_R1/tikz/TimeVarying/arrival_rate.txt};
\addplot[very thick, col1] file {Introduction/Version_R1/tikz/TimeVarying/offered_load.txt};
\legend{{$\lambda(t)$},$R(t)$}
\end{axis}
\end{tikzpicture}
\caption{Arrival rate and offered load functions}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 60,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,59)},anchor = north west}
]
\addplot[very thick] file {Introduction/Version_R1/tikz/TimeVarying/s_PSA.txt};
\addplot[very thick, col1] file {Introduction/Version_R1/tikz/TimeVarying/s_Jennings.txt};
\legend{PSA,MOL}
\end{axis}
\end{tikzpicture}
\caption{Staffing functions.}
\end{subfigure}
\caption{Time-varying parameters of a real-world emergency department.}
\label{fig:intro_example_arrival}
\end{figure}
Through simulation, we evaluate the delay probability as a function of time for $\varepsilon = 0.1,\, 0.3$ and 0.5.
In Figure \ref{fig:intr_timevarying_simulation_results} we see how the PSA approach fails to stabilize the performance of the queue, whereas the MOL method does stabilize around the target performance.
The erratic nature of the delay probability as a function of time can be explained by rounding effects of the staffing level.
Since this rather simple but elegant technique to address time-varying dimensioning is provably effective, we will adopt the underlying idea of the MOL method in various different settings in this thesis.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 0.5,0.02)},anchor = south west}]
\addplot[thick, col5] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e01_psa.txt};
\addplot[thick, col2] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e03_psa.txt};
\addplot[thick, col4] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e05_psa.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{PSA}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, col5] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e01_mol.txt};
\addplot[thick, col2] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e03_mol.txt};
\addplot[thick, col4] file {Introduction/Version_R1/tikz/TimeVarying/pdelay_e05_mol.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{MOL}
\end{subfigure}
\caption{Probability of delay under staffing functions obtained through PSA and MOL approximations.}
\label{fig:intr_timevarying_simulation_results}
\end{figure}
\section{Contributions}
\label{sec:intro_beyond}
We have explained how the QED regime can be used to dimension and staff large-scale service systems.
The basic concepts, however, were explained for the relatively simple $M/M/s$ and $M_t/M/s_t$ queue.
Many real-world service systems have essential features that are not captured by these elementary models.
We will now discuss some of these features and address the need to consider more involved models and extend the existing QED theory.
\subsection{Non-classical scaling regimes and pre-limit behavior}
\label{sec:intro_novel_scalings}
The QED theory is centered around the scaling relation $\sqrt{\lambda}(1-\rho_\lambda) \to \beta$, or equivalently $s_\lambda = \lambda + \beta \sqrt{\lambda} + o(\sqrt{\lambda})$, for $\lambda\to\infty$.
It is worthwhile to study how pre-limit behavior of many-server queues is affected when one deviates from the square-root staffing rule.
Consider a novel family of heavy-traffic scaling regimes, described in terms of the parameter $\eta$ for which we assume that
\begin{equation}
\label{eq:novel_scaling_rule}
\lambda^\eta (1-\rho_\lambda) \to \beta, \qquad \text{as } \lambda\to\infty,\ \beta > 0.
\end{equation}
The parameter $\eta \geq 0$ defines a whole range of possible scaling regimes, including the classic case $\eta = 1/2$, as well as the cases $\eta=0$ and $\eta=1$ investigated in Subsection \ref{sec:intro_many_server_regimes}.
In terms of a capacity sizing rule, the condition \eqref{eq:novel_scaling_rule} is tantamount to $s_\lambda = \lambda +\beta\,\lambda^{1-\eta}$.
This framework thus bridges the gap between the QD and QED regime if $\eta\in(0,1/2)$ and the QED and ED regime if $\eta\in(1/2,1)$, in the $M/M/s$ model.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\eta\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\eta\in(0,1/2)$ can be seen as \textit{moderate} heavy traffic: heavy-traffic conditions in which the full occupancy is reached more slowly, as a function of $\lambda$, than for classical heavy traffic. See \cite{Chang1996,Puhalskii1998,Puhalskii1999,Atar2012,Atar2014,Atar2015,Atar2016} for more details.
For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to \textit{extreme} heavy traffic due to a relatively small variability hedge.
We use the insights of Section \ref{sec:intro_QED_regime} and the connection of the QED scaling to the CLT to argue intuitively that the following trichotomy in the qualitative system behavior as $\lambda\to\infty$ holds under scaling \eqref{eq:novel_scaling_rule}.
For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, because the order of the variability hedge $\beta \lambda^{1-\eta}$ is greater than strictly necessary to accommodate the stochastic fluctuations in demand.
Scalings in which $\eta\in(1/2,\infty)$, have adverse behavior, since stochastic fluctuations are not accounted for sufficiently, so that the probability of delay converges to 1.
The value $\eta=1/2$ is therefore the tipping point, at which the delay probability converges to a limit between 0 and 1.
Above and below this critical value, the asymptotic performance of the queue flips to either one of the extremes.
In Chapter 2, we formalize this heuristic argument and conduct an asymptotic analysis to reveal the rate at which the limit of performance metrics is attained, depending on the parameters $\eta$ and $\beta$ and the system size $\lambda,{s_\lambda}$.
\subsection{Overdispersed arrivals}
\label{sec:intro_overdispersion}
Until now we have considered queueing systems with perfect knowledge on the model primitives, including the mean demand per time period. For large-scale service systems, the dominant assumption in the literature is that demand arrives according to a non-homogeneous Poisson process, which in practice translates to the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies of service systems shows that the variance of demand typically exceeds the mean significantly, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2003, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}. The feature that variability is higher than one expects from the Poisson assumption is referred to as \textit{overdispersion}.
Due to its inherent connection with the CLT, the dimensioning rule in \eqref{eq:square_root_staffing rule} relies heavily on the premise that the variance of the number of customers entering the system over a period of time is of the same order as the mean.
Subsequently, when stochastic models do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly in critical loading.
To deal with overdispersion, existing capacity sizing rules like the square-root staffing rule need to be modified in order to incorporate a correct hedge against (increased) variability.
Following our findings in Section \ref{sec:intro_characteristics}, we propose a capacity allocation rule similar to \eqref{eq:square_root_staffing rule} in which the original variability hedge is replaced by an amount that is proportional to the square-root of the variance of the arrival process.
In Chapter 3, we elaborate on this idea and show how to adapt the scaling of the queueing process appropriately to achieve QED-type behavior in the presence of overdispersion.
\subsection{Finite-size constraints}
The canonical examples in Section \ref{sec:intro_QED_regime} assume an infinite amount of waiting space.
Physical service systems, however, are sometimes limited in the number of customers that can be held in the system simultaneously.
For instance in a call center, the maximum number of clients in service or queueing is restricted by the number of available trunk lines \cite{Khudyakov2006}, while in the emergency department of a hospital, the number of beds constrains the number of patients that can be admitted \cite{YomTov2010}.
Depending on the practical setting and admission policy, if the maximum capacity, say $n$, is reached, newly arriving customers either leave the system immediately (blocking), reattempt getting access later (retrials) or queue outside the facility (holding).
In any case, expectations are that the queueing dynamics within the service facility are affected considerably in the presence of such additional capacity constraints.
We illustrate these implications through the $M/M/s/n$ queue, that is, the standard $M/M/s$ queue with additional property that a customer who finds upon arrival $n$ customers already present in the system, is deferred and considered lost.
To avoid trivialities, let $n\geq s$.
Since the expected workload reaching the servers is less than in the unconstrained scenario, one expects less congestion and resource utilization.
Consider the $M/M/{s_\lambda}/n_\lambda$ in the QED regime.
So, let $\lambda$ increase while ${s_\lambda}$ scales as ${s_\lambda}=\lambda+\beta\sqrt{\lambda} + o(\sqrt{\lambda})$.
We then ask how $n_\lambda$ should scale along with $\lambda$ and ${s_\lambda}$ to maintain the non-degenerate behavior as seen in Section \ref{sec:intro_QED_regime}.
We provide a heuristic answer.
Let $Q^{({s_\lambda},n_\lambda)}$ and $W^{({s_\lambda},n_\lambda)}$ denote the number of customers in the system and the waiting time in the $M/M/{s_\lambda}/n_\lambda$ queue in steady state.
Note through Proposition \ref{thm:intro_HW_stationary_distribution} that if there were no finite-size constraints, we would have, for $\lambda$ large,
\begin{align}
\mathbb{P}(Q^{({s_\lambda})}\geq n_\lambda)
&= \mathbb{P}\left(\frac{Q^{({s_\lambda})}-{s_\lambda}}{\sqrt{{s_\lambda}}} \geq \frac{n_\lambda-{s_\lambda}}{\sqrt{{s_\lambda}}}\right) \nonumber \\
&\to
\left\{
\begin{array}{ll}
g(\beta), & \text{if }n_\lambda = {s_\lambda} + o({s_\lambda}),\\
g(\beta)\,{\rm e}^{-\beta \gamma}, & \text{if } n_\lambda = {s_\lambda}+\gamma\sqrt{{s_\lambda}} + o(\sqrt{s_\lambda}),\\
0, & \text{if } n_\lambda = {s_\lambda}+\Omega(\sqrt{{s_\lambda}}),
\end{array}
\right.
\end{align}
as $\lambda\to\infty$ for some $\gamma>0$.
Here, the relation $u(\lambda) = \Omega(v(\lambda))$ implies $u(\lambda)/v(\lambda) >1$ for $\lambda\to\infty$.
Hence, asymptotically the finite-size effects only play a role if the extra variability hedge of $n_\lambda$ is of order $\sqrt{{s_\lambda}}$ (or equivalently $o(\sqrt{\lambda})$).
Furthermore, if the variability hedge is $o(\sqrt{\lambda})$, then we argue that asymptotically, all customers who do enter the system have probability of delay equal to zero.
More formally, under the \textit{two-fold scaling rule}
\begin{equation}
\label{eq:intro_twofold_scaling_rule}
\left\{
\begin{array}{ll}
{s_\lambda} = \lambda + \beta\sqrt{\lambda} + o(\sqrt{\lambda}),\\
n_\lambda = {s_\lambda} + \gamma \sqrt{{s_\lambda}} + o(\sqrt{\lambda}),
\end{array}
\right.
\end{equation}
it is not difficult to deduce that, see e.g. \cite{masseywallace},
\begin{equation}
\mathbb{P}(W^{({s_\lambda},n_\lambda)} > 0) \to \left( 1 + \frac{\beta\,\Phi(\beta)}{(1-{\rm e}^{-\beta\gamma})\varphi(\beta)}\right)^{-1}, \quad \text{as } \lambda\to\infty,
\end{equation}
which is strictly smaller than $g(\beta)$ in \eqref{fig:delay_probs_HW_MMs}, but still bounded away from both 0 and 1.
Furthermore, the buffer size of the queue is $n_\lambda-{s_\lambda} = \gamma\sqrt{{s_\lambda}}$, so that by Little's law, the expected waiting time of an admitted customer is $O(1/\sqrt{{s_\lambda}})$.
Even though resource utilization in the $M/M/{s_\lambda}/n_\lambda$ is less efficient than in the queue with unlimited waiting space, it can easily be shown that $\rho\to 1$ as $\lambda\to\infty$.
Hence, all three key characteristics of the QED regime are carried over to the finite-size setting if adhered to scaling \eqref{eq:intro_twofold_scaling_rule}.
On a process level, adding a capacity constraint translates to adding a reflection barrier to the normalized queue length process $X^{({s_\lambda},n_\lambda)} = (Q^{({s_\lambda},n_\lambda)} -{s_\lambda} ) /\sqrt{{s_\lambda}}$, at $\gamma$, as is illustrated by the sample paths of $X^{({s_\lambda},n_\lambda)}$ for three values of $\lambda$ in Figure \ref{fig:sample_paths_MMsn}.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {Introduction/Version_R1/tikz/SamplePaths_MMsn/lambda5.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale = 0.7]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {Introduction/Version_R1/tikz/SamplePaths_MMsn/lambda50.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda = 50$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}\centering
\begin{tikzpicture}[scale = 0.7]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel style={right},
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[thick] file {Introduction/Version_R1/tikz/SamplePaths_MMsn/lambda100.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}\centering
\begin{tikzpicture}[scale = 0.7]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel style={right},
ylabel = {$X^{({s_\lambda},n_\lambda)}$},
xscale=1,
yscale=1]
\addplot[] file {Introduction/Version_R1/tikz/SamplePaths_MMsn/lambda500.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda},n_\lambda)}(t)$ with $\lambda = 5,\, 50,\, 100$ and $500$ under scaling \eqref{eq:intro_twofold_scaling_rule} with $\beta=0.5$ and $\gamma = 1$.}
\label{fig:sample_paths_MMsn}
\end{figure}
\
It has been shown by \cite{masseywallace} that under \eqref{eq:intro_twofold_scaling_rule}
\begin{equation}
\label{eq:asymptotic_blocking_prob}
\sqrt{{s_\lambda}}\,\mathbb{P}({\rm block}) = \sqrt{{s_\lambda}}\, \mathbb{P}(Q^{({s_\lambda},n_\lambda)} = n_\lambda) \to f(\beta,\gamma), \quad \text{as } \lambda \to\infty,
\end{equation}
for a non-negative function $f$.
The idea of the two-fold scaling in \eqref{eq:intro_twofold_scaling_rule} can be extended to settings in which the interior is in fact a network of queues, rather than the single-station setting discussed here, see \cite{Khudyakov2006,YomTov2010,Tan2012} for examples of such \textit{semi-open} queueing networks.
When customers retry getting access after being blocked initially, the QED analysis becomes much more difficult, and no explicit limiting results are known.
Nevertheless, observe that the volume of blocked arrivals is by \eqref{eq:asymptotic_blocking_prob} of order $\sqrt{\lambda}$, the exact same magnitude as the variability hedge of both ${s_\lambda}$ and $n_\lambda$.
Therefore, retrials and holding customers have a non-negligible effect on the service levels within the facility in the QED regime.
This will be the topic of Chapters 4 and 5.
\subsection{Pre-limit behavior}
The results on queues in the QED regime discussed in Section \ref{sec:intro_QED_regime} are in two ways of an asymptotic nature.
First, the heavy-traffic limits prescribe the queueing dynamics for $\lambda,{s_\lambda}\to\infty$.
Real-world systems obviously do not experience infinite demand nor have infinite capacity, and hence the heavy-traffic limits only form an approximation for such finite-sized systems.
Although these approximations are qualitatively insightful, the asymptotic analyses do not reveal much about their accuracy with respect to actual performance.
For instance, we would like to know how fast the convergence takes place, and how inaccuracies in asymptotic approximations percolate into inaccuracies in pre-limit systems.
To answer such questions, it would be helpful to have an asymptotic estimate for the difference between the (scaled) queueing process and its limiting counterpart, to be able to judge the error made by relying on asymptotic as opposed to actual performance evaluation.
Characterization of the error term gives rise to so-called \textit{corrected diffusion approximations}~ \cite{Siegmund1978,Blanchet2006,Janssen2008}, which are refinements to heavy-traffic limits for finite systems, and are also related to \textit{universal approximations} \cite{Gurvich2014,Huang2016,Braverman2015,Braverman2015a}.
We will derive such corrected diffusion approximations in the context of the novel scaling regimes mentioned in Section \ref{sec:intro_novel_scalings} in Chapter 2.
Second, the bulk of queueing literature is concerned with the performance analysis and optimization of steady-state systems.
However, in practice, service systems certainty do not run infinitely long, which renders this assumption questionable.
Validation of the steady-state assumption is related to the \textit{relaxation time} of a queueing process \cite{Abate1987,Abate1988,relaxation,Leeuwaarden2011,Leeuwaarden2012,Gamarnik2013}, which prescribes the time it takes a system starting out of equilibrium to approximate its stationary distribution.
In case the relaxation time is small, stationary performance evaluation is likely to be accurate.
On the contrary, if the relaxation time is large, a time-dependent analysis of the queueing system is required in order to capture realistic behavior.
Subsequently, we can investigate the implications of applying staffing principles that are based on steady-state performance metrics in settings which are inherently transient over the planning period.
We will touch upon this topic in Chapter 6.
\section{Outline of the thesis}
The remainder of this thesis builds upon the ideas behind the QED scaling regime exhibited in this introductory chapter, and is organized as follows.
Chapter 2 is concerned with the analysis of the limiting behavior of queues in case one deviates from the square-root staffing principle as demand grows large.
Using the bulk-service queue together with the many-sources paradigm as a vehicle, we derive corrected diffusion approximations for the performance metrics of pre-limit systems in these alternative scaling regimes.
The work presented in Chapter 2 is based on \cite{Janssen2015}.
In Chapter 3, we also analyze the bulk-service queueing model, but with many correlated sources, so that demand becomes overdispersed.
As we alluded to in Section \ref{sec:intro_overdispersion}, this requires an alternative scaling of the queue length process and associated staffing rule.
This chapter exhibits the ideas of \cite{Mathijsen2016}.
In Chapter 4, we discuss how QED-type behavior prevails in simple settings in which the system size is finite, given appropriate capacity-sizing rules.
More specifically, we show how customer retrials can be incorporated heuristically into the performance analysis of finite-size systems in the QED regime.
The content of this chapter is based on \cite{Leeuwaarden2015} and \cite{Leeuwaarden2016}.
Building upon the insights gained in Chapter 4, we show in Chapter 5 how the approximation methods carry over to a more complex finite-size queueing system, inspired by delay analysis in a health care facility.
We show how the QED scaling limits for this model offer surprisingly accurate approximations for realistic model parameters in systems of small to moderate size, and develop a staffing algorithm for dimensioning such systems.
Chapter 5 is based on the ideas of \cite{Leeuwaarden2016a}.
Chapter 6 investigates the validity of a capacity allocation rule based on steady-state performance metrics in practical settings.
Namely, in realistic scenarios, the parameters of a queueing model are typically subject to change over the planning period.
This asks for a more elaborate transient analysis of the queue dynamics, and an adaptation of the staffing level.
In this chapter, we present how to do so appropriately in a single-server queueing model facing a L\'evy input process by prescribing a correction to the steady-state optimum, which has a square-root form.
This chapter is based on \cite{Mathijsen2016a}.
Chapter 7 presents the analysis of an inventory model with backlogs, perishable goods and consumer impatience.
This model resembles the inventory level of a blood bank, and can be regarded as a shot-noise model with both positive and negative jumps and exponential decay rates above and below zero.
Besides the derivation of the stationary distribution of the inventory level, we show how under appropriate scaling the process converges to an Ornstein-Uhlenbeck process.
The latter allows for a more tractable approximate analysis of the model in case the number of blood deliveries and demand is large.
Chapter 7 is based on \cite{Bar-Lev2015}.
\chapter{Introduction}
\begin{chapterstart}
Stochastic service systems describe settings in which users compete for service from scarce resources. Think of check-in lines at airports, waiting rooms in hospitals or queues in supermarkets, where the scarce resources are human manpower.
Next to these traditional settings, our increasingly digitalized society creates quite different types of resource sharing systems, such as the internet, wireless networks and cloud computing facilities.
In these virtual environments, geographical location does not play a restricting role on the system size, paving the way for the emergence of large-scale resource sharing networks.
This thesis serves to explain how to analyze and dimension large-scale systems in order to achieve economies-of-scale, by which we mean that the system is highly occupied and hence utilizes efficiently the expensive resources, while at the same time, the offered service levels remain high.
In this chapter, we give an overview of the available machinery that supports such principles and explain how this thesis contributes to the existing study of large-scale service systems. The fundamental law behind these mathematical techniques is the Central Limit Theorem (CLT) -- arguably one of the most important theorems in mathematics and science.
\end{chapterstart}
\newpage
\section{Service systems and queueing theory}
\subsection{Quality vs. Efficiency}
Large-scale service systems take many shapes and forms.
Classical examples of large-scale service systems include call centers \cite{Erlang1917,Palm1957,Whitt1999,Gans2003,Borst2004,Brown2005,Zeltyn2005,Bassamboo2009,Khudyakov2006} and communication systems \cite{Kleinrock1976,Anick1982,Kelly1985,Kleinrock2007,johanthesis}.
More recently, congestion-related issues in health care facilities and cloud-computing facilities have received much attention \cite{Armony2015,Green2007,YomTov2010,Gupta2007,Tan2012}.
In all settings, one can think of service systems as being composed of \textit{customers} and \textit{servers}.
In call centers, customers typically call to request help from one of the center's agents (servers).
In communication networks, the data packets are the customers and the communication channels are the servers.
In health care facilities, patients are the customers, and nurses/physicians are the servers.
The system scale may refer to either the size of the client base it caters to, or the magnitude of its capacity, or both, as is frequently the case.
Next tot the central notions of customers and servers, we view service systems are inherently stochastic, that is, subject to uncertainty.
Although arrival volumes can be anticipated to some extend over a certain planning horizon, for instance through historical data and forecasting methods, one cannot predict with certainty future arrival patterns.
Moreover, service requirements are typically random as well, adding more uncertainty.
This intrinsic stochastic variability is a predominant cause of delay experienced by customers in the system.
Due to the inherent randomness in both their arrival and service processes, stochastic models have proved instrumental in both in quantifying and improving the operational performance of service systems.
Queueing theory and stochastics provide the tools and machinery to describe and evaluate these service systems.
Queueing models are often able to capture and explain fundamental phenomena that are common across applications.
When evaluating the performance of service systems, an important model is the $M/GI/s$ queue, which we will refer to as the \textit{many-server} queue.
This model assumes that customers arrive to the queue according to a Poisson process of rate $\lambda$, and customer service times are mutually independent and identically distributed (i.i.d.) samples from the distribution of a non-negative random variable $B$.
The parameter $s$ represents the number of servers in the system, and hence restricts the number of simultaneous services.
In this thesis we restrict attention to the policy in which customers are handled \textit{first-come-first-served}.
In case $s=1$, we speak of a single-server queue.
First principles say that the queueing process is stable, that is, the number of customers does not explode as time evolves, if and only if the expected workload $R := \lambda\mathbb{E}[B]$ brought into the system per time unit is strictly less than the system capacity.
In other words, the \textit{utilization} of the queue, defined as $\rho := \lambda\mathbb{E}[B] / s$ should remain strictly below 1.
Naturally, a system manager prefers to operate at a utilization level close to 1, so that resources are used efficiently.
However, it is known that pushing the occupation levels to 100\% leads to an explosive increase in congestion, thereby reducing the quality of service (QoS) and also customer satisfaction.
These seemingly conflicting objectives give rise to a classical trade-off between customer satisfaction and costs of resources.
\subsection{Economies-of-scales}
Under the assumption that service times are exponentially distributed with mean $1/\mu$, the many-server queue reduces to the well-studied $M/M/s$ queue.
Despite its simplicity, the analysis of the $M/M/s$ queue explains mathematically the distinctive traits of queues in general, such as the non-linear effect of utilization on the queue size, and pooling effects.
Let $W^{(s)}$ denote the waiting time of a customer and $Q^{(s)}$ the queue length (including the customers in service) in the steady-state $M/M/s$ queue. Without loss of generality, we fix $\mu=1$.
A straightforward balance argument gives the stationary distribution:
\begin{equation}
\label{eq:MMs_stationary_distribution}
\pi_k := \mathbb{P}( Q^{(s)} = k )
= \left\{
\begin{array}{ll}
\pi_0\frac{R^k}{k!}, & \text{if } k\leq s, \\
\pi_0\frac{R^s}{s!}\,\rho^{k-s} & \text{if } k > s,
\end{array}
\right.
\end{equation}
where
\begin{equation*}
\pi_0 := \left( \sum_{k=0}^s \frac{R^k}{k!} + \frac{\rho}{1-\rho} \frac{R^s}{s!}\right)^{-1}.
\end{equation*}
Natural QoS indicators include the expected waiting time $\mathbb{E}[W^{(s)}]$ and the delay probability $\mathbb{P}(W^{(s)}>0)$.
Invoking Little's law and the PASTA (Poisson arrivals see time averages) property \cite{Wolff1982}, it follows that
\begin{equation}
\label{eq:MMs_wait}
\mathbb{P}(W^{(s)} > 0) = \frac{R^s}{s!} \left( (1-\rho) \sum_{k=0}^{s-1} \frac{R^k}{k!} + \frac{R^s}{s!} \right)^{-1},
\quad
\mathbb{E}[W^{(s)}] = \mathbb{P}(W^{(s)} > 0)\,\frac{1/s}{1-\rho}.
\end{equation}
From these formulae, it is readily seen that $\mathbb{P}(W^{(s)} > 0) \to 1$ and $\mathbb{E}[W^{(s)}] \to \infty$ as $\rho \uparrow 1$ . That is, increasing $\lambda$ to $s$, while keeping the latter fixed, leads to a system in which all customers are delayed before service, and the expected delay before reaching a server increases to infinity.
The $M/M/s$ queue also reveals the effect of \textit{resource pooling}.
To illustrate the operational benefits of sharing resources, we compare a system of $s$ separate $M/M/1$ queues, each serving a Poisson arrival stream with rate $\lambda<1$, against one $M/M/s$ queue facing arrival rate $\lambda s$.
The two systems thus experience the same workload and utilization, namely $\rho = \lambda$.
We fix the value of $\lambda$ and vary $s$.
Obviously, the waiting time and queue length distribution in the first scenario are unaffected by the parameter $s$, since there is no interaction between the single-server queues.
This lack of coordination allows for the possibility of having an idle server, while the total number of customers in the system exceeds $s$, therefore wasting resource capacity.
Such an event cannot happen in the many-server scenario, due to the central queue.
This central coordination improves QoS. Indeed Figure \ref{fig:waiting_time_pooling} shows that the reduction in expected waiting time can be substantial.
\begin{figure}
\centering
\begin{subfigure}{0.48\textwidth}
\small \centering
\input{./tikz_tex/Ewait_pooling.tex}
\caption{Expected waiting time}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\input{./tikz_tex/Pwait_pooling.tex}
\caption{Probability of delay}
\end{subfigure}
\caption{Effects of resource pooling in the $M/M/s$.}
\label{fig:waiting_time_pooling}
\end{figure}
So pooling kills two birds with one stone: QoS for customers improves and the system efficiency increases.
\subsection{Many-server scaling regimes}
\label{sec:intro_many_server_regimes}
Now that we know that economies-of-scale can be achieved, it is relevant to ask how to match capacity $s$ to a demand $\lambda$ in the setting where both $s$ and $\lambda$ become large.
The expressions in \eqref{eq:MMs_wait} provide a starting point for finding such demand-capacity relations, particularly when we apply asymptotic analysis for $s\to\infty$, \cite{Halfin1981,Borst2004,Reed2009}.
Asymptotic theory of many-server systems relies on the prerequisite that the limiting behavior of the service system is determined by the way in which capacity $s$ is adjusted to demand, assuming demand grows large.
We illustrate this idea by investigating typical sample paths of the queue length process $Q = \{Q(t),t\geq 0\}$ of an $M/M/s$ queue for increasing values of $\lambda$.
Sample paths of queueing processes are insightful, because congestion and server utilization are both visualized.
As an example, Figure \ref{fig:sample_path_small} depicts a sample path for $\lambda = 3$ and $s = 4$.
The number of customers queueing at time $t$ is given by $(Q(t)-s)^+$ with $(\cdot)^+ := \max\{0,\cdot\}$.
The number of idle servers is given by $(s-Q(t))^+$.
In Figure \ref{fig:sample_path_small}, the red and green area hence represent the cumulative queue length and cumulative number of idle server, respectively, over the given time period.
Bearing in mind the dual goals of QoS and efficiency, we want to minimize both of these areas simultaneously.
\begin{figure}[b!]
\centering
\input{./tikz_tex/sample_path_small.tex}
\caption{Sample path of the $M/M/s$ queue with $\lambda = 3$ and $s=4$.}
\label{fig:sample_path_small}
\end{figure}
Next, we conduct a similar sample path experiment for increasing values of $\lambda$.
Since $s > \lambda$ is required for stability, the value of $s$ needs to be adjusted accordingly.
We propose three scaling rules:
\begin{equation}
\label{eq:intro_three_scaling_rules}
s^{(1)}_\lambda = \left[ \lambda + \beta \right ], \qquad
s^{(2)}_\lambda = \left[ \lambda + \beta\sqrt{\lambda} \right], \qquad
s^{(3)}_\lambda = \left[ \lambda + \beta\,\lambda \right],
\end{equation}
for some $\beta>0$, where $[\cdot]$ denotes the rounding operator.
Note that these three rules differ in terms of increasing overcapacity $s-\lambda$.
Figure \ref{fig:sample_paths_lambda100} depicts typical sample paths of the queue length process for increasing values of $\lambda$ for the three scaling rules with $\beta = 0.5$.
\begin{figure}
\centering
\input{./tikz_tex/sample_paths_lambda10.tex}
\caption{Sample paths of the $M/M/s$ queue with $\lambda = 10,50$ and $100$ and $s$ set according to the three scaling rules in \eqref{eq:intro_three_scaling_rules}.}
\label{fig:sample_paths_lambda100}
\end{figure}
Observe that for all scaling rules, the stochastic fluctuations of the queue length processes relative to $\lambda$ decrease with the size of the system.
Moreover, the paths in Figure \ref{fig:sample_paths_lambda100} appear to become increasingly continuous in nature with increasing $\lambda$.
Of course, the actual sample path always consists of upwards and downward jumps of size 1, but we will show how proper centering and scaling of the queue length process indeed gives rise to a \textit{diffusion process} in the limit as $\lambda\to\infty$.
Although the difference in performance of the three regimes is not yet evident for relatively small $\lambda$, clear distinctive behavior occurs for large $\lambda$.
Under ${s_\lambda}^{(1)}$, the majority of customers is delayed and server idle time is low, since $\rho = (1+\beta/\lambda)^{-1} \to 1$ as $\lambda \to \infty$.
Systems dimensioned according to this rule value server efficiency over customer satisfaction and therefore this regime is in the literature also known as the \textit{efficiency-driven} (ED) regime \cite{Zeltyn2005}.
In contrast, the third scaling rule $s^{(3)}$ yields a constant utilization level $\rho = 1/(1+\beta)$, which stays away from 1, even for large $\lambda$.
Queues operating in this regime exhibit significant server idle times.
Moreover, for the particular realization of the queueing processes for $\lambda = 50$ and $\lambda=100$ none of the customers waits.
This customer-centered regime is known as the \textit{quality-driven} (QD) regime \cite{Zeltyn2005}.
The scaling rule $s^{(2)}_\lambda$ is in some ways a combination of the other two regimes.
First, we have $\rho = (1 +\beta/\sqrt{\lambda})^{-1} \to 1$ as $\lambda \to \infty$, which indicates efficient usage of resources as the system grows.
Nonetheless, the sample paths indicate that only a fraction of the customers is delayed, and if a queue is present, it seems to be of moderate size, which suggest good QoS.
This regime is therefore called \textit{quality-and-efficiency driven} (QED) regime.
Since this scaling regime and the related \textit{square-root staffing rule}
\begin{equation}
\label{eq:square_root_staffing rule}
s = \lambda + \beta\sqrt{\lambda}
\end{equation}
strikes the right balance between the two profound objectives of capacity allocation in service systems, we discuss in the next section the mathematical foundations of the QED regime and quantify the favorable properties revealed by Figure \ref{fig:sample_paths_lambda100}.
\section{The QED regime: two canonical examples}
\label{sec:intro_QED_regime}
We saw in Figure \ref{fig:waiting_time_pooling} the advantageous effect of resource pooling and economies-of-scale in many-server systems.
The driving force behind this fundamental mathematical insight is the Central Limit Theorem (CLT), arguably one of the most important theorems in mathematics and science in general.
\begin{theorem}[Central Limit Theorem, e.g. {\cite[Thm.~27.1]{Billingsley1995}}]
Suppose $X_1,X_2,\ldots,X_n$ is an independent sequence of random variables having mean $\mu$ and positive variance $\sigma^2$.
Then,
\[
\frac{\sum_{i=1}^n X_i - n\mu }{\sqrt{n}\sigma} {\;\buildrel{d}\over\Rightarrow\;} \mathcal{N}(0,1), \qquad \text{for }n\to\infty.
\]
where ${\;\buildrel{d}\over= \;}$ denote convergence in distribution and $\mathcal{N}(0,1)$ is the standard normal distribution.
\end{theorem}
Notice that the CLT does not pose any restrictions on the distribution of the samples, apart from its finite mean and variance, its statement is extremely powerful, and its consequences appear in many areas of science.
In this thesis provides the basis of the asymptotic study of many-server systems, as will become clear in this section.
Striking the proper balance between queueing delay and server efficiency asymptotically, i.e.~balancing the green and red areas in Figure \ref{fig:sample_paths_lambda100}, in mathematical terms boils down to choosing a service level $s_\lambda$ such that both $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ and $\mathbb{P}(Q^{(s_\lambda)} < s_\lambda)$ remain strictly smaller than 1 as $\lambda\to\infty$.
In other words, one would like to see that the delay probability $\mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda)$ converges to a non-degenerate limit $\alpha \in (0,1)$ as $\lambda\to\infty$.
To get a feel for the natural scale of the queue, we first examine the situation with unlimited capacity.
More precisely, let $Q^{(\infty)}$ be the number of customers in a steady-state $M/G/\infty$ queue with mean service requirement $\mathbb{E}[B]=1$.
Notice that in this infinite-server setting, $Q^{(\infty)}$ also represents the steady-state number of busy servers.
It is commonly known that $Q^{(\infty)}$ follows a Poisson distribution with mean $R$, the expected workload.
Moreover, if we assume that $\lambda$ is integer, then a Poisson random variable with rate $\lambda$ can be viewed as the sum of $\lambda$ i.i.d. Poisson random variables with rate 1.
In other words, $Q^{(\infty)} = \sum_{i=1}^\lambda P_i$, where the $P_i$, $i=1,2,\ldots,n$, has Poisson distribution with mean and variance 1.
The CLT thus gives
\begin{equation}
\label{eq:infinite_server_tail}
\mathbb{P}(Q^{(\infty)} \geq x_\lambda )
= \mathbb{P}\left(\frac{Q^{(\infty)} -\lambda }{\sqrt{\lambda}} \geq \frac{ x_\lambda - \lambda}{\sqrt{\lambda}} \right)
\approx 1-\Phi\left( \frac{x_\lambda-\lambda}{\sqrt{\lambda}} \right),
\end{equation}
where $\Phi$ denotes the cumulative distribution function of the standard normal distribution.
which converges to a constant value away from both 0 and 1 if and only if $(x_\lambda - \lambda)/\sqrt{\lambda} \to x \in \mathbb{R}$, or $x_\lambda = \lambda + x \sqrt{\lambda} + o(\sqrt{\lambda})$, as $\lambda\to\infty$.
It also shows that the leading order of the random variable describing the queue length is $\lambda$, while the stochastic fluctuations are of order $\sqrt{\lambda}$.
If we now pretend, for a moment, that the infinite-server queue serves as a good approximation for the many-server queue with $s_\lambda$ servers, then we derive through \eqref{eq:infinite_server_tail} that the steady-state probability of wait for ${s_\lambda} = \lambda +\beta\sqrt\lambda$ obeys the Gaussian approximation
\begin{equation}
\label{eq:infinite_server_approx_delay}
\mathbb{P}(W^{(s_\lambda)}>0) = \mathbb{P}(Q^{(s_\lambda)} \geq s_\lambda ) \approx 1-\Phi(\beta).
\end{equation}
Of course, the infinite-server system ignores the one thing that makes a queueing system unique, namely that a queue is formed when all servers are busy.
During these periods of congestion, customers will depart from a system with a finite number of servers at a slower pace than in its infinite-server counterpart.
So the approximation in \eqref{eq:infinite_server_approx_delay} is likely to overestimate the actual delay probability, and a more careful investigation of the queue length process in many-server settings is needed. Nevertheless, the infinite-server heuristic reveals that in a well-managed system, i.e. queues are of acceptable length, the size at which the system operates is of the order $\lambda$, with fluctuations of order $\sqrt{\lambda}$.
We shall now demonstrate through two canonical examples how these guessed natural scalings can be turned into mathematically rigorous statements.
Both examples which will play a key role in this thesis.
\subsection{The $M/M/s$ queue}
\label{sec:intro_MMsqueue}
\textbf{Converging delay probability}.
Let $Q^{(s)}$ denote the steady-state number of customers in an $M/M/s$ queue with arrival rate $\lambda$ and mean service requirement 1, of which the probability distribution is given in \eqref{eq:MMs_stationary_distribution}.
Halfin and Whitt \cite{Halfin1981} showed that, just as in the infinite-server setting, the delay probability in the $M/M/s$ queue converges under scaling \eqref{eq:square_root_staffing rule} to a value between 0 and 1.
Moreover, they showed that this is in fact the only scaling regime in which such a non-degenerate limit exists and identified its value.
Because this result serves as a key prerequisite, we include the result from \cite{Halfin1981} here and present a slightly modified proof.
\begin{proposition}[{\cite[Prop.~2.1]{Halfin1981}}]
\label{prop:HalfinWhitt_delay_probability}
The probability of delay in the $M/M/s_\lambda$ queue has the non-degenerate limit
\begin{equation}
\lim_{\lambda\to\infty} \mathbb{P}( W^{(s_\lambda)} > 0 ) = \left( 1+ \frac{\beta\,\Phi(\beta)}{\varphi(\beta)} \right)^{-1} =: g(\beta) \in (0,1),
\end{equation}
if and only if
\begin{equation}
\label{eq:HalfinWhitt_scaling}
\lim_{\lambda\to\infty} (1-\rho_{s_\lambda}) \sqrt{s_\lambda} \to \beta, \quad \beta > 0,
\end{equation}
where $\Phi$ and $\varphi$ denote the cumulative distribution function and the probability density function of the standard normal distribution, respectively.
\end{proposition}
\begin{proof}
Rewrite \eqref{eq:MMs_wait} as
\begin{equation}
\label{eq:proof_HW_0}
\mathbb{P}( Q^{(s_\lambda)} \geq s_\lambda )
= \left( 1 + (1-\rho_{{s_\lambda}})\frac{ \mathbb{P}({\rm Pois}(\lambda) < {s_\lambda}) }{\mathbb{P}({\rm Pois} (\lambda) = {s_\lambda})}\right) ^{-1}.
\end{equation}
Similar to \eqref{eq:infinite_server_tail} we find
\begin{align}
\mathbb{P}({\rm Pois}(\lambda) < {s_\lambda})
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < \frac{{s_\lambda}-\lambda}{\sqrt{\lambda}}\right) \nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\frac{{s_\lambda}}{\sqrt\lambda}\right)\nonumber\\
&= \mathbb{P}\left(\frac{{\rm Pois}(\lambda)-\lambda}{\sqrt{\lambda}} < (1-\rho_{{s_\lambda}})\,\sqrt{{s_\lambda}}\left(1+o(1)\right) \right) \to \Phi(\beta),
\label{eq:proof_HW_1}
\end{align}
for $\lambda\to\infty$.
Using Stirling's approximation gives
\begin{align*}
\mathbb{P}({\rm Pois}(\lambda)=s) &= {\rm e}^{-\lambda}\frac{\lambda^{{s_\lambda}}}{{s_\lambda}!}
\sim {\rm e}^{-\lambda} \lambda^{{s_\lambda}}\cdot \frac{1}{\sqrt{2\pi\,{s_\lambda}}} \left(\frac{\rm e}{{s_\lambda}}\right)^{{s_\lambda}} = \frac{1}{\sqrt{2\pi{s_\lambda}}}\,{\rm e}^{{s_\lambda}-\lambda - {s_\lambda}\log(\rho_{{s_\lambda}})}.
\end{align*}
Since $\log(\rho_{{s_\lambda}}) = -(1-\rho_{{s_\lambda}}) - \tfrac{1}{2}(1-\rho_{{s_\lambda}})^2 + o((1-\rho_{{s_\lambda}})^2)$ we find that
\begin{equation}
\label{eq:proof_HW_2}
\frac{ \mathbb{P}({\rm Pois}(\lambda) = s) }{ 1-\rho_{{s_\lambda}} }
= \frac{1}{(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}} \, \frac{{\rm e}^{ -\tfrac{1}{2}(1-\rho_{{s_\lambda}})^2{s_\lambda} + o\left((1-\rho_{{s_\lambda}})^2{s_\lambda}\right)}}{\sqrt{2\pi}} \to \frac{1}{\beta}\, \frac{{\rm e}^{{-}\tfrac{1}{2} \beta^2}}{\sqrt{2\pi}} = \frac{\varphi(\beta)}{\beta}.
\end{equation}
Substituting \eqref{eq:proof_HW_1} and \eqref{eq:proof_HW_2} into \eqref{eq:proof_HW_0} proves the result.
\end{proof}
Observe that $g(\beta)$ is a strictly decreasing function on $(0,\infty)$ with $g(\beta) \to 1$ as $\beta\to 0$ and $g(\beta)\to 0$ for $\beta\to\infty$.
Thus the entire range of delay probabilities is achievable in the QED regime, which will prove useful for the dimensioning of systems (see Subsection \ref{sec:intro_dimensioning}).
\begin{figure}
\centering
\input{./tikz_tex/halfin_whitt_accuracy.tex}
\caption{The delay probability $\mathbb{P}(Q^{({s_\lambda})} \geq {s_\lambda})$ with ${s_\lambda} = [ \lambda + \beta \sqrt{\lambda} ]$ for $\beta = 0.1,\ 0.5,\ 1$ as a function of $\lambda$.}
\label{fig:delay_probs_HW_MMs}
\end{figure}
Although Proposition \ref{prop:HalfinWhitt_delay_probability} is an asymptotic result for $\lambda\to\infty$, Figure \ref{fig:delay_probs_HW_MMs} shows that for various values of $\beta$, $g(\beta)$ serves as an accuracy approximation for the delay probability for relatively small $\lambda$.
Moreover, D'Auria \cite{DAuria2012} has proven that $\mathbb{P}(W^{({s_\lambda})}>0) \geq g(\beta)$ for all finite $\lambda$ under scaling \eqref{eq:HalfinWhitt_scaling}.
From Proposition \ref{prop:HalfinWhitt_delay_probability}, it also follows that under \eqref{eq:HalfinWhitt_scaling},
\begin{equation}
\label{eq:halfinwhitt_wait}
\sqrt{{s_\lambda}}\,\mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{P}(W^{({s_\lambda})}>0)}{(1-\rho_{s_\lambda})\sqrt{{s_\lambda}}} \to \frac{g(\beta)}{\beta} =: h(\beta), \qquad \text{ for }\lambda\to\infty,
\end{equation}
where we have used the characterization of $\mathbb{E}[W^{({s_\lambda})}]$ in \eqref{eq:MMs_wait}.
This implies that in the QED regime, the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$ as $\lambda\to\infty$.
By Little's law this implies that the expected queue length is $O(\sqrt{{s_\lambda}})$.
The theoretical results of the QED regime we presented here are based on steady-state queueing analysis.
But at the heart of the QED theory lies a much deeper result in which the entire queue-length process, over all points in time, converges to some other limiting process.
\\*
\noindent\textbf{Process-level convergence.}
Obtaining rigorous statements about stochastic-process limits poses considerable mathematical challenges.
Rather than presenting the deep technical details of the convergence results, we give a heuristic explanation of how the limiting process arises and what it should look like.
Having another look at the sample paths of the queue-length process $Q^{(s)}$ in Figure \ref{fig:sample_paths_lambda100} with scaling rule ${s_\lambda} = [\lambda + \beta \sqrt{\lambda}]$, the process appears to concentrate around the level ${s_\lambda}$.
As argued before, the stochastic fluctuations are of order $\sqrt{\lambda}$, or equivalently $\sqrt{{s_\lambda}}$.
For that reason, we consider the centered and scaled process
\begin{equation}
\label{eq:intro_scaled_queue_length_process}
X^{({s_\lambda})}(t) := \frac{ Q^{({s_\lambda})}(t) - {s_\lambda}}{\sqrt{{s_\lambda}}}, \qquad \text{ for\ all } t\geq 0,
\end{equation}
and ask what happens to this process as $\lambda\to\infty$.
First, we consider the expected drift conditioned on $X^{({s_\lambda})}(t) = x$.
When $x> 0$, this corresponds to a state in which $Q^{({s_\lambda})}>{s_\lambda}$ and hence all servers are occupied.
Therefore, the expected rate at which customers leave the system is ${s_\lambda}$, while the arrival rate remains $\lambda$, so that the expected drift of $X^{({s_\lambda})}(t)$ in $x>0$ satisfies
\[
\frac{\lambda - {s_\lambda}}{\sqrt{{s_\lambda}}} \to -\beta, \qquad \text{as }\lambda\to\infty,
\]
under scaling $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to \beta$ in \eqref{eq:HalfinWhitt_scaling}.
When $x\leq 0$, only ${s_\lambda} + x\sqrt{{s_\lambda}}$ servers are working, so that the net drift is
\[
\frac{\lambda - ({s_\lambda} + x\sqrt{{s_\lambda}} )}{\sqrt{{s_\lambda}}} \to -\beta-x, \qquad \text{as }\lambda\to\infty,
\]
Now, imagine what happens to the sample paths of $\{X^{({s_\lambda})}\}_{t\geq 0}$ as we increase $\lambda$.
Within a fixed time interval, larger $\lambda$ and ${s_\lambda}$ will trigger more and more events, both arrivals and departures.
Also, the jump size at each event epoch decreases as $1/\sqrt{{s_\lambda}}$ as a consequence of the scaling in \eqref{eq:intro_scaled_queue_length_process}.
As a result, within each time interval, there will be more events, each with a smaller impact, and in the limit as $\lambda\to\infty$, there will be infinitely many events of infinitesimally small impact.
This heuristic explanation suggests that the process $X^{({s_\lambda})}(t)$ converges to a stochastic-process limit, which is continuous and has infinitesimal drift ${-}\beta$ above zero and ${-}\beta-x$ below zero.
Figure \ref{fig:sample_paths_diffusion} visualizes more clearly how the suggested scaling limit arises with increasing $\lambda$ and ${s_\lambda}$.
\begin{figure}
\centering
\input{./tikz_tex/sample_path_diffusion.tex}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda})}(t)$ with $\lambda = 5$, $\lambda=5$ and $\lambda=500$ and ${s_\lambda} = [\lambda+0.5\sqrt{\lambda}]$.}
\label{fig:sample_paths_diffusion}
\end{figure}
The following theorem by Halfin and Whitt \cite{Halfin1981} characterizes this scaling limit more formally.
\begin{theorem}
\label{thm:Halfin_Whitt_diffusion}
Let $X^{({s_\lambda})}(0)\, {\;\buildrel{d}\over\Rightarrow\;} X(0) \in \mathbb{R}$ and $\sqrt{{s_\lambda}}(1-\rho_{{s_\lambda}})\to\beta$. Then for all $t\geq 0$,
\[
X^{({s_\lambda})}(t) {\;\buildrel{d}\over\Rightarrow\;} X(t),
\]
as $\lambda\to\infty$, where $X(t)$ is the diffusion process with infinitesimal drift $m(x)$ given by
\[
m(x) = \left\{
\begin{array}{ll}
-\beta, & \text{if }x> 0,\\
-\beta-x, & \text{if } x \leq 0
\end{array}\right.
\]
and infinitesimal variance $\sigma^2(x) = 2$.
\end{theorem}
The limiting diffusion process $\{X(t)\}_{t\geq 0}$ in Theorem \ref{thm:Halfin_Whitt_diffusion} is a combination of a negative-drift Brownian motion in the upper half plane and an Ornstein-Uhlenbeck (O-U) process in the lower half plane.
We refer to this hybrid diffusion process as the Halfin-Whitt diffusion.
Much is known for such diffusion processes with piecewise linear drift coefficient, see \cite{Leeuwaarden2012,Fralix2014}.
Its stationary distribution can for instance be derived, see e.g. \cite{BrowneWhitt1995}.
\begin{theorem}
\label{thm:intro_HW_stationary_distribution}
Let $X(t) {\;\buildrel{d}\over\Rightarrow\;} X(\infty)$ for a random variable $X(\infty)$ and $(1-\rho_{{s_\lambda}})\sqrt{{s_\lambda}}\to \beta$ for $\lambda\to\infty$.
Then
\begin{align}
\mathbb{P}(X(\infty) > 0 ) &= g(\beta),
\mathbb{P}(X(\infty) \geq x | X(\infty) > 0) &= {\rm e}^{-\beta x} ,\quad \text{for }x>0,\\
\mathbb{P}(X(\infty) \leq x | X(\infty) \leq 0 ) &= \frac{\Phi(\beta+x)}{\Phi(\beta)},\quad \text{for }x\leq 0.
\end{align}
\end{theorem}
\noindent
This result shows that as the system grows large, the $Q^{({s_\lambda})}(t)$ concentrates around ${s_\lambda}$, and the fluctuations are of order $\sqrt{{s_\lambda}}$.
Moreover, Theorem \ref{thm:intro_HW_stationary_distribution} iterates the limiting values for the delay probability and scaled expected delay. Namely,
\[ \mathbb{P}\big(W^{({{s_\lambda}})} > 0 \big) \rightarrow \mathbb{P}( X(\infty) > 0 ) = g(\beta)\]
and
\[ \sqrt{{s_\lambda}}\mathbb{E}[W^{({s_\lambda})}] \approx \frac{\mathbb{E}[ Q^{({s_\lambda})}]}{\sqrt{{s_\lambda}}} \rightarrow \mathbb{E}[X(\infty)] = \int_0^\infty g(\beta){\rm e}^{-\beta x} {\rm d} x = \frac{g(\beta)}{\beta},/\]
For obvious reasons, the QED regime is also referred to as the Halfin-Whitt regime, and both these names are used interchangeably in this thesis.
\subsection{The $M/D/s$ queue}
\label{sec:intro_discrete_model}
We next consider a many-server queue with deterministic service requirements equal to 1, a Poisson arrival process of rate $\lambda$ and ${s_\lambda}$ servers.
We let $Q^{({s_\lambda})}(t)$ be process describing the number of customers in the system and only examine the process at discrete time epochs $t=0,1,2,...$.
In our analysis we focus on the queue length process $Z^{({s_\lambda})}(t) := (Q^{({s_\lambda})}(t) - {s_\lambda})^+$.
Since we discretize time, the number of new arrivals per time period is given by the sequence of i.i.d. random variables $\{A_k\}_{k\geq 1}$, which has a Poisson distribution with rate $\lambda$.
At the start of the $k^{\rm th}$ period, $Z^{({s_\lambda})}(k)$ customers are waiting.
Because the service time of a customer is equal to the period length, all $\min\{Q^{({s_\lambda})},{s_\lambda}\}$ customers in service at the beginning of the period will have left the system by time $t=k+1$.
This implies that $\min\{Z^{({s_\lambda})},{s_\lambda}\}$ of the waiting customers are taken into service during period $k$, but could not possibly have departed before its end, due to the deterministic service times.
If $Z^{({s_\lambda})}<{s_\lambda}$, then additionally $\min\{ A_k , s-Z^{({s_\lambda})}(k) \}$ of the new arrivals are taken into service.
This yields a total of $A_k$ arrivals, and $\min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\}$ departures from the queue during period $k$, which gives the Lindley type recursion \cite{Lindley1952}, with $Z^{({s_\lambda})}(0) = 0$,
\begin{equation}
\label{eq:discrete_recursion}
Z^{({s_\lambda})}(k+1) = Z^{({s_\lambda})}(k) + A_k - \min\{Z^{({s_\lambda})}(k)+A_k,{s_\lambda}\} = \max\{ 0,Z^{({s_\lambda})}(k) + A_k - {s_\lambda} \}.
\end{equation}
The queue length process thus gives rise to a random walk with i.i.d. steps of size
$(A^{({s_\lambda})}-{s_\lambda})$, with a reflection barrier at zero. We can iterate the recursion in \eqref{eq:discrete_recursion} to find
\begin{align}
Z^{({s_\lambda})}(k+1) &= \max\left\{ 0 , Z^{({s_\lambda})}(k) + A_k-{s_\lambda} \right\} \nonumber\\
&= \max\left\{ 0 , \max\{ 0 , Z^{({s_\lambda})}(k-1) + (A_{k-1}-{s_\lambda})\} + (A_k-{s_\lambda})\} \right\}\nonumber \\
&= \max\left\{ 0 , (A_k-{s_\lambda}) , Z^{({s_\lambda})}(k-1) + (A_k-{s_\lambda}) + (A_{k-1}-{s_\lambda})\right\}\nonumber \\
&= \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_{k-i}-{s_\lambda})\Big\}
{\;\buildrel{d}\over= \;} \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j (A_i-{s_\lambda}) \Big\},
\label{eq:max_randomwalk}
\end{align}
where the last equality in distribution holds due to the duality principle for random walks, see e.g. \cite[Sec.~7.1]{Ross1996}.
For stability the expected step size satisfies $\mathbb{E}[A_k - {s_\lambda}] = \lambda-{s_\lambda} < 0$.
We use the shorthand notation for the partial sum $S_k := \sum_{i=1}^k (A_i-{s_\lambda})$.
Let $Z^{({s_\lambda})}(\infty):= \lim_{k\to\infty} Z^{({s_\lambda})}(k)$ denote the stationary queue length in this $M/D/s$ queue, which can be shown to exist under our assumptions.
The generating function (pgf) of $Z^{({s_\lambda})}(\infty)$ can then be expressed in terms of the pgf of the positive parts of the partial sum:
\begin{equation}
\label{eq:Spitzers_identity}
\mathbb{E}[ w^{Z^{({s_\lambda})}(\infty)} ]
= \exp\Big\{ - \sum_{k=1}^\infty \tfrac{1}{k}\, (1- \mathbb{E}[w^{S_k^+}]) \Big\},\qquad |w|\leq 1,
\end{equation}
From \eqref{eq:Spitzers_identity} we obtain for the mean queue length and empty-queue probability the expressions
\begin{align}
\mathbb{E}[Z^{({s_\lambda})}(\infty)] &= \sum_{k=1}^\infty \tfrac{1}{k}\, \mathbb{E}[ S_k^+ ],\nonumber\\
\mathbb{P}(Z^{({s_\lambda})}(\infty) = 0 ) &= \exp\Big\{ -\sum_{k=1}^\infty \tfrac{1}{k}\, \mathbb{P}( S_k^+ > 0 ) \Big\}.
\label{eq:spitzer_expressions}
\end{align}
Although explicit, the expressions in \eqref{eq:spitzer_expressions} reveal little of the structure of the queue length process.
Hence, we again turn to asymptotics. \\
\noindent\textbf{Gaussian random walk}.
\label{sec:intro_gaussian_random_walk}
We take another look at the identity in \eqref{eq:max_randomwalk}, and ask ourselves what happens if $\lambda$ grows large.
Since $\mathbb{E}[A_k-{s_\lambda}] = \lambda-{s_\lambda} = -\beta\sqrt{\lambda} + o(\sqrt{\lambda})$ under the QED scaling \eqref{eq:square_root_staffing rule}, it makes sense to consider the scaled queue length process $X^{({s_\lambda})}(k) := Z^{({s_\lambda})}(k)/\sqrt{\lambda}$ for all $k\geq 0$, with scaled steps $Y_k^{({s_\lambda})} := (A_k-{s_\lambda})/\sqrt{\lambda}$.
Dividing both sides of \eqref{eq:max_randomwalk} by $\sqrt{\lambda}$ then gives
\begin{equation}
X^{({s_\lambda})}(k+1) = \max_{0\leq j\leq k} \Big\{ \sum_{i=1}^j Y^{({s_\lambda})}_k \Big\}.
\end{equation}
Observe that $A_k \sim {\rm Pois}(\lambda)$.
Hence by the CLT
\begin{equation*}
Y^{({s_\lambda})}_k = \frac{ A_k - {s_\lambda} }{\sqrt\lambda} = \frac{A_k-\lambda}{\sqrt\lambda} - \beta \ {\;\buildrel{d}\over\Rightarrow\;} \ Y_k {\;\buildrel{d}\over= \;} \mathcal{N}(-\beta,1),
\end{equation*}
for $\lambda\to\infty$.
So by intuition, we expect the scaled queue length process to converge in distribution to a reflected random walk with normally distributed increments, i.e. a reflected \textit{Gaussian random walk}.
Indeed, it is easily verified that \cite{Janssen2008a},
\begin{equation}
X^{({s_\lambda})}(k)\ {\;\buildrel{d}\over\Rightarrow\;} \ M_\beta(k) := \max_{0\leq j\leq k} \Big\{\sum_{i=1}^j Y_j \Big\}, \qquad \lambda\to\infty.
\end{equation}
Let $M_\beta:= \lim_{k\to\infty} M_\beta(k)$ denote the all-time maximum of a Gaussian random walk.
It can be shown that $M_\beta$ almost surely exists and that $X^{({s_\lambda})}(\infty) := \lim_{k\to\infty} X^{({s_\lambda})}(k)$ ${\;\buildrel{d}\over\Rightarrow\;} M_\beta$ for instance by \cite[Prop.~19.2]{Spitzer1964} and \cite[Thm.~X6.1]{Asmussen2003}.
The following theorem can be proved using a similar approach as in \cite{Jelenkovic2004}.
(We prove this result in a more general setting in Chapter 3)
\begin{theorem}
Let $X^{({s_\lambda})}(\infty)$ be the scaled queue length in steady-state. If $(1-\rho_{{s_\lambda}})\sqrt{\lambda}\to\beta$, then as $\lambda\to\infty$,
\begin{enumerate}
\item[\normalfont (i)] $X^{({s_\lambda})}(\infty) {\;\buildrel{d}\over\Rightarrow\;} M_\beta$,
\item[\normalfont (ii)] $\mathbb{P}(X^{({s_\lambda})}(\infty) = 0) \to \mathbb{P}(M_\beta = 0)$,
\item[\normalfont (iii)] $\mathbb{E}[X^{({s_\lambda})}(\infty)^k] \to \mathbb{E}[M_\beta^k]$, for any $k>0$.
\end{enumerate}
\end{theorem}
The Gaussian random walk is well studied in \cite{Siegmund1978,Chang1997,Janssen2006,Blanchet2006,Janssen2006} and there is an intimate connection with Brownian motion.
The only difference, one could say is that Brownian motion is a continuous-time process, whereas the Gaussian random walk only changes at discrete points in time.
If $\{B(t)\}_{t\geq 0}$ is Brownian motion with drift $-\mu <0$ and infinitesimal variance $\sigma^2$ and $\{W(t)\}_{t = 0}^\infty$ is a random walk with $\mathcal{N}(-\mu,\sigma^2)$ steps and $B(0) = W(0)$, then $W$ can be regarded as the process $B$ embedded at equidistant time epochs.
That is, $W(t) {\;\buildrel{d}\over= \;} B(t)$ for all $t\in\mathbb{N}^+$.
For the maximum of both processes this coupling implies
\begin{equation}
\max_{k\in \mathbb{N}^+} W(k) = \max_{k\in \mathbb{N}^+} B(k) \leq_{\rm st}
\max_{t\in \mathbb{R}^+} B(t),
\label{eq:max_inequality}
\end{equation}
where $\leq_{\rm st}$ denotes stochastic dominance.
This difference in maximum is visualized in Figure \ref{fig:BrownianMotion_vs_GaussianRW}.
It is known that the all-time maximum of Brownian motion with negative drift $-\mu$ and infinitesimal variable $\sigma^2$ has an exponential distribution with mean $\sigma/2\mu$ \cite{Harrison1985}.
Hence, \eqref{eq:max_inequality} implies that $M_\beta$ is stochastically upper bounded by an exponential random variable with mean $1/2\beta$.
\begin{figure}
\centering
\begin{tikzpicture}[scale = 1.1 ]
\begin{axis}[
xmin = 0,
xmax = 10,
ymin = -2.2,
ymax = 5,
axis line style={->},
axis x line=middle,
axis y line=left,
xlabel={$t$},
ylabel={},
xscale=1,
yscale=1]
\addplot[gray] file {./tikz/Brownian_Motion_SamplePath/BM.txt};
\addplot[only marks, red] file {./tikz/Brownian_Motion_SamplePath/GW.txt};
\addplot[dashed] file {./tikz/Brownian_Motion_SamplePath/maxBM.txt};
\addplot[dotted,very thick, red] file {./tikz/Brownian_Motion_SamplePath/maxGW.txt};
\end{axis}
\end{tikzpicture}
\caption{Brownian motion and embedded Gaussian random walk with their respective running maxima.}
\label{fig:BrownianMotion_vs_GaussianRW}
\end{figure}
Despite this easy bound, precise results for $M_\beta$ are more involved. Let $\zeta$ denote the Rieman zeta function.
\begin{theorem}[{\cite[Thm.~1]{Chang1997}}]
For $0<\beta<2\sqrt{\pi}$,
\begin{equation}
\mathbb{P}(M_\beta = 0) = \sqrt{2}\beta\, \exp \left\{ \frac{\beta}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(1/2-l)}{l!(2l+1)} \left(\frac{-\beta^2}{2}\right)^l \right\}.
\end{equation}
\end{theorem}
In \cite{Janssen2006,Janssen2007,Janssen2009}, similar series expansions are derived for e.g. the mean and variance of the maximum of the Gaussian random walk.
\begin{theorem}[Thm.~2\&3, \cite{Janssen2006}]
For $0<\beta<2\sqrt{\pi}$,
\begin{equation}
\mathbb{E}[M_\beta] = \frac{1}{2\beta} + \frac{\zeta(1/2)}{\sqrt{2\pi}} + \frac{\beta}{4}
+ \frac{\beta^2}{\sqrt{2\pi}} \sum_{l=0}^\infty
\frac{\zeta(-1/2-l)}{l!(2l+1)(2l+2)} \left(\frac{-\beta^2}{2}\right)^l,
\end{equation}
\begin{equation}
{\rm Var}\, M_\beta =
\frac{1}{4\beta^2} - \frac{1}{4} - \frac{2\,\zeta(-1/2)}{\sqrt{2\pi}}\beta - \frac{\beta^2}{24} -
\frac{2\beta^3}{\sqrt{2\pi} } \sum_{l=0}^\infty
\frac{\zeta(-3/2-l)}{l!(2l+1)(2l+2)(2l+3)} \Big(\frac{-\beta^2}{2}\Big)^l.
\end{equation}
\end{theorem}
\subsection{Characteristics of the QED regime}
\label{sec:intro_characteristics}
Now that we have seen how the square-root staffing principle \eqref{eq:square_root_staffing rule} yields non-degenerate limiting behavior in two classical queueing models, we can elaborate on how the QED regime fosters three desirable properties.
The first property relates to the efficient usage of resources, expressed as:
\begin{equation}
\rho_{{s_\lambda}} = \frac{\lambda}{{s_\lambda}} = 1 - \frac{\beta}{\sqrt{{s_\lambda}}} + O\big(1/\lambda\big), \tag{Efficiency}
\end{equation}
where we used that ${s_\lambda} = O(\lambda)$.
The second property relates to good QoS:
\begin{equation}
\mathbb{E}[W^{({s_\lambda})}] = \frac{h(\beta)}{\sqrt{{s_\lambda}}} + o(1/\sqrt{{s_\lambda}}) \qquad \text{and} \qquad \mathbb{E}[W^{({s_\lambda})}] = \frac{\mathbb{E}[M_\beta]}{\sqrt{{s_\lambda}}} + O(1/\sqrt{{s_\lambda}}), \tag{QoS}
\end{equation}
in the $M/M/s$ and $M/D/s$ models, respectively.
Hence the expected waiting time vanishes at rate $1/\sqrt{{s_\lambda}}$.
The third distinctive property is the balance between QoS and efficiency:
\begin{equation}
\mathbb{P}(W^{({s_\lambda})}>0) \to g(\beta), \qquad \text{and} \qquad \mathbb{P}(W^{({s_\lambda})}>0) \to 1-\mathbb{P}(M_\beta=0), \tag{Balance}
\end{equation}
as ${s_\lambda} \to \infty$, both values laying within the interval (0,1), in the $M/M/s$ and $M/D/s$ models, respectively.
The three key properties, reformulated in a model-independent way are, with $s = \lambda +\beta\sqrt\lambda$ and $\rho = \lambda/s$,
\begin{align}
\rho &= 1-\beta/\sqrt\lambda + O(\lambda^{-1}), \tag{*} \\
\mathbb{E}[W] &= O(1/\sqrt\lambda), \tag{**}\\
\mathbb{P}(W>0) &\to \alpha \in(0,1). \tag{***}
\end{align}
Since the mathematical foundation for these properties comes from the CLT, we can expect the properties to hold for a much larger class of models.
These models should then be members of the same universality class (to which the CLT applies).
Let us again show this by example.
Consider a stochastic system in which demand per period is given by some random variable $A$, with mean $\mu_A$ and variance $\sigma_A^2<\infty$.
For systems facing large demand we propose to set the capacity according to the more general rule $s = \mu_A + \beta\sigma_A$, which consists of a minimally required part $\mu_A$ and a variability hedge $\beta\sigma$.
Assume that the workload brought into the system is generated by $n$ stochastically identical and independent sources.
Each source $i$ generates $A_{i,j}$ work in the $j$th period, with $\mathbb{E}[A_{i,j}] = \mu$ and ${\rm Var}\,\,A_{i,j} = \sigma^2$.
Then the total amount of work arriving to the system during one period is $A_j^{(n)} = \sum_{i=1}^n A_{i,j}$ with mean $n\mu$ and variance $n\sigma^2$.
Assume that the system is able to process a deterministic amount of work $s_n$ per period and denote by $U^{(n)}(j)$ the amount of work left over at the end of period $j$.
Then,
\begin{equation}
U(j+1) = \left( U^{(n)}(j) + A^{(n)}_j - s_n \right)^+.
\end{equation}
Given that $s_n > \mathbb{E}[A^{(n)}_1] = n\mu$, the stationary limit $U^{(n)} := \lim_{t\to\infty} U^{(n)}(t)$ exists and satisfies
\begin{equation}
U^{(n)} {\;\buildrel{d}\over= \;} \left( U^{(n)} + A^{(n)}_j - s_n \right)^+.
\label{eq:bulk_service_stationary_recursion}
\end{equation}
This framework is also known as the bulk service queue or the Anick-Sondhi-Mitra model \cite{Anick1982,Janssen2005,Janssen2008}.
In this scenario, increasing the system size is done by increasing $n$, the number of input flows.
As we have seen before, it requires a rescaling of the process $U^{(n)}$ by an increasing function $c(n)$, in order to obtain a non-degenerate scaling limit $U := \lim_{n\to\infty} U^{(n)}/c(n)$.
(We omit the technical details needed to justify the interchange of limits.)
From \eqref{eq:bulk_service_stationary_recursion} it becomes clear that the scaled increment
\begin{equation}
\frac{A^{(n)}_j - s_n}{c(n)} = \frac{\sum_{i=1}^n A_{i,j} - n\mu}{c(n)} + \frac{n\mu - s_n}{c(n)}
\end{equation}
only admits a proper limit if $c(n)$ is of the form $c(n) = O(\sqrt{n})$, by the virtue of the CLT, and $(s_n-n\mu)/c(n) \to \beta >0$ as $n\to\infty$.
Especially for $c(n) = \sigma\sqrt{n}$, this reveals that $U$ has a non-degenerate limit, which is equal in distribution to the maximum of a Gaussian random walk with drift -1 and variance 1, if
\[
s_n = n\mu+\beta \sqrt{n}\sigma + o(\sqrt{n}).
\]
Moreover, the results on the Gaussian random walk presented in Subsection \ref{sec:intro_gaussian_random_walk} are applicable to this model and the key features of the QED scaling carry over to this more general setting as well.
In conclusion, the many-sources framework shows that the QED scaling finds much wider applications than queueing models with Poisson input only.
\subsection{Related literature}
We now provide a partial overview on the literature on heavy-traffic analysis in queueing theory and the QED regime in particular.\\
\\*
\noindent\textbf{Conventional heavy-traffic}.
Before the formal introduction of the Halfin-Whitt scaling regime in 1981, see \cite{Halfin1981}, the existing literature on the asymptotic analysis of many-server queues mostly evolved around two types of scaling regimes.
The idea of studying a sequence of queues in which the utilization approaches 100\%, i.e.~heavy-traffic, was first laid out by Kingman in the 1960s.
In \cite{Kingman1961,Kingman1962} he showed how in the $GI/G/1$ queue, under mild conditional on the arrival and service processes, the scaled steady-state waiting time
distribution $(1-\rho)W^{(1)}$ to an exponentially distributed random variable.
The notion that heavily loaded systems admit a scaling limit that is remarkably simple compared to the otherwise intractable pre-limit queueing systems triggered a surge of research within the field of queueing theory in the 1960s and 1970s, see \cite{Borovkov1965,Iglehart1970,Brumelle1971,Newell1973,Kollerstrom1974,Kollerstrom1979,Whitt1974} among others.
These works conduct their asymptotic analysis in what we now call conventional heavy-traffic.
That is, the service times and number of servers are held fixed, while the arrival rate approaches the critical value from below.
A noteworthy result of these efforts is the extension of Kingman's findings to the $GI/G/s$, which finds that the scaled queue length $(1-\rho)Q^{(s)}$ converges in distribution to an exponential random variable with mean $(c_a^2+c_s^2)/2$, where $c_a$ and $c_s$ denote the coefficient of variation of the interarrival and service time distribution, respectively.
We remark that this limiting result is the key ingredient to the famous Kingman's formula:
\[
\mathbb{E}[W^{(1)}] \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2+c_s^2}{2} \cdot \mathbb{E}[B],
\]
which serves as an approximation to the expected waiting time in the single-server queue and of which the usage is now widespread.
The scaling limits reveal that in the conventional heavy-traffic regime, the expected waiting time explodes as $\rho\to 1$.
Hence, efficient usage of resources is achieved, at the expense of poor QoS.
An alternative regime that received much attention, see e.g. \cite{Iglehart1965,Borovkov1965,Iglehart1973,Iglehart1973a,Whitt1982}, fixed the service time distribution while increasing both the arrival rate $\lambda$ and the number of servers to infinity simultaneously, such that the ratio $\lambda/s$ is constant.
It has been shown that the sequence of queues under this scaling start resembling the behavior of infinite-server queues as $\lambda$ and $s$ grow.
That is, the probability of a customer finding a queue on arrival is negligible.
The sample paths in Figure \ref{fig:sample_paths_lambda100} are illustrative for this regime.
Since the utilization level $\rho$ remains strictly below 1 in the limit, this setting is typically not recognized as heavy-traffic.
Accordingly, server efficiency is not achieved in this case.
However, this regime offers excellent service levels, as customers experience virtually no wait.
As Halfin and Whitt spell out themselves, their novel regime in which service times are held fixed, and $\lambda$ and $s$ tend to infinite while satisfying $(1-\rho)\sqrt{s} \to \beta$, is a hybrid between the two aforementioned regimes.
Namely, it adopts the efficiency property of the conventional heavy-traffic scaling, and the good QoS levels from the resemblance with infinite-server queues.\\
\\*
\noindent
\textbf{The $G/G/s$ queue in the QED regime}.
Since the literature on queues in the QED regime is vast, we choose to give an overview of only the most relevant advances in the performance analysis of the many-server queues in the QED regime under the most general conditions, together with the extension that are rooted within practical service systems.
Whereas \cite{Halfin1981} was able to exploit the Markovian of the exponentially distributed service times, the heavy-traffic analysis of the $G/G/s$ queue requires fundamentally different approaches than Halfin and Whitt's.
For various service distribution classes, the convergence sequence of diffusion-scaled processes has been studied within finite time intervals.
Puhalskii \& Reiman \cite{Puhalskii2000} analyze the multi-class queue with phase-type service times in the Halfin-Whitt regime.
Heavy-traffic limits for queues in which service time distributions are lattice-based and/or have finite support are studied by Mandelbaum \& Momcilovic \cite{Mandelbaum2008} and Gamarnik \& Momcilovic \cite{Gamarnik2008}.
Other approaches by Kang, Kaspi \& Ramanan \cite{Kaspi2011,Kang2012,Kaspi2013}.
The most general class of distributions is considered by Reed \cite{Reed2009} and Puhalskii \& Reed \cite{Puhalskii2010}, who impose no assumption on the service time distribution except for the existence of the first moment.
Considerably less is known for the corresponding steady-state distribution of the $G/G/s$ queue in the QED regime.
Namely, under the assumption of general service time distributions, truly infinite-dimensional limits arise, since the Markovian nature of the service time and `age' process can no longer be exploited.
Works that have been able to characterize limiting behavior for the specific service time distribution classes include Jelenkovic et al.~\cite{Jelenkovic2004}, who assume deterministic service times, and Whitt \cite{Whitt2005}, who identifies the heavy-traffic limit in the case of hyperexponentially distributed service times.
Progress in the understanding of steady-state behavior of $G/G/s$ queues in the Halfin-Whitt regime has been facilitated by Gamarnik \& Goldberg \cite{Goldberg,Gamarnik2013a}, who perform their analysis under the mild assumption that the service distribution has finite $(2+\e)$ moment.
A significant advance has been made by Aghanjani \& Ramanan \cite{Aghajani2016}, who under the most general conditions identify the limit as the steady-state distirbution of infinite-dimensional Markov process, drawing upon previous results by Kang, Kaspi \& Ramanan \cite{Kaspi2011, Kang2012,Kaspi2013}
For an elaborate survey on the techniques required for analysis of $G/G/s$ queues, we refer the reader to \cite{Pang2007} and references therein.\\
\\*
\textbf{Model extensions}
A variety of extensions to the standard many-server queue can be considered.
A feature ubiquitous to service systems involving humans is customer abandonment \cite{Gans2003,Brown2005,Zeltyn2005,Mandelbaum2013}.
The $M/M/s+M$ queue introduced by Palm \cite{Palm1957}, also known as the Erlang-A model \cite{Garnett2002,Leeuwaarden2012}, acknowledges this feature by assigning every customer an exponentially distributed \textit{patience time} upon his arrival (denoted by $+M$ in the model definition).
If a customer has not yet started receiving service by the expiration of his patience, he leaves the system.
Note that abandonments render queues stable under any load.
Under QED scaling, the more general $G/G/s+G$ queue has received much attention under various modeling assumptions, see e.g. \cite{Garnett2002,Gans2003,Whitt2006,Mandelbaum2009,Zeltyn2005,Mandelbaum2012a,Kang2012,Dai2010,Reed2012,Jennings2012,Zhang2013}.
Noteworthy findings include the vanishing abandonment probability \cite{Garnett2002} and insensitivity of the patience time distribution as long as its density at 0, i.e.~the balking probability, is fixed, as the system grows large under QED scaling.
Neat overviews of queues with abandonment and their asymptotic counterpart are given by Zeltyn \& Mandelbaum \cite{Zeltyn2005} and Dai \& He \cite{Dai2012} and Ward \cite{Ward2012}.
Other features model extensions that have been studied in the QED regime include multiple customer classes, see e.g. \cite{Harrison2004,Atar2014,Gurvich2008,Gurvich2009,Tezcan2010}, or heterogeneous servers \cite{Armony2005,Armony2010,Mandelbaum2012b,Stolyar2010}.
These models are all interesting in their own respect and are fairly well-understood.
Therefore, we choose to focus in this thesis on a different set of extensions, which will be discussed in Section \ref{sec:intro_beyond}.
\section{Dimensioning}
\label{sec:intro_dimensioning}
We adopt the term \textit{dimensioning} used by Borst, Mandelbaum \& Reiman~\cite{Borst2004} to say that the capacity of a service system is adapted to the load in order to reach certain performance levels.
In \cite{Borst2004} dimensioning refers to the staffing problem in a large-scale call center and key ingredients are the square-root staffing rule in \eqref{eq:square_root_staffing rule} and the QED regime.
We now revisit the results in \cite{Borst2004} and its follow-up works to explain this connection to the QED regime.
\subsection{Constraint satisfaction}
\label{sec:intro_constraint}
Consider the $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$.
A classical dimensioning problem is to determine the minimum number of servers $s$ necessary to achieve a certain target level of service, say in terms of waiting time.
Suppose we want to determine the minimum number of servers such that the fraction of customer who are delayed in the queue is at most $\varepsilon\in(0,1)$.
Hence we should find
\begin{equation}
s^{*}_\lambda(\varepsilon) := \min \left\{s \geq \lambda\, |\, \mathbb{P}(W^{(s)}>0) \leq \varepsilon \right\}. \tag{A}
\end{equation}
But alternatively, we can use the QED framework, which says that under \eqref{eq:HalfinWhitt_scaling},\ \ $\lim_{s\to\infty} \mathbb{P}(W^{({s_\lambda})} > 0) = g(\beta)$ (see Proposition \ref{prop:HalfinWhitt_delay_probability}).
Then by (A) can be replaced by
\begin{equation}
s^{\rm srs}_\lambda(\varepsilon) = \lceil \lambda + \beta^*(\varepsilon) \sqrt{\lambda}\rceil, \tag{B}
\end{equation}
where $\beta^*(\varepsilon)$ solves
\begin{equation}
g(\beta^*) = \varepsilon. \tag{C}
\end{equation}
In Figure \ref{fig:MMs_staffing_levels} we plot the exact staffing level $s^*_\lambda(\varepsilon)$ and the heuristically obtained staffing level $s^{\rm srs}_\lambda(\varepsilon)$ as a functions of $\varepsilon$ for several loads $\lambda$.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda5_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\small
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda10_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda100_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 1,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to \varepsilon$},
ylabel = {},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/Constraint_Satisfaction/lambda500_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Constraint_Satisfaction/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(\varepsilon)$},$s^{\rm srs}_\lambda(\varepsilon)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500.$}
\end{subfigure}
\caption{Staffing levels corresponding to delay probability targets $\varepsilon$.}
\label{fig:MMs_staffing_levels}
\end{figure}
Observe that even for very small values of $\lambda$, the staffing function $s^{\rm srs}(\varepsilon)$ coincides with the exact solution for almost all $\varepsilon\in(0,1)$ and differs no more than by one server for all $\varepsilon$.
Borst et al.~\cite{Borst2004} recognized this in their numerical experiments too, and Janssen, van Leeuwaarden and Zwart \cite{Janssen2011} later confirmed this theoretically.
As the scale of the queue increases, these differences naturally cancel out.
Henceforth, we may argue that the staffing method via the square-root staffing principle performs close to optimal in systems that are `large enough'.
One could easily adapt this asymptotic procedure to fit the staffing problem with a constraint on the mean waiting time or the tail probability of the waiting time, e.g. $\mathbb{P}(W^{(s)}>T)$, which are asymptotically approximated by $h(\beta)/\sqrt{\lambda}$ and $g(\beta){\rm e}^{-\beta \sqrt{\lambda} T}$, respectively, but we do not go into details here.
\subsection{Optimization}
\label{sec:intro_optimization}
A similar line of reasoning holds when one turn form a constraint satisfaction problem to an optimization problem, for instance to strike the right balance between the costs for servers and costs incurred by customer dissatisfaction.
More specifically, assume a salary cost of $a$ per server per unit time, and a penalty cost of $q$ per waiting customer per unit time, yielding the total cost function
\[
\bar{C}_\lambda(s) := a\,s + q\,\lambda\mathbb{E}[W^{(s)}]
\]
and then ask for the staffing level $s$ that minimizes $\bar{C}_\lambda(s)$
in which we target the minimizing number of servers $s$.
Since $s>\lambda$, we have $\bar{C}_\lambda(s) > a\,\lambda$ for all feasible solutions $s$.
Moreover, the minimizing value of $\bar{C}_\lambda$ is invariant with respect to scalar multiplication of the objective function.
Hence we have to optimize
\[
C_\lambda(s) = r\,(s-\lambda) + \lambda\mathbb{E}[W^{(s)}], \qquad r = a/q.
\]
The exact solution to the staffing problem is denoted by $s^*_\lambda(r) := \arg\min_{s > \lambda} C_\lambda(s)$.
With ${s_\lambda} = \lambda + \beta\sqrt{\lambda}$ and the QED limiting results in \eqref{eq:halfinwhitt_wait}, we can replace this optimization problem by its asymptotic counterpart
\begin{align*}
\frac{C_\lambda({s_\lambda})}{\sqrt{\lambda}} = r\,\beta + \sqrt{\lambda} \mathbb{E}[W^{(s)}] \to r\,\beta + \frac{g(\beta)}{\beta} =: \hat{C}(\beta), \qquad \lambda\to\infty.
\end{align*}
Once again we obtain a limiting objective function that is easier to work with than its exact pre-limit counterpart.
Hence, in the spirit of the asymptotic staffing procedure in the previous subsection, we propose the following method to determine the staffing level that minimizes overall costs.
First, (numerically) compute the value $\beta^*(r) = \arg\min_{\beta>0} \hat{C}(\beta)$, which is well-defined, because the function $\hat{C}(\beta)$ is strictly convex for $\beta>0$.
Afterwards, set $s^{\rm srs}_\lambda(r) = [ \lambda + \beta^*(r) \sqrt{\lambda} ]$.
In Figure \ref{fig:MMs_staffing_levels_optimization} we compare the outcomes of this asymptotic staffing procedure against the true optima as a function of $r\in(0,\infty)$, for several values of $\lambda$.
The staffing levels $s^{\rm srs}_\lambda(r)$ and $s^*_\lambda(r)$ are aligned for almost all $r$, and differ no more than one server for all instances.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 5,
ymax = 12,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda5_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda5_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\small
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 10,
ymax = 19,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda10_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda10_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=10.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 100,
ymax = 125,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend pos = north east]
\addplot[very thick] file {./tikz/Optimization/lambda100_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda100_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=100.$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.72]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = 500,
ymax = 550,
axis line style={->},
axis lines = left,
xlabel = {\small $\to r$},
ylabel = {},
yscale = 0.8,
axis y discontinuity=crunch,
legend style = {at = {(axis cs:4.8,540)},anchor = north east}]
\addplot[very thick] file {./tikz/Optimization/lambda500_exact.txt};
\addplot[very thick, dashed, red] file {./tikz/Optimization/lambda500_asymptotic.txt};
\legend{{$s^*_\lambda(r)$},$s^{\rm srs}_\lambda(r)$}
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500.$}
\end{subfigure}
\caption{Optimal staffing levels corresponding as a function of $r = a/q$.}
\label{fig:MMs_staffing_levels_optimization}
\end{figure}
These numerical experiments show the power and value of insights gained from asymptotic analysis of queues in the QED regimes, even in small to moderate service systems.
\subsection{Time-varying dimensioning}
So far we have only considered queues in which the model primitives are constant over time.
In practice, though, the arrival rate can fluctuate and depends on the time of day, the day of the week, season or even larger time scales.
It is therefore more realistic to describe these mostly predictable fluctuations through $\lambda(t)$, which represents the instantaneous arrival rate of the arrival process at time $t\in \mathbb{R}$.
The existence of time-varying demand requires a re-evaluation of staffing levels throughout the planning horizon as well.
That is, the number of servers $s(t)$ becomes a function of time, rather then a constant and this clearly asks for an adaption of the dimensioning procedures in Subsections \ref{sec:intro_constraint} and \ref{sec:intro_optimization}.
We explain the concept of time-varying staffing and connection with the QED regime through the time-varying extension of the $M/M/s$ queue denoted by $M_t/M/s_t$ queue, where the subscript $t$ refers to the time-varying nature of both the arrival process and the staffing level.
In this setting, customers arrive according to a non-homogeneous Poisson process with rate function $\lambda(t)$ and customers have exponentially distributed service times with mean $1/\mu$.
Under a constraint satisfaction strategy, we aim to find the staffing function $s(t)$ such that the delay probability is at most $\varepsilon\in(0,1)$ for all $t$.
The analysis and optimization of time-varying many-server queueing systems is known to be intrinsically hard, but many approximation techniques and heuristic methods have been proposed throughout the years \cite{Green1991,Jennings1996}. (nog wat meer toevoegen)
A natural but naive approach is the \textit{pointwise-stationary approximation} (PSA) \cite{Green1991}, which evaluates the system at time $t$ as if it were in steady-state with instantaneous parameters $\lambda=\lambda(t)$, $\mu$ and $s = s(t)$.
Consequently, the analysis and optimization of queues is performed on steady-state performance metrics.
Variants of the PSA method include the \textit{simple-stationary approximation} (SSA) \cite{Green2001}, which uses the long-term (moving) average arrival rate instead of the instantaneous arrival rate, and the \textit{stationary-independent-period-by-period approximation} (SIPP) \cite{Green2001}, which splits the time-horizon into multiple intervals and performs steady-state analysis with the averaged parameters in each of these intervals, among others.
PSA performs well in slowly varying environments with relatively short service times \cite{Green1991,Whitt1991}.
However, when the the model parameters fluctuate significantly, as is often the case in real-life systems, the accuracy of PSA can be poor, as we will see in the numerical experiment at the end of this section.
The main reason why PSA, SSA and SIPP can fail to be accurate is that they neglect that customers actually reside in the system (being in service or waiting in the queue).
In contrast, staffing decisions should be based on the number of customers present in the system rather than the arrival rate at that particular time.
Jennings et al. \cite{Jennings1996} introduced a more sophisticated method that exploits the relation with infinite-server queues.
We explain their idea in the context of the $M_t/M/s_t$ queue.
Let $B_e$ denote the random variable describing the excess service time, which is equal to the original random variable in case $B$ has an exponential distribution.
Then by Eick et al. \cite{Eick1993}, the number of customers in the $M_t/M/\infty$ queue at time $t$ is Poisson distributed with mean
\begin{equation}
\label{eq:offered_load_eick}
R(t) = \mathbb{E}\left[ \lambda(t-B_e)\right] \mathbb{E}[B] = \int_0^\infty \lambda(t-u)\,\mathbb{P}(B>u)\, {\rm d}u = \int_0^\infty \lambda(t-u)\, {\rm e}^{-\mu u} \,{\rm d}u.
\end{equation}
We remark that this result holds for more general service time distributions.
Now, recall that in large systems in the QED regime, the expected delay is negligible (***).
Therefore, under these conditions, the many-server system may be approximated by the infinite-server approximation with offered load as in \eqref{eq:offered_load_eick}.
Accordingly, we can determine the staffing levels $s(t)$ for each $t$ based on steady-state $M/M/s$ measures with offered load $R=R(t)$.
Jennings et al. \cite{Jennings1996} proceed by exploiting the heavy-traffic results of Halfin-Whitt \eqref{eq:halfinwhitt_wait}.
In conjunction with the dimensioning scheme in Subsection \ref{sec:intro_constraint}, the authors propose to set
\begin{equation}
s(t) = \bigg\lceil R(t) + \beta^*(\varepsilon) \sqrt{R(t)} \bigg\rceil,
\end{equation}
where $\beta^*(\varepsilon)$ solves $g(\beta^*(\varepsilon)) = \varepsilon$.
Remark that the number of servers is rounded up to ensure that the achieved delay probability is indeed below $\varepsilon$.
This method was called in \cite{Jennings1996,Massey1994} to \textit{modified-offered-load} (MOL) approximation, and we adopt this term in this thesis.
Let us show that this approximation scheme works.
Figure \ref{fig:intro_example_arrival}(a) shows an arrival rate pattern $\lambda(t)$ and corresponding offered load function $R(t)$ for $\mu=1/2$.
This arrival rate stems for a real-world emergency department.
The resulting staffing level function based on the PSA and MOL approximations with $\varepsilon = 0.3$ are plotted in
Figure \ref{fig:intro_example_arrival}(b).
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 45,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {$\mathbb{P}(W>0)$},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/TimeVarying/arrival_rate.txt};
\addplot[very thick, red] file {./tikz/TimeVarying/offered_load.txt};
\legend{{$\lambda(t)$},$R(t)$}
\end{axis}
\end{tikzpicture}
\caption{Arrival rate and offered load functions.}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 60,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {$\mathbb{P}(W>0)$},
yscale = 0.8,
legend pos = north east]
\addplot[very thick] file {./tikz/TimeVarying/s_PSA.txt};
\addplot[very thick, red] file {./tikz/TimeVarying/s_Jennings.txt};
\legend{PSA,MOL}
\end{axis}
\end{tikzpicture}
\caption{Staffing functions.}
\end{subfigure}
\caption{Time-varying parameters of real-world emergency department.}
\label{fig:intro_example_arrival}
\end{figure}
Through simulation, we evaluate the delay probability as a function of time for $\varepsilon = 0.1,\, 0.3$ and 0.5.
In Figure \ref{fig:intr_timevarying_simulation_results} we see how the PSA approach fails to stabilize the performance of the queue, while the MOL method does stabilize around the target performance.
The erratic nature of the delay probability as a function of time can be explained by rounding effects of the staffing level.
Since this rather simple but elegant technique to address time-varying dimensioning is provably effective, we will adopt the underlying idea of the MOL method in various different settings in this thesis.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, red] file {./tikz/TimeVarying/pdelay_e01_psa.txt};
\addplot[thick, green] file {./tikz/TimeVarying/pdelay_e03_psa.txt};
\addplot[thick, blue] file {./tikz/TimeVarying/pdelay_e05_psa.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{PSA}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.8]
\begin{axis}[
xmin = 0,
xmax = 24,
ymin = 0,
ymax = 1,
axis line style={->},
axis lines = left,
xlabel = {\small $\to t$},
ylabel = {},
yscale = 0.8,
legend style = {at = {(axis cs: 23.5,0.98)},anchor = north east}]
\addplot[thick, red] file {./tikz/TimeVarying/pdelay_e01_mol.txt};
\addplot[thick, green] file {./tikz/TimeVarying/pdelay_e03_mol.txt};
\addplot[thick, blue] file {./tikz/TimeVarying/pdelay_e05_mol.txt};
\legend{{$\varepsilon=0.1$},{$\varepsilon=0.3$},{$\varepsilon=0.5$}}
\end{axis}
\end{tikzpicture}
\caption{MOL}
\end{subfigure}
\caption{Probability of delay under staffing functions obtained through PSA and MOL approximations.}
\label{fig:intr_timevarying_simulation_results}
\end{figure}
\section{Contributions}
\label{sec:intro_beyond}
We have explained how the QED regime can be used to dimension and staff large-scale service systems.
The basic concepts, however, where explained for the relatively simple $M/M/s$ and $M_t/M/s_t$ queue.
Many real-world service systems have essential features that are not captured by these elementary models.
We will now discuss some of these features and address the need to consider more involved models and extend the existing QED theory.
\subsection{Non-classical scaling regimes and pre-limit behavior}
\label{sec:intro_novel_scalings}
The QED theory is centered around the scaling relation $\sqrt{\lambda}(1-\rho_\lambda) \to \beta$, or equivalently $s_\lambda = \lambda + \beta \sqrt{\lambda} + o(\sqrt{\lambda})$, for $\lambda\to\infty$.
It is worthwhile to study how pre-limit behavior of many-server queues is affected when is deviated from this scaling regime.
We introduce a novel family of heavy-traffic scaling regimes, described in terms of the parameter $\eta$ for which we assume that
\begin{equation}
\label{eq:novel_scaling_rule}
\lambda^\eta (1-\rho_\lambda) \to \beta, \qquad \text{as } \lambda\to\infty,\ \beta > 0.
\end{equation}
The parameter $\eta \geq 0$ defines a whole range of possible scaling regimes, including the classic case $\eta = 1/2$, as well as the cases $\eta=0$ and $\eta=1$ investigated in Subsection \ref{sec:intro_many_server_regimes}.
In terms of a capacity sizing rule, the condition \eqref{eq:novel_scaling_rule} is tantamount to $s_\lambda = \lambda +\beta\,\lambda^{1-\eta}$.
This framework thus bridges the gap between the QD and QED regime if $\eta\in(0,1/2)$ and the QED and ED regime if $\eta\in(1/2,1)$, in the $M/M/s$ model.
Similar capacity sizing rules have been considered in \cite{Bassamboo2010,maman} for many-server systems with uncertain arrival rates. Hence, for $\alpha\in(0,1/2)$ the variability hedge is relatively large, so that the regime parameterized by $\alpha\in(0,1/2)$ can be seen as \textit{moderate} heavy traffic: heavy traffic conditions in which the full occupancy is reached more slowly, as a function of $\lambda$, than for classical heavy traffic. See \cite{Chang1996,Puhalskii1998,Puhalskii1999,Atar2012,Atar2014,Atar2015,Atar2016} for more details.
For opposite reasons the range $\eta\in(1/2,\infty)$ corresponds to \textit{extreme} heavy traffic due to a relatively small variability hedge.
We use the insights of Section \ref{sec:intro_QED_regime} and the connection of the QED scaling to the CLT to argue by intuition that the following trichotomy in the qualitative system behavior as $\lambda\to\infty$ holding under scaling \eqref{eq:novel_scaling_rule}.
For $\eta \in (0,1/2)$ the empty-system probability converges to $1$, because the order of the variability hedge $\beta \lambda^{1-\eta}$ is greater than strictly necessary to accommodate the stochastic fluctuations in demand.
Scalings in which $\eta\in(1/2,\infty)$, have adverse behavior, since stochastic fluctuations are not accounted for sufficiently, so that the probability of delay probability converges to 1.
The value $\eta=1/2$ is therefore the tipping point, at which the delay probability converges to a limit between 0 and 1.
Above and below this critical value, the asymptotic performance of the queue overturns to either one of the extremes.
In Chapter 2, we formalize this heuristic argument and conduct an asymptotic analysis to furthermore reveal the rate at which the limit of performance metrics is attained, depending on the parameters $\eta$ and $\beta$ and the system size $\lambda,{s_\lambda}$.
\subsection{Overdispersed arrivals}
\label{sec:intro_overdispersion}
Until now we have assumed queueing systems with perfect knowledge on the model primitives, including the mean demand per time period. For large-scale service systems, the dominant assumption in the literature is that demand arrives according to a non-homogeneous Poisson process, which in practice translates into the assumption that arrival rates are known for each basic time period (second, hour or day).
Although natural and convenient from a mathematical viewpoint, the Poisson assumption often fails to be confirmed in practice. A deterministic arrival rate implies that the demand over any given period is a Poisson random variable, whose variance equals its expectation. A growing number of empirical studies of service sysyems shows that the variance of demand typically exceeds the mean significantly, see \cite{Avramidis:2004, Bassamboo2010, Bassamboo2009, Brown2005, Chen2001, Gans2003, Gurvich2010, koolejongbloed, kimwhitt, maman, Mehrotra2010, Robbins2010, Steckley2009, Zan2012}. The feature that variability is higher than one expects from the Poisson assumption is referred to as \textit{overdispersion}.
Due to its inherent connection with the CLT, the dimensioning rule in \eqref{eq:square_root_staffing rule} relies heavily on the premise that the variance of the number of customers entering the system over a period of time is of the same order as the mean.
Subsequently, when stochastic models do not take into account overdispersion, resulting performance estimates are likely to be overoptimistic. The system then ends up being underprovisioned, which possibly causes severe performance problems, particularly in critical loading.
To deal with overdispersion new models are needed, scaling rules must be adapted, and existing capacity sizing rules need to be modified in order to incorporate a correct hedge against (increased) variability.
Following our findings in Subsection \ref{sec:intro_characteristics}, we propose a capacity allocation rule similar to \eqref{eq:square_root_staffing rule} in which the original variability hedge is replaced by an amount that is proportional to the square-root of the variance of the arrival process.
In Chapter 3, we elaborate on this idea and show how to adapt the scaling of the queueing process appropriately to achieve QED-type behavior in the presence of overdispersion.
\subsection{Finite-size constraints}
The canonical examples in Section \ref{sec:intro_QED_regime} assume an infinite amount of waiting space.
Physical service systems, however, are typically limited in the number of customers that can be held in the system simultaneously.
For instance in a call center, the maximum number of clients in service or queueing is restricted by the number of available trunk lines \cite{Khudyakov2006}, while in the emergency department of a hospital, the number of beds constrains the number of patients that can be admitted \cite{YomTov2010}.
Depending on the practical setting and admission policy, if the maximum capacity, say $n$, is reached, newly arriving customers can either leave the system immediately (blocking), reattempt getting access later (retrials) or queue outside the facility (holding).
In any case, expectations are that the queueing dynamics within the service facility are affected considerably in the presence of such additional capacity constraints.
We illustrate these implications through the $M/M/s/n$ queue, that is, the standard $M/M/s$ queue with additional property that a customer who finds upon arrival $n$ customers already present in the system, is deferred and considered lost.
To avoid trivialities, let $n\geq s$.
Since the expected workload reaching the servers is less than in unconstrained scenario, one expects less congestion and resource utilization.
Consider the $M/M/{s_\lambda}/n_\lambda$ in the QED regime.
That is, we increase $\lambda$ indefinitely and ${s_\lambda}$ scales as ${s_\lambda}=\lambda+\beta\sqrt{\lambda} + o(\sqrt{\lambda})$.
We then ask how $n_\lambda$ should scale along with $\lambda$ and ${s_\lambda}$ to maintain the non-degenerate behavior as seen in Section \ref{sec:intro_QED_regime}.
We provide a heuristic answer.
Let $Q^{({s_\lambda},n_\lambda)}$ and $W^{({s_\lambda},n_\lambda)}$ denote the number of customers in the system and the waiting time in the $M/M/{s_\lambda}/n_\lambda$ queue in steady state.
Note through Theorem \ref{thm:intro_HW_stationary_distribution} that if there were no finite-size constraints, we would have, for $\lambda$ large,
\begin{align}
\mathbb{P}(Q^{({s_\lambda})}\geq n_\lambda)
&= \mathbb{P}\left(\frac{Q^{({s_\lambda})}-{s_\lambda}}{\sqrt{{s_\lambda}}} \geq \frac{n_\lambda-{s_\lambda}}{\sqrt{{s_\lambda}}}\right) \nonumber \\
&\to
\left\{
\begin{array}{ll}
g(\beta), & \text{if }n_\lambda = {s_\lambda} + o({s_\lambda}),\\
g(\beta)\,{\rm e}^{-\beta \gamma}, & \text{if } n_\lambda = {s_\lambda}+\gamma\sqrt{{s_\lambda}} + o(\sqrt{s_\lambda}),\\
0, & \text{if } n_\lambda = {s_\lambda}+\Omega(\sqrt{{s_\lambda}}),
\end{array}
\right.
\end{align}
as $\lambda\to\infty$ for some $\gamma>0$.
Here, the relation $u(\lambda) = o(v(\lambda))$ implies that $u(\lambda)/v(\lambda) \to 0$ as $\lambda\to\infty$ and $u(\lambda) = \Omega(v(\lambda))$ implies $u(\lambda)/v(\lambda) >1$ for $\lambda\to\infty$.
Hence, asymptotically the finite-size effects only play a role if the extra variability hedge of $n_\lambda$ is of order $\sqrt{{s_\lambda}}$ (or equivalently $o(\sqrt{\lambda})$).
Furthermore, if the variability hedge is $o(\sqrt{\lambda})$, then we argue that asymptotically, all customers who do enter the system have zero probability of delay.
That is, asymptotically, we obtain a loss model.
More formally, under the \textit{two-fold scaling rule}
\begin{equation}
\label{eq:intro_twofold_scaling_rule}
\left\{
\begin{array}{ll}
{s_\lambda} = \lambda + \beta\sqrt{\lambda} + o(\sqrt{\lambda}),\\
n_\lambda = {s_\lambda} + \gamma \sqrt{{s_\lambda}} + o(\sqrt{\lambda}),
\end{array}
\right.
\end{equation}
it is not difficult to deduce that, see e.g. \cite{masseywallace},
\begin{equation}
\mathbb{P}(W^{({s_\lambda},n_\lambda)} > 0) \to \left( 1 + \frac{\beta\,\Phi(\beta)}{(1-{\rm e}^{-\beta\gamma})\varphi(\beta)}\right)^{-1}, \quad \text{as } \lambda\to\infty,
\end{equation}
which is strictly smaller than $g(\beta)$ as in \eqref{fig:delay_probs_HW_MMs}, but still bounded away from both 0 and 1.
Furthermore, the buffer size of the queue is $n_\lambda-{s_\lambda} = \gamma\sqrt{{s_\lambda}}$, so that by Little's law, the expected waiting time of an admitted customer is $O(1/\sqrt{{s_\lambda}})$.
Even though resource utilization in the $M/M/{s_\lambda}/n_\lambda$ is less than in the queue with unlimited waiting space, it can easily be shown that $\rho\to 1$ as $\lambda\to\infty$.
Hence, all three key characteristics of the QED regime are carried over to the finite-size setting if adhered to scaling \eqref{eq:intro_twofold_scaling_rule}.
On a process level, adding a capacity constraint translates to adding a reflection barrier to the normalized queue length process $X^{({s_\lambda},n_\lambda)} = (Q^{({s_\lambda},n_\lambda)} -{s_\lambda} ) /\sqrt{{s_\lambda}}$, at $\gamma$, as is illustrated by the sample paths of $X^{{s_\lambda},n_\lambda}$ for three values of $\lambda$ in Figure \ref{fig:sample_paths_MMsn}.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.66]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda5.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=5$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale = 0.66]
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda50.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda = 50$}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[scale = 0.66]
\centering
\begin{axis}[
xmin = 0,
xmax = 5,
ymin = -2.5,
ymax = 2.5,
axis line style={->},
axis x line=middle,
axis y line=left,
y label style={at={(axis cs: -0.2,0)}},
ylabel={$X^{({s_\lambda},n_\lambda)}(t)$},
xlabel style={right},
xscale=1,
yscale=1]
\addplot[] file {./tikz/SamplePaths_MMsn/lambda500.txt};
\addplot[dashed] coordinates{ (0,1) (5.2,1) };
\end{axis}
\end{tikzpicture}
\caption{$\lambda=500$}
\end{subfigure}
\caption{Sample paths of the normalized queue length process $X^{({s_\lambda},n_\lambda)}(t)$ with $\lambda = 5$, $\lambda=50$ and $\lambda=500$ under scaling \eqref{eq:intro_twofold_scaling_rule} with $\beta=0.5$ and $\gamma = 0.5$.}
\label{fig:sample_paths_MMsn}
\end{figure}
It has been shown by \cite{masseywallace} that under \eqref{eq:intro_twofold_scaling_rule}
\begin{equation}
\label{eq:asymptotic_blocking_prob}
\sqrt{{s_\lambda}}\,\mathbb{P}({\rm block}) = \sqrt{{s_\lambda}} \mathbb{P}(Q^{({s_\lambda},n_\lambda}) = n_\lambda) \to f(\beta,\gamma), \quad \text{as } \lambda \to\infty,
\end{equation}
for a non-negative function $f$.
The idea of the two-fold scaling in \eqref{eq:intro_twofold_scaling_rule} can be extended to settings in which the interior is in fact a network of queues, rather that the single-station setting discussed here, see \cite{Khudyakov2006,YomTov2010,Tan2012} for examples of such \textit{semi-open} queueing networks.
In case customers retry of hold upon being initially refused access to the system, the QED analysis becomes much more tricky, and no explicit limiting result are known.
Nevertheless, observe that the volume of blocked arrivals is by \eqref{eq:asymptotic_blocking_prob} of order $\sqrt{\lambda}$, the exact same magnitude as the variability hedge of both ${s_\lambda}$ and $n_\lambda$.
Therefore, retrials and holding customers have a non-negligible effect on the service levels within the facility in the QED regime.
This will be the topic of Chapters 4 and 5.
\subsection{Pre-limit behavior}
The results on queues in the QED regime discussed in Section \ref{sec:intro_QED_regime} are in two ways of an asymptotic nature.
First, the heavy-traffic limits prescribe the the queueing dynamics for $\lambda,{s_\lambda}\to\infty$.
Real-world systems obviously do not experience infinite demand nor have infinite capacity, and hence the heavy-traffic limits only form an approximation for such finite-sized systems.
Although these approximations are qualitatively insightful, these asymptotic analyses do not reveal much about their accuracy with respect to actual performance.
For instance, we would like to know how fast the convergence takes place, and how inaccuracies in asymptotic performance analyses relate to the staffing schemes in pre-limit systems.
To answer such questions, it would be helpful to have an asymptotic estimate for the difference between the (scaled) queueing process and its limiting counterpart, to be able to judge the error made by relying on asymptotic as opposed to actual performance evaluation.
Characterization of the error term gives rise to so-called \textit{corrected diffusion approximations}, \cite{Siegmund1978,Blanchet2006,Janssen2008}.
Correction diffusion approximations are refinements to the heavy-traffic limits and therefore give a more concise description of large but finite load, which makes them a useful tool in the study of large-scale service systems.
We will derive cprrected diffusion approximation in the context of the novel scaling regimes mentioned in Subsection \ref{sec:intro_novel_scalings} in Chapter 2.
Secondly, the bulk of literature on queues is concerned with the performance analysis and optimization of queueing systems, assuming they are in steady-state, which is tantamount to requiring $t\to\infty$.
However, in practice, service systems certainty do not run infinitely long, which renders this assumption questionable.
Validation of the steady-state assumption is related to the \textit{relaxation time} of a queueing process \cite{Abate1987,Abate1988,relaxation,Leeuwaarden2011,Leeuwaarden2012,Gamarnik2013}, which prescribes the time it takes a system starting out of equilibrium to approximate its stationary distribution sufficiently close.
In case the relaxation time is small, stationary performance evaluation is likely to be accurate.
On the contrary, if the relaxation time is the large, transient analysis of the queueing system is required in order to capture realistic behavior.
Subsequently, we can ask ourselves what are the implications of applying staffing principles that are based upon steady-state performance metrics in settings which are inherently transient over the planning period.
We will touch upon this topic in Chapter 6.
\section{Outline of the thesis}
The remainder of this thesis builds upon the ideas behind the QED scaling regime exhibited in this introductory chapter, and is organized as follows.
Chapter 2 is concerned with the analysis of the limiting behavior of queues in case one deviates from the square-root staffing principle as demand grows large.
Using the bulk-service queue together with the many-sources paradigm as a vehicle, we derive corrected diffusion approximations for the performance metrics of pre-limit systems in these alternative scaling regimes.
The work presented in Chapter 2 is based on \cite{Janssen2015}.
In Chapter 3, we also analyze the bulk-service queueing model, but with many correlated sources, so that demand appears to be overdispersed.
As we eluded to in Subsection \ref{sec:intro_overdispersion}, this requires an alternative scaling of the queue length process and associated staffing rule.
This chapter exhibits the ideas of \cite{Mathijsen2016}.
In Chapter 4 we discuss how QED-type behavior prevails in simple settings in which the system size is finite, given appropriate scaling of capacity levels.
More specifically, we show how customer retrials can be incorporated heuristically into the performance analysis of finite-size systems in the QED regime.
The content of this chapter is based on \cite{Leeuwaarden2015} and \cite{Leeuwaarden2016}.
Building upon the insights gained in Chapter 4, we show in Chapter 5 how the heuristic approximation methods carry over to a more complex finite-size queueing system, inspired by delay analysis in a health care facility.
We show how the QED scaling limits for this model offer surprisingly accurate approximations for realistic model parameter in systems of small to moderate size, which moreover gives rise to a means to dimension service systems of such type.
Chapter 5 is based on the ideas of \cite{Leeuwaarden2016a}.
Chapter 6 investigates the validity of a capacity allocation rule based on steady-state performance metrics in practical settings.
Namely, in realistic scenarios, the parameters of a queueing models are typically subject to change over the planning period.
This asks for a more elaborate transient analysis of the queue dynamics, and an adaption of the staffing level.
In this chapter, we present how to do so appropriately in a single-server queueing model facing a L\'evy input process by prescribing a correction to the steady-state optimum, which has square-root form.
This chapter is based on \cite{Mathijsen2016a}.
Chapter 7 presents the analysis of an inventory model with backlogs, perishable goods and consumer impatience.
This model resembles the inventory level of a blood bank, and can be regarded as a shot-noise model with both positive and negative jumps and exponential decay rates above and below zero.
Besides the derivation of the stationary distribution of the inventory level, we show how under appropriate scaling the process converges to an Ornstein-Uhlenbeck process.
The latter allows for a more tractable approximate analysis of the model in case the number of blood deliveries and demand is large.
Chapter 7 is based on \cite{Bar-Lev2015}.
\chapter*{Summary}
\vspace{-1cm}
\noindent\rule[0.5ex]{\linewidth}{2pt}
\section*{Asymptotic dimensioning of stochastic service systems}
Stochastic service systems describe situations in which customers
compete for service from scarce resources. Think of check-in
lines at airports, waiting rooms in hospitals or queues in supermarkets,
where the scarce resource is human manpower. Next
to these traditional settings, resource sharing is also important
in large-scale service systems such as the internet, wireless networks
and cloud computing facilities. In these virtual environments,
geographical conditions do not restrict the system size, paving the way for the emergence of large-scale
resource sharing networks.
This thesis investigates how to design large-scale systems in order to achieve the dual goal of operational efficiency and quality-of-service, by which we mean that the system is highly occupied and hence efficiently utilizes the expensive resources, while at the same time, the level of service, experienced by customers, remains high.
The intrinsic stochastic variability of arrival and service processes is the predominant cause of delays experienced by customers.
Queueing theory and stochastics provide the tools to describe and evaluate congestion in these systems.
An important insight obtained through queueing analysis is the effect of resource pooling for systems with many servers and corresponding economies-of-scale that can be achieved by increasing the scale of the system.
Although classical queueing theory allows for exact evaluation of the performance of queueing systems of moderate size,
exact analysis becomes intractable as demand $R$ and capacity $s$ become large. In those cases, one typically resorts to
asymptotic approximation techniques, such as heavy-traffic diffusion approximations: the analysis of a
sequence of queueing processes, scaled in space, in which the server utilization level approaches 100\%.
The resulting probabilistic limiting processes are easier to analyze. Moreover, the diffusion approximations have direct interpretations in
terms of the original systems and lead to tractable characterizations of their performance.
The heavy-traffic regime that plays a central role in this thesis is the Halfin-Whitt regime, also known as the Quality-and-Efficiency Driven (QED) regime, which dictates that capacity should be equal to the nominal demand plus an additional
variability hedge which is proportional to the square-root of the nominal load, i.e. $s = R + \beta\sqrt{R}$ for some $\beta>0$. The driving force behind this scaling regime is the central limit theorem (CLT).
The rule $s=R+\beta\sqrt{R}$, commonly known as the square-root staffing
principle, has been proved to secure both efficiency (utilization approaches 100\%) and quality-of-service, since the mean waiting time is negligible under this scaling as the system grows large.
Since the QED regime allows coexistence of the
two seemingly conflicting objectives in large-scale service systems, the paradigm has been implemented in a wide variety of
operational settings.
However, the standard QED regime fails to acknowledge features that play a dominant role in practice. This thesis
contributes to the existing literature by identifying these distinctive traits and showing how to account for them in a modified QED framework.
In Chapters 2 \& 3, we study how the limiting behavior of many-server queues is affected when one deviates from the standard square-root staffing principle.
In Chapter 2 we investigate a novel family of scaling regimes, in which the amount of overcapacity $s-R$ is not necessarily of the order $\sqrt{R}$, which gives rise to a novel family of heavy-traffic regimes and corresponding scaling limits.
Continuing our study of alternative scaling regimes, we investigate in Chapter 3 how to adapt the square-root staffing
paradigm in case the system faces demand patterns that are stochastically more volatile than anticipated.
This phenomenon is known as overdispersion and can be caused by e.g.~the existence of correlation between the sources generating demand, or uncertainty about the arrival volume.
In Chapters 4 \& 5, we review a family of queueing models in the QED regime in which the total number of customers that can reside in the system simultaneously is limited.
As a result, customers may be denied access in case they find a full
system on arrival.
This fraction of arrivals may either reattempt later or leave the system directly.
The impact of retrials on scaling rules in the QED regime is the focus of Chapter 4.
Since the volume of initially blocked customers is proportional to $\sqrt{R}$, that is, the same order as the variability hedge in the
staffing rule, retrials are prone to have a non-negligible effect on performance.
We propose a heuristic method for the performance analysis of these types
of queueing models with finite-size restrictions, which is based on a fixed-point equation. As a by-product this yields a two-fold square-root staffing principle, which prescribes a synchronous scaling for both the system capacity and waiting space.
Chapter 5 describes how these ideas can be applied in the context of an emergency department.
Chapter 6 studies a cost minimization problem in a single-server queue with non-stationary input.
The bulk of the queueing literature concerns performance analysis assuming that steady state is reached. However, the validity of this assumption in practice is questionable, for the simple fact that no service system runs infinitely long. Moreover,
system parameters, such as the arrival volume, are likely to change over time.
In this chapter, we characterize the error in performance metrics that follows from this transient nature of queues, and present a correction to the original staffing rule to account for the finite time horizon.
Finally, we analyze in Chapter 7 a specific stochastic service system: an inventory model of a blood bank with backlogs, perishable goods and consumer impatience.
We obtain the stationary distribution of the inventory level, and deduce under appropriate scaling the stochastic process limit in terms of a diffusion process.
This process limit allows for a more tractable approximate analysis of the model in case the number of blood deliveries and demand is large.
|
2,877,628,090,791 | arxiv | \section{Introduction}\label{sec:intro}
Mathematical models of epidemics have been studied extensively for over two centuries, providing insight into the process by which infectious diseases and viruses spread across human or other biological populations~\cite{review,hethcote2000mathematics,anderson1991_virusbook}. Models utilizing health compartments are classical, where each individual in a large population may be susceptible to the virus (S), infected with the virus and able to infect others (I), or removed with permanent immunity through recovery or death (R). Different diseases or viruses are modeled by including different compartments and specifying the possible transition paths between the compartments. Susceptible--Infected--Removed (SIR) and Susceptible--Infected--Susceptible (SIS) frameworks are common, while other compartments can be added to reflect latent or incubation periods for the disease, or otherwise provide a more realistic description of the epidemic process. Moving beyond single populations, network models of meta-populations have also been widely studied, where each node in the network represents a large population and links between nodes represent the potential for the virus to spread between populations~\cite{shuai2013epidemic_lyapunov,review,mei2017epidemics_review,brockmann2013epidemic_propagate}.
Recently, increasing attention has been directed to network models of epidemics involving two or more viruses~\cite{wang2019coevolution}. Depending on the problem scenario, the viruses may be cooperative; being infected with one virus makes an individual more vulnerable to infection from another virus~\cite{newman2013interacting,cai2015avalanche}. Alternatively, viruses may be competitive, whereby being infected with one virus can provide an individual with partial or complete protection from also being infected with another virus. For competitive models centered on the SIR framework, the literature often focuses on comparing the outbreak sizes, by characterizing the final number of removed individuals for the different viruses~\cite{newman2005threshold,poletto2013host,karrer2011competing,miller2013cocirculation,funk2010interacting,ahn2006epidemic}. In contrast, for competitive models utilizing the SIS framework, a central question is whether each virus will persist over time or become extinct~\cite{sahneh2014competitive,liu2019bivirus,castillo1989epidemiological,wei2013competing,watkins2016optimal,santos2015bi,carlos2,yang2017bi,van2014domination,santos2015bivirus_conference,pare2021multi,janson2020networked,wang2012dynamics,granell2013dynamical}. If a particular virus persist while others become extinct, it is said to have won the ``survival-of-the-fittest'' battle, and is also referred to as competitive exclusion~\cite{carlos2}. An important problem is to identify the winning virus for the given conditions. It is also crucial to understand when multiple viruses may persist in the meta-population, resulting in a state of ``coexistence''.
Our work considers one of the most popular models for competing epidemics, namely two viruses in the SIS framework. The two viruses, termed virus~$1$ and virus~$2$, spread across a two-layer meta-population network; each layer represents the possibly distinct contact networks for virus~$1$ and virus~$2$. An individual is infected by virus~$1$, or infected by virus~$2$, or not infected by either of the viruses. The competing nature implies that an individual infected by virus~$1$ cannot be infected by virus~$2$, and vice versa. Within each population, an infected individual that recovers from either virus will do so with no immunity, and then becomes susceptible again to infection from either virus.
Existing literature on bivirus networks has identified a variety of scenarios and conditions that specify the winning virus in the survival-of-the-fittest battle, regardless of the initial state~\cite{sahneh2014competitive,liu2019bivirus,santos2015bi,santos2015bivirus_conference,pare2021multi,janson2020networked}. In this paper, however, we address a key yet relatively unexplored question: \textit{are there conditions on the network such that either virus can prevail in the survival-of-the-fittest battle?} For a three-node network with a specialized tree structure, \cite{carlos2} presented a necessary and sufficient condition for either of the viruses to prevail, depending on the initial state of the network. However, the question has remained unanswered for networks with four or more nodes and general topology structure; the complexity arising from the coupled spreading dynamics of multiple nodes and two viruses makes it nontrivial to extend the approach in \cite{carlos2}.
The main contribution of this paper is to show that for any given finite number of nodes, there exist bivirus networks where at least one network layer has essentially arbitrary structure, for which either virus can survive depending on the network's initial state. Our epidemic spreading process is described by a deterministic continuous-time dynamical system~\cite{carlos2,sahneh2014competitive,santos2015bi,liu2019bivirus}. The main results are based on novel control-theoretic arguments, and begin by stating that under certain necessary and sufficient condition on the infection and recovery rates of the epidemic dynamics, the two equilibria associated with either virus winning the survival battle are both locally exponentially stable, extending the condition presented in~\cite{carlos2} for three-node networks. This ensures that there are initial states for which \textit{either} of the viruses can win the battle. While it is straightforward to check whether a given bivirus network satisfies the condition, the converse problems of existence and design of such networks are significantly more challenging.
Merely demonstrating the existence of bivirus networks with more than three nodes satisfying the aforementioned condition has remained an elusive challenge, because the condition is expressed implicitly as complex nonlinear functions of the infection and recovery rates. Further, there are no simple procedures to design or create the two network layers to satisfy the condition (a numerical example of which would resolve the existence question).
We prove that, given almost any network layer of one virus, there always exists a network layer of the second virus such that the resulting bivirus network satisfies the necessary and sufficient condition we identified. We subsequently operationalize the theoretical results by developing a robust four-step procedure, starting with an essentially arbitrary network layer, to construct the other network layer to satisfy the condition. This allows one to generate bivirus networks that have two possible survival-of-the-fittest outcomes. Numerical examples are presented to demonstrate the procedure, and shows that the bivirus network model can exhibit a rich and complex set of dynamical phenomena, verifying the theoretical findings in \cite{carlos2}, including the presence of an unstable equilibrium where, in each population, both viruses coexist. Our work offers insight into bivirus networks and the complex survival-of-the-fittest battles that unfold over them. \vspace{-6pt}
\section{Bivirus Network Model}\label{sec:model}
We consider a set of $n \geq 2$ nodes, $\mathcal{V} = \{1, \hdots, n\}$. Each node represents a well-mixed population of individuals with a large and constant size~\footnote{A well-mixed population means any two individuals in the population can interact with the same positive probability.}. Following the convention in the literature~\cite{sahneh2014competitive}, two viruses spread over a two-layer contact network represented by the graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}_A, \mathcal{E}_B)$, where $\mathcal{E}_A$ and $\mathcal{E}_B$ are the edge sets that determine the contact spreading network for virus~$1$ and virus~$2$, respectively.
We define $x_i(t) \in [0, 1]$ and $y_i(t) \in [0, 1]$, $t\in \mathbb{R}_+$, as the fraction of individuals in population $i \in \mathcal{V}$ infected with virus~$1$ and virus~$2$, respectively~\footnote{In some bivirus literature, node $i$ is taken to be a single individual, and $x_i$ and $y_i$ are the probabilities that individual $i$ is infected with virus~$1$ and virus~$2$, respectively. While the modeling context differs, the resulting system equations are identical to that in \eqref{eq:bivirus_dynamics}. Thus, the dynamical properties studied and the results presented in this paper are equally applicable to this alternative interpretation of the bivirus network model.}. The epidemic dynamics at node $i \in \mathcal{V}$ are given by
\begin{subequations}\label{eq:bivirus_node}
\begin{align}
\dot x_i(t) & = - x_i(t) + (1-x_i(t)-y_i(t))\sum_{j=1}^n a_{ij} x_j(t) \\
\dot y_i(t) & = - y_i(t) + (1-x_i(t)-y_i(t))\sum_{j=1}^n b_{ij} y_j(t).
\end{align}
\end{subequations}
By defining $x(t) = [x_1(t), \hdots, x_n(t)]^\top$ and $y(t) = [y_1(t), \hdots, y_n(t)]^\top$, we obtain the following bivirus dynamics for the meta-population network:
\begin{subequations}\label{eq:bivirus_dynamics}
\begin{align}
&\dot x(t) = - x(t) + (I-X(t)-Y(t))Ax(t) \label{eq:virus1_dynamics} \\
&\dot y(t) = - y(t) + (I-X(t)-Y(t))By(t), \label{eq:virus2_dynamics}
\end{align}
\end{subequations}
where $X = \diag(x_1, \hdots, x_n)$, and $Y = \diag(y_1, \hdots, y_n)$, and $I$ is the $n$-dimensional identity matrix. The nonnegative matrices $A =\{a_{ij}\}$ and $B = \{b_{ij}\}$ are the adjacency matrices capturing the edge weights for $\mathcal{E}_A$ and $\mathcal{E}_B$, respectively, i.e., $a_{ij} > 0$ and $b_{ij} > 0$ if and only if $(j,i) \in \mathcal E_A$ and $(j,i) \in \mathcal E_B$, respectively, where $(j,i)$ is the directed edge from node $j$ to node $i$. The system in \eqref{eq:bivirus_dynamics} has state variable $(x(t), y(t))$, and is in fact a mean-field approximation of a coupled Markov process that captures the SIS bivirus contagion process~\cite{sahneh2014competitive,santos2014bi,liu2019bivirus}. Note that we have taken the recovery rates for both viruses to be equal to unity for every population for the purposes of clarity. Importantly, this can actually be done without loss of generality when examining the stability properties of equilibria for the bivirus system (see Appendix~\ref{app:unit_rate} for the nontrivial argument). We suppose that both contact layers are strongly connected, which is a standard assumption in the literature~\footnote{A layer is strongly connected if and only if there is a path from any node $i$ to any other node $j$ that traverses just the edges of the layer.}, and is equivalent to both $A$ and $B$ being irreducible matrices~\cite{godsil2001algebraic}.
\begin{figure}
\centering
\subfloat[]{\def0.7\linewidth{0.7\linewidth}
\input{transitions_bivirus.pdf_tex}\label{fig:transitions_bivirus}}
\hfill
\subfloat[]{\def0.7\linewidth{0.8\linewidth}
\input{two_layer.pdf_tex}\label{fig:two_layer}}
\caption{Schematic of the compartment transitions and two-layer infection network. (a) Each individual exists in one of three health states: Susceptible ($S$), Infected with virus 1 ($I$, orange), or Infected with virus $2$, ($I$, purple). Arrows represent possible transition paths between health states. (b) The two-layer network through which the viruses can spread between populations (nodes). Note that the edge sets of the two layers do not need to match, so that virus $1$ can spread between two nodes but virus $2$ cannot, and vice versa. }
\label{fig:epidemic_schematic}
\end{figure}
It is known from \cite[Lemma~8]{liu2019bivirus} that $$\Delta = \{(x, y) \in \mathbb R^n_{\geq 0} \times \mathbb R^n_{\geq 0} : \vect 0_n \leq x + y \leq \vect 1_n \}$$ is a positive invariant set for the bivirus dynamics in \eqref{eq:bivirus_dynamics}, where $\vect 0_n$ and $\vect 1_n$ are the all-$0$ and all-$1$ column vectors of dimension $n$~\footnote{This implies that if the system \eqref{eq:bivirus_dynamics} has initial states $(x(0),y(0)) \in \Delta$, then $(x(t), y(t)) \in \Delta$ for all $t\geq 0$.}.
For two vectors $x = \{x_i\}$ and $y = \{y_i\}$ of the same dimension, the vector inequalities are entry-wise: $x \leq y \Leftrightarrow x_i \leq y_i$ for all~$i$, and $x < y \Leftrightarrow x_i < y_i$ for all~$i$. Given that $x_i$ and $y_i$ represent the fraction of population~$i$ infected with virus~$1$ and virus~$2$, respectively, we naturally consider \eqref{eq:bivirus_dynamics} exclusively in $\Delta$, and the positive invariance of $\Delta$ ensures that $x_i(t)$ and $y_i(t)$ retain their physical meaning in the context of the model for all $t\geq 0$.
With irreducible $A$ and $B$, there can be at most three equilibria in which at least one virus is extinct: the healthy equilibrium $(x = \vect 0_n, y = \vect 0_n)$, and two ``survival-of-the-fittest'' equilibria $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$, where $\vect 0_n < \bar x < \vect 1_n$ and $\vect 0_n < \bar y < \vect 1_n$~\cite{liu2019bivirus}. A necessary and sufficient condition for $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ to exist is that $\rho(A) > 1$ and $\rho(B) > 1$, respectively, where $\rho(\cdot)$ denotes the spectral radius. We assume that the aforementioned spectral radii condition holds throughout the paper. Then, $(\vect 0_n, \vect 0_n)$ is an unstable equilibrium (in fact, a repeller such that all trajectories starting in its neighborhood move away from it)~\cite{liu2019bivirus}.
Moreover, with $\rho(A) > 1$ and $\rho(B) > 1$, $\bar x$ and $\bar y$ correspond to the unique endemic equilibrium of the classical SIS model considering only virus~$1$ and only virus~$2$, respectively~\cite{fall2007epidemiological,Lajmanovich1976,janson2020networked,liu2019bivirus}. These two separate single virus systems are given by
\begin{subequations}\label{eq:single_SIS}
\begin{align}
\dot x(t)& =-x(t)+(I-X(t))Ax(t),\label{eq:v1} \\
\dot y(t)& =-y(t)+(I-Y(t))By(t).\label{eq:v2}
\end{align}
\end{subequations}
See Appendix~\ref{app:prelim} for a brief summary of the single virus system dynamics and additional details on \eqref{eq:bivirus_dynamics}.
{\bf Problem formulation.} In this paper, we study scenarios where either of the viruses can win the survival-of-the-fittest battle. Such scenarios are uncovered by examining the stability properties of the two equilibria $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ for the network dynamics in \eqref{eq:bivirus_dynamics}. If $(\bar x, \vect 0_n)$ is locally exponentially stable, then $\lim_{t\to\infty} (x(t), y(t)) = (\bar x, \vect 0_n)$ for all $(x(0), y(0))$ in some open set $U \in \intr(\Delta)$ with non-zero Lebesgue measure, where $\intr(\cdot)$ denotes the interior of the set. In context, for every initial state in $U$, virus~$1$ will win the survival-of-the-fittest battle. If $(\bar x, \vect 0_n)$ is unstable, then for almost all $(x(0), y(0))$, virus~$1$ will not emerge as the winner of the battle~\footnote{Unstable equilibria can have stable manifolds, but such manifolds always have zero Lebesgue measure.}. The same is true for virus~$2$, if we instead consider $(\vect 0_n, \bar y)$. Thus, this paper will study bivirus networks with conditions on $A$ and $B$ that ensure both $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are locally exponentially stable.
\section{Main Results}\label{sec:results}
The main results of this work are presented in four parts. We first provide the necessary and sufficient condition for both $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ to be locally exponentially stable. Then, we present an existence result which states that given almost any $A$ matrix, a corresponding $B$ matrix can be found to satisfy the required condition, followed by the four-step procedure for finding such a $B$ matrix. Finally, we present numerical examples a two-node and five-node network to highlight the multiple survival-of-the-fittest outcomes.
\subsection{Necessary and sufficient condition for stability}\label{ssec:ns_cond}
The local exponential stability and instability of $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ can be characterized by analysis of the Jacobian of the right hand side of \eqref{eq:bivirus_dynamics}, evaluated at the two equilibria. Let $\bar X = \diag(\bar x_1, \hdots, \bar x_n)$ and $\bar Y = \diag(\bar y_1, \hdots, \bar y_n)$. As detailed in Appendix~\ref{app:stability}, the stability and also instability of $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are determined precisely by the value of $\rho((I-\bar X)B)$ and $\rho((I-\bar Y)A)$, respectively. Specifically,
\begin{enumerate}
\item The equilibrium $(\bar x, \vect 0_n)$ is locally exponentially stable if and only if $\rho((I-\bar X)B)<1$.
\item The equilibrium $({\bf{0}}_n,\bar y)$ is locally exponentially stable if and only if $\rho((I-\bar Y)A)<1$.
\end{enumerate}
Notice that these inequalities involve $\bar X$ and $\bar Y$ which are a nonlinear function of $A$ and $B$, respectively. In other words, $\rho((I-\bar X)B)$ depends on $B$ explicitly and $A$ implicitly, and hence the stability property of $(\bar x, \vect 0_n)$ is tied to the complex interplay between the $A$ and $B$ matrices. The same is true for $(\vect 0_n, \bar y)$. Thus, if one were provided $A$ and $B$, it is straightforward to check if the conditions hold, as there are iterative algorithms to compute $\bar x$ and $\bar y$, e.g.~\cite[Theorem~4.3]{mei2017epidemics_review} or \cite[Theorem~5]{vanMeighem2009_virus}. However, the inverse problems of existence and design are significantly more difficult to address. First, for an arbitrary number of nodes, proving the \textit{existence} of a bivirus system satisfying the above inequalities has remained an elusive challenge one; it is not automatically guaranteed that there exist $A$ and $B$ which satisfy one let alone both of conditions $1$ and $2$ above. Second, no methods have been developed for designing bivirus networks with multiple survival-of-the-fittest outcomes. In the rest of this paper, we comprehensively address both of these issues.
\subsection{Existence of two stable survival equilibria}\label{ssec:main_thm}
We now present the main theoretical result of this paper, showing that given almost any $A$ matrix, one can find a $B$ matrix such that the two inequalities in Section~\ref{ssec:ns_cond} are satisfied.
To begin, consider $A$ with $\rho(A) > 1$. Recall that the single virus system in \eqref{eq:v1} has the unique endemic equilibrium $\vect 0_n < \bar x < \vect 1_n$. This implies that
\begin{equation}\label{eq:equi_A}
[I - (I - \bar X) A]\bar x = \mathbf{0}_n,
\end{equation}
or that $\bar x$ is a positive right eigenvector for the matrix $(I - \bar X) A$ associated with the simple eigenvalue at $1$, according to the Perron--Frobenius Theorem~\footnote{Note that $(I-\bar X)A$ is irreducible and nonnegative precisely because $A$ is irreducible and nonnegative, and $(I-\bar X)$ is positive diagonal.}. Let $B'$ be any other nonnegative and irreducible matrix such that
\begin{equation}
[I - (I - \bar X) B']\bar x = \mathbf{0}_n, \label{eq:equi_B'}
\end{equation}
which similarly implies that $\bar x$ is a positive right eigenvector for $(I - \bar X) B'$ associated to the simple eigenvalue at $1$.
By the Perron--Frobenius Theorem, let $u^\top$ and $v^\top$, respectively, be the positive left eigenvector of $(I - \bar X) A$ and $(I - \bar X) B'$ associated with the simple eigenvalue at $1$, normalized to satisfy $u^\top \bar x = v^\top \bar x = 1$. We require that $u$ and $v$ be linearly independent, and this can be achieved by selecting an appropriate $B'$ when given $A$. The existence of $u^\top$, $v^\top$, their linear independence, and detailed arguments in applying the Perron--Frobenius Theorem are provided in Appendix~\ref{app:main_proof} and Lemma~\ref{lem:useful}. In the sequel, Lemma~\ref{cor:main_paper} is presented, showing a procedure to select $B'$ when given $A$.
The main result follows, with proof in Appendix~\ref{app:main_proof}.
\begin{theorem} \label{thm:doubly_stable_method}
Suppose that $A$ and $B'$ are irreducible nonnegative matrices, with $\rho(A) > 1$ and $\rho(B^\prime) > 1$, that satisfy \eqref{eq:equi_A} and \eqref{eq:equi_B'}.
Suppose further that $u^\top$ and $v^\top$, as defined above, are linearly independent. Then there exists $\delta x \in \mathbb{R}^n$ with arbitrarily small Euclidean norm and satisfying
\begin{align}
u^{\top}[\bar X(I -\bar X)^{-1}] \delta x &> 0 \label{eq:deltax_1} \\
v^{\top}[\bar X(I -\bar X)^{-1}] \delta x &< 0. \label{eq:deltax_2}
\end{align}
Furthermore, there exists $\delta B \in \mathbb{R}^{n\times n}$ such that $B^\prime + \delta B$ is an irreducible nonnegative matrix, and $\delta B$ also satisfies
\begin{equation}\label{eq:delta_B}
\delta B \bar x=[(I- \bar X)^{-2}-B']\delta x.
\end{equation}
Then, with $B := B' + \delta B$, for the bivirus network in \eqref{eq:bivirus_dynamics}, both the survival-of-the-fittest equilibria $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are locally exponentially stable, and $\bar y = \bar x + \delta x+o(\delta)$.
\end{theorem}
Provided $\delta x$ and $\delta B$ are sufficiently small, the resulting bivirus network in \eqref{eq:bivirus_dynamics} is such that either virus~$1$ or virus~$2$ may actually win a survival-of-the-fittest battle, depending whether the initial states $(x(0), y(0))$ are in the region of attraction for $(\bar x, \vect 0_n)$ or $(\vect 0_n, \bar y)$, respectively. Our result does not exclude other limiting behavior, such as converging to a coexistence equilibrium where every population~$i$ has individuals infected with virus~$1$ and virus~$2$. This is because the regions of attraction for $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ together cannot cover all of $\intr(\Delta)$~\cite{chiang2015stability}.
In Theorem~\ref{thm:doubly_stable_method}, we require $A$ and $B^\prime$ to be irreducible nonnegative matrices such that, if they define the infection matrix for two separate single virus systems in \eqref{eq:v1} and \eqref{eq:v2}, then the two systems have the same endemic equilibrium. However, we further require that the left positive eigenvectors $u^\top$ and $v^\top$ be linearly independent. In the next subsection, we present one method of selecting $B'$ and $\delta x$ and $\delta B$ (although there may be other approaches).
\subsection{Systematic construction procedure}\label{ssec:procedure}
A procedure to systematically construct a bivirus network according to Theorem~\ref{thm:doubly_stable_method} is now presented. To begin, we provide a specific method for constructing a suitable $B'$, with proof given in Appendix~\ref{app:main_proof}.
Let $e_i$ be the $i$-th basis vector, with $1$ in the $i$-th entry and $0$ elsewhere.
\begin{lemma}\label{cor:main_paper}
Let $A$ be an irreducible nonnegative matrix fulfilling \eqref{eq:equi_A} for some $\bar{x}$ such that $\mathbf{1}_n > \bar x > \mathbf{0}_n$. For a fixed but arbitrary $i \in \mathcal{V}$, let $z^\top \neq \vect 0_n$ be chosen to satisfy $z^\top \bar x = 0$ and the $j$-th entry $z_j < 0$ only if $a_{ij} > 0$. Then, there exists a sufficiently small $\epsilon$ such that $B' := A + \epsilon e_i z^\top$ is an irreducible nonnegative matrix. Moreover, $B'$ fulfills the conditions in the hypothesis of Theorem~\ref{thm:doubly_stable_method}: $\rho(B') > 1$, \eqref{eq:equi_B'} is satisfied, and $u$ and $v$ are linearly independent.
\end{lemma}
{\bf Step 1.} Consistent with Theorem~\ref{thm:doubly_stable_method}, we begin by assuming that we are given an irreducible nonnegative matrix $A$ with spectral radius greater than 1.
Because we are given $A$, we are therefore also given $\bar x$ (which can be computed using iterative algorithms, see e.g.~\cite[Theorem~4.3]{mei2017epidemics_review} or \cite[Theorem~5]{vanMeighem2009_virus}). Construct the matrix $B^\prime = A + \epsilon e_i z^\top$ according to Lemma~\ref{cor:main_paper}.
{\bf Step 2.} With $u^\top$ and $v^\top$ as defined in Section~\ref{ssec:main_thm}, set $F = (I- \bar X)^{-2}-B'$ and $\tilde u^\top = u^{\top}\bar X(I -\bar X)^{-1}F^{-1}$ and $\tilde v^\top = v^{\top}\bar X(I -\bar X)^{-1}F^{-1}$. Note that $F$ is invertible and $F^{-1}$ is a positive matrix, as detailed in Appendix~\ref{app:main_proof}. Select two integers $j$ and $k$ for which $\tilde u_j/\tilde u_k > \tilde v_j/\tilde v_k$. This is possible since $u^\top$ and $v^\top$ are linearly independent. Select $\alpha > 0$ to satisfy
\begin{equation*}
\frac{\alpha \tilde u_j}{\tilde u_k} > 1 > \frac{\alpha \tilde v_j}{\tilde v_k},
\end{equation*}
noting that such an $\alpha$ can always be found. Identify one positive entry in each of the $j$th row and $k$th row of $B'$, say $b_{jp}^\prime$ and $b_{kq}^\prime$. Set $\beta \in (0, b_{kq}^\prime \bar x_q)$. Finally, define the vector $s \in \mathbb R^n$ which has zeros in every entry except $s_k = -\beta$ and $s_j = \alpha\beta$. Compute $\delta x = F^{-1}s$.
{\bf Step 3.} To obtain $\delta B$, set all of its entries to be equal to zero, except that $\delta b_{kq}^\prime = -\beta/\bar x_q$ and $\delta b_{jp}^\prime = \alpha \beta/\bar x_p$. Then, set $B = B^\prime + \delta B$.
{\bf Step 4.} (If necessary). Since the theoretical analysis uses arguments centered on perturbation methods (see Appendix~\ref{app:main_proof}), the $\delta x$ and $\delta B$ must be sufficiently small. If the selected values of $\alpha$ and $\beta$ do not satisfy the necessary and sufficient condition outlined in Section~\ref{ssec:ns_cond}, one can iterate the three steps and reduce the magnitude of $\beta$ until the resulting $B$ meets the condition.
If $A$ is a positive matrix, corresponding to an all-to-all connected virus~$1$ layer, then a more straightforward approach can be taken. We set $B'= A + \epsilon \vect 1_n z^\top$, with $z^\top \bar x = 0$ and $\epsilon$ sufficiently small to guarantee $B'$ is a positive matrix. Then, solve \eqref{eq:deltax_1} and \eqref{eq:deltax_2} for $\delta x$ using standard linear programming methods. Next, compute a solution $\delta B$ for \eqref{eq:delta_B} and apply a scaling constant to decrease the entries of $\delta B$ to ensure that $B = B'+\delta B$ remains a positive matrix. The challenge occurs when $A$ and $B'$ are not positive matrices, because any $\delta B$ satisfying \eqref{eq:delta_B} must have both positive and negative entries. This can be problematic if we obtain a solution $\delta B$ that has a negative entry where $B'$ has a zero entry but we also require $B$ to be nonnegative irreducible. The above four-step procedure resolves this issue, by producing a $\delta B$ whose single negative entry is in the same position corresponding to a positive entry in $B'$, and the former is smaller in magnitude than the latter.
In order to apply the four-step construction procedure, one requires knowledge of the infection matrix $A$ and the endemic equilibrium $\bar x$ associated with the single virus system \eqref{eq:v1}. It is important to stress that only knowledge of the single virus system is needed, as opposed to knowledge of any bivirus system. From knowledge of $A$ and $\bar x$, one would construct a suitable $B^\prime$, and subsequently compute $\delta x$ and $\delta B$ as necessary.
\begin{figure*} [!htb]
\subfloat[]{\includegraphics[width= 0.3\textwidth]{different_initconds_3_1.pdf}\label{fig:2a}}
\hfill
\subfloat[]{\includegraphics[width= 0.3\textwidth]{different_initconds_virus1wins.pdf}\label{fig:2b}}
\hfill
\subfloat[]{\includegraphics[width= 0.3\textwidth]{different_initconds_virus2wins.pdf}\label{fig:2c}}
\hfill
\\
\subfloat[]{\includegraphics[width= 0.3\textwidth]{different_turnoverrates.pdf}\label{fig:2d}}
\hfill
\subfloat[]{\includegraphics[width= 0.34\textwidth]{winner_graph.pdf}\label{fig:2e}}
\hfill
\subfloat[]{\includegraphics[width= 0.34\textwidth]{winner_graph_virus1faster.pdf}\label{fig:2f}}
\caption{The dynamics of the two-node case study of \eqref{eq:bivirus_dynamics}. In (a), the trajectories $(x_1(t), y_1(t))$ are shown for two different initial states (blue and red); virus 1 and virus 2 win the survival-of-the-fittest battle in the blue and red trajectories, respectively. In (b) and (c), the time evolution of $(x(t), y(t))$ is shown for the blue and red initial states in (a), respectively. In (d), we show the trajectories $(x_1(t), y_1(t))$ for virus $1$ and virus $2$ of the same speed (green, $\gamma = 1$) and virus 1 that is $1.2$ times faster relative to virus 2 (purple, $\gamma = 1.2$), for different initial states. The winning virus for different initial states is recorded when (e) virus 1 and virus 2 are the same speed and (f) when virus 1 is faster than virus 2, with $\gamma = 1.2$. Note the line where the boundaries of the two regions meet forms part of the stable manifold of the unstable coexistence equilibrium. } \label{fig:2}
\vspace{15pt}
\subfloat[]{\includegraphics[width= 0.4\textwidth]{PRE_5node_1_v2.pdf}\label{fig:irred_1wins}}
\hspace{33pt}
\subfloat[]{\includegraphics[width= 0.4\textwidth]{PRE_5node_2_v2.pdf}\label{fig:irred_2wins}}
\caption{The dynamics of the $n = 5$ node example (note the logarithmic scale of time, $t$, along the horizontal axis). In (a) and (b), the time evolution of $(x(t), y(t))$ shows two different initial states yielding two different survival-of-the-fittest outcomes.} \label{fig:irred}
\end{figure*}
\subsection{Case studies}
We now present two case studies to illustrate the procedure and the diverse limiting behavior that can be observed, including different survival-of-the-fittest outcomes. For the sake of clarity and simplicity, we first present a detailed case study of a network with two nodes, in Fig.~\ref{fig:2} and then another case study of a network with five nodes, in Fig.~\ref{fig:irred}~\footnote{Full code found at \url{https://github.com/lepamacka/bivirus_code} and \url{https://github.com/mengbin-ye/bivirus}.}.
For the the two-node example, the particular $A$ and $B$ matrices are reported in Appendix~\ref{app:2node}, and they give $\bar x = [0.8077, 0.8077]^\top$ and $\bar y = [0.7801, 0.8699]^\top$. In this particular example, we can also compute that there is a unique coexistence equilibrium, $(\tilde x, \tilde y)$, with $\tilde x = [0.5467, 0.4180]^\top$ and $\tilde y = [0.2418, 0.4101]^\top$, see Appendix~\ref{app:2node}.
Fig.~\ref{fig:2a}, shows that for two initial states in $\intr(\Delta)$ that are close together, different survival-of-the-fittest outcomes occur, with either virus 1 (blue) or virus 2 (red) winning. Figs.~\ref{fig:2b} and \ref{fig:2c} show the time evolution of the blue and red trajectories in Fig.~\ref{fig:2a}, respectively. It is notable that there is a rapid initial transient that takes the system to a point arbitrarily close to a curve that connects $(\bar x, \vect 0_n)$ to $(\vect 0_n, \bar y)$ and passes through the unstable coexistence equilibrium, followed by a slower convergence to the two survival equilibria.
Separating the time-scales of the two viruses can change the shape of the regions of attraction for $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$, but the local exponential stability property is unchanged, and thus both regions will always have non-zero Lebesgue measure. Time-scale separation can be easily achieved by introducing a parameter $\gamma > 0$ and modifying \eqref{eq:virus1_dynamics} to be
\begin{equation}\label{eq:virus1_faster}
\dot x(t) = \gamma\Big(- x(t) + (I-X(t)-Y(t))Ax(t)\Big).
\end{equation}
Adjusting $\gamma$ allows study of scenarios of interest where virus~1 has much faster or slower dynamics relative to virus 2. Fig.~\ref{fig:2d} shows the trajectories for the two viruses having the same speed (green) and virus 1 being faster than virus 2 (purple, $\gamma = 1.2)$. Thus, from the same initial condition, the virus that survives may depend on the relative speeds of the two viruses, but there are always two nontrivial regions of attraction for the two stable equilibria.
Fig.~\ref{fig:2e} indicates the initial states that lead to virus~1 or virus~2 winning the survival-of-the-fittest battle, for initial states constrained to satisfy $x_i(0)+y_i(0) = 0.01$ for $i = 1,2$. See Appendix~\ref{app:2node} for details on the simulation setup. Fig.~\ref{fig:2f} maps out the same region, but with virus~$1$ having the faster dynamics relative to virus~$2$. Consistent with the above, we see that adjusting the relative dynamics changes the shape of the region for which virus~$1$ or virus~$2$ wins the survival battle. It appears that as the dynamics of one virus becomes faster, the region of attraction increases in size, which accords with intuition. However, there are always initial states for either virus to win the battle and be the sole survivor.
Note that in this case study, the coexistence equilibrium $(\tilde x, \tilde y)$ is in fact unstable, as the associated Jacobian has one eigenvalue with positive real part.
It is known that the region of attraction for an equilibrium forms an open set~\cite{chiang2015stability}, and there are two locally stable equilibria (the two survival-of-the-fittest equilibria) and two unstable equilibria (the healthy state and the coexistence equilibrium). Thus, the boundaries of the regions of attraction for $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ do not belong to the region of attraction of either, and if the system is initialized at a common point of the two boundaries, then necessarily the trajectories do not converge to either stable equilibrium. In fact, the common boundary of the two regions of attraction forms the stable manifold of $(\tilde x, \tilde y)$. Initial states on this manifold would lead to convergence to $(\tilde x, \tilde y)$, providing a third outcome of the battle, where nodes~$1$ and $2$ each have individuals infected with both virus~$1$ and virus~$2$, different from an outcome where only one virus wins the survival battle. Since stable manifolds of unstable equilibria have zero Lebesgue measure~\cite{chiang2015stability}, this third outcome is unlikely to be encountered in practice, but not impossible.
We conclude with another case study with $n = 5$ nodes, with details of the setup reported in Appendix~\ref{app:3node}. For two different sets of initial states, Fig.~\ref{fig:irred_1wins} and Fig.~\ref{fig:irred_2wins} show that virus~$1$ and virus~$2$ win the survival-of-the-fittest battle, respectively. Notice the logarithmic scale of the $x$-axis.
\subsection{Relation to existing literature}
The contributions in this paper are illustrated in Fig.~\ref{fig:cool_figure}. Assuming networks endowed with an arbitrary topology, for survival-of-the-fittest battles, Regions~I--III (colored in blue) highlight the outcomes recorded in the existing literature, whereas Region~IV (colored in red) depicts the novel outcome presented in this paper. The spectral radii of $(I-\bar X)B$ and $(I-\bar Y)A$, denoted by $\rho_1$ and $\rho_2$, respectively, are depicted on the $y$ and $x$-axis. The local stability properties of $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are precisely determined by $\rho_1$ and $\rho_2$, respectively, see Appendix~\ref{app:stability} for details.
The existing literature has identified sufficient conditions on $A$ and $B$ that result in the bivirus network exhibiting the behavior depicted in Region~I or III, where virus~$2$ and virus~$1$, respectively, win the survival-of-the-fittest battle, see ~\cite{santos2015bi,santos2015bivirus_conference,janson2020networked,sahneh2014competitive,liu2019bivirus,carlos2}.
In fact, global stability of a specific survival-of-the-fittest equilibrium is secured in \cite{santos2015bi,santos2015bivirus_conference}. Sufficient conditions can also be identified for the bivirus network to be in Region II, where both survival equilibria are unstable~\cite{sahneh2014competitive,janson2020networked,carlos2}. For the particular case of a $3$-node network endowed with a tree topology that disallows self-loops, a necessary and sufficient condition for local exponential stability of each of the survival-of-the-fittest equilibria, corresponding to Region~IV, was identified in \cite{carlos2}. However, to the best of our knowledge, the literature has not considered bivirus networks in Region~IV for networks with arbitrary but finite number of nodes endowed with an arbitrary topology on either layer (and not necessarily tree).
The present paper addresses this gap by
identifying a necessary and sufficient condition for
a bivirus system to be in Region~IV, and then presents a procedure to obtain such a system.
\begin{figure}
\centering
\def0.7\linewidth{0.7\linewidth}
\input{cool_figure.pdf_tex}
\caption{Existing results (blue shaded regions) and new phenomena reported in this paper (red shaded region), characterized by $\rho_1$ and $\rho_2$, which are the spectral radii of $(I-\bar X)B$ and $(I-\bar Y)A$, respectively.
In Region~I, $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are unstable and locally exponentially stable, respectively. In Region~II, $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are both unstable. In Region~III, $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are locally exponentially stable and unstable, respectively. In Region IV, $(\bar x, \vect 0_n)$ and $(\vect 0_n, \bar y)$ are \textit{both} locally exponentially stable. \vspace{-11pt}
}
\label{fig:cool_figure}
\end{figure}
\section{Conclusion}
In summary, we explored a fundamental problem for competing epidemic spreading across a meta-population, using the deterministic SIS bivirus network model. We presented a necessary and sufficient condition on the two infection matrices $A$ and $B$, defining the network layers of virus~$1$ and virus~$2$, such that the winner of a survival-of-the-fittest battle depends nontrivially on the initial state of the bivirus network. We then provided a rigorous argument which demonstrated that for almost any $A$, there exists a $B$ such that the pair of matrices satisfied the aforementioned necessary and sufficient condition. Finally, we presented a systematic procedure to generate such a bivirus network, and studied two numerical examples. This paper significantly expands the known dynamical phenomena of the bivirus model, but should be considered as just a first important step for the epidemic modeling community to explore the diverse new outcomes that are now unlocked for competing epidemic spreading models. A key direction of our future work is to investigate models of three or more competing viruses (multivirus networks)~\cite{janson2020networked}, and to explore how the regions of attraction might change as a function of the relative speeds of the different virus dynamics. Our work did not provide theoretical conclusions on the presence or stability properties of coexistence equilibria, although our example demonstrated a unique unstable coexistence equilibrium; this is a current focus.
|
2,877,628,090,792 | arxiv | \section{Introduction} This paper is a continuation of the investigations of
the problem of restricted words enumeration from the author's previous papers~\cite{ja1,ja2,ja3},
where two functions $f_m$ and $c_m$ were defined as follows. For an initial arithmetic function $f_0$, the function $f_m,(m\geq 1)$ is the $m^\text{th}$ invert transform of $f_0$. The function $c_m(n,k)$ was defined as
\begin{equation}\label{cmnk}
c_m(n,k)=\sum_{i_1+i_2+\cdots+i_k=n}f_{m-1}(i_1)\cdot f_{m-1}(i_2)\cdots f_{m-1}(i_k),\end{equation}
where the sum is over positive $i_1,i_2,\ldots,i_k$.
For $m\geq 1$, the following formula holds:
\begin{equation}\label{suma1}f_m(n)=\sum_{k=1}^nc_m(n,k).\end{equation}
The functions $f_m$ and $c_m$ depend only on the initial function $f_0$, and are related to the enumeration of weighted compositions. Namely, if weights are $\{f_{m-1},f_{m-2}(2),\ldots\}$, then $f_m(n)$ is the number of all weighted compositions of $n$, and $c_m(n,k)$ is the number of weighted compositions of $n$ into $k$ parts.
In Janji\'c~\cite{ja1,ja2,ja3}, weighted compositions were related to
restricted words over a finite alphabet. For a given initial function $f_0$, we investigated restricted words counted by $f_m$ and $c_m$. In this paper, we reverse the problem. Namely, for a particular type of restricted words, we first find the initial function $f_0$ which count such words. We then derive formulas for $f_m$ and $c_m$ and give its combinatorial meanings in terms of restricted words.
We restate~\cite[Propositions 12]{ja3}, which will be used frequently in the paper.
\begin{proposition}\label{alf}
Assume that $f_{0}(1)=1$ and $m>1$. Assume next that, for $n\geq 1$, we have $f_{m-1}(n)$
words of length $n-1$ over a finite alphabet $\alpha$. Let $x$ be a letter which is not in $\alpha$. Then, $c_m(n,k)$ is the number of words of length $n-1$ over the alphabet $\alpha\cup\{x\}$ in which $x$ appears exactly $k-1$ times.
\end{proposition}
We also restate the result in~\cite[Proposition 6]{ja3}.
The following formula holds:
\begin{equation}\label{cm1}c_m(n,k)=\sum_{i=k}^n(m-1)^{i-k}{i-1\choose k-1}c_1(n,i),\;(1\leq k\leq n).\end{equation}
We consider the following five types of restricted words:
\begin{itemize}
\item[1.]
Words over the alphabet $\{0,1,\ldots,a-1,\ldots,m+a-1\}$,
such that no two adjacent letters from $\{0,1,\ldots,a-1\}$ are the same.
\item[2.]
Words over the alphabet
$\{0,1,\ldots,a-1,\ldots,a+m-1\}$ such that letters $0,1,\ldots,a-1$
avoid a run of odd length.
\item[3.] Words over the alphabet $\{0,1,\ldots,b-1,\ldots,m+a-1\}$
avoiding subwords of the form $0i,(i=1,\ldots,b)$.
\item[4.]
Words over the alphabet $\{0,1,\ldots,m+1\}$ such that $0$ and $1$ appear only as subwords of the form $1i$, where $i$ is a run of zeros of length at least $1$.
\item[5.]
Words over the alphabet $\{0,1,\ldots,m+1\}$ in which $0$ appears only in a run of even length, and $1$ appears only in a run the length of which is divisible by $3$.
\end{itemize}
We note that the initial function $f_0$ is defined by a linear homogenous recurrence in all cases.
\section{Case 1} To solve the problem posed in Case 1, we consider the following linear recurrence:
\begin{equation*}f_0(1)=1,f_0(a)=a,f_0(n+2)=(a-1)f_0(n+1),(n\geq 1),
\end{equation*}
where $a>0$.
It is easy to see that
\begin{equation*}f_0(n)=a(a-1)^{n-2},(n\geq 2).\end{equation*}
\begin{remark}
This formula appears in Birmajer at al.~\cite[Example 17]{bir}. Also, the case $a=1$ is considered in~\cite[Example 18]{ja3}.
\end{remark}
The function $f_0$ has the following combinatorial interpretation:
\begin{proposition} The number $f_0(n)$ is the number of words of length $n-1$ over the alphabet $\{0,1,\ldots,a-1\}$ such that no two adjacent letters are the same.
\end{proposition}
\begin{proof} We have $f_0(1)=1$, since only the empty word has length $0$.
Also, $f_0(2)=a$, since a word of length $1$ may consist of an arbitrary letter.
To obtain a word of length $n+2$, for $n>0$, we need to insert $a-1$ letters in front of each word of length $n+1$.
\end{proof}
As an immediate consequence of Janji\'c~\cite[Corollary 9]{ja1}, we obtain
\begin{corollary}\label{co1}
For $m\geq 0$, the following recurrence holds:
\begin{gather*}f_m(1)=1,f_m(2)=m+a,\\f_m(n+2)=(m+a-1)f_m(n+1)+mf_m(n),(n\geq 1).\end{gather*}
\end{corollary}
We next prove that $f_m$ counts the desired words.
\begin{proposition}
The number $f_m(n)$ is the number of words of length $n-1$ over
the alphabet $\{0,1,\ldots,a-1,a,\ldots,m+a-1\}$,
such that no two adjacent letters from $\{0,1,\ldots,a-1\}$ are the same.
\end{proposition}
\begin{proof}
We have $f_m(1)=1$, since only the empty word has length $0$. Also, $f_m(2)=m+a$
since a word of length $1$ may consist of any letter of the alphabet.
Assume that $n>2$. Consider a word of length $n+1$. In front of such a word, we insert a letter different from the first letter of the word. In this way, we obtain all words of length $n+2$ beginning with two different letters. The remaining words must begin with two same letters. Since there are $mf_m(n)$ such words, the statement is true.
\end{proof}
\begin{remark} The continued fraction $[f_0(1);f_0(2),f_0(3),\ldots]$ equals $\sqrt 2$. Also,
the sequence $f_1(1),f_1(2),\ldots,f_1(n)$ is the numerator of the $n$th convergent of $\sqrt 2$.
\end{remark}
Since $f_m(1)=1$, we may apply Proposition \ref{alf} to obtain
\begin{corollary} The number $c_m(n,k)$ is the number of words of length $n-1$ over
$\{0,1,\ldots,a-1,\ldots,m+a-1\}$ in which $k-1$ letters equal $m+a-1$, and no two letters from $\{0,1,\ldots,a-1\}$ are identical.
\end{corollary}
We next derive an explicit formula for $c_1(n,k)$.
\begin{proposition} We have
\begin{equation}\label{c1nk}c_1(n,n)=1,c_1(n,k)=\sum_{i=0}^{k-1}{k\choose i}{n-k-1\choose k-i-1}a^{k-i}(a-1)^{n-2k+i},(k<n).
\end{equation}
\end{proposition}
\begin{proof}
From (\ref{cmnk}), we firstly obtain $c_1(n,n)=1$. Assume that $k<n$. Since at most $k-1$ of $i_t,(t=1,2,\ldots,k)$ may equal $1$, then
\begin{gather*}c_1(n,k)=\sum_{i=0}^{k-1}{k\choose i}\sum_{j_1+j_2+\cdots+j_{k-i}=n-i}f(j_1)f(j_2)\cdots f(j_{k-i})\\
=\sum_{i=0}^{k-1}{k\choose i}a^{k-i}(a-1)^{n-2j+i}\sum_{j_1+j_2+\cdots+j_{k-i}=n-i}1\\
=\sum_{i=0}^{k-1}a^{k-i}(a-1)^{n-2k+i}{k\choose i}{n-k-1\choose k-i-1}.
\end{gather*}
\end{proof}
\begin{remark} Note that, in (\ref{c1nk}), terms in which $i<2k-n$ would equal zero.
\end{remark}
To obtain an explicit formula for $c_m(n,k)$, we use (\ref{cm1}).
We first extract the term for $i=n$ to obtain
\begin{equation*}c_m(n,k)=m^{n-k}{n-1\choose k-1}+\sum_{i=k}^{n-1}{i-1\choose k-1}c_1(n,i).\end{equation*}
It follows that
\begin{equation*}c_m(n,k)=m^{n-k}{n-1\choose k-1}+\sum_{i=k}^{n-1}\sum_{j=0}^{i-1}(m-1)^{i-k}a^{i-j}(a-1)^{n-2i+j}{i-1\choose k-1}{n-i-1\choose i-j-1}{i\choose j}.
\end{equation*}
Using (\ref{suma1}), we obtain the following formula for $f_m(n)$:
\begin{gather*}
f_m(n)=m^{n-1}+\sum_{k=1}^{n-1}\sum_{i=k}^{n-1}\sum_{j=0}^{i-1}(m-1)^{i-k}a^{i-j}(a-1)^{n-2i+j}{i-1\choose k-1}{i\choose j}{n-i-1\choose i-j-1}.
\end{gather*}
\section{Case 2}
Let $a$ be a positive integer. Define $f_0$ as follows:
\begin{equation}\label{r1}f_0(1)=1,f_0(2)=0, f_0(n+2)=af_0(n),(n\geq
1).\end{equation}
We firstly describe the restricted words counted by $f_0$.
\begin{proposition}\label{ll1} For $a>0$, the number $f_0(n)$ is the number of words of length $n-1$ over the alphabet $\{0,1,\ldots,a-1\}$ in which there are no runs of odd length.
\end{proposition}
\begin{proof}
Let $d(n)$ denote the number of words of length $n$, which we wish to
count. Firstly, $d(0)=1$ since only the empty word has length $0$. Next, $d(1)=0$ as there are no runs of length $1$.
Assume that $n>2$. A word of length $n$ must begin with two identical letters. Hence, there are $ad(n-2)$ such words.
We conclude that the following recurrence holds:
\[d(0)=1,d(1)=0, d(n)=ad(n-2),(n\geq 2),\]
which yields $d(n-1)=f_0(n),(n\geq 1)$.
\end{proof}
From (\ref{r1}), we easily obtain the following explicit formula for $f_0$:
\begin{equation*}
f_0(n)=\begin{cases}0,&\text{ if $n=2t$};\\a^t,&\text{ if $n=2t+1$}.
\end{cases}
\end{equation*}
\begin{corollary}\label{co2}
For $m\geq 0$, the following recurrence holds:
\begin{gather*}f_m(1)=1,f_m(2)=m,\\f_m(n+2)=mf_m(n+1)+af_m(n),(n\geq 1).\end{gather*}
\end{corollary}
\begin{proof}
The proof is a consequence of~\cite[Corollary 9]{ja1}.
\end{proof}
\begin{proposition}\label{pp5}
The number $f_m(n)$ is the number of words of length $n-1$ over the alphabet
$\{0,1,\ldots,a-1,\ldots,a+m-1\}$, such that letters $0,1,\ldots,a-1$
avoid runs of odd length.
\end{proposition}
\begin{proof}
We let $d(n)$ denote the number of desired words of length $n-1$. It is clear that $d(0)=1$ and $d(1)=m$.
A word of length $n+1$ may begin with a letter from $\{a,a+1,\ldots,a+m-1\}$. There are $md(n)$ such word.
If a word begins with a letter from $\{0,1,\ldots,a-1\}$, it must be followed by the same letter.
Hence, there are $ad(n-1)$ such words.
We conclude that $d(n)=f_m(n+1)$.
\end{proof}
Some well-known classes of numbers satisfy the recurrence from Corollary \ref{co2}. We give the appropriate combinatorial meaning for some of them.
\begin{enumerate}
\item The case $a=1,m=1$ concerns the Fibonacci numbers.
The number of binary words of length $n-1$ in which $0$ avoids a run of odd length is
$F_n$.
\item The case $a=1,m=2$ concerns the Pell numbers $P_n$. The number of ternary words of length $n-1$
in which $0$ avoids runs of odd length is $P_n$.
\item The case $a=2,m=1$ concerns the Jacobhstal numbers $J_n$. The number of ternary words of
length $n-1$ in which $0$ and $1$ avoid runs of odd length is $J_n$.
\end{enumerate}
From the combinatorial interpretation, we easily derive an explicit formula for $f_m(n)$.
\begin{proposition}\label{lrj} We have
\[f_m(n)=\sum_{j=0}^{\lfloor\frac {n-1}{2}\rfloor}m^{n-2j-1}a^{j}{n-1-j\choose j}.\]
\end{proposition}
\begin{proof}According to Proposition \ref{pp5}, in a word counted by $f_m$, the letters from $\{0,1,\ldots,a-1\}$ may appear only in pairs. There are $a$ such pairs.
We may choose $j,(0\leq j\leq \lfloor\frac{n-1}{2}\rfloor)$ pairs in a word of length $n-1$. These $j$ pairs may be chosen in ${n-j-1\choose j}$ ways. When we have chosen $j$ pairs from $\{0,1,\ldots,a-1\}$, the remaining $n-1-2j$ letters are from $\{a,a+1,\ldots,a+m-1\}$, which are $m$ in number.
\end{proof}
As a consequence, we obtain the following similar explicit formulas for the Fibonacci, Pell and Jacobsthal numbers:
\begin{gather*}F_n=\sum_{j=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}{n-j-1\choose j}, P_n=\sum_{j=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}2^{n-2j-1}{n-j-1\choose j},\\J_n=\sum_{j=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}2^{j}{n-j-1\choose j}
.\end{gather*}
\begin{corollary}
The number $c_m(n,k)$ is the number of words of length $n-1$ over the alphabet $\{0,1,\ldots,a-1,\ldots,a+m-1\}$ in which the letter $a+m-1$ appears $k-1$ times, and letters from $\{0,1,\ldots,a-1\}$ avoid runs of odd length.
\end{corollary}
\begin{proof} The proof follows from Proposition \ref{alf}.
\end{proof}
We now derive an explicit formula for $c_1(n,k)$.
\begin{proposition}\label{plr} The following equation holds:
\begin{equation*}c_1(n,k)=\begin{cases}a^{\frac{n-k}{2}}{\frac{n+k}{2}-1\choose k-1},&\text{ if $n-k$ is even};\\0,&\text{ if $n-k$ is odd }.\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}
Each term in (\ref{cmnk}) in which $i_t$ is even equals zero. Hence, (\ref{cmnk}) becomes
\begin{gather*}c_1(n,k)=\sum_{2j_1+1+2j_2+1+\cdots+2j_k+1=n}a^{j_1}\cdot a^{j_2}\cdots a^{j_k}\\=a^{\frac{n-k}{2}}\sum_{j_1+j_2+\cdots+j_k=\frac{n+k}{2}}1=a^{\frac{n-k}{2}}{\frac{n+k}{2}-1\choose k-1}.
\end{gather*}
\end{proof}
As a consequence of (\ref{suma1}), we obtain the following explicit formulas for the Fibonacci and Jacobsthal numbers:
\begin{gather*}F_{2n}=\sum_{k=1}^n{n+k-1\choose n-k},F_{2n-1}=\sum_{k=1}^n{n+k-2\choose n-k},\\
J_{2n}=\sum_{k=1}^n2^{n-k}{n+k-1\choose n-k},J_{2n-1}=\sum_{k=1}^n2^{n-k}{n+k-2\choose n-k}.
\end{gather*}
Furthermore, we derive an explicit formula for $c_2(n,k)$.
Using (\ref{cm1}), for even $n$, we obtain
\begin{gather*}c_2(2n,k)=\sum_{i=k}^{2n}{i-1\choose k-1}c_1(2n,i)=
\sum_{j=\lceil\frac k2\rceil}^{n}{2j-1\choose k-1}c_1(2n,2j)\\
=\sum_{j=\lceil\frac k2\rceil}^{n}a^{n-j}{2j-1\choose k-1}{n+j-1\choose n-j}.
\end{gather*}
For odd $n$, we have
\begin{gather*}c_2(2n-1,k)=\sum_{i=k}^{2n-1}{i-1\choose k-1}c_1(2n,i)=
\sum_{j=\lceil\frac {k+1}{2}\rceil}^{n}{2j-2\choose k-1}c_1(2n-1,2j-1)\\
=\sum_{j=\lceil\frac {k+1}{2}\rceil}^{n}a^{n-j}{2j-2\choose k-1}{n+j-2\choose n-j}.
\end{gather*}
In particular, for $a=1$, we obtain the following formulas for Pell numbers:
\begin{gather*}P_{2n}=\sum_{k=1}^{2n}\sum_{j=\lceil\frac k2\rceil}^{n}{2j-1\choose k-1}{n+j-1\choose n-j},\\
P_{2n-1}=\sum_{k=1}^{2n-1}\sum_{j=\lceil\frac{k+1}{2}\rceil}^{n}{2j-2\choose k-1}{n+j-2\choose n-j}.
\end{gather*}
\begin{remark} Using (\ref{cm1}), we may obtain an explicit formula for $c_m(n,k)$.
\end{remark}
\section{Case 3}
Let $a>b>0$ be integers. We define $f_0$ by the following recurrence:
\begin{equation}\label{re2}f_0(1)=1,f_0(2)=a, f_0(n+2)=af_0(n+1)-bf_0(n),(n\geq 1).\end{equation}
\begin{proposition}
The number $f_0(n)$
is the number of words of length $n-1$ over the alphabet $\{0,1,\ldots,a-1\}$, avoiding subwords $0i,(i=1,\ldots,b)$.
\end{proposition}
\begin{proof} We let $d(n)$ denote the number of the words of length $n-1$. Firstly, $d(0)=1$, since only the empty word has length $0$. Next, $d(1)=a$, since there are no restrictions on words of length $1$.
Assume that $n>1$. A word of length $n$ may begin with any letter. We have $a\cdot d(n-1)$ such words. From this number, we must subtract words which begin with subwords $0i,(i=1,2,\ldots,b)$.
Hence, $d(n)$ satisfies the same recurrence as $f_0(n)$, and the proposition is proved.
\end{proof}
\begin{example}
\begin{enumerate}
\item
If $a=2,b=1$, we have
\[f_0(1)=1,f_0(2)=2,f_0(n+2)=2f_0(n+1)-f_0(n),(n\geq 1),\]
which yields that $f_0(n)=n$. Hence, $n$ is the number of binary words of length $n-1$ avoiding subword $01$.
\item If $a=3,b=1$, we have
\[f_0(1)=1,f_0(2)=3,f_0(n+2)=3f_0(n+1)-f_0(n),(n\geq 1),\]
which is a well-known recurrence for the Fibonacci numbers $F_{2n}$. Hence,
$F_{2n}$ is the number of ternary words of length $n-1$ avoiding subword $01$.
\end{enumerate}
\end{example}
We now consider the particular case $a=b+1$.
\begin{corollary}
If $b>1$ and $a=b+1$, then
\begin{equation*}f_0(n)=\frac{b^n-1}{b-1}.\end{equation*}
\end{corollary}
\begin{proof}
We have $f_0(1)=1,f_0(2)=1+b=a$. Further,
\begin{equation*}f_0(n+2)=\frac{b^{n+2}-1}{b-1}.\end{equation*}
On the other hand, we have
\begin{equation*}
(b+1)f_0(n+1)-bf_0(n)=(b+1)\cdot\frac{b^{n+1}-1}{b-1}-b\cdot\frac{b^n-1}{b-1}=\frac{b^{n+2}-1}{b-1}.
\end{equation*}
\end{proof}
In particular, for $a=3,b=2$, we have $f_0(n)=2^n-1$, which yields
\begin{corollary} The Mersenne number $2^n-1$ is the number of ternary words of length $n-1$ avoiding $01$ and $02$.
\end{corollary}
Using~\cite[Corollary 9]{ja1}, we obtain
\begin{equation}
\label{ab1}f_m(1)=1,f_m(2)=m+a;f_m(n+2)=(a+m)f_m(n+1)-bf_m(n),(n\geq 1).\end{equation}
This means that $f_m$ counts the same sort of words as $f_0$, with $m+a$ instead of $a$.
Using Proposition \ref{alf}, we obtain the following combinatorial interpretation of $c_m(n,k)$.
\begin{corollary} The number $c_m(n,k)$ is the number of words of length $n-1$ over the alphabet $\{0,1,\ldots,b-1,b\ldots,m+a-1\}$ having exactly $k-1$ letters equal $m+a-1$ and avoiding subwords $0j,(j=1,2,\ldots,b)$.
\end{corollary}
We next derive an explicit formula for $c_1(n,k)$. A generating function for the sequence
$f_0(1),f_0(2),\ldots$ is $\frac{1}{bx^2-ax+1}$. According to~\cite[Equation (1)]{ja3}, we have \[\frac{x^k}{(bx^2-ax+1)^k}=\sum_{n=k}^\infty c_1(n,k)x^k.\]
The numbers $\alpha=\frac{a+\sqrt{a^2-4b}}{2b}$ and $\beta=\frac{a-\sqrt{a^2-4b}}{2b}$ are the solutions of the equation $bx^2-ax+1=0$.
\begin{proposition}\label{p10} We have
\[c_1(n,k)=\frac{1}{b^k}\sum_{j=0}^{n-k}\frac{1}{\alpha^{j+k}\beta^{n-j}}{n-j-1\choose k-1}{k+j-1\choose k-1}.\]
\end{proposition}
\begin{proof}
We expand $\frac{x^k}{b^k(\alpha-x)^k(\beta-x)^k}$ into powers of $x$.
Since \[\frac{1}{(\gamma -x)^k}=\sum_{i=0}^\infty{k+i-1\choose k-1}\frac{x^i}{\gamma^{i+k}},\] we easily obtain
\[\frac{x^k}{b^k(\alpha-x)^k(\beta-x)^k}=\sum_{i=0}^\infty
\left[\sum_{j=0}^i\frac{1}{b^k\alpha^{j+k}
\beta^{i-j+k}}
{k+j-1\choose k-1}{k+i-j-1\choose k-1}\right]x^{i+k},\]
and the statement follows by replacing $i$ by $n-k$.
\end{proof}
In the case $a=b+1$, we have $\alpha=1$ and $\beta=\frac 1b$.
Therefore, the following formula holds:
\begin{equation}\label{amb}c_1(n,k)=\sum_{i=0}^{n-k}
b^{n-k-i}{n-i-1\choose k-1}{k+i-1\choose k-1}.\end{equation}
Using (\ref{cmnk}), we obtain
\begin{identity}
\begin{equation*}
\sum_{i_1+i_2+\cdots+i_k=n}\left[\prod_{t=1}^k(b^{i_t}-1)\right]=\sum_{i=0}^{n-k}
b^{n-k-i}(b-1)^k{n-i-1\choose k-1}{k+i-1\choose k-1},
\end{equation*}
where $i_t>0,(t=1,2,\ldots,k)$.
\end{identity}
\begin{remark} Using (\ref{cm1}) and (\ref{suma1}), we obtain explicit formulas for $c_m(n,k)$ and $f_m(n)$.
\end{remark}
\section{Case 4}
We let $\mathcal R$ denote the condition given in this case.
We first solve the problem for binary words.
\begin{proposition}Let $f_0(n)$ be the number of binary words satisfying $\mathcal R$. Then,
\begin{enumerate}
\item $f_0(1)=1,f_0(2)=0,f_0(n+2)=f_0(n+1)+f_0(n),(n>1).$
\item For $n>1$, we have $f_0(n)=F_{n-2}$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item
We have $f_0(1)=1$, since the empty word has length $0$. Next, $f_0(2)=0$, since no words of length $1$ satisfy $\mathcal R$. Also, $f_0(3)=1$, since $10$ is the only word of length $2$ satisfying $\mathcal R$.
Next, $f_0(4)=1$, since $100$ is the only word of length $3$ satisfying $\mathcal R$. Assume that $n>1$. Then,
\begin{equation*}f_0(n+4)=f_0(n+2)+f_0(n+1)+\cdots,\end{equation*}
since a word of length greater than $3$ must begin with a subword of the form $10\ldots 0$.
Analogously, we obtain
\begin{equation*}f_0(n+5)=f_0(n+3)+f_0(n+2)+\cdots.\end{equation*}
Comparing these two equations, we get
\begin{equation*}f_0(n+5)=f_0(n+4)+f_0(n+3).\end{equation*}
\item
The formula follows from the preceding recurrence.
\end{enumerate}
\end{proof}
Since $f_0(1)=1$, and so $f_m(1)=1$, using Proposition \ref{alf} and (\ref{suma1}), we obtain the following combinatorial interpretations of $f_m$ and $c_m(n,k)$.
\begin{corollary}
\begin{enumerate}
\item
The number $c_m(n,k)$ is the number of words over the alphabet
$\{0,1,\ldots,m+1\}$ of length $n-1$ having $k-1$ letters equal $m+1$ and satisfying $\mathcal R$.
\item The number $f_m(n)$ is the number of words of length $n-1$ over
the alphabet $\{0,1,\ldots,m\}$ satisfying $\mathcal R$.
\end{enumerate}
\end{corollary}
We next derive an explicit formula for $c_1(n,k)$.
It is known that $c_1(n,k)$ is the coefficient of $x^n$ in the expansion of
$\left(\sum_{i=1}^\infty F_{i-2}x^i\right)^k$ into powers of $x$.
We consider the following auxillary initial function:
\begin{equation*}\overline{f}_0(1)=0,\overline{f}_0(n)=1,(n>1).\end{equation*}
From~\cite[Proposition 23]{ja1}, we obtain $\overline{f}_1(n)=F_{n-1}$.
It is proved in~\cite[Proposition 13]{ja2} that
\[\overline{c}_2(n,k)={n-k-1\choose k-1},\left(k=1,2,\ldots,\left\lfloor\frac n2\right\rfloor\right),\]
and $\overline{c}_2(n,k)=0$ otherwise.
Using~\cite[Proposition 6]{ja3} yields
\[\overline{c}_3(n,k)=\sum_{i=k}^{\left\lfloor\frac n2\right\rfloor}{i-1\choose k-1}{n-i-1\choose i-1}.\]
Hence,
\begin{equation}\label{xy}\left(\sum_{i=1}^\infty F_{i-1}x^i\right)^k=\sum_{n=k}^\infty\overline{c}_3(n,k)x^n.\end{equation}
We let $X$ denote $\sum_{i=1}^\infty F_{i-1}x^i$.
We have to expand the expression $\left(\sum_{i=1}^\infty F_{i-2}x^i\right)^k,$
which we denote by $Y$.
It follows that $Y=x^k(1+X)^k$. Hence,
\[Y=x^k\left(1+\sum_{i=1}^k{k\choose i}X^i\right)^k=\sum_{n=k}^\infty c_1(n,k)x^n.\]
Using (\ref{xy}) yields
\[Y=x^k+\sum_{i=1}^k\sum_{j=i}^\infty{k\choose i}\overline{c}_3(j,i)x^{j+k}.\]
It is easy to see that, in the case $j+k=n$, the coefficient of $x^n$ on the right-hand side of this equation equals $\sum_{i=1}^k{k\choose i}\overline{c}_3(n-k,i)$.
We thus obtain
\begin{proposition}\label{p16} The following equations hold:
\begin{gather*}c_1(k,k)=1,\\
c_1(n,k)=\sum_{i=1}^k\sum_{j=i}^{\left\lfloor\frac{n-k}{2}\right\rfloor}{k\choose i}{j-1\choose i-1}{n-k-j-1\choose j-1},(n>k).\end{gather*}
\end{proposition}
Using~\cite[Corollary 9]{ja1}, we easily obtain the following recurrence for $f_m$:
\[f_m(1)=1,f_m(2)=m,f_m(n+2)=(m+1)f_m(n+1)-(m-1)f_m(n).\]
Some particular cases are of note. In the case $m=1$, we obtain
\[f_1(1)=1,f_1(2)=1,f_1(n+2)=2f_1(n+1),(n>1),\]
which implies
\[f_1(1)=f_1(2)=1,f_1(n)=2^{n-2},(n>2).\]
We thus obtain the following property of powers of $2$.
\begin{corollary}
For $n\geq 2$, the number $2^{n-2}$ is the number of ternary words of length $n-1$ satisfying $\mathcal R$.
\end{corollary}
It yields that the following Euler-type identity holds:
\begin{identity} For $n>2$, the number of binary words of length $n-2$ is the
number of ternary words of length $n-1$, in which $0$ and $1$ appear only in a run of the
form $1i$, where $i$ is the run of zeros of length $i\geq 1$.
\end{identity}
From Propositions \ref{p16} and (\ref{suma1}), we obtain the following identity for the Mersen numbers:
\begin{identity}
\begin{equation*}2^{n-2}-1=\sum_{k=1}^{n}\sum_{i=1}^k\sum_{j=i}^{\left\lfloor\frac{n-k}{2}\right\rfloor}{k\choose i}{j-1\choose i-1}{n-k-j-1\choose j-1},(n>2).\end{equation*}
\end{identity}
We now consider the second particular case $m=2$. We have
\[f_2(1)=1,f_2(2)=2,f_2(n+2)=3f_2(n+1)-f_2(n),\] which is the recurrence
for Fibonacci numbers $F_{2n-1}$. We thus have
\begin{corollary}
The Fibonacci number $F_{2n-1}$ is the number of quaternary words of length $n-1$ satisfying $\mathcal R$.
\end{corollary}
Calculating values for $c_2(n,k)$, we obtain
\begin{identity}
\begin{equation*}F_{2n-1}=\sum_{k=1}^n\sum_{i=k}^{n}\sum_{t=0}^i\sum_{j=t}^{\left\lfloor\frac{n-i}{2}\right\rfloor}
{i-1\choose k-1}{i\choose t}{j-1\choose t-1}{n-i-j-1\choose j-1}.\end{equation*}
\end{identity}
\begin{remark} Using (\ref{cm1}) and (\ref{suma1}),
we obtain the explicit formulas for $c_m(n,k)$ and $f_m(n)$.
\end{remark}
\section{Case 5}
We let $\mathcal R$ denote the given condition. Again, we first consider the binary words.
\begin{proposition}
\begin{enumerate}
\item The following formula holds:
\begin{gather*}f_0(1)=1,f_0(2)=0,f_0(3)=1;\\
f_0(n+3)=f_0(n+1)+f_0(n),(n\geq 1).\end{gather*}
\item We have $f_0(n)=p_{n+2}$, where $p_n$ is the $n$th Padovan number.
\end{enumerate}
\end{proposition}
\begin{proof} The first statement is easy to prove. Since $(1)$ is essentially
the recurrence for the Padovan numbers, the statement $(2)$ is true.
\end{proof}
This means that the Padovan
number $p_{n+2}$ is the number of binary words
of length $n-1$ in which $0$ appears in runs of even length, while $1$ appears in
runs, the lengths of which are divisible by $3$. This means that the Padovan numbers count
the compositions into parts $2$ and $3$, which is a well-known.
\begin{corollary}\label{fib}
\begin{enumerate}
\item
The function $f_m$ satisfies the following recurrence:
\begin{gather*}f_m(1)=1,f_m(2)=m,f_m(3)=m^2+1,\\f_m(n+3)=mf_m(n+2)+f_m(n+1)+f_m(n),(n>1).\end{gather*}
\item Then, $f_m(n)$ is the number of words of length $n-1$ over $\{0,1,\ldots,m+1\}$ satisfying $\mathcal R$.
\item Also, $c_m(n,k)$ is the number of words of length $n-1$ over $\{0,1,\ldots,m+1\}$ having $k-1$ letters equal to $m+1$, and satisfying $\mathcal R$.
\end{enumerate}
\end{corollary}
\begin{proof}
The claim $(1)$ easily follows from~\cite[Theorem 6]{ja1}.
The claims $(2)$ and $(3)$ follow from Proposition \ref{alf}.
We add a short combinatorial proof for $(2)$.
Equation $f_m(1)=1$ means that the empty word satisfies $\mathcal R$. Further, $f_m(2)=m$
means that a word of length $1$ may consist of any letter except $0$ and $1$.
Next, $f_m(3)=m^2+1$ means that a word of length $2$ may consist of pairs
from $\{2,3,\ldots,m+1\}$, which are $m^2$ in number, plus the word $00$.
Finally, a word of length $n>2$ may begin with any letter from $\{2,3,\ldots,m+1\}$,
or from $00$, or from $111$.
\end{proof}
The case $m=1$ in Corollary \ref{fib} is the recurrence for Tribonacci numbers. Hence,
\begin{corollary} The sequence $1, 1, 2, 4, 7,\ldots$ of the Tribonacci numbers is the invert transform of the sequence $1,0,1,1,1,2,\ldots$ of the Padovan numbers.
Also, Tribonacci numbers count ternary words satisfying $\mathcal R$.
\end{corollary}
Finally, we calculate $c_1(n,k)$.
We define the arithmetic function $\overline{f}_0$ such that $\overline{f}_0(2)=\overline{f}_0(3)=1$, and $\overline{f}_0(n)=0$ otherwise. It follows from~\cite[Propositon 5]{ja2} that $\overline{c}_1(n,k)={k\choose n-2k}.$ Also, using~\cite[Theorem 6]{ja1}, we obtain
\begin{gather*}\overline{f}_1(1)=0,\overline{f}_1(2)=1,\overline{f}_1(3)=1,\\
\overline{f}_1(n+3)=\overline{f}_0(n+1)+\overline{f}_0(n).\end{gather*}
This implies that $\overline{f}_1(n)=f_0(n-1),(n>1)$. The sequence $f_0(1),f_0(2),\ldots$ is thus obtained by inserting $1$ at the beginning of the sequence
$\overline{f}_1(1),\overline{f}_1(2),\ldots$.
Using~\cite[Equation (10)]{ja3}, we obtain
\begin{equation*}\overline{c}_2(n,k)=\sum_{i=k}^n{i-1\choose k-1}\cdot{i\choose n-2\cdot i}.\end{equation*}
On the other hand,~\cite[Proposition 2]{ja3} yields
\begin{equation}\label{jjj}\left(\sum_{i=1}^\infty\overline{f}_1(i)x^i\right)^k=
\sum_{n=k}^\infty\overline{c}_2(n,k)x^n.\end{equation}
To obtain an explicit formula for $c_1(n,k)$, we need to expand the expression $X$ given by
$X=\left(\sum_{i=1}^\infty f_0(i)x^i\right)^k$ into powers of $x$.
We have \begin{equation*}X=\left(x+\sum_{i=2}^\infty f_0(i)x^i\right)^k=(x+xY)^k,\] where
$Y=\sum_{i=1}^\infty \overline{f}_1(i)x^i$.
Hence, \begin{equation*}X=x^k\sum_{i=0}^k{k\choose i}Y^i.\end{equation*}
Applying Equation(\ref{jjj}) yields
\begin{equation*}X=\sum_{i=0}^k{k\choose i}\sum_{j=i}^\infty\overline{c}_2(j,i)x^{j+k}.\end{equation*}
Taking $n=j+k$, we get
\begin{proposition} The following formula holds:
\begin{equation*}c_1(n,k)=\sum_{i=0}^k\sum_{j=i}^{n-k}{k\choose i}{j-1\choose i-1}{j\choose n-k-2j}.\end{equation*}
\end{proposition}
We thus obtain the following identity for the Tribonacci numbers $T_n$:
\begin{identity}
\begin{equation*} T_n=\sum_{k=1}^n\sum_{i=0}^k\sum_{j=i}^{n-k}{k\choose i}{j-1\choose i-1}{j\choose n-k-2j}.\end{equation*}
\end{identity}
\begin{remark} Using (\ref{cm1}) and (\ref{suma1}), we obtain explicit formulas for $c_m(n,k)$ and $f_m(n)$.
\end{remark}
|
2,877,628,090,793 | arxiv | \section{Introduction}
\label{sec:introduction}
Evolution has equipped microorganisms with a variety of motility patterns that allow them to explore their environment efficiently for various tasks.
For example, bacteria live in soil or larger host organisms where they search their environment for nutrients and surfaces to colonize.
A very common environmental constraint is confinement, both in the habitat of biological microswimmers and in the application domain of their artificial counterparts, \textit{e.g.}, micro-robots.
Bacteria are used for engineering applications in porous media such as crack sealing, soil stabilisation and contamination remediation~\cite{choi17a, phillips16a,mujah17a, priyadarshanee21a}.
It is envisioned that artificial micro-robots or micro-swimmers can in the future act as micro-surgeons and perform medical tasks inside human tissue~\cite{nelson10a, patra13a}.
In each case, the bacteria or micro-robots, from no on called agents, first have to traverse a highly confining, disordered porous environment before they can fulfill their function.
Self-propulsion is a necessary ingredient for the efficient exploration of such an environment, however, self-steering can improve the performance significantly.
Microorganisms can achieve directional control by changing the beating patterns and synchronisation of their propelling \textit{cilia}~\cite{josef06a} or \textit{flagella}~\cite{schwarz16a, gong20a}.
Many basic artificial microswimmers are unable to steer, especially if their propulsion mechanism relies on chemical reactions~\cite{paxton04a, howse07a, li14b}.
However, progress has been made in the control of individual artificial swimmers that are actuated by light~\cite{lozano16a, baeuerle20a, massana22a} or magnetic fields~\cite{cheang17a, fernandez20a, keicheang14a}, endowing them with a steering feature.
Biological microswimmers are known to possess various motility patterns~\cite{johansen02a, stocker12a}, \textit{i.e.}, strategies to use a combination of self-propulsion and self-steering to navigate through their environment:
\textit{P. aeruginosa}{} and many marine bacteria can reverse their locomotion and perform a run-and-reverse{} pattern in which they alternate between forward and backward swimming~\cite{johansen02a}.
Bacterium \textit{E. coli}{} interrupts its forward swimming mode ("run") with reorientation events ("tumble"), where the bacterium rotates before continuing to swim in a new direction~\cite{berg72a}.
\textit{V. alginolyticus}{} alternates between swimming forward, swimming backward and flicking its orientation by \SI{90}{\degree}, a pattern called run-reverse-flick~\cite{xie11a}.
In the following we will use these motility patterns as a starting point to investigate optimal strategies for porous media exploration and navigation.
The spreading behaviour of active particles with different motility patterns has been well studied in unconfined fluids~\cite{taktikos13a} and weakly confined environments~\cite{bechinger16a}.
Diffusive properties under strong confinement have also been the subject of a number of experimental and theoretical works:
Zeitz \textit{et al.} investigated in detail the mean-squared displacement of disk-like active Brownian particles (straight swimmers subject to rotational diffusion) in a porous environment close to the percolation threshold~\cite{zeitz17a}.
Bhattacharjee \& Datta tracked \textit{E. coli}{} cells in three-dimensional porous media and found that the bacterial trajectories cannot be identified as run-and-tumble{} anymore, but they rather found a sequence of hopping events through the channels, with the bacteria being intermittently trapped in small pores~\cite{bhattacharjee19a,bhattacharjee19b}.
Theoretical studies of run-and-tumble{}-swimmers in porous media find a maximal effective diffusivity by optimizing the duration of runs for specific pore configurations~\cite{reichhardt14a, licata16a,bertrand18a, irani22a}.
Similarly, numerical simulations of run-and-reverse{}-swimmers show that the optimal run length can be inferred from the distribution of the lengths of straight paths in a porous medium~\cite{kurzthaler21b}.
While the aforementioned works have optimized the parameters of specific patterns for porous media exploration, we will attempt here to optimize the motility pattern itself.
We study the qualitative features of different patterns when used by otherwise identical agents in various three-dimensional, disordered environments.
We cover the range of all relevant pore sizes from bulk fluid to confinement ranging down to the size of the micro-swimmer.
Using the insights gained from our analysis of biologically inspired motility patterns, we develop a new pattern which requires the agents to be capable of sensing their environment and making an intelligent decision.
This pattern, which can be deployed by artificial autonomous self-propelled agents, performs best across all investigated environments, and can be a basis for developing further optimal navigation strategies.
\section{Results}
\label{sec:results}
\subsection{Agent model and motility patterns}
\begin{figure}
\includegraphics[width=\linewidth]{bact.pdf}
\caption{Schematic representation of the geometry of the rigid self-propelled agent.}
\label{fig:model}
\end{figure}
We simulate $N=100$ individual active agents in three dimensions by modeling them as rod-like, rigid-body particles with length $l_\text{body} = \SI{2}{\micro\meter}$ and radius $r_\text{body}= \SI[parse-numbers = false]{1/3}{\micro\meter}$, as shown in \cref{fig:model}.
The agents perform translational and rotational Brownian motion and are subject to repulsive interactions with their porous environment.
Additionally, we apply time-dependent active forces and torques to achieve self-propulsion with speed $v_\text{swim}$ and self-steering with rotation rate $\omega_\text{act}$ (see Methods for a detailed description of the equations of motion).
The porous environment is modelled by randomly placing overlapping spherical obstacles with radius $R_\text{sphere} = 10 r_\text{body}$ within the simulation domain.
An example is shown in \cref{fig:porous_slice_local_thickness}, for the interaction potential between agents and obstacles, see Methods.
\begin{figure}
\includegraphics[width=\linewidth]{./local_thickness_slice_run_and_tumble_norelax_T_packing_fraction_1_1052631578947367_seed_6}
\caption{A two-dimensional slice through a typical randomly generated porous geometry with mean pore radius $r_\text{p} \approx \SI{2.6}{\micro\meter}$. The colors indicate the local thickness, see color bar.}
\label{fig:porous_slice_local_thickness}
\end{figure}
We create motility patterns by combining phases of self-propulsion and self-rotation, prescribing the durations of the phases and their temporal sequence.
In the following, we list the algorithms of the patterns used in this study, example trajectories are shown in \cref{fig:example_trajs}.
\begin{figure*}
\includegraphics[width=\linewidth]{./traj_time_color}
\caption{Two-dimensional projections of example trajectories without confinement for the four biologically inspired motility patterns.
The pictograms (not to scale) show the phases involved in the respective pattern, \textit{i.e.}, forward swimming, backward swimming and rotation.
The trajectory for reverse-when-stuck{} is not depicted, as this pattern reduces to straight swimming if there are no pores to get trapped in.
For easier distinction, the temperature is reduced by a factor of 60 compared to the simulations used in our analysis.}
\label{fig:example_trajs}
\end{figure*}
\paragraph{a) Straight swimming}
A constant force along the symmetry axis is applied, there is no active rotation.
The only source of randomness in the trajectory is the translational and rotational diffusion.
Aside from the anisotropic shape of the self-propelled agent, this pattern is an implementation of a 3D active Brownian particle (ABP).
\paragraph{b) Run-and-reverse}
With this motility pattern, agents swim at constant speed $v_\text{swim}$ ("run") but can reverse their swimming direction, realised by a change in sign of the self-propulsion velocity $v_\text{swim} \to - v_\text{swim}$.
The reversal algorithm thus implies that agents reverse their swimming propulsion and not the direction of their body, as observed in nature~\cite{hintsche17a, taylor74a}, and so no active torques are applied.
The durations $t_\text{run}$ of runs are commonly found to be exponentially distributed for bacteria~\cite{duffy97a, berg93a}, therefore we draw them from a distribution
\begin{equation}
p(t_\text{run}) = \frac{1}{\expval{\tRun}} \exp(-t_\text{run}/ \expval{\tRun}).
\label{eq:run_time_exp_distr}
\end{equation}
Here, $\expval{\tRun}$ is the average run duration, which is the only adjustable parameter of the run-and-reverse{} pattern.
\paragraph{c) Run-and-tumble}
The run-and-tumble{} pattern allows agents to swim straight ("run") and actively change their orientation ("tumble") at distinct times, a strategy employed by, \textit{e.g.}, \textit{E. coli}~\cite{berg72a}.
Our numerical algorithm follows Lee \textit{et al.}~\cite{lee19a}, we only give a brief summary here:
The durations $t_\text{run}, t_\text{tumble}$ of runs and tumbles are exponentially distributed and drawn from distributions analogous to \cref{eq:run_time_exp_distr}.
They are characterised by their respective means $\expval{\tRun}$ and $\expval{\tTumble}$.
The model for tumbling is based on the assumption that during a tumble the rods perform rotational Brownian motion with an increased rotational diffusion coefficient $D_\text{rot, tumble}$.
With this assumption, a distribution of tumble angles $\Theta_\text{tumble}$ can be calculated analytically for each tumble duration.
The tumble rotational diffusion coefficient together with the average tumble duration are the control parameters that determine the average angle of reorientation $\expval{\Theta_\text{tumble}}$.
In our implementation, a tumble duration is drawn, and the associated tumble angle distribution is calculated.
Then, a tumble angle is drawn from this distribution and an active torque with $\omega_\text{act} = \Theta_\text{tumble}/t_\text{tumble}$ is applied to the rod such that, in the absence of thermal noise and obstacles, the correct angle of rotation is achieved within the tumble duration.
The direction $\vu{n}$ of the active torque is orthogonal to the particle orientation $\vu{u}$, and its azimuthal angle in the particle frame of reference is drawn at random from $[0, 2\pi)$.
\paragraph{d) Run-reverse-flick}
This motility pattern can be found in marine bacteria \textit{V. alginolyticus}{}~\cite{xie11a} and combines elements from run-and-reverse{} and run-and-tumble.
Here, runs (of exponentially distributed durations $t_\text{run}$) are interrupted by both, reversals and flicks.
A flick is a tumble with a constant angle $\Theta_\text{flick} = \pi/2$ and duration $t_\text{flick}$.
Reversals and flicks occur in alternating fashion.
\paragraph{e) Reverse-when-stuck}
Leaving the realm of motility patterns that occur in nature, we propose a hypothetical optimal pattern for porous media navigation that combines straight swimming and reversals.
For this pattern, the agent must be endowed with sensing capabilities, a way to store a memory over a limited amount of time, and an intelligence unit to make simple decisions.
Together, these capabilities enable smart reactions to the environment beyond following a predetermined order of self-propulsion and -rotation.
Using a position sensor, the agent constructs a memory of its trajectory within a time frame $t_\text{memory}$.
If it did not move more than one rod length $l_\text{body}$ in that time, a reversal is triggered.
Upon reversal, the memory is reset.
This algorithm is used as a representative of the whole class of motility patterns in which the agent is able to sense if it is stuck in a pore.
A position sensor is not necessarily required, agents could also obtain this information from a sensor for swimming speed.
Mechanical sensors on the agent body or propulsion mechanism such as the ones found in bacteria~\cite{gordon19a} could determine a trapped state as well.
\subsection{Effective diffusivity} \label{subsec:effective-diffusivity}
From the scale of the different trajectories in \cref{fig:example_trajs} one can already get a qualitative understanding of how efficient agents can explore unconfined spaces depending on the strategy they employ.
To quantify the efficiency of exploration in both, unconfined space and porous media, we calculate the mean-squared displacement (\ensuremath{\text{MSD}})
\begin{equation}
\ensuremath{\text{MSD}}(t) = \frac{1}{N} \sum_{i=1}^{N} \frac{1}{\mathcal{T}-t} \int_{0}^{\mathcal{T}-t} \abs{\vb{r}_i(t'+t)-\vb{r}_i(t')}^2 \dd{t'},
\end{equation}
where $\mathcal{T}$ is the duration of the simulation, and $\vb{r}_i$ the center of mass position of agent $i$.
An example for run-and-reverse{} is shown in \cref{fig:msd_rnr_subdiff}.
\begin{figure}
\includegraphics[width=\linewidth]{./MSD_combined}
\caption{Mean-squared displacement of the run-and-reverse{} pattern at various mean pore sizes $r_\text{p}$.
The black lines indicate different scaling behaviours as a guide to the eye.}
\label{fig:msd_rnr_subdiff}
\end{figure}
It contains the qualitative features that are present in the \ensuremath{\text{MSD}} s for all motility strategies:
For short timescales it is super-diffusive with $\ensuremath{\text{MSD}}(t) \sim t^2$, where the ballistic contribution of self-propulsion dominates over random motion and interactions.
For intermediate timescales there is a sub-diffusive regime, \textit{i.e.}, $\ensuremath{\text{MSD}}(t) \sim t^\alpha$ with $\alpha<1$.
This is a result of trapped agents that spend significant time not moving in narrow pores, waiting for a random event to allow them to escape.
For long timescales, the motion is diffusive, \textit{i.e.}, $\ensuremath{\text{MSD}}(t) \sim 6 D_\text{eff} t$ with an effective diffusion coefficient $D_\text{eff}$.
This holds true without confinement, and also in porous media as long as the confinement is not strong enough to prohibit agents from moving altogether.
We use $D_\text{eff}$ as the key metric to rank the different motility patterns.
\Cref{fig:deff_vs_pore_size} shows $D_\text{eff}$ as a function of mean pore radius $r_p$ of the confining geometry.
\begin{figure}
\includegraphics[width=\linewidth]{./d_eff_vs_pore_size_5}
\caption{Effective diffusion coefficient as function of average pore size for all motility strategies.
The shaded areas denote one standard error of the mean over $N_\text{ensemble} = 7$ statistically independent simulations.}
\label{fig:deff_vs_pore_size}
\end{figure}
Without confinement (mean pore radius $r_\text{p} \to \infty$), straight swimming leads to a larger effective diffusion than any of the other patterns, but only by a factor of about 2 to 3.
This ratio is quite small considering that there are active reorientations in the other motility patterns while for the straight swimmers the rotational diffusion is the only source of deviation from ballistic motion.
Due to the small size of the particles, rotational diffusion has a strong effect on swimming regardless of the specific pattern: From the rotational friction coefficient $\gamma_\text{r}$ (see Methods on how we calculate this quantity) follows the typical timescale $\tau_\text{rot}$ for rotational diffusion via the Einstein-Smoluchowski relation
\begin{equation}
\tau_\text{rot} = \frac{1}{2D_\text{rot}} = \frac{\gamma_\text{r}}{2 \kB\temperature},
\end{equation}
where $D_\text{rot}$ is the rotational diffusion coefficient, $k_\text{B}$ the Boltzmann constant, and $T$ the temperature.
For the parameters of the agents simulated here (see Methods), we obtain $\tau_\text{rot} \approx \SI{0.7}{\second}$.
This timescale is comparable to the typical time $\expval{\tRun} = \SI{1}{\second}$ between active reorientations in non-straight swimming patterns, causing the relatively small ratio of $D_\text{eff}$ between the patterns when there is no confinement.
In the absence of obstacles, the effective diffusion coefficient of the straight swimmer can be calculated analytically.
It is then equivalent to the simple active Brownian particle (ABP), where the effective diffusion coefficient reads
\begin{equation}
D_\text{eff}^\text{ABP} = \frac{\kB\temperature}{\gamma_\text{t}} + \frac{1}{3} \tau_\text{rot} v_\text{swim}^2,
\label{eq:deff_abp}
\end{equation}
where $\gamma_\text{t}$ is the translational friction coefficient.
Here, we obtain $D_\text{eff} \approx \SI{183}{\micro\meter\squared\per\second}$ as seen in \cref{fig:deff_vs_pore_size}.
Without confinement, the other patterns show a smaller effective diffusivity than the straight swimmers, because in addition to rotational diffusion, they use active reorientations.
Since for run-and-tumble{} the average reorientation angle is $\expval{\Theta_\text{tumble}}\approx \SI{56}{\degree}$, it results in more persistent motion than run-and-reverse{} with a reorientation angle of $\SI{180}{\degree}$.
Run-reverse-flick{} performs slightly better than run-and-reverse{} because the flicks lead to less retracing of the trajectory compared to reversals.
For decreasing pore size, \textit{i.e.}, stronger confinement, agents that employ straight swimming are the first to become ineffective at navigating through their environment.
Even though the porous geometry is made of spheres, \textit{i.e.}, convex surfaces, overlap between them can generate concave pore shapes in which elongated swimmers get stuck.
Straight swimmers have to rely on thermal motion to randomly reorient themselves away from such pores to escape.
Escapes are additionally hindered by the constant forward propulsion that drives them into the pore, such that translational diffusion is very unlikely to lead to a displacement out of the pore.
The occurrence of concave, trapping pores happens at porosities where the average pore radius is still much larger than the size of the swimmer.
Only a few of such pores significantly decrease the effective diffusivity because straight swimmers can get trapped for long durations.
The next pattern to become ineffective is run-and-tumble, but there is a range of pore sizes where run-and-tumble{} outperforms straight swimming.
Here, tumble events make it possible to escape from pores where rotational diffusion is not strong enough to lead to sufficient reorientations.
Since the tumble angle is drawn from a distribution over $\qty[0, \pi]$, there is a probability for tumbles with $\Theta_\text{tumble} > \pi/2$, pointing the swimmer out of the pore and back to an open channel.
Yet, the pore size at which run-and-tumble{} becomes ineffective is still significantly larger than that of run-reverse-flick{} or run-and-reverse.
This is because swimmer reorientation and pore escape requires rotation of the elongated swimmer body in space, which can be suppressed by confinement.
To illustrate this point, we show the probability density of attempted tumble angles $\Theta_\text{tumble}$ and the actual angle $\tumbleAngle^*$ between start and finish of a tumble in \cref{fig:tumble_theta_target_vs_actual} for one typical simulation.
\begin{figure}
\includegraphics[width=\linewidth]{./tumble_angle_hist2d_target_vs_actl_run_and_tumble_norelax_T_packing_fraction_1_657894736842105_seed_0}
\caption{Joint probability density of attempted tumble angle $\Theta_\text{tumble}$ and actual tumble angle $\tumbleAngle^*$ between start and end of a tumble at mean pore radius $r_\text{p} \approx \SI{2}{\micro\meter}$.
The black line indicates $\tumbleAngle^* = \Theta_\text{tumble}$.}
\label{fig:tumble_theta_target_vs_actual}
\end{figure}
Without rotational Brownian motion or obstacles, there would only be non-zero values on the angle bisector of the coordinate axes with magnitude according to the distribution of attempted tumble angles.
However, in porous confinement, the deviation from $\Theta_\text{tumble} = \tumbleAngle^*$ is strongly asymmetric with the majority of actual tumble angles happening close to zero.
Most tumbling, especially for larger angles, is suppressed by confinement, leaving agents trapped in pores despite their attempts to escape.
To quantify this effect, we show the average actual tumble angle $\expval{\tumbleAngle^*}$ for different mean pore sizes in \cref{fig:actl_theta_vs_pore_size}.
\begin{figure}
\includegraphics[width=\linewidth]{./avg_tumble_vs_small_pores}
\caption{Three dimensionless quantities as a function of the mean pore size.
Blue: The mean actual tumble angle $\expval{\tumbleAngle^*}$ normalized by the mean attempted tumble angle $\expval{\Theta_\text{tumble}}$.
Orange: The fraction of pores smaller than the agent size, \textit{i.e.}, with $\tau(\vb{r}) < l_\text{body}$.
Green: The fraction of the void space that is occupied by the largest connected region with $\tau(\vb{r})>r_\text{body}$.
The shaded areas denote one standard error of the mean over $N_\text{ensemble} = 7$ statistically independent simulations.
For further explanations, see the main text.
}
\label{fig:actl_theta_vs_pore_size}
\end{figure}
The suppression of tumbles with decreasing mean pore radius starts around $r_\text{p} \approx \SI{5}{\micro\meter}$, the same value where $D_\text{eff}$ begins to drop significantly for run-and-tumble{} agents.
We note that at this mean pore size, only a relatively small fraction of pores has a smaller radius than the length $l_\text{body}$ of an agent (orange curve).
It is enough to cause a significant deviation of $\tumbleAngle^*$ from the target tumble angle $\Theta_\text{tumble}$ because self-propelled agents are much more likely to encounter the small, trapping pores than passive particles would be:
Active agents tend to be in contact with surfaces over long periods of time and slide along the pore walls due to their persistent motion.
This increases the chance of entering a location of strong confinement.
Run-and-reverse{} and run-reverse-flick{} swimmers can explore environments with average pore sizes ranging down to the size of a single agent.
This is not only because of the large probability of reversal events (certainty for run-and-reverse, 50\% for run-reverse-flick), but also because they lead to a guaranteed pore escape, unlike large tumble angles with run-and-tumble.
For example, a tumble with $\Theta_\text{tumble} = \pi$ is not equivalent to a reversal event in run-and-reverse.
In the former, there needs to be enough space to allow the rotation of the swimmer body whereas in the latter, the propulsion is reversed without affecting the swimmer orientation.
Run-reverse-flick{} reduces to run-and-reverse{} because flicks are geometrically suppressed just as tumbles are.
Its effective diffusivity is slightly larger than that of run-and-reverse{} because the smaller frequency of reversals allows the agents to move faster through open channels inbetween trapping pores.
Both patterns become ineffective at porous media exploration only when the available void space becomes disconnected and motion is only possible within a finite volume.
To quantify this, we calculate the volumes of connected regions with local thickness $\tau(\vb{r})>r_\text{body}$.
The local thickness represents the radius of the largest sphere that contains the point $\vb{r}$ and fits entirely in the void space between the obstacles as seen in \cref{fig:porous_slice_local_thickness}.
In \cref{fig:actl_theta_vs_pore_size} we show the ratio between the volume largest of these regions and the total void space $V_\text{void} = \phi L^3$, where $\phi$ is the porosity of the porous geometry and $L$ the length of the cubic simulation domain.
There is only one such region for $r_\text{p} \gtrsim \SI{2}{\micro\meter}$, but around $r_\text{p} \approx \SI{1.5}{\micro\meter}$ the void space splits into many smaller regions such that even for the larger ones there can be no more percolating motion through the simulation box.
Kurzthaler \textit{et al.}~\cite{kurzthaler21b} find that there is no significant difference between the effective diffusivity of run-and-tumble{} and run-and-reverse{} in porous media.
However, their implementation of run-and-tumble{} includes a $50\%$ chance of reversing when tumbling, so we would classify that pattern as run-and-tumble-or-reverse.
According to our observation of suppressed tumbles, this pattern will reduce to run-and-reverse{} when sufficiently confined, at which point our results are in agreement with theirs.
Run-and-reverse{} and run-reverse-flick{} are the best biologically inspired patterns for porous media exploration at very small pore sizes, but they do not perform well for larger porosities, where straight swimming is optimal.
This inspired the creation of the reverse-when-stuck{} pattern, combining the best features of straight swimming and run-and-reverse, especially propulsion reversal without rotation of the swimmer body.
As expected, it results in the largest effective diffusivity and therefore best exploration efficiency over the whole range of pore sizes.
At very small pore sizes, reverse-when-stuck{} performs better than run-and-reverse{} and run-reverse-flick, because the agent only performs reversals when they are needed for pore escape.
When it has found an open channel through the porous medium, it follows that channel until it gets stuck at the end without being interrupted by a randomly triggered reversal event.
\section{Conclusion}
\label{sec:conclusion}
We have performed Langevin dynamics simulations of rod-shaped, self-propelled and self-steered agents with various motility patterns in porous model geometries spanning a large range of porosities and pore sizes.
By quantifying their long-time, effective diffusivity, we evaluated their ability to explore these porous environments: At high porosity, \textit{i.e.}{}, large pore sizes, straight swimming performs best due to the absence of active rotation, hence the trajectories are ballistic for the longest possible time.
At intermediate pore sizes, run-and-tumble{} has the largest effective diffusivity.
Here, the agents can escape the pores by tumbling, which straight swimmers cannot, and they can explore the larger pore spaces with a more persistent motion than run-and-reverse{} or run-reverse-flick{}.
At very small pore sizes, rotation of the rods is suppressed by confinement, causing run-and-tumble{} swimmers to get stuck and preventing run-reverse-flick{} swimmers from flicking.
In this regime, run-and-reverse{} and run-reverse-flick{} swimmers can still escape small pores because their reversal mechanism enables them to reverse propulsion without rotation of the agent itself, making them the optimal biologically inspired motility patterns we considered here.
These results prompted us to develop a motility pattern that outperforms the biologically inspired patterns by endowing the active agents with memory and the ability to sense position (or velocity) for some time span, and an intelligence feature that makes a decisions based on this memory:
If the agent only reverses when it is stuck, defined as not moving more than its own length in its memory time, it can optimally explore open channels in the porous geometry while still being able to escape trapping pores.
With or without intelligence, we suggest that being able to reverse propulsion without rotation of the agent itself should be a high priority when designing active micro-agents for medical and engineering applications in confined spaces.
Only with this ability they can efficiently navigate the inevitably porous geometries in which they are deployed.
After all, the need for miniaturisation of agents in these applications arises from the highly confined environments in which their tasks are to be performed.
Furthermore, our results can serve as a basis for developing other optimized navigation strategies for specific environments.
\section{Methods}
\label{sec:model}
\subsection{Particle model}
The Langevin equations of motion for the particle positions $\vb{r}_i$ in three dimensions read
\begin{equation}
m \ddot{\vb{r}}_i = -\gamma_\text{t} \dot{\vb{r}}_i + \frac{v_\text{swim}}{\gamma_\text{t}} \vu{u}_i + \vb{F}(\vb{r}_i, \vu{u}_i) + \sqrt{2 \gamma_\text{t} \kB\temperature} \vb{\eta}^\text{t}_i.
\label{eq:langevin_trans}
\end{equation}
Here, $m$ is the particle mass, $\vu{u}$ a unit vector describing the particle orientation, $\vb{F}$ an external force from particle-boundary interactions, and $\vb{\eta}^\text{t}(t)$ a random noise vector with $\expval{\vb{\eta}^\text{t}} = \vb{0}$ and $\expval{\vb{\eta}^\text{t}_i(t)\vb{\eta}^\text{t}_j(t')} = \delta_{ij}\delta(t-t')\vb{1}$, where $\expval{\cdot}$ denotes an ensemble average and $\vb{1}$ the identity matrix in three dimensions.
For the particle orientations $\vu{u}_i$ we have analogously
\begin{align}
\outerdot{\vu{u}}_i = \vb*{\omega}_i \cross \vu{u}_i, \\
I \dot{\vb*{\omega}_i} = -\gamma_\text{r} \vb*{\omega}_i + \frac{\omega_\text{act}}{\gamma_\text{r}} \vu{n}_i + \vb{M}(\vb{r}_i, \vu{u}_i) + \sqrt {2 \gamma_\text{r} \kB\temperature} \vb{\eta}^\text{r}_i,
\label{eq:langevin_rot}
\end{align}
where $I$ is the particle moment of inertia tensor, $\vb*{\omega}_i$ the angular velocity vector, $\vu{n}_i$ a unit vector perpendicular to $\vu{u}_i$, $\vb{M}$ an external torque stemming from interactions, and $\vb{\eta}^\text{r}$ a noise term with the same properties as $\vb{\eta}^\text{t}$.
All simulations are performed using \textit{ESPResSo}~\cite{weik19a} to integrate the equations of motion.
Our active agents are constructed from $N_\text{beads} = 5$ point particles that are rigidly connected in a rod-like manner as shown in \cref{fig:model}.
Only the position and orientation of the central particle are propagated in time using \cref{eq:langevin_trans} to \cref{eq:langevin_rot}.
All other particles transfer their forces and torques to the central particle and are repositioned after propagating the central particle such that the rod behaves like a rigid body.
In our simulations we do not consider interactions between agents as we want to analyse only single agent properties.
The mass $m$ and moment of inertia $I$ are calculated by approximating the rods as cylinders with constant density $\rho$.
However, we show later that the exact values of $m$ and $I$ do not alter the physical behaviour of the agents.
To obtain the translational friction coefficient $\gamma_\text{t}$, we approximate the rods as a spheroids and use the results of Datta \& Srivastava~\cite{datta99a}.
Taking half of the rod length $l_\text{body}$ and the radius $r_\text{body}$ as the long and short half-axis, respectively, the friction coefficient is calculated \textit{via}{}
\begin{align}
e = \sqrt{1 - \qty(\frac{r_\text{body}}{l_\text{body}/2}) ^ 2}, \nonumber \\
\gamma_\text{t} = \frac{16 \pi e ^ 3l_\text{body}/2}{(1 + e ^ 2) \ln[(1 + e) / (1 - e)] - 2 e} \mu,
\end{align}
where $\mu$ is the dynamic viscosity of the surrounding medium.
Rotational Brownian motion has a strong influence on the dynamics of self-propelled particles as it sets the persistence of active, ballistic motion.
It is therefore vital to obtain a good estimate of the rotational friction coefficient $\gamma_\text{r}$ of our agents.
For rotations around equatorial axes (\textit{i.e.}, axes perpendicular to the symmetry axis) it is calculated from Perrin theory~\cite{perrin34a} \textit{via}
\begin{align}
p = \frac{l_\text{body}/2}{r_\text{body}}, \quad \xi = \frac{\sqrt{p ^ 2 - 1}}{p}, \nonumber \\
F_{eq} = \frac{2}{3} \frac{p ^{-2} - p ^ 2}{1 - (2 - p ^{-2})\text{atanh}\qty(\xi) / \xi}, \nonumber \\
\gamma_\text{r} = F_{eq} 8 \pi \mu \frac{l_\text{body} }{2} r_\text{body}^ 2.
\end{align}
Rotations around the axis of symmetry are neglected as they do not affect any observable of the system.
The self-propulsion and self-rotation that separates our model of active agents from passive colloids is determined by $v_\text{swim}$ and $\omega_\text{act}$.
According to the specific motility pattern, these terms can be constant or time dependent.
The motility pattern is evaluated and the active forces and torques are updated with a time step $\Delta t$.
It is an integer multiple of the simulation timestep $\delta t$ used in the velocity-Verlet scheme to integrate the equations of motion.
This reflects the difference in time scales between the Brownian motion and changes in motility, and speeds up simulations significantly.
\subsection{Porous media model}
\label{subsec:porous-media-model}
Inspired by the experimental setup of Bhattacharjee \& Datta~\cite{bhattacharjee19a}, we model the porous environment with spheres of radius $R_\text{sphere}$.
The spheres are placed randomly throughout the simulation box and fixed in space for the entire duration of the simulation.
As an approximation of hardcore repulsion, all individual particles of the swimmer rods interact with all spheres with a truncated and shifted purely repulsive Lennard-Jones potential
\begin{align}
V(r) = &4 \epsilon \left[ \left(\frac{\sigma}{r-R_\text{sphere}} \right)^{12} - \left(\frac{\sigma}{r-R_\text{sphere}} \right)^{6} + \epsilon \right] \nonumber \\
&\cross H(R_\text{sphere} + r_\text{body} - r),
\end{align}
where $r$ is the distance between the particle and the sphere center, $\epsilon$ the interaction strength, $\sigma = 2^{-\frac{1}{6}}r_\text{body}$ and $H(\cdot)$ the Heaviside step function.
All simulations are performed in a cubic, $L\cross L \cross L$ domain with periodic boundary conditions, where $L$ denotes the simulation box size.
The control parameter for the porous geometry is the number of spheres.
To analyse the porous geometry, we first use the positions of the spherical obstacles to generate a binary image of the pore space at a resolution of $\Delta x = \SI{0.25}{\micro\meter}$.
Then we use the \textit{porespy}~\cite{gostick19a} python package to obtain quantitative measures such as
porosity $\phi$, local thickness $\tau(\vb{r})$, and the pore size distribution.
\subsection{Parameter choice}
\label{subsec:parameter-choice}
To compare the motility patterns against each other, we choose the same physical parameters for all simulations: $l_\text{body} = \SI{2}{\micro\meter}$, $r_\text{body} = \SI[parse-numbers = false]{1/3}{\micro\meter}$, $v_\text{swim} = \SI{28}{\micro\meter\per\second}$, $T = \SI{300}{\kelvin}$, $\mu = \SI{8.9e-4}{\pascal\second}$, $R_\text{sphere} = \SI{5}{\micro\meter}$, $\epsilon = \kB\temperature$ and $L = \SI{80}{\micro\meter}$.
We set the average run times for run-and-reverse, run-and-tumble{} and run-reverse-flick{} to $\expval{t_{run}}= \SI{1}{\second}$, the average time of rotation for run-and-tumble{} and run-reverse-flick{} to $\expval{\tTumble} = t_\text{flick} = \SI{0.1}{\second}$ and the memory time for reverse-when-stuck{} to $t_\text{memory} = \SI{1}{\second}$.
For run-and-tumble, we set $D_\text{rot, tumble} = \SI{5}{\per\second}$, which results in $\expval{\Theta_\text{tumble}} \approx \SI{56}{\degree}$, close to values observed in \textit{E. coli}~\cite{berg72a}.
These might not be the optimal parameters for each of the patterns for all pore sizes, but they serve as a common ground for the evaluation of the pattern performance.
For agents of this size at the density of water $\rho_\text{water} = \SI{1e3}{\kilo \gram\per\meter\cubed}$, the diffusive relaxation time $\tau_\text{relax}= m/\gamma_\text{t} \approx \SI{7e-8}{\second}$ is very small compared to all other timescales of the system, so the dynamics is overdamped and the exact value of $m$ does not influence the physical behaviour.
In our simulations, we set the density of the agents to $\rho = 10^5 \rho_\text{water}$.
This still leaves the dynamics overdamped but allows us to choose larger time steps of $\Delta t=15\delta t=\SI{5e-3}{\second}$.
Simulations are performed with $N = 100$ agents and run for $\mathcal{T} = \SI{6000}{\second}$ to collect a sufficient amount of stochastic data, with an additional $\SI{600}{\second}$ warm-up phase before data collection starts.
They are repeated at least $N_\text{ensemble} = 7$ times with different random seeds, \textit{i.e.}, different geometries, particle starting positions and noise realisations.
Error quantifications shown in the previous sections represent the standard error of the mean over different simulations.
\section*{Data availability}
The data that support the findings of this
study are available from the corresponding author upon reasonable request.
\section*{Code availability}
The source code to reproduce the findings of this
study are available from the corresponding author upon reasonable request.
\input{main_wo_url.bbl}
\section*{Acknowledgments}
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project Number 327154368—SFB 1313.
We thank Prof. Sujit S. Datta for inspiring discussions.
\section*{Author contributions}
C.H. designed the overall study.
C.L. developed the numerical model, performed the simulations and analyzed the data.
C.L. and C.H. wrote the manuscript.
\section*{Competing interests}
The authors declare no competing interests
\end{document}
|
2,877,628,090,794 | arxiv | \section{Introduction}
Schreier formula for the rank of a subgroup of finite index
of a finitely generated free group $F$ is generalized to an arbitrary
(even infinitely generated) subgroup $H$ through the Schreier transversals
of $H$ in $F$. The rank formula may also be expressed in terms of the
cogrowth of $H$. \\
We introduce the rank-growth function $rk_{\dh}(i)$ of a subgroup $H$
of a finitely generated free group $F$. $rk_{\dh}(i)$ is defined to be
the rank of the subgroup of $H$ generated by elements of length
less than or equal to $i$ (with respect to the generators of $F$),
and it equals the
rank of the fundamental group of the subgraph of the cosets graph of $H$,
which consists of the paths starting at $1$ that are of length $\leq i$.
When $H$ is supnormal, i.e.
contains a non-trivial normal subgroup of $F$, we show that its rank-growth
is equivalent to the cogrowth of $H$. A special case of this is the known
result that a supnormal subgroup of $F$ is of finite index if and only
if it is finitely generated. In particular, when $H$ is normal then the
growth of the group $G=F/H$ is equivalent to the rank-growth of $H$. \\
A Schreier transversal forms a spanning tree of the cosets
graph of $H$, and thus its topological structure is of a contractible
spanning subcomplex of a simplicial complex. The $d$-dimensional
simplicial complexes that contain contractible spanning subcomplexes
have the homotopy type of a bouquet of $r$ $d$-spheres.
When these complexes are also $n$-regular then $r$ can be computed by
generalizing the rank formula (which applies to Schreier transversals)
to higher dimensions. \\
Let us remark that part of the results here apply in a similar form
also to Schreier transversals and Schreier bases of right ideals in
free group algebras (see \cite{Lew}, \cite{RR}, \cite{Ros2}).
\section{Generalized Schreier Formula}
Let $H$ be a non-trivial subgroup of a free group $F$.
By the Nielsen-Schreier Theorem $H$ is free too
(see, for example, \cite{Lyn}), and explicit free generators for
it can be given.
Suppose that $F$ is freely generated on a set $X$ (not necessarily
finite). The Cayley graph of $F$ (with respect to $X$) has the form of
a tree and is the universal covering of a space $\mathcal Q$ which
is a bouquet of $|X|$ loops ($| . |$ denotes cardinality throughout the paper).
The covering space of $\mathcal Q$ with regard to $H$ is
the {\em cosets graph} $\mathcal G$ of $H$, and it is obtained as the
quotient of the Cayley graph of $F$ under the left action of $H$.
Thus $H$ is the fundamental group of $\mathcal G$.
The set of vertices of $\mathcal G$ is the set of right
cosets of $H$ in $F$. A (double) edge which is labeled with $x \in X$
goes in the direction from the coset $Ht_1$ to the coset $Ht_2$ if
and only if $Ht_2=Ht_1x$, and it is labeled with
$x^{-1}$ in the direction from $Ht_2$ to $Ht_1$. This gives a
connected graph with $|X \cup X^{-1}|$ edges at each vertex.
It is more convenient to
label the vertices of $\mathcal G$ with specific coset representatives
in the following way. Let $\mathcal T$ be a spanning tree of $\mathcal G$.
The identity element $1$
is chosen to represent the coset $H$ and defined to be the root
of $\mathcal T$, and each other vertex is labeled with
the group element one gets by reading off the edge labels in
a path in $\mathcal T$ that starts at the root and ends at the given vertex.
We also denote by $\mathcal T$ the set of (the labels of) the
vertices $V({\mathcal T})$
of the tree $\mathcal T$, that is the coset representatives of $H$.
This set is a {\em Schreier transversal} for $H$ in $F$, which is
characterized by the property that every initial segment
of an element of $\mathcal T$ is also in $\mathcal T$.
For each $1 \neq w \in H$
there exist $u,v \in \mathcal T$ of maximal lengths such that $u$ is a prefix
of $w$ and $v$ is a prefix of $w^{-1}$. Since $t_1t_2^{-1} \notin H$ for
every $t_1 \neq t_2$ in $\mathcal T$, then $l(u)+l(v)<l(w)$, where $l$ denotes
the length of the (reduced) element in $F$. The Schreier generators
for $H$ with respect to $\mathcal T$ are those $w \in H$ for which
\begin{equation}
l(u)+l(v)= l(w)-1.
\end{equation}
Moreover, if $\phi$ is the coset map associated with $\mathcal T$ then
$H$ is freely generated by the non-trivial elements
\begin{equation}
tx(\phi(tx))^{-1},
\label{eqS10}
\end{equation}
where $t$ ranges over $\mathcal T$ and $x$ ranges over $X$ (see \cite{Lyn}).
This set is called a Schreier basis for $H$.
Since $tx = \phi(tx)$ only
when $tx \in \mathcal T$ then by (\ref{eqS10}) the rank of $H$ equals
the {\em cyclomatic number} of $\mathcal G$,
the cardinality of the ``missing'' edges in the directions of $X$
in $\mathcal T$, that is
\begin{equation}
\mbox{rank}(H) = |\{ e \in E({\mathcal G}) - E({\mathcal T}) \}|,
\label{eqS20}
\end{equation}
where $E({\mathcal G})$, $E({\mathcal T})$ denote the set of edges of
$\mathcal G$, $\mathcal T$ respectively.
This is because each edge is labeled with some $x \in X$ in exactly
one direction, and thus counted exactly once.
Suppose now that $F$ is finitely generated with $\mbox{rank}(F)=n$,
and $H$ is of finite index $m$ in $F$. Then $|E({\mathcal G})|=nm$ and
$|E({\mathcal T})|=m-1$ (since $\mathcal T$ is a tree).
By (\ref{eqS20})
we get that
\begin{equation}
\mbox{rank}(H) = 1 + (n-1)m.
\label{eqS23}
\end{equation}
This is Schreier Formula (see \cite{Lyn}). When $H$ is not
necessarily of finite index in $F$ and also not necessarily finitely
generated, we give in the proposition
below a formula that generalizes the above one. The rank is
computed on a Schreier transversal, and the simpler form of the
formula is given in Corollary~\ref{crS10}, which expresses the
rank in terms of the cogrowth (see below) of the subgroup.
The common way of computing the rank of the subgroup as a
limit of the ranks of the fundamental groups (the cyclomatic numbers)
of finite subgraphs deals with counting {\em edges}. Whereas, what we
are doing here is counting only {\em vertices}. \\
We use the following terminology and notation on graphs. A {\em path}
in a graph $\mathcal G$ is a sequence $v_0,e_1,v_1,e_2, \ldots$,
$v_i \in V({\mathcal G})$, $e_i \in E({\mathcal G})$, such that $e_i$
starts at the vertex $v_{i-1}$ and terminates at $v_i$. The
length of a path $v_0,e_1,v_1,e_2, \ldots, v_n$ is $n$.
A {\em simple path} is a path in which the vertices along it are distinct,
except possibly for the first and last one, in which case it is a
{\em simple closed path} or a {\em simple circuit}.
We assume that each path is {\em reduced}, i.e. it is not homotopic
to a shorter one when the initial and terminal vertices are kept fixed.
If ${\mathcal H} \subseteq {\mathcal G}$, i.e.
$\mathcal H$ is a collection of vertices
and edges of the graph $\mathcal G$, then
we denote by $< {\mathcal H} >$ the subgraph {\em generated} by $\mathcal H$.
It is the smallest
subgraph of $\mathcal G$ which contains $\mathcal H$.
That is, we add to $\mathcal H$ the
endpoint vertices of all the edges in $\mathcal H$. On the other hand, the
subgraph of $\mathcal G$ {\em induced} by $\mathcal H$ is the one whose
vertices are
those of $\mathcal H$ and whose edges are all the edges which join
these vertices
in $\mathcal G$. An induced subgraph is a subgraph which is induced by some
${\mathcal H} \subseteq {\mathcal G}$. If ${\mathcal H}_1, {\mathcal H}_2
\subseteq \mathcal G$
then ${\mathcal H}_1 - {\mathcal H}_2$ is the collection of
vertices $V({\mathcal H}_1) - V({\mathcal H}_2)$ and edges $E({\mathcal H}_1)
- E({\mathcal H}_2)$,
and it does not necessarily form a subgraph of $\mathcal G$,
even when ${\mathcal H}_1$
and ${\mathcal H}_2$ are subgraphs of $\mathcal G$. The {\em boundary}
of the subgraph $\mathcal H$ of $\mathcal G$ is
$\partial {\mathcal H} = {\mathcal H} \cap <{\mathcal G} - {\mathcal H}>$,
and its {\em interior} is $\dot{\mathcal H} = {\mathcal H} -
\partial {\mathcal H}$.
The {\em outer boundary} of $\mathcal H$ (in $\mathcal G$)
is the set of vertices
of ${\mathcal G} - {\mathcal H}$ which are adjacent to $\mathcal H$
in $\mathcal G$.
Assume now that each edge of $\mathcal G$ is labeled with some $x \in X$
in one direction and with $x^{-1} \in X^{-1}$ in the other direction.
Then we define $E^X_{out}({\mathcal H})$ to be the set of edges of
${\mathcal G} - {\mathcal H}$ whose initial vertices with respect to
the directions $X$ are in $\mathcal H$.
If ${\mathcal H}_i \subseteq {\mathcal G}$, $i=1,2, \ldots$, then $\mathcal H =
\liminf {\mathcal H}_i$ if the vertices of $\mathcal H$ are
$V({\mathcal H}) = \bigcup_{i \geq 1} \bigcap_{j \geq i} V({\mathcal H}_j)$,
and its edges are
$E({\mathcal H}) = \bigcup_{i \geq 1} \bigcap_{j \geq i} E({\mathcal H}_j)$.
Finally, let
$\alpha({\mathcal H})=
|\pi_0({\mathcal H})|$ be the cardinality of the (connected)
components of $\mathcal H$.
\begin{proposition}
Let $F$ be a free group of rank $n$ and let $H < F$. Let $\mathcal T$ be
a Schreier transversal for $H$ in $F$ and let ${\mathcal T}_i$ be finite
subgraphs of $\mathcal T$ such that ${\mathcal T} = \liminf {\mathcal T}_i$.
Then
\begin{equation}
\mbox{\em rank}(H) = \lim_{i \rightarrow \infty}
\left(
\alpha({\mathcal T}_i) + (n - 1)|V({\mathcal T}_i)| - \frac{1}{2}
\sum_{j=1}^{\alpha({\mathcal T}_i)} |V(\partial_{out}{\mathcal T}_{i,j})|
\right),
\label{eqS22}
\end{equation}
where, for a fixed $i$, $\partial_{out} {\mathcal T}_{i,j}$ is the outer
boundary (in $\mathcal T$) of the component ${\mathcal T}_{i,j}$
of ${\mathcal T}_i$, for $j= 1, \ldots, \alpha({\mathcal T}_i)$.
\label{prS40}
\end{proposition}
{\em Proof}.
If $H$ is of finite index $m$ in $F$ then there exists $i_0$
such that ${\mathcal T}_{i} = \mathcal T$ for every $i \geq i_0$, and
then $\alpha({\mathcal T}_i) = 1$, $|V({\mathcal T}_i)| = m$ and
$|V(\partial_{out}{\mathcal T}_i)| = 0$. Thus (\ref{eqS22}) reduces
to Schreier Formula.
Assume that $H$ is finitely generated but of infinite index.
Denote as before by $\mathcal G$ the cosets graph of $H$, which contains the
Schreier transversal tree $\mathcal T$. Let $C({\mathcal G})$ be the {\em core}
of $\mathcal G$ (see \cite{Sta}), that is the minimal deformation retract
of $\mathcal G$. It is the minimal connected subgraph of $\mathcal G$ which
contains all its simple circuits.
Since $H$ is finitely generated $C({\mathcal G})$ is
finite, and there exists $i_0$ such that, after possibly renaming the
components of each ${\mathcal T}_i$, $V(C({\mathcal G}))$ is contained in
$V({\mathcal T}_{i,1})$ for each $i \geq i_0$. Let us denote by
${\mathcal G}_{i,j}$ the subgraph of $\mathcal G$ induced by ${\mathcal T}_{i,j}$.
Then $E({\mathcal G}) - E({\mathcal T}) = E({\mathcal G}_{i,1}) -
E({\mathcal T}_{i,1})$, for each $i \geq i_0$, and by (\ref{eqS20}) the
cardinality of this set equals the rank of $H$. Hence it suffices to
show that for each $i \geq i_0$ and for each $j$
\begin{eqnarray}
|E({\mathcal G}_{i,j}) - E({\mathcal T}_{i,j})| &=&
|E({\mathcal G}_{i,j})| - |E({\mathcal T}_{i,j})|
\label{eqS27} \\
&=& 1+(n - 1)|V({\mathcal T}_{i,j})|
- \frac{1}{2} |V(\partial_{out}{\mathcal T}_{i,j})|.
\label{eqS28}
\end{eqnarray}
So assume $i \geq i_0$. Then $|E^X_{out}({\mathcal G}_{i,j})| =
|E^X_{out}({\mathcal T}_{i,j})| =
|V(\partial_{out}{\mathcal T}_{i,j})|/2$ for each $j$,
since all simple circuits of $\mathcal G$ are in ${\mathcal G}_{i,1}$.
Each vertex in $\mathcal G$ is the initial vertex of exactly $n$ edges
in the directions $X$. Therefore
\begin{equation}
|E({\mathcal G}_{i,j})|= n|V({\mathcal G}_{i,j})|-|E^X_{out}
({\mathcal G}_{i,j})|.
\end{equation}
As for ${\mathcal T}_{i,j}$, since it is a tree then
\begin{equation}
|E({\mathcal T}_{i,j})| = |V({\mathcal T}_{i,j})| -1.
\end{equation}
Substituting in (\ref{eqS27}) gives (\ref{eqS28}).
Assume now that $H$ is not finitely generated. Then, because in general
\begin{equation}
|E^X_{out}({\mathcal G}_{i,j})| \geq
|V(\partial_{out}{\mathcal T}_{i,j})|/2,
\end{equation}
we get that for each $i,j$
\begin{equation}
|E({\mathcal G}_{i,j}) - E({\mathcal T}_{i,j})| \leq
1+(n - 1) |V({\mathcal T}_{i,j})| - \frac{1}{2}
|V(\partial_{out}{\mathcal T}_{i,j})|.
\end{equation}
Since $\mbox{rank}(H) = \lim_{i \rightarrow \infty}
\left(
\sum_{j=1}^{\alpha({\mathcal T}_i)} |E({\mathcal G}_{i,j}) -
E({\mathcal T}_{i,j})|
\right) = \infty$,
equation~(\ref{eqS22}) follows.
\hfill $\Box$\\
We remark that instead of taking finite subgraphs ${\mathcal T}_i$ such
that ${\mathcal T} = \liminf {\mathcal T}_i$,
the rank formula can be clearly given
as the supremum, over all finite subgraphs of $\mathcal T$, of the expression
appearing in (\ref{eqS22}).
A special case of Proposition~\ref{prS40} is when each component
${\mathcal T}_{i,j}$ is a {\em ball}, that is its vertices are all the
vertices of $\mathcal T$ which lie at distance not greater than some fixed
$k$ from some fixed vertex. If $\mathcal H$ is a subgraph of $\mathcal G$ and
$|V({\mathcal G})| >1$ then we define $\delta({\mathcal H})$ to be the
number of components of $\mathcal H$ which consist of a single vertex,
i.e. balls of radius $0$. When $|V({\mathcal G})| =1$ then
$\delta({\mathcal G})$
is defined to be $0$.
When the ${\mathcal T}_i$ are concentric balls centered at the identity
$1$ then the values $|V({\mathcal T}_i)|$,
$i=0,1,2,\ldots$ relate to the {\em growth} function $\Gamma_{\dt}$
of $\mathcal T$, as is defined below.
By $l(g)$ we denote the {\em length} of $g \in F$, and we always assume
that the group elements are written in reduced form with respect to
the generating set $X$ of $F$. Then define
\begin{eqnarray}
\gamma_{\dt}(i) &=& |\{ v \in {\mathcal T} \mid l(v)=i \}|, \\
\Gamma_{\dt}(i) &=& |\{ v \in {\mathcal T} \mid l(v) \leq i \}|.
\end{eqnarray}
When $\mathcal T$ is a {\em minimal} Schreier transversal tree, that is
when it has also the property that every coset of $H$ is
represented by an element of minimal length, then $\Gamma_{\dt}(i)$
is the {\em cogrowth} function of $H$, relative to the generating
set of $F$, and is denoted by $\Gamma_{\dfh}(i)$ (see \cite{Ros1}).
We may look at $\Gamma_{\dfh}(i)$ as representing the ``volume''
of the ball of radius $i$ with center $1$ in the cosets graph of $H$
(with the metric induced by the word metric on $F$).
If, in addition, $H$ is a normal
subgroup of $F$ then the cogrowth function of $H$ equals the
growth function of the group $F/H$, relative to the the generating
set which is the canonical image of the generating set of $F$.
(In this case the Schreier transversal for $H$ which is minimal with
regard to a fixed ShortLex order on $F$ is also suffix-closed.)
\begin{corollary}
Let $F$ be a free group of rank $n$, let $H$ be a subgroup of $F$ and
let $\mathcal T$ be
a Schreier transversal for $H$ in $F$. Let ${\mathcal T}_i$ be induced
finite subgraphs of $\mathcal T$, whose components ${\mathcal T}_{i,j}$ are
balls, such that ${\mathcal T} = \liminf {\mathcal T}_i$. Then
\begin{eqnarray}
\mbox{\em rank}(H) &=& \lim_{i \rightarrow \infty}
\left(
\alpha({\mathcal T}_i) - \frac{1}{2} \delta({\mathcal T}_i)
+ (n - 1)|V({\mathcal T}_i)|
- \frac{2n-1}{2} |V(\partial {\mathcal T}_i)|
\right) \\
&=& \lim_{i \rightarrow \infty}
\left(
\alpha({\mathcal T}_i) - \frac{1}{2} \delta({\mathcal T}_i)
+ (n - 1)|V(\dot{\mathcal T}_i)|
- \frac{1}{2} |V(\partial {\mathcal T}_i)|
\right).
\end{eqnarray}
In particular,
\begin{eqnarray}
\mbox{\em rank}(H) &=& 1 + \lim_{i \rightarrow \infty}
\left(
(n-1)\Gamma_{\dt}(i) - \frac{1}{2} \gamma_{\dt}(i+1)
\right) \\
&=& 1 + \lim_{i \rightarrow \infty}
\left(
(n-1)\Gamma_{\dfh}(i) - \frac{1}{2} \gamma_{\dfh}(i+1)
\right).
\label{eqS35}
\end{eqnarray}
\label{crS10}
\end{corollary}
{\em Proof}.
The corollary follows from the fact that when the core $C({\mathcal G})$ is
finite then for each $i$ large enough every vertex
of $\partial {\mathcal T}_{i,j}$ is adjacent to $2n-1$ vertices of ${\mathcal T}
- {\mathcal T}_{i,j}$, unless ${\mathcal T}_{i,j}$
is a single vertex and then it
is adjacent to $2n$ vertices of ${\mathcal T} - {\mathcal T}_{i,j}$.
When $H$ is not finitely generated then we first notice that the
expression we calculate for each ball is non-negative. Secondly,
since ${\mathcal T} = \liminf {\mathcal T}_i$, then for every $r$ there exists
$i_r$ such that, for every $i \geq i_r$, ${\mathcal T}_i$ has a component
(ball) which contains the ball of radius $r$ around the identity.
But the expression calculated on these balls tends to infinity whenever
$H$ is of infinite rank, as shown below. This can also be concluded
directly from Proposition~\ref{prS40}.
\hfill $\Box$
\section{Rank-growth}
Given a Schreier transversal $\mathcal T$, let us define
\begin{eqnarray}
r_{\dt}(i) &=&
1+(n-1)\Gamma_{\dt}(i) - \frac{1}{2} \gamma_{\dt}(i+1) \\
&=& 1+\frac{2n-1}{2}\Gamma_{\dt}(i) - \frac{1}{2}\Gamma_{\dt}(i+1).
\label{eqS42}
\end{eqnarray}
$r_{\dt}(i)$ is an upper bound to the cyclomatic number of the subgraph
of $\mathcal G$ which is induced by the vertices of $\mathcal T$ of distance
at most $i$ from the root.
In case $\mathcal T$ is a minimal Schreier transversal then $r_{\dt}(i)$
is also denoted by $r_{\dh}(i)$:
\begin{equation}
r_{\dh}(i) = 1+(n-1)\Gamma_{\dfh}(i)-\frac{1}{2} \gamma_{\dfh}(i+1).
\label{eqS44}
\end{equation}
The sequence $r_{\dt}(i)$, $i=1,2, \ldots$ is non-decreasing. This is because
\begin{eqnarray}
r_{\dt}(i) - r_{\dt}(i-1) &=&
\frac{2n-1}{2}\gamma_{\dt}(i) - \frac{1}{2}\gamma_{\dt}(i+1),
\end{eqnarray}
and each vertex of $\mathcal T$ of level $i$ is adjacent to at most $2n-1$
vertices of level $i+1$. Thus $r_{\dt}(i)$ becomes eventually
constant if and only if either $\mathcal T$ is finite, or for some $i_0$
each vertex of $\mathcal T$ of level $i \geq i_0$ has
degree exactly $2n$, and this happens if and only if there are only
finitely many edges in $E({\mathcal G}) - E({\mathcal T})$, or equivalently when
$H$ is finitely generated.
It is interesting to know also the
rate in which the function $r_{\dt}(i)$ grows. A preorder is defined
on growth functions by
\begin{equation}
f_1(i) \preceq f_2(i) \ \mbox{$\Longleftrightarrow$} \
\exists c>0 \ \forall i \ [ f_1(i) \leq c f_2(ci) \ ].
\end{equation}
Then an equivalence relation is given by
\begin{equation}
f_1(i) \sim f_2(i) \ \mbox{$\Longleftrightarrow$} \ f_1(i) \preceq f_2(i)
\ \mbox{and} \ f_2(i) \preceq f_1(i).
\end{equation}
(we refer to \cite{Gri} for a survey on growth functions of groups
and to Gromov's \cite{Gro} rich and beautiful geometric theory.)
In Theorem~\ref{thS60} below we show that when the subgroup $H$ of the
free group $F$ is {\em supnormal}, i.e. contains a non-trivial subgroup
which is normal in $F$, then
for every Schreier transversal $\mathcal T$
of $H$, its growth function $\Gamma_{\dt}(i)$ is equivalent to
the function $r_{\dt}(i)$.
This implies that the cogrowth of $H$
is also equivalent to what we call the rank-growth of $H$.
We look at $H$ as the direct limit of the subgroups
\begin{equation}
H_i = < \{h \in H \mid l(h) \leq i\} >,
\end{equation}
where $l(h)$ is measured with respect to the generating set of $F$.
Then the {\em rank-growth} of $H$ (with respect to the generators of $F$) is
\begin{equation}
rk_{\dh} (i) = \mbox{rank} (H_i).
\end{equation}
Clearly, if we choose another generating set for $F$, we get an equivalent
rank-growth function. Notice that $H_i$
is the fundamental group of the subgraph of the cosets graph $\mathcal G$ of
of $H$ which contains all paths starting at $1$ of length $\leq i$.
Thus $rk_{\dh} (i)$ is a non-decreasing function. If we define
\begin{equation}
\rho_{\dh} (i) = \mbox{rank} (\pi_1({\mathcal B}_i)),
\end{equation}
where ${\mathcal B}_i$ is (the induced subgraph which is) the ball of radius
$i$ centered at the vertex $1$
of $\mathcal G$, then
\begin{equation}
\rho_{\dh} (i) = rk_{\dh} (2i+1).
\end{equation}
Therefore $rk_{\dh}(i)$ and $\rho_{\dh}(i)$ are equivalent.
Also $\rho_{\dh}(i) \sim r_{\dh}(i)$. In fact,
\begin{equation}
\rho_{\dh}(i) \leq r_{\dh}(i) \leq \rho_{\dh}(i+1).
\end{equation}
More precisely,
\begin{equation}
r_{\dh}(i) = \rho_{\dh}(i) +\frac{1}{2}(|E^X_{out}({\mathcal B}_i)| -
\gamma_{\dfh}(i+1)) \leq \rho_{\dh}(i+1).
\label{eqS54}
\end{equation}
\begin{theorem}
Let $H$ be a supnormal subgroup of a finitely generated free group $F$,
and let $\mathcal T$ be a Schreier transversal for $H$ in $F$. Then
\begin{equation}
r_{\dt}(i) \sim \Gamma_{\dt}(i).
\end{equation}
In fact, if $H$ is not necessarily supnormal but has the property that
$|F:N_{\df}(A)| < \infty$ for some non-trivial $A<H$ then
\begin{equation}
rk_{\dh}(i) \sim \Gamma_{\dfh}(i).
\end{equation}
\label{thS60}
\end{theorem}
{\em Proof}.
For every Schreier transversal of a subgroup of $F$ we have
$r_{\dt}(i) \preceq \Gamma_{\dt}(i)$. This follows immediately from
the definition of $r_{\dt}(i)$ - see (\ref{eqS42}).
Suppose now that $H$ is supnormal. Let $h$ be a non-trivial
element of a subgroup of $H$ which is normal in $F$, and let $m = l(h)$
(as usual, the length is with respect to the generators of $F$).
Then at every vertex $v$ of the cosets graph $\mathcal G$ of $H$, if we follow
the path defined by $h$ we form a circuit. Therefore at every vertex
of $\mathcal T$ of level at most $i$, by following the path defined by $h$
we reach a vertex of $\mathcal T$ of level at most $i+m$ where we must
stop because the next edge is missing. The number of these missing edges
is less then or equal to $r_{\dt}(i+m)$. Since at most
$m$ vertices are the starting point of a tour defined by $h$ which reaches
the same missing edge $h$ then
\begin{equation}
\Gamma_{\dt}(i) \leq m r_{\dt}(i+m).
\end{equation}
By the two inequalities we have
\begin{equation}
r_{\dt}(i) \sim \Gamma_{\dt}(i).
\end{equation}
Applying this result to a minimal Schreier transversal yields
\begin{equation}
rk_{\dh}(i) \sim r_{\dh}(i) \sim \Gamma_{\dfh}(i).
\end{equation}
The condition of $H$ being supnormal can be weakened. It suffices to
demand
that $H$ contains a non-trivial subgroup $A$ such that $|F:N_{\df}(A)|
< \infty$,
because then the cogrowth of $H$ is equivalent to the growth (with respect
to the generators of $F$) of the minimal coset representatives of
$H \cap N_{\df}(A)$ in $N_{\df}(A)$. Even more, we need only
the growth (again, with respect to the generators of $F$) of
$\{ g \in {\mathcal T} \mid gAg^{-1} \subseteq H \}$, where $\mathcal T$
is a minimal Schreier transversal for $H$,
to be equivalent to the cogrowth of $H$ in $F$.
\hfill $\Box$ \\
Since $\Gamma_{\dt}(i) \preceq \Gamma_{\dfh}(i)$ for every Schreier
transversal $\mathcal T$ of a subgroup $H$ of $F$, then by Theorem~\ref{thS60}
when $H$ is supnormal in $F$ then $r_{\dt}(i) \preceq rk_{\dh}(i)$.
We also notice that a special case of Theorem~\ref{thS60} is the known result
stating that a supnormal subgroup of a finitely generated group is of
finite index if and only if it is finitely generated. And when $H$ is
normal in $F$, then the growth $\Gamma_{\dg}(i)$ of the group $G=F/H$
is equivalent to the rank-growth of $H$ and to the growth of
\begin{equation}
r_{\dh}(i) =1+(n-1)\Gamma_{\dg}(i) - \frac{1}{2} \gamma_{\dg}(i+1).
\end{equation}
The growth of the subgroup $H$ is always exponential when it is of
rank greater than $1$, since it is free. But Grigorchuk showed
(\cite{Gri0}) that when $H$ is normal then its ``growth exponent''
$\limsup_{i \rightarrow \infty} \Gamma_{\dh}^{(\df)} (i)^{1/i} = 2n-1$,
if and only if $G=F/H$ is amenable,
(in fact, Grigorchuk \cite{Gri0} obtained more: a formula which connects the
growth exponent of $G$ with the spectral radius of a random walk on $G$),
where $n = \mbox{rank}(F)$ and
$\Gamma_{\dh}^{(\df)} (i)$ represents the growth of $H$ with respect
to the generators of $F$. (Recall that a group $G$ is amenable if there exists
an invariant mean on $B(G)$, the space of all bounded complex-valued
functions on $G$ with the sup norm $\parallel f \parallel_{\infty}$, see
\cite{Gre}). When $G$ is non-amenable then the growth
exponent of $H$ is less than $2n-1$. But then the group $G$ has exponential
growth, and we have shown that in this case the rank-growth of $H$ is
also exponential, i.e. the maximal possible (up to equivalence).
This seems at first sight contradictory. To illustrate this phenomenon
we may think of a tree, called $F$, that we prune its sides going from
bottom upward. The number of branches we cut is called (half) the rank
of $H$, the tree that is left after the pruning is called $G$, and
(part of) what we cut is called $H$. Then
the further the cut is from the periphery and closer to the middle
of the tree the larger $H$ is, the smaller $G$ is, and the rank of $H$
also becomes smaller since we cut towards the main branches.
Although the rank of the subgroup of a free group can be expressed, as
we have seen in Corollary~\ref{crS10}, in terms of the growth function of
any Schreier transversal of it, the growth function itself of one
Schreier transversal of an infinitely generated subgroup may in general
differ completely from that of another Schreier transversal.
This is shown in the next proposition.
\begin{proposition}
There exists a subgroup of the free group of rank $2$ with
exponential cogrowth which has a Schreier transversal $\mathcal T$
whose growth is $\Gamma_{\dt}(i) = i+1$.
\label{prS50}
\end{proposition}
{\em Proof}.
We will construct the cosets graph of such a subgroup inductively.
Let $c$ be a positive integer which is large enough. First we make a
simple circuit $\lambda_1$ of length $c$ that starts at the root $1$.
Then at the $n$-th step we construct a path $\lambda_n$ of length $2nc$,
whose vertices, apart from the initial and terminal ones, are new.
The initial vertex of $\lambda_n$ is the one before the last vertex
in the path $\lambda_{n-1}$. The terminal vertex of $\lambda_n$ is
chosen to be of minimal distance from the root among the vertices
whose degree is less than $4$.
If we delete the last edge of each path $\lambda_n$, then we get a
linear Schreier transversal $\mathcal T$, i.e. $\Gamma_{\dt}(n) = n+1$.
On the other hand, if we delete the middle edge of each $\lambda_n$,
then we get a tree ${\mathcal T}'$ with exponential growth,
because each vertex of it has degree $4$, except for a sequence of
vertices $v_n$ of distances $\geq cn$ respectively from the root. Since
the cogrowth function is greater than or equal to the growth function of any
Schreier transversal of the subgroup, the result follows.
\hfill $\Box$ \\
It is shown in \cite{Ros1} that when $H = H_1 \cap H_2$ the
cogrowth of $H$ satisfies
\begin{equation}
\Gamma_{\dfh}(i) \leq \Gamma_{\dfhA}(i) \Gamma_{\dfhB}(i) \ \ \
\mbox{for every $i$}.
\end{equation}
The rank-growth of the intersection of two subgroups behaves similarly.
\begin{proposition}
Let $H_1, H_2$ be non-trivial subgroups of a finitely generated
free group $F$ and let $H = H_1 \cap H_2$. Then
\begin{equation}
rk_{\dh}(i) \leq 1 + 2(rk_{\dhA}(i)-1)(rk_{\dhB}(i)-1)-\min(rk_{\dhA}(i),
rk_{\dhB}(i))
\end{equation}
for every $i$.
Hence
\begin{equation}
rk_{\dh}(i) \preceq rk_{\dhA}(i)rk_{\dhB}(i).
\end{equation}
\label{prS60}
\end{proposition}
{\em Proof}.
This follows immediately from the best general bound for the intersection
of finitely generated subgroups in free groups, which is due to
Burns (\cite{Bur}).
\hfill $\Box$
\section{The Generalized Word Problem}
Given a subgroup $H$ of a group $G$ it is interesting to know the
{\em distorsion} of $H$ with respect to $G$, that is a bound $f(i)$ of
the length, with respect to a finite set of generators of $H$, of an
element of $H$ whose length is $i$ with respect to a finite set of
generators of $G$ (see \cite{Gro}, \cite{Farb}).
When $F$ is free then it is known that every element of a subgroup $H$
of it, whose length is $i$ in $F$, has length at most $i$ with respect to a
Schreier basis of $H$ (or a Nielsen-reduced basis, which is no other than a
minimal Schreier basis), thus the distortion
is linear. A bit more precise description is obtained by using
$d(w,{\mathcal T})$, the distance of $w \in F$ from a Schreier
transversal $\mathcal T$. That is
\begin{equation}
d(w,{\mathcal T}) = \min \{ l(t^{-1}w) \mid t \in {\mathcal T} \},
\end{equation}
where $l$ denotes the length in $F$. Notice that $d(w,{\mathcal T}) \leq l(w)$
since $1 \in {\mathcal T}$. Then if $B_{\dt}$ is a corresponding
Schreier basis for the subgroup $H<F$ then every
$w \in F$ can be written in the form
\begin{equation}
w = b_{i_1}^{\varepsilon_1} \cdots b_{i_k}^{\varepsilon_k} \bar{w},
\end{equation}
with $b_{i_j} \in B_{\dt}$, $\varepsilon_j= \pm 1$ and $\bar{w} \in
\mathcal T$, such that $k \leq d(w,{\mathcal T})$.
To see it,
let $t \in {\mathcal T}$ be the maximal prefix of $w$ in $\mathcal T$, i.e.
$l(t^{-1}w) = d(w,{\mathcal T})$. If $t=w$ we are done. Otherwise, there
exists $x \in X \cup X^{-1}$, $X$ the generating set of $F$, such that
$b_{i_1}^{\varepsilon_1}=tx(\phi(tx))^{-1} \in B_{\dt} \cup B_{\dt}^{-1}$,
$\phi$ the coset map, and such that $w=txw'$ when written in reduced
form. Thus
\begin{equation}
w = b_{i_1}^{\varepsilon_1}\phi(tx)w'.
\end{equation}
But $d(\phi(tx)w',{\mathcal T}) \leq l(w') = d(w,{\mathcal T})-1$
and we proceed by induction.
The above shows that when we are given a
Schreier transversal $\mathcal T$ and a corresponding Schreier basis
$B_{\dt}$ for $H < F$ then it is possible to obtain algorithmically
a ``normal form'' modulo $H$ for every element of $F$, i.e. its coset
representative in $\mathcal T$, and this demonstrates the importance of
Schreier generating sets (whose shape and role is similar to those of
Gr\"obner bases for algebras, see \cite{Ros2}). Thus the generalized
word problem
for $H$ in $F$ is then solvable. In fact, whenever $G = <X \mid R>$,
and $H$ is a subgroup of $G$ generated by a set $S \subseteq F=<X>$,
then the generalized word problem for $H$ in $G$ is solvable when
the underlying set of the subgroup $<N,S>$ of $F$, where $N=<R>^{\df}$
is the normal closure of $R$ in $F$, as well as a set $\mathcal T$ of coset
representatives for $H$ in $G$, are recursively enumerable (r.e.) sets.
For, by listing the elements of $<N,S>$ and those of $\mathcal T$ we can list
all elements of $F$, and also find which coset is the coset $<N,S>$ in $F$.
Hence, both the set $<N,S>$ and its complement in $F$ are r.e. and
therefore $<N,S>$ is recursive.
\begin{proposition}
Let $G = <X \mid R>$ be a finitely generated group and let $H$ be a
subgroup of $G$ generated by a set $S$. Suppose also that the subgroup
$A=<N,S>$ of the free group $F=<X>$, where $N=<R>^{\df}$, is r.e.
Then the generalized word problem for $H$ in $G$ is solvable
whenever one of the functions $\Gamma_{\dfa}(i), r_{\da}(i)$ or
$rk_{\da}(i)$ is recursive.
\label{prS80}
\end{proposition}
{\em Proof}.
First we remark that $A$ is r.e. for example when $R$ and $S$ are r.e.
We construct inductively ${\mathcal B}_i$, the concentric
balls of radius $i$, of the cosets graph of $A$ in $F$.
We start with the vertex $1$. Then assuming that ${\mathcal B}_i$ was constructed,
we first extend it to level $i+1$ without forming new circuits. If the
number of vertices at level $i+1$ agrees with $\Gamma_{\dfa}(i+1)$ or
with $r_{\da}(i+1)$ or if $\pi_1({\mathcal B}_{i+1}) = rk_{\da}(2i+1)$ then
we are done. Otherwise, by listing the elements of $A$, each defining
a circuit in the cosets graph, we stop until we reach the desired values
of our functions.
\hfill $\Box$ \\ \\
\section{Contractable Spanning Subcomplexes}
If we look at Proposition~\ref{prS40} we see that it makes little use
of the group structure. It is mainly a statement about $m$-regular
graphs, i.e. graphs whose vertices have all the same degree $m$.
We may then try to generalize this theorem from such graphs to special
simplicial complexes. When $\CC$ is a simplicial complex
then we denote by $| {\CC} |$ the topological space of $\CC$ (as in
\cite{Spa}), and whenever we relate some topological properties to
$\CC$ they describe, in fact, those of $| {\CC} |$.
We Call a subcomplex $\DD$ of a $d$-dimensional simplicial complex
$\CC$ a {\em spanning} subcomplex if
\begin{description}
\item[(i)] $\DD$ contains ${\CC}^{(d-1)}$, the $(d-1)$-skeleton of $\CC$;
\item[(ii)] every principal simplex of $\DD$ (i.e. a simplex which is
not a face of another simplex of $\DD$ of higher dimension) is also
principal in $\CC$.
\end{description}
Then the analogue of a spanning tree in graph theory
is a spanning subcomplex $\DD$ whose topological space is contractible.
We call such a subcomplex a {\em contractible spanning subcomplex}.
In this case condition (ii) becomes redundant. This is because if
$\sigma_1^{d-1}$ is principal in $\DD$ but is a face of a $d$-simplex
$\sigma_2^d$ of $\CC$ then by considering the boundary of $| \sigma_2^d |$
in $| {\DD} |$ we get that $\pi_{d-1} (| {\DD} |)$ is not trivial and thus
$\DD$ is not contractible. Or we can look at the homology of $\DD$,
and see that $H_{d-1}({\DD})$ does not vanish since $\sigma_1^{d-1}$
does not appear in $B_{d-1}({\DD})$ but is a summand of a cycle.
We come now to the analogue of Proposition~\ref{prS40} for simplicial
complexes. The formula we give, however, is not so nice as the one
in the one-dimensional case, where all terms involve only the
zero-dimensional skeleton. We use the following additional
notation and definitions.
Let $\CC$ be a simplicial complex and let $\DD$ be a subcomplex of it.
The collection of $k$-simplices of $\CC$ is denoted by $F^k({\CC})$,
and its cardinality is denoted by $\beta_k({\CC})$. When $X$ is a collection of
simplices of $\CC$ we denote by $< X >$ the subcomplex generated by
$X$. If ${\DD}_1, {\DD}_2$ are subcomplexes of $\CC$ then ${\DD}_1 - {\DD}_2$
is the collection of simplices ${\DD}_1 - {\DD}_2 = \{ \sigma \mid \sigma
\in {\DD}_1, \ \sigma \notin {\DD}_2 \}$, and it does not necessarily form a
subcomplex. The {\em boundary} of the subcomplex $\DD$ of $\CC$ is
$\partial \DD = \DD \cap <{\CC} - {\DD}>$, and its {\em interior} is
$\dot{\DD} = {\DD} - \partial {\DD}$.
If $\sigma^k$ is a $k$-simplex of $\CC$ we define its {\em degree}
to be $\mbox{deg}(\sigma^k)=|\{\sigma^{k+1} \in
F^{k+1}({\CC}) \mid \sigma^k \subset \sigma^{k+1} \}|$. If $X$ is a
collection of simplices of $\CC$ then $\mbox{deg}(X)$ is the total
degree of the members of $X$. When all members of $X$ belong to a
subcomplex $\DD$ of $\CC$ and we want to compute the degree relative
to this subcomplex then we write it $\mbox{deg}_{\dd}(X)$.
When $\CC$ is of dimension $d$ then we say it is
$n$-{\em regular} if every $(d-1)$-simplex of it has degree $n$.
If ${\DD}_i$, $i \geq 1$, is a sequence of subcomplexes of $\CC$ then
we denote by $\liminf {\DD}_i$ the subcomplex of $\CC$ whose simplices
are $F^k(\liminf {\DD}_i) = \bigcup_{i \geq 1}
\bigcap_{j \geq i} F^k({\DD}_j)$, for every $k \geq 0$.
\begin{theorem}
Let $\CC$ be a countable $d$-dimensional simplicial complex
which contains a contractible spanning subcomplex $\DD$. Then
$| {\CC} |$ is homotopic to a bouquet of $r$ ($r$ can be $\infty$)
$d$-spheres. \\
If, in addition, $\CC$ is $n$-regular and
${\DD}_i$ are finite subcomplexes of $\DD$ such that $\DD = \liminf {\DD}_i$
then
\begin{equation}
r = \lim_{i \rightarrow \infty}
\left(
\frac{1}{d+1} (n \beta_{d-1}({\DD}_{i}) - \mbox{\em deg}_{<\dd-\dd_i>}(F^{d-1}
(\partial {\DD}_{i}))) - \beta_d({\DD}_{i}) \right).
\label{eqCSS15}
\end{equation}
\label{thCSS20}
\end{theorem}
{\em Proof}.
We define a contractible space to be homotopic to a bouquet of zero
$d$-spheres. So assume that $| {\CC} |$ is not contractible.
Since $| {\DD} |$ is contractible, $| {\CC} |$ is homotopic to $| {\CC} | /
| {\DD} |$.
If there are $r$ ($r$ can be $\infty$) $d$-simplices $\sigma^d$ which
are not in $\DD$, then since the boundary of $| \sigma^d |$ is in
$| {\DD} |$, we get that $| {\CC} |$ is homotopic to a bouquet of $r$
$d$-spheres (see also \cite{Bjo} for shellable complexes, where every
shellable complex contains a contractible spanning subcomplex but not
necessarily the other way round).
The second part of the proof is similar to that of Proposition~\ref{prS40}.
In fact we are only
dealing with the simplices of $\CC$ of dimensions $d-1$ and $d$,
and then we compute the number of $d$-simplices of ${\CC} - {\DD}$.
\hfill $\Box$ \\
We remark that in case each subcomplex ${\DD}_i$ in (\ref{eqCSS15}) has
contractible
connected components then $\beta_d({\DD}_i)$ may be expressed
in terms of the $\beta_j({\DD}_i)$, $j=0, \ldots, d-1$ using
the (topological) {\em Euler characteristic}
\begin{equation}
\chi({\DD}_i^{(d-1)}) = \sum_{i=0}^{d-1} (-1)^i \beta_i({\DD}_i^{(d-1)}).
\end{equation}
|
2,877,628,090,795 | arxiv | \section{Introduction}
Conceptual graphs (CGs)~\cite{chein_conceptual_2008} refer to a family
of formalisms of graph-based knowledge representation, close to
existing semantic web languages such as
RDF(S)~\cite{manola2004rdf,brickley2014rdf} and
OWL~\cite{mcguinness2004owl}. Their advantages include their data
modeling capacities, grounded on first-order logic (FOL) semantics, as
well as the possibility to manage knowledge through graph-based
operations. They differ from other graph-based semantic knowledge
representations by the clear distinction between ontological knowledge
and factual knowledge which ensures conformity of reasoning with FOL
formulas. CGs have many applications in research and industry, eg. in
security~\cite{fu2017privacy}, semi-structure data
modeling~\cite{varga2018conceptual}, software
development~\cite{vlasenko19}, clustering~\cite{perez2019review},
music~\cite{fowler2019john} or
decision-making~\cite{tremblay2017cognitive} to name a few. One drawback is the
major difficulty in designing a CG database without prior
expertise. It may be one of the reasons why there has been for a long
time the need of CG datasets of
quality~\cite{baget2006towards,croitoru2007conceptual}, in particular
for benchmarking. Indeed, existing CGs are either private
properties, small examples to
illustrate the formalism or specific use cases that only represent a
reduced part of the formalism, for instance with no ontological part.
As will be discussed in Section~\ref{sec:edla},
$T_{nat}$~\cite{baget_translation_2010} is a translation algorithm
from RDF(S)/OWL~\cite{manola2004rdf,brickley2014rdf,mcguinness2004owl}
to CGs. The quality of the resulting CG base depends on the quality of
the RDF(S)/OWL input base and only a part of the CG formalism is taken
into account. The notion of quality is here mainly understood as
the combination of two criteria:
variability, i.e. the fact that many datasets varying on several
characteristics can be generated from the same input, and
predictability, i.e. the fact that the characteristics of the
resulting database can be derived from the input without mining of the
base. Two additionnal criteria are used: expressiveness, i.e. how much of
the CG formalism is represented, and computational efficiency.
This paper proposes CG2A (Conceptual Graphs Generation Algorithm), an
algorithm generating a CG database from a set of constraints
corresponding to ontological knowledge. In essence, factual knowledge
is generated from the input ontological knowledge defined as a
vocabulary and a set of CGs with some label variables, called
$\gamma$-CGs.
The ontological knowledge thus constitutes an underlying model of the
generated dataset. A benefit is that the user has explicit
knowledge on datasets generated from this model, without
analysis of the generated datasets. It is inspired by the benchmark
generation process in the clustering community, where synthetic datasets are
generated from given data distributions that determine expected
results for a clustering algorithm running on these datasets. It has been used to validate cgSpan \cite{faci_cgSpan_21}, an algorithm proposed to mine frequent patterns in CGs.
In order to generate realistic datasets, without a total randomization
of labels and structure, ontological knowledge is
required as input. This corresponds to constraints on the generated CGs domain. Still
this input can be generated automatically from a reduced set of
numerical parameters based on three proposed extensions to the
algorithm, respectively automating the generation of the vocabulary,
the $\gamma$-CGs and the $\gamma$-CGs variables. The generated CGs
domain is therefore extended to all CGs that can be defined over the
ontological knowledge generated from the given set of numerical
parameters.
The use of these extensions requires further analysis to establish the
same quantity of ontological knowledge and thereby of expected
results, i.e. reduces predictability.
Consequently the CG2A version to use depends on the use case: on
one hand it is possible to define all input ontological knowledge or
reuse an existing one to represent a specific situation; on the other
hand the use of automatically generated input enables a swift CG
generation and leads to more variability.
Section~\ref{sec:edla} presents a short reminder about the Conceptual Graphs formalism, including the proposed $\gamma$-CGs, and a state of the art on CG datasets generation. Section~\ref{sec:CG2A} presents the proposed Conceptual Graphs Generation Algorithm as well as its randomzsation modules. Section~\ref{sec:expe} describes the conducted experimental study, detailing the proposed criteria, to measure numerically variability and efficiency, and to assess qualitatively immediate predictability and representativity of CGs formalism. Section~\ref{sec:concl} concludes the paper and discusses some directions for future works.
\section{State of the art}\label{sec:edla}
\subsection{Conceptual Graphs}\label{ssec:CG}
{Conceptual graphs~\cite{chein_conceptual_2008} are a family of formalisms for knowledge representation, made of ontological and factual knowledge. A CG is a bipartite graph representing factual knowledge referring to a vocabulary that represents the ontological knowledge.}
A vocabulary is a {5-tuple\textit{ $\mathcal{V}$ = $(T_C,T_R,\sigma,I,\tau)$}}.~$T_C$ and $T_R$, that respectively correspond to concept and relation types, are two partially ordered disjoint finite sets, where ordering corresponds to generalisation. $T_C$ contains a greatest element $\top$. Each relation type has an associated arity; which subdivides $T_R$ in subsets regrouping types of equal arity. $\sigma$ is a mapping associating a signature with each relation type, i.e. a function with constraints on the type of arguments, where a more specific relation type is mapped with a more restrictive signature respectively for each argument. $\sigma(r)$ returns $(t_1, \ldots, t_n)$ where $n$ is $r$ arity and the $t_i$ are elements of $T_C$.
For $c$ connected to $r$, $\sigma(r)(c)$ denotes the type restriction matching $c$. $I$ is a set of individual markers used to instantiate concept nodes.~$\tau$ is a mapping from $I$ to $T_C$ that defines the type instantiated by each individual marker.
A CG is a bipartite labeled multigraph represented as a 4-tuple
\textit{G = (C,R,E,label)} defined over such a
vocabulary~$\mathcal{V}$. $C$ and $R$ correspond to concept and
relation nodes, $E$ denotes the set of the edges connecting elements
of $C$ and~$R$. $label$ is a labelling function from $C$ to
$T_C\times I$ and from $R$ to $T_R$. For any $r\in R$,
$label(r)=t_r\in T_R$ is the type of $r$ and for any $c\in C$,
$label(c)=(t_c,i_c)\in T_C\times I$ where $t_c$ is the type of $c$
and $i_c$ is the optional individual marker of $c$.
We extend this formalism to represent CGs where some labels are
replaced with variables, named $\gamma$-Conceptual Graphs and
inspired by $\lambda$-BGs from the CG
formalism~\cite{chein_conceptual_2008}.~A $\gamma$-CG
$\Gamma=((v_1,D_1) \ldots (v_n,D_n)) G$, $n \geq 1$ is a conceptual
graph $G$ with $n$ variables $v_i$ and their respective domains
$D_i$. Each variable $v_i$ is assigned to a label of~$G$, either a
relation type label, concept type label or marker label. It is
illustrated in Fig.~\ref{fig:CG2AAutoVar} where $v_1$, $v_2$ and
$v_3$ are respectively assigned to a concept type, marker and
relation type. For a variable $v_i$ associated with a relation type
of $r$ with $label(r)=t_r$, its domain~$D_i$ is a subset of $T_R$
reduced to relation types of same arity, i.e.
$D_i = \{t \in~T_R, arity(t) = arity(t_r)\}$. For a variable $v_i$
associated with a concept type of $c$ with $label(c)=(t_c,i_c)$, its
domain $D_i$ is a subset of $T_C$ reduced to the concept types
respecting all constraints imposed by the signatures of connected
relation nodes, i.e.
$D_i = \{t \in~T_C, \forall~r \in~R, (c,r) \in~E, t \leq~
\sigma(r)(c)\}$. For a variable $v_i$ associated with a marker
$m_i$, the domain is a subset of $I$ reduced to markers of same or
more specific concept types, i.e.
$D_i = \{m \in~I, type(m) \leq type(m_i)\}$.
Finally we define a neighborhood as a node and its connected nodes, for instance a relation node and its connected concept nodes.
\subsection{Conceptual graphs data generation}
To the best of our knowledge, there is no CG dataset of quality
available. Available CG datasets are for instance based on
flat hierarchy, i.e. with no order defined between types, as the
conceptual graph data-set for
NLP/NLU~\cite{elseidy2014grami}\footnote{\url{https://github.com/alexge233/conceptual_graph_set}},
or even actually not consistent with the CG formalism. There exit CG
datasets in industry but they remain company property, as they may be
the result of intense work
and may contain private data.
CG datasets can be obtained as the result of
translation algorithms to generate them from a dataset respecting
another formalism. The main differences with a proper generation
algorithm are that the goal is different and that the resulting dataset
depends on the chosen input dataset and its formalism. $T_{3}$
and $T_{nat}$~\cite{baget_translation_2010} are existing algorithms
translating knowledge datasets expressed in the
RDF(S)/OWL~\cite{manola2004rdf,brickley2014rdf,mcguinness2004owl}
formalism to knowledge datasets expressed in the CG formalism. They
are implemented in CoGUi\footnote{\url{http://www.lirmm.fr/cogui/}}, a
tool to visualize and manipulate CGs. Their main validation criterion
is the equivalence between reasoning in RDF(S)/OWL before translation
and reasoning in the CGs formalism after translation: they aim at
ensuring that the same conclusions are deduced from the same premises
in both datasets, and that reasoning remains identical when translating
back to RDF(S)/OWL. In this regard, $T_{3}$ is a sound and complete
translation w.r.t. RDF(S) but not intuitive visually. Indeed, it represents the RDF(S)/OWL triplets
constituted of subject, object and predicate by a blank relation node
linking these three elements as concept nodes. As a consequence the
fact that relations in CGs correspond to relations between entities is
not represented. It is more intuitive to represent relations nodes
connecting concepts nodes as predicates linking subject and
object, as is the case of~$T_{nat}$.
In addition $T_{nat}$ focuses on exploiting the separation between background knowledge and factual knowledge by translating the predicates as binary relation nodes linking the subject and object, both translated as concept nodes. This translation ensures two properties that enable a better representation of CGs but hinders the reasoning equivalence. First, a \textit{separability condition} has to be satisfied by the input RDF(S)/OWL dataset: it states that any entity in the knowledge base appears either as a class, a property or an instance (in the RDF(S) sense). Otherwise the entity is considered ambiguous and different choices are made depending on the situation: if a violation of this separation requirement between classes and properties occurs, the {ambiguous predicates are ignored}: if a violation occurs between {classes and instances, or properties and instances}, the triples involving the ambiguous entity as an instance are ignored.
Second, a distinction between ontological and factual triples is performed to populate either the vocabulary or the conceptual graphs when a new triple is processed. This distinction stems from {the flexibility of RDF(S) that does not impose a clear distinction} between factual and ontological knowledge.
A particularity of GC databases constituted with~$T_{nat}$ is that only relations of arity~2 are built
because of RDF(S) restrictions. This drawback is minimised by the fact that relation of arity greater than 2 can always be brought back to a set of relations of arity 2, and conversely. This is immediate considering that CGs are graph-based representations of first-order logic formulas and that relations correspond to atomic formulas, which are 2-decomposable~\cite{jeavons1998constraints}.
\section{CG2A: generation from a set of constraints}\label{sec:CG2A}
CG2A is a three step algorithm that generates a CG dataset from
ontological knowledge. It ensures representativity of the CG formalism
as well as variability and immediate predictability of the generated
base characteristics. First CG2A generates a CG by randomly combining
input $\gamma$-CGs until reaching a specified minimum size. Then
variables are assigned random values from their respective
domains. Finally the nodes in the generated CGs with the same
individual marker are merged to increase the connectivity of the resulting CG. CG2A iterates until a specified number of CGs is reached.
This section first describes CG2A input and details its three steps. It then presents its extension modules automating the generation of input.
\subsection{Input}\label{ssec:input}
CG2A, in its default mode, takes five parameters. They include the number of CGs to be generated, \textit{maxCGs}, the minimum size, in number of nodes, for each generated CG, \textit{minSize}, and the maximum number of specializations to be operated on each variable assigned to a type label, \textit{maxSpe}. They are used in the stopping conditions of the algorithm. The two other parameters are a vocabulary $\mathcal{V}$ and a set of $\gamma$-CGs $\mathcal{G}$, as detailed hereinafter.
The vocabulary, as formally presented in Section~\ref{ssec:CG}, contains a hierarchy on concept types, a hierarchy on relation types and a set of signatures corresponding to the relation types in the hierarchy. The individual markers set is populated during generation when a concept node is instantiated. The set~$\mathcal{G}$ of $\gamma$-CGs are the components of the generated CGs. Compared to a classic CG, as described in Section~\ref{ssec:CG}, some labels are replaced with a variable referring to a list of values from the vocabulary. Thus $\gamma$-CGs are configurable constraints.
\subsection{Proposed algorithm}\label{ssec:CG2A}
Fig.~\ref{fig:CG2A} gives the pseudo-code of CG2A, commented below:
CG2A generates sets of CGs by randomly combining elements from the set~$\mathcal{G}$ of input $\gamma$-CGs into bigger CGs.
Let $G_c=(C_c,R_c,E_c,label_c)$ be the currently generated CG and
$\Gamma = ((v_1,D_1) \ldots (v_n,D_n)) G=(C,R,E,label)$ be a
$\gamma$-CG from the input set $\mathcal{G}$. First $G$ variables are
instantiated with values from their domains, and the ones assigned to
type labels are specialized from 0 to \textit{maxSpe} times using
hierarchies from~$\mathcal{V}$. Then $G_o = (C_c\uplus C, R_c\cup R,
E_c\uplus E, label_c\uplus label)$ is formed from the join of $G_c$
and $G$ where $\uplus$, based on coreferent nodes
merge~\cite{chein2004concept}, is a specific union whose differences
follow: if there are elements of $C_c$ with an individual marker
similar to one of $C$, only the most specialised is kept. Then
neighborhoods are merged so that both neighborhoods are connected to
the resulting node, i.e. elements of $E_c$ and $E$ corresponding to
the two merged nodes are reassigned to the resulting node.
\begin{figure}
Input: $\mathcal{V}$ = $(T_C,T_R,\sigma,I,\tau)$, $\mathcal{G}$,
\\\phantom{Input:} \textit{maxCGs}, \textit{minSize}, \textit{maxSpe}.
\begin{itemize}
\setlength\itemsep{0.5em}
\item Initialize $\mathcal{G}_o$ to an empty set
\item Iterate until size($\mathcal{G}_o$)$\geq$\textit{maxCGs}
\begin{itemize}
\setlength\itemsep{0.2em}
\item Initialize $G_c=(C_c,R_c,E_c,label_c)$ to an empty CG
\item Iterate until $size(G_c)\geq minSize$
\begin{enumerate}
\item Get $(v_1,\ldots,v_n)G=(C,R,E,label)$ in $\mathcal{G}$
\item Attribute value to each variable $v_i$
\item Specialize each type label var from 0 to \textit{maxSpe} times
\item $G_c = Join(G_c,G)$
\end{enumerate}
\item Add $G_c$ to $\mathcal{G}_o$
\end{itemize}
\item Return $\mathcal{G}_o$
\end{itemize}
\caption{Pseudo-code of the proposed CG2A.}
\label{fig:CG2A}
\end{figure}
This join operator is illustrated in Fig.~\ref{fig:CG2AFusion}, where
the node colour indicates their associated markers: the two green
nodes, resp. at the right end of the current CG and at the top of the
added CG, are merged. They are not necessarily of same type; the most
specific type is retained, indeed as illustrated in the example, the
connected signatures enforce a specialisation of this
type.
Without operator $\uplus$, the algorithm would obtain for each
generated CG a set of unconnected instantiated elements from
$\mathcal{G}$. The connectivity of the resulting CGs thus depends on
the number of common nodes. There are other techniques available for
graph fusion based on the join
operator~\cite{laudy2007high,chein2014conceptual}, but this simple
fusion operator based on coreferent nodes merge operator is sufficient
in this case.
CG2A stops CGs combinations upon reaching the desired minimum size,
\textit{minSize} and stops generation upon reaching the desired number
of generated CGs, \textit{maxCGs}. Since CGs of potentially several
nodes are added at the same time, the resulting CGs are typically
greater than \textit{minSize}.
The advantages of using $\gamma$-CGs instead of directly defining many
variants of a CG is that the process is automatic and that from one
designed $\gamma$-CG, many can be generated while keeping its
structure and its semantic. As a consequence CG2A guarantees
variability from one input as well as predictability thanks to
knowledge of input $\gamma$-CGs and their characteristics.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{CG2AFusion.pdf}
\caption{ CG join step in CG2A. In this representation, concept nodes are squares and relation nodes are circles.}
\label{fig:CG2AFusion}
\end{center}
\end{figure}
\subsection
Input generation to increase variability}\label{ssec:random}
This section presents three modules to generate automatically the
input so as to increase variability and ease the generation. All
mentioned numerical parameters can be replaced by mean
and standard deviation, and drawn from
the associated normal distribution.
\subsubsection{Automatic generation of vocabulary }\label{sssec:randVoc}
This module generates automatically the vocabulary~$\mathcal V$ from four
parameters: the desired depth of hierarchies both for concept and
relation types, the maximum number of children of each node of the
hierarchy and the number of individual markers for each concept
type. This generation is random, however the four parameters ensure a
number of fixed characteristics in the resulting vocabulary.
As illustrated on Fig.~\ref{fig:CG2AAutoVoc} and
\ref{fig:CG2AAutoSign}, for concepts and relations respectively, a
hierarchical structure is generated until the desired depth is reached
and random unique labels are assigned to each node of the hierarchy.
The hierarchy of concept types is a rooted tree with the most general
type $\top$, denoted "Top" in Fig.~\ref{fig:CG2AAutoVoc}. For each
concept type, a list of individual markers is generated. For relation
types, a top type is respectively defined for each arity, e.g. denoted
$T_3$ for the case of arity 3 illustrated in
Fig.~\ref{fig:CG2AAutoSign}. Then signatures are defined using the
previously generated hierarchy of concepts for each relation type,
with each relation top type having a default signature with only
$\top$ as concept type restriction. A more specific relation type has
a more restrictive signature, meaning that the specified restrictions
require an identical or more specific concept type. It is illustrated
in Fig.~\ref{fig:CG2AAutoSign} where at each step the hierarchy is
deepened and signatures are defined as identical or more restrictive
than signatures of more general relation types.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{CG2AAutoVoc.pdf}
\caption{ Automatic generation of a hierarchy of concept types, here performed in three steps.
Parameters are: Depth = 4; Maximum number of children = 3; Number of individual markers per type = 3.}
\label{fig:CG2AAutoVoc}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{CG2AAutoSign.pdf}
\caption{ Automatic generation of a hierarchy of relation types with their signatures, here performed in two steps.
Parameters are: Depth = 3; Maximum number of children = 3.}
\label{fig:CG2AAutoSign}
\end{center}
\end{figure}
\subsubsection{Automatic generation of input $\gamma$-CGs}\label{sssec:randCG}
This module generates automatically a set of input $\gamma$-CGs so as to define $\mathcal{G}$, as illustrated in Fig.~\ref{fig:CG2AAutoCG}. The generated $\gamma$-CGs actually have no defined variable, but as this module can be used independently, one can subsequently define variables manually or use automatic generation. This module takes as input a vocabulary~$\mathcal{V}$ (possibly generated automatically using the module described in the previous subsection~\ref{sssec:randVoc}) and two numbers: the number of $\gamma$-CGs to be generated and their minimum size. While designing CGs requires proficiency in the formalism and also requires to respect of the vocabulary constraints, only two numbers restrict the domain of $\gamma$-CGs generation.
The $\gamma$-CGs module is similar to applying CG2A to the set of signatures taken from the input vocabulary $\mathcal{V}$. The only difference is that each label is treated as a variable with no constraint, so that they all have a randomly attributed label and are randomly specialized.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{CG2AAutoCG.pdf}
\caption{Automatic generation of input $\gamma$-CGs. }
\label{fig:CG2AAutoCG}
\end{center}
\end{figure}
\subsubsection{Automatic generation of variables}\label{sssec:randVar}
This module generates variables in input $\gamma$-CGs. Instead of having to choose which labels are variables and their respective domains, this module takes a set of $\gamma$-CGs and a matching vocabulary as input, which can be generated with the previous modules, as well as five numbers: the numbers of concept types, relation types and individual marker variables per CG, the number of values per variable and the number of specialisations. It may be run even if variables have already been defined in the $\gamma$-CGs to increase their number.
First, for each CG, variables are attributed to a relation type, a
concept type or an individual marker. Then a list of values is
associated with each variable. Fig.~\ref{fig:CG2AAutoVar} illustrates
this operation with the variables $v_1,v_2$ and $v_3$. For a relation
type as $v_3$ in Fig.~\ref{fig:CG2AAutoVar}, the module chooses from
relations with the same arity and identical or less restrictive
concept types. For a concept type as $v_1$ in
Fig.~\ref{fig:CG2AAutoVar}, the module chooses from concept types
equal to or more specific than the ones compatible with the signatures of
the neighborhood. For an individual marker as $v_2$ in
Fig.~\ref{fig:CG2AAutoVar}, it chooses from individual markers with an
assigned concept type equal to or more specific than the concept node
type. Because of these restrictions, first relation type variables
are defined, then concept type variables and finally individual marker
variables. Then, all assigned variables corresponding to type labels
are specialized a number of times up to the number of specialisations
parameter.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{CG2AAutoVar.pdf}
\caption{ Automatic generation of $\gamma$-CGs variables.}
\label{fig:CG2AAutoVar}
\end{center}
\end{figure}
\section{Experimental study}\label{sec:expe}
This section presents the experimental study conducted to evaluate CG2A both per se and compared to existing translation techniques.
The input used in these experiments as well as the desired properties of variability, predictability, representativity and efficiency are presented in the following subsection. Then results are subsequently examined in view of each criterion.
\subsection{Experimental protocol}\label{ssec:prot}
We consider four criteria to optimize for
a data generation algorithm. First the algorithm has to enable
\emph{variability} in the generated data from one input. Indeed from a
unique ensemble of ontological knowledge the possibility to produce
datasets with various sizes and characteristics may be required to
assess the breadth of corresponding possible facts. Second it has to
provide a certain level of \emph{immediate predictability}. This means that
from a given data generation algorithm, denoted dGenA,
the expected results of a data mining algorithm, denoted dMinA,
run on a database generated by dGenA can be defined. Obviously the
expected results will differ depending on dMinA goal and whether all
that can be deduced from the dGenA input is relevant regarding this
goal. This criterion is essential to enable dMinA validation but is
difficult to quantify. Third the generation algorithm has to achieve
\emph{representativity}, i.e. exploit as much of the CG formalism as
possible. Any fragment that cannot be represented limits the use of
the generation algorithm to the subset of situations where this fragment is
unnecessary. Fourth the \emph{computational time}
has to be minimised. Indeed it may be crucial to
obtain quickly a set of examples datasets when given ontological
knowledge. Besides many generations may be required while testing
parameters variations to satisfy expectations.
These criteria are used to compare CG2A with $T_{nat}$ using the
following configuration. The input of $T_{nat}$ is an RDF(S)/OWL
dataset\footnote{\url{http://www.semanticbible.com/ntn/ntn-view.html}},
modified to resolve some issues when parsing for translation. The
modified dataset includes an ontology constituted of 39 concept types
and 35 relation types organized in a hierarchy with a depth of 5 and
between 1 and 3 children for each non-leaf type. An option proposed by
$T_{nat}$ to split CGs in several connected components has been used,
so that each resulting CG is a connected graph. Otherwise it results
in one big unconnected CG. The input for CG2A
matches the characteristics of~$T_{nat}$ produced base to ensure that
we mostly evaluate the influence of the algorithmic part rather than
the variability induced by parameters. Running $T_3$ on the same
dataset results in a unique CG of about 6000 vertices and no ontology
(other than RDF/RDF(S) knowledge). All relation nodes are the "triple"
relation node to connect elements of a triple. As it does not lead to
many CGs nor a proper ontology, $T_3$ is irrelevant for our concerns
and in consequence its results are not used in what follows.
The first row of Tab.~\ref{table1} displays the results of $T_{nat}$
and the following ones display the average results across 100 runs on CG2A and
its variants with each extension module individually (Auto Voc, Auto
$\gamma$CG and Auto Var respectively) and CG2A with all extension modules
(Full Auto).
\subsection{Variability results}
In order to assess numerically the notion of variability, we consider
the following criteria: the average size of generated CGs in number of
nodes, denoted~NbN, and the average number of unique labels, denoted
NbL, in one CG, both with their standard deviation, and the
distribution of relation arity from 1 to 3, denoted Ar1, Ar2 and Ar3.
These criteria are defined for the sake of comparison with existing
techniques providing CG datasets. They are not optimized or even considered by the
translation algorithm $T_{nat}$ that aims at maximizing conformity of reasoning between the input and
output databases. CG2A and its variants can represent
relation of arities greater than 3, but the use Ar1, Ar2 and Ar3 seems
to suffice for the presented experiments.
It can first be observed that the translation of $T_{nat}$ results in
a huge CG of hundreds of nodes and many CGs of a few nodes. This is
why in Tab.~\ref{table1} the standard deviation of NbN is
significantly more important than the average NbN, and the average NbL
is relatively small. Moreover Ar1 and Ar3 are zero, which is due to
the fact that the RDF(S)/OWL languages do not represent relations of
arities different than 2.
CG2A leads to significantly smaller standard deviations, and the resulting
CGs characteristics are close to the input parameters.
It is expected that the use of automatically generated input leads to
more variability in the expected results. It can be observed that,
indeed, the use of random vocabulary increases NbL standard
deviation and relation arities while the use of random CGs increases
the standard deviation of NbN and arities. The results of the fully
automated CG2A combine these consequences.
\begin{table*}[t]
\begin{center}
\vspace{1ex}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Test} & \textbf{NbN (avg. $\pm$ sd.)} & \textbf{NbL (avg. $\pm$ sd.)} & \textbf{Ar1} & \textbf{Ar2} & \textbf{Ar3}\\
\hline
$T_{nat}$ & 15.2 $\pm$ 321 & 3 $\pm$ 1 & 0 & 3 & 0\\
\hline
CG2A & 36.3 $\pm$ 4 & 22.5 $\pm$ 4 & 0.5 & 44 & 3\\
\hline
Auto Voc & 33 $\pm$ 3.5 & 55 $\pm$ 14 & 4 & 34 & 9\\
\hline
Auto $\gamma$CG & 39.9 $\pm$ 2 & 22 $\pm$ 4.1 & 6 & 22 & 31\\
\hline
Auto Var & 35.3 $\pm$ 4 & 32 $\pm$ 7 & 0.4 & 42 & 7\\
\hline
Full Auto & 35 $\pm$ 4 & 67 $\pm$ 17 & 8 & 33 & 26\\
\hline
\end{tabular}
\caption{Results for one run of $T_{nat}$ and different versions of CG2A averaged across 100 runs (see experimental protocol in Sec.~\ref{ssec:prot}).}
\label{table1}
\end{center}
\end{table*}
\subsection{Predictability results}
Predictability refers to the possibility to define a priori the
results a data mining algorithm is expected to obtain when run on a
generated data set. It can be put in balance with the cost of the
specific resources to deploy and efforts to undertake to define the
input, in particular the ontological knowledge.
$T_{nat}$ possesses the advantage that it generates a CG database from
a RDF(S)/OWL dataset, without requiring any prior
knowledge on this dataset. However this implies that without mining
the input dataset first, $T_{nat}$ cannot be considered as
predictable. As such, it does not meet the immediate predictability
aim.
CG2A can be considered as predictable as the generated CG are defined as combination of the input $\gamma$-CG that are defined over the input vocabulary: $\mathcal G$ together with $\mathcal V$ determine the expected results whose characteristics are known.
CG2A used with automatic generation of vocabulary or $\gamma$-CGs changes the nature of the expected results, that are defined in terms of their general characteristics rather than specific information.
As compared to CG2A, in the case of Auto~Voc, only the general characteristics of the vocabulary are known, its specificities are not. In the case of Auto~$\gamma$-CG, only the general characteristics of the components used to build the generated CG are known (number and size of the $\gamma$-CGs, as well as the signatures they are built on): Auto~$\gamma$-CG adds to the CG2A variation from $\gamma$-CGs to generated CGs another variation, from the signatures to the input $\gamma$-CGs.
CG2A used with automatic variable generation, Auto~Var, does not modify predictability
significantly. We consider that automatic variables slightly reduce
the immediate predictability by increasing variability.
The fully automated CG2A variant, that includes the three
random modules, combines their respective properties and defines expected results in terms of their general characteristics.
Overall, the results are significant when compared with $T_{nat}$:
there is much more variability with CG2A, and while CG2A has to cope
with a balance between variability and immediate predictability,
$T_{nat}$ does not enable immediate expected results.
\subsection{Representativity results}
As discussed earlier, $T_{nat}$ outputs CG databases only including relations of arity 2, that are unbalanced with respect to ontological or factual knowledge, i.e. that mostly comprehend one of the two types. Yet, if can be argued that the advantage of disposing of relations with varying arity only is a question of perspective or reformulation and that the lack of balance is due to the considered input RDF(S)/OWL knowledge bases. Similarly, the fact that $T_{nat}$ often results in bases constituted of one or two huge conceptual graphs and the rest containing only a few nodes is mostly due to the available RDF(S)/OWL bases, rather than the algorithm itself.
CG2A and its variants more naturally avoid these drawbacks. They enable the representation of most of the CG formalism as reminded in Sec.~\ref{sec:edla}. CG2A retains most of $T_{nat}$
advantages by using the CoGui formalism and adds the possibility to
generate a large proportion of relation nodes with various arities,
and to have both a wide vocabulary and a considerable quantity of CGs,
i.e. both ontological and factual knowledge. Besides when defining
the input, e.g. the characteristics of the vocabulary, the user can determine the extent of the CGs formalisms that is exploited, which is one main advantage of CG2A.
Generally speaking, CG2A ensures that the user can choose more precisely the characteristics of the resulting base.
\subsection{Efficiency results}
In the conducted experiments, depending on the stopping conditions parameters, most CG2A runs last less than one second and never exceed 5 seconds. The use of the automatic generation modules increases the computational time, with a factor~2, however the time spent to design input without these modules is not accounted for. $T_{nat}$ generation lasts much longer, with factor of three up to ten depending on the size of the input base. At some point if the input is too massive, $T_{nat}$ aborts so the tests could not be pursued.
\section{Conclusion}\label{sec:concl}
This papers proposes CG2A, an algorithm to produce Conceptual Graphs.
CG2A enables more variability in the generated dataset than any other known method as it offers a lot of variance in the size and labels of CGs as well as a reasonable proportion of relation nodes with an arity different than 2 in the generated CGs. As such, numerous different situations can be tested through the use of CG2A, either a strongly constrained domain to test a specific case or a more relaxed generation to test a broad variety of situations. In addition the CG formalism is well represented with deep hierarchies, variation in the signatures and various arities of relation nodes.
Finally, when using a different method to obtain a CG dataset, it is not possible to define expected results without first mining the dataset or having prior knowledge on this dataset.
Ongoing works aim at extending CG2A to generate more complex CGs, e.g. nested or fuzzy CGs. Another direction considers a different possibility in terms of predictability, expressed as a desired distribution over CGs parameters so as to ensure that the resulting dataset respects such a distribution.
\bibliographystyle{eusflat2021}
|
2,877,628,090,796 | arxiv | \section{Introduction}
\label{sec_intro}
Magnetic fluid hyperthermia (MFH) has gained wide interest in its applicability in medical sciences \cite{biomedical}. Enhancing specific absorption loss (SAR) by core-shell nanostructures, see e.g. \cite{boubeta,lee} or via anisotropy \cite{bertotti, poperechny} are also extensively studied. Furthermore, efficiency of circularly polarized field opposed to linearly polarized field is also investigated by models \cite{mazsipeter,cantillon-murphy,mazsijudit, raikher,denisov2006prb,sun,denisov2006prl} and experiments \cite{ahsen, jordan} as well. An interesting study of the chain formation of nanoparticles can be read in the paper of He \cite{he}, examining the possibility of data storage.
In MFH, the basic question is how to describe energy losses. In the literature relaxation and hysteresis losses are distinguished though both proportional to the area of hysteresis curve. See also the comment about it in the paper of Carrey \textit{et al.} \cite{carrey}, in which they also argue the necessity of distinction. The relaxation losses could originate either turning the single-domain particle with its magnetic momentum, called Brownian relaxation, or the nanoparticle itself is fixed but its magnetic momentum aligns to the external field, as in the case of N\'{e}el relaxation. There is a vivid dispute, whether both relaxations contribute to the losses and if so, to what extent. Also in the paper of Wang \textit{et al.}\cite{wang}, they compared SAR values for nanoparticles with and without polymerization, and they found no change in the SAR value excluding the possibility of Brownian relaxation. Furthermore from theoretical considerations, at typical size of nanoparticles up to few tens nanometers of diameters, depending on the experimental conditions, N\'{e}el relaxation is regarded to be the dominant process.
It is also a question, whether a single independent particle or statistical ensemble of particles should be taken into account. The single particle picture has the advantage of ab initio description, though in this case no temperature and therefore thermal fluctuation is included yet \cite{mazsipeter,mazsijudit}, while in the statistical picture the treatment is phenomenological \cite{shliomis,rosensweig1985}. Shliomis \cite{shliomis} provided a complex equation of motion of the magnetization from hydrodynamical considerations, but as we saw till now, radical simplifications were used leaving only the Debye-term of the equation, see e.g. paper of Cantillon-Murphy \textit{et al}. \cite{cantillon-murphy}. Stochasticity can also be included \cite{raikher} for a more elaborated picture of possible relaxation processes.
Aim of this paper is to test the validity of magnetization dynamics model with Larmor-precession term included \cite{jzsuzsa} in comparison with experimental results \cite{mehdaoui,wang,hergt,suto}. In the model the so-called Bloch-Bloembergen equation\cite{bloch, bloembergen} is rewritten for the single domain nanoparticle magnetization. We show that experimental results can be understood by the polidispersity of size of nanoparticles. Doe to analytic solution for both linearly and circularly polarized fields, the model is unique being valid without restrictions to any parameter value.
For example, in the paper of Mehdaoui \textit{et al.}\cite{mehdaoui}, a combined theoretical and experimental study was conducted, in which linear response theory and Stoner-Wohlfarth model were used, according to their range of validity. We provided reliable SAR values with our model both in the range of linear response theory and Stoner-Wohlfarth model.
For practical reason, we focus on hyperthermia application and carried out our analysis and evaluation at lower field strength and frequency region. In this case we can claim the equivalence of linearly and circularly polarized field for SAR. Finally, dominance of N\'{e}el relaxation with respect to Brownian relaxation is also discussed.
\section{Theoretical background}
\label{sec_theo}
\subsection{Basic equations of motion}
\label{subsec_eqs}
The basic equation of motion of the magnetization is given as,
\begin{equation}
d\bm{M}/dt = \gamma\bm{M} \times \bm{B}.
\label{dynamicsM}
\end{equation}
Here $\gamma$ is the gyromagnetic ratio with value $\gamma= - 1.76 \times 10^{11}$ Am$^2$/Js. It comes from eq.(\ref{dynamicsM}) that any change of the magnetization is perpendicular to $\bm{M}$ and for constant $\bm{B}$ the angle between the two vectors $\bm{M}$ and $\bm{B}$ is also constant. The solution is the precessing $\bm{M}$ around $\bm{B}$ for this equation of motion, yielding the \textit{Larmor precession}, where $\bm{\omega_L} = \gamma\bm{M}\times\bm{B}/M_{\bot}$. Here $M_{\bot}$ is the projection of $\bm{M}$ on the plane perpendicular to $\bm{B}$. The Larmor frequency is defined as a positive quantity, $\omega_L=|\gamma| B$.
Shliomis in $1974$ \cite{shliomis} has suggested the equation of exponential relaxation to be written for describing the behaviour of magnetic nanoparticles in ferrofluids at given conditions,
\begin{equation}
\frac{d\bm{M}}{dt}=-\frac{\bm{M}-\bm{M}_{eq}}{\tau},
\label{debye}
\end{equation}
\noindent where $\bm{M}$ is the average magnetization of the particles,
\begin{equation}
\bm{M}_{eq}=M_S {\cal L}\left( \frac{\mu_0 HM_d V}{kT} \right)\hat{\bm{e}}_H,
\label{meq}
\end{equation}
\noindent $V$ is the particle volume, $\hat{\bm{e}}_H=\bm{H}/H$ is the unit vector along $\bm{H}$ and $M_S$ is the saturation magnetization of the colloid $M_S=\phi M_d$, with volume fraction $\phi$ and the single-domain magnetic nanoparticle magnetization $M_d$. The Langevin function, ${\cal L}(x)=\coth(x)-1/x$, gives the magnitude of the magnetization in thermal equilibrium, where $M_{eq}$ must be an ensemble average and so is $\bm{M}$. The eq.(\ref{debye}) is the reduced form of the Shliomis relaxation equation \cite{shliomis} and often called Debye relaxation equation. Susceptibility is frequently approximated with Rosensweig's chord susceptibility \cite{rosensweig2002},
\begin{equation}
\chi_{ch}=\frac{M_S}{H_0} {\cal L}\left( \frac{\mu_0 H_0 M_d V}{kT} \right)
\label{chord}
\end{equation}
\noindent instead of the Langevin function appearing in eq.(\ref{meq}), where actual magnetic field is used contrary to its maximum.
The Debye relaxation equation, eq.(\ref{debye}), can be regarded as a simplified version of the Bloch-Bloembergen equation \cite{bloch, bloembergen}, which describes nuclear magnetic resonance experiments \cite{bloch} and ferromagnetic resonance \cite{bloembergen}. The latter does include the gyromagnetic torque, $\mu_0 \gamma \bm{M}\times\bm{H}$. The Bloch-Bloembergen alike formalism was used in paper \cite{jzsuzsa} for the description of the dynamics of magnetization of paramagnetic nanoparticles in ferrofluids. It is pointed out that at field strengths and frequencies used in MFH, the linearly and circularly polarized fields can be equivalent up to a normalization constant, which holds for the interested hyperthermia region in this paper.
\subsection{Model equations}
\label{subsec_bb}
Analytical solutions of Bloch-Bloembergen alike equations are briefly summarized here for circularly and linearly polarized fields:
\begin{equation}
\frac{d\bm{M}}{dt}=\mu_0\gamma \bm{M}\times\bm{H} -\frac{\bm{M}-\bm{M}_{eq}}{\tau},
\label{debyewlarmor}
\end{equation}
\noindent with $\bm{M}_{eq}$ as given in eq.(\ref{meq}). For details of the derivation of equations, see paper \cite{jzsuzsa}.
Setting rotating magnetic field, $H_x=H_0 \cos(\omega t); H_y=H_0 \sin(\omega t)$, the solution is:
\begin{equation}
\bm{M}(t)=M_{eq}(H_0)\frac{\omega}{\Omega^2}\frac{1}{1+(\Omega\tau)^2}
\begin{pmatrix}
\omega\cos(\omega t)+\Omega^2\tau\sin(\omega t)\\ \omega\sin(\omega t)-\Omega^2\tau\cos(\omega t)\\ -\omega_L
\end{pmatrix},
\label{analfinal}
\end{equation}
\noindent where $\Omega = \sqrt{\omega^2+\omega_L^2}$ and $\omega_L=\mu_0|\gamma|H_0$. Calculating the energy loss per cycle,
\begin{eqnarray}
E&=&-\mu_0 \int_0^{0+2\pi/\omega} \bm{M}\cdot\frac{d\bm{H}}{dt}\ dt\nonumber\\
&=&2\pi\mu_0 M_{eq}(H_0)H_0 \frac{\omega}{\Omega}\frac{\Omega\tau}{1+(\Omega\tau)^2}.
\label{energyCirc}
\end{eqnarray}
\noindent The specific absorption rate (SAR) is defined as energy loss for unit time and mass:
\begin{equation}
SAR \doteq \frac{E\omega}{2\pi\rho}=\frac{\mu_0 M_{eq}(H_0)H_0}{\rho}\frac{\omega^2}{\Omega}\frac{\Omega\tau}{1+(\Omega\tau)^2}.
\label{SARCirc}
\end{equation}
In case of linearly polarized field, $\bm{H}(t)=(0,0,H_0\cos(\omega t))$, the solution for $M_z(t)$ is found in the form of effective magnetization:
\begin{equation}
M_z^{eff}(t)=2M_S\sum_{m=1}^{\infty}\frac{1}{(m!)^2}B_{2m}\zeta^{2m-1}\frac{\cos(\omega t)+\omega\tau\sin(\omega t)}{1+(\omega\tau)^2},
\label{effsolz}
\end{equation}
\noindent where $B_{2m}$ and $\zeta^{2m-1}$ are the Bernoulli numbers [Ref.~\onlinecite{gradshteyn}, p.1040 and p.1045] and the argument of the Langevin function $\zeta \doteq \mu_0 H_0 M_d V/kT$ respectively.
The energy loss per cycle is
\begin{widetext}
\begin{equation}
E=-\mu_0 \int^{t_0+2\pi/\omega}_{t_0} \bm{M}\cdot\frac{d\bm{H}}{dt}dt=-\mu_0 \int^{t_0+2\pi/\omega}_{t_0} M_z^{eff}\cdot\frac{dH_z}{dt}dt=\pi\mu_0\chi_{loss}(\zeta)H_0\frac{\omega\tau}{1+(\omega\tau)^2},
\label{energyLin}
\end{equation}
\end{widetext}
and the corresponding SAR is
\begin{equation}
SAR \doteq \frac{E\omega}{2\pi\rho}=\frac{\mu_0 \chi_{loss}(\zeta)H_0}{2\rho}\omega\frac{\omega\tau}{1+(\omega\tau)^2}.
\label{SARLin}
\end{equation}
\section{Comparisons with experimental results and discussion}
\label{sec_results}
In this chapter our model predictions are compered to experimental data. All the parameters can be found in more detail in references. First of all, we highlight the paper of Mehdaoui \textit{et al.} \cite{mehdaoui}, because they included very detailed information on polydispersity of their samples and thorough analysis of their measurement. It provided great opportunity to test our model with polydispersity built in. Excellent agreement was found and it encouraged us to analize other resutls when polydispersity is mentioned but no additional data are given. Log-normal distribution was used like in the paper of Mehdaoui \textit{et al.} \cite{mehdaoui}, as particle size could take only non-negative values. Fitting log-normal distribution to experimental data, we can conclude not only the possible standard deviations from the average value but also might distinguish among samples fabricated by different processes.
Analyzing samples from Mehdaoui \textit{et al.}, see Table \ref{tab:mehdaoui}, we found excellent agreement both for SAR and standard deviation reproducing also the particle size distribution with our model. Mehdaoui \textit{et al.} used samples with mean radius $2.8$nm, $9.85$nm and $13.75$nm and found ferromagnetic behavior with $r\ge 7$nm. Table \ref{tab:mehdaoui} shows an other very important feature of our model providing reliable SAR values out of the range of linear response theory, where originally Stoner-Wohlfarth model was used instead. Accordingly we carried out analytical simulation for direct comparisons with experiments instead of numerical ones. Finally, we included polydispersity in our model.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
& $\mathbf{r= 2.8nm}$ & $\mathbf{r= 9.85nm}$ & $\mathbf{r= 13.75nm}$ \\
\hline
$SAR_{Mehdaoui} (W/g)$ & 20 & 290 & 40 \\
\hline
$SAR_{N} (W/g)$ & 21 & 291 & 42.5 \\
\hline
$\sigma_{Mehdaoui}$ & 0.2 & 0.09 & 0.2 \\
\hline
$\sigma_{calc}$ & 0.16 & 0.13 & 0.19 \\
\hline
$\pm\delta_{calc}(nm)$ & -1.2,\ +1.8 & -3.2, +5.3 & -6.3, 11.8 \\
\hline
\end{tabular}
\caption{SAR (W/g) values are compared for different nanoparticles sizes. SAR from paper Mehdaoui \textit{et al.} \cite{mehdaoui} ($SAR_{Mehdaoui}$) and SAR calculated taking N\'{e}el-relaxation time ($SAR_{N}$) only are shown. Log-normal distribution is taken. Their standard deviations are also given by Mehdaoui \textit{et al.} ($\sigma_{Mehdaoui}$) and by us ($\sigma_{calc}$). The possible deviations from the mean value ($\pm\delta_{calc}$) are given in the last row. }
\label{tab:mehdaoui}
\end{table}
In Table \ref{tab:nopoly} experimental SAR data from Wang \textit{et al.} \cite{wang}, Hergt \textit{et al.} \cite{hergt} and Suto \textit{et al.} \cite{suto} are compared to model predictions calculated without polydispersity. It can be seen, the predictions are at the order of experiments, but inconsistently under- or overestimating them. Data also show that taking into account only Brownian relaxation results in extremely poor agreement among model predictions and experiments. Including polydispersity in calculations results in drastic improvement in the consistency of experimental data and model prediction, the error is less than $5\%$ in most cases. In case of pure Brownian relaxation process the agreement is poor again. Based on these results, the scenario of pure Brownian relaxation process can be excluded. Whether Brownian relaxation process is still contributing to the relaxation processes or only N\'{e}el relaxation should be considered, cannot be decided. The first possibility slightly overestimates, the latter one underestimates the experiments.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
& \textbf{Wang \textit{et al.}} & \textbf{Hergt \textit{et al.}} & \textbf{Suto \textit{et al.}} \\
\hline
$SAR_{Paper} (W/g)$ & 123 & 45 & 22 \\
\hline
$SAR_{N} (W/g)$ & 3 & 19 & 30 \\
\hline
$SAR_{tot} (W/g)$ & 3 & 19 & 35 \\
\hline
$SAR_{B} (W/g)$ & 767 & 1313* & 5.5 \\
\hline
$r (nm)$ & 2.5 & 2.8 & 3.125 \\
\hline
\end{tabular}
\caption{Without polydispersity, SAR (W/g) values are compared for different papers. SAR from Wang \textit{et al.} \cite{wang}, Hergt \textit{et al.} \cite{hergt} and Suto \textit{et al.} \cite{suto} are listed. Samples were for Wang \textit{et al.} size of $r=5nm$ and for Suto \textit{et al.} the so-called $Sample A$. SAR calculated taking N\'{e}el-relaxation time ($SAR_{N}$) only, combined N\'{e}el- and Brownian-relaxation time ($SAR_{tot}$) and Brownian-relaxation time ($SAR_{B}$) only.
* No assumption for the value of hydrostatic volume is taken. }
\label{tab:nopoly}
\end{table}
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
& \textbf{Wang \textit{et al.}} & \textbf{Hergt \textit{et al.}} & \textbf{Suto \textit{et al.}} \\
\hline
$SAR_{Paper} (W/g)$ & 123 & 45 & 22 \\
\hline
$SAR_{N} (W/g)$ & 122 & 47 & 19 \\
\hline
$SAR_{tot} (W/g)$ & 129 & 47 & 23 \\
\hline
$SAR_{B} (W/g)$ & 146 & 146* & 5.8 \\
\hline
$r (nm)$ & 2.5 & 2.8 & 3.125 \\
\hline
$\sigma_{calc}$ & 0.27 & 0.14 & 0.25 \\
\hline
$\pm\delta_{calc}(nm)$ & -1.5,\ +6.5 & -1.0, +3.0 & -2.0, +5.5 \\
\hline
\end{tabular}
\caption{With polydispersity, SAR (W/g) values are compared for different papers. SAR from Wang \textit{et al.} \cite{wang}, Hergt \textit{et al.} \cite{hergt} and Suto \textit{et al.} \cite{suto} are listed. Samples were for Wang \textit{et al.} size of $r=5nm$ and for Suto \textit{et al.} the so-called $Sample A$. SAR calculated taking N\'{e}el-relaxation time ($SAR_{N}$) only, combined N\'{e}el- and Brownian-relaxation time ($SAR_{tot}$) and Brownian-relaxation time ($SAR_{B}$) only. Their standard deviations ($\sigma_{calc}$) are calculated by us only. The possible deviations from the mean value ($\pm\delta_{calc}$) are given in the last row.
* No assumption for the value of hydrostatic volume is taken. }
\label{tab:withpoly}
\end{table}
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
& $\mathbf{r= 5nm}$ & $\mathbf{r= 4nm}$ & $\mathbf{r= 3nm}$ \\
\hline
$SAR_{Wang} (W/g)$ & 123 & 78 & 50 \\
\hline
$SAR_{N} (W/g)$ & 122 & 76 & 47 \\
\hline
$SAR_{tot} (W/g)$ & 129 & 82 & 53 \\
\hline
$SAR_{B} (W/g)$ & 146 & 496 & 304 \\
\hline
$\sigma_{calc}$ & 0.27 & 0.36 & 0.47 \\
\hline
$\pm\delta_{calc}(nm)$ & -2.6,\ +7.4 & -2.0, +7.5 & -2.0, +8.5 \\
\hline
\end{tabular}
\caption{SAR (W/g) values are compared for different nanoparticles sizes. SAR from paper Wang \textit{et al.} \cite{wang} ($SAR_{Wang}$) and SAR calculated taking N\'{e}el-relaxation time ($SAR_{N}$) only, combined N\'{e}el- and Brownian-relaxation time ($SAR_{tot}$) and Brownian-relaxation time ($SAR_{B}$) only are shown. Log-normal distribution is taken. Their standard deviations ($\sigma_{calc}$) are calculated by us only. The possible deviations from the mean value ($\pm\delta_{calc}$) are given in the last row. }
\label{tab:wang}
\end{table}
Finally, we compared our model predictions to a series of data taken from Wang \textit{et al.} \cite{wang}. We experienced the same tendencies as before. The consideration of Brownian relaxation process exclusively did not explain the experimental data. Contrary, taking N\'{e}el relaxation alone, resulted in slight underestimation of measured values, the error was within $5\%$. It might worth be note, the absolute value of variation of the particle size from the mean value is nearly constant, which could be explained by having samples fabricated with the same technique. Contrary the samples of Mehdaoui \textit{et al.} were prepared differently, and according to it, their standard deviations of particle sizes differ markedly. The asymmetry of the variation of particle size from the mean value can be explained by the asymmetry of the log-normal distribution.
Wang \textit{et al.} also estimated the optimum size for reaching the maximum of SAR being at $r=9.15$nm, based on the linear response theory and Rosensweig's susceptibility. From our model, we got $r=9.21$nm for the optimum size to get resonance, which size is in excellent agreement with the estimations given by Wang \textit{et al.}. We note, that at the resonance the SAR is $6365 W/g$, see Fig.\ref{fig:janosfalvi_Fig1}. The same curve can be seen on logarithmic scale, Fig.\ref{fig:janosfalvi_Fig2}, to highlight the tendency for small nanoparticles.
\begin{figure}
\centering
\includegraphics{janosfalvi_Fig1.eps}
\caption{SAR is shown as a function of particle radius. Resonance can be observed at $9.21$ nm with $SAR=6365$ W/g. }
\label{fig:janosfalvi_Fig1}
\end{figure}
\section{Summary}
\label{sec_sum}
The aim of this paper was to show the effect of size polydispersity on the specific absorption loss in a magnetic nanoparticles containing media. For the first time, analytic calculations were carried out instead of numerical simulations, using a Bloch-Bloembergen alike model, developed and described detailed in paper \cite{jzsuzsa}. As a reference, the paper of Mehdaoui \textit{et al.} was used for direct comparison, including nanoparticles $r\ge 7$nm, the range of Stoner-Wohlfarth model beyond linear response theory. We found excellent agreement, see Table\ref{tab:mehdaoui}. Based on this agreement, we estimated the possible mean value and standard deviation of particle sizes for other experiments of Wang \textit{et al.}, Hergt \textit{et al.} and Suto \textit{et al.}, see Table\ref{tab:withpoly}. For the sake of comparison, the SAR values were calculated also without polydispersity, Table\ref{tab:nopoly}, and we found drastic improvement when polydispersity was included. Our analytical approach predicted the resonance in the experiment of Wang \textit{et al.}. After analyzing standard deviations from mean value of particle sizes, samples fabricated by different technologies might be distinguished, see experiments of Mehdaoui \textit{et al.} and Wang \textit{et al.} It can be concluded, the polydispersity is inevitable in experiments and it should be taken into account in models for calculating SAR as well. The case of pure Brownian relaxation is excluded. N\'{e}el relaxation was found to be essential in relaxation processes for paramagnetic nanoparticles at hyperthermia.
\begin{figure}[b]
\centering
\includegraphics{janosfalvi_Fig2.eps}
\caption{SAR is shown as a function of particle radius at logarithmic scale. Low SAR data are enhanced by logarithmic scale. Resonance can be observed at $9.21$ nm with $SAR=6365$ W/g.}
\label{fig:janosfalvi_Fig2}
\end{figure}
\section*{Acknowledgement}
One of the author, Zs.J., wishes to express special thanks to I. N\'andori, P.F. de Ch\^{a}tel and to colleagues in the lab for their valuable critical advices and continuous support. The authors acknowledge support from the Hungarian Scientific Research Fund (OTKA) No.101329.
|
2,877,628,090,797 | arxiv | \section{Introduction}
Computed tomography is widely used in various fields: medicine~\citep{kesminiene2018cancer}, precise measurements~\citep{buratti2018applications}, agriculture~\citep{mairhofer2017x}.
In tomography, the mutual trajectories of the sample, detector, and probe radiation source are usually considered known, since they are determined by the targeted movement of the setup components.
Most of the computed tomography reconstruction methods rely on the geometric accuracy of the instrument and the reliably known trajectories of all its parts~\citep{feldkamp1984practical, katsevich2004improved}
However, the realized trajectory differs from the desired one for various reasons (mechanical backlash, an error in measuring the angle of rotation of an object, thermal deformations, the slope of the sample relative to the axis of rotation), which negatively affects the quality of the reconstruction.
Thus, geometric errors are one of the main sources of reconstruction errors~\citep{ferrucci2015towards}.
To compensate for these deviations different calibration approaches are used. These approaches can be divided into two classes. The first class of geometric calibration methods is based on observations of a specific object with a known geometry~\citep{dewulf2013uncertainty, hermanek2017optimized, weiss2012geometric} or is based on reference measuring instruments~\citep{welkenhuyzen2014investigation, bircher2018geometry}. Geometric calibration is a laborious and expensive process, requiring the involvement of specialists with the appropriate qualifications.
Moreover, geometrical errors inherent even in a calibrated system can still have a negative effect on the quality of reconstruction, and the quality of calibration decreases with time. The second class of methods called online calibration refines trajectory directly during measurements by analyzing obtained projections~\citep{yang2017direct, xu2017simultaneous, zhang2014iterative, muders2014stable, chung2018tomosynthesis}. The online calibration methods do not require additional experiments and allow to compensate for geometric errors. Since circular motion estimation is of practical interest in tomography \citep{ferrucci2015towards} we focus on this particular type of motion. While there are existing datasets for camera motion estimation with images in visible spectrum \citep{hodan2017t, ovg2009multi, seitz2006comparison}, there is no analogous dataset containing X-ray images. Moreover, the application of classical computer vision techniques to digital X-ray images requires an analysis of the influence introduced by the different nature of such images to the quality of algorithms. Again, еhere is no known dataset allowing to assess the influence of the translucent world model applicable for X-ray images on the algorithms developed for the classical opaque world model.
We present a dataset with digital X-ray images of a plastic object. The purpose of the presented dataset is twofold. It allows to measure the performance of circular motion estimation algorithms on X-ray images as well as to study the difference in computer vision algorithms performance (e.g. keypoints detection and matching) while applied to visible and X-ray data.
\section{Data description}
The presented dataset consists of two parts: (i) images made in visible and (ii) X-ray spectra. The structure of the dataset is presented below:
\begin{minipage}{3in}
\dirtree{%
.1 xvcm\_dataset.
.2 xray.
.3 raw.
.4 dark.
.4 data.
.4 empty.
.3 preprocessed.
.4 downscaled.
.3 calib.
.2 visible.
.3 raw.
.3 preprocessed.
.4 cropped.
.3 calib.
}
\end{minipage}
Figure~\ref{fig:data} demonstrates samples images while Table~\ref{tab:params} presents the dataset parameters. The X-ray part of the dataset was collected at the X-ray microtomograph developed and operating at the Federal Research Center for Crystallography and Photonics of Russian Academy of Sciences~\citep{buzmakov2018laboratory, buzmakov2019laboratory} with the following parameters:
\begin{itemize}
\item exposure time -- 5 seconds;
\item radiation energy -- 17 keV;
\item monochromator -- pyrographite;
\item MoKa line.
\end{itemize}
While recording data the camera was stationary. The optical axis of the camera was perpendicular to the axis of object rotation. Such system setup is equivalent to the case of the circular camera motion around a stationary object (see Fig.~\ref{fig:circular_motion}).
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\includegraphics[width=1\linewidth]{visible_dataset.png}
\caption{visible spectrum}
\label{fig:Ng1}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\includegraphics[width=1\linewidth]{tomo_dataset.jpg}
\caption{X-ray spectrum}
\label{fig:Ng2}
\end{subfigure}
\caption{Examples of images presented in dataset for visible (a) and X-ray (b) spectrum.}
\label{fig:data}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\linewidth]{one_point_rotation.png}
\caption{Circular camera motion. $C$ and $C'$ stand for camera centers in different moments of time, $\alpha$ is a rotation angle, $A$ is a point belonging to the observed object.}
\label{fig:circular_motion}
\end{figure}
\subsection{Data preprocessing}
To compensate for X-ray detector noise the following prepossessing algorithm was applied:
\begin{equation}
\textrm{preprocessed\_image}_{i,j} = \frac{\textrm{data}_{i,j}-mean(\textrm{dark}_{i,j})}{mean(\textrm{empty}_{i,j})-mean(\textrm{dark}_{i,j})}
\end{equation}
where $i, j$ -- pixel coordinates, $mean()$ -- is the mean value of pixels in the given coordinates over the whole subfolder, three subfolders contain the following images:
\begin{itemize}
\item \texttt{dark} -- images taken in absence of X-ray radiation;
\item \texttt{empty} -- images taken in absence of object;
\item \texttt{data} -- images with rotating object.
\end{itemize}
Prerocessed images are stored in \texttt{xray\textbackslash preprocessed}.
Due to the Canon EOS 5D Mark III camera limitations object of interest occupies only a small part of visible images. Folder \texttt{visible\textbackslash preprocessed\textbackslash cropped} contains cropped images of the size 407 x 407 pixels. To make visible and X-ray images comparable the latest were downscaled to the comparable resolution (407 x 360 pixels) in a aspect ratio preserving way (folder \texttt{visible\textbackslash preprocessed\textbackslash cropped}).
\begin{table}
\caption{Parameters of visible and X-ray dataset parts}
\label{tab:params}
\begin{center}
\begin{tabular}[h]{|c|M{3.5cm}|c| }
\hline
\hline
\textbf{Parameter} & \textbf{Visible} & \textbf{X-ray} \\ \hline
Number of images & 400 & 400 \\ \hline
Rotation angle step & $0.5^\circ$ & $0.5^\circ$ \\ \hline
Resolution & 5760 x 3840 pixels (object occupies an area of approximately 400 x 400 pixels) & 3000 x 2650 pixels \\ \hline
Camera & Canon EOS 5D Mark III & Ximea xiRay11 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Calibration}
Both parts of the dataset are accompanied by camera calibration parameters. In case of visible data, calibration was performed with the OpenCV toolbox~\citep{bradski2000opencv} by detecting chessboard pattern. Calibration parameters as well as images used for calibration are stored in \texttt{visible\textbackslash calib} folder. In the case of X-ray data calibration was obtained by the known dimensions of the object were used to estimate camera calibration parameters. The calibration is stored at \texttt{xray\textbackslash calib}.
\section{Conclusion}
We presented the dataset containing both X-ray and visible images of the same rotating object. The presented dataset can be used while developing camera circular motion estimation techniques for online calibration methods. The second purpose of the presented dataset is the evaluation of the computer vision techniques applied to the X-ray data.
\bibliographystyle{unsrt}
|
2,877,628,090,798 | arxiv | \section{Introduction}
The $(k,d)$-hypersimplex, denoted by $\Delta _{d,k}$, is defined as the
convex hull of all ($0,1)$-vectors in $%
\mathbb{R}
^{d}$ whose nonzero elements sum to $k$. Hypersimplices are $(d-1)$%
-dimensional polytopes that appear in various algebraic and geometric
contexts (e.g., see [6]). The polytope $\Delta _{d,k}$ can also be defined
as a "slice" of the $(d-1)$-hypercube located between the two hyperplanes $%
\sum x_{i}$ $=d-1$ and $\sum x_{i}$ $=d$ in $%
\mathbb{R}
^{d}$. A classical result implied by the work of Laplace [7], is that the
normalized volume of this polytope equals the Eulerian number $A_{k,d-1\text{%
.}}$\ Hypersimplices have also appeared in the theory of characteristic
classes and\ Gr\"{o}bner bases (for more details on this and on polytopes in
general, see [3] and [12].) The graph of the hypersimplex $\Delta _{d,k}$,
denoted by $G_{d,k}$, is the graph consisting of the vertices and edges of $%
\Delta _{d,k}$. This graph is also known as the Johnson graph and $G_{d,k}$
provides an example of a family of "distance-regular" graphs of unbounded
diameter, which are also a special type of Coxeter graph [1].
For the case $k=1$, the graph $G_{d,1}$ is the complete graph $K_{d}$ whose
role is fundamental in Graph Theory. Compared to the complete graphs, the
properties of the closely related hypersimplex graphs are not very well
known. Here we show how many of the parameters of $K_{d}$ are extended to $%
G_{d,k}.$ For example, $K_{d}$ has $d$ vertices, is regular of degree $d-1$
and has diameter $1$. We show that $G_{d,k}$ has $\left(
\begin{array}{c}
d \\
k%
\end{array}%
\right) $ vertices, is regular of degree $k(d-k)$, and has diameter $k$, for
$k$ $\leq \frac{d}{2}$. We also characterize adjacency, show that $G_{d,k}$
is vertex transitive, obtain an explicit formula for the number of edges,
determine the clique number for $G_{d,k}$, and study varous connectivity
properties. In addition, since the number of vertices in $G_{d,k}$ is $%
\left(
\begin{array}{c}
d \\
k%
\end{array}%
\right) $, it is natural to ask how $G_{d,k}$ can be decomposed into
subgraphs whose vertex counts satisfy Pascal's Identity $\left(
\begin{array}{c}
d \\
k%
\end{array}%
\right) =$ $\left(
\begin{array}{c}
d-1 \\
k-1%
\end{array}%
\right) +\left(
\begin{array}{c}
d-1 \\
k%
\end{array}%
\right) $. We show that this leads to a recursive decomposition of\ $G_{d,k}$%
\ into self similar subgraphs. The paper concludes with a discussion of edge
expansion properties of $G_{d,k}$ and random walks on $G_{d,k}$ that may be
used to generate random subsets of $\{1,2,3,...,d\}$ of size $k$.
\bigskip
\bigskip
\section{\protect\bigskip The vertices and edges of $G_{d,k}$}
\bigskip
A polytope contained in $%
\mathbb{R}
^{d}$ is called $(0,1)$-\textit{valued} if all of its vertices are vectors
having coordinates that are all either $0$ or $1$. Given any convex polytope
$P$, two distinct vertices $x\neq y$ in $P$ are \textit{adjacent} if for
every $\lambda $\ satisfying $0<\lambda \ <1$, it holds that\ $\lambda
x+(1-\lambda )y$ can not be expressed as a convex combination of other
vertices in $P$. A graph is said to be \textit{regular} of degree $r$ if
every vertex in the graph has degree $r$. Let $x$ and $y$ be points in $%
\mathbb{R}
^{d}$, then $x\cdot y$ is the inner product $\dsum%
\limits_{i=1}^{d}x_{i}y_{i} $.
\bigskip
\bigskip
\textbf{Proposition 1} \textit{For} $1\leq k<d$ and $d\geq 4$:
(a) \textit{The number of vertices in }$G_{d,k}$ \textit{is} $\left(
\begin{array}{c}
d \\
k%
\end{array}%
\right) .$
(b) \textit{Two distinct vertices x and y of }$G_{d,k}$ \textit{are adjacent
if and only if }$x\cdot y=k-1$.\bigskip
\bigskip
\textbf{Proof.} (a) The count follows from the fact that there is an obvious
one-to-one correspondence between the subsets of $\{1,2,...,d\}$ with $k$
elements and the number of $0,1$ $d$-vectors with exactly $k$ ones.
(b) If $k=1$, then $G_{d,k}$ is the complete graph $K_{d}$ and the result
holds since all vertices in $K_{d}$\ are adjacent. So assume that $k\geq 2$,
and suppose that $x\cdot y<k-1$. Then there exists $p,$ $q,$ $r,$ $s\in
\{1,2,...,d\}$ such that $x_{p}=1$, $x_{q}=1$, $x_{r}=0$, $x_{s}=0$, and $%
y_{p}=0$, $y_{q}=0$, $y_{r}=1$, $y_{s}=1$. Define vertices $u$ and $v$ as
follows: $x_{i}=u_{i}$, for all $i$ except $i=q,$ $r$, which satisfy $%
u_{q}=0 $ and $u_{r}=1$, and $y_{i}=v_{i}$, for all $i$ except $i=q,$ $r$,
which satisfy $v_{q}=1$ and $v_{r}=0$. Then $\frac{1}{2}x+\frac{1}{2}y$ $=%
\frac{1}{2}u+\frac{1}{2}v,$ so $x$ and $y$ are not adjacent.
Next observe that $x\cdot y>k$ is impossible, and $x\cdot y$ $=k$ implies
that $x=y$, which is a contradiction. So suppose that $x\cdot y$ $=k-1$ and
that there exists $z^{1}$, $z^{2}$, ... , $z^{n}$ such that $\lambda
x+(1-\lambda )y=\dsum\limits_{j=1}^{n}\alpha _{j}z^{j}$. Since $x$ and $y$
both have exactly $k$ ones, $d-k$ zeroes, and $x\cdot y$ $=k-1$, $x$ and $y$
must be equal for all indices except for $2$. This implies that the $z^{j}$
are equal on all indices except for $2$, and hence $n=2$. Consequently, we
must have $\{x,y\}=\{z^{1},z^{2}\}$. \ $\blacksquare $
\bigskip
\bigskip
\textbf{Proposition 2} (a) \textit{The graph }$G_{d,k}$ \textit{is regular
of degree }$k(d-k)$.
\bigskip (b) \textit{The number of edges in} $G_{d,k}$ \textit{is} $\frac{d!%
}{2(k-1)!(d-k-1)!}.$
\bigskip
\textbf{Proof.} (a) Let $x$ be any vertex in $G_{d,k}$. By Proposition 1, a
vertex $y$ is adjacent to $x$ if and only if $x\cdot y=k-1$. So $y$ must
have $k-1$ of the $k$ ones in $x$. There are $k$ ways for this to happen. In
addition, one of the $d-k$ indices for which $x_{i}=0$ must be a one for $y$%
. There are $d-k$ ways for this to happen. Hence the number of vertices
adjacent to $x$ is $k(d-k)$.
(b) The count follows from the well known Handshaking Lemma [11] and the
fact that the sum of the vertex degrees in $G_{d,k}$\ is given by $\left(
\begin{array}{c}
d \\
k%
\end{array}%
\right) k(d-k)$. \ \ $\blacksquare $
\bigskip
\bigskip
A graph $G$ with vertices $V(G)$ is called \textit{vertex transitive} if
given any two vertices $x$ and $y$ there is an automorphism $%
f:V(G)\rightarrow V(G)$ such that $f(x)=y$. It is known that graphs of
Platonic solids and Archimedean solids are vertex transitive, as well as $%
K_{d}$ and the complete bipartite graph $K_{d,d}$. We now show that this
property is also true for $G_{d,k}$.
\bigskip
\bigskip \textbf{Proposition 3} \textit{For} $1\leq k<d,$ \textit{the graph }%
$G_{d,k}$ \textit{is vertex transitive}.
\bigskip
\textbf{Proof.} Given vertices $x$ $=(x_{1}x_{2}...x_{d})$ and $%
y=(y_{1}y_{2}...y_{d})$ define $f:V(G_{d,k})\rightarrow V(G_{d,k})$ as
follows. (An example illustrating the construction of $f$ is given
immediately after the proof.) If $x_{i}=y_{i}$, $f$ takes $i$ to $i$; i.e., $%
f$ takes the $i$'th digit in $x$ to the $i$'th digit in $f(x)$. Now consider
the set $D(x,y)=\{i:x_{i}\neq y_{i}\}$. Let $p$ be the number of elements in
$D(x,y)$ for which $x_{i}=0$, and let $q$ be the number for which $x_{i}=1.$
Then the number of elements in $D(x,y)$ for which $y_{i}=1$ must be $p$ and
the number for which $y_{i}=0$ is $q$. But $\sum x_{i}$ $=\sum y_{i}$, so we
must have $p=q,$ and hence, $D(x,y)$ contains $2p$ indices. Let $i_{1}$\ be
the smallest index for $x$ in $D(x,y)$ with $x_{i_{1}}=1$ and $i_{2}$ the
smallest index in $D(x,y)$ with $x_{i_{2}}$ $=0$. Let $j_{1}$\ be the
smallest index for $y$ in $D(x,y)$ with $y_{j_{1}}=0$ and $j_{2}$ the
smallest index with $y_{j_{_{2}}}=$ $1$. Then we define $f$ such that the $%
i_{1}$th digit in $x$ becomes the $j_{2}$th digit in $f(x)$, and the $i_{2}$%
th digit in $x$ becomes the $j_{1}$th digit in $f(x)$. Since $D(x,y)$ has an
even number of indices, we may repeat this step as often as necessary.
Observe that $f(x)=y$. In addition, $f$ is its own inverse. Hence $f(x)=y$
implies that $f(f(x))=f(y)$, or simply $x=f(y)$. This gives $x\cdot
y=f(y)\cdot f(x)$, so $f$ is an adjacency preserving automorphism. $\ \
\blacksquare $
\bigskip
To illustrate the construction of $f$ suppose that $x=(110101101000)$ and $%
y\ =(100110010110)$. Then $f$ is given by $%
f(x_{1}x_{2}...x_{12})=(x_{1}x_{5}x_{3}x_{4}x_{2}x_{8}x_{10}x_{6}x_{11}x_{7}x_{9}x_{12})
$ $=y$. Furthermore, if $z=(101010101010),$ then $f(z)=(111000001110)$.
\bigskip
\section{Connectivity Properties}
\bigskip
The \textit{distance} between any two vertices $x$ and $y$, in a graph $G$
is the number of edges in a shortest path joining $x$ to $y$. The \textit{%
diameter} of $G$, $\delta (G)$, is the maximum distance amongst all pair of
vertices in $G$. A graph is called \textit{distance-regular} if it is a
regular graph such that, given any two vertices $x$ and $y$ at any distance $%
i\leq \delta (G)$, the number of vertices adjacent to $y$ and at a distance $%
j$ from $x$ depends only on $i$ and $j$, and not on the particular vertices.
A graph $G$ is called $d$-\textit{connected} if for every pair of vertices $%
x $ and $y$ there exists $d$ disjoint paths joining $x$ to $y$. A graph is
called \textit{hamilton connected} if every pair of distinct vertices is
joined by a path that passes through every vertex of $G$ exactly once. A
subset of vertices $H$\ is called a \textit{clique} in $G$ if there is an
edge in $G$ between every pair of vertices in $H$. The cardinality of the
largest clique in G is called the \textit{clique number} of $G$. It is easy
to show that $G_{d,k}$\ is isomorphic to $G_{d,d-k}$\ so in the following
propositions we restrict $k$ such that $1\leq k\leq \frac{d}{2}.$
\bigskip
\textbf{Proposition 4} \textit{Let }$1\leq k\leq \frac{d}{2}.$
\bigskip
(a) \textit{Given any two vertices} $x$ \textit{and} $y$ \textit{in} $%
G_{d,k} $, \textit{the distance between} $x$ \textit{and} $y$ \textit{is} $%
k-x\cdot y $.
(b) \textit{The diameter of} $G_{d,k}$\ \textit{is} $k$.
(c) $G_{d,k}$\ \textit{is a distance-regular graph}.
\bigskip
\textbf{Proof}. (a) Let $x\neq y$ be given. If $x$ and $y$ are adjacent,
then by Proposition 1, the distance between $x$ and $y$ is $1=k-x\cdot y$.
So assume that $x$ and $y$ are not adjacent and hence, $x\cdot y<k-1$. Since
$x$ any $y$ both have $k$ ones and $d-k$ zeros, there exist indices $p$ and $%
q$ such that $x_{p}=1$, $y_{p}=0$, $x_{q}=0$ and $y_{q}=1$. Define the
vertex $z$ by $z_{i}=x_{i}$, for all $i$ except $p$ and $q$, where $z_{p}=0$%
, and $z_{q}=1$. Then $z$ has exactly $k$ ones and $x\cdot z=k-1$, so $x$
and $z$ are adjacent. Moreover, $x\cdot z=x\cdot y+1.$ We can repeat this as
often as necessary each time getting one step closer to $y$.
(b) Since $k\leq \frac{d}{2}$ there exists vertices $x$ and $y$ such that\ $%
x\cdot y=0$. By (a)\ this implies that the distance between $x$ and $y$ is $%
k $. Hence, the diameter must be $k$.
(c) Let $x$ and $y$ be vertices of $G_{d,k}$ whose distance is $i$. Then $x$
and $y$ have $k-i$ ones in common. Moreover, any vertex $z$ adjacent to $y$
at a distance $j$ from $x$, satisfies $y$ and $z$ have $k-1$ ones in common,
and $x$ and $z$ have $k-j$ ones in common. The number of vertices $z$
satisfying this depends only on $i$ and $j$. \ \ $\blacksquare $
\bigskip
\bigskip
\bigskip \textbf{Proposition 5} \ \textit{For} $2\leq k\leq \frac{d}{2}$:
(a) $G_{d,k}$ \textit{contains the complete graph} $K_{d-k+1}$ \textit{as a
subgraph}.
(b) \textit{The clique number of }$G_{d,k}$ \textit{is} $d-k+1.$
\bigskip
\bigskip
\textbf{Proof.} (a) Let $H$ be the subset of vertices whose first $k-1$
coordinates are all one. Then $H$ contains $d-k+1$ vertices, and every pair
of vertices $x$, $y$ in $H$\ satisfy $x\cdot y$ $=k-1$. Hence the subgraph
induced by $H$ must be $K_{d-k+1}$.
(b) Suppose, to obtain a contradiction, that the clique number of $G_{d,k}$\
is $w$ and $w>d-k+1$. Then there exists a subgraph isomorphic to $K_{p}$
where $p=d-k+2$. Let $x^{1},x^{2},...,x^{p}$ be the vertices of the subgraph.
If $x^{1},x^{2},...,x^{p}$\ all have $k-1$ ones in common, then without loss
of generality, we may assume that $x^{1},x^{2},...,x^{p}$ all have their
first $k-1$ digits equal to 1. Moreover, the last $d-(k-1)$ digits for\ each
of $x^{1},x^{2},...,x^{p}$ must all consist of zeros and exactly one 1.
Since the $x^{j}$ are all distinct, there are only $d-k+1$ possibilities for
this, implying that $p<d-k+2$, a contradiction.
Now suppose that $x^{1},x^{2},...,x^{p}$\ do not all have $k-1$ ones in
common, and that the first $k$ digits of $x^{1}$ are one. Observe that $k-1$
of the first $k$ digits of $x^{2},...,x^{p}$\ must be one since these
vertices must be adjacent to $x^{1}$. We show that no two of these vertices
have the same first $k$ digits. Suppose $x^{2}$ and $x^{3}$ have the same
first $k$ digits as illustrated below. Then, since $x^{1},x^{2},...,x^{p}$\
do not all have $k-1$ ones in common, there exists an $x^{4}$ whose first $k$
digits are different from those of $x^{2}$, also illustrated below. Notice
that $x_{k+1}^{4}$ $=1$, since $x^{4}$ and $x^{2}$ must be adjacent. But now
$x^{4}$ and $x^{3}$ are not adjacent.
$\bigskip $
$\qquad \qquad \qquad \ k-2\qquad k-1\qquad k\qquad k+1\qquad k+2$
$x^{2}=(1...1\qquad \ \ 1$ $\ \ \qquad \ \ 1$ $\ \ \ \qquad 0\ \qquad \ \ 1$
$\ \ \ \qquad \ \ 0\qquad \qquad 00...0)$
$x^{3}=(1...1\qquad \ \ 1$ $\ \ \qquad \ \ 1$ $\ \ \ \qquad 0$ $\qquad \ \
0\ \ \ \ \qquad \ \ 1\qquad \qquad 00...0)$
$x^{4}=(1...1\qquad \ \ 0$ $\ \ \qquad \ \ 1$ $\ \ \ \qquad 1\ \qquad \ \ 1$
$\ \ \ \qquad \ \ 0\qquad \qquad 00...0)$
\bigskip
Since $k-1$ of the first $k$ digits of $x^{2},...,x^{p}$\ must be $1$, and
no two of these \ vertices have the same first $k$ digits, $p$ must satisfy $%
p\leq k+1$. But $p=d-k+2$ implies that $d-k+2\leq k+1.$ A little algebra
gives $\frac{d+1}{2}\leq k$. However, $k\leq \frac{d}{2},$ which implies $%
\frac{d+1}{2}\leq \frac{d}{2}$ a contradiction. \ \ \ $\blacksquare $
\bigskip
\bigskip
\textbf{Proposition 6} \textit{For} $1\leq k\leq \frac{d}{2}:$
(a) $G_{d,k}$\ \textit{is }$(d-1)$-\textit{connected.}
(b) $G_{d,k}$\textit{\ is hamilton connected.}
\bigskip
\textbf{Proof}. (a)\ Balinski's Theorem [12] tells us that every $d$%
-dimensional polytope is $d$-connected. Since $\Delta _{d,k}$ is a $(d-1)$%
-dimensional polytope, it must be $(d-1)$-connected.
(b) Naddef and Pulleyblank [10] proved that if the graph of a $(0,1)$%
-polytope is bipartite, then it is a hypercube. Moreover, if the graph is
nonbipartite, then it is hamilton connected. Proposition 5 implies that $%
G_{d,k}$\ contains $K_{d-k+1}$ as a subgraph. Since $d-k+1\geq 3$, $G_{d,k}$
contains an odd cycle. Therefore, $G_{d,k}$ is not bipartite, and hence, is
hamilton connected. \ $\blacksquare $
\bigskip
\bigskip
\textbf{Proposition 7} \textit{For} $1\leq k\leq \frac{d}{2}$, $G_{d,k}$\
\textit{decomposes into} $G_{d-1,k}\cup G_{d-1,k-1}\cup E$, \textit{where} $%
E $ \textit{is a subgraph containing} $\frac{(d-1)!}{(k-1)!(d-k-1)!}$
\textit{edges that link} $G_{d-1,k}$ \textit{to} $G_{d-1,k-1}$.
\bigskip
\textbf{Proof}. Consider the subset of $(x_{1}x_{2}...x_{d})$ $\in
V(G_{d,k}) $ that satisfy $x_{1}=1$. These vertices must all satisfy $%
\dsum\limits_{i=2}^{d}x_{i}$ $=k-1$. Let $H_{1}$ be the subgraph induced by
these $\left(
\begin{array}{c}
d-1 \\
k-1%
\end{array}%
\right) $ vertices. Then $H_{1}$ is isomorphic to $G_{d-1,k-1}$. For given
any vertex $x$ in $H_{1}$ we can remove the first coordinate to obtain a
vertex $x^{\prime }$ in $V(G_{d-1,k-1})$. Moreover if $x$ and $y$ are
adjacent in $G_{d,k}$, then $x\cdot y=k-1$. The corresponding vertices $%
x^{\prime }$and $y^{\prime }$ in $G_{d-1,k-1}$ will be adjacent in $%
G_{d-1,k-1}$ since $x^{\prime }\cdot y^{\prime }=k-2.$
\bigskip
Now consider the subset of $V(G_{d,k})$ that satisfy $x_{1}=0$. These
vertices must all satisfy $\dsum\limits_{i=2}^{d}x_{i}$ $=k$, so there are $%
\left(
\begin{array}{c}
d-1 \\
k%
\end{array}%
\right) $ such vertices. Let $H_{0}$ be the subgraph induced by these
vertices. Then an argument similar to the above shows that $H_{0}$ is
isomorphic to $G_{d-1,k}$.
The formula for the number of edges in $G_{d,k}$ given in Proposition 3 can
be used to obtain the equation below, which can then be used to find $%
\left\vert E\right\vert .$
\[
\frac{d!}{2(k-1)!(d-k-1)!}=\frac{(d-1)!}{2(k-1)!(d-k-2)!}+\frac{(d-1)!}{%
2(k-2)!(d-k-1)!}+\left\vert E\right\vert
\]%
$\blacksquare $
\bigskip
\section{Random walks and the expansion of G$_{d,k}$}
\bigskip
We have demonstrated that $G_{d,k}$ is a tractable graph and many of the
well known graph attributes and parameters of the complete graph $K_{d}$ may
be extended to $G_{d,k}$. In [4], [5] and [8] random walks on the graphs of $%
(0,1)$-polytopes were investigated as a potential algorithm for random
generation of combinatorial objects. In the case of the hypersimplex, the
vertices of $G_{d,k}$\ can be used to represent subsets of $\{1,2,...,d\}$
of size $k$ as follows. Given a vertex $x$, $i$ is in subset $S$ if and only
if $x_{i}=1$. The adjacency criterion given in Proposition 1 allows us to
generate a random neighbor. For given a vertex $x$, generate two random
integers between $1$ and $d$, say $r$ and $s$, until $x_{r}+x_{s}=1$. Then
whichever of $x_{r}$ or $x_{s}$ is equal to $1$ we change to $0$, and
whichever is $0$ we change to 1. Starting with any vertex we may repeat this
process a large number of times. The result is a randomly generated vertex
corresponding to subset of size $k$. We note that there are other known
algorithms to generate random subsets of size $k$ (e.g., see [9]) but
advantages of the above algorithm is that it is easy to code and also an
interesting application of a random walk.
\bigskip
Surprisingly perhaps, the success of the above algorithm is known to depend
on the "edge expansion" properties of $G_{d,k}$. Given a graph $G=(V,E)$,
the \textit{edge expansion} of $G$, denoted $\chi (G),$ is defined as
\[
\chi (G)=min\left\{ \frac{\left\vert \delta (U)\right\vert }{\left\vert
U\right\vert }:U\subset V,\text{ }U\neq \emptyset ,\text{ }\left\vert
U\right\vert \leq \frac{\left\vert V\right\vert }{2}\right\}
\]%
where $\delta (U)$ is the set of all edges with one end node in $U$ and the
other one in $V-U$. The edge expansion rate for graphs of polytopes with $%
(0,1)$-coordinates has been recently studied and is an important parameter
for a variety of reasons [4]. In the case of random walks on graphs, "good"
edge expansion implies that the process converges to its limiting
distribution as rapidly as possible [4]. It is known that the hypercube, $%
Q_{n}$ has edge expansion $1$, and has been conjectured that all $(0,1)$%
-polytopes have edge expansion at least $1$ [8]. In [5] it was shown that
the the graph $G_{d,k}$\ has expansion rate at least 1.
\bigskip
When a graph is regular, algebraic graph theory [2] can be used to help
study its expansion rate. If $A$ is the adjacency matrix \ of an $n$-vertex
graph $G$, then $A$ has $n$ real eigenvalues which we denote by $\lambda
_{0}\geq \lambda _{1}\geq \cdots \geq \lambda _{n-1}.$ If $G$\ is a regular
graph with degree $r$, then it is known that $\lambda _{0}=r$, and a result
of Cheeger tells us that $\frac{r-\lambda _{1}}{2}\leq \chi (G)\leq \sqrt{%
2r(r-\lambda _{1})}$ (for a proof, see [4])$.$ For example, the eigenvalues
of the adjacency matrix of $K_{d}$ are $d-1,-1,-1,\ldots \ ,\ -1,$ and
hence, $\frac{d}{2}\leq \chi (K_{d})$. By Proposition 2, we know that the
adjacency matrix associated with $G_{d,k}$ has $\lambda _{0}=r=k(d-k)$. To
investigate the expansion rate of $G_{d,k}$ we need the following
proposition [1].
\bigskip
\textbf{Proposition 8} \textit{For} $1\leq k\leq \frac{d}{2},$ \textit{the
eigenvalues of }$G_{d,k}$\ \textit{are given by }$\lambda
_{j}=(k-j)(d-k-j)-j $, \textit{for }$j=0,1,...,k$, \textit{with
multiplicities} $m_{j}=$ $\left(
\begin{array}{c}
d \\
j%
\end{array}%
\right) -\left(
\begin{array}{c}
d \\
j-1%
\end{array}%
\right) .$
\bigskip
\textbf{Proposition 9} \textit{For }$1\leq k\leq \frac{d}{2},$ \textit{the
edge expansion of }$G_{d,k}$\ \textit{satisfies} $\frac{d}{2}\leq \chi
(G_{d,k})\leq \sqrt{2dk(d-k)}.$
\bigskip
\textbf{Proof}. By Proposition 8, we see that $\lambda _{1}$ $%
=(k-1)(d-k-1)-1 $. Since $r=k(d-k)$, we have that $r-\lambda _{1}\ =d$. If
we now apply Cheeger's Theorem, then $\frac{r-\lambda _{1}}{2}=$ $\frac{d}{2}%
\leq \chi (G_{d,2})\leq \sqrt{2r(r-\lambda _{1})}=$\ $\sqrt{2dk(d-k)}$. $\ \
\blacksquare $
\bigskip
This again extends a property of $K_{d}$, and it is interesting to note that
the lower bound $\frac{d}{2}\leq \chi (G_{d,k})$ is independent of $k$. It
was shown in [5] that $1\leq \chi (G_{d,k}),$ which confirms the conjecture
of Mihail for this special case of hypersimplices. Proposition 9 provides an
improved lower bound and that implies the family of graphs $G_{d,k}$\ has
very good expansion. Consequently, the algorithm mentioned above should be
able to efficiently generate good random subsets.
\bigskip
\bigskip
|
2,877,628,090,799 | arxiv | \section{Introduction}
\subsection{The microscopic onset of irreversibility}
Recently the phenomenon of \emph{probabilistic hysteresis} has been introduced \cite{dimer, trimer, quantum_dimer_I}: A slow, cyclic sweep of an external parameter can lead to a state that is very different from the initial state, even though the external parameter is tuned back to its initial value. This phenomenon can be interpreted as the microscopic analogue of macroscopic irreversible processes as e.g. cooking an egg and then cooling it down again to the initial temperature. As everyday experience tells us, the initial raw egg will not be recovered. From the adiabatic theorem, however, one would rather expect the system to follow a stationary state as long as the parameter sweep is sufficiently slow, so that it would return to its initial state again when the cycle of the sweep is complete. While one might not expect the adiabatic theorem to apply to macroscopic systems, which typically have many low-frequency degrees of freedom, it should be possible to attain the adiabatic limit in sufficiently small systems. The possibility of this egg-cooking kind of irreversibility even in a dissipationless microscopic system is therefore surprising.
In \cite{dimer, trimer} we have shown how this irreversibility can occur due to the crossing of a separatrix in phase space, where adiabaticity breaks down even for arbitrarily slow change of the control parameter. This is the case because the orbital period diverges at the separatrix and so the criterion for the validity of the adiabatic theorem can never be met, no matter how slow the variation of the external parameter might be. In the integrable system of \cite{dimer} this results in a finite probability to either return to the initial state or to a state with much higher energy (thus \emph{probabilistic} hysteresis); in the chaotic system of \cite{trimer} the return probability is typically very close to zero.
\subsection{Quantum-classical correspondence}
In our previous work this phenomenon was identified in two specific models for trapped ultracold atom systems, namely the Bose-Hubbard dimer \cite{dimer} and trimer \cite{trimer}, in a semiclassical mean-field approximation. Although in classical Hamiltonian evolution adiabaticity can fail even in the quasi-static limit of infinitely slow sweep rate, this quasi-static limit \emph{must} be adiabatic under fully quantum-mechanical evolution, because quantum energy level splittings always remain non-zero in this system (i.e. there can never be any exact degeneracies). This means that the classical and adiabatic limits do not commute, as has been noted in the literature \cite{Wu,Berry}. In the true quantum adiabatic limit, therefore, the sweep process is necessarily fully reversible and hysteresis is absent.
How slow does the sweep have to be, though, to actually reach this true quantum adiabatic limit? It turns out that even for quite small total particle numbers $N=\mathcal O(10)$, and for any remotely realistic values of the other system parameters, the sweep time has to be quite unrealistically slow: anywhere from several years up to many times the age of the universe \cite{quantum_dimer_I}. In \cite{quantum_dimer_I} we used Landau-Zener theory to describe the cyclic sweep process in the quantum version of the Bose-Hubbard dimer system that we had previously studied in \cite{dimer}, and compared a wide range of different sweep rates. We found that for not too small total particle numbers and a broad range of sweep rates the basic classical picture of two qualitatively different final states being reached probabilistically is recovered quantum mechanically. The quantum probability to recover the initial state (the \emph{return probability}), however, was found to depend sensitively on the sweep rate; it oscillates around the constant classical quasi-static value, with finite frequency and significant amplitude, even for very large particle numbers. We confirmed numerically that this non-classical oscillation of the return probability with sweep rate disappears only if, in addition to having large $N$, the initial quantum state is not a single energy eigenstate but a mixed state with a sufficient energy width.
\subsection{A quantum phase space picture}
These results may have shed some light on the role of quantum mechanics in the microscopic onset of irreversibility, but they have not clarified that role as well as one might wish, because quantum and classical probabilistic hysteresis have been described in such different terms. In the classical system the process is quite clear in phase space \cite{dimer}; it combines incompressible phase-space flow under Liouville's theorem, topological change of energy surfaces as they merge and separate, and effective ergodization through very fine swirling of initially coarse distributions. The return probability in the slow-sweep limit could even be computed analytically by applying Kruskal's theorem \cite{dimer}. The quantum phenomenon was in contrast described in terms of a sequence of Landau-Zener transitions between adiabatic quantum many-body eigenstates. The clear classical explanation for probabilistic hysteresis was hard to discern in this sequence, and the recovery of the classical return probability through a combination of many Landau-Zener transitions seemed to be a sheer numerical conspiracy.
To gain more insight into quantum irreversibility we therefore turn in this paper to a phase space representation of quantum dynamics, since the classical phase space picture is clear and quantum phase space methods have often proven to be very useful in understanding quantum-classical correspondence \cite{Polkovnikov_TWA, Witthaut2, Weinbub, Takahashi, Mahmud, Torres-Vega}. In particular we will use the Husimi quasi-probabiltiy function as a representation for the quantum states in phase space and compare its full quantum evolution to the semiclassical Truncated Husimi approximation \cite{Vermersch, Witthaut2, Trimborn}. While this quantum phase space description brings us closer to an analytical understanding of the purely numerical results of \cite{quantum_dimer_I} for the return probability, it also shows why quantum-classical correspondence can still break down for large total particle numbers, resulting in strong quantum effects. This is especially surprising because a naive argument suggests that the quantum correction term scales like $1/N$ (see Sec.~\ref{Husimi}). Furthermore the Husimi description allows us to distinguish the two qualitatively different effects of quantum noise and quantum interference. Finally, the Husimi phase space description will provide an intuitive understanding of why the true quantum adiabatic limit is so extremely hard to reach.
\subsection{The quantum Bose-Hubbard dimer}
The two-site Bose-Hubbard ``dimer'' system has been realized experimentally in ultracold atom systems \cite{Oberthaler1, Oberthaler2}; well before this achievement its theoretical study had already been extensive. Several previous works \cite{Wu2, Zobay, Liu, Wu4, Jona-Lasinio,Yang} have even specifically addressed slow parameter sweeps in this model. As discussed in more detail in \cite{quantum_dimer_I}, however, these earlier papers have used some typically quantum terminology (such as ``tunneling probability'') but have actually been restricted to the classical, mean-field version of the problem, and so do not really bear on our current topic.
In contrast to the two-state nonlinear Schr\"odinger evolution of mean-field theory, the full $N$-particle quantum many-body system of the Bose-Hubbard dimer concerns an $(N+1)$-component wave function evolving under a \emph{linear} Schr\"odinger equation. This problem has been studied for a single non-cyclic sweep in \cite{Korsch, Trimborn3, Wu, Chen}; it has been shown that the many-body Landau-Zener probability for a diabatic transition between the quantum levels goes to zero in the adiabatic limit of infinitely slow sweep rate, in accordance with the quantum adiabatic theorem. While this means that the mean-field and adiabatic limits do not commute, as mentioned above, it has also been demonstrated \cite{Korsch,Trimborn3,Wu} that for a fixed slow but finite sweep rate the ``Landau-Zener probability'' (i.e. the ratio of $\braket{n_2}$ to $\braket{n_1}$) approaches the mean-field value quite rapidly with increasing $N$, with good quantum-classical correspondence already for $N=\mathcal O (10)$.
As we have numerically demonstrated in \cite{quantum_dimer_I}, however, quantum-classical correspondence is more subtle than this for the phenomenon of probabilistic hysteresis, for two main reasons. Firstly, probabilistic hysteresis can occur for a finite range of initial states, not only the initial ground state, and the simple correspondence of the ground state turns out to be a special case. Secondly the scenario of probabilistic hysteresis includes a second, backward sweep, which begins from the excited state that was created by the forward sweep even when the initial state was the ground state. All cases of probabilistic hysteresis therefore turn out to involve significant quantum interference effects which are not captured by the semiclassical approximation and which persist even in the limit of very large $N$.
\subsection{Structure of the paper}
The rest of the paper is organized as follows: In Sec.~II we present the Hamiltonian and the sweep protocol that we will study and we also briefly review the semiclassical description of the sweep process. We will show how hysteresis and the finite return probability in the classical adiabatic limit can be understood by considerations in phase space. In Sec.~III we first introduce appropriate coherent states and the Husimi function. We then show how the semiclassical evolution in phase space and the evolution of the quantum Husimi function are related. We demonstrate that the classical ergodization mechanism fails to produce quantum ergodization, but instead induces the breakdown of quantum-classical correspondence, and is thus responsible for the strong quantum effects that occur, for single initial energy eigenstates, even at large total particle numbers. We also show how ergodization in the quantum system can be restored by a different mechanism so that for a finite initial energy width the semiclassical results are recovered after all. After a brief discussion of the entropy generated in the classical and quantum sweep process we show in Sec.~IV that the quantum adiabatic limit, in which the return probability is always one, can be understood as macroscopic quantum tunneling of a large number of atoms through the separatrix energy barrier and is therefore exponentially slow. We then proceed to Sec.~V where we summarize our main results.
\section{Setup and semiclassical description}
\subsection{Setup}
Our system is the two-mode Bose-Hubbard system with attractive interaction $U<0$ and tunneling rate $\Omega$. The two modes have a time-dependent energy detuning $\Delta(t)$, which will be our control parameter. The system Hamiltonian therefore reads
\begin{equation}
\hat H = -\frac{\Omega}{2}(\hat{a}^{\dagger}_{1}\hat{a}_{2}+\hat{a}^{\dagger}_{2}\hat{a}_{1})+\frac{U}{2}(\hat{n}_{1}^{2}+\hat{n}_{2}^{2})+\frac{\Delta(t)}{2}(\hat{n}_{1}-\hat{n}_{2}), \label{eq:H}
\end{equation}
where the bosonic operators $\hat a_{1,2}^{\dagger}$ ($\hat a_{1,2}$) create (destroy) a boson in the respective mode 1 or 2 and the number operators $\hat n_{1,2}=\hat a_{1,2}^{\dagger} \hat a_{1,2}$ are defined as usual. In this paper we choose units such that $\hbar=1$ and measure $\Delta$, $U$, energy and time in units defined by $\Omega$. The total particle number operator $\hat N=\hat n_1+\hat n_2$ commutes with the Hamiltonian, so that the total particle number given by its eigenvalue $N$ is conserved.
Our protocol consists of slowly ($T\gg \Omega^{-1}$) sweeping the energy detuning $\Delta(t)$ from a negative value $\Delta_I$ at the initial time $t=-T$ to the larger value $\Delta_0$ at $t=0$ (forward sweep) and then back again to $\Delta_I$ at the final time $t=+T$ (backward sweep):
\begin{equation}
\Delta(t)= \Delta_{I}\frac{|t|}{T}+\Delta_{0}\left(1-\frac{|t|}{T}\right), \qquad \Delta_0>\Delta_I.
\label{eq:sweep}
\end{equation}
We will study the evolution of a quantum state during this cyclic sweep; as initial states we will choose either a low-lying instantaneous energy eigenstate of the Hamiltonian with fixed $\Delta=\Delta_I$, or else a narrow microcanonical ensemble of such eigenstates. We then ask the question: With what probability is the initial state recovered at the final time, after the slow forward-and-back cycle of $\Delta$?
\subsection{Semiclassical picture}
\label{semiclassical}
The semiclassical description \cite{dimer} of the quantum sweep process is obtained by evolving an ensemble of initial phase space points that represent the initial quantum state under the mean-field equations of motion (Truncated Wigner or Truncated Husimi approximation). These equations of motion can be derived from the mean-field Hamiltonian
\begin{equation}
H=-\Omega \sqrt{p_0^2-p^2} \cos(q)+U\left(p_0^2+p^2\right)+\Delta(t) p,
\label{eq:H_cl}
\end{equation}
where $(q,p)$ are canonical coordinates representing the relative phase and particle imbalance, respectively \cite{dimer}. As always in these types of mean-field systems the system behavior can be characterized by the single parameter $u=UN/\Omega$ since Hamiltonians for different total particle numbers but same $u$ can be mapped onto each other by trivial rescaling.
For the following discussion let us assume as our initial state a microcanonical ensemble (i.e. a complete thin shell of fixed energy) at the initial detuning $\Delta_I$. Our reasoning may then be applied to arbitrary phase space distributions where the probability depends only on energy, by viewing them as consisting of a large number of narrow microcanonical ensembles. We sample this initial phase space density with a finite number of points and evolve each point under the mean-field equations of motion derived from Eq.~(\ref{eq:H_cl}). Since our sweep is slow compared to $\Omega^{-1}$ the classical adiabatic theorem can be applied unless the orbital period deviates significantly from $\Omega^{-1}$. This is not the case in the subcritical case $|u|<1$ (which means $u>-1$ for our attractive negative $U$) and so the action of each trajectory is an adiabatic invariant \cite{Goldstein}. This means that the orbits deform and their energies change during the forward sweep, but in a way that keeps their enclosed phase space area constant. During the backward sweep the same deformation happens in reverse and the initial and final ensemble coincide; consequently the return probability is one.
In the supercritical case $|u|>1$ ($u<-1$), on the other hand, there is an unstable fixed point in a certain $\Delta$ range. The energy contour running through this unstable fixed point, the \emph{separatrix}, divides the phase space into three mutually exclusive regions that we label $A_u$, $A_l$ and $A_o$ (see Fig.~\ref{fig:classical_phase_space}). Here and in the rest of the paper we consider $u=-3$ but other supercritical values give similar results.
\begin{figure*}
\centering
\subfloat[Initial state]{\includegraphics[width=.32\textwidth]{Fig1a.pdf}}
\subfloat[End of forward sweep]{\includegraphics[width=.32\textwidth]{Fig1b.pdf}}
\subfloat[Final state]{\includegraphics[width=.32\textwidth]{Fig1c.pdf}}
\caption{(Reproduced from Fig.~1 of Ref.~\cite{quantum_dimer_I}.) Evolution of a classical ensemble consisting of 2000 points (black dots) in phase space. The gray lines show adiabatic energy contours; the dashed black line is the separatrix that divides phase space into the regions $A_u$, $A_l$ and $A_o$. Because adiabaticity breaks down when the separatrix is crossed between (a) and (b) and between (b) and (c), only a finite fraction of the ensemble returns to the initial energy shell, so that probabilistic hysteresis occurs. For a clearer graphical presentation we have chosen the canonical coordinates $q'=\arctan\left(p/\left(\sqrt{p_0^2-p^2}\cos(q)\right)\right)$, $p'=-\sqrt{p_0^2-p^2}\sin(q)$ here. }
\label{fig:classical_phase_space}
\end{figure*}
When the separatrix first forms the entire ensemble resides within the upper lobe of the separatrix $A_u$---see Fig.~\ref{fig:classical_phase_space}(a). However, this separatrix lobe shrinks during the forward sweep so that at some point it meets the ensemble. As the separatrix approaches the ensemble, the orbital period of the ensemble members grows and finally diverges, since the orbital period associated with the separatrix itself is infinite. This means that the adiabatic theorem no longer holds at the separatrix, no matter how slow the parameter sweep may be. As a result the trajectories change their enclosed actions by leaving the upper separatrix lobe and moving into the only growing phase space region, namely the lower separatrix lobe. The continued growth of this lower lobe then means that the separatrix expands away from the ensemble, and so adiabaticity is restored to the ensemble again for the rest of the forward sweep (Fig.~\ref{fig:classical_phase_space}(b)).
During the backward sweep, then, adiabaticity continues to hold until the now-shrinking lower lobe of the separatrix $A_l$ hits the ensemble, at the same value of the detuning at which the separatrix crossing occurred in the forward sweep. Adiabaticity breaks down again, just as during the forward sweep, but now the upper lobe $A_u$ and the outside region $A_o$ are both growing, so that the members of the ensemble can go into either of these phase space regions. Kruskal's theorem (see \cite{LC} and references therein), which is derived from Liouville's theorem, gives the proportion of the parts of the ensemble going to the upper lobe $A_u$ and the outside region $A_o$. Since adiabaticity holds once again after the separatrix has been crossed for the second time, the part of the ensemble that went to the upper separatrix lobe ends up in the same energy shell in which it started initially. The rest of the ensemble, however, ends up with a much higher energy than it had initially, so that the final state consists of two well-separated sub-ensembles (Fig.~\ref{fig:classical_phase_space}(c)). We then define the return probability $P_{\mathrm{ret}}$ as the fraction of ensemble members that returned to the initial energy shell.
It is important to note that in the semiclassical system a well-defined quasi-static limit exists, even though adiabaticity is necessarily broken at some point, in the sense that the return probability settles quickly to the value predicted by Kruskal's theorem once the sweep rate falls below a certain finite range. Further reducing the sweep rate does not alter the return probability further.
Depending on the system parameters and on the energy of the initial ensemble, the return probability of a finite-width ensemble can range between almost zero and almost one. The essentially-zero return probability that is familiar from macroscopic systems for all initial conditions is not present in our simple integrable system; it can, however, be realized in a similar trimer system that allows chaotic dynamics \cite{trimer}. The general relationships between quantum chaos, quantum ergodicity, and irreversibility \cite{Zhang2,Peres} clearly require much further study; as a first step toward understanding probabilistic hysteresis as a particularly simple form of microscopic irreversibility, here we will consider only the quantum version of the integrable dimer system, and leave the quantum version of the non-integrable trimer for future work.
\section{Quantum Phase Space Picture}
\subsection{Husimi function}
\label{Husimi}
The classical phenomenon of probabilistic hysteresis concerns ensemble evolution in phase space, rather than the motion of individual phase space points. The classical evolution of the phase space density $\rho(q,p)$ is given by the Liouville equation, which for the Hamiltonian (\ref{eq:H_cl}) reads
\begin{equation}
\begin{split}
\dot \rho&=\frac{\partial \rho}{\partial p} \frac{\partial H}{\partial q}-\frac{\partial \rho}{\partial q} \frac{\partial H}{\partial p}\\
&=\Omega \sqrt{p_0^2-p^2} \sin(q) \frac{\partial \rho}{\partial p}\\
&-\left(2Up+\Delta+\frac{\Omega p}{\sqrt{p_0^2-p^2}}\cos(q)\right) \frac{\partial \rho}{\partial q}.
\end{split}
\label{eq:Liouville}
\end{equation}
In order to compare directly with this classical evolution, therefore, we must also formulate the quantum evolution in phase space, using a quantum quasi-probability function. In particular we use the Husimi function $Q$ \cite{Husimi}, which is defined as the probability to find the quantum system in a coherent state $\ket \Gamma$
\begin{equation}
Q(\Gamma,t)=|\braket{\Gamma|\psi(t)}|^2=\braket{\Gamma|\hat \rho(t)|\Gamma}.
\label{eq:Husimi_def}
\end{equation}
The definition of the Husimi function is motivated by the fact that coherent states are the most localized quantum states in phase space, in the sense that they minimize the uncertainty product of the phase space variables, so that they are the closest quantum analogues of classical phase space points. Despite this intuitive meaning of the Husimi function, one cannot simply interpret $Q$ as a probability distribution in phase space, because it does not necessarily give the correct marginal distributions if one of the phase space variables is integrated out. The Husimi function does provide a complete description of the quantum state, however, in the sense that all information about the quantum state can in principle be extracted from $Q$.
Using the von Neumann equation for the evolution of the density operator $\hat \rho$, the time evolution of the Husimi function is given by
\begin{equation}
\begin{split}
\dot Q&=\braket{\Gamma|\dot {\hat \rho}|\Gamma}=\tr \left(\dot{\hat \rho} \ket \Gamma \bra \Gamma\right)\\
&=\tr\left(i \hat \rho \hat H \ket \Gamma \bra \Gamma-i\ket \Gamma \bra \Gamma \hat H \hat \rho \right).
\label{eq:Qdot}
\end{split}
\end{equation}
Because the symmetry group of the Bose-Hubbard dimer is $SU(2)$ and it is therefore equivalent to a spin system with $s=p_0=N/2$ \cite{Schwinger, Sakurai}, the appropriate generalized coherent states are the so-called SU(2) coherent states \cite{Arecchi, Gilmore, Narducci, Narducci2, Trimborn, Witthaut2}
\begin{equation}
\begin{split}
\ket{\Gamma}&=\ket{\theta,\phi}\\&
=\sum_{n=0}^N \sqrt{{N}\choose {n}} \left(\cos \frac{\theta}{2}\right)^{n} \left(\sin \frac{\theta}{2} e^{i\phi}\right)^{N-n} \ket{n,N-n},
\end{split}
\end{equation}
where $(\theta,\phi)$ are the angles in a spherical coordinate system, namely the Bloch sphere. The classical canonical coordinates $(q,p)$, which have already been used in Sec.~II, are given by $q=\phi$ and $p=N/2 \cos(\theta)$; they map the Bloch sphere onto a flat phase space.
Defining the angular momentum operators
\begin{equation}
\begin{split}
\hat L_x&=\frac 12 \left(\hat a_1^{\dagger} \hat a_2+ \hat a_2^{\dagger} \hat a_1\right),\\
\hat L_y&=\frac i2 \left(\hat a_2^{\dagger} \hat a_1- \hat a_1^{\dagger} \hat a_2\right),\\
\hat L_z&=\frac 12 \left(\hat a_1^{\dagger} \hat a_1-\hat a_2^{\dagger} \hat a_2\right),
\end{split}
\end{equation}
and $\hat L_\pm=\hat L_x\pm i \hat L_y$, it then follows that the action of the $\hat L_\alpha$ operators on the coherent state projector $\ket \Gamma \bra \Gamma$ can be represented by differential operators $\mathcal D(\hat L_\alpha)$, such that \cite{Narducci, Narducci2, Zhang}
\begin{equation}
\begin{split}
\hat L_+ \ket \Gamma \bra \Gamma&=\mathcal D(\hat L_+) \ket \Gamma \bra \Gamma\\
&=e^{i \phi} \left(\frac{N}{2}\sin \theta +\frac{i}{2} \tan \frac{\theta}{2} \partial_\phi - \sin^2 \frac{\theta}{2} \partial_\theta \right) \ket \Gamma \bra \Gamma,\\
\hat L_- \ket \Gamma \bra \Gamma&=\mathcal D(\hat L_-) \ket \Gamma \bra \Gamma\\
&=e^{-i \phi} \left(\frac{N}{2}\sin \theta -\frac{i}{2} \cot \frac{\theta}{2} \partial_\phi + \cos^2 \frac{\theta}{2} \partial_\theta \right) \ket \Gamma \bra \Gamma,\\
\hat L_z \ket \Gamma \bra \Gamma&=\mathcal D(\hat L_z) \ket \Gamma \bra \Gamma\\
&= \left(\frac{N}{2}\cos \theta +\frac{i}{2} \partial_\phi - \frac 12 \sin \theta \partial_\theta \right) \ket \Gamma \bra \Gamma.
\label{eq:D_operators}
\end{split}
\end{equation}
In terms of these $\hat L_\alpha$ operators the Hamiltonian (\ref{eq:H}) can now be rewritten as
\begin{equation}
\hat H=-\frac{\Omega}{2}\left(\hat L_++\hat L_-\right)+U\left(\hat L_z^2+\frac{\hat N^2}{4}\right)+\Delta \hat L_z.
\label{eq:H_L}
\end{equation}
Using Eq.~(\ref{eq:Qdot}), Eq.~(\ref{eq:D_operators}) and Eq.~(\ref{eq:H_L}) we finally find
\begin{equation}
\begin{split}
\dot Q(\theta,\phi)&=i \tr \left(\mathcal D(\hat H) \ket \Gamma \bra \Gamma \hat \rho-\mathcal D(\hat H)^*\ket \Gamma \bra \Gamma \hat \rho\right)\\
&=-2 \imag \left[\mathcal D(\hat H)\right] \tr \left(\ket \Gamma \bra \Gamma \hat \rho \right)\\
&=\bigg[\frac{\Omega}{2}\cos(\phi)\left(\tan \frac{\theta}{2}-\cot \frac{\theta}{2}\right)\partial_\phi-\Omega \sin(\phi)\partial_\theta\\
& \qquad-UN\left(\cos \theta -\frac 1N\sin \theta \partial_\theta\right)\partial_\phi-\Delta \partial_\phi \bigg] Q(\theta,\phi)
\end{split}
\label{eq:Q_evolution}
\end{equation}
where $UN$ is of order one and the term containing second-order derivatives is therefore suppressed by a factor of $1/N$.
Using $\phi=q$, $\theta=\arccos(2p/N)$ we can also express Eq.~(\ref{eq:Q_evolution}) in $(q,p)$ as
\begin{equation}
\begin{split}
\dot Q(q,p)=&\Omega \sqrt{p_0^2-p^2} \sin(q) \frac{\partial Q(q,p)}{\partial p}\\
&-\left( 2Up+\Delta+\frac{\Omega p}{\sqrt{p_0^2-p^2}}\cos(q)\right) \frac{\partial Q(q,p)}{\partial q}\\
&-UN\left(\frac{p_0}{N}-\frac{p^2}{p_0 N}\right)\frac{\partial^2 Q(q,p)}{\partial q \partial p}.
\end{split}
\label{eq:Husimi}
\end{equation}
Comparing Eq.~(\ref{eq:Husimi}) to Eq.~(\ref{eq:Liouville}), we see that the evolution of the Husimi function is given by the classical Liouville equation plus a correction term containing second-order derivatives. Neglecting this last term leads to a ``Truncated Husimi approximation'' \cite{Witthaut2, Vermersch, Trimborn}, which, in analogy to the more familiar Truncated Wigner approximation \cite{Blakie_TWA, Polkovnikov_TWA, Sinatra_TWA}, can be understood as including quantum effects to first order. More specifically, quantum noise is modeled by sampling the initial conditions for the classical evolution from the quantum Husimi function, but quantum interference between different trajectories is neglected. Since $p$ and $p_0$ are of order $N$ and $UN$ is of order one, the quantum correction term seems to vanish in the classical limit ($N \to \infty$ with $UN$ held fixed), since the derivative with respect to $p$ comes with an additional factor of $1/N$. One might therefore naively expect to recover the classical phase space evolution for large $N$. While this argument is often invoked, we will show in the following that this is not necessarily the case if a separatrix is involved in the classical evolution.
\subsection{Classical and quantum ergodization}
In the (semi-)classical description the finite return probability can be predicted accurately by Kruskal's theorem, as outlined in Sec.~\ref{semiclassical}. Since Kruskal's theorem makes statements about a phase space volume (more specifically: an energy shell) and how it is split up among multiple growing phase space regions, it is a crucial assumption in this application of Kruskal's theorem that the actual phase space density of the ensemble is uniform in this phase space volume, i.e. $\rho(q,p)=\rho(E)$ is an ergodic phase space distribution. In our case this means that one assumes that the energy shell in the lower separatrix lobe is filled with a uniform reduced density after the separatrix has been crossed. Of course this can only be true in some kind of approximate sense, because the phase space volume occupied by the ensemble is required to stay constant by Liouville's theorem. What actually happens is that as the initially uniform microcanonical ensemble crosses the separatrix and spills into its lower lobe, the exact phase space density rapidly develops an extremely fine ``swirling'' structure, in which the phase space volume which is actually occupied by the ensemble does indeed remain constant, but it is distributed within the larger phase space volume in extremely fine threads \cite{dimer}. The ``coarse-grained'' phase space density is thus uniform but reduced. For slower sweep rates the swirling becomes steadily finer, so that this approximation becomes perfect in the quasi-static limit.
It is not obvious how this ergodization mechanism could be realized in the quantum evolution, however. While the Husimi function does \emph{not} have to obey Liouville's theorem, so that ergodization might conceivably be even more effective quantum mechanically than it is classically, it turns out that quantum-classical correspondence breaks down precisely because of the swirling discussed above.
\subsubsection{Single initial quantum eigenstate}
Fig.~\ref{fig:N=1000_single} (right column) shows the evolution of the Husimi function for $N=1000$, $\Delta_0/\Omega=-\Delta_I/\Omega=2$, $T=5000\Omega^{-1}$ and a single energy eigenstate as the initial state (we have chosen the 37th eigenstate as in \cite{quantum_dimer_I}), along with the classical evolution of the same initial phase space density under Eq.~(\ref{eq:Liouville}) (left column). For the evolution of the Husimi function we solve the Schr\"odinger equation with the Hamiltonian (\ref{eq:H}) numerically and then calculate the Husimi function via Eq.~(\ref{eq:Husimi_def}). For the classical evolution we use the method of characteristics \cite{Courant, Sarra} to solve the Liouville equation, i.e. we evolve individual phase space patches (to which some probability is assigned by the initial conditions) under the Hamiltonian equations of motion. Note that every Husimi function is in principle a valid classical phase space density, since it is non-negative and normalizable, so that in particular the initial Husimi function is also a valid initial condition for Eq.~(\ref{eq:Liouville}). The corresponding classical evolution is then the Truncated Husimi approximation of the full quantum evolution. (Another reasonable choice for the classical initial phase space density would be a classical microcanonical distribution with energy boundaries between the 36th and 37th and 37th and 38th quantum energy eigenvalues. Since the initial Husimi function is already quite microcanonical this choice would lead to a very similar evolution.) For a better visual presentation we have again used the canonical coordinates that we used in Fig.~\ref{fig:classical_phase_space},
\begin{equation}
\begin{split}
q'&=\arctan\left(\frac{p}{\sqrt{p_0^2-p^2}\cos(q)}\right)\\
p'&=-\sqrt{p_0^2-p^2}\sin(q),
\end{split}
\end{equation}
which simply correspond to a rotation of the Bloch sphere before the mapping from $(\theta,\phi)$ to $(q,p)$ is performed. Note also that we have normalized the Husimi function according to $\int \mathrm d q' \mathrm dp' \; Q(q',p')=1$.
\begin{figure}
\includegraphics[width=.45\textwidth]{Fig2.pdf}
\caption{Evolution of the Husimi function corresponding to a single initial state (37th adiabatic energy eigenstate, right) compared to the classical Liouville dynamics with the same initial phase space distribution (left). After the separatrix is crossed the two evolutions are quite different, despite the large total particle number. While the classical phase space density is ``grainy'' due to swirling (see text), the Husimi function is smooth, but far from ergodic, in that it has bright and dark spots. The system parameters are $N=1000$, $u=-3$, $\Delta_0/\Omega=-\Delta_I/\Omega=2$ and $T=5000\Omega^{-1}$.}
\label{fig:N=1000_single}
\end{figure}
Before the separatrix is crossed around $\Omega t=-1500$, the two evolutions are very similar, as expected, because the quantum correction term is suppressed by $1/N$. After the separatrix has been crossed the classical phase space density spreads almost uniformly into a larger phase space volume along an energy contour that is determined by the initial action \cite{dimer}. The classical phase space density actually has the extremely fine swirling structure mentioned above, but this structure is not fully resolved in our simulation with a finite sampling of phase space and therefore appears as seemingly random black dots spread through the energy shell. The swirling is so fine that a phase space volume much smaller than the Heisenberg limit $\hbar$ ($=1$ in our units) has to be resolved to reveal it. Fig.~\ref{fig:swirling} shows a part of the panels with high resolution, at the end of the forward sweep ($t=0$).
\begin{figure}
\centering
\subfloat[classical phase space density]{\includegraphics[width=.24\textwidth]{Fig3a.pdf}}
\subfloat[Husimi function]{\includegraphics[width=.24\textwidth]{Fig3b.pdf}}
\caption{Higher resolution image of Fig.~\ref{fig:N=1000_single} at $t=0$ showing the swirling of the classical phase space density (a) and the Husimi function (b). The region shown here corresponds to approximately one pixel in Fig.~\ref{fig:N=1000_single} and is much smaller than $\hbar$.}
\label{fig:swirling}
\end{figure}
Note that the phase space area shown in this figure is $0.006 \hbar$ and corresponds to approximately one pixel in Fig.~\ref{fig:N=1000_single}. For slower sweep rates the swirling becomes even finer, approaching a uniform but reduced phase space density, as discussed above. Consequently the swirling mechanism is responsible for (coarse-grained) ergodization. We emphasize again that this ergodization mechanism is crucial for the explanation of irreversibility in the classical limit, since Kruskal's theorem assumes an ergodic phase space distribution. Details can be found in \cite{dimer}. The fineness of the swirling also demonstrates why, even though the classical evolution is deterministic, we speak of \emph{probabilistic} hysteresis: to guarantee a reversible evolution the initial conditions would have to be tuned to be within phase space volumes of comparable size to the very fine scale of the swirling. For the sweep rate and particle number presented here this would mean controlling the initial phase space location of the system on scales much smaller than the Heisenberg limit. Since this is clearly impossible, experiments would show effectively random run-to-run alternations between the two final outcomes even if the evolution were classical.
The Husimi function, on the other hand, while also being localized on an energy contour, does not spread uniformly along this contour. Instead it forms a rapidly changing, more or less localized pattern (see also Fig.~\ref{fig:entropy}), so that there is \emph{no} ergodization. In the Landau-Zener picture \cite{quantum_dimer_I} (where the sweep is considered as a series of Landau-Zener crossings) this is an interference effect of the many involved adiabatic eigenstates. In the phase space picture, however, the classical ergodization mechanism via fine swirling breaks down despite the large particle number. The reason for this is that as soon as the classical swirling structure even begins to develop, the second derivatives in the quantum correction term in Eq.~(\ref{eq:Husimi}) become extremely large, because the swirling introduces steep gradients in $Q$ between the high and low probability stripes, as shown in Fig.~\ref{fig:swirling}. Therefore the same swirling mechanism that leads to ergodization of the classical phase space density is also directly responsible for the breakdown of quantum-classical correspondence. What turns out to happen quantum mechanically is that the sub-$\hbar$ sized swirling structure does not develop in the Husimi function.
With the breakdown of the classical ergodization mechanism there is no reason to expect ergodization of the Husimi function, and it is indeed absent as we have confirmed numerically. The observed localization of the Husimi function is a purely quantum phenomenon and has dramatic effects on the evolution in phase space, despite the large particle number $N$ that would at first sight suggests good quantum-classical correspondence.
During the backward sweep both the classical phase space density and the Husimi function split into two well separated parts, corresponding to the returning and non-returning fraction. However, due to the oscillatory behavior of the Husimi function the return probability $P_{\mathrm{ret}}$ at the end of the sweep depends sensitively on the sweep rate, in contrast to the classical return probability, see Fig.~\ref{fig:p_N=1000_single}. The return probability in the quantum case is defined in analogy to the classical return probability as the phase space integral of the Husimi function over the inner ring in Fig.~\ref{fig:N=1000_single} at $t=T$. This integral defines the return probability unambiguously, quantum mechanically as well as classically, because in both cases the inner and outer rings are well separated for large $N$.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Fig4.pdf}
\caption{Dependence of the return probability $P_{\mathrm{ret}}$ on the total sweep time $2 T$ for $N=1000$, $u=-3$, $\Delta_0/\Omega=-\Delta_I/\Omega=2$ and a single initial state. The red line shows the quantum return probability if the initial state is the 37th eigenstate of the initial Hamiltonian. The black line shows the corresponding classical return probability for an initial phase space density equal to the initial Husimi function. The slight variation of the classical return probability is due to our finite phase space resolution (sampling error).}
\label{fig:p_N=1000_single}
\end{figure}
In \cite{quantum_dimer_I} it was shown that the forward sweep leads to a superposition of adiabatic eigenstates. In the phase space picture this superposition is responsible for the rapid dynamics of the Husimi function. Since the return probability depends on the shape and localization of the Husimi function when it crosses the separatrix, the superposition of adiabatic eigenstates generally leads to a non-trivial dependence of the return probability on the total sweep time. The almost periodic dependence shown in Fig.~\ref{fig:p_N=1000_single} is due to the fact that only a relatively small number of adiabatic eigenstates, within a narrow range of adiabatic energies, have an appreciable amplitude. With weak nonlinearity in our system, the adiabatic eigenfrequencies of these superposed states are nearly even spaced within their narrow energy range, and thus nearly commensurate, so that the periodic maxima and minima of the return probability as a function of sweep rate are in fact similar to simple two-state Ramsey fringes.
\subsubsection{Initial ensemble of quantum eigenstates}
If we start with a microcanonical ensemble of quantum states initially, instead of with a single energy eigenstate, we obtain the evolution displayed in Fig.~\ref{fig:N=1000_ensemble}. Our ensemble contains 20 consecutive initial adiabatic eigenstates, chosen in such a way that the mean energy of the ensemble is essentially the same as the energy of the single state in Fig.~\ref{fig:N=1000_single}.
\begin{figure}
\includegraphics[width=.45\textwidth]{Fig5.pdf}
\caption{Evolution of the Husimi function corresponding to a microcanonical ensemble of 20 initial states (right) compared to the classical Liouville dynamics with the same initial phase space distribution (left). The oscillations of the Husimi function seen in Fig.~\ref{fig:N=1000_single} are strongly suppressed, so that we observe much better quantum-classical correspondence in the return probability. Parameters are the same as in Fig.~\ref{fig:N=1000_single}. The 20 quantum states are the initial 28th to 47th states, so that the mean energy of the ensemble is essentially the same as in Fig.~\ref{fig:N=1000_single}.}
\label{fig:N=1000_ensemble}
\end{figure}
Because the energy width is still small due to the large particle number, which makes the spacing between quantum energy eigenstates small enough that 20 states is a narrow range of eigenvalues, the classical evolution of this ensemble is almost indistinguishable from the classical evolution in Fig.~\ref{fig:N=1000_single}. The evolution of the Husimi function before the separatrix is crossed also remains very similar to the evolution shown in Fig.~\ref{fig:N=1000_single}.
For even this narrow 20-state microcanonical ensemble, however, the oscillations of the Husimi function that were found for a single initial state after the separatrix had been crossed are now suppressed, and ergodization is restored to a good approximation. The classical swirling structure is also present in the classical case with the larger initial energy width, and this still destroys naive quantum-classical correspondence as explained above, by inducing a large quantum correction term. In the 20-state quantum ensemble, however, a new quantum ergodization mechanism has emerged. The Husimi function of a mixed state is simply the weighted sum of the Husimi functions of the pure states that have been mixed (recall the definition of the Husimi function Eq.~(\ref{eq:Husimi_def})). For our microcanonical ensemble the Husimi function is therefore the average of many Husimi functions like the one shown in Fig.~\ref{fig:N=1000_single}. In each of these Husimi functions the localized dark and bright patches at any given time appear at different locations that depend strongly on energy. Averaging over energy therefore averages out the bright and dark patches, yielding an evenly ergodic total Husimi function.
Once quantum ergodization is established, the quantum evolution of the finite-width ensemble shows much better agreement with the semiclassical phase space evolution, because now the quantum correction term in Eq.~(\ref{eq:Husimi}) really does remain small (of order $1/N$). In particular the return probability loses its high sensitivity to the sweep rate and approaches the classical value, see Fig.~\ref{fig:p_N=1000_ensemble}. With no fine swirling, the quantum evolution of the Husimi function is approximately Liouvillian for large $N$, and so Kruskal's theorem then also holds approximately for the Husimi function when the separatrix is encountered during the backward sweep and the return probability is determined.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Fig6.pdf}
\caption{Dependence of the return probability $P_{\mathrm{ret}}$ on the total sweep time $2 T$ for $N=1000$, $u=-3$, $\Delta_0/\Omega=-\Delta_I/\Omega=2$ and an ensemble of 20 initial states (28th to 47th energy eigenstates) with mean energy similar to the 37th state (red line). The black line shows the corresponding classical return probability for an initial phase space density equal to the initial Husimi function. The oscillations of the return probability are much smaller than in Fig.~\ref{fig:p_N=1000_single} and are expected to vanish completely for larger initial energy width.}
\label{fig:p_N=1000_ensemble}
\end{figure}
It is therefore clear that the classical limit is not obtained simply by letting $N\to \infty$ with a single quantum state, but an ensemble with finite energy width is needed for good quantum-classical correspondence of the return probability in probabilistic hysteresis, as was also found in \cite{quantum_dimer_I}. In the quantum phase space formalism this can be explained by the fact that an ensemble containing enough quantum states effectively smears out the localized individual Husimi functions, so that the total Husimi function becomes effectively ergodized like the classical phase space density, albeit for quite different reasons. For their different reasons, the finely swirled exact classical phase space density and the energy-averaged Husimi function both behave very similarly to a smooth ergodized phase space density. The Husimi function for sufficient initial energy width and the classical phase space density thus effectively behave very similarly to each other, up to the small quantum-classical discrepancies of order $1/N$ that one naively expects from the Liouville and Husimi evolution equations.
\subsection{Entropy}
In a macroscopic system ergodization leading to irreversibility is associated with the growth of entropy. Microscopically, however, the phase space volume that is occupied by a classical ensemble is invariant under Hamiltonian time evolution---and so is the entropy. This is a direct consequence of Liouville's theorem, which is often expressed in the statement that classical phase space flow is like the flow of an incompressible fluid. The entropy in which one is normally interested, however, is some coarse-grained entropy, which is usually defined in reference to a limited resolution in phase space or to some implicit time-averaging. We realize this coarse-graining at every instant by time-averaging the fine-grained phase space density to obtain the coarse-grained density $\rho_c$
\begin{equation}
\rho_c(q',p')=\lim_{\widetilde T \to \infty} \frac{1}{\widetilde T} \int_0^{\widetilde T} \mathrm dt \; \rho(q',p';t),
\label{eq:rho_cg}
\end{equation}
where the time dependence of the phase space density on the right side is due to the evolution under the frozen Hamiltonian with fixed $\Delta$. Note that if the system is described in action-angle coordinates this provides coarse-graining in the angle coordinate only, but not in the action (or, equivalently, energy). Coarse-graining thus smears out the fine swirling structure found in the classical evolution, so that the coarse-grained entropy should increase when the separatrix is crossed and swirling occurs.
In the quantum system, on the other hand, the von Neumann entropy remains constant, since we do not trace out any degrees of freedom and the evolution is unitary. Therefore the question arises: what quantum entropy corresponds to the classical coarse-grained entropy? As we have already seen there is a close analogy between the classical coarse-grained phase space density and the Husimi function, so that one natural quantum analogue of the coarse-grained entropy is the \emph{Wehrl entropy} \cite{Wehrl,Wehrl2}
\begin{equation}
S_W=-k_B \int \mathrm dq' \mathrm dp' \; Q(q',p') \log\left[Q(q',p')\right].
\end{equation}
Because the flow of the Husimi density in phase space is \emph{not} incompressible, this entropy \emph{can} increase during the evolution, even without coarse-graining. Fig.~\ref{fig:entropy} shows the Wehrl entropy for the two cases discussed above, of a single initial eigenstate (black) and a 20-state microcanonical ensemble (blue). Note that the Wehrl entropy for the ensemble of states is always higher than the Wehrl entropy for the single quantum state, simply because the initial Husimi function is wider.
\begin{figure}
\includegraphics[width=.45\textwidth]{Fig7.pdf}
\caption{Wehrl entropy $S_W$ corresponding to the quantum evolutions shown in Fig.~\ref{fig:N=1000_single} (black) and Fig.~\ref{fig:N=1000_ensemble} (blue). While the oscillation of the Husimi function in the case of a single initial state leads to a strong oscillation of the Wehrl entropy, the oscillations of the Wehrl entropy in the case of an initial ensemble of states are suppressed. The red line shows the coarse-grained entropy of the classical simulation of Fig.~\ref{fig:N=1000_ensemble} for comparison. For further comparison, the magenta line shows the von Neumann entropy of the time-averaged quantum density matrix (see text). Since the definitions of the Wehrl and classical entropies allow an arbitrary constant shift from the phase space measure, we have used this to set all three entropies equal for the mixed initial state.}
\label{fig:entropy}
\end{figure}
In the case of a single initial state (black), where there is no quantum ergodization, the Wehrl entropy can be used to quantify the localization of the Husimi function. The strong and fast oscillations shown in Fig.~\ref{fig:entropy} for $t\gtrsim -1500 \Omega^{-1}$ therefore confirm what we have already seen for a few discrete values of $t$ in Fig.~\ref{fig:N=1000_single}: fast ``collapse and revival'' of the Husimi function instead of smooth ergodization.
In the case of the ensemble of initial states (blue) these oscillations of the Wehrl entropy are much smaller, reflecting the smoothness of the Husimi function observed in Fig.~\ref{fig:N=1000_ensemble}. Does this Wehrl entropy then correspond to the classical coarse-grained entropy? To answer this question we also show the coarse-grained classical entropy, obtained from the classical coarse-grained phase space density Eq.~(\ref{eq:rho_cg}) of Fig.~\ref{fig:N=1000_single}, as a red line in Fig.~\ref{fig:entropy}. The red and blue lines agree initially, until the separatrix crossing, because as long as there is no swirling quantum and classical evolution correspond closely at this $N$, and because without swirling of the initially ergodic classical ensemble, coarse-graining has no effect on it.
The classical entropy (red) increases in two steps and stays essentially constant in between. The first step occurs around $\Omega t=-1500$ when the separatrix is crossed during the forward sweep and the coarse-grained density fills the larger phase space volume in the growing lower separatrix lobe. The second step around $\Omega t=3500$ is due to the same mechanism: After $\Omega t=2500$, where $\Delta$ is negative again, the phase space region outside the separatrix is shrinking. This means that the outer shell, representing the non-returning fraction, crosses the separatrix again when it makes the transition from the figure-eight shape shown in the second-to-last panel of Fig.~\ref{fig:N=1000_ensemble} to the ellipsoidal shape shown in the last panel. In the same way as when the separatrix was crossed during the forward sweep, the outer shell merges with another empty energy shell (which is leaving the lower separatrix lobe), so that the phase space density decreases and entropy increases. This transition does not influence the return probability, because which trajectories return to the initial state has already been decided much earlier and this phase space dilution process involves only the part of the phase space density that did not return, anyway.
The Wehrl entropy for the ensemble of states does not fully agree with the classical entropy, but shows additional features that are due to its finite minimum width. When the separatrix is crossed around $\Omega t=-1500$ the Wehrl entropy increases to a value above the classical entropy, simply because the Husimi function for our large but finite $N$ is a little wider than the classical phase space density (see Fig.~\ref{fig:N=1000_ensemble}). When the separatrix is encountered again during the backward sweep around $\Omega t=1500$, the classical phase space density and the Husimi function split into two parts, the outer figure-eight shaped shell and the inner ellipsoidal shell. While the classical occupied phase space volume stays constant during this transition, so that the entropy also does not change, the outer shell becomes much thinner than the Husimi function can ever be. The greater width of the Husimi function compared to the classical phase space distribution then leads to a small additional increase of the Wehrl entropy that has no classical counterpart. In general a finite width Husimi function will always have a larger (Wehrl) entropy than the corresponding classical distribution, because the Husimi function has blurred edges in comparison to the classical phase space density. This also means that the Wehrl entropy can decrease if the shape of the classical distribution is deformed in a such a way that the length of the edges decreases. This is the case when the outer shell crosses the separatrix around $\Omega t=3500$, and apparently the decrease of the Wehrl entropy in this process is almost compensated by the increase of the entropy that was expected from the classical considerations.
While the Wehrl entropy for our large but still finite particle number $N$ does therefore not fully agree with the classical entropy, a somewhat better agreement can be found for the alternative entropy
\begin{equation}
S=-k_B\sum_n p_n \log(p_n)
\label{eq:cg_vN_entropy}
\end{equation}
with
\begin{equation}
p_n=\braket{n|\hat \rho|n}
\end{equation}
where $\ket n$ are the adiabatic eigenstates. This entropy, which was introduced by von Neumann \cite{vonNeumann} for the case of a pure quantum state, can be understood as the more widely known mixed-state von Neumann entropy $-k_B \mathrm{Tr}(\hat{\rho}\ln\hat{\rho})$ of the time-averaged density matrix \begin{equation}
\hat \rho_c=\lim_{\widetilde T\to \infty} \frac{1}{\widetilde T}\int_0^{\widetilde T} \mathrm dt \; \hat \rho(t),
\end{equation}
where $\hat \rho(t)$ on the right side is again given by the evolution under the frozen Hamiltonian with fixed detuning $\Delta$. The off-diagonal terms in the density matrix oscillate because the quantum phases of the adiabatic eigenstates evolve at different speeds, and because the phase difference between different adiabatic eigenstates $\ket n$, $\ket m$ is thus effectively random, the off-diagonal terms are averaged out. In fact, due to the slowness of the sweep compared to the time scale on which the adiabatic phases change, the averaging can also be done for finite $\widetilde T$ and $\hat \rho(t)$ given by evolution under the actual time-dependent Hamiltonian. The same reasoning was the motivation for the incoherent Landau-Zener approximation in \cite{quantum_dimer_I}.
The entropy Eq.~(\ref{eq:cg_vN_entropy}) is thus conceptually similar to the classical coarse-grained entropy, where time-averaging smeared out the classical swirling structure. It and its generalizations to other bases besides energy eigenstates have been shown to satisfy analogs of the Boltzmann $H$-theorem \cite{Han,Hu} and have been proposed elsewhere as useful tools for analyzing quantum-classical correspondence \cite{Fang}. Since the entropy Eq.~(\ref{eq:cg_vN_entropy}) may therefore also be considered as an alternative quantum entropy, we plot it in Fig.~\ref{fig:entropy}; it shows better agreement with the classical coarse-grained entropy compared to the Wehrl entropy. Note that the classical and Wehrl entropies in Fig.~\ref{fig:entropy} have both been shifted by a fixed amount, as one is always free to do by changing the size of the elementary phase space cell, so that the blue, red and magenta curves start out at the same value of $S=k_B\log(20)$, since we start with a microcanonical ensemble of 20 quantum states. The von Neumann entropy of the time-averaged density matrix then increases whenever the evolution of one of the 20 initial states leads to a superposition of multiple adiabatic eigenstates, which can, for large $N$, only happen close to where the separatrix is crossed classically, as has been discussed in \cite{quantum_dimer_I}. We may therefore say that the quantum analogue of classical swirling is the quantum superposition of adiabatic eigenstates.
To summarize, in the case of sufficient initial energy width corresponding patterns of plateaus and jumps can be seen in the quantum and classical entropies. Precise agreement between quantum and classical entropies even in the large-$N$ limit is hard to confirm, however. This is partly due to the basic fact, ultimately due to the uncertainty principle, that probability density in phase space is just not really well-defined in quantum mechanics. Whatever kind of quantum quasi-probability function one may define in phase space, the width of this cannot be guaranteed to coincide exactly with the width of any classical phase space density, which can become arbitrarily thin. In particular the Husimi function of a given quantum state cannot be considered to represent the probability distribution in phase for that quantum state, since integrating the Husimi function over position or momentum will in general not yield the correct probability distribution of the remaining canonical observable in that state. It is still true that every quantum Husimi function is valid as \emph{a} classical ensemble in phase space, but the Husimi function of a quantum microcanonical ensemble is \emph{not} a classical microcanonical ensemble, and this makes the comparison between the Wehrl entropy and the entropy of the time-averaged density matrix difficult. Even though correspondence of the return probabilities is restored, quantum-classical correspondence is thus not perfect, even at large $N$ and sufficient energy width for quantum ergodization.
Besides the dramatic failure of quantum-classical correspondence for large particle numbers and a single initial state, there is also the more expected failure at low particle numbers. In particular the adiabatic limit in the quantum system is very different from the classical adiabatic limit, in that the evolution is reversible in the former limit whereas irreversibility persists in the latter limit. We will show in the next section how the quantum adiabatic limit appears in the Husimi phase space formulation.
\section{Reversibility by Macroscopic Quantum Tunneling}
After having demonstrated how the classical limit of the return probability can emerge from the quantum phase space description for large particle numbers, sufficient energy width, and very but finitely slow sweep rate, we now consider the case of ultimately slow sweep rates, in which the quantum adiabatic limit of completely reversible evolution is always different from the classical quasi-static limit of probabilistic hysteresis. As has been shown in \cite{quantum_dimer_I}, even for quite small $N$ this quantum adiabatic limit requires quite unrealistically slow sweep rates, because the energy gaps with respect to which the sweep has to be adiabatic are exponentially small in $N$. For $N=10$, however, we can at least reach the quantum adiabatic limit with a numerical simulation, in the sense that the system remains in the same adiabatic eigenstate with probability $>0.99$. We can achieve this for the same Hamiltonian parameters as in Fig.~\ref{fig:N=1000_single} with $T=10^8 \Omega^{-1}$. For $N=20$ and the same Hamiltonian parameters, on the other hand, the total sweep time $2T$ already has to be on the order of $10^{15} \Omega^{-1}$ to obtain a fully reversible evolution, which would be around 30 years if $\Omega$ were in the experimentally typical MHz regime, or even 30000 years for $\Omega$ in the experimentally feasible kHz regime. The question of why the quantum adiabatic limit is so insanely hard to reach for higher $N$ can actually be answered by the numerically achievable case of $N=10$.
Fig.~\ref{fig:Husimi_N=10} shows the evolution of the Husimi function for $N=10$ and $T=10^8 \Omega^{-1}$, where the initial state is the ground state at $\Delta_I/\Omega=-2$, meaning that initially almost all atoms are in the first mode and the Husimi function is localized in the upper half of the phase space $q'>0$.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{Fig8.pdf}
\caption{Evolution of the Husimi function in the reversible quantum adiabatic limit for $N=10$, $T=10^8 \Omega^{-1}$, $u=-3$ and $\Delta_0/\Omega=-\Delta_I/\Omega=2$ during the forward sweep, where the initial state is the ground state. The panels for the backward sweep are essentially identical to the panels shown, but in reverse order. Reversibility in this extreme case is restored by macroscopic quantum tunneling: All atoms tunnel collectively through the separatrix, which is indicated by the dashed white line, in the forward and backward sweep. Accordingly, the Husimi function also tunnels through the separatrix: instead of continuously flowing through the separatrix as in Fig.~\ref{fig:N=1000_single} and Fig.~\ref{fig:N=1000_ensemble}, the Husimi function simply fades away on one side of the separatrix and grows on the other side, without ever having visible support on the separatrix itself. Because the system always stays in a single adiabatic eigenstate, interference effects like those in Fig.~\ref{fig:N=1000_single} are absent.}
\label{fig:Husimi_N=10}
\end{figure}
We find that around $\Delta\sim 0$ ($t\sim -T/2$), where the energy difference between the two lowest energy eigenstates becomes minimal, the Husimi function \emph{tunnels} into the lower half of the phase space. At this time most of the corresponding classical orbits would still be far away from the separatrix and would therefore stay in the upper separatrix lobe, until a much larger detuning is reached and they are touched by the shrinking separatrix lobe. This tunneling through an energetic barrier (and through the separatrix in phase space) is an example of \emph{macroscopic quantum tunneling} \cite{Owerre, Leggett, Takagi, Leggett2}: in the adiabatic limit all atoms tunnel from the first mode to the second mode collectively during the forward sweep around $\Delta\sim 0$. During the backward sweep the same tunneling occurs again, so that the evolution is reversible. Because the energetic barrier is very high, this macroscopic tunneling is so extremely slow that it only plays a role for extremely slow sweep rates. For the classical phase space distribution, in contrast, tunneling through an energetic barrier is not possible at all, no matter how slow the sweep may be. This tunneling through a separatrix, which is only possible quantum mechanically, is the reason for the incommutability of the semiclassical and adiabatic limits.
Because \emph{all} particles have to tunnel through the barrier it is also clear that the sweep time needed to reach the adiabatic limit quickly increases with $N$. For higher initial energy the energetic barrier is lower and a smaller number of particles has to tunnel ($N-2(i-1)$ for the initially $i$-th state). As long as the total particle number is not very small, however, the effect of macroscopic tunneling can still be neglected for realistic sweep rates until the Husimi function comes close to the separatrix and the energetic barrier is very low. In this case the only practical consequence of macroscopic tunneling is that the crossing of the separatrix happens in a slightly larger $\Delta$ range than would be expected classically.
\section{Conclusion}
In conclusion we have provided a phase space description of how irreversibility in the form of probabilistic hysteresis occurs in a dissipationless quantum system. In particular we have used the Husimi function to show that the quantum evolution is closely related to the classical evolution, with the usual $1/N$ dependence of the quantum correction term, but in spite of this scaling, which naively suggests good quantum-classical correspondence for large $N$, we have found that the particular ergodization mechanism due to ``swirling'' in the classical evolution leads to a breakdown of quantum-classical correspondence. This inhibits quantum ergodization of the Husimi function, precisely at the point where classically the separatrix is crossed and irreversibility begins its onset. This quantum lack of ergodization has the result that the quantum return probability oscillates around the semiclassical value if the sweep time is varied, even for very large $N$.
The classical limit for the return probability thus only emerges in the full quantum evolution if there is a specifically quantum ergodization mechanism. Such an ergodization mechanism appears naturally if an ensemble of initial quantum states is considered instead of a single state, because the Husimi function in this case is the superposition of the Husimi functions of the individual pure states, each of which is quite localized but rapidly oscillating. The classical return probability in probabilistic hysteresis is determined under Kruskal's theorem by the combination of Liouvillian evolution and effective ergodization, and so the classical return probability is only recovered from the quantum evolution if, besides the usual mean-field limit $N\to \infty$, a finite initial energy width is also allowed, since in this case Kruskal's theorem can also be applied to the Husimi evolution to a good approximation.
Even for large $N$ and sufficient initial energy width for quantum ergodization, quantum-classical correspondence is not perfect, though. While to the naked eye the Husimi function and the classical phase space density appear almost indistinguishable in this case, the comparison of the Wehrl entropy and the classical coarse-grained entropy reveals that the Husimi function still has distinctive quantum features. In particular the Husimi function always has a finite width due to the uncertainty principle, so that even at large $N$ it can be considerably wider than the corresponding classical phase space density. Good correspondence between the quantum and classical return probabilities still appears for finite $N$, however, because the classical return probability for thin energy shells only depends weakly on energy itself, so that the additional width of the Husimi function does not change the return probability significantly. It remains an open question whether the entropy discrepancies due to thinness eventually vanish in the true classical limit of $N \to \infty$ or for larger initial energy width. After all, the Husimi function cannot be interpreted as a true probability distribution in phase space.
In the opposite limit of small particle numbers we have shown that it is macroscopic quantum tunneling that is responsible for the different behavior of the quantum and classical systems in the quasi-static limit of infinitely slow sweep rate. Unlike the classical system, the quantum system can tunnel through the separatrix---and can thereby remain in the same adiabatic energy eigenstate throughout the whole forward-and-back sweep cycle, so that reversibility is restored. Since the energetic barrier is high, however, and the number of atoms that have to tunnel is of order $N$ for not too high initial energy, this tunneling is so very slow that for realistic sweep rates it plays no role unless the total particle number is very low ($N\lesssim 20$).
In final summary, we have offered a complementary viewpoint on quantum probabilistic hysteresis to the description in terms of Landau-Zener crossings that was obtained in \cite{quantum_dimer_I}. We have identified why quantum-classical correspondence breaks down in probabilistic hysteresis (fine classical swirling) and why the classical and quantum adiabatic limits are different (macroscopic quantum tunneling). Furthermore the phase space picture that we have presented in this paper leads us at least to the conjecture that the exact unitarity of quantum evolution, which is a close analog to the Liouvillian incompressibility of classical phase space flow, must be a fundamentally robust point of correspondence between quantum and classical dynamics, such that given some form of ergodization in either kind of dynamics, conclusions like those of the classical Kruskal's theorem must emerge in both cases. A fully quantum analog of Kruskal's theorem thus becomes a natural goal for future study; it might be approached by returning to the Landau-Zener picture of our previous paper \cite{quantum_dimer_I}.
\acknowledgements
The authors acknowledge support from State Research Center OPTIMAS and the Deutsche Forschungsgemeinschaft (DFG) through SFB/TR185 (OSCAR), Project No. 277625399.
|
2,877,628,090,800 | arxiv | \section{Reorthogonalization of LIOMs} \label{app1}
For clarity, the derivation of the true perturbation, $H^{\perp}_{\Delta}$, has been carried out for the grand canonical averaging, $\langle ... \rangle$, when the occupations of the Anderson states are independent and the products $Q^{(2)}_{\alpha,d}$ are mutually orthogonal and normalized, i.e. $\langle Q^{(2)}_{\alpha,d} Q^{(2)}_{\alpha',d'} \rangle=\delta_{\alpha,\alpha'} \delta_{d,d'} $.
In order to study larger systems and to minimize the finite-size effects, the numerical calculations presented in the main text have been carried out in a subspace with a fixed number of fermions, $N=L/2$. Then, the Anderson LIOMs introduced in Eq.~(\ref{h0diag}) in the main text are not independent because
$\sum_{\alpha=1}^L Q_{\alpha}$=const. As a consequence, the products of the Anderson LIOMs, $Q^{(2)}_{\alpha,d}$, are not orthonormal within this subspace. In order to apply the orthogonal projections, see Eq.~(\ref{proj}) in the main text, one needs to reorthogonalize the
set $\{Q^{(2)}_{\alpha,d}\}$. To this end we solve the eigenproblem
\begin{equation}
\sum_{\alpha'=1}^{L} \sum_{d'=1}^{d_{max}} \langle Q^{(2)}_{\alpha,d} Q^{(2)}_{\alpha',d'} \rangle V_{(\alpha',d'),\gamma} = \lambda_{\gamma} V_{(\alpha,d),\gamma}
\end{equation}
for the real symmetric matrix built out of all scalar products of $Q^{(2)}_{\alpha,d}$, i.e., we solve the eigenproblem for $\langle Q^{(2)}_{\alpha,d} Q^{(2)}_{\alpha',d'} \rangle$. Here, $V_{(\alpha,d),\gamma}$ is an orthogonal matrix and the eigenvalues are positive, $\lambda_{\gamma} > 0$, for $d_{\max} < L/2-1$. We introduce a new set of LIOMS
\begin{equation}
q_{\gamma}=\sum_{\alpha=1}^{L} \sum_{d=1}^{d_{max}} \frac{1}{\sqrt{ \lambda_{\gamma}}} V_{(\alpha,d),\gamma} Q^{(2)}_{\alpha,d} \;, \label{lintra}
\end{equation}
which are normalized and mutually orthogonal
\begin{eqnarray}
\langle q_{\gamma} q_{\gamma'} \rangle &=& \sum_{\alpha,\alpha'=1}^{L} \sum_{d,d'=1}^{d_{max}} \frac{ V_{(\alpha,d),\gamma} \langle Q^{(2)}_{\alpha,d} Q^{(2)}_{\alpha',d'} \rangle V_{(\alpha',d'),\gamma'}}
{\sqrt{ \lambda_{\gamma}} \sqrt{ \lambda_{\gamma'}}} \nonumber \\
&=& \delta_{\gamma,\gamma'} \frac{ \lambda_{\gamma'}}{\sqrt{ \lambda_{\gamma}} \sqrt{ \lambda_{\gamma'}}} = \delta_{\gamma,\gamma'}.
\end{eqnarray}
The new set of orthonormal Anderson LIOMs, $\{ q_{\gamma} \} $, should be used in the Eq.~(\ref{proj}) in the main text instead of $\{ Q^{(2)}_{\alpha,d} \}$. We note that
that the linear transformation in Eq.~(\ref{lintra}) of local $Q^{(2)}_{\alpha,d}$ leads to local $q_{\gamma}$. Therefore the reorthogonalization
does not spoil locality of the products of Anderson LIOMs. \\
\ \\
\section{Bounds for projected operators} \label{app2}
Here, we study in more details the properties of the projected operators, $N^{\perp}_i$, and establish a lower bound on their norm.
To this end we use the many-body Anderson states $| \vec{\alpha} \rangle=|\alpha_1,\alpha_2,... ,\alpha_L \rangle$ and introduce an auxiliary operator
\begin{equation}
\tilde{N}_i=N_i-\sum_{\vec{\alpha}} \langle \vec{\alpha}|N_i| \vec{\alpha} \rangle \; |\vec{\alpha} \rangle \langle \vec{\alpha} |. \label{fulproj}
\end{equation}
where all diagonal matrix elements have been eliminated. We note that the projection in Eq.~(\ref{proj}) in the main text eliminates the diagonal matrix elements of $N_i^{\perp}$ only partially, hence it is intuitively clear that
$ ||N_i^{\perp} || \ge || \tilde{N}_i|| $. Below we present a formal proof of this lower bound on $ ||N_i^{\perp} ||$.
We rewrite Eq.~(\ref{proj}) in the main text as
\begin{equation}
N_i=N^{\perp}_i + \sum_{\beta,d} \langle Q^{(2)}_{\beta,d} N_i \rangle Q^{(2)}_{\beta,d} \label{ss1}
\end{equation}
and note that $Q^{(2)}_{\beta,d}$ are diagonal in the many-body Anderson basis, hence
\begin{equation}
\sum_{\vec{\alpha}} \langle \vec{\alpha}| Q^{(2)}_{\beta,d} | \vec{\alpha} \rangle \; |\vec{\alpha} \rangle \langle \vec{\alpha}| =Q^{(2)}_{\beta,d} \;. \label{ss2}
\end{equation}
Putting Eqs.~(\ref{ss1}) and~(\ref{ss2}) into the right-hand side of Eq.~(\ref{fulproj}) one obtains the identity
\begin{equation}
N^{\perp}_i = \tilde{N}_i + \sum_{\vec{\alpha}} \langle \vec{\alpha}|N^{\perp}_i | \vec{\alpha} \rangle \; |\vec{\alpha} \rangle \langle \vec{\alpha}|. \label{ss3}
\end{equation}
In the Anderson basis, $ \tilde{N}_i$ has only off-diagonal matrix elements whereas the second term, $ \sum_{\vec{\alpha}} \langle \vec{\alpha}|N^{\perp}_i | \vec{\alpha} \rangle \; |\vec{\alpha} \rangle \langle \vec{\alpha}|$, is
diagonal. Moreover, the squared Hilbert-Schmidt norm can be explicitly written as a sum of squares of all matrix elements, $ ||...||^2= \frac{1}{Z} \sum_{\vec{\alpha},\vec{\alpha}'} | \langle \vec{\alpha}| ... | \vec{\alpha}' \rangle|^2$. The latter two properties combined with Eq.~(\ref{ss3})
imply that
\begin{equation}
||N^{\perp}_i ||^2=|| \tilde{N}_i ||^2+ || \sum_{\vec{\alpha}} \langle \vec{\alpha}|N^{\perp}_i | \vec{\alpha} \rangle \; |\vec{\alpha} \rangle \langle \vec{\alpha}| \; ||^2 \ge || \tilde{N}_i ||^2.
\end{equation}
Consequently, one obtains the lower bound
\begin{equation}
||N^{\perp}_i ||^2 \ge || \tilde{N}_i ||^2, \label{bound0}
\end{equation}
which holds true for arbitrary distance $d_{max}$ in Eq. (\ref{proj}) in the main text.
\begin{figure}[!b]
\includegraphics[width=1.0\columnwidth]{figS1.png}
\caption{Results for chain with $L=14$ sites and $L/2$ fermions. Points show $||N_i^{\perp} ||$ and $||\tilde{N}_i || $ for various sites $i$, disorder realizations and $W$. In (a), (b), (c) and (d) we use, respectively, the maximal distance
$d_{max}=2,3,4$ and 5, see Eq. (\ref{proj}) in the main text.
}
\label{figs1}
\end{figure}
Figure \ref{figs1} shows correlations between norms of $N_i^{\perp}$ and $ \tilde{N}_i$ for various $d_{max}$. Each point in this plot shows result for a single site $i$ and a single realization of disorder. These results not only confirm the
bound $ ||N_i^{\perp} || \ge || \tilde{N}_i||$ from Eq.~(\ref{bound0}) but also demonstrate that for strong disorder (i.e., for small $||N_i^{\perp} ||$) the projected operator can be well approximated as $N_i^{\perp} \simeq \tilde{N}_i$. Comparing Figs.~\ref{figs1}(a)-\ref{figs1}(d) one observes that the larger is distance $d_{max}$ the better is the approximation, however for strong disorder it is accurate already for $d_{max}=2$. It means that the conserved part of the interaction, $N_i$, is dominated by the projections of $N_i$ on $Q^{(2)}_{\alpha,d}$, i.e., by the products of the Anderson LIOMs corresponding to the neighboring orbitals with $d=1$ or $d=2$.
Next, we discuss the origin of the bound $|| N^{\perp}_i ||\ge 1/(8W)^2$ which was observed from the numerical data shown in the main text in Fig. \ref{fig1}. To this end we show that $|| \tilde{N}_i ||\ge 1/(8W)^2$ and
then make use of the inequality (\ref{bound0}). For simplicity we assume that the single-particle wave functions, $u_{i \alpha}$, are real,
we express the operator $N_i$ in the Anderson basis
\begin{eqnarray}
&& N_i = \left(a^{\dagger}_{i} a_{i}- \frac{1}{2}\right) \left(a^{\dagger}_{i+1} a_{i+1}-\frac{1}{2}\right) \label{nia} \\
&&=\sum_{\alpha,\beta,\gamma,\eta} u_{i \alpha} u_{i \beta} (a^{\dagger}_{\alpha} a_{\beta}-\frac{\delta_{\alpha,\beta}}{2}) u_{i+1 \gamma} u_{i+1 \eta} (a^{\dagger}_{\gamma} a_{\eta}-\frac{\delta_{\gamma,\eta}}{2}) \;, \nonumber
\end{eqnarray}
and recall that $\tilde{N}_i$ represents the off-diagonal contribution to Eq.~(\ref{nia}).
The more localized are the Anderson wave-functions the smaller are the off-diagonal contributions to Eq.~(\ref{nia}).
Then, it is useful to study a two-site problem for the Anderson Hamiltonian [see Eq.~(\ref{h0}) in the main text] with extreme values of the disorder potentials $\epsilon_{1,2}=\pm W$
\begin{equation}
\left(
\begin{array} {cc}
W & \frac{1}{2} \\
\frac{1}{2} & -W \\
\end{array}
\right)
\left(
\begin{array} {c}
u_{1 \alpha} \\
u_{2 \alpha} \\
\end{array}
\right)
=\varepsilon_{\alpha}
\left(
\begin{array} {c}
u_{1 \alpha} \\
u_{2 \alpha} \\
\end{array}
\right).
\label{sw2}
\end{equation}
Direct calculations show that $\lim_{W \to \infty} u_{11}=\lim_{W \to \infty} u_{22}=1$, whereas,
\begin{equation}
\lim_{W \to \infty} W u_{i \alpha }= \pm \frac{1}{4}, \quad \quad i \ne \alpha. \label{s2}
\end{equation}
We assume that the lower bound on $|| \tilde{N}_i || ^2$ denoted as $ ||N^{\rm bound}_i ||^2$, can be obtained via introducing single-particle wave-functions from Eq.~(\ref{sw2}) into Eq.~(\ref{nia}).
In other words, $N^{\rm bound}_i $ corresponds to the most localized orbitals, $u_{i \alpha}$ on two sites.
Namely, we assume that for each site $i$ there is a single Anderson state denoted as $\alpha(i)$
such that $u_{i \alpha(i)}$, is of the order $O(1)$ and one other state, $\alpha'(i) \ne \alpha(i)$, for which $u_{i \alpha'(i)}=\pm \frac{1}{4W} $
\begin{equation}
u_{i \beta} \simeq \delta_{\beta,\alpha(i)} \pm \frac{1}{4W} \delta_{\beta,\alpha'(i)}.
\end{equation}
Then, one finds the leading (with respect to $1/W$) contributions to the off-diagonal part of $N^{\rm bound}_i $,
\begin{eqnarray}
N^{\rm bound}_i &\simeq & \pm \frac{1}{4W} [a^{\dagger}_{\alpha'(i)}a_{\alpha(i)} + {\rm H.c.}][n_{\alpha(i+1)}-\frac{1}{2}] \nonumber \\
&& \pm \frac{1}{4W}[n_{\alpha(i)}-\frac{1}{2}] [a^{\dagger}_{\alpha'(i+1)}a_{\alpha(i+1)} + {\rm H.c.}]\;. \nonumber \\ \label{bound1}
\end{eqnarray}
Using the identity $[n_{\alpha(i)}-\frac{1}{2}] | \vec{\alpha} \rangle =\pm \frac{1}{2} | \vec{\alpha} \rangle $ one may simplify Eq.~(\ref{bound1})
\begin{eqnarray}
N^{\rm bound}_i &\simeq & \pm \frac{1}{8W} [a^{\dagger}_{\alpha'(i)}a_{\alpha(i)} + {\rm H.c.}] \nonumber \\
&& \pm \frac{1}{8W} [a^{\dagger}_{\alpha'(i+1)}a_{\alpha(i+1)} + {\rm H.c.}]. \nonumber \\ \label{bound2}.
\end{eqnarray}
The resulting $N^{\rm bound}_i$ is a sum of two hopping terms and the squared Hilbert-Schmidt norm of each term equals $\frac{1}{2} \frac{1}{(8W)^2} $. Finally, we find the inequalities
\begin{equation}
||N^{\perp}_i ||^2 \ge ||\tilde{N}_i ||^2 \ge ||N^{\rm bound}_i ||^2=1/(8W)^2.
\end{equation}
Fig.~\ref{fig1} in the main text demonstrates that $1/(8W)^2$ very accurately reproduces the minimum of $||N^{\perp}_i ||^2 $ obtained from numerical simulations.
\begin{figure}[!]
\includegraphics[width=1.0\columnwidth]{figS2.pdf}
\caption{Distributions $f$ of the eigenstate-to-eigenstate fluctuations $\langle \delta A \rangle$, which is calculated at single lattice site and different realizations of disorder, see also Fig.~\ref{fig4} of the main text.
(a,c,e) The standard model $H$, and (b,d,f) the rescaled model $\tilde{H}$. }
\label{figs2}
\end{figure}
\section{Nearest level spacings} \label{app3}
In Fig.~\ref{fig2} in the main text we studied the statistical properties of the ratio of nearest level spacings, shortly the gap ratio~\cite{oganesyan07}.
For a target many-body eigenstate $|E_m\rangle$, the gap ratio is defined as
\begin{equation} \label{def_rm}
r_m = \frac{{\rm min}\{\delta E_m, \delta E_{m-1}\}}{{\rm max}\{ \delta E_m, \delta E_{m-1} \}} \;,
\end{equation}
where $\delta E_m = E_{m+1} - E_m$ is the level spacing.
We then averaged $r_m$ over the middle third of the energy spectrum to obtain $r_E$, and we plotted the cumulative distribution function (over different disorder realizations) in the main panels of Figs.~\ref{fig2}(a) and~\ref{fig2}(b).
In the inset of Fig.~\ref{fig2}(b) we the plotted a probability density function $P(r)$ that includes results for $r_m$ from Eq.~(\ref{def_rm}) obtained at different disorder realizations, as well as different target eigenstates $|E_m\rangle$ at a fixed disorder realization.
The latter are again obtained from the middle third of the energy spectra.
Results are compared to the analytical predictions for the Poisson distribution~\cite{oganesyan07},
\begin{eqnarray}
P(r) = \frac{2}{(1+r)^2} \;,
\end{eqnarray}
see the dash-dotted line the inset of Fig.~\ref{fig2}(b), and for the GOE~\cite{atas2013},
\begin{equation}
P(r) = \frac{27}{4} \frac{r + r^2}{(1+r+r^2)^{5/2}} \;,
\end{equation}
see the dashed line in the inset of Fig.~\ref{fig2}(b).
\section{Matrix elements of observables} \label{app4}
In the main text we studied the dependence of fluctuations of the diagonal matrix elements on the disorder strength $W$ and the system size $L$.
We calculated the average eigenstate-to-eigenstate fluctuations $\langle \delta A \rangle$, as defined in Eq.~(\ref{eefluct}) in the main text, for site occupations at single lattice site and different realizations of disorder.
The resulting density plots are shown in Fig.~\ref{fig4} in the main text.
Figure~\ref{figs2} shows the distribution of $\langle \delta A \rangle$ obtained at the disorder strengths $W=2, 20, 100$ and system sizes $L=10,12,14,16$. Results in the left column were obtained for the standard model, $H$, whereas the right column shows results for the rescaled model, $\tilde{H}$. In the former case at large $W$, the distributions become narrower upon increasing $L$ and their center remains pinned at $\langle \delta A \rangle=1$. This can be interpreted as a violation of the ETH. However in the rescaled model the most probable values of $\langle \delta A \rangle$, see the positions of maxima of the probability density functions, decrease with $L$. The latter result suggests that the rescaled model remains ergodic at least at disorders as large as $W=100$.
\end{document}
|
2,877,628,090,801 | arxiv | \section{Introduction}
During the past decades, two-dimensional (2D) gravitational models continue attracting the attention of theorists for a variety of reasons. First of all, the field equations obtained in many 2D gravity models are simple enough to allow a rigorous analysis of some difficult issues of gravitational theory, such as the quantization of gravity~\cite{Henneaux1985,Alwis1992}, gravitational collapse~\cite{VazWitten1994,VazWitten1996}, black hole evaporation~\cite{CallanGiddingsHarveyStrominger1992,BilalCallan1993,RussoSusskindThorlacius1992,RussoSusskindThorlacius1992a,RussoSusskindThorlacius1993}, see~\cite{Brown1988,Thorlacius1995,GrumillerKummerVassilevich2002} for comprehensive reviews on early works. Second, a number of very different approaches of quantum gravity all hint that at very short distances space-time becomes effectively two dimensional~\cite{AmbjornJurkiewiczLoll2005,Horava2009b,MureikaStojkovic2011,AnchordoquiDaiFairbairnLandsbergEtAl2012,Stojkovic2013,Loll2020}. Here, the dimensions that are reduced can be effective, spectral, topological or the usual dimensions~\cite{Carlip2017}. Recently, the studies of the Sachdev-Ye-Kitaev (SYK) model \cite{SachdevYe1993,Kitaev2015} also lead to a resurgence of interest in 2D gravity~\cite{AlmheiriPolchinski2015,MaldacenaStanfordYang2016,MaldacenaStanford2016,Jensen2016}, see~\cite{Rosenhaus2018,Sarosi2018,Trunin2020} for pedagogical introductions.
Since the Einstein tensor vanishes identically in two dimensions, the Einstein-Hilbert action cannot be used to describe 2D gravity. An economical solution to this problem is to introduce a dilaton field. Many different 2D dilaton gravity models have been proposed and studied so far. The simplest action for 2D dilaton gravity is the Jackiw-Teitelboim (JT) action~\cite{Jackiw1985,Teitelboim1983}
\begin{eqnarray}
S_{J T}=\frac{1}{\kappa} \int d^{2} x \sqrt{-g} \varphi(R+\Lambda),
\end{eqnarray}
where the dilaton $\varphi$ plays the role of a Lagrangian multiplier. $\kappa$ and $\Lambda$ are the gravitational coupling and the cosmological constant, respectively. Two other famous actions for 2D dilaton gravity are the Mann-Morsink-Sikkema-Steele (MMSS) action, which generalize the JT action by giving the dilaton a kinetic term~\cite{MannMorsinkSikkemaSteele1991}
\begin{eqnarray}
\label{MMSSgra}
S_{\textrm{MMSS}}=\frac{1}{\kappa}\int{d^2x}\sqrt{-g}\left[ -\frac{1}{2}(\nabla\varphi)^2 +\varphi R +\Lambda\right],
\end{eqnarray}
and the Callan-Giddings-Harvey-Strominger (CGHS) action~\cite{CallanGiddingsHarveyStrominger1992}:
\begin{eqnarray}
S_{\mathrm{CGHS}}=\frac{1}{2\pi} \int d^{2} x \sqrt{-g}\left\{e^{-2 \varphi}\left[R+4 (\nabla\varphi)^2 +4\Lambda^2 \right]-\frac12(\nabla \phi )^2\right\},
\end{eqnarray}
where $\phi$ is a massless scalar matter field.
A comprehensive review of 2D dilaton gravity models and their applications in black hole physics and quantum gravity can be found in Ref. \cite{GrumillerKummerVassilevich2002}.
It is a natural idea to extend the discussion on 2D dilaton gravity to other classical solutions such as topological solitons, which could be produced by cosmic phase transitions~\cite{VilenkinShellard2000}. As the simplest topological soliton solution, kink (or domain wall) has been extensively studied in 4D cosmology~\cite{Vachaspati2006} and 5D thick brane world models~\cite{DzhunushalievFolomeevMinamitsuji2010,Liu2018}. In the case of two dimensions, previous works have revealed close connections between kinks and 2D black holes~\cite{ShinSoh1995,JohngShinSoh1996,GegenbergKunstatter1998,Cadoni1998}, or naked singularities~\cite{VazWitten1994,VazWitten1996,VazWitten1995,YanQiu1998,YanWangTao2001}.
In 1995, an exact 2D self-gravitating sine-Gordon kink solution without curvature singularity was found by St\"otzel, in the MMSS gravity model~\cite{Stoetzel1995}. In addition to the kink configuration of the scalar field, the metric solution~\cite{Stoetzel1995} describes a 2D asymptotic anti de-Sitter (AdS$_2$) geometry. This property reminds us the thick brane solutions found in asymptotic AdS$_5$ geometry~\cite{SkenderisTownsend1999,DeWolfeFreedmanGubserKarch2000,Gremm2000}.
The aim of the present work is to reveal similarities between 2D self-gravitating kinks and 5D thick brane worlds.
The organization of the paper is as follows. In Sec.~\ref{SecModel}, we give a brief review of St\"otzel's model, and show that for static solutions, the field equations can be written as a group of first-order differential equations by introducing the so called superpotential. With the superpotential formalism, one can easily generate exact self-gravitating kink solutions by chosen proper superpotentials. We will discuss two analytical solutions in Sec.~\ref{SecSolution}. Then, in Sec.~\ref{SecSability} we give a complete analysis to the linear stability of the solutions. To our knowledge, no such analysis was done before. In a recent work~\cite{IzquierdoFuertesGuilarte2020}, the authors considered the linear perturbations around self-gravitating kink solutions in 2D MMSS gravity. However, they expand the metric around the Minkowski metric rather than the asymptotic AdS$_2$ metric solution. Finally, we offer in Sec.~\ref{SecConc} some concluding remarks.
\section{The model and the superpotential formalism}
\label{SecModel}
The action of St\"otzel's model~\cite{Stoetzel1995} contains an MMSS gravity part along with a canonical real scalar $\phi$:
\begin{eqnarray}
\label{action}
S=\frac{1}{\kappa}\int{d^2x}\sqrt{-g}\left[ -\frac{1}{2}\partial ^{\mu}\varphi \partial _{\mu}\varphi +\varphi R +\Lambda +\kappa \mathcal{L}_{\text{m}}\right],
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{L}_{\text{m}}=-\frac{1}{2}\partial ^{\mu}\phi \partial _{\mu}\phi -V(\phi)
\end{eqnarray}
is the Lagrangian density of the scalar field.
After variation, one immediately obtains the Einstein equations
\begin{eqnarray}
\label{eqEinstein}
\nabla _{\mu}\varphi \nabla _{\nu}\varphi +2\nabla _{\mu}\nabla _{\nu}\varphi -\frac{1}{2}g_{\mu \nu}\left( \nabla _{\lambda}\varphi \nabla ^{\lambda}\varphi +4\nabla _{\lambda}\nabla ^{\lambda}\varphi -2\Lambda\right) =-\kappa T_{\mu \nu},
\end{eqnarray}
the dilaton equation
\begin{eqnarray}
\label{eqDilatonxcoord}
\nabla _{\lambda}\nabla ^{\lambda}\varphi +R =0,
\end{eqnarray}
and the scalar field equation
\begin{eqnarray}
\label{EqEOM}
\nabla ^{\mu}\nabla _{\mu}\phi -\frac{dV}{d\phi}=0.
\end{eqnarray}
The energy-momentum tensor in Eq.~\eqref{eqEinstein} is defined as
\begin{eqnarray}
T_{\mu \nu}&=&g_{\mu \nu}\mathcal{L}_{\text{m}}-2\frac{\delta \mathcal{L}_{\text{m}}}{\delta g^{\mu \nu}}\nonumber\\
&=&\partial _{\mu}\phi \partial _{\nu}\phi -\frac{1}{2}g_{\mu \nu}\left(\partial ^{\alpha}\phi \partial _{\alpha}\phi +2V \right).
\end{eqnarray}
To obtain self-gravitating kink solution, St\"otzel used the following metric
\begin{eqnarray}
\label{metricX}
ds^2=-e^{2A(x)}dt^2+dx^2.
\end{eqnarray}
Similar metric ansatz is also used in 5D brane world models with non-factorizable geometry~\cite{RandallSundrum1999a,RandallSundrum1999}, therefore, we will follow the terminology of brane world theory and call the function $A(x)$ as the warp factor. As a convention, the derivative with respect to $x$ will always be denoted as a subscript, for example, $\phi_x\equiv d\phi/dx.$
Substituting metric \eqref{metricX} into the Einstein equations \eqref{eqEinstein}, one obtains
\begin{eqnarray}
\label{EinEq1}
2 A_x \varphi_x -2 \varphi_{xx}- \varphi_x^2&=&\kappa \phi _x^2,\\
\label{EinEq2}
A_x \varphi_x + \varphi_{xx} &=& \Lambda- \kappa V.
\end{eqnarray}
The equations of motion for the dilaton and the scalar fields read
\begin{eqnarray}
\label{Req}
-2 A_{xx}-2 A_{x}^2+\varphi_{xx}+A_{x} \varphi_{x}=0.
\end{eqnarray}
and
\begin{eqnarray}
\label{EoM}
A_{x} \phi_{x}+\phi_{xx}=\frac{dV}{d\phi},
\end{eqnarray}
respectively. Note that only three of the above equations are independent. For example, Eq.~\eqref{EoM} can be derived by using Eqs. \eqref{EinEq1}-\eqref{Req}.
At a first glance, Eqs. \eqref{EinEq1}-\eqref{EoM} constitute a complicate nonlinear differential system, and finding their solutions seems to be a formidable task. But the study of brane world models has taught us a lesson on how to solve such system by means of superpotential method, which rewrites second-order differential equations, such as Eqs. \eqref{EinEq1}-\eqref{EoM}, into some first-order ones~\cite{SkenderisTownsend1999,DeWolfeFreedmanGubserKarch2000,Gremm2000}.
To construct a superpotential formalism for the present model, we first note that the combination of Eqs. \eqref{EinEq2} and \eqref{Req} leads to an expression of $V$ in terms of cosmological constant and warp factor:
\begin{eqnarray}
\label{eqV}
\kappa V=\Lambda-2A_{xx}-2A_{x}^2.
\end{eqnarray}
Taking the derivative of the above equation and eliminating $dV/d\phi$ by using Eq. \eqref{EoM}, one obtains a relation between $A$ and $\phi$:
\begin{eqnarray}
\label{EqAphi}
A_{xxx}+2A_{x} A_{xx}=-\frac{1}{2}\kappa( A_{x}\phi_{x}^2+ \phi_{xx} \phi_{x}).
\end{eqnarray}
The superpotential method starts with an assumption that the first-order derivative of $\phi$ equals to a function of $\phi$ itself, namely, the superpotential $W(\phi)$ via the following equation:
\begin{eqnarray}
\label{EqPhiW}
\phi_{x}=\frac{dW}{d\phi}.
\end{eqnarray}
Under this assumption, one can testify that Eq. \eqref{EqAphi} supports a very simple special solution:
\begin{eqnarray}
\label{EqAW}
A_{x}=-\frac14 \kappa W.
\end{eqnarray}
Then, Eq. \eqref{eqV} enables us to write $V$ in terms of superpotential:
\begin{eqnarray}
\label{EqVW}
V=\frac{1}{2}\left(\frac{dW}{d\phi}\right)^2-\frac{1}{8}\kappa W^2+\frac{\Lambda}{\kappa}.
\end{eqnarray}
Finally, the general solution of Eq.~\eqref{Req} gives a simple relation between dilaton and warp factor:
\begin{eqnarray}
\varphi=2A+\beta \int e^{-A} dx+\varphi_0, \nonumber
\end{eqnarray}
where $\beta$ and $\varphi_0$ are just two integral constants. Since the field equations only contain the derivatives of the dilaton, the value of $\varphi_0$ is unimportant to the solution of other variables, and can be taken as $\varphi_0=0$.
Besides, to consist with Eq. \eqref{EinEq1}, $\beta$ must be set as zero, so
\begin{eqnarray}
\label{eqAVarphi}
\varphi=2A.
\end{eqnarray}
Eqs.~\eqref{EqPhiW}-\eqref{eqAVarphi} constitute the first-order superpotential formalism of the present model. Exact kink solutions can be derived by choosing proper superpotentials. The freedom of choosing a superpotential comes from the fact that there are four unknown variables ($A, \phi, \varphi$ and $V$) but only three independent equations. Taking a superpotential amounts to specifying one of the four unknown variables.
\section{Exact solutions}
\label{SecSolution}
In this section, we show how to use the superpotential formalism to derive exact self-gravitating kink solutions. We first reproduce St\"otzel's solution and then report a new solution.
\subsection{Reproducing St\"otzel's solution}
In fact, the superpotential formalism presented in last section has been derived and used, although unconsciously, by St\"otzel~\cite{Stoetzel1995}. Instead of choosing a superpotential $W(\phi)$, St\"otzel started with the Sine-Gordon potential
\begin{eqnarray}
V(\phi )=2m^2\sin ^2\frac{\phi}{2}.
\end{eqnarray}
He observed that when $\kappa= \frac{\lambda }{4 m^2-\lambda }$, Eq. \eqref{EqVW} surports two solutions of the superpotential:
\begin{eqnarray}
\label{SolW}
W_{\pm}=\pm 2 \sqrt{4 m^2-\lambda } \cos \left(\frac{\phi }{2}\right),
\end{eqnarray}
where $0<\lambda\equiv \frac{2\Lambda}{\kappa}<4 m^2$. The solutions of $\phi(x)$ corresponds to $W_{-}$ could be obtained by integrating Eqs. \eqref{EqPhiW}, and the result turns out to be the sine-Gordon kink~\cite{Stoetzel1995}:
\begin{eqnarray}
\label{SolPhi}
\phi_K(x)=4 \arctan\left( e^{ M (x-x_0)}\right).
\end{eqnarray}
Here $x_0$ is an integral constant that represents the position of the kink, and will be set to zero from now on. The constant $M$ is defined as $M\equiv \frac{1}{2} \sqrt{4 m^2-\lambda }$. Obviously, $M\in(0,m)$.
The solution corresponds to $W_{+}$ is an antikink
\begin{eqnarray}
\phi_{\bar{K}}(x)=4 \arctan\left( e^{-M x}\right),
\end{eqnarray}
which is similar as the kink in many aspects. Thus, we will focus on the kink solution only, and eliminate the subscript $K$ from now on.
Plugging the solutions of $W(\phi)$ and $\phi(x)$ into Eq.~\eqref{EqAW}, one immediately obtains the expression of the warp factor:
\begin{eqnarray}
A(x)=A_0-\frac{\lambda}{4 M^2}\ln (2 \cosh (M x)),
\end{eqnarray}
which further reduces to~\cite{Stoetzel1995}
\begin{eqnarray}
A(x)&=&-\frac{\lambda}{4 M^2}\ln \cosh (M x)\nonumber\\
&=&-\kappa\ln \cosh (M x)
\end{eqnarray}
after taking integral constant $A_0=\frac{\lambda}{4 M^2}\ln 2$. Obviously, this warp factor describes an asymptotic AdS$_2$ geometry. Finally, the dilaton field reads
\begin{eqnarray}
\varphi(x)&=&2A(x)=-2\kappa\ln \cosh (M x).
\end{eqnarray}
The profiles of $\phi$, $A$ and $\varphi$ are plotted in Fig.~\ref{figSine}.
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{FigSineGordon.pdf}
\caption{The shapes of some important variables in St\"otzel's solution, incluting (a) scalar field, (b) warp factor and the dilaton field. The parameters are taken as $\kappa=1$, $m=\sqrt{2}$ and $\lambda=4$, therefore $M=1$ and $\Lambda=2$. }
\label{figSine}
\end{figure}
\subsection{A polynomial superpotential}
As shown repeatedly in the study of 5D thick brane models, it is quite easy to construct exact self-gravitating kink solutions once the superpotential formalism is established. In the following discussions, we will take $\Lambda=0$ for simplicity, as it can be absorbed into the definition of $V(\phi)$.
Consider a simple polynomial potential with parameter $c$ \cite{EtoSakai2003,TakamizuMaeda2006,BazeiaMenezesRocha2014}
\begin{eqnarray}
W=c+\phi \left(1-\frac{\phi ^2}{3}\right).
\end{eqnarray}
It has two minima at $\phi_{\pm}=\pm1$, where $W(\phi_{\pm})=\pm\frac23 +c$. With this superpotential, one obtains~\cite{BazeiaMenezesRocha2014}
\begin{eqnarray}
\phi (x)&= & \tanh (x),\\
\varphi(x)&=&2A(x),\\
A(x)&=&\frac{1}{24} \kappa \left[-6 c x+\text{sech}^2(x)-4 \ln (\cosh (x))-1\right],\\
V(\phi)&=&-\frac{1}{72} \kappa \left(-3 c+\phi ^3-3 \phi \right)^2+\frac{1}{2} \left(\phi ^2-1\right)^2.
\end{eqnarray}
The asymptotic behaviors of the warp factor and the scalar potential are
\begin{eqnarray}
A_\pm(x)&=&-\frac14 \kappa W(\phi_\pm) x=-\frac14 \kappa (\frac23\pm c)|x|,\\
V_\pm&=&-\frac{1}{72} (3 c\pm2)^2 \kappa.
\end{eqnarray}
Depending on the value of $c$, there are four different situations~\cite{BazeiaMenezesRocha2014}:
\begin{enumerate}
\item $c=0$: In this case, the kink connects two equivalent AdS$_2$ spaces symmetrically, and $V_+=V_-=-\frac{1}{18} \kappa$.
\item $0<|c|<\frac23$: The kink connects two distinct AdS$_2$ spaces.
\item $|c|=\frac23$: The kink connects an AdS$_2$ space and a 2D Minkowski space (M$_2$) asymmetrically. This situation is of particular interesting when considering kink collision in asymptotical AdS space-time~\cite{TakamizuMaeda2006,OmotaniSaffinLouko2011}.
\item $|c|>\frac23$: The warp factor diverges at one side of the kink.
\end{enumerate}
The behavior of $e^{A}$ for different values of $c$ has been plotted in Fig.~\ref{FigPolyWarp}. Obviously, for $c\neq 0$, the warp factor is asymmetric.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{FigPolyWarp.pdf}
\caption{Plots of warp factor $e^{A(x)}$ of the polynomial model with $\kappa=1$.}
\label{FigPolyWarp}
\end{figure}
\section{Linear stability analysis}
\label{SecSability}
In this section, we discuss the linear stability of the self-gravitating kink solutions against small perturbations. This issue has been studied extensively in 5D brane world models~\cite{DeWolfeFreedmanGubserKarch2000,Giovannini2001a,Giovannini2002,Giovannini2003,ZhongLiu2013}, but remains untouched in the case of 2D. The reducing of dimensions and the introducing of dilaton field make it impossible to analyze linear stability of 2D self-gravitating kinks by simply copying the stability analysis of 5D thick branes. For example, there are no vector and tensor perturbation in 2D, so the traditional scalar-vector-tensor decomposition~\cite{Giovannini2002,ZhongLiu2013} is no longer needed. Beside, in 2D there is no way to eliminate the non-minimal gravity-dilaton coupling by using conformal transformation.
It is convenient to discuss the linear stability in the conformal flat coordinates
\begin{eqnarray}
\label{MetricRCoord}
ds^2=e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu,
\end{eqnarray}
where $r$ is defined through $dr\equiv e^{-A(x)}dx$. For simplicity, we use a prime and an overdot to represent the derivatives with respect to $r$ and $t$, respectively.
In this coordinates, the Einstein equations take the following form:
\begin{eqnarray}
\label{EqEinstrCoord}
\kappa \phi '^2&=&4 A'\varphi '-2 \varphi ''-\varphi '^2,\\
\varphi '' &=& e^{2A} (\Lambda -\kappa V).
\end{eqnarray}
The equation of motion for the scalar and dilaton fields are
\begin{eqnarray}
\phi ''&=&e^{2 A} \frac{dV}{d\phi},
\end{eqnarray}
and
\begin{eqnarray}
\label{eqVphiAr}
\varphi ''&=&2 A'',
\end{eqnarray}
respectively. Obviously, the general solution of Eq.~\eqref{eqVphiAr} is $\varphi=2A+\beta r+\varphi_0$, but as stated before, we will take $\beta=0=\varphi_0$.
Equation \eqref{EqAphi} becomes
\begin{eqnarray}
2 A'''-4 A' A''+\kappa \phi ' \phi ''=0,
\end{eqnarray}
which, after integration, gives
\begin{eqnarray}
\label{EqAandphiPrim}
A''- {A'} ^2+\frac14 \kappa {\phi ' }^2 =0,
\end{eqnarray}
where the integral constant has been taken as zero.
Now, let us consider small field perturbations around an arbitrary static background solution $\{\varphi(r), \phi(r), g_{\mu\nu}(r)\}$:
\begin{eqnarray}
\varphi(r)+\delta\varphi(r,t),\quad \phi(r)+\delta\phi(r,t),\quad g_{\mu\nu}(r)+\delta g_{\mu\nu}(r,t).
\end{eqnarray}
We also define
\begin{eqnarray}
\delta g_{\mu\nu}(r,t)\equiv e^{2A(r)} h_{\mu\nu}(r,t),
\end{eqnarray}
for convenience.
In the linear perturbation analysis of cosmological or brane world models, one usually decompose $h_{\mu\nu}$ into scalar, vector and tensor sectors~\cite{MukhanovFeldmanBrandenberger1992,KodamaSasaki1984}. Each sector can be discussed independently. In the present case, we have only one spatial dimension and no such decomposition is needed. So we will directly deal with the components of the metric perturbation
\begin{eqnarray}
h_{\mu\nu}=\left(
\begin{array}{cc}
h_{00}(r,t) & \Phi (r,t) \\
\Phi (r,t) & h_{rr}(r,t) \\
\end{array}
\right),
\end{eqnarray}
where we have renamed $h_{01}=h_{10}$ as $\Phi$, and $h_{11}$ as $h_{rr}$.
To the first order, the perturbation of the metric inverse is given by
\begin{eqnarray}
\delta g^{\mu \nu}=-e^{-2A} h^{\mu \nu}.
\end{eqnarray}
Note that the indices of $h$ are always raised or lowered with $\eta_{\mu\nu}$, thus,
\begin{eqnarray}
h^{\mu \nu}\equiv \eta^{\mu\rho}\eta^{\nu\sigma}h_{\rho\sigma}=\left(
\begin{array}{cc}
h_{00} & -\Phi \\
-\Phi & h_{rr} \\
\end{array}
\right).
\end{eqnarray}
After linearization, the Einstein equations \eqref{eqEinstein} lead to three nontrivial perturbation equations, namely,
the $(0,0)$ component:
\begin{eqnarray}
\label{PertEq0}
&&2A'\delta \varphi '-2A'\varphi ' h_{rr}-2\delta \varphi ''-\delta \varphi '\varphi '+h_{rr}'\varphi '\nonumber\\
&+&2h_{rr}\varphi ''
+\frac{1}{2}h_{rr}\varphi '^2
=\kappa\left( \phi '\delta \phi '+ \phi ''\delta \phi -\frac{1}{2} \phi '^2 h_{rr}\right) ,
\end{eqnarray}
the $(0,1)$ or $(1,0)$ components:
\begin{eqnarray}
\label{PertEq1}
&&2A'\delta \varphi -2 \delta \varphi '- \varphi '\delta \varphi + \varphi '{h_{rr}}=\kappa \phi '\delta \phi,
\end{eqnarray}
and the $(1,1)$ component:
\begin{eqnarray}
\label{PertEq2}
2A'\delta \varphi '-2A'\varphi 'h_{rr}-\delta \varphi '\varphi '-2\ddot{\delta}\varphi +\frac{1}{2}h_{rr}\varphi '^2+\Xi \varphi '=\kappa \left( \phi '\delta \phi '-\phi ''\delta \phi -\frac{1}{2}\phi '^2h_{rr} \right).
\end{eqnarray}
Here we have defined a new variable $\Xi\equiv 2\dot{\Phi}-{h}_{00}'$.
One can testify that after using background equations \eqref{EqEinstrCoord}-\eqref{eqVphiAr}, Eq. \eqref{PertEq0} reduces to Eq. \eqref{PertEq1}.
Another independent equation comes from the perturbation of the scalar equation of motion:
\begin{eqnarray}
\label{PertEq3}
-\ddot{\delta}\phi +\delta \phi ''+2A'\frac{\phi ''}{\phi '}\delta \phi -\frac{\phi '''}{\phi '}\delta \phi -\frac{1}{2}\phi ' h_{rr}'-\phi ''h_{rr}+\frac{1}{2} \phi '\Xi=0.
\end{eqnarray}
One can also linearize the dilaton equation \eqref{eqDilatonxcoord}, but it does not offer new information further.
Therefore, we have three independent perturbation equations, i.e., \eqref{PertEq1}-\eqref{PertEq3}. But one should note that the perturbation variables are not all independent. The invariance of the dynamical equations under coordinate transformations
\begin{eqnarray}
x^\mu\to \tilde{x}^\mu=x^\mu+\xi^\mu(r, t)
\end{eqnarray}
induces an invariance of the linear perturbation equations \eqref{PertEq1}-\eqref{PertEq3} under the following gauge transformations:
\begin{eqnarray}
\label{EqhmnTrans}
\Delta h_{\mu\nu} &\equiv& \widetilde{h}_{\mu\nu}-h_{\mu\nu} =-2 \xi_{(\mu, \nu)}
-2 \eta_{\mu,\nu} A' \xi^{1} ,\\
\Delta \delta \phi &\equiv& \widetilde{\delta \phi }-\delta \phi = - \phi^{\prime}\xi^{1},\\
\Delta \delta \varphi &\equiv& \widetilde{\delta \varphi}-\delta \varphi = - \varphi^{\prime} \xi^{1}.
\end{eqnarray}
The components of $h_{\mu\nu}$ transform as
\begin{eqnarray}
\Delta h_{00}&=&2\partial _t\xi ^0+2A' \xi ^1,
\\
\Delta \Phi&=&-\partial _t\xi ^1+\partial _r\xi ^0,
\\
\Delta h_{rr}&=&-2\partial _r \xi ^1-2 A' \xi ^1,
\end{eqnarray}
which means that the variable $\Xi= 2\dot{\Phi}-{h}_{00}'$ should transforms as
\begin{eqnarray}
\Delta \Xi=-2\left[\ddot{\xi}^1+\left(A'\xi^1\right)'\right].
\end{eqnarray}
We see that the gauge degree of freedom $\xi^0$ has been canceled.
The residual gauge degree of freedom in $\xi^1$ allows us to eliminate one of the perturbation variables. Here we simply take $\delta\varphi=0$, with which Eq. \eqref{PertEq1} reduces to
\begin{eqnarray}
\varphi '{h_{rr}}=\kappa \phi '\delta \phi,
\end{eqnarray}
and Eq. \eqref{PertEq2} becomes
\begin{eqnarray}
-2A'\varphi 'h_{rr}
+\frac{1}{2}h_{rr}\varphi '^2+\Xi \varphi '=\kappa \left( \phi '\delta \phi '-\phi ''\delta \phi -\frac{1}{2}\phi '^2h_{rr} \right).
\end{eqnarray}
After eliminating $h_{rr}$ and $\Xi$, equation \eqref{PertEq3} can be written as a wave equation of $\delta\phi$:
\begin{eqnarray}
\label{eqDelPhi}
\ddot{\delta \phi }-\delta \phi ''+ V_{\text{eff}}(r)\delta \phi=0,
\end{eqnarray}
where the effective potential reads
\begin{eqnarray}
V_{\text{eff}}(r)=4A''-2A'\frac{\phi ''}{\phi '}-\varphi ''+2\left( \frac{\varphi ''}{\varphi '} \right) ^2-2\frac{\varphi '''}{\varphi '}+\frac{\phi '''}{\phi '}.
\end{eqnarray}
Using Eqs. \eqref{eqVphiAr}-\eqref{EqAandphiPrim}, one can obtain an useful identity:
\begin{eqnarray}
\varphi''=\frac{\varphi'''}{\varphi'}+\frac{\phi''}{\phi'}\varphi'-2\frac{\phi''}{\phi'}\frac{\varphi''}{\varphi'},
\end{eqnarray}
which enable us to rewrite the effective potential as
\begin{eqnarray}
V_{\text{eff}}=\frac{\phi '''}{\phi '}
-2\frac{\phi ''}{\phi '}\frac{\varphi ''}{\varphi '}
+2\left( \frac{\varphi ''}{\varphi '} \right) ^2
-\frac{\varphi '''}{\varphi '},
\end{eqnarray}
or, in a more compact form
\begin{eqnarray}
V_{\text{eff}}=\frac{f ''}{f}, \quad \textrm{with} \quad f\equiv \frac{\phi '}{\varphi '}.
\end{eqnarray}
If we take $\delta\phi=\psi(r)e^{iwt}$, Eq.~\eqref{eqDelPhi} becomes a Schr\"odinger-like equation of $\psi(r)$:
\begin{eqnarray}
-\psi ''+ V_{\text{eff}} \psi=w^2 \psi.
\end{eqnarray}
It is interesting to note that the Hamiltonian operator are factorizable:
\begin{eqnarray}
\hat{H}=-\frac{d^2}{dr^2}+V_{\text{eff}}=\hat{\mathcal{A}}\hat{\mathcal{A}}^\dagger,
\end{eqnarray}
with
\begin{eqnarray}
\mathcal{A}=\frac{d}{d r}+\frac{{f}'}{f},
\quad \mathcal{A}^{\dagger}=-\frac{d}{d r}+\frac{{f}'}{f}.
\end{eqnarray}
According to the theory of supersymmetric quantum mechanics~\cite{CooperKhareSukhatme1995}, the eigenvalues of a factorizable Hamiltonian operator are semipositive definite, namely, $w^2\geq 0$. Therefore, static kink solutions are stable against linear perturbations. The zero mode ($w_0=0$) satisfies $ \mathcal{A}^{\dagger}\psi_0(r)=0$, and the solution reads
\begin{eqnarray}
\psi_0(r)\propto f=\frac{\phi '}{\varphi '}=\frac{\phi '}{2A '}.
\end{eqnarray}
Obviously, for any solution with a non-monotonic warp factor, $\psi_0(r)$ diverges at the extrema of $A$, and would be unnormalizable. Since it is not always possible to obtain the explicit expression of $x(r)$, it is useful to transform $V_{\text{eff}}$ back to the $x$-coordinates:
\begin{eqnarray}
V_{\text{eff}}(x)&=&e^{2A}\left(A_x\frac{f_x}{f}+\frac{f_{xx}}{f}\right),
\end{eqnarray}
with $f(x)=\phi_x/\varphi_x$.
It should be note that the stability analysis presented so far are rather general and does not depend on the specific form of the solution, but only on the general form of the metric \eqref{MetricRCoord} and of the action \eqref{action}.
Now, we move on to the specific solutions. For St\"otzel's sine-Gordon model and the polynomial model, the effective potentials read
\begin{eqnarray}
V_{\text{eff}}(x)&=&M^2 \cosh ^{-2 \kappa }(M x) \left[\kappa +2 \text{csch}^2(M x)+1\right],
\end{eqnarray}
and
\begin{eqnarray}
V_{\text{eff}}(x)&=&
\frac{\exp\left[\frac{1}{12} \left(-6 c x+\text{sech}^2(x)-1\right)\right]}{{12 \sqrt[3]{\cosh (x)} \left[3 c+\tanh (x) \left(\text{sech}^2(x)+2\right)\right]^2}} \left\{-\text{sech}^2(x) \left[296\right.\right. \nonumber\\
&+&\left.\left. 702 c^2+\left(27 c^2-424\right) \text{sech}^2(x)+118 \text{sech}^4(x)
+\text{sech}^6(x)
+\text{sech}^8(x)
\right]\right. \nonumber\\
&+&\left.18 c \tanh (x) \left[3 c^2
+23 \text{sech}^4(x)-32 \text{sech}^2(x)+36\right]+540 c^2+208\right\},
\end{eqnarray}
respectively. For the later case, we have taken $\kappa=1$, for simplicity.
The profiles of the $V_{\text{eff}}(x)$ are depicted in Fig.~\ref{figVeff}. For St\"otzel's model, we take $m=\sqrt{2}$, $\Lambda=2\kappa$ such that $M\equiv \frac{1}{2} \sqrt{4 m^2-\frac{2\Lambda}{\kappa} }=1$, while keep $\kappa$ as a free parameter. We see that $V_{\text{eff}}$ is positive and divergent at $x=0$ for $\kappa=0.2$, 1 and 3.
For the polynomial model, we take $c=0$, 1/3, 2/3 and 1 as examples. We see that $V_{\text{eff}}(x)$ diverges at $x=0$ for both $c=0$ and 1/3, while blows up at $x\to -\infty$ if $c=1$, but becomes finite when $c=2/3$.
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{FigVeff.pdf}
\caption{Plots of $V_{\text{eff}}(x)$. For polynomial model with $c=2/3$, $V_{\text{eff}}(x)$ becomes finite, and approaches to $4 \sqrt[3]{2} e^{-\frac{1}{12}}\approx 4.637$ as $x\to-\infty$.}
\label{figVeff}
\end{figure}
It is worth to mention that in many 5D thick brane models the effective potentials of the scalar perturbation also have singularities, and the corresponding scalar zero modes are usually unnormalizable. Without normalizable scalar zero modes, these models are free of the problem of long range scalar fifth force~\cite{Giovannini2002,Giovannini2001a,ZhongLiu2013}. For the 2D self-gravitating kink solutions considered in this paper, however, we find an unusual situation where the zero mode might be normalizable, namely, the polynomial model with $c>2/3$. In this case, the zero mode reads
\begin{eqnarray}
\psi_0(x)=\mathcal{N}\frac{\phi_x}{2A_x}=-\mathcal{N}\frac{6 \text{sech}^2(x)}{3 c + \tanh (x) \left(\text{sech}^2(x)+2\right)},
\end{eqnarray}
where $\mathcal{N}$ is the normalization constant, and we have taken $\kappa=1$. The normalization of zero mode requires
\begin{eqnarray}
1&=&\int_{-\infty}^{+\infty} dr \psi_0^2(r)
=\mathcal{N}^2 \int_{-\infty}^{+\infty} dx e^{-A} \left(\frac{\phi_x}{2A_x}\right)^2.
\end{eqnarray}
The integration can be done numerically, for instance, taking $c=1$, 1.2 and 1.5 we obtain $|\mathcal{N}|\approx$ 0.334, 0.446 and 0.598, respectively. Plots of $\psi_0(x)$ is depicted in Fig. \ref{FigZeroMode.pdf}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{FigZeroMode.pdf}
\caption{Plots of $\psi_0(x)$ for the polynomial model with $\kappa=1$, $c=1$, 1.2 and 1.5.}
\label{FigZeroMode.pdf}
\end{figure}
\section{Summary and outlook}
\label{SecConc}
In this work, we revisited smooth self-gravitating kink solutions of a type of 2D dilaton gravity proposed by Mann et al.~\cite{MannMorsinkSikkemaSteele1991}. We first showed that exact kink solutions can be constructed with the aid of a first-order superpotential formalism \eqref{EqPhiW}-\eqref{eqAVarphi} of the dynamical equations. This formalism has already been derived and used by St\"otzel in 1995, for 2D self-gravitating sine-Gordon model~\cite{Stoetzel1995}, but its virtue was not completely appreciated until the advent of 5D thick brane world models. After reproducing St\"otzel's solution~\cite{Stoetzel1995}, we reported another kink solution generated by a polynomial superpotential used in some 5D brane world models~\cite{EtoSakai2003,TakamizuMaeda2006,BazeiaMenezesRocha2014}.
The main contribution of the present work, however, is a general analysis on the stability of static kink solutions under small linear perturbations. After eliminating the redundant gauge degrees of freedom, we derived a Schr\"odinger-like equation for the physical perturbation. We found that the Hamiltonian operator can be factorized as $\hat{H}=\hat{\mathcal{A}}\hat{\mathcal{A}}^\dagger$, which implies the stability of the solutions. Besides, the zero mode takes the form $\psi_0(r)\propto f\equiv\frac{\phi '}{\varphi '}=\frac{\phi '}{2A'}$, which diverges at the extrema of $A$. For St\"otzel's model, the zero mode is not normalizable, because the symmetric solution of the warp factor corresponds to a singularity of $\psi_0(r)$ at $r=0$. For the polynomial model, however, the zero mode is normalizable provides $c>2/3$.
It is natural to ask if the superpotential formalism and stability analysis of the present work can also be extended to other 2D dilaton gravity models, such as the CGHS model~\cite{CallanGiddingsHarveyStrominger1992} or other more general models~\cite{IkedaIzawa1993,TakahashiKobayashi2019}. The superpotential formalism also makes it possible discuss the application of the present model to the study of holographic RG flow~\cite{BianchiFreedmanSkenderis2001,KiritsisNittiSilvaPimenta2017}. We will leave these questions to our future works.
\section*{Acknowledgements}
This work was supported by the National Natural Science Foundation of China (Grant Nos.~11847211, 11605127), Fundamental Research Funds for the Central Universities (Grant No.~xzy012019052), and China Postdoctoral Science Foundation (Grant No.~2016M592770).
\providecommand{\href}[2]{#2}\begingroup\raggedright |
2,877,628,090,802 | arxiv | \section{Introduction}
Given a finite set $\Omega$, a coloring of $\Omega$ is a map $\chi: \Omega\to \{1, -1\}$, and $\chi(A) = \sum_{x\in A}\chi(x).$ For a family $\mathcal{A}$ of subsets of $\Omega$, the discrepancy of $\mathcal{A}$ is defined to be
\[\disc(\mathcal{A}): = \min_{\chi} \max_{A \in \mathcal{A}} |\chi(A)|,\]
where the minimum is over all colorings of $\Omega$. Let $\mathcal{A}_1$ be the family of arithmetic progressions contained in $[N]:=\{1,\ldots,N\}$.
Roth \cite{Roth} showed using Fourier analysis that there is an absolute constant $c > 0$ such that
\[\disc(\mathcal{A}_1) \geq cN^{\frac 14}. \]
In the other direction, Beck \cite{Beck} proved that
\[\disc(\mathcal{A}_1) \leq CN^{\frac14}(\log N)^{\frac 52}\]
for some absolute constant $C$, thereby showing that Roth's lower bound is sharp up to a polylogarithmic factor. Finally, Matou\v{s}ek and Spencer \cite{MS} removed the polylogarithmic factor and resolved this problem of determining the discrepancy up to a constant factor.
It is natural to study the generalization of this problem to higher dimensions. An {\it arithmetic progression in $d$ dimensions} is a set of the form
\[\AP_d(\mathbf{a}, \mathbf{b}, l) := \{\mathbf{a} + i \mathbf{b}: i= 0,1,\dots, l-1\}\]
where $\mathbf{a},\mathbf{b} \in \mathbb Z^d$ with $\mathbf{b} \not = \mathbf{0}$, and $l\in \mathbb N$. Here $\mathbf{b}$ is the common difference of the arithmetic progression. Let $\mathcal{A}_d$ be the set of arithmetic progressions in $d$ dimensions that are subsets of $[N]^d$.
The quantity we are interested in is
\[\disc(\mathcal{A}_d): = \min_{\chi: [N]^d \to \{1,-1\} } \max_{A \in \mathcal{A}_d} |\chi(A)|, \]
where $\chi(A) = \sum_{x\in A}\chi(x)$. Valk\'o \cite{Valko} proved that there exist constants $c = c(d), C=C(d)$ such that
\[ cN^{\frac{d}{2d+2}} \leq \text{disc}(\mathcal{A}_d) \leq CN^{\frac{d}{2d+2}}(\log N)^{\frac{5}{2}}. \]
Valk\'o's proof of the lower bound extends Roth's proof, while the upper bound adapts Beck's proof. The problem of estimating the discrepancy of higher dimensional
arithmetic progressions is further discussed in \cite{DSW}.
In this paper, we remove the polylogarithmic factor in the upper bound and thus determine the quantity up to a constant factor dependent on $d$.
\begin{theorem}\label{thm:main}For all positive integers $N$ and $d$,
we have
\[\disc(\mathcal{A}_d) = \Theta_d(N^{\frac{d}{2d+2}}). \]
\end{theorem}
The general proof strategy is similar to that in the paper by Matou\v{s}ek and Spencer \cite{MS}. However, new ideas are needed to make the strategy work.
In particular, we need to overcome some difficulties arising from geometric aspects which requires delicate analysis and tools like Minkowski's theorem and the Lenstra-Lenstra-Lov\'{a}sz basis reduction algorithm.
It is natural to study the generalization of the problem to the discrepancy of arithmetic progressions in grids of side lengths that are not necessarily equal. Given positive integers $N_1, \dots, N_d$, let $\Omega = [N_1]\times \cdots \times [N_d] \subseteq \mathbb Z^d$ and $\mathcal{A}_\mathbf{N}$ be the set of arithmetic progressions in $d$ dimensions that are subsets of $\Omega$, where $\mathbf{N} = (N_1, \dots, N_d)$. The discrepancy is defined in a similar way,
\[\disc(\mathcal{A}_\mathbf{N}):= \min_{\chi: \Omega\to \{ 1, -1\}}\max_{A\in \mathcal{A}_\mathbf{N}} |\chi(A)|.\]
In the proof of Theorem \ref{thm:main}, we shall see that we will have to consider more generally grids with side lengths of comparable size (see \eqref{eqn:almost-cubes}).
\begin{theorem}\label{thm:almost-cubes}
For any positive integer $d$ and positive integers $N_1, \dots, N_d$, if $\delta > 0$ satisfies that
\begin{equation}\label{eqn:almost-cubes}
N_1\cdots N_d \leq \left(\min_{1\leq i\leq d}N_i\right)^{d+1-\delta},
\end{equation}
then there exist positive constants $c_d, C_d$ such that for $\mathbf{N} = (N_1, \dots, N_d)$,
\[c_d\left(N_1\cdots N_d\right)^\frac{1}{2d+2} \le \disc(\mathcal{A}_{\mathbf{N}})
\leq C_d\cdot \frac{1}{\delta}\left(N_1\cdots N_d\right)^\frac{1}{2d+2}.
\]
\end{theorem}
We remark that Theorem~\ref{thm:almost-cubes} implies Theorem~\ref{thm:main} by choosing $\delta =1$ and $N_1 = N_2 = \cdots = N_d = N$.
The lower bound in Theorem~\ref{thm:almost-cubes} holds even without condition \eqref{eqn:almost-cubes}. Theorem~\ref{thm:rectangular} gives a more general lower bound, and a matching upper bound up to a sub-logarithmic factor for general grids of differing side lengths. The lower bound in Theorem~\ref{thm:rectangular} implies the lower bound in Theorem~\ref{thm:almost-cubes} by taking $I = [d]$ in the maximum.
The proof of lower bound uses Fourier analytic tools.
\begin{theorem}\label{thm:rectangular}
For any positive integer $d$ and $\mathbf{N} = (N_1, \dots, N_d)$ where the $N_i$'s are positive integers whose product is at least three, there exist positive constants $c_d, C_d$ such that
\begin{equation}\label{eqn: general rectangular}
c_d\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{2|I|+2} \le \disc(\mathcal{A}_{\mathbf{N}})
\leq C_d \frac{\log (N_1\cdots N_d)}{\log \log (N_1\cdots N_d)} \cdot \max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^{\frac{1}{2|I|+2}}.
\end{equation}
Here by convention if $I = \emptyset$ then $\prod_{i\in I}N_i = 1$.
\end{theorem}
\begin{remark}
Since $\disc(\mathcal A_\mathbf{N})$ does not depend on the order of the $N_i$'s, we may assume without loss of generality that $N_1\geq N_2\geq \cdots \geq N_d$. In this case \[\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^{\frac{1}{2|I|+2}} = \max_{1\leq k\leq d}\left(\prod_{i=1}^k N_i\right)^\frac{1}{2k+2}.\]
\end{remark}
\noindent {\bf Organization.}
In Section~\ref{sec:decomposition}, we show how to efficiently decompose arithmetic progressions into ``canonical'' arithmetic progressions and provide an upper bound on the number of such canonical arithmetic progressions with a given size. This implies that a coloring which has low discrepancy in canonical arithmetic progressions of each possible size also achieves low discrepancy for all arithmetic progressions (see Lemma \ref{lem:decomposition} for details). We prove the upper bound in Theorem~\ref{thm:almost-cubes} in Section~\ref{sec: first coloring} by showing the existence of a coloring which has low discrepancy in canonical arithmetic progressions of each possible size. In the proof we use an improved bound on the number of canonical arithmetic progressions with each given size, the proof of which is deferred to Section~\ref{sec:geometry}. We finally study the case of grids with different side lengths in the last three sections. We prove the lower bound of Theorem~\ref{thm:rectangular} in Section~\ref{sec: rec-lower} and the upper bound of Theorem~\ref{thm:rectangular} in Section~\ref{sec: rec-upper}. Finally, we have some concluding remarks in Section~\ref{sec: conclusion and open problem}, including a conjecture that the lower bound for the
the discrepancy for grids in Theorem \ref{thm:rectangular} is tight up to the constant factor.
\begin{notations}
Throughout the paper, all logarithms are base $e$ unless specified.
We generally assume that $d$ is fixed, except in Section~\ref{sec:geometry} where the proof relies on an induction argument on $d$.
We use symbols $c, c_1, c_2, C_0, C, C^{*}$ to denote positive absolute constants, and $c_d, C_d$ to denote those that only depend on $d$.
We use notation $f = O_d(g)$ if there exists a positive constant $C_d$ so that $f \leq C_dg$.
\end{notations}
\section{Decomposition}\label{sec:decomposition}
Let $N_1, \dots, N_d$ be positive integers, $\Omega = [N_1]\times \cdots \times [N_d]$, and $\mathbf{N} = (N_1, \dots, N_d)$.
To find a coloring giving low discrepancy,
the general idea is to apply a partial coloring lemma (specifically
Lemma \ref{lem:partial-coloring-lemma}, whose proof uses the entropy method)
to repeatedly partially color $\Omega$ until we get a full coloring of $\Omega$ with low discrepancy. At each stage, we color a constant fraction of the remaining uncolored elements,
until we get a full coloring of $\Omega$ with low discrepancy. To accomplish this, we show that for any $X\subseteq \Omega$, there is a partial coloring of $X$ with low discrepancy. Once we have this statement, we may apply this with $X$ as $\Omega$ in the initial iteration to get a partial coloring of $\Omega$, and then pick $X$ as the set of uncolored elements of $\Omega$ in later iterations. Hence more generally the set family we need to consider is $(X, \mathcal A_X)$ where $\mathcal A_X := \{A\cap X: A\in \mathcal A_\mathbf{N}\}.$
The family of sets $\mathcal A_X$ is too large if we want to apply
Lemma \ref{lem:partial-coloring-lemma}
on $(X, \mathcal A_X)$ directly. Instead we apply it to a small subfamily $\mathcal C_X \subseteq \mathcal A_X$ so that any set in $\mathcal A_X$ can be efficiently decomposed into sets in $\mathcal C_X$.
For each $\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}$, we may partition elements in $X$ into congruence classes modulo $\mathbf{b}$. For each congruence class $I = \{\mathbf{x}\in X: \mathbf{x} \equiv \mathbf{a} \pmod \mathbf{b}\}$, since distinct elements in $I$ differ by nonzero multiples of $\mathbf{b}$, and their dot products with $\mathbf{b}$ differ by nonzero multiples of $\|\mathbf{b}\|^2\ne 0$, we may order elements in $I$ by their dot products with $\mathbf{b}$. Write $I = \{\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_l\}$, where the subscripts respect the ordering and $l = |I|$. Now any set in $\mathcal A_X$ can be written as $\{\mathbf{x}_u: i\leq u \leq j\}$ for some $(\mathbf{b}, I, i, j)$.
We use the following decomposition. For each $\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}$ and congruence class $I$ modulo $\mathbf{b}$, we consider sets of the form $\{\mathbf{x}_u: (j-1)s + 1 \leq u \leq js\}$ where $s = 2^t$ is a power of $2$, and $1\leq j\leq \lfloor l/s\rfloor$. Let $\mathcal C_X$ be the collection of such sets for all $(\mathbf{b}, I)$. All sets in $\mathcal C_X$ are of sizes powers of $2$.
\begin{lemma}\label{lem:decomposition}
Let $b: \mathbb N \to (0, \infty)$ be a function. If $\chi$ is a partial coloring of $X$ so that
\[|\chi(S)| \leq b(|S|)\]
for all $S\in \mathcal C_X$, then
\[|\chi(A)| \leq 2\sum_{s: s = 2^t}b(s)\]
for all $A\in \mathcal A_X$.
\end{lemma}
\begin{proof}
For any $A\in \mathcal A_X$, we know that it can be written as $A_0 \cap X$ for some arithmetic progression $A_0\in \mathcal A_\mathbf{N}$. Let $\mathbf{b}$ be the common difference of $A_0$, and let $I$ be intersection of $X$ and the congruence class mod $\mathbf{b}$ containing $A_0$. Then $A$ is a subset of $I$. Moreover, as we describe in the procedure above, if we order elements in $I = \{\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_l\}$ by their dot product with $\mathbf{b}$, we know that $A$ must be in the form $\{\mathbf{x}_u: i\leq u \leq j\}$. We write $A = A_1\setminus A_2$ where $A_1 = \{\mathbf{x}_u: 1\leq u \leq j\}$ and $A_2 = \{\mathbf{x}_u: 1\leq u \leq i-1\}$. Then we know that $A_1$ can be written as a disjoint union of sets in $\mathcal C_X$ of different sizes $A_1 = S_1 \cup S_2 \cup \dots \cup S_t$ using the binary representation of $j$, where $t$ is the number of digits $1$ in the representation. Also note that all sets in $\mathcal C_X$ are of sizes powers of $2$, so we have
\[|\chi(A_1)| = \left|\sum_{k=1}^t \chi(S_k)\right| \leq \sum_{k=1}^t |\chi(S_k)| \leq \sum_{s: s = 2^t}b(s).\]
We may prove the similar inequality for $\chi(A_2)$ by replacing $j$ with $i-1$. Combining them we get
\[|\chi(A)| = |\chi(A_1) - \chi(A_2)| \leq 2\sum_{s: s = 2^t}b(s).\]
\end{proof}
To apply the partial coloring lemma, Lemma \ref{lem:partial-coloring-lemma}, to $(X, \mathcal C_X)$, we need to estimate the number of sets of each size, and pick each $\Delta_S$ appropriately. Let $s = 2^t$ be any power of $2$, we define $f(s, X)$ to be the number of sets of size $s$ in $\mathcal C_X$. Note that $f(1, X) = |X|$.
For a positive integer $s$, a finite set $X \subseteq \mathbb Z^d$ and a vector $\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}$, let $U^d(X, \mathbf{b}, s)$ denote the set of all $x\in X$, whose residue class mod $\mathbf{b}$ contains at least $s$ elements in $X$, or formally $\{x'\in X: x'\equiv x \pmod \mathbf{b}\}$ is of size at least $s$. The following inequality shows how these sets $U^d$ relate to the quantity $f(s, X)$.
\begin{lemma}\label{lem:counting-subsets-by-U} Let $X\subseteq \Omega$, and $s$ be a power of $2$. Then
\begin{equation}\label{eqn:counting-subsets-by-U}f(s, X) \leq \frac{1}{s}\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}}|U^d(X, \mathbf{b}, s)|.\end{equation}
\end{lemma}
\begin{proof}We would like to estimate the number of sets in $\mathcal C_X$ of size $s$. For each $\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}$, we partition $X$ into the congruence classes modulo $\mathbf{b}$ which we denote by $I_1, \dots, I_t$. By definition, $U^d(X, \mathbf{b}, s)$ is the disjoint union of all $I_k$ that contains at least $s$ elements.
Each set of size $s$ in $\mathcal C_X$ lies entirely in some $I_k$ for some appropriate choice of $\mathbf{b}$ and $I_k$. The number of such sets in $I_k$
is at most $\lfloor |I_k|/s\rfloor$.
Therefore if we sum over all congruence classes, the number of sets in $\mathcal C_X$ of size $s$ for any fixed $\mathbf{b}$ is
\[\sum_{k=1}^t \lfloor |I_k|/s\rfloor \leq \sum_{k: |I_k| \geq s} \frac{|I_k|}{s} = \frac{|U^d(X, \mathbf{b}, s)|}{s}.\]
Summing over all possible common differences $\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}$, we obtain \eqref{eqn:counting-subsets-by-U}. \end{proof}
Hence we need to estimate the sum of the numbers of elements in these $U^d$ sets. We have the following simple upper bound.
\begin{lemma}\label{lem:counting-U-1}
For any $s\geq 2$ and $X\subseteq [N_1]\times \cdots \times [N_d]$, we have
\[\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}}|U^d(X, \mathbf{b}, s)|\leq |X| \cdot \prod_{i=1}^d \left(4\frac{N_i}{s}+1\right).\]
\end{lemma}
\begin{proof}
We focus on those $\mathbf{b}$ for which $U^d(X, \mathbf{b}, s)$ is nonempty. If $U^d(X, \mathbf{b}, s)$ is nonempty, then we know that for each $i$, the $i$-th coordinate of $\mathbf{b}$ is in the interval $(-\frac{N_i}{s-1}, \frac{N_i}{s-1})$. Therefore, the number of nonempty $U^d(X, \mathbf{b}, s)$ is at most
\[\prod_{i=1}^d \left(\frac{2N_i}{s-1}+1\right) \leq \prod_{i=1}^d \left(4\frac{N_i}{s}+1\right).\]
Applying the trivial bound $|U^d(X, \mathbf{b}, s)| \leq |X| = m$, we get the desired inequality.
\end{proof}
By combining Lemma~\ref{lem:counting-subsets-by-U} and Lemma~\ref{lem:counting-U-1} above, the following upper bound on $f(s, X)$ holds.
\begin{corollary}\label{cor:counting-subsets-1}
For any $1\leq s \leq \min_{1\leq i\leq d}N_i$, we have
\[f(s, X) \leq 5^d\frac{N_1\cdots N_d}{s^{d+1}}|X|.\]
\end{corollary}
\begin{proof}
When $s = 1$, we have $f(1, X) = |X|$ and the inequality clearly holds. In the remaining cases, $2\leq s \leq \min_{1\leq i\leq d}N_i$, we would like to apply Lemma~\ref{lem:counting-subsets-by-U} and Lemma~\ref{lem:counting-U-1}. As $s\leq N_i$ for each $1\leq i\leq d$, it follows that
\[f(s, X) \leq \frac{1}{s} \cdot |X|\prod_{i=1}^d \left(4\frac{N_i}{s}+1\right) \leq 5^d\frac{N_1\cdots N_d}{s^{d+1}}|X|.\]\end{proof}
We remark that in Section~\ref{sec:geometry} we prove Lemma \ref{lem:counting-subsets-3} which, together with Lemma \ref{lem:counting-subsets-by-U}, gives a better upper bound on $f(s,X)$ than Corollary \ref{cor:counting-subsets-1} for a certain range of $s$.
\section{Proof of the upper bound in Theorem \ref{thm:almost-cubes}}\label{sec: first coloring}
We use the following version of the partial coloring lemma which was first proved by Matou\v{s}ek and Spencer in \cite{MS}, to show the existence of a partial coloring that colors a constant fraction of elements of a set system with low discrepancy.
\begin{lemma}[Section 4.6 in \cite{Mat}]\label{lem:partial-coloring-lemma}
Let $(\Omega, \mathcal C)$ be a set system on $n$ elements, and let a number $\Delta_S > 0$ be given for each set $S\in \mathcal C$. Suppose that
\begin{equation}\label{eqn:partial-coloring-lemma}
\sum_{S\in \mathcal C:S\ne \emptyset}g\left(\frac{\Delta_S}{\sqrt{|S|}}\right) \leq \frac{n}{5}
\end{equation}
where
\begin{equation}\label{eqn:def-g}
g(\lambda) = \begin{cases} 10e^{-\lambda^2/4} & \textrm{if }\lambda \geq 2, \\ 10\log(1 + 2\lambda^{-1}) & \textrm{if } 0 < \lambda < 2. \end{cases}
\end{equation}
Then there exists a partial coloring $\chi$ that assigns $\pm1$ to at least $n/10$ variables (and $0$ to the rest), satisfying $|\chi(S)| \leq \Delta_S$ for each $S\in \mathcal C$.
\end{lemma}
We apply Lemma~\ref{lem:partial-coloring-lemma} to the set system $(X, \mathcal C_X)$ defined in Section~\ref{sec:decomposition}. In \eqref{eqn:def-g} we intentionally choose $g$ to be monotonically decreasing. We further show the following property of $g$.
\begin{lemma}\label{lem:choice-of-b}
Let $d\in \mathbb N$ and $c = 10d+2400$. Let $K$ be a positive real number, and let $b: \mathbb N\to (0, \infty)$ be defined as
\begin{equation}\label{eqn:def-b-as-K}
b(s) = \begin{cases}c\sqrt{s} \cdot \left(sK^{-\frac{1}{d+1}}\right)^{-1} &\mbox{if } s \geq K^{\frac{1}{d+1}} \\ c\sqrt{s} \cdot \left(sK^{-\frac{1}{d+1}}\right)^{-0.1} &\mbox{if } s < K^{\frac{1}{d+1}}\end{cases}.\end{equation}
Then for $g$ as defined in \eqref{eqn:def-g}, we have
\[\sum_{i = 0}^\infty \frac{K}{2^{i(d+1)}}g\left(\frac{b(2^i)}{2^{i/2}}\right) \leq 1.\]
\end{lemma}
\begin{proof}
Let $s_i = 2^i$ and $\tau_i = 2^iK^{-\frac{1}{d+1}}$, where $i$ takes nonnegative integer values. Now we may rewrite
\[\frac{b(2^i)}{2^{i/2}} = \begin{cases}c\tau_i^{-1} &\mbox{if } \tau_i \geq 1 \\ c\tau_i^{-0.1} &\mbox{if } \tau_i < 1\end{cases}.\]
Since the right hand side only depends on $\tau_i$, we denote it as $\lambda(\tau_i)$. Therefore we have
\begin{equation}\label{eqn:rewrite-sum-as-tau}\sum_{i = 0}^\infty \frac{K}{2^{i(d+1)}}g\left(\frac{b(2^i)}{2^{i/2}}\right) = \sum_{i=0}^\infty \tau_i^{-d-1}g(\lambda(\tau_i)) = \sum_{\tau_i < 1}\tau_i^{-d-1}g(c\tau_i^{-0.1}) + \sum_{\tau_i \geq 1}\tau_i^{-d-1}g(c\tau_i^{-1}).\end{equation}
We bound two terms on the right hand side of \eqref{eqn:rewrite-sum-as-tau} separately. For the first term, note that $\{\tau_i < 1\}$ can be seen as a geometric sequence with ratio $1/2$. For any $x > 0$ and positive integer $t$, we have $e^x \geq \frac{x^t}{t!}$. Thus by setting $x = \tau_i^{-0.2}$ and $t = 5d+10$ we get that the series
\[\sum_{\tau_i < 1}\tau_i^{-d-1}e^{-\tau_i^{-0.2}} \leq \sum_{\tau_i < 1}\tau_i^{-d-1}\cdot (5d+10)!\tau_i^{0.2\cdot (5d+10)} = (5d+10)!\sum_{\tau_i<1}\tau_i \leq 2(5d+10)!.\]
We denote $c_1 = 2(5d+10)!$. Note that $c = 10d+2400$ satisfies $c^2/4 > 1 + \log (20c_1)$. For $\tau < 1$, we have $c\tau^{-0.1} > 2$, so $g(c\tau^{-0.1})$ uses the branch $g(\lambda) = 10e^{-\lambda^2/4}$. Therefore
\[g(c\tau^{-0.1}) = 10e^{-\frac{c^2\tau^{-0.2}}{4}} \leq 10e^{-(1+\log (20c_1))\tau_i^{-0.2}} \leq 10e^{-\tau^{-0.2} - \log 20c_1} = \frac{1}{2c_1}e^{-\tau^{-0.2}}.\]
Using this, we may bound the first term on the right hand side of \eqref{eqn:rewrite-sum-as-tau} as following
\begin{equation}\label{eqn:tau-first-term}
\sum_{\tau_i < 1}\tau_i^{-d-1}g(c\tau_i^{-0.1}) \leq \sum_{\tau_i < 1} \tau_i^{-d-1}\cdot \frac{1}{2c_1}e^{-\tau_i^{-0.2}} \leq \frac{1}{2c_1}\cdot c_1 = \frac{1}{2}.
\end{equation}
Now we bound the second term on the right hand side of \eqref{eqn:rewrite-sum-as-tau}. As $\{\tau_i \geq 1\}$ forms a geometric sequence with ratio $2$ and $\log(1+2\tau_i)\leq 3\tau_i$ when $\tau_i \geq T > 1$, we have for $T = 240$,
\[\sum_{\tau_i > T} \tau_i^{-d-1}\log(1+2\tau_i) \leq \sum_{\tau_i > T}\tau_i^{-d-1} \cdot 3\tau_i = 3\sum_{\tau_i > T}\tau_i^{-d} \leq 3\sum_{\tau_i > T}\tau_i^{-1} \leq \frac{6}{T} = \frac{1}{40}.\]
For any $c = c(d) > 1$ and $\tau\geq T$, as $g$ is monotonically decreasing, we have $g(c\tau^{-1}) \leq g(\tau^{-1}) = 10\log(1+2\tau)$ where we use the branch $g(\lambda) = 10\log(1+2\lambda^{-1})$ as $\tau^{-1} < 2$. Hence
\begin{equation}\label{eqn:tau-second-term-second-half}\sum_{\tau_i \geq T}\tau_i^{-d-1}g(c\tau_i^{-1}) \leq \sum_{\tau_i \geq T}\tau_i^{-d-1} \cdot 10\log(1+2\tau_i) \leq 10\cdot \frac{1}{40} = \frac{1}{4}.\end{equation}
Finally, for $\tau_i < T$, we have $g(c\tau^{-1}) \leq g(cT^{-1})$. As $g(10)\leq \frac{1}{8}$ and $c > 2400 = 10T$, $g(cT^{-1}) \leq \frac{1}{8}$. Consequently,
\begin{equation}\label{eqn:tau-second-term-first-half}
\sum_{1\leq \tau_i < T}\tau_i^{-d-1}g(c\tau_i^{-1}) \leq \sum_{1\leq \tau_i < T}\tau_i^{-d-1}\cdot \frac{1}{8} < \frac{1}{4}.
\end{equation}
Here in the last inequality we use that $\{\tau_i: 1\leq \tau_i < T\}$ forms a geometric series with ratio $2$ and the initial term is at least $1$.
Substituting the bounds \eqref{eqn:tau-first-term}, \eqref{eqn:tau-second-term-second-half}, and \eqref{eqn:tau-second-term-first-half} into \eqref{eqn:rewrite-sum-as-tau}, we obtain the desired inequality.
\end{proof}
We choose the function $b$ of the form specified in Lemma~\ref{lem:choice-of-b} as it has a good summation property over powers of $2$. This is illustrated by the following lemma.
\begin{lemma}\label{lem:sum-b}
Let $d$ be a positive integer, and $c, K$ be positive real numbers. Let $b: \mathbb N\to (0, \infty)$ be the function defined in \eqref{eqn:def-b-as-K}. Suppose that $u < v$ are positive real numbers. Then
\[\sum_{s: s = 2^t \in [u, v]}b(s) \leq 5cK^\frac{1}{2d+2}\min\left(\left(vK^{-\frac{1}{d+1}}\right)^{0.4}, \left(uK^{-\frac{1}{d+1}}\right)^{-0.5}\right).\]
\end{lemma}
\begin{proof}
Here we assume that $s$ takes value over powers of $2$. Note that
\[b(s) = \min\left(c\sqrt{s}\cdot \left(sK^{-\frac{1}{d+1}}\right)^{-1}, c\sqrt{s}\cdot \left(sK^{-\frac{1}{d+1}}\right)^{-0.1} \right).\]
Using that $b(s) \leq c\sqrt{s}\cdot \left(sK^{-\frac{1}{d+1}}\right)^{-0.1}$, we get
\[\sum_{u\leq s \leq v} b(s) \leq cK^{\frac{1}{10(d+1)}} \sum_{u\leq s \leq v}s^{0.4} \leq cK^{\frac{0.1}{d+1}}v^{0.4}\left(\sum_{j \leq 0}2^{0.4j}\right) \leq 5cK^\frac{1}{2d+2}\left(vK^{-\frac{1}{d+1}}\right)^{0.4}.\]
Similarly using $b(s) \leq c\sqrt{s}\cdot \left(sK^{-\frac{1}{d+1}}\right)^{-1}$, we have
\[\sum_{u\leq s \leq v} b(s) \leq cK^{\frac{1}{d+1}} \sum_{u\leq s \leq v}s^{-0.5} \leq cK^{\frac{1}{d+1}}u^{-0.5}\left(\sum_{j \geq 0}2^{-0.5j}\right) \leq 5cK^\frac{1}{2d+2}\left(uK^{-\frac{1}{d+1}}\right)^{-0.5}.\]
\end{proof}
Combining the lemmas above with Corollary~\ref{cor:counting-subsets-1}, we derive below Proposition~\ref{prop:first-partial-coloring}, which is slightly weaker than what we need to prove Theorem~\ref{thm:main}. Indeed, if we iteratively apply Proposition~\ref{prop:first-partial-coloring} to the remaining uncolored elements, we get $\disc(\mathcal A_d)=O_d(N^{\frac{d}{2d+2}}\log N)$.
\begin{proposition}\label{prop:first-partial-coloring}
Let $d\in \mathbb N$ and $c = 500d+10^6$. For any $X\subseteq [N]^d$, there exists a partial coloring $\chi: X \to \{-1, 0, 1\}$ that assigns $\pm1$ to at least $|X|/10$ elements in $X$ such that
\[\max_{A_0\in \mathcal A_d} |\chi(A_0\cap X)| \leq cN^{\frac{d}{2d+2}}.\]
\end{proposition}
\begin{proof}
Here we treat $d$ as a constant. Suppose that $|X| = m$. Let $b: \mathbb N\to (0, \infty)$ be determined later. We want to apply Lemma \ref{lem:partial-coloring-lemma} to the set system $(X, \mathcal C_X)$ to find a partial coloring $\chi: X\to \{0, \pm 1\}$ that assigns $\pm 1$ to at least $m/10$ elements, and that $|\chi(S)| \leq b(|S|)$ for any $S\in \mathcal C_X$. By Lemma~\ref{lem:partial-coloring-lemma}, it suffices to ensure that $b$ satisfies the inequality
\begin{equation}\label{eqn:prop-first-target}
\sum_{s: s = 2^t \leq N} f(s, X) g\left(\frac{b(s)}{\sqrt{s}}\right) \leq m/5.\end{equation}
By Corollary~\ref{cor:counting-subsets-1}, we know that $f(s, X) \leq 5^d \frac{N^dm}{s^{d+1}}.$ It now suffices to show
\begin{equation}\label{eqn:prop-first-target-converted}
\sum_{s: s = 2^t \leq N} 5^d \frac{N^dm}{s^{d+1}} g\left(\frac{b(s)}{\sqrt{s}}\right) \leq m/5.
\end{equation}
Let $K = 5^{d+1}N^d$. By Lemma~\ref{lem:choice-of-b}, if we set $b$ as in \eqref{eqn:def-b-as-K} with $c_1 = 10d+2400$
\begin{equation}\label{eqn:prop-first-def-b}
b(s) = \begin{cases}c_1\sqrt{s} \cdot \left(sK^{-\frac{1}{d+1}}\right)^{-1} &\mbox{if } s \geq K^{\frac{1}{d+1}} \\ c_1\sqrt{s} \cdot \left(sK^{-\frac{1}{d+1}}\right)^{-0.1} &\mbox{if } s < K^{\frac{1}{d+1}}\end{cases},
\end{equation}
then \eqref{eqn:prop-first-target-converted} is satisfied. Therefore by Lemma~\ref{lem:partial-coloring-lemma}, we know that there exists a partial coloring $\chi$ that assigns $\pm 1$ to at least $m/10$ elements, and that $|\chi(S)| \leq b(|S|)$ for any $S\in \mathcal C_X$. By Lemma~\ref{lem:decomposition}, we know that for any $A\in \mathcal A_X$, $|\chi(A)| \leq 2\sum_{s:s=2^t}b(s)$. Now we have
\[2\sum_{s: s=2^t \leq N}b(s) \leq 2\sum_{s: s=2^t \in [1, K^{\frac{1}{d+1}}]} b(s) + 2\sum_{s: s=2^t \in [K^{\frac{1}{d+1}}, N+1]} b(s) \leq 20c_1K^\frac{1}{2d+2}.\]
In the second inequality above we use Lemma~\ref{lem:sum-b} for $(u, v) = (1, K^{\frac{1}{d+1}})$ and $(u, v) = (K^{\frac{1}{d+1}}, N)$ respectively. Note that $K^\frac{1}{2d+2} = 5^{\frac{d+1}{2d+2}}N^{\frac{d}{2d+2}}=\sqrt{5}N^{\frac{d}{2d+2}}$. Hence we conclude that we can always find partial coloring $\chi$ that assigns $\pm 1$ to at least $m/10$ elements in $X$, and satisfies
\[\max_{A_0\in \mathcal A_d} |\chi(A_0\cap X)| = \max_{A\in \mathcal A_X} |\chi(A)| \leq cN^{\frac{d}{2d+2}},\]
where $c = 20\sqrt{5}c_1 < 500d + 10^6$.
\end{proof}
Not surprisingly, to improve on the bound above, we need to improve on Corollary~\ref{cor:counting-subsets-1}. In particular, we will show in Section~\ref{sec:geometry} that the following holds.
\begin{restatable}{lemma}{counting}\label{lem:counting-subsets-3}
There exists an absolute constant $C_0$ such that the following holds. Let $d$ be a positive integer. Given positive integers $N_1, N_2, \dots, N_d$ satisfying $N_1\cdots N_d \leq (\min_{1\leq i\leq d} N_i)^{d+1-\delta}$ for some $\delta\in (0, 1]$, suppose that $X\subseteq [N_1]\times \cdots \times [N_d]$ is of size $m$. Letting $\rho = \frac{m}{N_1\cdots N_d}$, if integer $s$ satisfies $(N_1\cdots N_d)^{\frac{1}{d+1}}\rho^\frac{\delta}{4^d(d+1)} \leq s \leq (\min_{1\leq i\leq d} N_i)\rho^\beta$ for some $\beta \in (0, 1/2)$, then
\begin{equation}\label{eqn:counting-subsets-3}\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}} |U^d(X, \mathbf{b}, s)| \leq C_02^{d^3}5^d\frac{mN_1N_2\cdots N_d}{s^d} \cdot \rho^{\frac{\min(\beta, \delta)}{4^d\cdot(d+2)!}}.\end{equation}
\end{restatable}
Assuming Lemma~\ref{lem:counting-subsets-3}, we prove the following improvement on Proposition~\ref{prop:first-partial-coloring}.
\begin{proposition}\label{prop:partial-coloring-almost-cube}
Let $d$ be a positive integer. There exist constants $C_d$ and $c_d$ such that the following holds. Let $N_1, N_2, \dots, N_d$ be positive integers satisfying $N_1\cdots N_d \leq (\min_{1\leq i\leq d} N_i)^{d+1-\delta}$ for some $\delta\in (0, 1]$. For any nonempty $X\subseteq [N_1]\times \cdots \times [N_d]$, there exists a partial coloring $\chi: X\to \{-1, 0, 1\}$ that assigns $\pm 1$ to at least $|X|/10$ elements in $X$, and
\[\max_{A_0\in \mathcal{A}_\mathbf{N}}|\chi(A_0\cap X)| \leq C_d(N_1\cdots N_d)^{\frac{1}{2d+2}}\cdot \left(\frac{|X|}{N_1\cdots N_d}\right)^{c_d\delta}.\]
\end{proposition}
\begin{proof}
The general proof strategy is the same as in the proof of Proposition~\ref{prop:first-partial-coloring}. Without loss of generality we may assume $N_1\leq \cdots \leq N_d$. We would like to find a function $b: \mathbb N \to (0, \infty)$ such that
\begin{equation}\label{eqn:prop-almost-cube-target}
\sum_{s: s = 2^t \leq N_d} f(s, X) g\left(\frac{b(s)}{\sqrt{s}}\right) \leq |X|/5.\end{equation}
Here we sum only over $s\leq N_d$ as there is no congruence class of $X\subseteq [N_1]\times \cdots \times [N_d]$ of length greater than $N_d$.
To estimate $f(s, X)/|X|$, we know that Corollary~\ref{cor:counting-subsets-1} gives an upper bound for $1\leq s \leq N_1$. For $N_1 < s \leq N_d$, we apply Lemma~\ref{lem:counting-subsets-by-U} and Lemma~\ref{lem:counting-U-1} and get that
\begin{equation}\label{eqn:large-s}
\frac{f(s, X)}{|X|} \leq \frac{1}{s}\prod_{i=1}^d\left(4\frac{N_i}{s}+1\right) \leq \frac{1}{s}\cdot 5\frac{N_d}{s}\cdot \prod_{i=1}^{d-1}5\frac{N_i}{N_1} = \frac{5^dN_1\cdots N_d}{N_1^{d-1}}\cdot \frac{1}{s^2}.
\end{equation}
Here we use the inequalities $4\frac{N_d}{s}+1\leq 5\frac{N_d}{s}$ as $s \leq N_d$, and $4\frac{N_i}{s}+1\leq 5\frac{N_i}{N_1}$ as $s > N_1$ and $N_i\geq N_1$. Now combining with Lemma~\ref{lem:counting-subsets-3} applied with $\beta = \frac{\delta}{4(d+1)^2}$, we derive the following inequality:
\begin{equation}\label{eqn:piecewise-up-for-f/m}
\frac{f(s, X)}{|X|} \leq \begin{cases}
\frac{1}{s^2}\frac{5^dN_1\cdots N_d}{N_1^{d-1}} & \mbox{if }N_1 < s \leq N_d \\
\frac{1}{s^{d+1}}5^dN_1\cdots N_d & \mbox{if } 1\leq s < (N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\mbox{ or }N_1\rho^\frac{\delta}{4(d+1)^2}< s \leq N_1\\
\frac{1}{s^{d+1}}C_dN_1\cdots N_d\rho^{\frac{\delta}{4^{d+1}(d+1)^2\cdot (d+2)!}} & \mbox{if }(N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\leq s \leq N_1\rho^\frac{\delta}{4(d+1)^2}
\end{cases}
\end{equation}
for $\rho := \frac{|X|}{N_1\cdots N_d}$. Here we applied \eqref{eqn:large-s} on the first range, Corollary~\ref{cor:counting-subsets-1} on the second, and Lemma~\ref{lem:counting-subsets-3} on the third with $C = C_02^{d^3}5^d$ for some absolute constant $C_0$.
We denote $K_1 = 15\frac{5^dN_1\cdots N_d}{N_1^{d-1}}$, $K_2 = 15\cdot 5^dN_1\cdots N_d$, and $K_3 = 15CN_1\cdots N_d\rho^{\frac{\delta}{4^{d+1}(d+1)^2\cdot (d+2)!}}$ for simplicity. Applying Lemma~\ref{lem:choice-of-b} three times, if we define with $c_1 = 2410$ and $c_2 = 10d+2400$
\begin{equation}\label{eqn:b1}
b_1(s) = \begin{cases}c_1\sqrt{s} \cdot \left(sK_1^{-\frac{1}{2}}\right)^{-1} &\mbox{if } s \geq K_1^{\frac{1}{2}} \\ c_1\sqrt{s} \cdot \left(sK_1^{-\frac{1}{2}}\right)^{-0.1} &\mbox{if } s < K_1^{\frac{1}{2}}\end{cases},
\end{equation}\label{eqn:b23}
and for $i = 2$ and $3$
\begin{equation}
b_i(s) = \begin{cases}c_2\sqrt{s} \cdot \left(sK_i^{-\frac{1}{d+1}}\right)^{-1} &\mbox{if } s \geq K_i^{\frac{1}{d+1}} \\ c_2\sqrt{s} \cdot \left(sK_i^{-\frac{1}{d+1}}\right)^{-0.1} &\mbox{if } s < K_i^{\frac{1}{d+1}}\end{cases},
\end{equation}
then as $s$ taking values in powers of $2$, we have the following three inequalities:
\begin{align}
\label{eqn:b1-entropy}
\sum_{N_1 < s\leq N_d}\frac{K_1}{s^2}g\left(\frac{b_1(s)}{\sqrt{s}}\right) &\leq 1,\\
\label{eqn:b2-entropy}
\sum_{1\leq s < (N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}}\frac{K_2}{s^{d+1}}g\left(\frac{b_2(s)}{\sqrt{s}}\right) + \sum_{N_1\rho^\frac{\delta}{4}< s \leq N_1}\frac{K_2}{s^{d+1}}g\left(\frac{b_2(s)}{\sqrt{s}}\right)&\leq 1,\\
\label{eqn:b3-entropy}
\sum_{(N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\leq s \leq N_1\rho^\frac{\delta}{4}}\frac{K_3}{s^{d+1}}g\left(\frac{b_3(s)}{\sqrt{s}}\right)&\leq 1.
\end{align}
If we apply \eqref{eqn:piecewise-up-for-f/m} to \eqref{eqn:b1-entropy}, \eqref{eqn:b2-entropy}, and \eqref{eqn:b3-entropy}, we derive that
\begin{align}
\label{eqn:b1-entropy-1}
\sum_{N_1 < s\leq N_d}\frac{f(s, X)}{|X|}g\left(\frac{b_1(s)}{\sqrt{s}}\right) &\leq \frac{1}{15},\\
\label{eqn:b2-entropy-1}
\sum_{1\leq s < (N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}}\frac{f(s, X)}{|X|}g\left(\frac{b_2(s)}{\sqrt{s}}\right) + \sum_{N_1\rho^\frac{\delta}{4(d+1)^2}< s \leq N_1}\frac{f(s, X)}{|X|}g\left(\frac{b_2(s)}{\sqrt{s}}\right)&\leq 1,\\
\label{eqn:b3-entropy-1}
\sum_{(N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\leq s \leq N_1\rho^\frac{\delta}{4(d+1)^2}}\frac{f(s, X)}{|X|}g\left(\frac{b_3(s)}{\sqrt{s}}\right)&\leq 1.
\end{align}
Therefore, we may define $b$ in terms of $b_1, b_2, b_3$ as following so that \eqref{eqn:prop-almost-cube-target} is satisfied:
\begin{equation}
b(s) = \begin{cases}
b_1(s) & \mbox{if }N_1 < s \leq N_d \\
b_2(s) & \mbox{if } 1\leq s < (N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\mbox{ or }N_1\rho^\frac{\delta}{4(d+1)^2}< s \leq N_1\\
b_3(s) & \mbox{if }(N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\leq s \leq N_1\rho^\frac{\delta}{4(d+1)^2}
\end{cases}.
\end{equation}
Hence by Lemma~\ref{lem:partial-coloring-lemma}, we conclude that there exists $\chi:X\to \{-1, 0, 1\}$ that assigns $\pm 1$ to at least $|X|/10$ elements in $X$ such that $|\chi(S)|\leq b(|S|)$ for any $S\in \mathcal{C}_X$. By Lemma~\ref{lem:decomposition}, we know that $|\chi(A)| \leq 2\sum_s b(s)$ for any $A\in \mathcal{A}_X$. We shall estimate $\sum_s b(s)$ on five intervals using Lemma~\ref{lem:sum-b}:
\begin{align}
\label{eqn:int-1}
\sum_{N_1 < s \leq N_d}b_1(s) &\leq 5c_1K_1^{\frac{1}{2}}N_1^{-\frac{1}{2}},\\
\label{eqn:int-2}
\sum_{1\leq s < (N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}}b_2(s) & \leq 5c_2K_2^\frac{1}{10d+10}(N_1\cdots N_d)^\frac{2}{5d+5}\rho^\frac{2\delta}{5\cdot 4^d(d+1)},\\
\label{eqn:int-3}
\sum_{N_1\rho^\frac{\delta}{4(d+1)^2}< s \leq N_1}b_2(s) &\leq 5c_2K_2^\frac{1}{d+1}N_1^{-\frac{1}{2}}\rho^{-\frac{\delta}{8(d+1)^2}},\\
\label{eqn:int-4}
\sum_{(N_1\cdots N_d)^\frac{1}{d+1}\rho^\frac{\delta}{4^d(d+1)}\leq s < K_3^{\frac{1}{d+1}}}b_3(s) & \leq 5c_2K_3^\frac{1}{2d+2},\\
\label{eqn:int-5}
\sum_{K_3^{\frac{1}{d+1}} \leq s \leq N_1\rho^\frac{\delta}{4(d+1)^2}}b_3(s) &\leq 5c_2K_3^\frac{1}{2d+2}.
\end{align}
Note that $N_1 \geq (N_1\cdots N_d)^\frac{1}{d+1-\delta}$. As $X$ is nonempty, we have $\rho \geq \frac{1}{N_1\cdots N_d}$. For \eqref{eqn:int-1} we have
\begin{equation}\label{eqn:int-1-sim}
5c_1K_1^{\frac{1}{2}}N_1^{-\frac{1}{2}} \leq 5^{\frac{d}{2}+2}c_1(N_1\cdots N_d)^{\frac{1}{2}\left(1-\frac{d}{d+1-\delta}\right)}\leq 5^{\frac{d}{2}+2}c_1(N_1\cdots N_d)^{\frac{1}{2d+2}}\rho^{\frac{d\delta}{2(d+1)(d+1-\delta)}}.
\end{equation}
Here in the first inequality we use $\sqrt{15\cdot 5^d}\leq 5^{\frac{d}{2}+1}$ and $N_1 \geq (N_1\cdots N_d)^\frac{1}{d+1-\delta}$, and in the second we use $\rho \geq \frac{1}{N_1\cdots N_d}$. Similarly we can estimate the right hand sides of \eqref{eqn:int-2}, \eqref{eqn:int-3}, \eqref{eqn:int-4}, \eqref{eqn:int-5} as follows:
\begin{equation}\label{eqn:int-2-sim}
5c_2K_2^\frac{1}{10d+10} = 5c_2\cdot (5^d15)^\frac{1}{10d+10}(N_1\cdots N_d)^\frac{1}{2d+2}\rho^\frac{2\delta}{5\cdot 4^d(d+1)} \leq 10c_2(N_1\cdots N_d)^\frac{1}{2d+2}\rho^\frac{2\delta}{5\cdot 4^d(d+1)},
\end{equation}
\begin{equation}\label{eqn:int-3-sim}
\begin{split}
5c_2K_2^\frac{1}{d+1}N_1^{-\frac{1}{2}}\rho^\frac{-\delta}{8(d+1)^2} &\leq 50c_2(N_1\cdots N_d)^{\frac{1}{2d+2}} (N_1\cdots N_d)^{\frac{1}{2d+2}- \frac{1}{2d+2-2\delta}}\rho^\frac{-\delta}{8(d+1)^2}\\
& \leq
50c_2(N_1\cdots N_d)^{\frac{1}{2d+2}}\rho^{\frac{\delta}{2(d+1)(d+1-\delta)}-\frac{\delta}{8(d+1)^2}} \\
& \leq 50c_2(N_1\cdots N_d)^\frac{1}{2d+2}\rho^\frac{\delta}{4(d+1)^2},
\end{split}
\end{equation}
\begin{equation}\label{eqn:int-45-sim}
5c_2K_3^\frac{1}{2d+2}\leq 10c_2C^\frac{1}{2d+2}(N_1\cdots N_d)^\frac{1}{2d+2}\rho^\frac{\delta}{2\cdot 4^{d+1}(d+1)^3\cdot (d+2)!}.
\end{equation}
Now we define $c_d = \frac{1}{2\cdot 4^{d+1}(d+1)^3\cdot (d+2)!}$. The exponents of $\rho$ in \eqref{eqn:int-1-sim}, \eqref{eqn:int-2-sim}, \eqref{eqn:int-3-sim} and \eqref{eqn:int-45-sim} are at least $c_d\delta$. Putting \eqref{eqn:int-1-sim} into \eqref{eqn:int-1}, \eqref{eqn:int-2-sim} into \eqref{eqn:int-2}, \eqref{eqn:int-3-sim} into \eqref{eqn:int-3}, and \eqref{eqn:int-45-sim} into \eqref{eqn:int-4} and \eqref{eqn:int-5} and summing them together, we derive that
\[\sum_{s}b(s) \leq \left(5^{\frac{d}{2}+2}c_1 + 10c_2 + 50c_2 + 10c_2C^\frac{1}{2d+2} + 10c_2C^\frac{1}{2d+2}\right)\cdot (N_1\cdots N_d)^\frac{1}{2d+2}\rho^{c_d\delta}.\]
Hence if we set $C_d = 2\left(5^{\frac{d}{2}+2}c_1 + 60c_2+20c_2C^\frac{1}{2d+2}\right)$, then for any $X$ there exists $\chi:X\to \{-1, 0, 1\}$ that assigns $\pm 1$ to at least $|X|/10$ elements in $X$ such that
\[\max_{A_0\in \mathcal{A}_\mathbf{N}}|\chi(A_0\cap X)| = \max_{A\in \mathcal{A}_X}|\chi(A)|\leq C_d(N_1\cdots N_d)^\frac{1}{2d+2}\left(\frac{|X|}{N_1\cdots N_d}\right)^{c_d\delta},\]
as desired.
\end{proof}
\begin{remark}
The argument gives $C_d = 2^{O(d^2)}$ and $c_d = 2^{-O(d\log d)}$.
\end{remark}
\begin{corollary}\label{cor: almost cube partial color}
Let $d$ be a positive integer. There exist constants $C_d$ and $c_d$ such that the following holds. Let $N_1, N_2, \dots, N_d$ be positive integers satisfying $N_1\cdots N_d \leq (\min_{1\leq i\leq d} N_i)^{d+1-\delta}$ for some $\delta\in (0, 1]$. For any nonempty $X\subseteq [N_1]\times \cdots \times [N_d]$, there exists a coloring $\chi: X\to \{-1, 1\}$ with
\[\max_{A_0\in \mathcal{A}_\mathbf{N}}|\chi(A_0\cap X)| \leq C_d\frac{1}{\delta}(N_1\cdots N_d)^{\frac{1}{2d+2}}\cdot \left(\frac{|X|}{N_1\cdots N_d}\right)^{c_d\delta}.\]
\end{corollary}
\begin{proof}
Let $C_d', c_d'$ be the constants in Proposition~\ref{prop:partial-coloring-almost-cube}. Then we set $c_d = \min(c_d', 1)$. Starting from $X_1 = X$, for each $i\geq 0$, we partially color $X_i$ by using Proposition~\ref{prop:partial-coloring-almost-cube} and let $X_{i+1}$ be the set of uncolored points. Suppose that we do this for $K$ iterations such that $X_{K+1} = \emptyset$. Since $|X_i| \leq (0.9)^{i-1}|X|$ it follows that the total discrepancy of this coloring is at most
\[\sum_{i=1}^K C_d'(N_1\cdots N_d)^\frac{1}{2d+2}\left(\frac{|X_i|}{N_1\cdots N_d}\right)^{c_d\delta} \leq (N_1\cdots N_d)^\frac{1}{2d+2}\left(\frac{|X|}{N_1\cdots N_d}\right)^{c_d\delta}\cdot C_d'\sum_{i=1}^K \left((0.9)^{c_d\delta}\right)^{i-1}.\]
Finally noting that $\delta \leq 1$, we have $1-(0.9)^{c_d\delta} \geq \frac{c_d\delta}{10}$ as $c_d\delta \in (0, 1]$. Therefore we have
\[\sum_{i=1}^K \left((0.9)^{c_d\delta}\right)^{i-1} \leq \frac{1}{1-(0.9)^{c_d\delta}} \leq \frac{10}{c_d\delta}.\]
We may set $C_d = \frac{10C_d'}{c_d}$ to get the desired inequality.
\end{proof}
\begin{proof}[Proof of the upper bound of Theorem~\ref{thm:almost-cubes}]
By using Corollary~\ref{cor: almost cube partial color}
and taking $X = [N_1]\times \cdots \times [N_d]$, we have $\disc(\mathcal{A}_\mathbf{N}) \leq C_d\frac{(N_1\cdots N_d)^{\frac{1}{2d+2}}}{\delta}$ for some $C_d = 2^{O(d^2)}$ which proves Theorem~\ref{thm:almost-cubes}.
\end{proof}
\section{A better estimate on the number of sets in the decomposition}\label{sec:geometry}
In this section, we prove Lemma~\ref{lem:counting-subsets-3}, which improves upon Lemma~\ref{lem:counting-U-1}.
Recall that $U^d(X, \mathbf{b}, s)$ is the set of elements in $X$ for which there are at least $s$ elements of $X$ in the same residue class mod $\mathbf{b}$.
To prove Lemma~\ref{lem:counting-subsets-3}, we induct on $d$.
We need the following two results regarding lattice points.
\begin{lemma}[Minkowski's theorem, see e.g.~Section III.2.2 in \cite{Cassels}]\label{thm:Minkowski}
Let $X\subseteq \mathbb R^d$ be a point set of volume $V(X)$ which is symmetric about the origin and convex. Let $\Gamma\subseteq \mathbb R^d$ be a $d$-dimensional lattice of determinant $\det(\Gamma)$. If $V(X) > 2^d\det(\Gamma)$, then $X\cap \Gamma$ contains a pair of distinct points $\pm \mathbf{x}$.
\end{lemma}
\begin{lemma}[Lenstra-Lenstra-Lov\'asz Basis Reduction \cite{LLL}]\label{thm:LLL}
Let $\Gamma\subseteq \mathbb R^d$ be a $d$-dimensional lattice of determinant $\det(\Gamma)$. There exists a basis $\mathbf{x}_1, \dots, \mathbf{x}_d$ of $\Gamma$ such that
\[\det(\Gamma) \leq \prod_{i=1}^n \|\mathbf{x}_i\|_2 \leq 2^{\frac{d(d-1)}{4}}\det(\Gamma).\]
\end{lemma}
\begin{remark}
The inequality above is true for any reduced basis (see \cite[Proposition 1.6]{LLL}); the existence of which is guaranteed by an algorithm which transforms any given basis to a reduced one (see \cite[Proposition 1.26]{LLL}). It will not be important for us what the definition of a reduced basis is.
\end{remark}
\begin{corollary}\label{cor:LLL}
Let $\Gamma\subseteq \mathbb R^d$ be a $d$-dimensional lattice. Suppose that $V_0 > 0$ is a real number such that the following holds: for every set $P\subseteq \mathbb R^d$ of volume $V_d(P) > V_0$ that is symmetric about the origin and convex, $P\cap \Gamma$ contains a nonzero point. Then there exists a basis $\mathbf{x}_1, \dots, \mathbf{x}_d$ of $\Gamma$ such that
\[\prod_{i=1}^n \|\mathbf{x}_i\|_2 \leq 2^{\frac{d(d-1)}{4}-d}V_0.\]
\end{corollary}
\begin{proof}
Let $\mathbf{y}_1, \dots, \mathbf{y}_d$ be a basis of $\Gamma$. Note that the fundamental parallelepiped of $\Gamma$ defined as $\{\mathbf{a}\cdot \mathbf{y}: \mathbf{a}\in (0, 1)^{d-1}\}$ where $\mathbf{y} := (\mathbf{y}_1, \dots, \mathbf{y}_d)$ contains no nonzero vector of $\Gamma$. By translation, there is no nonzero vector of $\Gamma$ in $P:= \{\mathbf{a}\cdot \mathbf{y} : \mathbf{a}\in (-1, 1)^{d-1}\}$, and from the condition we know that $V_d(P) \leq V_0$. Note that $V_d(P) = 2^d\det(\Gamma)$, so $\det(\Gamma) \leq 2^{-d}V_0$. By Lemma~\ref{thm:LLL}, there exists a basis $\mathbf{x}_1, \dots, \mathbf{x}_d$ such that
\[\prod_{i=1}^n\|\mathbf{x}_i\|_2 \leq 2^{\frac{d(d-1)}{4}}\det(\Gamma) \leq 2^{\frac{d(d-1)}{4}-d}V_0,\]
which completes the proof.
\end{proof}
Before we start to prove Lemma~\ref{lem:counting-subsets-3}, we introduce some standard notation. For a map $\phi: X\to Y$ and subsets $A\subseteq X$ and $B\subseteq Y$, let $\phi(A)$ be the set of images $\{\phi(a): a\in A\}\subseteq Y$, and let $\phi^{-1}(B)$ be the set of preimages $\{x\in X: \phi(x)\in B\}\subseteq X$.
We next prove a geometric lemma which shows that, given a vector $\mathbf{b} \in \mathbb{Z}^d$ whose coordinates have greatest common divisor $1$, there is a linear map from $\mathbb Z^d$ to $\mathbb Z^{d-1}$ which has full rank with null space generated by $\mathbf{b}$, and maps a grid into another grid with similar size.
\begin{lemma}\label{lem:projection-coprime}
Let $d \geq 2$ be a positive integer, and $N_1, N_2, \dots, N_d$ be positive integers. Suppose that $\mathbf{b} = (b_1, \dots, b_d)\in \mathbb Z^d$ is a nonzero vector satisfying that $\operatorname{gcd}(b_1, \dots, b_d) = 1$ and $|b_i| \leq N_i$ for all $1\leq i\leq d$. Then there exists a linear map $f_\mathbf{b}: \mathbb Z^{d} \to \mathbb Z^{d-1}$ so that the following two conditions holds.
\begin{enumerate}
\item \label{item:nullspace-projection-coprime}For any $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb Z^d$, $f_\mathbf{b}(\mathbf{x}_1) = f_\mathbf{b}(\mathbf{x}_2)$ if and only if $\mathbf{x}_1 - \mathbf{x}_2 = k\mathbf{b}$ for some $k\in \mathbb Z$.
\item \label{item:range-projection-coprime}There exist positive integers $N_1^*, N_2^*, \dots, N_{d-1}^* \geq \min_{1\leq i\leq d} N_i$ so that
\begin{equation}\label{eqn:range-projection-coprime} \frac{1}{2} N_1\cdots N_d \cdot \left(\max_{1\leq i\leq d}\frac{|b_i|}{N_i}\right)\leq N_1^*\cdots N_{d-1}^* \leq 2^{d^2} N_1\cdots N_d \cdot \left(\max_{1\leq i\leq d}\frac{|b_i|}{N_i}\right)\end{equation}
and $f_\mathbf{b}([N_1]\times \cdots \times [N_d])\subseteq [N_1^*] \times \cdots \times [N_{d-1}^*]$.
\end{enumerate}
\end{lemma}
\begin{proof}
We may write $f_\mathbf{b}(\mathbf{x}) = M\mathbf{x} + \mathbf{v}$ for some $M\in \mathbb Z^{(d-1)\times d}$ and $\mathbf{v}\in \mathbb Z^{d-1}$ to be chosen later. Condition \eqref{item:nullspace-projection-coprime} says that $M$ is full rank, with null space generated by $\mathbf{b}$.
We regard the rows of $M$ as vectors $\mathbf{r}_1, \dots, \mathbf{r}_{d-1}\in \mathbb Z^d$. For each $1\leq j\leq d-1$, if we write $\mathbf{r}_j = (r_{j1}, \dots, r_{jd})$, then we define $\mathbf{r}^*_j := (r_{j1}N_1, \dots, r_{jd}N_d) \in \Lambda$ where $\Lambda := N_1\mathbb Z\times \cdots \times N_d\mathbb Z$. Condition \eqref{item:nullspace-projection-coprime} is equivalent to saying that the vectors $\mathbf{r}_j$ in $\mathbb Z^d$ for $1\leq j\leq d-1$ are linearly independent and $\mathbf{r}_j \cdot \mathbf{b} = 0$ for all $1\leq j\leq d-1$. In terms of $\mathbf{r}^*_j$, this is equivalent to $\mathbf{r}^*_j\cdot \mathbf{b}^* = 0$ for $\mathbf{b}^* := (\frac{b_1}{N_1}, \dots, \frac{b_d}{N_d})$, and $\mathbf{r}^*_1, \dots, \mathbf{r}^*_{d-1}$ are linearly independent vectors. The following claim allows us to find these vectors whose product of $\ell_2$-norms is small.
\begin{claim}\label{claim:find-short-basis}
There exists linearly independent vectors $\mathbf{r}^*_1, \dots, \mathbf{r}^*_{d-1}\in \Lambda$ that satisfy $\mathbf{r}^*_j\cdot \mathbf{b}^* = 0$ for each $1\leq j\leq d-1$, and
\begin{equation}\label{eqn:LLL}\prod_{j=1}^{d-1}\|\mathbf{r}^*_j\|_2 \leq 2^{\frac{(d-1)(d-2)}{4}} \cdot \|\mathbf{b}^*\|_2N_1N_2\cdots N_d,\end{equation}
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:find-short-basis}]
Consider the subspace of $\mathbb R^d$ defined by $\langle \mathbf{b}^*\rangle^\perp := \{\mathbf{x}\in \mathbb R^d: \mathbf{x} \cdot \mathbf{b}^* = 0\}$ which has dimension $d-1$. The intersection $\Lambda^* := \Lambda \cap \langle \mathbf{b}^*\rangle^\perp$ is a lattice in $\langle \mathbf{b}^*\rangle^\perp$. As $\mathbf{b}\in \mathbb Z^d$ is a nonzero vector with integer entries, there exist linearly independent vectors $\mathbf{r}_1, \dots, \mathbf{r}_{d-1}\in \mathbb Z^d$ that satisfy $\mathbf{r}_j\cdot \mathbf{b} = 0$ for each $1\leq j\leq d-1$. Hence we can find $d-1$ linearly independent vectors $\mathbf{r}_j^*\in \Lambda\cap \langle \mathbf{b}^*\rangle^\perp$ defined by $\mathbf{r}_j^* = (r_{j1}N_1,\dots, r_{jd}N_d)$ where $\mathbf{r}_j = (r_{j1}, \dots, r_{jd})$ for $1\leq j\leq d-1$. The linear independence of $\{\mathbf{r}_j^*\}_{j=1}^{d-1}$ follows from the linear independence of $\{\mathbf{r}_j\}_{j=1}^{d-1}$, while $\mathbf{r}_j^*\cdot \mathbf{b}^* = \mathbf{r}_j \cdot \mathbf{b} = 0$ for each $1\leq j\leq d-1$. Thus we conclude that $\Lambda^*$ is a $(d-1)$-dimensional lattice.
We next consider some geometric properties of $\Lambda^*$ as a subset of the $(d-1)$-dimensional Euclidean space $\langle \mathbf{b}^*\rangle^\perp$. Let $P$ be a subset of $\langle \mathbf{b}^*\rangle^\perp$ that is symmetric about the origin and convex. Now we consider the set $X := \{\mathbf{p} + a\mathbf{b}^*: \mathbf{p}\in P, a\in (-1/\|\mathbf{b}\|_2^2, 1/\|\mathbf{b}\|_2^2)\}$. We could equivalently phrase this as: $X$ contains all the point $\mathbf{x}$ that satisfies $\mathbf{x}\cdot \mathbf{b}^* \in (-1, 1)$, and the projection of $\mathbf{x}$ onto the hyperspace $\langle\mathbf{b}^*\rangle$ is in $P$. Therefore from this geometric interpretation we know that $V_d(X) = \frac{2}{\|\mathbf{b}\|_2}V_{d-1}(P)$ where $V_d$ denotes the $d$-dimensional volume. Meanwhile, we see that as $P$ and $\{a\mathbf{b}^*: a\in (-1/\|\mathbf{b}\|_2^2, 1/\|\mathbf{b}\|_2^2)\}$ are both convex and symmetric about the origin, their Minkowski sum $X$ is also convex and symmetric about the origin. By Minkowski's theorem, Lemma \ref{thm:Minkowski}, if $V_d(X) > 2^d\det(\Lambda)$, then there is a nonzero point $\mathbf{x}\in \Lambda\cap X$. Note that any point $\mathbf{x}\in \Lambda$ satisfies that $\mathbf{x}\cdot \mathbf{b}^*$ is an integer, while any point $\mathbf{b}\in X$ satisfies that $\mathbf{b}\cdot \mathbf{b}^* \in (-1, 1)$, we know that if $\mathbf{x} \in \Lambda\cap X$, then $\mathbf{x}\in \langle \mathbf{b}^*\rangle^\perp$. Note that $\Lambda^* = \Lambda\cap \langle \mathbf{b}^*\rangle^\perp$ and $P = X\cap \langle \mathbf{b}^*\rangle^\perp$, this means $\mathbf{x} \in \Lambda^*\cap P$. In summary, if $V_{d-1}(P) > 2^{d-1}\|\mathbf{b}\|_2\det(\Lambda) = 2^{d-1}\|\mathbf{b}\|_2N_1\cdots N_d$, then $P\cap \Lambda^*$ contains a nonzero point.
Therefore we may apply Corollary~\ref{cor:LLL} with dimension $d-1$, lattice $\Gamma$, and $V_0=2^{d-1}\|\mathbf{b}\|_2N_1\cdots N_d$, and we obtain that there exists a basis $\mathbf{r}_1^*,\dots, \mathbf{r}_{d-1}^*$ such that
\begin{equation*}\prod_{j=1}^{d-1}\|\mathbf{r}^*_j\|_2 \leq 2^{\frac{(d-1)(d-2)}{4}} \cdot \|\mathbf{b}^*\|_2N_1N_2\cdots N_d,\end{equation*}
so we have these linearly independent vectors as expected.
\end{proof}
From the set of vectors $\{\mathbf{r}_j^*\}_{j=1}^{d-1}$ whose existence is guaranteed by Claim~\ref{claim:find-short-basis}, we obtain the set of vectors $\{\mathbf{r}_j\}_{j=1}^{d-1}$, which are the row vectors of $M$. Then condition \eqref{item:nullspace-projection-coprime} is satisfied, since they are $d-1$ linearly independent vectors in $\mathbb Z^d$ and satisfy that $\mathbf{r}_j^*\cdot \mathbf{b}^* = 0$ for each $1\leq j\leq d-1$.
For each $1\leq j\leq d-1$, we know that for any $\mathbf{x} = (x_1, \dots, x_d)\in \mathbb Z^d$, $(M\mathbf{x})_j = \mathbf{r}_j\cdot \mathbf{x}$. Note that $\mathbf{r}_j = (r_{j1}, \dots, r_{jd})$. We have that whenever $\mathbf{x}\in [N_1]\times \cdots \times [N_d]$,
\begin{equation}\label{eqn:imageisingrid}|\mathbf{r}_j\cdot \mathbf{x}| = \left|\sum_{i = 1}^d r_{ji}x_i\right| \leq \sum_{i=1}^d |r_{ji}N_i| = \|\mathbf{r}^*_j\|_1.\end{equation}
Let $N_j^* = 3\|\mathbf{r}^*_j\|_1$ and $\mathbf{v} = (2\|\mathbf{r}^*_j\|_1)_{j=1}^{d-1}$. Observe that this choice of parameters together with (\ref{eqn:imageisingrid}) ensures that $f_\mathbf{b}([N_1]\times \cdots \times [N_d])\subseteq [N_1^*]\times \cdots \times [N_{d-1}^*]$. For each $1\leq j\leq d-1$, as $\mathbf{r}_j\in \mathbb Z^d$ is nonzero, we have $r_{ji}$ is nonzero for some $i$, and hence $N^*_j \geq \|\mathbf{r}^*_j\|_1 \geq |r_{ji}N_i| \geq N_i \geq \min_{1\leq i\leq d}N_i$.
Also, by condition \eqref{item:nullspace-projection-coprime}, elements in $f_\mathbf{b}^{-1}(\mathbf{x}^*)$ differ by multiples of $\mathbf{b}$. Note that there are at most $\frac{1}{\|\mathbf{b}^*\|} + 1 \leq \frac{2}{\|\mathbf{b}^*\|}$ such elements in $[N_1]\times \cdots \times [N_d]$ for each fixed $\mathbf{x}^*\in [N_1^*]\times \cdots \times [N_{d-1}^*]$. We have
\[\frac{2}{\|\mathbf{b}^*\|}N_1^* \cdots N_{d-1}^* \geq N_1\cdots N_d.\]
It remains to show the other half of the inequality in \eqref{eqn:range-projection-coprime}. With $N_j^*$ as above, we have
\[N_1^*\cdots N_{d-1}^* = \prod_{j=1}^{d-1}3 \|\mathbf{r}^*_j\|_1 \leq \prod_{j=1}^{d-1}3\sqrt{d} \|\mathbf{r}^*_j\|_2 \leq 3^{d-1}(\sqrt{d})^{d-1} 2^{\frac{(d-1)(d-2)}{4}}\|\mathbf{b}^*\|_2N_1N_2\cdots N_d.\]
Using that $\|\mathbf{b}^*\|_2\leq \sqrt{d} \|\mathbf{b}^*\|_{\infty}$, we have
\[3^{d-1}(\sqrt{d})^{d-1} 2^{\frac{(d-1)(d-2)}{4}}\|\mathbf{b}^*\|_2N_1N_2\cdots N_d \leq 2^{d^2}N_1N_2\cdots N_d\|\mathbf{b}^*\|_{\infty},\]
so condition \eqref{item:range-projection-coprime} is also satisfied. Here we use that $3^{d-1}(\sqrt{d})^{d} 2^{\frac{(d-1)(d-2)}{4}} \leq 2^{d^2}$ for $d\geq 2$.\end{proof}
For any $\mathbf{b} = (b_1, \dots, b_d)\in \mathbb Z^d\setminus \{\mathbf{0}\}$ whose entries are not coprime, we may apply the lemma above to $\mathbf{b}/\operatorname{gcd}(b_1, \dots, b_d)$ instead. This gives the following corollary.
\begin{corollary}\label{cor:projection}
Let $d \geq 2$ be an integer, and $N_1, N_2, \dots, N_d$ be positive integers. Suppose that $\mathbf{b} = (b_1, \dots, b_d)\in \mathbb Z^d$ is a nonzero vector satisfying that $\lambda = \lambda(\mathbf{b}):= \max_{1\leq i\leq d} \frac{|b_i|}{\operatorname{gcd}(b_1, \dots, b_d)N_i} \leq 1$. Then there exists a linear map $f_\mathbf{b} \in \mathbb Z^{(d-1)\times d}$ so that the following two conditions holds.
\begin{enumerate}
\item \label{item:nullspace-projection}For any $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb Z^d$, $f_\mathbf{b}(\mathbf{x}_1) = f_\mathbf{b}(\mathbf{x}_2)$ if and only if $\mathbf{x}_1 - \mathbf{x}_2 = k\mathbf{b}$ for some $k\in \mathbb Q$.
\item \label{item:range-projection}There exist positive integers $N_1^*, N_2^*, \dots, N_{d-1}^* \geq \min_{1\leq i\leq d} N_i$ so that $\frac{1}{2}\leq \frac{N_1^*\cdots N_{d-1}^*}{\lambda N_1\cdots N_d} \leq 2^{d^2}$ and $f_\mathbf{b}([N_1]\times \cdots \times [N_d])\subseteq [N_1^*] \times \cdots \times [N_{d-1}^*]$.
\end{enumerate}
\end{corollary}
The linear map in the corollary above is the main tool for reduction from $\mathbb Z^d$ to $\mathbb Z^{d-1}$. We have the following simple relation between $f_\mathbf{b}$ and $U^d(X, \mathbf{b}, s)$.
\begin{lemma}\label{lem:preimage-of-projection}
Let $d\geq 2$ be an integer, and $N_1, \dots, N_d$ be positive integers. Let $X\subseteq [N_1]\times \cdots \times [N_d]$, let $\mathbf{b} = (b_1, \dots, b_d)\in \mathbb Z^d\setminus \{\mathbf{0}\}$ with $\lambda = \lambda(\mathbf{b}):= \max_{1\leq i\leq d} \frac{|b_i|}{\operatorname{gcd}(b_1, \dots, b_d)N_i} \leq 1$. Let $s \leq \min_{i}N_i$ be a positive integer. Suppose that $f_\mathbf{b}: \mathbb Z^d\to \mathbb Z^{d-1}$ is a linear map satisfying \eqref{item:nullspace-projection} in Corollary~\ref{cor:projection}. Then each element in $f_\mathbf{b}(U^d(X, \mathbf{b}, s))$ has at least $s$ and at most $\frac{2}{\lambda}$ preimages under $f_\mathbf{b}$ in $U^d(X, \mathbf{b}, s)$.
\end{lemma}
\begin{proof}
as $f_\mathbf{b}$ satisfies \eqref{item:nullspace-projection}, we know that if $f_\mathbf{b}(\mathbf{x}_1) = f_\mathbf{b}(\mathbf{x}_2)$, then $\mathbf{x}_1 - \mathbf{x}_2$ is a multiple of $\mathbf{b}/\operatorname{gcd}(b_1, \dots, b_d)$. From the definition of $\lambda$, we see that each element in $\mathbb Z^{d-1}$ has at most $\frac{1}{\lambda}+1 \leq \frac{2}{\lambda}$ preimages in $U^d(X, \mathbf{b}, s)\subseteq [N_1]\times \cdots \times [N_d]$.
Note that each element in $U^d(X, \mathbf{b}, s)$ is in a residue class mod $\mathbf{b}$ of size at least $s$. Again by condition \eqref{item:nullspace-projection}, elements in the residue class mod $\mathbf{b}$ get mapped to the same element by $f_\mathbf{b}$. Therefore, every element in $f_\mathbf{b}(U^d(X, \mathbf{b}, s))$ has at least $s$ preimages.
\end{proof}
Using Lemma~\ref{lem:preimage-of-projection} we derive the following lemma, which bounds the size of the intersection of two $U^d$ sets.
\begin{lemma}\label{lem:single-intersection}
Let $d\geq 2$ be a positive integer, and $N_1, \dots, N_d$ be positive integers. Let $X\subseteq [N_1]\times \cdots \times [N_d]$, let $\mathbf{b} = (b_1, \dots, b_d)\in \mathbb Z^d\setminus \{\mathbf{0}\}$ with $\lambda = \lambda(\mathbf{b}):= \max_{1\leq i\leq d} \frac{|b_i|}{\operatorname{gcd}(b_1, \dots, b_d)N_i} \leq 1$. Let $s^* \leq s \leq \min_{i}N_i$ be positive integers. Suppose that $f_\mathbf{b}: \mathbb Z^d\to \mathbb Z^{d-1}$ satisfies condition \eqref{item:nullspace-projection} in Corollary~\ref{cor:projection}, and $\mathbf{b}'\in \mathbb Z^d$ satisfies that $f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})$. Then
\begin{equation}\label{eqn:single-intersection}
|U^d(X, \mathbf{b}, s) \cap U^d(X, \mathbf{b}', s)| \leq \frac{s^*-1}{s}|U^d(X, \mathbf{b}', s)| + \frac{2}{\lambda}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*)|.
\end{equation}
\end{lemma}
\begin{proof}
For simplicity let $X_1 = U^d(X, \mathbf{b}, s)$. Note that $f_\mathbf{b}(X_1)$ is a subset of $\mathbb Z^{d-1}$.
Partition the set $U^d(X, \mathbf{b}', s)$ into nonempty residue classes mod $\mathbf{b}'$. By definition, we know that each such residue class contains at least $s$ elements, so there are at most $\frac{1}{s}|U^d(X, \mathbf{b}', s)|$ such residue classes. For each such residue class, there are two cases:
\begin{itemize}
\item the residue class contains at most $s^* - 1$ elements in $X_1\cap U^d(X, \mathbf{b}', s)$;
\item the residue class contains at least $s^*$ elements in $X_1\cap U^d(X, \mathbf{b}', s)$.
\end{itemize}
We next upper bound the size of $X_1 \cap U^d(X, \mathbf{b}', s)$. The number of elements in $X_1\cap U^d(X, \mathbf{b}', s)$ contained in a residue class of the first case is at most $\frac{s^*-1}{s}|U^d(X, \mathbf{b}', s)|$. It remains to estimate the number of elements contained in a residue class of the second case.
We first show that, if $\mathbf{x}$ is an element of $X_1\cap U^d(X, \mathbf{b}', s)$, whose residue class mod $\mathbf{b}'$ contains at least $s^*$ elements in $X_1\cap U^d(X, \mathbf{b}', s)$, then
\begin{equation}\label{eqn: f_b(x)}
f_\mathbf{b}(\mathbf{x})\in U^{d-1}(f_\mathbf{b}(X_1), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*).
\end{equation}
Let $I = \{\mathbf{x}_1, \dots, \mathbf{x}_k\}$ be the residue class mod $\mathbf{b}'$ of $X_1\cap U^d(X, \mathbf{b}', s)$ with $\mathbf{x} = \mathbf{x}_1$. Suppose that $I$ contains at least $s^*$ elements. Note that if $\mathbf{x}_i = t\mathbf{b}' + \mathbf{x}_j$ for some integer $t\ne 0$, then
$f_\mathbf{b}(\mathbf{x}_i) = f_\mathbf{b}(\mathbf{x}_j) + t(f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}))$. As $f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})$, we know that $f_\mathbf{b}(I) = \{f_\mathbf{b}(\mathbf{x}_i): 1\leq i\leq k\}$ consists of $k\geq s^*$ distinct elements, whose pairwise differences are multiples of $(f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}))$. In particular, there are at least $k \geq s^*$ elements in $f_\mathbf{b}(X_1)$ that are congruent to $f_\mathbf{b}(\mathbf{x})$ mod $(f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}))$. This proves \eqref{eqn: f_b(x)}.
Note that by Lemma~\ref{lem:preimage-of-projection}, each element in $U^{d-1}(f_\mathbf{b}(X_1), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*)$ has at most $\frac{2}{\lambda}$ preimages in $X_1\cap U^d(X, \mathbf{b}', s)$. Therefore, the number of elements contained in a residue class of the second case is at most $\frac{2}{\lambda}|U^{d-1}(f_\mathbf{b}(X_1), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*)|$. Putting these together, we have \eqref{eqn:single-intersection}.
\end{proof}
\begin{lemma}\label{lem:sum-over-intersection-n-dim}
Let $d\geq 2$ be a positive integer and $N_1, \dots, N_d, s$ be positive integers with $\min_{1\leq i\leq d} N_i > s \geq 2$. Let $X\subseteq [N_1]\times \cdots \times [N_d]$, $\lambda_0\in (0, \frac{1}{s-1})$ and $B\subseteq [-\frac{N_1}{s-1}, \frac{N_1}{s-1}] \times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$ such that each $(b_1, \dots, b_d)\in B$ is a nonzero integer point satisfying that $\lambda_0 \leq \max_{1\leq i\leq d} \frac{|b_i|}{\operatorname{gcd}(b_1, \dots, b_d)N_i}$. Let $\mathbf{b}$ be an integer point in $B$, and $f_\mathbf{b}:\mathbb Z^d\to \mathbb Z^{d-1}$ be a map satisfying condition \eqref{item:nullspace-projection} in Corollary~\ref{cor:projection}, and $s^*\leq s$ be a positive integer. Then we have
\begin{equation}\begin{split}\label{eqn:sum-over-intersection-n-dim}
\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| & \leq \frac{4}{s\lambda_0}|U^d(X, \mathbf{b}, s)| + \frac{s^*-1}{s}\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}', s)| \\
& \quad + \frac{12}{s\lambda_0^2}\sum_{\mathbf{b}^*\in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)|.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Let us denote $\mathbf{b} = (b_1, \dots, b_d)$. We first give an upper bound on the number of elements $\mathbf{b}'\in B$ with $f_\mathbf{b}(\mathbf{b}') = f_\mathbf{b}(\mathbf{0})$. By condition \eqref{item:nullspace-projection}, we know that all such $\mathbf{b}'$ are given by $k \mathbf{b}/\operatorname{gcd}(b_1, \dots, b_d)$ for $k \in \mathbb Z$. If $k \mathbf{b}/\operatorname{gcd}(b_1, \dots, b_d)$ is contained in $B\subseteq [-\frac{N_1}{s-1}, \frac{N_1}{s-1}] \times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$, then we know that $\frac{|k|\cdot |b_i|}{\operatorname{gcd}(b_1, \dots, b_d)} \leq \frac{N_i}{s-1}$ for all $i$. Hence each choice of $k$ with $k \mathbf{b}/\operatorname{gcd}(b_1, \dots, b_d)\in B$ satisfies that
\[|k| \leq \min_{1\leq i\leq d} \frac{N_i\operatorname{gcd}(b_1, \dots, b_d)}{|b_i|(s-1)} \leq \frac{1}{(s-1)\lambda_0} \le \frac{2}{s\lambda_0}.\]
Therefore, the number of such $\mathbf{b}'\in B$ is at most $\frac{4}{s\lambda_0}$ (noting that $k\ne 0$). Thus we have
\begin{equation}\label{eqn:sum-over-intersection-n-dim-zero-part}
\sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}') = f_\mathbf{b}(\mathbf{0})}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| \leq \sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}') = f_\mathbf{b}(\mathbf{0})}|U^d(X, \mathbf{b}, s)| \leq \frac{4}{s\lambda_0}|U^d(X, \mathbf{b}, s)|.
\end{equation}
Now we consider the summation over all $\mathbf{b}'\in B$ with $f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})$. For each such $\mathbf{b}'$, by Lemma~\ref{lem:single-intersection},
\begin{equation}\label{eqn:two-U-intersection-n-dim}
|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| \leq \frac{s^*-1}{s}|U^d(X, \mathbf{b}', s)| + \frac{2}{\lambda}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*)|.
\end{equation}
Observe that
\[\sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})}|U^d(X, \mathbf{b}', s)| \leq \sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}', s)|.\]
Now, note that $\mathbf{b}'\in B \subseteq [-\frac{N_1}{s-1}, \frac{N_1}{s-1}] \times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$. Thus, by a similar argument as above, for each $\mathbf{b}^*\in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}$, the number of $\mathbf{b}'\in B$ with $f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}) = \mathbf{b}^*$ is at most $\frac{2}{(s-1)\lambda_0} + 1 \leq \frac{6}{s\lambda_0}$. Therefore we have
\[\sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}')\ne f_\mathbf{b}(\mathbf{0})}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), f_\mathbf{b}(\mathbf{b}') - f_\mathbf{b}(\mathbf{0}), s^*)| \leq \frac{6}{s\lambda_0} \cdot \sum_{\mathbf{b}^* \in \mathbb Z\setminus 0}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)|.\]
We sum \eqref{eqn:two-U-intersection-n-dim} over all $\mathbf{b}'\in B$ with $f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})$. Combining it with \eqref{eqn:sum-over-intersection-n-dim-zero-part}, we have
\begin{equation}\label{eqn:lem-subset-counting-sub-n-dim}\begin{split}
\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| & = \sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}') = f_\mathbf{b}(\mathbf{0})}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| \\
& \quad + \sum_{\mathbf{b}'\in B: f_\mathbf{b}(\mathbf{b}') \ne f_\mathbf{b}(\mathbf{0})}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)|\\
& \leq \frac{4}{s\lambda_0}|U^d(X, \mathbf{b}, s)| + \frac{s^*-1}{s}\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}', s)|\\
& \quad + \frac{12}{s\lambda_0^2}\sum_{\mathbf{b}^* \in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)|.
\end{split}
\end{equation}
This establishes the desired inequality \eqref{eqn:sum-over-intersection-n-dim}. \end{proof}
Lemma \ref{lem:sum-over-intersection-n-dim} is a useful bound for those $\mathbf{b}$ for which $\lambda(\mathbf{b})$ (as defined in Lemma \ref{lem:single-intersection}) is not too small. The following lemma shows that there are not many choices of $\mathbf{b}$ for which $\lambda(\mathbf{b})$ is small.
\begin{lemma}\label{lem:number-theory-2}
Let $d\geq 2$ be a positive integer, $n_1, \dots, n_d\in \mathbb N$ and $\epsilon \in (0, 1)$. If $\frac{1}{\epsilon} \leq n_i$ for all $1\leq i\leq d$, then there are at most $6^d\epsilon n_1\cdots n_d$ nonzero points $(b_1, b_2, \dots, b_d)\in [-n_1, n_1]\times \cdots \times [-n_d, n_d]$ with $|b_i/\operatorname{gcd}(b_1, \dots, b_d)| \leq \epsilon n_i$ for all $1\leq i\leq d$.
\end{lemma}
\begin{proof}
For each $i$, the number of tuples with $b_i = 0$ is given by
\[\prod_{j\ne i}(2n_j+1) \leq 3^dn_1\cdots n_d\cdot \frac{1}{n_i} \leq 3^d\epsilon n_1\cdots n_d.\]
Hence the number of such tuples with at least one zero entry is at most $3^dd\epsilon n_1\cdots n_d$.
We may next only consider the tuples with nonzero entries. Suppose that $\operatorname{gcd}(b_1, \dots, b_d) = k$. For any given $k$, we know that $b_i' = b_i/k$ satisfies that $|b_i'| \leq n_i/k$. From the problem condition, we further know that $|b_i'|\leq \epsilon n_i$. Hence when $k$ is fixed, the number of such tuples is at most $(2\epsilon)^d n_1\cdots n_d$ if $k\leq \frac{1}{\epsilon}$, and $(2/k)^dn_1\cdots n_d$ if $k > \frac{1}{\epsilon}$. Thus summing over $k$, we know that the number of such tuples $(b_1, b_2, \dots, b_d)$ is at most
\[\sum_{1\leq k \leq \frac{1}{\epsilon}} 2^d\epsilon^dn_1\cdots n_d + \sum_{\frac{1}{\epsilon} < k} 2^d\frac{n_1\cdots n_d}{k^d} \leq 2^d\epsilon^{d-1} n_1\cdots n_d + 2^d\cdot 2\epsilon^{d-1} n_1\cdots n_d = 3\cdot 2^d\epsilon^{d-1} n_1\cdots n_d.\]
Therefore the total number of such tuples is at most
\[3^dd\epsilon n_1\cdots n_d + 3\cdot 2^d\epsilon^{d-1} n_1\cdots n_d \leq 6^d\epsilon n_1\cdots n_d. \]
\end{proof}
If $s\geq 2$ and $U^d(X, \mathbf{b}, s)$ is nonempty, then $\mathbf{b}\in [-\frac{N_1}{s-1}, \frac{N_1}{s-1}] \times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$. In the lemma above we pick $n_i = \frac{N_i}{s-1}$. Then the lemma above gives an upper bound on the number of nonzero points $\mathbf{b}$ whose $\lambda$ value (as defined in Lemma \ref{lem:single-intersection}) is at most $\frac{\epsilon}{s-1}$.
Finally, we need the following lemma.
\begin{lemma}\label{lem:sum-of-intersection}
Let $X$ be a set of size $m > 0$. Let $\{A_i\}_{i\in I}$ be a family of subsets of $X$ over indices $i\in I$. Then we have
\[\sum_{i, j\in I}|A_i\cap A_j| \geq \frac{1}{m}\left(\sum_{i\in I}|A_i|\right)^2.\]
\end{lemma}
\begin{proof}
We count the number of tuples in $T = \{(x, i, j)\in X\times I\times I: x\in A_i\cap A_j\}$. Note that if we fix $i$ and $j$, the number of choices for $x$ is exactly $|A_i\cap A_j|$. Hence we have $|T| = \sum_{i, j\in I}|A_i\cap A_j|.$
For each $x\in X$, let $m_x = |\{i\in I: x\in A_i\}|$, the number of sets $A_i$ that contain $x$. First we see that $\sum_{x\in X}m_x = \sum_{i\in I}|A_i|$.
Moreover, when counting $X$, once we fix $x\in X$, the number of choices for $(i, j)$ is $m_x^2$, so we have $ |T| = \sum_{x\in X}m_x^2.$
By the Cauchy-Schwarz inequality, we have
\[m\sum_{i, j\in I}|A_i\cap A_j| = m|T| = |X|\left(\sum_{x\in X}m_x^2\right) \geq \left(\sum_{x\in X}m_x\right)^2 = \left(\sum_{i\in I}|A_i|\right)^2.\]
This gives the desired inequality.
\end{proof}
We next prove Lemma~\ref{lem:counting-subsets-3}. The proof is by induction on $d$. The following proposition handles the base case $d = 1$ and is due to
Matou\v sek and Spencer \cite{MS}.
\begin{proposition}[Proposition 4.1 in \cite{MS}]\label{prop:MS-4.1}There exists an absolute constant $C$ such that the following holds. For positive integers $N$ and $m$, if $X\subseteq [N]$ is a subset of size $m$ and $s\geq 5\sqrt{m}$, then
\[\sum_{b\in \mathbb Z \setminus \{0\}}|U^1(X, b, s)| \leq C\frac{N^{\frac12}m^{\frac32}}{s}.\]
\end{proposition}
Now we have all the tools to set up the proof of Lemma~\ref{lem:counting-subsets-3}. We first describe the proof idea.
Let us consider a fixed $d \geq 2$ with the induction hypothesis that the statement holds for $d-1$. Since we are to run an induction, the crux of the proof is to apply the induction hypothesis. Corollary~\ref{cor:projection} enables us to project the $d$-dimensional set $X\subseteq [N_1]\times \cdots \times [N_d]$ to a $(d-1)$-dimensional set $X^*\subseteq [N_1^*]\times \cdots \times [N_{d-1}^*]$ for some set $X^*$ and integers $N_j^*$ for $1\leq j\leq d-1$. Let $\rho = \frac{|X|}{N_1\cdots N_d}$ and $\rho^* = \frac{|X^*|}{N_1^*\cdots N_{d-1}^*}$ be the densities of the sets in the grids that they are subsets of. Fix $\epsilon = \rho^\gamma$ where $\gamma$ is chosen to be the exponent of $\rho$ in \eqref{eqn:counting-subsets-3}. As Lemma~\ref{lem:number-theory-2} says that only $O_d(\epsilon)$-fraction of $\mathbf{b}$'s satisfy $\lambda(\mathbf{b}) < \frac{\epsilon}{s}$, it allows us to only focus on $\mathbf{b}$ with $\lambda(\mathbf{b})$ roughly $\frac{1}{s}$, off by a factor of at most $O_d(\epsilon) = O_d(\rho^\gamma)$. It follows that we can estimate both $N_1^*\cdots N_{d-1}^*$ and $X^* = f_\mathbf{b}(U^d(X, \mathbf{b}, s))$ within a factor of $\rho^{O_d(\gamma)}$ by applying Corollary~\ref{cor:projection} and Lemma~\ref{lem:preimage-of-projection} respectively. Thus we can estimate $\rho^*$ within a factor of $\rho^{O_d(\gamma)}$. Finally we would like to apply Lemma~\ref{lem:sum-over-intersection-n-dim} and combine that with Lemma~\ref{lem:sum-of-intersection} to get the desired bound. Together these two lemmas give us an upper bound on the sum of the sizes of the $U^d(X,\mathbf{b},s)$ over all $\mathbf{b}$ in a carefully chosen set $B\subseteq \mathbb{Z}^d$ in which the $i$-th coordinate is at most $N_i/(s-1)$ in absolute value for each $i$.
In the bound in Lemma~\ref{lem:sum-over-intersection-n-dim}, it is not hard to see that the first term on the right hand side is of lower order. By choosing $s^* = s\rho^\gamma$, we can save a factor of $\rho^\gamma$ in the second term. For the third term, we apply the induction hypothesis to save a factor of $(\rho^*)^{\gamma^*}$ (where the bound is expressed in terms of $|X^*|$, $N_1^*\cdots N_{d-1}^*$ and $\rho^*$). Since we can estimate each of them within a factor of $\rho^{O_d(\gamma)}$, we conclude that we save a factor of $\rho^{\gamma^* - O_d(\gamma)}$ in the third term. We can make $\gamma$ small enough so that $\gamma^* - O_d(\gamma) \geq \gamma$. In summary we save a factor of $O_d(\rho^\gamma)$ in all three terms, which is exactly what we need in Lemma~\ref{lem:counting-subsets-3}.
We next recall the statement of Lemma~\ref{lem:counting-subsets-3} and prove it.
\counting*
\begin{proof}[Proof of Lemma~\ref{lem:counting-subsets-3}]
Let $C_0 > 1$ be an absolute constant that satisfies Proposition~\ref{prop:MS-4.1}. We prove by induction on $d$ that the statement holds for $C = C_0\cdot 5^d\cdot 2^{d^3}$. We may assume $N_1 = \min_{1\leq i\leq d}N_i$.
If $m = 0$ or if $N_1 = \cdots = N_d = 1$, then the statement trivially holds. Hence we may assume that $m \geq 1$ and $N_1\cdots N_d > 1$. Therefore, noting that $\frac{1}{d+1} - \frac{\delta}{4^d(d+1)} > 0$, we have that $s$ satisfies
\begin{equation}\label{eqn:lower-bound-s}
s \geq (N_1\cdots N_d)^{\frac{1}{d+1}}\rho^{\frac{\delta}{4^d(d+1)}} = (N_1\cdots N_d)^{\frac{1}{d+1} - \frac{\delta}{4^d(d+1)}}m^{\frac{\delta}{4^d(d+1)}} \geq (N_1\cdots N_d)^{\frac{1}{d+1} - \frac{\delta}{4^d(d+1)}}.
\end{equation}
It follows that $s\geq 2$. Also note that $\rho \leq 1$. We know that $2\leq s \leq \min_{1\leq i\leq d}N_i$. Hence we have
\begin{equation}
\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}}|U^d(X, \mathbf{b}, s)|\leq |X| \cdot \prod_{i=1}^d \left(4\frac{N_i}{s}+1\right) \leq m \cdot \prod_{i=1}^d 5\frac{N_i}{s} = 5^d\frac{mN_1\cdots N_d}{s^d}
\end{equation}
by Lemma~\ref{lem:counting-U-1}. Therefore the statement holds if $C\rho^{\frac{\min(\beta, \delta)}{4^d(d+2)!}} \ge 5^d$. Hence we may assume that $\rho^{\frac{\min(\beta, \delta)}{4^d(d+2)!}} < \frac{5^d}{C} = 2^{d^3}C_0 < 2^{d^3}$, or equivalently $\rho < 2^{-\frac{(d+2)!\cdot 4^dd^3}{\min(\beta, \delta)}} =: \rho_0$.
We prove the base case $d = 1$ using Proposition~\ref{prop:MS-4.1}. Note that $\frac{\delta}{4^d(d+1)} = \frac{\delta}{8} \leq 1/8$ and $\frac{\min(\beta, \delta)}{4^d(d+2)!} \leq 1/24$. As we assumed that $\rho \leq \rho_0 = 2^{-\frac{24}{\min(\beta, \delta)}} < 5^{-\frac{8}{3}}$, we have $\sqrt{N_1}\rho^{\frac{\delta}{4^d(d+1)}} \geq \sqrt{N_1\rho} \rho^{-\frac{3}{8}}\geq 5\sqrt{m}.$
If $s\geq \sqrt{N_1}\rho^{\frac{\delta}{4^d(d+1)}} \geq 5\sqrt{m}$, then by Proposition~\ref{prop:MS-4.1} we have the desired inequality for $d = 1$
\[\sum_{b\in \mathbb Z\setminus \{\mathbf{0}\}}|U^1(X, b, s)| \leq C_0\frac{mN_1}{s}\rho^\frac{1}{2} \leq C\frac{mN_1}{s}\rho^\frac{\min(\beta, \delta)}{4^d\cdot (d+2)!}.\]
We next show the desired bound for $d \geq 2$, assuming the induction hypothesis for $d^* = d - 1$.
Let $\gamma = \frac{\min(\beta, \delta)}{4^{d}\cdot (d+2)!}$ and $\epsilon = \rho^\gamma$. As $|U^d(X, \mathbf{b}, s)|$ is zero if $\mathbf{b}$ is not a nonzero integer point in $[-\frac{N_1}{s-1}, \frac{N_1}{s-1}]\times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$, we may ignore these points in the summation. Let $B_0$ be the set of nonzero integer points in $[-\frac{N_1}{s-1}, \frac{N_1}{s-1}]\times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$. Let $B_1$ be the set of points $\mathbf{b} = (b_1, \dots, b_d)$ in $B_0$ for which $|b_i/\operatorname{gcd}(b_1, \dots, b_d)| \leq \epsilon \frac{N_i}{s-1}$ for all $1\leq i\leq d$. Let $B_2$ be the set of nonzero integer points $\mathbf{b}$ in $B_0\setminus B_1$ for which $|U^d(X, \mathbf{b}, s)| \leq \epsilon m$. Let $B = B_0\setminus (B_1\cup B_2)$. Therefore
\begin{equation}\label{eqn:partition-sum}
\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}} |U^d(X, \mathbf{b}, s)| = \sum_{\mathbf{b}\in B_1} |U^d(X, \mathbf{b}, s)| + \sum_{\mathbf{b}\in B_2} |U^d(X, \mathbf{b}, s)| + \sum_{\mathbf{b}\in B} |U^d(X, \mathbf{b}, s)|.
\end{equation}
We estimate each term on the right hand side of \eqref{eqn:partition-sum}. As $s \leq N_{1}\rho^\beta \leq N_i\rho^\beta$, we have $n_i:= \frac{N_i}{s-1} \geq \rho^{-\beta} \geq \rho^{-\gamma} = \frac{1}{\epsilon}$. Hence we may apply Lemma~\ref{lem:number-theory-2} and conclude that
\[|B_1| \leq 6^d \epsilon n_1\cdots n_d = 6^d\epsilon \frac{N_1\cdots N_d}{(s-1)^d} \leq 12^d\epsilon \frac{N_1\cdots N_d}{s^d} = 12^d \rho^\gamma \frac{N_1\cdots N_d}{s^d}.\]
For each $\mathbf{b}\in B_1$, we have $U^d(X, \mathbf{b}, s) \subseteq X$, so $|U^d(X, \mathbf{b}, s)| \leq m$. It follows that
\begin{equation}\label{eqn:partition-sum-term-1}
\sum_{\mathbf{b}\in B_1} |U^d(X, \mathbf{b}, s)|\leq |B_1|m \leq 12^d\rho^\gamma\frac{mN_1\cdots N_d}{s^d}.
\end{equation}
As $B_0\subseteq [-\frac{N_1}{s-1}, \frac{N_1}{s-1}]\times \cdots \times [-\frac{N_d}{s-1}, \frac{N_d}{s-1}]$ and $s\leq N_1\rho^\beta \leq N_i$, we have
\[|B_0| \leq \prod_{i = 1}^d \left(2\frac{N_i}{s-1} + 1\right) \leq \prod_{i = 1}^d \left(4\frac{N_i}{s} + 1\right) \leq 5^d\frac{N_1\cdots N_d}{s^d}.\]
Therefore as $B_2\subseteq B_0$, we have that
\begin{equation}\label{eqn:partition-sum-term-2}
\sum_{\mathbf{b}\in B_2} |U^d(X, \mathbf{b}, s)| \leq |B_2| \cdot \epsilon m \leq 5^d\frac{N_1\cdots N_d}{s^d} \cdot \epsilon m = 5^d\rho^\gamma\frac{mN_1\cdots N_d}{s^d}.
\end{equation}
Observe that $B\subseteq B_0$, and it follows
\begin{equation}\label{eqn:bound-B-size}
|B|\leq |B_0| \leq 5^d \frac{N_1\cdots N_d}{s^d}.
\end{equation}
Here \eqref{eqn:partition-sum-term-2} gives an upper bound on the second term in \eqref{eqn:partition-sum}.
Finally we bound the third term. By Lemma~\ref{lem:sum-of-intersection}, noting that $\{U^d(X, \mathbf{b}, s)\}_{\mathbf{b}\in B}$ is a family of subsets of $X$, we have
\begin{equation}\label{eqn:partition-sum-term-3-intersection}
\sum_{\mathbf{b}\in B}\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| \geq \frac{1}{m}\left(\sum_{\mathbf{b}\in B}|U^d(X, \mathbf{b}, s)\right)^2.
\end{equation}
We next give an upper bound on $\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)|$ for fixed $\mathbf{b}\in B$ by Lemma~\ref{lem:sum-over-intersection-n-dim}. Before we can apply it, we need to make a few preparations to ensure that the conditions are satisfied.
Since we have excluded elements in $B_1$, for any $\mathbf{b} = (b_1, \dots, b_d) \in B$ there exists some index $i$ for which $|b_i/\operatorname{gcd}(b_1, \dots, b_d)| > \epsilon n_i = \epsilon \frac{N_i}{s-1}$. This implies that
\begin{equation}\label{eqn:bound-lambda}
\lambda_\mathbf{b} := \max_{1\leq i\leq d} \frac{|b_i|}{\operatorname{gcd}(b_1, \dots, b_d)N_i} \geq \frac{\epsilon}{s-1} > \frac{\epsilon}{s}.
\end{equation}
Meanwhile, as $|b_i| \leq \frac{N_i}{s-1}$ for each $i$, we know that $\lambda_\mathbf{b} \leq \frac{1}{s-1} \leq \frac{2}{s}$. Therefore $\lambda_\mathbf{b}\in (\frac{\epsilon}{s}, \frac{2}{s}]$ for all $\mathbf{b}\in B$. Hence $B$ satisfies the conditions in Lemma~\ref{lem:sum-over-intersection-n-dim} for $\lambda_0 := \frac{\epsilon}{s}$.
We fix an arbitrary $\mathbf{b}\in B$. By Corollary~\ref{cor:projection} there exists a linear map $f_\mathbf{b}: \mathbb Z^d\to \mathbb Z^{d-1}$ that satisfies conditions \eqref{item:nullspace-projection} and \eqref{item:range-projection}. Let $s^* = \lceil\epsilon s\rceil$. Now we know that $\mathbf{b}$, $f_\mathbf{b}$, and $s^*$ satisfy the conditions in Lemma~\ref{lem:sum-over-intersection-n-dim}. We apply it and get
\begin{equation}\begin{split}\label{eqn:single-sum}
\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| & \leq \frac{4|U^d(X, \mathbf{b}, s)|}{s\lambda_0} + \frac{s^*-1}{s}\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}', s)| \\
& \quad + \frac{12}{s\lambda_0^2}\sum_{\mathbf{b}^*\in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)|.
\end{split}
\end{equation}
For the third term on the right hand side of \eqref{eqn:single-sum}, we would like to apply the induction hypothesis.
To do this, we need to verify the various conditions in the statement by proving following claims.
As $f_\mathbf{b}$ satisfies condition \eqref{item:range-projection} in Corollary~\ref{cor:projection}, there exist positive integers $N_1^*, \dots, N_{d-1}^*$ such that $f_\mathbf{b}([N_1]\times \cdots \times [N_d]) \subseteq [N_1^*]\times \cdots \times [N_{d-1}^*]$,
\begin{equation}\label{eqn:bound-product-N^*-as-lambda}
\frac{1}{2}\lambda_\mathbf{b} N_1\cdots N_d \leq N_1^* \cdots N_{d-1}^* \leq 2^{d^2}\lambda_\mathbf{b} N_1\cdots N_d,
\end{equation}
Let $M := \min_{1\leq i\leq d-1}N_i^* \geq N_1$. As $\lambda_\mathbf{b}\in (\frac{\epsilon}{s}, \frac{2}{s}]$, we have
\begin{equation}\label{eqn:bound-product-N^*}
\frac{\epsilon}{2s} N_1\cdots N_d \leq N_1^* \cdots N_{d-1}^* \leq \frac{2^{d^2+1}}{s}N_1\cdots N_d.
\end{equation}
Let $X^* = f_\mathbf{b}(U^d(X, \mathbf{b}, s))$. As $f_\mathbf{b}$ satisfies \eqref{item:range-projection} in Corollary~\ref{cor:projection}, we have $X^* \subseteq [N_1^*]\times \cdots \times [N_{d-1}^*]$. Let $m^* := |X^*|$ and $\rho^* := m^*/(N_1^*\cdots N_{d-1}^*)$.
The following claims allow us to apply the induction hypothesis to $(N_j^*)_{j=1}^{d-1}$, $X^*$, and $s^*$.
\begin{claim}\label{claim:condition-N^*}
$N_1^*\cdots N_{d-1}^* \leq M^{d-\delta/3}.$
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:condition-N^*}]
Note that $N_1\leq M$, so we have that
\begin{equation}\label{eqn:upper-bound-N^*-product}N_1\cdots N_d\leq N_1^{d+1-\delta} \leq M^{d+1-\delta}.\end{equation}
Since $m\geq 1$, we know that $M^{d+1} \geq N_1\cdots N_d \geq \frac{1}{\rho} > 2^\frac{(d+2)!\cdot 4^dd^3}{\min(\beta, \delta)}$ and so $M \geq 2^{\frac{16(d+2)d^3}{\delta}} > 2^{\frac{12(d^2+1)}{\delta}}$.
Combining \eqref{eqn:upper-bound-N^*-product} with \eqref{eqn:lower-bound-s} and \eqref{eqn:bound-product-N^*}, we have
\begin{equation}\label{eqn:upper-bound-N^*-product-by-N^*}
N_1^*\cdots N_{d-1}^* \leq 2^{d^2}\lambda_\mathbf{b} N_1\cdots N_d \leq 2^{d^2+1} (N_1\cdots N_d)^{\frac{d}{d+1}+\frac{\delta}{4^d(d+1)}} \leq 2^{d^2+1}M^{\frac{d+1-\delta}{d+1}(d+\frac{\delta}{4^d})}.
\end{equation}
For the exponent of $M$ on the right hand side in \eqref{eqn:upper-bound-N^*-product-by-N^*}, we know that
\[\frac{d+1-\delta}{d+1}\left(d+\frac{\delta}{4^d}\right) = d+\frac{\delta}{4^d} - \delta\frac{d+\frac{\delta}{4^d}}{d+1} < d + \frac{\delta}{4} - \frac{\delta}{2} = d - \frac{\delta}{3} - \frac{\delta}{12}.\]
Therefore, we can simplify \eqref{eqn:upper-bound-N^*-product-by-N^*} and get
\[N_1^*\cdots N_{d-1}^* \leq 2^{d^2+1}M^{d - \frac{\delta}{3} - \frac{\delta}{12}} \leq M^{d - \frac{\delta}{3}}\]
as expected, where in the last inequality we use that $M \geq 2^{\frac{12(d^2+1)}{\delta}}$.
\end{proof}
By Claim~\ref{claim:condition-N^*}, we can apply the induction hypothesis to $X^* \subseteq [N_1^*]\times \cdots\times [N_{d-1}^*]$ and $\delta^* = \delta/3$. It remains to show that $s^*$ is also in the desired range.
\begin{claim}\label{claim:condition-s^*}$(N_1^*\cdots N_{d-1}^*)^\frac{1}{d} (\rho^*)^{\frac{\delta/3}{4^{d-1}d}} \leq s^* \leq M(\rho^*)^{\beta}.$
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:condition-s^*}]
By Lemma~\ref{lem:preimage-of-projection}, we have \begin{equation}\label{eqn:bound-m^*} \frac{\lambda_\mathbf{b}}{2}|U^d(X, \mathbf{b}, s)| \leq m^* \leq \frac{1}{s}|U^d(X, \mathbf{b}, s)|.\end{equation}
Note that $\epsilon = \rho^{\gamma}$ and that $\rho = \frac{m}{N_1\cdots N_d}$. As $|U^d(X, \mathbf{b}, s)| \leq m$, combining with \eqref{eqn:bound-product-N^*}, we have
\begin{equation}\label{eqn:upper-bound-rho^*}\rho^* = \frac{m^*}{N_1^*\cdots N_{d-1}^*} \leq \frac{\frac{m}{s}}{\frac{\epsilon}{2s}N_1\cdots N_d} = \frac{2}{\epsilon}\frac{m}{N_1\cdots N_d} = 2\rho^{1-\gamma}.\end{equation}
Recall that $\gamma = \frac{\min(\beta, \delta)}{4^d\cdot (d+2)!} \leq \frac{\delta}{12\cdot 4^dd}$, and so
\[\frac{\delta}{3\cdot 4^{d-1}d}(1-\gamma) > \frac{\delta}{3\cdot 4^{d-1}d}-\gamma = \frac{\delta}{4^dd}+ \frac{\delta}{3\cdot 4^{d}d} - \gamma \geq\frac{d+1}{d}\cdot \frac{\delta}{4^d(d+1)} + \gamma + \frac{\delta}{6\cdot 4^dd}.\]
As a result, raising both sides of $(N_1\cdots N_d)^{\frac{1}{d+1}}\rho^{\frac{\delta}{4^d(d+1)}} \leq s$ to the $\frac{d+1}{d}$-th power, we have
\[s\rho^\gamma \geq s^{-\frac{1}{d}} \cdot (N_1\cdots N_d)^{\frac{1}{d}} \rho^{\frac{d+1}{d}\frac{\delta}{4^d(d+1)}+\gamma} \geq 2^{-\frac{d^2+1}{d}-\frac{\delta}{3\cdot 4^{d-1}d}}\rho^{-\frac{\delta}{6\cdot 4^dd}}\left(\frac{2^{d^2+1}}{s}N_1\cdots N_d\right)^\frac{1}{d} (2\rho^{1-\gamma})^{\frac{\delta}{3\cdot 4^{d-1}d}}.\]
Note that $s^* \geq \epsilon s = \rho^\gamma s$. By \eqref{eqn:upper-bound-rho^*}, we have $\rho^* \leq 2\rho^{1-\gamma}$. Combining these with \eqref{eqn:bound-product-N^*}, we have
\begin{equation}\label{eqn:claim-s^*-1}s^* \geq s\rho^\gamma \geq 2^{-\frac{d^2+1}{d}-\frac{\delta}{3\cdot 4^{d-1}d}}\rho^{-\frac{\delta}{6\cdot 4^dd}}(N_1^*\cdots N_{d-1}^*)^{\frac{1}{d}}(\rho^*)^{\frac{\delta}{3\cdot 4^{d-1}d}} \geq (N_1^*\cdots N_{d-1}^*)^{\frac{1}{d}}(\rho^*)^{\frac{\delta}{3\cdot 4^{d-1}d}}.\end{equation}
Here in the last inequality we use that $\rho^{-\frac{\delta}{6\cdot 4^dd}} \geq \rho^{-2\gamma} \geq 2^{2d^3} \geq 2^{\frac{d^2+1}{d} + \frac{\delta}{3\cdot 4^{d-1}d}}$.
Note that from our choice of $\mathbf{b}\in B$, $|U^d(X, \mathbf{b}, s)|$ is at least $\epsilon m$. Therefore combining with \eqref{eqn:bound-m^*} we have $m^*\geq \frac{\epsilon}{2}\lambda_\mathbf{b} m$. Combining this with \eqref{eqn:bound-product-N^*-as-lambda} and $\epsilon = \rho^{\gamma}$, we have
\begin{equation}\label{eqn:lower-bound-rho^*}
\rho^* = \frac{m^*}{N_1^*\cdots N_{d-1}^*} \geq \frac{\frac{\epsilon}{2}\lambda_\mathbf{b} m}{2^{d^2}\lambda_\mathbf{b} N_1\cdots N_d} \geq 2^{-d^2-1}\cdot \epsilon\rho = 2^{-d^2-1}\rho^{1+\gamma}.
\end{equation}
By \eqref{eqn:lower-bound-s}, we know that $\epsilon s \geq \rho^\gamma \cdot (m/\rho)^{\frac{1}{d+1}-\frac{\delta}{4^d(d+1)}} \geq \rho^{-\frac{1}{d+1}+\frac{\delta}{4^d(d+1)} + \gamma}$. As $\frac{\delta}{4^d(d+1)}+\gamma < \frac{1}{d+1}$ and $\rho < 1$, we have $\epsilon s > 1$, so $s^* = \lceil\epsilon s\rceil\leq 2\epsilon s$. Note that $s \leq N_1\rho^\beta \leq M\rho^\beta$. Therefore, we have
\begin{equation}\label{eqn:claim-s^*-2}
s^* \leq 2\epsilon s \leq 2\rho^\gamma M\rho^{\beta} = 2^{1+\beta(d^2+1)}\rho^{\gamma - \beta\gamma} M (2^{-d^2-1}\rho^{1+\gamma})^\beta \leq M (\rho^{*})^\beta,
\end{equation}
where in the last inequality we use \eqref{eqn:lower-bound-rho^*} and that $\rho^{-\gamma + \beta\gamma} \geq \rho^{\gamma/2} > 2^{d^3/2} > 2^{1+\beta(d^2+1)}$ for $d\geq 2$.
\end{proof}
We conclude that the conditions for the induction hypothesis are satisfied by $d^* = d-1$, $(N_i^*)_{i=1}^{d-1}$, $\delta^* = \delta/3$, $\beta^* = \beta$, $X^* = f_\mathbf{b}(U^d(X, \mathbf{b}, s))$, and $s^* = \lceil \epsilon s\rceil$. Applying the induction hypothesis we get
\begin{equation}\label{eqn:apply-induction-1}
\sum_{\mathbf{b}^*\in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)| \leq C^*\frac{N_1^*\cdots N_{d-1}^*m^*}{(s^*)^{d-1}}(\rho^*)^{\frac{\min(\beta, \delta/3)}{4^{d-1}(d+1)!}},
\end{equation}
where $C^* = C_0\cdot 5^{d-1}\cdot 2^{(d-1)^3}$. Using \eqref{eqn:bound-product-N^*}, \eqref{eqn:bound-m^*}, \eqref{eqn:upper-bound-rho^*}, and that $s^* \geq \rho^\gamma s$, we have
\begin{equation}\label{eqn:apply-induction-2}C^*\frac{N_1^*\cdots N_{d-1}^*m^*}{(s^*)^{d-1}}(\rho^*)^{\gamma^*} \leq C^* \frac{2^{d^2+1}N_1\cdots N_d}{\rho^{(d-1)\gamma}s^{d+1}}\rho^{(1-\gamma)\gamma^*}|U^d(X, \mathbf{b}, s)|\end{equation}
where for simplicity we denote $\gamma^* := \frac{\min(\beta, \delta/3)}{4^{d-1}\cdot (d+1)!} \geq \frac{\min(\beta, \delta)}{3\cdot 4^{d-1}\cdot (d+1)!} = \frac{4(d+2)}{3}\gamma$. Note that exponent of $\rho$ on the right hand side of \eqref{eqn:apply-induction-2} satisfies $(1-\gamma)\gamma^* - (d-1)\gamma = \gamma^* - (d-1+\gamma^*)\gamma > 3\gamma$. Combining this with the inequalities \eqref{eqn:apply-induction-1} and \eqref{eqn:apply-induction-2}, we have
\begin{equation}
\sum_{\mathbf{b}^*\in \mathbb Z^{d-1}\setminus \{\mathbf{0}\}}|U^{d-1}(f_\mathbf{b}(U^d(X, \mathbf{b}, s)), \mathbf{b}^*, s^*)| \leq 2^{d^2+1}C^* \rho^{3\gamma}\frac{N_1\cdots N_d}{s^{d+1}}|U^d(X, \mathbf{b}, s)|.
\end{equation}
Put this into \eqref{eqn:single-sum}. Note that $\lambda_0 = \frac{\epsilon}{s} = \rho^\gamma / s$ and that $\frac{s^*-1}{s}\leq \frac{\epsilon s}{s} = \rho^\gamma$. We have
\begin{equation}\label{eqn:single-sum-simplified}\begin{split}
\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| & \leq 4\rho^{-\gamma}|U^d(X, \mathbf{b}, s)| + \rho^\gamma\sum_{\mathbf{b}'\in B}|U^d(X, \mathbf{b}', s)| \\
& + 12\cdot 2^{d^2+1}C^* \rho^{\gamma}\frac{N_1\cdots N_d}{s^{d}}|U^d(X, \mathbf{b}, s)|.
\end{split}
\end{equation}
We sum \eqref{eqn:single-sum-simplified} over $\mathbf{b}\in B$.
Recall that $\gamma = \frac{\min(\beta, \delta)}{4^d\cdot (d+2)!} < \frac{\beta}{4}$. We have
\[\frac{N_1\cdots N_d}{s^d} \geq \frac{N_1^d}{s^d} \geq \rho^{-d\beta} > \rho^{-2\gamma},\]
and it follows that $\rho^{-\gamma} < \rho^\gamma\frac{N_1\cdots N_d}{s^d}$. Using this and \eqref{eqn:bound-B-size}, we have
\[\begin{split}\sum_{\mathbf{b}, \mathbf{b}'\in B}|U^d(X, \mathbf{b}, s)\cap U^d(X, \mathbf{b}', s)| & \leq \left(4\rho^{-\gamma} + \rho^\gamma |B| + 12\cdot 2^{d^2+1}C^*\rho^\gamma \frac{N_1\cdots N_d}{s^d}\right)\sum_{\mathbf{b}\in B}|U^d(X, \mathbf{b}, s)|\\
& \leq \left(4 + 5^d + 12\cdot 2^{d^2+1}C^*\right)\rho^\gamma \frac{N_1\cdots N_d}{s^d}\sum_{\mathbf{b}\in B}|U^d(X, \mathbf{b}, s)|.\end{split}\]
Combine this with \eqref{eqn:partition-sum-term-3-intersection}, we have
\begin{equation}\label{eqn:partition-sum-term-3}
\sum_{\mathbf{b}\in B}|U^d(X, \mathbf{b}, s)| \leq \left(4 + 5^d + 12\cdot 2^{d^2+1}C^*\right)\rho^\gamma \frac{N_1\cdots N_d}{s^d}m.
\end{equation}
Substituting in the bounds \eqref{eqn:partition-sum-term-1}, \eqref{eqn:partition-sum-term-2}, and \eqref{eqn:partition-sum-term-3} into \eqref{eqn:partition-sum}, we have
\[\sum_{\mathbf{b}\in \mathbb Z^d\setminus \{\mathbf{0}\}} |U^d(X, \mathbf{b}, s)| \leq \left(12^d + 5^d + 4 + 5^d + 12\cdot 2^{d^2+1}C^*\right)\rho^\gamma \frac{mN_1\cdots N_d}{s^d}.\]
Finally, we show that the sum of the additive constant above is at most $C$.
Recall that $C^* = C_0\cdot 5^{d-1}\cdot 2^{(d-1)^3}$ and $C = C_0\cdot 5^d\cdot 2^{d^3}$, we know that $24\cdot 2^{d^2}C^* \leq \frac{3}{5}\cdot 5\cdot 2^{3d^2-3d+1}C^* = \frac{3C}{5}$ as $d\geq 2$. For other terms, we have $4\leq 5^d \leq 12^d \leq 2^{d^3} = \frac{C}{5^dC_0} \leq \frac{C}{25}$ and it follows that $12^d + 5^d + 4 + 5^d + 12\cdot 2^{d^2+1}C^* \leq \frac{4}{25} C + \frac{3}{5}C < C$. Thus the statement holds for $d$. Therefore we conclude that the statement holds for all positive integer $d$.\end{proof}
\section{A proof of the lower bound in Theorem~\ref{thm:rectangular}}\label{sec: rec-lower}
We prove the following result, which is the lower bound on the discrepancy in Theorem~\ref{thm:rectangular}.
\begin{theorem}\label{thm:rectangular-lower}
For any positive integer $d$, there exists a constant $c_d>0$ such that the following holds. For positive integer $N_1,\ldots,N_d$, letting $\mathbf{N} = (N_1, \dots, N_d)$, we have
\[\disc(\mathcal{A}_{\mathbf{N}}) \geq c_d\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{2|I|+2}.\]
Here by convention if $I = \emptyset$ then $\prod_{i\in I}N_i = 1$.
\end{theorem}
Roth \cite{Roth} proved the case $d=1$, and Valk\'o \cite{Valko} proved the case that the $N_i$'s are equal. Similar to these previous results, our proof uses Fourier analysis. We first set up some notations. Let $f, g: \mathbb Z^d \to \mathbb C$ be two functions that each has finite support. The Fourier transform $\widehat{f}: [0, 1]^d\to \mathbb C$ is given by $\widehat{f}(\mathbf{r}) = \sum_{\mathbf{x}\in \mathbb Z^d}f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot \mathbf{r}}$. The convolution $f*g: \mathbb Z^d\to \mathbb C$ is given by $f*g(\mathbf{x}) = \sum_{\mathbf{r}\in \mathbb Z^d}f(\mathbf{r})g(\mathbf{x}-\mathbf{r})$, which also has finite support. With these notations, we have the convolution identity $\widehat{f*g} = \widehat{f}\cdot \widehat{g}$ and Parseval's identity
\[\sum_{\mathbf{x}\in \mathbb Z^d}f(\mathbf{x})\overline{g(\mathbf{x})} = \int_{[0, 1]^d} \widehat{f}(\mathbf{r}) \overline{\widehat{g}(\mathbf{r})}\,d\mathbf{r}.\]
In the proof of Theorem~\ref{thm:rectangular-lower} below, for a vector $\mathbf{x} \in \mathbb{Z}^d$, we let $\mathbf{x}_i$ denote the $i^{\textrm{th}}$ coordinate of $\mathbf{x}$.
\begin{proof}[Proof of Theorem~\ref{thm:rectangular-lower}]
We take $c_d = \frac{6^{-d/2}}{2}$ Let $\Omega = [N_1]\times \cdots \times [N_d]\subseteq \mathbb Z^d$. Fix any $\chi: \Omega \to \{1, -1\}$. Let $T = \max_{A\in \mathcal{A}_\mathbf{N}}|\chi(A)|$. It suffices to show that $T \geq c_d\max_{I\subseteq [d]}\left(\prod_{i\in S}N_i\right)^\frac{1}{2|I|+2}$.
For $\chi: \Omega\to \{1, -1\}$, we may extend it to a function $\chi: \mathbb Z^d \to \{-1, 0, 1\}$ by assigning $0$ to $\mathbb Z^d\setminus \Omega$. Clearly $\chi$ takes nonzero values on $N_1\cdots N_d$ points. Hence we may apply Parseval's identity and get
\begin{equation}\label{eqn:chi-norm}
\int_{[0, 1]^d} \widehat{\chi}(\mathbf{r}) \overline{\widehat{\chi}(\mathbf{r})}\,d\mathbf{r} = \sum_{\mathbf{x}\in \mathbb Z^d}\chi(\mathbf{x})\overline{\chi(\mathbf{x})} = N_1\cdots N_d.
\end{equation}
Let $L$ be a positive integer and $D_1, \dots D_d$ be nonnegative integers to be determined later. For each $\mathbf{b}\in \mathbb Z^d\setminus \mathbf{0}$ satisfying that $\mathbf{b}_i\in [-D_i, D_i]$ for $1\leq i\leq d$, let $g_\mathbf{b}: \mathbb Z^d\to \mathbb C$ be the indicator function of the set $\{0, \mathbf{b}, \dots, (L-1)\mathbf{b}\}$. Now we have for each $\mathbf{x}\in \mathbb Z^d$,
\[g_\mathbf{b}*\chi(\mathbf{x}) = \sum_{t=0}^{L-1}\chi(\mathbf{x} - t\mathbf{b}) = \chi(\Omega\cap \{\mathbf{x} - t\mathbf{b}: 0\leq t < L\}).\]
Since $\Omega\cap \{\mathbf{x} - t\mathbf{b}: 0\leq t < L\}$ is an arithmetic progression contained in $\Omega$, it is a set in $\mathcal{A}_\mathbf{N}$, so $|g_\mathbf{b}*\chi(\mathbf{x})| \leq T$. This is true for all $\mathbf{x}\in \mathbb Z^d$. Also note that $|g_\mathbf{b}*\chi(\mathbf{x})|$ is nonzero only when $\mathbf{x} - t\mathbf{b}\in \Omega$ for some $0\leq t < L$. In this case we have $\mathbf{x}_i \in [1-LD_i, N_i+LD_i]$ for each $1\leq i\leq d$. Therefore, $g_\mathbf{b}*\chi$ is nonzero on at most $\prod_{i=1}^d(N_i+2LD_i)$ points in $\mathbb Z^d$. We have
\begin{equation}\label{eqn:upper-L2}
\sum_{\mathbf{x}\in \mathbb Z^d}{g_\mathbf{b} * \chi}(\mathbf{x}) \overline{{g_\mathbf{b} * \chi}(\mathbf{x})} = \sum_{\mathbf{x}\in \mathbb Z^d}\left|{g_\mathbf{b} * \chi}(\mathbf{x})\right|^2 \leq T^2\prod_{i=1}^d(N_i+2LD_i).\end{equation}
By the convolution identity and Parseval's identity, we have
\begin{equation}\label{eqn:upper-Parseval}
\sum_{\mathbf{x}\in \mathbb Z^d}{g_\mathbf{b} * \chi}(\mathbf{x}) \overline{{g_\mathbf{b} * \chi}(\mathbf{x})} = \int_{[0, 1]^d}\widehat{g_\mathbf{b} * \chi}(\mathbf{r}) \overline{\widehat{g_\mathbf{b} * \chi}(\mathbf{r})}\,d\mathbf{r} = \int_{[0, 1]^d}\widehat{g_\mathbf{b}}(\mathbf{r})\widehat{\chi}(\mathbf{r}) \overline{\widehat{g_\mathbf{b}}(\mathbf{r})\widehat{\chi}(\mathbf{r})}\,d\mathbf{r}
\end{equation}
Combining \eqref{eqn:upper-L2} and \eqref{eqn:upper-Parseval}, we get that for any nonzero $\mathbf{b}\in [-D_1, D_1] \times \dots \times [-D_d, D_d]$
\begin{equation}\label{eqn:single-b}
\int_{[0, 1]^d}\left|\widehat{g_\mathbf{b}}(\mathbf{r})\right|^2\widehat{\chi}(\mathbf{r}) \overline{\widehat{\chi}(\mathbf{r})}\,d\mathbf{r} \leq T^2 \prod_{i=1}^d(N_i + 2LD_i).
\end{equation}
Let $A$ be the set of integer points in $[0, D_1] \times \cdots \times [0, D_d]$ and $B$ be the set of nonzero integer points in $[-D_1, D_1] \times \cdots \times [-D_d, D_d]$. Clearly any two distinct points in $A$ have their difference in $B$. The number of points in $B$ is at most $\prod_{i=1}^d(2D_i+1)$. Hence if we sum over $\mathbf{b}\in B$ in \eqref{eqn:single-b}, we get
\begin{equation}\label{eqn:sum-b}
\int_{[0, 1]^d}\left(\sum_{\mathbf{b}\in B}\left|\widehat{g_\mathbf{b}}(\mathbf{r})\right|^2\right)\widehat{\chi}(\mathbf{r}) \overline{\widehat{\chi}(\mathbf{r})}\,d\mathbf{r} \leq T^2 \prod_{i=1}^d(N_i + 2LD_i)(2D_i+1)
\end{equation}
Fix any $\mathbf{r}\in [0, 1]^d$. By the pigeonhole principle, there exists two distinct $\mathbf{a}, \mathbf{a}'\in A$ such that the fractional parts of $\mathbf{a}\cdot \mathbf{r}$ and $\mathbf{a}'\cdot \mathbf{r}$ differ by at most $1/|A|$. Hence for any $\mathbf{r}$ we can find $\mathbf{b}'\in B$ (we shall take $\mathbf{b}' = \mathbf{a}-\mathbf{a}'$ or $\mathbf{b}' = \mathbf{a}'-\mathbf{a}$) such that the fractional part of $\mathbf{b}'\cdot \mathbf{r}$ is in $[0, 1/|A|]$. If $L \leq \frac{|A|}{2} = \frac{1}{2}\prod_{i=1}^d (D_i+1)$, then for any $\mathbf{r}\in [0, 1]^d$,
\[\sum_{\mathbf{b}\in B}\left|\widehat{g_\mathbf{b}}(\mathbf{r})\right|^2 \geq \left|\widehat{g_{\mathbf{b}'}}(\mathbf{r})\right|^2 = \left|\sum_{t=0}^{L-1}e^{-t2\pi i\mathbf{b}'\cdot \mathbf{r}}\right|^2 \geq \frac{4}{\pi^2}L^2.\]
Put this into \eqref{eqn:sum-b} and combine with \eqref{eqn:chi-norm}. We conclude that for any positive integer $L$ and nonnegative integers $D_1, \dots, D_d$ such that $L\leq \frac{(D_1+1)\cdots (D_d+1)}{2}$, then
\begin{equation}\label{eqn:Fourier-lower-unsimplified}
T^2\prod_{i=1}^d(N_i+2LD_i)(2D_i+1) \geq \frac{4}{\pi^2}L^2\prod_{i=1}^dN_i.
\end{equation}
Let $R = \max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{2|I|+2}$. If $R \leq 2$, the statement is trivial as $c_d = \frac{6^{-d/2}}{2} \leq 1/2$. Therefore we may assume that $R > 2$ and the maximum in the definition of $R$ is achieved by some nonempty $I\subseteq [d]$. For each $j\in I$, we have $R \geq \left(\prod_{i\in I\setminus \{j\}}N_i\right)^\frac{1}{2|I|}$, so $N_j \geq R^2$. With these properties, we may now choose the values of $L$ and $D_1, \dots, D_d$. We set $L = \lfloor R^2/2\rfloor$, $D_{i} = \lfloor \frac{N_{i}}{R^2}\rfloor$ for $i\in I$, and $D_j = 0$ for each $j\notin I$.
Since
\[\frac{(D_1+1)\cdots (D_d+1)}{2} \geq \frac{\prod_{i\in I}N_i}{2R^{2|I|}} = \frac{R^2}{2} \geq L,\]
we can apply \eqref{eqn:Fourier-lower-unsimplified} to these variables. For $j\notin I$, as $D_j = 0$, we have $(N_j+2LD_j)(2D_j+1) = N_j$. For $i\in I$, since $N_i \geq R^2$, we have $N_i/R^2 \geq D_i \geq 1$, so
\[(N_i+2LD_i)(2D_i+1) \leq \left(N_i + 2\cdot \frac{R^2}{2} \cdot \frac{N_i}{R^2}\right)\cdot 3\frac{N_i}{R^2} = 6\frac{N_i^2}{R^2}.\]
Put these into \eqref{eqn:Fourier-lower-unsimplified}. Note that $L \geq \frac{R^2}{2}$. We have
\[T^2 \prod_{i\in I}6\frac{N_i^2}{R^2} \cdot \prod_{j\notin I}N_j \geq \frac{R^4}{4}\prod_{i=1}^dN_i.\]
Also note that $\prod_{i\in I}N_i = R^{2|I|+2}$. We conclude that $T \geq \frac{6^{-\frac{|I|}{2}}}{2}R \geq c_dR$.
\end{proof}
\section{A proof of the upper bound in Theorem~\ref{thm:rectangular}}\label{sec: rec-upper}
In this section, we aim to generalize the upper bound in Theorem~\ref{thm:almost-cubes} to all grids of differing side lengths. The following lemma allows us to remove dimensions of short side lengths.
\begin{lemma}\label{lem:slice}
Let $d\geq 2$ be a positive integer and $N_1, \dots, N_d$ be positive integers. Then for $\mathbf{N} = (N_1, \dots, N_d)$ and $\mathbf{N}' = (N_1, \dots, N_{d-1})$, we have
\[\disc(\mathcal A_{\mathbf{N}}) \leq \max\left(\disc(\mathcal A_{\mathbf{N}'}), \sqrt{6N_d\log(2N_1\cdots N_d)}\right).\]
\end{lemma}
\begin{proof}
Firstly we may choose an optimal coloring $\chi'$ for the grid $[N_1]\times \cdots \times [N_{d-1}]$ that achieves discrepancy $\disc(\mathcal A_{\mathbf{N}'})$.
We extend this coloring to a coloring $\chi: [N_1]\times \cdots \times [N_d] \to \{1, -1\}$ by the following procedure. Take $N_d$ i.i.d. Rademacher random variables $v(i)$ for $1\leq i\leq N_d$ (i.e. $\Pr(v(i) = 1) = \Pr(v(i) = -1) = \frac{1}{2}$). Now we set $\chi(x_1, \dots, x_d) = \chi'(x_1, \dots, x_{d-1})v(x_d)$ for any $(x_1, \dots, x_d)\in [N_1]\times \cdots \times [N_d]$.
Now we analyze $\chi(S)$ for $S\in \mathcal A_\mathbf{N}$. Let $(k_1, \dots, k_d)$ be the common difference of arithmetic progression $S$. If $k_d = 0$, then all elements in $S$ share the same $d$-th coordinate $x_d$, so we may write $S$ as $S'\times \{x_d\}$, where $S'$ is also an arithmetic progression with common difference $(k_1, \dots, k_{d-1})$. By our construction of $\chi$, we have $|\chi(S)| = |\chi'(S')v(x_d)| \leq \disc(\mathcal A_{\mathbf{N}'})$.
Otherwise if $k_d\ne 0$, then all elements in $S$ have distinct $d$-th coordinates, and $|S|\leq N_d$. Since $\chi'$ is deterministic, we know that $\chi(S)$ is a summation of $|S|$ i.i.d. Rademacher random variables. Now by the Chernoff bound (e.g. see Theorem A.1.1 and Corollary A.1.2 in \cite{AS}), we have
\[\Pr(|\chi(S)| > \sqrt{6N_d\log(2N_1\cdots N_d)}) \leq 2e^{-\frac{6N_d\log(2N_1\cdots N_d)}{2|S|}} \leq \frac{1}{4}(N_1\cdots N_d)^{-3} < (N_1\cdots N_d)^{-3},\]
where in the last inequality we use that $|S| \leq N_d$. Finally we apply the union bound on all $S$ with $k_d\ne 0$. Clearly there are $N_1\cdots N_d$ ways to pick the first element in the arithmetic progression, and at most $N_1\cdots N_d$ ways to pick the last element, and at most $N_d$ ways to choose $|S|$ (as $1\leq |S| \leq N_d$). Once these are chosen, then clearly $S$ is determined as the common difference in the last coordinate is determined. Hence the total number of distinct $S$ in $\mathcal A_\mathbf{N}$ with $k_d\ne 0$ is at most $(N_1\cdots N_d)^3$. By union bound, we conclude that there exists a choice of $v$ such that $|\chi(S)|\leq \sqrt{6N_d\log(2N_1\cdots N_d)}$ for all $S\in \mathcal{A}_{\mathbf{N}}$ with distinct $d$-th coordinates.
In summary, we conclude that there is a choice of $\chi:[N_1]\times \cdots \times[N_d] \to \{1,-1\}$ so that
\[\max_{S\in \mathcal{A}_\mathbf{N}}|\chi(S)| \leq \max\left(\disc(\mathcal A_{\mathbf{N}'}), \sqrt{6N_d\log(2N_1\cdots N_d)}\right).\]
Note that $\disc(\mathcal{A}_\mathbf{N})$ is defined as the minimum over all $\chi$, so we have the desired inequality.
\end{proof}
\begin{proof}[Proof of the upper bound in Theorem~\ref{thm:rectangular}]
Suppose that $N_1 \geq N_2 \geq \cdots \geq N_d \geq 1 =: N_{d+1}$. Assume that $N_1$ is sufficiently large to avoid triviality.
Let $R_i = \left(\prod_{j=1}^i N_j\right)^{\frac{1}{i+1}}$ for $1\leq i\leq d$. Clearly
\[\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{|I|+1} = \max_{1\leq i\leq d} R_i.\]
Now we take $t$ to be the first index $1\leq i\leq d$ such that $R_i > \frac{N_{i+1}}{(\log(N_1\cdots N_d))^{\frac{1}{2}}}$. By repeatedly applying Lemma~\ref{lem:slice}, for $\mathbf{N}' = (N_1, \dots, N_t)$, we have
\begin{equation}\label{eqn: upper disc}
\disc(\mathcal{A}_\mathbf{N}) \leq \max\left(\disc(\mathcal{A}_{\mathbf{N}'}), 4\sqrt{N_{t+1}\log(N_1\cdots N_d)}\right).
\end{equation}
By our choice of $t$, we have $4\sqrt{N_{t+1}\log(N_1\cdots N_d)} \leq 4\sqrt{R_t}(\log (N_1\cdots N_d))^{\frac{3}{4}}$.
Also we have $N_t \geq R_{t-1}(\log(N_1\cdots N_d))^{\frac{1}{2}}$, so $N_t \geq R_t(\log(N_1\cdots N_d))^{\frac{t}{2(t+1)}} \geq R_t(\log(N_1\cdots N_d))^{\frac{1}{4}}$. Consequently, we may pick $\delta = O\left(\frac{\log (N_1\cdots N_d)}{\log\log(N_1\cdots N_d)}\right)$ so that $N_t^{t+1-\delta} = R_t^{t+1}$. By Theorem~\ref{thm:almost-cubes} we have
\[\disc(\mathcal A_{\mathbf{N}'}) = O_d\left(\sqrt{R_t}\frac{\log (N_1\cdots N_d)}{\log\log (N_1\cdots N_d)}\right).\]
This completes the proof by invoking \eqref{eqn: upper disc}.
\end{proof}
\begin{remark}
The above proof gives that we can take $C_d = 2^{O(d^3)}$ in Theorem~\ref{thm:rectangular}.
\end{remark}
\section{Concluding remarks }\label{sec: conclusion and open problem}
Theorem~\ref{thm:rectangular} determines $\disc(\mathcal A_\mathbf{N})$ up to a constant factor for many $\mathbf{N}$'s. However,
even when $d=2$, there is a regime where the upper and lower bounds are not within a constant factor. As a special case, let $\mathbf{N} = (N, \sqrt{N}(\log N)^k)$ for $k \geq \frac{3}{2}$ and large $N$. Theorem~\ref{thm:rectangular} yields a lower bound of $\Omega(N^{\frac{1}{4}}(\log N)^{\frac{k}{6}})$ and an upper bound of $O(N^{\frac{1}{4}}{(\log N)^{\frac{k}{6}+1}}{(\log\log N)^{-1}})$. If we apply Lemma~\ref{lem:slice}
and the Matou\v{s}ek-Spencer theorem in one dimension \cite{MS}, we get a weaker upper bound of $O(N^{\frac{1}{4}}(\log N)^{\frac{k+1}{2}})$. In some other regimes, such as when $0 < k<\frac{3}{2}$ in the above example, Lemma~\ref{lem:slice} and \cite{MS} gives a better upper bound than Theorem~\ref{thm:rectangular}, yet it is still not within a constant factor from the lower bound.
It is interesting to know if the sub-logarithmic factor in the upper bound of Theorem~\ref{thm:rectangular} can be removed or not. We conjecture that it can be and the lower bound is tight.
\begin{conjecture}
For any integer $d\geq 1$, let $\mathbf{N} = (N_1, N_2, \cdots, N_d)$ where $N_1,\dots, N_d$ are positive integers. Then
\[\disc(\mathcal{A}_\mathbf{N}) = \Theta_d\left(\max_{I\subseteq [d]}\left(\prod_{i\in I}N_i\right)^\frac{1}{2|I|+2}\right).\]
\end{conjecture}
|
2,877,628,090,803 | arxiv | \section{Introduction}
The observation of the Mott-insulator state of a degenerate Bose gas
in an optical lattice \cite{Greiner2002a} has opened an exciting
new avenue for investigating strongly correlated condensed matter
systems. Ensuing experiments (e.g. \cite{Greiner2002b,Mandel2003a,Widera2004a})
have investigated a wide variety of phenomena making optical lattices
one of the leading systems for studying quantum atom optics.
In the initial experimental report, two key pieces of evidence were
given for the Mott insulator state: the loss of phase coherence, and
the appearance of a gap in the excitation spectrum. More recent experimental work has used Bragg spectroscopy \cite{Schori2004a,Stoferle2004a} to study the equilibrium properties of this system. While this type of probing reveals the appearance of a gap in the excitation spectrum, experimental applications of this technique operate well-beyond the linear response regime, making direct comparison with theory difficult. \footnote{The observable for Bragg spectroscopy in optical lattices is the energy transferred to the system. This observable is rather difficult to measure accurately, and to obtain a signal experiments necessarily add a large amount of energy, well-beyond the linear response limit.}
In this paper we propose and theoretically analyse a Raman spectroscopy scheme for probing Mott insulator states in the lattice system. The formalism we present is sufficiently general to include the excitation of atoms to higher vibrational bands where they should be easily discernible from the unscattered atoms in time-of-flight analysis. We apply this formalism to a uniform lattice and show how the correlated nature of the Mott insulator state can give rise to localized states in the excited bands when the lattice is sufficiently deep, and that the resonant frequency of these localized states is sensitive to the local filling factor in the optical lattice.
\section{Formalism} The system of interest, which we will refer
to as system 1, is a degenerate collection of bosonic atoms populating
the lowest vibrational band of an optical lattice well-characterized
by the Bose-Hubbard Hamiltonian
\begin{equation}
\hat{H}_{1}=\epsilon_{0}\sum_{j}\hat{n}_{j}-J\sum_{\langle i,j\rangle}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{U_{11}\alpha_{w}}{2}\sum_{j}\hat{n}_{j}(\hat{n}_{j}-1),\label{eq:HBH}\end{equation}
where $\hat{a}_{j}$ is a bosonic operator that annihilates an atom
from the Wannier state $w_{j}(\mathbf{x})$ centered on lattice site
$j$, with $\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$ the respective
number operator. The quantity $J$, known as the tunneling matrix
element, characterizes the tunneling between lattice sites and is
determined from band structure calculations \cite{Jaksch1998a,Blakie2004a}.
Interactions between particles are described by the matrix element
$\alpha_{w}\equiv\int d^{3}\mathbf{x}\,|w_{j}(\mathbf{x})|^{4}$ and
the coefficient $U_{11}=4\pi a_{11}\hbar^{2}/m,$ where $a_{11}$
is the s-wave scattering length for collisions between atoms in internal
state $1$. We restrict our attention here to the translationally
invariant system (i.e. neglect the influence of external trapping
potentials in addition to the optical lattice), and the constant $\epsilon_{0}$
characterizes the mid-point energy of the ground band.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9cm]{Fig1.eps}
\caption{\label{cap:exptdiag} Schematic diagram of the spectroscopic process. Atoms in the ground band of system 1 are transfered into the initially unoccupied system 2 by a two-photon Raman process through a virtual level $|v\rangle$. A residual energy difference $\hbar\omega_{0}$ and momentum kick $\hbar\mathbf{q}$ (not shown) are transfered to the atoms by each scattering event.
}
\end{center}
\end{figure}
In this work, we consider a scheme for probing the properties of system
1, by an internal state changing Raman transition, of the form used
in Ref. \cite{Hagley1999a} to output couple an atom laser. A theory for this type of process in an optical lattices has also been proposed by Konabe \emph{et al.} \cite{Konabe2004a}\footnote{We note that the primary difference of our treatment to that of Konabe \emph{et al.}, is that we include interactions between scattered and unscattered atoms, which are essential for the main results we present here.}.
This is in contrast to a recent theory \cite{Menotti2003a,Roth2004a,Oosten2005a,Rey2005a,Batrouni2005a,Pupillo2006a} and experiments \cite{Schori2004a,Stoferle2004a} that have considered internal state preserving transitions to perform spectroscopy -- known as Bragg scpectroscopy \cite{Blakie2002a}. A schematic diagram of this coupling is shown in Fig. \ref{cap:exptdiag}. Following the formalism in Ref. \cite{blakie2003a} this type
of coupling is described by an interaction term of the form\begin{equation}
\hat{V}=\frac{V_{p}}{2}\theta(t)\sum_{j}\int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})\hat{a}_{j}w_{j}(\mathbf{x})e^{i(\mathbf{q}\cdot\mathbf{x}-\omega_{0}t)}+{\rm H.c},\label{eq:Hpert}\end{equation}
where $\hat{\psi}_{2}^{\dagger}$ is the field operator for bosonic
atoms in the second internal state, and the quantities $\hbar\mathbf{q}$
and $\hbar\omega_{0}$, specify the momentum and energy transfer of
the Raman process respectively. For simplicity we define $\hbar\omega_{0}$
to be the excess energy transferred over the internal state energy
difference. We take the amplitude of the Raman coupling to be of strength
$V_{p}$ beginning at $t=0,$ and of duration $T_{p}$. We will assume
that the internal states 1 and 2 are different hyperfine ground states
for which we can neglect any collisional spin evolution (see \cite{Widera2005a,Hall1998a}).
To arrange such a Raman probe experimentally, a pair of light fields (in addition to those used to create the lattice) with appropriately chosen polarizations to couple the states of interest will need to be applied to the atoms in the lattice. While such a probe could be focused down to address a few sites in a lattice, our theoretical development here is for the case where the Raman fields uniformly illuminate the system and have no spacial selectivity.
Our interest lies in the linear response regime, where only a small
portion of the atoms are scattered into internal state 2. The evolution
of these atoms in internal state 2, which we will refer to as system
2, is described by the Hamiltonian \begin{eqnarray}
\hat{H}_{2} & = & \int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})(H_{{\rm sp}}+U_{12}\sum_{j}\hat{n}_{j}|w_{j}(\mathbf{x})|^{2})\hat{\psi}_{2}(\mathbf{x}),\label{eq:H2}\end{eqnarray}
where
\begin{equation}H_{{\rm sp}}=p^{2}/2m+V_{{\rm ext}}(\mathbf{x}),\end{equation}
is the single
particle Hamiltonian with $V_{{\rm ext}}(\mathbf{x})$ the external
potential (taken to be the same lattice potential experienced by atoms
in system 1), and $U_{12}=4\pi a_{12}\hbar^{2}/m$ characterizes the
interactions between particles in systems 1 and 2. Within the validity
regime of linear response it is permissible to neglect interactions
between atoms in system 2, as their density will remain low.
System 2 is initially in the vacuum state, and the spectroscopic signal
to be measured is the number of atoms in system 2 after the Raman
pulse. This is a convenient experimental observable as detection techniques
can readily distinguish between atoms in different internal states
\cite{Hall1998a}. Additionally when the coupling produces atoms in
an excited vibrational band, the excited atoms can be differentiated
by their behavior upon expansion from the lattice \cite{Greiner2001a,Denschlag2002a}. These properties of the observable should enable the system response to be measured for small amounts of Raman excitation, allowing the system to be probed in the linear response regime.
The expression for the number of excited atoms, derived using linear
response theory, is\begin{equation}
\left\langle \hat{N}_{2}\right\rangle =\frac{\pi V_{p}^{2}T_{p}}{2\hbar^{2}}\int_{-\infty}^{+\infty}d\omega R(\mathbf{q},\omega)\,\frac{2\sin^{2}([\omega-\omega_{0}]T_{p}/2)}{\pi T_{p}(\omega-\omega_{0})^{2}},\label{eq:LinRespforN2}\end{equation}
where we have introduced the correlation function
\begin{eqnarray} R(\mathbf{q},\omega)=\frac{1}{2\pi}\int dt\int d^3\mathbf{x}\int d^3\mathbf{x}^\prime\,e^{i\mathbf{q}\cdot(\mathbf{x}^\prime-\mathbf{x})+i\omega t}\label{eq:RQWGen}\\ \times\sum_{ij}w_i^*(\mathbf{x})w_j(\mathbf{x}^\prime)\left\langle\hat{a}_i^\dagger(t)\hat{\psi}_2(\mathbf{x},t)\hat{\psi}_2^\dagger(\mathbf{x}^\prime,0)\hat{a}_j(0)\right\rangle_{\rm{Eq}}. \nonumber\end{eqnarray}In
this expression the time dependence of the operators is in an interaction
picture with respect to unperturbed Hamiltonians $\hat{H}_{1}+\hat{H}_{2}$,
and the expectation is taken on the initial equilibrium ensemble.
We will now show how to evaluate the correlation function $R(\mathbf{q},\omega)$,
and how it relates to the properties of the atoms in system 1. We
take the initial density matrix to be $\rho=\rho_{1}\otimes\rho_{2}^{{\rm vac}}$,
where $\rho_{1}$ is the degenerate lattice state we are interested
in probing and $\rho_{2}^{{\rm vac}}$ is the initial vacuum state
for system 2. We can then show that without further approximation,
$R(\mathbf{q},\omega)$ can be written in the form
\begin{eqnarray}
R(\mathbf{q},\omega)&=&\frac{1}{2\pi}\int dt\int d^{3}\mathbf{x}\int d^{3}\mathbf{x}^{\prime}e^{i\mathbf{q}\cdot(\mathbf{x}^{\prime}-\mathbf{x})+i\omega t}\label{eq:RQ}\\&& \sum_{Q\, ij}\left\langle Q\left|\hat{a}_{i}^{\dagger}(t)\hat{a}_{j}(0)\rho_{1}\right|Q\right\rangle \left\langle {\rm 0}_{2}\left|\hat{\psi}_{j}^{Q}(\mathbf{x},t)\hat{\psi}_{2}^{\dagger}(\mathbf{x}^{\prime},0)\right|0_{2}\right\rangle w_{i}^{*}(\mathbf{x})w_{j}(\mathbf{x}^{\prime}),\nonumber\end{eqnarray}
where the variable $Q$ represents a trace carried
out over the number state basis for the operators $\{\hat{a}_{j}\}$,
i.e. $Q\leftrightarrow|\ldots,n_{l-1}^{Q},n_{l}^{Q},\ldots\rangle$,
and $|0_{2}\rangle$ represents the vacuum state of system 2. The
operator $\hat{\psi}_{j}^{Q}(\mathbf{x},t)$ is $\hat{\psi}_{2}(\mathbf{x})$
evaluated in an interaction picture with respect to the Hamiltonian\begin{equation}
\hat{H}_{j}^{Q}\equiv\int d^{3}\mathbf{x}\hat{\psi}_{2}^{\dagger}(\mathbf{x})(H_{{\rm sp}}+U_{12}\sum_{l}(n_{l}^{Q}-\delta_{lj})|w_{l}(\mathbf{x})|^{2})\hat{\psi}_{2}(\mathbf{x}).\label{eq:HjQ}\end{equation}
We note that in deriving Eq. (\ref{eq:RQ}) we have made use of the
result \begin{equation}
e^{i\hat{H}_{2}t/\hbar}\hat{\psi}_{2}(\mathbf{x})e^{-i\hat{H}_{2}t/\hbar}\hat{a}_{j}(0)|Q\rangle=\hat{a}_{j}(0)|Q\rangle\hat{\psi}_{j}^{Q}(\mathbf{x},t),\label{eq:HiQresult}\end{equation}
where the $\delta_{lj}$ term in Eq. (\ref{eq:HjQ}) arises from commuting
$\hat{a}_{j}$ with $\exp(i\hat{H}_{2}t/\hbar)$, and we have made
the replacement $\hat{n}_{l}\to n_{l}^{Q}$ as $|Q\rangle$ are number
states.
We refer to $\hat{H}_{j}^{Q}$ as the $Q$-defect Hamiltonian for
system 2, as it arises from the removal of a system 1 atom from site
$j$ of the number state $Q$. As the system 1 atoms form an effective
potential for those in system 2, the removal of an atom at site $j$
due to the Raman excitation creates a potential hole (i.e. the $n_{l}^{Q}-\delta_{lj}$
term in Eq. (\ref{eq:HjQ})). This defect plays a key role in the
response spectrum as we will demonstrate later. Since $\hat{H}_{j}^{Q}$
is a quadratic Hamiltonian it can be diagonalized by numerical methods
to obtain its eigenvectors $\{\phi_{jm}^{Q}(\mathbf{x})\}$ and eigenvalues
$\{\hbar\omega_{jm}^{Q}\}$, i.e. $\hbar\omega_{jm}^{Q}\phi_{jm}^{Q}(\mathbf{x})=\hat{H}_{jQ}\phi_{jm}^{Q}(\mathbf{x}),$
where $m$ is the quantum number specifying the state and $j$ labels
the defect location. We note that in the limit $U_{12}\to0$, the
$\phi_{jm}^{Q}$ reduce to the Bloch states of $H_{{\rm sp}}$
and $m$ becomes the quasimomentum and band index. We obtain $\hat{\psi}_{j}^{Q}(\mathbf{x},t)=\sum_{m}\phi_{jm}^{Q}(\mathbf{x})\hat{b}_{jm}^{Q}e^{-i\omega_{jm}^{Q}t},$
where \textbf{$\hat{b}_{jm}^{Q}$} is a bosonic annihilation operator
and arrive at the expression
\begin{equation}
R(\mathbf{q},\omega)=\frac{1}{2\pi}\int dt\, e^{i\omega t}\sum_{Q}\sum_{ijm}c_{ij}^{Q}(t)A_{mjj}^{Q}A_{mij}^{Q*}e^{-i\omega_{jm}^{Q}t},\label{eq:RQ2}\end{equation}
where we have defined
\begin{eqnarray}A_{mij}^{Q}&\equiv&\int d^{3}\mathbf{x}\,\phi_{jm}^{Q*}(\mathbf{x})e^{i\mathbf{q}\cdot\mathbf{x}}w_{i}(\mathbf{x}),\\
c_{ij}^{Q}(t)&\equiv&\langle Q|\hat{a}_{i}^{\dagger}(t)\hat{a}_{j}(0)\rho_{1}|Q\rangle.\end{eqnarray}
Eqs. (\ref{eq:RQ}) and (\ref{eq:RQ2}) represent the key results
of this work, and we now briefly comment on the physical process they
describe. Fundamentally, $R(\mathbf{q},\omega)$ characterizes the
excitation spectrum of atoms from system 1 into system 2. In the context
of ultra-cold gases, a result similar to our starting point {[}Eq.
(\ref{eq:LinRespforN2}){]} has been given by Luxat \emph{et al.}
\cite{Luxat2002a} for the case of a harmonically trapped Bose gas
(also see \cite{Choi2000a,Girardeau2001a}). However their treatment
neglects interactions between atoms in different hyperfine states
and assumes the single particle correlation functions for the atoms
in systems 1 and 2 are independent. The extension of this theory to the optical lattice has been provided by Konabe \emph{et al.} \cite{Konabe2004a}.
For the current experiments with Rubidium atoms in a deep
optical lattice these approximations are not tenable. In particular,
as we noted previously, the lattice site from which the atom is excited
acts as the localizing defect and accounting for correlations between
the systems (as we have done in Eq. (\ref{eq:RQ})) is essential.
We note that our formalism [i.e. Eqs. (\ref{eq:RQ}) and (\ref{eq:RQ2})] is quite general. As long as the dominant number states of the many-body state are known, e.g. through exact diagonalization or Matrix Product Decomposition techniques (e.g. esee \cite{Clark2004a}), then the Raman response can be determined. In the following sections we consider the application of our formalism for two special cases. The results we present in the next section are calculated for 1D systems for numerical convenience, though our interest is in the regime where the scattering between particles is three-dimensional and well-described by a contact interaction.
\section{Single Site Limit}
The limiting case of
a single tightly confining harmonic well of frequency $\omega_{{\rm ho}}$
is a useful approximation to a deep lattice in the regime where tunneling
between sites can be neglected. This limit shows the main physical
features of Raman spectroscopy and provides a useful approximation
to the full solution. Assuming that interaction shifts are small compared
to the oscillator energy $\hbar\omega_{{\rm ho}}$, we may approximate
the modes of the system as being harmonic oscillator eigenstates $\{\varphi_{m}(\mathbf{x})\}$,
with respective energies $\{\epsilon_{m}=\hbar\omega_{{\rm ho}}(m+1/2)\}$.
For the single site case our formalism maps according to
\begin{eqnarray}
w_{j}(\mathbf{x})&\to&\varphi_{0}(\mathbf{x}),\\
\phi_{jm}^{Q}(\mathbf{x})&\to&\varphi_{m}(\mathbf{x}),\\
A_{mij}^{Q}&\to& A_{m}\equiv\int d^{3}\mathbf{x}\,\varphi_{m}^{*}(\mathbf{x})e^{i\mathbf{q}\cdot\mathbf{x}}\varphi_{0}(\mathbf{x}),\\
\hbar\omega_{jm}^{Q}\to\hbar\omega_{m}^{n}&=&\epsilon_{m}+U_{12}\alpha_{0m}(n-1)\end{eqnarray}
where $\alpha_{0m}\equiv\int d^{3}\mathbf{x}\,|\varphi_{0}(\mathbf{x})|^{2}|\varphi_{m}(\mathbf{x})|^{2}$.
Since the Hamiltonian for system 1 reduces to $\hat{H}_{1}\to\epsilon_{0}\hat{n}+U_{11}\alpha_{w}\hat{n}(\hat{n}-1)/2$
in this limit, we have replaced $Q$ by $n$ (i.e. the single site many-body number states are just single mode number states $|n\rangle$) and evaluated the correlation
function as
\begin{equation}c_{ij}^{Q}(t)\to c_{ij}^{n}(t)=n\exp(i[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]t/\hbar).\end{equation}
Using Eq. (\ref{eq:RQ2}) with $\rho_{1}=\sum_{nn^{\prime}}|n\rangle\rho_{nn^{\prime}}\langle n^{\prime}|$,
we obtain\begin{eqnarray}
R(\mathbf{q},\omega)&=&\sum_{n}\rho_{nn}\, n|A_{m}|^{2}\delta(\omega_{{\rm res}}(n)-\omega),\label{eq:RqwSS} \\
\hbar\omega_{{\rm res}}(n)&=&\epsilon_{m}-\epsilon_{0}+[U_{12}\alpha_{0m}-U_{11}\alpha_{w}](n-1).\end{eqnarray}
We have also assumed that the probe only couples to a particular excited
state $m$, with the other states sufficiently far detuned that their
contribution is negligible.
Eq. (\ref{eq:RqwSS}) shows that the response of the system is proportional
to $\rho_{nn}$, and most notably, if $[U_{12}\alpha_{0m}-U_{11}\alpha_{w}]\ne0$,
then the response frequency is linearly dependent on the value of
$n$ (i.e. the number of atoms at the site). In this regime the Raman
spectrum reveals the number distribution at the site. For the case
of $^{87}$Rb (the atom of primary interest in bosonic optical lattice
experiments) the interactions between the relevant hyperfine states
are approximately degenerate (i.e. $U_{11}\approx U_{12}$) , so that
the difference $\alpha_{0m}-\alpha_{w}$ will be the primary factor
in determining the magnitude of the number dependent shift. As an
immediate consequence we note that for the case $U_{11}=U_{12}$ and
a Raman pulse coupling to ground vibrational state of system 2, then
the spectrum will be independent of $n$ (i.e. for $m=0$ we have
$\alpha_{0m}=\alpha_{w}$). Therefore to obtain a number dependent
spectral response with $^{87}$Rb will require scattering into excited
vibrational states of system 2.
\section{Uniform Mott Insulator} We now consider
the more general case of a translationally invariant lattice with
$N_{s}$ sites and periodic boundary conditions. We assume that the number of atoms in the system is commensurate with the number of lattice sites and
system is deep in the Mott insulating regime, where $U_{11}\alpha_{w}\gg J$. In this regime
the many-body ground state is well approximated as $|Q_{n}\rangle=|\ldots,n,n,\ldots\rangle$
(i.e a definite number of atoms at each site). In calculating the
response of the system to the Raman probe according to Eq. (\ref{eq:RQ}),
the summation over $Q$ reduces to this single state. The next order
correction to the translationally Mott state is particle hole states
\cite{Rey2005a}, which contribute to the ground state with an amplitude
$J/U_{11}\alpha_{w}\ll1$, and can be neglected.
To obtain an approximation for the temporal correlation function of
system 1 in state $|Q_{n}\rangle$ we ignore the tunneling term in
$\hat{H}_{1}$. This approximation amounts to neglecting particle
tunneling between sites in system 1 over the time scale of the Raman
probe, and should be a good approximation in deep lattices. In this
limit the correlation function is
\begin{equation}c_{ij}^{Q_{n}}(t)=n\,\exp(i[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]t/\hbar)\delta_{ij},
\end{equation}
and making use of the translational invariance in evaluating Eq. (\ref{eq:RQ2})
we obtain
\begin{eqnarray}
R(\mathbf{q},\omega)&=&\sum_{m}n\, N_{s}|A_{mll}^{Q_{n}}|^{2}\delta(\omega-\omega_{{\rm res}}^{Q_{n}m}),\label{eq:RqLatt}\\
\omega_{{\rm res}}^{Q_{n}m}&=&\omega_{lm}^{Q_{n}}-[\epsilon_{0}+U_{11}\alpha_{w}(n-1)]/\hbar.\end{eqnarray}
Note that due to translational invariance the precise value of $l$
used in Eq. (\ref{eq:RqLatt}) is unimportant. %
\begin{figure}[tb]
\begin{center}
\includegraphics[width=12cm]{Fig2.eps}
\caption{\label{cap:SpecComp} The percentage of atoms excited after Raman
excitation in a 1D lattice of depth (a) $V_{D}=10E_{R}$ (b) $V_{D}=30E_{R}$.
In each plot the spectrum of Mott insulator states with filling factors
of $n=1$ (dark line), $n=2$ (medium line) and $n=3$ (light line)
are shown. The respective single site results for $m=1$ are given
as dashed lines. Parameters are $q=\pi/\lambda$, $T_{p}=1.5$ms,
$V_{p}=0.05E_{R}$, $\lambda=850$nm, $N_{s}=51$, transverse confinement
taken to be harmonic with $f_{\perp}\approx37.6$kHz, and $a_{11}=a_{12}=5.29$nm. }
\end{center}
\end{figure}
We now calculate the response spectrum $\left\langle \hat{N}_{2}\right\rangle $
using Eqs. (\ref{eq:LinRespforN2}) and (\ref{eq:RqLatt}) for a translationally
invariant 1D lattice. We take the lattice potential to be $V_{{\rm ext}}(x)=V_{D}\cos^{2}(x/\lambda),$
arising from counter-propagating lasers fields of wavelength $\lambda$,
and we specify the lattice depth $V_{D}$ in units of $E_{R}=h^{2}/2m\lambda^{2}$.
The calculation is performed for typical $^{87}$Rb parameters (e.g.
see Ref. \cite{Greiner2002a}) to demonstrate the practicality of
our probing scheme. We will not consider the influence of $\mathbf{q}$
on the spectrum in this paper%
\footnote{Generally large $q$ favors coupling to higher bands, though certain
choices of $\mathbf{q}$ can suppress coupling efficiency, e.g for
$\mathbf{q}=\mathbf{0}$ the response from the first band is zero
by symmetry.%
}, and for simplicity we have taken $\mathbf{q}$ as being identical
to a reciprocal lattice for our calculations.
The results we have obtained are shown in Fig. \ref{cap:SpecComp}.
For the case of $V_{D}=10E_{R}$ (Fig. \ref{cap:SpecComp}(a)) the
Raman response is shown over a frequency range that includes resonant
coupling to the first two excited bands. The superimposed graphs show
the spectra for Mott insulating states of various filling factors
and clearly exhibit frequency shifts proportional to $n$. Additionally,
we notice that in the first excited band ($12$kHz - $18$kHz), the
dominant response peak occurs at the low end of the spectral feature
(e.g. the peak at $\sim15.5$kHz for $n=1$), adjacent to a broad
base. This is feature is also seen in the 2nd excited band ($24$kHz
- $36$kHz), however the peaked feature is much less dominant relative
to the broad base. The peak originates from an excited band state
that is partially localized above the defect site. This localization
leads to a strong coupling matrix element $A_{mll}^{Q_{n}}$, and
as this state resides at the defect, its energy is lower than the
other states in the excited band. We refer to this state as the defect
state, which is clearly identified as the most strongly excited feature
at the bottom each spectral band. Since the localization is not perfect, the
other states in the excited band have appreciable amplitude at the
defect site. These states give rise to the broad though more weakly
excited, band of states above the resonant peak. In the 2nd excited
band, the same features are seen, however as the effective tunneling
in this band is much higher, and the defect state is much less localized.
In Fig. \ref{cap:SpecComp}(b) the response spectrum is shown for
a lattice of depth $V_{D}=30E_{R}$. At this depth the response from
the first excited band states have shifted up to $30$kHz, and the
second excited band has moved out of the frequency range considered.
In contrast to the spectrum in Fig. \ref{cap:SpecComp}(a), we only
notice the resonant peak due to the defect state, without a discernible
broad base of band states. This arises because at this depth the tunneling
rate in the excited band is sufficiently small that the defect state
a becomes completely localized. The other states in the excited band
are necessarily orthogonal to the defect state and so have vanishing
coupling matrix elements $A_{mll}^{Q_{0}}$.
For comparison in Figs. \ref{cap:SpecComp}(a) and (b) we also show
the single site predictions for the spectrum calculated using expression
(\ref{eq:RqwSS}) with $\omega_{{\rm ho}}$ chosen to match the effective
trap frequency at the lattice site minima. The frequency location
of the response spectra compares badly with the full lattice solution,
arising primarily from the inadequacy of the harmonic approximation
for accurately predicting band structure. The location of the spectra
can be easily corrected for by calculating the term $\epsilon_{m}-\epsilon_{0}$
in the expression for $\omega_{{\rm res}}(n)$ using the non-interacting
band structure result. However, the single site approximation captures
many of the salient features of the full lattice solution, such as
the magnitude of the $n$-dependent shift in the the response spectrum.
In the strongly-localized defect limit (Fig. \ref{cap:SpecComp}(b)),
the single site approximation quantitatively predicts the response
amplitude as the role of the band states can be neglected.%
\begin{figure}[tb]
\begin{center}
\includegraphics[width=12cm]{Fig3.eps}
\caption{\label{cap:SpecDefect} (a) Spectrum of excited states versus lattice
depth. (b) The bandwidth of the first excited band (solid) and the
on-site interaction (dashed) versus lattice depth. Other parameters
as in Fig. \ref{cap:SpecComp}.}
\end{center}
\end{figure}
To quantify the emergence of the defect state we examine the spectrum
of $\omega_{{\rm res}}^{Q_{n}m}$ as a function of $V_{D}$ in Fig.
\ref{cap:SpecDefect}(a). For $V_{D}\gtrsim13.5E_{R}$ a single defect
state is observed to drop below the first excited band. The condition
for the emergence of a strongly localized defect state is that the
energy reduction from localization above the defect in system 1 is
large compared to the effective inter-site tunneling in the excited
band. To verify this criterion we compare the excited bandwidth (characterizing
the excited band tunneling rate) and energy reduction at the defect,
approximated by $U_{11}\alpha_{w}$. These two energy scales are seen
to cross at $V_{D}\approx13.5E_{R}$. Finally, we note the validity
condition for linear response treatment of Raman spectroscopy. This
requires the number of atoms excited to remain small compared to the
total number of atoms in the system. In terms of the Raman parameters
this condition is given as $nV_{p}^{2}T_{p}^{2}/4\hbar^{2}\ll1$,
where $n$ is the average number of atoms per site.
We briefly comment on the relationship of the 1D calculations presented here to an equivalent system in a 3D lattice. The primary difference in applying our formalism to a fully three-dimensional system is that Eq. (\ref{eq:HjQ}) will need to be solved in 3D. For the case of coupling to excited bands, the tunneling between lattice sites can occur in all directions, but is dominated along the direction in which the vibrational excitation has occurred. This direction will be parallel to the direction of the momentum transfer in the Raman coupling, which we take to be parallel to a lattice vector. The tunneling in the orthogonal directions will be given by the ground band tunneling rate which is typically much smaller. This suggests that the additional shifts in a fully 3D lattice will be of order the ground state tunneling rate and will thus contribute small corrections to the results presented here. A full study in 3D will be the subject of a future investigation.
\section{Conclusions and outlook}
In this work we have proposed and analysed a Raman spectroscopy technique for probing the properties of quantum degenerate bosons in the ground band of an optical lattice.
We have observed that for sufficiently deep lattices, localized states in higher vibrational bands play an important role in the system response, and shifts in resonant frequency of excitation are sensitive to the number of particles per site.
While our main study has considered the case of a perfect Mott insulator
in a translationally invariant lattice, our results suggest that in
the limit of strongly localized defect states the response of the
system is well-described by the single site result, and thus only
depends on the local number distribution at each lattice site. The
Raman spectrum may therefore be a useful method for measuring the relative portion
of system 1 at sites with filling factor $n$ atoms.
We speculate that in the strongly localized limit, the homogeneity of the lattice is rather unimportant, with the existence of the defect states arises from the large difference in the effective potential energy between the site where the particle is excited and the neighbouring site. This suggests that many of the predictions we have made here, and in particular the results of the 1-site model, should qualitatively apply to inhomogeneous lattices, such as the combined harmonic and optical lattice potentials made in experiments. In this case, as well as in the superfluid limit (where significant number fluctuations exist), significant corrections may arise from resonances between neighbouring sites that would allow the defect to be localized over several sites. However, it seems reasonable to expect that these resonances would contribute to the broad background in the Raman spectrum, and the sharp features from Mott-insulating regions (if present) would be clearly visible. Characterizing the role of superfluid fluctuations and the external confining potential will be the subject of future work.
\ack PBB would like to acknowledge valuable
discussions with Trey Porto, Crispin Gardiner, and thanks the University
of Otago for supporting this research.
\section*{References}
\bibliographystyle{unsrt}
|
2,877,628,090,804 | arxiv | \section{Introduction}
The transfer of quantum information between harmonic oscillators coupled by linear interactions has a variety of applications, including communication in linear networks~\cite{Chudzicki10}, cooling of mechanical resonators~\cite{Wang11, Machnes12}, and frequency conversion~\cite{Tian10, Stannigel10, Regal11, Safavi11b, Tian12, Wang12}. For cooling mechanical oscillators, transferring the entropy of the mechanical oscillator to a superconducting or optical oscillator is the most effective method known to-date~\cite{Wilson-Rae07, Marquardt07, Tian09, Schliesser08, Teufel11b}. The faster the transfer can be performed the lower the achievable temperature, or equivalently the higher the resulting purity of the prepared state~\cite{Wang11, Machnes12}. Frequency conversion is achieved by transferring the state of an oscillator at one frequency to an oscillator at a different frequency. While here we focus on communication in linear networks, our results are applicable to all the above applications. Speed is important for obvious reasons in communication and computation, and it is of additional importance in quantum technologies because of the ever-present effects of environmental noise that degrade quantum states over time. Here we are concerned with finding time-dependent control protocols that will transfer information at the maximum speed, a problem more generally referred to as ``time-optimal'' control~\cite{Anandan90, Giovannetti03, Margolus98, delCampo13, Carlini06, Carlini07, Hegerfeldt13}.
Two oscillators A and B that are coupled by a linear interaction are described by the Hamiltonian
\begin{equation}
H = H_0 + V_{\mbs{I}},
\end{equation}
with
\begin{eqnarray}
H_0 & = & \hbar \omega_a a^\dagger a + \hbar \omega_b b^\dagger b , \\
V_{\mbs{I}} & = & \hbar g x_a x_b ,
\end{eqnarray}
where $x_a = a + a^\dagger$, $x_b = b + b^\dagger$, $w_a$ and $w_b$ are the respective frequencies of A and B, and $g$ is the linear coupling rate~\footnote{There is no need to choose the more general form of the linear interaction in which $x_a = e^{- i \theta} a + e^{i \theta}a^\dagger$, $x_b = e^{- i \phi} b + e^{i \phi}b^\dagger$, since varying $\theta$ and $\phi$ merely shifts the relative phases of the oscillators.}. Given this interaction it is not obvious how to engineer a unitary transformation that will transfer an arbitrary state of A to B. Note that for the purposes of information transfer, an operation transfers an arbitrary state $|\psi\rangle$ of A to B if for every $|\psi\rangle$ the oscillator B finishes in the state $U |\psi\rangle$ where $U$ is unitary that does not depend on $|\psi\rangle$. If A and B form a closed system, then this unitary must swap their respective states (up to the local evolution of either oscillator) because unitary operations are reversible; B's state has to go somewhere when replaced by A's.
A swap between A and B can be obtained in an especially simple way if the coupling rate $g$ is much smaller than the frequencies $\omega_a$ and $\omega_b$. To do this one first modulates the coupling rate at the difference frequency $\Delta = |\omega_a - \omega_b|$ so that the interaction Hamiltonian becomes $V_I = g\cos(\Delta t) x_a x_b$. Moving into the interaction picture with respect to $H_0$ and making the rotating-wave approximation the Hamiltonian becomes \begin{equation}
H_{\mbs{I}} = \hbar g (b^\dagger a + a^\dagger b) ,
\end{equation}
where the subscript denotes the interaction picture. Remarkably this interaction swaps the states of the resonators in a time $\tau_{\mbs{RWA}} = \pi/g$. The speed of this method of state-swapping is restricted, however, by the requirement that $g \ll \mbox{min}(\omega_a,\omega_b)$, and thus $\tau_{\mbs{RWA}} \gg T_{\mbs{slow}}$ where $T_{\mbs{slow}}$ is the period of the slower of the two oscillators.
In 2011 Wang \textit{et al.}~\cite{Wang11} and Machnes \textit{et al.}~\cite{Machnes12} showed that a swap between two linear oscillators could be achieved within a single oscillation of the slower oscillator by using a time-dependent modulation of the coupling constant $g$. The technique of engineering unitary operations by changing the Hamiltonian with time is an important one within the toolset of \textit{quantum control}~\cite{Brif10, Schulte05}, and a given prescription for varying the Hamiltonian is called a \textit{control protocol}. While this technique is powerful, finding control protocols to perform a given task is a highly complex problem, and one for which numerical search methods are often essential~\cite{Kosloff89, Schulte05, Grace07, Haidong09, Doria11, Nimbalkar12, Ashhab12, Huang14}.
In the case of two coupled oscillators the unitary operations that can be realized by varying the coupling rate $g$ are those within the algebra generated by the three operators $a^\dagger a$, $b^\dagger b$, and $x_a x_b$, as these are the three operators that are effectively combined with different weights by the variation. Since the Hamiltonian of the slower oscillator is essential for generating a swap, we can expect that its frequency will limit the swapping rate --- there must be enough time for this Hamiltonian to make a non-trivial contribution to the dynamics. By using a numerical search Wang \textit{et al.}~\cite{Wang11} found a control protocol that performed a swap in a little over half the period $T_{\mbs{slow}}$. By starting the numerical search with an approximate analytical protocol Machnes \textit{et al.}~\cite{Machnes12} were able to obtain a swap in a little over a quarter of $T_{\mbs{slow}}$. It appeared unlikely from these results that significantly shorter swap times were possible.
Here we show that, while the frequency of the slow oscillator does ultimately limit the speed of a swap operation, high-fidelity swaps can be performed in times significantly shorter that those previously known. Finding these fast protocols was made possible by a ``path-tracing'' technique that we describe below. While path tracing itself is not new --- it was introduced by Moore-Tibbets~\textit{et al.}~\cite{Moore12b} as a fast method to find protocols when one wishes to scan across protocol durations --- its ability to readily solve problems that are virtually impossible otherwise has not been previously demonstrated. An optimization problem becomes hard when local minima and/or saddle points become so dense that gradient search methods get trapped at these points with high probability. In finding previously undiscovered state-swapping protocols for oscillators we demonstrate two key properties of quantum control: i) quantum control problems that are easy for long protocol times~\cite{Wu08} can become very hard as the protocol duration is reduced, with the result that time-optimal quantum control problems can be similarly hard; ii) path tracing is able to solve at least some of these hard problems.
\section{The Method of Path Tracing}
\label{pathtrace}
Before we describe path tracing we introduce some terms and definitions. A numerical search method is a procedure for finding a local minimum of a function. If the local minimum found is also a global minimum then the search has solved the minimization problem. To find a control protocol that achieves a specific unitary transformation $V$ we use a search method to minimize a quantity, $\varepsilon$, that measures the difference between $V$ and the unitary generated by the protocol. For example, if the protocol generates the unitary $U(T)$ over a time $T$, then
\begin{equation}
\varepsilon = 1 - \mbox{Tr}[V^\dagger U(T)]
\end{equation}
is a good measure of the difference between $U(T)$ and $V$. Because the search method must minimize $\epsilon$ it cannot simultaneously minimize the time that the protocol takes, and this is an essential difficulty in time-optimal control. Let us define a ``good protocol'' as one with an error $\epsilon$ that is below some threshold $\varepsilon_{\mbs{t}} \ll 1$. The standard approach to finding a protocol with minimum time is to perform a number of independent searches, where each search uses a different fixed time $T$. In this way one obtains protocols for a range of durations. It is then hoped that of the good protocols so obtained, the one with the smallest value of $T$ is near-optimal. For problems for which this procedure works well, one finds that above a critical time $T_{\mbs{min}}$ the resulting error $\varepsilon$ is zero (to within machine precision), and below this time the error steadily increases away from zero. The fact that the behavior of $\varepsilon$ is a continuous function of $T$ provides confidence that $T_{\mbs{min}}$ is the minimum time for which perfect protocols can be obtained. For the problems considered here we do not find this behavior; instead there is a critical time $T_{\mbs{cr}}$ at which the error obtained by the search jumps abruptly from zero to a large value. This behavior is a clear sign that the search is being trapped in local minima below $T_{\mbs{cr}}$.
Even when the above procedure works well, it is certainly cumbersome. To increase the efficiency of the process Moore-Tibbets~\textit{et al.}~\cite{Moore12b} introduced the following procedure. i) Start with a duration $T_0$ that is expected to be larger than the minimum time $T_{\mbs{min}}$ and obtain a protocol $p(T_0)$. ii) Reduce the duration to $T_1 = T_0 - \Delta T$ and perform a new search, but this time starting the search at the protocol $p(T_0)$ found in i), where this protocol is now allowed to run only for the duration $T_1$. In particular, we simply scale the protocol $p(T_0)$ along the time axis so that it takes time $T_1$, producing a slightly modified unitary transformation. iii) Repeat this procedure, on each iteration reducing the time by $\Delta T$, and starting the search for this new duration with the protocol found in the previous step. In this way, each time we search for a new protocol we can expect to be starting the search at a protocol that is close to the one we seek, thus reducing the search time. We refer to this procedure as ``path tracing'' because we expect the sequence of protocols obtained to trace a continuous path in protocol space. Moore-Tibbets~\textit{et al.} applied this method to problems for which the standard method works well. They found that their path-tracing method was indeed significantly faster than the standard method, successfully tracing a path of minimum error as a function of duration, $\varepsilon_{\mbs{min}}(T)$. The curve given by $\varepsilon_{\mbs{min}}(T)$ is called the fidelity-time tradeoff frontier.
We write the Hamiltonian for a control problem in the form
\begin{equation}
H(t) = H_0 + \sum_{j=1}^M \lambda_j(t) H_j ,
\end{equation}
in which $H_0$ and $\{H_j\}$ are fixed and $\{\lambda_j(t)\}$ is the set of parameters that are available to be varied as functions of time. These functions are referred to as the control functions, and the rate at which they must be changed with time in order to implement a given control protocol is an important practical consideration. Because of this, in searching for good control protocols we wish to bound this rate. We must also discretize the control functions so that we have a finite set of parameters over which to minimize $\varepsilon$. If we discretize each of the control functions using $N$ parameters, then we may write
\begin{equation}
\lambda_j(t) = \sum_k \lambda_{jk}f_{jk}(t), \;\;\;\; k = 1,\ldots, N ,
\end{equation}
in which $f_{jk}(t)$ are fixed functions of time, and $\{ \lambda_{jk} \}$ is the set of $MN$ parameters over which to optimize. A natural way to discretize the control functions when limiting their rate of change (bandwidth) is to represent each by a truncated Fourier series. Here we instead use a simpler discretization which is effectively equivalent. We make the control functions piecewise-constant on $N$ time segments each of length $T/N$. This choice is ideal for explorations because it allows us to solve for the evolution using matrix exponentiation, which provides a robust solution without any time-stepping error, and is relatively fast for dimensions that are not too large. While the discontinuities in the control functions mean that technically they have infinite bandwidth, experience shows that the existence of a piecewise-constant protocol implies the existence of one or more continuous protocols that have the same error and the same number of degrees of freedom $\lambda_{jk}$ (see, e.g.~\cite{Wang11}). If we represent the control functions using a Fourier series, then the number of degrees of freedom is $MN$, where in this case $N$ is the number of terms in the Fourier series. The minimum bandwidth is then approximately $N/(2T)$. Finding a piecewise-constant protocol with $N$ segments thus implies that there exists a similar protocol with a bandwidth $\sim N/(2T)$. In performing the path-tracing procedure we will keep the number of intervals, $N$, fixed, thus decreasing the length of each interval as we reduce $T$.
\begin{figure}[t]
\begin{center}
\onefigure[width=8.2cm]{fig1.eps}
\vspace{-0.9cm}
\end{center}
\caption{(Color online) Protocols that swap the states of two linearly-coupled oscillators. The slower oscillator has period $\tau = 2\pi/\omega$. (a) The error $\varepsilon$ as a function of duration, $T$. Red: protocols found using independent gradient searches for each of 2000 values of the duration, $T$. Light blue: protocols found using a gradient search to trace a single path through decreasing time, using 2000 points on the path. Dark blue: protocols with the least error of those found by tracing 11 paths. A total of 64845 points were used in tracing these 11 paths, for an average of 5895 points per path. (b) The five control parameters as a function $T$ for the protocols whose errors are shown in the light blue curve in (a).}
\label{fig1}
\end{figure}
\section{State-transfer in linear networks}
\label{apps}
We now apply the path-tracing method to the transfer of a quantum state from one harmonic oscillator to another. We consider two configurations: in the first the oscillators A and B are directly coupled, and in the second each is instead coupled to a third oscillator, C. The numerical simulation of these scenarios is greatly facilitated by the fact that Gaussian states remain Gaussian under linear evolution. Because of this, if the mean values of the canonical coordinates are zero, we need only keep track of their second moments. For two linearly coupled oscillators with respective annihilation operators $a$ and $b$, the equation of the motion for joint second moments is given by \begin{equation}
\frac{dC}{dt} = A C + C A^{\mbs{t}} ,
\end{equation}
where
\begin{equation}
C \equiv \langle \mathbf{v}^{\mbox{\scriptsize t}}\mathbf{v}\rangle \;\;\;\; \mbox{with} \;\;\;\; \mathbf{v} \equiv (a,a^\dagger,b,b^\dagger) ,
\end{equation}
and
\begin{equation}
A = \left( \begin{array}{cccc} \omega & 0 & -i\lambda_1(t) g & -i\lambda_1(t) g \\ 0 & -\omega & i\lambda_1(t) g & i \lambda_1(t) g \\ -i\lambda_1(t) g & -i\lambda_1(t) g & \omega & 0 \\ i\lambda_1(t) g & i\lambda_1(t) g & 0 & -\omega \end{array} \right) ,
\end{equation}
in which the single control parameter is $\lambda_1$. Since the evolution is unitary, a transfer of von Neumann entropy $S$ is equivalent to the transfer of $S$ qubits of quantum information~\cite{MikeandIke}. To simulate this transfer we therefore start oscillator A in a mixed Gaussian state (we choose the state with $\langle a \rangle = \langle a^\dagger \rangle = 0$ and $\langle a^\dagger a \rangle = 1$, which has von Neumann entropy $S = 2~\mbox{bits}$) and oscillator $B$ in the ground state. The condition for successful transfer is that oscillator A is left in the ground state so that all the entropy is stored in B. Since the oscillators are undriven the ground state is the only accessible pure state, thus in achieving the swap the control protocol is free to apply any local unitary to either oscillator.
For the numerical search we define the error as $\varepsilon = \langle a^\dagger a \rangle$, divide the duration $T$ into $N=5$ segments (giving five parameters $\lambda_{1k}$, $k=1,\ldots,5$), and use the BFGS gradient search method~\cite{Nocedal06}. The results are shown in Fig.~\ref{fig1}. We see that the standard search method finds no good protocols below $T \approx \tau/3$ where $\tau \equiv 2\pi/\omega$ (with a brief exception at $T \approx 0.08 \tau$). When we instead trace a single path a very broad trade-off frontier is revealed in which the achievable error increases slowly as $T$ is reduced. We also show the trade-off frontier that results from taking the best protocols over 11 paths. That the resulting frontier is smooth and that we find no paths that are significant outliers are indications that it may be the true frontier for this control task. If we set our error tolerance at $\varepsilon = 10^{-4}$ then we can perform the swap in just under $T = \tau/20$.
\begin{figure}[t]
\begin{center}
\onefigure[width=8.2cm]{fig2.eps}
\vspace{-0.9cm}
\end{center}
\caption{(Color online) Protocols that swap the states of two oscillators that are linearly coupled via a third oscillator. The slower oscillator has period $\tau = 2\pi/\omega$. (a) The error $\varepsilon$ as a function of duration, $T$, for protocols found by tracing a single path through decreasing time, using 900 points on the path. (b) The 20 control parameters as a function of $T$ for the protocols in (a).}
\label{fig2}
\end{figure}
In Fig.~\ref{fig2} we show results for the task of transferring a state from A to B when the two oscillators are linearly coupled via a third oscillator C. Note that we can perform this operation using two state-swapping operations for directly coupled oscillators (swap A with C and then B with C). We are therefore interested in whether it is possible to perform the transfer A to B in less than twice the time of the previous protocol with no more than twice the error. For this scenario we start B and C in the ground state and the ideal final state is that A and C are both in the ground state. This time we allow the controller to vary the coupling rates between both pairs of oscillators, and use 10 time segments for the a total of 20 control parameters. In this case independent searches can take so long that it is impractical to obtain protocols in this way. Path tracing is much faster, and we see from the single path shown in Fig.~\ref{fig2}, that if we allow an error of $\varepsilon = 2\times 10^{-4}$, the time taken by the fastest protocol that achieves this error is approximately $T = 2\tau/20$. This is no faster than two swap protocols performed in sequence.
\begin{figure}[t]
\begin{center}
\onefigure[width=8.2cm]{fig3.eps}
\vspace{-0.9cm}
\end{center}
\caption{(Color online) Protocols that generate an entangled two-mode squeezed state, given in the main text, between two oscillators linearly coupled via a third oscillator. The slower oscillator has period $\tau = 2\pi/\omega$. (a) The error $\varepsilon$ as a function of duration, $T$. Red: protocols found using independent gradient searches for each of 400 values of $T$. Light blue: protocols found by tracing a single path through decreasing time, using 1350 points on the path. (b) The 20 control parameters as a function $T$ for the protocols whose errors are shown in the light blue curve in (a).}
\label{fig3}
\end{figure}
\section{Generation of entanglement}
As an additional example of the use of path tracing, we consider the generation of entanglement between two oscillators that are again coupled only via their interactions with a third oscillator. This control problem provides an especially clear demonstration of trapping in local minima as $T$ is reduced. For this task all three oscillators start in the ground state and we wish to prepare a two-mode squeezed state between oscillators A and B. This state is defined by
\begin{equation}
\langle a^\dagger a\rangle = \langle b^\dagger b\rangle = \frac{\cosh(2r)}{2}, \;\;\;\;\; \langle a b^\dagger \rangle = \frac{\sinh(2r)}{2},
\end{equation}
and has an entanglement of
\begin{equation}
E = (1+\lambda)\ln(1+\lambda) - \lambda\ln\lambda
\end{equation}
with $\lambda = \sinh^2r$. We define the error as
\begin{eqnarray}
\varepsilon & = & \left(\langle a^\dagger a\rangle - \frac{\cosh(2r)}{2}\right)^{\!\! 2} + \left(\langle b^\dagger b\rangle - \frac{\cosh(2r)}{2}\right)^{\!\! 2} \nonumber \\
& & + \left(\langle a b^\dagger \rangle - \frac{\sinh(2r)}{2}\right)^{\!\! 2} \!\! .
\end{eqnarray}
We choose $r=2$ corresponding to an entanglement of $5.2$ bits. We plot the results in Fig.~\ref{fig3} from which we see that as $T$ is reduced the independent searches, shown in red, jump between poor solutions and good solutions, where trapping in the poor solutions dominates for $T \lesssim 0.01 \tau$. In comparison, the traced path usually remains closer to the good solutions; for small $T$ it climbs repeatedly out of good solutions, but continually drops back into them, something that the independent searches fail to do.
\section{Summary}
\label{conc}
Here we have used the method of path-tracing, a simple procedure that can be executed easily with any gradient search method, to find previously unknown control protocols for linear quantum networks. In doing so we have shown that path tracing is extremely powerful, finding fast protocols that appear to be out of reach of previous methods. While we have considered here purely oscillator-based problems in the ultra-strong coupling regime, path tracing may well change what is possible with quantum control across a wide range of problems. Preliminary investigations indicate that problems in which sets of qubits are used to interface with, or control the dynamics of resonators become very hard when maximal speed is sort, and are thus natural candidates for the application of this method.
\acknowledgments In the early part of this work KJ was partially supported by the NSF Project Nos.\ PHY-1005571 and PHY-1212413. HR was partially supported by the ARO MURI grant W911NF-11-1-0268, and RBW by the NSFC Grant Nos.\ 60904034, 61374091, and 61134008. XW acknowledges support from the NSF Project No.\ CCF-1350397.
|
2,877,628,090,805 | arxiv | \section{Introduction}
\label{sec:introduction}
The ANTARES neutrino telescope is located at a depth of 2475 m
in the Mediterranean Sea, roughly \mbox{40 km}
offshore from Toulon in France.
Its main objective is the observation of
extraterrestrial neutrinos.
Relativistic charged leptons produced by neutrino interactions in
and around the detector produce
Cherenkov light in the sea water.
This light is detected by an array of photomultiplier
tubes, allowing the muon direction to be reconstructed.
Although the ANTARES detector is optimised for upward going particle
detection, the most abundant signal is due to atmospheric
downward going muons \cite{Coll2009,Coll2010,Coyle} produced
in the particle showers induced by the
interactions of cosmic-rays in the atmosphere.
In order to reduce this background the Earth is
used as a filter, restricting the search
for cosmic neutrinos to sources in the Southern sky.
The processes contributing to the energy loss of a muon in water
include ionization, \mbox{$e^+e^-$ pair} production,
bremsstrahlung, and photonuclear
interactions \cite{PDG,Groom,Klimush,Gaisser}.
Below about \mbox{1 TeV}, the muon
energy loss is dominated by the continuous ionization process.
Above about \mbox{1 TeV}, the muon energy loss
is characterised by large energy fluctuations and
discrete bursts. These bursts originate from
pair production and bremsstrahlung (electromagnetic showers).
The photonuclear interaction processes are less frequent and in the following
no distinction is made between photonuclear induced showers
and electromagnetic showers.
The average muon energy loss per unit track length due to
these electromagnetic showers increases linearly with
the energy of the muon \cite{PDG}.
A reconstruction algorithm is presented to
identify electromagnetic showers induced by highly energetic muons
with the ANTARES detector. The shower reconstruction algorithm relies
on the identification of increased photon emission along the muon trajectory.
Counting electromagnetic showers along muon tracks to give an
estimate of the muon energy is called the \textit{pair meter} method \cite{Kokoulin, Gandhi}.
The estimate of the energy of muons is important for many research
topics. For example, an alternative method for estimating
the energy of muons based on the occurrence rate of repeated measured photons
on the photomultiplier tubes \cite{flux} has been used
by the ANTARES Collaboration to search
for a diffuse flux of cosmic high energy muon neutrinos.
Moreover, the angular resolution
of the muon trajectory
could benefit from a precise
discrimination of photon emission mechanisms along the
estimated track.
A similar measurement technique as the one presented in this
article has been
published recently by the Super-Kamiokande Collaboration
and used to select a sample of upward going muons with
energies above a TeV \cite{Desai}.
\section{The ANTARES detector}
\label{detector}
A detailed description of the ANTARES detector
is given elsewhere \cite{time,ARS,DAQ,Ant2}.
The full detector consists of twelve vertical
lines approximately \mbox{$450$ m} in height
equipped with a total of 885 photomultiplier tubes (PMTs).
The lines,
separated from each other by about \mbox{$65$ m}, are
anchored to the sea floor by a dead weight and held taut by a buoy located
at the top. The instrumented
part of the line starts \mbox{100 m} above the sea floor and consists
of 25 floors with a separation distance of
\mbox{14.5 m} along the line. The distance from the highest floor to the
sea surface is around 2000 m. A floor consists of three
PMTs pointing downward at an angle of $45^{\circ}$ with respect
to the vertical direction, in order to maximise the detection efficiency of upward going
tracks.
ANTARES is operating in the so called
all-data-to-shore mode: all signals above a charge threshold (typically 0.3 photoelectrons)
are digitised offshore
and sent to
shore to be processed in a computer farm.
This farm applies a set of trigger criteria in order to separate muon-induced Cherenkov
light from background light. The main sources of background light are the decay of $^{40}$K
nuclei and the bioluminescence from organisms in the sea water.
\section{Algorithm for shower identification}
\label{algorithm}
The technique of the electromagnetic shower identification aims
at distinguishing
Cherenkov photons emitted continuously along the muon track, hereafter
called \textit{muon Cherenkov photons},
from the Cherenkov photons induced by electromagnetic showers, hereafter
called \textit{shower photons}.
Because of the short radiation length in water \mbox{($X_0=35$ cm)}, these
showers rarely extend more than a few meters and can be considered point-like
light sources for the ANTARES detector.
The electromagnetic showers are identified by an excess of photons
above the continuous baseline of Cherenkov photons emitted by
a minimum ionizing muon.
The shower identification algorithm consists of two steps.
The first step allows the identification and reconstruction of
muon tracks. In the second step, a distinct shower candidate is identified by
a cluster of measured photons at a particular point along the muon path.
The criteria to isolate this cluster are determined
using a simulation code based
on Corsika \cite{Corsika}.
\subsection{Simulation}
\label{simulation}
Cosmic-ray interactions in the atmosphere,
including atmospheric shower development, were
simulated with Corsika for primary energies between
\mbox{1 TeV} and \mbox{$10^5$ TeV}, and incident angles between zero
(vertical downward going) and 85 degrees.
The primary cosmic-ray composition and flux model employed
is a simplified version of the H\"orandel parametrisation\footnote{The primary composition of the flux is subdivided into only five mass groups, namely proton, helium, nitrogen, magnesium and iron.} \cite{Hoerandel}.
The chosen hadronic interaction
model is QGSJET \cite{QGSJET}. The result of the Corsika simulation
is a set of muon tracks
with their position, arrival time and momentum given at the surface of the sea.
Typically, a single interaction leads to many muons at the sea surface.
These muons are propagated through water. The discrete energy losses
at high energies, the
Cherenkov light production and propagation, including scattering,
and the response of the detector
are simulated using a dedicated simulation package \cite{Km3_0, Bru}.
The muon propagation is performed by MUSIC
\cite{MUSIC} in steps of 1 m. If
the energy loss of the muon over the step exceeds a given threshold \mbox{(1 GeV)},
an electromagnetic shower is simulated and shower photons are emitted.
If the energy loss of
the muon over the step
is below the threshold, only muon Cherenkov photons are simulated.
The simulation package also uses tables generated from
\mbox{GEANT 3} \cite{GEANT} that
parameterise the arrival time
and the amount of light detected by individual PMTs. These tables
take into account the measured properties of the water at the ANTARES site,
the angular dependence of the acceptance of the PMT and also the position,
distance and orientation of the PMT with respect to a given muon track.
The optical background is assumed to be constant at a rate of
60 kHz \cite{Escoffier} on each PMT.
\subsection{Algorithm}
\label{algorithm1}
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\includegraphics[width=13.5cm]{figure1.eps}
\caption[]{Schematic view of muon Cherenkov light detection. The thick line represents the muon trajectory, the thin line the path of Cherenkov light and the thin dashed line the path of shower light. The muon goes through a reference point $(r, z, t)$. The Cherenkov light is emitted at an angle $\theta_{\rm CK}$ with respect to the muon track at point $\zeta_i^{\rm CK}$ and is detected by a PMT as a hit at point $(r_i, z_i, t_i)$. The shower light is emitted at point $\zeta_i$ and is detected by the same PMT at a different time.}
\label{fig:calc}
\end{figure}
The shower identification algorithm proceeds in several steps.
First, the muon trajectory must be determined. This is done using a
standard tracking algorithm \cite{line1, Ronald} that provides an estimate of
the direction and position of the muon at a given time.
In what follows, a hit is taken to be a photomultiplier signal exceeding a charge threshold of 0.3 photoelectrons~\cite{PMT}.
Using the configuration in Figure \ref{fig:calc}, the expected Cherenkov photon arrival time $t^{\rm CK}_i$ for each hit $i$ is calculated as
\begin{equation}
t^{\rm CK}_i = t + \frac{1}{c} \Big(z_i-z - \frac {r_i}{\tan \theta_{\rm CK}}\Big) + \frac{n}{c} \frac{r_i}{\sin \theta_{\rm CK}},
\label{Eq:CKlight}
\end{equation}
where $t$ is the time at which the muon passes point $(r,z)$, $\frac{c}{n}$ is the group velocity of light
in water ($n$ = 1.38 is the group refractive index
for ANTARES water), $\theta_{\rm CK}$ is the Cherenkov angle for a relativistic muon in water
($\theta_{\rm CK}\sim42^o$)
and $r_i$ is the perpendicular distance between the
muon trajectory and the PMT.
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\includegraphics[width=15.0cm]{figure2.eps}
\caption[Sc]{Time residuals for the measured photon arrival times relative to the calculated arrival times of Cherenkov photons coming from reconstructed muon tracks in a Monte Carlo sample. Contributions are shown for the direct and scattered photons originating from the muon as well as from the showers. Also shown are background photons. The three vertical lines define the early time interval (between -20 ns and 20 ns) and the late time interval (between 20 ns and 200 ns). The enhancement at \mbox{45 ns} is due to an effect of the PMT read-out electronics, which has been included in the simulation.}
\label{fig:timeresidual}
\end{figure}
The fitted trajectory can be used to
characterize hits by their arrival times into
three groups: early hits that are
predominantly due to Cherenkov photons, late hits that are
mainly due to scattered and shower photons, and extremely
early or late hits that can safely be assumed to be due to background.
Figure \ref{fig:timeresidual}
shows time residuals ($t_i-t^{\rm CK}_i$) for muon energies between 100 GeV
and 100 TeV generated by the simulation
described in Section \ref{simulation}.
Direct hits have a roughly Gaussian
distribution (with a long tail of late hits) with a peak
at zero time residual and a full width at half maximum of $\sim$20 ns.
Early hits ($|t_i-t^{\rm CK}_i| < t_{min}, t_{min}= 20$ ns)
contain mostly muon Cherenkov photons whose emission positions
along the muon track are given by
\begin{equation}
\zeta_i^{\rm CK}=z_i- z - \frac {r_i}{\tan \theta_{\rm CK}}.
\label{equ:cvpos}
\end{equation}
The variation in the arrival time of these Cherenkov hits can
be attributed to the dispersion of light in the sea water.
Note that Equation \ref{equ:cvpos} is used to determine
the emission point of all photons leading to early hits, even shower
photons that may not be emitted at the Cherenkov angle.
Late hits ($t_{min} < t_i-t^{\rm CK}_i < t_{max}, t_{max}= 200$ ns)
contain the
largest fraction of hits due to shower photons. The value of $t_{max}$
has been taken to be the point at which a hit is equally likely to be
due to a shower photon as to a background photon.
These shower photons may not necessarily be emitted at the Cherenkov angle
from the muon track. Therefore the emission angle is left as a free parameter
and, with the photon emission taking
place at $\zeta_i$ (see Figure \ref{fig:calc}),
the hit time is given by
\begin{eqnarray}
t_i = t + \frac{\zeta_i-z}{c} + \frac{n}{c} \sqrt{ r_i^2 + (z_i- \zeta_i )^2 }.
\label{equ:showertime}
\end{eqnarray}
Equation (\ref{equ:showertime}) has two distinct
solutions, $\zeta_i^{+}$ and $\zeta_i^{-}$.
Extremely late or early hits ($t_i > t_{max}$ or $t_i <-t_{min}$)
are assumed to be background hits and are rejected.
All calculated $\zeta_i^{\rm CK}$, $\zeta_i^{+}$ and $\zeta_i^{-}$ positions
along the muon track are collected in a one dimensional histogram.
The shower position is identified by the localised increase of
the number of emitted photons along the reconstructed muon trajectory,
identified by a peak in the histogram.
If the two solutions $\zeta_i^{+}$ and $\zeta_i^{-}$ are found in different peaks,
the shower identification procedure will ignore the solution in the smaller peak.
\begin{figure}[]
\setlength{\unitlength}{1cm}
\centering
\begin{picture}(15.5,14.5)
\put(-1.0,0.0){\epsfig{file=figure3.eps,width=17.0cm,clip=}}
\put(0.2,15.1){\textbf{\textsf{(a)}}}
\put(8.7,15.1){\textbf{\textsf{(b)}}}
\put(0.2,7.3){\textbf{\textsf{(c)}}}
\put(8.7,7.3){\textbf{\textsf{(d)}}}
\end{picture}
\caption[Sc]{Display of an atmospheric muon event. The first three panels (a)(b)(c) show, for each line, the altitude of the photomultiplier tube for each associated photon (crosses) as a function of the arrival time of the photon. The origin on the z-axis corresponds to the middle of the line and the time offset is chosen with respect to the time of the first detected photon compatible with the muon trajectory. The muon track (solid line) and two electromagnetic \mbox{showers} (curved dotted lines) are reconstructed. The black squares indicate identified photons which are used in the muon trajectory reconstruction. The empty circles around the crosses indicate photons used in the shower reconstruction. The bottom right plot (d) shows the number of detected photons projected along the muon trajectory. The peaks correspond to the reconstructed shower positions indicated by the triangles.}
\label{fig:EvDisplay}
\end{figure}
An example of the application of this procedure to data can be seen in Figure \ref{fig:EvDisplay}. The bottom right panel of this figure
shows the emission points of photons along the muon trajectory, as determined by the
solutions of Equation \ref{equ:cvpos} and Equation \ref{equ:showertime}.
Two excesses are visible and are attributed to two electromagnetic showers.
Each of the other three panels shows a height versus time diagram of data obtained from
one of the detector lines.
The result of the muon trajectory reconstruction for a relativistic
muon in water together with results of the shower
identification are also displayed.
A downward going muon with
two electromagnetic showers is thereby identified.
Using the fitted shower positions from the shower identification
algorithm, a prediction is made for the arrival time of the shower light.
The dotted curves in Figure \ref{fig:EvDisplay} show the expected photon arrival time under the assumption
of a spherical light emission from the reconstructed position of the shower.
As can be seen from \mbox{Figure \ref{fig:EvDisplay}}, most, but not all, photons
are associated with the muon track fit. Many photons that do not
comply with the muon track fit are associated to shower photons.
The photons not associated with the muon track
or associated showers can be attributed to random
background photons due to radioactive $^{40}\mathrm{K}$ decays
and bioluminescence.
\section{Selection and performance}
\label{simulationandselection}
The selection and performance of the shower identification
algorithm has been studied
with the simulation described in Section \ref{simulation}.
\subsection{Muon and shower selection}
\label{sec:showermuonselect}
The shower identification algorithm is applied to
well reconstructed muon tracks that have the potential
to produce a detectable shower. Two criteria have been used to select
such tracks. First, the track length is required to be at
least 125 m. The track length is taken to be the distance
between the emission points of the photons that gave rise to
the earliest and latest hit used in the muon track
reconstruction. Second, the muon trajectory reconstruction
is required to have used at least 12 hits.
These criteria select about 65\% of
all reconstructed (downward going) muon tracks.
The shower-induced photon emission along the muon
trajectory results in localised peaks as shown in Figure \ref{fig:EvDisplay}d.
The task to identify a shower is then reduced to a one dimensional
peak finding algorithm whose result can be characterised by
three parameters, namely the center, width and height of the
identified peak.
Potential peaks are identified through the subtraction of
the constant photon background, as determined by a sensitive
nonlinear peak clipping algorithm. This algorithm tracks the baseline
of a spectrum by comparing the value of each data point with
the average value of neighboring data points, taking the baseline
to be the smaller value. Further details can be found in \cite{Morhac}.
For each potential peak, the number of hits is integrated
in a $\pm 5$ m window around the peak center.
All hits are assumed to be single photons.
Only peaks
having at least 10 hits above Cherenkov photon baseline
in this window of 10 m are selected (typically yielding peaks with three sigma significance).
The number of baseline hits is defined as the
average number of hits along the track times the window of 10 m.
In order to suppress wrongly identified showers,
hits from at least
five different floors
are required for each peak.
\subsection{Performance of the shower identification method}
\label{sec:performance}
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\includegraphics[width=8.5cm]{figure4.eps}
\caption[Sc]{Average number of detectable showers which have shower photons detected on at least five different floors per atmospheric muon event as a function of the muon energy.}
\label{fig:showervsenergy}
\end{figure}
Figure \ref{fig:showervsenergy} shows the number of detectable showers,
coming from well reconstructed muons,
that have shower photons detected on at least five different floors
per atmospheric muon event as a function
of the muon energy.
The atmospheric muon events are usually muons in a
bundle with an average multiplicity around 3.3~\cite{Corsika}.
The muon energy in Figure \ref{fig:showervsenergy} refers
to the muon with the largest energy in the bundle.
These muons have an average energy of \mbox{1.2 TeV}
and their mean generated shower energy is around 120 GeV.
Muons with at least one identified shower
have on average 2.5 times higher energy than
muons without an identified shower.
The event selection and algorithm has been tuned to count the number
of showers with a high level of purity,
even at the expense of efficiency.
In order to study the efficiency and purity of the reconstruction,
the Monte Carlo truth information is consulted to determine
whether the result of the shower reconstruction corresponds
to any actual shower and, if so, whether that shower has been well
reconstructed. A reconstructed shower is said to have correctly
identified an underlying shower if its position is determined to
within 25 m of the true generation point and if 25\% of the hits in the
reconstructed shower are truly due to photons produced by the underlying
shower. Here, a hit is in the reconstructed shower if its projected emission
point along the muon track is within 5 m of the reconstructed shower
position (see Figure \ref{fig:EvDisplay}d).
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\begin{picture}(15.5,8.5)
\put(-1.5,0.3){\epsfig{file=figure5a.eps,width=8.5cm,clip=}}
\put(6.5,0.3){\epsfig{file=figure5b.eps,width=8.5cm,clip=}}
\put(0.1,7.1){\textbf{\textsf{(a)}}}
\put(8.1,7.1){\textbf{\textsf{(b)}}}
\end{picture}
\caption[Sc]{(a) Efficiency and (b) purity as a function of the shower energy for reconstructed showers obtained with a Monte Carlo sample of atmospheric muons.}
\label{fig:showereff}
\end{figure}
The efficiency with which showers are correctly identified is given
by the ratio of the number of well identified showers to the total
number of simulated showers that give rise to hits in
at least five different floors, i.e.
all showers that may reasonably be expected to be reconstructed.
The shower identification efficiency ranges from 5\% at low shower energy
($\sim$300 GeV) to 30\% at high shower energy ($\sim$5 TeV), as shown in Figure \ref{fig:showereff}a.
The purity of the reconstructed shower sample is given by the
ratio of the number of correctly identified showers to the number
of all reconstructed showers and is shown in Figure \ref{fig:showereff}b.
The shower purity ranges from 40\% at low
shower energy ($\sim$300 GeV) to 90\% at high shower energy ($\sim$1 TeV). At
even higher shower energies, the purity decreases, reaching 60\% at 30 TeV.
Such showers are mainly produced by very high energy muons. The
density of photons along the trajectory of such a muon is
great enough that an excess of photons due to a particular shower becomes
difficult to observe.
\section{Comparison between data and Monte Carlo simulations}
A sample of data corresponding to 47.3 days of data
taking between January and December 2007 has been used to study the behavior of the shower identification algorithm.
During this period the detector comprised five lines.
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\begin{picture}(15.5,8.5)
\put(-1.5,0.3){\epsfig{file=figure6a.eps,width=8.5cm,clip=}}
\put(6.5,0.3){\epsfig{file=figure6b.eps,width=8.5cm,clip=}}
\put(0.1,7.1){\textbf{\textsf{(a)}}}
\put(8.1,7.1){\textbf{\textsf{(b)}}}
\end{picture}
\caption[Sc]{(a) Photon emission angle and (b) relative number of hits (note that a minimal number of ten hits is required by the identification algorithm) used in the shower identification for the data and the Corsika simulation.}
\label{fig:showers1}
\end{figure}
The Corsika simulation (including the simplified H\"orandel flux) was scaled by a factor 0.83 to normalise
the simulated muon rate to the measured muon rate for the selected tracks.
Figure \ref{fig:showers1}a shows
the angular distribution of the
shower photons with respect to the muon direction.
The shape of the distribution is determined
by detector effects and the cuts used in the analysis. The peak
around 42 degrees comes from shower photons emitted
at the Cherenkov angle through showers oriented in the direction of the muon
and whose emission points have been calculated using Equation \ref{equ:cvpos}, whereas the other hits have been calculated using Equation \ref{equ:showertime}.
Figure \ref{fig:showers1}b shows the hit multiplicity distribution of
selected showers. With the final set of cuts applied,
the average number of hits associated to an identified shower is around 14.
As the quantity of Cherenkov light produced by an electromagnetic shower increases linearly with the muon energy,
counting the number of hits in one shower provides a first order estimate of its energy.
The data distributions agree reasonably
well with the Corsika simulation.
\begin{table}[tb]
\begin{center}
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
Number of identified showers & 0 & 1 & 2 & 3+ \\
\hline
Source of uncertainty & \multicolumn{4}{|c|}{Variation in [\%]} \\
\hline
Background rate & 0.1 & 1 & 12 & 14 \\
Minimal shower energy & 0.1 & 3 & 6 & 5 \\
PMT angular acceptance & 1.2 & 18 & 30 & 3 \\
Absorption length & 1.3 & 17 & 39 & 77 \\
\hline
Total systematic uncertainty [\%] & $\pm$ 1.8 & $\pm$ 25 & $\pm$ 51 & $\pm$ 78 \\
\hline
\end{tabular}
\end{center}
\caption[]{Variation in the number of identified showers as the values of selected Monte Carlo parameters are changed. The systematic uncertainty is estimated by varying the background rate, the energy threshold to produce photons from electromagnetic shower light, the PMT angular acceptance and the water absorption length (see text).}
\label{tab:systematic}
\end{table}
The simulation has also been used to evaluate
systematic uncertainty on the number of identified showers.
\mbox{Table \ref{tab:systematic}} shows the systematic uncertainty
determined by varying parameters in the simulation \cite{Coll2010}.
The measured detector
background rate is around 60 kHz on average \cite{Escoffier}.
The systematic error arising from the uncertainty of this background rate is estimated by
repeating the analysis with a background rate of 50 kHz and with a background rate of 120 kHz (row three in Table \ref{tab:systematic}).
The values are the percentage variation with respect to the values from the default simulation.
Uncertainties arising from the
energy threshold to produce hits from electromagnetic showers
or hits from muon Cherenkov light
is estimated by varying the threshold $\pm 50$\% from its default value
of \mbox{1 GeV} (row four in Table \ref{tab:systematic}).
Uncertainty on the angular acceptance of the optical modules is estimated
by taking a different parametrization of the PMT angular acceptance \cite{Coll2010} (row five in Table \ref{tab:systematic}).
The water properties
are taken into account by varying the
absorption length by $\pm 20$\% around the measured value \cite{transmission} (row six in Table \ref{tab:systematic}).
All systematic uncertainties are added in quadrature.
The largest contributions to the systematic error arise from uncertainties
on the PMT angular acceptance and on the water absorption length.
When decreasing
the absorption length, fewer showers are identified, since more photons are absorbed in the water
before they reach the PMTs. On the other hand, the systematic studies
show evidence that the shower algorithm
is robust against large variations of the background, because showers emit
a light density much bigger than that of the optical background.
The muon event rate as a function of the number of identified
showers is shown
in \mbox{Figure \ref{fig:showers}}.
The distribution shows
the results for data and the Corsika based simulation with no correction for the
identification efficiency.
Also shown is the systematic uncertainty for the simulation.
For the data points, only the statistical errors
are shown.
As can be seen, about 4\% of the
selected muon tracks have one well identified shower.
There is agreement between data and Monte Carlo over five orders of magnitude.
\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\centering
\includegraphics[width=8.5cm]{figure7.eps}
\caption[Sc]{Muon event rate as a function of the shower multiplicity for data (points) and the Corsika simulation (line) with no correction for the identification efficiency. The systematic error for the simulation is given by the height of the grey bands. Only statistical errors are shown for the data points.}
\label{fig:showers}
\end{figure}
\section{Conclusions}
\label{conclusion}
A method to identify electromagnetic showers emitted by
muons has been developed, characterised and
applied to the downward going muon data taken with
the ANTARES detector.
This algorithm exploits the different emission characteristics
of shower-induced and primary muon-induced Cherenkov photons.
The shower light emission is localised at discrete
points along the muon trajectory, whereas the traversing muon continuously
emits Cherenkov photons under a constant and known angle relative to the muon
trajectory.
The essential element of the algorithm is
the projection and identification of photon vertices along the muon
track with a subsequent peak finding algorithm.
The performance of the identification
algorithm has been validated using a sample of simulated
atmospheric muon events and agreement was found in the number of identified showers
between data and simulations.
With the development and validation of this electromagnetic shower multiplicity
estimator, important new information becomes available
for physics analysis. In particular the method establishes a first step
towards a new energy estimator.
With the application of this method, it can be concluded that stochastic
energy loss has been observed in ANTARES.
\section*{Acknowledgments}
The authors acknowledge the financial support of the funding agencies:
Centre National de la Recherche Scientifique (CNRS), Commissariat
\'a l'\'ene\-gie atomique et aux \'energies alternatives (CEA), Agence
National de la Recherche (ANR), Commission Europ\'eenne (FEDER fund
and Marie Curie Program), R\'egion Alsace (contrat CPER), R\'egion
Provence-Alpes-C\^ote d'Azur, D\'e\-par\-tement du Var and Ville de
La Seyne-sur-Mer, France; Bundesministerium f\"ur Bildung und Forschung
(BMBF), Germany; Istituto Nazionale di Fisica Nucleare (INFN), Italy;
Stichting voor Fundamenteel Onderzoek der Materie (FOM), Nederlandse
organisatie voor Wetenschappelijk Onderzoek (NWO), the Netherlands;
Council of the President of the Russian Federation for young scientists
and leading scientific schools supporting grants, Russia; National
Authority for Scientific Research (ANCS), Romania; Ministerio de Ciencia
e Innovaci\'on (MICINN), Prometeo of Generalitat Valenciana and MultiDark,
Spain. We also acknowledge the technical support of Ifremer, AIM and
Foselev Marine for the sea operation and the CC-IN2P3 for the computing facilities.
\bibliographystyle{report}
|
2,877,628,090,806 | arxiv | \section{Introduction}
Potentials define a specific concept in physics. They predict the evolution of a system from a variational principle. Such principles span many scientific fields from mechanics, electromagnetism and optics to control theory, thermodynamics and statistical physics. A variational principle elegantly summarizes the method used to solve a problem into the extremization of the appropriated cost function, for instance the action in mechanics \cite{Book_Landau1976}, the optical path length in optics \cite{Book_Born1999_vol}, or the thermodynamic potential in statistical physics \cite{Callen1985_vol}. The underlying idea is to explore all possibilities, including non physical ones, to find the physical solution from the extremum of the cost function.
In statistical physics, a thermodynamic potential is a state function of the thermodynamic variables. The latter specify a coarse-grained representation of the state of a system including a large number of degrees of freedom. Thermodynamic variables come in conjugated pairs: in each pair, one variable is free and one is constrained according to the environmental conditions. The EQ thermodynamic potentials proceed from the Legendre transformation of either energy or entropy. This transformation, at the core of the theory's dual structure, allows us to interchange the free and constrained variables. The thermodynamic state is reached at the extremum of the thermodynamic potential. There, the mean free variables are functions of the constrained ones. Beyond the mean description, the potential also predicts the statistics of the free thermodynamic variables, either by generating their cumulants, or from its connection with the asymptotic probability of the free variables.
Statistical physics provides a microscopic foundation to thermodynamics and a method to describe equilibrium systems. In the last decades, the large deviation theory \cite{Oono1989_vol99,Touchette2009_vol478} has modernized our understanding of statistical physics and accounted for its successes. More recently, it has received a growing interest thanks to its applications to NE systems, for instance in glasses \cite{Garrahan2009_vol42, Turci2011_vol94, Speck2012_vol109, Nemoto2014_vol2014}, biological systems \cite{Hertz1999_vol15, Lacoste2008_vol78, Ritort2008_vol137} or rare events sampling \cite{Giardina2011_vol145}.
Clearly, one step toward understanding NE phenomena starts with the derivation of a NE thermodynamic potential verifying most of the aforementioned properties. With this in mind, many authors have shed light on the structure of statistical physics for NE Markov processes. Oono and Eyink considered that Large Deviation Functions (LDF) could represent NE potentials \cite{Oono1989_vol99, Eyink1996_vol54, Eyink1998_vol130}. On this basis, Oono and Paniconi proposed a phenomenological framework to study NE steady states \cite{Oono1998_vol}. For NE continuous processes, Bertini \textit{et al.} developed the macroscopic fluctuation theory describing the statistics of density and current fluctuations in Non-Equilibrium Stationary States (NESS) \cite{Bertini2002_vol107,Bertini2015_vol87}. Bodineau and Derrida used an additivity principle to predict those fluctuations in diffusive systems \cite{Bodineau2004_vol92,Derrida2007_vol}.
For discrete processes, Lecomte, Appert-Rolland and van Wijland introduced a dynamical partition function and the corresponding topological pressure identified as a LDF \cite{Lecomte2005_vol95, Lecomte2007_vol127}. Baule and Evans explored these ideas using a path entropy with the aim of finding rules constraining the dynamics of fluids under continuous shear \cite{Evans2010_vol51,Baule2008_vol101, Baule2010_vol2010}. Monthus proposed a similar approach, but involving the maximization of a trajectory-based relative entropy in the presence of constraints \cite{Monthus2011_vol2011}. Using large deviation theory, Maes and Neto{\v c}n{\'y} \cite{Maes2008_vol82} found a canonical structure and obtained the LDF of occupation and current probabilities from a variational approach based on the LDF of occupation and transition probabilities. A key step was the introduction of an EQ reference process to highlight that EQ fluctuations naturally appear when studying NE fluctuations. From another perspective, Nemoto and Sasa have shown that a Cumulant Generating Function (CGF) also proceeds from a variational principle, strengthening the dual structure of the theory \cite{Nemoto2011_vol}.
\begin{center}
\begin{table*}
\caption{Relationship between the various stochastic processes and NE ensembles. The EQ reference process conditioned on the energy currents $j$, activities $f$ and occupations $p$ generates the trajectories of the systems in the NE micro-canonical ensemble. The process with mean energy currents, activities and occupations equal to the constrained values of the conditioned process is the driven process. The path probabilities of the driven process are asymptotically equivalent to the path probability of the NE process and of the canonical reference process. The CGF of $j,f$ and $p$ for the NE process is exactly the same as the CGF of the EQ reference process (up to a translation), i.e spontaneous rare fluctutions of the EQ process are associated to typical realizations of the NE process. The NE process generates the trajectories of the systems in the meta-canonical ensemble. This ensemble includes the systems that are put out-of-equilibrium by gradients of temperatures imposed by heat reservoirs. From the equivalence between the conditioned reference process with the driven reference process and the NE process, we conclude that the NE micro-canonical and meta-canonical ensembles are equivalent. \label{structure}
}
\begin{tikzpicture}
[node distance=2.0cm,
start chain=going right, ]
\node[punktchain, join, ] (NeCanEns) {NE metacanonical \\ ensemble $\Gamma(a,b,m)$};
\begin{scope}[start branch=venstre, every join/.style={->, thick, shorten <=1pt}, ]
\node[punktchain, on chain=going below, node distance=1.0cm ] (EqProcess) {EQ reference process $z$ with generator ${\bm{k}}$};
\begin{scope}[start branch=venstre, every join/.style={->, thick, shorten <=1pt}, ]
\node[punktchain, on chain=going below, node distance=1.0cm ] (NeProcess) {NE process $\bar z$ with generator $\bar {\bm{k}}$ };
\node[punktchain, on chain=going right, join ] (CondNeProcess) {Conditioned NE process $\bar z | j,f,p$ };
\node[punktchain, on chain=going right, join ] (DrivenNeProcess) {Driven NE process with generator $\bar \bm{K}$};
\end{scope}
\node[punktchain, join, ] (CondEqProcess) {Conditioned reference \\ process $z | j,f,p$} ;
\node[punktchain, join, ] (DrivenEqProcess) {Driven reference \\ process with generator $\bm{K}$} ;
\end{scope}
\node[punktchain, ] (NeMicroCanEns) {NE micro-canonical ensemble \\ $L(j,f,p)$};
\node[punktchain, ] (CanProcess) {Canonical reference \\ process with generator $\bm{\mathcal{K}}$ };
\draw[|-,-|,<->, thick, ] (DrivenNeProcess.north) -- (DrivenEqProcess.south);
\draw[|-,-|,<->, thick, ] (NeMicroCanEns.west) -- (NeCanEns.east);
\draw[|-,-|,<->, thick, ] (DrivenEqProcess.east) --(13.5,-2.02)--(13.5,-5.3)--(0,-5.3)-- (NeProcess.south);
\draw[|-,-|,<-, thick, ] (NeCanEns.west) -|(-2.5,-4.035)-- (NeProcess.west);
\draw[|-,-|,<-, thick, ] (NeMicroCanEns.south) -- (CondEqProcess.north);
\draw[|-,-|,<->, thick, ] (EqProcess.south) -- (NeProcess.north);
\draw[|-,-|,->, thick, ] (EqProcess.north) -- (NeCanEns.south);
\draw[|-,-|,<->, thick, ] (CanProcess.south) -- (DrivenEqProcess.north);
\draw[tuborg] let
\p1=(NeCanEns.west), \p2=(NeMicroCanEns.east) in
($(\x1,\y1+2.5em)$) -- ($(\x2,\y2+2.5em)$) node[above, midway] {Ensembles of NE systems};
\begin{scope}
\path (DrivenNeProcess) -- (DrivenEqProcess) node [midway, left] { Generators mapping };
\path (NeMicroCanEns) -- (NeCanEns) node [midway, above] { Ensemble};
\path (NeMicroCanEns) -- (NeCanEns) node [midway, below] { equivalence};
\path (EqProcess.north) -- (NeCanEns.south) node [midway, right] {CGF built from the EQ process};
\path (EqProcess) -- (CondEqProcess) node [midway, above] { \begin{tabular}{c} \raisebox{-1ex}{Micro-} \\ \raisebox{-1ex}{canonical}
\end{tabular}};
\path (EqProcess) -- (CondEqProcess) node [midway, below] { conditioning};
\path (NeProcess) -- (CondNeProcess) node [midway, above] { \begin{tabular}{c} \raisebox{-1ex}{Micro-} \\ \raisebox{-1ex}{canonical}
\end{tabular}};
\path (NeProcess) -- (CondNeProcess) node [midway, below] { conditioning};
\path (CondNeProcess) -- (DrivenNeProcess) node [midway, above] { Optimization};
\path (CondNeProcess) -- (DrivenNeProcess) node [midway, below] { problem};
\path (CondEqProcess) -- (DrivenEqProcess) node [midway, above] { Optimization};
\path (CondEqProcess) -- (DrivenEqProcess) node [midway, below] { problem};
\path (EqProcess) -- (NeProcess) node [midway, right] { Tilted-operators mapping (${\bm{\kappa}} \longleftrightarrow \bar {\bm{\kappa}} $)};
\path (13.5,-5.3)--(0,-5.3) node [midway, below, sloped] { Similarity transformation $\bm{K}= |\bm{\pi}| \cdot \bar {\bm{k}} \cdot |\bm{\pi}|^{-1} \;\; \Rightarrow \;\;\; $Asymptotic path equivalence };
\path (NeMicroCanEns.south) -- (CondEqProcess.north) node [midway, right] {Generates the path ensemble};
\draw (-2.5,-3.2)--(-2.5,-1) node [midway, above, sloped] {Generates the path ensemble};
\path (CanProcess.south) -- (DrivenEqProcess.north) node [anchor=center, text width=3.5cm, auto, midway] {Asymptotic path equivalence};
\end{scope}
\end{tikzpicture}
\end{table*}
\end{center}
More recently, Chetrite and Touchette proposed a general framework for both continuous and discrete processes: they found that a conditioned Markov process is ensemble-equivalent to a condition-free process called the driven (or auxiliary) process, but also to an exponentially tilted process called the canonical process \cite{Chetrite2015_vol16,Chetrite2013_vol111}. This later process is defined by exponentially weighting the probability of each trajectory with a weight depending on a functional $v$ of the stochastic process. This weighting procedure, is analogous to the definition of the canonical ensemble from the superposition of micro-canonical ensembles using a Boltzmann weight. On the other hand, the conditioned Markov process assumes that the variable $v$ is constrained to a given value. Finally, the driven process has a dynamics defined such that the mean value of $v$ is equal to the imposed value in the conditioned process. A systematic method of constructing this driven process from a variational approach was provided in Ref. \cite{Chetrite2015_vol2015}. A construction of the canonical process was also proposed by Giardin\'a, Kurchan and Peliti in Ref.~\cite{Giardina2006_vol96} for classical systems and by Garrahan and Lesanovsky in Ref.~\cite{Garrahan2010_vol104} for dissipative quantum systems. Jack and Sollich constructed a driven process for classical systems in Ref.~\cite{Jack2010_vol184}. The questions of the validity of the path ensemble equivalence has recently been studied in Ref.~\cite{Szavits-Nossan2015_vol2015}
Despite all these results, the structure of NE statistical physics is incomplete as regards to EQ statistical physics. For instance, the identification of the relevant coarse-grained degrees of freedom, i.e. the NE thermodynamic variables, is still missing. Accordingly, no general definition exists for stationary NE thermodynamic potentials. To progress in this direction, focusing on continuous-time Markov chains and stationary processes, we consider the following questions: can we describe the NE fluctuations of a system from the fluctuations of the \emph{same} system at EQ? If yes, can we define meaningful NE thermodynamic potentials using the variables involved in this correspondence? We positively answer these two questions by finding an exact mapping between the statistics of EQ and NE processes. This mapping involves, among others, the affinities of the NE process and some dynamical biases. The later parameter enables to dilate the energy barriers separating the various states of the system. The variables conjugated to the affinities and the dynamical biases are, respectively, the energy currents and the activities of the exchanges with the environment. The existence of a simple mapping when considering the appropriated couples of conjugated variables suggests that a complete canonical structure for NE statistical physics exists. With respect to previous works on conditioned Markov processes, our main contribution is to identify the constrains that does not modify the system dynamics, apart from changing the temperatures of the heat reservoirs. Accordingly, we define two ensembles of NE systems: the meta-canonical ensemble where the constrained variables are the affinities, and the NE micro-canonical ensemble where the constrained variables are the energy currents. We prove the equivalence of these ensembles and derive the NE thermodynamic potentials conjugated by Legendre transformation. We also obtain the NE equations of state connecting the conjugated variables.
Our results and the structure of the theory are summarized in Table~\ref{structure}. Accordingly, the outline of the paper is as follows. We start by studying the fluctuations of an EQ reference process in Sec.~\ref{EQfluctuation} whose material corresponds to the middle row of Table~\ref{structure}. The definition of the EQ reference process and an introduction to large deviation theory are provided in Secs.~\ref{DefEQprocess} and \ref{LDtimeAVGvar}. After these introductory subsections, we look for an asymptotic approximation of the probability of the energy currents, activities and occupations of the systems states. Since we are dealing with an EQ system, no mean energy current exists. However, rare spontaneous fluctuations may produce non-zero energy currents and some arbitrary activities and occupations. We seek the probability of these events from an optimization problem: given that some energy currents $j$, activities $f$, and occupations $p$ are observed, defining the conditioned reference process, which process (called the driven reference process) reproduces these conditioned values $j$, $f$ and $p$ as typical values? We construct this driven process in Sec.~\ref{EQ-LDF} and obtain the LDF of $j$, $f$ and $p$. We use this result to derive the corresponding scaled CGF from a variational approach in Sec.~\ref{EQ-CGF}.
We switch to the study of the fluctuations of a NE process in Sec.~\ref{NEfluctuations}. This section corresponds to the third row of Table~\ref{structure}, which is obtained following exactly the same path as for the EQ reference process, except that we start with a NE process as defined in Sec.~\ref{DefNEprocess}: we look for the NE driven process that will typically reproduce the arbitrary energy currents $j$ and activities $f$ imposed in the NE conditioned process. Our first main result is to connect, in Sec.~\ref{MappingNEonEQ}, the EQ reference process and the NE process, and as a consequence, also to connect their associated driven processes (see the vertical arrows in Table~\ref{structure}). Our second main result is to prove, in Sec.~\ref{EnsembleEquivalence}, the asymptotic equivalence between the path probabilities of the driven reference process with the NE process. This equivalence is at the core of the aforementioned equivalence between the NE micro-canonical ensemble and the meta-canonical ensemble. In Sec.\ref{discussion}, we comment the structure of the theory starting with a short summary in Sec.~\ref{CanonicalStructure}. We discuss the symmetries of the NE potentials and the connection with close-to-equilibrium and far-from-equilibrium perturbation theory in Secs.~\ref{NEsymmetries} and \ref{NEresponse} respectively. We end by illustrating our work on a two-level model in Sec.\ref{Example}.
For the sake of simplicity, we focus on systems exchanging only energy with heat reservoirs. The generalization of our results to include matter, volume or other extensive variable exchanges with reservoirs is straightforward \cite{VandenBroeck2014_vol418}.
\section{Equilibrium fluctuations}
\label{EQfluctuation}
\subsection{Definition of the EQ reference process}
\label{DefEQprocess}
We consider an \emph{EQ reference process} corresponding to a physical system modeled by a continuous-time Markov chain with a finite number $M$ of discrete states. This system exchanges energy with $\chi $ heat reservoirs labeled by $\nu =1 \dots \chi$ at inverse temperatures $ \beta_1 = 1/ (k_B T_1)$, with $ k_B =1 $ the Boltzmann constant, see Fig.~\ref{fig10}. The reference process is at EQ, i.e. all the heat reservoirs share the same inverse temperature $\beta_1 $. We use several heat reservoirs to allow different mechanisms of energy exchange. As a result, some rare events with net energy flow from one heat reservoir to another may occur. The system states are generically denoted $x$, $y$ and $z$. The state at time $\tau$ is $z(\tau)$. A system state trajectory during time interval $[0,t]$ is denoted $[z]$. This trajectory includes the state $z(\tau)$ at all time $\tau \in [0,t]$ and the label $\nu(\tau)$ of the reservoir providing the energy at each change of state in the trajectory.
\begin{figure}
\includegraphics[width=6.5cm]{./fig10.pdf}
\caption{System with $M=6$ states connected to $\chi = 2$ heat reservoirs at the same temperature $T_1 = T_2 $ for the EQ reference process, or at different temperature $T_1 \neq T_2$ for the NE process. \label{fig10}}
\end{figure}
\begin{figure}
\includegraphics[width=7cm]{./fig11.pdf}[h]
\caption{Energy lanscape for the $x \leftrightarrow y$ transition. The discrete states $x$ and $y$ represent the locations of the minima in the energy landscape. Changing the dilatation factor $l_1$ modify the height of all energy barriers for the EQ reference process. \label{fig11}}
\end{figure}
The energy of state $x$ is $\epsilon_x$. The probability per unit time of switching from state $y$ to state $x$ exchanging the energy $\epsilon_x-\epsilon_y$ with reservoir $\nu$ is given by the Arrhenius transition rates
\begin{equation}
k_{xy}^{\nu} \equiv \gamma_{xy}^{\nu} e^{-\beta_1 (\epsilon_x-\epsilon_y)/2 - \beta_1 l_1 d_{xy}}. \label{defEQrates}
\end{equation}
We have introduced the symmetric matrices $ \bm{\gamma}^{\nu} $, whose $(x,y)$ element yields the coupling with reservoir $ \nu $ for a transition from $ y $ to $ x $. The $(x,y)$ element of the symmetric matrix $\bm{d}$ represents the height of the energy barrier that must be crossed when the system switches between states $y$ and $x$, see Fig.~\ref{fig11}. The dimensionless parameter $l_1$ is a \emph{dilatation factor} that enables to modify the height of the energy barriers ($l_1 =1$ implies no dilatation).
The transition rates defined in Eq.~(\ref{defEQrates}) verify for all $\nu$ the local detailed balance relation
\begin{equation}
\ln \frac{k^{\nu}_{xy}}{k^{\nu}_{yx}} = - \beta_1(\epsilon_x-\epsilon_y),
\end{equation}
which ensures that the system will reach EQ \cite{VanKampen2007_vol}.
The reference probability per unit time of escaping from state $y$, given that energy is exchanged with reservoir $\nu$, is denoted
\begin{equation}
\lambda^{\nu}_y \equiv \sum_{x\neq y} k^{\nu}_{xy} = - k^{\nu}_{yy},
\end{equation}
such that each column of the matrix $ {\bm{k}}^{(\nu )}$ sums to zero as required for continuous time Markov chains. The reference transition rate matrix $ {\bm{k}} \equiv \sum_\nu {\bm{k}}^{(\nu )}$ returns the transition probabilities per unit time disregarding the reservoir involved in the energy exchanges. Similarly, $ \lambda \equiv \sum_\nu \lambda^{(\nu )} $ is the total escape-rate vector. As a convention, we drop the subscripts of vector or matrix elements to refer to the whole vector or matrix and use bold face letters for matrices. We denote the ensemble average over all trajectories $[z]$ generated with dynamics corresponding to $ {\bm{k}}$ with the brackets $\l \dots \r_{{\bm{k}}} $.
\subsection{Large deviations of empirical time-averaged variables}
\label{LDtimeAVGvar}
Throughout the paper, we assume that the long-time statistics of time-extensive variables obey a large deviation principle. For instance, let $z(t)$ be the position at time $t$ of a random walker on a one-dimensional circular lattice and $X[z]$ the number of steps the walker takes during the trajectory $[z]$. We remark that the variable $X$ is a functional of the trajectory $[z]$ that is a realization of a stochastic process. $X$ is not a random variable in itself. When $X$ is not evaluated on a trajectory, it refers either to the physical variable ``number of step'' or to a numerical value of this variable. The variable $X[z]$ is \emph{time-extensive} since $X[z]+X[z']=X[z,z']$, where $[z,z']$ denote the trajectory made with $[z]$ followed by $[z']$. Then, the number of transitions typically increases with time. Accordingly, $v[z]=X[z]/t$ is the number of steps per unit time and is regarded as a \emph{time-averaged} variable. At long time, it converges to the step frequency of the walker. The probability of $ v[z] = v $, i.e. that the time-averaged number of steps $v[z]$ takes the value $v$ at long time $t$, is $P_t (v) \simeq e^{-t I(v)}$.
The function $I$ is called a large deviation function (LDF). It is non-negative and vanishes at $v = \l v[z] \r_{{\bm{k}}}$, denoting that
the ensemble average value is the most likely time-averaged $v$. Small (respectively large) deviations correspond to the time-averaged number of steps that are close-to (respectively far-from) the ensemble average value. These events become exponentially unlikely with increasing time for ergodic systems. The convexity of the LDF ensures that a large deviation is less likely than a small fluctuation.
Following, we introduce the empirical time-averaged variables used to derive our central results. We name \emph{empirical variables} those that are defined from experimental observations of the system and that usually depend on the observed trajectory $[z]$.
First, we define the empirical occupation in $x$ by
\begin{equation}
p_x[z] \equiv \frac{1}{t} \int_0^t \mathrm{d} \tau \delta_{x,z(\tau)},
\end{equation}
where $\delta$ is the Kronecker symbol. Given the probability of each state being gathered into the column vector $ p=(p_1, \cdots, p_M)^\dag $, the Shannon entropy $s =s(p)$ is
\begin{equation}
s(p)\equiv - \sum_x p_x \ln p_x = - ( p^\dag \cdot \ln p) , \label{def:AvgS}
\end{equation}
and the energy $e=e(p)$ is
\begin{equation}
e(p) \equiv \sum_x \epsilon_x p_x \label{def:AvgE} = \epsilon^\dag \cdot p,
\end{equation}
with the central dot denoting the matrix product and $\dag$ the transposition.
The time-averaged energy along trajectory $ [z] $ can be written
$ e[z] = e( p[z]) $, and similarly for the entropy.
Second, we define the empirical transition probability from $y$ to $x$ induced by reservoir $\nu$
\begin{equation}
\omega^{\nu}_{xy}[z] \equiv \frac{1}{t} \sum_{\tau \in [0,t]} \delta_{x,z(\tau+d\tau)}\delta_{y,z(\tau)} \delta_{\nu, \nu(\tau)},
\label{def:TimeAvgOmega}
\end{equation}
where the sum is over all time $\tau$ at which the system changes from state $z(\tau)$ to state $ z(\tau+d\tau) $, exchanging energy with reservoir $ \nu(\tau) $. Given a transition probability $ \omega^{\nu}_{xy} $ for each possible transitions, the current of energy received from reservoir $\nu$ by the system is
\begin{equation}
j_\nu(\bm{\omega}) \equiv \frac{1}{2}\sum_{x,y} \left( \omega^{\nu}_{xy} -\omega^{\nu}_{yx} \right)(\epsilon_x-\epsilon_y).
\label{def:TimeAvgJ}
\end{equation}
Its empirical value during trajectory $ [z] $ is written $ j_\nu[z] = j_\nu (\bm{\omega}[z]) $. These time-averaged currents describe the anti-symmetric part of fluctuations since they change sign upon time-reversal of the trajectories. On the contrary, the weighted frequency of interaction with reservoir $ \nu $, named \emph{activity} for short and written
\begin{equation}
f_\nu(\bm{\omega}) \equiv \frac{1}{2}\sum_{x,y\neq x} \left( \omega^{\nu}_{xy} +\omega^{\nu}_{yx} \right )d_{xy},
\label{def:TimeAvgF}
\end{equation}
describes the symmetric part of fluctuations. Indeed, $f_\nu[z] = f_\nu ( \bm{\omega}[z] )$ does not change sign upon time-reversal of the trajectory $[z]$.
When the activity is low (high), the system either changes of state less (more) frequently or mostly switches between states with low (high) $d_{xy}$.
The term ``activity'' was proposed to qualify the symmetric part of the fluctuations in \cite{Lecomte2007_vol127, Baiesi2009_vol103, Baiesi2009_vol137, Baerts2013_vol88, Polettini2015_vol48}, see also references therein. Let us finally remark that, in the definitions of the energy currents and activities, the one-half factor is just a symmetry factor since we can sum over transitions disregarding their directions ($\sum_{x,y}$) or for only one direction ($\sum_{x>y}$). Half of the first sum is equivalent to the second sum.
\subsection{LDF of energy currents, activities and occupation from a variational approach}
\label{EQ-LDF}
At long time $ t $, the probability of observing an empirical transition probability $\bm{\omega}[z]=\bm{\omega}$ and an empirical occupation $p[z]=p$ is
\begin{equation}
P_t(\bm{\omega},p ) \underset{t \rightarrow \infty}{\simeq} e^{-t I(\bm{\omega},p)}. \label{AsymPJumpOccup}
\end{equation}
From the work of Maes and Neto$\check{c}$n\'y \cite{Maes2008_vol82}, Wynants \cite{Wynants2010_vola} or Bertini \textit{et al.} \cite{Bertini2015_vol51}, the LDF $I(\bm{\omega},p)$ of the empirical transition probabilities and occupations for the continuous-time Markov chain with generator ${\bm{k}}$ is
\begin{equation}
I(\bm{\omega},p) = \sum_{x,y \neq x, \nu} \left[ k^{\nu}_{xy}p_y - \omega^{\nu}_{xy} + \omega^{\nu}_{xy} \ln \frac{\omega^{\nu}_{xy} }{ k^{\nu}_{xy} p_y} \right ],
\label{LDF:OccupJump}
\end{equation}
where the sum is over $\nu$ from $1$ to $\chi$ and all couples $(x,y)$ such that $x \neq y$.
The derivation of Eqs.~(\ref{AsymPJumpOccup}-\ref{LDF:OccupJump}) is reproduced in Appendix \ref{LDFoccupationandjump}.
In Ref.~\cite{Maes2008_vol82}, the LDF of the occupation and probability current was obtained from a constrained optimization problem constructed with $I(\bm{\omega},p)$. This procedure, called ``contraction'' \cite{Oono1989_vol99}, is equivalent at the level of probabilities to marginalize $P_t(\bm{\omega},p)$ to obtain the probability of currents and occupations. We now proceed to the contraction of $I(\bm{\omega},p)$ to obtain the LDF of energy currents, activities and occupations denoted $L(j,f,p)$. The long-time asymptotic approximation of the probability $P_t(j,f,p)$ that $j[z]=j$, $f[z]=f$ and $p[z]=p$ at time $t$ will then read
\begin{equation}
P_t(j,f,p) \underset{t \rightarrow \infty}{\simeq} e^{ -t L(j,f,p)}. \label{ProbaAsymptotic}
\end{equation}
We prove in Appendix \ref{AppendixCountStat} a sharper approximation of this probability that involves a pre-exponential factor dictating the thermodynamic behavior: it leads to the statistics of the usual EQ thermodynamic variables that only depend on the system state, such as energy for instance. It is the first correction to the exponential decay of non-typical time-extensive variables after a long time. This prefactor was first obtained in Ref. \cite{Polettini2015_vol48}, but we provide in Appendix \ref{AppendixCountStat} a logically independent derivation (though restricted to the large time limit) that involves some results of Sec.\ref{NEfluctuations}.
At long time, the energy current $j $, activity $f$ and occupation $p$ mainly appear thanks to the most likely event producing them. The probability of this event is associated with smaller values of $I(\bm{\omega},p)$ with $\bm{\omega}$ constrained by the value of the energy currents and activity. For this reason, and in virtue of the contraction principle, we minimize $I(\bm{\omega},p)$ under the energy currents constraint
\begin{equation}
j_\nu=j_\nu(\bm{\omega}), \label{CurrentConstraints}
\end{equation}
for $\nu > 1$, because current conservation imposes $j_1 = -\sum_{\nu\neq 1} j_\nu $. We also impose the activity constraint
\begin{equation}
f_\nu=f_\nu(\bm{\omega}). \label{TrafficConstraints}
\end{equation}
In addition, the probability currents should be compatible with the conservation of the norm of the occupation vector, i.e. for all $y$
\begin{equation}
\sum_{x ,\nu} \left(\omega^{\nu}_{xy}- \omega^{\nu}_{yx}\right) = 0 \label{ConsNormProba}.
\end{equation}
To perform our optimization problem, we use the following cost function
\begin{multline}
\mathcal{F}(\bm{\omega},p) = I(\bm{\omega},p) + \sum_{\nu } a_\nu [j_\nu - j_\nu(\bm{\omega})] \\
+ \sum_{\nu } b_\nu [f_\nu - f_\nu(\bm{\omega})] + \sum_{x,y,\nu} \u_y \left( \omega^{\nu}_{xy}- \omega^{\nu}_{yx} \right) \label{OptimizationFunctional}
\end{multline}
where $a_\nu$, $ b_\nu $ and $ \u_y $ are Lagrange multipliers that will be chosen to satisfy the constraints of Eqs.~(\ref{CurrentConstraints}-\ref{ConsNormProba}).
We choose $a_1= 0$ so as not to constrain the current $j_1$ that is already set by the current conservation law. We now minimize the function $\mathcal{F}$ with respect to $\bm{\omega}$, calculating $\partial \mathcal{F} / \partial \omega^{\nu}_{xy} = 0$ to get
\begin{equation}
0 = \ln \frac{\omega^{\nu}_{xy} }{ k^{\nu}_{xy} p_y} - a_\nu(\epsilon_x-\epsilon_y) - b_\nu d_{xy} + (\u_y-\u_x), \label{eq:OptimJump}
\end{equation}
where we have used Eq.~(\ref{LDF:OccupJump}) and Eqs.~(\ref{CurrentConstraints}-\ref{TrafficConstraints}). Therefore, the optimal transition probability in terms of the Lagrange multipliers satisfies
\begin{equation}
\omega^{\nu}_{xy} = K^{\nu}_{xy} p_y, \label{def:BiasedRates}
\end{equation}
where we have introduced $ \bm{K}^{\nu} = \bm{K}^{\nu}(a,b,\u) $, the transition probability for mechanism $ \nu $ divided by the empirical occupation of the state before transition. Its off-diagonal elements are
\begin{equation}
K^{\nu}_{xy} \equiv k^{\nu}_{xy} e^{a_\nu (\epsilon_x-\epsilon_y)+b_\nu d_{xy} + \u_x-\u_y} , \label{BiasRate}
\end{equation}
or more explicitly using Eq.~\ref{defEQrates}
\begin{equation}
K^{\nu}_{xy} = \gamma_{xy}^{\nu} e^{-(\beta_1/2-a_\nu) (\epsilon_x-\epsilon_y) - ( \beta_1 l_1 - b_\nu ) d_{xy}+ \u_x-\u_y}, \label{ExplicitBiasRate}
\end{equation}
and the diagonal elements are
\begin{equation}
K^{\nu}_{yy} = - \sum_{x\neq y} K^{\nu}_{xy} \equiv - \Lambda^{\nu}_y , \label{DiagBiasRate}
\end{equation}
such that any column of any matrix $ \bm{K}^{\nu} $ sums to zero. We remark that the matrices $ \bm{K}^{\nu}$ satisfy a modified detailed-balance relation
\begin{equation}
\ln \frac{ K^{\nu}_{xy}}{ K^{\nu}_{yx}} = (2a_\nu- \beta_1)(\epsilon_x-\epsilon_y) + 2(\u_x-\u_y). \label{NEdetailbalance}
\end{equation}
In this local detailed balance, the Lagrange multiplier $a_\nu$ biases the inverse temperatures $\beta_1$ to make typical the energy exchanges corresponding to the energy currents constraint. The reservoir $\nu$ behaves as if it had the temperature $\beta_\nu \equiv \beta_1-2a_\nu$ in order to satisfy the current constraint. Thus, the variable
\begin{equation}
2 a_\nu = \beta_1-\beta_\nu
\end{equation} is an \emph{affinity} \cite{Book_Prigogine1955,Book_Donder1927,Tome2015_vol91}, also called \emph{thermodynamic force} \cite{Andrieux2007_vol127,Andrieux2007_vol}. Notice that $a_1=0$ as required. The similarity between Eqs.~(\ref{defEQrates}) and (\ref{ExplicitBiasRate}) indicates that we can also introduce new dilatation factors $l_\nu$ such that the \emph{dynamical bias}
\begin{equation}
b_\nu \equiv \beta_1 l_1 -\beta_\nu l_\nu
\end{equation}
gives the modification of the dynamics in order to satisfy the activity constraint. Finally, we call the variable $\u$ the \emph{drift} because it acts like a force biasing each transition.
The explicit solution $ \bm{\omega} $ of our variational problem $ \mathrm{d} \mathcal{F} = 0 $ is now almost reached. The next step is to use the constraints of Eqs.~(\ref{CurrentConstraints}-\ref{ConsNormProba}) to obtain the Lagrange multipliers. More explicitly the constraint equations are
\begin{eqnarray}
j_{\nu}&=& \frac{1}{2}\sum_{x,y} \left( K^{\nu}_{xy}p_y - K^{\nu}_{yx}p_x \right)(\epsilon_x-\epsilon_y), \label{EqOfStatej} \\
f_{\nu}&=& \frac{1}{2}\sum_{x,y} \left( K^{\nu}_{xy}p_y + K^{\nu}_{yx}p_x \right)d_{xy}, \label{EqOfStatef} \\
0 &=& \bm{K} \cdot p, \label{StationaryEq}
\end{eqnarray}
where $\bm{K} = \sum_\nu \bm{K}^{\nu}$ is the generator of the \emph{driven reference process} \cite{Chetrite2015_vol16,Chetrite2015_vol2015}.
For the third equation, the conservation law of the probability current of Eq.~(\ref{ConsNormProba}) is reformulated as a requirement that the empirical occupation $p$ is the stationary probability of the continuous-time Markov chain with rate matrix $\bm{K}= \bm{K}(a,b,\u)$. Inverting these three equations gives the vectors $a$, $b$ and $u$ as a function of $(j,f,p)$.
The final step to obtain the asymptotic probability of energy currents, activities and occupations is to write the LDF of Eq.~(\ref{LDF:OccupJump}) at the optimal transition probability of Eq.~(\ref{def:BiasedRates}). This leads to
\begin{equation}
L(j,f,p) = a^\dag \cdot j + b^\dag \cdot f + \left( \lambda- \Lambda \right)^\dag \cdot p , \label{LDFExtensive}
\end{equation}
where we have used the anti-symmetry of $\epsilon_x-\epsilon_y$ or symmetry of $d_{xy}$ in the exchange of $x$ and $y$ to make explicit the dependence in $ j $ and $ f $. We also used Eq.~(\ref{ConsNormProba}) to get rid of the term involving $\u_x-\u_y$.
\subsection{Scaled CGF of energy currents, activities and occupations from a variational approach}
\label{EQ-CGF}
In the previous section, we have obtained the LDF $L$ from the solution of an optimization problem. From now on, and for the remainder of the paper, we assume the convexity of the LDF. Our aim here is to derive the scaled CGF conjugated to $L$ from a variational approach, using the fact that LDF and scaled CGF are conjugated by Legendre transformation \cite{Oono1989_vol99,Touchette2009_vol478,Chetrite2015_vol2015}. On the way, we obtain useful properties associated to the canonical structure.
The scaled CGF of the energy current, activity and occupation is defined by
\begin{equation}
\Gamma(a',b',m') \equiv \lim_{t \rightarrow \infty} \frac{1}{t} \ln \l e^{t(a'^\dag\cdot j[z] + b'^\dag\cdot f[z] + m'^\dag \cdot p[z])} \r_{{\bm{k}}}, \label{DefCGF}
\end{equation}
and is the Legendre transformation of $L$
\begin{equation}
\Gamma(a',b',m') = \max_{p,j,f} \left[ a'^\dag \cdot j + b'^\dag \cdot f + m'^\dag \cdot p - L(p,j,f) \right ].
\end{equation}
The maximum on $j$ and $f$ is reached for $a'=a$ and $b'=b$, and the scaled CGF becomes
\begin{equation}
\Gamma(a,b,m') = \max_{ p \,|\, \bm{K} \cdot p =0} \left[ (m'+ \Lambda - \lambda)^\dag \cdot p \right],
\end{equation}
where the maximum is taken over all occupations with the Lagrange multiplier $\u$ in the generators of the driven process $ \bm{K}$ tuned such that $ \bm{K} \cdot p =0$. An alternative expression of the scaled CGF of energy current, activity and occupation is
\begin{equation}
\Gamma(a,b,m') = \max_{ u } \left[ (m'+ \Lambda- \lambda)^\dag \cdot p \right] \label{VariationalCGF}
\end{equation}
with $p$ the stationary probability associated to $ \bm{K}$. From the optimal drift $\u=\u(a,b,m')$ realizing the maximum in Eq.~(\ref{VariationalCGF}), we introduce the \emph{escape weight} $m\equivm(a,b,\u)$ giving the value of $m'$ for given $(a,b,\u)$. In Eq.(G14) of the appendix of reference \cite{Nemoto2011_vol}, Nemoto and Sasa gave the scaled CGF of energy current from a variational expression analogous to our Eq.~(\ref{VariationalCGF}). We recover their result taking $b =0$ and $m'=0$. We further comment Eq.~(\ref{VariationalCGF}) noticing that the maximum is reached for $\u$ satisfying
\begin{equation}
\Gamma(a,b,m)= m_y + \Lambda_y - \lambda_y \label{exitrule},
\end{equation}
for all $y$. This equation allows us to derive the following escape-rate rule
\begin{equation}
m_y + \Lambda_y - \lambda_y = m_x + \Lambda_x - \lambda_x,
\end{equation}
that can be related to the exit rate constraint of Refs.~\cite{Chetrite2015_vol16, Baule2008_vol101, Baule2010_vol2010} taking $m = 0$. To prove Eq.~(\ref{exitrule}), we introduce the \emph{tilted operator} ${\bm{\kappa}} ={\bm{\kappa}}(a,b,m) $ for the EQ reference process
\begin{eqnarray}
\kappa_{yy} &\equiv& -\sum_{x\neq y, \nu} k^{\nu}_{xy} + m_y, \label{DressedOperatorDiag}\\
\kappa_{xy} &\equiv& \sum_{\nu} k^{\nu}_{xy} e^{a_\nu(\epsilon_x-\epsilon_y)+b_\nu d_{xy}} . \label{DressedOperator}
\end{eqnarray}
The generator of the driven reference process $ \bm{K}$ is connected to this tilted operator by
\begin{equation}
K_{xy} = e^{\u_x} \kappa_{xy} e^{-\u_y}
- \left(m_y + \Lambda_y- \lambda_y \right) \delta_{xy}. \label{kappaToK}
\end{equation}
Using this equation and $ \bm{K} \cdot p = 0$, we find
\begin{equation}
\sum_y e^{\u_x} \kappa_{xy} e^{-\u_y} p_y = \left(m_x + \Lambda_x- \lambda_x \right) p_x,
\end{equation}
after summing over $x$ and maximizing over $\u$, it follows from Eq.~(\ref{VariationalCGF}) that the drift giving the maximum satisfies
\begin{equation}
\sum_{x,y} e^{\u_x} \kappa_{xy} e^{-\u_y} p_y = \Gamma.
\end{equation}
By definition \cite{Lebowitz1999_vol95}, the scaled CGF $ \Gamma(a,b,m)$ is the highest eigenvalue of $ {\bm{\kappa}}$. Then $\pi_x \equiv \pm e^{\u_x} / Z(\u) $ is the normalized left eigenvector of ${\bm{\kappa}}$ with $Z(\u)$ a normalization constant such that $ \sum_x \pi_x = 1$.
The vector $r \equiv \bm{\pi}^{-1} \cdot p $ is a right eigenvector with $\bm{\pi}_{xy} \equiv \pi_x \delta_{xy}$. Its norm is set by $\sum_x \pi_x r_x = \sum_x p_x = 1$. Notice that we cannot determine from the values of $\u$ the sign of each component of the vectors $\pi$ and $r$, but their $x$ components share the same sign. Now, summing Eq.~(\ref{kappaToK}) over $x$ leads to Eq.~(\ref{exitrule}) since $\sum_x K_{xy} = 0$ and $\sum_x e^{\u_x} \kappa_{xy} e^{-\u_y} = \Gamma$.
Then, the optimal drift $\u = \u(a,b,m)$, leading to the maximum in Eq.~(\ref{VariationalCGF}), is simply obtained from the left eigenvector of the tilted operator by $ \ln |\pi_x| = u_x - \ln |Z(\u)|$ up to a constant that plays no role, since only differences of drifts matter. The drift makes the escape-rate rule holds true and, using Eqs.~(\ref{LDFExtensive}) and (\ref{exitrule}), leads to the Legendre structure that one expects for LDFs and scaled CGFs. Finally, from Eqs.~(\ref{exitrule}) and (\ref{kappaToK}), we recover the results of Refs.~\cite{Jack2010_vol184, Chetrite2015_vol16, Chetrite2013_vol111, Chetrite2015_vol2015} in which the generator $\bm{K}$ of the driven process corresponds to the Doob's transformation of the tilted operator ${\bm{\kappa}}$
\begin{equation}
K_{xy} = |\pi_x| \kappa_{xy} |\pi_y|^{-1}
- \Gamma \delta_{xy}.
\end{equation}
Notice that in Refs.~\cite{Chetrite2015_vol16, Chetrite2013_vol111, Chetrite2015_vol2015}, the right eigenvector of the tilted operator is used in the Doob's transformation instead of the left one, since the tilted operator in these references is the adjoint of ${\bm{\kappa}}$.
\section{Non-equilibrium fluctuations}
\label{NEfluctuations}
\subsection{Definition of the NE process}
\label{DefNEprocess}
The \emph{NE process} is defined by the rate matrices $ \bar {\bm{k}}^{\nu} = \bar {\bm{k}}^{\nu}(a_\nu,b_\nu)$ associated to energy exchanges with each reservoir $\nu$ at different temperatures $\beta_\nu =\beta_1 -2a_\nu $ and with different dilatation factors $l_\nu$ related to dynamical bias by $b_\nu = \beta_1 l_1 - \beta_\nu l_\nu$. The elements of the rate matrices are
\begin{equation}
\bar k_{xy}^{\nu} \equiv \gamma_{xy}^{\nu} e^{-(\beta_1/2-a_\nu) (\epsilon_x-\epsilon_y) - (\beta_1 l_1 - b_\nu) d_{xy}} \label{defNErates}.
\end{equation}
Accordingly, the escape rate $\bar \lambda^{\nu}_y= \bar \lambda^{\nu}_y(a_\nu,b_\nu)$ from state $y$ is
\begin{equation}
\bar \lambda^{\nu}_y \equiv \sum_{x\neq y} \bar k^{\nu}_{xy} = -\bar k^{\nu}_{yy}.
\end{equation}
We define a total rate matrix by $ \bar {\bm{k}} \equiv \sum_\nu \bar {\bm{k}}^{\nu}$ and a total escape-rate vector by $\bar \lambda \equiv \sum_\nu \bar \lambda^{\nu}$. These rates are functions of the affinities and dynamical bias; their analogs for the reference process are recovered at the point of vanishing of $a$ and $b$, namely $ {\bm{k}}= \bar {\bm{k}} (0,0)$ and $ \lambda = \bar \lambda(0,0)$. For the NE process, the state at time $\tau$ is $\bar z(\tau)$. A system state trajectory during time interval $[0,t]$ is denoted $[\bar z]$. The ensemble average over all trajectories $[\bar z]$ generated with dynamics corresponding to $ \bar {\bm{k}}$ is $\l \dots \r_{\bar {\bm{k}}} $.
\subsection{Mapping typical NE fluctuations on rare EQ fluctuations \label{MappingNEonEQ}}
We now connect the energy currents, activities and occupations statistics for the EQ process with the statistics of the same variables for the stationary NE process. This mapping involves the \emph{escape-rate change} $c=c(a,b)$ defined by
\begin{equation}
c \equiv \lambda - \bar \lambda, \label{EscapeRateChange}
\end{equation}
that is zero at vanishing affinities and dynamical biases. We emphasize that $c$ cannot be adjusted independently of $a$ and $b$. This means that the affinity and the dynamical bias are the central variables in determining the NESS reached by the system.
To connect EQ and NE fluctuations, one needs to redo all the calculations of sections \ref{EQ-LDF} and \ref{EQ-CGF}, but for the NE process, introducing the NE scaled CGF $\bar \Gamma = \bar \Gamma(\bar a, \bar b, \bar m) $ and LDF $\bar L=\bar L(j,f,p)$, the NE tilted operator $\bar {\bm{\kappa}} = \bar {\bm{\kappa}}(\bar a, \bar b, \bar m)$, the generator of the NE driven process $\bar \bm{K} = \bar \bm{K}(\bar a, \bar b, \bar \u) $ and associated escape rate $\bar \Lambda=\bar \Lambda (\bar a, \bar b, \bar \u)$, the affinity increment $2\bar a $, the dynamical bias increment $\bar b$, the NE drift $\bar \u$ and the NE escape weight $\bar m = \bar m(\bar a, \bar b, \bar \u)$, all denoted with a bar to distinguish them from their equivalent for the EQ reference process. One obtains all these objects replacing $ k$ by $ \bar k$ and the Lagrange multipliers $(a,b,\u) $ by $ (\bar a,\bar b, \bar \u)$ in all the definitions. For instance, for the NE tilted operator, we have
\begin{eqnarray}
\bar \kappa_{yy} &\equiv& -\sum_{x\neq y, \nu} \bar k^{\nu}_{xy} + \bar m_y, \label{NEDressedOperatorDiag}\\
\bar \kappa_{xy} &\equiv& \sum_{\nu} \bar k^{\nu}_{xy} e^{\bar a_\nu(\epsilon_x-\epsilon_y)+\bar b_\nu d_{xy}} . \label{NEDressedOperator}
\end{eqnarray}
Notice that we call $2\bar a$ an affinity ``increment'' since we already deal with a NE process: a deviation from the typical current is associated with an ``increase'' of affinity that will make this fluctuation typical. For the same reason, the dynamical bias $ \bar b$ is also qualified as an increment.
The mapping between EQ and NE fluctuations now comes from the connection between the EQ and NE tilted operators
\begin{equation}
\bar {\bm{\kappa}}(\bar a ,\bar b, \bar m) = {\bm{\kappa}}(\bar a + a,\bar b +b,\bar m +c), \label{DresOpmapppingNEtoEQ}
\end{equation}
that we obtain by comparing Eqs.~(\ref{DressedOperatorDiag}-\ref{DressedOperator}) with Eqs.~(\ref{NEDressedOperatorDiag}-\ref{NEDressedOperator}).
Hence, the same symmetry exists between the eigenvalues and between the eigenvectors: the \emph{full} spectrum of the two operators is connected. In particular, the scaled CGFs are connected by
\begin{equation}
\bar \Gamma(\bar a ,\bar b, \bar m) = \Gamma(\bar a + a,\bar b +b,\bar m +c), \label{CGFmappingNEtoEQ}
\end{equation}
and, from the Legendre transformation, the LDFs verify
\begin{equation}
\bar L(j,f,p) = L(j,f,p)- a^\dag\cdot j - b^\dag\cdot f - c^\dag \cdot p. \label{LDFmappingNEtoEQ}
\end{equation}
The left eigenvectors of the tilted operators satisfy
\begin{equation}
\bar \pi(\bar a ,\bar b, \bar m)= \pi(\bar a + a,\bar b +b,\bar m +c) \label{PimappingNEtoEQ}
\end{equation}
or equivalently
\begin{equation}
\bar \u(\bar a ,\bar b, \bar m)= \u(\bar a + a,\bar b +b,\bar m +c). \label{drifmappingNEtoEQ}
\end{equation}
The mapping also holds for the right eigenvectors and this leads to
\begin{equation}
\bar p(\bar a ,\bar b, \bar m)= p(\bar a + a,\bar b +b,\bar m +c). \label{ProbamappingNEtoEQ}
\end{equation}
Finally, the generators of the driven processes also verify
\begin{equation}
\bar \bm{K}(\bar a, \bar b,\bar \u) = \bm{K}(\bar a + a,\bar b +b, \u),
\end{equation}
where $\bar \u$ and $ \u$ are respectively the left- and right-hand sides of Eq.~(\ref{drifmappingNEtoEQ}).
Thus, the EQ and NE processes are tightly connected and one can focus on the EQ process' fluctuations only:
Eq.~(\ref{CGFmappingNEtoEQ}) shows that the statistics of energy currents, activities and occupations for any NE process with affinity $2a$ and dynamical bias $b$ is known from the statistics of the same variables computed for the EQ process. Indeed, the derivatives of Eq.~(\ref{CGFmappingNEtoEQ}) with respect to $\bar a$, $\bar b$ or $\bar m$ evaluated in $(\bar a,\bar b,\bar m) = (0,0,0)$ yields the NE cumulants of the energy currents, activities and occupations from the scaled CGF for the EQ reference process, e.g. for $j_\nu$ we have
\begin{equation}
\l j_\nu[\bar z] \r_{\bar {\bm{k}}} = \frac{\partial \bar \Gamma}{\partial \bar a_\nu}(0,0,0) = \frac{\partial \Gamma}{\partial a_\nu}(a,b,c).
\end{equation}
Notice that evaluating Eq.~(\ref{CGFmappingNEtoEQ}) at the point of vanishing of $(\bar a ,\bar b, \bar m)$ returns by definition of a scaled CGF
\begin{equation}
0 = \bar \Gamma(0,0,0) = \Gamma(a,b,c) , \label{AVGpoint}
\end{equation}
for all $a$ and $b$, with $c = (\lambda-\bar \lambda)$. Accordingly, the total derivatives of $ \Gamma(a,b,c)$ with respect to $a$ or $b$ also vanish exactly such that $ \Gamma$ remains constant and equal to zero in the direction $(a,b,c)$. We call the subspace where $\Gamma$ vanishes the \emph{physical system subspace}: each point $(a,b,c)$ in this subspace defines a precise physical process with affinity $2a$ and dynamical bias $b$. The function $ \Gamma$ includes the full thermodynamic information on any system defined with the same energy levels $\epsilon$, coupling matrices $\bm{\gamma}^{\nu}$ and energy barriers $\bm{d}$ (up to a reservoir specific dilatation), and so does the LDF $L$. One simply changes the degree of NE or the type of dynamics, encoded into the dilatation factors, by moving into the physical system subspace.
We end by remarking that the idea of mapping EQ and NE fluctuations was first proposed by Andrieux in Refs. \cite{ Andrieux2012_vola, Andrieux2012_vol}, but for the statistics of energy currents only. However, this mapping had no concrete application since the NE statistics of the currents were needed to define the EQ dynamics involved in the mapping. On the contrary, the mapping of Eq.~(\ref{CGFmappingNEtoEQ}) and (\ref{LDFmappingNEtoEQ}) is explicit, with the price that, when comparing with Ref. \cite{ Andrieux2012_vola, *Andrieux2012_vol}, the EQ statistics of activities and occupations must be known in addition to the energy currents statistics.
\subsection{Asymptotic equivalence of the driven reference process and the NE process \label{EnsembleEquivalence}}
We now discuss the asymptotic equivalence of the driven reference process and the NE process. We first prove the equality of their escape rates and on the way give a slightly simplified expression of $L$. Using this result, we demonstrate the equivalence of the path probabilities of the driven reference process and the NE process.
From Eqs.~(\ref{exitrule}) and (\ref{AVGpoint}), we find $c + \Lambda -\lambda = 0$. This leads with Eq.~(\ref{EscapeRateChange}) to the equality of the escape rates of the driven reference process and the NE process
\begin{equation}
\Lambda= \bar \lambda, \label{DrivEQandNEescapeEquality}
\end{equation}
even though these two processes are different in general due to the drift $\u$, i.e. $K_{xy} \neq \bar k_{xy} $ if $x\neq y$. As a consequence, the LDF is written as
\begin{equation}
L(j,f,p) = a^\dag \cdot j + b^\dag \cdot f + c^\dag \cdot p.
\end{equation}
The equality of the escape rates indicates that the driven reference process and NE process look alike. Their generators are connected by the similarity transformation
\begin{equation}
\bm{K} = |\bm{\pi}| \cdot \bar {\bm{k}} \cdot | \bm{\pi} |^{-1},
\end{equation}
that follows from the comparison of Eqs.~(\ref{ExplicitBiasRate}) and (\ref{defNErates}). We denote $|\bm{\pi}|$ the positive and diagonal matrix obtained by taking the absolute value of the elements of $\bm{\pi}$. The equality of the diagonal part of the Markov matrices of the two processes is granted by Eq.(\ref{DrivEQandNEescapeEquality}). From this similarity transformation, one can show the asymptotic equality of the path probabilities associated to each process
\begin{equation}
\P_{\bm{K}}[y] \underset{t \rightarrow \infty}{\simeq} \P_{\bar {\bm{k}}}[y],
\end{equation}
for any trajectory $[y]$. We have defined the path probabilities knowing the initial state $y(0)$
\begin{equation}
\P_{\bar {\bm{k}}}[y] \equiv \exp \left( -\int_0^t \mathrm{d} \tau \bar \lambda_{y(\tau)} \right) \prod_{\tau\in [0,t]}\bar k^{\nu(\tau)}_{y(\tau+d\tau)y(\tau)},
\end{equation}
for the NE process and
\begin{equation}
\P_{\bm{K}}[y] \equiv \exp \left( -\int_0^t \mathrm{d} \tau \Lambda_{y(\tau)} \right) \prod_{\tau\in [0,t]} K^{\nu(\tau)}_{y(\tau+d\tau)y(\tau)},
\end{equation}
for the driven reference process.
In these equations, the product applies for all times $\tau$ at which the system changes of state during the trajectory $[y]$, with $y(\tau)$ (respectively $y(\tau+d\tau)$) the system state before (respectively after) the transition at time $\tau$. The exponential terms appearing in these two path probabilities are equal. Concerning the product terms, they differ from boundary terms only
\begin{eqnarray}
\prod_{\tau} K^{\nu(\tau)}_{y(\tau+d\tau)y(\tau)} &=& \prod_{\tau} |\pi_{y(\tau+d\tau)}| \bar k^{\nu(\tau)}_{y(\tau+d\tau)y(\tau)} |\pi_{y(\tau)}| ^{-1}, \nonumber \\
&=& |\pi_{y(t)}| \left( \prod_{\tau} \bar k^{\nu(\tau)}_{y(\tau+d\tau)y(\tau)} \right) |\pi_{y(0)}| ^{-1}. \nonumber \\
\end{eqnarray}
Then, the path probabilities of the driven reference process and NE process verify
\begin{equation}
\lim_{t\rightarrow \infty} \frac{1}{t} \ln \frac{\P_{\bar {\bm{k}}} [y]}{\P_{\bm{K}} [y]} =0,
\end{equation}
and are asymptotically equivalent \cite{Chetrite2015_vol16}. Since the driven reference process is the dynamics that typically reproduces the conditioned reference process, we conclude that there is an ensemble equivalence between the NE process and the conditioned reference process. This central result is similar to the path-ensemble equivalence derived in Refs. \cite{Chetrite2015_vol16,Chetrite2013_vol111}. In Appendix \ref{EnsembleEquivalence2}, we show that the NE process is asymptotically equivalent to the canonical process that is defined by exponentially weighting each trajectory, even though these two processes are not exactly identical.
\section{Discussion and general summary \label{discussion}}
\begin{table*}
\caption{EQ and NE thermodynamic potentials. \label{summary}}
\begin{tabular}{lcccc}
\hline
\hline
Ensemble & \quad Micro-canonical \quad & \quad Canonical \quad & \quad NE micro-canonical \quad & \quad Meta-canonical \quad \\
\hline
Potential & $s = -\sum_x p_x \ln p_x$ & $\varphi = -\ln \l \exp{(-\beta_1 e[z])} \r_{{\bm{k}}}$ & $L(j,f,p)$ & $\Gamma (a,b,m)$ \\
Variational principle & max & min & max & max \\
Free variables & $ \beta_1$ & $e$ & $ a $, $ b $, $m$ & $ j $, $ f $, $p$ \\
Constrained variables & $ e $ & $ \beta_1 $ & $ j $, $ f $ , $p$ & $ a $, $ b $, $m$ \\
Physical system subspace&--- &--- & $m(j,f,p) = c\big(a(j,f,p),b(j,f,p)\big)$ & $m =c(a,b)$ \\
No dilatation space &--- &--- & $b(j,f,p) = 2a(j,f,p)$ & $b=2a$ \\
Legendre structure & \multicolumn{2}{c}{$s + \varphi = \beta_1 e $ }
& \multicolumn{2}{c}{$L + \Gamma = a^\dag \cdot j + b^\dag \cdot f + m^\dag \cdot p $ }\\
\hline
\hline
\end{tabular}
\end{table*}
In Sec.\ref{EQfluctuation}, we have studied the fluctuations of an EQ system exchanging energy with several heat reservoirs at the same temperature. We have seen that energy may spontaneously flow from one reservoir to another, even if it does not on average. Each of these current fluctuations has been associated to a temperature difference that would typically reproduce it. Similarly, we have shown that a fluctuation of the activity of the exchanges with each reservoir would be typically reproduced by dilating the appropriated energy barriers. From these observations, we have identified two couples of conjugated variables and provided the corresponding LDF and CGF from a variational approach.
In Sec.\ref{NEfluctuations}, we have considered the fluctuations of the system defined in Sec.\ref{EQfluctuation}, but driven out-of-equilibrium by temperature differences between the heat reservoirs. We have found an exact mapping between the statistics of the energy currents, activities and occupations for the EQ and NE systems. We have also discussed the asymptotic equivalence of the trajectory ensembles generated by the conditioned EQ process and the NE process. From the existence of the mapping between EQ and NE systems, we have concluded that the study of a NE system amounts to the calculation of the probability of rare fluctuations of the same system at EQ. Now that the distinction between the dynamical fluctuations of EQ and NE systems has been dispelled, we come back to the results of Sec.\ref{EQfluctuation} and summarize the canonical structure satisfied by the two ensembles of NE systems.
\subsection{Summary of the NE canonical structure}
\label{CanonicalStructure}
The ensemble of systems in contact with several heat reservoirs at different temperatures is called the \emph{meta-canonical ensemble}. The trajectories of the systems in the meta-canonical ensemble are generated by the NE process with generator $\bar {\bm{k}}$.
All the systems in this ensemble have the same energy levels $\epsilon_x$, and the same dynamical parameters, i.e. energy barriers $d_{xy}$ and couplings with the heat reservoirs $\gamma^\nu_{xy}$. By convention, the heat reservoir of smallest temperature is the reference reservoir ($\nu=1$) such that all the affinities $2a_\nu = \beta_1 - \beta_\nu$ are positive. Notice that the temperature of the reference reservoir sets the energy scale and has no physical relevance. On the opposite, the affinities $a_\nu$ are the central variables of the meta-canonical ensemble that are set by the environmental constraints. The affinities are naturally conjugated to the energy currents. However, we know from the previous sections that considering $(a,j)$ as the unique couple of conjugated variables does not afford to study all NE systems from the same NE potential. Intuitively, a change of an affinity also impacts the system activity and the occupation of the various states. Hence, we have introduced additional intensive variables to take into account these effects separately: the dynamical biases connected to the dilatation factors of the energy barriers and the escape weights modifying the escape probability of each state. These two intensive variables cannot be adjusted independently of the affinities if we want to avoid a change of the system dynamics: no dilatation should be applied to the energy barriers ($l_\nu =1$ for all $\nu$) yielding to dynamical biases that are equal to the affinities ($b = 2 a $); the dynamics should conserve the norm of the occupation vector imposing that an affinity must be associated with an escape weight equal to the escape-rate change $m=c(a,2a)$. Therefore, in the meta-canonical ensemble, the environment sets the affinity vector $a$ which in turn constrains the dynamical intensive variables, namely the dynamical biases and the escape weights. The NE potential of the meta-canonical ensemble is the CGF of energy currents, activities and occupations $\Gamma(a,b,m)$. It vanishes for all $a$ when $b=2a$ and $m=c(a,2a)$, but its partial derivatives with respect to $a$, $b$ and $m$ produces all the NESS cumulants of energy currents, activities and occupations for any affinity. For instance, the thermodynamic behavior follows from the NE equations of state
\begin{align}
\left. \frac{\partial \Gamma}{ \partial a_\nu } \right|_{a_{\smallsetminus \nu},b,m } &= j_\nu, \label{CanonicalRelNESS1} \\
\left. \frac{\partial \Gamma}{ \partial b_\nu } \right|_{a,b_{\smallsetminus \nu},m } &= f_\nu, \label{CanonicalRelNESS2} \\
\left. \frac{\partial \Gamma}{ \partial m_x } \right|_{a,b,m_{\smallsetminus x} } &= p_x, \label{CanonicalRelNESS3}
\end{align}
where the subscripts on the vertical bars indicate variables that remain constant when taking the partial derivative. We denote $ a_{\smallsetminus \nu} $ the vector $ a $ without the $ \nu $\textit{th} component. The cumulants of EQ thermodynamic variables are obtained with the NESS occupations defined by $p^*=p(a,2a,c)$ that only depend on $\chi-1$ affinities. The mean energy in the NESS is $ \l e[\bar z] \r_{\bar {\bm{k}}} = e( p^*) $, and the mean entropy is $ \l s[\bar z] \r_{\bar {\bm{k}}} =s(p^*)$.
The ensemble of systems conditioned on the energy currents they received from their environment is called the \emph{NE micro-canonical ensemble}. The trajectories of the systems in this ensemble are generated by the EQ reference process with generator ${\bm{k}}$ filtrated to achieve the condition on the energy currents. The physical implementation of systems in the NE micro-canonical ensemble would require the existence of energy sources with no fluctuations. These sources will very likely not exist in practice \bibnote{Beyond the case of energy sources, one can imagine an experimental setup where a matter flux or a rate of volume increase can be controlled exactly.}, even though this problem is not specific to NE ensembles (see for instance page 83 of Ref.~\cite{Callen1985_vol} for an example in EQ thermodynamic theory). If we assume that an energy current can be imposed from the outside, the activities and the occupations must take precise values so that the system can sustain the energy current.
On the opposite, the conjugated intensive variables become free to fluctuate. The relationship between currents, activities and occupations is obtained from the correspondence between the conjugated variables $(j,f,p)$ and $(a,b,m)$, as summarized in Table \ref{summary}. The NE micro-canonical potential is the LDF $L(j,f,p)$ and the statistics of the intensive variables $(a,b,m)$ follows from its partial derivative
\begin{align}
\left. \frac{\partial L }{ \partial j_\nu } \right|_{ j_{\smallsetminus \nu},f,p} &= a_\nu, \label{microCanonicalRelNESS1} \\
\left. \frac{\partial L }{ \partial f_\nu } \right|_{ j,f_{\smallsetminus \nu},p } &= b_\nu, \label{microCanonicalRelNESS2} \\
\left. \frac{\partial L }{ \partial p_x } \right|_{ j,f,p_{\smallsetminus x} } &= m_x. \label{microCanonicalRelNESS3}
\end{align}
We proved in sections \ref{EQfluctuation} and \ref{EnsembleEquivalence} the equivalence of the ensembles of trajectories generated by the NE process and the conditioned EQ reference process assuming that the NE potentials are convex. Accordingly, the meta-canonical ensemble and NE micro-canonical ensembles are ensemble equivalent. In other words, systems submitted to temperature gradients are equivalent, at the thermodynamic level, to systems subjected to stationary energy injection (and extraction). By construction, the NE potentials are conjugated by Legendre transformation
\begin{eqnarray}
L(j,f,p) + \Gamma(a,b,m) &=& a^\dag \cdot j + b^\dag \cdot f + m^\dag \cdot p, \label{LegL}
\end{eqnarray}
and the NE stationary state can be obtained from a variational approach.
If we consider $ a^\dag \cdot j + b^\dag \cdot f + m^\dag \cdot p - \Gamma(a,b,m) $ as the potential $ L $ that would be obtained from Eq.~(\ref{LegL}) by assuming the independence of the conjugated variables, then the NESS affinity, dynamical bias and escape weight reached by the system at constant imposed energy current $ j $, activity $f$, and occupation $p$ maximize this potential in the subspace of constant $ (j,f,p) $:
\begin{equation}
(a,b,m)= \underset{a,b,m| j,f,p }{\arg\!\max} \left[ a^\dag \cdot j + b^\dag \cdot f + m^\dag \cdot p - \Gamma(a,b,m) \right]
\end{equation}
which are exactly Eqs.~(\ref{CanonicalRelNESS1}-\ref{CanonicalRelNESS3}).
The same argument holds the other way around.
If we consider $ a^\dag \cdot j +b^\dag \cdot f + m^\dag \cdot p - L(j,f,p) $ as the potential $ \Gamma $ that would be obtained from Eq.~(\ref{LegL}) assuming the independence of the conjugated variables, then the NESS energy currents, activities and occupations reached by the system at constant imposed affinity $ a $, dynamical bias $b$, and escape weight $m$ maximize this potential in the subspace of constant $ (a,b,m) $:
\begin{equation}
(j,f,p) = \underset{j,f,p | a,b,m }{\arg\!\max} \left[ a^\dag \cdot j + b^\dag \cdot f + m^\dag \cdot p - L(j,f,p) \right]
\end{equation}
which are exactly Eqs.~(\ref{microCanonicalRelNESS1}-\ref{microCanonicalRelNESS3}).
\subsection{Symmetries of the NE potentials \label{NEsymmetries}}
The metacanonical potential is even under the sign change of all affinities. We prove in Appendix \ref{FT} that this symmetry leads to the fluctuation theorem (FT), a fundamental result regarding the asymptotic statistics of entropy production first studied in Refs. \cite{Bochkov1981_vol106a, Evans1994_vol50, Gallavotti1995_vol74}. Another fundamental symmetry is obtained from the equality of second derivatives of the NE potentials. This symmetry is the NE equivalent of the Maxwell relations and reads as
\begin{equation}
\frac{\partial^2 \Gamma}{ \partial h_{\alpha} \partial h'_{\alpha'} } = \frac{\partial^2 \Gamma}{ \partial h'_{\alpha'} \partial h_{\alpha} } \quad \mbox{and} \quad
\frac{\partial^2 L}{ \partial v_{\alpha} \partial v'_{\alpha'} } = \frac{\partial^2 L}{ \partial v'_{\alpha'} \partial v_{\alpha} }
\label{MaxwellRelations}
\end{equation}
where $h$ and $h'$ are two vectors in $(a,b,m)$ and similarly $v$ and $v'$ in $(j,f,p)$.
The subscripts $\alpha$ and $\alpha'$ indicate two arbitrary components of these vectors.
At EQ, Maxwell's relations deeply constrain the number of EQ response coefficients that should be introduced to completely describe a system. Here, they constrain the derivatives of the non-linear functions giving, for instance, the currents in terms of the affinities. In the close-to-EQ limit, Eq.~(\ref{MaxwellRelations}) implies that the linear response matrix is symmetric, or in other words it implies the Onsager reciprocity relations \cite{Onsager1931_vol37,Andrieux2004_vol121}, as we will see in the next section.
\subsection{NE linear response theory \label{NEresponse}}
We study the linear response of a system in an arbitrary NESS and further perturbed by a change of temperature $\beta_\nu \rightarrow \beta_\nu' = \beta_\nu + \Delta \beta_\nu $ or of dilatation factor $l_\nu \rightarrow l_\nu'= l_\nu + \Delta l_\nu$. More precisely, we want to determine the change of the energy currents and activities when the half affinities $ a_\nu = (\beta_1 - \beta_\nu)/2 $ and dynamical biases $ b_\nu = (\beta_1 l_1 - \beta_\nu l_\nu) $ are slightly changed to the new values $ a_\nu + \Delta a_\nu $ and $ b_\nu + \Delta b_\nu$. We assume that $ l_1 $ and $ \beta_1 $ do not change during the perturbation. Then, the perturbations are written as
\begin{eqnarray}
\Delta a_\nu &=& - \Delta \beta_\nu/2, \\
\Delta b_\nu &=& \left( - \beta_\nu + 2\Delta a_\nu \right) \Delta l_\nu + 2 l_\nu \Delta a_\nu \nonumber \\
& \simeq & -\beta_\nu \Delta l_\nu +2 l_\nu \Delta a_\nu,
\end{eqnarray}
at linear order. We remark that the dynamical biases change when perturbing the affinities, but the converse is not true.
A Taylor expansion of the meta-canonical potential $ \Gamma $ gives the following quadratic function
\begin{multline}
\Gamma( a+\Delta a, b+\Delta b, m+\Delta m) = \\
\Gamma( a, b, m) + \sum_{h=a,b,m} \Delta h^\dag \cdot \nabla_h \Gamma \\
+ \frac{1}{2}\underset{h'=a,b,m}{\sum_{h\;=a,b,m}} \Delta h^\dag \cdot \nabla_{hh'} \Gamma \cdot \Delta h', \label{QuadraticApproxGamma}
\end{multline}
where $\Delta m$ is not yet specified. We have used the short notations for the derivatives of the meta-canonical potential
\begin{eqnarray}
\left( \nabla_h \Gamma \right)_\alpha &\equiv& \frac{\partial \Gamma}{\partial h_\alpha }\left( a, b, m \right) , \\
\left( \nabla_{hh'} \Gamma \right)_{\alpha\alpha'} &\equiv& \frac{\partial^2 \Gamma}{\partial h_\alpha \partial h'_{\alpha'}} \left( a, b, m \right).
\end{eqnarray}
The perturbation induces a variation $\Delta j$ of the energy currents, $\Delta f$ of the activities or $\Delta p$ of the occupation. Taking the partial derivative of Eq.~(\ref{QuadraticApproxGamma}) with respect to $ \Delta a $, $ \Delta b $, or $ \Delta m$ and evaluated in $\Delta m = \Delta c $, with $\Delta c$ the variation of the escape-rate change due to the perturbation, leads to the linear response equation
\begin{equation} \left(
\begin{array}{c}
\Delta j \\
\Delta f \\
\Delta p
\end{array} \right)
\simeq \left[
\begin{array}{ccc}
\nabla_{aa} \Gamma & \nabla_{ab} \Gamma & \nabla_{am} \Gamma \\
\nabla_{ba} \Gamma & \nabla_{bb} \Gamma & \nabla_{bm} \Gamma \\
\nabla_{m a} \Gamma & \nabla_{m b} \Gamma & \nabla_{m\m} \Gamma \\
\end{array} \right]
\cdot \left(
\begin{array}{c}
\Delta a \\
\Delta b \\
\Delta c
\end{array} \right). \label{NELinearResponse}
\end{equation}
From Eq.~(\ref{MaxwellRelations}), the response matrix above is symmetric even close to an arbitrary NESS. However, the chain rule yields
\begin{equation}
\Delta c = \nabla_{a}c \cdot \Delta a + \nabla_{b}c \cdot \Delta b,
\end{equation}
and the variation of the currents and activities becomes
\begin{align}
\Delta j =& \left( \nabla_{aa} \Gamma + \nabla_{a m} \Gamma \cdot \nabla_{a}c \right) \cdot \Delta a \nonumber \\ & \qquad\qquad + \left( \nabla_{ab} \Gamma + \nabla_{a m} \Gamma \cdot \nabla_{b}c \right) \cdot \Delta b. \label{CurrResp} \\
\Delta f =& \left( \nabla_{ab} \Gamma + \nabla_{bm} \Gamma \cdot \nabla_{a}c \right) \cdot \Delta a \nonumber \\ & \qquad\qquad + \left( \nabla_{bb} \Gamma + \nabla_{bm} \Gamma \cdot \nabla_{b}c \right) \cdot \Delta b. \label{FrenResp}
\end{align}
The response matrix defined from Eqs.~(\ref{CurrResp}-\ref{FrenResp}) is no longer symmetric in general as already emphasized in former works on NE linear response theory \cite{Baiesi2009_vol137, Baiesi2009_vol103,Verley2011_vol93,Verley2011_vol10, Speck2006_vol74, Chetrite2008_vol2008, Chetrite2009_vol137, Seifert2010_vol89}.
The second derivatives of the meta-canonical potential appearing in Eq.~(\ref{CurrResp}) are
\begin{align}
\left ( \nabla_{aa} \Gamma\right)_{\nu \nu'} &= \lim_{t\rightarrow \infty} t \left\{ \l j_\nu[\bar z] j_{\nu'}[\bar z] \r_{\bar {\bm{k}}} - \l j_\nu[\bar z] \r_{\bar {\bm{k}}} \l j_{\nu'}[\bar z] \r_{\bar {\bm{k}}} \right\} , \nonumber \\
\left ( \nabla_{ab} \Gamma\right)_{\nu \nu'} &= \lim_{t\rightarrow \infty} t \left\{ \l j_\nu[\bar z] f_{\nu'}[\bar z] \r_{\bar {\bm{k}}} - \l j_\nu[\bar z] \r_{\bar {\bm{k}}} \l f_{\nu'}[\bar z] \r_{\bar {\bm{k}}}\right\} , \nonumber \\
\left ( \nabla_{am} \Gamma\right)_{\nu x} &= \lim_{t\rightarrow \infty} t \left\{ \l j_\nu[\bar z] p_{x}[\bar z] \r_{\bar {\bm{k}}} - \l j_\nu[\bar z] \r_{\bar {\bm{k}}} \l p_{x}[\bar z] \r_{\bar {\bm{k}}}\right\} , \label{2ndeDerivGamma}
\end{align}
and correspond respectively to the current-current, the current-activity and the current-occupation covariances in the unperturbed NESS \bibnote[time]{We have multiplied the covariances by a factor $t$ since we consider time-averaged variables; we would have divided by $t$ if we were considering the time-integrated variables.}. In addition to the above covariances, the response functions include another contribution involving the derivatives of the escape-rate change $c$. Since the escape-rate change satisfies
\begin{eqnarray}
- \frac{\partial c_x}{\partial a_\nu} &=& \sum_y \bar k^{\nu}_{yx} (\epsilon_y - \epsilon_x),
\end{eqnarray}
the unperturbed mean-occupation multiplied by this derivative returns the unperturbed mean energy current
\begin{equation}
- \sum_{x}\frac{\partial c_x}{\partial a_\nu} \l p_{x}[\bar z] \r_{\bar {\bm{k}}} = \sum_{x,y} \bar k^{\nu}_{yx} \l p_{x}[\bar z] \r_{\bar {\bm{k}}} (\epsilon_y - \epsilon_x) = \l j_\nu[\bar z] \r_{\bar {\bm{k}}} .
\end{equation}
Therefore, the response to the affinity perturbation is
\begin{multline}
\left( \nabla_{aa} \Gamma + \nabla_{a m} \Gamma \cdot \nabla_{a}c \right)_{\nu\nu'} \\
= \lim_{t \rightarrow \infty} t \left\{ \l \vphantom{\frac{1}{1}} j_\nu[\bar z] j_{\nu'}[\bar z] \r_{\bar {\bm{k}}} - \l j_\nu[\bar z] \frac{\partial }{\partial a_{\nu'}} \left( p^\dag[\bar z] \cdot \bar \lambda \right) \r_{\bar {\bm{k}}}\right\}. \label{AffResponse}
\end{multline}
As expected, the response has an additive structure with an equilibriumlike part given by a currents correlation function, and a NE part corresponding to a current and traffic-excess correlation function. We call traffic-excess the derivative of the empirical escape rate $p^\dag[\bar z] \cdot \bar \lambda $ with respect to the perturbed variable \cite{Baiesi2009_vol137, Baiesi2009_vol103}. Similarly, the response of the energy current to a perturbation of the dynamical bias in the second line of Eq.~(\ref{CurrResp}) has two parts with an activity-current correlation function and a current-traffic excess correlation function.
As regards the perturbation of an EQ system, i.e. all $\beta'_\nu $ are close to the reference inverse temperature $ \beta_1$, one recovers the Yamamoto--Zwanzig formula expressing the response coefficients to a temperature perturbation from the covariances of energy currents \cite{Yamamoto1960_vol33,Zwanzig1965_vol16}. In order to see this, let us first consider a reference system at EQ only perturbed by a change of the dilatation factors, i.e. $ \Delta a = 0$ and $ \Delta b = -\beta_1 \Delta l $. Thanks to Eq.~(\ref{CurrResp}), the variation of the energy currents is written as
\begin{equation}
\Delta j = \left( \nabla_{ab} \Gamma + \nabla_{am} \Gamma \cdot \nabla_{b}c \right) \cdot \Delta b = 0.
\end{equation}
It vanishes for any perturbations $ \Delta b $ since no mean energy current exists at EQ. Thus, we find
\begin{equation}
\nabla_{ab} \Gamma + \nabla_{am} \Gamma \cdot \nabla_{b}c = 0,
\end{equation}
if the derivatives are taken in $a=0$. This removes the contribution due to the dynamical bias from the EQ response. Another contribution disappears in the close-to-equilibrium limit due to the decoupling between occupations and energy currents \cite{Wynants2010_vola}. Indeed, from the symmetry of the meta-canonical potential with sign change of the affinities, namely $\Gamma(a,b,m) = \Gamma(-a,b,m)$, we have
\begin{equation}
\frac{\partial^2 \Gamma}{ \partial a_{\nu} \partial m_x }(a,b,m) = - \frac{\partial^2 \Gamma}{ \partial a_{\nu} \partial m_x } (-a,b,m).
\end{equation}
Accordingly, $\nabla_{am} \Gamma = 0$ if the derivatives are taken in $a=0$. From the third line of Eq.~(\ref{2ndeDerivGamma}), we can conclude that the energy currents and occupations are decoupled. The Yamamoto--Zwanzig formula follows from Eq.~(\ref{CurrResp})
\begin{equation}
\Delta j = \frac{\nabla_{aa} \Gamma}{2} \cdot (\beta_1 - \beta'),
\end{equation}
where $\nabla_{aa} \Gamma$ is given in the first line of Eq.~(\ref{2ndeDerivGamma}) with EQ averages $\l \cdots \r_{{\bm{k}}}$ instead of the NE averages $\l \cdots \r_{\bar {\bm{k}}}$. Therefore, we recover the Onsager reciprocity relations from the NE Maxwell-relations.
\section{Illustrative example: a two-level system \label{Example}}
We now illustrate our results on a two-level system with states $z=1,2$ and mechanisms $\nu=1,2, \cdots, \chi$ enabling energy exchanges with $\chi$ different heat reservoirs. The coupling strength with reservoir $\nu$ is denoted $\gamma_\nu$ in this section since it is not a matrix but a vector when there are only two states. The energy states are $\epsilon_1$ and $ \epsilon_2$. Let $\epsilon_\pm = \epsilon_1\pm \epsilon_2$ to shorten notations. The transition rate matrix of the EQ reference process for each mechanism $\nu$ is
\begin{equation}
{\bm{k}}^{\nu} = \left[ \begin{matrix}
-\gamma_\nu e^{ \frac{\beta_1 \epsilon_-}{2}} &
\gamma_\nu e^{ -\frac{\beta_1 \epsilon_-}{2}} \\
\gamma_\nu e^{ \frac{\beta_1 \epsilon_-}{2}} &
-\gamma_\nu e^{ -\frac{\beta_1 \epsilon_-}{2}}
\end{matrix} \right] \label{ExRefProcess}
\end{equation}
where we assume vanishing dilatation factors $l_1$, see Eq.~(\ref{defEQrates}).
The rate matrices for the NE system are
\begin{equation}
\bar {\bm{k}}^{\nu} = \left[ \begin{matrix}
-\gamma_\nu e^{ (\beta_1/2 - a_\nu) \epsilon_- + b_\nu \epsilon_+} &
\gamma_\nu e^{ -(\beta_1/2 - a_\nu) \epsilon_- + b_\nu \epsilon_+} \\
\gamma_\nu e^{ (\beta_1/2 - a_\nu) \epsilon_- + b_\nu \epsilon_+} &
-\gamma_\nu e^{ -(\beta_1/2 - a_\nu) \epsilon_- + b_\nu \epsilon_+}
\end{matrix} \right], \label{ExNEProcess}
\end{equation}
if we chose $d_{12}= \epsilon_+$.
The escape-rate changes for this model are
\begin{eqnarray}
c_1 &=& \sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2} \left( 1-e^{ - a_\nu \epsilon_- + b_\nu \epsilon_+} \right), \label{EscChange} \\
c_2 &=& \sum_\nu \gamma_\nu e^{-\beta_1 \epsilon_-/2} \left( 1-e^{ a_\nu \epsilon_- + b_\nu \epsilon_+} \right).
\end{eqnarray}
The tilted operator $ {\bm{\kappa}} = {\bm{\kappa}}(a,b,m)$ for the EQ reference system is
\begin{equation}
{\bm{\kappa}}= \! \left[ \begin{matrix} \displaystyle
-\sum_\nu\gamma_\nu e^{ \frac{\beta_1 \epsilon_-}{2}} +m_1 & \displaystyle
\sum_\nu \gamma_\nu e^{-(\beta_1/2 - a_\nu) \epsilon_- +b_\nu \epsilon_+} \\
\displaystyle \sum_\nu \gamma_\nu e^{ (\beta_1/2 - a_\nu) \epsilon_- +b_\nu \epsilon_+} & \displaystyle
-\sum_\nu \gamma_\nu e^{ -\frac{\beta_1 \epsilon_-}{2}} +m_2
\end{matrix} \right].
\end{equation}
The highest eigenvalue of this matrix is the meta-canonical potential
\begin{multline}
\Gamma = -\sum_\nu \gamma_\nu \cosh \left(\beta_1 \epsilon_-/2 \right)+\frac{m_1+m_2}{2} \\
+ \sqrt{ \hat \gamma^2+ \sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{(a_\nu- a_{\nu'} )\epsilon_- + (b_\nu + b_{\nu'})\epsilon_+}},
\end{multline}
where we have introduced
\begin{equation}
\hat \gamma \equiv -\sum_\nu \gamma_\nu\sinh \left( \frac{\beta_1 \epsilon_-}{2} \right)+\frac{m_1-m_2}{2}. \label{hatgamma}
\end{equation}
The meta-canonical potential $\Gamma$ provides the statistics of $j_\alpha $ the energy current flowing from the $\alpha$th reservoir toward the system and of $f_\alpha$ the activity induced by the $\alpha$th mechanism. From direct derivation of $\Gamma$ with respect to $a_{\alpha}$, $b_{\alpha}$ or $m_z$ the energy current coming from reservoir $\alpha > 1 $ is
\begin{equation}
j_\alpha = \frac{\sum_\nu \epsilon_-\gamma_\alpha\gamma_\nu e^{(b_\nu+b_\alpha) \epsilon_+} \sinh \left[ ( a_\alpha - a_\nu) \epsilon_- \right]}{\sqrt{\hat \gamma^2+ \sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{( a_\nu- a_{\nu'} )\epsilon_- + (b_\nu + b_{\nu'})\epsilon_+} }}, \label{meanj}
\end{equation}
\\*
the activity for the transitions induced by mechanism $\alpha$ is
\begin{equation}
f_\alpha = \frac{\sum_{\nu} \epsilon_+\gamma_\alpha \gamma_\nu e^{(b_\nu + b_\alpha) \epsilon_+} \cosh\left[ ( a_\alpha - a_\nu) \epsilon_- \right]}{\sqrt{\hat \gamma^2+ \sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{( a_\nu- a_{\nu'} )\epsilon_-/ + (b_\nu + b_{\nu'})\epsilon_+} }},
\label{meanf}
\end{equation}
and the occupation of state $z$ is
\begin{equation}
p_z = \frac{1}{2}+ \frac{(\delta_{z,1}-\delta_{z,2}) \hat \gamma /2 }{\sqrt{ \hat \gamma ^2 + \sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{(a_\nu- a_{\nu'} )\epsilon_- + (b_\nu + b_{\nu'})\epsilon_+}}} \label{NESSprob}
\end{equation}
where $\hat \gamma$ of Eq.~(\ref{hatgamma}) is evaluated in $m = c(a,2a)$, and taking $b=2a$ to obtain the mean values of $j$, $f$ and $p$ in the NESS with affinity $2a$.
Deriving once more with respect to $ a_{\alpha'}$, $b_{\alpha'}$ or $m_{z'}$ leads to the symmetric response matrix, see Eq.~(\ref{NELinearResponse}).
The left and right eigenvectors of ${\bm{\kappa}}$ associated to the eigenvalue $\Gamma $ are respectively $\pi$ and $r=\bm{\pi}^{-1} \cdot p$. We find for the two-level model
\begin{widetext}
\begin{eqnarray}
\pi_1 &=& \frac{ \sum_\nu \gamma_\nu e^{ (\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+}}{\sum_\nu \gamma_\nu e^{ (\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+} +\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma }, \label{pi1}\\
\pi_2 &=& \frac{ \sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma }{\sum_\nu \gamma_\nu e^{ (\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+} +\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma }, \label{pi2} \\
r_1 &=& \frac{ \left( \sum_\nu \gamma_\nu e^{ -(\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+} \right) \left( \sum_\nu \gamma_\nu e^{ (\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+} +\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma \right) }{\sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{(a_\nu- a_{\nu'} )\epsilon_- + (b_\nu + b_{\nu'})\epsilon_+} +\left(\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma \right)^2}, \\
r_2 &=& \frac{ \left( \sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma \right) \left( \sum_\nu \gamma_\nu e^{ (\beta_1/2-a_\nu) \epsilon_- +b_\nu \epsilon_+} +\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma \right) }{\sum_{\nu,\nu'} \gamma_\nu \gamma_{\nu'} e^{(a_\nu- a_{\nu'} )\epsilon_- + (b_\nu + b_{\nu'})\epsilon_+} +\left(\sum_\nu \gamma_\nu e^{ \beta_1 \epsilon_-/2 }-m_1 +\Gamma \right)^2}. \label{rightEigVec}
\end{eqnarray}
\end{widetext}
\begin{figure*}
\begin{center}
\begin{tabular}{lll}
\includegraphics[width=6cm]{./fig7.pdf}
& \includegraphics[width=6cm]{./fig3.pdf} & \includegraphics[width=6cm]{./fig5.pdf} \\ && \\
\includegraphics[width=6cm]{./fig8.pdf}
& \includegraphics[width=6cm]{./fig4.pdf} & \includegraphics[width=6cm]{./fig6.pdf}\\
\end{tabular}
\end{center}
\caption{ (a) meta-canonical potential for various $(a_2,m_2)$ with $b=0$, (b) energy current, (c) energy, (e) activity and (f) entropy as a function of the affinity $a_2$ and the dynamical bias $b_2$. (d) NE microcanonical potentials for the energy current $j_2$ and the activity $f_2$ after a contraction on $f_1$, $p_1$ and $p_2$. Other parameters are $b_1 = 0$, $\beta_2 = \beta_1- 2 a_2 $ , $\gamma_2 = 0.5$, $\epsilon_1=1$ and $\epsilon_2=0.5$. For all figures, $\beta_1 = 1$ set the energy scale and $\gamma_1 =1$ the time scale. The variables $a$ and $b$ are in unit of $1/\beta_1$, the variables $j$ and $f$ are in unit of $\gamma_1/\beta_1$, and finally $L$, $\Gamma$, and $m$ are in unit of $\gamma_1$. \label{fig3}}
\end{figure*}
We can now illustrate the consistency of the theory: from Eqs.~(\ref{pi1}-\ref{rightEigVec}) and the product $\bm{\pi} \cdot r$, we recover the NESS probability of Eq.~(\ref{NESSprob}) obtained from derivation of the meta-canonical potential; Eqs.~(\ref{pi1}-\ref{pi2}) allow us to compute the drift $\u$ to get the current and activity of Eqs.~(\ref{meanj}-\ref{meanf}) from Eqs.~(\ref{EqOfStatej}-\ref{EqOfStatef}) knowing the NESS probability.
We turn to the discussion of the properties of the two-level system with $\chi =2$ heat reservoirs in the light of Fig.~\ref{fig3} obtained from our analytic results. For simplicity, we chose $b_1=0$.
We set the energy scale and the time scale taking respectively $\beta_1=1$ and $\gamma_1 =1$. Fig.~(\ref{fig3}a) shows that the meta-canonical potential is a symmetric function of the affinity $a_2$ and is strictly convex. From this symmetry, one should not conclude that the energy current $j_2$ is an anti-symmetric function of $a_2$. Indeed, the energy current comes from the derivative of the meta-canonical potential with respect to $a_2$ evaluated in $m = c$ that has no particular symmetry when changing the sign of $a_2$.
The absolute value of the energy current $|j_2|$ and the activity $f_2$ always increases with the absolute value of the affinity $|a_2|$ at given dynamical bias $b_2$, see Fig.~(\ref{fig3}b) and Fig.~(\ref{fig3}e). A decrease of $|j_2|$ with increasing $|a_2|$ would mean that the system has negative response for some affinities. Such a behavior is not expected for a simple two-level model. Another general trend is that $|j_2|$ and $f_2$ increases with $b_2$. Indeed, a higher dynamical bias increases the value of the transition rates corresponding to $\nu=2$, if one has $\epsilon_+>0$, see Eq.~(\ref{ExNEProcess}). Then, a high dynamical bias accelerates the dynamics associated to reservoir $\nu=2$, whereas a small one slows it down, letting the reference dynamics associated to reservoir $\nu=1$ dominates in the transition rate matrix. Therefore, in the limit of low dynamical bias with respect to the affinity, the system approaches the EQ state at temperature $\beta_1$, with current $j_2$ and activity $f_2$ decreasing to zero.
To represent the NE micro-canonical potential $L$, one has to focus on the statistics of some specific variables by contraction: this step consists in evaluating the NE micro-canonical potential at the mean value of the disregarded variables, for instance $f_1$, $p_1$ and $p_2$ in the case of Fig.~(\ref{fig3}d). However, it is much more convenient to obtain $L(j_2,f_2)$ directly from a parametric plot of $(j_2,f_2,L)$ with $(a_2,b_2)$ being the parameters and taking $b_1=0$. In this way, we have obtained Fig.~(\ref{fig3}d) showing the NE micro-canonical potential as a convex function of $(j_2,f_2)$. This function is undefined in the regions corresponding to low activities in comparison to the energy current. The explanation is that a current can only flow if some minimal activity holds, i.e. if the system changes state regularly enough.
Finally, the system energy $e$ and Shannon entropy $s$ are, in our framework, functions of the affinity and dynamical bias. We see in Fig.~(\ref{fig3}c) and (\ref{fig3}f) that these functions have a very similar shape in a large area corresponding to the EQ limit. The dimensionless free energy of the reference system at temperature $\beta_1$ is $\varphi = \beta_1 e-s $ and should reach its minimum value for low affinity $|a_2|$ or low dynamical bias $b_2$. There, since $\beta_1=1$, the system energy and entropy differ only in the value of the dimensionless free energy of the EQ reference system. On the contrary, at high affinity $|a_2|$, most of the time the system is either in energy state $\epsilon_1=1$ for positive $a_2$, or $\epsilon_2=0.5$ for negative $a_2$. The system is driven to a state where the entropy is lower than at EQ and the NE mean energy is moved away from the EQ mean value for the reference process.
\section{Conclusion}
In this paper, we have established that the asymptotic probability of the energy currents, the activities and the occupations in a NE process proceeds from the long-time statistics of the same variables at EQ. We have connected the affinities of the NE process, the dynamical biases and the escape-rate changes to constraints imposed on the EQ reference process, respectively on the energy currents, on the activities and on the occupations of each state. This connection is the analog of the ensemble equivalence between the canonical and micro-canonical ensembles of EQ statistical physics for which the temperature of the heat reservoir is associated to an energy constraint. We have argued that the mapping between EQ and NE fluctuations allows us to distinguish the reduced set of variables which play a key role in the description of NESSs.
Beyond the understanding of the structure of NE statistical physics, phenomenological and/or operational methods must be developed to compute the NE potentials of real complex systems. In this regard, it was shown that efficient algorithms exist to compute the scaled cumulants of currents \cite{Wachtel2015_vol92} or to simulate samples of rare trajectories \cite{Giardina2011_vol145}. A promising technique for macroscopic systems relies on the saddle point approximation of a path integral producing the cumulant generating function \cite{Book_Ross2008}. This calculation leads to a dynamical problem with a small number of degrees of freedom compared to the original problem. Solving this dynamical problem seems easier than finding the highest eigenvalue of a large tilted operator.
\section*{Acknowledgement}
I thank M. Polettini and D. Lacoste for their useful comments on the manuscript and U. Dimitrijevi\'c for her careful rereading. I also thank A. Engel and A. Ran\c con for the enlightening discussions concerning respectively the large deviation theory (Appendix \ref{LDFoccupationandjump}) and the calculation of asymptotic probabilities (Appendix \ref{AppendixCountStat}).
|
2,877,628,090,807 | arxiv | \section{Introduction}
The binary erasure channel (BEC) is a very well known channel of \emph{communication} which was introduced by Elias in 1955 \cite{Elias1955}. Due to its simplicity, it has been a starting point to design new coding schemes and analyze the properties of codes \cite{Arikan2009, Di2002}. In addition, coding schemes for BEC are still being actively researched since BEC is a very good model of the Internet \cite{Luby2001, Luby2002, Shokrollahi2006}.
In BEC, as shown in Fig.~\ref{fig:BEC}, the channel input $X \in \{0, 1\}$ is binary and the channel output $Y=\{0, 1, \varepsilon\}$ is ternary. It is assumed that the decoder knows the locations of erased bits denoted by $\varepsilon$. The capacity of the BEC with erasure probability $\alpha$ is given by \cite{Elias1955, Cover2006}
\begin{equation}\label{eq:BEC_capacity}
C_{\mathrm{BEC}} = 1 - \alpha.
\end{equation}
Elias \cite{Elias1955} showed that random codes of rates arbitrarily close to $C_{\textrm{BEC}}$ can be decoded on the BEC with an exponentially small error probability using maximum likelihood (ML) decoding. In the case of BEC, ML decoding of linear codes is equivalent to solving linear equations \cite{Elias1955, Shokrollahi2006}.
The binary defect channel (BDC) also has a long history. The BDC was introduced to model computer memory for \emph{storage} by Kuznetsov and Tsybakov in 1974 \cite{Kuznetsov1974}. At that time, erasable and programmable read only memories (EPROM) and random access memories (RAM) were modeled by the BDC \cite{Kuznetsov1974}. Recently, BDC has received renewed attention for nonvolatile memories such as flash memories and phase change memories (PCM) \cite{Hwang2011a, Jagmohan2010a, Lastras-Montano2010}. In addition, BDC is theoretically important since write once memories (WOM), write unidirectional memories (WUM), and some other constrained memories can be considered as special cases of the BDC \cite{Kuzntsov1994}.
As shown in Fig.~\ref{fig:BDC}, BDC has a ternary channel state $S \in \{0, 1, \lambda\}$ whereas the channel input $X$ and the channel output $Y$ are binary. The state $S=0$ corresponds to a stuck-at 0 defect that always outputs a 0 independent of its input value, the state $S=1$ corresponds to a stuck-at 1 defect that always outputs a 1, and the state $S = \lambda$ corresponds to a normal cell that outputs the same value as its input. The probabilities of these states are $\beta / 2$, $\beta / 2$ (assuming a symmetric defect probability), and $1 - \beta$, respectively \cite{ElGamal2011, Heegard1983:capacity, Heegard1983}.
It is known that the capacity is $1 - \beta$ when both the encoder and the decoder know the defect information. If the decoder is aware of the defect locations, then the defects can be regarded as erasures so that the capacity is $1 - \beta$ \cite{ElGamal2011, Heegard1983:capacity}. On the other hand, Kuznetsov and Tsybakov assumed that the encoder knows the defect information such as locations and stuck-at values of defects and the decoder does not have any information of defects \cite{Kuznetsov1974}. It was shown that the capacity is also $1 - \beta$ even if only the encoder knows the defect information \cite{ElGamal2011, Heegard1983:capacity}. Thus, the capacity of the BDC is given by
\begin{equation}\label{eq:BDC_capacity}
C_{\mathrm{BDC}} = 1 - \beta.
\end{equation}
The capacity of the BDC can be achieved by \emph{random binning} when only the encoder knows the defect information \cite{Heegard1983:capacity, ElGamal2011}. The practical coding scheme is the \emph{additive encoding} which masks defects by adding a carefully selected binary vector \cite{Kuznetsov1974, Tsybakov1975b}. Masking defects is to make a codeword whose values at the locations of defects match the stuck-at values of the defects at those locations.
Heegard proved that the additive encoding (formulated as an optimization problem) and ML decoding can achieve the capacity of a channel that has both defects and random errors \cite{Heegard1983}. Note that the additive encoding masks defects and the ML decoding corrects random errors.
In \cite{Kim2013}, an upper bound on the probability of masking failure was derived when the additive encoding is accomplished by solving the linear equations instead of solving the optimization problem. The derived upper bound is based on the weight distribution of the underlying codes. Based on the upper bound of \cite{Kim2013}, we will show that the additive encoding can achieve $C_{\mathrm{BDC}}$ by using random linear codes and solving a system of linear equations. In addition, numerical results show that structured linear codes such as Bose, Chaudhuri, and Hocquenghem (BCH) codes are good choices since their performance is not far from $C_{\mathrm{BDC}}$.
In Section~\ref{sec:duality}, the BEC and the BDC will be discussed separately. Their channel properties, capacities, capacity achieving coding schemes and upper bounds on the probability of failure will be discussed comprehensively. Afterwards, we will investigate the duality of the BEC and the BDC. Basically, an erasure $\varepsilon$ is \emph{neither} 0 nor 1. In contrast, a stuck-at value (i.e., defect value) is \emph{either} 0 or 1. Also, the \emph{decoder} corrects erasures in the BEC and the \emph{encoder} masks defects in the BDC. Both channels have similar capacities as shown in \eqref{eq:BEC_capacity} and \eqref{eq:BDC_capacity}. In addition, both capacities can be achieved by solving the linear equations.
However, as we will show later in this paper, the BEC and the BDC have some important differences. The linear equations for the BEC can be described by an \emph{overdetermined} system and there is only one solution that corrects all erasures. Meanwhile, the linear equations for the BDC correspond to an \emph{underdetermined} system which allows several solutions that mask defects. In addition, the solution of linear equations for the BEC is the estimate of message or the estimate of erased bits, whereas the solution for the BDC is the parity or the the codeword.
In addition, the minimum distance and the weight distribution of the coding scheme for the BEC are controlled by the \emph{parity check matrix}, whereas the minimum distance and the weight distribution of the coding scheme for the BDC come from the \emph{generator matrix}. Because of these duality properties, the upper bound on the probability of decoding failure for the BEC and the upper bound on the probability of masking failure for the BDC have interesting similarities and differences. Considering that the BEC is a channel model for digital \emph{communication} and the BDC is a channel model for digital \emph{storage}, the duality is meaningful.
In Section~\ref{sec:BDEC}, the binary defect and erasure channel (BDEC) will be introduced. As shown in Fig.~\ref{fig:BDEC}, the BDEC has both erasures (with erasure probability $\alpha$ for a normal cell) and defects (with defect probability $\beta$). The capacity of the BDEC is given by
\begin{equation}\label{eq:BDEC_capacity}
C_{\mathrm{BDEC}} = \left(1 - \alpha \right)\left(1 - \beta \right).
\end{equation}
We will show that the capacity of the BDEC can be achieved by a coding scheme that combines the encoding of the BDC and the decoding of the BEC. This proposed coding scheme for BDEC has two separated redundancy parts: one for correcting erasures and the other for masking defects. In order to minimize the probability of failure of correcting erasures and masking defects, the redundancy allocation between these two redundancy parts should be optimized.
During our proof that the proposed coding scheme achieves the capacity $C_{\mathrm{BDEC}}$, lower bounds on these two redundancy components (achieving the capacity) can be obtained. However, these lower bounds may not be of much help in determining the redundancy allocation when the codeword length is finite.
Thus, we will investigate redundancy allocation for the BDEC. First, the optimal redundancy allocation is obtained by simulations. Then, we will derive the upper bound on the probability of failure for a finite codeword length and obtain the estimate of the optimal redundancy allocation by minimizing this upper bound instead of the probability of failure. Same methodology has been applied for the channel that has both defects and random errors in \cite{Kim2013:plbc} and it was shown that the estimated redundancy allocation matches the optimal one very well. From the numerical results, we will show that this method to minimize the upper bound works well for the BDEC as well as the channel of defects and random errors.
The rest of the paper is as follows. Section~\ref{sec:duality} discusses the duality between erasures and defects. In Section~\ref{sec:BDEC}, the BDEC will be introduced and the redundancy allocation for the BDEC will be investigated. After discussing the numerical results in Section~\ref{sec:numerical_results}, we conclude in Section~\ref{sec:conclusion}.
\section{Duality between Erasures and Defects} \label{sec:duality}
\subsection{Binary Erasure Channel}
For the BEC, the codeword most likely to have been transmitted is the one that agrees with all of received bits that have not been erased. If there is more than one such codeword, the decoding may lead to a failure. Thus, the following simple coding scheme was proposed in \cite{Elias1955}.
\emph{Encoding:} A message $\mathbf{m}\in \{0, 1\}^k$ is encoded to a corresponding codeword $\mathbf{c} \in \{0, 1\}^n$ by $\mathcal{C} = \left\{\mathbf{c} = G \mathbf{m} \mid \mathbf{m} \in \{0, 1\}^k \right\}$ where $\mathcal{C}$ is a set of codewords and the generator matrix $G$ is an $n \times k$ matrix over $\{0, 1\}$ such that $\rank(G)=k$. Note that the code rate $R = \frac{k}{n}$.
\emph{Decoding:} Let $g: \mathbf{y} \in \{0, 1, \varepsilon \}^n \rightarrow \mathcal{C}$ denote the decoding rule. If the channel output $\mathbf{y}$ is identical to one and only one codeword on the unerased bits, the decoding succeeds. If $\mathbf{y}$ matches completely with several codewords on the unerased bits, the decoder chooses one of them randomly. Note that there exists at least one codeword that matches with $\mathbf{y}$ on the unerased bits \cite{Elias1955}.
We will define a random variable $D$ as follows.
\begin{equation}
D =
\begin{cases}
0, & \mathbf{c} \ne \widehat{\mathbf{c}}\text{ (decoding failure)}; \\
1, & \mathbf{c} = \widehat{\mathbf{c}}\text{ (decoding success)}
\end{cases}
\end{equation}
where $\widehat{\mathbf{c}}$ is the estimated codeword produced by the decoding rule of $g$.
The minimum distance $d_{\mathrm{min}}$ of $\mathcal{C}$ is given by
\begin{align} \label{eq:BEC_dmin}
d_{\mathrm{min}} &= \underset{
\substack{
\mathbf{c} \ne \mathbf{0} \\
H^T \mathbf{c}= \mathbf{0}
}}
{\text{min }} \|\mathbf{c}\|
\end{align}
where the parity check matrix $H$ is an $n \times (n-k)$ matrix such that $H^{T}G = \mathbf{0}$ (superscript $T$ denotes transpose). Also, $\| \cdot \|$ represents the Hamming weight of a vector. Due to \eqref{eq:BEC_dmin}, any $d_{\mathrm{min}} - 1$ rows of $H$ are linearly independent. If $e < d_{\mathrm{min}}$, all $e$ erasures will be successfully corrected, which will be shown in Lemma~\ref{lemma:BEC_UB}.
The decoding rule of $g$ can be described by the following linear equations \cite{Elias1955}.
\begin{equation}\label{eq:BEC_decoder_LE}
G^{\mathcal{V}} \widehat{\mathbf{m}} = \mathbf{y}^{\mathcal{V}}
\end{equation}
where $\widehat{\mathbf{m}}$ is the estimate of $\mathbf{m}$ and $\mathcal{V}=\left\{j_1,\cdots, j_v\right\}$ indicates the locations of $v$ unerased bits. We use the notation of $\mathbf{y}^{\mathcal{V}}=\left(y_{j_1}, \cdots, y_{j_v}\right)^T$ and $G^{\mathcal{V}}=\left[ \mathbf{g}_{j_1}^T, \cdots, \mathbf{g}_{j_v}^T \right]^T$ where $\mathbf{g}_j$ is the $j$-th row of $G$. Note that $G^{\mathcal{V}}$ is a $v \times k$ matrix.
In addition, we can represent the decoding rule $g$ by the parity check matrix $H$ instead of the generator matrix $G$ as follows.
\begin{equation}\label{eq:BEC_decoder_LE_H_1}
H^T \mathbf{\widehat{c}} = \left(H^{\mathcal{E}} \right)^T \widehat{\mathbf{c}}^{\mathcal{E}} + \left(H^{\mathcal{V}} \right)^T \widehat{\mathbf{c}}^{\mathcal{V}} = \mathbf{0}
\end{equation}
where $\mathcal{E}=\left\{i_1,\cdots, i_e\right\}$ indicates the locations of $e$ erased bits such that $\mathcal{E} \cup \mathcal{V} = \left\{ 1,2,\ldots, n \right\}$ and $n=e+v$. Note that $\widehat{\mathbf{c}}^{\mathcal{E}}=\left(\widehat{c}_{i_1}, \cdots, \widehat{c}_{i_e}\right)^T$, $\widehat{\mathbf{c}}^{\mathcal{V}}=\left(\widehat{c}_{j_1}, \cdots, \widehat{c}_{j_v}\right)^T$, $H^{\mathcal{E}}=\left[ \mathbf{h}_{i_1}^T, \cdots, \mathbf{h}_{i_e}^T \right]^T$ and $H^{\mathcal{V}}=\left[ \mathbf{h}_{j_1}^T, \cdots, \mathbf{h}_{j_v}^T \right]^T$ where $\mathbf{h}_i$ is the $i$-th row of $H$.
From the channel model of BEC, it is clear that $\widehat{\mathbf{c}}^{\mathcal{V}} = \mathbf{y}^{\mathcal{V}} = \mathbf{c}^{\mathcal{V}}$ and we have to estimate the erased bits of $\mathbf{c}$, i.e., $\widehat{\mathbf{c}}^{\mathcal{E}}$. Thus, \eqref{eq:BEC_decoder_LE_H_1} can be represented by the following linear equations.
\begin{equation}\label{eq:BEC_decoder_LE_H}
\left(H^{\mathcal{E}} \right)^T \widehat{\mathbf{c}}^{\mathcal{E}} = \mathbf{q}
\end{equation}
where $\mathbf{q} = \left(H^{\mathcal{V}} \right)^T \mathbf{c}^{\mathcal{V}}$. $\widehat{\mathbf{m}}$ can be obtained from $\widehat{\mathbf{c}}$. Note that $\left(H^{\mathcal{E}}\right)^T$ is a $(n - k) \times e$ matrix.
Because of the weak law of large numbers, we can claim that $nC_{\mathrm{BEC}} - \epsilon \le v = n - e \le nC_{\mathrm{BEC}} + \epsilon$ with high probability for sufficiently large $n$. Assuming that $R < C_{\mathrm{BEC}} - \epsilon$, we can claim that \eqref{eq:BEC_decoder_LE} and \eqref{eq:BEC_decoder_LE_H} are \emph{overdetermined} because of $v > k$ and $n - k > e$.
Since $\dim \left( \mathcal{C} \right) = k$, there exists exactly one solution of \eqref{eq:BEC_decoder_LE} so long as $\rank \left(G^{\mathcal{V}}\right) = k$. If $\rank \left(G^{\mathcal{V}}\right) < k$, there are several solutions, which may result in decoding failure. Similarly, there exists exactly one solution of \eqref{eq:BEC_decoder_LE_H} so long as $\rank \left( H^{\mathcal{E}}\right) = e$. Otherwise, there are several solutions, which may result in decoding failure.
The following Lemma and its proof have been known in coding theory community.
\begin{lemma}[\cite{Barg2014}] \label{lemma:BEC_UB} The upper bound on the probability of decoding failure of the decoding rule $g$ is given by
\begin{equation}\label{eq:BEC_UB}
P\left(D=0 \mid |\mathcal{E}|=e \right) \le \frac{\sum_{w=d_{\mathrm{min}}}^{e}{A_w \binom{n-w}{e-w}}}{\binom{n}{e}}
\end{equation}
where $A_w$ is the weight distribution of $\mathcal{C}$. Also, $\mathcal{E}$ represents the set of erased locations in the channel output vector $\mathbf{y}$ and $|\mathcal{E}|=e$ is the number of erasures in $\mathbf{y}$.
\end{lemma}
\begin{IEEEproof}
Without loss of generality, we can assume that the all-zero codeword $\mathbf{0}$ has been transmitted and there exists a nonzero codeword $\mathbf{c}$ of Hamming weight $w$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$ where $\Psi_w(\mathbf{c})=\left\{ i \mid c_i \ne 0 \right\}$ denotes the locations of nonzero elements of $\mathbf{c}$ and $\mathcal{E}=\left\{i_1, \cdots, i_e \right\}$ denotes the locations of $e$ erasures. From the given decoding rule, $\mathbf{y}$ agrees with two codewords $\mathbf{0}$ and $\mathbf{c}$ on unerased bits, which may result in decoding failure. Meanwhile, if there is no nonzero codeword $\mathbf{c}$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$, then $\mathbf{y}$ agrees with only $\mathbf{0}$ on the unerased bits and the decoding succeeds.
For a nonzero $\mathbf{c}$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$, the number of possible $\mathcal{E}$ is $\binom{n-w}{e-w}$. Due to double counting, the number of possible $\mathcal{E}$ which results in decoding failure will be less than or equal to $\sum_{w=d_{\mathrm{min}}}^{e}{A_w \binom{n-w}{e-w}}$. Since the number of all possible $\mathcal{E}$ such that $|\mathcal{E}|=e$ is $\binom{n}{e}$, the upper bound on $P\left(D=0 \mid |\mathcal{E}|=e \right)$ is given by \eqref{eq:BEC_UB}.
\end{IEEEproof}
From the upper bound in Lemma~\ref{lemma:BEC_UB}, it is clear that $P\left(D=0 \mid |\mathcal{E}|=e \right) = 0$ for $e < d_{\mathrm{min}}$. The following Lemma shows that $P\left(D=0 \mid |\mathcal{E}|=e \right)$ can be obtained exactly for $d_{\mathrm{min}} \le e \le d_{\mathrm{min}} + \left\lfloor \frac{d_{\mathrm{min}}-1}{2} \right\rfloor$ where $\left\lfloor x \right\rfloor$ represents the largest integer not greater than $x$.
\begin{lemma} \label{lemma:BEC_exact} For $e \le d_{\mathrm{min}} + t$ where $t = \left\lfloor \frac{d_{\mathrm{min}}-1}{2} \right\rfloor$, $P\left( D=0 \mid |\mathcal{E}|=e \right)$ is given by
\begin{equation} \label{eq:BEC_exact}
P\left(D=0 \mid |\mathcal{E}|=e \right) = \frac{1}{2} \cdot \frac{\sum_{w=d_{\mathrm{min}}}^{e}{A_w \binom{n-w}{e-w}}}{\binom{n}{e}}.
\end{equation}
\end{lemma}
\begin{IEEEproof}
Without loss of generality, we can assume that the all-zero codeword $\mathbf{0}$ has been transmitted and suppose that there exists \emph{only one} nonzero codeword $\mathbf{c}$ of Hamming weight $w$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$. Since there are only two possible candidates such as $\mathbf{0}$ and $\mathbf{c}$ to guess the transmitted codeword, \eqref{eq:BEC_exact} is true. Thus, we need to show that there exists only one nonzero codeword $\mathbf{c}$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$ for $d_{\mathrm{min}} \le e \le d_{\mathrm{min}} + t$.
Suppose that there are two nonzero codewords $\mathbf{c}_1, \mathbf{c}_2 \in \mathcal{C}$ such that $\|\mathbf{c}_1\| = w_1$ and $\|\mathbf{c}_2\| = w_2$ where $d_{\mathrm{min}} \le w_1 \le w_2$. The locations of nonzero elements of $\mathbf{c}_1$ and $\mathbf{c}_2$ are given by
\begin{align}
\Psi_{w_1}\left(\mathbf{c}_1\right)&= \left\{ i_{1,1},\ldots,i_{1, w_1} \right\}, \\
\Psi_{w_2}\left(\mathbf{c}_2\right)&= \left\{ i_{2,1},\ldots,i_{2, w_2} \right\}.
\end{align}
Let $\Psi_{\alpha} = \left\{i_1, \ldots, i_{\alpha} \right\}$ denote $\Psi_{\alpha} = \Psi_{w_1}\left(\mathbf{c}_1\right) \cap \Psi_{w_2}\left(\mathbf{c}_2\right)$. Then $\Psi_{w_1}\left(\mathbf{c}_1\right)$ and $\Psi_{w_2}\left(\mathbf{c}_2\right)$ are given by
\begin{align}
\Psi_{w_1}\left(\mathbf{c}_1\right)&= \Psi_{\alpha} \cup \left\{ i_{1,1}',\ldots,i_{1, \beta_1}' \right\}, \\
\Psi_{w_2}\left(\mathbf{c}_2\right)&= \Psi_{\alpha} \cup \left\{ i_{2,1}',\ldots,i_{2, \beta_2}' \right\}
\end{align}
where $i_{1, j_1}'$ for $j_1 \in \left\{1, \ldots, \beta_1 \right\}$ and $i_{2, j_2}'$ for $j_2 \in \left\{1, \ldots, \beta_2 \right\}$ are the reindexed locations of nonzero elements of $\mathbf{c}_1$ and $\mathbf{c}_2$ that are mutually disjoint with $\Psi_{\alpha}$. Note that $\left\{ i_{1,1}',\ldots,i_{1, \beta_1}' \right\} \cap \left\{ i_{2,1}',\ldots,i_{2, \beta_2}' \right\} = \emptyset $, $\beta_1 = w_1 - \alpha$ and $\beta_2 = w_2 - \alpha$.
Due to the property of linear codes, $\mathbf{c}_3 = \mathbf{c}_1 + \mathbf{c}_2$ is also a codeword of $\mathcal{C}$, i.e., $\mathbf{c}_3 \in \mathcal{C}$ and $\|\mathbf{c}_3\|=\beta_1 + \beta_2$. Also, the following conditions should hold because of the definition of $d_{\mathrm{min}}$.
\begin{align}
\alpha + \beta_1 &\ge d_{\mathrm{min}}, \\
\alpha + \beta_2 &\ge d_{\mathrm{min}}, \\
\beta_1 + \beta_2 &\ge d_{\mathrm{min}}
\end{align}
Thus, we can claim that $2 \left( \alpha + \beta_1 + \beta_2 \right) \ge 3 d_{\mathrm{min}}$, which results in $\alpha + \beta_1 + \beta_2 \ge d_{\mathrm{min}} + \left\lfloor \frac{d_{\mathrm{min}} + 1}{2} \right\rfloor = d_{\mathrm{min}} + t +1 $ since $\alpha + \beta_1 + \beta_2$ is an integer.
If there exist two codewords $\mathbf{c}_1$ and $\mathbf{c}_2$ such that $\Psi_{w_1}\left(\mathbf{c}_1\right) \subseteq \mathcal{E}$ and $\Psi_{w_2}\left(\mathbf{c}_2\right) \subseteq \mathcal{E}$ (i.e., $\Psi_{w_1}\left(\mathbf{c}_1\right) \cup \Psi_{w_2}\left(\mathbf{c}_2\right) \subseteq\mathcal{E}$), it means that $e \ge \alpha + \beta_1 + \beta_2 \ge d_{\mathrm{min}} + t + 1$. Thus, for $d_{\mathrm{min}} \le e \le d_{\mathrm{min}} + \left\lfloor \frac{d_{\mathrm{min}} - 1}{2} \right\rfloor = d_{\mathrm{min}} + t$, there exists at most one nonzero codeword $\mathbf{c}$ such that $\Psi_w(\mathbf{c}) \subseteq \mathcal{E}$.
\end{IEEEproof}
\begin{theorem} \label{thm:BEC_bound} $P\left(D = 0 \mid |\mathcal{E}|=e\right)$ is given by
\begin{numcases}{P\left(D = 0 \mid |\mathcal{E}|=e\right)=}
0 & for $e < d_{\mathrm{min}}$,
\\
\frac{1}{2} \cdot \frac{\sum_{w=d_{\mathrm{min}}}^{e}{A_{w} \binom{n-w}{e-w}}}{\binom{n}{e}} & for $d_{\mathrm{min}} \le e \le d_{\mathrm{min}}+t$,
\\
\le \frac{\sum_{w=d_{\mathrm{min}}}^{e}{A_{w} \binom{n-w}{e-w}}}{\binom{n}{e}} & for $ e > d_{\mathrm{min}}+t$.
\end{numcases}
\end{theorem}
\begin{IEEEproof}The proof comes from the definition of $d_{\mathrm{min}}$ in \eqref{eq:BEC_dmin}, Lemma~\ref{lemma:BEC_UB} and Lemma~\ref{lemma:BEC_exact}.
\end{IEEEproof}
\begin{theorem}[\cite{Barg2014}] The decoding rule of $g$ is a capacity achieving scheme. \label{thm:BEC}
\end{theorem}
\begin{IEEEproof}
The decoding failure probability is given by
\begin{align}
P\left(D=0\right)& = P\left(D=0, |\mathcal{E}| \le n(\alpha + \epsilon) \right) + P\left(D=0, |\mathcal{E}| > n(\alpha + \epsilon) \right) \label{eq:BEC_CAS_pf_1}\\
& \le \sum_{e=1}^{n(\alpha + \epsilon)}{P(D=0, |\mathcal{E}|=e)} + \epsilon' \\
& \le \sum_{e=1}^{n(\alpha + \epsilon)}{P(D=0 \mid |\mathcal{E}|=e)} + \epsilon' \label{eq:BEC_conditional_prob}\\
& \le \sum_{e=1}^{n(\alpha + \epsilon)}{\frac{\sum_{w=d_{\mathrm{min}}}^{e}{A_w \binom{n-w}{e-w}}}{\binom{n}{e}}} + \epsilon' \label{eq:BEC_CAS_pf_UB} \\
& \le \frac{n}{2^{n-k} }\sum_{e=1}^{n(\alpha + \epsilon)}{\frac{\sum_{w=d_{\mathrm{min}}}^{e}{\binom{n}{w} \binom{n-w}{e-w}}}{\binom{n}{e}}} + \epsilon' \label{eq:BEC_CAS_pf_Aw} \\
& \le \frac{n}{2^{n-k} }\sum_{e=1}^{n(\alpha + \epsilon)}{\frac{\sum_{w=d_{\mathrm{min}}}^{e}{\binom{e}{w} \binom{n}{e}}}{\binom{n}{e}}} + \epsilon' \label{eq:BEC_CAS_binom} \\
& \le \frac{n}{2^{n-k} }\sum_{e=1}^{n(\alpha + \epsilon)}{\sum_{w=d_{\mathrm{min}}}^{e}{\binom{e}{w}}} + \epsilon' \\
& \le \frac{n}{2^{n-k} }\sum_{e=1}^{n(\alpha + \epsilon)}{2^{e}} + \epsilon' \\
& \le n^2 2^{k-n} (\alpha + \epsilon) 2^{n (\alpha + \epsilon)} + \epsilon' \\
& = n^2 (\alpha + \epsilon) 2^{n \left\{R - (1 - \alpha) + \epsilon \right\}} + \epsilon' \label{eq:BEC_CAS_capacity}
\end{align}
where we assume that $n(\alpha + \epsilon)$ is an integer without loss of generality in \eqref{eq:BEC_CAS_pf_1}. Also, \eqref{eq:BEC_CAS_pf_UB} follows from \eqref{eq:BEC_UB} in Lemma~\ref{lemma:BEC_UB}. \eqref{eq:BEC_CAS_pf_Aw} follows from the fact that there exists an $\left[n, k\right]$ binary linear code whose weight distribution is bounded by \cite{Barg2014}
\begin{equation} \label{eq:Aw_UB}
A_w \le \frac{n}{2^{n-k}} \binom{n}{w}.
\end{equation}
Also, \eqref{eq:BEC_CAS_binom} follows from $\binom{n}{w} \binom{n-w}{e-w} = \binom{e}{w} \binom{n}{e}$.
If $R < 1 - \alpha - \epsilon = C_{\mathrm{BEC}} - \epsilon$ and $n$ is sufficiently large, \eqref{eq:BEC_CAS_capacity} goes to zero. Thus, the decoding rule of $g$ achieves $C_{\mathrm{BEC}}$.
\end{IEEEproof}
\begin{remark} \label{remark:BEC} We can show that the decoding rule $g$ is a capacity achieving scheme without considering the weight distribution of codes. If each element of $G$ in \eqref{eq:BEC_decoder_LE} is selected uniformly at random from $\left\{0, 1\right\}$,
\begin{equation} \label{eq:BEC_CAS_G}
P\left( \rank \left(G^{\mathcal{V}}\right) < k \right) = \frac{2^k}{2^v} = 2^{-n(C_{\mathrm{BEC}} - R)}.
\end{equation}
Similarly, if each element of $H$ in \eqref{eq:BEC_decoder_LE_H} is selected uniformly at random from $\left\{0, 1\right\}$,
\begin{equation} \label{eq:BEC_CAS_H}
P\left( \rank \left(H^{\mathcal{E}}\right) < e \right) = \frac{2^e}{2^{n-k}} = 2^{-n(C_{\mathrm{BEC}} - R)}.
\end{equation}
If $R < C_{\mathrm{BEC}}$ and $n$ is sufficiently large, both \eqref{eq:BEC_CAS_G} and \eqref{eq:BEC_CAS_H} go to zero. Thus, the decoding rule of $g$ can achieve $C_{\mathrm{BEC}}$ by solving either \eqref{eq:BEC_decoder_LE} or \eqref{eq:BEC_decoder_LE_H}.
\end{remark}
\begin{remark} The computational complexity of \eqref{eq:BEC_decoder_LE} is $\mathcal{O}\left(k^3\right)$ where $k = nR$. Also, the computational complexity of \eqref{eq:BEC_decoder_LE_H} is $\mathcal{O}\left(e^3\right)$ where $e = n \alpha$. Though both complexities are eventually $\mathcal{O}\left(n^3\right)$, we can choose one of them to reduce the computational complexity. If $\alpha < 0.5$ and $R > 0.5$, the computational complexity of \eqref{eq:BEC_decoder_LE_H} is less than that of \eqref{eq:BEC_decoder_LE}.
\end{remark}
\subsection{Binary Defect Channel}
We will use the notations of \cite{Heegard1983, Kim2013} with slight modifications for the BDC. We define an additional variable ``$\lambda$'' (denoting the defect-free state) and the channel state $S \in \{0, 1, \lambda\}$. Let ``$\circ$'' denote the operator $\circ:\{0, 1\} \times \left\{0, 1, \lambda\right\} \rightarrow \{0, 1\}$ by
\begin{equation}\label{eq:circ_operator}
x \circ s =
\begin{cases}
x, & \text{if } s = \lambda ; \\
s, & \text{if } s \ne \lambda.
\end{cases}
\end{equation}
An $n$-cell memory with defects is modeled by
\begin{equation}\label{eq:BDC_channel_model}
\mathbf{y} = \mathbf{x} \circ \mathbf{s}
\end{equation}
where $\mathbf{x} \in \left\{0, 1\right\}^n$ is the channel input vector and $\mathbf{y} \in \left\{0, 1\right\}^n$ is the channel output vector. Also, $\mathbf{s} \in \left\{0, 1, \lambda \right\}^n$ is the channel state vector which has the information of defect locations and stuck-at values. Note that $\circ$ is the vector component-wise operator. The number of defects is equal to the number of non-$\lambda$ components in $\mathbf{s}$. The number of errors due to defects is given by
\begin{equation}\label{eq:BDC_num_errors}
\| \mathbf{x} \circ \mathbf{s} - \mathbf{x} \|.
\end{equation}
As shown in Fig.~\ref{fig:BDC},
\begin{equation}\label{eq:BDC_random_channel_model}
\begin{aligned}
P(S=s)& =
\begin{cases}
1-\beta, & s = \lambda; \\
\frac{\beta}{2}, & s = 0; \\
\frac{\beta}{2}, & s = 1.
\end{cases}
\end{aligned}
\end{equation}
In \cite{Heegard1983}, Heegard discussed additive encoding and defined the $[n,k,l]$ partitioned linear block code (PLBC) which consists of a pair of linear subspaces $\mathcal{C}_1 \subset \{0, 1\}^n$ and $\mathcal{C}_0 \subset \{0, 1\}^n$ of dimension $k$ and $l$ such that $\mathcal{C}_1 \cap \mathcal{C}_0 =\{ \mathbf{0}\}$. Then the direct sum is given by
\begin{equation}\label{eq:direct_sum}
\mathcal{C} \triangleq \mathcal{C}_1 + \mathcal{C}_0 = \{ \mathbf{c} = \mathbf{c}_1 + \mathbf{c}_0 | \mathbf{c}_1 \in \mathcal{C}_1 , \mathbf{c}_0 \in \mathcal{C}_0 \}.
\end{equation}
\emph{Encoding:} A message $\mathbf{m} \in \{0, 1\}^k$ is encoded to a corresponding codeword $\mathbf{c}$ as follows.
\begin{equation}
\mathbf{c}= \mathbf{c}_1 + \mathbf{c}_0 = G_1 \mathbf{m} + G_0 \mathbf{d}
\end{equation}
where $\mathbf{c}_1 = G_1 \mathbf{m}$ and $\mathbf{c}_0 = G_0 \mathbf{d}$. The generator matrix for $\mathbf{c}_1$ is $G_1 = \left[ I_k \quad 0_{k,l} \right]^T$ where $I_k$ is the $k$-dimensional identity matrix and $0_{k,l}$ is the zero matrix with size of $k \times l$. Also, the generator matrix for $\mathbf{c}_0$ is $G_0$ which is an $n \times l$ matrix. Note that $k + l = n$.
Since the channel state vector $\mathbf{s}$ is available at the encoder, the encoder should choose $\mathbf{d} \in \{0, 1\}^l$ judiciously. The optimal parity $\mathbf{d}$ is chosen to minimize the number of errors due to defects, i.e., $ \|(\mathbf{c} \circ \mathbf{s}) - \mathbf{c} \|$.
\emph{Decoding:} The decoder estimates the message $\mathbf{m}$ as follows.
\begin{equation} \label{eq:BDC_decoding}
\widehat{\mathbf{m}} = \widetilde{G}_1^T \mathbf{y} = H_0^T \mathbf{y}
\end{equation}
where $\widehat{\mathbf{m}}$ is the estimate of $\mathbf{m}$ and the channel output vector $\mathbf{y} = \mathbf{c} \circ \mathbf{s}$ is given by \eqref{eq:BDC_channel_model}. The message inverse matrix $\widetilde{G}̃_1$ is defined as an $n \times k$ matrix such that $\widetilde{G}̃_1^T G_1=I_k$, and $\widetilde{G}̃_1^T G_0 =0_{k,l}$ \cite{Heegard1983}. For the BDC, the message inverse matrix $\widetilde{G}̃_1$ defined by Heegard will be the systematic parity check matrix $H_0$ since it satisfies two conditions for the message inverse matrix.
For convenience, we will define a random variable $M$ as follows.
\begin{equation}
M =
\begin{cases}
0, & \|(\mathbf{c} \circ \mathbf{s}) - \mathbf{c} \| \ne 0 \text{ (masking failure)}; \\
1, & \|(\mathbf{c} \circ \mathbf{s}) - \mathbf{c} \| = 0 \text{ (masking success)}
\end{cases}
\end{equation}
The minimum distance $d_0$ of an $[n,k,l]$ PLBC is given by \cite{Tsybakov1975b, Heegard1983}
\begin{align} \label{eq:BDC_dmin}
d_0 &= \underset{
\substack{
\mathbf{c} \ne \mathbf{0} \\
G_0^T \mathbf{c}= \mathbf{0}
}}
{\text{min }} \|\mathbf{c}\|
\end{align}
which means that any $d_0 - 1$ rows of $G_0$ are linearly independent. If $u < d_0$, all $u$ defects will be masked and $\|(\mathbf{c} \circ \mathbf{s})-\mathbf{c} \|=0$ (i.e., $M = 1$), which will be shown in Lemma~\ref{lemma:BDC_UB}.
The encoder knows the channel state vector $\mathbf{s}$ and tries to minimize $ \|(\mathbf{c} \circ \mathbf{s}) - \mathbf{c} \|$ by choosing $\mathbf{d}$ judiciously. The encoding of PLBC includes an implicit optimization problem which can be formulated as follows \cite{Heegard1983, Lastras-Montano2010, Hwang2011a}.
\begin{align}
\mathbf{d}^* & = \underset{\mathbf{d}}{\text{argmin }} \left\|G_0^{\mathcal{U}} \mathbf{d} + G_1^{\mathcal{U}} \mathbf{m} - \mathbf{s}^{\mathcal{U}} \right\| \\
&= \underset{\mathbf{d}}{\text{argmin }} \left\| G_0^{\mathcal{U}} \mathbf{d} + \mathbf{b}^{\mathcal{U}} \right\| \label{eq:BDC_opt_problem}
\end{align}
where $\mathcal{U}=\left\{i_1,\cdots,i_u \right\}$ indicates the set of locations of $u$ defects and $\mathbf{b} = G_1 \mathbf{m} - \mathbf{s}$. Thus, $\mathbf{b}^{\mathcal{U}}$ is given by
\begin{equation}\label{eq:BDC_b}
\mathbf{b}^{\mathcal{U}} = G_1^{\mathcal{U}} \mathbf{m} - \mathbf{s}^{\mathcal{U}}
\end{equation}
where $s^{\mathcal{U}}=\left(s_{i_1},\cdots,s_{i_u}\right)^T$, $G_0^{\mathcal{U}}=\left[\mathbf{g}_{0,i_1}^T,\cdots,\mathbf{g}_{0,i_u}^T \right]^T$, and $G_1^{\mathcal{U}}=\left[\mathbf{g}_{1,i_1}^T,\cdots,\mathbf{g}_{1,i_u}^T \right]^T$. Note that $\mathbf{g}_{0,i}$ and $\mathbf{g}_{1,i}$ are the $i$-th rows of $G_0$ and $G_1$ respectively. Also, $\left\|G_0^{\mathcal{U}} \mathbf{d}+ \mathbf{b}^{\mathcal{U}} \right\|$ represents the number of errors due to defects which is equivalent to \eqref{eq:BDC_num_errors}.
By solving the optimization problem of \eqref{eq:BDC_opt_problem}, the number of errors due to defects will be minimized. However, the computational complexity for solving \eqref{eq:BDC_opt_problem} is exponential, which is impractical \cite{Hwang2011a}.
Instead of solving the impractical optimization problem, $C_{\mathrm{BDC}}$ can be achieved by solving the following system of linear equations \cite{Jagmohan2010a}.
\begin{equation}\label{eq:BDC_encoder_LE}
G_0^{\mathcal{U}} \mathbf{d} = \mathbf{b}^{\mathcal{U}}
\end{equation}
Because of the weak law of large numbers, we can claim that $n C_{\mathrm{BDC}} - \epsilon \le n - u \le n C_{\mathrm{BDC}} + \epsilon$ with high probability for sufficiently large $n$. For $R < C_{\mathrm{BDC}} - \epsilon$, \eqref{eq:BDC_encoder_LE} is \emph{underdetermined} since $G_0^{\mathcal{U}}$ is a $u \times (n - k)$ matrix. If \eqref{eq:BDC_encoder_LE} has at least one solution, the masking succeeds since $\left\| G_0^{\mathcal{U}} \mathbf{d} + \mathbf{b}^{\mathcal{U}} \right\|=0$.
In \cite{Kim2013}, an upper bound on the probability of masking failure was derived and numerical results showed that the upper bound is tight and the performance of partitioned Bose, Chaudhuri, Hocquenghem (PBCH) codes is not far from $C_{\textrm{BDC}}$. We will show that the additive encoding achieves $C_{\mathrm{BDC}}$ by using the upper bound in \cite{Kim2013}, which explains why the performance of PBCH codes is good. The PBCH code is a special class of PLBC and its generator matrices and minimum distances can be designed by a similar method such as standard BCH codes \cite{Heegard1983, Kim2013:plbc}
First, we will present the upper bound on the probability of masking failure for $u$ defects through the following Lemma~\ref{lemma:BDC_lower_upper_bounds},~\ref{lemma:BDC_UB},~\ref{lemma:BDC_exact}, and Theorem~\ref{thm:BDC_bound}.
\begin{lemma}[{\cite{Kim2013}}] \label{lemma:BDC_lower_upper_bounds} The lower and upper bounds on $P(M=0||\mathcal{U}|=u)$ is given by
\begin{equation} \label{eq:BDC_lower_upper_bounds}
\begin{aligned}
\frac{1}{2} \cdot P \left( \rank \left( G_0^{\mathcal{U}} \right) < u \mid |\mathcal{U}|=u \right) &\le P\left(M=0 \mid |\mathcal{U}|=u\right) \\
& \le P \left( \rank \left( G_0^{\mathcal{U}} \right) < u \mid |\mathcal{U}|=u \right).
\end{aligned}
\end{equation}
\end{lemma}
\begin{IEEEproof}
\eqref{eq:BDC_encoder_LE} has at least one solution if and only if
\begin{equation} \label{eq:BDC_LE_sol_exist}
\rank \left( G_0^{\mathcal{U}} \right) = \rank \left( G_0^{\mathcal{U}} \mid \mathbf{b}^{\mathcal{U}} \right)
\end{equation}
where $\left( G_0^{\mathcal{U}} \mid \mathbf{b}^{\mathcal{U}} \right)$ is the augmented matrix.
If $\rank \left( G_0^{\mathcal{U}} \right) = u$, \eqref{eq:BDC_encoder_LE} has at least one solution since \eqref{eq:BDC_LE_sol_exist} holds. Thus, $P\left(M=0 \mid |\mathcal{U}|=u\right)=0$.
If $\rank \left( G_0^{\mathcal{U}} \right) = u - j$ for $1 \le j \le u$, the last $j$ rows of the row reduced echelon form of $G_0^{\mathcal{U}}$ are zero vectors. In order to satisfy the condition of \eqref{eq:BDC_LE_sol_exist}, the last $j$ elements of the column vector $\mathbf{b}^{\mathcal{U}}$ should also be zeros. The probability that the last $j$ elements of the column vector $\mathbf{b}^{\mathcal{U}}$ are zeros is $\frac{1}{2^j}$ since $P(S=0 \mid S \ne \lambda) = P(S=1 \mid S \ne \lambda) = \frac{1}{2}$. Thus, $P\left(M=0 \mid |\mathcal{U}|=u\right)$ is given by
\begin{equation} \label{eq:BDC_exact}
P\left(M=0 \mid |\mathcal{U}|=u\right) = \sum_{j=1}^{u}{\frac{2^j-1}{2^j}P \left( \rank \left( G_0^{\mathcal{U}} \right) =u - j \mid |\mathcal{U}|=u \right)}
\end{equation}
which results in \eqref{eq:BDC_lower_upper_bounds}.
\end{IEEEproof}
\begin{lemma}[\cite{Kim2013}] \label{lemma:BDC_UB} The upper bound on $P(M=0||\mathcal{U}|=u)$ is given by
\begin{equation}\label{eq:BDC_UB}
P\left(M=0 \mid |\mathcal{U}|=u \right) \le \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}}
\end{equation}
where $B_{w}$ is the weight distribution of $\mathcal{C}_{0}^{\perp}$ (i.e., the dual code of $\mathcal{C}_0$).
\end{lemma}
\begin{IEEEproof}
Suppose that there exists a nonzero codeword $\mathbf{c}^{\perp} \in \mathcal{C}_{0}^{\perp}$ of Hamming weight $w$. Note that $G_0$ is the parity check matrix of $\mathcal{C}_{0}^{\perp}$. Let $\Psi_w(\mathbf{c}^{\perp})=\left\{ i \mid c_i^{\perp} \ne 0 \right\}$ denote the locations of nonzero elements of $\mathbf{c}^{\perp}$ and $\mathcal{U} =\left\{i_1,\ldots,i_u \right\}$ denote the locations of $u$ defects.
If $\Psi_w(\mathbf{c}^{\perp}) \subseteq \mathcal{U}$, $\rank \left( G_0^{\mathcal{U}} \right) < u$. The reason is that $G_0^{\Psi_{w}(\mathbf{c}^{\perp})}$ is a submatrix of $G_0^{\mathcal{U}}$ and the rows of $G_0^{\Psi_{w}(\mathbf{c}^{\perp})}$ are linearly dependent since $G_0^T \mathbf{c}^{\perp} = \mathbf{0}$.
For any $\mathbf{c}^{\perp}$ such that $\Psi_w(\mathbf{c}^{\perp}) \subseteq \mathcal{U}$, the number of possible $\mathcal{U}$ is $\binom{n-w}{u-w}$. Due to double counting, the number of $\mathcal{U}$ which results in $\rank \left( G_0^{\mathcal{U}} \right) < u$ will be less than or equal to $\sum_{w=d_{0}}^{u}{B_w \binom{n-w}{u-w}}$. Since the number of all possible $\mathcal{U}$ such that $|\mathcal{U}|=u$ is $\binom{n}{u}$,
\begin{equation} \label{eq:BDC_upper_bound_rank}
P\left(\rank \left( G_0^{\mathcal{U}} \right) < u \mid |\mathcal{U}|=u \right) \le \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}}.
\end{equation}
By \eqref{eq:BDC_lower_upper_bounds} and \eqref{eq:BDC_upper_bound_rank}, the upper bound on $P\left(M=0 \mid |\mathcal{U}|=u \right)$ is given by \eqref{eq:BDC_UB}.
\end{IEEEproof}
From the upper bound in Lemma~\ref{lemma:BDC_UB}, it is clear that $P\left(M=0 \mid |\mathcal{U}|=u \right) = 0$ for $u < d_{0}$. It is worth mentioning that the upper bound on $P\left(M=0 \mid |\mathcal{U}|=u\right)$ for the BDC is similar to the upper bound on $P\left(D=0 \mid |\mathcal{E}|=e\right)$ for the BEC presented in Lemma~\ref{lemma:BEC_UB}.
Similar to Lemma~\ref{lemma:BEC_exact}, the following Lemma shows that $P\left(M=0 \mid |\mathcal{U}|=u \right)$ can be obtained exactly for $d_0 \le u \le d_0 + \left\lfloor \frac{d_0-1}{2} \right\rfloor$.
\begin{lemma}[\cite{Kim2013}] \label{lemma:BDC_exact}For $u \le d_0 + t_0$ where $t_0 = \left\lfloor \frac{d_0 - 1}{2} \right\rfloor$, $P\left(M=0 \mid |\mathcal{U}|=u \right)$ is given by
\begin{equation}
\label{eq:BDC_exact_condition}
P\left(M=0 \mid |\mathcal{U}|=u \right) = \frac{1}{2} \cdot \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}}.
\end{equation}
\end{lemma}
\begin{IEEEproof}
The proof has two parts. First, we will show that
\begin{equation} \label{eq:BDC_exact_condition_pf1}
P\left(\rank \left( G_0^{\mathcal{U}} \right) < u \mid |\mathcal{U}|=u \right) = \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}}
\end{equation}
for $u \le d_0 + t_0$, which means that there is no double counting in \eqref{eq:BDC_upper_bound_rank}. Second, we will prove that
\begin{equation} \label{eq:BDC_exact_condition_pf2}
P\left(\rank \left( G_0^{\mathcal{U}} \right) < u \mid |\mathcal{U}|=u \right) = P\left(\rank \left( G_0^{\mathcal{U}} \right) = u-1 \mid |\mathcal{U}|=u \right)
\end{equation}
for $u \le d_0 + t_0$, which means that $P\left(\rank \left( G_0^{\mathcal{U}} \right) \le u-2 \mid |\mathcal{U}|=u \right)=0$.
Then, $P\left(M=0 \mid |\mathcal{U}|=u \right)$ is given by
\begin{align}
P\left(M=0 \mid |\mathcal{U}|=u \right) &= \frac{1}{2} \cdot P \left( \rank \left( G_0^{\mathcal{U}} \right) = u - 1 \mid |\mathcal{U}|=u \right) \label{eq:BDC_exact_condition_step1}\\
&= \frac{1}{2} \cdot \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}} \label{eq:BDC_exact_condition_step2}
\end{align}
where \eqref{eq:BDC_exact_condition_step1} follows from \eqref{eq:BDC_exact} and \eqref{eq:BDC_exact_condition_pf2}. Also, \eqref{eq:BDC_exact_condition_step2} follows from \eqref{eq:BDC_exact_condition_pf1}.
1) Proof of \eqref{eq:BDC_exact_condition_pf1}
Suppose that there are two nonzero codewords $\mathbf{c}_1^{\perp}, \mathbf{c}_2^{\perp} \in \mathcal{C}_0^{\perp}$ such that $\|\mathbf{c}_1^{\perp}\| = w_1$ and $\|\mathbf{c}_2^{\perp}\| = w_2$. Without loss of generality, we can assume that $d_0 \le w_1 \le w_2$. The locations of nonzero elements of $\mathbf{c}_1^{\perp}$ and $\mathbf{c}_2^{\perp}$ are given by
\begin{align}
\Psi_{w_1}\left(\mathbf{c}_1^{\perp}\right)&= \left\{ i_{1,1},\ldots,i_{1, w_1} \right\} \\
\Psi_{w_2}\left(\mathbf{c}_2^{\perp}\right)&= \left\{ i_{2,1},\ldots,i_{2, w_2} \right\}.
\end{align}
Let $\Psi_{\alpha}=\left\{i_1, \ldots, i_{\alpha} \right\}$ denote $\Psi_{\alpha} = \Psi_{w_1}\left(\mathbf{c}_1^{\perp}\right) \cap \Psi_{w_2}\left(\mathbf{c}_2^{\perp}\right)$. Then $\Psi_{w_1}\left(\mathbf{c}_1^{\perp}\right)$ and $\Psi_{w_2}\left(\mathbf{c}_2^{\perp}\right)$ are given by
\begin{align}
\Psi_{w_1}\left(\mathbf{c}_1^{\perp}\right)&= \Psi_{\alpha} \cup \left\{ i_{1,1}',\ldots,i_{1, \beta_1}' \right\} \\
\Psi_{w_2}\left(\mathbf{c}_2^{\perp}\right)&= \Psi_{\alpha} \cup \left\{ i_{2,1}',\ldots,i_{2, \beta_2}' \right\}
\end{align}
where $i_{1, j_1}'$ for $j_1 \in \left\{1, \ldots, \beta_1 \right\}$ and $i_{2, j_2}'$ for $j_2 \in \left\{1, \ldots, \beta_2 \right\}$ are the reindexed locations of nonzero elements of $\mathbf{c}_1^{\perp}$ and $\mathbf{c}_2^{\perp}$ that are mutually disjoint with $\Psi_{\alpha}$. Note that $\left\{ i_{1,1}',\ldots,i_{1, \beta_1}' \right\} \cap \left\{ i_{2,1}',\ldots,i_{2, \beta_2}' \right\} = \emptyset $, $\beta_1 = w_1 - \alpha$ and $\beta_2 = w_2 - \alpha$.
Due to the property of linear codes, $\mathbf{c}_3^{\perp} = \mathbf{c}_1^{\perp} + \mathbf{c}_2^{\perp}$ is also a codeword of $\mathcal{C}_0^{\perp}$, i.e., $\mathbf{c}_3^{\perp} \in \mathcal{C}_0^{\perp}$ and $\|\mathbf{c}_3^{\perp} \|=\beta_1 + \beta_2$. Also, the following conditions should hold because of the definition of $d_0$.
\begin{align}
\alpha + \beta_1 &\ge d_0 \\
\alpha + \beta_2 &\ge d_0 \\
\beta_1 + \beta_2 &\ge d_0
\end{align}
Thus, we can claim that $2 \left( \alpha + \beta_1 + \beta_2 \right) \ge 3 d_0$, which results in $\alpha + \beta_1 + \beta_2 \ge d_0 + \left\lfloor \frac{d_0 + 1}{2} \right\rfloor = d_0 + t_0 + 1$ since $\alpha + \beta_1 + \beta_2$ is an integer.
For double counting in \eqref{eq:BDC_upper_bound_rank}, there should exist at least two codewords $\mathbf{c}_1^{\perp}$ and $\mathbf{c}_2^{\perp}$ such that $\Psi_{w_1}\left(\mathbf{c}_1^{\perp}\right) \cup \Psi_{w_2}\left(\mathbf{c}_2^{\perp}\right) \subseteq\mathcal{U}$. It means that double counting occurs only if $u \ge \alpha + \beta_1 + \beta_2 \ge d_0 + t_0 + 1$. Thus, there is no double counting for $u \le d_0 + t_0$. For $u \le d_0 + t_0$, there exists at most one codeword $\mathbf{c}^{\perp}$ such that $\Psi_{w}\left( \mathbf{c}^{\perp} \right) \subseteq \mathcal{U}$.
2) Proof of \eqref{eq:BDC_exact_condition_pf2}
It is clear that $\rank \left( G_0^{\mathcal{U}}\right) = u - 1$ if and only if there exists only one nonzero codeword $\mathbf{c}^{\perp}$ such that $\Psi_{w}\left( \mathbf{c}^{\perp} \right) \subseteq \mathcal{U}$. Note that $\rank \left( G_0^{\mathcal{U}}\right) < u - 1$ if and only if $\mathcal{U}$ includes the locations of nonzero elements of at least two nonzero codewords. We have already shown that there exists at most one nonzero codeword $\mathbf{c}^{\perp}$ such that $\Psi_{w}\left( \mathbf{c}^{\perp} \right) \subseteq \mathcal{U}$ for $u \le d_0 + t_0$.
\end{IEEEproof}
Similar to the upper bound on $P\left( D = 0 \mid |\mathcal{E}|=e \right)$ in Theorem~\ref{thm:BEC_bound} for the BEC, we can provide the upper bound on $P\left( M = 0 \mid |\mathcal{U}|=u \right)$ for the BDC as follows.
\begin{theorem}[\cite{Kim2013}] \label{thm:BDC_bound} $P\left(M = 0 \mid |\mathcal{U}|=u\right)$ is given by
\begin{numcases}{P\left(M = 0 \mid |\mathcal{U}|=u\right)=}
0 & for $u < d_0$,
\\
\frac{1}{2} \cdot \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}} & for $d_0 \le u \le d_0+t_0$, \label{eq:BDC_exact_condition_thm}
\\
\le \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}} & for $ u > d_0+t_0$.
\end{numcases}
\end{theorem}
\begin{IEEEproof}The proof comes from the definition of $d_0$ in \eqref{eq:BDC_dmin}, Lemma~\ref{lemma:BDC_UB} and Lemma~\ref{lemma:BDC_exact}.
\end{IEEEproof}
Comparing the upper bound on $P\left( D = 0 \mid |\mathcal{E}|=e \right)$ for the BEC and $P\left( M = 0 \mid |\mathcal{U}|=u \right)$ for BDC, the duality of erasures and defects can be seen. The expressions for both upper bounds in Theorem~\ref{thm:BEC_bound} and Theorem~\ref{thm:BDC_bound} are very similar. The one difference is in the minimum distances such as $d_{\textrm{min}}$ and $d_{0}$. For the definition of $d_{\textrm{min}}$, $H$ is the parity check matrix in \eqref{eq:BEC_dmin}. Meanwhile, $G_0$ is the parity check matrix in \eqref{eq:BDC_dmin}. The other difference comes from the weight distributions such as $A_w$ and $B_w$. Note that $A_w$ is the weight distribution of $\mathcal{C}$ and $B_w$ the weight distribution of $\mathcal{C}_0^{\perp}$.
The following Theorem shows that the capacity of the BDC can be achieved by an encoding scheme based on solving the linear equations \eqref{eq:BDC_encoder_LE}.
\begin{theorem} The encoding scheme of solving the linear equations \eqref{eq:BDC_encoder_LE} is a capacity achieving scheme. \label{thm:BDC}
\end{theorem}
\begin{IEEEproof}
The masking failure probability is given by
\begin{align}
P\left( M=0 \right) & = P\left(M=0, U \le n(\beta + \epsilon) \right) + P\left(M=0, U > n(\beta + \epsilon) \right) \label{eq:BDC_CAS_pf_1}\\
& \le \sum_{u=1}^{n(\beta + \epsilon)}{P(M=0, |\mathcal{U}|=u)} + \epsilon' \\
& \le \sum_{u=1}^{n(\beta + \epsilon)}{P(M=0 \mid |\mathcal{U}|=u)} + \epsilon' \\
& \le \sum_{u=1}^{n(\beta + \epsilon)}{ \frac{\sum_{w=d_{0}}^{u}{B_{w} \binom{n-w}{u-w}}}{\binom{n}{u}}} + \epsilon' \label{eq:BDC_CAS_pf_UB} \\
& \le \frac{n}{2^{n-k} }\sum_{u=1}^{n(\beta + \epsilon)}{\frac{\sum_{w=d_{0}}^{u}{\binom{n}{w} \binom{n-w}{u-w}}}{\binom{n}{u}}} + \epsilon' \label{eq:BDC_CAS_pf_Bw} \\
& \le \frac{n}{2^{n-k} }\sum_{u=1}^{n(\beta + \epsilon)}{\frac{\sum_{w=d_{0}}^{u}{\binom{u}{w} \binom{n}{u}}}{\binom{n}{u}}} + \epsilon' \label{eq:BDC_CAS_binom} \\
& \le \frac{n}{2^{n-k} }\sum_{u=1}^{n(\beta + \epsilon)}{\sum_{w=d_{0}}^{u}{\binom{u}{w}}} + \epsilon' \\
& \le \frac{n}{2^{n-k} }\sum_{u=1}^{n(\beta + \epsilon)}{2^{u}} + \epsilon'\\
& \le n^2 2^{k-n} (\beta + \epsilon) 2^{n (\beta + \epsilon)} + \epsilon' \\
& = n^2 (\beta + \epsilon) 2^{n\left\{R - \left(1 - \beta \right) + \epsilon\right\}} + \epsilon' \label{eq:BDC_CAS_capacity}
\end{align}
where we assume that $n(\beta + \epsilon)$ is an integer without loss of generality in \eqref{eq:BDC_CAS_pf_1}. Also, \eqref{eq:BDC_CAS_pf_UB} follows from Lemma~\ref{lemma:BDC_UB}. \eqref{eq:BDC_CAS_pf_Bw} follows from the fact that $\mathcal{C}_{0}^{\perp}$ is an $[n, k]$ linear code and the upper bound on the weight distribution of $\mathcal{C}_{0}^{\perp}$ can be obtained from \eqref{eq:Aw_UB}. \eqref{eq:BDC_CAS_binom} follows from the fact that $\binom{n}{w} \binom{n-w}{u-w} = \binom{u}{w} \binom{n}{u}$.
If $R < 1 - \beta - \epsilon = C_{\mathrm{BDC}} - \epsilon$ and $n$ is sufficiently large, \eqref{eq:BDC_CAS_capacity} goes to zero. Thus, the additive encoding with solving the system of linear equations of \eqref{eq:BDC_encoder_LE} achieves the channel capacity of BDC.
\end{IEEEproof}
It is well known that random binning is a capacity achieving scheme for the BDC. The encoding of random binning is as follows \cite{Heegard1983:capacity, ElGamal2011}: Randomly partition the $2^n$ sequences into $2^{nR}$ equal size subsets (or bins) and associate a different message with each bin. When the $i$-th message is to be stored, search the $i$-th bin for a sequence (or codeword) $\mathbf{c}$ such that $\mathbf{c} \circ \mathbf{s} = \mathbf{c}$. The decoding is to choose the index of the bin that the channel output vector $\mathbf{y}$ belongs to.
The encoding of random binning can be described by the following linear equations \cite{Jagmohan2010a}.
\begin{equation} \label{eq:BDC_random_binning_1}
H_0^T \mathbf{c} = \mathbf{m}
\end{equation}
where $\mathbf{c}$ will be chosen to satisfy $\mathbf{c} \circ \mathbf{s} = \mathbf{c}$. We can see that the linear equations for the encoding of random binning is equivalent to \eqref{eq:BDC_decoding} which represents the decoding of additive encoding, which shows \emph{the duality between additive encoding and random binning}.
\eqref{eq:BDC_random_binning_1} can be modified into
\begin{align}\label{eq:BDC_random_binning_2}
H_0^T \mathbf{c} & = \left(H_0^{\mathcal{U}} \right)^T \mathbf{c}^{\mathcal{U}} + \left(H_0^{\mathcal{W}} \right)^T \mathbf{c}^{\mathcal{W}} \\
& = \mathbf{m}
\end{align}
where $\mathcal{U}=\left\{i_1,\cdots, i_u\right\}$ indicates the locations of stuck-at defects and $\mathcal{W} = \left\{j_1,\cdots, j_w\right\}$ represents the locations of normal cells such that $\mathcal{U} \cup \mathcal{W} = \left\{ 1,2,\ldots, n \right\}$. Note that $\mathbf{c}^{\mathcal{U}}=\left(c_{i_1}, \cdots, c_{i_u}\right)^T$, $\mathbf{c}^{\mathcal{W}}=\left(c_{j_1}, \cdots, c_{j_w}\right)^T$, $H_0^{\mathcal{U}}=\left[ \mathbf{h}_{0, i_1}^T, \cdots, \mathbf{h}_{0, i_e}^T \right]^T$ and $H_0^{\mathcal{W}}=\left[ \mathbf{h}_{0, j_1}^T, \cdots, \mathbf{h}_{0, j_w}^T \right]^T$ where $\mathbf{h}_{0, i}$ is the $i$-th row of $H_0$. Since $\mathbf{s}^{\mathcal{U}}$ is known to the encoder, the encoder of random binning can set $\mathbf{c}^{\mathcal{U}} = \mathbf{s}^{\mathcal{U}}$. Thus, the random binning can be described by
\begin{equation}\label{eq:BDC_encoder_LE_binning}
\left(H_0^{\mathcal{W}} \right)^T \mathbf{c}^{\mathcal{W}} = \mathbf{m}'
\end{equation}
where $\mathbf{m}' = \mathbf{m} - \left(H_0^{\mathcal{U}} \right)^T \mathbf{s}^{\mathcal{U}}$. The solution of \eqref{eq:BDC_encoder_LE_binning} represents the codeword elements of normal cells. Note that $\left(H_0^{\mathcal{W}} \right)^T$ is a $k \times (n - u)$ matrix. Thus, \eqref{eq:BDC_encoder_LE_binning} is also \emph{underdetermined} for $R < C_{\mathrm{BDC}} - \epsilon$.
\begin{remark} \label{remark:BDC} We can show that both additive encoding and random binning are capacity achieving scheme by the same method in Remark~\ref{remark:BEC}. If each element of $G_0$ in \eqref{eq:BDC_encoder_LE} is selected uniformly at random from $\left\{0, 1\right\}$,
\begin{equation} \label{eq:BDC_CAS_G}
P\left( \rank \left(G_0^{\mathcal{U}}\right) < u \right) = \frac{2^u}{2^{n-k}} = 2^{-n(C_{\mathrm{BDC}} - R)}.
\end{equation}
Similarly, if each element of $H_0$ in \eqref{eq:BDC_encoder_LE_binning} is selected uniformly at random from $\left\{0, 1\right\}$,
\begin{equation} \label{eq:BDC_CAS_H}
P\left( \rank \left(H_0^{\mathcal{W}}\right) < u \right) = \frac{2^k}{2^{n-u}} = 2^{-n(C_{\mathrm{BDC}} - R)}.
\end{equation}
If $R < C_{\mathrm{BDC}}$ and $n$ is sufficiently large, both \eqref{eq:BDC_CAS_G} and \eqref{eq:BDC_CAS_H} go to zero. Thus, both additive encoding and random binning achieve $C_{\mathrm{BDC}}$ by solving linear equations.
\end{remark}
\begin{remark} The computational complexity of additive encoding of \eqref{eq:BDC_encoder_LE} is $\mathcal{O}\left(u^3\right)$ where $u = n\beta$. Also, the computational complexity of random binning of \eqref{eq:BDC_encoder_LE_binning} is $\mathcal{O}\left(k^3\right)$ where $k = n R$. Though both complexities are $\mathcal{O}\left(n^3\right)$, we can claim that \emph{additive encoding is better than random binning} since $\beta$ is generally very small for storage systems, i.e., $u \ll k$.
\end{remark}
\subsection{Duality between Erasures and Defects}
We will discuss the duality of erasures and defects which is summarized in Table~\ref{tab:duality}. In the BEC used for \emph{communication}, the channel input $X \in \left\{0, 1 \right\}$ is binary and the channel output $Y = \left\{ 0, 1, \varepsilon \right\}$ is ternary where the erasure $\varepsilon$ is \emph{neither} 0 nor 1. In the BDC used for \emph{storage}, the channel state $S \in \left\{0, 1, \lambda \right\}$ is ternary whereas the channel input and output are binary. The ternary channel state $S$ informs whether the given cells are stuck-at defects or normal cells. The stuck-at value is \emph{either} 0 or 1.
The expressions for capacities of both channels are quite similar as shown in \eqref{eq:BEC_capacity} and \eqref{eq:BDC_capacity}. In the BEC, the \emph{decoder} corrects erasures by using the information of locations of erasures, whereas the \emph{encoder} masks the defects by using the information of defect locations and stuck-at values in the BDC.
The capacity achieving scheme of the BEC can be represented by the linear equations based on the generator matrix $G$ of \eqref{eq:BEC_decoder_LE} or the linear equations based on the parity check matrix $H$ of \eqref{eq:BEC_decoder_LE_H}. Both linear equations are \emph{overdetermined}. The solution of the linear equations based on $G$ is the \emph{estimate of message} $\widehat{\mathbf{m}}$ and there should be only one $\widehat{\mathbf{m}}$ for decoding success. Also, the solution of the linear equations based on $H$ is the \emph{estimate of erased bits} $\widehat{\mathbf{c}}^{\mathcal{E}}$ which should be only one $\widehat{\mathbf{c}}^{\mathcal{E}}$ for decoding success.
On the other hand, the capacity achieving scheme of the BDC can be described by \emph{underdetermined} linear equations. The \emph{additive encoding} can be represented by the linear equations based on the generator matrix $G_0$ of \eqref{eq:BDC_encoder_LE} whose solution is the \emph{parity} $\mathbf{d}$. Also, the \emph{random binning} can be represented by the linear equations based on the parity check matrix $H_0$ of \eqref{eq:BDC_encoder_LE_binning} whose solution is the \emph{codeword elements of normal cells} $\mathbf{c}^{\mathcal{W}}$. Unlike the coding scheme of the BEC, there can be several solutions of $\mathbf{d}$ or $\mathbf{c}^{\mathcal{W}}$ that matches all stuck-at defects.
We can see the duality between erasures and defects by comparing the solution $\widehat{\mathbf{m}}$ of \eqref{eq:BEC_decoder_LE} and the solution $\mathbf{d}$ of \eqref{eq:BDC_encoder_LE}, i.e., \emph{message} and \emph{parity}. Note that coding schemes of \eqref{eq:BEC_decoder_LE} and \eqref{eq:BDC_encoder_LE} are based on the generator matrix. In addition, we can compare the duality of \emph{codeword elements of erasures} and \emph{codeword elements of normal cells} from \eqref{eq:BDC_encoder_LE} and \eqref{eq:BDC_encoder_LE_binning} which are coding schemes based on the parity check matrix.
In the BEC, the minimum distance $d_{\mathrm{min}}$ is defined by the \emph{parity check matrix} $H$, whereas the minimum distance $d_0$ of additive encoding for the BDC is defined by the \emph{generator matrix} $G_0$. The upper bound on the probability of decoding failure given $e$ erasures is dependent on the weight distribution of $\mathcal{C}$, whereas the upper bound on the probability of masking failure given $u$ defects is dependent on the weight distribution of $\mathcal{C}_0^{\perp}$.
If $A_w = B_w$ and $e = u$, it is clear that the upper bound on $P\left( D=0 \mid |\mathcal{E}|=e \right)$ is same as the upper bound on $P\left( M=0 \mid |\mathcal{U}|=u \right)$ by Theorem~\ref{thm:BEC_bound} and Theorem~\ref{thm:BDC_bound}. In particular, the following Theorem shows the equivalence of the failure probabilities (i.e., the probability of correction failure of erasures and the probability of masking failure of defects).
\begin{theorem} If $A_w = B_w$ and $\alpha = \beta$, then the probability of decoding failure of the BEC is same as the probability of masking failure of the BDC.\label{thm:BEC_BDC_failure}
\end{theorem}
\begin{IEEEproof}
Without loss of generality, we can assume that the all-zero codeword $\mathbf{0}$ has been transmitted through the BEC. If there is only one nonzero codeword such that $\Psi(\mathbf{c}_1) \subseteq \mathcal{E}$ where $\mathcal{E}$ indicates the location of $e$ erasures, the decoding success probability $P \left(D=1 \mid |\mathcal{E}|=e\right)=\frac{1}{2}$ since the decoder chooses between $\mathbf{0}$ and $\mathbf{c}_1$ randomly.
If there are two nonzero codewords $\mathbf{c}_1$ and $\mathbf{c}_2$ such that $\Psi(\mathbf{c}_1) \subseteq \mathcal{E}$ and $\Psi(\mathbf{c}_2) \subseteq \mathcal{E}$, it is clear that $\Psi(\mathbf{c}_3) \subseteq \mathcal{E}$ where $\mathbf{c}_3 = \mathbf{c}_1 + \mathbf{c}_2$. Since the decoder chooses one codeword from $\left\{\mathbf{0} , \mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3 \right\}$ randomly, $P \left(D=1 \mid |\mathcal{E}|=e \right)=\frac{1}{4}$. Similarly, if there are three codewords $\mathbf{c}_i$ such that $\Psi(\mathbf{c}_i) \subseteq \mathcal{E}$ for $i=1, 2, 3$ and $\mathbf{c}_1 + \mathbf{c}_2 \ne \mathbf{c}_3$, $P \left(D=1 \mid |\mathcal{E}|=e \right)=\frac{1}{8}$ since the decoder randomly chooses a codeword among $\left\{\mathbf{0} , \mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3, \mathbf{c}_1 + \mathbf{c}_2, \mathbf{c}_1 + \mathbf{c}_3, \mathbf{c}_2 + \mathbf{c}_3, \mathbf{c}_1 +\mathbf{c}_2 + \mathbf{c}_3 \right\}$. Generalizing this observation, we can claim that
\begin{equation} \label{eq:duality_BEC}
P \left(D=1 \mid |\mathcal{E}|=e \right)=\frac{1}{2^j}
\end{equation}
if $\Psi(\mathbf{c}_i) \subseteq \mathcal{E}$ for $i = 0,1\ldots,2^j-1$ and $\mathbf{c}_0 = \mathbf{0}$.
It is clear there are at least $j$ codewords $\mathbf{c}^{\perp} \in \mathcal{C}_0^{\perp}$ such that $G_0^T \mathbf{c}^{\perp} = \mathbf{0}$ and $\Psi(\mathbf{c}^{\perp}) \subseteq \mathcal{U}$ for $\rank\left( G_0^{\mathcal{U}} \right) = u - j$. From these $j$ codewords, we can list $2^j$ codewords $\mathbf{c}^{\perp} \in \mathcal{C}_0^{\perp}$ such that $G_0^T \mathbf{c}^{\perp} = \mathbf{0}$ and $\Psi(\mathbf{c}^{\perp}) \subseteq \mathcal{U}$. Since the last $j$ elements of the column vector $\mathbf{b}^{\mathcal{U}}$ in \eqref{eq:BDC_LE_sol_exist} should be zeros for masking success, we can claim that
\begin{equation} \label{eq:duality_BDC}
P\left(M=1 \mid |\mathcal{U}|=u \right) = \frac{1}{2^j}
\end{equation}
if $\Psi(\mathbf{c}_i^{\perp}) \subseteq \mathcal{U}$ for $i = 0,1\ldots,2^j-1$. It is assumed that the distribution of each element of $\mathbf{b}^{\mathcal{U}}$ is uniform since $P(S=0 \mid S \ne \lambda) = P(S=1 \mid S \ne \lambda) = \frac{1}{2}$.
If $\alpha = \beta$, the number of erasures $|\mathcal{E}|$ and the number of defects $|\mathcal{U}|$ follow an identical binomial distribution. If $A_w = B_w$, the codeword set $\mathcal{C}$ for the BEC and the dual codeword set $\mathcal{C}_0^{\perp}$ for the BDC are also identical. Thus, we can claim that $P(D = 1 \mid |\mathcal{E}|=e) = P(M = 1 \mid |\mathcal{U}|=u)$ by \eqref{eq:duality_BEC} and \eqref{eq:duality_BDC}.
Since
\begin{align}
P(D=1) &= \sum{P\left( |\mathcal{E}| = e \right)}{P(D = 1 \mid |\mathcal{E}|=e)}, \\
P(M=1) &= \sum{P\left( |\mathcal{U}| = u \right)}{P(U = 1 \mid |\mathcal{U}|=u)},
\end{align}
we can claim that $P(D=1) = P(M=1)$.
\end{IEEEproof}
In Section~\ref{subsection:BEC_BDC}, we show that numerical results confirm the duality between erasures and defects.
From channel properties, capacities, capacity achieving schemes, their upper bounds, and their failure probability, we have demonstrated the duality between erasures and defects.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Duality between BEC and BDC}
\label{tab:duality}
\centering
{\small
\hfill{}
\begin{tabular}{|c|c|c|}
\hline
& BEC & BDC \\ \hline \hline
Channel property & Ternary output $Y \in \{0, 1, \varepsilon \}$ & Ternary state $S \in \{0, 1, \lambda\}$ \\ \hline
Value & Erasure $\varepsilon$ is neither ``0'' nor ``1'' & Defect is either ``0'' or ``1'' \\ \hline
Capacity & $C_{\mathrm{BEC}} = 1 - \alpha$ \: \eqref{eq:BEC_capacity} & $C_{\mathrm{BDC}} = 1 - \beta$ \: \eqref{eq:BDC_capacity} \\ \hline
Channel information & Locations & Locations and stuck-at values \\ \hline
Correcting / masking & Decoder corrects erasures & Encoder masks defects \\ \hline
\multirow{2}{*}{Linear equation} & $G^{\mathcal{V}} \widehat{\mathbf{m}} =\mathbf{y}^{\mathcal{V}}$\: \eqref{eq:BEC_decoder_LE} & $G_0^{\mathcal{U}} \mathbf{d} = \mathbf{b}^{\mathcal{U}}$ \:\eqref{eq:BDC_encoder_LE} \\
& $\left(H^{\mathcal{E}} \right)^T \widehat{\mathbf{c}}^{\mathcal{E}} = \mathbf{q}$ \ \eqref{eq:BEC_decoder_LE_H} & $ \left(H_0^{\mathcal{W}} \right)^T \mathbf{c}^{\mathcal{W}} = \mathbf{m}'$ \ \eqref{eq:BDC_encoder_LE_binning} \\ \hline
\multirow{2}{*}{Solution of linear equation} & $\widehat{\mathbf{m}}$ (estimate of message) or & $\mathbf{d}$ (parity) or \\
& $\widehat{\mathbf{c}}^{\mathcal{E}}$ (estimate of erased bits) & $\mathbf{c}^{\mathcal{W}}$ (codeword elements of normal cells) \\ \hline
Type of linear equation & Overdetermined & Underdetermined \\ \hline
\multirow{2}{*}{Minimum distance} & $d_{\text{min}} = \min\{ \|\mathbf{c}\|: H^T \mathbf{c} = \mathbf{0}, \mathbf{c} \ne \mathbf{0} \}$ & $d_{0} = \min\{ \|\mathbf{c}\|: G_0^T \mathbf{c} = \mathbf{0}, \mathbf{c} \ne \mathbf{0} \}$ \\
& If $e < d_{\text{min}}$, $e$ erasures are corrected. & If $u < d_0$, $u$ defects are masked. \\ \hline
Upper bound on& \multirow{2}{*}{Theorem~\ref{thm:BEC_bound}} & \multirow{2}{*}{Theorem~\ref{thm:BDC_bound}} \\
probability of failure & & \\ \hline
Probability of failure & \multicolumn{2}{c|}{If $A_w = B_w$ and $\alpha = \beta$, then $P(D = 0) = P (M = 0)$ (Theorem~\ref{thm:BEC_BDC_failure}) } \\ \hline
\end{tabular}}
\hfill{}
\end{table}
\section{Binary Defect and Erasure Channel}\label{sec:BDEC}
\subsection{Binary Defect and Erasure Channel}
Considering the duality between erasures and defects, we now introduce the BDEC which has both erasures and defects. As shown in Fig.~\ref{fig:BDEC}, the probability of defects are defined by \eqref{eq:BDC_random_channel_model}, and normal cells behave as the BEC with parameter $\alpha$. The capacity of the BDEC was given by \eqref{eq:BDEC_capacity}.
In order to mask defects and correct erasures, the following two cases will be considered.
\begin{itemize}
\item Case 1: Only the decoder has knowledge of both defects and erasures.
\item Case 2: The encoder has only knowledge of defects and the decoder has only knowledge of erasures.
\end{itemize}
In case 1, the decoder can regard defects as erasures. Then, a fraction $\left(1 - \beta \right)\left(1 - \alpha \right)$ of bits are unerased. The coding scheme for the BEC can achieve the channel capacity of \eqref{eq:BDEC_capacity} since the BDEC is equivalent to the BEC with parameter $1 - \left(1 - \beta \right)\left(1 - \alpha \right)$.
In case 2, the encoder masks the defects by the additive encoding and the decoder corrects the erasures. The proposed coding scheme for the BDEC combines the encoding of the BDC and the decoding of the BEC.
\emph{Encoding:} A message $\mathbf{m} \in \{0, 1\}^k$ is encoded to a codeword $\mathbf{c}= G_1 \mathbf{m} + G_0 \mathbf{d}$. Note that $G_1$ is an $n \times k$ generator matrix and $G_0$ is an $n \times l$ generator matrix such that $n > k + l$. Two generator matrices are used to correct erasures and mask defects. First, $G_1$ encodes a message $\mathbf{m}$ into $G_1 \mathbf{m}$ for correcting erasures. Next, the defects will be masked by $G_0 \mathbf{d}$. The parity for masking defects $\mathbf{d}$ will be chosen by solving \eqref{eq:BDC_encoder_LE}. The encoding can be represented by
\begin{equation}\label{eq:BDEC_encoder}
\mathbf{c} = \widetilde{G} \begin{bmatrix} \mathbf{m} \\ \mathbf{d} \end{bmatrix}
\end{equation}
where $\widetilde{G} = \left[ G_1 \quad G_0 \right]$ is an $n \times (k + l)$ matrix. Note that $r = n - k - l$ is the number of parity bits for correcting erasures and $l$ is the number of parity bits for masking defects.
\emph{Decoding:} The decoding of BDEC can be done by solving the following linear equations.
\begin{equation}\label{eq:BDEC_decoder}
\widetilde{G}^{\mathcal{V}} \begin{bmatrix} \widehat{\mathbf{m}} \\ \widehat{\mathbf{d}} \end{bmatrix} = \mathbf{y}^{\mathcal{V}}
\end{equation}
where $\mathcal{V}=\left\{j_1,\cdots, j_v\right\}$ indicates the locations of $v$ unerased bits. We use the notation of $\mathbf{y}^{\mathcal{V}}=\left(y_{j_1}, \cdots, y_{j_v}\right)^T$ and $\widetilde{G}^{\mathcal{V}}=\left[ \widetilde{\mathbf{g}}_{j_1}^T, \cdots, \widetilde{\mathbf{g}}_{j_v}^T \right]^T$ where $\mathbf{g}_j$ is the $j$-th row of $\widetilde{G}$. By solving \eqref{eq:BDEC_decoder}, we can obtain the estimate of message $\mathbf{m}$ and the estimate of parity $\mathbf{d}$, i.e., $\widehat{\mathbf{m}}$ and $\widehat{\mathbf{d}}$. Note that \eqref{eq:BDEC_decoder} is equivalent to \eqref{eq:BEC_decoder_LE}.
Also, the decoding can be done by solving the following linear equations based on the parity check matrix $\widetilde{H}$ instead of \eqref{eq:BDEC_decoder}.
\begin{equation}
\left(\widetilde{H}^{\mathcal{E}} \right)^T \widehat{\mathbf{c}}^{\mathcal{E}} = \mathbf{q}'
\end{equation}
where $\mathbf{q}' = \left(\widetilde{H}^{\mathcal{V}} \right)^T \mathbf{y}^{\mathcal{V}}$.
The weight distribution of the coding scheme is defined as a pair of sets $\left(A_{1,w}, B_{0,w} \right)$. $A_{1, w}$ is the weight distribution of the $[n, k+l]$ linear block code with the generator matrix $\widetilde{G} = \left[G_1 \quad G_0 \right]$ and the parity check matrix $\widetilde{H}$. Also, $B_{0, w}$ is the weight distribution of the $[n, k+r]$ linear block code with parity check matrix $G_0$ \cite{Heegard1983}. Thus, \eqref{eq:Aw_UB} will be modified into
\begin{align}
A_{1,w} & \le \frac{n}{2^{n - (k+l)}} \binom{n}{w}, \label{eq:BDEC_A1w_UB}\\
B_{0,w} & \le \frac{n}{2^{n - (k+r)}} \binom{n}{w}. \label{eq:BDEC_B0w_UB}
\end{align}
Also, a pair of minimum distances $(d_0 , d_1)$ are defined, where $d_0$ represents the minimum distance for masking defects and $d_1$ is the minimum distance for correcting erasures in the BDEC. $d_0$ is same as \eqref{eq:BDC_dmin} and $d_1$ is given by
\begin{align}
d_1 &= \underset{
\substack{
\mathbf{m} \ne \mathbf{0} \\
\widetilde{H}^T \mathbf{c}= \mathbf{0}
}}
{\text{min }} \|\mathbf{c}\| \label{eq:BDEC_d1}
\end{align}
Note that $d_1$ is greater than or equal to the minimum distance of the $[n,k+l]$ linear block code with parity check matrix $\widetilde{H}$, while $d_0$ is the minimum distance of the $[n,k+r]$ linear block code with the parity check matrix $G_0$ \cite{Heegard1983}.
In case 2, the encoder solves the linear equations of \eqref{eq:BDC_encoder_LE} in order to determine the parity $\mathbf{d}$ for masking defects. Also, the decoder solves the linear equations of \eqref{eq:BDEC_decoder} to estimate $\mathbf{m}$. Thus, it is clear that this coding scheme is a combination of the coding scheme for the BEC and the coding scheme for the BDC. We will now prove that this proposed coding scheme is a capacity achieving scheme.
\begin{theorem} The proposed coding scheme achieves the capacity of the BDEC. The encoding and the decoding are represented by \eqref{eq:BDEC_encoder} and \eqref{eq:BDEC_decoder}, respectively. \label{thm:BDEC_CAS}
\end{theorem}
\begin{IEEEproof}
We can see that
\begin{align}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) & = P\left(M=0, D=1\right) + P\left(M=0, D=0\right) + P\left(M=1, D=0\right) \\
& = P\left(M=0\right) + P\left(M=1, D=0\right). \label{eq:BDEC_P_failure}
\end{align}
First, we will derive the upper bound on $P\left(M=0\right)$, which is similar to Theorem~\ref{thm:BDC}. The only difference is that $B_{0, w}$ of \eqref{eq:BDEC_B0w_UB} should be used instead of $B_w$. Thus, \eqref{eq:BDC_CAS_capacity} will be changed into
\begin{equation}
P(M=0) \le n^2 (\beta + \epsilon) 2^{n \left\{ \frac{k+r}{n} - (1 - \beta) + \epsilon \right\}} + \epsilon'. \label{eq:BDEC_CAS_UB_M}
\end{equation}
Next, the upper bound on $P\left(M = 1, D = 0\right)$ will be derived.
\begin{align}
& P\left(M = 1, D = 0\right) \\
& = P \left(M = 1, D = 0, U \le n(\beta + \epsilon), E \le n\left\{(1 - \beta) \alpha + \epsilon \right\} \right) + \epsilon' \\
& = \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{P \left(M = 1, D = 0, |\mathcal{U}|=u, |\mathcal{E}|=e \right)}} + \epsilon' \\
& = \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{P(|\mathcal{U}|=u) P(M = 1 \mid |\mathcal{U}|=u) P(|\mathcal{E}|=e \mid M = 1, |\mathcal{U}|=u)}} \nonumber \\
& \qquad \qquad \qquad \quad \ \cdot P(D = 0 \mid M = 1, |\mathcal{U}|=u, |\mathcal{E}|=e) + \epsilon' \label{eq:BDEC_CAS_chain_rule} \\
& \le \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{P(D = 0 \mid M = 1, |\mathcal{U}|=u, |\mathcal{E}|=e) }} + \epsilon' \\
& \le \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ \frac{ \sum_{w = d_1}^{e}{A_{1,w} \binom{n-u-w}{e-w}} }{\binom{n-u}{e}} }} + \epsilon' \label{eq:BDEC_CAS_UB}\\
& \le \frac{n}{2^{n - (k+l)}} \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ \frac{ \sum_{w = d_1}^{e}{ \binom{n}{w} \binom{n-u-w}{e-w}} }{\binom{n-u}{e}} }} + \epsilon' \label{eq:BDEC_CAS_UB_given_e} \\
& \le \frac{n}{2^{n - (k+l)}} \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ \sum_{w = d_1}^{e}{\binom{e}{w}} \frac{\binom{n}{w}}{\binom{n-u}{w}} }} + \epsilon' \label{eq:BDEC_CAS_binom} \\
& \le \frac{n}{2^{n - (k+l)}} \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ \sum_{w = d_1}^{e}{\binom{e}{w}}}} + \epsilon' \label{eq:BDEC_CAS_lemma_binom} \\
& \le \frac{n}{2^{n - (k+l)}} \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ 2^e }} + \epsilon' \\
& \le \frac{n}{2^{n - (k+l)}} \sum_{u=0}^{n(\beta + \epsilon)}{\sum_{e=0}^{n((1 - \beta) \alpha + \epsilon)}{ 2^{n((1 - \beta) \alpha + \epsilon)} }} + \epsilon' \\
& \le n^3 \left\{ (\beta + \epsilon) + \frac{1}{n} \right\} \left\{\left( \left( 1 - \beta \right) \alpha + \epsilon \right) + \frac{1}{n}\right\} { 2^{n \left\{\frac{k+l}{n} - 1 + (1 - \beta) \alpha + \epsilon \right\}} } + \epsilon' \label{eq:BDEC_CAS_UB_D}
\end{align}
where we assume that $n(\beta + \epsilon)$ and $n((1 - \beta) \alpha + \epsilon)$ are integers without loss of generality. \eqref{eq:BDEC_CAS_chain_rule} follows from the chain rule. \eqref{eq:BDEC_CAS_UB} follows from the modification of \eqref{eq:BEC_UB} where all the defects are successfully masked and we do not need to consider the defects. Also, $A_{1, w}$ of \eqref{eq:BDEC_A1w_UB} has been used instead of $A_w$. \eqref{eq:BDEC_CAS_binom} follows from ${\binom{n-u-w}{e-w}}/{\binom{n-u}{e}} = {\binom{e}{w}}/{\binom{n-u}{w}}$. In addition, \eqref{eq:BDEC_CAS_lemma_binom} follows from ${\binom{n}{w}}/{\binom{n-u}{w}}\le 1$ for $0 \le u \le n$.
By \eqref{eq:BDEC_CAS_UB_M} and \eqref{eq:BDEC_CAS_UB_D}, we can see that the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ goes to zero if $n$ is sufficiently large and the following two conditions hold.
\begin{align}
\frac{k+r}{n} & < \left(1 - \beta \right) - \epsilon \label{eq:BDEC_CAS_C1} \\
\frac{k+l}{n} & < 1 - \left(1 - \beta \right) \alpha - \epsilon \label{eq:BDEC_CAS_C2}
\end{align}
From the sum of \eqref{eq:BDEC_CAS_C1} and \eqref{eq:BDEC_CAS_C2},
\begin{equation}
R = \frac{k}{n} < (1 - \beta)(1 - \alpha) - 2 \epsilon = C_{\mathrm{BDEC}} - 2\epsilon.
\end{equation}
Thus, the proposed coding scheme achieves the channel capacity of BDEC.
\end{IEEEproof}
\subsection{Redundancy Allocation of BDEC}
The proposed coding scheme for the BDEC requires two generator matrices, namely $G_0$ for masking defects and $G_1$ for correcting erasures, which results in two parts of redundancy. Since the number of parity bits for masking defects and for correcting erasures are $l$ and $r$ respectively, the total redundancy is $l+r = n-k$ and the code rate is $R = k/n$.
The fact that the redundancy can be divided into two parts leads to the problem of redundancy allocation. The objective is to find an optimal redundancy allocation between $l$ and $r$ in order to minimize $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$. The problem of redundancy allocation can be formulated as follows \cite{Kim2013:plbc}.
\begin{equation}\label{eq:BDEC_opt_problem}
\begin{aligned}
( \widehat{l}, \widehat{r} ) = \; & \underset{(l, r)}{\text{argmin}} & & P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) \\
& \text{subject to} & & l+r = n-k \\
& & & 0 \le l \le n-k \\
& & & 0 \le r \le n-k
\end{aligned}
\end{equation}
Not surprisingly, the optimal redundancy allocation depends on the BDEC parameters $\alpha$ and $\beta$. For the BDC (i.e., $\alpha$ = 0), we should allot all redundancy to masking defects and the optimal redundancy allocation will be $(l^{*}, r^{*})=(n-k, 0)$. Meanwhile, the optimal redundancy allocation for the BEC (i.e., $\beta$ = 0) will be $(l^{*}, r^{*})=(0, n-k)$, which is same as the result of \cite{Kim2013:plbc}.
When the BDEC has both defects and erasures (i.e., $\alpha \ne 0$ and $\beta \ne 0$), it is not straightforward to obtain the optimal redundancy allocation $(l^{*}, r^{*})$. Without an expression for $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ as a function of $\left(l, r\right)$, this optimization problem cannot be solved. Unfortunately, it is difficult to obtain the exact mathematical expression for $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$.
Alternatively, we can obtain $(l^*, r^*)$ via Monte-Carlo simulations. However, to find $(l^{*}, r^{*})$ by simulations requires significant computations, especially for a low $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$. Thus, we will consider an estimate $(\widehat{l}, \widehat{r})$ which minimizes the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ instead of $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$.
For sufficiently large $n$, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ was already derived in Theorem~\ref{thm:BDEC_CAS} since the upper bound is the sum of \eqref{eq:BDEC_CAS_UB_M} and \eqref{eq:BDEC_CAS_UB_D}. From \eqref{eq:BDEC_CAS_C1} and \eqref{eq:BDEC_CAS_C2} in Theorem~\ref{thm:BDEC_CAS}, the required redundancy $\left(l, r\right)$ for achieving the capacity can be given by
\begin{align}
l & > n \left( \beta + \epsilon \right) \label{eq:BDEC_CAS_C1_modified}, \\
r & > n \left\{ \left(1 - \beta \right) \alpha + \epsilon \right\}. \label{eq:BDEC_CAS_C2_modified}
\end{align}
However, these asymptotic results are not useful to choose the redundancy allocation of $\left(l, r\right)$ for a finite length code. Thus, we will derive the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ for a finite $n$.
We assume that the weight distributions $A_{1,w}$ and $B_{0,w}$ can be approximated by the binomial distribution as follows.
\begin{align}
A_{1,w} & \cong 2^{-r} \binom{n}{w} \label{eq:BDEC_A1w}\\
B_{0,w} & \cong 2^{-l} \binom{n}{w} \label{eq:BDEC_B0w}
\end{align}
which hold for random codes. In addition, the weight distribution of BCH codes can be approximated by the above binomial distribution \cite{Macwilliams1977theory}. By using \eqref{eq:BDEC_A1w} and \eqref{eq:BDEC_B0w} instead of \eqref{eq:BDEC_A1w_UB} and \eqref{eq:BDEC_B0w_UB}, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ for a finite $n$ will be derived in the following Theorem.
\begin{theorem} \label{thm:RA_UB}For a finite $n$, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BDEC is given by
\begin{equation} \label{eq:RA_UB}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) \le 2^{-l} \left(1 + \beta \right)^n + 2^{-r} \left\{ 1 + \alpha \left(1 - \beta \right) \right\}^n.
\end{equation}
\end{theorem}
\begin{IEEEproof}
The proof for the finite $n$ is similar to the proof of Theorem~\ref{thm:BDEC_CAS}.
First, the upper bound on $P(M=0)$ is given by
\begin{align}
P\left(M=0\right) & = \sum_{u=0}^{n}{P(|\mathcal{U}|=u) P\left(M=0 \mid |\mathcal{U}|=u \right)} \label{eq:RA_chain_rule1} \\
& \le \sum_{u = d_0}^{n}{\binom{n}{u} \beta^u \left(1 - \beta \right)^{n-u} \frac{\sum_{w=d_0}^{u}{B_{0, w} \binom{n-w}{u-w}}}{\binom{n}{u}} } \label{eq:RA_B_1} \\
& = 2^{-l} \sum_{u = d_0}^{n}{\beta^u \left(1 - \beta \right)^{n-u} \sum_{w=d_0}^{u}{\binom{n}{w} \binom{n-w}{u-w}}} \label{eq:RA_B_2} \\
& = 2^{-l} \sum_{u = d_0}^{n}{\beta^u \left(1 - \beta \right)^{n-u} \sum_{w=d_0}^{u}{\binom{u}{w} \binom{n}{u}}} \label{eq:RA_binom_1} \\
& \le 2^{-l} \sum_{u = 0}^{n}{\binom{n}{u} \beta^u \left(1 - \beta \right)^{n-u} \sum_{w = 0}^{u}{\binom{u}{w} }} \\
& = 2^{-l} \sum_{u = 0}^{n}{\binom{n}{u} \left(2\beta\right)^u \left(1 - \beta \right)^{n-u}} \\
& = 2^{-l} \left(1 + \beta \right)^n \label{eq:RA_M}
\end{align}
where \eqref{eq:RA_B_1} follows from \eqref{eq:BDC_UB} in Lemma~\ref{lemma:BDC_UB} and \eqref{eq:RA_B_2} comes from \eqref{eq:BDEC_B0w}. Also, \eqref{eq:RA_binom_1} follows from $\binom{n}{w} \binom{n-w}{u-w} = \binom{u}{w} \binom{n}{u}$.
Next, the upper bound on $P\left(M=1, D=0\right)$ is given by
\begin{align}
& P\left(M=1, D=0\right) \nonumber \\
& = \sum_{u=0}^{n}{ \sum_{e=0}^{n-u}{ P\left( M=1, D=0, |\mathcal{U}|=u, |\mathcal{E}|=e \right)}} \\
& \le \sum_{u=0}^{n}{ \sum_{e=0}^{n-u}{ P\left( |\mathcal{U}|=u \right) P\left( |\mathcal{E}|=e \mid |\mathcal{U}|=u \right) P\left( D=0 \mid M=1, |\mathcal{U}|=u, |\mathcal{E}|=e \right)}} \label{eq:RA_chain_rule2}\\
& \le \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \sum_{e=0}^{n-u}{ \binom{n-u}{e} \alpha^e \left( 1-\alpha \right)^{n-u-e} \frac{ \sum_{w=d_1}^{e}{A_{1, w} \binom{n-u-w}{e-w}} }{\binom{n-u}{e}}}} \label{eq:RA_A_1} \\
& = 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \sum_{e=0}^{n-u}{ \binom{n-u}{e} \alpha^e \left( 1-\alpha \right)^{n-u-e} \frac{ \sum_{w=d_1}^{e}{\binom{n}{w} \binom{n-u-w}{e-w}} }{\binom{n-u}{e}}}} \label{eq:RA_A_2} \\
& = 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \sum_{e=0}^{n-u}{ \binom{n-u}{e} \alpha^e \left( 1-\alpha \right)^{n-u-e} \sum_{w=d_1}^{e}{\binom{e}{w}\frac{\binom{n}{w}} {\binom{n-u}{w}}}}} \label{eq:RA_binom} \\
& \le 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \sum_{e=0}^{n-u}{ \binom{n-u}{e} \alpha^e \left( 1-\alpha \right)^{n-u-e} \sum_{w=d_1}^{e}{\binom{e}{w}} }}\\
& \le 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \sum_{e=0}^{n-u}{ \binom{n-u}{e} (2\alpha)^e \left( 1-\alpha \right)^{n-u-e} }}\\
& \le 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left( 1-\beta \right)^{n-u} \left( 1+ \alpha\right)^{n-u}}\\
& = 2^{-r} \sum_{u=0}^{n}{ \binom{n}{u} \beta^u \left\{ \left( 1-\beta \right)\left( 1+ \alpha\right) \right\}^{n-u} }\\
& = 2^{-r} \left\{1 + \alpha \left( 1- \beta\right) \right\}^{n} \label{eq:RA_D}
\end{align}
where \eqref{eq:RA_chain_rule2} follows from the chain rule and $P\left( M=1 \mid |\mathcal{E}|=e, |\mathcal{U}|=u \right) \le 1$. \eqref{eq:RA_A_1} follows from \eqref{eq:BEC_UB} in Lemma~\ref{lemma:BEC_UB} and \eqref{eq:RA_A_2} comes from \eqref{eq:BDEC_A1w}. In addition, \eqref{eq:RA_binom} is similar to \eqref{eq:BDEC_CAS_binom}.
Finally, \eqref{eq:RA_UB} is obtained from \eqref{eq:BDEC_P_failure}, \eqref{eq:RA_M}, and \eqref{eq:RA_D}.
\end{IEEEproof}
From Theorem~\ref{thm:RA_UB}, the upper bounds on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BEC and the BDC for a finite $n$ can be derived as follows.
\begin{corollary} \label{cor:BEC}For a finite $n$, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BEC is given by
\begin{align}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) & \le 2^{-r} \left( 1 + \alpha \right)^n \label{eq:BEC_UB_finite}, \\
\log_2{P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)} & \le n \left\{R - 1 + \log_2{(1+\alpha)} \right\}. \label{eq:BEC_UB_finite_log}
\end{align}
\end{corollary}
\begin{IEEEproof}
It is clear that $P(M=0)=0$ and $\beta = 0$ for the BEC. By \eqref{eq:BDEC_P_failure} and \eqref{eq:RA_D}, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BEC is given by
\begin{equation*}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) = P\left(M=1, D=0\right) \le 2^{-r} \left( 1 + \alpha \right)^n.
\end{equation*}
Also, \eqref{eq:BEC_UB_finite_log} can be obtained by taking the logarithm.
\end{IEEEproof}
\begin{corollary} For a finite $n$, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BDC is given by
\begin{align}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) & \le 2^{-l} \left( 1 + \beta \right)^n \label{eq:BDC_UB_finite} \\
\log_2{P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)} & \le n \left\{R - 1 + \log_2{(1+\beta)} \right\}. \label{eq:BDC_UB_finite_log}.
\end{align}
\end{corollary}
\begin{IEEEproof}
It is clear that $P(M=1, D=0)=0$ and $\alpha = 0$ for the BDC. By \eqref{eq:BDEC_P_failure} and \eqref{eq:RA_M}, the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ of the BDC is given by
\begin{equation*}
P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right) = P\left(M=0\right) \le 2^{-l} \left( 1 + \beta \right)^n.
\end{equation*}
Also, \eqref{eq:BDC_UB_finite_log} can be obtained by taking the logarithm.
\end{IEEEproof}
Since $( \widehat{l}, \widehat{r} )$ minimizes the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$, the optimization problem in \eqref{eq:BDEC_opt_problem} is given by
\begin{equation}\label{eq:BDEC_opt_problem_estimate}
\begin{aligned}
( \widehat{l}, \widehat{r} ) = \; & \underset{(l, r)}{\text{argmin}} & & 2^{-l} \left(1 + \beta \right)^n + 2^{-r} \left\{ 1 + \alpha \left(1 - \beta \right) \right\}^n \\
& \text{subject to} & & l+r = n-k \\
& & & 0 \le l \le n-k \\
& & & 0 \le r \le n-k
\end{aligned}
\end{equation}
where the objective function is the upper bound on $P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)$ for a finite $n$. This objective function is intuitively reasonable since $\beta$ is the probability of defects and $\alpha(1-\beta)$ is the probability of erasures. If $\beta \ge \alpha(1-\beta)$, we have to allot more redundancy for masking defects, i.e., $l \ge r$. Otherwise, we should allot more redundancy for correcting erasures. For $\alpha = 0$ or $\beta = 0$, we do not need to consider the above optimization problem since the solution of the BEC or the BDC is straightforward.
If the codeword length $n$, the information length $k$ and the channel parameters such as $\alpha$ and $\beta$ are given, the solution $( \widehat{l}, \widehat{r} )$ of the above optimization problem can be readily obtained. For example, we will consider $\left[ n = 1023, k=923, l \right]$ PBCH codes. All possible redundancy allocation candidates of PBCH codes are presented in Table~\ref{tab:plbc}. Since there are only 11 redundancy allocation candidates in Table~\ref{tab:plbc}, we can readily obtain the $( \widehat{l}, \widehat{r} )$ that minimizes the objective function of \eqref{eq:BDEC_opt_problem_estimate}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{All Possible Redundancy Allocation Candidates of $\left[ n = 1023, k=923, l \right]$ PBCH Codes}
\label{tab:plbc}
\centering
{\small
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Code & {$l$} & {$r$} & {$d_0$} & {$d_1$} & Notes \\ \hline \hline
0 & 0 & 100 & 0 & 21 & Only correcting erasures \\ \hline
1 & 10 & 90 & 3 & 19 &\\ \hline
2 & 20 & 80 & 5 & 17 &\\ \hline
3 & 30 & 70 & 7 & 15 &\\ \hline
4 & 40 & 60 & 9 & 13 &\\ \hline
5 & 50 & 50 & 11 & 11 & \\ \hline
6 & 60 & 40 & 13 & 9 &\\ \hline
7 & 70 & 30 & 15 & 7 &\\ \hline
8 & 80 & 20 & 17 & 5 &\\ \hline
9 & 90 & 10 & 19 & 3 &\\ \hline
10& 100 & 0 & 21 & 0 & Only masking defects\\ \hline
\end{tabular}}
\end{table}
In addition, the objective function is \emph{convex} if we assume that $l$ and $r$ are real values. Since the optimization problem is convex, we can derive the solution $(\widetilde{l}, \widetilde{r})$ of \eqref{eq:BDEC_opt_problem_estimate} by \emph{Karush-Kuhn-Tucker} (KKT) conditions.
\begin{numcases}{(\widetilde{l}, \widetilde{r})=}
(0, n-k), & if $2^{-l} \left(1 + \beta \right)^n \le 2^{-r} \left\{ 1 + \alpha \left(1 - \beta \right) \right\}^n$; \label{eq:BDEC_opt_sol_con1}
\\
(n-k, 0), & if $2^{-l} \left(1 + \beta \right)^n \ge 2^{-r} \left\{ 1 + \alpha \left(1 - \beta \right) \right\}^n$; \label{eq:BDEC_opt_sol_con2}
\\
\left( \check{l}, \check{r} \right), & \text{otherwise} \label{eq:BDEC_opt_sol_con3}
\end{numcases}
where $\left( \check{l}, \check{r} \right)$ is given by
\begin{align}
\check{l} &= \frac{1}{2} \left\{ n \left( 1 + \log_2{\left(\frac{1+\beta}{1 + \alpha (1 - \beta)}\right)} \right) - k \right\}, \label{eq:BDEC_opt_sol_1} \\
\check{r} &= \frac{1}{2} \left\{ n \left( 1 - \log_2{\left(\frac{1+\beta}{1 + \alpha (1 - \beta)}\right)} \right) - k \right\}. \label{eq:BDEC_opt_sol_2}
\end{align}
The details of derivation are given in Appendix. \eqref{eq:BDEC_opt_sol_con1} and \eqref{eq:BDEC_opt_sol_con2} are easy to see. Also, \eqref{eq:BDEC_opt_sol_1} and \eqref{eq:BDEC_opt_sol_2} are intuitively reasonable since $\check{l} \ge \check{r}$ for $\beta \ge \alpha(1 - \beta)$. If $\beta < \alpha(1 - \beta)$, $\check{l} < \check{r}$.
In Section~\ref{subsection:redundancy_allocation}, the numerical results show that $(\widehat{l}, \widehat{r})$ and $(\widetilde{l}, \widetilde{r})$ match $(l^*, r^*)$ very well.
\section{Numerical Results}\label{sec:numerical_results}
\subsection{BEC and BDC} \label{subsection:BEC_BDC}
The numerical results for the BEC and the BDC will be presented. For the BEC, the \emph{generator matrices} of BCH codes are used for $G$ of \eqref{eq:BEC_decoder_LE}. For the BDC, the PBCH codes are used, so the \emph{parity check matrices} of BCH codes are used for $G_0$ of \eqref{eq:BDC_encoder_LE} \cite{Heegard1983}. Thus, $A_w$ for the BEC and $B_w$ for the BDC are same.
Fig.~\ref{fig:BEC_numerical} shows the probability of decoding failure (i.e., $P(D=0)$) and its upper bound. Also, Fig.~\ref{fig:BDC_numerical} shows the probability of masking failure (i.e., $P(M=0)$) and its upper bound. The upper bounds are given by \eqref{eq:BEC_UB_finite} and \eqref{eq:BDC_UB_finite}. Since $\alpha = \beta = 0.1$, the upper bound for the BEC is same as the upper bound for the BDC.
Note that the slope of the upper bound on $ \log_2{P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)}$ is $n$ as shown in Fig.~\ref{fig:BEC_numerical} and Fig.~\ref{fig:BDC_numerical}, which can be explained by \eqref{eq:BEC_UB_finite_log} and \eqref{eq:BDC_UB_finite_log}. Also, the x-axis intercepts (i.e., $R$ for $ \log_2{P\left( \widehat{\mathbf{m}} \ne \mathbf{m} \right)} = 0$) are $1 - \log_2{\left(1 + \alpha\right)}$ and $1 - \log_2{\left(1 + \beta\right)}$ for each channel.
Fig.~\ref{fig:BEC_BDC_numerical} compares the probability of decoding failure of the BEC and the probability of masking failure of the BDC. By Fig.~\ref{fig:BEC_BDC_numerical}, we can see that $P(D=0) = P(M=0)$ if $A_w = B_w$ and $\alpha = \beta$, which confirms the duality in Theorem~\ref{thm:BEC_BDC_failure}.
\subsection{Redundancy Allocation for BDEC} \label{subsection:redundancy_allocation}
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{BDEC with the Same $C_{\textrm{BDEC}}=0.95$}
\label{tab:channel}
\centering
{\small
\hfill{}
\begin{tabular}{|c|c|c|c|}
\hline
Channel & {$\alpha$} & {$\beta$} & {Notes} \\ \hline \hline
1 & 0.0500 & 0 & BEC \\ \hline
2 & 0.0404 & 0.0100 & \\ \hline
3 & 0.0306 & 0.0200 & \\ \hline
4 & 0.0253 & 0.0253 & \\ \hline
5 & 0.0200 & 0.0306 & \\ \hline
6 & 0.0100 & 0.0404 & \\ \hline
7 & 0 & 0.0500 & BDC \\ \hline
\end{tabular}}
\hfill{}
\end{table}
In order to discuss the redundancy allocation for BDEC, we will consider multiple BDECs in Table~\ref{tab:channel} whose capacities are $C_{\textrm{BDEC}}=0.95$. For these channels, we apply $\left[ n = 1023, k=923, l \right]$ PBCH codes whose all possible redundancy allocation candidates are presented in Table~\ref{tab:plbc}.
Fig.~\ref{fig:redundancy_allocation_simulation} shows the simulation results for the channels of Table~\ref{tab:plbc}. The simulation results of channel 1 (BEC) and channel 7 (BDC) are incomplete due to their impractical computational complexities. However, it should be obvious that the optimal redundancy allocation for channel 1 (BEC) will be $\left(l^*, r^* \right) = (0, 100)$. The more defects a channel has, the larger $l$ is expected to be for the optimal redundancy allocation. Eventually, the optimal redundancy allocation for channel 7 (BDC) will be $\left(l^*, r^* \right) = (100, 0)$. The optimal $l^*$ for all channels of Table~\ref{tab:channel} can be obtained from Fig.~\ref{fig:redundancy_allocation_simulation}, which are presented in the second column of Table~\ref{tab:redundancy}. The optimal $r^*$ can be obtained by $r^* = n - k - l^*$ \cite{Kim2013:plbc}.
To find the optimal redundancy allocation $(l^*, r^*)$ by simulation requires significant computations. Therefore, we will try to estimate the redundancy allocation from \eqref{eq:BDEC_opt_problem_estimate} instead of the simulation for estimating the optimal redundancy allocation.
First, we can readily obtain the $( \widehat{l}, \widehat{r} )$ that minimizes the objective function of \eqref{eq:BDEC_opt_problem_estimate} for each channel since there only 11 redundancy allocation candidates in Table~\ref{tab:plbc}. The estimate $\widehat{l}$ for all channels can be obtained from Fig.~\ref{fig:redundancy_allocation_UB}. The estimate $\widehat{l}$ for all channels are presented in the third column of Table~\ref{tab:redundancy}. Note that $\widehat{r} = n - k - \widehat{l}$. Table~\ref{tab:redundancy} shows that the estimate $(\widehat{l}, \widehat{r})$ matches the optimal redundancy allocation $(l^*, r^*)$ very well.
Next, $(\widetilde{l}, \widetilde{r})$ can be calculated by \eqref{eq:BDEC_opt_sol_con1}$\sim$\eqref{eq:BDEC_opt_sol_2} assuming that $l$ and $r$ are real values. The solution $\widetilde{l}$ are presented in the last column of Table~\ref{tab:redundancy}. Table~\ref{tab:redundancy} shows that the optimal $l^*$ is the nearest one from $\widetilde{l}$ considering the possible redundancy allocation candidates in Table~\ref{tab:plbc}
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Optimal Redundancy Allocation $l^*$ and its Estimate $\widehat{l}$ and $\widetilde{l}$}
\label{tab:redundancy}
\centering
{\small
\begin{tabular}{|c|c|c|c|}
\hline
Channel & {$l^*$} & {$\widehat{l}$} & {$\widetilde{l}$} \\ \hline \hline
1 & 0 & 0 & 0 \\ \hline
2 & 30 & 30 & 28.4 \\ \hline
3 & 40 & 40 & 42.8 \\ \hline
4 & 50 & 50 & 50.5 \\ \hline
5 & 60 & 60 & 58.1 \\ \hline
6 & 70 & 70 & 72.2 \\ \hline
7 & 100& 100& 100 \\ \hline
\end{tabular}}
\end{table}
\section{Conclusions}\label{sec:conclusion}
The duality of erasures and defects was revealed. The erasures are corrected by the decoder and the defects are masked by the encoder. The duality holds for channel capacities, capacity achieving schemes, minimum distances, upper bounds on probabilities of failure, and probabilities of failure. By using the upper bounds on the probability of failures, it was proved that the capacities of the BEC and the BDC can be achieved by solving overdetermined linear equations and underdetermined linear equations, respectively.
Also, the BDEC was introduced, which has both erasures and defects. The capacity of the BDEC can be achieved by the coding scheme that combines the coding schemes of the BEC and the BDC.
In addition, we investigated the redundancy allocation for the BDEC. The optimal redundancy allocation was obtained by simulations. In order to reduce the computation complexity, we proposed two methods to estimate the optimal redundancy allocation based on the upper bound on failure probability. The numerical results showed that the estimates of redundancy allocation match the optimal redundancy allocation well.
|
2,877,628,090,808 | arxiv | \section{Introduction}
The role of the gas in the formation and evolution process of early-type galaxies is still not fully understood. For example, recent studies have shown a large complexity in the gas structures in these systems (e.g. Morganti et al. 2006, Sarzi et al. 2006, Combes et al 2007) despite their often unspectacular optical appearance. In addition, some sources show nuclear activity while others do not. Cold-gas structures represent a fossil record of the formation and evolution of early-type galaxies. In particular, gas found on kiloparsec scales can be used to trace the evolution of the host galaxy (e.g. major merger vs. small accretions). In that respect, neutral hydrogen is an important tracer of these events as it often extends the dust and optical (disk) structure by a factor of two or more.
Close to the centre of galaxies ($<100$~pc), (cold) gas also plays a crucial role as it can provide the fuel that is needed to make the central black hole active. It is often believed that mergers are important in driving gas to the centre (see e.g. Hibbard \& van Gorkom 1996, Barnes 2002), but recent studies of radio galaxies have shown that the activity in some galaxies may be associated with the (slow) accretion of gas from the ISM/IGM (e.g. Best et al. 2005).
Despite the need for large statistical investigations to understand (and classify) the different mechanisms at work (e.g. mergers, interactions, accretion, AGN activity etc.), it is indispensable to study close-by objects in great detail with very high linear resolution. For example, the circumnuclear region around the black hole ($< 100$~pc) can only be resolved in the most nearby galaxies to a degree that is needed to understand the accretion/fueling process.
By far the closest radio-loud early-type galaxy is Centaurus~A (NGC~5128) at a distance of only 3.8~Mpc\footnote{At this distance 1\hbox{$^\prime$} ~corresponds to $\sim 1.1$~kpc.} (Harris 2009). Cen~A has been studied in all possible wavelength regimes with very high linear resolution (for an overview see the review by Israel (1998) and the other contributions to this volume). However, it is crucial to compare Cen~A with other sources to check whether it is a typical example of its class, or whether it is unusual w.r.t. other radio-loud, low-luminosity sources.
In this paper we discuss whether Cen~A is special as seen from the neutral hydrogen perspective. That is, do the Cen~A properties (such as e.g. \textup{H\,{\mdseries\textsc{i}}}~ mass, morphology, kinematics etc.) differ from other early-type galaxies and does Cen~A share properties with radio galaxies that have comparable luminosity? In Sect.~2 we give a brief description of the \textup{H\,{\mdseries\textsc{i}}}~ morphology and kinematics on kpc and sub-kpc scales. Section~3 compares Cen~A with other nearby early-type galaxies and in Sect.~4 Cen~A is compared to a complete sample of nearby radio galaxies. We summarize our comparison in Sect.~5.
\section{Cen~A as seen in neutral hydrogen}
\subsection{The \textup{H\,{\mdseries\textsc{i}}}~ large-scale morphology and kinematics}
Van Gorkom et al. (1990) performed the first interferometric observations to map the \textup{H\,{\mdseries\textsc{i}}}~ in emission and absorption, they showed that the \textup{H\,{\mdseries\textsc{i}}}~ follows the prominent, warped dust lane (e.g. Nicholson et al. 1992, Quillen et al. 2006) and that most of the \textup{H\,{\mdseries\textsc{i}}}~ rotates around the centre. Some hints of unsettled gas were found in the outer SE parts of the disk suggesting that some of the gas has not yet settled into regular rotating orbits.
\textup{H\,{\mdseries\textsc{i}}}~ well outside the disk was discovered by Schiminovich et al. (1994). A number of \textup{H\,{\mdseries\textsc{i}}}~ clouds located 10 to 15~kpc from the nucleus form a partial ring structure with a smooth N-S velocity gradient, rotating perpendicular to the gas in the dust lane, but in the same sense as the main stellar body of the galaxy. The prominent (warped) dust and gas disk in the central region, the outer optical shell structure, together with the partial \textup{H\,{\mdseries\textsc{i}}}~ ring structure indicate that Cen~A is likely the product of a recent accretion of a small gas-rich spiral galaxy with a larger elliptical galaxy as already suggested by Baade \& Minkowski 1954) and Tubbs (1980).
We have recently performed (with the ATCA) new, higher resolution observations with better sensitivity to study the circumnuclear region and disk kinematics in more detail (C. Struve et al. in prep.). The new observations have allowed to better separate emission and absorption and to detect new faint features. Figure~\ref{dss.m0} shows the large-scale \textup{H\,{\mdseries\textsc{i}}}~ emission of Centaurus~A, Fig.~\ref{fig2} is a zoom-in and shows the emission and absorption of the disk. Our main results are:
\begin{itemize}
\item \textup{H\,{\mdseries\textsc{i}}}~ is detected in emission along the dust lane with a diameter of 13~kpc. The total \textup{H\,{\mdseries\textsc{i}}}~ mass in emission is M$_{\textup{H\,{\mdseries\textsc{i}}}~}=4.6\cdot 10^8$~{M$_{\odot}$} . Absorption is detected against the nucleus, the northern jet and against the southern lobe.
\item Tilted-ring modeling shows that the inner, modestly warped ($<30$\hbox{$^\circ$} ) 5~kpc disk in radius can be well described by a set of tilted circular rings that explain the morphology and kinematics. The rotation curve quickly rises in the central part and remains essentially flat thereafter with only a very modest decline with radius. At larger radii asymmetries in the morphology and kinematics are present and the gas has not yet settled into regular rotating orbits.
\item Our tilted-ring model also describes the dust disk detected by Spitzer Quillen et al (2006) and the stellar ring recently discovered by Kainulainen et al. (2009).
\item We detect no emission/absorption down to about 3.6~mJy~beam$^{-1}$ in the inner 1~kpc (except for the central beam) in agreement with observations at other frequencies (Nicholson et al. 1992, Quillen et al. 2006).
\item There is no need for non-circular motions for $1<r<5$~kpc.
\item Based on the regular rotation for $r<5$~kpc, we estimate the age of the disk to be a few times $10^8$~yr and hence the event that produced the disk is too old to explain the current phase of activity ($<10^7$~yr). However, the disk could have approximately the same age as estimated for the northern middle lobe (Saxton et al. 2001).
\item A significant fraction of the absorption detected against the core (see below) cannot be explained by the rotating large-scale disk and hence this absorbing gas must be located close to the nucleus.
\item Outside the disk we discovered two additional clouds that could be part of the partial ring structure discovered by (Schiminovich et al. 1994).
\end{itemize}
\subsection{The circumnuclear region}
Dedicated \textup{H\,{\mdseries\textsc{i}}}~ absorption studies of the nucleus previously have shown that the absorption against the nucleus is solely redshifted (e.g. van der Hulst et al. 1983, Sarma et al. 2002). This was taken as evidence that the absorbing \textup{H\,{\mdseries\textsc{i}}}~ might fall towards the black hole, potentially providing the fuel that is needed for the nuclear activity. However, our new observations (C. Struve et al. in prep.; Morganti et al. 2008) show the existence of blueshifted absorption and that the redshifted absorption is significantly broader ($\Delta v_{\rm{absorp}}^{\rm{total}} = 400$~{km~s$^{-1}$} ) than previously measured. The broad absorption component was previously missed due to insufficient bandwidths. Morganti et al. (2008) suggest that the nuclear \textup{H\,{\mdseries\textsc{i}}}~ absorption could be interpreted as a circumnuclear disk which would be the atomic counterpart of the observed CO disk (Liszt 2001, Espada et al. 2009).
Jones et al. (1996) have shown that the core at VLBI scales is visible at 8.4~GHz, but at lower frequencies the core becomes self-absorbed. Hence, the central \textup{H\,{\mdseries\textsc{i}}}~ absorption detected with the ATCA occurs against an extended structure. New Long Baseline Array observations show a similar velocity width (compared to the ATCA observations) of the deepest part of the absorption ($\Delta v_{\rm{absorp}}^{\rm{deep}} \approx 70$~{km~s$^{-1}$} ) against the bright part of the beginning VLBI jet (projected distance $\sim 1$~pc from the core). However, this absorbing gas could be part of the kpc-scale disk as it is close in velocity to the systemic velocity of Cen~A. A further analysis is needed to clarify the physical origin of this absorption.
\section{Comparison with other early-type galaxies}
Although early-type galaxies used to be perceived to be gas poor, different gas phases are in fact detected in many objects, provided deep observations are available. Ionised gas has recently been found in $75$\% in early-type galaxies (Sarzi et al. 2006) and molecular gas was detected in up to 54\% of the observed sources (e.g. Combes et al. 2007, Flaquer et al. 2008). The presence of ionised, molecular and atomic gas in Cen~A is therfore not surprising (for a summary of the gas properties in Cen~A see Morganti 2009).
Also in \textup{H\,{\mdseries\textsc{i}}}~ more than 50\% of non-cluster galaxies are detected with \textup{H\,{\mdseries\textsc{i}}}~ masses between $10^6$ and $10^{10}$~{M$_{\odot}$} ~(Morganti et al. 2006, Oosterloo et al. 2007). A variety of gas morphologies is present, ranging from very extended (up to 200~kpc) regularly rotating disk/ring structures of low column density (Oosterloo et al. 2007) to long tidal tails and barely resolved blobs ($<4$~kpc diameter) (Morganti et al. 2006). Disks in early-type galaxies can form simultaneously with their host, from major mergers of gas-rich galaxies (see e.g. Hibbard \& van Gorkom 1996, Barnes 2002). In those mergers, about half the gas is quickly funneled to the centre, partly consumed in a burst of star formation and partly settling in a nuclear central disk (Bournaud et al. 2005). The other half is ejected to large distances but might remain bound to the merger remnant and eventually will fall back settling in a large-scale disk. Compared to other galaxies, the presence of a large-scale rotating \textup{H\,{\mdseries\textsc{i}}}~ disk with unsettled gas in the outer parts of the disk (at $r>5$~kpc), as well as the partial ring structure at larger distances from the nucleus is not unusual.
The observed, essentially flat and only mildly decreasing \textup{H\,{\mdseries\textsc{i}}}~ rotation curve is a commonly observed phenomenon in other early-type (disk) galaxies (see e.g. Noordermeer et al. 2007). In addition, some early-type galaxies (e.g. NGC~3108, ESO~381-47, IC~2006) have ring structures, i.e. they have a depression of \textup{H\,{\mdseries\textsc{i}}}~ towards the nucleus (Oosterloo et al. 2002, Donovan et al. 2009, Franx et al. 1994). Therefore, neither the rotation curve nor the radial \textup{H\,{\mdseries\textsc{i}}}~ surface density distribution of Cen~A is unusual when compared with other early-type galaxies.
One of the well known characteristic of the gas/dust disk in Cen~A is the warped structure. While the origin of warps (e.g. mergers, interactions, accretion of gas from the intergalactic medium) is still under debate and may differ from object to object (for a discussion see e.g. Briggs 1990, Garc{\'{\i}}a-Ruiz et al. 2002), it is an observational fact that most disk galaxies are at least mildly warped in \textup{H\,{\mdseries\textsc{i}}}~ (Garc{\'{\i}}a-Ruiz et al. 2002). Most extended \textup{H\,{\mdseries\textsc{i}}}~ disks/rings in early-type galaxies are warped, e.g. in IC~4200 or ESO~381-47 (Serra et al. 2006, Donovan et al. 2009). In some cases, the warping amplitude can be large (i.e. approaching 90\hbox{$^\circ$} ) and also start well within the optical disk as is observed in NGC~2685 (J{\'o}zsa et al. 2009). Considering the merger history of Cen~A, it is therefore not unusual that the gas disk in Cen~A is warped (Sect.~2), nor the warping amplitude is spectacularly high. It is rather its proximity and the orientation on the sky that make Cen~A appear as a peculiar warped object.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45, angle=270]{struve3.ps}
\caption{Distances of the radio sources of the complete sample of nearby radio galaxies (Emonts 2006, B.H.C. Emonts et al. in prep.). Cen~A ($D=3.8$~Mpc) is located a factor $>4$ closer than the closest sample source and a factor $\sim 26$ than the average sample source ($\langle D_{\rm{sample}} \rangle =101$~Mpc).}
\label{distances}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1.2]{struve4.ps}
\caption{Total \textup{H\,{\mdseries\textsc{i}}}~ mass detected in emission plotted against the linear size of the radio sources of the complete sample of radio galaxies (B.H.C. Emonts et al. in prep.). The crosses denote the \textup{H\,{\mdseries\textsc{i}}}~ detections, the filled triangles the upper limits of the non-detections. The open square represents Cen~A.}
\label{segregation}
\end{center}
\end{figure}
\section{Comparison with other radio galaxies}
In the previous section we have shown that Cen~A has an \textup{H\,{\mdseries\textsc{i}}}~ morphology and kinematics that is also found in a number of other early-type galaxies and that in this respect Cen~A is not special. However, it is still important to compare Cen~A with other radio sources as it could be an exceptional radio galaxy.
Radio activity is a short lived (sometimes recurrent) phenomenon (typically believed to be $<10^8$~yr) in the evolution of an early-type galaxy. It is therefore natural that the number density of radio galaxies in the (local) universe is significantly lower than that of early-type galaxies in general. This makes Cen~A a unique object as it can be observed in great detail because it is much closer to us than any other radio source with comparable radio luminosity. Figure~\ref{distances} shows the distances of a complete sample of nearby low-luminosity radio sources (Emonts 2006, B.H.C. Emonts et al. in prep.). The distance advantage of Cen~A (and hence the linear resolution) is evident from the figure as even the closest sample source is located more than 4 times as far away as Cen~A. The sample covers all nearby ($z<0.41$) sources of a large fraction of the northern sky with $F_{\rm{408MHz}}>0.2$~Jy with the restriction that cluster members are excluded as the probability of an \textup{H\,{\mdseries\textsc{i}}}~ detection in those environments is low. For details of the sample and the analysis we refer the reader to (Emonts 2006, Emonts et al. 2007).
Six out of the 21 sample sources ($=29$\%) are detected in emission. The reason for the slightly lower detection rate (compared to radio-quiet early-type galaxies, $>50$\%) is that the upper \textup{H\,{\mdseries\textsc{i}}}~ mass limits of the non-detections are higher. Therefore, the difference is believed to be the result of observational limitations and not to be an intrinsic feature of radio galaxies (Emonts 2006, B.H.C. Emonts et al. in prep.). Similar to the sample of early-type galaxies, a variety of \textup{H\,{\mdseries\textsc{i}}}~ structures is found for the radio galaxies, including (settling) disk structures. However, the striking result from this analysis is that large amounts of \textup{H\,{\mdseries\textsc{i}}}, rotating in regular disks, are only detected around {\sl compact} sources (typically $<10$~kpc, Emonts et al. 2007). Extended radio sources --- comparable to Cen~A --- do only show modest amounts of \textup{H\,{\mdseries\textsc{i}}}~ (few times $10^8$~{M$_{\odot}$} ), Fig.~\ref{segregation} (see also Emonts et al. 2007). If Cen~A would be located at a distance of $70$~Mpc the peak of the emission would correspond to about $4\sigma_{\rm{rms}}$ resulting in a blob like structure, extended over 1-2 beams. In addition, the deep part of the absorption would be detected. However, at larger distances ($D > 70$~Mpc), the emission of Cen~A would not have been detected. Therefore, the results from the statistical sample are not in conflict with Cen~A's \textup{H\,{\mdseries\textsc{i}}}~ mass.
Cen~A, with its 650-kpc continuum structure (see e.g. Feain 2009), is believed to have gone through multiple phases of AGN activity. Recurrent radio activity is also found in a number of other sources (see e.g. Schoenmakers et al. 2000). We note that at least one \textup{H\,{\mdseries\textsc{i}}}~ detected compact radio sample source shows a 250~kpc relic structure at the sub-mJy level (B2~0258+35; C. Struve et al. in prep.), showing that also some of these compact sources had previous phases of AGN activity. Therefore, also the radio continuum structure of Cen~A does not appear unusual compared to other radio galaxies.
\textup{H\,{\mdseries\textsc{i}}}~ in absorption has been detected in a large number of radio galaxies (e.g. Vermeulen et al. 2003, Morganti et al. 2005). Initially, only absorption profiles redshifted (relative to the systemic velocity) were found and the absorbing gas clouds were seen as evidence for infall towards the nuclear region, potentially providing the fuel for the AGN (van Gorkom et al. 1989). This picture has changed in recent years with the availability of more sensitive and broader-band observations revealing also blueshifted absorption (Vermeulen et al. 2003, Morganti et al. 2005). In some cases, the \textup{H\,{\mdseries\textsc{i}}}~ absorption is centred on the systemic velocity of the galaxy and is often interpreted as a circumnuclear disk/torus, (see e.g. Conway \& Blanco 1995, van Langevelde et al. 2000, Peck \& Taylor 2001) which is actually in agreement with theoretical predictions (e.g. Maloney et al. 1996). The \textup{H\,{\mdseries\textsc{i}}}~ ATCA observations of Cen~A are in agreement with a circumnuclear structure (see Morganti et al. 2008) which is further supported by the existence of a molecular circumnuclear disk (Liszt 2001, Neumayer et al. 2007).
\section{Summary}
In order to understand Centaurus~A in the context of galaxy formation and evolution, we have compared the \textup{H\,{\mdseries\textsc{i}}}~ properties of Cen~A with early-type and radio galaxies. The \textup{H\,{\mdseries\textsc{i}}}~ mass, its distribution and the mainly settled kinematics is commonly found in other early-type/radio galaxies. The current phase of AGN activity is not connected to the recent merger which is in line with recent results for a sample of radio galaxies. The absorption against the nucleus is red- and blueshifted with respect to the systemic velocity and is in agreement --- as is also the case in other sources --- with a circumnuclear \textup{H\,{\mdseries\textsc{i}}}~ disk/torus structure. Hence, Centaurus~A seems to be --- from an \textup{H\,{\mdseries\textsc{i}}}~ perspective --- a typical galaxy of its class.
\section*{Acknowledgments}
This research was supported by the EU Framework 6 Marie Curie Early Stage Training programme under contract number MEST-CT-2005-19669 ``ESTRELA''.
|
2,877,628,090,809 | arxiv |
\section{Conclusion}
We showed a design for a quantization guided JPEG artifact correction network. Our single network is able to
acheive state-of-the-art results, beating methods which train a different network for
each quality level. Our network relies only on information that is available at inference time, and solves
a major practical problem for the deployment of such methods in real-world scenarios.
\section{Introduction}
The JPEG image compression algorithm~\cite{wallace1992jpeg} is ubiquitous in modern computing. Thanks to
its high compression ratios, it is extremely popular in bandwidth constrained applications. The JPEG algorithm is a lossy compression algorithm, so by
using it, some information is lost for a corresponding gain in saved space. This is most noticable for low quality settings
For highly space-constrained scenarios, it may be desirable to use aggressive compression. Therefore,
algorithmic restoration of the lost information, referred to as artifact correction, has been well studied both in classical literature
\cite{foi2006pointwise, jancsary2012loss}, and in the context of deep neural networks
\cite{zhang2017beyond, liu2018multi, dong2015compression, cavigelli2017cas, wang2016d3}.
While these methods have enjoyed academic success, their practical application is limited by a single
architectural defect:
they train a single model per JPEG quality level. The JPEG quality level is an
integer between 0 and 100, where 100 indicates very little loss of information and 0
indicates the maximum loss of information. Not only is this expensive to train and deploy, but the quality setting is not known at inference time (it
is not stored with the JPEG image~\cite{wallace1992jpeg}) making it impossible to use these models in
practical applications.
We solve this problem by creating a single model that uses quantization data, which is stored in the JPEG file, to
perform restoration. Our CNN model processes the image entirely in the DCT~\cite{ahmed1974discrete} domain. While
previous works have recognized that the DCT domain is less likely to spread quantization errors
\cite{wang2016d3, zhang2018dmcnn}, DCT domain-based models alone have historically not been successful unless
combined with pixel domain models (so-called ``dual domain'' models). Inspired by recent methods
\cite{ehrlich2019deep, deguerre2019fast,dong2015compression,gueguen2018faster}, we formulate fully DCT domain
regression. This allows our model to be parameterized by the quantization matrix,
an $8 \times 8$ matrix that directly determines the quantization applied to each DCT coefficient. We develop
a novel method for parameterizing our network called Convolution Filter Manifolds, an
extension of the Filter Manifold technique~\cite{kang2016crowd}. By adapting our network weights to the input
quanitzation matrix, our single network is able to handle a wide range of quality settings. Finally, since JPEG images are stored in
the YCbCr color space, with the Y channel containing more information than the subsampled color channels, we
use the reconstructed Y channel to guide the color channel reconstructions to achieve a better result on
color images. As in~\cite{zini2019deep}, we observe that using the Y channel in this way achieves good color correction
results. Finally, since regression results for artifact correction are often blurry, as a result of lost
texture information, we fine-tune our model using a GAN loss specifically designed to restore texture. This allows
us to generate highly realistic reconstructions. See Figure \ref{fig:header} for an overview of the correction flow. Our
focus is on the restoration of full color JPEG images.
To summarize, our contributions are:
\begin{enumerate}
\item A single model for artifact correction of JPEG images at any quality, parameterized by the
quantization matrix, which is state-of-the-art in color JPEG artifact correction and competitive for
grayscale artifact correction.
\item A formulation for fully DCT domain image-to-image regression.
\item Convolutional Filter Manifolds for parameterizing CNNs with spatial side-channel information.
\end{enumerate}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/header_2.pdf}
\vspace{-0.1in}
\caption{\textbf{Correction process.} Excerpt from ICB RGB 8bit dataset ``hdr.ppm''. Input was compressed at quality 10.}
\label{fig:header}
\vspace{-0.1in}
\end{figure}
\section{Experiments}
\vspace{-0.05in}
We validate the theoretical discussion in the previous sections with experimental results. We first describe the datasets
we used along with the training procedure we followed. We then show artifact correction results and compare them with previous state-of-the-art methods. Finally, we perform an ablation study. Please see our supplementary material for further
results. In the following section, we stress that the models we are comparing against train a bespoke model for each JPEG quality
setting they test on while we train only a single model.
\begin{table}[t]
\centering
\caption{\textbf{Color Artifact Correction Results.} PSNR / PSNR-B / SSIM format. Best result in bold, second best underlined. JPEG column gives input error. For ICB, we used the RGB 8bit dataset}
\footnotesize
\renewcommand{\arraystretch}{1.2}
\renewcommand{\tabcolsep}{1.2mm}
\resizebox{0.95\columnwidth}{!}{
\begin{tabular}{@{}l *{7}{c}@{}}
\toprule
Dataset & Quality & JPEG & ARCNN\cite{dong2015compression} & MWCNN \cite{liu2018multi} & IDCN \cite{zheng2019implicit} & DMCNN \cite{zhang2018dmcnn} & Ours \\
\midrule
\multirow{3}{*}{Live-1} & \csvreader[late after line=\\]{data/comparisons/live1_color.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{BSDS500} & \csvreader[late after line=\\]{data/comparisons/BSDS500_color.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{ICB} & \csvreader[late after line=\\]{data/comparisons/ICB-RGB8.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\bottomrule
\end{tabular}
}
\label{tab:colorresults}
\vspace{-0.05in}
\end{table}
\vspace{-0.05in}
\subsection{Experimental Setup}
\subsubsection{Datasets and Metrics.} For training, we use the DIV2k and Flickr2k~\cite{agustsson2017ntire} datasets. DIV2k consists of 900 images, and the Flickr2k dataset contains 2650 images. We preextract $256 \times 256$ patches from these images taking 30 random patches from each image and compress them using quality in $[10, 100]$ in steps of 10. This gives a total training set of 1,065,000 patches.
For evaluation, we use the Live1~\cite{sheikh2006statistical, sheikh2006live}, Classic-5~\cite{foi2006pointwise},
BSDS500~\cite{arbelaez2010contour}, and ICB datasets~\cite{icb}. ICB is a new dataset which provides 15 high-quality lossless images designed specifically to measure compression quality. It is our hope that the community will gradually begin including ICB dataset results. Where previous works have provided code and models, we reevaluate their methods and provide results here for comparison. As with all prior works, we report PSNR, PSNR-B~\cite{tadala2012novel}, and SSIM~\cite{wang2004image}.
\vspace{-0.1in}
\subsubsection{Implementation Details.} All training uses the Adam \cite{kingma2014adam} optimizer with a batch size of 32 patches. Our network is implemented using the PyTorch \cite{NEURIPS2019_9015} library. We normalize the DCT coefficients using per-frequency and per-channel mean and standard deviations. Since the DCT coefficients are measurements of different signals, by computing the statistics per-frequency we normalize the distributions so that they are all roughly the same magnitude. We find that this greatly speeds up the convergence of the network. Quantization table entrys are
normalized to [0, 1], with 1 being the most quantization and 0 the least. We use libjpeg~\cite{libjpeg} for compression with the baseline quantization setting.
\vspace{-0.1in}
\subsubsection{Training Procedure.}
As described in Section~\ref{sec:app:train}, we follow a staged training approach by first training the Y channel or grayscale artifact correction network, then training the color (CbCr) channel network, and finally training both networks using the GAN loss.
For the first stage, the Y channel artifact correction network, the learning rate starts at $1 \times 10^{-3}$ and decays by a factor of 2 every 100,000 batches. We stop training after 400,000 batches. We set $\lambda$ in Equation~\ref{eq:regressionloss} to 0.05.
For the next stage, color artifact correction network, all color channels are restored. The weights for the Y channel network are initialized from the previous stage and frozen during training. The color channel network weights are trained using a cosine annealing learning rate schedule \cite{loshchilov2016sgdr} decaying from $1 \times 10^{-3}$ to $1 \times 10^{-6}$ over 100,000 batches.
Finally, we train both Y and color channel artifact correction networks (jointly referred to as the generator model) using a GAN loss to improve qualitative textures. The generator model weights are initialized to the pre-trained models from the previous stages. We use the DCGAN \cite{radford2015unsupervised} discriminator. The model is trained for 100,000 iterations using cosine annealing~\cite{loshchilov2016sgdr}
with the learning rate starting from $1 \times 10^{-4}$ and ending at $1 \times 10^{-6}$. We set $\gamma$ and $\nu$ in Equation~ \ref{eq:ganloss} to $5 \times 10^{-3}$ and $1 \times 10^{-2}$ respectively.
\vspace{-0.1in}
\subsection{Results: Artifact Correction}
\subsubsection{Color Artifact Correction.}
We report the main results of our approach, color artifact correction, on Live1, BSDS500, and ICB in Table~\ref{tab:colorresults}. Our model consistently outperforms recent baselines on all datasets. Note that of all the approaches, only ours and IDCN~\cite{zheng2019implicit} include native processing of color channels. For the other models, we convert input images to YCbCr and process the channels independently. We would like to highlight that all the baseline models trained a \emph{unique} network for \emph{each} JPEG quality factor, whereas our approach trains a \emph{single} model to handle \emph{all} quality factors.
For quantitative comparisons to more methods on Live-1 dataset, at compression quality 10, refer to Figure \ref{fig:compare}. We present qualitative results from a mix of all three datasets in Figure~\ref{fig:gan} (``Ours''). Since our model is not restricted by which quality settings it can be run on, we also show the increase in PSNR for qualities 10-100 in Figure \ref{fig:ipsnr}. As expected, we see the most improvement in PSNR values for low-quality settings.
\begin{table}[t]
\centering
\caption{\textbf{Y Channel Correction Results.} PSNR / PSNR-B / SSIM format, the best result is highlighted in bold, second best is underlined. The JPEG column gives with input error of the images. For ICB, we used the Grayscale 8bit dataset. We add Classic-5, a grayscale only dataset.}
\footnotesize
\renewcommand{\arraystretch}{1.2}
\renewcommand{\tabcolsep}{1.2mm}
\resizebox{0.95\columnwidth}{!}{
\begin{tabular}{@{}l *{7}{c}@{}}
\toprule
Dataset & Quality & JPEG & ARCNN\cite{dong2015compression} & MWCNN \cite{liu2018multi} & IDCN \cite{zheng2019implicit} & DMCNN \cite{zhang2018dmcnn} & Ours \\
\midrule
\multirow{3}{*}{Live-1} & \csvreader[late after line=\\]{data/comparisons/live1.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{Classic-5} & \csvreader[late after line=\\]{data/comparisons/classic5.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{BSDS500} & \csvreader[late after line=\\]{data/comparisons//BSDS500.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{ICB} & \csvreader[late after line=\\]{data/comparisons//ICB-GRAY8.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\bottomrule
\end{tabular}
}
\label{tab:yresults}
\vspace{-0.1in}
\end{table}
\begin{figure}[t]
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{ccccc}
Original & JPEG & IDCN Q=10 & IDCN Q=20 & Ours \\
\includegraphics[width=0.2\linewidth]{figures/generalize/original.png} &
\includegraphics[width=0.2\linewidth]{figures/generalize/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/generalize/idcn_10.png} &
\includegraphics[width=0.2\linewidth]{figures/generalize/idcn_20.png} &
\includegraphics[width=0.2\linewidth]{figures/generalize/ours.png}
\end{tabular}}
\vspace{-0.1in}
\caption{\textbf{Generalization Example}. Input was compressed at quality 50. Please zoom in to view details.}
\label{fig:gen}
\vspace{-0.15in}
\end{figure}
\begin{table}[t]
\centering
\caption{\textbf{Generalization Capabilities}. Live-1 dataset (PSNR / PSNR-B / SSIM)}
\footnotesize
\renewcommand{\arraystretch}{1.2}
\renewcommand{\tabcolsep}{1.2mm}
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{@{} *{5}{c}@{}}
\toprule
Model Quality & Image Quality & JPEG & IDCN \cite{zheng2019implicit} & Ours \\
\midrule
10 & \multirow{2}{*}{50} & 30.91 / 28.94 / 0.905 & 30.19 / 30.14 / 0.889 & \multirow{2}{*}{\textbf{32.78 / 32.19 / 0.932}} \\
20 & & 30.91 / 28.94 / 0.905 & 31.91 / 31.65 / 0.916 & \\
\midrule
10 & 20 & 27.96 / 25.77 / 0.837 & 29.25 / 29.08 / 0.863 & \textbf{29.92 / 29.51 / 0.882} \\
20 & 10 & 25.60 / 23.53 / 0.755 & 26.95 / 26.24 / 0.804 & \textbf{27.65 / 27.40 / 0.819} \\
\bottomrule
\end{tabular}
}
\label{tab:genresults}
\vspace{-0.15in}
\end{table}
\begin{figure}[t]
\begin{minipage}[t][][b]{0.48\linewidth}
\centering
\vspace{0pt}
\includegraphics[height=0.7\linewidth]{plots/increase.pdf}
\vspace{-0.08in}
\caption{\textbf{Increase in PSNR on color datasets.} For all three datasets we show the average improvement in PSNR values on qualities 10-100. Improvement drops off steeply at quality 90.}
\label{fig:ipsnr}
\end{minipage}
\hfill
\begin{minipage}[t][][b]{0.48\linewidth}
\centering
\vspace{0pt}
\includegraphics[height=0.65\linewidth]{plots/compare.pdf}
\vspace{-0.08in}
\caption{\textbf{Comparison for Live-1 quality 10.} Where
code was available we reevaluated, otherwise we used published numbers. If no color results were published we
used Y channel results.}
\label{fig:compare}
\end{minipage}
\vspace{-0.15in}
\end{figure}
\begin{figure}[t]
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{ccccccc}
Original & JPEG & IDCN & Ours & Ours-GAN \\
\includegraphics[width=0.2\linewidth]{figures/comparison/artificial/original.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/artificial/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/artificial/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/artificial/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/artificial/gan.png} \\
\includegraphics[width=0.2\linewidth]{figures/comparison/nightshot/original.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/nightshot/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/nightshot/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/nightshot/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/nightshot/gan.png} \\
\includegraphics[width=0.2\linewidth]{figures/comparison/lamb/original.jpg} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lamb/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lamb/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lamb/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lamb/gan.png} \\
\includegraphics[width=0.2\linewidth]{figures/comparison/lighthouse/original.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lighthouse/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lighthouse/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lighthouse/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/lighthouse/gan.png} \\
\includegraphics[width=0.2\linewidth]{figures/comparison/flower/original.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/flower/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/flower/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/flower/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/flower/gan.png} \\
\includegraphics[width=0.2\linewidth]{figures/comparison/womanhat/original.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/womanhat/jpeg.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/womanhat/idcn.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/womanhat/ours.png} &
\includegraphics[width=0.2\linewidth]{figures/comparison/womanhat/gan.png} \\
\end{tabular}}
\vspace{-0.05in}
\caption{\textbf{Qualitative Results.} All images were compressed at Quality 10. Please zoom in to view details.}
\label{fig:gan}
\vspace{-0.05in}
\end{figure}
\begin{table}[t]
\footnotesize
\renewcommand{\arraystretch}{1.1}
\renewcommand{\tabcolsep}{1mm}
\centering
\begin{minipage}[t][][b]{0.47\linewidth}
\centering
\caption{\textbf{GAN FID Scores.}}
\resizebox{0.6\columnwidth}{!}{
\begin{tabular}{@{}l *{3}{c}@{}}
\toprule
Dataset & Quality & Ours & Ours-GAN \\
\midrule
\multirow{3}{*}{Live-1} & \csvreader[late after line=\\]{data/fid/fid_live1.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{BSDS500} & \csvreader[late after line=\\]{data/fid/fid_bsds500.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{3}{*}{ICB} & \csvreader[late after line=\\]{data/fid/fid_icb.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\bottomrule
\end{tabular} }
\label{tab:fid}
\end{minipage}
\hfill
\begin{minipage}[t][][b]{0.51\linewidth}
\centering
\caption{\textbf{Ablation Results.} (refer to Section~\ref{sec:res:ablation} for details)}
\resizebox{0.75\columnwidth}{!}{
\begin{tabular}{@{}L{1.8cm} L{1.8cm} *{7}{c}@{}}
\toprule
Experiment & Model & PSNR & PSNR-B & SSIM \\
\midrule
\multirow{3}{*}{CFM} & \csvreader[late after line=\\]{data/ablation/cmf.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{2}{*}{Subnetworks} & \csvreader[late after line=\\]{data/ablation/bvf.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\midrule
\multirow{2}{*}{Fusion} & \csvreader[late after line=\\]{data/ablation/ft.csv}{}{\csviffirstrow{}{&}\csvlinetotablerow}
\bottomrule
\end{tabular} }
\label{tab:ablation}
\end{minipage}
\end{table}
\vspace{-0.1in}
\subsubsection{Intermediate Results on Y Channel Artifact Correction.}
Since the first stage of our approach trains for grayscale or Y channel artifact correction, we can also compare the intermediate results from this stage with other approaches. We report results in Table~\ref{tab:yresults} for Live1, Classic-5, BSDS500, and ICB. As the table shows, intermediate results from our model can match or outperform previous state-of-the-art models in many cases, consistently providing high SSIM results using a single model for all quality factors. Again, all the baselines in Table \ref{tab:yresults} use different models, each trained specifically for the given quality factor. We want to emphasize that our main goal is color artifact correction, and these are intermediate results from our approach.
\vspace{-0.1in}
\subsubsection{GAN Correction}
\label{sec:res:gan}
Finally, we show results from our model trained using GAN correction. We use model interpolation~\cite{wang2018esrgan}
and show qualitative results for the interpolation parameter ($\alpha$) set to 0.7 in Figure \ref{fig:gan}. (``Ours-GAN'') Notice that the GAN loss is able to restore texture to blurred, flat regions and sharpen edges, yielding a more visually pleasing result. We provide additional qualitative results in the supplementary material. Note that we do not show error metrics using the GAN model as it produces higher quality images, at the expense of quantitative metrics, by adding texture details that are not present in the original images. We instead show
FID scores for the GAN model compared to our regression model in Table~\ref{tab:fid}, indicating that the GAN model generates significantly more
realistic images.
\vspace{-0.1in}
\subsection{Results: Generalization Capabilities}
\vspace{-0.05in}
A major advantage of our method, as we have highlighted repeatedly, is that it uses a single model to correct JPEG images at any quality, while prior works train a model for each quality factor. Therefore, we explore if other methods are capable of generalizing or if they really require this ensemble of quality-specific models. To evaluate this generalization capability, we use our closest competitor and prior state-of-the-art, IDCN~\cite{zheng2019implicit}. IDCN does not provide a model for quality higher than 20, so we explore if their model generalizes by using their quality 10 and quality 20 models to correct
quality 50 Live-1 images. We also test using the quality 20 model to correct quality 10 images and using the quality
10 model to correct quality 20 images. These results are shown in Table~\ref{tab:genresults} along with our result for comparison.
As the table shows, the choice of model is critical for IDCN, and there is a significant quality drop when choosing the wrong model. Neither their quality 10 nor their quality 20 model is able to effectively correct images that it was not trained on, scoring significantly lower than if the correct model were used. At quality 50, the quality 10 model produces a result worse than the input JPEG, and the quality 20 model makes only a slight improvement. In comparison, our single model provides consistently better results across image quality factors. We stress that the quality setting is not stored in the JPEG file, so a deployed system has no way to pick the correct model. Since our model adapts to the quantization matrix, which is stored with the JPEG file, this limitation is alleviated. We show
an example of a quality 50 image and artifact correction results in Figure \ref{fig:gen}.
\vspace{-0.1in}
\subsection{Design and Ablation Analysis}
\label{sec:res:ablation}
\vspace{-0.05in}
In this section, we ablate many of our design decisions and observe their effect on network accuracy. The results are reported in Table~\ref{tab:ablation}, we report all three metrics on quality 10 classic-5.
\vspace{-0.05in}
\paragraph{Implementation details:} For all ablation experiments, we keep the number of parameters approximately the same between tested models to alleviate the concern that a network performs better simply because it has a higher capacity. All models are trained for 100,000 batches on the grayscale training patch set using cosine
annealing~\cite{loshchilov2016sgdr} from a learning rate of $1 \times 10^{-3}$ to $1 \times 10^{-6}$.
\vspace{-0.05in}
\subsubsection{Importance of CFM layers.} We emphasized the importance of adaptable weights in the CFM layers, which can be adapted using the quantization matrix. However, there are other simpler methods of using side-channel information. We could simply concatenate the quantization matrix channelwise with the input, or we could ignore the quantization matrix altogether. As shown in the ``CFM'' experiment in Table~\ref{tab:ablation}, the CFM unit performs better than both of these alternatives by a considerable margin. We further visualize the filters learned by the CFM layers and the underlying embeddings in the supplementary material which validate that the learned filters follow a manifold structure.
\vspace{-0.05in}
\subsubsection{BlockNet \textit{vs}.\ FrequencyNet.} We noted that the FrequencyNet should not be able to perform without a preceding BlockNet because high-frequency information will be zeroed out from the compression process. To test this claim, we train individual BlockNet and FrequencyNet in isolation and report the results in Table~\ref{tab:ablation} (``Subnetworks''). We can see that BlockNet alone attains significantly higher performance than FrequencyNet alone.
\vspace{-0.05in}
\subsubsection{Importance of the fusion layer.} Finally, we study the necessity of the fusion layer presented. We posited that the fusion layer was necessary for gradient flow to the early layers of our network. As demonstrated in Table~\ref{tab:ablation} (``Fusion''), the network without fusion fails to learn, matching the input PSNR of classic-5 after full training, whereas the network with fusion makes considerable progress.
\section{Our Approach}
\vspace{-0.05in}
Our goal is to design a single model capable of JPEG artifact correction at any quality. Towards this, we formulate an architecture, parameterized by the quantization matrix, using the first principles of the JPEG compression algorithm.
Recall that a JPEG quantization matrix captures the amount of rounding applied to DCT coefficients and is indicative of information lost during compression. A key contribution of our approach is utilizing this quantization matrix directly to guide the restoration process using a fully DCT domain image-to-image regression network. JPEG stores color data in the YCbCr colorspace. The compressed Y channel is much higher quality (generally, twice the resolution) compared to CbCr channels since human perception is less sensitive to finer color details than to brightness details. Therefore, we follow a staged approach: first restoring artifacts in the Y channel and then using the restored Y channel as guidance to restore the CbCr channels.
An illustrative overview of our approach is presented in Figure~\ref{fig:overview}.
Next, we present building blocks utilized in our architecture in $\S$\ref{sec:app:bb}, that allow us to parameterize our model using the quantization matrix and operate entirely in the DCT domain. Our Y channel and color artifact correction networks are described in $\S$\ref{sec:app:y} and $\S$\ref{sec:app:c} respectively, and finally the training details in $\S$\ref{sec:app:train}.
\subsection{Building Blocks}
\label{sec:app:bb}
\vspace{-0.01in}
By creating a single model capable of JPEG artifact correction at any quality, our model solves a significantly harder problem than previous works. To solve it, we parameterize our network using the $8 \times 8$ quantization matrix available with every JPEG file. We first describe Convolutional Filter Manifolds (CFM), our solution for adaptable convolutional kernels parameterized by the quantization matrix. Since the quantization matrix encodes the amount of rounding per each DCT coefficient, this parameterization is most effective in the DCT domain, a domain where CNNs have previously struggled. Therefore, we also formulate artifact correction as fully DCT domain image-to-image regression and describe critical frequency-relationships-preserving operations.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figures/overview.pdf}
\caption{\textbf{Overview.} We first restore the Y channel of the input image, then use the restored Y channel to correct
the color channels which have much worse input quality.}
\label{fig:overview}
\vspace{-0.15in}
\end{figure}
\vspace{-0.04in}
\subsubsection{Convolutional Filter Manifold (CFM).}
Filter Manifolds~\cite{kang2016crowd} were introduced as a way to parameterize a deep CNN using side-channel scalar data. The method learns a manifold of convolutional kernels, which is a function of a scalar input. The manifold is modeled as a three-layer multilayer perceptron. The input to this network is the scalar side-channel data, and the output vector is reshaped to the shape of the desired convolutional kernel and then convolved with the input feature map for that layer.
Recall that in the JPEG compression algorithm, a quantization matrix is derived from a scalar quality setting to determine the amount of rounding to apply, and therefore the amount of information removed from the original image. This quantization matrix is then stored in the JPEG file to allow for correct scaling of the DCT coefficients at decompression time. This quantization matrix is then a strong signal for the amount of information lost. However, the quantization matrix is an $8 \times 8$ matrix with spatial structure, applying the Filter Manifold technique to it has the same drawbacks as processing images with multilayer perceptrons,
\textit{e}.\textit{g}., a large number of parameters and a lack of spatial relationships.
To solve this, we propose an extension to create Convolutional Filter Manifolds (CFM), replacing the multilayer perceptron by a lightweight three-layer CNN. The input to the CNN is our quantization matrix, and the output is reshaped to the desired convolutional kernel shape and convolved with the input feature map as in the Filter Manifold method. For our problem, we follow the network structure in Figure~\ref{fig:cmf} for each CFM layer. However, this is a general technique and can be used with a different architecture when spatially arranged side-channel data is available.
\begin{figure}[t]
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/cmf.pdf}
\caption{\textbf{Convolutional Filter Manifold}, as used in our network. Note that the convolution with the input
feature map is done with stride-8.}
\label{fig:cmf}
\end{minipage}
\hspace{0.03\linewidth}
\begin{minipage}{0.47\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/coef.pdf}
\caption{\textbf{Coefficient Rearrangement}. Frequencies are arranged channelwise giving an image with 64 times the number of channels
at $\frac{1}{8}$th the size. This can then be convolved with 64 groups per convolution to learn per-frequency filters.}
\label{fig:coef}
\end{minipage}
\vspace{-0.12in}
\end{figure}
\vspace{-0.05in}
\subsubsection{Coherent Frequency Operations.}
In prior works, DCT information has been used in dual-domain models~\cite{wang2016d3,zhang2018dmcnn}. These models used standard $3 \times 3$ convolutional kernels with U-Net~\cite{ronneberger2015u} structures to process the coefficients. Although the DCT is a linear map on image pixels~\cite{smith1994fast, ehrlich2019deep}, ablation studies in prior work show that the DCT network alone is not able to surpass even classical artifact correction techniques.
Although the DCT coefficients are arranged in a grid structure of the same shape as the input image, that spatial structure does not have the same meaning as pixels. Image pixels are samples of a continuous signal in two dimensions. DCT coefficients, however, are samples from different, orthogonal functions and the two-dimensional arrangement indexes them. This means that a $3 \times 3$ convolutional kernel is trying to learn a relationship not between spatially related samples of the same function as it was designed to do, but rather between samples from completely unrelated functions. Moreover, it must maintain
this structure throughout the network to produce a valid DCT as output. This is the root cause of CNNs poor performance on DCT coefficients for image-to-image regression, semantic segmentation, and object detection (Note that this should not affect whole image classification performance as in~\cite{gueguen2018faster,ghosh2016deep}).
A class of recent techniques~\cite{deguerre2019fast,lo2019exploring}, which we call Coherent Frequency Operations for their preservation of frequency relationships, are used as the building block for our regression network. The first layer is an $8 \times 8$ stride-8 layer~\cite{deguerre2019fast}, which computes a representation for each block (recall that JPEG blocks are non-overlapping $8\times8$ DCT coefficients). This block representation, which is one eighth the size of the input, can then be processed with a standard CNN.
The next layer is designed to process each frequency in isolation. Since each of the 64 coefficients in an $8\times8$ JPEG block corresponds to a different frequency, the input DCT coefficients are first rearranged so that the coefficients corresponding to different frequencies are stored channelwise (see Figure~\ref{fig:coef}). This gives an input, which is again one eighth the size of the original image, but this time with 64 channels (one for each frequency), this was referred to as Frequency Component Rearrangement in~\cite{lo2019exploring}. We then use convolutions with 64 groups to learn per-frequency convolutional weights.
Combining these two operations (block representation using $8\times8$ 8-stride and frequency component rearrangement) allows us to match state-of-the-art pixel and dual-domain results using only DCT coefficients as input and output.
\vspace{-0.05in}
\subsection{Y Channel Correction Network}
\label{sec:app:y}
Our primary goal is artifact correction of full color images, and we again leverage the JPEG algorithm to do this. JPEG stores color data in the YCbCr colorspace. The color channels, which contribute less to the human visual response, are both subsampled and more heavily quantized. Therefore, we employ a larger network to correct only the Y channel, and a smaller network which uses the restored Y channel to more effectively correct the Cb and Cr color channels.
\begin{figure}[t]
\begin{minipage}[t]{0.32\linewidth}
\centering
\includegraphics[height=0.9\linewidth]{figures/blocknet_abhinav.pdf}
\caption{\textbf{BlockNet.} Both the block generator and decoder
are parameterized by the quanitzation matrix.}
\label{fig:blocknet}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\linewidth}
\centering
\includegraphics[height=\linewidth]{figures/frequencynet_abhinav.pdf}
\caption{\textbf{FrequencyNet.} Note that the 256 channels in the
RRDB layer actually compute 4 channels per frequency.}
\label{fig:frequencynet}
\end{minipage}
\hfill
\begin{minipage}[t]{0.32\linewidth}
\centering
\includegraphics[height=\linewidth]{figures/fusion_abhinav.pdf}
\caption{\textbf{Fusion subnetwork.} Outputs from all three subnetworks are fused to produce the final residual.}
\label{fig:fusion}
\end{minipage}
\vspace{-0.1in}
\end{figure}
\vspace{-0.05in}
\subsubsection{Subnetworks.}
Utilizing the building blocks developed earlier, our network design proceeds in two phases: block enhancement, which learns a quantization invariant representations for each JPEG block, and frequency enhancement, which tries to match each frequency reconstruction to the regression target. These phases are fused to produce the final residual for restoring the Y channel. We employ two purpose-built subnetworks: the block network (BlockNet) and the frequency network (FrequencyNet). Both of these networks can be thought of as separate image-to-image regression models with a structure inspired by ESRGAN~\cite{wang2018esrgan}, which allows sufficient low-level information to be preserved as well as allowing sufficient gradient flow to train these very deep networks. Following recent techniques \cite{wang2018esrgan}, we remove batch normalization layers. While recent works have largely replaced PReLU~\cite{he2015delving} with LeakyReLU~\cite{maas2013rectifier, wang2018esrgan,galteri2017deep,galteri2019deep}, we find that PReLU activations give much higher accuracy.
\vspace{-0.05in}
\paragraph{BlockNet.} This network processes JPEG blocks to restore the Y channel (refer to Figure~\ref{fig:blocknet}). We use
the $8 \times 8$ stride-8 coherent frequency operations to create a block representation. Since this layer is computing a block representation from all the
input DCT coefficients, we use a Convolutional Filter Manifold (CFM) for this layer
so that it has access to quantization information. This allows the layer to learn the quantization table entry to DCT coefficient correspondence with the goal to output a quantization-invariant block representation. Since there is a one to one correspondence between the quantization table entry and rounding applied to a DCT coefficient, this motivates our choice to operate entirely in the DCT domain. We then process these quantization-invariant block representations with Residual-in-Residual Dense Blocks (RRDB) from~\cite{wang2018esrgan}. RRDB layers are an extension of the commonly used residual block~\cite{he2016deep} and define several recursive and highly residual layers. Each RRDB has 15 convolution layers, and we use a single RRDB for the block network with 256 channels. The network terminates with another
$8 \times 8$ stride-8 CFM, this time transposed, to reverse the block representation back to its original form so that it can be used for later tasks.
\vspace{-0.05in}
\paragraph{FrequencyNet.}
This network, shown in Figure \ref{fig:frequencynet}, processes the individual frequency coefficients using the Frequency Component Rearrangement technique (Figure~\ref{fig:coef}). The architecture of this network is similar to BlockNet. We use a single $3 \times 3$ convolution to change the number of channels from the 64 input channels to the 256 channels used by the RRDB layer. The single RRDB layers processes feature maps with 256 channels and 64 groups yielding 4 channels per frequency. An output $3 \times 3$ convolution transforms the 4 channel output to the 64 output channels, and the coefficients are rearranged back into blocks for later tasks.
\vspace{-0.1in}
\subsubsection{Final Network.}
The final Y channel artifact correction network is shown in Figure~\ref{fig:ychannel}. We observe that since the FrequencyNet processes frequency coefficients in isolation if those coefficients were zeroed out by the compression
process then it can make no attempt at restoring them (since they are zero valued they would be set to the layer bias). This is common with high frequencies by design, they have larger quanitzation table entries and they contribute less to the human visual response. We, therefore, lead with the BlockNet to restore high frequencies. We then pass the result to the FrequencyNet, and its result is then processed by a second block network to restore more information. Finally, a three-layer fusion network (see Figure~\ref{fig:fusion} and~\ref{fig:ychannel}) fuses the output of all three subnetworks into a final result. Allowing all three subnetworks to contribute to the final result in this way allows for better gradient flow. The effect of fusion, as well as the three subnetworks, is tested in our ablation study. The fusion output is treated as a residual and added to the input to produce the final corrected coefficients for the Y channel.
\begin{figure}[t]
\begin{minipage}[t]{0.43\linewidth}
\centering
\includegraphics[width=0.9\columnwidth]{figures/ynetwork_abhinav.pdf}
\caption{\textbf{Y Channel Network.} We include two copies of the BlockNet, one to perform early restoration of
high frequency coefficients, and one to work on the restored frequencies. All three
subnetworks contribute to the final result using the fusion subnetwork.}
\label{fig:ychannel}
\end{minipage}
\hfill
\begin{minipage}[t]{0.54\linewidth}
\centering
\includegraphics[width=0.9\columnwidth]{figures/colornet_abhinav.pdf}
\caption{\textbf{Color Channel Network.} Color channels are downsampled, so the block representation is upsampled using a learned upsampling.
The Y and color channel block representations are then concatenated to guide the color channel restoration. Cb and Cr channels are processed independently with the
same network.}
\label{fig:color}
\end{minipage}
\vspace{-0.15in}
\end{figure}
\vspace{-0.1in}
\subsection{Color Correction Network}
\label{sec:app:c}
\vspace{-0.05in}
The color channel network (Figure~\ref{fig:color}) processes the Cb and Cr DCT coefficients. Since the color channels are subsampled with respect to the Y channel by half, they incur a much higher loss of information and lose the structural information which is preserved in the Y channel. We first compute the block representation of the downsampled color channel coefficients using a CFM layer, then process them with a single RRDB layer. The block representation is then upsampled using a $4 \times 4$ stride-2 convolutional layer. We compute the block representation of the restored Y channel, again using a CFM layer. The block representations are concatenated channel-wise and processed using a single RRDB layer before being transformed back into coefficient space using a transposed $8 \times 8$ stride-8 CFM. By concatenating the Y channel restoration, we give the network structural information that may be completely missing in the color channels. The result of this network is the color channel residual. This process is repeated individually for each color channel with a single network learned on Cb and Cr. The output residual is added to nearest-neighbor upsampled input coefficients to give the final restoration.
\vspace{-0.1in}
\subsection{Training}
\label{sec:app:train}
\vspace{-0.05in}
\subsubsection{Objective.} We use two separate objective functions to train, an error loss and a GAN loss. Our error loss is based on prior works which minimize the $l_1$ error of the result and the target image. We additionally
maximize the Structural Similarity (SSIM)~\cite{wang2004image} of the result since SSIM is generally regarded as a closer metric to human perception than PSNR. This gives our final objective function as
\begin{equation}
\label{eq:regressionloss}
\mathcal{L}_{\text{JPEG}}(x, y) = \|y - x\|_1 - \lambda\text{SSIM}(x, y)
\end{equation}
where $x$ is the network output, $y$ is the target image, and $\lambda$ is a balancing hyperparameter.
A common phenomenon in JPEG artifact correction and superresolution is the production of a blurry or textureless result.
To correct for this, we fine tune our fully trained regression network with a GAN loss. For this objective, we use the relativistic
average GAN loss $\mathcal{L}^{Ra}_G$~\cite{jolicoeur2018relativistic}, we use $l_1$ error to prevent the image from moving too far away from the regression result, and we use preactivation network-based loss~\cite{wang2018esrgan}. Instead of a perceptual loss that tries to keep the outputs close in ImageNet-trained VGG feature space used in prior works, we use a network trained on the MINC dataset~\cite{bell2015material},
for material classification. This texture loss provided only marginal benefit in ESRGAN \cite{wang2018esrgan} for super-resolution. We find it to be critical in our task for restoring texture to blurred regions, since JPEG compression destroys these fine details. The texture loss is defined as
\begin{equation}
\mathcal{L}_{\text{texture}}(x, y) = \|\text{MINC}_{5,3}(y) - \text{MINC}_{5,3}(x)\|_1
\end{equation}
where MINC$_{5,3}$ indicates that the output is from layer 5 convolution 3.
The final GAN loss is
\begin{equation}
\label{eq:ganloss}
\mathcal{L}_\text{GAN}(x, y) = \mathcal{L}_{\text{texture}}(x, y) + \gamma\mathcal{L}^{Ra}_{G}(x, y) + \nu\|x - y\|_1
\end{equation}
with $\gamma$ and $\nu$ balancing hyperparameters. We note that the texture restored using the GAN model is, in general,
not reflective of the regression target at inference time and actually produces worse numerical results than the regression model despite the images looking more realistic.
\vspace{-0.07in}
\subsubsection{Staged Training.} Analogous to our staged restoration, Y channel followed by color channels, we follow a staged training approach. We first train the Y channel correction network using $\mathcal{L}_{\text{JPEG}}$. We then train the color correction network using $\mathcal{L}_{\text{JPEG}}$ keeping the Y channel network weights frozen. Finally, we train the entire network (Y and color correction) with $\mathcal{L}_\text{GAN}$.
\section{Prior Work}
\vspace{-0.05in}
Pointwise
Shape-Adaptive DCT~\cite{foi2006pointwise} is considered a standard classical technique which uses thresholded
DCT coefficients reconstruct local estimates of the input signal. Yang~\textit{et al}. ~\cite{yang2000blocking} use a
lapped transform to approximate the inverse DCT on the quantized coefficients.
More recent techniques use convolutional neural networks~\cite{lecun1990handwritten, sutskever2012imagenet}.
ARCNN~\cite{dong2015compression}, was a straightforward regression model inspired
by superresolution techniques, L4/L8~\cite{svoboda2016compression} continue this work. CAS-CNN~\cite{cavigelli2017cas} add hierarchical skip connections and a
multi-scale loss function. Liu~\textit{et al}. ~\cite{liu2018multi} use a wavelet-based network
for general denoising and artifact correction, which is extended by Chen~\textit{et al}. ~\cite{chen2018dpw}. Galteri~\textit{et al}. ~\cite{galteri2017deep} use a GAN formulation to
achieve more visually appealing results. S-Net~\cite{zheng2018s} introduces a scalable architecture that can produce
different quality outputs based on the desired computation complexity. Galteri~\textit{et al}. ~\cite{galteri2019deep} extend their
GAN work with an ensemble of GANs where each GAN in the ensemble is trained to correct artifacts
of a specific quality level much as in prior work. They train an auxiliary network to classify the image into
the quality level that it was compressed with. The resulting quality level classification is used to pick a GAN from the ensemble to use for the final artifact correction. Zhang~\textit{et al}. ~\cite{zhang2017beyond} formulate general image denoising
using residual networks. Zhang~\textit{et al}. ~\cite{zhang2020residual} use a dense residual formulation for image enhancement. Tai~\textit{et al}.
\cite{tai2017memnet} use persistent memory in their restoration network.
Liu~\textit{et al}. ~\cite{liu2015data} introduce the dual domain idea in the sparse coding setting. Guo and Chao~\cite{guo2016building}
use convolutional autoencoders for both domains. DMCNN~\cite{zhang2018dmcnn} extends this with novel DCT rectifier to
constrain DCT errors. Zheng~\textit{et al}. ~\cite{zheng2019implicit} target color images and use an implicit DCT layer
to compute DCT domain loss using pixel level information. D3~\cite{wang2016d3} extends Liu~\textit{et al}. ~\cite{liu2015data}
by using a feed-forward formulation for certain parameters which were assumed in~\cite{liu2015data}. DCSC uses
convolution features in their sparse coding scheme~\cite{fu2019jpeg}.
\section{Further Analysis}
In this section we provide futher analysis of our model. We start by examining the Convolution Filter Manifold layers in more detail, providing visualizations of what they learn in order to better understand their contribution to our result. Next, we examine model interpolation in more detail by showing qualitative comparisons for varying interpolation strengths between the regression and GAN model. We then conduct a study that shows how much space can be saved by storing low quality JPEG images and using our method to restore them. We then examine the frequency domain qualitative results and show that our GAN model is capabile of generating images that have more high frequency content than the regression model alone. We conclude by examining the runtime throughput of our model compared to the other methods we tested against.
\input{supplement_sections/understanding}
\input{supplement_sections/model_interpolation}
\input{supplement_sections/equivilent_quality}
\input{supplement_sections/frequency}
\input{supplement_sections/runtime}
\input{supplement_sections/qualitative_results}
\input{supplement_sections/jpeg}
\subsection{Equivalent Quality}
One major motivation for JPEG artifact correction is that space or bandwidth can be saved by transmitting a small
low quality JPEG and algorithmically correcting it before display. We explore how effective our model is at this
by computing the equivalent quality JPEG file for a restored image. Our argument is that a system can get the
storage space savings of the lower quality JPEG and the visual fidelity of a higher quality JPEG by
using our model.
To show this we use the Live-1 dataset. For qualities in [10, 50] in steps of 10, we compute the average increase in
JPEG quality incurred by our model. We do this by compressing the input image at higher and higher qualities until we
find the first quality with SSIM greater than or equal to our restoration's SSIM. We then save the low quality JPEG
and the equivalent quality JPEG and measure the size difference in kilobytes. We average the quality increase and
space savings over the entire dataset, to show the amount of space saved by using our method over using the higher quality JPEG
directly. This result is shown in Figure \ref{fig:eq_qual_live1}. We also show qualitative examples for several images in
Figure \ref{fig:eq_qual_res}. Note that because the SSIM measure is not perfect, often our model outputs images that look
better than the equivalent quality JPEG.
\begin{figure}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{plots/equivalent_quality.pdf}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{plots/space_saving.pdf}
\end{minipage}
\caption{Equivalent quality and space savings for Live-1 dataset.}
\label{fig:eq_qual_live1}
\end{figure}
\begin{figure}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{ccc}
Input & Equivalent Quality JPEG & Ours \\
\includegraphics[width=0.3\linewidth]{figures/eq_qual/monarch/input.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/monarch/eq.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/monarch/ours.png} \\
Quality: 50 &
Quality: 85 &
29.5kB Saved \\ \\
Input & Equivalent Quality JPEG & Ours \\
\includegraphics[width=0.3\linewidth]{figures/eq_qual/bikes/input.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/bikes/eq.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/bikes/ours.png} \\
Quality: 30 &
Quality: 58 &
46.8kB Saved \\ \\
Input & Equivalent Quality JPEG & Ours \\
\includegraphics[width=0.3\linewidth]{figures/eq_qual/dolls/input.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/dolls/eq.jpg} &
\includegraphics[width=0.3\linewidth]{figures/eq_qual/dolls/ours.png} \\
Quality: 40 &
Quality: 78 &
25kB Saved
\end{tabular}
}
\caption{Equivalent quality visualizations. For each image we show the input JPEG, the JPEG with equivalent SSIM
to our model output, and our model output.}
\label{fig:eq_qual_res}
\end{figure}
\subsection{Model Interpolation}
Here we show more model interpolation results. Model interpolation creates a new model by linearly interpolating the GAN
and regresion model parameters as follows
\begin{equation}
\Theta_I = (1 - \alpha)\Theta_R + \alpha\Theta_G
\end{equation}
where $\Theta_I$ are the interpolated parameters, $\Theta_R$ are the regression model parameters and $\Theta_G$ are the
GAN model parameters with $\alpha \in [0, 1]$ being the interpolation parameter. The new model blends the result of the GAN and regression results. We observe that using the GAN model alone
can introduce artifacts (see Figure \ref{fig:model_interp}), blending the models in this way helps surpress those artifacts. Note that in this scheme, $\alpha=0$ gives the regression model and $\alpha = 1$ gives the GAN
model. Model interpolation has been shown to produce cleaner results than image interpolation, and has the
added benefit of not needing to run two models to produce a result. In Figure \ref{fig:model_interp} we show the model interpolation
results for $\alpha \in \{0.0, 0.7, 0.9, 1.0\}$ for several images from
the Live-1 dataset. This figure also serves as additional qualitative results for our method. These results were
generated from quality 10 JPEGs.
\begin{figure}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Regression & $\alpha=0.7$ \\
\includegraphics[width=0.4\linewidth]{figures/merp/fountain/reg.png} &
\includegraphics[width=0.4\linewidth]{figures/merp/fountain/7.png} \\
$\alpha=0.9$ & GAN \\
\includegraphics[width=0.4\linewidth]{figures/merp/fountain/9.png} &
\includegraphics[width=0.4\linewidth]{figures/merp/fountain/gan.png} \\ \\
Regression & $\alpha=0.7$ \\
\includegraphics[width=0.5\linewidth]{figures/merp/capitol/reg.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/capitol/7.png} \\
$\alpha=0.9$ & GAN \\
\includegraphics[width=0.5\linewidth]{figures/merp/capitol/9.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/capitol/gan.png}
\end{tabular}
}
\caption{Model interpolation results 1/2}
\end{figure}
\begin{figure}
\ContinuedFloat
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Regression & $\alpha=0.7$ \\
\includegraphics[width=0.5\linewidth]{figures/merp/woman/reg.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/woman/7.png} \\
$\alpha=0.9$ & GAN \\
\includegraphics[width=0.5\linewidth]{figures/merp/woman/9.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/woman/gan.png} \\ \\
Regression & $\alpha=0.7$ \\
\includegraphics[width=0.5\linewidth]{figures/merp/buildings/reg.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/buildings/7.png} \\
$\alpha=0.9$ & GAN \\
\includegraphics[width=0.5\linewidth]{figures/merp/buildings/9.png} &
\includegraphics[width=0.5\linewidth]{figures/merp/buildings/gan.png}
\end{tabular}
}
\caption{Model interpolation results 2/2}
\label{fig:model_interp}
\end{figure}
\subsection{Understanding Convolutional Filter Manifolds}
\begin{figure}
\begin{minipage}{0.49\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.33\linewidth]{figures/weights/q=10c=0.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=10c=15.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=10c=30.png} \\
\includegraphics[width=0.33\linewidth]{figures/weights/q=50c=0.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=50c=15.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=50c=30.png} \\
\includegraphics[width=0.33\linewidth]{figures/weights/q=100c=0.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=100c=15.png} & \includegraphics[width=0.33\linewidth]{figures/weights/q=100c=30.png}
\end{tabular}
\caption{\textbf{CFM Weight Visualization.} Horizontal axis shows different channels of the weight, vertical
axis shows quality. Quality levels shown are Top: 10, Middle: 50, Bottom: 100.}
\label{fig:weight_vis}
\end{minipage}
\hspace{0.01\linewidth}
\begin{minipage}{0.49\linewidth}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.33\linewidth]{figures/activations/q=10c=0.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=10c=15.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=10c=30.png} \\
\includegraphics[width=0.33\linewidth]{figures/activations/q=50c=0.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=50c=15.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=50c=30.png} \\
\includegraphics[width=0.33\linewidth]{figures/activations/q=100c=0.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=100c=15.png} & \includegraphics[width=0.33\linewidth]{figures/activations/q=100c=30.png}
\end{tabular}
\caption{\textbf{Images Which Maximally Activate CFM Weights.} Horizontal axis shows
different channels from the weight, vertical axis shows quality. Quality levels shown are Top: 10, Middle: 50, Bottom: 100.}
\label{fig:max_activation}
\end{minipage}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{plots/tsne.pdf}
\caption{\textbf{Embeddings for Different CFM Layers.} 3 channels are taken from each embedding, color shows JPEG quality
setting that produced the input quantization matrix. Circled
points indicate quantization matrices that were seen during training.}
\label{fig:quality_tsne}
\end{figure}
CFM layers are both our largest departure
from a vanilla CNN and also quite important to learning quality invariant features, so it is a natural result to try to visualize
their operation. In Figure \ref{fig:weight_vis}, we
compute the final $8 \times 8$ convolution weight for different quality levels. The quality levels, on the vertical axis, are
10, 50, and 100. The horizontal axis shows three different channels from the weight. What we see makes intuitive sense:
the filters in different channels have different patterns, but for the same channel, the pattern is roughly the same as the
quality increases. Furthermore, the filter response becomes smaller as the quality increases since the filters have
to do less ``work'' to correct a high quality JPEG.
Next we visualize compression artifacts learned by the weight. To do this we find the image that maximally activates a single channel of the
CFM weight. The result
of this is shown in Figure \ref{fig:max_activation}. Again the horizontal axis shows different channels of the weight and the vertical axis
shows quality levels 10, 50, and 100. The result shows clear images of
JPEG artifacts. At quality 10, the local blocking artifacts are extremely prominant. By
quality 50, the blocking artifacts are suppressed, while structural artifacts remain. The qualtiy 100 images are almost untouched, leaving only the
input noise pattern. It makes sense that
quality 100 filters are only minmally activated since there is not much correction to do on a quality 100 JPEG. Note that we only show Y
channel response for this figure and that Figures \ref{fig:weight_vis} and \ref{fig:max_activation} use the same channels from the same
layer.
Finally we examine the manifold structure of the CFM. We claim in Section 3.1 (and the name implies) that the CFM learns a smooth manifold
of filters through quantization space. If this is true, then a quality 25 quantization matrix should generate a weight halfway inbetween
a qualty 20 and a quality 30 one. To show that this happens, we generate weights
for all 101 quanitzation matrices (0 to 100 inclusive) and then compute t-SNE
embeddings to reduce the dimensionality to 2. We plot 3 channels from the weight embeddings with the quality level that was used to
generate the weight given
as the color of the point. This plot is shown in Figure \ref{fig:quality_tsne}. What see is a smooth line through the space starting from
dark (low quality) to bright (high quality) showing that the CFM has not only separated the different quality levels but has
ordered them as well.
Futhermore we see that the low quality filters are separated in space, indicating that they are quite different (and
perform different functions), a property that is important for effective neural networks. As the quality increases and the problem becomes
easier, the filters tend to converge on a single point where they are all doing very little to correct the image.
\subsection{Frequency Domain Analysis}
In this section we show results in the DCT frequency domain. A well known phenomenon of JPEG compression is the removal
of high frequency information. To check how well our model restores this information, we take the Y channel from several
images and show the colormapped DCT of the original image, the JPEG at quality 10, the image as restored by our regression
model, and the image restored by our GAN model. Next, for each image, we plot the probability that each of the 15 spatial
frequencies in a DCT block are set (\textit{e}.\textit{g}., has a magnitude greater than 0). This is shown in Figure \ref{fig:freq}. While our regression model is able to fill in
high frequencies, our GAN model nearly matches the original images in terms of frequency saturation. Additionally since our
network operates in the DCT domain, these outputs serve as an interesting qualitative result.
\begin{figure}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Original & Plot \\
\includegraphics[width=0.3\linewidth]{figures/frequencies/woman/womanhat.png} & \includegraphics[width=0.7\linewidth]{plots/frequencies_woman.pdf}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccc}
DCT & JPEG Q=10 & Regression & GAN \\
\includegraphics[width=0.25\linewidth]{figures/frequencies/woman/original.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/woman/compressed.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/woman/regression.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/woman/gan.png}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Original & Plot \\
\includegraphics[width=0.5\linewidth]{figures/frequencies/bikes/bikes.png} & \includegraphics[width=0.5\linewidth]{plots/frequencies_bikes.pdf}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccc}
DCT & JPEG Q=10 & Regression & GAN \\
\includegraphics[width=0.25\linewidth]{figures/frequencies/bikes/original.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/bikes/compressed.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/bikes/regression.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/bikes/gan.png}
\end{tabular}
}
\caption{Frequency domain results 1/2.}
\label{fig:freq}
\end{figure}
\begin{figure}
\ContinuedFloat
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Original & Plot \\
\includegraphics[width=0.3\linewidth]{figures/frequencies/lighthouse/lighthouse.png} & \includegraphics[width=0.7\linewidth]{plots/frequencies_lighthouse.pdf}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccc}
DCT & JPEG Q=10 & Regression & GAN \\
\includegraphics[width=0.25\linewidth]{figures/frequencies/lighthouse/original.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/lighthouse/compressed.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/lighthouse/regression.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/lighthouse/gan.png}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cc}
Original & Plot \\
\includegraphics[width=0.5\linewidth]{figures/frequencies/parrots/parrots.png} & \includegraphics[width=0.5\linewidth]{plots/frequencies_parrots.pdf}
\end{tabular}
}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccc}
DCT & JPEG Q=10 & Regression & GAN \\
\includegraphics[width=0.25\linewidth]{figures/frequencies/parrots/original.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/parrots/compressed.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/parrots/regression.png} & \includegraphics[width=0.25\linewidth]{figures/frequencies/parrots/gan.png}
\end{tabular}
}
\caption{Frequency domain results 2/2.}
\end{figure}
\section{JPEG Compression Algorithm}
Since the JPEG algorithm is core to the operation of our method, we describe it here in detail. Where the JPEG standard is ambiguous or lacking in guidance, we defer to the Independent JPEG Group's libjpeg software.
\subsubsection{Compression}
JPEG compression starts with an input image in RGB color space (for grayscale images the procedure is the same using only the Y channel equations) where each pixel uses the 8-bit unsigned integer represenation (\textit{e}.\textit{g}., the pixel value is an integer in [0, 255]). The image is then converted to the YCbCr color space using the full 8-bit
represenation (pixel values again in [0, 255], this is in contrast to the more common ITU-R BT.601 standard YCbCr color conversion) using the equations:
\begin{align}
Y = 2.99R + 0.587B + 0.114G \\
Cb = 128 - 0.168736R - 0.331264B + 0.5G \nonumber \\
Cr = 128 + 0.5R - 0.418688B - 0.081312G \nonumber
\end{align}
Since the DCT will be taken on non-overlapping $8 \times 8$ blocks, the image is then padded in both dimensions to a multiple of 8. Note that if the color channels will be chroma subsampled, as is usually the case, then the image must be padded to the scale factor of the smallest channel times 8 or the subsampled channel will not be an even number of blocks. In most cases, chroma subsampling will be by half, so the image must be padded to a multiple of 16, this size is referred to as the minimum coded unit (MCU), or macroblock size. The padding is always done by repeating the last pixel value on the right and bottom edges. The chroma channels can now be subsampled.
Next the channels are centered around zero by subtracing 128 from each pixel, yielding pixel values in [-128, 127]. Then the 2D Discrete type 2 DCT is take on each non-overlapping $8 \times 8$ block as follows:
\begin{align}
D_{i, j} = \frac{1}{4}C(i)C(j)\sum_{x=0}^7\sum_{y=0}^7P_{x,y}\cos\left[\frac{(2x+1)i\pi}{16}\right]\cos\left[\frac{(2y+1)j\pi}{16}\right] \\
C(u) = \left\{\begin{array}{lr}
\frac{1}{\sqrt{2}} & u = 0 \\
1 & \text{otherwise}
\end{array}\right. \nonumber
\end{align}
Where $D_{i,j}$ gives the coefficient for frequency $i, j$, and $P_{x,y}$ gives the pixel value for image plane $P$ at position pixel position $x,y$. Note that $C(u)$ is a scale factor that ensures the basis is orthonormal.
The DCT coefficients can now be quantized. This follows the same procedure for the Y and color channels but with different quanitzation tables. We encourage readers to refer to the libjpeg software for details on how the quantization tables are computed given the scalar quality factor, an integer in [0, 100] (this is not a standardized process). Given the quantization tables $Q_Y$ and $Q_C$, the quanized coeffcients of each block are computed as:
\begin{align}
Y'_{i, j} = \text{truncate}\left[\frac{Y_{i, j}}{Q_{Y_{i,j}}}\right] \\
Cb'_{i, j} = \text{truncate}\left[\frac{Cb_{i, j}}{Q_{C_{i,j}}}\right] \nonumber \\
Cr'_{i, j} = \text{truncate}\left[\frac{Cr_{i, j}}{Q_{C_{i,j}}}\right] \nonumber
\label{eq:quant}
\end{align}
\
The quantized coefficients for each block are then vectorized (flattened) using a zig-zag ordering (see Figure \ref{fig:zig}) that is designed to place high frequencies further towards the end of the vectors. Given that high frequencies have lower magnitude and are more heavily quanitized, this usually creates a run of zeros at the end of each vector. The vectors are then compressed using run-length encoding on this final run of zeros (information prior to the final run is not run-length encoded.). The run-length encoded vectors are then entropy coded using either huffman coding or arithmetic coding and then written to the JPEG file along with associated metadata (EXIF tags), quantization tables, and huffman coding tables.
\begin{figure}
\centering
\begin{minipage}{0.4\linewidth}
\resizebox{\columnwidth}{!}{
\includegraphics{figures/zigzag.pdf}
}
\end{minipage}
\caption{\textbf{Zigzag Ordering}}
\label{fig:zig}
\end{figure}
\subsubsection{Decompression}
The decompression algorithm largely follows the reverse procedure of the compression algorithm. After reading the raw array data, huffman tables, and quantization tables, the entropy coding, run-length coding, and zig-zag ordering is reversed. We reiterate here that the JPEG file does not store a scalar quality from which the decompressor is expected to derive a quanitzation table, the decompressor reads the quanitzation table from the JPEG file and uses it directly, allowing any software to correctly decode JPEG files that were not written by it.
Next, the $8 \times 8$ blocks are scaled using the quantization table:
\begin{align}
Y_{i, j} = Y'_{i, j}Q_{Y_{i,j}} \\
Cb_{i, j} = Cb'_{i, j}Q_{C_{i,j}} \nonumber \\
Cr_{i, j} = Cr'_{i, j}Q_{C_{i,j}} \nonumber
\end{align}
There are a few things to note here. First, if dividing by the quantization table entry during compression (Equation \ref{eq:quant}) resulted in a fractional part (the result was not an integer), that fractional part was lost during truncation and the scaling here will recover an integer near to the true coefficient (how close it gets depends on the magnitude quantization table entry). Next, if the division in Equation \ref{eq:quant} resulted in a number in [0, 1), then that coeffient would be truncated to zero and is lost forever (it remains zero after this scaling process). This is the \textit{only} source of loss in JPEG compression, however it allows for the result to fit into integers instead of floating point numbers, and it creates larger runs of zeros which leads to significantly larger compression ratios.
Next, the DCT process for each block is reversed using the 2D Discrete type 3 DCT:
\begin{align}
P_{x, y} = \frac{1}{4}\sum_{i=0}^7\sum_{j=0}^7C(i)C(j)D_{i,j}\cos\left[\frac{(2x+1)i\pi}{16}\right]\cos\left[\frac{(2y+1)j\pi}{16}\right] \\
C(u) = \left\{\begin{array}{lr}
\frac{1}{\sqrt{2}} & u = 0 \\
1 & \text{otherwise}
\end{array}\right. \nonumber
\end{align}
and the blocks are arranged in their correct spatial positions. The pixel values are uncentered (adding 128 to each pixel value), and the color channels are interpolated to their original size. Finally, the image is converted from YCbCr color space to RGB color space:
\begin{align}
R = Y + 1.402(Cr - 128) \\
G = Y - 0.344136(Cb - 128) -0.714136(Cr - 128) \nonumber \\
B = Y + 1.772(Cb - 128) \nonumber
\end{align}
and cropped to remove any block padding that was added during compression. The image is now ready for display.
\section{Qualitative Results}
In this section we show qualitative results on Quality 10 and 20 images for our regression network. These results
are in Figure \ref{fig:qual}.
\begin{figure}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{ccc}
JPEG Q=10 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/jpeg10.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/ours10.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/original.png} \\
JPEG Q=20 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/jpeg20.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/ours20.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/ocean/original.png} \\
JPEG Q=10 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/house/jpeg10.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/house/ours10.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/house/original.png} \\
JPEG Q=20 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/house/jpeg20.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/house/ours20.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/house/original.png}
\end{tabular}
}
\caption{Qualitative results 1/2. Live-1 images.}
\end{figure}
\begin{figure}
\ContinuedFloat
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{ccc}
JPEG Q=10 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/night/jpeg10.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/night/ours10.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/night/original.png} \\
JPEG Q=20 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/night/jpeg20.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/night/ours20.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/night/original.png} \\
JPEG Q=10 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/jpeg10.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/ours10.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/original.png} \\
JPEG Q=20 & Ours & Original \\
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/jpeg20.jpg} &
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/ours20.png} &
\includegraphics[width=0.3\linewidth]{figures/qual/bridge/original.png}
\end{tabular}
}
\caption{Qualitative results 2/2. ICB images.}
\label{fig:qual}
\end{figure}
\subsection{Runtime analysis}
We show the runtime inference performance of our network compared to the other networks we ran against. We measure FPS
on our NVIDIA Pascal GPU for 100 720p ($1280 \times 720$) frames and plot frames per second vs SSIM increase for quality 10
Live-1 images in Figure \ref{fig:fps}. We do not include ARCNN in this figure as the authors do not provide
GPU accelerated inference code. For grayscale
only models we only use single channel test images (we not not run the model three times as would be required to produce
an RGB output).
\begin{figure}
\centering
\begin{minipage}{0.7\linewidth}
\centering
\includegraphics[width=\linewidth]{plots/fps.pdf}
\end{minipage}
\caption{Increase in SSIM vs FPS. Our result is highlighted.}
\label{fig:fps}
\end{figure}
|
2,877,628,090,810 | arxiv | \section{Introduction}
The recent discovery of a short gamma-ray burst (GRB) associated with a gravitational-wave event
from a binary neutron star merger (e.g., \citealp{abbott2017}) has sparked renewed interest in these fascinating
phenomena. GRBs have been studied for decades and great progress has been made in understanding them
(see, e.g., \citealp{kumarandzhang} for a recent review). However, a general consensus on the prompt variable
gamma-ray generation mechanism in GRBs has not been reached. This issue continues to be of utmost
importance, especially as we enter the era of multi-messenger astronomy with the discovery of
more gravitational-wave signals. Reaching a complete picture of the particular systems that produce
a GRB in connection with gravitational-wave signals will only be reached once the issue of the GRB prompt emission is settled.
The non-thermal quality of the prompt GRB emission spectrum naturally suggests that the radiation is
produced by synchrotron emission from a power-law distribution of electrons (e.g., \citealp{katz1994, reesandmeszaros1994, sarietal1996}). However, the synchrotron mechanism faces the well-known
``line of death" problem (e.g., \citealt{Preece1998}) observed in many bursts (e.g., \citealt{Preece2000,Ghirlanda2002,Kaneko2006,Nava2011}).
Namely, in the majority of bursts, the measured low-energy spectral slope is significantly harder than expected for synchrotron in the fast cooling regime (which is the expected cooling regime in prompt GRBs). A fast cooling synchrotron slope would also over-produce optical and X-ray emission as compared with upper limits from observations during the prompt phase \citep{BP2014}.
Furthermore, in almost half of the cases, the observed low-energy spectral slope is even harder than the slow cooling spectral slope, where $F_{\nu}\propto \nu^{1/3}$. Nevertheless, the large energy coverage of the {\it Fermi} and {\it Swift} satellites have allowed for broad time-dependent spectral analysis of GRB prompt emission and a more comprehensive understanding in which the synchrotron mechanism might not be ruled out (e.g., \citealt{Guiriec2015,Burgess2017,Oganesyan2017b,Oganesyan2017a,2017Ravasio}).
GRB models in which the radiation is produced at or close to the photosphere, are a natural way of producing harder spectra (e.g., \citealt{Goodman1986,Thompson1994,Meszaros2000,Giannios2006,Peer2006,Beloborodov2010,Lazzati2010,Ryde2010,Giannios2012,Pe'er2012,Pe'er2015}).
These models may also include a significant synchrotron component \citep{BG2017} which due to the small radius of the emitting region, could become self-absorbed near the X-ray band and thus be consistent with the upper limits on the prompt optical and X-rays. The main concern however with all of these models has to do with reproducing the observed variability of GRB light-curves For $R\approx R_{\rm ph}\lesssim 10^{13}$cm, the typical dynamical time-scale is expected to be $t_{\rm dyn}=R/2c\Gamma^2\lesssim 0.01$ s, almost two orders of magnitude smaller than the observed variability (e.g., \citealt{Fishman1995,Norris1996,Quilligan2002}). The variability and temporal evolution in these models must then be provided by the central engine activity or the propagation of the jet through the stellar envelope. An additional concern has to do with the ``early steep decay" radiation observed at the end of the prompt phase in many GRBs (e.g, \citealt{Tagliaferri2005}), where the X-ray luminosity of the burst is seen to decline as a power-law with a decay index between 3 to 5. This decline is naturally accounted for when $t_{\rm dyn}\approx t_{\rm v}$ by a purely geometrical effect, high-latitude emission \citep{KP2000}. Of course, shallower declines are also possible, if the time-scale for the shutting down of the engine and/or the dissipation process are long enough \citep{BarniolDuran2009,Fan2005}\footnote{In fact, declines that are more rapid than the regular high-latitude emission are also possible if the prompt radiation is anisotropic in the co-moving frame \citep{BarniolDuran2016,BG2016}.}. Instead, for photospheric-like models, the high latitude emission decays too fast ($t_{\rm dyn}\ll t_{\rm GRB}$) and the early steep decline must be produced by the shutting down of the central engine, by the dissipation process, or by some combination of the two. As shown by \cite{BGM2017}, at least in the case of magnetar central engines, where more robust predictions can be made, this does not seem to occur naturally.
In this paper we focus on the synchrotron mechanism as the origin of the variable prompt $\gamma$-ray emission in GRBs.
We revisit it by obtaining several general constraints on any synchrotron GRB model based on typical observed
properties of the prompt emission of GRBs. We focus on ``marginally fast cooling" conditions \citep{Daigne2011} that can allow for a hard low-energy spectral slope, while maintaining high efficiency. By marginal fast cooling we mean that electrons cool on a timescale similar to the dynamical time at the source. This paper partially follows the work of \cite{pawananderin, pazandtsvi, BP2014}
and we recover many of their results. However, we make use of results of particle-in-cell (PIC) simulations
\citep{SironiSpitkovsky2014,Kagan2015,Guo2015,SKL2015,Sironi2015,Werner2016} to guide our efforts. In particular, as these studies show that a significant fraction of particles are expected to be accelerated to large energies in both shocks and reconnection, we can strongly limit the energy per particle at the emitting region, and strongly disfavour non-magnetic jets as the origin of the required emission. Furthermore, as PIC simulations of highly magnetized dissipation regions show that the emitting plasmoids may exhibit relativistic motions compared to the bulk frame of the jet (the jet co-moving frame), we relax the assumption made in previous studies that the emitters' Lorentz factor (LF) equals the bulk LF. Indeed we find that $\Gamma_{\rm em}>\Gamma_b$, i.e. some relativistic motion of the emitters, relative to the bulk frame, is necessary under these conditions.
The paper is organized as follows. In \S \ref{sec:General} we discuss general constraints on the required conditions for synchrotron to account for the prompt emission, in terms of the energy per particle, the cooling regime, implications on the jet composition and LF of the emitting material and the contribution of Inverse Compton. Motivated by our results in this section, we turn in \S \ref{sec:Reconnection} to discuss magnetic reconnection models. In these models, the allowed parameter range is further constrained and, given the large required values of the magnetization upstream of the emitting region, the particle spectra may become harder than $dN/d\gamma\propto \gamma^{-2}$. Nonetheless, self-consistent solutions that satisfy all the observational constraints are still available. In \S \ref{sec:Dis} we explore a variant of the basic model, and also provide a general discussion of the spectral shape and compare our results with previous studies. We present our conclusions in \S \ref{sec:conclusion}.
\section{General constraints}
\label{sec:General}
Let us consider some general constraints on the synchrotron emission mechanism in the context of the prompt emission of GRBs. We assume that the energy available per electron is $E$, and that the thermal (or random) LF of these electrons is given by
\begin{equation}
\gamma_{\rm e} \equiv \frac{E}{m_{\rm e} c^2} = 2 \times 10^3 E_{\rm GeV},
\label{gamma_e}
\end{equation}
where $E_{\rm GeV}$ is the energy in units GeV, $m_{\rm e}$ is the electron mass and $c$ is the speed of light.
We initially focus on two extreme situations. First, we consider the instantaneous injection of this energy to the electrons. Second, we consider a simple model in which electrons receive this energy continuously over a dynamical time: the ``slow heating" model. We note that when referring to the thermal (or random) LF of the electrons relative to the emitting region we will use $\gamma$, whereas the emitting region is assumed to be moving at a LF $\Gamma_{\rm em}$ towards the observer.
\subsection{Instantaneous injection}
\label{sec:inst}
We assume here that electrons are accelerated instantaneously by some mechanism (e.g., ``one-shot" shock acceleration). Both relativistic shocks and magnetic reconnection can accelerate particles ``instantaneously" (i.e. over a time-scale much shorter than the dynamical time) and over a broad range of LFs. In shocks, the power-law index of the particles' LF distribution is typically expected to be $p>2$ (e.g., \citealt{Heavens1988,Bednarz1998,Achterberg2001}). This leads to a distribution in which the minimal value of $\gamma$ dominates both the total energy stored in the electrons (which scales as $E_{\rm tot}\propto\gamma^{2-p}$) and their overall number (which scales as $N_{\rm tot}\propto\gamma^{1-p}$). The same holds true also for magnetic reconnection models with $\sigma \lesssim 10$ (e.g., \citealt{Cerutti2012,SironiSpitkovsky2014,Guo2014,Melzani2014}). Here we explore instantaneous acceleration with $p>2$. We turn to more gradual heating models in \S \ref{sec:slowheat} and to models with $p<2$ in \S \ref{sec:particledist}.
Under these assumptions, each electron gains an energy $E$ and therefore its instantaneously attained LF is
\begin{equation}
\gamma_{\rm i} = \gamma_{\rm e},
\label{gamma_i}
\end{equation}
given by equation (\ref{gamma_e}). For general acceleration mechanisms (i.e., not necessarily instantaneous), this will serve as an upper limit on the LF that electrons can achieve for a given energy per particle, $E$. Electrons with LF $\gamma_{\rm i}$ in a co-moving magnetic field strength $B_{\rm em}$ radiate via the synchrotron process at a characteristic energy given by
\begin{equation}
\nu_{\rm p} = \frac{e B_{\rm em} \gamma_{\rm i}^2 \Gamma_{\rm em}}{2 \pi m_{\rm e} c},
\end{equation}
Observationally, the peak of the GRB in the source frame is approximately $\nu_{\rm p} \sim 300$ keV (e.g., \citealt{Preece2000,Kaneko2006,Nava2011}). Furthermore compactness arguments show that $\Gamma_{\rm em}\geq 100$ (e.g, \citealt{Fenimore1993,Woods1995,Lithwick2001}). With these constraints, the magnetic field is
\begin{equation}
B_{\rm em} \approx \frac{2.6 \times 10^{11} \nu_{\rm p,5.5}}{\gamma_{\rm i}^2 \Gamma_{\rm em,2}} \approx \frac{(6 \times 10^4 \, {\rm G}) \, \nu_{\rm p,5.5}}{\Gamma_{\rm em,2} E_{\rm GeV}^2},
\label{B_field}
\end{equation}
where we made use of equation (\ref{gamma_i}) and $\nu_{\rm p,5.5}$ is the (source frame) peak frequency in units of 300 keV. We have adopted here the usual convention ($Q_n = Q / (10^n {\rm cgs})$).
The (source frame) variability time of a single pulse in the GRB prompt emission light curve is of the order of $t_{\rm v} \sim 0.5$ s (e.g., \citealt{Fishman1995,Norris1996,Quilligan2002}). The cooling LF is defined as the LF for which the synchrotron cooling time is equal to the dynamical time. Here, the dynamical time in the co-moving frame is $t_{\rm v}'\sim t_{\rm v} \Gamma_{\rm em}$ and thus
\begin{eqnarray}
\gamma_{\rm c} &=& \frac{6 \pi m_{\rm e} c}{\sigma_{\rm T} B_{\rm em}^2 t_{\rm v} \Gamma_{\rm em}}=4 \times 10^{-3}\, \frac{\Gamma_{\rm em,2} E_{\rm GeV}^4}{t_{\rm v,0.5} \, \nu_{\rm p,5.5}^2 }.
\label{cooling_LF}
\end{eqnarray}
Comparing $\gamma_{\rm c}$ with $\gamma_{\rm i}$ we find
\begin{equation}
\frac{\gamma_{\rm c}}{\gamma_{\rm i}} \approx 2\times 10^{-6}\frac{\Gamma_{\rm em,2} E_{\rm GeV}^3}{t_{\rm v,0.5} \, \nu_{\rm p,5.5}^2 }.
\end{equation}
Synchrotron models for prompt GRBs require that $\gamma_{\rm c}/\gamma_{\rm i}\approx 1$. This is because $\gamma_{\rm i}\gtrsim \gamma_{\rm c}$ is needed in order to account for the large observed efficiency of GRBs (e.g., \citealt{Fan2006,Beniamini2015,Beniamini2016}), while $\gamma_{\rm c} \gtrsim \gamma_{\rm i}$ is needed in order to avoid a low-energy spectral slope that is too soft as compared with observations, i.e the ``line of death" problem (e.g., \citealt{Preece2000,Ghirlanda2002,Kaneko2006,Nava2011}), and an excess of X-ray and optical emission as compared with observational limits \citep{BP2014}. While this can be achieved for some choice of $\Gamma_{\rm em}, E_{\rm GeV}$ (see purple region in figure \ref{figgcgm} and also \citealt{Daigne2011,pazandtsvi}), it is not clear why this would be the case in most GRBs. This issue could be resolved in a slow heating scenario, since particles maintain their energy (and thus emitting frequency) over a long time and therefore the fast cooling spectrum and excess low-energy emission can be avoided.
Various studies have considered such continuous acceleration models in the past (e.g., \citealt{Ghisellini1999,gianniosandspruit2005,pawananderin,asano2009,Fan2010,Daigne2011,BP2014}; see also discussion in \S \ref{sec:compare}). In what follows, we reconsider this possibility in a slightly different context.
\subsection{Slow heating}
\label{sec:slowheat}
In the opposite extreme case, instead of being accelerated instantaneously, particles could be heated ``slowly" over a dynamical time. In this case, the heating rate is
\begin{equation}
\dot{\epsilon_{\rm h}} = \frac{E}{\Gamma_{\rm em} t_{\rm v}} \approx 3\times 10^{-5} \frac{E_{\rm GeV}}{\Gamma_{\rm em,2} \, t_{\rm v,0.5}} \mbox {erg s}^{-1}.
\label{eq:heatrate}
\end{equation}
The last expression assumes that all the particles are heated throughout the entirety of the dynamical time. In \S \ref{sec:shortheat} we explore the possibility of acceleration over a shorter timescale, in \S \ref{sec:intermediate} we explore the possibility of multiple acceleration episodes, and in \S \ref{sec:particledist}, we explore models with $p<2$, in which effectively only a selective fraction of particles are heated.
As they are heated, particles also cool via the synchrotron process with a cooling rate that is given by
\begin{equation}
\dot{\epsilon_{\rm c}} = \frac{1}{6 \pi} \sigma_{\rm T} c \gamma_{\rm s}^2 B_{\rm em}^2 \approx 10^{-15} \gamma_{\rm s}^2 B_{\rm em}^2
\approx \frac{6.7 \times 10^{7} \, \mbox {erg s}^{-1}}{\Gamma_{\rm em,2}^2 \gamma_{\rm s}^2},
\label{eq:coolrate}
\end{equation}
where $\gamma_{\rm s}$ is their electron LF. The last expression was obtained using equation (\ref{B_field}), therefore, it incorporates the constraint on the observed synchrotron peak energy. We now set the cooling rate equal to the heating rate $\dot{\epsilon_{\rm h}} = \dot{\epsilon_{\rm c}}$ and find
\begin{equation}
\gamma_{\rm s} \approx 1.5 \times 10^6 \frac{ t_{\rm v,0.5}^{1/2} \, \nu_{\rm p,5.5}}{\Gamma_{\rm em,2}^{1/2} E_{\rm GeV}^{1/2}}.
\label{gamma_s}
\end{equation}
Given that $\gamma_{\rm e}$ is the LF that corresponds to the energy available per electron (and also the LF attained in the instantaneous injection case, $\gamma_{\rm i}$), the {\it true} electron LF in the slow heating case cannot be larger than $\gamma_{\rm i}$ (see \S \ref{sec:inst}).
Since $\gamma_e\propto E$ while $\gamma_s\propto E^{-1/2}$, there is a minimum `transition' energy, $E_{\rm tr}$, for which $\gamma_s\leq \gamma_e$ and slow heating solutions become available. $E_{\rm tr}$ and its corresponding (maximal) LF, $\gamma_{\rm tr}$, are
\begin{eqnarray}
E_{\rm tr} &\approx& 83 \, {\rm GeV}\frac{t_{\rm v,0.5}^{1/3} \, \nu_{\rm p,5.5}^{2/3} }{\Gamma_{\rm em,2}^{1/3} }
\label{eq:Etr}
\end{eqnarray}
\begin{eqnarray}
\gamma_{\rm tr} &\approx& 1.7 \times 10^5 \frac{t_{\rm v,0.5}^{1/3} \, \nu_{\rm p,5.5}^{2/3}}{\Gamma_{\rm em,2}^{1/3}}\equiv\gamma_{\rm i}^{1/3}\gamma_{\rm s}^{2/3}, \label{eq:gtr}
\end{eqnarray}
which weakly depend on the observables and $\Gamma_{\rm em}$ (in \S \ref{sec:revisedlimits} we provide slightly revised versions of these equations, taking into account our constraints on $\Gamma_{\rm em}$, derived in \S \ref{sec:luminosity}). In figure \ref{fig1} we present the electrons LF in the instantaneous injection case (equation (\ref{gamma_e})) and in the slow heating case (equation (\ref{gamma_s})).
When $E=E_{\rm tr}$ the cooling LF, defined in equation (\ref{cooling_LF}), is $\gamma_{\rm c} = \gamma_{\rm tr}$ (see figure \ref{figgcgm}). The reason is that at $E_{\rm tr}$, by definition, $\gamma_{\rm s} = \gamma_{\rm i}$, and since equation (\ref{gamma_s}) both (i) incorporates the constraint on the observed peak synchrotron energy (equation (\ref{B_field})), and (ii) assumes that heating occurs in a dynamical time, then it follows that $\gamma_{\rm c} = \gamma_{\rm i}$ and thus $\gamma_{\rm c} = \gamma_{\rm s} = \gamma_{\rm tr}$, when $E=E_{\rm tr}$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{Diagram.png}
\end{center}
\caption{The typical LF of the electrons radiating at the observed peak synchrotron energy, $\gamma_{\rm peak}$, versus the available energy per electron, $E$. Both the instantaneous and the slow heating cases are shown. All solutions must satisfy $\gamma_{\rm peak}\leq\gamma_i$ or else they would require a LF larger than allowed by the available energy per electron. In particular, slow heating solutions are not available below $E_{\rm tr}$. Similarly, no self-consistent solutions are available for $\gamma_{\rm peak}>\gamma_{\rm s}$ (see \S \ref{sec:intermediate}). Below the slow heating line, electrons are fast cooling due to synchrotron radiation. Viable synchrotron models for the GRB emission lie close to the solid ``slow heating" line, where electrons are only marginally fast cooling. }
\label{fig1}
\end{figure}
\subsection{Intermediate regime}
\label{sec:intermediate}
Consider now the intermediate regime between instantaneous acceleration and continuous heating. In this regime the particles can be thought of as having undergone $f>1$ acceleration episodes, with each acceleration boosting them to a LF
\begin{equation}
\tilde{\gamma}_{\rm i}=\gamma_{\rm i}/f.
\end{equation} This allows for solutions characterized by a LF smaller than $\gamma_{\rm e}$, but without necessarily invoking continuous acceleration (see grey lines in figure \ref{figgcgm}).
The shorter time between consecutive acceleration episodes effectively reduces the available cooling time by a factor $f$ and thus increases $\gamma_{\rm c}$ by the same factor \citep{pawananderin,BP2014} as compared with equation (\ref{cooling_LF}). Below, we examine the allowed parameter space and implications corresponding to solutions with $f\geq 1$.
Consider first the case in which electrons are fast cooling. In this case the peak of the synchrotron spectrum is produced by electrons with $\tilde{\gamma}_{\rm i}$. Since $\gamma_{\rm c}\propto B_{\rm em}(\gamma_{\rm peak})^{-2}\propto \gamma_{\rm peak}^{4}$ and since when $\gamma_{\rm peak}=\gamma_{\rm s}$ we have $\gamma_{\rm c}=\gamma_{\rm s}$ (see \S \ref{sec:slowheat}), we can re-write equation (\ref{cooling_LF}) as a scaling relation $\gamma_{\rm c}=(\gamma_{\rm peak}/\gamma_{\rm s})^4 \gamma_{\rm s}$. It follows that
\begin{equation}
\frac{\gamma_{\rm c}}{\gamma_{\rm peak}}=\bigg(\frac{\gamma_{\rm peak}}{\gamma_{\rm s}}\bigg)^3\ll1.
\label{eq:gcgi}
\end{equation}
Therefore, solutions with $\gamma_{\rm peak}=\tilde{\gamma}_{\rm i}<\gamma_{\rm s}$ are strongly fast cooling, even when the acceleration is done in multiple episodes.
For $\tilde{\gamma}_i=\gamma_{\rm s}$, electrons radiating at the peak become slow cooling, i.e. $\tilde{\gamma}_i=\gamma_{\rm peak}=\gamma_{\rm s}=\gamma_{\rm c}$. At these conditions
equation (\ref{gamma_e}) no longer directly constrains the minimum injected LF, $\gamma_{\rm m}$, as electrons with $\gamma_{\rm c}$ carry more energy overall than those with $\gamma_{\rm m}$ (assuming the slope of the electrons' injected number spectrum, $dN/d\gamma\propto \gamma^{-p}$, is $p<3$). Therefore solutions with $\tilde{\gamma}_i=\gamma_{\rm s}$ can be either marginally fast or slow cooling. Nonetheless, strongly slow cooling solutions are disfavoured as the radiative efficiency of the GRB is reduced significantly at those conditions (the efficiency scales as $(\gamma_{\rm c}/\gamma_{\rm m})^{2-p}<1$), while GRB observations show that the radiative efficiency of GRBs must be large. Notice that solutions with $\tilde{\gamma}_i=\gamma_{\rm peak}>\gamma_{\rm s}$ are impossible. This is because: (a) No fast cooling solutions exist in this regime (if $\gamma_{\rm peak}>\gamma_{\rm s}$ then it is not possible to have $\gamma_{\rm c}<\gamma_{\rm peak}$, see equation (\ref{eq:gcgi})), (b) For slow cooling:
\begin{equation}
\tilde{\gamma}_i=\bigg( \frac{\gamma_{\rm c}}{\gamma_{\rm m}}\bigg)^{1-p}\gamma_{\rm c}<\gamma_{\rm c},
\end{equation}
while as shown above, slow cooling requires $\gamma_{\rm c}=\gamma_{\rm s}$. Combining these two relations we have $\tilde{\gamma}_i<\gamma_{\rm s}$ contrary to our initial assumption. Thus, no solutions exist with $\gamma_{\rm peak}>\gamma_{\rm s}$. In particular, this implies that $\gamma_{\rm tr}$ is the largest allowed value of the electrons' LF (see figure \ref{fig1}) and correspondingly $B_{\rm em}(\gamma_{\rm tr}) $ is the lowest allowed value of the magnetic field.
Figure \ref{figgcgm} depicts the value of $\gamma_{\rm c}/\gamma$ in the $\gamma-E$ plane. Allowed solutions correspond to $\gamma_{\rm peak}\leq\gamma_{\rm s}$, for which electrons are fast (or at best, marginally fast) cooling. We define here the marginally fast cooling regime as $0.1<\gamma_c/\tilde{\gamma}_i<1$ (shown as a purple region in figure \ref{figgcgm}). This is in accord with recent observational results suggesting a synchrotron cooling break with $\nu_{\rm p}/\nu_{\rm c}\approx 5-80$ \citep{Oganesyan2017b}. Since for $\tilde{\gamma}_i\gg\gamma_c$ the GRB spectrum is in strong contradiction with observations ('fast cooling line of death'), we focus on the {\it marginally fast cooling solutions} \citep{Daigne2011} with $\tilde{\gamma}_{\rm i}\approx\gamma_{\rm c}$, which implies $\gamma_{\rm peak}\approx\gamma_{\rm s}$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.39]{gcgmFC.png}
\end{center}
\caption{ $\log_{10}(\gamma_{\rm c}/\gamma)$ as implied by the association $\nu_{\rm p}=\nu(\mbox{max}(\gamma,\gamma_{\rm c}))$. Self consistent solutions are only available when $\nu_{\rm p}=\nu(\gamma)$, i.e. when electrons are fast cooling. Grey lines depict the injected LF of typical electrons, given that they are accelerated $f$ times within $t_v$. Marginally fast cooling solutions require $\gamma\approx\gamma_{\rm s}$. Below this line, the spectrum is in strong contradiction with observations. The purple shaded region depicts the allowed parameter space for marginally fast cooling synchrotron solutions (taken here as $0.1<\gamma_c/\tilde{\gamma}_i<1$). The upper limit on $E_{\rm GeV}$ arises from the requirement that IC is sub-dominant as compared with synchrotron (see \S \ref{sec:IC}).}
\label{figgcgm}
\end{figure}
\subsection{Implications for the jet's composition}
\label{sec:MatterorMag}
The initial energy per baryon in the jet, $\eta$, can be related to $E\Gamma_{\rm em}/(m_p c^2)$, leading to $\eta \approx 110 \Gamma_{\rm em,2} E_{\rm{ GeV}}$, and demonstrating that large initial amounts of energy per baryon are required (given that $E>E_{\rm tr}$); see equation (\ref{eq:Etr}). Notice that the lower limits become even larger, by a factor of $m_p/m_e$, for a pair dominated outflow. In \S \ref{sec:luminosity}, we show that the observed luminosity implies that the LF of the emitting material must be very large, $\Gamma_{\rm em} \approx 1.8\times 10^4E_{\rm{ GeV}}^{-1/3}$, resulting in even larger energies per baryon at the base of the jet $\eta \approx 2\times 10^4 E_{\rm{ GeV}}^{2/3}$.
At the emission region, the energy per particle in the co-moving frame depends on the nature of the jet (baryonic or Poynting flux). For a baryonic jet in which dissipation proceeds through internal shocks we expect $E=\epsilon_e g m_pc^2\lesssim$ GeV (where $\epsilon_e\leq 1$ is the fraction of the total energy that goes to accelerating electrons and $g\sim 2-3$ is a numerical factor that depends on the relative LF of the colliding shells). However, as shown above, this leads to strongly fast cooling conditions with $\gamma_{\rm c}/\gamma_{\rm i}\lesssim 10^{-5}$. This in turn translates to a spectral slope that is in stark contradiction with observations. The large required values of $E$, suggest that the jet must be magnetically dominated. Assuming that the magnetic energy can be efficiently dissipated at the emitting region, $E$ is related to the magnetization of the jet in the upstream of the radiation zone, $\sigma_{\rm up}=B_{\rm b}^2/4\rho' c^2$ (where $B_{\rm b}$ is the magnetic field, and $\rho'$ is the matter density, both measured in the bulk frame):
\begin{equation}
E\leq \epsilon_e \sigma_{\rm up} m_p c^2=0.17 \epsilon_{e,0.2}\sigma_{\rm up} \rm{ GeV}
\label{Esigma}
\end{equation}
where $\epsilon_{e,0.2}=\epsilon_e/0.2$. Assuming $\epsilon_e\approx 0.2$, this means that $\sigma_{\rm up}\gtrsim 100$ is required in order to have $E\gtrsim 20$ GeV, as needed to account for marginally fast cooling solutions (see equation (\ref{eq:Etr}) and also equation (\ref{eq:Etrplas}) below).
Therefore, marginally fast cooling synchrotron implies jets that have both a very large amount of energy per baryon at their base, {\it and} just before the emission zone.
\subsection{Observed Luminosity - Determining the Lorentz factor of the emitting material}
\label{sec:luminosity}
The observed isotropic equivalent luminosity of GRBs, $L_{\rm rad}\approx 10^{52} \mbox{erg s}^{-1}$ introduces a constraint on the Lorentz factor of the emitting material, $\Gamma_{\rm em}$. We focus here on emitting regions that are quasi spherical in the co-moving frame. In appendix \ref{sec:planar} we show that the conclusion regarding the LF of the emitting material holds also for planar geometry, as applicable for instance to internal shock models.
Consider a spherical emitting region in the co-moving frame. In its own frame, the emitting region grows with time at a speed $v_g<c$. The region grows over the dynamical time-scale, which is observationally related to the variability time-scale $t_{\rm v}'$. In marginally fast cooling conditions, the radiated energy, is proportional to the energy of electrons in this volume. As usual, we denote by $\epsilon_e,\epsilon_B$ the fractions of the dissipated energy deposited in electrons and magnetic fields respectively. In the co-moving frame, the emission lasts over a period $t_{\rm v}'=\Gamma_{\rm em}t_{\rm v}$. The observed luminosity is then
\begin{equation}
L_{\rm obs}=\Gamma_{\rm em}^4 L'=\Gamma_{\rm em}^4 e_e'\frac{4\pi t_{\rm v}'^3v_g^3}{3t_{\rm v}'} =\Gamma_{\rm em}^6 \frac{\epsilon_e}{\epsilon_B}\frac{4\pi}{3}\frac{B_{\rm em}^2}{4\pi} t_{\rm v}^2v_g^3,
\end{equation}
where $e_e'\equiv \frac{\epsilon_e}{\epsilon_B}\frac{B_{\rm em}^2}{4\pi}$ is the co-moving energy density in electrons.
Using the magnetic field implied by the condition of marginally fast cooling $B_{\rm em}=B_{\rm em}(\gamma=\gamma_{\rm s})$, we get an estimate of the Lorentz Factor:
\begin{equation}
\Gamma_{\rm em}=1.5\times 10^4 \bigg(\frac{\epsilon_B}{\epsilon_e}\bigg)^{1/6}v_{g,.3c}^{-1/2}\frac{\nu_{\rm p,5.5}^{1/3} L_{52}^{1/6}}{E_{\rm GeV}^{1/3}},
\label{eq:Gammaspher}
\end{equation}
where $v_{g,.3c}\equiv v_g/0.3c$. Interestingly the dependence on the observed parameters, and on $\epsilon_B/\epsilon_e$ is very weak. Efficiency consideration imply $10^{-2}<\epsilon_B/\epsilon_e<10$ ($\epsilon_e$ has to be large enough to tap a significant of energy from the electrons, while $\epsilon_B$ cannot be too small in order to avoid most of this energy being deposited in an unobserved SSC component, see equation (\ref{eq:upperE}) in \S \ref{sec:IC} and \citealp{pazandtsvi}). We find that even for weakly magnetized jets, unless $E_{\rm GeV}>10^3$, the LF of the emitting material is $\Gamma_{\rm em}\gtrsim 700$, which is incompatible with estimates of the jet's bulk LF of typical bursts, as obtained for instance by early afterglow peaks (e.g., \citealp{Liang2010,Lu2012,Ghirlanda2012}). We note however that this derivation does not explicitly assume that $\Gamma_{\rm em}$ is the bulk Lorentz factor, $\Gamma_b$. In fact, this consideration shows that $\Gamma_{\rm em}\gg \Gamma_b$. Thus, {\it marginally fast cooling requires relativistic motions in the bulk frame.}
A lower limit on the emission radius arises from requiring that the emitting blob has sufficient time to grow, i.e., from requiring that the co-moving expansion time $t_{\rm exp}'= R/c\Gamma_b$ of the shell is larger than $t_{\rm v}'$, the lifetime of the emitting region. This leads to
\begin{eqnarray}
& R\geq 2 \Gamma_{\rm em} \Gamma_b c t_{\rm v} \nonumber \\ &=1.8 \times 10^{17} \frac{\Gamma_{b,2.5} \nu_{\rm p,5.5}^{1/3} L_{52}^{1/6}t_{\rm v,0.5}}{E_{\rm GeV}^{1/3}v_{g,.3c}^{1/2}}\bigg(\frac{\epsilon_B}{\epsilon_e}\bigg)^{1/6} \mbox{cm.}
\label{eq:rspher}
\end{eqnarray}
Using the value of $E_{\rm GeV}$ inferred by the requirement for marginally fast cooling solutions, the emitting radius in equation (\ref{eq:rspher}) is quite large but still marginally consistent with upper limits implied by the deceleration radius ($R\lesssim 10^{17}\mbox{cm}$).
In equation (\ref{eq:rspher}) we have assumed that $\Gamma_{\rm em}\gg \Gamma_b$, which effectively suggests relativistic motion in the bulk frame. If this is not the case, and instead $\Gamma_{\rm em}=\Gamma_b$, then the lower limit on the LF of the emitting material as given by equation (\ref{eq:Gammaspher}) would lead to a very large lower limit on the radius:
\begin{equation}
R\geq 2 \Gamma_{\rm em}^2 ct_{\rm v} =1.2\times 10^{19} \bigg(\frac{\epsilon_B}{\epsilon_e}\bigg)^{1/3}\frac{\nu_{\rm p,5.5}^{2/3}L_{52}^{1/3}}{E_{\rm GeV}^{2/3}v_{g,.3c}}t_{\rm v,0.5} \mbox{ cm}.
\end{equation}
As mentioned above, these values are inconsistent with upper limits implied by the deceleration radius (even when redshift corrections are applied). This consideration, together with the estimates of $\Gamma_{\rm em}$ above lead us to conclude that $\Gamma_{\rm em}\gg \Gamma_b$ is required, and therefore that relativistic motion in the bulk frame is inevitable unless the energy per particle is unrealistically large: $E>10^3$ GeV. We show below (\S \ref{sec:IC}) that values of $E\gtrsim700$GeV (in models involving an acceleration of a significant fraction of the electrons) are in fact inconsistent with the synchrotron models considered in this paper, as they would lead to excess IC emission and cooling. Furthermore, as will be shown in \S \ref{sec:relmotion}, in magnetic reconnection models there are two additional independent considerations that both reach the same outcome of relativistic motion in the bulk frame.
\subsection{Optical depth of the emitting region and Synchrotron self-Compton}
\label{sec:IC}
The conditions required to account for a balanced heating synchrotron model for GRBs' prompt emission discussed in \S \ref{sec:General} impose constraints on $\tau$, the optical depth of the emitting region to Thomson scatterings. $\tau$ can be related to the co-moving electron number density, $n'$, of the emitting region via \citep{Abramowicz1991}
\begin{equation}
\tau=\int \Gamma_{\rm em}(1-\beta_{\rm em})\sigma_T n' dr \approx \sigma_T n' \frac{v_g t_v \Gamma_{\rm em}}{2},
\end{equation}
where the integral is over the length of the emitting region and we have used $\Gamma_{\rm em} \gg 1$. Using the constraint for $\Gamma_{\rm em}$ in equation (\ref{eq:Gammaspher}), we use equation (\ref{gamma_s}) to obtain $\gamma_{\rm s}$ (equation (\ref{eq:gammasplasmoid}) below) and $B_{\rm em}=B(\gamma_{\rm s})$. $n'$ can be related to the energy density in electrons and the energy of an individual electron at any given time:
\begin{equation}
n'=\frac{\epsilon_e}{\epsilon_B}\frac{B_{\rm em}^2}{4\pi}\frac{1}{\gamma_{\rm s} m_e c^2}=2 \times 10^{-3} \frac{\epsilon_{e,0.2}^{11/12}E_{\rm GeV}^{7/3}L_{52}^{1/12}}{\epsilon_B^{11/12}\nu_{\rm p,5.5}^{17/6}t_{\rm v,0.5}^{2.5}v_{g,.3c}^{1/4}}\mbox{ cm}^{-3}.
\end{equation}
With this, the corresponding optical depth is then
\begin{equation}
\tau=6\times 10^{-14}\frac{\epsilon_{e,0.2}^{3/4}E_{\rm GeV}^{2}L_{52}^{1/4}v_{g,.3c}^{1/4}}{\epsilon_B^{3/4}\nu_{\rm p,5.5}^{5/2}t_{\rm v,0.5}^{1.5}} \ll 1.
\end{equation}
Clearly $\tau \ll 1$ for any reasonable value of $E_{\rm GeV}$. This leads to important conclusion, that if there is also a photospheric component present in GRBs' prompt emission, it cannot originate from the same location as the synchrotron emission in the balanced-heating models considered in this paper.
The optical depth can also be related to the Compton-$Y$ parameter, which in the Thomson regime is given by
\begin{equation}
\label{eq:Y}
Y_{\rm Th}\approx \tau \gamma_{\rm s}^2= 10^{-3} \frac{\epsilon_{e,0.2}^{11/12}E_{\rm GeV}^{4/3}L_{52}^{1/12}v_{g,.3c}^{3/4}}{\epsilon_B^{11/12}\nu_{\rm p,5.5}^{5/6}t_{\rm v,0.5}^{1/2}}\ll 1.
\end{equation}
Since typically $\gamma_{\rm s}h\nu_p>\Gamma_{\rm em}m_e c^2$, $Y_{\rm Th}$ may be further suppressed due to the Klein Nishina effect. Assuming marginally fast cooling, where the slope of $F_{\nu}$ below the peak is $\nu^{1/3}$ (see discussion in \S \ref{sec:specshape}), this suppression factor can be approximated by \citep{Ando2008}:
\begin{eqnarray}
\label{zetaKN}
&\zeta_{\rm KN} \approx \min\bigg[\bigg(\frac{\Gamma_{\rm em} m_e c^2}{\gamma_{\rm s} h \nu_p}\bigg)^{\frac{4}{3}},1\bigg]\\ \nonumber &=\min\bigg[0.16\frac{L_{52}^{1/3}\epsilon_B^{1/3}}{\epsilon_{e,0.2}^{1/3}\nu_{\rm p,5.5}^{2}t_{\rm v,0.5}^{2/3}v_{g,.3c}},1\bigg]
\end{eqnarray}
The final value of the Compton parameter is then $Y=Y_{\rm Th}\zeta_{\rm KN}$. The power of the synchrotron self-Compton and the cooling time of electrons due to IC, are both reduced by a factor $Y$ as compared with the synchrotron power and cooling time. Equations (\ref{eq:Y}) and (\ref{zetaKN}) imply that in order to keep the IC suppression small ($Y\lesssim 1$), and maintain high efficiency of the sub-MeV synchrotron component as required by observations (e.g, \citealt{Fan2006,Beniamini2015,Beniamini2016}), we must put an upper limit on the allowed value of\footnote{We assume in the following derivation that $\zeta_{KN}\leq 1$ applies. The dependence on parameters changes slightly if this is not the case, but the typical energies are similar.} $E$:
\begin{equation}
\label{eq:upperE}
E\lesssim 700 \frac{\epsilon_B^{7/16}\nu_{\rm p,5.5}^{17/8}t_{\rm v,0.5}^{5/4}v_{g,.3c}^{3/16}}{\epsilon_{e,0.2}^{7/16}L_{52}^{5/16}}\mbox{GeV.}
\end{equation}
When this condition is satisfied, the assumptions that synchrotron dominates the energy release rate and the observed emission are indeed self-consistent.
\subsection{Resulting constraints on the energy per particle and particles' LF}
\label{sec:revisedlimits}
Combining the value of $\gamma_{\rm s}$ as implied by equation (\ref{gamma_s}) with the estimate of $\Gamma_{\rm em}$ given by equation (\ref{eq:Gammaspher}), we find
\begin{equation}
\label{eq:gammasplasmoid}
\gamma_{\rm s}=1.1\times 10^5 t_{\rm v,0.5}^{1/2}\nu_{\rm p,5.5}^{5/6} L_{52}^{-1/12}E_{\rm GeV}^{-1/3}\epsilon_{e,0.2}^{1/12}\epsilon_B^{-1/12}v_{g,.3c}^{1/4}.
\end{equation}
Plugging this back into equations (\ref{eq:Etr}) and (\ref{eq:gtr}) we can rewrite the limits on the minimal allowed energy per particle, the corresponding electrons' (maximal) LF and the (minimum) allowed value for the magnetic field:
\begin{eqnarray}
E_{\rm tr}\approx 20 t_{\rm v,0.5}^{3/8}\nu_{\rm p,5.5}^{5/8}L_{52}^{-1/16}\epsilon_{e,0.2}^{1/16}\epsilon_B^{-1/16}v_{g,.3c}^{3/16} \mbox{GeV}
\label{eq:Etrplas}
\end{eqnarray}
\begin{eqnarray}
\gamma_{\rm tr}\approx 4\times 10^4 t_{\rm v,0.5}^{3/8}\nu_{\rm p,5.5}^{5/8}L_{52}^{-1/16}\epsilon_{e,0.2}^{1/16}\epsilon_B^{-1/16}v_{g,.3c}^{3/16}
\label{eq:gtrplas}
\end{eqnarray}
\begin{eqnarray}
B_{\rm em}\approx 2 t_{\rm v,0.5}^{-5/8}\nu_{\rm p,5.5}^{-3/8}L_{52}^{-1/16}\epsilon_{e,0.2}^{1/16}\epsilon_B^{-1/16}v_{g,.3c}^{3/16}\mbox{G}.
\label{eq:Btrplas}
\end{eqnarray}
Given the upper limit on the energy per particle implied by equation (\ref{eq:upperE}), we find that the energy per particle (and therefore also the typical LF and the magnetic field) has a range of roughly one and a half orders of magnitude, $20\lesssim E_{\rm GeV} \lesssim 700$, in which it can account for balanced heating synchrotron solutions, as discussed in this paper (the allowed parameter ranges are increased somewhat if we allow for $0.1<\gamma_{\rm c}/\gamma_{\rm s}<1$ as discussed in \S \ref{sec:intermediate}).
\section{Magnetic reconnection models}
\label{sec:Reconnection}
Motivated by the large energy per baryon required by continuous heating or marginally fast cooling models (see \S \ref{sec:MatterorMag}), we consider here specific constraints for magnetically dominated jets.
We focus here on emission from plasmoids, quasi-spherical regions of plasma that have strong magnetic fields and highly energetic particles. These are expected to be the main sources of emission in reconnection models. Such a model has been applied to account for the observed fast flares from blazar jets from active galactic nuclei (e.g., \citealt{Giannios2013,mariaetal2016}). Previous studies have suggested that these plasmoids could be moving relativistically compared to the bulk frame, as indeed implied by \S \ref{sec:luminosity}.
\cite{mariaetal2016} assume that the particles
are accelerated instantaneously once they are injected in the plasmoid.
Here, we also consider the possibility that the particles can undergo a slower
injection of energy, or ``heating", while they reside in the plasmoid during
major merger events. We assume here that a pulse in the GRB
light-curve arises from the merger of two plasmoids. Naturally,
the most luminous pulses will correspond to the merger of some of the
largest plasmoids. When two large plasmoids merge into one, the whole
structure relaxes to a new MHD equilibrium. This equilibrium
is reached after an Alfven crossing time, which up to a numerical
factor of order $\lesssim 3$ is of the order of the light-crossing time.
This is also the time during which the energy is released. Such a
merger could excite Alfvenic turbulence, which dissipates energy and heats up
the electrons until they reach a LF, $\gamma_{\rm s}$, where the synchrotron
cooling is balanced by heating (see, e.g., \citealt{Thompson1994}).
Magnetic reconnection simulations find that plasmoids grow in size with velocity $v_g\approx 0.3c$ (e.g., \citealt{Guo2015,Liu2015}). Furthermore, the conditions in the plasmoid can be approximated by $\epsilon_B=1, \epsilon_e=0.2$ \citep{Sironi2015}.
As will be shown below, these values constrain the degree of relativistic motion in the bulk frame as well as the
power-law distribution of accelerated electrons.
\subsection{Relativistic motion in the bulk frame}
\label{sec:relmotion}
We have seen in \S \ref{sec:luminosity} that the observed luminosity constrains the LF of the emitting material, $\Gamma_{\rm em}$. The large required values of $\Gamma_{\rm em}$, suggest relativistic motion of the emitting material in the bulk frame, with
\begin{equation}
\label{eq:Gammapdef}
\Gamma' \approx \Gamma_{\rm em}/\Gamma_b.
\end{equation}
Here we show that $\Gamma'$ is strongly restricted by the available Poynting luminosity of the jet, $L_B$.
The (isotropic equivalent) Poynting luminosity of the jet at a radius $R$ is given by:
\begin{equation}
\label{eq:LB}
L_B=4\pi R^2\Gamma_b^2\frac{B_{\rm b}^2}{4\pi}c
\end{equation}
where $B_b$ is the magnetic field in the bulk frame. Since this is the source of energy that feeds the emitters, we must have that $L_B \gtrsim \langle L_{\rm rad}\rangle$. Furthermore, since the `filling factor' of GRB light-curves is of order unity (i.e. there are no prolonged episodes where the luminosity dips below the values typically seen during the $\gamma$-ray pulses), we conclude that $\langle L_{\rm rad}\rangle \approx \frac{1}{2}L_{\rm rad}$. At the same time, efficiency considerations impose an upper limit on $L_B$. Requiring an efficiency $\gtrsim 0.1$, then $L_B\lesssim 5 L_{\rm rad}$. We define a dimensionless parameter $C_L\equiv L_B/L_{\rm rad}$. The considerations above imply $0.5\lesssim C_L \lesssim 5$.
As shown above, equation (\ref{eq:rspher}) puts a lower limit on the emitting radius for the case of a spherical geometry. At the same time, the radius cannot be much larger, as the prompt emission radius must be smaller than the deceleration radius. Furthermore, the same expression equals (rather than just providing a limit on) the emitting radius for the case of planar geometry (see equation (\ref{eq:rplanar})). We can thus once more define a dimensionless parameter $C_R\geq 1$, which represents the emission radius in units of $2\Gamma_{\rm em}\Gamma_b c t_{\rm v}$.
Putting all of this together and making use of equations (\ref{eq:Gammaspher}), (\ref{eq:Gammapdef}), (\ref{eq:LB}) we have
\begin{equation}
\Gamma'=4.6\frac{C_R^{1/2}}{C_L^{1/4}} \bigg(\frac{\epsilon_e}{\epsilon_B}\bigg)^{-1/2} \bigg(\frac{B_{\rm b}}{B_{\rm em}}\bigg)^{1/2} v_{g,.3c}^{-3/4}.
\label{eq:Gammaprime}
\end{equation}
Magnetic reconnection models suggest that plasmoids are accelerated in parallel to the original orientation of the reconnecting field lines. This implies that the magnetic field of the emitters as seen from the bulk is not relativistically boosted as compared with the field in the emitters' frame.
Pressure balance between the plasmoids and their surroundings then implies that $B_{\rm b}\approx B_{\rm em}$. Using this value, as well as $\epsilon_B=1, \epsilon_e=0.2$ we find $\Gamma'\approx 10 \frac{C_R^{1/2}}{C_L^{1/4}}$.
An additional constraint on $\Gamma'$ and $E$ can be obtained by considering the magnetization in the upstream of the emitting region. Since $E$ is the energy per electron in the emitters' frame, $\Gamma' E$ is the energy in the bulk frame and thus equation (\ref{Esigma}) is re-written as
\begin{eqnarray}
E \Gamma'=0.17 \bigg(\frac{\epsilon_e}{0.2}\bigg) \sigma_{\rm up} \rm{ GeV}
\label{Esigmaplas}
\end{eqnarray}
Motivated by results of analytic models and PIC simulations of magnetic reconnection, we relate $\Gamma'$ to the magnetization as $\Gamma'=\sigma_{\rm up}^n$, with $0\leq n\leq 1$. For $n=0$ this parametrization reduces back to the case of no relativistic motion in the bulk frame, while reconnection models suggest that $n$ may be as large as 0.5 \citep{Lyubarsky2005}. We consider some representative values for $n$ below. We now use equation (\ref{Esigmaplas}) to relate $\Gamma'$ to $E_{\rm GeV}$ and plug the results into equations (\ref{eq:Gammaspher}), (\ref{eq:Gammapdef}). Solving for $E_{\rm GeV},\Gamma'$ as functions of $\Gamma_{\rm b}$, this leads to
\begin{equation}
E_{\rm GeV}\!=\!(63^{6-6n}0.17^{6n} \epsilon_{e,0.2}^{7n-1}\epsilon_B^{1-n}L_{52}^{1-n}\nu_{p,5.5}^{2-2n}\Gamma_{\rm b,2.5}^{6n-6}v_{g,.3c}^{3n-3})^{1\over 4n+2},
\label{eq:EGeVsigma05}
\end{equation}
which reduces to
\begin{eqnarray}
\label{eq:EGeVfinal}
& E_{\rm GeV}\!=\!6 \epsilon_{e,0.2}^{5/8}\epsilon_B^{1/8}L_{52}^{1/8}\nu_{p,5.5}^{1/4}\Gamma_{\rm b,2.5}^{-3/4}v_{g,.3c}^{-3/8} \mbox{ for }n\!=\!1/2\\ \nonumber
&E_{\rm GeV}\!=\!200 \epsilon_{e,0.2}^{1/4}\epsilon_B^{1/4}L_{52}^{1/4}\nu_{p,5.5}^{1/2}\Gamma_{\rm b,2.5}^{-3/2}v_{g,.3c}^{-3/4} \mbox{ for }n\!=\!1/4
\end{eqnarray}
and therefore:
\begin{eqnarray}
\label{eq:Gammapfinal}
& \Gamma'\!=\!35 \epsilon_{e,0.2}^{-3/8}\epsilon_B^{1/8}L_{52}^{1/8}\nu_{p,5.5}^{1/4}\Gamma_{\rm b,2.5}^{-3/4}v_{g,.3c}^{-3/8} \mbox{ for }n\!=\!1/2\\ \nonumber
&\Gamma'\!=\! 10 \epsilon_{e,0.2}^{-1/4}\epsilon_B^{1/12}L_{52}^{1/12}\nu_{p,5.5}^{1/6}\Gamma_{\rm b,2.5}^{-1/2}v_{g,.3c}^{-1/4} \mbox{ for }n\!=\!1/4.
\end{eqnarray}
These results are consistent with the lower limits on $E_{\rm GeV}$ implied by the requirement on marginally fast solutions (\S \ref{sec:revisedlimits}) and with the upper limits implied by requirements on the IC cooling (\S \ref{sec:IC}). Interestingly, for $n=1/4$, we find $\Gamma'\approx 10$. These limits on $\Gamma'$ are consistent with the independent constraints imposed by equation (\ref{eq:Gammaprime}), as well as with expectations from relativistic turbulence or `mini-jets' models (e.g., \citealp{LB2003,Lyutikov2006,KN2009,Lazar2009,Giannios2009, zhangandzhang2014, BarniolDuran2016}) and with constraints on the variability time-scale \citep{BG2016}. These values lead to very large values of the upstream magnetization,
\begin{eqnarray}
\label{eq:sigmafinal}
& \sigma_{\rm up}\!=\!1100 \epsilon_{e,0.2}^{-3/4}\epsilon_B^{1/4}L_{52}^{1/4}\nu_{p,5.5}^{1/2}\Gamma_{\rm b,2.5}^{-3/2}v_{g,.3c}^{-3/4} \mbox{ for }n\!=\!1/2\\ \nonumber
&\sigma_{\rm up}\!=\! 1.1\times 10^4 \epsilon_{e,0.2}^{-1}\epsilon_B^{3/4}L_{52}^{3/4}\nu_{p,5.5}^{2/3}\Gamma_{\rm b,2.5}^{-2}v_{g,.3c}^{-1} \mbox{ for }n\!=\!1/4.
\end{eqnarray}
We explore the implication of these large values below.
\subsection{Particle energy distribution}
\label{sec:particledist}
So far our analysis assumes that the power-law distribution of the radiating particles is such that a characteristic Lorentz factor $\gamma_{\rm e}$ dominates both in terms of the total particles' energy {\it and} number. This is a common expectation in shocks where $p>2$. However, for the extreme magnetization $\sigma_{\rm up}$ inferred for the jet (see \S \ref{sec:MatterorMag}), this assumption is likely to break down. In this Section, we explore the implications of this decoupling.
PIC simulations of reconnection find that the slope of the electrons' LF distribution, $p$, depends sensitively on $\sigma_{\rm up}$ \citep{SironiSpitkovsky2014,Kagan2015,Guo2015,Werner2016}. These simulations find that for $\sigma_{\rm up}\gtrsim 10$, the spectra become hard, with $p<2$. For $1<p<2$, the number of electrons is dominated by the lowest LF electrons, $\gamma_{\rm min}$, while the total energy instead is dominated by the highest energy LF electrons, $\gamma_{\rm max}$. This means that the LF $\gamma_e$ associated with the energy per particle (equation (\ref{gamma_e})) is smaller than the LF of particles contributing to the peak of the emission, which for instantaneous acceleration is at $\gamma_i=\gamma_{\rm max}$. Assuming $\gamma_{\rm max}\gg \gamma_{\rm min}$, we can rewrite equation (\ref{gamma_i}) as
\begin{eqnarray}
&\gamma_{\rm e} m_e c^2\! =\!\frac{\int_{\gamma_{\rm min}}^{\gamma_{\rm max}}\frac{dN}{d\gamma}\gamma m_e c^2 d\gamma}{\int_{\gamma_{\rm min}}^{\gamma_{\rm max}}\frac{dN}{d\gamma}d\gamma} \! \rightarrow \! \gamma_e\!=\!\frac{p-1}{2-p}\bigg(\frac{\gamma_i}{\gamma_{\rm min}}\bigg)^{1-p}\! \gamma_i.
\end{eqnarray}
Assuming $\gamma_{\rm min}\approx 1$ as the most extreme case (motivated also by PIC simulations), we obtain
\begin{equation}
\gamma_i=\bigg( \frac{2-p}{p-1}\gamma_e\bigg)^{1\over 2-p}.
\label{eq:gammaip}
\end{equation}
As an example, for $p\approx 1.5$ (as found in simulations for $\sigma_{\rm up}\geq 50$), equation (\ref{eq:gammaip}) reduces to $\gamma_i=\gamma_e^2$. Since the energy is dominated by particles with $\gamma_i$ that are a small fraction of the total number of particles, they must be accelerated to larger energies as compared with the $p>2$ case (where $\gamma_i=\gamma_e$, see equation (\ref{gamma_i})) in order to achieve the same energy per particle. This also implies that in order to achieve a balance between heating and cooling rates, the heating rate of these particles is increased by $\gamma_i/\gamma_e$ as compared with equation (\ref{eq:heatrate}).
The result is:
\begin{equation}
\gamma_s=6.6\times 10^7 \bigg(\frac{2000(2-p)E_{\rm GeV}}{p-1}\bigg)^{1\over 2(p-2)}\frac{\nu_{\rm p,5.5}t_{\rm v,0.5}^{1/2}}{\Gamma_{\rm em,2}^{1/2}},
\end{equation}
which for $p=1.5$ simplifies to:
\begin{equation}
\gamma_s=3\times 10^4\frac{\nu_{\rm p,5.5}t_{\rm v,0.5}^{1/2}}{E_{\rm GeV}\Gamma_{\rm em,2}^{1/2}}.
\end{equation}
This leads to a decrease of the minimum allowed energy per particle ($E_{\rm tr}$) and the corresponding LF ($\gamma_{\rm tr}$) for balanced heating solutions. Their new values are (for $p=1.5$):
\begin{eqnarray}
& E_{\rm tr}=0.2 t_{\rm v,0.5}^{1/6}\nu_{\rm p,5.5}^{1/3} \Gamma_{\rm em,2}^{-1/6}\mbox{GeV} \nonumber \\
& \gamma_{\rm tr}=1.3\times 10^5 t_{\rm v,0.5}^{1/3}\nu_{\rm p,5.5}^{2/3} \Gamma_{\rm em,2}^{-1/3}.
\label{eq:gammatrp15}
\end{eqnarray}
The reduced values of $\gamma_{\rm s}$ (as compared with $p>2$) lead to stronger values of the magnetic field, and therefore reduce the estimate of $\Gamma_{\rm em}$ given by equation \ref{eq:Gammaspher} (for $p=1.5$):
\begin{equation}
\Gamma_{\rm em}=1.6\times 10^3 \frac{\nu_{\rm p,5.5}^{1/3} L_{52}^{1/6}\epsilon_B^{1/6}}{\epsilon_{e,0.2}^{1/6}E_{\rm GeV}^{2/3}v_{g,.3c}^{1/2}}.
\label{eq:Gammap15}
\end{equation}
Combining this with $E \Gamma'\propto \sigma_{\rm up}$ (equation \ref{Esigmaplas}), $\Gamma'=\sigma_{\rm up}^n$ and equation (\ref{eq:Gammapdef}) we obtain
\begin{equation}
E_{\rm GeV}=5^{3(1-n)\over n+2}0.17^{3n\over n+2}\epsilon_{e,0.2}^{7n-1 \over 2(n+2)}\epsilon_B^{1-n\over 2n+4}\Gamma_{\rm b,2.5}^{-3(1-n)\over n+2}\nu_{\rm p,5.5}^{1-n\over n+2}L_{52}^{1-n \over 2(n+2)}v_{g,.3c}^{3n-3 \over 2n+4},
\end{equation}
which reduces to
\begin{eqnarray}
\label{eq:EGeVp15}
& E_{\rm GeV}\!=\!0.9 \epsilon_{e,0.2}^{1/2}\epsilon_B^{1/10} L_{52}^{1/10}\nu_{p,5.5}^{1/5}\Gamma_{\rm b,2.5}^{-3/5}v_{g,.3c}^{-3/10} \mbox{ for }n\!=\!1/2 \nonumber \\
&E_{\rm GeV}\!=\!2.7 \epsilon_{e,0.2}^{1/6}\epsilon_B^{1/6}L_{52}^{1/6}\nu_{p,5.5}^{1/3}\Gamma_{\rm b,2.5}^{-1}v_{g,.3c}^{-1/2} \mbox{ for }n\!=\!1/4
\end{eqnarray}
and:
\begin{eqnarray}
\label{eq:Gammapp15}
& \Gamma'\!=\!5 \epsilon_{e,0.2}^{-0.5}\epsilon_B^{1/10}L_{52}^{1/10}\nu_{p,5.5}^{1/5}\Gamma_{\rm b,2.5}^{-3/5}v_{g,.3c}^{-3/10} \mbox{ for }n\!=\!1/2\\ \nonumber
&\Gamma'\!=\! 2.5 \epsilon_{e,0.2}^{-0.28}\epsilon_B^{1/18}L_{52}^{1/18}\nu_{p,5.5}^{1/9}\Gamma_{\rm b,2.5}^{-1/3}v_{g,.3c}^{-1/6} \mbox{ for }n\!=\!1/4.
\end{eqnarray}
These values are consistent with the condition $E_{\rm GeV}\geq E_{\rm tr}$. In addition, they demonstrate that even for $p\approx 1.5$, the plasmoids are expected to be moving at least mildly relativistically in the bulk frame. It is interesting to note that these values of $E,\Gamma',n$ correspond to $31 \lesssim \sigma_{\rm up}\lesssim 45$ which is indeed consistent with the value of $p\approx 1.5$ assumed here. One concern with this scenario however, is that due to the increased energy density implied for a given $E$ as compared with the case of $p>2$, the required $E$ in order for IC cooling to be sub-dominant to synchrotron (equation (\ref{eq:upperE})) is reduced.
Following the same procedure described in \S \ref{sec:IC} for $p>2$ (but using the revised values for $\gamma_s$, $\Gamma_{\rm em}$ appropriate for $p=1.5$) we find
\begin{equation}
E<2 \frac{\epsilon_B^{21/96}\nu_{\rm p,5.5}^{51/96}t_{\rm v,0.5}^{7/16}v_{g,.3c}^{3/32}}{\epsilon_{e,0.2}^{21/96}L_{52}^{5/32}}\mbox{GeV}.
\end{equation}
These values are considerably smaller than the equivalent limits for $p>2$ although still consistent with the energies implied by equation (\ref{eq:EGeVp15}), assuming $n\approx 1/2$.
An additional limit on the electrons' LFs, $\gamma_s$, is obtained by equating the Larmour acceleration time with the energy loss time due to synchrotron \citep{deJager1996}\footnote{A related limit arises from requiring that the size of the emitting region must be smaller than the Larmour radius of the highest energy particles. This consideration results in the same scaling for $\gamma_{\rm L}$, but reduced by a factor $(v_g/c)^{1/2}\approx 0.5$, and does not change the qualitative conclusion below.}:
\begin{equation}
\gamma_{\rm L}\approx 4\times 10^7B_{\rm em}^{-1/2}\approx 2.6\times 10^6 t_{\rm v,0.5}^{1/2}\nu_{\rm p,5.5}^{1/2} E_{\rm GeV}^{-1}.
\end{equation}
Since these values are more than an order of magnitude above $\gamma_{\rm tr}$ (given by equation (\ref{eq:gammatrp15})) and since balanced heating solutions have $\gamma_s<\gamma_{\rm tr}$, the Larmour limits are consistent with the picture presented here.
\section{Discussion}
\label{sec:Dis}
\subsection{Shorter heating times}
\label{sec:shortheat}
Consider a variant on the slow heating model presented in this paper where each particle undergoes heating for only a fraction $\alpha\leq 1$ of $t_{\rm v}$. This scenario implies that particles experience a balance between heating and cooling for a time $\alpha t_{\rm v}$, after which acceleration ceases and they spend the rest of the dynamical time in fast cooling conditions. Even though the particles maintain a balance between heating and cooling for only a small part of the dynamical time, the overall spectrum emitted by those particles will resemble a slow cooling slope rather than a fast cooling one. The reason is that the overall energy emitted by the particles is $E$, while at the end of their heating they have a LF $\gamma_{\rm s}$ and, as can be seen by figure \ref{fig1}, for slow heating solutions $\gamma_{\rm s}m_ec^2\ll E$. Thus, although the particles may spend a large amount of time in fast cooling conditions, only a small portion of their total emitted energy is released at this stage. These shorter lived `balanced heating' conditions can therefore satisfy the requirement on the low-energy spectral slope while maintaining a large efficiency. It is thus interesting to consider how it can affect the results presented in this paper.
In order to obtain the same energy per particle, $E$, the heating rate, $\dot{\epsilon_{\rm h}}$, is increased by $\alpha^{-1}$ as compared with equation (\ref{eq:heatrate}), while the cooling rate given by equation (\ref{eq:coolrate}) remains the same. The result is that $\gamma_{\rm s}$ is reduced by $\alpha^{1/2}$ and $E_{\rm tr},\gamma_{\rm tr}$ are both reduced by $\alpha^{1/3}$ as compared with their values for the slow heating case given by equations (\ref{eq:Etr}) and (\ref{eq:gtr}). Faster heating also reduces somewhat the constraint on $\Gamma_{\rm em}$ obtained by equation (\ref{eq:Gammaspher}): $\Gamma_{\rm em}\propto B_{\rm em}(\gamma_{\rm s})^{-1/3}\propto \gamma_{\rm s}^{-2/3}\propto \alpha^{1/3}$. At face value, it would seem that $\alpha \ll 1$ could thus mitigate the requirement on relativistic motion in the bulk frame. However, solutions with $\alpha \ll 1$ require very strong magnetic fields $B(\gamma_s)\propto \alpha^{-1}$, and as a result correspond to a huge Poynting luminosity. Assuming $B_{\rm b}=B_{\rm em}$ (see \ref{sec:relmotion}) and taking the minimum allowed radius as implied by variability (equation (\ref{eq:rspher}); results become more constraining for larger radii) we obtain that $L_B\propto R^2 B^2 \propto \alpha^{-4/3}$ while $L_{\rm rad}$ is unchanged. Therefore, balanced heating with small values of $\alpha$ will lead to extremely inefficient bursts with $L_B \gg L_{\rm rad}$. Assuming $L_B<5 L_{\rm rad}$, we find that $\alpha \gtrsim 0.3$. We conclude that balanced heating solutions require a heating time that is not much smaller than the dynamical time.
\subsection{Spectral shape}
\label{sec:specshape}
We have discussed in this work the characteristic energies of the particles that contribute to the peak of the $\gamma$-ray emission regardless of the specific particle acceleration mechanism. The considerations that we made here are generic requirements on a synchrotron signal such that it would not result in a strongly fast cooling spectrum (which will be in strong contradiction with observations). The exact shape of the spectrum however, could still be affected by the details of the particle acceleration mechanism. The effects of slow heating acceleration mechanisms on the particle spectrum have been discussed in the literature by various authors (e.g., \citealt{asano2009,Brunetti2016,Xu2017}). We refer the reader to those papers for a more in depth discussion of how acceleration can modify the particle spectrum.
A major consideration regarding the acceleration mechanism, has to do with the rate at which particles of different energies are energized. Since the energy loss rate via synchrotron (as well as in IC if the Klein-Nishina effect can be neglected) scales as $P_{\rm cool}\propto \gamma^2$, this implies that in order for energy balance to hold for particles of all energies, one should also have $P_{\rm heat}\propto \gamma^2$. This condition may be difficult to obtain in practice. This is because in Fermi type II acceleration, the energy gain rate scales as $P_{\rm heat}\propto D/\gamma$, where $D$ is the diffusion coefficient. $D$ is then expected to scale as $\gamma^n$ with $n=2$ for small scale MHD turbulence, $n=5/3$ for Kolmogorov turbulence, or at the limit of fastest acceleration, or $n=1$ for strong turbulence (also known as the Bohm limit). The energy gain rate is therefore expected to be constant or decreasing with $\gamma$ and either way is softer (as a function of $\gamma$) than $P_{\rm cool}$. The implication is that if energy balance is maintained for particles with $\gamma\approx \gamma_s$, then particles with an initial LF $\gamma<\gamma_{\rm s}$ will heat faster than they cool, while particles with $\gamma>\gamma_{\rm s}$ cool faster than they heat. This would lead to an eventual bunching up of particles around $\gamma_s$.
Furthermore, this demonstrates that in order to initially accelerate particles to $\gamma>\gamma_{\rm s}$, there must, in fact be two distinct acceleration processes taking place. The first, creating the $dN/d\gamma\propto \gamma^{-p}$ distribution on a short time-scale, and the second, slowly heating the electrons such that cooling balances acceleration at $\gamma=\gamma_{\rm s}.$ If the initial acceleration process is not present, no particles will reach $\gamma>\gamma_{\rm s}$, and the spectrum would cut-off sharply beyond the sub-MeV peak, contrary to observations.
As particles above $\gamma_{\rm s}$ (assuming that such particles exist, i.e., that $p>2$) are essentially in fast cooling conditions, their cooling would result in a spectrum $F_{\nu}\propto \nu^{-p/2}$ for $\nu>\nu_{\rm syn}(\gamma_s)$ (where $p$ is the slope of the initial particle spectra). This is similar to the spectrum from instantaneously accelerated electrons radiating at $\nu>\max(\nu_m,\nu_c)$. Particles with $\gamma\ll \gamma_{\rm s}$ will heat up to $\gamma_{\rm s}$ over a dynamical time-scale. Since the heating rate is a decreasing (or at most flat) function of the electrons LF, electrons starting at $\gamma\ll \gamma_{\rm s}$ will spend a short time (compared to $t_{\rm v}$) at their initial LFs. As a result most of the emission from these electrons will take place after they reach $\gamma\approx \gamma_{\rm s}$. Therefore, at the synchrotron frequencies corresponding to $\gamma\approx \gamma_{\rm s}$, the spectrum will be dominated by the classical $F_{\nu}\propto \nu^{1/3}$ contribution of electrons at $\gamma\approx \gamma_{\rm s}$. As $\gamma$ approaches $\gamma_{\rm s}$ the ratio of the heating to the dynamical time becomes closer to the unity, and the emitted spectrum becomes slightly softer than $\nu^{1/3}$. The softness of the spectrum in this range is always limited however by $F_{\nu}\propto \nu^{1-p\over 2}$ which is the slow cooling spectrum emitted by a stationary (unheated) distribution of the type $dN/d\gamma \propto \gamma^{-p}$.
\subsection{Comparison to other studies}
\label{sec:compare}
We have focused in this work on the required conditions at the emitting region needed to account for marginally fast cooling of the $\gamma$-ray emitting electrons via synchrotron. This topic has been studied in the literature by different authors (e.g., \citealt{Ghisellini1999,pawananderin,Daigne2011,pazandtsvi,BP2014}).
These studies can be divided into three groups. First, e.g., \cite{Ghisellini1999,Peer2006,Giannios2008} discussed continuous heating in the context of photospheric models, where the electrons are sub or at most mildly relativistic. At these conditions, synchrotron photons are self absorbed and the emission is dominated by multiple IC scatterings of the synchrotron seed. The second group of studies consider general synchrotron models \citep{pawananderin,pazandtsvi,BP2014}. In these, if the acceleration is instantaneous, the resulting parameter space is characterized by a large radius ($R\approx 6\times 10^{16}$cm), a large Lorentz factor of the emitting material ($\Gamma_{\rm em}\approx2000$), a large electrons' LF ($\gamma_e\approx 10^5$), and weak magnetic fields ($B_{\rm em}\approx$ few $G$). Introducing re-acceleration, increases the required magnetic field and therefore reduces somewhat $\gamma_e$. These results are consistent with the parameter ranges found in the current study. Furthermore, since the required number of electrons is significantly decreased in this case (since it scales as $f^{-1}$ and $f\gg1$), the implied energy per electron associated with these solutions is larger than for the instantaneous case. This, as well, is consistent with the findings reported here (although the formulation of the problem was quite different in the earlier studies).
The third group of studies \citep{Daigne2011} and section 4 of \cite{pazandtsvi}, also considered synchrotron solutions, but specifically in the context of internal shocks. In this case, since the jet is baryonic and the energy per electrons is $\lesssim 0.2$GeV (same as equation (\ref{Esigma}) but for $\sigma_{\rm up}\rightarrow1$), slow heating conditions are not possible unless only a small fraction of the electrons $\xi \lesssim 10^{-2}$ are accelerated to relativistic energies by the shock. Since these conditions are not supported by PIC simulations of acceleration in shocks which demonstrate that practically all electrons undergo heating behind the shock front \citep{SKL2015} (as well as in reconnection, see \citealt{Sironi2015}), we have assumed that $\xi=1$ in the current work.
\section{Conclusions}
\label{sec:conclusion}
The synchrotron mechanism has been widely discussed for the prompt phase of GRBs. Previous studies have shown that the physical conditions at the emitting region typically lead to the electrons cooling via synchrotron on a very short time-scale (as compared with the dynamical one). This results in a low-energy spectral slope that is in strong contention with observations. This problem may be overcome if the electrons' energy losses due to synchrotron are balanced by a continuous source of heating, leading to `marginally fast cooling' electrons ($\nu_c \approx \nu_m$). Here we revisit the model and derive some general constrains on any synchrotron model based on basic observed properties of GRBs: the characteristic sub-MeV energy where the emission peaks, the hardness of the slope below the peak, the characteristic luminosity and the variability time-scale of GRBs.
If the peak emission is dominated by the majority of the particles (as expected in shock models or low $\sigma$ reconnection), the emitting region has to be characterized by $\Gamma_{\rm em} \gtrsim 3000$, well in excess of what is inferred for the bulk jet motion from afterglow modeling. Several independent constraints indicate that emitters have to be characterized by fast motion ($\Gamma'\sim10$) in the rest frame of the jet and that, at the emitting region, the jet must be in the high-$\sigma$ regime (where the energy per electron can reach $E\gtrsim20$GeV). In such a regime, the particle distribution is hard so that most of the energy is injected in a minority of the particles and the constraints on the bulk motions are somewhat relaxed. Synchrotron-only models work for $R>10^{16}$cm, $\sigma\sim 30-50$, $\Gamma'\sim$ several.
These results can be used as a basis for future PIC simulations of magnetic reconnection in GRBs. Since simulations are extremely computationally demanding, this work may prove to be critical to narrow down the possible parameter space that could lead to the observed properties of GRBs.
\section*{Acknowledgements}
We thank Jonathan Granot, Pawan Kumar, Maxim Lyutikov, Lara Nava and Tsvi Piran for useful discussions and comments.
DG acknowledges support from NASA through the grants NNX16AB32G and NNX17AG21G issued through the Astrophysics Theory Program.
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keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.
\section{Sections and lists}
Sections and lists are generally the same as in the standard \LaTeX\ classes.
\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
\subsection{Subsection}
\subsubsection{Subsubsection}
\paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.
Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Lists}
Two forms of lists can be used in MNRAS -- numbered and unnumbered.
For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.
For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.
\section{Mathematics and symbols}
The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.
Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.
\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:
\begin{verbatim}
\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb' |
2,877,628,090,811 | arxiv | \section{\label{sec:intro}Introduction}
Bisphosphonates (BPs) are widely used as the most powerful drug for protecting bone and treating bone disease such as osteoporosis and other metabolic bone disorders~\cite{Ebetino,Bartl,Coxon,Russell08,Russell11}. From the 1960s onward, a great number of extensive studies on BPs with respect to the development and clinical treatment have been carried out, but the research field on BPs is still very active, being continued to evolve towards a better understanding of treatment mechanisms and a finding of more potent BP on the basis of it. In this context, zoledronic acid (or zoledronate, ZOD), whose chemical name is [1-hydroxy-2-(1H-imidazol-1) ethylidene] bisphosphonic acid or 2-(imidazol-1-yl)-1-hydroxy-ethane-1,1-diphosphonic acid, was developed as a third-generation BP and was approved for the oral treatment of bone disease in 2012 in DPR Korea after the success of its synthesis in our own way by two of the authors (Yong-Man Jang and Song-Un Kim).
BPs are characterized by two phosphonate groups, central carbon atom in between, and two side groups R1 and R2, while human bone is composed of complex array of hydroxyapatite (HAP, \ce{Ca5(PO4)3OH}) crystallites with nano sizes ranging from 30 to 200 nm embedded within the collagen matrix~\cite{Hassenkam,Henning,Elliot}. It is well established that the function of BPs to inhibit bone resorption is originated with their ability of binding strongly to bone mineral, that is, HAP crystal, and stiff resistance to chemical and enzymatic hydrolysis~\cite{Russell08}. In more detailed explanation, the P-C-P backbone of BPs made from two phosphonate groups and carbon atom has a high binding affinity for the compositions of HAP -- tetrahedral PO$_4$ groups, OH groups and Ca ions -- with an additional contribution from the R1 side group (in most cases including ZOD, it is -OH)~\cite{Russell08,Russell}. Moreover, the P-C-P group of BPs is considerably more resistant to the dissolution of HAP crystal than the P-O-P group of pyrophosphate, of which BPs are stable structural analogues~\cite{Kontecka,Fleich,Rogers,Roelofs,Dunford}.
Varying the another side group R2 of BPs can result in differences in antiresorptive potency with several orders of magnitude. It was observed that more potent BPs posses a primary, secondary or tertiary nitrogen atom in the R2 side chain~\cite{Nancollas,Lawson}. At present, the most potent antiresorptive BPs include a heterocyclic R2 side chain containing a nitrogen atom like risedronate and zoledronate~\cite{Green}. The structure-activity relationship analysis showed that the strong binding affinity of zoledronate for HAP is related to its 3D shape and atomic orientation, indicating an important role of 3D shape of nitrogen-containing BP and the orientation of its nitrogen in binding affinity for HAP~\cite{Russell08}.
Atomistic modeling and simulations are a powerful tool to describe the structural characteristics of BPs and bone mineral HAP and the interaction between them at atomic scale, as proved in materials science and molecular science through a vast number of applications. For instance, molecular dynamics (MD) simulations based on the well-constructed classical force field have provided the structural information and energetics of bone mineral, HAP crystal and its surfaces~\cite{Bhowmik,Zhu, Matsunaga, Barrios}. Bhowmik et al.~\cite{Bhowmik} obtained the structural parameters of monoclinic HAP crystal and its surface energetics using the consistent valence force field (CVFF), presenting a valuable description of the interaction between polyacrylic acid and HAP. The surface energetics of HAP crystalline surfaces using {\it ab initio} density functional theory (DFT) calculations within the generalized gradient approximation (GGA) for the exchange-correlation functional has been studied by Zhu and Wu~\cite{Zhu}, testing the effects of slab thickness, vacuum width between slabs and surface relaxation on surface energy. Barrios~\cite{Barrios} has investigated the interaction between collagen protein and HAP surface by using a combination of computational techniques, DFT and classical MD methods. Duarte et al.~\cite{Duarte} performed molecular mechanics simulations for molecular structures of 18 novel hydroxyl- and amino-bisphosphonates to examine the interaction between BPs and hydroxyapatite and to extract relating structural characteristics of BPs and their affinities for the mineral, which are in agreement with {\it in vitro} and {\it in vivo} studies for some of the studied BPs. To the best our knowledge, however, investigation on surface adsorption of zoledronate on HAP surface with its detailed atomistic structure based on quantum mechanics is still scarce, in spite of such extensive theoretical studies of BPs and HAP surface, and we believe that {\it ab initio} study on this phenomenon should definitely contribute to a better understanding of the interaction of zoledronate with bone mineral at atomic and electronic scale.
In this paper, we carry out systematic {\it ab initio} DFT calculations for zoledronate molecule, hydroxyapatite bulk and surface, and surface adsorption of zoledronate molecule on hydroxyapatite surface. Our special focus is placed on the adsorption of zoledronate molecule on hydroxyapatite surface, providing the adsorption energy and atomistic structures of surface complexes, and an insight how charge transferring is occurred in the event of adsorption. This is aimed to get a reliable insight for bone protection effect of zoledronate at electronic scale. In the following, we first describe a computational method in Sec.~\ref{sec:method}, present the results for structural parameters and electronic properties of hydroxyapatite bulk and surface, zoledronate molecule, and for binding of zoledronate to hydroxyapatite surface in Sec.~\ref{sec:result}, and finally give our conclusions in Sec.~\ref{sec:con}.
\section{\label{sec:method}Computational Method}
For the DFT calculations in this work, we have employed \textsc{SIESTA} code\cite{SIESTA} which solves numerically Kohn-Sham equation within DFT~\cite{Hoh1964,Koh1965} using a localized numerical basis set, namely pseudo atomic orbitals (PAO), and pseudopotentials for describing the interaction between ionic core (nucleus plus core electrons) and valence electrons. The BLYP GGA functional (the Becke exchange functional~\cite{Becke} in conjunction with the Lee-Yang-Parr correlation functional~\cite{LYP}) was used for exchange--correlation interaction between electrons. For all the atoms, Troullier-Martins~\cite{TMpseudo} type norm-conserving pseudopotentials were generated within local density approximation (LDA)~\cite{PZlda}, and checked carefully. The basis sets used in this work were the DZP type (double $\zeta$ plus polarization). The mesh size of grid, which is controlled by energy cutoff to set the wavelength of the shortest plane wave represented on the grid, has taken a value of 200 Ry. Non-fixed atoms were allowed to relax until the forces converge less than 0.02 eV/\AA.
We first determine the crystal lattice parameters of bulk HAP by performing structural relaxation with atomic coordinates optimization using the conjugate gradient scheme and appropriate Monkhorst-Pack k-points. Through a comparison of the lattice constants with the experimental values, we have a confidence of the above mentioned selection for calculation parameters. Structural optimization for isolated ZOD molecule is performed as well, and this for several conformations to select the most stable one, of which total energy is the lowest among studied conformations.
We then select the interesting surface index as (001) on the basis of experimental evidence that this face provides binding sites for many ionic species~\cite{Barrios09,Wierzbicki}. In order to simulate the surface within a three-dimensional simulation code, we use two-dimensional periodic slabs, which have a thickness of atomic layers of 2 crystal unit cells plus 30~\AA~vacuum layer, allowing the atoms in one half of a crystal unit cell at each side of the slab to relax. These structural parameters in the supercell can give well-converged result: increasing the thickness of slab to 3 and 4 crystal unit cells and the vacuum width up to 50~\AA~result in variations of surface energy smaller than $\pm$0.05 J/m$^2$. To check the stability of the surface, the surface formation energies are calculated approximately as the difference between the total energies of the slab and the corresponding bulk crystal:
\begin{equation}
\gamma=\left(E_\text{slab}-\frac{N_\text{slab}}{N_\text{bulk}}E_\text{bulk}\right)/2A, \label{eq:gamma}
\end{equation}
where $N_\text{slab}$ and $N_\text{bulk}$ are the numbers of atoms in the surface slab and in the bulk unit cell, $A$ is the slab surface area, and $E_\text{slab}$ and $E_\text{bulk}$ are the total energies of the slab and bulk structures, respectively~\cite{yucj}. The sign of surface formation energy is a test of surface stability: positive (negative) means energy should be provided (released) in order to create a surface. We note that there might be several possible cutting planes for a particular Miller index while guaranteeing neutrality of the surface charge perpendicular to the surface\cite{tasker}.
Final step of the work is to simulate the adsorption of ZOD molecule on the relevant relaxed HAP (001) surfaces. In order to have the rough, initial surface adsorption structure, we use classical molecular mechanics (MM) method. In this MM simulation, General Utility Lattice Program (GULP)~\cite{GULP} is utilized. For describing the interaction between atoms, we use the Dreiding force field, where the potential energy is described as the sum of the contributions resulting from the bonded interactions (bond stretching, bond bending, and torsions) and from the non-bonded interactions (e.g., electrostatic interaction, and van der Waals interaction)~\cite{Leach}. After making a guess for the initial state of the adsorption complex, we also carry out atomic relaxation for the surface complex -- ZOD molecule and surface atoms in the slab -- to obtain a stable final state. Then, the adsorption energy can be calculated as follows:
\begin{equation}
E_\text{ads}=\frac{1}{N_\text{mol}}\left[E_\text{mol-slab}-(E_\text{slab}+N_\text{mol}E_\text{mol})\right],
\label{eq:Ehydro}
\end{equation}
where $N_\text{mol}$ is the number of adsorbed molecules, and $E_\text{mol-slab}$ and $E_\text{mol}$ are the total energies of the adsorbed surface and of the isolated molecule, respectively. The adsorption energy can be either negative or positive: negative (positive) means energy should be released (provided) during the adsorption, indicating that the adsorption is (not) spontaneous exothermic (endothermic) process.
\section{\label{sec:result}Results and discussion}
\subsection{\label{subsec:bulk}Bulk hydroxyapatite}
The crystal structure of hydroxyapatite is hexagonal with space group $P6_3/m$, which contains a formula unit \ce{Ca10(PO4)6(OH)2}. In fact, there must be four hydroxyl groups in the unit cell with the $P6_3/m$ space group, each oxygen atom of hydroxyl group with 1/2 occupancy~\cite{Barrios}. To make the DFT calculations enable, therefore, we have made a model with full occupancies for the hydroxyl groups, by assigning alternate 0 and 1 occupancies to these hydroxyl groups, which results in the change of the space group from $P6_3/m$ to $P6_3$. However, the modified model has a net electric polarization contrary to the experiment which shows zero polarization, because all the hydroxyl groups in the model are oriented in the same direction. To mimic the real structure where exists disorder in the relative orientation of the parallel OH channels, so that electric polarizations are compensated each other, we have created a supercell containing two unit cells in the $[100]$ direction, and assigning opposite orientations to the two OH channels in the supercell. The crystal structure with antiparallel hydroxyl groups in a double unit cell compared to the original hexagonal structure is monoclinic with $P2_1$ space group. We have used this structure in this work, thus allowing a direct comparison of the simulation results with experimentally determined surfaces.
\begin{figure}[!ht]
\includegraphics[clip=true,scale=0.28]{fig1-a.eps} \\ \vspace{1pt} \hspace{16pt}
\includegraphics[clip=true,scale=0.265]{fig1-b.eps}
\caption{\label{fig:bulk}(Color online) Top view (a) and side view (b) of supercell of bulk hydroxyapatite crystal with monoclinic $P2_1$ space group, which is doubled the unit cell in $[100]$ direction so as to compensate the electric polarization by assigning opposite orientations to the two OH channels in hexagon surrounded by triangle of Ca1 ions. (a) top view and (b) side view. (Ca: Green, P: purple, O: red, and H: white)}
\end{figure}
The full optimization of the structure, allowing the cell shape and volume, and ionic positions to relax, was performed. We have checked the convergence with respect to the special k-points; increasing the k-points set from $(1\times2\times3)$ to $(2\times4\times6)$ leads to the change between the total energies as about 5 meV, thus indicating the safe use of $(1\times2\times3)$ set. Figure~\ref{fig:bulk} depicts the fully optimized supercell of bulk HAP in this work. The calculated structural parameters and chemical bonding properties of bulk HAP are given in Table~\ref{tab:optlattice}. The lattice constants ($a$=9.348, $b$=9.352, and $c$=6.955 \AA) of the fully optimized structure are in good agreement with the experimental values ($a$=$b$=9.43, $c$=6.891 \AA)~\cite{Kim, Posner} (less than 1.0\% error) as well as with the earlier DFT results~\cite{Leeuw02, Ellis, Barrios}. Note that $a$ is the half of lattice constant of the supercell.
\begin{table}[!ht]
\begin{center}
\small
\caption{\label{tab:optlattice}Structural parameters of bulk hydroxyapatite crystal structure with monoclinic $P2_1$ space group, compared with experimental results.}
\begin{tabular}{lcc}
\toprule
&\multicolumn{2}{c}{Lattice parameters} \\
\cline{2-3}
& This work & Experiment$^a$ \\
\cline{2-3}
$a, b, c$ (\AA) & 9.348, 9.352, 6.955 & 9.430, 9.430, 6.891 \\
$\alpha, \beta, \gamma$ ($^\circ$) & 90, 90, 119.86 & 90, 90, 120 \\
\hline
& \multicolumn{2}{c}{Average of bond lengths (\AA)} \\
\cline{2-3}
P$-$O & 1.578 & 1.540 \\
Ca1$-$O & 2.423 & 2.478 \\
Ca1$-$O$_\text{H}$ & 2.339 & 2.354\\
Ca2$-$O & 2.553 & 2.556 \\
\hline
& \multicolumn{2}{c}{Average of bond angles ($^\circ$)} \\
\cline{2-3}
O$-$P$-$O & 109.44 & 109.45 \\
O$-$Ca1$-$O & 98.62 & \\
O$-$Ca2$-$O & 99.40 & \\
\bottomrule
\end{tabular} \\
{\raggedright
$^a$ Ref.~\cite{Kim}\\
}
\end{center}
\end{table}
\normalsize
As shown in Figure~\ref{fig:bulk}, there are two different Ca sites (denoted as Ca1 and Ca2), four oxygen sites (O1, O2, and O3 of phosphate tetrahedron, and O$_\text{H}$ of hydroxyl group), one P site, and one H site. The Ca1 atom is coordinated to seven oxygen ions, which are six oxygens of different five \ce{PO4} groups and one oxygen of OH group, while the Ca2 atom has a ninefold coordination with oxygen ions situated in six different \ce{PO4} tetrahedra. The three Ca1 atoms also form a triangle at the same plane normal to the $c$ axis, whose center is occupied by an OH group, and the two triangles form a hexagonal screw configuration. The calculations yield average bond lengths as 1.578~\AA~for P-O bond, 2.423~\AA~for Ca1$-$O, 2.339~\AA~for Ca1$-$O$_\text{H}$, and 2.553~\AA~for Ca2$-$O, compared to the corresponding experimental values of 1.54, 2.478, 2.354, and 2.556~\AA, respectively. The similar agreement is obtained in the bond angles, as shown in Table~\ref{tab:optlattice}. These comparisons confirm that our computational parameters are reasonably satisfactory for a good assessment of surface calculations.
\subsection{\label{subsec:surf}Hydroxyapatite (001) Surface}
In this work, we have selected the Miller indices of HAP surface as (001), since there exists experimental evidence that the (001) surface is the most stable among several surfaces with low indices and provides binding sites for many ionic species~\cite{Leeuw04}. As we use the supercell doubled in the [100] direction as a unit cell for bulk HAP, the (001) surface unit cell is an oblique (2$\times$1) surface cell and there might be two kinds of terminations; (1) 2Ca2$-($6PO$_4\cdot$6Ca1$\cdot$2OH)$-$2Ca2, denoted as Type (I), and (2) (3PO$_4\cdot$3Ca1$\cdot$OH)$-$4Ca2$-($3PO$_4\cdot$3Ca1$\cdot$OH), denoted as Type (II). They are repeated in the [001] direction, assigned to building layer, and belong to the Tasker (II)-type surface~\cite{tasker} with no electric dipole moment perpendicular to the surface.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.45]{fig2.eps}
\caption{\label{fig:surf}(Color online) Top view (upper panel) and side view (lower panel) of optimized structure of hydroxyapatite (001) surface with (2$\times$1) surface unit cell, vacuum thickness of 30 \AA, 4 building layers, and termination 2Ca2$-($6PO$_4\cdot$6Ca1$\cdot$2OH)$-$2Ca2 (a) and termination (3PO$_4\cdot$3Ca1$\cdot$OH)$-$4Ca2$-($3PO$_4\cdot$3Ca1$\cdot$OH) (b). The top and bottom layers are allowed to relax, while the layers between dashed lines are fixed at their crystalline positions. (Ca: green, P: purple, O: red, H: white)}
\end{center}
\end{figure*}
\begin{table}[!ht]
\begin{center}
\small
\caption{\label{tab:surfene}Surface energies (J/m$^2$) according to vacuum thickness and building layers of hydroxyapatite (001) surfaces with (2$\times$1) surface unit cell. The values in parenthesis represent the number of atoms.}
\begin{tabular}{lllll}
\toprule
\multicolumn{5}{c}{Type (I); 2Ca2$-($6PO$_4\cdot$6Ca1$\cdot$2OH)$-$2Ca2} \\
\cline{2-5}
& Size & Unrelaxed & Relaxed & Ref.$^a$ \\
\cline{2-5}
\multirow{4}{*}{Vacuum (\AA)} & 20 & 1.460 & 1.208 & \\
& 30 & 1.461 & 1.205 & \\
& 40 & 1.448 & 1.197 & \\
& 50 & 1.449 & 1.205 & \\
\cline{2-5}
\multirow{3}{*}{Layer} & 4 (176) & 1.461 & 1.205 & \\
& 6 (264) & 1.474 & 1.218 & \\
& 8 (352) & 1.496 & & \\
\hline
\multicolumn{5}{c}{Type (II); (3PO$_4\cdot$3Ca1$\cdot$OH)$-$4Ca2$-($3PO$_4\cdot$3Ca1$\cdot$OH)} \\
\cline{2-5}
\multirow{3}{*}{Layer} & 4 (176) & 2.084 & 1.514 & 1.01 \\
& 6 (264) & 2.057 & 1.506 & \\
& 8 (352) & 2.078 & & \\
\bottomrule
\end{tabular} \\
{\raggedright $^a$ Ref.~\cite{Barrios} \\}
\end{center}
\end{table}
\normalsize
The HAP (001) surface has been modelled using three-dimensional periodic supercell (slab) with two equivalent surfaces at bottom and top side. To ensure no interaction between bottom surface of the above image slab and top surface of the present slab across the vacuum region, the vacuum region must be wide enough. And the atomic layer, which is consisted of surface layer allowed to relax and crystal layer fixed at its crystalline position, should be thick enough so that the two surfaces of each slab do not interact through the crystal layer. To check the convergence according to the vacuum region, we tested 20, 30, 40, and 50 \AA~thickness, and confirmed that there is no distinct change between 20 and 50 \AA~thick vacuums (Table~\ref{tab:surfene}). Therefore, the vacuum region of 30 \AA~thickness will be used in the following calculations. The thickness of the slab is usually expressed in terms of a number of building layers, where one layer contains 44 atoms. The four layers (two bulk unit cells, 176 atoms), six layers (three unit cells, 264 atoms), and eight layers (four unit cells, 352 atoms) were tested with the vacuum width of 30 \AA, and the change of surface energies between 4 layers slab and 8 layers slab is only 0.05 J/m$^2$. Thus we will use a slab with 4 building layers in the study of surface adsorption.
As listed in Table~\ref{tab:surfene}, the surface energy of Type (II) surface is slightly higher than that of Type (I) surface, indicating the Type (I) terminated surface is more favorable than the Type (II) terminated surface. We see that the surface relaxation in the Type (I) surface is not really much compared to the Type (II) surface, since the difference between unrelaxed and relaxed surface energies in the former case is not remarkable contrary to the latter case. Since there is no data available in the literature for Type (I), we only compared the calculated Type (II) surface energy with the previous SIESTA work~\cite{Barrios}.
Figure~\ref{fig:surf} shows the fully relaxed atomistic structure of the HAP (001) surfaces modelled by a slab with vacuum thickness of 30 \AA~and four building layers. It is observed that the coordination number (CN) of Ca1 atom changes from 7 in bulk to 6, and CN of Ca2 from 9 to 6 in the Type (I) termination, while in the Type (II) termination the CN of Ca1 atom changes from 7 to 6, and Ca2 atom from 9 to 5, and 6. The coordination environments around the surface oxygen atoms (O1, O2, O3, and O$_\text{H}$) are also changed from the their bulk environment, while P atoms are still fully surrounded by four oxygen atoms like in bulk.
Table~\ref{tab:surfstruct} shows the bond lengths of cation-oxygen and bond angles of oxygen-cation-oxygen at the top surface layer.
\begin{table}[!ht]
\begin{center}
\small
\caption{\label{tab:surfstruct}Bond lengths of cation-oxygen and bond angles of oxygen-cation-oxygen at hydroxyapatite (001) surface with coordination numbers (CN) of cations.}
\begin{tabular}{lllll}
\toprule
\multicolumn{5}{c}{Type (I); 2Ca2$-($6PO$_4\cdot$6Ca1$\cdot$2OH)$-$2Ca2} \\
\cline{2-5}
& & \multicolumn{2}{c}{Angle ($^\circ$)} & \\
\cline{3-4}
& Length (\AA) & Range & Average & CN \\
\cline{2-5}
P & 1.583 & 103.47$\sim$116.23 & 109.39 & 4 \\
Ca1 & 2.352 & 64.54$\sim$155.26 & 101.50 & 6 \\
Ca2 & 2.434 & 62.45$\sim$137.08 & 94.01 & 6 \\
\hline
\multicolumn{5}{c}{Type (II); (3PO$_4\cdot$3Ca1$\cdot$OH)$-$4Ca2$-($3PO$_4\cdot$3Ca1$\cdot$OH)} \\
\cline{2-5}
P & 1.585 & 102.07$\sim$117.07 & 109.22 & 4 \\
Ca1 & 2.420 & 60.97$\sim$159.10 & 99.72 & 6 \\
Ca1 & 2.343 & 64.73$\sim$160.44 & 102.36 & 6 \\
Ca2 & 2.325 & 66.52$\sim$147.60 & 101.91 & 5 \\
Ca2 & 2.414 & 60.91$\sim$170.66 & 98.61 & 6 \\
Ca2 & 2.380 & 61.52$\sim$154.20 & 99.52 & 6 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\normalsize
In the case of Type (II) surface, we consider the two kinds of Ca1 and three kinds of Ca2 surface atoms. In both cases, the averages (1.583, 1.585 \AA) of bond lengths of P$-$O are a bit expanded compared with the bulk (1.578 \AA), and the averages of bond angles get smaller than in the bulk. The bond lengths of Ca2$-$O are clearly contracted at the surface; 2.434 in Type (I), and 2.325, 2.380, and 2.414 \AA~in Type (II), which are all smaller than the bulk value 2.553 \AA. From these observations, it can be concluded that the undercoordinated surface Ca atoms and O atoms can be favorable adsorption sites.
\subsection{\label{subsec:zol}Zoledronic acid molecule}
To simulate an isolated molecule, we have used a cubic supercell with lattice constants of $a=b=c=50$~\AA, which has been proved to be enough to prevent the artificial interaction between adjacent molecules. There might be four different conformations, as presented in Figure~\ref{fig:zol}. We can make a distinction between the conformations according to the directions of two pairs of OH groups attached to the two P atoms; (1) ZOD$_\text{Bout}$ for the case where the directions of both OH pairs are outward against the nitrogen heterocyclic ring, (2) ZOD$_\text{Bin}$ for the case of inward directions of both pairs, (3) ZOD$_\text{Nin}$ for the case of inward direction of one pair in the same side of nitrogen atom, and (4) ZOD$_\text{Nout}$ for the case of outward direction of one pair in the N side. After the total energy minimizations to get the optimized structure of molecule, we have compared the total energies of four conformations. The calculations tell us the most stable structure is just the third case, ZOD$_\text{Nin}$, which will be used in the following adsorption study.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.65]{fig3-a.eps} \\
\includegraphics[clip=true,scale=0.40]{fig3-b.eps} \\
\includegraphics[clip=true,scale=0.40]{fig3-c.eps}
\caption{\label{fig:zol}(Color online) Optimized structures of zoledronic acid with four different conformations, (1) ZOD$_\text{Bout}$, (2) ZOD$_\text{Bin}$, (3) ZOD$_\text{Nin}$, and (4) ZOD$_\text{Nout}$. (C: grey, P: purple, N: blue, O: red, H: white)}
\end{center}
\end{figure}
The optimized bond lengths and bond angles related with P atoms in ZOD$_\text{Nin}$ conformation are listed in Table~\ref{tab:zol}. The averages of P$-$O bond lengths are 1.597 \AA~in P1 side and 1.586 \AA~in P2 side, which are similar to those of bulk HAP (1.578 \AA) and of HAP (001) surface (1.583 \AA). The averages of O$-$P$-$O bond angles (112.11 and 111.42$^\circ$) are also close to those of bulk HAP (109.44$^\circ$) and HAP surface (109.39$^\circ$). It is found that those values of P2 side are slightly closer to the bulk and surface values than P1 side, so that P2 side are more favorable to binding with HAP due to the closer structural similarity.
\begin{table}[!ht]
\begin{center}
\small
\caption{\label{tab:zol}Optimized bond lengths and angles related with phosphorus atoms in zoledronic acid conformation, ZOD$_\text{Nin}$. Ave. means average.}
\begin{tabular}{ll|ll}
\toprule
\multicolumn{4}{c}{P$_1$ related} \\
\hline
\multicolumn{2}{c|}{Bond length (\AA)} & \multicolumn{2}{c}{Bond angle ($^\circ$)} \\
\hline
P1$=$O1 & 1.495 & O1$=$P1$-$O2 & 117.26 \\
P1$-$O2 & 1.654 & O1$=$P1$-$O3 & 115.10 \\
P1$-$O3 & 1.641 & O1$=$P1$-$C1 & 115.22 \\
P1$-$C1 & 1.902 & O2$-$P1$-$O3 & 103.97 \\
& & O2$-$P1$-$C1 & 102.96 \\
& & O3$-$P1$-$C1 & 100.12 \\
& Ave. & O$-$P1$-$O & 112.11 \\
& Ave. & O$-$P1$-$C1 & 106.10 \\
\hline
\multicolumn{4}{c}{P$_2$ related} \\
\hline
P2$=$O4 & 1.495 & O4$=$P2$-$O5 & 120.01 \\
P2$-$O5 & 1.622 & O4$=$P2$-$O6 & 117.35 \\
P2$-$O6 & 1.641 & O4$=$P2$-$C1 & 99.93 \\
P2$-$C1 & 1.934 & O5$-$P2$-$O6 & 96.91 \\
& & O5$-$P2$-$C1 & 112.73 \\
& & O6$-$P2$-$C1 & 110.40 \\
& Ave. & O$-$P2$-$O & 111.42 \\
& Ave. & O$-$P2$-$C1 & 107.69 \\
\hline
& & P1$-$C1$-$P2 & 98.65 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\normalsize
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.42]{fig4.eps}
\caption{\label{fig:homo}(Color online) Molecular orbitals of zoledronic acid conformation, ZOD$_\text{Nin}$.}
\end{center}
\end{figure}
Figure~\ref{fig:homo} shows the computed isodensity surfaces for the highest occupied molecular orbitals (HOMO) and lowest unoccupied molecular orbitals (LUMO) of ZOD (conformation ZOD$_\text{Nin}$). While the HOMO and HOMO-1 are localized on the heterocyclic ring containing nitrogen atom, LUMO and LUMO+1 are on the phosphonate groups. This leads to the intramolecular charge separation upon excitation. The change in electronic distribution between HOMO and LUMO indicates that two phosphonate groups can play a role of electron acceptor in the chemical reaction, while the heterocyclic ring containing nitrogen atom could be a donor.
\subsection{\label{subsec:ads}Adsorption of zoledronic acid on hydroxyapatite (001) surface}
To begin with adsorption, we have utilized GULP to obtain rough structure of ZOD absorbed on HAP (001) surface, where Dreiding forcefield was adopted. Simulated annealing was performed, increasing the temperature from 300 K to 10000 K and subsequently decreasing back to 300 K with the temperature interval of 50 K. Using the obtained rough structure as the starting structure, we then performed atomic relaxation with SIESTA code to get the final optimized structure of zoledronic acid adsorbed HAP (001) surface.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.45]{fig5.eps}
\caption{\label{fig:ads}(Color online) Top view (upper panel) and side view (lower panel) of optimized atomistic structure of zoledronic acid adsorption complexes on hydroxyapatite (001) surfaces with Type (I) (a) and Type (II) (b) terminations at 0.25 ML coverage. (Ca: green, P: purple, O: red, N: blue, H: white)}
\end{center}
\end{figure*}
In this work we have tried to simulate the adsorption of ZOD molecule at 0.5 ML (one molecule on (2$\times$1) surface cell) and at 0.25 ML coverages (one molecule on (2$\times$2) surface cell). As mentioned above, there are two possible terminations in the HAP (001) surface, and therefore four kinds of simulation tasks were carried out. We have used the slab model with 4 building layers and vacuum thickness of 30 \AA, which guarantee to give a reliable adsorption energy. To make a systematic error small, the pristine surface energies for (2$\times$2) surface slabs were also calculated.
\begin{table}[!ht]
\begin{center}
\small
\caption{\label{tab:ads-ene}Calculated adsorption energies (kJ/mol) of zoledronic acid on hydroxyapatite (001) surface.}
\begin{tabular}{llll}
\toprule
& \multicolumn{2}{c}{Coverage} \\
\cline{2-3}
& 0.5 ML (2$\times$1) & 0.25 ML (2$\times$2) \\
\hline
Type (I) & $-$388.44 & $-$246.70 \\
Type (II) & $-$439.49 & $-$268.34 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\normalsize
In Table~\ref{tab:ads-ene}, the adsorption energies are listed. All the adsorption energies are negative, indicating that all the adsorptions are exothermic reactions. We see that the adsorption on Type (II) surface is slightly more favorable than on Type (I) at both 0.5 ML and 0.25 ML, although the Type (I) surface formation from the bulk needs more energy than the Type (II) surface.
Figure~\ref{fig:ads} shows the optimized atomistic structure of ZOD adsorption complexes on HAP (001) surfaces at 0.25 ML coverage. It is observed that in the Type (I) surface the hydrogen atom of ZOD's phosphonate group moved to oxygen atom of HAP surface's phosphate group, making formation of additional OH group on the surface, and thus indicating that the adsorption is kind of chemisorption caused by proton exchange. There are several hydrogen bonds between ZOD's phosphonate OH group and oxygen atoms of the surface's phosphate groups, and vice versa. It is also found that Ca1 and Ca2 atoms with 6 coordination number (CN) make bonds with oxygen and nitrogen atoms of ZOD, respectively. Such bonding was expected based on the surface relaxation analysis and HOMO-LUMO distribution of isolated ZOD molecule. In the case of Type (II) surface, there are also hydrogen bonds between the surface and ZOD, but the move of hydrogen atom is not observed, which indicates that the adsorption is kind of physisorption. Nevertheless, the magnitude of adsorption energy on Type (II) surface is larger than that on Type (I) surface.
To make it clear the charge transfer occurred during the adsorption, we show the electron density difference, $\rho_\text{surf+mol}-(\rho_\text{surf}+\rho_\text{mol})$, plots in Figure~\ref{fig:diffrho}, where (a) is for the isosurface figure with the value of $\pm$0.04 eV/\AA$^3$, and (b) and (c) are the contour plots on the planes around hydrogen bond and Ca cation, respectively. These figures clearly give the evidence of charge transferring at the event of adsorption, from hydrogen of phosphate group of ZOD molecule to oxygen of PH$_4$ group of HAP surface making formation of hydrogen bond, and from Ca of HAP surface to oxygens of phosphonate group forming the coordinate bond. Therefore the adsorption of ZOD on the HAP (100) surface is surely chemisorption.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.3]{fig6-a.eps} \\
\includegraphics[clip=true,scale=0.23]{fig6-b.eps}
\includegraphics[clip=true,scale=0.23]{fig6-c.eps}
\caption{\label{fig:diffrho}(Color online) Electronic charge density difference of zoledronic acid adsorption complex on hydroxyapatite (001) surface. (a) isosurface figure with the value of $\pm$0.04 eV/\AA$^3$ (yellow for + and blue for - value), (b) contour figure on the plane containing hydrogen bond between O of phosphonate group of HAP surface and hydrogen of phosphate group, and (c) contour figure around Ca cation. (Ca: green, P: purple, O: red, N: blue, H: white)}
\end{center}
\end{figure*}
We calculated the density of states (DOS) of electrons, total and partial DOS shown in Figure~\ref{fig:dos} and atom projected and partial DOS in Figure~\ref{fig:lpdos}. In Figure~\ref{fig:dos} we see the energy spectrum of isolated ZOD molecule (a), the DOS of ZOD molecule that was adsorbed on HAP surface (b), the DOS of HAP (100) surface (c), and the whole complex system comprised of the adsorbed ZOD molecule and HAP (100) surface. Some hybridization between ZOD and HAP surface electrons is shown in the figures. Which atoms cause the charge transferring at the adsorption? To make an answer to this question, we carefully investigate the atom projected partial DOS. The 1s peak of hydrogen of isolated ZOD molecule placed over 0 eV (red line in Figure~\ref{fig:dos} (a) panel) is remarkably weakened after the adorption (blue line in (a) panel), indicating the loss of electrons from hydrogen and the gain by oxygen, and thus the formation of hydrogen bond between hydrogen atom of ZOD and oxygen of HAP surface. We also see the hybridizations between hydrogen 1s state and oxygen 2p state in (a) and (b) panels, and between oxygen 2s state and calcium 3p state in (c) and (d) panels.
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.55]{fig7.eps}
\caption{\label{fig:dos}(Color online) Total and partial density of states in isolated zoledronic acid (a), adsorbed zoledronic acid (b), hydroxyapatite (100) surface (c) and zoledronic acid adsorption complex on the surface (d).}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[clip=true,scale=0.55]{fig8.eps}
\caption{\label{fig:lpdos}(Color online) Atom projected partial density of states; (a) for 1s electrons of hydrogen in different situations, (b) for 2s and 2p electrons of oxygen in different situations, (c) for 2s and 2p states of oxygens surrounding Ca cation in ZOD adsorbed HAP surface, and (d) for 4s and 3p of calcium forming the coordinate bond on HAP surface.}
\end{center}
\end{figure}
\section{\label{sec:con}Conclusion}
In conclusion, we have attempted to make a modeling of adsorption of zoledronic acid on hydroxyapatite (001) surface to get an atomistic insight of bone protection. The systematic study has been performed, from hydroxyapatite bulk and surface, and zoledronic acid to adsorption of the molecule on the (001) surface. We have carried out the structural optimizations and atomic relaxations of the bulk, molecule, surface, and adsorption complexes on the surfaces, and obtained the structural informations and energetics.
Using the three-dimensional periodic supercell model, we determined the stable conformation of the molecule and calculated the molecular orbitals. It was concluded that two phosphonate groups can play a role of electron acceptor in the chemical reaction, while the heterocyclic ring containing nitrogen atom can be a electron donor. After verifying the validity of computational parameters of SIESTA work through the bulk hydroxyapatite calculation, surface modeling and relaxations were performed, and surface formation energies were calculated for two kinds of (001) surface terminations, which are about 1.2 and 1.5 J/m$^2$. Subsequently, the adsorption of zoledronic acid on the relaxed surface was studied, obtaining the surface binding structure and calculating the adsorption energies. We found that the surface Ca atoms and oxygen atoms of phosphate groups can form surface bond including hydrogen bond and coordinate bond with nitrogen, hydrogen, and oxygen atoms of zoledronic acid. The calculated adsorption energies are about -260 kJ/mol at 0.25 ML coverage and -400 kJ/mol at 0.5 ML coverage, indicating the strong binding affinity of zoledronic acid to hydroxyapatite surface.
We have made an interpretation of such strong binding affinity through the analysis of atomistic structures, electron density difference, the density of states and Hirshfeld charges of atoms relevant to surface binding. The results showed that bond lengths and bond angles of phosphonate group of zoledronic acid are similar to those of phosphate group of hydroxyapatite bulk and surface, indicating the strong binding affinity related to the structural similarity. It was also found that the charge transferring is occurred mainly on the side of phosphonate group in one kind surface, while in another surface it is occurred on both imidazol ring containing nitrogen atom and phosphonate group of zoledronic acid.
\section*{\label{ack}Acknowledgments}
The simulations have been carried out on the HP Blade System c7000 (HP BL460c) that is owned and managed by Faculty of Materials Science, Kim Il Sung University. This work was supported from the Commette of Education (grant number 02-2014), DPR Korea.
\section*{\label{note}Notes}
The authors declare that they have no conflict of interest.
\bibliographystyle{elsarticle-num-names}
|
2,877,628,090,812 | arxiv | \section{Introduction}
The field-antifield BV formalism \cite{BV} is a
Lagrangian path integral method to quantize
general gauge theories (important early contributions are
\cite{ZJ}-\cite{WH}). It has been shown to work for
an ever increasing number of models. In the
BV formalism one introduces antifields with opposite
Grassmann parities to all field and ghost
variables. It involves in a crucial way also an
antibracket operation and a nilpotent differential
$\Delta$-operator. The understanding of the
formalism was further deepened in
\cite{Vol}-\cite{Schw} and in \cite{BT93}-\cite{Trip}.
In the approach of the latter papers a
coordinate invariant general covariant formulation
was developed. The field-antifield variables are
here considered as arbitrary coordinates on an
antisymplectic manifold ${\cal M}$. (The standard BV
formalism may then be viewed as formulated in
terms of antisymplectic Darboux coordinates.) In
this formalism the geometric coordinate invariant
properties is formally demonstrated.
Furthermore, the formalism specifies the conditions
a consistent invariant measure density has to
satisfy. It involves also new ingredients like a hypergauge formulation
\cite{BT93,Trip} and a multilevel formalism
\cite{BT93-2}. Gauge invariance is demonstrated in
general terms. Among the further generalizations
are deformed $\Delta$-operators in \cite{BT94-1}
and an Sp(2) version in \cite{Trip,GTrip}.
In \cite{BT93} it was also shown
how antisymplectic second class constraints may be introduced and
treated consistently within this formalism.
In this paper we continue this set of formal
generalizations with still another one. Here we
show that the path integral may be formulated
on a large antisymplectic manifold also in the
presence of antisymplectic first class constraints.
All required conditions are shown to be formally
satisfied. This is therefore a major
further generalization of the general covariant BV
formalism. The beautiful general mathematical
structure of the BV formalism is thereby further extended.
However, it remains to demonstrate the
existence of examples which satisfy the generalizations
suggested by the obtained formal results.
Although we expect them to exist this is certainly a
nontrivial issue. Anyway the formal results
suggest alternative formulations which could turn
out to be useful. Particularly the results of the
present paper could allow for formulations with
specific global symmetries which are preferable
for some reasons. At a more speculative level
our results show the existence of new types of gauge
theories in an antisymplectic quantum theory in the spirit of
\cite{Vol}-\cite{Khud}.
In section 2 we recapitulate some basic properties of
the general covariant BV formalism. In section
3 and appendices A and B we present our formulation
and its formal properties. In section 4 and
appendix C we consider then a generalized conversion
of antisymplectic second class constraints
into
corresponding first class
ones by means of an extension of the
field-antifield manifold
${\cal M}$. This provides for an explicit formal verification
of the formalism. Throughout the paper we
make use of deWitt's compact notation which reduces
the treatment to a finite dimensional one. In
principle all functionals may be either local or nonlocal.
\setcounter{page}{1}
\section{Basics of general covariant BV formalism}
The basic object in field-antifield quantization is the nilpotent fermionic second
order differential operator
\begin{eqnarray}
&&\Delta\equiv\frac{1}{2}(-1)^{{\varepsilon}_{A}}\rho^{-1}{\partial}_{A}\rho E^{AB}{\partial}_{B},
\e{1}
where ${\partial}_A$ are derivatives with respect to local coordinates
${\Gamma}^A$, $A=1,\ldots,2N$, on an antisymplectic manifold ${\cal M}$. Their
Grassmann parities are ${\varepsilon}({\Gamma}^A)\equiv{\varepsilon}_{A}\in\{0,1\}$. (${\Gamma}^A$
are generalized fields and antifields.) $\rho({\Gamma})$ is a measure
density and $E^{AB}$ an odd metric tensor with the properties:
$E^{AB}=E^{BA}(-1)^{{\varepsilon}_{A}+{\varepsilon}_{B}+{\varepsilon}_{A}{\varepsilon}_{B}}$ and
${\varepsilon}(E^{AB})={\varepsilon}_{A}+{\varepsilon}_{B}+1$.
Another basic object in the field-antifield formalism is the antibracket given by
\begin{eqnarray}
&&(F,G)\equiv (-1)^{{\varepsilon}_F}\Delta(FG)-(-1)^{{\varepsilon}_F}(\Delta F)G-F\Delta
G=F{\stackrel{\lea}{\partial}}_AE^{AB}{\partial}_BG
\e{2}
for arbitrary functions $F,G$ on ${\cal M}$. It satisfies
\begin{eqnarray}
&&{\varepsilon}((F,G))={\varepsilon}_F+{\varepsilon}_G+1,\quad(F,G)=-(G,F)(-1)^{({\varepsilon}_F+1)({\varepsilon}_G+1)},\nonumber\\
&&(F,GH)=(F,G)H+(-1)^{{\varepsilon}_G({\varepsilon}_F+1)}G(F,H),\nonumber\\
&&((F,G),H)(-1)^{({\varepsilon}_F+1)({\varepsilon}_H+1)}+{\rm cycle}(F,G,H)\equiv0,\nonumber\\
&&\Delta(F,G)=(\Delta F,G)+(-1)^{({\varepsilon}_F+1)}(F,\Delta G).
\e{3}
The measure density $\rho$ satisfies also
\begin{eqnarray}
&&(-1)^{({\varepsilon}_A+{\varepsilon}_C)}\rho^{-1}{\partial}_A \rho E^{AB}{\partial}_B\rho^{-1}{\partial}_C\rho
E^{CD}=0.
\e{4}
All these properties follow from the nilpotency of the
$\Delta$-operator \r{1}, {\em i.e.\ }
$\Delta^2=0$.
On ${\cal M}$ we assume the existence of a quantum master action $W$ satisfying
\begin{eqnarray}
&&\Delta e^{{i\over \hbar}W}=0\quad{\Leftrightarrow}\quad\frac{1}{2}(W,W)=i\hbar\Delta W.
\e{5}
In terms of $W$ there is another nilpotent second order differential
operator
$\sigma_W$ defined by \cite{MH}
\begin{eqnarray}
&&\sigma_WF\equiv {\hbar\over i}e^{-{i\over\hbar}W}\Delta
e^{{i\over\hbar}W}F=(W,F)-i\hbar\Delta F
\e{6}
which satisfies $\sigma_W^2=0$ and
\begin{eqnarray}
&&\sigma_W(F,G)=(\sigma_W F,G)+(-1)^{{\varepsilon}_F+1}(F,\sigma_W G),\nonumber\\
&&\sigma_W FG=(\sigma_W F)G+(-1)^{{\varepsilon}_F}F(\sigma_W
G)-i\hbar(-1)^{{\varepsilon}_F}(F,G).
\e{7}
Given two solutions $W$ and $X$ of the master equation \r{5}, one has
\begin{eqnarray}
&&[\sigma_W,\sigma_X]F=-\frac{1}{2}\left((-W+X,-W+X),F\right).
\e{71}
The path integral
in this generalized BV formalism is given by
\begin{eqnarray}
&&Z=\int\exp{\left\{{i\over\hbar}[W+X]\right\}}\rho[d{\Gamma}][d\eta],
\e{8}
where $W$ is the above quantum master action and $X$ a
hyper gauge-fixing master
action which also satisfies the quantum master equation
\r{5}. $\eta^a$ are second
level Lagrange multipliers \cite{BT94-1,BT94-2} with no
corresponding antifields. This
means that
${\cal M}$ is viewed as containing first level
Lagrange multipliers ${\lambda}^a$ and their
antifields
${\lambda}^*_a$ ($\{{\Gamma}^A\}=\{{\Gamma}_0^A,{\lambda}^a, {\lambda}^*_a\}$) with the
Grassmann parities
${\varepsilon}(\eta^a)={\varepsilon}({\lambda}^*_a)={\varepsilon}({\lambda}^a)+1$. The actions
$W$ and
$X$ have then the form
\begin{eqnarray}
&&W=W_0({\Gamma}_0)+{\lambda}^*_a\eta^a,\quad X=G_a({\Gamma}_0){\lambda}^a+\ldots,
\e{9}
where $G_a$ are hyperconstraints that fixes the antifields in
$\{{\Gamma}_0^A\}$. Also
$W_0$ satisfies the master equation \r{5}.
Under these conditions one may show that
the path integral
\r{8} is independent of the precise form of the hypergauge
conditions $G_a$
($X$-independence)
\cite{Trip}.
If we introduce some constraints $\Theta^\alpha=0$,
$\alpha=1,\ldots,2n<2N$, on ${\cal M}$
such that $E^{\alpha\beta}\equiv (\Theta^\alpha,\Theta^\beta)$ is
invertible, then we may
define a ``Dirac" antibracket by the expression \cite{BT93}
\begin{eqnarray}
&&(F,G)_{({\cal D})}\equiv(F,G)-(F,\Theta^\alpha)E_{\alpha\beta}
(\Theta^\beta,G),
\e{10}
where $E_{\alpha\beta}$ is the invers to $E^{\alpha\beta}$. Obviously
\begin{eqnarray}
&&(F,G)_{({\cal D})}=F{\stackrel{\lea}{\partial}}_AE^{AB}_{({\cal D})}{\partial}_BG,
\quad E^{AB}_{({\cal D})}\equiv
E^{AB}-E^{AC}({\partial}_C\Theta^\alpha) E_{\alpha\beta}
(\Theta^\beta{\stackrel{\lea}{\partial}}_D)E^{DB}.
\e{11}
Since $(F,\Theta^\alpha)_{({\cal D})}=0$ for any $F$ the
metric $E^{AB}_{({\cal D})}$ is
degenerate on ${\cal M}$. However, even in terms of such a metric
there is a consistent
path integral and it is given by \cite{BT93}
\begin{eqnarray}
&&Z_{({\cal D})}=\int\exp{\left\{{i\over\hbar}[W+X]\right\}}
\prod_\alpha{\delta}(\Theta^\alpha)\rho_{({\cal D})}[d{\Gamma}][d\eta],
\e{12}
where now $W$ and $X$ satisfy the quantum master equations
\r{5} with $\Delta$
replaced by
\begin{eqnarray}
&&\Delta_{({\cal D})}\equiv \frac{1}{2}(-1)^{{\varepsilon}_{A}}
\rho_{({\cal D})}^{-1}{\partial}_{A}\rho_{({\cal D})}
E_{({\cal D})}^{AB}{\partial}_{B}.
\e{13}
Thus, for antisymplectic second class constraints
$\Theta^\alpha=0$ there is a
consistent formulation already. We shall now propose a
consistent formulation for
corresponding first class constraints.
\section{Field-antifield formalism with first class constraints}
Let us call $T_\alpha=0$
antisymplectic first
class constraints provided $T_\alpha$ satisfy
\begin{eqnarray}
&&(T_\alpha,T_\beta)=T_{\gamma} U_{\;\alpha\beta}^{{\gamma}}.
\e{14}
In the presence of such constraints
we propose the following path integral
\begin{eqnarray}
&Z_{T}&=\int\exp{\left\{{i\over\hbar}[W+X]\right\}}
\prod_\alpha{\delta}(T_\alpha)\prod_\beta{\delta}(\chi^\beta)\:{1\over{\rm
sdet}(\chi^{\gamma},T_{\delta})}\rho({\Gamma})[d{\Gamma}][d\eta]=\nonumber\\
&&=
\int\exp{\left\{{i\over\hbar}
[W+X+{\overline{{\cal C}}}_\alpha(\chi^\alpha,T_\beta){\cal C}^\beta+
T_\alpha\pi^\alpha+\xi_\alpha\chi^\alpha]\right\}}d\mu,
\nonumber
\\
&d\mu&\equiv\rho({\Gamma})[d{\Gamma}][d\eta][d\pi][d\xi][d{\cal C}][d{\overline{{\cal C}}}],
\e{15}
where $\rho({\Gamma})$ is a gauge independent
measure density,
and where $\chi^\alpha=0$ are
gauge-fixing conditions to
$T_\alpha=0$, {\em i.e.\ }
$(\chi^{\gamma},T_{\delta})$ is required to be invertible.
The Grassmann parities of the
field variables in \r{15} are
\begin{eqnarray}
&&{\varepsilon}(\pi^\alpha)={\varepsilon}({\cal C}^\alpha)={\varepsilon}({\overline{{\cal C}}}_\alpha)={\varepsilon}_\alpha
\equiv{\varepsilon}(T_\alpha),\quad{\varepsilon}(\xi_\alpha)={\varepsilon}(\chi^\alpha)={\varepsilon}_\alpha+1.
\e{151}
The
first class constraints
$T_\alpha$ are in addition to \r{14} required to satisfy
\begin{eqnarray}
&&\sigma_W T_\alpha=
T_\beta P^\beta_{\;\alpha}-i\hbar
U_{\;\beta\alpha}^\beta(-1)^{{\varepsilon}_\beta},\nonumber\\
&&\sigma_X T_\alpha=
T_\beta Q^\beta_{\;\alpha}-i\hbar
U_{\;\beta\alpha}^\beta(-1)^{{\varepsilon}_\beta},
\e{16}
which also may be viewed as conditions on $W$
and $X$.
A general representation of the path
integral \r{15} is given in appendix A.
The path integral $Z_T$ is invariant under the supertransformation
\begin{eqnarray}
&&{\delta}{\Gamma}^A=({\Gamma}^A,T_\alpha){\cal C}^\alpha\mu,
\e{17}
where $\mu$ is an odd constant. It leads to
\begin{eqnarray}
&&{\delta}
T_\alpha=T_{\gamma} U_{\;\alpha\beta}^{\gamma}{\cal C}^\beta\mu,\quad{\delta}\chi^\alpha
=(\chi^\alpha,T_\beta){\cal C}^\beta\mu,\nonumber\\
&&{\delta}({\overline{{\cal C}}}_\alpha(\chi^\alpha,
T_\beta){\cal C}^\beta)=\frac{1}{2}{\overline{{\cal C}}}_\alpha(\chi^\alpha,T_{\delta})
U_{\;\beta{\gamma}}^{\delta}{\cal C}^{\gamma}{\cal C}^\beta(-1)^{{\varepsilon}_\beta}+\frac{1}{2}
T_{\delta}{\overline{{\cal C}}}_\alpha(\chi^\alpha,
U_{\;\beta{\gamma}}^{\delta}){\cal C}^{\gamma}{\cal C}^\beta(-1)^{{\varepsilon}_\beta}
\e{18}
and
\begin{eqnarray}
&&{\delta}(W+X)=2i\hbar(\Delta
T_\alpha- U_{\;\beta\alpha}^\beta(-1)^{{\varepsilon}_\beta}){\cal C}^\alpha\mu+
T_\beta(P^\beta_{\;\alpha}+Q^\beta_{\;\alpha}){\cal C}^\alpha\mu
\e{19}
from
\r{16}.
Furthermore, it gives rise to the following Jacobian
\begin{eqnarray}
&&J=1+2(\Delta T_\alpha){\cal C}^\alpha\mu.
\e{20}
All these contributions from the transformation \r{17} in the
integrand of $Z_T$ are
compensated by the transformations
\begin{eqnarray}
&{\delta}{\cal C}^{\delta}=&-\frac{1}{2} U_{\;\beta{\gamma}}^{\delta}{\cal C}^{\gamma}
{\cal C}^\beta(-1)^{{\varepsilon}_\beta}\mu,\nonumber\\
&{\delta}\pi^{\delta}=&-\frac{1}{2}{\overline{{\cal C}}}_\alpha(\chi^\alpha,
U_{\;\beta{\gamma}}^{\delta}){\cal C}^{\gamma}{\cal C}^\beta(-1)^{{\varepsilon}_\beta}\mu-\nonumber\\ &&-
U_{\;\alpha\beta}^{\delta}{\cal C}^\beta\pi^\alpha
(-1)^{{\varepsilon}_\alpha}\mu-(P^{\delta}_{\;\alpha}+Q^{\delta}_{\;\alpha}){\cal C}^\alpha\mu,\nonumber\\
&{\delta}{\overline{{\cal C}}}_\alpha=&\mu\xi_\alpha,
\e{21}
together with the resulting contributions from the corresponding Jacobians
\begin{eqnarray}
&&{\delta}{\cal C}^{\delta}{{\stackrel{\lea}{\partial}}\over{\partial}{\cal C}^{\delta}}(-1)^{{\varepsilon}_{\delta}}=-
U_{\;\beta{\gamma}}^\beta{\cal C}^{\gamma}(-1)^{{\varepsilon}_\beta}\mu,\quad
{\delta}\pi^{\delta}{{\stackrel{\lea}{\partial}}\over{\partial}\pi^{\delta}}(-1)^{{\varepsilon}_{\delta}}
=- U_{\;\beta{\gamma}}^\beta{\cal C}^{\gamma}(-1)^{{\varepsilon}_\beta}\mu.
\e{22}
The path integral $Z_T$ is also independent of the gauge-fixing
functions $\chi^\alpha$.
To see this consider the shift
\begin{eqnarray}
&&\chi^\alpha{\rightarrow}\chi^\alpha+{\delta}\chi^\alpha
\e{23}
in $Z_T$. It is in fact exactly compensated
by the transformation \r{17} with the
choice
\begin{eqnarray}
&&\mu={i\over\hbar}{\overline{{\cal C}}}_\alpha{\delta}\chi^\alpha,
\e{24}
since when compared to the previous
transformation with $\mu$ constant this choice
gives rise to the following additional contribution to the Jacobian
\begin{eqnarray}
&{\delta}
J&\equiv({\Gamma}^A,T_\alpha){\cal C}^\alpha{i\over\hbar}
\left({\overline{{\cal C}}}_\beta{\delta}\chi^\beta{\stackrel{\lea}{\partial}}_A\right)
(-1)^{{\varepsilon}_A}+{i\over\hbar}(-1)^{{\varepsilon}_\alpha}
{{\partial}\over{\partial}{\overline{{\cal C}}}_\alpha}({\overline{{\cal C}}}_\beta{\delta}\chi^\beta)\xi_\alpha=
\nonumber\\&&=
-{i\over\hbar}{\overline{{\cal C}}}_\beta({\delta}\chi^\beta,T_\alpha)
{\cal C}^\alpha-{i\over\hbar}\xi_\alpha{\delta}\chi^\alpha.
\e{25}
\section{Conversion and the Abelian case}
We shall now apply and verify the general formulation above. We
consider then a generalized conversion of antisymplectic second class
constraints into corresponding first class ones. Within the
ordinary Hamiltonian formalism the conversion mechanism has been
formulated in
general terms in
\cite{BF,BFF} (see also
\cite{FS}). It has been applied to many models.
One interesting application is the
new approach to geometric quantization in \cite{FL} which is mainly based
on
\cite{BFF}. In the following application to the field-antifield formalism a new
ingredient
appears since we not only have antibrackets, which
corresponds to Poisson brackets,
but also the nilpotent differential $\Delta$-operator.
Consider the second class constraints
$\Theta^\alpha=0$, $\alpha=1,\ldots,2n<2N$, on ${\cal M}$ which by definition are
such that $E^{\alpha\beta}\equiv
(\Theta^\alpha,\Theta^\beta)$ is invertible. We now
convert these constraints into
abelian first class constraints by extending the
original antisymplectic manifold ${\cal M}$. Introduce therefore
the additional field-antifield coordinates
$\Phi^\alpha$ with the Grassmann parities
${\varepsilon}(\Phi^\alpha)={\varepsilon}(\Theta^\alpha)={\varepsilon}_\alpha$.
On the resulting extended manifold,
${\cal M}{_{\rm ext}}\equiv{\cal M}\oplus\{\Phi^\alpha\}$, we define then an extended
antibracket with the
extended metric
\begin{eqnarray}
&&({\Gamma}^A,{\Gamma}^B){_{\rm ext}}=E^{AB},
\quad({\Gamma}^A,\Phi^\alpha){_{\rm ext}}=0,\quad(\Phi^\alpha,\Phi^\beta){_{\rm ext}}=\omega^{\alpha\beta},
\e{26}
where $\omega^{\alpha\beta}$ is an odd invertible constant matrix.
On ${\cal M}{_{\rm ext}}$ we may then define first class constraints $T^\alpha$ satisfying
\begin{eqnarray}
&&(T^\alpha,T^\beta){_{\rm ext}}=0, \quad T^\alpha|_{\Phi=0}=\Theta^\alpha.
\e{27}
These functions may be
constructed perturbatively with the ansatz
\begin{eqnarray}
&&T^\alpha({\Gamma},\Phi)=\Theta^\alpha({\Gamma})+\sum_{n=1}^\infty{1\over
n!}\Phi^{\beta_n}\cdots\Phi^{\beta_1}
X_{\beta_1\cdots\beta_n}^{\qquad\alpha}({\Gamma}).
\e{28}
We may furthermore construct gauge
invariant functions
$\overline{G}({\Gamma},\Phi)$ to any function $G({\Gamma})$ by the conditions
\begin{eqnarray}
&&(\overline{G},T^\alpha){_{\rm ext}}=0,\quad\overline{G}|_{\Phi=0}=G.
\e{29}
Also these conditions may be solved
perturbatively with an ansatz of the form
\begin{eqnarray}
&&\overline{G}({\Gamma},\Phi)=G({\Gamma})+\sum_{n=1}^\infty{1\over
n!}\Phi^{\beta_n}\cdots\Phi^{\beta_1}Y_{\beta_1\cdots\beta_n}({\Gamma}).
\e{30}
In appendix B it is shown that
\begin{eqnarray}
&&(\overline{F},\overline{G}){_{\rm ext}}|_{\Phi=0}=(F,G)_{({\cal D})},
\e{31}
where the right-hand side is the Dirac antibracket \r{10}.
One may also show that (see below)
\begin{eqnarray}
&&\Delta{_{\rm ext}}\overline{G}|_{\Phi=0}=\Delta_{({\cal D})}G
\e{32}
provided
\begin{eqnarray}
&&\Delta{_{\rm ext}} T^\alpha=0.
\e{36}
$\Delta{_{\rm ext}}$ is the corresponding
$\Delta$-operator to \r{1} on the extended
manifold
${\cal M}{_{\rm ext}}$ with the metric \r{26} and with
a measure density $\rho({\Gamma},\Phi)$
satisfying \r{36}. To show
\r{31} it is sufficient to solve \r{28} and \r{30}
up to the first order in $\Phi$
as is shown in appendix B, while \r{32} requires
a solution up to second order.
The
gauge independent path integral \r{15} is here given by
\begin{eqnarray}
&&Z_{T}=\int\exp{\left\{{i\over\hbar}[{W}{_{\rm ext}}+{X}{_{\rm ext}}]\right\}}
\prod_\alpha{\delta}(T^\alpha)\prod_\beta{\delta}(\chi_\beta)\:{1\over{\rm
sdet}(\chi_{\gamma},T^{\delta})}\rho({\Gamma},\Phi)[d{\Gamma}][d\Phi][d\eta],\nonumber\\
\e{33}
where ${W}{_{\rm ext}}$ and ${X}{_{\rm ext}}$ satisfy the quantum master equations
\begin{eqnarray}
&&\Delta{_{\rm ext}} e^{{i\over\hbar}{W}{_{\rm ext}}}=0,\quad
\Delta{_{\rm ext}} e^{{i\over\hbar}{X}{_{\rm ext}}}=0.
\e{34}
According to \r{16} we have additional conditions like (see appendix A)
\begin{eqnarray}
&&({W}{_{\rm ext}},T^\alpha){_{\rm ext}}=i\hbar\Delta{_{\rm ext}}
T^\alpha,\quad({X}{_{\rm ext}},T^\alpha){_{\rm ext}}=i\hbar\Delta{_{\rm ext}} T^\alpha.
\e{35}
They are easy to solve for measure densities satisfying \r{36}
since these conditions then reduce to
\begin{eqnarray}
&&(W{_{\rm ext}},T^\alpha){_{\rm ext}}=(X{_{\rm ext}},T^\alpha){_{\rm ext}}=0.
\e{37}
This implies that ${W}{_{\rm ext}}$ and ${X}{_{\rm ext}}$ in this case
are gauge invariant extensions
of
${W}$ and ${X}$ defined by
\begin{eqnarray}
&&{W({\Gamma})}\equiv{W}{_{\rm ext}}|_{\Phi=0},\quad {X({\Gamma})}\equiv{X}{_{\rm ext}}|_{\Phi=0},
\e{38}
which means that ${W}{_{\rm ext}}=\overline{W}$ and ${X}{_{\rm ext}}=\overline{X}$.
It follows now that the path integral \r{33} in
{\em e.g.\ } the gauge $\chi_\alpha=\omega_{\alpha\beta}\Phi^\beta$ reduces to
\begin{eqnarray}
&&Z_{T}=\int\exp{\left\{{i\over\hbar}[W+X]\right\}}
\prod_\alpha{\delta}(\Theta^\alpha)\:{1\over{\rm
sdet}(X_{\gamma}^{\;{\delta}})}\rho({\Gamma},0)[d{\Gamma}][d\eta],
\e{39}
which when compared to the second class expression
$Z_{({\cal D})}$ in \r{12} requires the
boundary condition
\begin{eqnarray}
&&\rho({\Gamma},0)=\rho_{({\cal D})}({\Gamma})\:{\rm sdet}(X_{\gamma}^{\;{\delta}}),
\e{40}
where $X_{\gamma}^{\;{\delta}}$ is the first order coefficient in
the expansion \r{28}.
That $W$ and $X$ satisfy the appropriate master equations
follows from \r{31} and
\r{32}.
Another equivalent but more explicit and transparent way
to derive the equivalence
between
\r{12} and \r{33} is to first construct gauge invariant coordinates
${\overline{\Gamma}}^A({\Gamma},\Phi)$ defined by
\begin{eqnarray}
&&({\overline{\Gamma}}^A,T^\alpha){_{\rm ext}}=0,\quad {\overline{\Gamma}}^A|_{\Phi=0}={\Gamma}^A,
\e{41}
which also may be solved by a
perturbative ansatz like \r{30}. Then we have
$\overline{G}=G({\overline{\Gamma}})$ for any gauge invariant function and
\begin{eqnarray}
&&({\overline{\Gamma}}^A,{\overline{\Gamma}}^B){_{\rm ext}}|_{\Phi=0}=E^{AB}_{({\cal D})}.
\e{42}
Thus, the gauge invariant functions lives on
the submanifold of ${\cal M}{_{\rm ext}}$ spanned by
${\overline{\Gamma}}^A$. The same is true for the
$\Delta$-operator as will be shown below.
It is convenient to change coordinates on
${\cal M}{_{\rm ext}}$ from $\{{\Gamma},\Phi\}$ to
$\{{\overline{\Gamma}},\Phi\}$. In terms of these coordinates we have
\begin{eqnarray}
&&(\overline{F},\overline{G}){_{\rm ext}}=F({\overline{\Gamma}}){\stackrel{\lea}{\overline{\partial}}}_A
\overline{E}^{AB}{\overline{\partial}}_BG({\overline{\Gamma}}),\nonumber\\
&&\overline{E}^{AB}\equiv({\overline{\Gamma}}^A,{\overline{\Gamma}}^B){_{\rm ext}},
\e{43}
where ${\overline{\partial}}_A$ are derivatives with respect to ${\overline{\Gamma}}^A$.
Furthermore, we have
\begin{eqnarray}
&&{\Delta}{_{\rm ext}}=\frac{1}{2}(-1)^{{\varepsilon}_A}\overline{\rho}^{-1}
{\overline{\partial}}_A\overline{\rho}\overline{E}^{AB}{\overline{\partial}}_B+
\frac{1}{2}(-1)^{{\varepsilon}_A}\overline{\rho}^{-1}{\overline{\partial}}_A\overline{\rho}
\overline{E}^{A\beta}{\overline{\partial}}_\beta+\nonumber\\
&&+\frac{1}{2}(-1)^{{\varepsilon}_\alpha}\overline{\rho}^{-1}{\overline{\partial}}_\alpha
\overline{\rho}\overline{E}^{\alpha B}{\overline{\partial}}_B+
\frac{1}{2}(-1)^{{\varepsilon}_\alpha}\overline{\rho}^{-1}{\overline{\partial}}_\alpha\overline{\rho}
\overline{E}^{\alpha\beta}{\overline{\partial}}_\beta,
\e{44}
where ${\overline{\partial}}_\alpha$ are derivatives with respect to
$\Phi^\alpha$ while keeping ${\overline{\Gamma}}^A$
fixed, and where $\overline{\rho}$ is related to
$\rho$ through the formula
\begin{eqnarray}
&&\overline{\rho}({\overline{\Gamma}},\Phi)\:{\rm sdet}({\partial}_A{\overline{\Gamma}}^B)=\rho({\Gamma},\Phi).
\e{45}
The ``bar"-metric in \r{44} is given in \r{43} and
\begin{eqnarray}
&&\overline{E}^{A\beta}\equiv({\overline{\Gamma}}^A,\Phi^\beta){_{\rm ext}}=
{\overline{\Gamma}}^A{\stackrel{\lea}{\partial}}_\alpha\omega^{\alpha\beta},\quad
\overline{E}^{\alpha B}\equiv(\Phi^\alpha,{\overline{\Gamma}}^B){_{\rm ext}}=
\omega^{\alpha\beta}{\partial}_\beta{\overline{\Gamma}}^B,\nonumber\\
&&\overline{E}^{\alpha \beta}\equiv(\Phi^\alpha,
\Phi^\beta){_{\rm ext}}=\omega^{\alpha\beta}.
\e{46}
Since the Jacobi identities yield
\begin{eqnarray}
&&(\overline{E}^{AB},T^\alpha){_{\rm ext}}=0,
\e{48}
we have also
\begin{eqnarray}
&&\overline{E}^{AB}={E}^{AB}_{({\cal D})}({\overline{\Gamma}})
\e{49}
due to \r{42}.
The $\Delta$-operator expression \r{44} may be decomposed as follows
\begin{eqnarray}
&&{\Delta}{_{\rm ext}}=\overline{\Delta}+K^\alpha{\overline{\partial}}_\alpha,\nonumber\\
&&\overline{\Delta}=
\frac{1}{2}(-1)^{{\varepsilon}_A}\overline{\rho}^{-1}{\overline{\partial}}_A
\overline{\rho}\overline{E}^{AB}{\overline{\partial}}_B+
\frac{1}{2}(-1)^{{\varepsilon}_\alpha}\overline{\rho}^{-1}({\overline{\partial}}_\alpha
\overline{\rho}\overline{E}^{\alpha
B}){\overline{\partial}}_B.
\e{47}
Obviously ${\Delta}{_{\rm ext}}\overline{F}=\overline{\Delta}\,\overline{F}$
for any gauge invariant
function $\overline{F}$.
Furthermore, since the condition
${\Delta}{_{\rm ext}}
T^\alpha\equiv{\Delta}{_{\rm ext}}\Theta^\alpha({\overline{\Gamma}})=
\overline{\Delta}\Theta^\alpha({\overline{\Gamma}})=0$ implies that
\begin{eqnarray}
&&\frac{1}{2}(-1)^{{\varepsilon}_\alpha}\overline{\rho}^{-1}({\overline{\partial}}_\alpha\overline{\rho}\overline{E}^{\alpha
B})=\frac{1}{2}(-1)^{{\varepsilon}_A}F_A\overline{E}^{AB}
\e{50}
for any function $F_A$, we have
\begin{eqnarray}
&&\overline{\Delta}=
\frac{1}{2}(-1)^{{\varepsilon}_A}({\overline{\partial}}_A+F_A+({\overline{\partial}}_A\ln\overline{\rho}))\overline{E}^{AB}{\overline{\partial}}_B.
\e{51}
Therefore, if we restrict $F_A$ such that $F_A+({\overline{\partial}}_A\ln\overline{\rho})$
only depends on
${\overline{\Gamma}}^A$ then also $\overline{\Delta}$ only depends on
${\overline{\Gamma}}^A$ due to \r{49}. The nilpotency of
$\Delta{_{\rm ext}}$ requires then that
$\overline{\Delta}$ is
nilpotent which in turn implies \cite{BT94-2}
\begin{eqnarray}
&&F_A+({\overline{\partial}}_A\ln\overline{\rho})={\overline{\partial}}_A\ln\widetilde{\rho}({\overline{\Gamma}}).
\e{52}
This result may equivalently be expressed as follows:
In order for the measure density
$\rho$ to satisfy
\r{36} it should be such that
\begin{eqnarray}
&&\rho[d{\Gamma}][d\Phi]=\overline{\rho}[d\overline{{\Gamma}}][d\Phi]=
\widetilde{\rho}(\overline{{\Gamma}})[d\overline{{\Gamma}}][d
T^*]
\e{521}
for any measure density $\widetilde{\rho}(\overline{{\Gamma}})$ where
$T^*_\alpha$ satisfies
\begin{eqnarray}
&&(T^\alpha,T^*_\beta){_{\rm ext}}={\delta}_\beta^\alpha,
\quad(T^*_\alpha,T^*_\beta){_{\rm ext}}=0,\quad{\varepsilon}(T^*_\alpha)=
{\varepsilon}_\alpha+1.
\e{522}
We assert that there is a solution of the form
\begin{eqnarray}
&&T^*_\alpha({\Gamma},\Phi)=\Theta^*_\alpha({\Gamma})+\sum_{n=1}^\infty{1\over
n!}\Phi^{\beta_n}\cdots
\Phi^{\beta_1}X^*_{\beta_1\cdots\beta_n\alpha}({\Gamma}).
\e{523}
where also the function $\Theta^*_\alpha({\Gamma})$
is to be determined (see appendix C).
Obviously
$\Delta{_{\rm ext}}\overline{G}|_{\Phi=0}=
\overline{\Delta}\,\overline{G}|_{\Phi=0}=\Delta_{({\cal D})}G$ in
agreement with the assertion \r{32}, provided
\begin{eqnarray}
&&\widetilde{\rho}({\overline{\Gamma}})=\rho_{({\cal D})}({\overline{\Gamma}}),
\e{53}
where $\rho_{({\cal D})}$ is the Dirac measure density in \r{12} and \r{13}.
Thus, the $\Delta$-operator
$\overline{\Delta}$ is just a gauge invariant
extension of
$\Delta_{({\cal D})}$ on
${\cal M}{_{\rm ext}}$.
It should also be mentioned that the
transformation \r{521} with the identification
\r{53} is consistent with the boundary
condition \r{40} due to the relation
\begin{eqnarray}
&&(T^*_\alpha,\Phi^\beta){_{\rm ext}}|_{\Phi=0,\Theta=0}
=-\frac{1}{2}(X^{-1})_\alpha^{\;\beta},
\e{54}
which follows from \r{522} and \r{523}
to lowest order in $\Phi^\alpha$ (see
formula \r{c9} in appendix C).
\vspace{1cm}
\newpage
{\bf Acknowledgements}\\
I.A.B. thanks Klaus Bering and Poul Damgaard
for stimulating discussions at an early
stage of this work. I.A.B. would also like to thank
Lars Brink for very
warm hospitality at the Institute of Theoretical
Physics, Chalmers and G\"{o}teborg
University. The work is partially supported by grant INTAS-RFBR 95-0829.
The work of I.A.B. is also
supported by grants
INTAS 93-2058, INTAS 93-0633, RFBR 96-01-00482,
RFBR 96-02-17314, and NorFa 97.40.002-O.\\ \\
\def\thesection.\arabic{equation}{\thesection.\arabic{equation}}
\setcounter{section}{1}
\setcounter{equation}{0}
\renewcommand{\thesection}{\Alph{section}}
\noindent
{\Large{\bf{Appendix A}}}\\ \\
{\bf Invariant formulation of the path integral $Z_T$}\\ \\
Let us extend the antisymplectic manifold ${\cal M}$
in section 2 by three sets of field-antifield
pairs:
$\{{\cal C}^*_\alpha,{\cal C}^\alpha;
\pi_\alpha^*,\pi^\alpha;{\overline{{\cal C}}}^*_\alpha,{\overline{{\cal C}}}^{\alpha}\}$. Their
Grassmann parities are
\begin{eqnarray}
&&{\varepsilon}{({\cal C}^\alpha)}={\varepsilon}{(\pi^\alpha)}=
{\varepsilon}{({\overline{{\cal C}}}^{*}_\alpha)}={\varepsilon}_{\alpha}\equiv{\varepsilon}{(T^\alpha)},
\quad{\varepsilon}{({\cal C}^*_\alpha)}={\varepsilon}{(\pi_\alpha^*)}={\varepsilon}{({\overline{{\cal C}}}^\alpha)}={\varepsilon}_{\alpha}+1.
\e{a1}
On this
extended manifold, $\widetilde{{\cal M}}$, we define the bosonic charge
\begin{eqnarray}
&&\Omega=\left(
T_\alpha{\cal C}^\alpha-\frac{1}{2}{\cal C}^*_\alpha
U^\alpha_{\;\beta{\gamma}}{\cal C}^{\gamma}{\cal C}^\beta(-1)^{{\varepsilon}_\beta}
+\ldots\right)-\pi_\alpha^*{\overline{{\cal C}}}^{\alpha},
\e{a2}
where $T_\alpha$ satisfies the algebra \r{14}
and where the dots indicate terms
independent of
$\pi^\alpha$ and
${\overline{{\cal C}}}^*_\alpha$ such that
$\Omega$ satisfies
\begin{eqnarray}
&&(\Omega,\Omega)=0,
\e{a3}
where the antibracket from now on is the
extended one on $\widetilde{{\cal M}}$. An invariant
path integral
may then be written as
\begin{eqnarray}
&&Z_T=\int \exp{\left\{{i\over\hbar}[\widetilde{W}+
\widetilde{X}+(\Psi,\Omega)]\right\}}\widetilde{\rho}
[d{\Gamma}][d\eta][d{\cal C}][d{\cal C}^*][d\pi][d\pi^*][d{\overline{{\cal C}}}][d{\overline{{\cal C}}}^*],
\e{a4}
where $\widetilde{W}$ and $\widetilde{X}$
satisfy the extended master equations
\begin{eqnarray}
&&\widetilde{\Delta}e^{{i\over\hbar}\widetilde{W}}=0,\quad\widetilde{\Delta}
e^{{i\over\hbar}\widetilde{X}}=0,
\e{a5}
where in turn $\widetilde{\Delta}$ is
the nilpotent $\Delta$-operator \r{1} extended to
$\widetilde{{\cal M}}$ with $\widetilde{\rho}$. The objects
$\Omega$,
$\widetilde{W}$ and
$\widetilde{X}$ are in addition required to satisfy
\begin{eqnarray}
&&\sigma_{\widetilde{W}}\Omega=\sigma_{\widetilde{X}}\Omega=0.
\e{a7}
The gauge-fixing charge $\Psi$ is odd and is of the form
\begin{eqnarray}
&&\Psi={\overline{{\cal C}}}^*_\alpha\chi^\alpha-{\cal C}^*_\alpha\pi^\alpha.
\e{a8}
The solutions of \r{a5} are
\begin{eqnarray}
&&\widetilde{W}=W({\Gamma})-{\cal C}^*_\alpha
P^\alpha_{\;\beta}{\cal C}^\beta+\ldots,\quad\widetilde{X}=X({\Gamma})-{\cal C}^*_\alpha
Q^\alpha_{\;\beta}{\cal C}^\beta+\ldots,
\e{a6}
The equations for $W$ and $X$ reduce to \r{5} if and only if
$P^\alpha_{\;\alpha}=Q^\alpha_{\;\alpha}=0$. Otherwise $W$ and $X$ generalize
to satisfy the modified equations with supertrace
``anomalies" in their right-hand sides (see appendix B).
The formulation given in section 2
corresponds therefore to the first case. Notice that
eq.\r{16} is obtained from
\r{a7} when \r{a6} is inserted, and that with the choice \r{a8}
the path integral \r{a4} reduces to
\r{15} after the identifications
$\pi^*_\alpha\equiv\xi_\alpha$,
${\overline{{\cal C}}}_\alpha^*\equiv{\overline{{\cal C}}}_\alpha$, and provided
$\widetilde{\rho}={\rho}({\Gamma})$ and
$\chi^\alpha$ only depends on ${\Gamma}^A$.
The path integral \r{a4} is invariant
under the following transformation
\begin{eqnarray}
&&{\delta}\widetilde{{\Gamma}}^{\cal A}=(\widetilde{{\Gamma}}^{\cal A},\Omega)\mu,
\e{a9}
where
$\widetilde{{\Gamma}}^{{\cal A}}\equiv\{{\Gamma}^A;{\cal C}^*_\alpha,{\cal C}^\alpha;
\pi_\alpha^*,\pi^\alpha;{\overline{{\cal C}}}^{*}_{\alpha},{\overline{{\cal C}}}^\alpha\}\in\widetilde{{\cal M}}$ and
where $\mu$ is an odd constant.
The contribution to the Jacobian
\begin{eqnarray}
&&J-1=2(\widetilde{\Delta}\Omega)\mu
\e{a10}
is compensated by corresponding
terms from \r{a7}. The path integral \r{a4} is
also independent of $\Psi$ since
$\Psi{\rightarrow}\Psi+{\delta}\Psi$ is compensated by the
additional contribution to the Jacobian
from the transformation \r{a9} with the
choice
$\mu=i{\delta}\Psi/\hbar$. Furthermore,
\r{a4} is independent of $\widetilde{X}$ which
contains the hypergauge-fixing.
To see this consider the transformation
\begin{eqnarray}
&&{\delta}\widetilde{{\Gamma}}^{\cal A}=(\widetilde{{\Gamma}}^{\cal A},-
\widetilde{W}+\widetilde{X})\xi+{\hbar\over
i}(\widetilde{{\Gamma}}^{\cal A},\xi),
\e{a11}
where $\xi$ is an odd infinitesimal
function satisfying the condition
\begin{eqnarray}
&&(\xi,\Omega)=0.
\e{a12}
The Jacobian of \r{a11} is
\begin{eqnarray}
&&J-1=2(-\widetilde{\Delta}\widetilde{W}+
\widetilde{\Delta}\widetilde{X})\xi+2{\hbar\over
i}\widetilde{\Delta}\xi+(-\widetilde{W}+\widetilde{X},\xi).
\e{a13}
Summing up the total contribution from
\r{a11} in \r{a4} one finds after use of the
master equations \r{a5} that what remains
may be viewed as the following
transformations
\begin{eqnarray}
&&{\delta}\Psi=(\Psi,-\widetilde{W}+\widetilde{X})\xi+{\hbar\over i}(\Psi,\xi),\quad
{\delta}\widetilde{X}=2{\hbar\over i}\sigma_{\widetilde{X}}\xi.
\e{a14}
However, since $Z_T$ is independent of $\Psi$
as was shown above only ${\delta}\widetilde{X}$
remains. Notice that $({\delta}\widetilde{X},\Omega)=0$
in consistency with \r{a7}.
One may also notice that \r{a4} is invariant under
general anticanonical transformations of the form
\begin{eqnarray}
&&{\delta}\widetilde{{\Gamma}}^{\cal A}=(\widetilde{{\Gamma}}^{\cal A},G)
\e{a15}
for any odd infinitesimal function $G$ provided
$\widetilde{W}$ and $\widetilde{X}$ transform
according to
\begin{eqnarray}
&&{\delta}\widetilde{W}=\sigma_{\widetilde{W}}G,
\quad{\delta}\widetilde{X}=\sigma_{\widetilde{X}}G.
\e{a16}
This may be used to demonstrate the
existence of the above formulation. Consider the
abelian case
\begin{eqnarray}
&&\Omega_{\rm Abel}=
t_\alpha{\cal C}^\alpha-\pi_\alpha^*{\overline{{\cal C}}}^{\alpha},
\quad (\Omega_{\rm Abel},\Omega_{\rm Abel})=0.
\e{a17}
Let the master actions here satisfy $\widetilde{W}=
{W}$ and $\widetilde{X}={W}$. (This
implies that
\r{a7} reduces to
\r{35}.) We define then $\Omega$, $\widetilde{W}$,
$\widetilde{X}$ in terms of a finite
transformation of $\Omega_{\rm Abel}$, ${W}$,
${X}$ of the form \r{a15}-\r{a16}, {\em i.e.\ }
\begin{eqnarray}
&&\Omega\equiv\exp{(G,\cdot)}\Omega_{\rm Abel},\quad
e^{{i\over\hbar}\widetilde{W}}=
e^{[\widetilde{\Delta},G]}e^{{i\over\hbar}{W}},\quad
e^{{i\over\hbar}\widetilde{X}}=
e^{[\widetilde{\Delta},G]}e^{{i\over\hbar}{X}}.
\e{a18}
Obviously $(\Omega,\Omega)=0$, and if
$W$, $X$ satisfies \r{a7} with $\Omega_{\rm
Abel}$ then $\widetilde{W}$, $\widetilde{X}$ satisfy
\r{a7} with $\Omega$. In this manner
the non-abelian case is obtained by
choosing the anticanonical generator $G$ in the
following form
\begin{eqnarray}
&&G={\cal C}^*_\alpha \Lambda^\alpha_{\;\beta}{\cal C}^\beta,
\e{a19}
where the matrix $e^\Lambda$ changes
effectively the constraint basis.\\ \\
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\noindent
{\Large{\bf{Appendix B}}}\\ \\
{\bf The path integral \r{15} with anomalous master actions}\\ \\
Consider the path integral \r{15} in section 2
where $T_\alpha$ satisfies the algebra \r{14} and the
properties \r{16}. According to appendix A the
master actions $W$ and $X$ do no longer
satisfy the master equation \r{5} if
$P^\alpha_{\;\alpha}\neq0$ and $Q^\alpha_{\;\alpha}\neq0$.
The appropriate generalized master equations in the latter case are
\begin{eqnarray}
&&(i\hbar\Delta +P^\alpha_{\;\alpha}) e^{{i\over \hbar}W}=0\quad{\Leftrightarrow}\quad
\frac{1}{2}(W,W)=i\hbar\Delta W-i\hbar P^\alpha_{\;\alpha},\nonumber\\
&&(i\hbar\Delta +Q^\alpha_{\;\alpha}) e^{{i\over \hbar}X}=0\quad{\Leftrightarrow}\quad
\frac{1}{2}(X,X)=i\hbar\Delta X-i\hbar Q^\alpha_{\;\alpha}.
\e{b1}
Since a consistent theory requires $\Delta$ to be
nilpotent we have from \r{b1} the
consistency conditions
\begin{eqnarray}
&&\Delta(P^\alpha_{\;\alpha} e^{{i\over \hbar}W})=0\quad{\Leftrightarrow}\quad
\sigma_W P^\alpha_{\;\alpha}=0,\nonumber\\
&&\Delta (Q^\alpha_{\;\alpha} e^{{i\over \hbar}X})=0\quad{\Leftrightarrow}\quad
\sigma_X Q^\alpha_{\;\alpha}=0.
\e{b2}
The proof of the invariance under \r{17} as well
as the independence of the gauge
fixing function $\chi^\alpha$ given in \r{23}-\r{25}
are still valid in this generalized
case. We may also prove the independence of the
gauge fixing action $X$ following the
argument of appendix A in a reduced form. We perform then the
following change of integration variables in the path integral \r{15}:
\begin{eqnarray}
&{\delta}\Gamma^A=&(\Gamma^A,-W+X)\mu+{\hbar\over
i}(\Gamma^A,\mu),\quad{\delta}{\cal C}^\alpha=
(P^\alpha_{\;\beta}-Q^\alpha_{\;\beta})\,{\cal C}^\beta
\mu+{\hbar\over i}R^\alpha_{\;\beta}\,{\cal C}^\beta,\nonumber\\
&{\delta}\pi^\alpha=&(P^\alpha_{\;\beta}-Q^\alpha_{\;\beta})\,\pi^\beta \mu+{\hbar\over
i}R^\alpha_{\;\beta}\,\pi^\beta+{\hbar\over
i}{\overline{{\cal C}}}_\beta\,(\chi^\beta,R^\alpha_{\;{\gamma}})\,{\cal C}^{\gamma}+\nonumber\\&&+{\overline{{\cal C}}}_\beta\,
(\chi^\beta,P^\alpha_{\;{\gamma}}
-Q^\alpha_{\;{\gamma}})\,{\cal C}^{\gamma}
\mu,
\e{b3}
where $\mu$ is an odd function which satisfies the condition
\begin{eqnarray}
&&(\mu, T_\alpha)=T_\beta R^\beta_{\;\alpha},
\e{b4}
which in turn determines $R^\beta_{\;\alpha}$.
The change of integration variables \r{b3}
in
\r{15} results in the following change in $X$:
\begin{eqnarray}
&&{\delta} X={2\hbar\over i}\left[\sigma_X \mu-i\hbar
R^\alpha_{\;\alpha}(-1)^{{\varepsilon}_\alpha}\right],
\e{b5}
together with the following variation in $\chi^\alpha$:
\begin{eqnarray}
&&{\delta}\chi^\alpha=(\chi^\alpha,-W+X)\mu+{\hbar\over i}(\chi^\alpha, \mu),
\e{b6}
which is inessential as the previous proof of
$\chi$-independence remains valid.
The change ${\delta} X$ in \r{b5} is the most general one
compatible with the allowed
variation
\begin{eqnarray}
&&{\delta} Q^\alpha_{\;\alpha}={2\hbar\over i}\left[\sigma_X
R^\alpha_{\;\alpha}(-1)^{{\varepsilon}_\alpha}+(Q^\alpha_{\;\alpha},
\mu)\right]
\e{b7}
in the modified equation for $X$ given in \r{b1}.
This variation is the trace of the
matrix variation
\begin{eqnarray}
&&{\delta} Q^\alpha_{\;\beta}={2\hbar\over i}\left[\sigma_X
R^\alpha_{\;\beta}(-1)^{{\varepsilon}_\alpha}+(Q^\alpha_{\;\beta},
\mu)+Q^\alpha_{\;{\gamma}} R^{\gamma}_{\;\beta}-R^\alpha_{\;{\gamma}} Q^{\gamma}_{\;\beta}+i\hbar
S^{\alpha{\gamma}}_{{\gamma}\beta}\right],
\e{b8}
where in turn $S^{\alpha{\gamma}}_{{\gamma}\beta}$ is determined by the relation
\begin{eqnarray}
&&(R^{\gamma}_{\;\alpha}, T_\beta)-(R^{\gamma}_{\;\beta},
T_\alpha)(-1)^{({\varepsilon}_\alpha+1)({\varepsilon}_\beta+1)}+(-1)^{{\varepsilon}_{\gamma}}(\mu,
U^{\gamma}_{\;\alpha\beta})-\nonumber\\
&&-R^{\gamma}_{\;{\delta}} U^{\delta}_{\;\alpha\beta}+U^{\gamma}_{\;\alpha{\delta}}
R^{\delta}_{\;\beta}-U^{\gamma}_{\;\beta{\delta}}
R^{\delta}_{\;\alpha}(-1)^{({\varepsilon}_\alpha+1)({\varepsilon}_\beta+1)}+T_{\delta}
S^{{\delta}{\gamma}}_{\alpha\beta}(-1)^{{\varepsilon}_{\gamma}}=0,
\e{b9}
which is a compatibility condition to \r{b4}.
In terms of ${\delta} Q^\alpha_{\;\beta}$ we
have
\begin{eqnarray}
&&({\delta} X,T_\alpha)=T_\beta \,{\delta} Q^\beta_{\;\alpha}.
\e{b10}
The main part of these formulas may be derived from
\r{a11}-\r{a14} with the ansatz
\begin{eqnarray}
&&\xi=\mu+{\cal C}^*_\alpha R^\alpha_{\;\beta}{\cal C}^\beta+{1\over
4}{\cal C}^*_\alpha{\cal C}^*_\beta S^{\beta\alpha}_{{\delta}{\gamma}}
{\cal C}^{\gamma}{\cal C}^{\delta}(-1)^{{\varepsilon}_{\delta}}+\ldots.
\e{b11}
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\noindent
{\Large{\bf{Appendix C}}}\\ \\
{\bf Proof of formula \r{31}}\\ \\
Inserting the ansatz \r{28} into \r{27} one
finds to the zeroth order in $\Phi^\alpha$ the
condition
\begin{eqnarray}
&&(\Theta^\alpha,\Theta^\beta)\equiv
E^{\alpha\beta}=-(-1)^{{\varepsilon}_{\gamma}(1+{\varepsilon}_\alpha)}
X_{\gamma}^{\;\alpha}\omega^{{\gamma}\rho}X_\rho^{\;\beta}
\e{c1}
for the first order coefficients
$X_\beta^{\;\alpha}({\Gamma})$ in \r{28}. This implies that
\begin{eqnarray}
&&E_{\alpha\beta}=-(-1)^{{\varepsilon}_\beta(1+{\varepsilon}_{\gamma})}(X^{-1})_\alpha^{\;\rho}
\omega_{\rho{\gamma}}(X^{-1})_\beta^{\;{\gamma}},
\e{c2}
where $(X^{-1})_\alpha^{\;\rho}$
is the inverse to $X_\beta^{\;\alpha}$ in the sense
\begin{eqnarray}
&&(X^{-1})_\alpha^{\;{\gamma}}X_{\gamma}^{\;\beta}=
X_\alpha^{\;{\gamma}}(X^{-1})_{\gamma}^{\;\beta}={\delta}_\alpha^\beta.
\e{c3}
For the gauge invariant functions
$\overline{F}$ and $\overline{G}$ in \r{31} we have to the
first order in
$\Phi$
\begin{eqnarray}
&&\overline{F}=F({\Gamma})+\Phi^\alpha Y_\alpha(F)+O(\Phi^2),\quad\overline{G}=
G({\Gamma})+\Phi^\alpha Y_\alpha(G)+O(\Phi^2),
\e{c4}
where
\begin{eqnarray}
&Y_\alpha(G)&=-(-1)^{{\varepsilon}_\alpha(1+{\varepsilon}_G)}(G,
\Theta^{\gamma})(X^{-1})_{\gamma}^{\;\beta}\omega_{\beta\alpha}
=\nonumber\\
&&=-(-1)^{{\varepsilon}_{\gamma}(1+{\varepsilon}_\beta)}
\omega_{\alpha\beta}(X^{-1})_{\gamma}^{\;\beta}(\Theta^{\gamma},G).
\e{c5}
These expressions and \r{c2} imply now
\begin{eqnarray}
&&(\overline{F},\overline{G}){_{\rm ext}}|_{\Phi=0}=(F,G)+(-1)^{{\varepsilon}_\alpha(1+{\varepsilon}_F)}
Y_\alpha(F)\omega^{\alpha\beta}Y_\beta(G)=\nonumber\\
&&=(F,G)-(F,\Theta^\alpha)E_{\alpha\beta}
(\Theta^\beta,G)\equiv(F,G)_{({\cal D})}.
\e{c6}\\ \\
\noindent
{\bf Equation for $\Theta^*_\alpha$ in \r{523}}\\ \\
In parallel to \r{28}, if one inserts
the ansatz \r{523} into eq.\r{522} one gets
to the zeroth order in $\Phi^\alpha$ the equations
\begin{eqnarray}
&&(\Theta^\alpha,\Theta^*_\beta)+(-1)^{{\varepsilon}_\mu(1+{\varepsilon}_\alpha)}
X^{\;\alpha}_\mu\omega^{\mu\nu}X^*_{\nu\beta}={\delta}^\alpha_\beta,
\e{c7}
\begin{eqnarray}
&&(\Theta^*_\alpha,
\Theta^*_\beta)+(-1)^{{\varepsilon}_\mu{\varepsilon}_\alpha}
X^*_{\mu\alpha}\omega^{\mu\nu}X^*_{\nu\beta}=0.
\e{c8}
Solving \r{c7} for $X^*_{\alpha\beta}$ one finds
\begin{eqnarray}
&&X^*_{\alpha\beta}= - \left({\delta}_\beta^{\gamma}+
(\Theta^*_\beta, \Theta^{\gamma}) \right)
(X^{-1})_{\gamma}^{\;{\delta}}
\omega_{{\delta}\alpha}(-1)^{{\varepsilon}_\alpha{\varepsilon}_\beta},
\e{c9}
which when inserted into \r{c8} results in
the following equation for $\Theta^*_\alpha$:
\begin{eqnarray}
&&(\Theta^*_\alpha,\Theta^*_\beta)_{({\cal D})}=-E_{\alpha\beta}-
(\Theta^*_\alpha,\Theta^{\gamma})E_{{\gamma}\beta}+
E_{\alpha{\gamma}}(\Theta^{\gamma},\Theta^*_\beta).
\e{c10}
The solution is of the form
\begin{eqnarray}
&&\Theta^*_\alpha=\frac{1}{2}\Theta^{\gamma} E_{{\gamma}\alpha}+O(\Theta^2).
\e{c11}
To confirm the existence of $T^*_\alpha$ satisfying \r{522}
we notice that for abelian second class
constraints $\theta^\alpha({\Gamma})=0$ satisfying
$(\theta^\alpha,\theta^\beta)=
-\omega^{\alpha\beta}$ we have the following explicit
conversion formula for corresponding abelian
first class constraints $t^\alpha$ and $t^*_\alpha$:
\begin{eqnarray}
&&t^\alpha=\theta^\alpha+\Phi^\alpha,\quad
t^*_\alpha=-\frac{1}{2}(\theta^{\gamma}-\Phi^{\gamma})\omega_{{\gamma}\alpha}
\e{c12}
The general abelian functions $T^\alpha$ and $T^*_\alpha$ are then obtained
from $t^\alpha$ and $t^*_\alpha$ through the formula
\begin{eqnarray}
&&T^\alpha=\exp{(G,\cdot){_{\rm ext}}}\,t^\alpha,
\quad T^*_\alpha=\exp{(G,\cdot){_{\rm ext}}}\,t^*_\alpha
\e{c13}
where $G({\Gamma},\Phi)$ is an odd function.
\newpage
|
2,877,628,090,813 | arxiv | \section{Supplementary material for "Vaporization dynamics of a dissipative quantum liquid"}
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\section{The current operator}
The physical fermion field is decomposed into right and left moving excitations as
$\Psi(x)=e^{ik_F x}\Psi_R(x) +e^{-ik_F x}\Psi_L(x)$ with $k_F$ the Fermi wavenumber.
The current operator can be rewritten as $j(x)=j_0+\exp(i2k_Fx)j_{2k_F}+\exp(-i2k_Fx)j_{2k_F}^+$, where the
long ($j_0$) and short ($j_{2k_F}$) wavelength current operators are determined from Eq. (15) in the main text in the continuum limit as
\begin{gather}
j_0(x)\sim \Psi_R^+(x)\Psi_R(x)-\Psi_L^+(x)\Psi_L(x),\\
j_{2k_F}\sim \Psi_R^+(x)\partial_x\Psi_L(x)-\left[\partial_x\Psi_R^+(x)\right]\Psi_L(x).
\end{gather}
Notably, the second expression contains an additional gradient compared to the first one, which increases its scaling dimension by one and is
considered to be more irrelevant than the long wavelength, $q\sim 0$
term in equilibrium.
We assume that this classification remains also valid for the early time dynamics of Lindblad description, and therefore retain only $j_0$ in the jump operator.
This is verified by comparing bosonization to numerics.
\section{Time evolution of the single particle density matrix}
The single particle density matrix is defined in Eq. (7) of the main text.
Following standard steps \cite{giamarchi,cazalillaprl}, we obtain
\begin{gather}
\frac{G(x,t)}{G_\mathrm{nonint}(x)}=
\exp\left(-\sum_{q>0}\frac{4\pi}{Lq}n^B_q(t)\left(1-\cos(qx)\right)\right),
\label{eq:green3}
\end{gather}
where
\begin{gather}
G_\mathrm{nonint}(x)=\frac{i}{2\pi(x+i \alpha)}
\end{gather}
is the non-interacting single particle density matrix and
\begin{gather}
n^{B}_q(t) =\mathrm{Tr}\left[\rho(t)b_q^{+}b_q\right]
\end{gather}
is the instantaneous number of $b$-bosons which describe the elementary excitations of the non-interacting system. After Bogoliubov transformation, the number of $b$-bosons is expressed as
\begin{gather}
n^{B}_q(t) = \frac{v}{\tilde v}n_q(t) - \frac{g_2}{\tilde v}\mathrm{Re}\,m_q(t) + \frac{1}{2}\left(\frac{v}{\tilde v}-1\right)
\label{eq:nB}
\end{gather}
where $n_q(t) = \mathrm{Tr}\left[\rho(t)d_q^+d_q\right]$ and $m_q(t) =\mathrm{Tr}\left[\rho(t)d_q^+d_{-q}^+\right]$. After substituting $n_q^B(t)$ into Eq. \eqref{eq:green3}, the integral of the term with $\frac{1}{2}\left(\frac{v}{\tilde v}-1\right)$ leads to a power-law function of $x$. This function (together with $G_\mathrm{nonint}(x)$) results in the interacting correlation function
\begin{gather}
G_0(x)=\frac{i}{2\pi(x+i\alpha)}\left(\frac{\alpha}{\sqrt{x^2+\alpha^2}}\right)^{\frac{K+K^{-1}}{2}-1}
\end{gather}
which also equals the single particle density matrix in the initial state. The time dependence is described in the first two terms of
Eq. \eqref{eq:nB} which, after all, end up in Eq. (8) of the main text.
\section{Time evolution of entropy}
In this section, the time dependence of the von Neumann entropy is studied. At any time instant, the system consists of two Bose gases for each $q>0$ quantum numbers.
Therefore, the entropy is defined as
\begin{gather}
S(t) = -\mathrm{Tr}\left[\rho(t)\ln\rho(t)\right]=\nonumber \\
= 2\sum_{q>0}\left[(N_q(t)+1)\ln(N_q(t)+1) - N_q(t)\ln N_q(t)\right]
\end{gather}
where $N_q(t) = \mathrm{Tr}\left[\rho(t)\tilde{b}_q^+(t)\tilde{b}_q(t)\right]$ is the number of bosons $\tilde{b}$ which diagonalize the instantaneous density matrix.
To calculate $N_q(t)$ and the entropy, we determine how the operators $\tilde{b}_q(t)$ are related to the operators $d_q$ which diagonalize the interacting Hamiltonian.
In terms of the operators $d_q$, the density matrix is expressed as
\begin{gather}
\rho(t)=\prod_{q>0}r_q(t)e^{c_q(t)K_{q,-}}e^{-2\ln(\nu_q(t)+1)K_{q,0}} e^{c_q(t)^*K_{q,+}}
\label{eq:rhodef}
\end{gather}
where the operators $K_{q,+}=K_{q,-}^{+} = d_q^+d_{-q}^+$ and $K_{q,0} =\frac{d_q^+d_q + d_{-q}d_{-q}^+}{2}$ obey the commutation relations of an $su(1,1)$ algebra.
Note that all the time dependence is incorporated into the functions $c_q(t)$ and $\nu_q(t)$. The prefactor is set to
$r_q(t)=(\nu_q(t)^2-|c_q(t)|^2)/(\nu_q(t)+1)$ in order to ensure the unit trace in each wavenumber channel. It can be shown that the functions are related to the expectation values $n_q(t)$ and $m_q(t)$, which are obtained in Eqs. (6) of the main text, by
\begin{gather}
\nu_q(t) = \frac{n_q(t)}{n_q(t)^2-|m_q(t)|^2}\quad c_q(t) = \frac{m_q(t)}{n_q(t)^2-|m_q(t)|^2}\,.
\label{eq:nuc}
\end{gather}
To diagonalize the exponent of \eqref{eq:rhodef}, first we rewrite the product of the three exponentials in a single exponential by using the commutation rules of the $su(1,1)$ algebra \cite{solomon,gilmore}.
\begin{gather}
\rho(t)=\prod_{q>0}r_q(t)\,e^{\frac{\Omega_q}{\sqrt{1-|s_q|^2}}\left(s_q K_{q,-}+2K_{q,0} + s_q^*K_{q,+}\right)}
\end{gather}
where
\begin{gather}
\Omega_q = \left|\textmd{acosh}\left(\frac{1}{2(n_q(t)+n_q(t)^2-|m_q(t)|^2)}+1\right)\right|
\label{eq:omegatdef}
\end{gather}
and
\begin{gather}
s_q = -\frac{2m_q(t)}{1+2n_q(t)}
\label{eq:sdef}
\end{gather}
are both time dependent.
Since the exponent of the density matrix is quadratic in the bosonic annihilation and creation operators, it can be diagonalized by the Bogoliubov transformation
\begin{gather}
\left[\begin{array}{c} \tilde{b}_{q}(t) \\ \tilde{b}_{-q}(t)^{+}\end{array}\right] = \left[\begin{array}{cc} u_q(t) & v_q(t) \\ v_q(t)^{*} & u_q(t) \end{array}\right]
\left[\begin{array}{c} d_{q} \\ d_{-q}^{+}\end{array}\right]
\label{eq:bogoliubov}
\end{gather}
where
\begin{gather}
u_q(t)=\frac{1}{\sqrt{2}}\sqrt{\frac{1}{\sqrt{1-|s_q(t)|^2}}+1} \\
v_q(t)=\frac{s_q(t)^{*}}{\sqrt{2}|s_q(t)|}\sqrt{\frac{1}{\sqrt{1-|s_q(t)|^2}}-1}
\end{gather}
leading to $\rho(t)\sim e^{-\Omega_q(t)\left( \tilde{b}_{q}(t)^+ \tilde{b}_{q}(t) + \tilde{b}_{-q}(t)^{+}\tilde{b}_{-q}(t)\right)}$. For the entropy, we have to calculate the expectation value of the number of bosons $\tilde{b}$. Substituting the Bogoliubov coefficients, we obtain
\begin{gather}
N_q(t) = \mathrm{Tr}\left[\rho(t)\tilde{b}_q^+(t)\tilde{b}_q(t)\right] = \nonumber \\
= (u_q^2+|v_q|^2)n_q(t) + 2\mathrm{Re}\left(u_q v_q m_q(t)\right) + |v_q|^2 = \nonumber \\
= \sqrt{\left(n_q(t) + \frac{1}{2}\right)^2-|m_q(t)|^2}\,\,-\frac{1}{2}
\end{gather}
\section{Spin-flip correlation function}
Eq. (14) in the main text is equivalent to the Heisenberg XXZ chain and can also be rewritten in terms of hard core bosons\cite{giamarchi}.
Then, the hard core boson equals time Green's function or the spin flip correlation function\cite{pollmannxxz,giamarchi} is
\begin{gather}
C(x,t)=\frac{(-1)^x}{2\pi\alpha}\mathrm{Tr}\left[\rho(t)e^{-i\Theta(x)}e^{i\Theta(0)}\right]
\end{gather}
where
\begin{gather}
\Theta(x) = i\sum_{q\neq 0}\sqrt{\frac{\pi}{2L|q|}}e^{iqx}\left(b_q^+ - b_{-q} \right)
\end{gather}
and the density matrix is given by Eqs. \eqref{eq:rhodef} and \eqref{eq:nuc}. Evaluating the trace, we obtain
\begin{gather}
\ln\frac{C(x,t)}{C_0(x)}=-\sum_{q>0}\frac{4\pi\sin^2\left(\frac{qx}{2}\right)}{LKq}\left(n_q(t) + \mathrm{Re}\,m_q(t)\right)
\end{gather}
where
\begin{gather}
C_0(x)=
\frac{(-1)^x}{2\pi\alpha}\left(\frac{\alpha}{\sqrt{x^2+\alpha^2}}\right)^{\frac{1}{2K}}
\end{gather}
is the initial correlation function.
Using the results in Eqs. (6) of the main text, the sum over wavenumbers can be carried out analytically as
\begin{gather}
\ln\frac{C(x,t)}{C_0(x)}=-\frac{\gamma}{2\pi K^2\tilde v}\left(\frac{\alpha 2\tilde v t}{\alpha^2+x^2}+I\left(\frac{\tilde v t}{\alpha},\frac{x}{\alpha}\right)\right)
\end{gather}
where $I(y,z)$ is defined after Eq. (9) in the main text.
In the scaling limit, i.e., when $2\tilde v t\gg \alpha$ and $x\gg\alpha$,
\begin{gather}
C(x,t)=C_0(x)e^{-\frac{\gamma t}{\pi K^2 \alpha}}\left\{\begin{array}{lc}
e^{-\frac{\gamma}{4K^2\tilde v }} & \textmd{ for }2\tilde v t\ll x\\
1 & \textmd{ for } x\ll 2\tilde v t
\end{array}\right.
\end{gather}
decays exponentially with time.
\end{document}
|
2,877,628,090,814 | arxiv | \section*{Introduction}
The first demonstration of the stable 3D optical trapping of micron-scale particles was in the 1980s \cite{Ashkin1986}, and since then there has been an explosion of research using ``optical tweezers'', to the point that they are an off-the-shelf tool for physical and biological scientists. Using this system, it is possible to control and track the motion of mesoscopic objects with astounding precision. The first investigation of a microscopic thermodynamic process with an optically trapped particle was the realization of a Brownian ratchet \cite{Faucheux1995} and there was a strong increase of activity following the foundation of stochastic thermodynamics \cite{SiefertReview} and the discovery of fluctuation theorems such as the Jarzynski equality, with Seifert describing trapped colloidal particles as ``\emph{the} paradigm for the field (of stochastic thermodynamics)''.
So, what makes the trapped microparticle such a good platform for thermodynamic studies? First and foremost, its characteristic energy is comparable to that of the thermal fluctuations of the bath $\sim \kB \T{env}$. These small particles in harmonic optical potentials are simple, and considering only the centre-of-mass motion is for most cases sufficient to fully describe their behaviour\footnote{Recent work with levitated nanoparticles also considers rotational degrees of freedom \cite{Kuhn2017}.}. Having few degrees of freedom enhances the relative role of thermal fluctuations via the central limit theorem: energy fluctuations of a system with $N$ degrees of freedom can be quantified by comparing the variance $\sigma^2 \propto N$ to the mean $\langle U \rangle \propto N$ of an extensive macroscopic quantity $U$, such as the total energy. For large $N$, $\langle U \rangle \gg \sigma$, whereas for small $N$, $\langle U \rangle \sim \sigma$ \cite{Gieseler2014}, illustrating the dominant role of fluctuations in systems with few degrees of freedom.
Thus, with optical trap depths $>10^4\,$K and optical spring constants of $\sim \pN/\mum$, the motion of micron-sized particles is sensitive to thermal fluctuations, but not destructively so.
The ability to dynamically alter the potential landscape in which the particle moves is also key to their application in studying thermodynamics. This can involve changing the depth of the optical potential, to realize compression stages in heat engines \cite{Schmiedl2008} or to speed-up equilibration \cite{Martinez2016}, or creating geometries with multiple stable trapping sites to test information thermodynamics \cite{Berut2012}.
The majority of thermodynamic studies with optically trapped particles involve \emph{colloidal} particles: objects suspended in a liquid. In contrast, this chapter will consider \emph{levitated} nanoparticles, that is particles trapped in a gas or vacuum. It is somewhat experimentally more challenging than working in liquid, requiring deeper optical potentials due to reduced viscous damping, and loss of the particles from the trap at low pressures is a common problem.
Why work in this challenging regime at all, if the colloidal system has been so successful? Firstly, working in a gaseous environment gives us access to \emph{underdamped} dynamics, as opposed to the overdamped dynamics typically observed in a liquid.
The underdamped regime is of fundamental interest, since the inertia of a particle plays a role in the dynamics, whereas it can be mostly ignored in overdamped systems.
Secondly, and motivated by the subject matter of this book, the underdamped regime allows one to make the connection to the even more fundamental unitary evolution of quantum mechanical systems.
In addition, there is the potential to study quantum physics with these mesoscopic objects \cite{Chang2010}. The observation of quantum phenomena with levitated nanoparticles absolutely requires working in a good vacuum, since collisions with gas molecules cause rapid heating and decoherence. \\
This chapter is intended as a pedagogical introduction to the dynamics of optically levitated nanoparticles with a focus on the study of single particle thermodynamics. Much of the work studying thermodynamics with nano- and micro-particles has taken place in liquid, and this chapter will avoid reviewing this impressive body of work, focussing instead on studies of thermodynamics with nanoparticles levitated in a gas. For a recent literature review we refer the reader to Ref. \cite{EntropyReview}. The authors will discuss extensions into the quantum regime where relevant throughout the chapter.
Section~\ref{sec:background} gives a detailed review of the stochastic and deterministic forces acting on an optically levitated nanoparticle, including a discussion of heating due to optical absorption. Section~\ref{sec:Brownian} describes the Brownian motion of a levitated particle, which will highlight the role of this system as a paradigm for studying stochastic thermodynamics. Section~\ref{sec:time_dep} will detail the utility of sculpting time-dependent potentials for trapped particles, in particular the ability to create effective baths and non-thermal states. Finally, section~\ref{sec:thermodynamics} will review and discuss recent experimental progress in realising important thermodynamics processes with levitated nanoparticles.
\section{The trapped nanoparticle system}
\label{sec:background}
A particle with radius $a \sim 100\,$nm has, generally speaking, of order $(a/a_o)^3\approx 10^{10}$ degrees of freedom, where $a_o$ is the size of the atoms making up the particle. However, for most practical purposes we characterize excitations within the particle by its internal temperature $\T{int}$ and the particle's external degrees of freedom, like its position $\mathbf{r}$, which describes its centre-of-mass motion, and its orientation.
In the context of single particle thermodynamics, the most relevant degree of freedom is the particle's position. Thus, we will focus our attention on this degree of freedom and only briefly mention the others in their relationship to the center-of-mass motion.
The equations of motion for the centre-of-mass can be well described classically and are given by Newton's second law
\begin{equation}
\ddot{\mathbf{r}}(t) \,+\, \boldsymbol{\g{CM}} \!\; \dot{\mathbf{r}} =\; \frac{1}{m }\left[ \Ffluct(t)
+ \mathbf{F}_{\rm det}(\mathbf{r}, t)
\right ],
\label{eq:NewtonLaw}
\end{equation}
where $m$ is the particle's and $\g{CM}$ the momentum damping rate, as discussed in detail below. We have isolated the contributions to the forces that act on the particle into stochastic forces $\Ffluct$ and deterministic forces $\mathbf{F}_{\rm det}$.
In the following section we discuss the origin of these forces and how they can be controlled in an experiment. This will lead to an effective description of the particle as a Brownian particle in a potential landscape, where the shape of the potential and the thermal bath can be controlled experimentally. This model is at the heart of many stochastic processes, which can therefore be simulated with this platform.
\begin{figure}[hbt]
\includegraphics[width=0.9\textwidth]{Schematic}
\caption{a.) An optical trap is formed by a tightly focused laser beam. In a single beam trap, the laser propagates along z. The non-conservative scattering force $\mathbf{F}_{\rm scat}$ acts along z while the gradient force points towards the maximum laser intensity, where the thick red lines on the figure represent the profile of a focussed light beam. In addition, the particle can experience torque and rotate in the trap. The orientation of a particle with a single symmetry axis is characterized by the two angles $\phi$ and $\theta$. The particle also feels a force $\mathbf{F}_{\rm g}$ due to gravity.
b.) The gradient force forms an optical potential with depth $U_0$. The particle is trapped at the bottom of the potential and oscillates at frequency $\wo$. For large oscillation amplitudes, the nonlinearity of the potential is characterized by the Duffing parameter $\xi$.
\label{fig:Schematic}}
\end{figure}
\subsection{Stochastic forces\label{sec:stochastic_force}}
The interaction of the particle with its environment has mechanical (collisions with air molecules) and radiative (blackbody and scattering) contributions. These interactions lead to dissipation acting on the center-of-mass motion $\g{CM}^\noise$ and are the source of the random forces acting on the particle. The strength of the random forces is characterized by their power spectral densities $\Sff{\noise}$. For most practical purposes, they can be considered as frequency independent (white noise), that is the autocorrelation functions of the stochastic forces are $\langle \mathcal{F}_{\noise}(t) \mathcal{F}_{\noise}(t')\rangle =2\pi \Sff{\noise}\delta(t-t')$. After a time $\approx 1/\g{CM}$, where $\g{CM} = \sum_\noise\g{\noise}$ is the total damping rate, the center-of-mass motion of the particle reaches an effective thermal equilibrium, which is characterized by an effective temperature through the fluctuation-dissipation relation:
\begin{equation}
\label{eqn:temp_def}
\T{CM} = \frac{\pi\Sff{} }{\kB m \g{CM} },
\end{equation}
where $\Sff{} = \sum_\noise\Sff{\noise}$ is the total force spectral density, $m$ the mass of the particle, and $\kB$ is Boltzmann's constant.
Below we describe the individual contributions. They are:
collisions with air molecules ($\noise =$ gas), radiation damping ($\noise =$ rad), feedback or cavity damping ($\noise =$ fb), stochastic driving ($\noise =$ drive), and in the quantum regime noise driving wavefunction collapse ($\noise =$ CSL).
\subsubsection{Gas damping}
For pressures higher than $\sim 10^{-6}\,\rm mbar$, the dominant contribution to the stochastic forces is due to collisions with surrounding air molecules, and the damping rate is given by \cite{Beresnev:1990tv}
\begin{equation}\label{eq:gas_damping}
\frac{\g{gas}}{2\pi} = 3\mu_v \frac{a}{m}\frac{0.619}{0.619+\Kn}\left(1+c_K\right),
\quad
\Sff{gas} = \frac{m\kB \T{gas}}{\pi}\g{gas},
\end{equation}
where $c_K = 0.31 \Kn\left/\left(0.785+1.152\Kn+\Kn^2\right)\right.$, $\mu_v$ is the viscosity coefficient, which for a dilute gas is $\mu_v = 2\sqrt{\mgas \kB\T{gas}}\left/ 3\sqrt{\pi} \crosssectiongas \right.$ and $\Kn = \bar{l}/a$ is the Knudsen number for the free mean path $\bar{l} = \kB \T{gas}\left/(\sqrt{2}\crosssectiongas\Pg)\right.$, $\crosssectiongas = \pi \dm^2$, $\dm = 0.372\,\rm nm$ is the diameter of the air molecules and $\mgas$ their mass.
For high pressures (where $\Kn\ll 1$), the interaction with the gas is so strong that the particle motion is heavily damped and its internal temperature $\T{int}$ and centre-of-mass temperature $\T{CM}$ quickly thermalize with the gas temperature $\T{gas}$. In this regime, the damping becomes independent of pressure $\g{CM}/2\pi\approx 3 a \mu_v/m$, as predicted by Stokes' law.
For decreasing pressure, the mean free path of the gas molecules increases (e.g. $\bar{l}\sim 60\,\mu$m at 1\,mbar). As a consequence, the particle no longer thermalizes with the gas since the impinging gas molecules no longer carry away enough thermal power to balance the optical absorption from the trapping laser. Due to the increased internal temperature $\T{int}$ of the particle, the average energy of the gas molecule after a collision with the particle increases. The process by which a surface exchanges thermal energy with a gas is called accommodation, which is characterized by the accommodation coefficient,
\begin{equation}
\label{eqn:accomm}
\acccoeff = \frac{\T{em} - \T{gas}}{\T{int} - \T{gas}},
\end{equation}
where $\T{int}$ is the temperature of the surface, $\T{gas}$ the temperature of the impinging gas molecules and $\T{em}$ the temperature of the gas molecules emitted from the surface. Accommodation quantifies the fraction of the thermal energy that the colliding gas molecule removes from the surface, such that $\acccoeff =1$ means that the molecule fully thermalizes with the surface. Since the mean free path in a dilute gas is long, one can safely assume that a molecule that comes from the particle surface will not interact again with the particle before thermalizing with the environment. Consequently, we can consider the particles that impinge on the particle surface and those that leave the surface as being in equilibrium with two different baths with temperatures corresponding to the temperature of the environment and the particle surface, respectively. Therefore, we get an additional contribution to the damping from the emerging hot molecules
\begin{equation}\label{eq:gas_damping_2bath}
\frac{\g{em}}{2\pi} = \frac{1}{16}\sqrt{\frac{\T{em}}{\T{gas}}}\g{gas} ,\quad
\Sff{em} =
\frac{m\kB}{\pi}\left[\acccoeff\T{int} +(1-\acccoeff\right)\T{gas}]\g{em},
\end{equation}
as experimentally observed by Millen \emph{et al.} \cite{Millen2014}. Note that $\T{em}$ can be calculated using eqn.~\eqref{eqn:accomm}. In addition to this noise contribution, the internal temperature of the particle can also cause deterministic forces to act on the particle's centre-of-mass motion through the photophoretic effect, where absorbing particles are repelled from the optical intensity maxima \cite{Lewittes1982}. However, since photophoretic forces require a temperature gradient across the particle, they vanish for sub-wavelength particles, which are mostly used in vacuum trapping experiments (typical trapping laser wavelengths range from 532\,nm to 1550\,nm).
For pressures below $\Pg =0.57 \kB \T{gas} \left/\crosssectiongas a\right.\approx 54.4\,{\rm mbar} \times (a/ \mu_v {\rm m})^{-1}$, where the mean free path is much larger than the radius of the particle ($\Kn\gg 1$), the damping becomes linear in pressure
\begin{equation}\label{eq:gas_damping_lin}
\frac{\g{gas}}{2\pi} = \frac{3}{\pi\sqrt{2}}\frac{\mu_v\crosssectiongas}{\kB \T{gas} \rho}\frac{\Pg}{a},
\end{equation}
where $\rho$ is the density of the particle. The two expressions eqns.~\eqref{eq:gas_damping} \& \eqref{eq:gas_damping_lin} differ by less than 10\% for $\Kn\gg 1$, with the discrepancy due to numerical accuracy when calculating the constant factors in eqn.~\eqref{eq:gas_damping} \cite{Beresnev:1990tv}. The total damping due to the hot particle with the gas environment is $\g{em}+\g{gas} =2\pi \cg \Pg / a$, where typically $\cg\approx 50\, \Hz (\mum/ \mbar)$.
When considering operation in the quantum regime, it is absolutely necessary to work under extremely good vacuum conditions, as collisions with gas molecules cause rapid decoherence and heating out of the ground state.
As an example, a 100\,nm radius silica nanosphere in a room temperature gas experiences $\g{gas} \sim\,$MHz at atmospheric pressures, and $\g{gas} \sim\,$mHz at $10^{-6}\,$mbar pressures.
So far, we have considered spherical particles with translational degrees of freedom. However, in general the particle has some anisotropy and is free to rotate within the trap. The orientation of the particle with respect to the trap (see fig.~\ref{fig:Schematic}) is described by the angles $(\phi, \theta)$, where $\phi$ is the angle between the $x$-axis and the projection onto the $x-y$ plane, and $\theta$ is the angle between the particle axis and the $z$-axis. The particle axis is usually defined along its symmetry axis and represented here by the vector $\mathbf{m}$. For anisotropic particles, e.g. a cylinder, the friction term is different along each of the axes, and depends upon the alignment $\mathbf{m}$ of the particle. As a consequence, the friction coefficient has to be replaced by a tensor $\boldsymbol{\Gamma}$ and the damping in a direction $\mathbf{s}$ is given by $\boldsymbol{\Gamma}\cdot \mathbf{s}$. In the low pressure regime, the friction tensor of the translational degrees of freedom for a particle with a single symmetry axis can be derived analytically \cite{Martinetz2018}. As an example, for a cylinder of diameter $d$
\begin{equation}
\label{eqn:cylinder_damp}
\frac{\boldsymbol{\Gamma}_\text{trans}}{2\pi} =
6\sqrt{2}\frac{\mu_v\crosssectiongas}{\kB \T{gas} \rho}\frac{\Pg}{d}
\left (2 - \frac{1}{2}\acccoeff + \frac{\pi}{4}\acccoeff \right )
\left ( \mathbb{I} - \frac{8 -6\acccoeff +\pi\acccoeff}{8 -2\acccoeff +\pi\acccoeff}\mathbf{m}\otimes\mathbf{m}\right ).
\end{equation}
For the rotational degrees of freedom we find that the damping is isotropic and given by
\begin{subequations}
\begin{equation}
\label{eqn:sphere_rot_damp}
\frac{\g{rot}^\text{sphere}}{2\pi} = \frac{30 \acccoeff}{8\pi\sqrt{2}}\frac{\mu_v\crosssectiongas}{\kB \T{gas} \rho}\frac{\Pg}{a},
\end{equation}
for a sphere and
\begin{equation}
\label{eqn:cylinder_rot_damp}
\frac{\g{rot}^\text{cyl}}{2\pi} =
6\sqrt{2}\frac{\mu_v\crosssectiongas}{\kB \T{gas} \rho}\frac{\Pg}{d}
\left (2 - \frac{1}{2}\acccoeff + \frac{\pi}{4}\acccoeff \right ),
\end{equation}
for a cylinder.
\end{subequations}
\subsubsection{Noise from optical fields}
\label{sec:photon}
At very low pressure ($\leq 10^{-6}\,\rm mbar$), gas damping rates become extremely small and photon shot noise starts to dominate \cite{Jain2016a}. Photon shot noise is a consequence of the particulate nature of light. Photons arrive at discrete times, where the number of photons arriving per time interval $\Delta t$ is given by $\sqrt{\Delta t \Popt\left/\hbar \wopt \right.} $, where $\Popt$ and $\wopt$ are the optical power and frequency, respectively. The recoil from the fluctuating number of phonons impinging on the nanoparticle can be modelled as an effective bath with the characteristics
\begin{equation}\label{eq:radiation_damping}
\frac{\g{rad}}{2\pi} = \dpcoeff\frac{\Pscat}{2\pi m c^2}
\quad{\rm and}\quad
\Sff{rad} = \dpcoeff \frac{\hbar \omega \Pscat}{2\pi c^2},
\end{equation}
where $\dpcoeff$ depends on the direction of motion of the particle with respect to the polarization of the laser and is $\dpcoeff=2/5$ for motion along the direction of polarization and $\dpcoeff=4/5$ for motion perpendicular to the polarization. The scattered power is $\Pscat = \scatcross \Iopt$, where $\scatcross = |\alpha|^2\kopt^4/6\pi\epsilon_0^2$ with $\alpha$ the particle polarizability, and $\Iopt$ is the laser intensity. The effective temperature of this bath can be calculated via eqn.~\eqref{eqn:temp_def}.
The noise processes described so far are present in any experiment with optically levitated nanoparticles in high vacuum. In addition, random forces and damping can be introduced through external fields that are under experimental control. Importantly, since energy can be injected or extracted from the particle, i.e. it is not in a thermal equilibrium, the fluctuation-dissipation relation does not have to hold and the effective damping and temperatures can be controlled independently.
For instance, by parametric feedback damping (see also section~\ref{sec:non-thermal}), the temperature alone is not sufficient to give a full description of the bath. Ideal feedback cooling damps the particle motion at a rate $\g{fb}$ without adding any fluctuating forces, thus $\Sff{fb}=0$ and it is therefore referred to as cold damping.
Similarly, cavity cooling up-converts the particle energy to optical frequencies, which are effectively at zero temperature because $\hbar \omega \gg \kB \T{env}$ in a room temperature environment. Conversely, fluctuations of the trapping or additional control fields only add fluctuating forces without providing damping. Hence, $\g{drive} = 0$ and $\Sff{drive} = \q^2 S_{\rm qq}$, where $\q$ is the coupling parameter to the control field and $S_{\rm qq}$ its spectral density.
This can be realized for example with fluctuating electric fields, where $\q$ corresponds to the charge on the particle.
A real feedback signal is noisy, since any measurement is accompanied by noise, and this will heat the particle motion without providing additional damping. In addition, one has to consider correlations between the measurement signal and the particle motion when interpreting the measurement result, in particular when using a linear feedback signal.
\subsubsection{Noise due to wavefunction collapse}
There are a class of theories, called \emph{collapse models}, which aim to phenomenologically explain why we do not observe superposition states of macroscopic objects \cite{BassiReview}. These models invoke a (classical) noise field, which acts upon particles in a mass-dependent way, to ensure localization of the wavefunction. There are various proposed forms of the noise, including white noise fields which violate conservation of energy, to coloured and dissipative noise with a finite temperature (suggested to be between 0.1-10\,K). The noise induces a type of Brownian motion on the centre-of-mass or alignment of the particle, which in principle can be observed.
For example, a model of wavefunction collapse known as Continuous Spontaneous Localization (CSL), predicts a random force with $\Sff{CSL} = \lambda_{\rm CSL}(\hbar/r_{\rm CSL})^2\alpha_{\rm CSL}$, where the parameters $\lambda_{\rm CSL}$ and $r_{\rm CSL}$ are a phenomenological rate and length scale, respectively.
The factor $\alpha_{\rm CSL}$ is mass and geometry dependent, and as an example is proportional to the mass $m^{2/3}$ for a sphere. For a thorough discussion of this process, see the review by Bassi \emph{et al.} \cite{BassiReview}.
\subsection{Deterministic forces}
In addition to stochastic forces, which we described in section~\ref{sec:stochastic_force}, the particle is also subject to deterministic forces. They are gravity $\F{g} = m \mathbf{g}$, electric forces $\F{e} = q\mathbf{E}$ if the particle carries a charge $q$, magnetic forces $\F{mag} = \nabla(\boldsymbol{\mu}\cdot\mathbf{B})$ if the particle has a magnetic dipole moment $\boldsymbol{\mu}$ and optical forces $\F{opt}$. Most experiments with levitated particles in vacuum use optical forces to create a stable trap. This gives a great deal of flexibility since optical fields can be controlled very well in both intensity and position, allowing the creation of almost arbitrary fluctuating force fields. Particles that are much smaller than the wavelength $\lambdaopt$ of the trapping laser $a \kopt \ll 1$, where $\kopt = 2\pi/\lambdaopt$, can be treated as dipoles in the Rayleigh approximation. The polarizability for a particle with volume $V$ is thereby given by
\begin{equation}
\label{eqn:polarisability}
\polar{0} = \eo V \boldsymbol{\chi},
\end{equation}
where the total susceptibility of the particle $\boldsymbol{\chi} = \boldsymbol{\chi}_e\left(1+\boldsymbol{N} \boldsymbol{\chi}_e\right)^{-1}$ depends on the material via the material susceptibility $\boldsymbol{\chi}_e$ and on its geometry through the depolarization tensor $\boldsymbol{N}$, which in general are both rank-2 tensors. However, for isotropic materials, the material susceptibility simplifies to a scalar $\chi_e$ and similarly for a sphere the depolarization tensor is isotropic and simplifies to a scalar $N = 1/3$. Thus, for a sphere we recover the Clausius-Mossotti relation $\chi = 3(\epsilon_p-1)/(\epsilon_p+2)$, where we use $\epsilon_p= 1+\chi_e$.
For a particle with a uniaxial anisotropy, e.g. a cylinder, the susceptibility $\boldsymbol{\chi} = \text{diag}(\chi_\|, \chi_\perp, \chi_\perp)$, has a component $\chi_\parallel$ parallel and a component $\chi_\perp$ perpendicular to the symmetry axis. For example, the depolarization tensor of a cylinder is $\boldsymbol{N} = \text{diag}(0, 1/2,1/2)$ in the frame of the cylinder, where the cylinder axis is along the $x$-axis. Consequently, $\chi_{\|} = \er - 1$, $\chi_{\perp} = 2(\er - 1)/(\er +1)$ for a cylinder with isotropic $\er$.
This means the maximal polarizability of a cylinder is $(\er+2)/3$ times higher than for a sphere of the equivalent volume. For silica, this is a factor of 2, whereas it is a factor of 4.6 for silicon.
In general, the particle reacts to the total field, that is the sum of the incident and the scattered field. The total field is the self consistent solution to Maxwell's equations and has to be calculated generally with numerical methods.
However, for a spherical particle, the modified polarizability
\begin{equation}\label{eq:polarisability}
\polar{} = \polar{0}\left(1-i\frac{\kopt^3\alpha_0}{6\pi\epsilon_0}\right)^{-1},
\end{equation}
accounts for the radiation reaction of the particle to its own scattered field, such that the induced polarization due to a field $\mathbf{E}_0$ is $\mathbf{P} = \alpha \mathbf{E}_0$. We introduce $\alpha'$ and $\alpha''$ to refer to the real and imaginary part of the polarisability, respectively.
Knowing the polarizability, we can calculate the optical force for sub-wavelength particles in the Rayleigh approximation. The optical force has conservative and non-conservative contributions
\begin{equation}\label{eq:Fopt}
\F{opt} =
\alpha'\nabla I_0/4 +\totcross\left[
\mathbf{S}/c+
c \nabla\times \mathbf{L}\right],
\end{equation}
where the total cross-section $\totcross =\alpha''\kopt/\epsilon_0$ is the sum of the absorption and scattering cross-sections. The optical intensity at the field maximum $I_0$ is related to the field $E_0$ through $I_0 = c\epsilon_0E_0^2/4$. The first term in eqn.~\eqref{eq:Fopt} is a conservative force. It pulls particles with a high refractive index relative to their surroundings toward the region of maximum light intensity.
In an optical tweezers, this is the focal volume of the light beam.
The second term is the non-conservative scattering force, which has two contributions: the radiation pressure term, which is proportional to the time averaged Poynting vector $\mathbf{S}=\left<\mathbf{E}\times\mathbf{H}^*\right>$, $\mathbf{H}$ being the magnetic field, and a curl force associated to the non-uniform distribution of the time averaged spin density of the light field $\mathbf{L} = -i\epsilon_0\left<\mathbf{E}\times\mathbf{E}^*\right>\left/4\wopt\right.$, $\langle\dots\rangle$ being a time average. The curl force is zero for a plane wave but can be significant for a tightly focused beam in an optical tweezers\footnote{For a Gaussian beam with waist $\waist_0$, we estimate the curl force as $F_{\rm curl} \approx -2 F_{\rm scat}\left/\waist_0^2 \kopt^2\right.$, where $F_{\rm scat} \sim \sigma_{\rm tot}\Popt\left/\waist_0^2 c\right.$.
}. However, since $\alpha''/\alpha'\propto a^3$, the non-conservative forces vanish for small particles and we will neglect them in the following discussion.
\subsubsection{Optical potential}
The conservative force in eqn.~\eqref{eq:Fopt} can be expressed as the gradient of a potential $\F{opt} = -\nabla U_{\rm opt}$. Even for a tightly focused laser beam, the optical intensity distribution is to a good approximation described by a transverse Gaussian profile and a Lorentzian profile along the direction of beam propagation.
Thus, for a single focused laser beam the optical potential reads
\begin{equation}\label{eq:potential}
U_{\rm opt}(\mathbf{r}) =
\frac{-\Uopt{}}{1+(z/z_0)^2}\exp\left[
-\frac{2}{1+(z/z_0)^2}\left(
\frac{x^2}{\waist_x^2}+
\frac{y^2}{\waist_y^2}
\right)\right],
\end{equation}
where $\Uopt = \alpha' E_0^2\left/4\right.$ is the potential depth, $\mathbf{r}$ is the position of the particle, $\waist_x$, $\waist_y$ denote the transverse extent of the focus, and we define a longitudinal waist $\waist_z$ via the Rayleigh range $z_0 = \waist_z/\sqrt{2} \approx \pi\waist_0^2/\lambdaopt$, which gives the depth of focus. Note that for tightly focused laser beams, as are commonly used in optical trapping, the field distribution is slightly elongated along the direction of polarization of the incident field.
Integrating the Poynting vector over a cross-section that is transverse to the direction of propagation allows us to relate the field intensity at the centre of the focus $E_0^2$ to the optical power of the trapping laser
$
\Popt = \int_s \left<\mathbf{S}\right>\mathbf{\d s} = \pi c\epsilon_0 \waist_0^2 E_0^2/4
$, where $ \waist_0^2 = \waist_y \waist_x$. At the bottom of the potential, the centre-of-mass motion is harmonic, with frequencies
\begin{eqnarray}\label{eqn:frequencies_translational}
\w_{i} &= &2 \sqrt{\frac{\chi}{c\pi\rho }}\frac{\sqrt{\Popt}}{\waist_0 \waist_i},
\end{eqnarray}
along the three directions ($q = x, y, z$). For larger oscillation amplitudes, the motion becomes anharmonic, and the nonlinear coefficients can be obtained from higher derivatives of the optical potential (see section~\ref{sec:non-lin-BM}).
\subsubsection{Rotation}
As we have already seen in section~\ref{sec:stochastic_force}, the light matter interaction is more complicated for anisotropic particles, since it depends upon the alignment of the object relative to the polarization axis of the field\footnote{For a thorough treatment, including optical scattering, see \cite{Kuhn2017} and references therein.}.
The induced polarization is $\mathbf{P} = \boldsymbol{\alpha}\mathbf{E}$.
Consequently, the particle experiences an optical torque
\begin{equation}
\mathbf{N}_\text{opt} = \langle \mathbf{P} \times \mathbf{E}^*\rangle,
\end{equation}
which aligns the particle along the polarization axis. For small deflections from the polarization axis the angular motion is harmonic. For a cylinder of length $l$ the frequencies are
\begin{equation}\label{eqn:frequencies_rotational}
\Omega_{\theta} = \sqrt{\frac{24 \Popt\chi_{\|}}{ \pi \rho c \waist_0^2 l^2} \left( \frac{\Delta\chi}{\chi_{\|}} + \frac{(\kopt l)^2}{12}\right)}, \quad
\Omega_{\phi}=\sqrt{\frac{24 \Popt \Delta\chi}{ \pi \rho c \waist_0^2 l^2}},
\end{equation}
where the term $(\kopt l)^2/12$ is a correction term that accounts for the particle's finite
extension.
In contrast to linearly polarized light, the polarization axis of circularly polarized light rotates at the optical frequency. This is too fast for the particle to follow. Nonetheless, light scattering transfers the angular momentum of the light to a particle with polarization anisotropy, which can originate from the intrinsic birefringence of the particle or from the anisotropic shape of the particle (c.f. eqn.~\eqref{eqn:polarisability}). The torque that results from angular momentum transfer is for a cylinder \cite{Kuhn2017}:
\begin{equation}
N_\phi= \frac{\Delta\chi l^2 d^4 \kopt^3}{96 c \waist_{0}^2} \left [\Delta \chi \eta_1(\kopt l) +\chi_\perp \eta_2(\kopt l) \right ] \Popt,
\end{equation}
where the functions $\eta_{1,2}(\kopt l)$ are given by
\begin{eqnarray}
\eta_1(\kopt l) & = & \frac{3}{4} \int_{-1}^1 \mathrm{d}s~(1 - s^2) \mathrm{sinc}^2 \left ( \frac{\kopt l s}{2} \right ), \notag\\
\eta_2(\kopt l) & = & \frac{3}{8} \int_{-1}^1 \mathrm{d}s~(1 - 3s^2) \mathrm{sinc}^2 \left ( \frac{\kopt l s}{2} \right ).
\end{eqnarray}
For short rods, $\kopt l \ll 1$, one has $\eta_1 \simeq 1$ while $\eta_2 \simeq 0$. The rotational frequency $\Omega_{\rm rot}$ is given by the balance between the torque $N_\phi$ and the damping $\g{rot}^{\rm cyl}$, such that $\Omega_{\rm rot} = N_\phi/(I\g{rot}^{\rm cyl})$, where $I$ is the moment of inertia $I = ml^2/12$. We note that this analysis is true in the Rayleigh-Gans approximation, where
$\koptl(\er -1)\ll 1$ and $\pi \kopt^2 d^2(\er -1)\ll 1$.
\subsection{Internal temperature}
\label{sec:internalT}
In the previous section we discussed the behaviour of a hot sphere levitated in gas and how its internal temperature $\T{int}$ couples to the centre-of-mass motion, which we characterized by its centre-of-mass temperature $\T{CM}$. In this section we will consider the process by which an optically levitated nanoparticle heats up.
Following Bateman \emph{et al.} \cite{Bateman2014} and Chang \emph{et al.} \cite{Chang2010}, the interaction between a sub-wavelength ($a<\lambdaopt$) sphere of radius $a$ and a light field of frequency $\omL$, is governed by the complex polarisability $\polar{}$.
The frequency dependent permittivity is related to the complex refractive index through $\eom{} = n(\omega)^2$. While $\polar{0}'$ determines the optical potential, $\polar{0}''$ determines optical absorption, with absorption cross-section $\abscross = \polar{0}'' \kopt/\eo$.
The bulk temperature depends on several competing processes: heating through absorption of the trapping light $\omL$, optical absorption of blackbody radiation with a spectral absorption rate $\rho_{\rm abs}$, and cooling through blackbody emission at a spectral emission rate $\rho_{\rm emis}$ and through energy exchange with the background gas. The blackbody spectral rates are given by:
\begin{eqnarray}
\label{eqn:absemis}
\rho_{\rm abs}(\ombb) &=& \frac{(\ombb/(\pi c))^2\abscross(\ombb)}{\exp(\hbar\ombb/(\kB \T{env}))-1}, \notag \\
\rho_{\rm emis}(\ombb, \T{int}) &=& \left ( \frac{\ombb}{\pi c}\right )^2 \abscross(\ombb) \exp \left (-\frac{\hbar\ombb}{\kB \T{int}}\right ),
\end{eqnarray}
where $\T{env}$ is the ambient temperature of the environment, and $\T{int}$ the surface temperature of the sphere (which we assume is equal to the bulk temperature). Following Chang \emph{et al.} \cite{Chang2010}, we can integrate across the blackbody spectrum to find the rate at which the sphere absorbs or emits blackbody energy:
\begin{eqnarray}
\label{eqn:bbenergy}
\dot{E}_{\mathrm{abs}}^{\mathrm{bb}} &= \frac{24 \xi_R(5)}{\pi^2\eo c^3 \hbar^4} \alpha_{\rm bb}'' (\kB \T{env})^5 \notag \\
\dot{E}_{\mathrm{emis}}^{\mathrm{bb}} &= -\frac{24 \xi_R(5)}{\pi^2\eo c^3 \hbar^4} \alpha_{\rm bb}'' (\kB \T{int})^5,
\end{eqnarray}
where $\xi_R(5) \approx 1.04$ is the Riemann zeta function, and $\polar{bb}$ is averaged over the blackbody spectrum, such that for silica, which is the material most commonly used in optical levitation experiments, $\polar{bb}'' \approx 4\pi\eo a^3 \times 0.1$ \cite{Chang2010}. These energy absorption and emission processes lead to decoherence when operating in the quantum regime, and for this reason it may be desirable to work in a cryogenic environment, or to work with internally cold particles.
Next we consider the cooling power due to collisions with gas molecules, again following \cite{Chang2010}:
\begin{equation}
\label{eqn:gascool}
\dot{E}_{\mathrm{gas}} = -\acccoeff \sqrt{\frac{2}{3\pi}}(\pi a^2)\vth \frac{\gamma_{\rm sh} + 1}{\gamma_{\rm sh} - 1}\left (\frac{\T{int}}{\T{gas}}-1 \right) \Pg,
\end{equation}
where $\vth$ is the mean thermal velocity of the impinging gas molecules, $\gamma_{\rm sh} = 7/5$ is the specific heat ratio of a diatomic gas, and for most experiments $\T{gas} \equiv \T{env}$. This expression holds in the Knudsen regime ($\bar{l}\gg a$). Combining all of these leads to a rate equation that describes $\T{int}$:
\begin{equation}
\label{eqn:intemp}
m\hcap \frac{\d\T{int}}{\d t} = \Iopt\abscross + \dot{E}_{\mathrm{gas}} + \dot{E}_{\mathrm{abs}}^{\mathrm{bb}} + \dot{E}_{\mathrm{emis}}^{\mathrm{bb}},
\end{equation}
where $\hcap$ is the specific heat capacity for the particle material and $\Iopt$ is the light intensity. Using eqn.~\eqref{eqn:intemp}, one can calculate the steady-state temperature of a sphere levitated in vacuum. It is also the case that the refractive index $n(\omega)$, and hence the permittivity $\eom{}$, of the levitated particle varies with the bulk temperature, but since for silica it varies by only $1\%$ over 2000\,K, we ignore this effect here. To avoid absorption, pairings of nanoparticle material and trapping wavelengths should be carefully chosen, for example working with pure silicon particles and telecoms wavelengths ($\sim 1500\,$nm).
In fig.~\ref{fig:pressure}, the variation in $\T{int}$ and $\T{CM}$ with pressure is shown, for silica spheres trapped in an optical tweezer under realistic experimental conditions. We can identify three regimes: At high pressures, the cooling power of the surrounding gas is sufficient to counter any heating due to optical absorption, and the particle surface and centre-of-mass temperatures thermalize to the environmental temperature. At low pressures, the surface temperature increases, due to reduced gas cooling power (eqn.~\eqref{eqn:gascool}) and at ultra low pressure, the centre-of-mass motion thermalizes with the photon shot noise \cite{Jain2016a}.
It should be noted that all centre-of-mass heating mechanisms pose a problem to operating in the quantum regime. Because of this, most proposals for testing quantum physics with optically trapped particles involve switching off the light fields after state-preparation, and letting the particle drop. This may not be desirable for thermodynamics experiments which require long interrogation times. For this reason, some proposals consider magnetic or electric levitation, and even operation in space.
\begin{figure}[b]
\begin{center}
{\includegraphics{thermo_T_fig.pdf}}
\caption{\label{fig:pressure} \textbf{The variation in particle dynamics with pressure.} Variation in the surface temperature $\T{int}$, centre-of-mass temperature $\T{CM}$, and damping rate $\g{CM}$ with pressure for a) $a = 10\,$nm, and b) $a = 100\,$nm silica spheres, with $\T{gas} \equiv \T{env} = 300\,$K. These dynamics are due to the balance between optical absorption, blackbody absorption and emission (eqn.~\eqref{eqn:bbenergy}), photon recoil heating (eqn.~\eqref{eq:radiation_damping}), and cooling due to collisions with gas molecules (eqn.~\eqref{eqn:gascool}). This figure assumes a sphere trapped with a realistic laser intensity of $6\times10^{11}\,$W\,m$^{-2}$ with a wavelength of 1550nm. The optical trap depth is a) $U_0/\kB = 520\,$K and b) $U_0/\kB = 5\times10^5\,$K. For silica we use a complex refractive index $n = 1.45 +(2.5\times10^{-9})i$ \cite{Bateman2014}, material density $\rho = 2198\,$kg\,m$^{-3}$, and we assume the surrounding gas is $N_2$, with a corresponding surface accommodation coefficient $\acccoeff = 0.65$.}
\end{center}
\end{figure}
\section{Brownian motion}
\label{sec:Brownian}
Besides its important role in the development of the foundations of physics, today the Brownian particle serves as an exemplary model to describe a variety of stochastic processes in many fields, including physics, finance and biology. Brownian motion in non-equilibrium systems is of particular interest because it is directly related to the transport of molecules and cells in biological systems. Important examples include Brownian motors, active Brownian motion of self-propelled particles, hot Brownian motion, and Brownian motion in shear flows.
Despite its importance, the first experimental observation of ballistic Brownian motion had to wait a century until Li \emph{et al.}'s seminal work with optically levitated microparticles \cite{Li2010}.
This result already highlights the importance of the levitated particle system for studying thermodynamics.\\
In this section we will discuss the basics of Brownian motion. We will mainly treat the aspects that are necessary for understanding the following discussion of thermodynamics with levitated nanoparticles. For details on the theory of Brownian motion we refer the reader to the work of Ornstein, Uhlenbeck and Wang \cite{Uhlenbeck:1930tn, Wang:1945wd}.
First we will consider the motion of a free Brownian particle. Then we will add the confining potential and discuss important concepts such as a the power spectral density. Then we will include higher order (nonlinear) terms of the trapping potentials and discuss how they impact the power spectra.
Finally, we go to the opposite extreme case where the particle is cooled to extremely low energies such that quantum effects have to be included.
\subsection{Free Brownian motion}
The three motional degrees of freedom of a free particle are decoupled and without loss of generality it suffices to discuss a single coordinate $q(t)$ ($q = x,y, z$).
When coupled to a thermal bath at temperature $\T{CM}$ with rate $\g{CM}$, the equation of motion is given by the Langevin equation
\begin{equation}\label{eq:FreeLangevin}
\ddot{q} \,+\, \g{CM} \!\; \dot{q} =\; \mathcal{F}_q(t)/m,
\end{equation}
where $\mathcal{F}_q(t)/m =\sqrt{2\kB \T{CM}\g{CM}/m}\,\whn(t)
$ and $\whn(t)$ is a normalised white-noise process with $\langle\whn(t)\rangle = 0$, $\langle\whn(t)\whn'(t)\rangle = \delta(t-t')$. Here $\delta(t-t')$ is the Dirac delta function.
Since $\mathcal{F}_q(t)$ is a random process, $q(t)$ is also a random variable, such that each trajectory starting from the same initial conditions is different. However, the mean and variance for an ensemble of particles are well defined and are identical to the values for a single particle measured over a long time by virtue of the ergodic theorem.
Since the average force is zero, the mean particle position is also zero $\langle q(t)\rangle = 0$.
Its variance, or mean-square displacement, is given by
\begin{equation}\label{eq:pos_variance_free}
\var{q}(t)= \langle \left[q(t)- q(0)\right]^2\rangle = \frac{2\kB \T{CM}}{m\g{CM}^2}\left[\g{CM} t-1+e^{-\g{CM} t}\right].
\end{equation}
At long time scales ($t\gg1/\g{CM}$), the variance is the same as that predicted by Einstein's theory of diffusion $ \var{q}(t) = 2 D t $ where $D = \kB \T{CM}\left/m\g{CM}\right.$ is the diffusion coefficient. This regime is truly random in the sense that the particle trajectory is fractal and, therefore, is continuous but not differentiable. At short time scales ($t \ll 1/\g{CM}$), the dynamics of a Brownian particle is dominated by its inertia and its trajectory is ballistic. In this regime, the variance grows quadratically in time $ \var{q}(t) = (\kB\T{CM}/m) t^2 $ as expected for a free particle.
\subsection{Harmonic Brownian motion}
Under the influence of trapping forces, the particle will be localised about its equilibrium position. For small displacements, the trap can be approximated by a three-dimensional harmonic potential. As before, we limit our discussion to a single coordinate. The equation of motion for a harmonically trapped Brownian particle is \cite{Wang:1945wd}
\begin{equation}\label{eq:HarmonicLangevin}
\ddot{q} \,+\, \g{CM} \!\; \dot{q} +\wo^2 q=\; \sqrt{2\kB \T{CM}\, \g{CM}/m}\,\whn(t).
\end{equation}
Due to the confinement provided by the trap the variance does not grow unbounded. Instead, the particle oscillates in the trap at the characteristic frequency $\tilde{\w} = \sqrt{\wo^2-\g{CM}^2/4}$. For the optical potential eqn.~\eqref{eq:potential}, the trap frequency $\wo$ is given by eqn.~\eqref{eqn:frequencies_translational}.
We distinguish between three cases, the overdamped ($\wo \ll \g{CM}$), the critically damped ($\wo \approx \g{CM}$) and underdamped case ($\wo \gg \g{CM}$). This stochastic equation of motion has been studied in detail by Ornstein and Uhlenbeck \cite{Uhlenbeck:1930tn} and we summarise their results here. The variance of the position of a Brownian particle in an under-damped harmonic trap is
\begin{equation}\label{eq:pos_variance_underdamped}
\var{q}(t)= \frac{2\kB \T{CM}}{m\wo^2}\left[1-e^{-\frac{1}{2}\g{CM} t}\left(\cos(\tilde{\w} t) +\frac{\g{CM}}{2\tilde{\w}}\sin(\tilde{\w} t)\right)\right].
\end{equation}
In the over-damped harmonic trap, set $\tilde{\w}\to i \tilde{\w}$. In a critically damped harmonic trap, set $\tilde{\w}\to 0$. The position autocorrelation function is related to the variance as follows
\begin{equation}\label{eq:autocorrelation}
\corr{q}{q}
= \frac{\kB \T{CM}}{m\wo^2} - \frac{1}{2}\var{q}(t)
\end{equation}
The velocity autocorrelation function is given by
\begin{equation}\label{eq:vel_corr}
\corr{v}{v}= \frac{\kB \T{CM}}{m}e^{-\frac{1}{2}\g{CM} t}\left(\cos(\tilde{\w} t) -\frac{\g{CM}}{2\tilde{\w}}\sin(\tilde{\w} t)\right),
\end{equation}
and an experimental verification of this form is shown in fig.~\ref{fig:BrownianMotion}. In addition, position and velocity are correlated and the position-velocity correlation function is given by
\begin{equation}\label{eq:velpos_corr}
\corr{q}{v}=\corr{v}{q}= \frac{\kB \T{CM}}{m \tilde{\w}}e^{-\frac{1}{2}\g{CM} t}\sin(\tilde{\w} t).
\end{equation}
\begin{figure}[hbt]
\includegraphics[width=0.8\textwidth]{BrownianMotion}
\caption{\textbf{First experimental observation of the instantaneous velocity of a Brownian particle.}
a) The mean-square displacement for short times is proportional to $t^2$, a signature of ballistic motion.
b) The normalised velocity autocorrelation functions for different pressures in perfect agreement with eqn.~\eqref{eq:vel_corr}.
Figures taken from \cite{Li2010} with permission from Science.
\label{fig:BrownianMotion}}
\end{figure}
According to the Wiener-Khinchin theorem, the position autocorrelation function is the Fourier transform of the power spectral density $S_{qq}(\w) = \int_{-\infty}^\infty \corr{q}{q}e^{i\w t}\d t$, which for eqn.~\eqref{eq:HarmonicLangevin} is given by
\begin{equation}\label{eq:PSD_position}
S_{qq}(\w) = |\chi(\w)|^2 \Sff{}(\w)
= \frac{\g{CM} \kB \T{CM}\left/\pi m\right.}{(\w^2-\wo^2)^2+\g{CM}^2\w^2}
\end{equation}
where
$
\chi(\w) = m^{-1}\left[\w^2-\wo^2+i\g{CM}\w\right]^{-1}
$
is the response function or susceptibility of a harmonic oscillator. In the underdamped regime, the frequency spectrum of the autocorrelation function is strongly peaked around the trap frequency $\wo$, where as when overdamped the frequency spectrum is broad, as shown in fig.~\ref{fig:PSD}a). The power spectral density (PSD) is a useful tool in experiments with harmonic oscillators, since the dynamics of the oscillator can be separated from (spectrally distant) noise. An analysis of the power spectral density allows one to extract the center of mass temperature of the oscillator and the damping rate, as is clear from eqn.~\eqref{eq:PSD_position}.
\subsection{Nonlinear Brownian motion}
\label{sec:non-lin-BM}
Until now, we have only considered small deviations from the equilibrium position, where the potential is harmonic. However, the actual trapping potential is nonlinear. For the transverse directions, the lowest order nonlinear term is a cubic or Duffing nonlinearity in the equation of motion due to the symmetry of the trap.
Along the direction of propagation of the trapping laser, the scattering force breaks the symmetry and we also get a quadratic term. Similarly, gravity breaks the symmetry along the $y$-axis (see fig.~\ref{fig:Schematic}).
However, due to the smallness of the quadratic nonlinearity we will neglect it and focus our discussion on the Duffing term. Including the latter, the equation of motion for a single coordinate reads
\begin{equation}\label{eq:DuffingLangevin}
\ddot{q}_i \,+\, \g{i} \!\; \dot{q}_i +\w_i^2 q_i + \w_i^2\left(\sum_j\xi_{ij} q_j^2\right) q_i=\; \sqrt{2\kB \T{CM}\,\g{i}/m}\,\whn(t),
\end{equation}
where we have re-introduced the indices for a clearer notation. From the optical potential eqn.~\eqref{eq:potential} we find that $\xi_{ij} \sim -\waist_j^{-2}$. As a consequence, the oscillation frequency becomes a function of the oscillation amplitude and is red shifted by \cite{Gieseler2013}
\begin{equation}\label{eq:freq_shift}
\Delta\w_{\rm i} =\frac{3}{8}\w_i \sum_j \xi_{ij} \a_{\rm j}^2,
\end{equation}
where $\a_i$ is the instantaneous oscillation amplitude. In the low damping regime ($\w\gg\g{CM}$), the amplitude $\a_i$ and phase $\phi_i$ are quasi-static over many oscillation periods $2\pi/\w$ and only change significantly over times scales on the order of the relaxation time $2\pi/\g{CM}$.
Hence, the position can be written as $q_i(t) = \a_i(\tau)\cos \left[\w t + \phi_i(\tau)\right]$, with $2\pi/\w \ll \tau\ll 2\pi/\g{CM}$, where $\tau$ represents the slow timescale of the amplitude and phase evolution.
The frequency shift due to changes in the oscillation amplitudes is also known as self-phase modulation ($j=i$) and cross-phase modulation ($j\neq i$). To resolve the nonlinear frequency shift originating from thermal motion, the nonlinear contribution must be larger than the linear one, resulting in the condition
\begin{equation}\label{eq:R_nonlinear_shift}
\mathcal{R} = \frac{\Delta \w_{\rm NL}}{\g{CM}} = \frac{3\xi Q \kB \T{CM}}{4\w^2 m }\gg 1,
\end{equation}
where $Q = \w/\g{cm}$ is the quality factor. If this condition is fulfilled, the power spectral density (PSD) is no longer given by eqn.~\eqref{eq:PSD_position}. Instead, the harmonic oscillator PSD is now weighted with the probability to find the particle with a certain energy $E$ and the resulting PSD
\begin{equation}\label{eq:PSD_NL}
S_{\rm NL}(\w) =\int_0^\infty\rho(E)S_L(\w, E)\d E,
\end{equation}
is no longer symmetric, as shown in fig.~\ref{fig:PSD}b). The energy distribution is given by the Gibbs distribution $\rho(E) = Z^{-1}\exp(-E/\kB \T{CM})$ with $Z = \int \rho(E) \d E = \kB \T{CM}$
and the spectra $S_L(\w, E) = E \g{CM} \left/\pi m\wo^2\right.[(\w^2-\hat{\w}(E)^2)^2+\g{CM}^2\w^2]^{-1}$ are shifted to $\hat{\w}_0(E) = \wo+3\xi / (4 m\wo) E$.
Notably, due to the Gibbs distribution weighting term, the symmetry of the thermally driven spectra is opposite to the frequency response of the driven Duffing oscillator.
\subsection{Quantum Brownian motion}
While the nonlinear aspects of the potential are only relevant for large excitations, the opposite extreme, when the center of mass temperature is of the order of a single quantum of motion $\kB \T{CM}\approx \hbar \w_0$, is of particular importance since quantum effects can no longer be neglected.
In the quantum regime, the position autocorrelation eqn.~\eqref{eq:autocorrelation} contains the product of time-evolved operators $\langle \hat{q}(t) \hat{q}(0)\rangle$, which do not commute.
As a result, the spectrum
\begin{equation}
\label{eq:PSD_quantum}
S_{Q}(\w) %
= \frac{\hbar/\pi}{1-\exp\left(-\frac{\hbar \w}{\kB \T{CM}}\right)} {\rm Im} \chi(\w)
= \frac{\hbar\w m\g{eff}/\pi}{1-\exp\left(-\frac{\hbar \w}{\kB \T{eff}}\right)} |\chi(\w)|^2
\end{equation}
is asymmetric in frequency, where the PSD at positive frequencies is a factor $\exp(\hbar \wo\left/\kB \T{CM} \right.)$ higher than the PSD at negative frequencies, as shown in fig.~\ref{fig:PSD}c). The positive-frequency part of the spectral density is a measure of the ability of the oscillator to absorb energy, while the negative-frequency part is a measure of the ability of the oscillator to emit energy.
Therefore, we can understand the positive frequency part of the spectral density as being related to stimulated emission of energy into the oscillator, while the negative-frequency part is related to the emission of energy by the oscillator.
Typically, the motional frequencies of a levitated particle are $\sim 100\,\rm kHz$. Therefore, the required temperature is a few micro-kelvin and therefore out of reach for cryogenic techniques and one has to resort to active cooling techniques. Recent experiments using feedback cooling have already attained motional occupations of a few tens of phonons \cite{Jain2016a}.
\begin{figure}[hbt]
\includegraphics[width=\textwidth]{PSD}
\caption{\textbf{Position power spectral densities (PSD) in different regimes.}
a.) In the overdamped regime (quality factor $Q = \w/\g{cm} = 1/2$, red) the PSD has its maximum at $\w=0$ and falls off for higher frequencies. In the underdamped regime ($Q$ = 10, black), the PSD is peaked around the resonance frequency $\wo$ and has a linewidth of $\g{} \approx \wo/Q$.
b) For even higher $Q$, nonlinear effects can broaden the linewidth and instead of the expected narrow harmonic oscillator PSD (black, $Q$ = 100, eqn.~\eqref{eq:PSD_position}), the observed PSD is highly asymmetric (red, $\xi = 5\,\rm \mu m^{-2}$, $T=300\,\rm K$, eqn.~\eqref{eq:PSD_NL})
c) When the motion is cooled near the quantum ground state (here $\T{CM} = 10\,\rm \mu$K), nonlinear effects are negligible. Instead, quantum features lead to an asymmetric PSD, where the PSD at positive frequencies (red) is by a factor $\exp(\hbar \wo / \kB\T{CM}.)$ higher than the PSD at negative frequencies (blue).
\label{fig:PSD}}
\end{figure}
\section{Time dependent potentials}
\label{sec:time_dep}
So far we have considered only static trapping potentials $U_{\rm opt} \equiv U_{\rm opt}(\mathbf{r})$, where the trapping laser power is constant. However, through modulation of the trapping beam intensity, we can make the optical potential time-dependent. This is particularly useful when studying non-equilibrium dynamics. From eqn.~\eqref{eqn:frequencies_translational} it follows that a change in optical power $\delta \Popt(t)$ changes the trap frequency by $\w = \wo(1+\emod(t)/2)$, where $\emod(t) = \delta \Popt(t) \left/ \Poptm\right.$ and $\Poptm$ is the mean optical power. The equation of motion under this parametric modulation is given by
\begin{equation}\label{eq:ParametricLangevin}
\ddot{q} \,+\, \g{CM} \!\; \dot{q} +\wo^2\left[1+\emod_0 \cos(\omegamod t)+\xi q^2 \right]q=\; \sqrt{2\kB \T{CM}\g{CM}/m}\,\whn(t) .
\end{equation}
Energy is most effectively exchanged between the trapping laser and the particle if the modulation $\emod(t) = \emod_0\cos(\omegamod t)$ occurs at twice the trapping frequency $\omegamod\approx 2\w_0$. The flow of energy is thereby determined by the relative phase $\phimod$ between the particle oscillation and the laser intensity modulation\footnote{Note that $\phimod$ doesn't appear in the expression for $\emod(t)$, since the modulation serves as the time reference and $\phimod$ is the phase of the particle with respect to the modulation.}. If the modulation is in-phase, energy is extracted (cooling), while the motion is excited when the modulation is out-of-phase (heating). For $\emod_0>1/2Q$, where $Q = \omega_0/\g{cm}$ is the motional quality factor, energy is pumped into the system faster than can be extracted through dissipation. For a harmonic trap this would lead to a steady increase in energy. However, due to the nonlinear Duffing term, the oscillation frequency of the particle shifts away from the energy matching condition, which limits the maximum oscillation amplitude.
Without active stabilization of the modulation phase with respect to the particle motion, the relative phase is random. Therefore, to achieve cooling the phase needs to be actively stabilized, for instance with a phase-locked-loop \cite{Jain2016a}.
\subsection{Effective potentials and non-equilibrium steady states}
\label{sec:effective_potential}
As we discussed in Sec.~\ref{sec:non-lin-BM}, the particle motion is described by a slowly varying evolution of the phase and amplitude and a fast modulation at frequency $\w$.
In many cases we are primarily interested in the slow evolution of the energy or amplitude and it is, thus, advantagous to work with the effective equations of motion for the energy instead of considering to full particle dynamics.
This strategy allows us to define effective potentials for the energy and to derive an effective temperature for the particle center-of-mass motion.
For convenience we introduce the position $q$ and momentum $p=m\dot q$ differential equations of motion for a particle in a time-dependent potential
\begin{subequations}
\begin{align}
\d q&=\frac{p}{m}\d t, \label{equ:SDE1}\\
\d p &=\left[-m\wo^2q-\g{CM} p+\emod_0 m \Omega^2_0 \cos(\omegamod t)q\right]\d t+\sqrt{2m\g{CM} \kB \T{CM}}\,\d W.
\label{equ:SDE2}
\end{align}
\end{subequations}
Here, $W(t)$ is the Wiener process with $\langle W(t) \rangle = 0$, $\langle W(t) W(t') \rangle = t'-t$. Note that $\langle W^2(t) \rangle = t$ for any time $t\ge 0$ and, thus, for an infinitesimal time interval $\d t$ one has $\langle ({\rm dW)^2}\rangle=\d t$. The white noise $\whn(t)$ appearing in the random force can be viewed as the time derivative of the Wiener process, $\whn(t)= \d W(t)/\d t$. The total energy of the particle in one dimension is given by
\begin{equation}\label{eq:Energy}
E(q,p)=\frac{1}{2}m\wo^2 q^2+ \frac{p^2}{2m}+\frac{1}{4}\xi m\wo^2 q^4.
\end{equation}
To avoid multiplicative noise, i.e. a noise term that depends on the current value of the energy, we consider the square root of the energy rather than the energy itself,
\begin{equation}
\sqrtE(q, p)=\sqrt{E(q, p)}.
\end{equation}
At low friction, the amplitude $\a$ and phase $\phimod$ with respect to the driving force are quasi-constant, and the particle performs an undisturbed harmonic oscillation evolving according to
\begin{equation}
q(t)=\a \cos (\Omega t + \phimod) \qquad \qquad p(t)=-m\Omega \a\sin(\Omega t + \phimod),
\end{equation}
where the amplitude of the oscillation is related to $\sqrtE$ by $\a=\sqrt{2/m}(\sqrtE/\Omega)$. Note that the oscillation frequency $\w$ is not necessarily the same as the frequency $\wo$ of the unperturbed harmonic oscillator. For instance, for strong modulation the particle motion entrains with the modulation and $\w \approx \omegamod/2$ \cite{Gieseler:2014wt}.
Applying Ito's formula for the change of variables to $\sqrtE(q, p)$ and integrating over an oscillation period, we find that the change $\d \sqrtE$ during a short time interval is given by a Langevin equation for a \emph{fictitious} overdamped Brownian particle
\begin{equation}\label{eq:stochastic_epsilon}
\d \sqrtE = \geff^{-1}f(\sqrtE)\d t+ \sqrt{2 \kB \T{CM}\left/\geff\right.} \d W
\end{equation}
with damping $\geff = 4/\g{CM}$, moving through an effective potential
\begin{equation}\label{eq:potential_epsilon}
\Ueff(\sqrtE)=\sqrtE^2
-\kB \T{CM} \ln \sqrtE
+\frac{\sqrtE^2\emod_0\wo^2 \sin(2\phimod)}{2\g{CM}\w },
\end{equation}
under the influence of an external force $f(\sqrtE) = - \d \Ueff(\sqrtE)/\d \sqrtE$. Note that due to the integration over one oscillation period, this equation has $\sqrtE$ as its only time dependent variable, while the dependence on other variables has been removed. By virtue of this isomorphism with over-damped Brownian motion, one can then immediately infer that eqn.~\eqref{eq:stochastic_epsilon} samples the distribution
$
P_\sqrtE(\sqrtE) \propto \exp\left\{-\beta \Ueff(\sqrtE)\right\},
$
where $\beta = 1\left/\kB \T{CM}\right.$. Equation~\eqref{eq:stochastic_epsilon} implies that the time evolution of $\sqrtE$ can be viewed as a Brownian motion in the high friction limit.
A small real friction $\g{CM}$ corresponds to large effective friction $\geff$ determining the time evolution of $\sqrtE$ and, thus, the energy $E$ of the oscillator.
Interestingly, the \emph{fictitious} Brownian particle of the \emph{time-dependent} optical potential can exhibit similar dynamics to the \emph{real} Brownian particle in a \emph{static} optical potential \cite{Ricci:2017eh}.
Changing variables from $\sqrtE$ to $E=\sqrtE^2$ and applying Ito's formula, we finally obtain the probability density function of the energy,
$
P_E(E) = \frac{1}{Z} \exp\left\{-\beta' E\right\}
$,
where $\beta' = 1/\kB\T{CM}'$ with effective temperature
\begin{equation}\label{eq:Teff}
\T{CM}' = \T{CM}\left(1+\frac{\emod_0\wo^2 \sin(2\phimod)}{2\g{CM}\w}\right)^{-1}.
\end{equation}
Equation~\eqref{eq:Teff} states that parametric modulation of the trapping potential results in an effective temperature change of the environment, where the particle centre-of-mass temperature changes from $\T{CM}$ to $\T{CM}'$. For $-\pi/2<\phimod<0$, $\T{CM}'>\T{CM}$, that is the particle motion is heated, while for $0<\phimod<\pi/2$, $\T{CM}'<\T{CM}$ and the particle motion is cooled. The rate at which the particle thermalizes with this effective bath is $\g{CM}' = \g{CM} \left(\T{CM}/\T{CM}'-1\right)$, where the largest rates are achieved at $\phimod = -\pi/4$ and $\phimod = \pi/4$, for heating and cooling respectively. If the relative phase between the particle motion and the modulation $\phimod$ is not stabilized actively, for example through implementing a phase-locked loop fed back onto the trapping laser intensity, the particle motion will self-lock to $\phimod = -\pi/4$. Thus, a effective hot bath can be implemented easily by a simple modulation of the trapping laser at $\omegamod\approx 2\wo$.
The change of variables also yields the corresponding stochastic differential equation for the energy
\begin{equation}
{\rm d} E = \left[-\g{CM} (E - \kB \T{CM})
-\frac{\eta \wo E^2}{2m \w^2}
-\frac{E\emod\wo^2 \sin(2\phi)}{2\w}
\right]{\rm d}t
+\sqrt{2E \g{CM} \kB \T{CM}}{\rm d}W.
\label{equ:SDE_energy}
\end{equation}
In contrast to the stochastic equation of motion for $\sqrtE$, here the noise is multiplicative, i.e., its amplitude is energy dependent.
It is predicted that by engineering an effective cold bath, it is possible to cool the motion of a levitated nanoparticle to its motional quantum ground state $\kB\T{CM} \leq \hbar\w$, for operation in the quantum regime. One method is to use the passive feedback provided by optical cavity cooling, with firm predictions of reaching the quantum ground-state \cite{Chang2010}. The thermal occupation of an optical cavity at room temperature is extremely low, with a photon occupation of $n_{\rm ph} = \sqrt{\kB\T{env}/\hbar\omL}\ll 1$, which forms the effective low temperature bath. Using active feedback cooling $\sim 100\, \mu$K temperatures have been achieved, corresponding to a phonon occupancy of $\sim 20$ \cite{Jain2016a}.
\subsubsection{Non-thermal states}
\label{sec:non-thermal}
The modulation of the trapping potential gives rise to a non-conservative force that allows us to inject and extract energy from the particle. Since there is a continuous flow of energy, the particle is no longer in thermal equilibrium. Surprisingly, under the appropriate conditions we can describe the particle as in thermal equilibrium with an effective bath (c.f. eqn.~\eqref{eq:Teff}). However, this description breaks down when we heat the particle ($\phimod = -\pi/4$) above the threshold condition
$
\emod > 2\Qf^{-1}\sqrt{1+\Qf^2\left(2-\omegamod/\wo \right)^2}\approx 2\Qf^{-1}
$, where the approximation is exact at parametric resonance $\omegamod = 2\wo$. Then the effective temperature diverges and the motion transitions from a thermal state to a coherent oscillation, which is phase-locked to the modulation source, similar to the lasing condition of an optical oscillator.
A more subtle non-equilibrium steady state, which can no longer be described an effective thermal bath can be achieved by parametric feedback modulation of the form $\emod_\text{fb}(t) = -(\fb/\wo) q(t)\dot{q}(t)$, where $\fb$ parametrizes the feedback strength. This leads to a parametric modulation at the parametric resonance condition, while ensuring a phase that is optimized for extracting energy from the mechanical mode. However, in contrast to the previous case with constant modulation amplitude, here the modulation amplitude is proportional to the particle energy $\emod_0 \propto \a^2\propto E$. As a consequence, the particle feels a nonlinear friction force with $\g{NL}\propto E$.
The probability distribution for the energy, including the position dependent feedback term $\fb$ and parametric modulation with constant amplitude $\emod_0$, is then given by
\begin{equation}
P_E(E) = \frac{1}{Z} \exp\left\{-\beta\left[\left(1+\frac{\emod_0\wo^2 \sin(2\phimod )}{2\g{CM}\w}\right)E
+\frac{\eta \wo}{4m\g{CM} \w^2}E^2 \right]\right\},
\label{eq:energy_distribution}
\end{equation}
where the normalization factor $Z=\int P_E(E){\rm d}E$ is given by
\begin{equation}
Z = \sqrt{\frac{\pi m\g{CM} \w^2}{\beta \eta \wo}}
h \left(\sqrt{\frac{\beta m\g{CM} \w^2}{\eta \wo}}\left(1+\frac{\emod_0\wo^2 \sin(2\phimod)}{2\g{CM}\w}\right) \right),
\end{equation}
and the function $h(x)$ is defined as
$
h(x)=\exp(x^2){\rm erfc}(x)
$
and ${\rm erfc}(x)$ is the complementary error function. Thus, the energy distribution is that of an equilibrium system with effective energy
\begin{equation}
H=\left[1+\frac{\emod_0\wo^2 \sin(2\phimod)}{2\g{CM}\w}\right]E+\frac{\eta \wo}{4m\g{CM} \w^2}E^2.
\end{equation}
While the term proportional to $E^2$ is caused by the feedback cooling, the term proportional to $E$ is affected only by the parametric modulation.
Since, for low friction, the energy of the oscillator is essentially constant over many oscillation periods, the full phase-space density $P_{q, p}$ can be obtained by averaging the micro-canonical distribution $P_m(q,p;\tilde{E}) = g^{-1}(\tilde{E})\delta\left[E(q,p)-\tilde{E}\right]$ over the energy distribution eqn.~\eqref{eq:energy_distribution}. For low friction constants and small feedback strength, this linear superposition of micro-canonical distributions is valid even under non-equilibrium, conditions and we obtain
\begin{equation}\label{eq:Position_distribution_SS}
P_{q, p}(q, p) =\frac{\wo}{2\pi} P_E(E(q,p)),
\end{equation}
where the $E(q,p)$ is given by eqn.~\eqref{eq:Energy}, and we approximated the micro-canonical density of states with the density of states for the harmonic oscillator $g(\tilde{E}) \approx 2\pi/\wo$, that is we neglect the Duffing term of the potential in $g(\tilde{E})$. Note, however, that while we have neglected the Duffing term in the expression for the density of states, it is included in the energy appearing in the argument of the exponential on the right-hand side of the above equation~\eqref{eq:energy_distribution}.
\subsubsection{Thermal squeezing}
A big advantage when using a levitated oscillator over conventional nano-mechanical oscillators is that the mechanical frequency can be changed by changing the power of the trapping laser. This allows one to perform unconfined time-of-flight measurements, to create physically large superposition states \cite{Bateman2014}, and to prepare squeezed states. In the quantum regime, squeezing enables one to push the fundamental quantum uncertainty below the standard quantum limit. A thermal state can be squeezed to reduce the uncertainty in one of the quadratures at the expense of anti-squeezing the other. While classical thermal squeezing of a levitated particle has been observed experimentally \cite{Rashid:2016hp}, squeezing below the standard quantum limit remains elusive. In the following we discuss how a change in laser power leads to squeezing.
A sudden change, or quench, in power of the trapping beam changes the mechanical frequency to a new value $\w \to \ws$. Thus, the time evolution of the position and momentum of the harmonic oscillator is given by
\begin{subequations}
\begin{align}
q(\tau \ws) &= X_Q\cos(\ws \tau)+P_Q\frac{\w}{\ws }\sin(\ws \tau)\\
p(\tau \ws) &= m P_Q \w \cos(\ws \tau)-m X_Q\ws\sin(\ws \tau),
\end{align}
\end{subequations}
where we introduced the position and momentum quadratures $X_Q = q(0)$ and $P_Q = -\dot{q}(0)/\w$ to denote the position and velocity at the time of the quench. After a time $\tau$, the power is switched back to its original value and we find that the phase space distribution for position and momentum is
\begin{align}\label{eq:Position_distribution_SQ}
P_{q,p}^\text{sq}(q, p, \tau) &= \frac{\w \beta}{2\pi} \times \\
&\exp\left[-\beta
\frac{1}{2}m\Omega^2
\left(
\left[X_Q \cos(\ws\tau)+e^{2r}P_Q \sin(\ws\tau)\right]^2
+
\left[P_Q \cos(\ws\tau)-e^{-2r}X_Q \sin(\ws\tau)\right]^2 \right)
\right]\nonumber
\end{align}
where we introduced the squeezing parameter $r = \frac{1}{2}\log(\w/\ws)$. Therefore, the squeezing pulse of duration $\tau$ leads to a non-Gaussian state, with correlations between position and momentum.
However, at times $\tau = \pi/ 2\ws$, the exponent in eqn.~\eqref{eq:Position_distribution_SQ} simplifies and we find that the position quadrature is squeezed by a factor $\exp(-2r)$, while the momentum quadrature is anti-squeezed by $\exp(2r)$.
Due to the reduced width of the squeezed distribution along a particular direction, this kind of state preparation allows one to reduce the measurement uncertainty. However, to be actually useful, the error introduced by the anti-squeezing of the momentum quadrature should not overwhelm the squeezing of the position quadrature.
Note that in contrast to the distribution eqn.~\eqref{eq:Position_distribution_SS}, which describes a steady state distribution, i.e. it does not depend on the observation time, eqn.~\eqref{eq:Position_distribution_SQ} is defined at a specific time (right after the application of the squeezing pulse). From this distribution, the system will relax back into thermal equilibrium as described in section \ref{sec:Relaxation}.
\section{Thermodynamics}
\label{sec:thermodynamics}
In this final section we will discuss the application of levitated nanoparticles to problems in stochastic thermodynamics and highlight some relevant experimental results.
\subsection{Kramers escape and turnover}
We have discussed the dynamics of a particle confined within a potential, and subject to fluctuating forces from the environment. Due to the stochastic nature of the imparted force, there is a probability that the particle will gain enough energy to escape the potential, even when it is confined by a potential much deeper than $\kB\T{CM}$, in a process known as Kramers escape. This form of ``classical tunnelling'' appears in a diverse range of physical systems, importantly including chemical reaction rates, protein folding, atomic transport in optical lattices and molecular diffusion at solid-liquid interfaces \cite{Rondin2017}.
The Kramers' escape rate is given by
\begin{equation}\label{eq:KramersRate}
\R{K} = \R{0} \exp\left(-\frac{U_{\rm opt}}{\kB \T{CM}}\right)
\end{equation}
where $\R{0}$ is the attempt frequency and $U_{\rm opt}$ is the barrier height. From the Boltzmann factor in eqn.~\eqref{eq:KramersRate} it follows that such a transition is exponentially suppressed if the potential is much deeper that the thermal energy $U_{\rm opt}\gg \kB \T{CM}$.
Closely related to the Kramers escape is the Kramers turnover problem, which describes the tunnelling between two potential minima as the friction is varied. This is often more relevant in physical situations, describing the transitions between two protein configurations, for example. Kramers found \cite{Kramers1940} that in the underdamped regime, the transition rate increases with \emph{increasing} friction, and that in the overdamped regime the transition rate increases with \emph{decreasing} friction, with the transition region labelled the turnover. Fifty years later, a theory was developed that linked the two regimes \cite{Melnikov1991}. The first quantitative measurement of Kramers turnover was achieved using a levitated nanoparticle hopping between two potential wells formed by focussed laser beams. In this experiment, the friction rate was varied over many orders of magnitude through a change in the gas pressure $\Pg$ \cite{Rondin2017}.
We consider the hopping between two metastable potential wells that are separated by a barrier. The local principal axes are labeled $i=x',y',z'$ and the potential extrema are labeled $p = A,B,C$, as illustrated in fig.~\ref{fig:Kramers}a). Since the particle is not lost but recaptured in the second well, this problem is much more convenient to study experimentally than stochastic escape. A double well potential can be created by using two tightly focused laser beams, where the intensity and exact relative position of the two foci determines the height of the barrier. The hopping rates between the two wells is determined by the local curvatures of the potential at the extrema.
In the overdamped regime, the hopping from well $A$ to well $C$ via the barrier $B$ is given by Kramers' law. For a three-dimensional optical potential its dependency on the potential parameters is given by
\begin{equation}\label{eq:R_AC_HD}
R_{A\to C}^{\rm HD} = \frac{1}{2\pi} \prod_{i\in\{x',y',z'\}} \frac{\w_i^A}{|\w_i^B|} \left[ \sqrt{|\w_B^S|^2+\frac{\g{CM}^2}{4}}-\frac{\g{CM}}{2}\right] e^{-\frac{U^A}{\kB\T{CM}}}
\approx \frac{1}{2\pi} \frac{\w^A\w^B}{\g{CM}} e^{-\frac{U^A}{\kB\T{CM}}} \ ,
\end{equation}
where $\w^p_i$ are the three frequencies at the three extrema along the local principal axis ($x',y',z'$) and $\w_B^S$ is the purely imaginary frequency of the saddle point \cite{Rondin2017}. The approximation holds in the limit of high damping $\g{CM}\gg\w_B$ and one dimensional motion.
In the underdamped regime, on the other hand, the rate is limited by the slow transfer of energy between the particle and its environment. This leads to a hopping rate that is proportional to $\g{CM}$
\begin{equation}\label{eq:R_AC_LD}
R^{\rm LD}_{A\to C}=\frac{\g{CM} S^A}{2\pi}\frac{\w^A}{\kB\T{CM}} e^{-\frac{U^A}{\kB\T{CM}}}.
\end{equation}
where $S^p=4\int_{r_p}^{r_B}\sqrt{2m(U_B-U(\mathbf{r}))} \d r$ is the particle action over one oscillation period in well $p$ and is measured along the minimum energy path of the potential. These two limiting cases were already derived by Kramers \cite{Kramers1940}. In the transition region, such a simple analytical formula does not exist. Instead the rates depend on the depopulation factor
\begin{equation}\label{eq:depopulation_factor}
\Upsilon(\delta) = \exp \left[\frac{1}{\pi} \int_{0}^{\infty}\ln\left\{ 1-\exp\left[-\frac{\delta}{k_B T} \left(x^2 +\frac{1}{4}\right) \right] \right\}\frac{\mathrm d x}{x^2 +\frac{1}{4}} \right] \ ,
\end{equation}
where $\delta$ is the energy loss parameter. Generally, the estimation of the energy loss parameter is quite challenging. However, for memory-free friction, the energy loss is well approximated by $\delta = \g{CM} S_p$.
To account for the difference in transition rates from well A to C versus well C to A we need to multiply the transition rates by a factor $\prod_{p=A,B}\Upsilon(\g{CM} S^p)/\sum_{p=A,B}\Upsilon(\g{CM} S^p)$ and arrive at the general expression for the hopping rate
\begin{equation}\label{eq:R_hopping}
R (\g{CM}) = \frac{\Upsilon(\g{CM} S^A) \Upsilon(\g{CM} S^C)}{\Upsilon(\g{CM} S^A+\g{CM} S^C)}\left[ R_{A\to C}^{\rm HD}+ R_{C\to A}^{\rm HD}\right]\, .
\end{equation}
Figure~\ref{fig:Kramers}b) shows the limiting cases in the high and low damping regime, and the full solution for arbitrary damping. In addition, the figure includes experimental data from Rondin \emph{et al.} which, using an optically levitated nanoparticle, presents the first quantitative measurement of the Kramers rate across the turnover \cite{Rondin2017}.
\begin{figure}[hbt]
\includegraphics[width=0.8\textwidth]{Kramers}
\caption{\textbf{Measurement of the Kramers turnover with a levitated nanoparticle.}
a.) Illustration of a particle in a (generally asymmetric) bistable potential. The hopping rate $R$ between the wells $A, C$ depends upon the local potential $U$, and the background pressure $\Pg$.
b.) Data illustrating the first experimental observation of Kramers' turnover, taken from \cite{Rondin2017}. Also marked is the full theory from \cite{Melnikov1991} (solid line), the turnover point which depends on the comparison of the damping rate $\g{CM}$ and the harmonic trap frequency at the crossing point $\Omega_{\rm B}$ (dashed line), and the limiting cases as predicted by Kramers \cite{Kramers1940} (dot-dashed lines).
\label{fig:Kramers}}
\end{figure}
\subsection{Relaxation \label{sec:Relaxation}}
In the steady-state, a trapped particle samples the distribution eqn.~\eqref{eq:energy_distribution}, which depends on experimental parameters, such as the average power of the trapping laser, and the rate and depth of any modulation of the optical potential. Hence, under a non-adiabatic change of the parameters, the systems relaxes into a new steady state.
The Fokker-Planck equation that describes the time evolution of the probability density function $P_E(E, t)$, including feedback and modulation, is given by
\begin{equation}\label{eq:FokkerPlanckEnergy}
\frac{\partial P_E(E, t)}{\partial t}=\frac{\partial }{\partial E}\left[\g{CM} (E - \kB T)
+\frac{\eta \wo E^2}{2m \Omega^2}
+\frac{E\emod_0\wo^2 \sin(2\phimod)}{2\w}\right]P_E(E, t)
+{\g{CM} \kB T}\frac{\partial^2 }{\partial E^2}E P_E(E, t).
\end{equation}
In general it is non-trivial to find an analytic solution to eqn.~\eqref{eq:FokkerPlanckEnergy}.
Amazingly, in the absence of feedback cooling ($\fb = 0$), the equation of motion for the energy corresponds to the
Cox-Ingersoll-Ross model for interest rates, for which the exact analytical solution is given by the Noncentral Chi-squared distribution \cite{Salazar:2016ey}
\begin{equation}
P_E(E|E_0, t) = c_t e^{-c_t(E+E_0e^{-\g{CM} t})}I_0(2c_t\sqrt{E E_0 e^{-\g{CM} t}}),
\end{equation}
where $c_t = \beta\left(1-e^{-\g{CM} t}\right)^{-1}$, $I_0(x)$ is the modified Bessel function of the first kind and $E_0$ is the initial energy, i.e $P_0(E|E_0) = \delta(E-E_0)$. As expected, the equilibrium distribution $P_\infty(E|E_0) = \beta\exp(-\beta E)$ does not depend on the initial conditions and is given by the Maxwell-Boltzmann distribution at temperature $\T{CM} = 1/(\kB \beta)$.
If the system is initially prepared at $t = 0$ in a steady state with energy distribution $P_0(E_0)$, the energy distribution after time $t$ is
\begin{equation}
P_E(E, t) = \int_0^\infty P_E(E|E_0, t)P_0(E_0)\d E_0.
\end{equation}
For an initial Maxwell-Boltzmann distribution, corresponding to a thermal equilibrium distribution at temperature $\T{init}$, the energy distribution at time $t$ is also a Maxwell-Boltzmann distribution
\begin{equation}
P^\text{MB}_E(E, t) = \beta(t) e^{-\beta(t)E},
\end{equation}
with time dependent temperature
\begin{equation}
\T{CM}(t) = \T{\infty}+(\T{init}-\T{\infty})e^{-\g{CM} t}.
\end{equation}
Note that the initial temperature $\T{init}$ and final temperature $\T{\infty}$ can be controlled in the experiment by modulation of the trapping laser (feedback cooling), as discussed in section~\ref{sec:non-thermal} and demonstrated by Gieseler \emph{et al.} \cite{Gieseler2014}. Explicitly, a levitated nanoparticle can be cooled via feedback to a centre-of-mass temperature $\T{CM}$ far below the ambient temperature $\T{env}$. Once the feedback modulation is switched off, the particle will thermalize with the environment (in general via collisions with surrounding gas), at an average rate $\g{CM}$, which can be controlled by varying the gas pressure.
The rate at which the particle relaxes to the new equilibrium state can also be accelerated by using time-dependent potentials \cite{Martinez2016}.
\subsection{Fluctuation theorems}
As a system relaxes to a thermal equilibrium, the dynamics satisfy detailed balance with respect to the equilibrium distribution, and the time reversibility of the underlying dynamics implies that the transient fluctuation theorem
\begin{equation}
\frac{P(-\dS)}{P(\dS)} = e^{-\dS},
\end{equation}
for the relative entropy change $\dS = \beta \Q+\Delta \entropy$ (or Kullback-Leibler divergence) holds. The quantity $\Delta \entropy = \entropy(t)-\entropy(0)$ is the difference in trajectory dependent entropy $\entropy(t) = -\ln P_0(u(t)) $ between the initial and the final states of the trajectory $u(t)$. The relative entropy change $\Delta \mathcal{S}$ is defined as the logarithmic ratio of the probability $P[u(t)]$
to observe a certain trajectory $u(t)$ and the probability $P[u^*(t)]$ of the time reversed trajectory $u^*(t)$,
\begin{equation}
\Delta \mathcal{S} = \ln \frac{P[u(t)]}{P[u^*(t)]}.
\end{equation}
Here, $u(t)$ denotes an entire trajectory of length $t$ including position and momentum of the oscillator and
$u^*(t)$ denotes the trajectory that consists of the same states visited in reverse order with inverted momenta.
$\Q$ is the heat absorbed by the bath at reciprocal temperature $\beta$.
Because no work is done on the system, the heat $\Q$ exchanged along a trajectory equals the energy lost by the system, $\Q = -(E_t-E_0)$ where $E_0$ and $E_t$ are the energy at the beginning and at the end of the stochastic trajectory.
Note that the fluctuation theorem holds for any time $t$ at which $\dS $ is evaluated, and it is not required that the system has reached the equilibrium distribution at time $t$.
In general, the steady distribution $P_0(u(t))$ necessary to compute $\Delta\entropy$ is unknown. However, from the distribution derived for our model eqn.~\eqref{eq:Position_distribution_SS}, we find that for the relaxation from a non-equilibrium steady state generated by nonlinear feedback of strength $\fb$ and parametric modulation of strength $\emod$, the relative entropy change is given by
\begin{equation}
\Delta \mathcal{S}=
-\beta \frac{\emod\wo^2 \sin(2\phimod)}{2\Gamma\Omega}\left[E_t-E_0\right]
-\beta \frac{\eta \wo}{4m\Gamma \Omega^2}\left[E^2_t-E^2_0\right].
\end{equation}
Thus, our stochastic model allows us to express the relative entropy change during a relaxation trajectory in terms of the energy at the beginning and the end of that trajectory. This model was verified using a levitated nanoparticle by Gieseler \emph{et al.} \cite{Gieseler2014}, when starting from a variety of non-equilibrium steady states.
In addition to the fluctuation statistics during relaxation between steady states, one can also consider fluctuations during different protocols, e.g. during a full thermodynamic cycle or while driving the particle with an external force $f(t)$ as was done by Hoang et al. \cite{Hoang:2018fn}, who verified a differential fluctuation theorem for the work $W = -\int_0^\tau \dot{f}(t) q(t) \d t$
\begin{equation}
\frac{P\left(-W, u^*(t)\right)}{P\left(W, u(t)\right)}
= e^{-\beta(W-\Delta F)}.
\end{equation}
The differential fluctuation theorems can be integrated to yield a series of well known fluctuation theorems, such as the Jarzynski equality, the Crooks fluctuation theorem and the Hummer-Szabo relation. Thus, by verifying the underlying differential fluctuation theorem, the validity of the integral fluctuation theorem is implied. Importantly, the fluctuation theorems are valid for arbitrarily-far-from-equilibrium processes. Both detailed and integral fluctuation theorems allow the estimation of equilibrium free energy changes from nonequilibrium protocols and have found applications in determining the free energies of DNA molecules.
For a detailed review see Refs.~\cite{SeifertReview}.
\subsection{Heat Engines}
\label{sec:engines}
Technology is continuously miniaturizing, but as we pass below the micro-scale the challenge is not limited to the difficulty in constructing small devices. Once the work performed per duty cycle of an engine becomes comparable to the thermal energy of the piston, it is possible for the engine to run in reverse for short times, due to the fluctuating nature of energy transfer with the heat bath. This is exactly the scale at which biological systems operate, and a regime which levitated nano- and micro-particles have access to.
To apply work to a trapped particle, one must either change (via a control parameter $\lambda_c(t)$) the trapping potential $U(q,\lambda_c)$, or apply an external force $f(q,\lambda_c)$, in which case the incremental work $\d\W$ reads:
\begin{equation}
\label{eqn:WI}
\d\W = (\partial U / \partial \lambda_c)\,\d\lambda_c + f\, \d q,
\end{equation}
with an associated heat increment:
\begin{equation}
\d\Q = F\,\d q,
\end{equation}
where $F(q, \lambda_c) = -(\partial U)/(\partial q) +f$ is the total force acting on the particle, due to both the potential $U$ and the external force $f$. Importantly, the external force $f$ accounts for deterministic and stochastic contributions. Hence, along a trajectory $u(0) \to u(\tau)$:
\begin{eqnarray}
\label{eqn:totalWQ}
&\W(q(t)) = \int_0^\tau [ (\partial U / \partial \lambda_c) \dot{\lambda_c} + f\dot{q}]\,\d t \notag \\
&\Q(q(t)) = \int_0^\tau \dot{\Q}\,\d t = \int_0^\tau F \dot{q}\,\d t.
\end{eqnarray}
\begin{figure}[b]
\begin{center}
{\includegraphics{HeatEngine.pdf}}
\caption{\label{fig:HeatEngine} \textbf{Stochastic heat engine.} This figure illustrates an engine cycle to realize a stochastic heat engine. A particle is confined by a potential $U(q,\lambda_c)$, and coupled to a hot/cold heat bath of temperature $\T{H/C}$. The shaded curves illustrate the position probability distribution of the particle. The cycle is explained in the text. The inset below step 1) illustrates an example optimum protocol for realising the compression step for an underdamped heat engine. The trap is expanded by lowering the spring constant from a value $\ks{i}$ to $\ks{f}$, with the most efficient protocol involving sharp parameter variations (adapted from \cite{Dechant2014}).}
\end{center}
\end{figure}
Next, we will explicitly apply this to the case of a heat engine. We discuss heat engines since they are an extremely useful machine. An engine, or motor, converts one type of energy into mechanical work, and a heat engine specifically converts heat flow between two reservoirs into mechanical work, particularly useful since heat is often generated as a waste product. Schmiedl \& Seifert gave the first full description of a colloidal stochastic heat engine \cite{Schmiedl2008}, which was realized by Blickle \& Bechinger a few years later \cite{Blickle2011}. The engine operates under the following cyclic process (as illustrated in fig.~\ref{fig:HeatEngine}): 1) an isothermal transition at temperature $\T{H}$ with a time dependent variation of $U(q,t)$ to extract work $\W<0$; 2) an instantaneous reduction in temperature $\T{H}\to \T{C}$, where no heat is exchanged with the bath (adiabatic); 3) an isothermal transition at $\T{C}$ with a time dependent variation of $U(q,t)$ and $\W>0$; 4) an instantaneous increase in temperature $\T{C} \to \T{H}$.
For a harmonically confined particle $U(q,t) = \ks{}(t)q(t)^2/2$, where $\ks{}$ is the trap stiffness, our control parameter $\lambda_c(t) \equiv \ks{}(t)$. Other choices of $\lambda_c$ could include a movement of the trap centre. Reducing $\ks{}$ corresponds to an expansion $\langle W\rangle <0$, as the confinement is weakened. Following references \cite{Schmiedl2008, Dechant2017}, it is convenient to analyse this scenario by considering the equations of motion for the variance $\var{q}(t) \equiv \langle q^2(t) \rangle$, with the equation of motion:
\begin{equation}
\label{eqn:sigmax}
\dvar{q} = -m\mu\ddvar{q} -2\mu\ks{}(t)\var{q} +2m\mu \var{v},
\end{equation}
where $\mu = 1/(m\g{CM})$, $\var{v}(t) \equiv \langle \dot{q}^2(t) \rangle$, and also noting for the harmonic oscillator that the frequency $\w(t)= \sqrt{\ks{}(t)/m}$. This can be simplified in the overdamped regime $\g{CM} >> \w$, first by removing the inertial term $\propto \ddvar{q}$, and secondly by assuming that the state is always thermal $\var{v} = \kB \T{CM}/m$ \cite{Dechant2017}. This yields the overdamped equation of motion,
\begin{equation}
\label{eqn:sigmax_od}
\dvar{q} = -2\mu \ks{}(t)\var{q} + 2\mu \kB{} \T{CM}.
\end{equation}
Using eqn.~\eqref{eqn:totalWQ} we find for the total work $\W$ along an isothermal trajectory at $\T{CM}$ from time $t_{\rm i} \to t_{\rm f}$,
\begin{equation}
\W(\ks{}(t)) = \int_{t_{\rm i}}^{t_{\rm f}} \dot{\ks{}}\frac{\var{q}}{2}\,\d t,
\end{equation}
where it is evident that the work done on the particle depends on the rate at which the potential is changed. For an instantaneous change in spring constant, where the position distribution of the particle does not have time to change, the work done is simply $\Delta \W = \frac{1}{2}[\ks{}(t_{\rm f}) - \ks{}(t_{\rm i})]q(t_{\rm i})^2$. More generally, through solving eqn.\ \eqref{eqn:sigmax_od} for $\ks{}(t)$ one finds the full expression for the work along the trajectory:
\begin{equation}
\W(\ks{}(t)) = \frac{1}{4\mu}\int_{t_{\rm i}}^{t_{\rm f}} \frac{(\dvar{q})^2}{\var{q}} - \frac{1}{2}\T{CM}[\ln\var{x}]_{t_{\rm i}}^{t_{\rm f}} + \frac{1}{2}[\ks{}(t)\var{q}]_{t_{\rm i}}^{t_{\rm f}}.
\end{equation}
Hence, using this expression, by monitoring the motion of a colloidal particle as it undergoes the cyclic heat engine, one can extract the work statistics. We leave a full discussion of the heat and entropy statistics to other sources, for example Spinney \& Ford \cite{FordBook}.
How does this discussion of heat engines change in the underdamped regime? In eqn.~\eqref{eqn:sigmax_od} we simplified the equation of motion in the overdamped regime, such that the position was independent of the velocity. This simplification allows one to analytically construct protocols (the way in which $\lambda_c$ changes over time) that maximize the efficiency of a stochastic heat engine \cite{Schmiedl2008}. The overdamped efficiency of a microscopic heat engine can even exceed the Curzon-Ahlborn efficiency-at-maximum-power limit $\eta^* = 1-\sqrt{\T{C}/\T{H}}$ for macroscopic engines \cite{Schmiedl2008}.
An analytic solution is not known in the underdamped case, where the position and velocity variables cannot be separated, and numerical methods must be used, which find that the efficiency of the underdamped stochastic heat engine is bounded by $\eta^*$ \cite{GomezMarin2008}. In both regimes, the optimum protocols call for instantaneous jumps in the control parameter\footnote{Instantaneous changes minimize work dissipation, since they minimize the time over which a particle is accelerating \cite{GomezMarin2008}.} $\lambda_c$, as illustrated in the inset to fig.~\ref{fig:HeatEngine}. In the overdamped regime, a particle reacts slowly to changes in $\lambda_c$, whereas in the underdamped regime it reacts rapidly. Hence, although in theory the overdamped efficiency may be higher, practically it may be easier to realize optimum work extraction cycles with an underdamped engine.
To realize an underdamped heat engine, one has to engineer a coupling to an effective heat bath (since by definition an underdamped system is weakly coupled to the environment). Such a coupling is described in detail in section~\ref{sec:effective_potential}. Dechant \emph{et al.} \cite{Dechant2014} propose to realize an underdamped heat engine through a combination of optical cavity cooling and thermalization with residual gas. Another option would be to levitate a charged particle in a Paul trap and provide the heat bath via noise applied to nearby electrodes \cite{Martinez2015, Goldwater2018}, which may be more suitable for operation in the quantum regime.
\section{Acknowledgements}
(JM): This project is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 803277), and by EPSRC New Investigator Award EP/S004777/1. \\
(JG): This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 655369.
\bigskip
\bibliographystyle{unsrt}
|
2,877,628,090,815 | arxiv | \section{Introduction} \label{sec:intro}
Fast radio bursts (FRBs) are bright transients that last roughly a millisecond \citep{Lorimer2007, Thornton2013}. Based on their large dispersion measures and the observed redshifts of several of their host galaxies, most of the detected FRBs are believed to originate at extragalactic distances \citep{Chatterjee2017, Keane2016, Tendulkar2017}. Recently, the Canadian Hydrogen Intensity Mapping Experiment (CHIME), the Five-hundred-meter Aperture Spherical radio Telescope (FAST), and other surveys have been reporting many new bursts \citep{Fonseca2020, Li2019}.
Some FRBs have been found to repeat, while others have only been detected once, and we do not have enough information to know whether the two types are different populations or whether all FRBs will eventually repeat {\citep{Caleb2019, James2019}}. In this paper, we will focus on the known repeaters in an attempt to understand some of the underlying properties of the engine that powers the bursts.
Two FRBs have been found to be modulated on particularly long periods. CHIME detected a 16 day periodicity in FRB 180916, with bursts arriving in a four day phase window, {i.e. several bursts were detected over the course of four days and none were reported the other 12 days for several periods of this FRB} \citep{CHIME2020, Marthi2020}. The other repeater, FRB 121102, is the focus of this \textit{Letter}. It is the most studied repeating FRB and was found to have a period of 157 days, with an 88 day active phase \citep{Rajwade2020}. {The period was found such that every reported observation of a burst from FRB 121102 fits within an active period and every non-detection fits within an inactive period.} However, there was not enough data collected to conclusively demonstrate that there could not be a smaller modulation period than that reported (see Figure 2 of \citealt{Rajwade2020}). Bursts from FRB 121102 originate in a star-forming region on the outskirts of a dwarf galaxy at redshift z = 0.193 \citep{Chatterjee2017, Tendulkar2017, Bassa2017, Marcote2017}.
Many models have been proposed to explain the sources producing the FRBs, and the most popular one for repeating FRBs describes them as pulses from magnetars, which are neutron stars with extremely strong magnetic fields \citep{Munoz2020, Katz2020, Levin2020}. This is motivated in part by the recently discovered FRB originating in a Galactic source \citep{Scholz2020, Bochenek2020}, which demonstrates some periodicity as well \citep{Grossan2020}. However, the luminosity of this FRB was $\sim 10^3 $ times too small for it to be of the same population as the ones detected at cosmological distances \citep{Margalit2020, Beniamini2020}. {Other theoretical models that could explain periodicities involve orbital motion in binary systems \citep{CHIME2020, Rajwade2020} or the precession of neutron stars \citep{Levin2020}.}
The organization of this \textit{Letter} is as follows. In section 2 we present the data set we used and our methods for data analysis. We describe the resulting fits to each data set in section 3, and finally the implications of these findings in section 4.
\section{Methods}
{Although there exist many sets of data on bursts from FRB 121102 \citep{Spitler2016, Scholz2016, Scholz2017, Michilli2018, Gajjar2018, Gourdji2019, Oostrum2020, Caleb2020, Cruces2020}, most sets have between 10 and 25 detected bursts in any single observation period, too few for any firm statistical inference.} However, the Breakthrough Listen group was able to detect 72 additional bursts to the original 21 reported by \cite{Gajjar2018} using machine learning methods, totaling 93 points in a five hour period \citep{Zhang2018}. This is the data set we will be focusing on.
Any observation period must have started at a specific time, which was unlikely to be the start of the set of bursts, so we include a timescale $ t_0 $, which roughly characterizes the appearance time of the first unseen burst in the series of bursts under consideration, among our floating parameters. In analyzing the data, we used the scipy optimization {\footnotesize CURVE\_FIT} package in python\footnote{ \url{https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve\_fit.html} } to develop a prediction for where the FRBs should land given each value of $ t_0 $. To find the standard deviation of these predictions, we binned the data points, evening out the Poisson fluctuations within each bin, and compared the bins against the resulting fit. We applied this algorithm to find the best fit curve and its error for each data set.
\begin{figure}[h]
\epsscale{1.1}
\plotone{fig1.jpeg}
\caption{Cumulative number of bursts as a function of observed time in seconds. The orange solid curve represents the logarithmic fit to the data points from \cite{Zhang2018} based on equation (1), with best fit values of $\alpha=18.1$ and $t_0=121$s. The green dashed line shows an example of a periodic signal, which is unable to fit the data. The difference in their goodness of fit is illustrated by their reduced $ \chi^2 $ values, as we have $ \chi_{\nu}^2 = 7.37 $ for the orange solid curve and $ \chi_{\nu}^2 = 1040 $ for the green dashed line.}
\centering
\label{fig:curve}
\end{figure}
\section{Results}
The best fit for the Breakthrough Listen data \citep{Zhang2018} is presented in Figure \ref{fig:curve}. We fit the data to the formula,
\begin{equation}
N(<t) = \alpha\ln(t/t_0),
\end{equation}
for $ t \gtrsim t_0 $, with $ N $ being the number of bursts and $ t $ the time since the start of the observation period. Our best fit values are $ \alpha=18.1 ^{+0.630}_{-0.663} $ and $ t_0=121 ^{+20.6}_{-20.3} $ s. The first in this series of 93 bursts started when $ N=1 $ at $ t=e^{1/\alpha}t_0 = 128^{+21.5}_{-21.2} $ s. Since the duration of a single burst is $ \sim $ 1 ms, this implies an initial duty cycle of $ \sim 10^{-5}$.
{The orange solid curve in Figure \ref{fig:curve} represents the logarithmic fit to the data points from \cite{Zhang2018} based on equation (1), while the green dashed curve represents a periodic underlying signal. The difference in their goodness of fit is illustrated by their reduced $ \chi^2 $ values, as we have $ \chi_{\nu}^2 = 7.37 $ for the logarithmic fit and $ \chi_{\nu}^2 = 1040 $ for the periodic fit.}
Since
\begin{equation}
N(<t) = \alpha\ln(t/t_0) = \alpha\ln(t) - \alpha\ln(t_0),
\end{equation}
taking the derivative with respect to $ \ln(t) $ yields
\begin{equation}
\alpha = \frac{dN}{d\ln(t)} = t\frac{dN}{dt},
\end{equation}
so our final result for the burst rate of FRBs is
\begin{equation}
\frac{dN}{dt} = \frac{\alpha}{t}.
\end{equation}
Figure \ref{fig:colorplot} shows the confidence contours in the ($ \alpha,\ t_0$) plane, where the colors represent the standard deviation. The plot demonstrates how closely correlated $ \alpha $ and $ t_0 $ are.
\begin{figure}
\epsscale{1.2}
\plotone{fig2.jpeg}
\caption{Contour plot demonstrating the impact of the initial timescale of the bursts ($t_0$) on the value of $\alpha$ in equation (1), where the colors represent the standard deviation, applied to the data points from \cite{Zhang2018}.}
\centering
\label{fig:colorplot}
\end{figure}
We also ran a simulation to check whether we could fit the rest of the existing data (from \citealt{Zhang2018}, \citealt{Rajwade2020}, and \citealt{Cruces2020}) to the pattern shown in our results. In our simulations, we assumed the theoretical data would appear in sets of logarithmic curves, and that the starts of these sets would either be constant (separated by 0.2 to 10 days) or random (selected from a range of 0-$N$ days where $N$ takes on integers 1 through 5). We started from MJD 57991.41, which is the start date of the data published by \cite{Zhang2018}, and we created a model set of data points that consists of curves resembling the orange solid line in Figure \ref{fig:curve}. One caveat lies in the possibility that the $ \alpha $ and $ t_0 $ found in Figure \ref{fig:curve} may be frequency dependent, but for the sake of this analysis we adopt a single frequency independent value for them. {Under this assumption, we were able to simultaneously compare detections at different frequencies and sensitivities by computing the error between the measured MJD and the closest predicted MJD.}
In the constant separation scenario, this model set of data points was repeated every fixed period of time, which we varied as a free parameter to minimize the error relative to the time-tags of the detected bursts. In the second scenario, our free parameter was the number of days the random increment could be selected from, and the starts of each set of bursts were accordingly separated by random increments from that range. When the random increment is selected from 4 or less days, the error is small enough that we should not rule out this model.
A histogram of the simulation for the constant separation scenario is shown in Panels A and B of Figure \ref{fig:sim}, where the red represents real data, the black represents non-detection periods, and the gray represents the simulated set of points. The violet vertical lines demonstrate the area of Panel A that is magnified in Panel B. For each real observed data point, we found the closest simulated point and used the difference between the two time tags as our error. Our final error per point was the norm of this set of errors (the square-root of the sum of the squares divided by the number of points), and this final result is shown by the red curve in Panel C of Figure \ref{fig:sim} for various separations from 5 hours to 10 days. All separations under a day gave an error better than 0.1, so we were not able to conclusively find the best possible separation between sets of bursts. We also created a random set of observations that had a uniform probability of appearing within the active periods mentioned in \cite{Rajwade2020} and \cite{Cruces2020}, which is shown by the blue curve in Panel C of Figure \ref{fig:sim}.
We also compared our modeled simulation against the non-detection periods published by \cite{Rajwade2020} and \cite{Cruces2020}, in which no bursts were detected. Whenever the model predicts a burst in such a non-detection period, the prediction is at least off by the duration of time between the closer edge of the non-detection period and the predicted burst, so we included this as additional error in our model. We have found that there is no information in these non-detection periods in terms of error optimization.
\begin{figure*}
\epsscale{1.15}
\plotone{fig3.jpeg}
\caption{A simulation of a fit to the data from \cite{Zhang2018, Rajwade2020, Cruces2020}. Panels A and B are examples of a case with a steady separation of 1.3 days between sets, where the red represents real data, the black represents non-detection periods, and the gray represents the simulated set of points. The violet vertical lines demonstrate the area of Panel A that is magnified in Panel B. Panel C demonstrates the error as a function of time separation between sets of bursts for the complete activity cycle, corresponding to Panel A. The red again represents real data and the blue curve represents a random set of observations that had a uniform probability of appearing within the active periods mentioned in \cite{Rajwade2020} and \cite{Cruces2020}.}
\centering
\label{fig:sim}
\end{figure*}
\section{Discussion and Conclusions}
An implication of these findings is that FRB 121102 is not periodic, but rather it follows a logarithmic repetition pattern, where the rate falls off as one over time for each set of bursts. This could plausibly explain the vacant, inactive regions in Figure 2 of \cite{Rajwade2020} since beyond a certain amount of time, there are so few bursts that the probability of measuring one goes to zero. However, if $ t_0 $ is on the order of hundreds of seconds, there may be many smaller sets of bursts within each period reported by \cite{Rajwade2020}. Another possibility is that FRB 121102 emits equal amounts of energy per log-time, but the emission pattern still follows the overall \cite{Rajwade2020} modulation periodicity of 157 days, thereby fitting the rest of the observations found over the past four to five years. More data and monitoring of the source is necessary to come to a more firm conclusion.
Another implication is that this lack of periodicity is not consistent with the idea that repeating FRBs are a type of pulsar \citep{Beniamini2020, Munoz2020}, rather they might be caused by a different type of phenomenon that involves an equal amount of energy output per log time. This could be indicative of a process with a characteristic timescale of $ t_0 $. For example, the source of the FRB could be an object that charges up and then discharges with equal amounts of energy output per logarithmic time interval.
{A potential concern with the model is that the data from \cite{Cruces2020} contains a set of 24 observations in a 7 hour period in which the rate significantly increases in the second half of the observations. Assuming the sensitivity of the data collection methods remained constant, this would imply a change in the intrinsic rate of bursting, unless within that 7 hour period one cycle of bursts ends and another begins.}
A few remaining open questions are the effect of the detection threshold of the observations as there might have been many fainter bursts that were missed, possibly affecting the pattern we found, and the potential effect of frequency on the underlying burst pattern.
In conclusion, we have shown that some repeating FRBs may not send periodic bursts, rather the bursts could be arriving at a logarithmic rate. Our equation (1) can be tested by upcoming data on repeating FRBs in the near future \citep{Li2019, Hashimoto2020}.
A roughly constant burst rate in log time was previously noted by \cite{Wadiasingh2019}.
\acknowledgments This work was supported in part by a grant from the Breakthrough Prize Foundation (for A.L.) and by a summer internship from Harvard's Institute for Theory and Computation (for E.T.). We thank Dunc Lorimer, Kshitij Aggarwal and Julian Munoz for insightful comments on an early draft of the paper. We also thank the anonymous referee, whose important suggestions improved the clarity of the manuscript.
\newpage
\bibliographystyle{aasjournal}
|
2,877,628,090,816 | arxiv | \section{Introduction}
Chinese calligraphy is a very unique visual art and an important manifestation
of Chinese ancient culture which is popular with many people in the
world. Writing a pleasing calligraphy work is so difficult that it
always takes the writer many years to learn from the famous calligraphers'
facsimiles. Is there a way to synthesize calligraphy with specified
style expediently? We will explore an effective and efficient approach
for calligraphy synthesis in this paper.
Automatic calligraphy synthesis is a very challenging problem due
to the following reasons: 1) Various Chinese calligraphy styles. A
Chinese character usually has thousands of calligraphy styles which
vary from the shapes of component and the styles of strokes; 2) Deformations
between the standard font image and calligraphy image. The standard
font image and calligraphy image for the same character are only similar
in relative layout of radicals of character but different in the layout
and style of strokes.
Recently, there are some attempts \cite{xu2005automatic,xu2009automatic}
to synthesize calligraphy automatically, which first extract strokes
from some known calligraphy characters and then some strokes are selected
and assembled into a new calligraphy character. The above mentioned
methods are largely dependent on the effect of strokes extraction.
However, the stroke extraction technology does not always work well
when the Chinese character is too complex or the character is written
in a cursive style (Fig. 1(b)) where the strokes are hard to separate
and have to be extracted artificially \cite{xu2007intelligent}.
Considering there are some shortcomings in stroke assemble based methods,
we treat the calligraphy generation as an image-to-image translation
problem and propose a new method which can generate calligraphy with
a specified style from a standard Chinese font (\textit{i.e.} Hei
Font) directly without extracting strokes of characters. Over the
past few years, many network architectures have been proposed and
applied to different image-to-image tasks. However, those networks
are all designed to handle the pixel-to-pixel problems, such as semantic
segmentation, and poor performance is achieved when there are deformations
between the input and target images (Fig. 1(c)).
To overcome these problems, we propose a deep neural network based
model which consists of two subnets. The first one is an encoder-decoder
network acting as image transfer, which encodes an input standard
font image to a feature representation and then decodes the feature
representation to a calligraphy image with specified style. The encoder-decoder
network with similar architecture has been used in \cite{hinton2006reducing}
and show that the feature representation is likely to compress the
image content. This network architecture is sufficient to reconstruct
an image. But considering that in our task the input images and output
images only have the same relative layout among radicals but are different
in the layout and style of strokes, it is hard for an encoder-decoder
network to yield vivid calligraphy images. So besides the transfer
who captures the layout of input standard font image, we also use
another encoder-decoder network acting as autoencoder which inputs
and reconstructs calligraphy images to guide the transfer to learn
the detailed stroke information from autoencoder's low level features.
Finally, we train the two subnets together with reconstruct loss and
adversarial loss to make the output look real.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.4]{img/introduction}\caption{\label{fig_introdunction}(a) Synthesizing Chinese calligraphy images
from the standard font images. (b) Some calligraphy images in cursive
style whose strokes are hard to separate. (c) From left to right:
the standard font image, the output of UNet \cite{isola2016image},
the output of our method and the target image. The poor performance
is achieved by UNet while our method gets desired result. }
\par\end{centering}
\end{figure}
\begin{figure*}[!]
\centering{}\includegraphics[bb=35bp 0bp 1134bp 426bp,scale=0.45]{img/models}\caption{\label{fig_model}The architecture of the proposed method. The upper
part of image is our transfer network and the lower part of the architecture
is the supervise network which is an autoencoder. The whole network
is trained end-to-end and the supervise network is used to supervise
the low feature of transfer network's decoder in training phase.}
\end{figure*}
In summary, the contributions of this paper are two aspects: Firstly,
we propose a neural network based method which can end-to-end synthesize
calligraphy images with specified style from standard Chinese font
images. Compared to some baseline methods, our approach achieves the
best results with more realistic details. Secondly, we establish a
large-scale dataset for Chinese calligraphy synthesis collected from
the Internet. The dataset composes of 4 calligraphy styles and each
style contains about 7000 calligraphy images.
\section{Related Work\label{sec:Related-Work}}
\subsubsection{Chinese Calligraphy Synthesis}
In the past few years, many works on Chinese calligraphy synthesis
have been proposed. In \cite{xu2005automatic}, Xu et al. propose
a method based on shape analysis technique and hierarchical parameterization
to automatically generate novel artistically appealing Chinese calligraphy
artwork from existing calligraphic artwork samples for the same character.
Xu et al. \cite{xu2009automatic} propose a method to parameterize
stroke shapes and character topology, and successfully transfer font
Style Kai into a specific users\textquoteright{} handwriting style
by choosing the most appropriate character topology and stroke shapes
for a character. Different from the above mentioned methods which
follow the stroke extraction and stroke assembly pipeline, we input
a standard font image to our model and output a calligraphy image
directly.
\subsubsection{Image-to-Image Translation}
Image-to-image translation is an extensive concept which covers many
tasks such as edge/contour extraction \cite{xie15hed,shen2016object},
semantic segmentation \cite{long2015fully,noh2015deconv-segmantation},
artistic style transfer \cite{johnson2016perceptual,chen2016faststyletransfer},
image colorization \cite{luan2007natural,zhang2016colorful} et al.
in computer vision field. However, in those tasks, image-to-image
translation problems are often formulated as pixel-to-pixel translation
problem, where the input images and target images have the same underlying
structure and without any deformations. In this paper, we focus on
another scenario in image-to-image translation where there are deformations
between the input and target images. To be specific, in our calligraphy
synthesis task, the input standard font images and target calligraphy
images only have the similar relative layout among radicals of the
same characters but are different in the position and style of strokes.
\subsubsection{Generative Adversarial Networks}
Generative Adversarial Networks is proposed by \cite{goodfellow2014generative}
which has attracted great interest from the computer vision and machine
learning community and has a rapid development \cite{mirza2014cgan,radford2015dcgan,denton2015laplaciangan,chen2016infogan,arjovsky2017wgan}.
GAN is not only used in unsupervised learning such as generate an
image from random noise vector but also used with some image-to-image
translation tasks \cite{isola2016image,pathak2016context} to make
the output look real. Like \cite{isola2016image,pathak2016context},
we train our image transfer using an adversarial loss as well as the
reconstruction loss between the output images and target images to
generate desirable results. To learn the deformation between the input
images and target images, we also reconstruct the low level features
of our transfer supervised by the low level feature from an autoencoder.
\section{Proposed Method\label{sec:Proposed-Method}}
In this section, we describe in detail the proposed method. As shown
in Fig. \ref{fig_model}, our module consists of two encoder-decoder
networks which have similar network structure and can be trained together
in an end-to-end way. We refer to the two subnets as \textit{Supervise
Network} and \textit{Transfer Network} respectively, as \textit{Transfer
Network} is used to transfer a standard font image to a calligraphy
image with specified style, and \textit{Supervise Network} can provide
supervision information for \textit{Transfer Network} in training
stage. Details of\textcolor{blue}{{} }the two subnets are discussed
below.
\begin{figure}
\centering{}\includegraphics[bb=0bp 0bp 960bp 540bp,scale=0.25]{img/dataset}\caption{\label{fig_dataset}Some calligraphy images from the proposed benchmark.
This dataset contains 4 styles calligraphy images written by 4 famous
calligraphers in ancient China.}
\end{figure}
\subsection{Supervise Network}
The supervise network is an autoencoder network. The encoder consists
of a series of Convolution-BatchNorm-LeakyReLU \cite{radford2015dcgan}
blocks which takes a calligraphy image as input and produces a $C\times1\times1$
latent feature representation of that image, where $C$ is the dimension
of the latent feature. The decoder is stacked by a series of Deconvolution-BatchNorm-ReLU
\cite{radford2015dcgan} blocks, which takes the latent feature representation
from encoder and outputs an image which is similar to the input image.
The architecture of the supervise network is a simple CNN based encoder-decoder
network but has skip connections between each layer $i$ and layer
$n-i$ as \cite{isola2016image}, where $n$ is the total number of
layers of supervise network. The skip connections are essential for
the supervise network to output images with photo-realistic details.
We have experimented and verified that the simple encoder-decoder
network can only output images with the rough layout but almost lost
all stroke information, but our supervise network can generate correct
strokes as input images. We argue that the feature maps of the bottleneck
layer in the simple encoder-decoder lost fine details of input images
but the spatial structure is kept, and that skip connections can provide
the decoder with detailed information.
\subsection{Transfer Network}
The transfer network is also a CNN based encoder-decoder network which
inputs a standard font image and generates a calligraphy-like image.
The encoder and decoder are similar as the supervise network which
is composed by a series of Convolution-BatchNorm-LeakyReLU and Deconvolution-BatchNorm-ReLU
blocks respectively, but there is a little difference in skip connections.
Chinese characters have diverse and complicated layouts and are hard
to transform to calligraphy image from standard font image even the
two images have the same layout. Instead of concatenating the feature
outputted by layer $i$ and layer $n-i$ directly, we use a residual
block \cite{he2016deep} to connect layer $i$ and layer $n-i$ and
sum the feature yielded by the residual block and layer $n-i$ to
enhance the capacity to learn the minute difference between the spatial
architecture of standard font and specified calligraphy images.
The standard font image and the corresponding calligraphy image always
have the same character component structure but vary greatly in the
layout and style of strokes. The high level features in encoder carry
layout information of the input standard font images, but it is not
enough to generate calligraphy images with clear strokes and specified
style when the model is only supervised by the target calligraphy
image. So we use the above supervise network to guide the transfer
network. Let $S=\{s_{1},s_{2},...,s_{k}\}$ and $T=\{t_{1},t_{2},...,t_{k}\}$
denote the low level feature representations yielded by supervise
network and transfer network's decoder respectively. We use $s_{j}$
to supervise $t_{j}$ in order to guide the decoder of transfer network
to learn the feature representations which carry the layout and style
of strokes, layer by layer.
Generative Adversarial Network (GAN) is recently proposed by \cite{goodfellow2014generative}
and has been widely used in image generation tasks \cite{larsen2015autoencoding,wu20163dlearning,zhang2016stackgan,isola2016image}.
Adversarial loss has the effect of learning the same distribution
of the ground truth distribution, which can make the output images
look real. \cite{isola2016image,pathak2016context} have shown that
an image transfer with an adversarial loss can output much sharper
results than the one only with L1 loss. We can adjust our transfer
network to a GAN framework easily with an additional discriminative
model. We treat transfer network as generator $G$ and use a deep
network as discriminative model $D$ following \cite{radford2015dcgan}.
In our work, the generator $G$ is optimized to output images which
have the same distribution as truth calligraphy images by generating
images that are difficult for the discriminator $D$ to differentiate
from real images. Meanwhile, $D$ is conditioned on the input standard
font images and optimized to distinguish real images and fake images
generated by $G$.
\subsection{End-to-End Joint Training}
We train the two subnets jointly in an end-to-end way. Given a pair
of training sample $(x,y)$ which is composed of a standard font image
$x$ and a calligraphy image $y$ for the same character.
For the supervise network, we take calligraphy image $y$ as input
and the objective is to reconstruct $y$. We use L1 loss as our reconstruction
loss rather than L2 loss as L1 tends to yield sharper and cleaner
image. Let $A(.)$ be the supervise network, the objective of the
supervise network can be expressed as:
\begin{center}
$\mathcal{L}_{supervise}=\mathbb{E}_{y\AC p_{data}(y)}[||y-A(y)||_{1}]$
\par\end{center}
For the transfer network, we input standard font image $x$ and take
calligraphy font image $y$ as ground truth. We also reconstruct the
low level feature $S$ from the supervise network. We define the reconstruction
loss function as:
\begin{center}
$\mathcal{L}_{reconstruct-1}=\mathbb{E}_{t_{1}\AC p_{data}(t_{1})}[||t_{1}-s_{1}||_{1}]$
\par\end{center}
\begin{center}
...
\par\end{center}
\begin{center}
$\mathcal{L}_{reconstruct-(k)}=\mathbb{E}_{t_{(k)}\AC p_{data}(t_{(k)})}[||t_{(k)}-s_{(k)}||_{1}]$
\par\end{center}
\begin{flushleft}
$\mathcal{L}_{reconstruct}=\lambda_{1}\times\mathcal{L}_{reconstruct-1}+...+\lambda_{k}\times\mathcal{L}_{reconstruct-(k)}$
\par\end{flushleft}
Besides, we define our adversarial loss as:
\begin{flushleft}
$\mathcal{L}_{adversarial}=\mathbb{E}_{y\sim pdata(y)}[\log D(x,y)]+\mathbb{E}_{x\sim pdata(x)}[\log(1-D(x,G(x)))]$
\par\end{flushleft}
So, our final objective is:
\begin{flushleft}
$G^{*}=\arg\underset{G}{\min}\underset{D}{\max}\mathcal{L}_{adversarial}+\lambda_{s}\mathcal{L}_{supervise}+\lambda_{r}\mathcal{L}_{reconstruct}$
\par\end{flushleft}
\subsection{Implementation details}
In this paper, all images are scaled to 256 \texttimes{} 256 and converted
to binary images before being fed into the model. In addition, we
employ data augmentation to artificially enlarge the dataset for the
purpose of reducing overfitting. We flip the image horizontally with
probability of 0.5.
The encoder of supervise network and transfer network both have 8
stacked Convolution-BatchNorm-LeakyReLU blocks, which yield 1 \texttimes{}
1 latent feature representations of the input calligraphy images and
standard font images respectively. The decoder of supervise network
and transfer network both have 7 stacked Deconvolution-BatchNorm-ReLU
blocks and followed by a Deconvolution layer which will generate 256
\texttimes{} 256 binary images. All Convolution and Deconvolution
layers in the above mentioned part have 4\texttimes 4 kernel size
and 2\texttimes 2 stride. The residual block in transfer net consists
of Convolution, Batch normalization and ReLU layers as \cite{he2016deep}
and only exists between the layers whose feature map size are $2\times2$
and $4\times4$ of encoder and decoder. The architecture of D is adapted
from \cite{radford2015dcgan}. 7 stacked Convolution-BatchNorm-ReLU
blocks are used and followed by a convolution layer and output the
probability of the input images like real.
The method was implemented in Torch \cite{collobert2011torch7}. In
our experiment, we supervise the decoder of transfer net from the
layer with feature map size $16\times16$ to $128\times128$ and set
$\lambda_{1}...\lambda_{k}$ to 1, and set $\lambda_{s}$ and $\lambda_{r}$
to 100. We choose initial learning rate of 0.002 and train the proposed
model end-to-end with Adam \cite{kingma2014adam} optimization method.
This model was trained with batch size set to 16 until the output
tends to be stable in training phase. When testing, only the transfer
network is used to generate calligraphy images. We also use a median
filter to denoise the output image as a post-process method to make
the results cleaner.
\section{Experiments\label{sec:Experiments}}
In this section, we propose a new benchmark for Chinese calligraphy
generation and evaluate our algorithm on the proposed dataset. Besides,
we also compare our method with other neural network based image translation
methods to prove the effectiveness of our approach.
\subsection{Dataset}
As far as we know, there are no existing public datasets for Chinese
calligraphy images generation with specified style. Therefore we propose
a new dataset for calligraphy images automatic generation collected
from the Internet. This dataset contains 4 subsets which are written
by 4 famous calligraphers in ancient China, namely Mi Fu, Zhao Zhiqian,
Liu Gongquan and Shen Yinmo in different style. Some samples from
4 subsets are shown in Fig .\ref{fig_dataset}. What we can see is
that the styles of 4 subsets vary from one to another and cover a
few categories, such as running script, official script and regular
script. As shown, running script shows enormous shape transformation.
Official script exhibits wide and flat shapes. Its characters are
usually horizontally long and vertically short. Regular script is
more clear and neat which is mostly similar to printed fonts. Each
subset in our proposed benchmark contains about 7000 images and is
split into two set: training set and validation set. We randomly select
6000 images as training set and the rest images are treated as validation
set for each style and ensure that the training set and validation
set have no overlap in characters.
For convenience, we call this dataset Chinese Calligraphy Synthesis-4(CCS-4).
\begin{figure}[t]
\centering{}\includegraphics[bb=0bp 240bp 960bp 1000bp,scale=0.55]{img/results1}\caption{\label{fig_result} The results of the baseline methods as well as
our method on the 4 subset of the proposed benchmark.}
\end{figure}
\subsection{Baseline Methods}
\subsubsection{Rewrite }
Rewrite \cite{rewrite} is a neural style transfer for Chinese font
which is effective to transfer a typographic font to another stylized
typographic font. It is a simple top-down Convolution network with
big convolution kernel size and $1\times1$ stride. Each Convolution
layer in Rewrite is followed by Batch Normalization layer and a ReLu
layer. The architecture of Rewrite is stacked by some above mentioned
convolution blocks and end up with a $2\times2$ MaxPooling layer
then followed by a Dropout layer and a Sigmoid layer. The network
is minimized by L1 loss and total variation loss.
\subsubsection{Encoder-Decoder Network}
Encoder-Decoder network is an effective image-to-image translation
model and has been widely used and studied for many image translation
tasks, such as semantic segmentation \cite{noh2015deconv-segmantation},
edge extraction \cite{yang2016edge-extraction}, image colorization
\cite{isola2016image}, image restoration \cite{mao2016image-restoration}
and image style transfer \cite{chen2016faststyletransfer} \textit{etc}.
We use the architecture proposed by \cite{isola2016image} as a baseline
and train the model with L1 loss and adversarial loss.
\subsubsection{UNet}
UNet is proposed by \cite{ronneberger2015u} and is an extension of
the simple encoder-decoder method. In \cite{isola2016image}, skip
connections are used to connect encoder and decoder, based on the
fact that in an image translation task, the input and output differ
in surface appearance, but both are renderings of the same underlying
structure. Besides, the network is also optimized with L1 loss and
adversarial loss.
\subsection{Evaluation and Discussions}
We evaluate our proposed method as well as other baselines on the
CCS-4 dataset. We use the font style Hei as character\textquoteright s
standard font for each method, as Hei font has the least style structures,
even thickness and reduced curves. So using Hei font as character's
standard font can avoid the similarity between input font style and
output calligraphy style, which may increase the difficulty of calligraphy
generation but can evaluate the robustness and effectiveness of the
evaluated methods.
\subsubsection{Qualitative Results}
We train a model for every above mentioned baseline methods as well
as our method on the four subsets individually. We show some samples
generated by the baseline methods and our proposed method in Fig.
\ref{fig_result}. Our method achieves the best results on all the
subsets.
Specifically, Rewrite and Encoder-Decoder method achieve the worst
results. The images generated by Rewrite and Encoder-Decoder only
have the right spatial structure of Chinese character component but
the layout and style of strokes are far from satisfactory. As Fig.
\ref{fig_result} shows, In Mi Fu and Zhao Zhiqian subsets, almost
all strokes can not match the ground truth strokes.
The UNet method achieves a better result than the Encoder-decoder
result. Some results are close to ground truth both in global style
and local stroke style but have a small part of wrong strokes on the
Zhao Zhiqian, Liu Gong quan and Shen Yinmo subsets. However, the results
on Mi Fu subset is a little unsatisfactory. We argue that the layout
of strokes in Zhao Zhi qian, Liu Gongquan and Shen Yinmo are very
similar to the strokes of the standard font, which is much easier
to transform for the translation invariance of CNN. But there are
significant differences between the standard font and Mi Fu calligraphy
which may be too hard for UNet to learn.
The best results are obtained by our method. Even evaluated on Mi
Fu subset, our model can still generate images with the similar style
of the global image and the local stroke. In some scenes, such as
the input character has complex stroke structure, our method still
can handle well.
\subsubsection{Effect of The Supervise Network}
\begin{figure}
\includegraphics[bb=50bp 50bp 1000bp 350bp,scale=0.24]{img/supervise}\caption{\label{fig_supervise}The results of our methods with/without supervise
network evaluated on Liu Gongquan subset.}
\end{figure}
In practice, low level features are hard to learn well in ordinary
autoencoder models, so we add Supervise Network in our model as a
reference to guide the transfer network to learn detail layout and
style of strokes. In Fig. \ref{fig_supervise}, we compare our model
with the one without Supervise network while other parts maintain
the same design. In the aspect of perceptual quality of the generated
font images, our model beats the one without Supervise network which
holds a general structure of the character while having some wrong
fine details.
\subsubsection{Effect of The Adversarial Loss}
\begin{figure}
\includegraphics[bb=50bp 50bp 1100bp 350bp,scale=0.24]{img/adversarial_1}\caption{\label{fig_adversarial}The results of our methods with/without adversarial
loss evaluated on Liu Gongquan subset.}
\end{figure}
From the results shown in the Fig. \ref{fig_adversarial}, we can
see that the introduction of GAN Loss helps to improve the quality
of the generated image. It is obvious that there are more valid vivid
details of the characters are added and the generated image tends
to be more sharp with much less blur. Take the first column as example,
we can see that the generated image is similar with the ground image
in shape layout but loses some details. However, after adding GAN
Loss, the image generated is more sharp and detailed. We can draw
the conclusion that GAN Loss helps the generator mitigate the blur
and add the details which cannot be captured only with the L1 Loss.
\subsubsection{Analysis of The Standard Font}
\begin{figure}
\includegraphics[scale=0.4]{img/kai_result1}\caption{\label{fig:kaifont}The results of our method with Kai Font as standard
font. }
\end{figure}
Our approach achieves desirable results when we use Hei Font as the
standard font. Here, we use a different font as our standard font
to explore the affect of the standard font. As shown in Fig. \ref{fig:kaifont},
we use Kai font as the standard font, and our model can still output
photo-realistic calligraphy images which shows the robustness of our
method.
\section{Conclusion}
In this paper, we propose a model consisting of two subnets: \textit{transfer
network} and \textit{supervise network} to synthesize Chinese calligraphy.\textit{
}The transfer network can transfer a standard font image to a calligraphy
image with specified style and the supervise network can supervise
the transfer network to learn detailed stroke information. The two
subnets can be trained together. Compared with recent Chinese calligraphy
generation works, our approach can generate Chinese calligraphy images
from standard font images directly. Besides, we also establish a benchmark
for Chinese calligraphy generation and evaluate our method as well
as other baseline approaches on the proposed dataset. our method achieves
the best results. In the future, we will investigate methods that
can handle large deformation between the input target images and expand
our method to more general problems.
\bibliographystyle{plain}
|
2,877,628,090,817 | arxiv | \section{Introduction}\label{s:intro}
Type Ia supernovae (SNIa) are the result of the thermonuclear disruption of carbon-oxygen white dwarf (WD) stars, due to their destabilization by a companion star in a binary or a ternary system \citep{2000hil,2011how,2012kaz,2014mao}. The spectra and light curves of SNIa can be recorded starting shortly after the explosion \citep{2013zhe,2017hos,2018mie} and continuing until a few years later \citep{2018gra,2018jac,2018mag}, if the luminosity and the distance to the event allow it. Besides providing valuable insights into the systematics of
SNIa \citep[][to cite only a few]{1993bran,1997fil,2002poz,2005how,2006jam,2008ars,2009bran},
these data allow researchers to constrain the properties of the explosion, from fundamental parameters, such as the ejected mass, explosion energy, and synthesized mass of $^{56}$Ni \citep{2006stt,2014sca}, to second-order details, such as the ejected amounts of other radioactive isotopes, notably $^{57}$Ni and $^{55}$Fe \citep{2016gra,2017dim,2017sha,2018yan,2019li}.
Whereas observational data are usually reported together with their error bars, there is scarce knowledge of the effects of the diverse sources of uncertainty related to supernova models. Indeed, \citet{2012bra,2013pkh,2019brab} proved that the nucleosynthesis of one-dimensional SNIa models is robust with respect to variations in individual reaction rates, at least during the explosion phase.
Currently, two competing models may account for the bulk of SNIa. In one model, the exploding WD is close to the Chandrasekhar-mass limit and the disruption is total; therefore, the mass of the ejecta is fixed a priori. The most accepted explosion mechanism of massive WDs is delayed detonation
\citep[DDT,][]{1991kho,2019pol},
in which the thermonuclear burning wave propagates subsonically at first and turns into a detonation later, typically one or two seconds after thermal runaway. In the other model, the mass of the exploding WD is substantially less than the Chandrasekhar limit, the burning wave propagates as a detonation from the very beginning, and the instability leading to the detonation may be a consequence of the burning of a thin helium layer, accumulated on top of the WD after accretion from a secondary star, or it may be due to the merging or collision with another degenerate star
\citep[e.g. a second WD;][]{1994woob,2010woo,2010sim,2013kus,2018she}.
Although there is not consensus about the level of asymmetry involved in SNIa explosions, most polarization measurements are indicative of small deviations from spherical symmetry \citep{1996wan,2001how,2010mau,2010maeb,2013mau}. One-dimensional models are able to account for the properties of SNIa in the visible \citep{1996hoe,1997nug,2011tan,2013blo,2017hoe} and gamma bands \citep[][see \citealp{2014die,2016ise} for a different view]{2014chu,2015chu}, and those of their remnants in the X-ray and radio bands \citep[e.g.][]{2006bad,2011lop,2018mar}. However, to understand SNIa it is necessary to simulate the explosion in three dimensions, either to account for hydrodynamical instabilities and turbulence \citep[e.g.][]{2004ple,2006rop,2006bra,2007kasb}, or to address the inherent asymmetrical configuration of colliding or merging WDs.
The computational requirements posed by three-dimensional hydrodynamics make it difficult to incorporate complex nuclear networks to follow the release of nuclear energy along with the propagation of the flame. Usually, multi-dimensional supernova codes need to model the flame by making use of simplified nuclear kinetics, with the goals of giving an accurate rate of nuclear energy generation and computing the explosion in a reasonable time. Afterwards, the thermodynamic trajectories of the integration nodes can be post-processed with the aid of a large nuclear network to obtain more reliable values of the supernova yields \citep[e.g.][]{1986thi,2010bra,2016tow,2017leu}.
Although, recently, several groups have designed algorithms to incorporate large nuclear networks into multi-dimensional models of SNIa \citep{2015pap,2019kusb}, most research still follows the post-processing approach.
In this work, I address the question of the performance of simplified nuclear networks with respect to the reproduction of the correct nuclear yields\footnote{In the present work, by correct nuclear yields I mean those obtained with the same supernova code without the simplifications described in the text.}. After a brief explanation of the methodology used (Sect.~\ref{s:2}), the first point I treat is the definition of a strategy to follow the properties of matter in a state of nuclear statistical equilibrium (NSE, Sect.~\ref{s:3}). In the following section, I test several simplified nuclear networks for the accuracy of the nucleosynthesis obtained through post-processing (Sect.~\ref{s:4}).
Two more sections address the performance of simplified nuclear networks for high-metallicity WD progenitors and the effect of different criteria for switching on and off NSE routines. A final section is dedicated to the conclusions of the present work.
\section{Methodology}\label{s:2}
The simulations presented in this work are performed with the same supernova code described in \citet{2019bra}, where extensive details of the method of computation can be found. The code integrates the hydrodynamic evolution using a large nuclear network, solves the Saha equations for NSE, when applicable, and calculates the neutronization rate and associated neutrino energy losses at each time step, computing the weak interaction rates using the NSE composition. The hydrodynamics is followed in one dimension, assuming spherical symmetry, but the nucleosynthetic processes through which matter passes are
qualitatively\footnote{The precise thermodynamic histories of mass shells may be a function of the dimensionality of the model.}
the same as in any three-dimensional SNIa simulation, hence it is appropriate to assess the accuracy of the computed nucleosynthesis. Furthermore, the supernova models generated by this code have been compared successfully with observed optical spectra (S. Blondin, private communication), gamma-ray emission from SN2014J \citep{2014chu,2015chu,2016ise} and the X-ray spectra of supernova remnants \citep{2006bad}.
The default nuclear network used here is the same as in \citet{2012bra} and can be found in column BM-P in Table~\ref{t:1}. This network includes all stable isotopes up to molybdenum, but the code allows to define different nuclear networks by simply listing the species to be followed. The nuclear reactions linking these species are included automatically in the network, together with a basic set formed by the fusion reactions triple-$\alpha$, $^{12}\mathrm{C}+^{12}\mathrm{C}$, $^{16}\mathrm{O}+^{16}\mathrm{O}$, and $^{12}\mathrm{C}+^{16}\mathrm{O}$. All reaction rates are taken from the JINA REACLIB\footnote{http://groups.nscl.msu.edu/jina/reaclib/db/.} compilation \citep{2010cyb},
in the version of November 6, 2008.
As references, I have selected two explosion models suitable for normal-luminosity SNIa, characterized by
an ejected mass of $^{56}$Ni about $M(^{56}\mathrm{Ni})\sim0.5 - 0.7~\mbox{$\mathrm{M_{\odot}}$} $. The first model, sub-$M_\mathrm{Ch}$, is a central detonation of a sub-Chandrasekhar WD of mass $M_\mathrm{WD} = 1.06~\mbox{$\mathrm{M_{\odot}}$}$ \citep[model 1p06\_Z9e-3\_std in][]{2019bra}. The second one, $M_\mathrm{Ch}$, is a delayed detonation of a massive WD with central density $\rho_\mathrm{c} = 3\times10^9$~\gccb and a deflagration-to-detonation transition density $\rho_\mathrm{DDT}=2.4\times10^7$~\gccb \citep[model ddt2p4\_Z9e-3\_std in][]{2019bra}. Both models assume an initial composition made of equal masses of \isotope{12}{C} and \isotope{16}{O}, contaminated with \isotope{22}{Ne} as appropriate for progenitor metallicity $0.009$ and other metals from sodium to indium in solar proportions with respect to \isotope{22}{Ne}.
\begin{table}
\caption{Nuclear networks for the convergence study.}
\label{t:1}
\begin{tabular}{lccc}
\hline\hline
\noalign{\smallskip}
& BM-P & netAKh & nse7 \\
$Z$ & $A_\mathrm{min}-A_\mathrm{max}$ & $A_\mathrm{min}-A_\mathrm{max}$ & $A_\mathrm{min}-A_\mathrm{max}$ \\
\hline\hline
\noalign{\smallskip}
n & 1 - 1 & 1 - 1 & 1 - 1 \\
H & 1 - 4 & 1 - 1 & 1 - 3 \\
He & 3 - 9 & 4 - 4 & 3 - 6 \\
Li & 4 - 11 & - & 6 - 7 \\
Be & 6 - 14 & - & 7 - 10 \\
B & 7 - 17 & - & 10 - 11 \\
C & 8 - 20 & 12 - 13 & 11 - 14 \\
N & 10 - 21 & 13 - 13 & 13 - 15 \\
O & 12 - 23 & 16 - 16 & 15 - 18 \\
F & 14 - 25 & - & 17 - 19 \\
Ne & 16 - 27 & 20 - 22 & 19 - 23 \\
Na & 18 - 34 & 23 - 23 & 21 - 25 \\
Mg & 20 - 35 & 23 - 26 & 23 - 28 \\
Al & 22 - 36 & 27 - 27 & 25 - 30 \\
Si & 24 - 38 & 27 - 32 & 27 - 33 \\
P & 26 - 40 & 30 - 33 & 29 - 35 \\
S & 28 - 42 & 31 - 36 & 30 - 37 \\
Cl & 30 - 44 & 35 - 37 & 32 - 39 \\
Ar & 32 - 46 & 36 - 41 & 34 - 42 \\
K & 34 - 49 & 39 - 43 & 37 - 45 \\
Ca & 36 - 51 & 40 - 46 & 38 - 48 \\
Sc & 38 - 52 & 41 - 47 & 41 - 51 \\
Ti & 40 - 54 & 43 - 50 & 43 - 53 \\
V & 42 - 56 & 45 - 52 & 45 - 55 \\
Cr & 44 - 58 & 47 - 56 & 47 - 57 \\
Mn & 46 - 60 & 49 - 60 & 49 - 59 \\
Fe & 49 - 63 & 51 - 62 & 50 - 62 \\
Co & 51 - 65 & 53 - 61 & 52 - 64 \\
Ni & 53 - 69 & 56 - 64 & 54 - 66 \\
Cu & 55 - 71 & 57 - 65 & 56 - 68 \\
Zn & 57 - 78 & 59 - 66 & 58 - 70 \\
Ga & 61 - 81 & - & 61 - 72 \\
Ge & 63 - 83 & - & 64 - 74 \\
As & 65 - 85 & - & 69 - 75 \\
Se & 67 - 87 & - & 75 - 75 \\
Br & 69 - 90 & - & - \\
Kr & 71 - 93 & - & - \\
Rb & 73 - 99 & - & - \\
Sr & 77 - 100 & - & - \\
Y & 79 - 101 & - & - \\
Zr & 81 - 101 & - & - \\
Nb & 85 - 101 & - & - \\
Mo & 87 - 101 & - & - \\
Tc & 89 - 101 & - & - \\
Ru & 91 - 101 & - & - \\
Rh & 93 - 101 & - & - \\
Pd & 95 - 101 & - & - \\
Ag & 97 - 101 & - & - \\
Cd & 99 - 101 & - & - \\
In & 101 - 101 & - & - \\
\hline\hline
\end{tabular}
\end{table}
\begin{table*}
\caption{Nucleosynthetic indicators.}
\label{t:2}
\centering
\begin{tabular}{llcccccccccccccc}
\hline\hline
\noalign{\smallskip}
& & $a_1$ & $a_2$ & $b_1$ & $b_2$ & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $d_1$ & $d_2$ & $d_3$ & $e_1$ & $e_2$ & $e_3$ \\
& & (\%) & (\%) & (\%) & (\%) & (\%) &(\%) & (\%) & (\%) & (\%) &(\%) & (\%) & (\%) & (\%) & (\%) \\
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{Convergence study} & \multicolumn{14}{c}{} \\
Network & Model & \multicolumn{14}{c}{} \\
netAKh & $M_\mathrm{Ch}$ & 0.3 & 0.2 & 0.4 & 0.0 & 1.5 & 1.5 & 0.3 & 0.4 & 1.5 & 6.1 & 1.3 & 1.5 & 3.9 & 6.1 \\
nse7 & $M_\mathrm{Ch}$ & 0.0 & 0.0 & 0.0 & 0.0 & 0.3 & 0.3 & 0.0 & 0.0 & 0.2 & 0.7 & 1.4 & 0.2 & 0.5 & 1.4 \\
netAKh & sub-$M_\mathrm{Ch}$ & 0.3 & 0.0 & 0.3 & 0.3 & 0.7 & 0.8 & 0.2 & 0.2 & 1.2 & 1.8 & 0.8 & 1.2 & 1.8 & 1.6 \\
nse7 & sub-$M_\mathrm{Ch}$ & 0.0 & 0.0 & 0.0 & 0.0 & 0.3 & 0.3 & 0.1 & 0.1 & 0.0 & 1.4 & 0.3 & 0.0 & 1.3 & 1.4 \\
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{NSE table interpolation} & \multicolumn{14}{c}{} \\
Interpolator & Model & \multicolumn{14}{c}{} \\
linlog & $M_\mathrm{Ch}$ & 0.2 & 0.6 & 11 & 7.6 & 4.3 & 23 & 0.9 & 4.4 & 7.4 & 12 & 0.8 & 22 & 100 & 360 \\
linlin & $M_\mathrm{Ch}$ & 0.0 & 0.0 & 0.1 & 0.0 & 1.0 & 8.3 & 0.2 & 1.5 & 0.9 & 4.7 & 0.3 & 8.2 & 32 & 82 \\
loglog & $M_\mathrm{Ch}$ & 0.2 & 0.6 & 13 & 7.1 & 4.3 & 23 & 0.9 & 4.4 & 7.4 & 12 & 0.8 & 22 & 101 & 355 \\
poly & $M_\mathrm{Ch}$ & 0.1 & 0.5 & 13 & 7.4 & 4.8 & 28 & 0.9 & 5.0 & 7.9 & 13 & 1.9 & 24 & 120 & 490 \\
spline & $M_\mathrm{Ch}$ & 0.0 & 0.0 & 0.1 & 0.0 & 0.1 & 0.2 & 0.0 & 0.1 & 0.1 & 0.2 & 0.4 & 0.6 & 0.7 & 0.8 \\
pe$\nu$n & $M_\mathrm{Ch}$ & 0.0 & 0.1 & 0.3 & 0.1 & 0.2 & 0.6 & 0.1 & 0.1 & 0.3 & 0.3 & 1.4 & 0.8 & 1.1 & 4.2 \\
den$\times10$ & $M_\mathrm{Ch}$ & 0.0 & 0.1 & 0.4 & 0.0 & 0.2 & 0.5 & 0.1 & 0.1 & 0.3 & 0.3 & 0.6 & 0.8 & 1.0 & 3.1 \\
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{Simplified networks} & \multicolumn{14}{c}{} \\
Network & Model & \multicolumn{14}{c}{} \\
iso7 & $M_\mathrm{Ch}$ & 0.5 & 9.4 & 86 & 29 & 58 & 120 & 12 & 23 & 64 & 620 & 14 & 310 & 1500 & 5900 \\
$13\alpha$ & $M_\mathrm{Ch}$ & 3.7 & 4.3 & 70 & 27 & 43 & 93 & 7.4 & 17 & 49 & 380 & 8.8 & 220 & 910 & 3100 \\
net21 & $M_\mathrm{Ch}$ & 4.3 & 3.2 & 6.4 & 6.5 & 8.9 & 8.4 & 3.5 & 4.1 & 16 & 17 & 12 & 17 & 18 & 14 \\
iso7 & sub-$M_\mathrm{Ch}$ & 0.8 & 6.1 & 26 & 31 & 33 & 43 & 11 & 15 & 52 & 140 & 9.8 & 92 & 170 & 320 \\
$13\alpha$ & sub-$M_\mathrm{Ch}$ & 1.2 & 3.9 & 21 & 27 & 23 & 30 & 7.5 & 11 & 40 & 93 & 7.4 & 66 & 110 & 180 \\
net21 & sub-$M_\mathrm{Ch}$ & 1.4 & 1.1 & 0.0 & 2.4 & 3.6 & 3.7 & 1.4 & 1.6 & 5.0 & 9.4 & 4.0 & 5.0 & 6.1 & 9.5 \\
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{High metallicity progenitor} & \multicolumn{14}{c}{} \\
Network & Model & \multicolumn{14}{c}{} \\
net21 & $M_\mathrm{Ch}$ & 3.8 & 4.5 & 12 & 4.4 & 11 & 12 & 3.7 & 7.4 & 19 & 19 & 17 & 21 & 20 & 31 \\
net23 & $M_\mathrm{Ch}$ & 6.0 & 2.5 & 2.8 & 9.0 & 15 & 18 & 4.2 & 7.6 & 28 & 29 & 22 & 28 & 69 & 26 \\
net21 & sub-$M_\mathrm{Ch}$ & 2.4 & 5.9 & 9.4 & 6.2 & 16 & 18 & 6.8 & 9.8 & 20 & 29 & 22 & 25 & 30 & 32 \\
net23 & sub-$M_\mathrm{Ch}$ & 2.9 & 0.1 & 1.0 & 4.5 & 5.5 & 6.5 & 1.7 & 3.1 & 7.6 & 11 & 7.0 & 10 & 13 & 10 \\
\hline
\noalign{\smallskip}
\multicolumn{2}{l}{Transition NSE $\leftrightarrows$ net21} & \multicolumn{14}{c}{} \\
Condition & Model & \multicolumn{14}{c}{} \\
$T_\mathrm{NSE}=5.5\times10^9$~K & $M_\mathrm{Ch}$ & 4.3 & 3.2 & 6.4 & 6.5 & 8.9 & 8.4 & 3.5 & 4.1 & 16 & 17 & 12 & 17 & 18 & 14 \\
$T_\mathrm{NSE}=5.0\times10^9$~K & $M_\mathrm{Ch}$ & 3.5 & 1.6 & 10 & 7.0 & 5.2 & 5.3 & 1.9 & 2.4 & 8.4 & 7.9 & 8.3 & 9.6 & 14 & 8.3 \\
$T_\mathrm{out}=4\times10^9$~K & $M_\mathrm{Ch}$ & 3.3 & 0.5 & 14 & 8.0 & 4.3 & 5.1 & 1.2 & 1.9 & 9.7 & 9.7 & 4.8 & 12 & 20 & 16 \\
$T_\mathrm{out}=3\times10^9$~K & $M_\mathrm{Ch}$ & 3.3 & 1.1 & 1.4 & 1.6 & 19 & 40 & 2.9 & 8.9 & 21 & 100 & 5.1 & 70 & 260 & 530 \\
$\rho_\mathrm{NSE0}=10^8$~\mbox {{\rm g~cm$^{-3}$}} & $M_\mathrm{Ch}$ & 4.4 & 3.3 & 4.1 & 4.6 & 8.8 & 8.3 & 3.6 & 4.2 & 17 & 17 & 12 & 17 & 18 & 15 \\
$\rho_\mathrm{NSE0}=4\times10^7$~\mbox {{\rm g~cm$^{-3}$}} & $M_\mathrm{Ch}$ & 4.3 & 3.2 & 6.7 & 6.3 & 8.9 & 8.4 & 3.5 & 4.1 & 17 & 17 & 12 & 17 & 19 & 14 \\
\hline\hline
\end{tabular}
\end{table*}
As a first test of convergence of the supernova code with respect to the size of the nuclear network, I have re-computed models $M_\mathrm{Ch}$ and sub-$M_\mathrm{Ch}$ with two different nuclear networks, both of them sufficiently large to give accurate nuclear energy generation rates. The first network (netAKh) is that employed in the SNIa models computed by Alexei Khokhlov and reported in \citet{2013blo,2017blo}. It uses 144 isotopes, including all stable isotopes between neon and copper, with the exception of \isotope{48}{Ca}. The second network (nse7) includes 260 isotopes and was introduced by \citet{2019kus} and used in their study of the structure of detonation waves in SNIa. It includes all stable isotopes up to arsenic, with the exception of \isotope{76}{Ge}. Table~\ref{t:1} shows both networks.
To assess the accuracy of the nucleosynthetic yields I use a set of fourteen indicators, all of them expressed as per cent relative differences between the quantities obtained in a test model with respect to the results in the reference hydrodynamic model that uses the default reaction network. Therefore, I compare sub-Chandrasekhar models with model 1p06\_Z9e-3\_std, and Chandrasekhar-mass models with ddt2p4\_Z9e-3\_std. The indicators are as follows:
\begin{itemize}
\item the discrepancy of the final kinetic energy, $a_1=\Delta K/K$,
\item the discrepancy of the ejected mass of $^{56}$Ni, \mbox{$a_2=\Delta M(^{56}\mathrm{Ni})/M(^{56}\mathrm{Ni})$},
\item the discrepancy of the ejected mass of $^{57}$Ni , \mbox{$b_1=\Delta M(^{57}\mathrm{Ni})/M(^{57}\mathrm{Ni})$},
\item the discrepancy of the ejected mass of $^{55}$Fe, \mbox{$b_2=\Delta M(^{55}\mathrm{Fe})/M(^{55}\mathrm{Fe})$},
\item a measure of the discrepancy, $c_1$, based on the average of the squared deviations of the logarithm of the ejected mass of the elements,
\begin{equation}
\sigma_\mathrm{log,ele}=\sqrt{\frac{1}{N_\mathrm{ele}}\sum\log^2\left[\frac{M'(Z)}{M(Z)}\right]}\,,
\end{equation}
\begin{equation}
c_1=10^{\sigma_\mathrm{log,ele}} - 1\,,
\end{equation}
\item a measure of the discrepancy, $c_2$, based on the average of the squared deviations of the logarithm of the ejected mass of the isotopes,
\begin{equation}
\sigma_\mathrm{log,iso}=\sqrt{\frac{1}{N_\mathrm{iso}}\sum\log^2\left[\frac{M'(^{A}Z)}{M(^{A}Z)}\right]}\,,
\end{equation}
\begin{equation}
c_2=10^{\sigma_\mathrm{log,iso}} - 1\,,
\end{equation}
\item a measure of the discrepancy, $c_3$, based on a weighted average of the squared deviations of the logarithm of the ejected mass of the elements,
\begin{equation}
\sigma_\mathrm{wm,ele}=\sqrt{\sum\omega(Z)\log^2\left[\frac{M'(Z)}{M(Z)}\right]}\,,
\end{equation}
\begin{equation}
c_3=10^{\sigma_\mathrm{wm,ele}} - 1\,,
\end{equation}
\item a measure of the discrepancy, $c_4$, based on a weighted average of the squared deviations of the logarithm of the ejected mass of the isotopes,
\begin{equation}
\sigma_\mathrm{wm,iso}=\sqrt{\sum\omega(^{A}Z)\log^2\left[\frac{M'(^{A}Z)}{M(^{A}Z)}\right]}\,,
\end{equation}
\begin{equation}
c_4 = 10^{\sigma_\mathrm{wm,iso}} - 1\,,
\end{equation}
\item the maximum relative discrepancy, $d_1$, in the mass of the elements with final yield $M(Z)\ge10^{-3}$~\mbox{$\mathrm{M_{\odot}}$},
\item the maximum relative discrepancy, $d_2$, in the mass of the elements with final yield in the range $10^{-3}>M(Z)\ge10^{-6}$~\mbox{$\mathrm{M_{\odot}}$},
\item the maximum relative discrepancy, $d_3$, in the mass of the elements with final yield in the range $10^{-6}>M(Z)\ge10^{-12}$~\mbox{$\mathrm{M_{\odot}}$},
\item the maximum relative discrepancy, $e_1$, in the mass of the isotopes with final yield $M(^{A}Z)\ge10^{-3}$~\mbox{$\mathrm{M_{\odot}}$},
\item the maximum relative discrepancy, $e_2$, in the mass of the isotopes with final yield in the range $10^{-3}>M(^{A}Z)\ge10^{-6}$~\mbox{$\mathrm{M_{\odot}}$}, and
\item the maximum relative discrepancy, $e_3$, in the mass of the isotopes with final yield in the range $10^{-6}>M(^{A}Z)\ge10^{-12}$~\mbox{$\mathrm{M_{\odot}}$}.
\end{itemize}
The quantities $M(Z)$ and $M(^{A}Z)$ are the masses, in \mbox{$\mathrm{M_{\odot}}$}, of element $Z$ and isotope $^{A}Z$ in the reference model, $M'$ stands for the same quantities for the test model, $N_\mathrm{ele}$ and $N_\mathrm{iso}$ are, respectively, the number of different elements and isotopes ejected and the weighting functions are defined as:
\begin{equation}
\omega(Z)=\frac{1}{\left[\log^2M(Z)\right]\times\sum\left[1/\log^2M(Z)\right]}\,,
\end{equation}
\begin{equation}
\omega(^{A}Z)=\frac{1}{\left[\log^2M(^{A}Z)\right]\times\sum\left[1/\log^2M(^{A}Z)\right]}\,.
\end{equation}
With these definitions, indicators $c_3$ and $c_4$ provide a measure of the mean deviation of the most abundant species, while indicators $c_1$ and $c_2$ give a measure of the deviation of the yields of all species. Indicators $a_2$, $b_1$, and $b_2$ are evaluated from the ejected masses of the isotopes 100~s after thermal runaway. All the indicators from $c_1$ to $e_3$ refer to the elemental or isotopic yields of isotopes between carbon and krypton after radioactive decays.
For reference, the kinetic energy and masses of radioactive isotopes ejected in the two reference models are the following: in model ddt2p4\_Z9e-3\_std, $K=1.42\times10^{51}$~erg, $M(^{56}\mathrm{Ni})=0.685$~\mbox{$\mathrm{M_{\odot}}$}, $M(^{57}\mathrm{Ni})=7.15\times10^{-3}$~\mbox{$\mathrm{M_{\odot}}$} and $M(^{55}\mathrm{Fe})=9.65\times10^{-3}$~\mbox{$\mathrm{M_{\odot}}$}; in model 1p06\_Z9e-3\_std, $K=1.32\times10^{51}$~erg, $M(^{56}\mathrm{Ni})=0.664$~\mbox{$\mathrm{M_{\odot}}$}, $M(^{57}\mathrm{Ni})=1.01\times10^{-2}$~\mbox{$\mathrm{M_{\odot}}$} and $M(^{55}\mathrm{Fe})=3.21\times10^{-3}$~\mbox{$\mathrm{M_{\odot}}$}.
The thermodynamic trajectories obtained with the netAKh and nse7 networks have been fed to a post-processing nuclear code that uses the same network and reaction rates as in the BM-P network\footnote{The same strategy has been applied to all the calculations presented in the following sections.}.
Table~\ref{t:2} show the results in the rows under the header ``Convergence study``. The agreement with the results of the hydrodynamic calculation using the default network is very satisfactory. The direct measure of the nuclear energy released, that is the final kinetic energy, is reproduced in all four calculations to better than 0.3\%. The yields of the radioactive isotopes are reproduced to within 0.4\% with both networks and, in particular, the yield of $^{56}$Ni to better than 0.2\%.
The nucleosynthesis of stable isotopes and elements also converges, where both netAKh and nse7 obtain similar ratings. The mean deviation of the most abundant elements and isotopes is $\le0.4$\%, while that representative of all ejected species lies in the range 0.3-1.5\%. Finally, the maximum deviation of elements and isotopes whose yield is larger than $10^{-3}$~\mbox{$\mathrm{M_{\odot}}$} is $\le1.5$\%, while that of the remaining elements and isotopes with yields $\ge10^{-12}$~\mbox{$\mathrm{M_{\odot}}$} is less than $\sim6$\%.
\section{Nuclear statistical equilibrium}\label{s:3}
One of the key ingredients of simulations of SNIa explosions is the treatment of NSE in matter burnt at high density. The composition of matter in NSE can be calculated by solving a set of Saha equilibrium equations linking the abundances of all isotopes to two arbitrarily chosen abundances (or combinations thereof), which play the role of independent variables, plus two closure relationships that account for the conservation of baryon number and the electrical neutrality of matter. The procedure is usually iterative, which makes it inefficient for a multi-dimensional hydrodynamic computation of a supernova explosion. Therefore, in this sort of simulation, it is usual to rely on interpolation of a table of NSE states, pre-computed on a net of density, $\rho$, temperature, $T$, and electron mole number, $Y_\mathrm{e}$, nodes, the denser the better. Usually, the table gives the main properties of matter in NSE, including nuclear binding energy, mean molar number, neutronization rate and neutrino energy loss rate.
One example of this kind of table of NSE properties is given in \citet{2009se2}. Recently, \citet{2019bra} reported that the final yields computed using their NSE table might disagree by order of magnitude for some isotopes with respect to those obtained computing the NSE state properties on the fly in the hydrodynamical calculation. The discrepancy did not affect significantly either the total energy release, i.e. the final kinetic energy, or the ejected mass of $^{56}$Ni, and was attributed to the interpolation procedure applied to obtain the NSE properties out of the $\rho$, $T$, and $Y_\mathrm{e}$ table nodes.
Here, I argue that the culprit for the discrepancy just mentioned is relying on the interpolation at the density nodes. To illustrate the situation, Fig.~\ref{f:1} shows the neutrino energy loss-rate, $\varepsilon_\nu$ in a sample of the \citet{2009se2} table nodes of $\rho$, $T$, and $Y_\mathrm{e}$ for typical values during a supernova explosion. While the energy-loss rate changes smoothly between consecutive temperature and electron mole number nodes, the dependence on density is more complex and the values of $\varepsilon_\nu$ change by four orders of magnitude between $\rho$ node numbers 12 and 20 ($\rho=2\times10^8$~\gccb and $\rho=2\times10^{10}$~\mbox {{\rm g~cm$^{-3}$}}). This huge change makes the results of the interpolation in density sensitive to the interpolation procedure. Indeed, the plot of $\varepsilon_\nu$ versus density in between nodes 12 and 20 in Fig.~\ref{f:1} is suggestive of a linear dependence between $\log\varepsilon_\nu$ and $\log\rho$ (the table nodes are equispaced in $\log\rho$).
\begin{figure}
\includegraphics[width=\columnwidth]{figura123.eps}
\caption{Sample of variation of the neutrino energy loss-rate as function of density in the range $2\times10^5-2\times10^{10}$~\gccb (at fixed $T=9\times10^9$~K and $Y_\mathrm{e}=0.5$~mol~g$^{-1}$, red dots), temperature in the range $3\times10^9-1.05\times10^{10}$~K (at fixed $\rho=2\times10^9$~\gccb and $Y_\mathrm{e}=0.5$~mol~g$^{-1}$, green downward triangles) and electron mole number in the range $0.4400-0.5025$~mol~g$^{-1}$ (at fixed $\rho=2\times10^9$~\gccb and $T=9\times10^9$~K, blue upward triangles) in the interpolation nodes of the NSE table.
}
\label{f:1}
\end{figure}
To test the impact of the NSE table interpolation scheme I have chosen different interpolants and computed the difference in NSE properties with respect to those obtained by solving the NSE Saha equilibrium equations. The interpolants used and the designations given are the following:
\begin{itemize}
\item linear interpolation of NSE properties, for example $\varepsilon_\nu$ or the neutronization rate $\dot{Y}_\mathrm{e}$, with respect to $\log\rho$ (linlog);
\item linear interpolation with respect to $\rho$ (linlin);
\item linear interpolation of $\log\dot{Y}_\mathrm{e}$ and $\log\varepsilon_\nu$ with respect to $\log\rho$ (loglog);
\item third order polynomial of the NSE properties, for example $\dot{Y}_\mathrm{e}$, with respect to $\log\rho$ (poly);
\item cubic spline fitting the NSE properties with respect to $\rho$ (spline);
\item a physically motivated interpolation function (pe$\nu$n).
\end{itemize}
The last interpolator is motivated by the dominant role of protons in the neutronization rate of NSE matter in SNIa models \citep{1985ful,2000brc,2019brab}. Hence it seems natural to interpolate using the same function that describes the dependence of the p(e$^{-}$,$\nu$)n rate or the associated neutrino energy emission rate on density \citep{1985ful}. The effective log($ft$)-values characterizing electron captures by protons are almost constant, whereas the rate dependence on $\rho$ is given by the so-called modified phase space factor \citep[Eqs.~3 and 6 in][]{1985ful}, $I_\mathrm{e}$. The neutrino energy emission rate depends on $\rho$ through the appropriate phase-space factor \citep[Eq.~7 in][]{1985ful}, $J_\mathrm{e}^\nu$. Instead of computing the relativistic Fermi integrals that appear in the definition of these space factors, I calculate approximate values taking advantage of Eqs.~15 in \citet{1985ful}.
\begin{figure*}
\includegraphics[width=\textwidth]{pltintlogrho+.eps}
\caption{Relative error between the exact neutronization rate, $\dot{Y}_\mathrm{e}$, and that computed by interpolation on a NSE table, for two values of the electron mole number, $Y_\mathrm{e}=0.50$~mol~g$^{-1}$ (left column) and $Y_\mathrm{e}=0.47$~mol~g$^{-1}$ (right column). The results are shown for different interpolation schemes, from top to bottom: linear interpolation of $\dot{Y}_\mathrm{e}$ versus $\log\rho$, linear interpolation of $\dot{Y}_\mathrm{e}$ versus $\rho$ and linear interpolation of $\log\dot{Y}_\mathrm{e}$ versus $\log\rho$. The relative error is colour coded according to the colour bar at the top of the plot. The dashed lines show the density and temperature at which the mass shells of model $M_\mathrm{Ch}$ start experiencing electron captures in NSE (left column) and at the time they reach an electron mole number $Y_\mathrm{e}=0.47$~mol~g$^{-1}$ (right column).
}
\label{f:2}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{pltsplinden+.eps}
\caption{Same as Fig.~\ref{f:2} but for different interpolation schemes, from top to bottom: third order polynomial of $\dot{Y}_\mathrm{e}$ versus $\log\rho$, cubic spline of $\dot{Y}_\mathrm{e}$ versus $\rho$ and pe$\nu$n.
}
\label{f:3}
\end{figure*}
Figures \ref{f:2} and \ref{f:3} show the relative error between $\dot{Y}_\mathrm{e}$ obtained using the different interpolants and the exact neutronization rate obtained solving the NSE Saha equilibrium equations. The error goes to zero at the table nodes, but can reach up to 1, i.e. 100\% error, at low densities and high temperatures. The dashed lines show the most relevant combination of density and temperature for the SNIa $M_\mathrm{Ch}$ model in two conditions: when electron captures start in NSE and the neutronization rate is maximum (left columns) and when a value of $Y_\mathrm{e}=0.47$~mol~g$^{-1}$ is attained, a condition that is only reached in the innermost $\sim0.15$~\mbox{$\mathrm{M_{\odot}}$} of the WD (right columns). The WD mass shells go through $\rho$-$T$ conditions for which the relative error in $\dot{Y}_\mathrm{e}$ is as high as $5-20$~\% for the most simple interpolators: linlog, linlin, loglog and poly. On the other hand, the upper bound on the maximum error in the same conditions is $\sim3-5$~\% when either the spline or the pe$\nu$n interpolator is used.
Table~\ref{t:2} shows the impact of the different interpolators on the final yields of the $M_\mathrm{Ch}$ model, under the heading ''NSE table interpolation``. The nucleosynthetic results confirm the intuition gained with Figs.~\ref{f:2} and \ref{f:3}: the most accurate interpolators are the cubic spline and pe$\nu$n, which perform almost equally well. All interpolants lead to accurate values of the final kinetic energy and just negligible errors in the mass of $^{56}$Ni synthesized. The relevant differences appear when one looks into the nucleosynthesis of less abundant species. For instance, using linlog, loglog and poly leads to errors in the ejected mass of $^{57}$Ni and $^{55}$Fe of about 10\%, average errors in the isotopic yields of $20-30$\% (indicator $c_2$), and order-of-magnitude errors in the yields of some isotopes with yields smaller than $10^{-3}$~\mbox{$\mathrm{M_{\odot}}$} (indicators $e_2$ and $e_3$). On the other hand, the maximum isotopic error obtained with the cubic spline interpolator is 0.8~\% and that obtained with pe$\nu$n is 4.2~\%. The behaviour of interpolator linlin is intermediate between the two groups above.
Table~\ref{t:2} also shows the result of increasing the resolution of the NSE table by up to ten times more density nodes (''den$\times$10``), with $\Delta\log\rho=0.025$. With a table of this size, even using a linlog interpolant gives very good results, comparable to both the spline and the pe$\nu$n interpolators with the original table with $\Delta\log\rho=0.25$.
In the calculations reported in the following sections, I work with simplified nuclear networks and use a table to obtain the properties of NSE matter, with the aim of testing as faithfully as possible the strategies commonly used in many SNIa explosion models. In these tests, it is important that the treatment of NSE matter does not introduce additional errors, beside those attributable to simplified networks. For this purpose, I use the cubic spline interpolator and the NSE table with $\Delta\log\rho=0.25$.
\section{Simplified nuclear networks}\label{s:4}
\begin{table}
\caption{Simplified nuclear networks.
\label{t:3}}
\centering
\begin{tabular}{lcccc}
\hline\hline
\noalign{\smallskip}
& iso7 & $13\alpha$ & net21 & net23 \\
$Z$ & $A$ & $A$ & $A$ & $A$ \\
\hline\hline
\noalign{\smallskip}
n & - & - & 1 & 1 \\
p & - & - & 1 & 1 \\
He & 4 & 4 & 4 & 4 \\
C & 12 & 12 & 12 & 12 \\
O & 16 & 16 & 16 & 16 \\
Ne & 20 & 20 & 20 & 20,22 \\
Mg & 24 & 24 & 24 & 24,25 \\
Si & 28 & 28 & 28 & 28 \\
S & - & 32 & 32 & 32 \\
Ar & - & 36 & 36 & 36 \\
Ca & - & 40 & 40 & 40 \\
Ti & - & 44 & 44 & 44 \\
Cr & - & 48 & 48 & 48 \\
Fe & - & 52 & 52,53,54, & 52,53,54 \\
& & & 55,56 & 55,56 \\
Co & - & - & 55,56 & 55,56 \\
Ni & 56 & 56 & 56 & 56 \\
\hline\hline
\end{tabular}
\end{table}
\begin{figure*}
\includegraphics[width=\textwidth]{indipp.png}
\caption{Accuracy of the nucleosynthesis obtained with the net21 network as compared to model ddt2p4\_Z6p75e-2\_std. {\bf Top:} final ejected mass of stable isotopes in the reference model. {\bf Bottom:} percentage error of the mass yield of each isotope when the simplified network is used, with respect to the reference model.
}
\label{f:4}
\end{figure*}
Because of the huge difference between the size of a WD and the width of thermonuclear burning waves, multi-dimensional simulations of SNIa need to allocate most memory and CPU resources to solve the hydrodynamic equations over as large a range of length-scales as possible \citep[e.g.][]{2003gam}. In turn, the nuclear kinetics must be solved with a reduced nuclear network, with the goal that the nuclear energy must be released as faithfully as possible, as if a complete nuclear network were used. Other compositional properties that affect the equation of state, such as the electron mole number and the mean molar weight, must also be reproduced accurately in order not to change the explosion development.
In this section, I show the impact of the use of reduced nuclear networks on the nucleosynthesis of both the $M_\mathrm{Ch}$ and the sub-$M_\mathrm{Ch}$ models in one dimension.
First, I test two small networks, designed iso7 and $13\alpha$, that are widely used in multi-dimensional simulations of SNIa, plus a slightly larger network designed to improve the nucleosynthetic results, named net21. Table~\ref{t:3} shows the composition of each network.
In these calculations, all nuclear reactions linking species present in the network directly are accounted for and the rates are taken from the REACLIB compilation (see Sect.~\ref{s:2}) unless otherwise stated here.
The iso7 network, introduced by \citet{2000tim} as a simplification of the nine-isotope reaction network described in detail in Table 1 of \citet{1986woo}, has been used in 2D simulations of DDT models of exploding WDs \citep[e.g.][]{2018leu}. This network is crafted for efficient computation of nuclear energy generation in multi-dimensional calculations of explosive burning stages from carbon-burning onwards. It assumes two quasi-equilibrium groups of isotopes, the silicon group and the iron group, and hard-wires the nucleosynthetic flows between both groups in a single step that link the abundances of \isotope{28}{Si} and $^{56}$Ni .
This step is computed making use of Eqs.~6-8 of \citet{2000tim}.
\citet{2000tim} warned that the iso7 and $13\alpha$ networks (and, in general, any $\alpha$-network) might give energy generation rates wrong by order of magnitude if $Y_\mathrm{e}\lesssim0.49$~mol~g$^{-1}$.
The $13\alpha$ network, introduced by \citet{1986mue} and later by \citet{1995liv} as a simplification of a larger network described in \citet{1978wea}, has been used in SPH simulations of merging and colliding WDs \citep{2014ras,2015dan}. The version of the $13\alpha$ network used in the present work follows \citet{1999tim} and \citet{2000tim}, where it is applied a special treatment to the links between $\alpha$-nuclei from magnesium onwards: above a temperature of $2.5\times10^9$~K, the flows from $(\alpha,\mathrm{p})$ reactions, followed by $(\mathrm{p},\gamma)$, are added to the flows from $(\alpha,\gamma)$. It is important to note that, in this version of the $13\alpha$ network\footnote{See also http://cococubed.asu.edu/code\_pages/burn\_helium.shtml}, the link from an $\alpha$-nucleus, \isotope{2Z}{Z}, to the next one, \isotope{2Z+4}{(Z+2)}, through $(\alpha,\mathrm{p})$ reactions takes into account the possibility that $(\mathrm{p},\alpha)$ follows instead of $(\mathrm{p},\gamma)$:
\begin{equation}\label{e:0}
^{2Z}\mathrm{Z} \rightleftarrows ^{2Z+3}\mathrm{(Z+1)} \rightleftarrows ^{2Z+4}\mathrm{(Z+2)}\,.
\end{equation}
For instance, in the conversion of \isotope{28}{Si} into \isotope{32}{S} through \isotope{31}{P}, the four reactions that follow have to be considered besides $^{28}\mathrm{Si}(\alpha,\gamma)^{32}\mathrm{S}$,
\begin{equation}
^{28}\mathrm{Si} \rightleftarrows ^{31}\mathrm{P} \rightleftarrows ^{32}\mathrm{S}\,.
\end{equation}
Assuming that the abundance of the intermediate nucleus, \isotope{31}{P} in this example, is established by the equilibrium of the direct and reverse reactions in Eq.~\ref{e:0}, the overall rate of change of the molar fraction of \isotope{2Z+4}{(Z+2)}, $Y\left[^{2Z+4}\mathrm{(Z+2)}\right]$, from \isotope{2Z}{Z} is given by
\begin{equation}
R_{(\alpha,\gamma)}^\mathrm{eff}
Y\left[^{2Z}\mathrm{Z}\right] Y_{\alpha} -
R_{(\gamma,\alpha)}^\mathrm{eff}
Y\left[^{2Z+4}\mathrm{(Z+2)}\right]\,,
\end{equation}
where the effective rates, $R_{(\alpha,\gamma)}^\mathrm{eff}$ and $R_{(\gamma,\alpha)}^\mathrm{eff}$, are
\begin{equation}\label{e:1}
R_{(\alpha,\gamma)}^\mathrm{eff} = R_{(\alpha,\gamma)} + R_{(\alpha,\mathrm{p})}\frac{R_{(\mathrm{p},\gamma)}}{R_{(\mathrm{p},\alpha)} + R_{(\mathrm{p},\gamma)}}\,
\end{equation}
and
\begin{equation}\label{e:2}
R_{(\gamma,\alpha)}^\mathrm{eff} = R_{(\gamma,\alpha)} + R_{(\gamma,\mathrm{p})}\frac{R_{(\mathrm{p},\alpha)}}{R_{(\mathrm{p},\alpha)} + R_{(\mathrm{p},\gamma)}}\,,
\end{equation}
and $R_{(\alpha,\mathrm{p})}$, $R_{(\mathrm{p},\alpha)}$, $R_{(\gamma,\mathrm{p})}$, and $R_{(\mathrm{p},\gamma)}$ are the true rates of the reactions \isotope{2Z}{Z}$\rightarrow$\isotope{2Z+3}{(Z+1)}, \isotope{2Z+3}{(Z+1)}$\rightarrow$\isotope{2Z}{Z}, \isotope{2Z+4}{(Z+2)}$\rightarrow$\isotope{2Z+3}{(Z+1)}, and \isotope{2Z+3}{(Z+1)}$\rightarrow$\isotope{2Z+4}{(Z+2)}, respectively.
As before, all true rates have been computed as in the REACLIB compilation.
In both networks, iso7 and $13\alpha$, the electron mole number in the initial model is $Y_\mathrm{e}=0.5$~\mbox {{\rm mol~g$^{-1}$}}. The electron mole number is allowed to change during NSE, due to electron captures, but is kept fixed when the composition is computed by integration of the nuclear network, because it does not include weak interactions.
Table~\ref{t:2} shows the accuracy of the nucleosynthesis obtained by post-processing the thermodynamic trajectories belonging to these two networks, under the heading ''Simplified networks``. Generally, the performance of the simplified networks in the $M_\mathrm{ch}$ models is worse than in the sub-$M_\mathrm{Ch}$ models. The kinetic energy is reproduced reasonably with the simplified networks, slightly better with iso7 than with $13\alpha$, and the error in the yield of \isotope{56}{Ni} is within 6-10\% when iso7 is used but $\sim4\%$ with $13\alpha$.
The errors in the yields of radioisotopes $^{55}$Fe and $^{57}$Ni, which are often constrained observationally by the late-time light curves of SNIa, are about 30\%, with the exception of $^{57}$Ni in $M_\mathrm{Ch}$, the abundance of which is wrong by nearly an order of magnitude. These errors are comparable with the maximum deviation obtained in the yields of the most abundant elements, $d_1\sim 40 - 60\%$, while the maximum error in the abundance of the isotopes (indicators $e_1$ to $e_3$) may be up to several orders of magnitude. Among those with the largest yields, the isotopes that present the maximum deviation are mostly part of the iron group: \isotope{40}{Ca}, \isotope{52,53}{Cr}, \isotope{55}{Mn}, \isotope{57}{Fe}, and \isotope{60,62}{Ni}.
The net21 network is an extension of the $13\alpha$ network
that includes additional isotopes of the iron group plus free protons and neutrons (its full composition can be seen in Table~\ref{t:3}), in order to obtain a more reliable representation of the nucleosynthesis of the most deficient nuclides from the results of the iso7 and $13\alpha$ networks.
The net21 network makes use of the same hard-wiring of rates as in Eqs.~\ref{e:1} and \ref{e:2}, with the exception of the chain
\begin{equation}
^{52}\mathrm{Fe} \rightleftarrows ^{55}\mathrm{Co} \rightleftarrows ^{56}\mathrm{Ni}\,.
\end{equation}
Since \isotope{55}{Co} is included explicitly in the network, the effective rates between \isotope{52}{Fe} and \isotope{56}{Ni} given by Eqs.~\ref{e:1} and \ref{e:2} are substituted by the corresponding true rates, $R_{(\alpha,\gamma)}$ and $R_{(\gamma,\alpha)}$.
When using the net21 network in the hydrocode, the errors in the kinetic energy and the yield of \isotope{56}{Ni} after post-processing are about $1 - 5\%$. On the other hand, the errors in the yields of $^{55}$Fe and $^{57}$Ni are less than $7\%$ and the maximum error in the predicted abundance of any isotope or element is less than 20\% (indicators $d_1$ to $e_3$).
The errors in the post-processed nucleosynthesis after using the net21 network in the hydrocode are comparable to, although slightly larger than, those obtained with the large networks used in the convergence study, netAkh and nse7. Therefore, the performance of net21 would barely be improved with other simplified networks based on no more than a few tenths of nuclides. I have experimented with larger networks, including a group of CNO isotopes (\isotope{13,14}{C}, \isotope{14}{N}, \isotope{17}{O}) and the intermediate species in Eq.~\ref{e:0}, from \isotope{27}{Al} to \isotope{51}{Mn}, with no significant improvement in the performance over that of network net21. I have also probed reducing the number of iron-group isotopes in the network, but the accuracy of the nucleosynthesis was worse than with net21.
\section{Nuclear post-processing for high metallicity progenitors}\label{s:5}
In this section, I test the accuracy of the post-processed nucleosynthesis with respect to the initial metallicity of the progenitor star. Network net21, as well as iso7 and $13\alpha$, is not capable to describe an initial composition of carbon-oxygen material with an excess of neutrons over protons. This is because the initial metallicity of the progenitor star is encoded, at the time of formation of a carbon-oxygen WD, in the abundance of \isotope{22}{Ne} \citep{2003tim}, which is not a part of any of the simplified networks discussed in the previous section. Hence one may wonder whether the accuracy of the net21 network degrades for high metallicity progenitors.
Here, I introduce a new network, net23, that complements the net21 network with the inclusion of \isotope{22}{Ne} and \isotope{25}{Mg} (Table~\ref{t:3}). As just explained, the presence of \isotope{22}{Ne} serves the purpose of building initial models with non-zero neutron-excess, as it is done in the hydrodynamic models that use the full network (see Sect.~\ref{s:2}). The isotope \isotope{25}{Mg} provides a simple route for the burning of \isotope{22}{Ne} through \isotope{22}{Ne}$\left(\alpha,\mathrm{n}\right)$\isotope{25}{Mg}$\left(\alpha,\mathrm{n}\right)$\isotope{28}{Si}.
For this test, I have selected an initial metallicity of the WD as high as $Z=0.0675$. Table~\ref{t:2} shows the results of the post-processed nucleosynthesis using networks net21 and net23, for both the $M_\mathrm{Ch}$ and the sub-$M_\mathrm{Ch}$ models, under the heading ''High metallicity progenitor``. Network net21 performs slightly better in the $M_\mathrm{Ch}$ model, whereas net23 does better in the sub-$M_\mathrm{Ch}$ model, but the overall accuracy of both networks is similar. When the nucleosynthesis errors in the high metallicity calculations are compared with those in the $Z=0.009$ models, using the same networks, the results are slightly better at low metallicity, but not significantly different.
Figure~\ref{f:4} shows the percentage error in the prediction of the abundances of the stable isotopes between carbon and krypton after post-processing the thermodynamic trajectories obtained with the net21 network, compared with those using the default network in the hydrocode. The largest error, 31\%, belongs to \isotope{48}{Ca}, the abundance of which is below $10^{-6}$~\mbox{$\mathrm{M_{\odot}}$}, while all other isotopes are predicted with errors smaller than $\sim25\%$.
\section{Transition from the nuclear network to NSE and vice versa}
The simultaneous inclusion in a simulation of a simplified network and an NSE routine, as described in Sect.~\ref{s:3}, raises the question of the criteria for the transition between both treatments. The transition between the nuclear network and NSE is usually defined in terms of the temperature, where different values are used for assuming NSE, $T_\mathrm{NSE}$, and leaving it, $T_\mathrm{out}$. In the hydrocode used in this work, a third parameter, $\rho_\mathrm{NSE0}$, allows acceleration of the burning of shells hit by a deflagration front.
Silicon exhaustion is a milestone for achieving NSE. It is reached at a temperature somewhere in between $\sim5\times10^9$~K and $\sim6\times10^9$~K with a slight dependence on density \citep[e.g., Fig.~20 in][]{1973woo}. In the calculations presented in this section, I have adopted a unified value of $T_\mathrm{NSE}$, independent of matter density\footnote{In the hydrocode using the default network, there are two different values of the minimum temperature to achieve NSE, depending on density; see \citet{2019bra} for further details}, as detailed in Table~\ref{t:2}.
The freezing-out of nuclear reactions when NSE matter cools is more complex, because at low densities the abundance of free particles (especially $\alpha$ particles) may be sufficiently large to affect the composition significantly.
\citet{2017har} studied the impact of the value of $T_\mathrm{out}$ in the context of core-collapse supernovae.
Usual values of the temperature at which matter is assumed to leave NSE lie in the range from $\sim2\times10^9$~K to $\sim5\times10^9$~K. Again, in the calculations presented in this section, there is a single value of $T_\mathrm{out}$, independent of density, at variance with the method adopted in the hydrocode using the default network \citep{2019bra}.
Following the passage of a deflagrative front, shells are assumed to achieve NSE if their density is larger than the third parameter, $\rho_\mathrm{NSE0}$. By default, I adopt conservative values for the three parameters: $T_\mathrm{NSE}=6\times10^9$~K, $T_\mathrm{out}=5\times10^9$~K and $\rho_\mathrm{NSE0}=8\times10^7$~\mbox {{\rm g~cm$^{-3}$}}.
As can be seen in Table~\ref{t:2}, the precise value of $T_\mathrm{NSE}$ does not affect the error in the post-processed nucleosynthesis of the $M_\mathrm{Ch}$ model significantly, as long as it is in the range from $\sim5\times10^9$~K to $\sim6\times10^9$~K\footnote{Recall that the errors reported in this subsection have to be compared with the reference model, that is the net21 $M_\mathrm{Ch}$ model under the heading ''Simplified networks``.}.
On the other hand, when the threshold for leaving NSE, $T_\mathrm{out}$, takes on a value between $4\times10^9$~K and $5\times10^9$~K the accuracy of the nucleosynthesis is satisfactory, but when this parameter goes down to $3\times10^9$~K the results worsen: for instance, the maximum error in the predicted isotopic yields increases by orders of magnitude.
Finally, the accuracy of the nucleosynthesis is not affected by the value of $\rho_\mathrm{NSE0}$, at least within the range explored in this work and presented in Table~\ref{t:2}, $\rho_\mathrm{NSE0}=4\times10^7$~\gccb to $10^8$~\mbox {{\rm g~cm$^{-3}$}}.
As explained in Appendix B4 of \citet{2019bra}, detonated matter is not assumed to be in NSE if its density is below $\rho_\mathrm{NSE0}$, irrespective of its temperature. In practice, it implies that, in the present models, freeze-out from NSE occurs at low entropy. Therefore, the effect of the NSE parameters just described limits to the so-called normal or particle-poor freeze-out. On the other hand, it is remarkable that alpha-rich freeze-out is managed by integration of the net21 network with very good accuracy, in spite of its small size.
\section{Conclusions}\label{s:conclusions}
Three-dimensional hydrodynamical simulations are necessary to predict the outcome of several explosion scenarios and compare them with SNIa observational data. Whereas great efforts have been made to obtain reliable nucleosynthetic yields and explore their dependence on a number of simulation parameters \citep[e.g.][]{2010mae,2013sei},
until now very few works have been published addressing the accuracy of the resulting nucleosynthesis with respect to the use of simplified nuclear networks \citep{2015pap}\footnote{https://trace.tennessee.edu/utk\_graddiss/3454/}.
In the present work, I use a supernova code, capable of integrating the nuclear kinetic equations using a large nuclear network and the hydrodynamical equations simultaneously, as a benchmark for testing the results of several simplifying assumptions related to nuclear kinetics. These simplifications are related to the use of a small nuclear network and the use of tabulated properties of matter in nuclear statistical equilibrium. I define a set of 14 indicators related to the accuracy of the nucleosynthesis.
The method used in this work does not allow testing of all the strategies currently used in multi-dimensional simulations of SNIa to follow the nuclear energy generation-rate accurately during a thermonuclear explosion. For instance, several studies \citep[e.g.][]{2004tra,2012dub,2017leu} use Lagrangian tracer particles smartly distributed through the simulated space to advect the thermodynamic properties of the underlying Eulerian cells. Nucleosynthesis is then obtained following a post-processing step on the tracer particles, the number of which is much less than the original Eulerian nodes of the simulation. Other studies \citep[e.g.][]{2007cal,2010fin,2016tow} adopt a nuclear energy generation rate linked to the nature of the burning wave (whether a detonation or a deflagration) and the density of fuel.
I propose to use published tables of NSE properties (neutronization rate, mean molar number, mean nuclear binding energy and neutrino energy loss-rate, as functions of density, temperature and electron molar number)
but putting great care into interpolation between their values at the tabulated density points. The interpolation in temperature and electron mole number is not so critical and can simply be linear. A cubic spline interpolation in density gives the most precise results, the accuracy of which is better than 1\% in all 14 nucleosynthetic indicators.
Alternatively, a simple linear interpolation on an NSE table with high resolution in density, $\Delta\log\rho=0.025$, may give almost as accurate results as the cubic spline interpolation of the standard NSE table with $\Delta\log\rho=0.25$.
I have tested several simplified nuclear networks for the accuracy of the nucleosynthesis obtained after post-processing, compared with the nucleosynthesis resulting directly from the supernova code when the default, large nuclear network is used. Short networks (iso7 and 13$\alpha$) are able to give an accurate yield of $^{56}$Ni after post-processing, but can fail by an order of magnitude in predicting the ejected mass of even mildly abundant species ($>10^{-3}$~\mbox{$\mathrm{M_{\odot}}$}). I find that a network of 21 species, net21, reproduces the nucleosynthesis of Chandrasekhar and sub-Chandrasekhar explosions nicely, their average errors being better than 10\% for the most abundant elements and isotopes (yields larger than $10^{-3}$~\mbox{$\mathrm{M_{\odot}}$}) and better than 20\% for the whole set of stable elements and isotopes followed in the model.
In these explosion models, the NSE state is switched on and off according to three criteria based on local temperature and density.
The temperature at which it can be safely assumed that matter will achieve NSE can adopt any value between $5\times10^9$~K and $6\times10^9$~K without affecting the accuracy of the post-processed nucleosynthesis significantly. For normal freeze-out of NSE, equilibrium abundances can be assumed for temperatures in excess of $\sim4\times10^9$~K. However, if the NSE state is kept until a temperature as low as $3\times10^9$~K, the resulting post-processed nucleosynthesis can be almost an order of magnitude wrong even for isotopes with yield $>10^{-3}$~\mbox{$\mathrm{M_{\odot}}$}.
On the other hand, the alpha-rich freeze-out of NSE is managed satisfactorily by the net21 network, in spite of its small size.
\section*{Acknowledgements}
This work has been supported by the MINECO-FEDER grants AYA2015-63588-P and PGC2018-095317-B-C21.
\bibliographystyle{mnras}
|
2,877,628,090,818 | arxiv | \section{Introduction}
Product distribution strategy has a significant impact on the supply chain performance. According to \cite{apte2000effective}, 30\% of an item price is incurred in the distribution process. Hence, lots of companies are currently trying to develop new distribution strategies to efficiently manage their product flow. Cross-docking, as a logistic policy in the distribution part of supply chain, is implemented to improve transport and delivery activities by consolidating the products. Cross-dock is a distribution facility in which the goods arriving by the inbound trucks are unloaded, sorted, consolidated based on their destination and loaded onto the outbound trucks. Inside the cross-dock, the products can be delivered directly to their related outbound trucks or stored in the temporary storage but not more than 24h. On the one hand, warehouses allow companies to make economies in transportation (by making full truckloads), reduce the effect of demand variability and execute additional operations closer to final customers \citep{dolgui2010supply}. On the other hand, cross-docking has the potential to eliminate the storage and retrieval operation functions of a traditional warehouse. As it provides the best of the warehousing and direct delivery strategy, cross-docking has become a popular distribution strategy in practice \citep{Rijal2019}.
The primary purpose of the cross dock is to enable a consolidation of differently sized shipments with the same destination to full truckloads, so that economies in transportation costs can be realized \citep{boysen2010cross}. Indeed, cross docking can decrease distribution network costs by reducing products' delivery time, material handling, and inventory holding costs \citep{kaboudani2018vehicle, theophilus2019truck, zarandi2016constraint}. In the cross-docking operations, an efficient transshipment process is required where inbound and outbound truckloads are synchronized, so that intermediate storage inside the terminal is kept low and on-time deliveries are ensured \citep{boysen2010cross}.
Various decision problems, arising in cross-docking, have been studied in the literature. They could be classified into strategic, tactical and operational problems. Vehicle scheduling, as an operational level optimization problem, is an important concern in the cross-dock planning which is largely considered by researchers. In the classical cross-docking operations, each product is dedicated to a specific outbound truck; But with interchangeability considerations, two products of the same type can be replaced in the predefined load of an outbound truck. Unlike classical interchangeability, in the studied problem, the loads of the outbound trucks are not predefined. Hence, this article presents a new extension of the product interchangeability in which, the load of the outbound trucks and the destination of each truck must be determined by solving the problem.
In the cross-docks where the product interchangeability is allowed, called post-distribution cross-docking, any arrived pallet must be attributed to an outbound truck (product-truck allocation) or to a destination (product-destination allocation). The studied truck scheduling problem incorporates both mentioned allocations simultaneously. Therefore, it deals with two synchronized interchangeabilities, called product-truck-destination allocation, in a post distribution cross-docking environment. The integration of these synchronized assignment problems with both inbound and outbound trucks scheduling is the main contribution of this study. Moreover, in the literature, the time windows are mostly defined for the outbound trucks; But in this research, they are assigned to the customers' orders under a JIT principle.
\cite{boysen2010scheduling} proved that the truck scheduling in a cross-dock facility with one inbound door and one outbound door is an NP hard problem. This paper studies a truck scheduling with multiple inbound and outbound doors combined with two synchronized assignment problems. It can be concluded that the presented problem is also NP hard. Consequently, a matheuristic approach, based on the decomposition of the formulation, is presented to solve the real size problem instances. Afterward, an adaptive heuristic algorithm is provided to improve the performance of the matheuristic.
The rest of this paper is organized in five sections as follows. After a brief literature review in section \ref{literature}, the studied problem is presented in section \ref{problem description}. The problem is formalized mathematically in section \ref{math model}. The solution approaches (matheuristic and heuristic methods) are explained in section \ref{solution approaches}. The results of the computational experiments are summarized in section \ref{results}. Finally, the conclusion of the paper and the directions for the future studies are presented in section \ref{conclusion}.
\section{Literature}\label{literature}
Cross-docking, as a product distribution strategy that serves as an intermediate node between suppliers and customers in the distribution networks \citep{mousavi2014location, joo2013scheduling}, has received increasing attention by the researchers over the last two decades. The decision problems in the cross-docking contain a large domain of researches. From strategic to operational, from design to scheduling, from network to local problems, are decided in the cross-dock optimization domain. These decision problems have been classified from the different point of views in the previous researches presented by \cite{Ladier2016}, \cite{buijs2014synchronization}, \cite{van2012cross} and \cite{boysen2010cross}. In this paper, the operational level problems inside the cross-dock environment are targeted.
Various operational level problems in the cross-dock planning have been studied by the researchers. Transportation planning, dock-door assignment, Vehicle Routing Problem (VRP) and vehicle scheduling are the main optimization problems in this domain.
For example, a dock-door assignment problem was studied by \cite{Nassief2018} in which the unloading and loading times were considered. Three new MIP formulations were presented and solved by LP relaxation and column generation algorithm. A transportation problem in the cross-dock network was presented by \cite{Musa2010} in which the direct shipment was possible. The objective was to minimize the total transportation cost by determining the best fleet dispatching and consolidation plans. A vehicle routing problem considering split delivery and time window for suppliers/cross-docks/retailers was presented by \cite{wang2017two}, where Less-than-TrackLoad (LTL) was allowed. Two extended approaches based on the simulated annealing and tabu search algorithms were proposed to minimize the total transportation cost.
In recent years, vehicle scheduling problem and its extensions have received more attention from the researchers. In this way, \cite{schwerdfeger2018just} studied an outbound truck scheduling problem in a specific type of cross-dock network, namely one-to-one. They seek to minimize the number of trucks for the parts deliveries regarding to the JIT intervals demanded by the plant. A heuristic algorithm based on the decomposition of the formulation is presented and compared with binary search and an exact method. \cite{Boysen2010} presented a truck scheduling problem without temporary storage in a cross-dock of food industry distribution. They studied three objectives separately where they developed a dynamic programming and a simulated annealing algorithm to solve this problem. \cite{Chen2009} aimed to minimize the makespan in a cross-dock system. They considered the cross-dock scheduling as a two machines flow-shop problem in which the variable loading/unloading times depended on the products (jobs). They proposed four heuristic algorithms containing two dynamic algorithms to solve the problem.
Moreover, many studies have been carried out on the combinatorial optimization problems in cross-docking. Dock-door assignment combined with VRP and vehicle scheduling are the frequent combinations in this way. For example, \cite{ENDERER201730} presented a combinatorial problem by integrating dock-door assignment and vehicle routing (DAVRP). The assignment of suppliers to the inbound doors and the routing from the outbound doors to the destinations were targeted with aiming to minimize the handling and transport cost. For the combination of vehicle scheduling with door assignment, \cite{Wisittipanich2017} presented a truck scheduling problem, comprising truck sequencing and door assignment simultaneously. The objective of their problem was to minimize the total operational time (makespan). They proposed a novel variant of Particle Swarm Optimization (PSO), namely GLNPSO to solve this problem. \cite{Rijal2019} presented another combinatorial problem in which the truck scheduling and dock-door assignment are solved simultaneously. They analyzed the mixed-mode dock-doors in a U-shape cross-docking problem in which a soft time windows for the truck departures and a temporary storage are considered. The objective was to minimize a combination of the costs, containing: inside traveling, storage and tardiness penalty. An ALNS algorithm was proposed to solve this problem. In another research, \cite{Kusolpuchong2019} studied a truck scheduling and dock-door assignment problem in a temporary storage allowed cross-dock. They proposed a genetic algorithm to optimize three objectives as travel distance of material handling, makespan and maximum storage level. The heterogeneous trucks (three sizes), LTL and the loading/unloading times were taken into account in their problem.
In various vehicle scheduling studies, a time windows was considered to the departure time of the outbound trucks. For example, in the truck scheduling model presented by \cite{Serrano2017}, a predefined time windows for the arrival trucks is considered as a soft constraint and the objective was to minimize the penalty of the violating these contracted times. A hard time windows for the outbound trucks was taken into account in the truck scheduling door assignment problem presented by \cite{Molavi2018}. Their model proposes the schedules for both inbound and outbound trucks and minimizes the transport cost and the penalty of lateness of delayed shipments by employing a hybrid heuristic algorithm composed of GA and Variable Neighborhood Search (VNS).
The simultaneous consideration of multi-period planning and departure time windows have been studied in a truck scheduling research where \cite{Ghomi2020} proposed an inbound truck scheduling model on a multi-period horizon of time with a temporary storage and variable arrival times for the inbound trucks. A fixed time windows was predefined for the departure of outbound trucks. In another research, \cite{Shahramfard2019} presented a multi-period truck scheduling and door assignment in a cross-dock with regarding to the material handling constraints and assignment of forklifts to the dock-doors. The time window constraints for both inbound and outbound trucks are considered in their schedules calculations. The waiting time for the outbound trucks together with the delayed outbound trucks and holding cost are considered as the objectives and four multi-objective meta-heuristic algorithms are proposed and compared.
The product-truck allocation is another decision problem in the cross-docking domain which arises where the product interchangeability is possible. That means each outbound truck has a list of products and its can be provided by every inbound truck (supplier) under a post-distribution strategy. \cite{tang2010pre} and \cite{yan2009pre} compared the cross-docking operations under post-distribution and pre-distribution considerations. Only a few papers have studied the product-truck allocation (post-distribution considerations) in their truck scheduling. The studies presented by \cite{tootkaleh2016cross}, \cite{assadi2016differential}, \cite{liao2013simultaneous} and \cite{lee2012genetic} considered post-distribution issues in their vehicle planning problems.
In this paper, a post distribution truck scheduling problem is presented in which the load and the destination of the outbound trucks must be determined (product-truck-destination allocation) and a specific type of time windows (just-in-time) is imposed on the customer demands rather than on the trucks. In order to illustrate the contributions of this study, a comparison with most relevant papers is carried out in table \ref{tab:literature}. All of the mentioned papers incorporated an assignment problem in their post-distribution truck scheduling problems (whether product to truck or product to destination); whereas in this study two interrelated assignments are integrated and synchronized.
\begin{table}
\caption{\label{tab:literature}Related literature on post-distribution truck scheduling in cross-docking}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{ccccccccc}
\hline
Paper & Product allocation & Arrival time & Departure condition & Split d. & LTL & T. storage & Objective & Solution approach\tabularnewline
\hline
\cite{assadi2016differential} & Product-Truck & Both trucks & JIT-Trucks & No & No & Infinite & Earliness-Tardiness & DE-SA\tabularnewline
\cite{bellanger2013three} & Product-Dest. & Zero & No & No & No & Infinite & Makespan & B\&B\tabularnewline
\cite{ladier2018crossdock} & Product-Dest. & Both trucks & TW-Destinations & Yes & No & Infinite & Inventory Level & TS-IP Decomposition\tabularnewline
\cite{larbi2011scheduling} & Product-Dest. & Inbound & No & Yes & Yes & Infinite & Handling cost & Stochastic heuristic\tabularnewline
\cite{lee2012genetic} & Product-Truck & Zero & No & Yes & Yes & Infinite & Throughput & GA\tabularnewline
\cite{liao2020integrated} & Product-Truck & Zero & TW-Destinations & No & No & Infinite & Earliness-Tardiness & ILS-Greedy search\tabularnewline
\cite{liao2013simultaneous} & Product-Truck & Zero & JIT-Trucks & No & No & Infinite & Tardiness & ACO-Hybrid DE\tabularnewline
\cite{nasiri2018incorporating} & Product-Truck & Zero & TW-Trucks & Yes & No & Infinite & Delivery costs & TSSA\tabularnewline
\cite{Serrano2017} & Product-Dest. & Inbound & No & No & No & Limited & Arrival times & MILP-CPLEX\tabularnewline
\cite{shahmardan2020truck} & Product-Truck & Zero & No & No & Yes & infinite & Makespan & LR-SA\tabularnewline
\cite{shakeri2012robust} & Product-Truck & Zero & Non & No & No & Limited & Makespan & 2phase heuristic\tabularnewline
\cite{tootkaleh2016cross} & Product-Truck & Zero & TW-Trucks & No & No & Infinite & Inventory cost & Constructive heuristic\tabularnewline
\cite{Wisittipanich2017} & Product-Truck & Zero & No & No & No & Infinite & Makespan & Adaptive PSO\tabularnewline
\textbf{Our paper} & \textbf{Product-Truck-Dest.} & \textbf{Inbound} & \textbf{JIT-Demands} & \textbf{Yes} & \textbf{Yes} & \textbf{Limited} & \textbf{Load waiting} & \textbf{Hybrid matheuristic}\tabularnewline
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\section{Problem description}\label{problem description}
A set of inbound trucks containing the pallets of different products come from the suppliers and arrive to the cross-dock in the estimated times. Under a JIT strategy, each customer needs a number of these products in a specific time windows. A set of outbound trucks must carry the needed products to the customers during the relevant intervals of time. The number of pallets to be transported from an inbound truck (supplier) to an outbound truck (destination) are not predefined; Therefore, the outbound trucks can be fulfilled through any arrived pallets. In other words, the product interchangeability is allowed. The objective is to minimize the total storage time of the pallets inside the cross-dock. The best solution for a pallet is to be unloaded from its related inbound truck and loaded to its allocated outbound truck at the same period. For that purpose, the inbound and outbound trucks must be docked simultaneously. In this case, the waiting time for the pallet will be zero.
The time windows of the customer demands must be respected. Note that, the time windows are assigned to the products rather than to the outbound trucks. Therefore, all of the pallets assigned to an outbound truck have the same due date, which determines the leaving time of that truck.
Furthermore, the outbound trucks can leave the cross-dock with less-than-truckload (LTL) if there is no more pallet available. In this study, the split shipment is possible, where a destination can be served by different trucks. The problem is simplified by some assumptions:
\begin{itemize}
\item The containers which are loaded on a specific outbound truck belong to the same destination.
\item The truck and the pallets are homogeneous.
\item The un/loading and transshipment times inside the cross-dock are independent of the load size and product type and is the same for all trucks and products.
\item The number of outbound trucks is limited by the constraints, and the transportation time between the cross-dock and every destination is ignored.
\item A working day is composed of a number of time periods.
\item The time during which a truck is docked to the cross-dock (loading/unloading time) in one period.
\subsubsection*{Interchangeability conditions:}
\item The load of the outbound trucks can be provided by every inbound trucks (suppliers).
\item Each outbound truck can be attributed to any destination
\end{itemize}
The problem is to determine two schedules for the inbound and outbound trucks, and the number of pallets to be transported from each inbound truck to every outbound truck. In other words, in addition to the truck scheduling, the assignment of the suppliers to the destinations for each product type on every outbound trucks is determined. It signifies two synchronized allocations. The allocation of the received products to the outbound trucks and the allocation of the outbound trucks to the destinations.
Figure \ref{Fig-Problem} provides an example for the planning of two periods. Five product types are shown by different colors ($P_{1}$, $P_{2}$,..., $P_{5}$). Two inbound trucks ($i_{1}$ and $i_{3}$) arrive at the first period but they are not docked to the cross-dock. There are three destinations ($D_{1}$, $D_{2}$, $D_{3}$), and each one needs a list of the products under a JIT strategy. In the first period no demand must be delivered; But in the second period, 8 pallets of the first product and 7 pallets of the third product must be delivered to the first destination and 9 pallets of the second product must be shipped to the second destination. In order to meet these demands, two inbound trucks ($i_{1}$ and $i_{2}$) are docked to the inbound doors and unloaded. At the same time, two outbound trucks ($j_{1}$ and $j_{3}$) are docked to the outbound side and pooled by the received pallets. The first outbound truck contains 9 pallets of product 2 and that is assigned to destination 2. The transshipment between every pair of trucks are shown in the figure for this small example.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{ProblemDescription2.pdf}
\caption{Problem description example}
\label{Fig-Problem}
\end{figure}
The best solution for a pallet is to be unloaded from its related inbound truck, and loaded to its allocated outbound truck at the same period. For this purpose, both inbound and outbound trucks must be docked simultaneously. In this case, the waiting time of the pallet will be zero.
An integrated mathematical model, which involves truck scheduling and product-truck-destination allocation, is presented for this post-distribution cross-docking problem.
\subsection{Mathematical model}\label{math model}
A mixed-integer mathematical model is proposed for the aforementioned problem. The objective is to minimize the waiting time of the pallets inside the cross-dock. It means the time interval length between unloading a pallet from inbound truck and loading it on the outbound truck. The parameters, variables, and problem formulation are presented as follows.
\subsubsection{Parameters and variables}
The sets and the parameters which are used to formulate the problem mathematically are presented in table \ref{table param}. The integer and binary variables, which are employed to model the problem, are listed in table \ref{table var}.
\begin{flushright}
\begin{table}[h]
\caption{Parameter and set notation}
\label{table param}
\small
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{ll}
\hline
\textbf{Sets} & \tabularnewline
$I$ & Set of inbound trucks; $I=\{1,2,...,m\}$;\tabularnewline
$J$ & Set of outbound trucks; $J=\{1,2,...,n\}$;\tabularnewline
$P$ & Set of product types; $P=\{1,2,...,k\}$;\tabularnewline
$D$ & Set of destinations; $D=\{1,2,...,f\}$;\tabularnewline
$T$ & Set of time periods; $T=\{1,2,...,r\}$;\tabularnewline
\textbf{Parameters} & \tabularnewline
$m$ & Total number of inbound trucks; \tabularnewline
$n$ & Total number of outbound trucks; \tabularnewline
$k$ & Total number of product types; \tabularnewline
$f$ & Total number of destinations (customers); \tabularnewline
$r$ & Total number of time periods during a working day; \tabularnewline
$ID$ & Total number of inbound doors; \tabularnewline
$OD$ & Total number of outbound doors;\tabularnewline
$C$ & Capacity of the trucks; (homogeneous trucks);\tabularnewline
$E_{i}$ & Arriving time of the inbound trucks;\tabularnewline
$L_{ip}$ & Total number of pallets of product $p$ in inbound truck $i$;\tabularnewline
$R_{pdt}$ & Number of pallets $p$ needed by destination $d$ in period $t$ (JIT customer demands);\tabularnewline
$PC$ & Penalty cost for each non-delivered pallet of a customer order;\tabularnewline
\multirow{1}{*}{$RB_{pdt}$} & $\begin{cases}
=1,\text{ if customer d needs product p in period t;}\\
=0,\text{ otherwise.}
\end{cases}$\tabularnewline
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\par\end{flushright}
\begin{flushleft}
\begin{table}[h]
\caption{Decision variables}
\label{table var}
\small
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{ll}
\hline
$S_{ijpd}$ & Number of pallets $p$ delivered to destination $d$ by outbound truck $j$ which is supplied by inbound truck $i$\tabularnewline
\multirow{1}{*}{$SB_{ijt}$} & $\begin{cases}
=1,\text{ if there is a shipment from inbound truck i to outbound truck j at time period t; }\\
=0,\text{ otherwise;}
\end{cases}$\tabularnewline
\multirow{1}{*}{$y_{jt}$} & $\begin{cases}
=1\text{ if outbound truck \ensuremath{j} is docked to the cross-dock (is loaded) in period \ensuremath{t}; }\\
=0,\text{ otherwise;}
\end{cases}$\tabularnewline
\multirow{1}{*}{$h_{it}$} & $\begin{cases}
=1\text{ if inbound truck \ensuremath{i} is docked to the cross-dock (is unloaded) in period \ensuremath{t}; }\\
=0,\text{ otherwise;}
\end{cases}$\tabularnewline
\multirow{1}{*}{$q_{jd}$} & $\begin{cases}
=1\text{ if outbound truck \ensuremath{j} is allocated to destination \ensuremath{d}; }\\
=0,\text{ otherwise;}
\end{cases}$\tabularnewline
\multirow{1}{*}{$WB_{jpd}$} & $\begin{cases}
=1,\text{ if outbound truck \ensuremath{j} carries pallets \ensuremath{p} to destination \ensuremath{d};}\\
=0,\text{ otherwise;}
\end{cases}$\tabularnewline
$V_{pd}$ & Number of pallets $p$ which are requested and not supplied to destination $d$; (uncovered demands)\tabularnewline
$St_{t}$ & Number of pallets remained inside the cross-dock at the end of period $t$;\tabularnewline
\hline
\end{tabular}
\end{adjustbox}
\end{table}
\end{flushleft}
\subsubsection{Problem formulation}
The objective function and the constraints of the studied problem is formulated as follows.
\begin{equation}
Minimize\:Z=\sum_{i\in I}\sum_{j\in J}(((\sum_{t\in T}t\times y_{jt})\times (\sum_{t\in T}SB_{ijt}))-((\sum_{t\in T}t\times h_{it})\times (\sum_{t\in T}SB_{ijt})))+PC\times\sum_{p\in P}\sum_{d\in D}V_{pd}\label{eq:1}
\end{equation}
$Subject\:to:$
\begin{align}
&\sum_{d\in D}\sum_{j\in J}S_{ijpd}\leq L_{ip} \qquad &\forall i\in I,\:p\in P\label{eq:2}\\
&\sum_{i\in I}\sum_{p\in P}\sum_{d\in D}S_{ijpd}\leq C \qquad &\forall j\in J\label{eq:4}\\
&\sum_{i\in I}\sum_{j\in J}S_{ijpd}+V_{pd}=\sum_{t\in T}R_{pdt}\qquad &\forall d\in D,\:p\in P\label{eq:5}\\
&(\sum_{p\in P}\sum_{d\in D}S_{ijpd})/\sum_{p\in P}L_{ip}\leq\sum_{t\in T}SB_{ijt}\leq\sum_{p\in P}\sum_{d\in D}S_{ijpd} \qquad &\forall i\in I,\:j\in J\label{eq:6}\\
&\sum_{t\in T}SB_{ijt}\leq1\qquad &\forall i\in I,\:j\in J\label{eq:8}\\
&\sum_{i\in I}h_{it}\leq ID\qquad &\forall t\in T\label{eq:9}\\
&\sum_{t\in T}h_{it}=1\qquad &\forall i\in I\label{eq:10}\\
&\sum_{j\in J}y_{jt}\leq OD\qquad &\forall t\in T\label{eq:12}\\
&\sum_{t\in T}y_{jt}\leq1\qquad &\forall j\in J\label{eq:13}\\
&\sum_{t\in T}t\times h_{it}\leq (\sum_{t\in T}t\times SB_{ijt})+(r\times(1-\sum_{t\in T}SB_{ijt}))\qquad &\forall i\in I,\:j\in J\label{eq:17}\\
&\sum_{t\in T}t\times SB_{ijt}\leq \sum_{t\in T}t\times y_{jt}\qquad &\forall i\in I,\:j\in J\label{eq:18}\\
&(\sum_{i\in I}S_{ijpd})/C \leq WB_{jpd}\leq \sum_{i\in I}S_{ijpd}\qquad &\forall j\in J,\:p\in P,\:d\in D\label{eq:20}\\
&\sum_{p\in P}WB_{jpd}/r\leq q_{jd}\qquad &\forall j\in J,\:d\in D\label{eq:30}\\
&\sum_{d\in D}q_{jd}\leq1\qquad &\forall j\in J\label{eq:31}\\
&(\sum_{t\in T}t\times y_{jt})\times WB_{jpd}=(\sum_{t\in T}t\times RB_{pdt})\times WB_{jpd}\qquad &\forall j\in J,\:p\in P,\:d\in D\label{eq:22}\\
&Capacity \: constraints:& \nonumber \\
&St_{t}=St_{t-1}+(\sum_{i\in I}(h_{it}\times\sum_{p\in P}L_{ip})-\sum_{j\in J}(y_{jt}\times\sum_{p\in P}\sum_{d\in D}\sum_{i\in I}S_{ijpd}))\qquad &\forall t\in T \; | \; St_{0}=0\
\label{eq:36}\\
&St_{t}\leq ID\times C\qquad &\forall t\in T\label{eq:37}
\end{align}
The objective function (equation \eqref{eq:1}) minimizes the waiting time of the pallets which must be shipped from inbound trucks to the outbound trucks inside the cross-dock, while minimizing the undelivered demands by adding a penalty cost per pallet. The formulation is nonlinear which must be linearized. For this purpose, the equation is divided to three parts among which two first parts are nonlinear.
The first part ($(\sum_{t\in T}t\times y_{jt})\times (\sum_{t\in T}SB_{ijt})$) can be defined as a set of integer variables ($qy_{ij}$), indicating the docking period of the outbound truck $j$, if there is any shipment from inbound truck $i$ to outbound truck $j$. If there is no shipment between two trucks, $qy_{ij}$ equals zero. Similarly for the inbound trucks, the second part of the objective function ($(\sum_{t\in T}t\times h_{it})\times (\sum_{t\in T}SB_{ijt})$) indicates the docking period of inbound truck $i$, if this truck contains the products to be shipped to outbound truck $j$. This part of the equation can be defined as another set of integer variables ($qh_{ij}$).
Aforementioned nonlinear parts of the objective function ($qy_{ij}=\sum_{t\in T}t\times y_{jt})\times (\sum_{t\in T}SB_{ijt})$ and $qh_{ij}=\sum_{t\in T}t\times h_{it})\times (\sum_{t\in T}SB_{ijt})$) are linearized by using two sets of inequalities \eqref{eq:28} and \eqref{eq:26}, respectively.
\begin{equation}
\begin{cases}
qy_{ij}\leq r\times (\sum_{t\in T}SB_{ijt})\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times y_{jt})-qy_{ij}\leq r\times(1-(\sum_{t\in T}SB_{ijt}))\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times y_{jt})-qy_{ij}\geq E_{i}-(1-(\sum_{t\in T}SB_{ijt}))\qquad & \forall i\in I,\:j\in J
\end{cases}\label{eq:28}
\end{equation}
\begin{equation}
\begin{cases}
qh_{ij}\leq r\times (\sum_{t\in T}SB_{ijt})\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times h_{it})-qh_{ij}\leq r\times(1-(\sum_{t\in T}SB_{ijt}))\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times h_{it})-qh_{ij}\geq E_{i}-(1-(\sum_{t\in T}SB_{ijt}))\qquad & \forall i\in I,\:j\in J
\end{cases}\label{eq:26}
\end{equation}
Constraints \eqref{eq:2} guarantee that the sum of the products delivered to the customers does not exceed the products carried to the cross-dock. Constraints \eqref{eq:4} limit the load of each outbound truck by its capacity. Constraints \eqref{eq:5} express
that the load delivered to a customer equals the load that customer requires, except in cases in which supplying the whole order is not feasible. Inequalities \eqref{eq:6} define a set of binary decision variables ($SB_{ijt}$) indicating that if $\sum_{p\in P}\sum_{d\in D}S_{ijpd}>0$, it means there is a shipment from inbound truck $i$ to outbound truck $j$, hence $\sum_{t\in T}SB_{ijt}=1$, whereas if $\sum_{p\in P}\sum_{d\in D}S_{ijpd}=0$, hence $\sum_{t\in T}SB_{ijt}=0$. Constraints \eqref{eq:8} ensure that the shipment from an inbound truck to an outbound truck (if there is any) is occurred in one period of time.
Constraints \eqref{eq:9} guarantee that the number of inbound trucks which are docked simultaneously does not exceed the number of inbound doors. Equation \eqref{eq:10} signifies that all of the inbound trucks must be docked during a working-day. Similarly, constraints \eqref{eq:12} limit the number of outbound trucks which are docked at each period by the number of outbound doors. Constraints \eqref{eq:13} show that it is not necessary that all of the outbound trucks are docked and they are used if needed. Constraints \eqref{eq:17} guarantee that, if there is any shipment from inbound truck $i$ to outbound truck $j$, this shipment occurs after docking time of the inbound truck. Constraints \eqref{eq:18} guarantee that all of the shipments to an outbound truck is occurred before or during the docking period of the outbound truck.
Constraints \eqref{eq:20} define another set of binary variables showing whether outbound truck $j$ transports product $p$ to destination $d$, or not. Constraints \eqref{eq:30} ensure that an outbound truck can transport the pallets only to its attributed destination. Constraints \eqref{eq:31} limit the outbound trucks to visit only one destination.
Under a JIT strategy, constraints \eqref{eq:22} indicate that an outbound truck is pooled (docked) in the period, which is imposed by the customer order. The right side of this equation is non-linear which can be defined as a set of variables ($WY_{jpd}$) announcing the docking period of outbound truck $j$, if this truck transports product $p$ to destination $d$, otherwise, it equals zero ($WY_{jpd}=(\sum_{t\in T}t\times y_{jt})\times WB_{jpd}$). This non-linear equation is linearized by employing the following inequalities:
\begin{equation}
\begin{cases}
WY_{jpd}\leq r\times WB_{jpd}\qquad & \forall j\in J,\:p\in P,\:d\in D\\
(\sum_{t\in T}t\times y_{jt})-WY_{jpd}\leq r\times(1-WB_{jpd})\qquad & \forall j\in J,\:p\in P,\:d\in D\\
(\sum_{t\in T}t\times y_{jt})-WY_{jpd}\geq0\qquad & \forall j\in J,\:p\in P,\:d\in D
\end{cases}\label{eq:23}
\end{equation}
\subsubsection*{Capacity constraints:}
At the end of each period, the total number of pallets remained in the temporary storage is obtained by equation \eqref{eq:36}. The temporary storage in period $t$ is calculated by adding the total number of pallets unloaded from the inbound trucks, to the temporary storage of the precedent period and subtracting the total number of pallets loaded onto the outbound trucks. In equation \eqref{eq:36}, the third part of the right side of the formulation ($\sum_{j\in J}(y_{jt}\times\sum_{p\in P}\sum_{d\in D}\sum_{i\in I}S_{ijpd})$) is nonlinear. For the linearization, a set of integer variables ($LJ_{jt}$) are defined which are equivalent to this part of the equation ($LJ_{jt}=y_{jt}\times\sum_{p\in P}\sum_{d\in D}\sum_{i\in I}S_{ijpd}$). $LJ_{jt}$ and present the total number of pallets loaded onto outbound truck $j$ in period $t$. It's related nonlinear formulation is linearized by adding the following inequalities:
\begin{equation}
\begin{cases}
LJ_{jt}\leq C\times y_{jt}\qquad & \forall j\in J,\:t\in T\\
\sum_{p\in P}\sum_{d\in D}\sum_{i\in I}S_{ijpd}-LJ_{jt}\leq C\times(1-y_{jt})\qquad & \forall j\in J,\:t\in T\\
\sum_{p\in P}\sum_{d\in D}\sum_{i\in I}S_{ijpd}-LJ_{jt}\geq0\qquad & \forall j\in J,\:t\in T
\end{cases}\label{eq:33}
\end{equation}
Constraints \eqref{eq:37} limit the temporary storage level that can not exceed a constant value which is calculated by multiplying the number of inbound doors by the capacity of a truck.
After formalizing the problem mathematically, a solution procedure is proposed to this problem. The steps and the algorithms of the solution procedure are explained in the next section.
\section{Solution method}\label{solution approaches}
In order to solve the studied problem, firstly, a set of symmetry breaking constraints are added to the mathematical model to make that more efficient. At the second step, a two-step matheuristic method is presented which is based on the decomposition of the formulation. As the first step of the matheuristic is still complex, a adaptive heuristic is proposed as an alternative to solve this step of matheuristic. Hence, a hybrid matheuristic, composed of an adaptive heuristic (for the first step) and a mathematical model for the second step, is presented to solve the studied combinatorial optimization problem.
\subsection{Symmetry breaking}
In any solution of the above-mentioned mathematical model, three attributes are determined for each outbound truck as: the load, the docking period and the destination. Solution 1 is a part of an entire solution of the problem in which $Y_{j}$ and $D_{j}$ imply respectively the docking period and the assigned destination of outbound truck $j$.
$Solution\:1\::\:\begin{cases}
Y_{1}=A_{1},...,Y_{i}=A_{i},...,Y_{j}=A_{j},...,Y_{n}=A_{n};\\
D_{1}=B_{1},...,D_{i}=B_{i},...,D_{j}=B_{j},...,D_{n}=B_{n};
\end{cases}$
Since the destinations and the loads of the outbound trucks are not predefined, thus another solution could be easily produced by simultaneously substituting the docking periods and the destinations of two trucks. As an example, solution 2 shows a part of a solution derived from solution 1 and obtained by replacing the docking period and the destination of truck $i$ by the ones of truck $j$.
$Solution\:2\::\:\begin{cases}
Y_{1}=A_{1},...,Y_{i}=A_{j},...,Y_{j}=A_{i},...,Y_{n}=A_{n};\\
D_{1}=B_{1},...,D_{i}=B_{j},...,D_{j}=B_{i},...,D_{n}=B_{n};
\end{cases}$
Because the outbound trucks are homogeneous, i.e. they have the same capacity, cost, entering time, availability and speed, ..., solution 2 (and any other solution obtained in the same way) results in the same objective value as solution 1. Hence, in each solution node of the problem, a group of symmetric solutions is raised that causes an exceedingly vast branch and bound three. Consequently, by keeping the values of other variables, every permutation of the outbound trucks ($n!$) over theirs docking periods and destinations simultaneously, gives a symmetric solution.
In order to avoid or decrease the symmetric solutions, appropriate inequalities called ``symmetry breaking constraints'' could be imposed to the model. In this paper, two categories of the symmetry breaking inequalities are proposed to take away all or a large number of symmetric solutions. Considering an ordering strategy to assign the trucks to the cross-dock is the first issue which is taken into account to handle the symmetries. In this way, inequality \eqref{eq:SB1} is imposed to the problem, knowing that it does not touch the solutions nodes and the feasible space.
\begin{equation}
Y_{1}\leq Y_{2}\leq...\leq Y_{n} \label{eq:SB1}
\end{equation}
This inequality keeps only one or a small number of solutions at each node. It signifies that outbound truck $i$ is docked to the cross-dock before or at the same period as the outbound truck $j$, if $i<j\;(i,j\in J)$.
By employing these constraints, a large number of symmetric solutions are removed, but there still remains some sub-symmetries inside each planning period. For $n$ outbound trucks, $n!$ possible permutations signifies that in each node, there exists $n!$ ($\prod_{1\leq i\leq n}i$) different solutions; But by applying inequality \eqref{eq:SB1}, this number is reduced to : $n'_{1}!\times n'_{2}!\times...\times n'_{r}!$, in which, $n'_{t}$ implies the number of outbound trucks assigned to time period $t$ $(t\in T)$, such that $n'_{1}+n'_{2}+...+n'_{r}=n$.
\textbf{\textit{Lemma 1}} : $If\:(n'_{1}+n'_{2}+...+n'_{r}=n)$ $then:$ $(\prod_{1\leq i\leq n'_{1}}i)\times(\prod_{1\leq i\leq n'_{2}}i)\times...\times(\prod_{1\leq i\leq n'_{r}}i)\leq(\prod_{1\leq i\leq n}i)$.
\textbf{\textit{Lemma 2}} : $If\:(n>0)$ $then:(\prod_{1\leq i\leq n'_{1}}i)\times(\prod_{1\leq i\leq n'_{2}}i)\times...\times(\prod_{1\leq i\leq n'_{r}}i)>0$.
Two above-mentioned lemmas signify that the number of symmetric solutions is decreased while keeping at least one solution in each node, i.e. these inequalities do not touch the nodes and feasible space of the problem.
The second category of Symmetry Breaking Constraints (SBC) is to avoid the sub-symmetries inside each planning period and it concerns the assignment of the trucks to the destinations. Even by taking into account the inequality \eqref{eq:SB1}, each solution node still contains multiple sub-symmetry groups. Inside a planning period, any permutation of the assigned trucks $(J'_{t}\subset J)$ over their destinations results in the same solution node and objective value. Similar to the first category of SBC, an ordering system is defined to keep one or a small number of solutions of each node and remove the others. This ordering inequality is defined as follows:
\begin{equation}
D_{i}<D_{j}\qquad\forall t\in T\;and\;i,j\in\{J'_{t}\mid i<j\};\label{eq:SB2}
\end{equation}
This inequality signifies that among the trucks which are docked simultaneously, the truck with smaller number is assigned to the destination with smaller number.
Under mixed-integer linear programming conditions, two mentioned SB inequalities are formulated as follows:
\begin{equation}
Y_{j}-Y_{j-1}\geq0\qquad\forall j\in\{J|\;j\geq1\}\label{eq:38}
\end{equation}
\begin{equation}
(\sum_{d\in D}d\times q_{jd})\times y_{jt}\leq((\sum_{d\in D}d\times q_{gd})\times y_{gt})+f\times(1-y_{gt})\qquad\forall j,g\in \{J|\;j<g\},\:t\in T\label{eq:42}
\end{equation}
Constraints \eqref{eq:38} guarantee that the outbound trucks are assigned to the cross-dock by increasing order of their numbers. Constraints \eqref{eq:42} define the assignment of the trucks to the destinations for the trucks which are docked at the same period. By these constraints, the smaller truck number is assigned to the smaller destination number. The left side of this inequality is nonlinear which must be linearized. For this purpose, a set of integer variables ($DT_{jt}$) is defined which is equivalent to the left side of inequality \eqref{eq:42} : $DT_{jt}=(\sum_{d\in D}d\times q_{jd})\times y_{jt}$. This equation is linearized by using inequalities \eqref{eq:41} as the follows:
\begin{equation}
\begin{cases}
DT_{jt}\leq f\times y_{jt}\qquad & \forall j\in J,\:t\in T\\
\sum_{d\in D}d\times q_{jd}-DT_{jt}\leq f\times(1-y_{jt})\qquad & \forall j\in J,\:t\in T\\
\sum_{d\in D}d\times q_{jd}-DT_{jt}\geq0\qquad & \forall j\in J,\:t\in T
\end{cases}\label{eq:41}
\end{equation}
In the same manner, the first part of the right side of inequality \eqref{eq:42} ($(\sum_{d\in D}d\times q_{gd})\times y_{gt}$) is linearized by considering the variables $DT_{gt}$.
By adding SBCs (equations \eqref{eq:38} and \eqref{eq:42}), the efficiency of the model is significantly improved; But in order to solve the real size problem instances, the more efficient solution approaches are needed. Regarding to the complexity of the model, the matheuristic method seems to be a suitable method to solve the studied problem. Hence, a two-step matheuristic algorithm is proposed and explained as follows.
\subsection{Matheuristic method}
A matheuristic method based on the decomposition of the formulation has been applied to solve the studied combinatorial optimization problem. By this method, the complex problem is decomposed to two sub-problems where the first sub-problem is solved and it's decision variables are used as the parameters to the second sub-problem. The complexity of the problem has been significantly decreased by using this decomposition. For the studied problem, the scheduling of the outbound trucks, the assignment of the outbound trucks to the destinations and the load (number of pallets of each product) of the outbound trucks is solved at the first step. Actually, all the variables related to the outbound of cross-dock and the destinations are considered in the first sub-problem.
In the second step, load and dock time of the outbound trucks are used as the parameters to schedule the inbound trucks and to determine the number of pallets that must be shipped from the inbound trucks to each outbound truck. Actually, the assignment of the product/supplier to the trucks, and accordingly, to the destinations are determined in this step. All constraints related to the inbound part of cross-dock, the load of the inbound trucks and the shipment times and amounts from the inbound trucks to the outbound trucks are considered in this sub-problem.
\subsubsection{Step 1 : Outbound scheduling and destination assignment}
Most of the variables, which are used in the first sub-problem, are the same as the variables of the main problem defined in the previous section. Moreover, two new sets of variables are applied to build the connection between the first and the second models. These variables are defined as follows :
\begin{itemize}
\item $x_{jp}$ : Number of pallets of product $p$ which are placed in outbound truck $j$.
\item $U_{jpdt}$ : Number of pallets of product $p$ which are placed in outbound truck $j$, if this truck is assigned to destination $d$ and is docked (loaded) at the cross-dock in period $t$.
\end{itemize}
Two mentioned sets of variables transform the necessary information from the first model to solve the second model in an optimal manner.
In this sub-problem, the constraints related to the demands, outbound trucks and doors, product and truck assignment (assignment of the products to the outbound trucks and assignment of these trucks to the destinations) are considered. The objective of this step is to minimize the outstanding demands.
The mathematical model of the first sub-problem is presented as follows :
\begin{equation}
Minimize\:Z1=\sum_{p\in P}\sum_{d\in D}\sum_{t\in T}V_{pdt},\label{eq:301}
\end{equation}
$Subject\:to:$
Assignment of the products and outbound trucks :
\begin{equation}
W_{jpd}= x_{jp}\times q_{jd}\qquad\forall j\in J,\:p\in P,\:d\in D\label{eq:302}
\end{equation}
\begin{equation}
\sum_{d\in D}q_{jd}\leq1\qquad\forall j\in J\label{eq:131}
\end{equation}
Schedule of the outbound trucks :
\begin{equation}
U_{jpdt}=W{jpd}\times y_{jt}\qquad\forall j\in J,\:p\in P,\:d\in D,\:t\in T\label{eq:303}
\end{equation}
\begin{equation}
\sum_{t\in T}y_{jt}\leq1\qquad\forall j\in J\label{eq:113}
\end{equation}
\begin{equation}
\sum_{t\in T}y_{jt}=\sum_{d\in D}q_{jd}\qquad\forall j\in J\label{eq:305}
\end{equation}
Demand/JIT constraints :
\begin{equation}
\sum_{j\in J}U_{jpdt}+V_{pdt}=R_{pdt}\qquad\forall p\in P,\:d\in D,\:t\in T\label{eq:105}
\end{equation}
Outbound doors/Trucks constraints :
\begin{equation}
\sum_{j\in J}y_{jt}\leq OD\qquad\forall t\in T\label{eq:112}
\end{equation}
\begin{equation}
\sum_{p\in P}x_{jp}\leq C\qquad\forall j\in J\label{eq:104}
\end{equation}
\begin{equation}
\sum_{t\in T}\sum_{d\in D}U_{jpdt}=x_{jp}\qquad\forall j\in J,\:p\in P\label{eq:304}
\end{equation}
The objective function of the model (equation \eqref{eq:301}) minimizes the demands which are not satisfied punctually under JIT strategy. That means the total number of pallets of different products which are not delivered at the demanded time to the costumers. Constraints \eqref{eq:302} and \eqref{eq:131} are related to the assignment of the pallets to the outbound trucks and also assignment of the outbound trucks to the destinations. Constraints \eqref{eq:302} define the destination and the load of each trucks and constraints \eqref{eq:131} guarantee that each truck is assigned to maximum one destination.
Constraints \eqref{eq:303}, \eqref{eq:113} and \eqref{eq:305} are concerned with the schedule of outbound trucks. Constraints \eqref{eq:303} define the time at which each truck leaves the cross-dock. Constraints \eqref{eq:113} restrain the outbound truck to be dock to the cross-dock in one and only one period. Constraints \eqref{eq:305} signify that if a truck is assigned to a destination, it must be docked to the cross-dock (must be loaded).
The next category of the constraints are related to the demands under a JIT strategy. Constraints \eqref{eq:105} imply that the outbound trucks ship the pallets to the destinations at the times and in the amounts which are precised and predefined by the customers (destinations); Except for the demands or a part of demands which are not satisfied ($V_{pdt}$).
Three next sets of constraints (\eqref{eq:112}, \eqref{eq:104} and \eqref{eq:304}) are to define the outbound doors and outbound trucks restrictions. Constraints \eqref{eq:112} signify that the number of trucks docked to the cross-dock at the same times must not exceed the number of outbound doors. Constraints \eqref{eq:104} limit the load (number of pallets) of each outbound truck by it's capacity. Constraints \eqref{eq:304} is to calculate the load (number and type of pallets) which is placed in each outbound truck.
By solving this step, the value of certain variables are obtained. $x_{jp}$, $q_{jd}$ and $y_{jt}$ are the results of the first model, that show the load, destination and docking period of the outbound trucks.
\subsubsection{Step 2 : inbound scheduling and load assignment}
The value of the variables which are obtained by solving the first model ($x_{jp}$, $q_{jd}$ and $y_{jt}$), are used as the parameters in this step (second sub-problem). The constraints related to the inbound trucks (supply constraints), inbound doors and shipment from inbound trucks to outbound trucks are considered in the second model.
Apart from the main variables, three other sets of variables are defined in this section in order to manipulate (confirm, adjust or modify) the solutions resulted from the first step and obtain the best value for the objective function. These variables are defined as follows :
\begin{itemize}
\item $A_{ijp}$ : (Decision variables) The final number of pallets of product $p$ shipped from inbound truck $i$ to outbound truck $j$
\item $B_{ip}$ : Number of pallets of product $p$ from inbound truck $i$ which are not shipped to any outbound truck.
\item $G_{jp}$ : Number of pallets of product $p$ which are assigned to outbound truck $j$ at the first step, but by considering the inbound constraints, they can not be arrived to the outbound truck.
\end{itemize}
Objective of this sub-problem is the same as the objective of the main problem. This step of the matheuristic is formalized mathematically as follows :
\begin{equation}
Minimize\:Z=\sum_{i\in I}\sum_{j\in J}(((\sum_{t\in T}t\times y_{jt})\times \sum_{t\in T}SB_{ijt})-((\sum_{t\in T}t\times h_{it})\times \sum_{t\in T}SB_{ijt}))+PC\times\sum_{i\in I}\sum_{p\in P}B_{ip}\label{eq:201}
\end{equation}
$Subject\:to:$
Supply constraints :
\begin{equation}
\sum_{j\in J}A_{ijp}+B_{ip}= L_{ip}\qquad\forall i\in I,\:p\in P\label{eq:202}
\end{equation}
\begin{equation}
\sum_{i\in I}A_{ijp}+G_{jp}=x_{jp}\qquad\forall j\in J,\:p\in P\label{eq:205}
\end{equation}
Inbound doors and trucks constraints
\begin{equation}
\sum_{i\in I}h_{it}\leq ID\qquad\forall t\in T\label{eq:209}
\end{equation}
\begin{equation}
\sum_{t\in T}h_{it}=1\qquad\forall i\in I\label{eq:210}
\end{equation}
Schedule of inbound trucks and shipments inside cross-dock :
\begin{equation}
(\sum_{p\in P}A_{ijp})/\sum_{p\in P}L_{ip}\leq\sum_{t\in T}SB_{ijt}\leq\sum_{p\in P}A_{ijp}\qquad\forall i\in I,\:j\in J\label{eq:206}
\end{equation}
\begin{equation}
\sum_{t\in T}SB_{ijt}\leq1\qquad\forall i\in I,\:j\in J\label{eq:208}
\end{equation}
\begin{equation}
\sum_{t\in T}t\times h_{it}\leq (\sum_{t\in T}t\times SB_{ijt})+(r\times(1-\sum_{t\in T}SB_{ijt}))\qquad\forall i\in I,\:j\in J\label{eq:217}
\end{equation}
\begin{equation}
\sum_{t\in T}t\times SB_{ijt}\leq \sum_{t\in T}t\times y_{jt}\qquad\forall i\in I,\:j\in J\label{eq:218}
\end{equation}
Capacity constraints :
\begin{equation}
St_{t}=St_{t-1}+(\sum_{j\in J}(y_{jt}\times\sum_{p\in P}\sum_{i\in I}A_{ijp})-\sum_{i\in I}(h_{it}\times\sum_{p\in P}L_{ip}))\qquad \forall t\in T \; | \; St_{0}=0\
\label{eq:236}
\end{equation}
\begin{equation}
St_{t}\leq ID\times C\qquad\forall t\in T\label{eq:237}
\end{equation}
Similar to the objective of the main model, the objective function of the model aims to minimize the overall non-satisfied demands by considering both inbound (supply) and outbound constraints while minimizing the waiting times for the unloaded pallets. It means, for each shipment between an inbound and an outbound truck, the time between unloading the pallets from inbound and loading the pallets into the outbound trucks is targeted to be minimized. A part of it's formulation is nonlinear $((\sum_{t\in T}t\times h_{it})\times \sum_{t\in T}SB_{ijt})$, which is linearized as follows :
\begin{equation}
\begin{cases}
qh_{ij}\leq r\times \sum_{t\in T}SB_{ijt}\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times h_{it})-qh_{ij}\leq r\times(1-\sum_{t\in T}SB_{ijt})\qquad & \forall i\in I,\:j\in J\\
(\sum_{t\in T}t\times h_{it})-qh_{ij}\geq E_{i}-(1-\sum_{t\in T}SB_{ijt})\qquad & \forall i\in I,\:j\in J
\end{cases}
\end{equation}
First category of the constraints (\eqref{eq:202} and \eqref{eq:205}) implies the supply restrictions of the cross-dock. Constraints \eqref{eq:202} guarantee that the shipments from an inbound trucks does not exceed the load of the truck. By these constraints, the number of pallets of each product which are not delivered to the outbound trucks (destinations) is calculated. Similarly, constraints \eqref{eq:205} signify that the number of pallets of each product which are shipped to each outbound truck does not exceed the number which is calculated in the first step ($x_{jp}$). The values of the variables $G_{jp}$ is calculated by these constraints. The positive value of $G$ means that it is not possible to response to all of the demands which are programmed in step one.
Similar to the main model, constraints \eqref{eq:209} and \eqref{eq:210} are related to the number of inbound doors and the time period at which the inbound trucks are docked. Next category of the constraints (\eqref{eq:206} to \eqref{eq:218}) are concerned with the scheduling of the inbound trucks and shipment times inside the cross-dock (between inbound and outbound trucks). Constraints \eqref{eq:206} imply that if there is a shipment from an inbound to an outbound truck, a time period must be assigned to this shipment. Constraints \eqref{eq:208} signify that the shipment from an inbound to an outbound truck, if there is any, is occurred in one and only one period of time. Constraints \eqref{eq:217} guarantee that the shipment from an inbound truck is occurred at the same period or after docking period of the truck. Similarly, constraints \eqref{eq:218} ensure that the shipments between two trucks is occurred at the same period or before docking period of the outbound truck. Finally, as presented in the main model, the capacity constraints are imposed to this step of the solution approach.
The decomposition of the main problem into two sub-problems decrease significantly the complexity of the problem. But a large number of variables and constraints still remain in the first sub-problem. For this reason, an adaptive heuristic method is proposed which could be employed instead of the mathematical model.
\subsection{Adaptive heuristic}
A heuristic algorithm is developed to solve the first sub-problem and provide the good feasible solutions. The obtained feasible solution is employed as a starting point for the solver to help that in converging more quickly to the optimal solution. The heuristic algorithm aims, on the one hand, to assign the outbound trucks to the destinations and periods, and on the other hand, to fill up the trucks according to the demands of their assigned destinations. The steps of the heuristic are summarized in \autoref{Fig-Heuristic} and presented as follows:
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Heuristic2.pdf}
\caption{Heuristic algorithm}
\label{Fig-Heuristic}
\end{figure}
\subsubsection{Pre-processing and calculations}
The first step of the algorithm concerns with the definition and the calculation of three auxiliary parameters which are obtained from the parameters of the main model. These parameters are needed for the next steps and they are
listed as follows:
\begin{itemize}
\item The total demand of every destination at each time period : $RP_{dt}=\sum_{p\in P}R_{pdt};\;\forall d\in D,\:t\in T;$
\item The number of trucks needed for every destination in each time period
: $ZP_{dt}=Ciel(RP_{dt}/C);\;\forall d\in D,\:t\in T;$
\item The total number of trucks needed in each time period : $ZT_{t}=\sum_{p\in P}ZP_{dt};\;\forall t\in T;$
\end{itemize}
In addition, the detailed demand of each destination ($R_{pdt}$) and the total number of outbound trucks $n$ are used to schedule and assign the outbound trucks in the next steps.
\subsubsection{Scheduling and assignment of outbound trucks to the destinations}
The algorithm starts by assigning trucks to the destinations that need products at the first time period in ascending order of the truck numbers and destination numbers. Considering that the trucks are homogeneous and they have no restriction or priority, they can be assigned to any destination at any time period. Note that, a truck could not be assigned to more than one destination, but a destination could need more that one truck in a time period. In the same manner, the trucks are attributed to the destinations over next periods. In this way, a number of trucks equal to ``$ZT_{t}$'', are programmed to be docked in period $t$. This step of the algorithm is terminated by assigning a needed number of trucks to the last required destination in the last planning period.
Actually, by having the total number of pallets which is needed by each destination at every time period ($RP_{dt}$), the number of trucks needed to deliver these pallets ($ZP_{dt}$) is determined. The first truck (group of trucks) is attributed to the first destination in the first period (if this destination needs a shipment). The next truck (group of trucks) is assigned to the next destination which needs a shipment in the same planning period and so on until the demand of the last destination. The same procedure is employed to assign the remained trucks to the destinations demands at the next periods, until any demand is attributed to a truck. At the end of this step, the destination of each outbound truck ($q_{jd}$), the time period in which the trucks are docked ($y_{jt}$) and the number of pallets placed in each outbound truck ($\sum_{p\in P}x_{jp}$) are determined, but the type of pallets (products) is unknown. The procedure of this step is summarized in algorithm \ref{Alg-S1}.
\begin{algorithm}
\caption{Scheduling and assignment of outbound trucks to the destinations}
\label{Alg-S1}
{\footnotesize{}}%
\begin{tabular}{lc}
{\footnotesize{}$j=0;$} & \tabularnewline
{\footnotesize{}$for(t\in T)\{for(d\in D)\{$ } & \tabularnewline
{\footnotesize{}$\quad BP_{dt}=RP_{dt};$} & {\footnotesize{}//initializing the auxiliary variables}\tabularnewline
{\footnotesize{}$\quad while(BP_{dt}>0)\{$} & \tabularnewline
{\footnotesize{}$\qquad y_{jt}=1;\;d_{jd}=1;\;LJ_{j}=min\{C,BP_{dt}\}$;} & {\footnotesize{}//updating the decision variables }\tabularnewline
{\footnotesize{}$\qquad BP_{dt}=BP_{dt}-LJ_{j};\;j=j+1;\}$ } & {\footnotesize{}//updating the auxiliary variables}\tabularnewline
{\footnotesize{}$\}\}$} & \tabularnewline
\end{tabular}{\footnotesize\par}
\end{algorithm}
\subsubsection{Assignment of loads to the outbound trucks}
In the previous step, according to the total demand of the destinations in each period, the total number of pallets attributed to each truck is determined. By considering the detailed destination demands ($R_{pdt}$), in this step, the type of pallets which must be placed in each truck is decided. The number of pallets of each type in a truck is related to the demands of the destination to which the truck is assigned. In the studied JIT distribution system, each truck is filled with the needed pallets at each period. Hence, a group of trucks may be assigned to the same destination in different planning periods, but the load of each one depends on the period to which it is assigned and the demand of the attributed destination in that period ($R_{pdt}$).
The algorithm starts with the first planning period, first destination and first product type. It evaluates if the related parameter $R$ is positive or not. If $R>0$, then, it finds the first truck which is assigned to this destination in this planning period. If the truck has a free capacity $B_{j}>0$, a load of the considered product is attributed to that $x_{jp}=min\{B_{j}\:\&\:R_{pdt}\}$. Subsequently, the remaining capacity of the truck and destination demand are revised. If the truck capacity is filled before the destination demand, then, the next truck assigned to the destination in current planning period must be found and pooled similarly. Otherwise, the algorithm passes to the next product, ordered by the same destination in the current planning period. This procedure continues until the assigned trucks are filled with all needed products for theirs designated destinations during the first period. For the next periods, the pallets of products (loads or shipments) are assigned to the trucks in the same manner. The details of the load of each truck, i.e. number of pallets of each product ($x_{jp}$) are determined in this step. Algorithm \ref{Alg-S2} shows the general procedure of this step in summary.
\begin{algorithm}
\caption{Assignment of loads to the outbound trucks}
\label{Alg-S2}
{\footnotesize{}}%
\begin{tabular}{lc}
{\footnotesize{}$for(t\in T)\{for(d\in D)\{for(p\in P)\{$} & \tabularnewline
{\footnotesize{}$\quad B_{pdt}=R_{pdt};$} & {\footnotesize{}//initializing the auxiliary variables}\tabularnewline
{\footnotesize{}$\quad for(j\in J)\{$} & \tabularnewline
{\footnotesize{}$\qquad if(y_{jt}>0\:and\:q_{jd}>0\:and\:B_{pdt}>0\:and\:BL_{j}>0)\{$} & \tabularnewline
{\footnotesize{}$\quad\qquad x_{jp}=min\{B_{pdt},BL_{j}\};\;d_{jd}=1;\;LJ_{j}=min\{C,BP_{dt}\}$;} & {\footnotesize{}//updating the decision variables }\tabularnewline
{\footnotesize{}$\quad\qquad BL_{j}=BL_{j}-x_{jp};\;B_{pdt}=B_{pdt}-x_{jp};\}$} & {\footnotesize{}//updating the auxiliary variables}\tabularnewline
{\footnotesize{}$\}\}$\}} & \tabularnewline
\end{tabular}{\footnotesize\par}
\end{algorithm}
\subsubsection{Feasibility evaluation}
Three previous steps prepare a solution that satisfies all destinations demands at their needed periods. This solution may be unfeasible because of the limited number of outbound doors. Actually, the number of trucks that can be docked to the cross-dock simultaneously (in the same period) must not be more than the number of doors. Otherwise, a number of trucks must be ignored in the violating periods. That signifies the non-delivered loads. Hence, the algorithm chooses the trucks with minimum loads to be removed.
Mathematically, the number of trucks needed in each period ($ZT_{t}$) was calculated in the second step of the algorithm. If $ZT_{t}>OD$, then, among the pooled trucks in the current period, algorithm finds the truck with minimum load ($min_{j\in J^{t}}\{L_{j}\}$); Where $J^{t}$ is a subset of $J$ that signifies the trucks which are docked in period $t$, and $L_{j}=\sum_{p\in P}x_{jp}$is the total pallets placed in truck $j$. This cycle is repeated until the number of assigned truck is equal to the number of door for every period ($ZT_{t}=OD$). Consequently, a feasible solution is obtained that satisfies the maximum number of destinations demands. After removing some trucks of the periods violating the outbound door constraint, the lists of the scheduled trucks in each period ($y_{jt}$) and the assigned truck to each destination ($q_{jd}$) are revised.
\section{Computational results}\label{results}
In order to evaluate the solution approaches, the main (integrated) model is analyzed in the first step. Then, the symmetry breaking constraints are added to the model to investigate the impact of these constraints on the efficiency of the model. In the second step, the matheuristic method are evaluated by solving various instances and that is compared with the best scenario of the first step (integrated model with symmetry breaking constraints). In view of the complexity of the first step of matheuristic to be solved by CPLEX, an adaptive heuristic algorithm is employed to solve that. In this way, two scenarios are proposed. In the first one, the solution of the heuristic method is injected as a primal (initial) solution for the first step of matheuristic, then that is solved by Cplex. In the second scenario, the heuristic algorithm is employed as a solution method to the first step and the solutions are directly transferred to the second step of matheuristic. These two scenarios are analyzed and are compared with each other and with the classical matheuristic in which both steps are solved by Cplex. Note that, in the numerical experiments, a working day is decomposed to $5$ periods of $2$ hours. In a multi-commodity logic, the products which must be distributed across the studied cross-dock are classified into five categories. Hence, two parameters are constant in all of the studied instances ($r=5;\;k=5$).
\subsection{Impact of the symmetry braking constraints}
To evaluate the impact of the symmetry braking constraints, a sum of $23$ small and medium size instances are solved with and without these constraints. A time limit of $10800$ seconds is imposed to the solver. The results of the experiments are presented in table \ref{tab:SB Impact}.
\begin{table}
\caption{\label{tab:SB Impact}Impact of the Symmetry Breaking constraints}
\scriptsize
$r:Periods=5$
$k:Product\;types=5$
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{cccccccccccc}
\hline
\multirow{2}{*}{DB} & \multirow{2}{*}{Ins} & \multirow{2}{*}{IT-OT-D-ID-OD} & \multicolumn{4}{c}{With Symmetry Breaking} & & \multicolumn{4}{c}{Without Symmetry Breaking}\tabularnewline
\cline{4-7} \cline{5-7} \cline{6-7} \cline{7-7} \cline{9-12} \cline{10-12} \cline{11-12} \cline{12-12}
& & & CPU & Nodes & BFS & Status & & CPU & Nodes & BFS & Status\tabularnewline
\hline
\multirow{3}{*}{1} & 1.1 & 5 - 8 - 3 - 3 - 3 & 1 & 7857 & 1304{*} & Opt & & 8 & 14236 & 1304{*} & Opt\tabularnewline
& 1.2 & 5 - 7 - 3 - 3 - 3 & <1 & 1037 & 13{*} & Opt & & 31 & 72160 & 13{*} & Opt\tabularnewline
& 1.3 & 5 - 9 - 3 - 3 - 3 & 10 & 6903 & 1105{*} & Opt & & 35 & 91452 & 1105{*} & Opt\tabularnewline
\hline
\multirow{4}{*}{2} & 2.1 & 6 - 11 - 4 - 4 - 4 & 13 & 9507 & 14{*} & Opt & & >10800 & 1.54e+6 & 14 & 40.05\%\tabularnewline
& 2.2 & 6 - 9 - 4 - 4 - 4 & 7 & 5446 & 1205{*} & Opt & & 14 & 16304 & 1205{*} & Opt\tabularnewline
& 2.3 & 6 - 10 - 4 - 4 - 4 & 10 & 6291 & 214{*} & Opt & & 12 & 16579 & 214{*} & Opt\tabularnewline
& 2.4 & 6 - 11 - 4 - 4 - 4 & 62 & 121824 & 1510{*} & Opt & & >10800 & 336159 & 1510 & 0.66\%\tabularnewline
\hline
\multirow{4}{*}{3} & 3.1 & 7 - 11 - 4 - 4 - 4 & 19 & 5930 & 1905{*} & Opt & & 415 & 346174 & 1905{*} & Opt\tabularnewline
& 3.2 & 7 - 12 - 4 - 4 - 4 & 44 & 42589 & 1608{*} & Opt & & >10800 & 397552 & 1608 & 0.37\%\tabularnewline
& 3.3 & 7 - 10 - 4 - 4 - 4 & 14 & 7192 & 614{*} & Opt & & 129 & 221713 & 614{*} & Opt\tabularnewline
& 3.4 & 7 - 9 - 4 - 4 - 4 & 5 & 6117 & 1103{*} & Opt & & 6 & 6919 & 1103{*} & Opt\tabularnewline
\hline
\multirow{4}{*}{4} & 4.1 & 8 - 12 - 5 - 5 - 5 & 638 & 1.89e+6 & 3208{*} & Opt & & >10800 & 960857 & 3208 & 0.25\%\tabularnewline
& 4.2 & 8 - 14 - 5 - 5 - 5 & 59 & 41205 & 410{*} & Opt & & >10800 & 279852 & 410 & 2.03\%\tabularnewline
& 4.3 & 8 - 12 - 5 - 5 - 5 & 9 & 9230 & 308{*} & Opt & & >10800 & 594711 & 308 & 2.60\%\tabularnewline
& 4.4 & 8 - 13 - 5 - 5 - 5 & 13 & 5112 & 218{*} & Opt & & >10800 & 1.00e+6 & 218 & 7.19\%\tabularnewline
\hline
\multirow{4}{*}{5} & 5.1 & 9 - 13 - 6 - 6 - 6 & 2152 & 3.58e+6 & 1604{*} & Opt & & >10800 & 308970 & 1604 & 19.89\%\tabularnewline
& 5.2 & 9 - 17 - 6 - 6 - 6 & 44 & 42926 & 10{*} & Opt & & >10800 & 239477 & 10 & 100\%\tabularnewline
& 5.3 & 9 - 15 - 6 - 6 - 6 & 280 & 50119 & 616{*} & Opt & & >10800 & 201172 & 616 & 99.68\%\tabularnewline
& 5.4 & 9- 17 - 6 - 6 - 6 & 139 & 43869 & 514{*} & Opt & & >10800 & 197311 & 514 & 2.72\%\tabularnewline
\hline
\multirow{4}{*}{6} & 6.1 & 10 - 18 - 7 - 7 - 7 & 159 & 66644 & 507{*} & Opt & & >10800 & 193237 & 507 & 100\%\tabularnewline
& 6.2 & 10 - 20 - 7 - 7 - 7 & >10800 & 1.01e+6 & 1105 & 0.27\% & & >10800 & 114208 & 1106 & 100\%\tabularnewline
& 6.3 & 10 - 16 - 7 - 7 - 7 & 3249 & 643603 & 806{*} & Opt & & >10800 & 168406 & 806 & 62.78\%\tabularnewline
& 6.4 & 10 - 15 - 7 - 7 - 7 & 1149 & 953575 & 617{*} & Opt & & >10800 & 283059 & 617 & 2.76\%\tabularnewline
\hline
\end{tabular}
\end{adjustbox}
\end{table}
In this table, ``DB'' means the data-base, ``Ins'' represents the instances where ``IT'', ``OT'', ``D'', ``ID'' and ``OD'' signify the number of Inbound Trucks, the number of Outbound Trucks, the number of Destinations, the number of Inbound Doors and the number of Outbound Doors respectively. In each data-base, the number of inbound trucks, destinations, inbound doors and outbound doors are fixed, the orders and their needed JIT periods are simulated and the number of outbound trucks are calculated based on the received orders. For the small and medium size instances, IT varies between 5 and 10 trucks, OT is from $7$ to $20$ trucks, and D, ID and OD vary between $3$ and $7$.
Two solution approaches are compared by their CPU times (in second), best feasible solutions (BFS) and the status of the solutions. If the solution approach attains the optimal solution in the considered time limit, the status of the solution is optimal (opt), if not, the status column presents the gap between lower and upper bounds of the solutions found in the time limit in percentage.
Note that the penalty of the undelivered pallets is considered to be $100$ for any pallet. As shown in table \ref{tab:SB Impact}, symmetry breaking constraints increase the efficiency of the model significantly. Without SBC, the solver attains the optimal solution for only 7 instances, whereas with SBC the solver reach to the optimal solution for 22 instances among 23, within the defined time limit. Even in the cases where both scenarios give the optimal solution, the number of nodes and computational time are decreased with the SBCs. As an example, for instance 3.1, the solver analyzed $340\times10^{3}$ nodes during $415$ second to attain the optimal solution, whereas with SBCs, $5900$ nodes are analyzed in $19$ seconds. Therefore, the model with SBCs overcomes the model without the SBCs over all studied instances.
Given the exponentially increasing of the computational time by the size of instances, even with SBCs, the generic solver is not suitable for the large size instances. For this reason, a matheuristic method is proposed and evaluated in the next section.
\subsection{Numerical results of matheuristic method}
A set of $39$ instances of different sizes (small, medium and large instances) are employed to evaluate the matheuristic method. Three essential parameters of the problem, which determine the size of the instances, are as follows: the number of inbound trucks, the number of outbound trucks and the number of destinations. The number of inbound trucks varies from $5$ to $80$, the variation range for the outbound trucks is between $7$ and $110$, where the number of destinations varies from $3$ to $40$ to generate the instances. In order to evaluate the matheuristic, the result of the best scenario of the previous analysis (Cplex with SBC) is selected to be compared with the matheuristic results. The proposed two-step matheuristic is coded by c++ language and both steps are solved by ILOG CPLEX. The time limit considered for each step is $5400$ seconds and for the complete solution procedure is $10,800$ seconds. The numerical results of the matheuristic are presented in table \ref{tab:MathResults} and compared with the results of CPLEX.
\begin{table}
\caption{\label{tab:MathResults}Results of the matheuristic method}
\scriptsize
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{cccccccccccccccc}
\hline
\multirow{3}{*}{Ins} & \multirow{3}{*}{IT-OT-D} & \multicolumn{4}{c}{Cplex with Symmetry Breaking} & & \multicolumn{9}{c}{2-step Matheuristic method}\tabularnewline
\cline{8-16} \cline{9-16} \cline{10-16} \cline{11-16} \cline{12-16} \cline{13-16} \cline{14-16} \cline{15-16} \cline{16-16}
& & \multicolumn{4}{c}{} & & \multicolumn{3}{c}{Step1 (Cplex)} & & \multicolumn{3}{c}{Step2 (Cplex)} & \multirow{2}{*}{BFS} & \multirow{2}{*}{E$_{max}$}\tabularnewline
\cline{3-6} \cline{4-6} \cline{5-6} \cline{6-6} \cline{8-10} \cline{9-10} \cline{10-10} \cline{12-14} \cline{13-14} \cline{14-14}
& & CPU & Nodes & BFS & Status & & CPU & Nodes & Status & & CPU & Node & Status & & \tabularnewline
\hline
1.1 & 5 - 8 - 3 & 1 & 7857 & 1304{*} & Opt & & <1 & 33 & Opt & & <1 & 0 & Opt & 1309 & 0.38\%{*}\tabularnewline
1.2 & 5 - 7 - 3 & <1 & 1037 & 13{*} & Opt & & <1 & 0 & Opt & & <1 & 0 & Opt & 13{*} & Opt\tabularnewline
1.3 & 5 - 9 - 3 & 10 & 6903 & 1105{*} & Opt & & <1 & 3 & Opt & & <1 & 0 & Opt & 1106 & 0.09\%{*}\tabularnewline
2.1 & 6 - 11 - 4 & 13 & 9507 & 14{*} & Opt & & <1 & 0 & Opt & & <1 & 544 & Opt & 14{*} & Opt\tabularnewline
2.2 & 6 - 9 - 4 & 7 & 5446 & 1205{*} & Opt & & <1 & 5 & Opt & & <1 & 0 & Opt & 1206 & 0.08\%{*}\tabularnewline
2.3 & 6 - 10 - 4 & 10 & 6291 & 214{*} & Opt & & <1 & 0 & Opt & & <1 & 35 & Opt & 214{*} & Opt\tabularnewline
2.4 & 6 - 11 - 4 & 62 & 121824 & 1510{*} & Opt & & <1 & 270 & Opt & & <1 & 529 & Opt & 1512 & 0.13\%{*}\tabularnewline
3.1 & 7 - 11 - 4 & 19 & 5930 & 1905{*} & Opt & & <1 & 45 & Opt & & <1 & 183 & Opt & 1908 & 0.15\%{*}\tabularnewline
3.2 & 7 - 12 - 4 & 44 & 42589 & 1608{*} & Opt & & <1 & 0 & Opt & & <1 & 4114 & Opt & 1609 & 0.06\%{*}\tabularnewline
3.3 & 7 - 10 - 4 & 14 & 7192 & 614{*} & Opt & & <1 & 0 & Opt & & <1 & 369 & Opt & 618 & 0.64\%{*}\tabularnewline
3.4 & 7 - 9 - 4 & 5 & 6117 & 1103{*} & Opt & & <1 & 33 & Opt & & <1 & 164 & Opt & 1105 & 0.18\%{*}\tabularnewline
4.1 & 8 - 12 - 5 & 638 & 1.89e+6 & 3208{*} & Opt & & <1 & 0 & Opt & & 1.22 & 4661 & Opt & 3209 & 0.03\%{*}\tabularnewline
4.2 & 8 - 14 - 5 & 59 & 41205 & 410{*} & Opt & & <1 & 0 & Opt & & 1.13 & 3176 & Opt & 410{*} & Opt\tabularnewline
4.3 & 8 - 12 - 5 & 9 & 9230 & 308{*} & Opt & & <1 & 0 & Opt & & <1 & 208 & Opt & 308{*} & Opt\tabularnewline
4.4 & 8 - 13 - 5 & 13 & 5112 & 218{*} & Opt & & <1 & 461 & Opt & & <1 & 81 & Opt & 222 & 1.83\%{*}\tabularnewline
5.1 & 9 - 13 - 6 & 2152 & 3.58e+6 & 1604{*} & Opt & & <1 & 153 & Opt & & 2.33 & 4109 & Opt & 1604 & 0.16\%{*}\tabularnewline
5.2 & 9 - 17 - 6 & 44 & 42926 & 10{*} & Opt & & <1 & 0 & Opt & & 2.23 & 6025 & Opt & 12 & 20\%{*}\tabularnewline
5.3 & 9 - 15 - 6 & 280 & 50119 & 616{*} & Opt & & <1 & 0 & Opt & & 3.44 & 4106 & Opt & 619 & 0.48\%{*}\tabularnewline
5.4 & 9 - 17 - 6 & 139 & 43869 & 514{*} & Opt & & <1 & 0 & Opt & & 1.03 & 358 & Opt & 516 & 0.38\%{*}\tabularnewline
6.1 & 10 - 18 - 7 & 159 & 66644 & 507{*} & Opt & & <1 & 0 & Opt & & 6.92 & 7274 & Opt & 508 & 0.19\%{*}\tabularnewline
6.2 & 10 - 20 - 7 & >10800 & 1.01e+6 & 1105 & 0.27\% & & 2.22 & 560 & Opt & & 4.81 & 5663 & Opt & 1106 & 0.09\%\tabularnewline
6.3 & 10 - 16 - 7 & 3249 & 643603 & 806{*} & Opt & & <1 & 0 & Opt & & 2.89 & 4238 & Opt & 806{*} & Opt\tabularnewline
6.4 & 10 - 15 - 7 & 1149 & 953575 & 617{*} & Opt & & <1 & 0 & Opt & & <1 & 193 & Opt & 620 & 0.48\%{*}\tabularnewline
7.1 & 12 - 20 - 8 & >10800 & 3.49e+6 & 1206 & 0.43\% & & 3.42 & 0 & Opt & & 18.33 & 14046 & Opt & 1206{*} & 0\%\tabularnewline
8.1 & 15 - 22 - 9 & >10800 & 710370 & 930 & 1.89\% & & 1.47 & 196 & Opt & & 4.20 & 3966 & Opt & 931 & 0.10\%\tabularnewline
9.1 & 17 - 24 - 10 & >10800 & 136119 & 19 & 64.18\% & & 1.30 & 0 & Opt & & 449.81 & 110571 & Opt & 20 & 5.2\%\tabularnewline
10.1 & 20 - 24 - 10 & >10800 & 309978 & 1315 & 1.14\% & & 2.55 & 1046 & Opt & & >10800 & 303973 & 0.48\% & 1316 & 0.07\%\tabularnewline
11.1 & 22 - 37 - 12 & >10800 & 38353 & 807 & 0.87\% & & 26.06 & 0 & Opt & & >10800 & 451043 & 1.07\% & 809 & 0.24\%\tabularnewline
12.1 & 25 - 38 - 15 & >10800 & 12928 & 2818 & 100\% & & 80.77 & 5522 & Opt & & >10800 & 139045 & 0.20\% & \textbf{2714} & \textbf{-3.69\%}\tabularnewline
13.1 & 27 - 45 - 15 & >10800 & 14853 & 2919 & 100\% & & 198.31 & 4384 & Opt & & >10800 & 1.18e+6 & 0.17\% & \textbf{1803} & \textbf{-38.23\%}\tabularnewline
14.1 & 30 - 49 - 15 & >10800 & 14254 & 12136 & 95.06\% & & >5400 & 121143 & 34.04\% & & >5400 & 229241 & 0.02\% & \textbf{5701} & \textbf{-53.02\%}\tabularnewline
15.1 & 32 - 54 - 18 & >10800 & 15910 & 11601 & 100\% & & 1039.34 & 19571 & Opt & & >10800 & 284247 & 0.17\% & \textbf{1803} & \textbf{-84.45\%}\tabularnewline
16.1 & 35 - 56 - 19 & >10800 & 7772 & 10520 & 100\% & & 2823.49 & 6498 & Opt & & >10800 & 857371 & 100\% & \textbf{6} & \textbf{-99.94\%}\tabularnewline
17.1 & 40 - 63 - 20 & >10800 & - & - & - & & 3497.91 & 33876 & Opt & & >10800 & 427966 & 0.16\% & \textbf{3105} & -\tabularnewline
18.1 & 45 - 67 - 23 & >10800 & - & - & - & & >5400 & 15661 & 92.16\% & & >5400 & 321555 & 0.14\% & \textbf{5107} & -\tabularnewline
19.1 & 50 - 79 - 25 & >10800 & - & - & - & & >5400 & 5764 & 100\% & & >5400 & 334438 & 0.08\% & \textbf{5104} & -\tabularnewline
20.1 & 60 - 94 - 30 & >10800 & - & - & - & & >5400 & 3955 & 100\% & & 1451.55 & 188938 & Opt & \textbf{10701} & -\tabularnewline
21.1 & 70 - 109 - 35 & >10800 & - & - & - & & >5400 & 112 & 100\% & & 1.34 & 0 & Opt & \textbf{118910} & -\tabularnewline
22.1 & 80 - 115 - 40 & >10800 & - & - & - & & >5400 & 1385 & 100\% & & 20.88 & 0 & Opt & \textbf{74303} & -\tabularnewline
\hline
\end{tabular}
\end{adjustbox}
\end{table}
As shown in the table, by employing the SBCs and Cplex solver, the optimal solutions are found for 22 small and medium instances in the limited time available (shown by {*}). For 5 medium instances, the solver finds the efficient lower and upper bounds for the solutions (Gaps are less than 2\%). But for 12 large instances, Cplex solver whether is not able to find two bounds (6 instances) or is not able to finds an acceptable lower bound (Gaps are greater than 64\%). The BFS column of the matheuristic method shows that among 22 instances for which the optimal solutions are known, 6 instances are solved
to optimality (shown by {*}). The $E_{max}$ column of the table shows the status of the solutions obtained by matheuristic. If the solution is not optimal, that shows the gaps between the found solution and the optimal solution. Note that, if the optimal solution is available, the gaps are shown together with a {*}, if not, the $E_{max}$ presents the difference between the solutions found by CPLEX and matheuristic method. For 11 large instances, the matheuristic method attains a solution better than Cplex (shown by bold), where for the other instances the gaps are mostly less than 1\% expect 3 instances (Instance 4.4 : 1.83\%, Instance 9.1 : 5.2\% and Instance 5.2 : 20\%).
Table \ref{tab:MathResults} shows that by using the matheuristic method the complexity of the problem decreases greatly but the quality of the solutions do not decrease significantly. The number of analyzed nodes and the CPU times illustrate the efficiency of the proposed solution method. As an example, for instance 6.3, Cplex analyzed more than 600 thousands nodes during more than 3 thousands seconds to find the optimal solution whereas matheuristic analyze about 4 thousands nodes within 3 seconds to reach the same solution as CPLEX (optimal solution).
As the first step of the proposed matheuristic has a significant impact on the quality of the solution, this method is not very suitable where the solution of the first step is not reliable. For the five largest instances, the solver can not attains an optimal solution or a solution with a small gap in the first step. Hence, the proposed classical matheuristic is not a qualified approach to solve these sizes of the problem. For this reason, a heuristic algorithm is presented to solve the first step of the matheuristic.
\subsection{Warmed-up matheuristic vs hybrid matheuristic}
The heuristic algorithm, which is proposed for the first step of matheuristic method, provides two scenarios to solve the problem. The first scenario is to apply the heuristic method as an alternative to solve the first step instead of CPLEX. In the second scenario, the solutions obtained by the heuristic method are used as the primal solution to be injected to CPLEX. In this way, the first step is solved by CPLEX, but from a good starting solution obtained by heuristic. These two scenarios are compared to the classical matheuristic method (CPLEX-CPLEX) by 39 studied instances. The results of the classical matheuristic method is presented in table \ref{tab:MathResults}. Table \ref{tab:WU-Hybride} shows the computational results of two aforementioned alternatives for the first step (1. CPLEX with a primal solution - CPLEX, 2. Heuristic - CPLEX).
\begin{flushleft}
\begin{table}
\caption{\label{tab:WU-Hybride}Results of matheuristic with Warm-Up and hybrid
matheuristic}
\scriptsize
\begin{adjustbox}{max width=\textwidth}
\begin{raggedright}
\begin{tabular}{cccccccccccccccccccc}
\hline
\multirow{3}{*}{Ins} & \multicolumn{9}{c}{Matheuristic with Warm-Up} & & \multicolumn{9}{c}{Hybrid matheuristic}\tabularnewline
\cline{2-10} \cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10} \cline{12-20} \cline{13-20} \cline{14-20} \cline{15-20} \cline{16-20} \cline{17-20} \cline{18-20} \cline{19-20} \cline{20-20}
& \multicolumn{3}{c}{Step1 (Warmed Up Cplex)} & & \multicolumn{3}{c}{Step2 (Cplex)} & \multirow{2}{*}{BFS} & \multirow{2}{*}{E$_{max}$} & & \multicolumn{3}{c}{Step1 (Heuristic)} & & \multicolumn{3}{c}{Step2 (Cplex)} & \multirow{2}{*}{BFS} & \multirow{2}{*}{E$_{max}$}\tabularnewline
\cline{2-4} \cline{3-4} \cline{4-4} \cline{6-8} \cline{7-8} \cline{8-8} \cline{12-14} \cline{13-14} \cline{14-14} \cline{16-18} \cline{17-18} \cline{18-18}
& CPU & Nodes & Status & & CPU & Node & Status & & & & CPU & Sol & E$_{m}$ & & CPU & Node & Status & & \tabularnewline
\hline
1.1 & <1 & 0 & Opt & & <1 & 0 & Opt & 1304{*} & Opt & & <1 & 13{*} & Opt & & <1 & 0 & Opt & 1304{*} & Opt\tabularnewline
1.2 & <1 & 0 & Opt & & <1 & 0 & Opt & 13{*} & Opt & & <1 & 0{*} & Opt & & <1 & 0 & Opt & 13{*} & Opt\tabularnewline
1.3 & <1 & 3 & Opt & & <1 & 0 & Opt & 1105{*} & Opt & & <1 & 11{*} & Opt & & <1 & 0 & Opt & 1105{*} & Opt\tabularnewline
2.1 & <1 & 0 & Opt & & <1 & 536 & Opt & 14{*} & Opt & & <1 & 0{*} & Opt & & <1 & 536 & Opt & 14{*} & Opt\tabularnewline
2.2 & <1 & 35 & Opt & & <1 & 0 & Opt & 1206 & 0.08\%{*} & & <1 & 12{*} & Opt & & <1 & 0 & Opt & 1206 & 0.08\%{*}\tabularnewline
2.3 & <1 & 0 & Opt & & <1 & 48 & Opt & 214{*} & Opt & & <1 & 2{*} & Opt & & <1 & 48 & Opt & 214{*} & Opt\tabularnewline
2.4 & <1 & 202 & Opt & & <1 & 594 & Opt & 1511 & 0.06\%{*} & & <1 & 15{*} & Opt & & <1 & 594 & Opt & 1511 & 0.06\%{*}\tabularnewline
3.1 & <1 & 311 & Opt & & <1 & 240 & Opt & 1906 & 0.05\%{*} & & <1 & 19{*} & Opt & & <1 & 240 & Opt & 1906 & 0.05\%{*}\tabularnewline
3.2 & <1 & 788 & Opt & & <1 & 1576 & Opt & 1609 & 0.06\%{*} & & <1 & 16{*} & Opt & & <1 & 1576 & Opt & 1609 & 0.06\%{*}\tabularnewline
3.3 & <1 & 0 & Opt & & <1 & 387 & Opt & 617 & 0.48\%{*} & & <1 & 6{*} & Opt & & <1 & 387 & Opt & 617 & 0.48\%{*}\tabularnewline
3.4 & <1 & 40 & Opt & & <1 & 66 & Opt & 1104 & 0.09\%{*} & & <1 & 11{*} & Opt & & <1 & 66 & Opt & 1104 & 0.09\%{*}\tabularnewline
4.1 & <1 & 5 & Opt & & <1 & 1510 & Opt & 3209 & 0.03\%{*} & & <1 & 32{*} & Opt & & <1 & 1510 & Opt & 3209 & 0.03\%{*}\tabularnewline
4.2 & <1 & 23 & Opt & & 1.64 & 3078 & Opt & 410{*} & Opt & & <1 & 4{*} & Opt & & 1.48 & 3078 & Opt & 410{*} & Opt\tabularnewline
4.3 & <1 & 0 & Opt & & <1 & 391 & Opt & 308{*} & Opt & & <1 & 3{*} & Opt & & <1 & 391 & Opt & 308{*} & Opt\tabularnewline
4.4 & <1 & 1187 & Opt & & <1 & 788 & Opt & 220 & 0.91\%{*} & & <1 & 2{*} & Opt & & <1 & 788 & Opt & 220 & 0.91\%{*}\tabularnewline
5.1 & 1.00 & 3441 & Opt & & 1.70 & 1696 & Opt & 1605 & 0.06\%{*} & & <1 & 16{*} & Opt & & 1.48 & 1696 & Opt & 1605 & 0.06\%{*}\tabularnewline
5.2 & <1 & 0 & Opt & & 1.77 & 3667 & Opt & 11 & 10\%{*} & & <1 & 0{*} & Opt & & 1.67 & 3667 & Opt & 11 & 10\%{*}\tabularnewline
5.3 & <1 & 184 & Opt & & 3.00 & 1534 & Opt & 618 & 0.32\%{*} & & <1 & 6{*} & Opt & & 2.78 & 1534 & Opt & 618 & 0.32\%{*}\tabularnewline
5.4 & 2.89 & 4070 & Opt & & <1 & 245 & Opt & 515 & 0.19\%{*} & & <1 & 5{*} & Opt & & <1 & 245 & Opt & 515 & 0.19\%{*}\tabularnewline
6.1 & 2.81 & 5585 & Opt & & 9.16 & 7054 & Opt & 509 & 0.39\%{*} & & <1 & 5{*} & Opt & & 9.27 & 7054 & Opt & 509 & 0.39\%{*}\tabularnewline
6.2 & 2.19 & 1876 & Opt & & 2.67 & 1539 & Opt & 1105 & 0\% & & <1 & 11{*} & Opt & & 2.77 & 1539 & Opt & 1105 & 0\%\tabularnewline
6.3 & <1 & 758 & Opt & & 4.00 & 3014 & Opt & 806{*} & Opt & & <1 & 8{*} & Opt & & 3.91 & 3014 & Opt & 806{*} & Opt\tabularnewline
6.4 & <1 & 0 & Opt & & <1 & 934 & Opt & 618 & 0.16\%{*} & & <1 & 6{*} & Opt & & <1 & 934 & Opt & 618 & 0.16\%{*}\tabularnewline
7.1 & 29.86 & 21443 & Opt & & 14.83 & 22344 & Opt & 1206 & 0\% & & <1 & 12{*} & Opt & & 24.44 & 22344 & Opt & 1206 & 0\%\tabularnewline
8.1 & <1 & 2135 & Opt & & 2.41 & 1440 & Opt & 931 & 0.10\% & & <1 & 9{*} & Opt & & 2.53 & 1440 & Opt & 931 & 0.10\%\tabularnewline
9.1 & <1 & 0 & Opt & & 258.86 & 87012 & Opt & 20 & 5.26\% & & <1 & 0{*} & Opt & & 257.09 & 87012 & Opt & 20 & 5.26\%\tabularnewline
10.1 & >5400 & 415941 & 100\% & & >5400 & 196686 & 0.57\% & 1315 & 0\% & & <1 & 13{*} & Opt & & >10800 & 415798 & 0.49\% & 1315 & 0\%\tabularnewline
11.1 & >5400 & 536493 & 72.84\% & & >5400 & 163957 & 0.99\% & 808 & 0.12\% & & <1 & 8{*} & Opt & & >10800 & 314531 & 0.82\% & 807 & 0\%\tabularnewline
12.1 & 71.61 & 10483 & Opt & & >10800 & 254186 & 0.13\% & 2713 & -3.72\% & & <1 & 27{*} & Opt & & >10800 & 246762 & 0.13\% & 2713 & -3.72\%\tabularnewline
13.1 & >5400 & 320026 & 16.67\% & & >5400 & 347553 & 0.22\% & 1804 & -38.19\% & & <1 & 18{*} & Opt & & >10800 & 377520 & 0.22\% & 1804 & -38.19\%\tabularnewline
14.1 & >5400 & 171603 & 89.77\% & & >5400 & 200136 & 0.09\% & 5705 & -52.99\% & & <1 & 57 & - & & >10800 & 240900 & 0.09\% & 5705 & -52.99\%\tabularnewline
15.1 & >5400 & 266308 & 100\% & & >5400 & 157615 & 0.39\% & 1807 & -84.42\% & & <1 & 18{*} & Opt & & >10800 & 178702 & 0.39\% & 1807 & -84.42\%\tabularnewline
16.1 & <1 & 0 & Opt & & 6711.59 & 287102 & Opt & 5 & -99.95\% & & <1 & 0{*} & Opt & & 6733.58 & 287039 & Opt & 5 & -99.95\%\tabularnewline
17.1 & >5400 & 36725 & 100\% & & >5400 & 144137 & 0.26\% & 3108 & - & & <1 & 31{*} & Opt & & >10800 & 175891 & 0.26\% & 3108 & -\tabularnewline
18.1 & 2701.16 & 170295 & Opt & & >10800 & 130821 & 0.22\% & \textbf{5011} & - & & <1 & 50 & - & & >10800 & 140860 & 0.22\% & \textbf{5011} & -\tabularnewline
19.1 & >5400 & 34217 & 100\% & & >5400 & 168164 & 0.12\% & \textbf{4305} & - & & <1 & 43 & - & & >10800 & 182593 & 0.12\% & \textbf{4305} & -\tabularnewline
20.1 & >5400 & 10494 & 100\% & & >5400 & 244870 & 0.12\% & \textbf{2403} & - & & <1 & 24 & - & & >10800 & 247599 & 0.12\% & \textbf{2403} & -\tabularnewline
21.1 & >5400 & 1742 & 100\% & & >5400 & 61024 & 0.06\% & \textbf{7104} & - & & <1 & 71 & - & & >10800 & 95676 & 0.04 & \textbf{7103} & -\tabularnewline
22.1 & >5400 & 1652 & 100\% & & >5400 & 94137 & 0.09\% & \textbf{6506} & - & & <1 & 65 & - & & >10800 & 131366 & 0.09\% & \textbf{6506} & -\tabularnewline
\hline
\end{tabular}
\par\end{raggedright}
\end{adjustbox}
\end{table}
\par\end{flushleft}
By regarding to the best feasible solutions found by these two scenarios and compare them with the classical matheuristic ones presented in table \ref{tab:MathResults}, it can be concluded that both hybrid methods are more efficient than the classical method. The optimal solutions have been found over 8 instances among 22 known optimal solutions. The efficiency of the methods over other instances is evaluated by two parameters : 1. Best found solution (BFS), 2. Deviation from the optimal ($E_{max}$). As shown in table \ref{tab:WU-Hybride}, the BFSs and the deviations of these new solution approaches are smaller than the classical matheuristic ones over most of the instances. But the efficiency of these methods is illustrated by analyzing the five largest instances (18.1 - 22.1) in which the quality of solutions are much more better than matheuristic.
Furthermore, table \ref{tab:WU-Hybride} shows that the heuristic method, proposed for step 1, attains the optimal solution for all of instances for which the optimal solution is known. The best feasible solutions found by both scenarios and their related deviations from the optimal are the same expect 2 instances (11.1 and 21.1) in which the quality of the solutions found by ``hybrid matheuristic'' are better than the ``warmed up matheuristic'' ones. Hence, the hybrid method overcomes the warmed up matheuristic over all of the studied instances.
As a conclusion, by an efficient mathematical modeling containing SBCs and CPLEX, only the small size instances are solvable (Ins 1.1 to 6.4). For the medium size instances (Ins 7.1 to 17.1), in some instances the hybrid method reaches a better solution and gap and in some other instances the classical method attains the best feasible solutions. Whereas for the large instances (Ins 18.1 to 22.1), the solutions obtained by hybrid approach are much more better than the solutions attained by the classical method. Therefore, the proposed hybrid matheuristic solution approach is very suitable to solve the medium and large sizes instances of the presented truck scheduling
and load assignment problem.
\section{Conclusions and future research opportunities}\label{conclusion}
A novel operational level optimization problem in the cross-dock planning domain is presented in this study. The studied problem concerns both inbound and outbound scheduling combined with two interrelated (synchronized) assignment problems as : The assignment of products to outbound trucks and the assignment of outbound trucks to destinations. The consideration of the JIT customer demands is another contribution of this research. An integrated mathematical model is presented and strengthened by adding the symmetry breaking constraints. A decomposition based matheuristic and a hybrid matheuristic algorithms are proposed to solve this combinatorial optimization problem. The instances of different sizes are solved by CPLEX. The computational results show that the symmetry breaking constraints can significantly improve the mathematical model. But even by the SBCs, CPLEX is not able to solve the large size instances. Therefore, a two-step matheuristic based on the decomposition of the formulation is proposed to decompose the integrated model into two smaller mathematical models. The numerical results show that the matheuristic is very efficient for the medium size instances, while the first step of matheuristic remains intractable to be solved by CPLEX. Therefore, an adaptive heuristic is proposed for the first step to form a hybrid matheuristic. The numerical results confirm a significant improvement on the quality of solutions for the large instances when applying the hybrid matheuristic as the solution approach.
The uncertainty considerations on the arrival times of the trucks to the cross-dock, integrating VRP and split deliveries, adapting an efficient exact algorithm and developing an on line optimization approach could be considered as the future researches.
\bibliographystyle{elsarticle-harv}
|
2,877,628,090,819 | arxiv | |
2,877,628,090,820 | arxiv | \section{Introduction}
It has been demonstrated that one can improve the sensitivity and precision of many classical measurement techniques using various quantum states of light~\cite
PhysRevA.44.3266,Ribeiro:97,Hayat:99,Treps940,Giovannetti1330,PhysRevA.77.053807,Plick_2010,Brida:2010lh,nphoton.2012.300,nphoton.2012.346,TAYLOR20161,Whittaker_2017,s41598-017-06545-w,doi:10.1063/5.0009681} \textcolor{black}{(For instance, the experimental work reported in Ref.~\cite{PhysRevA.44.3266} is a sub-shot-noise measurement of an intensity modulation on one of the quantum-correlated twin beams, and the intensity is modulated by adjusting the transmission of the beam from a liquid-crystal cell)}. Most prominently, sub-shot-noise detection of changes in optical phase have been demonstrated in interferometers using quantum light~\cite{Anderson:17,Plick_2010_1,Kolkiran:08,PhysRevLett.107.113603} and have been implemented for gravitational wave detection~\cite{nphys2083}.
Although a straightforward readily attainable approach to achieve desired performances of a classical measurement is to simply increase the photon flux of the probe light to yield a greater signal-to-noise ratio, it has been proven unfeasible whenever one faces limits on the brightness of the optical probes, for instance, in the case where samples can be altered or damaged by the probe light~\cite{TAYLOR20161,doi:10.1111/php.12572}. It is therefore highly desirable to optimize measurement sensitivity with a fixed amount of input photon flux~\cite{TAYLOR20161}. It is also important to note that for measurement schemes where the sensitivity itself varies with parameters of the measured sample it is possible for the sensitivity to be degraded, potentially requiring either prior knowledge about the optical sample or the addition of a feedback servo loop to ensure a sub-shot-noise performance~\cite{nphoton.2010.268,Yonezawa1514,nphoton.2015.139}.
Since the intensity measurement of an idealized laser fluctuates with a Poisson distributio
, it is therefore used to define the shot-noise limit (SNL) in optical measurements, and it can only be reached in classical experiments once all other sources of noise are removed. For a direct measurement of optical transmission, the number of photons that pass through a sample is used to estimate the sample's absorption $\alpha$, and thus the estimation sensitivity $\Delta \alpha$ is determined by the SNL. \textcolor{black}{One of the most popular approaches that allow for a sub-SNL measurement of an unknown sample's absorption is to use quantum-correlated beams of photons~\cite{Whittaker_2017,s41598-017-06545-w}.
For practical applications, the reduction of noise between quantum-correlated beams of photons generated with spontaneous parametric down-conversion (SPDC)
is widely adopted because of the implementation simplicity and the robustness of quantum nature against the introduction of an absorbing sample~\cite{Iskhakov:16}. In particular, such technique has been implemented in the context of imaging, where a charge-coupled-device (CCD) camera is usually employed to acquire sub-SNL measurement in the spatial domain by detecting correlated photons altogether in the same image captured by the camera~\cite{PhysRevA.77.053807,Brida:2010lh,ncomms3426,Samantaray:2017ff,PhysRevA.95.053849,PhysRevA.100.063828}.
With the inclusion of a spatially absorbing sample, it has been shown that correlated photons can be used to suppress noise in imaging objects to a degree that out-performs classical measurement using an equally efficient detection~\cite{Brida:2010lh,Knyazev:19}. Since absorption measurement is the most versatile tool for many applications in spectroscopy, metrology, chemistry and biology, improving the measurement sensitivity is thus indisputably beneficial to both science and engineering communities.
It is therefore absolutely valuable for experiments to be performed to observe clear quantum advantages that gained by using quantum states of light in absorption measurements.}
\textcolor{black}{Indeed, quantum advantages in absorption measurements have been demonstrated in a series of experiments carried out with photon pairs generated with SPDC in nonlinear crystals, Refs.~\cite{Ribeiro:97,nphoton.2012.300,Whittaker_2017,s41598-017-06545-w} are some prominent examples. It should be noted that SPDC is not the only source of quantum light, another important quantum source is the squeezed light produced with four-wave mixing (FWM) in atomic vapors or optical fibers~\cite{doi:10.1021/acsphotonics.9b00250}. In fact, squeezed light has been extensively studied for its advantage in phase measurement since the early prediction of Caves~\cite{PhysRevD.23.1693}, some prominent experiments are reported in Refs.~\cite{nphys2083,Gupta:18,Anderson:17,Yonezawa1514}. However, there are hardly any experimental demonstrations using squeezed light to achieve sub-SNL absorption measurement \textcolor{black}{since the seminal work done by Polzik~\textit{et al.}~\cite{PhysRevLett.68.3020}}. In this article, we report a practically realizable experimental scheme using squeezed light for direct absorption measurement. We use intensity squeezed beams generated with a seeded FWM process as the source to demonstrate clear quantum advantages over the SNL. Note that Moreau \textit{et al.}~\cite{s41598-017-06545-w} report a quantum advantage of 0.9~dB using SPDC in direct absorption measurement, while we report a higher quantum advantage of more than 1.2~dB for weak absorption levels ($\leq10\%$) as shown below.}
\textcolor{black}{Our experimental scheme is straightforward - a seeded FWM atomic vapor cell together with an electron-multiplying charge-coupled-device (EMCCD) camera comprise the bulk of what is needed to acquire a sub-SNL absorption measurement. Information containing absorption of the sample being measured can be readily obtained by simply integrating the images captured by the EMCCD camera, no homodyne/lock-in or logic coincidence is required. Our scheme therefore is very applicable in many circumstances where sub-SNL absorption measurement is highly desirable.
We also provide in this article a theoretical model to analyze and gain insights into the experimental observations.}
\section{Results}
\subsection{\textcolor{black}{Theoretical analysis of the quantum advantage for measurement sensitivity}}
\textcolor{black}{Our intensity squeezed light is generated with the FWM process in an atomic $^{85}$Rb vapor cell~\cite{Dowran:18,PhysRevA.95.023803,Anderson:17,Li:17,Pooser:15,Clark:2014vf}. The atomic medium possesses a large third-order electric susceptibility $\chi^{(3)}$, and when appropriately chosen laser light `seeds' the medium, `twin beams', also known as the `probe' and `conjugate' beams, are produced. The theoretical modeling of the twin beams generation in the FWM process is complex, as in the experiment one deals with the probe and conjugate beams of finite bandwidth. In fact, the bandwidth of the twin beams in our scheme is merely $\sim20$~MHz~\cite{Clark:2014vf,Glasser2012a}, which is much narrower compared to what one generates with SPDCs. Therefore, we can recover many of the aspects of our observations in terms of a theoretical model based on an equivalent \textit{single-mode} description of the probe and conjugate beams~\cite{Li:17}. In brief, we use the single-mode approximation and designate $\hat{a}$ and $\hat{b}$ as the mode operators for the probe and conjugate beams respectively, the final operators after detection can therefore be expressed as}
\begin{equation}
\begin{aligned}
\begin{split}
\hat{a}_f = \sqrt{\eta_p}\{\sqrt{1-\alpha}[(\text{cosh} r)\hat{a} + e^{i\theta}(\text{sinh} r)\hat{b}^{\dagger}]+i\sqrt{\alpha}\hat{\nu_{\alpha}}\} \\
+ i\sqrt{1-\eta_p}\hat{\nu_p},\\
\end{split}
\\ {\hat{b}^{\dagger}}_f= \sqrt{\eta_c}[(\text{cosh} r)\hat{b}^{\dagger} + e^{-i\theta}(\text{sinh} r)\hat{a}]
-i\sqrt{1-\eta_c}\hat{\nu}^{\dagger}_c,
\end{aligned}
\end{equation}
\textcolor{black}{where $r$ is the squeezing parameter of the FWM, $\theta$ is the relative phase between the twin beams (approximately, $\theta\cong 2\pi \times 2\nu_{HF} \times L/c$, where 2$\nu_{HF}$ is the frequency difference between the twin beams and $\nu_{HF}$ is the hyperfine splitting in the electronic ground state of $^{85}$Rb shown in Fig.~\ref{Setup}(b), $L$ is the vapor cell length and $c$ is the speed of light), $1-\eta_p$ and $1-\eta_c$ are the optical losses including imperfect detection quantum efficiencies in the probe and conjugate beam paths respectively, $\alpha$ is the absorption we are interested in measuring, and $\hat{\nu_p}$, $\hat{\nu_c}$ and $\hat{\nu_{\alpha}}$ are the vacuum/noise operators. When a coherent state $\vert\beta\rangle$, $\beta=\vert\beta\vert e^{i\phi}$, where $\phi$ is the input phase, seeds mode $a$, and only vacuum fluctuations $|0\rangle$ seed mode $b$, then the input state can be written as $\vert\beta, 0, 0, 0, 0\rangle$, where the third, fourth and fifth zeros are the inputs for the vacuum/noise operators $\hat{\nu_p}$, $\hat{\nu_c}$ and $\hat{\nu_{\alpha}}$ respectively. Although not trivial, it is fairly straightforward to calculate the number operators $\hat {N_a} = \hat{a}^{\dagger}_f\hat{a}_f$ and $\hat{N_b}=\hat{b}^{\dagger}_f\hat{b}_f$ for the probe and conjugate beams after detection. Since the sample is placed in the probe beam, and the conjugate beam is used as a reference, we adopt
the photon counts difference $\langle \hat{S_{\alpha}} \rangle= \langle \hat {N_a}-\hat {N_b} \rangle$ as our measurement signal. Note that this double-beam approach is commonly implemented in imaging and spectroscopy applications involving weak absorption~\cite{Samantaray:2017ff,Brida:2010lh}, because it enables the cancellation of classical super-Poissonian noise and provides a direct measurement of the absorption by instantaneous comparison with the unperturbed reference beam. The measurement sensitivity,}
\begin{equation}
\begin{aligned}
\Delta \alpha= \frac{\sqrt{\langle\Delta^2 \hat{S_{\alpha}}\rangle}}{|\partial_\alpha \langle \hat{S_{\alpha}} \rangle|},
\label{DEL}
\end{aligned}
\end{equation}
\textcolor{black}{can then be readily obtained. In this article we define the quantum advantage as the ratio of the sensitivity enabled by the squeezed light, $\Delta \alpha_{\text{sqz}}$, to the one acquired from the coherent light, $\Delta \alpha_{\text{snl}}$, with the same amount of average photon numbers $\langle N_a \rangle$ and $\langle N_b \rangle$ as the twin beams:}
\begin{equation}
\begin{aligned}
\text{Quantum Advantage (dB)} = 10\times\text{log}_{10}\frac{\Delta \alpha_{\text{sqz}}}{\Delta \alpha_{\text{snl}}}.
\label{QD}
\end{aligned}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{Theory.pdf}
\caption{
\textcolor{black}{Theoretical prediction for the quantum advantage (Qu. Adv.) for absorption $\alpha = 5\%$ as a function of optical transmission in the probe beam path $\eta_p$ and conjugate beam path $\eta_c$. The squeezing parameter $r=1.1$ corresponds to 6.5~dB two-mode squeezing.}
\label{Theory}}
\end{center}
\end{figure}
\textcolor{black}{In Fig.~\ref{Theory} we plot the theoretical quantum advantage for absorption $\alpha = 5~\%$ as a function of \textcolor{black}{optical transmission} in the probe beam path $\eta_p$ and conjugate beam path $\eta_c$. The squeezing parameter $r=1.1$, which is calculated from the two-mode squeezing of 6.5~dB~\cite{Li:17} measured by near-perfect photodiodes (see Fig.~\ref{Setup}(c) and Ref.~\cite{Li:20} for further details of the squeezing measurement). It is clear noticeable from the graph that if one could manage to curb the optical loss in both beam paths to be within 10~\%, more than 3~dB quantum advantage for the measurement sensitivity would be readily achievable.}
\subsection{Experimental demonstration of the quantum advantage}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\linewidth]{Setup.pdf}
\caption{
(a) Experimental setup in which a seeded $^{85}$Rb vapor cell produces strong quantum-correlated twin beams via FWM. The twin beams are separated from the pump by a $\sim$~$2\times10^5$~$:1$ polarizer after the cell. The probe beam passes through an absorption `sample' (i.e., a combination of a $\lambda/2$ plate and a PBS) while the conjugate beam serves as a reference, before they are focused onto an EMCCD camera. The camera is enclosed in a light-proof box with filters mounted to block ambient light. The AOM in the probe beam path is used to pulse the twin beams with 2~$\mu$s FWHM and duty cycle of $1/12$. PBS: polarizing beam splitter, PM fiber: polarization-maintaining fiber. (b) Level structure of the D1 transition of $^{85}$Rb atom. The optical transitions are arranged in a double-$\Lambda$ configuration, where $\nu_p$, $\nu_c$ and $\nu_1$ stand for probe, conjugate and pump frequencies, respectively, fulfilling $\nu_p$ + $\nu_c$ = $2\nu_1$. The width of the excited state in the level diagram represents the Doppler broadened line. $\Delta$ is the one-photon detuning, $\delta$ is the two-photon detuning, and $\nu_{\text{HF}}$ is the hyperfine splitting in the electronic ground state of $^{85}$Rb. \textcolor{black}{(c) Measured intensity-difference noise power spectrum for the squeezed twin beams (blue line) and for the SNL (red line), obtained with a radio frequency spectrum analyzer (with resolution and video bandwidth of 300~kHz and 100~Hz, respectively). A squeezing of 6.5~dB is achieved.}
\label{Setup}}
\end{center}
\end{figure}
\begin{figure*}[htbp]
\includegraphics[width=0.75\textwidth]{Noise.pdf}
\caption{ \textcolor{black}{(a) Time sequencing of the pump and twin beams. The pulse duration of 2~$\mu$s and duty cycle of 1/12 is realized by pulsing the probe beam with an AOM. The CW pump beam is present all the time. (b) Typical images of the twin beams with absorption $\alpha=3~\%$ captured by the EMCCD camera. \textcolor{black}{This subfigure is the `real life' version of subfigure (a). It is an image of four consecutive pulses with the pulse width and duty cycle shown in subfigure (a).} (c) Temporal photon counts fluctuations of the probe $N_p(t)$ and conjugate $N_c(t)$ obtained by integrating the photon counts in the cropped regions in (b). Clear similarities can be observed between the twin beams. (d) The strong noise reduction in the subtraction as opposed to the summation of the $N_p(t)$ and $N_c(t)$ depicted in (c) showcases strong correlations between them.}
\label{Method}}
\end{figure*}
The experimental setup and the respective $^{85}$Rb atomic level structure are shown in Fig.~\ref{Setup}(a) and (b). The atomic medium is pumped by a strong ($\sim 500$~mW) narrow-band continuous-wave (CW) laser at frequency $\nu_1$ ($\lambda = 795$~nm) with a typical linewidth $\Delta \nu_1 \sim 100$~kHz. Applying an additional weak ($\sim$~10~nW) coherent seed beam
at frequency $\nu_p = \nu_1 - (\nu_{HF}+\delta)$, where $\nu_{HF}$ and $\delta$ are the hyperfine splitting in the electronic ground state of $^{85}$Rb and the two-photon detuning respectively in Fig.~\ref{Setup}(b) (further experimental details can be found in Ref.~\cite{Li:20}), two pump photons are converted into a pair of twin photons, namely `probe $\nu_p$' and `conjugate $\nu_c$' photons, adhering to the energy conservation $2 \nu_1 = \nu_p + \nu_c$ (see the level structure in Fig.~\ref{Setup}(b)). The resulting twin beams are strongly quantum-correlated and are also referred to as bright two-mode squeezed light~\cite{PhysRevA.78.043816}. As can be seen from Fig.~\ref{Setup}(c), the twin beams exhibit a intensity-difference squeezing of 6.5~dB measured by balanced photodiodes (see Ref.~\cite{Li:20} for further details on the squeezing measurement), which is indicative of strong quantum correlations~\cite{PhysRevA.78.043816}.
After the $^{85}$Rb vapor cell, the pump and the twin beams are separated by a second polarizer, with $\sim$~$2\times10^5$~$:1$ extinction ratio for the pump. The probe beam transverses through a combination of a $\lambda/2$ plate and a PBS, acting as an absorption sample, while the conjugate beam serves as a reference. The twin beams are then focused onto an EMCCD camera (Andor iXon Ultra 897). The EMCCD camera is enclosed in a light-proof box with filters installed at the entrance to block ambient light photons from entering the camera. The acousto-optic modulator (AOM) in the probe beam path is used to pulse the beam with 2~$\mu$s duration (FWHM) and duty cycle of $1/12$. Since the CW pump beam is present all the time, the conjugate beam is therefore also pulsed as a result of the FWM process. The time sequencing of the pump and the twin beams are shown in Fig.~\ref{Method}(a) as the red strap, and the blue and green squares respectively.
We acquire the temporal behavior of the twin beams through the use of the \textit{kinetic} mode of the EMCCD camera. The EMCCD has $512\times512$ pixels with each pixel size of 16~$\mu$m$\times$16~$\mu$m. We focus the twin beams on the camera with an $1/e^2$ beam diameter of $\sim 50~\mu$m, occupying roughly 3 pixels as shown in Fig.~\ref{Method}(b). The temperature of the EMCCD is kept low ($<-65^\circ$C) to curb the thermal noise contributions.
The rest of the EMCCD camera settings can be found in Ref.~\cite{Li:20}.
For each absorption $\alpha$ (acquired by changing the angle of the $\lambda/2$ plate), we capture 200 kinetic series, i.e., 200 frame sequences, with each frame having 35 pairs of probe and conjugate images containing the desired absorption information. We then crop a $10\times$10 pixel region around the maximum-intensity area in each probe and conjugate images, large enough to enclose their respective full beam profiles (see Fig.~\ref{Method}(b)), we thus can obtain the average total number of photons in the probe beam $\bar{N}_p$ and in the conjugate beam $\bar{N}_c$ by integrating photon counts in the cropped regions.
\textcolor{black}{The measurement signal $S_{\alpha}$ is defined as the photon number difference between the probe and conjugate beams:}
\begin{equation}
\begin{aligned}
S_{\alpha} \equiv \bar{N}_{p} - \bar{N}_{c} = (1-\alpha)\bar{N}_{p0} - \bar{N}_{c},
\label{Sig}
\end{aligned}
\end{equation}
\textcolor{black}{where $\bar{N}_{p0}$ and $\bar{N}_p$ are the average numbers of photons in the probe beam before and after the faint absorber respectively, and $\bar{N}_{c}$ is the average number of photons in the conjugate beam. Factoring out $\alpha$, we have}
\begin{equation}
\begin{aligned}
\alpha = -\frac{1}{\bar{N}_{p0}} S_{\alpha} + \frac{S_0}{\bar{N}_{p0}},
\label{Alpha}
\end{aligned}
\end{equation}
\textcolor{black}{where $S_0 \equiv \bar{N}_{p0} - \bar{N}_{c}$ is the photon number difference of the twin beams without the presence of the absorber, which can be treated as a characteristic of the quantum light source itself.}
\textcolor{black}{Also, the relation between the uncertainties of absorption $\alpha$ and the measurement signal $S_{\alpha}$ can be derived from the error propagation formula (see Eq.~(\ref{DEL})):}
\begin{equation}
\begin{aligned}
\Delta \alpha= \frac{\Delta S_{\alpha} }{|\partial_\alpha {S_{\alpha}}|} = \frac{1}{\bar{N}_{p0}}\Delta S_{\alpha},
\label{Del}
\end{aligned}
\end{equation}
\textcolor{black}{where $|\partial_\alpha {S_{\alpha}}| = \bar{N}_{p0}$ is obtained from Eq.~(\ref{Sig}).}
\textcolor{black}{Therefore following Eqs.~(\ref{Alpha}) and (\ref{Del}), the absorption $\alpha$ and its sensitivity $\Delta \alpha$ can be readily obtained from the measurements of $S_{\alpha}$ and $\Delta S_{\alpha}$.}
\textcolor{black}{In Fig.~\ref{ActualMeasurement}, we plot the actual absorption $\alpha$ as a function of the measurement signal $S_{\alpha}$. The inset in Fig.~\ref{ActualMeasurement} is a zoom-in view of \textcolor{black}{the data points with absorption $\alpha<10$~\%} to illustrate the sizes of uncertainties of these two quantities, i.e., $\Delta S_{\alpha}$ on the $x$-axis and $\Delta \alpha$ on the $y$-axis.} \textcolor{black}{In the experiment, we observe $1.3\pm 0.2$~dB quantum advantage in terms of $\Delta S_{\alpha}$ when comparing to shot-noise limited classical measurements for faint absorption levels (see Fig.~\ref{Del}). Due to the fact that $\Delta \alpha \propto \Delta S_{\alpha}$ with $1/\bar{N}_{p0}$ be the proportionality constant (see Eq.~(\ref{Del})), this greater than 1~dB quantum advantage should also translate to $\Delta \alpha$ when compared to its shot-noise limited classical counterparts.}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{ActualMeasurement.pdf}
\caption{
\textcolor{black}{Actual absorption $\alpha$ as a function of the measurement signal $S_{\alpha}$ defined in Eq.~(\ref{Sig}). The inset is a zoom-in view of \textcolor{black}{the data points with absorption $\alpha<10$~\%} to illustrate the sizes of \textcolor{black}{the uncertainties on the $x$-axis, $\Delta S_{\alpha}$, and the uncertainties on the $y$-axis, $\Delta {\alpha}$}.}
\label{ActualMeasurement}}
\end{center}
\end{figure}
For measurements of the quantum noise reduction between the twin beams, we adopt an algorithm originally developed in the spatial domain~\cite{PhysRevA.95.053849,PhysRevA.100.063828} but re-deriving it in the temporal domain. As shown in Fig.~\ref{Method}(c), the temporal photon counts fluctuations of the probe beam $N_p(t)$ and conjugate beam $N_c(t)$ are acquired by integrating photon counts in the cropped $10\times$10 pixel regions for 7000 pairs of twin-beam images during 170~ms.
As expected, strong correlations between the photon counts fluctuations of the twin beams can be observed in Fig.~\ref{Method}(c) and manifested in Fig.~\ref{Method}(d) through the subtraction and addition of these two modes. The quantum noise reduction characterization, $\sigma$, in the temporal domain reads
\begin{equation}
\begin{aligned}
\sigma\equiv\frac{\langle\Delta^2[(N_p(t+\delta t)-N_p(t))-\eta(N_c(t+\delta t)-N_c(t))]\rangle_t}{\langle N_p(t+\delta t)+ N_p(t) + \eta N_c(t+\delta t) + \eta N_c(t)\rangle_t},
\end{aligned}
\label{sigma}
\end{equation}
where $N_p(t+\delta t)-N_p(t)$ and $N_c(t+\delta t)-N_c(t)$ are the subtractions of photon counts in the cropped regions in two successive probe and conjugate images with time interval of $\delta t = 24$~$\mu$s. Since intensities of the twin beams are inherently imbalanced due to the seed power and different transmissions through the vapor cell~\cite{Li:17}, a scaling factor $\eta=0.95$, which is obtained by taking the ratio between the conjugate and probe photon counts in the analysis regions without the presence of the absorption sample, is applied to the conjugate mode to rescale its photon count before the two modes are subtracted.
\textcolor{black}{Note that each image is involved in averaging over the spatial intensity profile of the beam, and the scaling factor effectively balances not only any differences in the averaging of the beam intensity profiles but also the intensity fluctuations. The subtraction of two successive images leads to the cancellation of the low-frequency portion of classical noise so that the rest of fluctuations are in the shot-noise-limited regime~\cite{PhysRevA.95.053849,PhysRevA.100.063828}.} The numerator of Eq.~(\ref{sigma}) represents the temporal variance of the intensity-difference noise between the probe and conjugate pulses. The denominator gives the mean photon counts for the probe and conjugate pulses used for the analysis and represents the shot noise. For coherent state pulses $\sigma =1$, which corresponds to the SNL, while for thermal light or other classical states $\sigma >1$. Temporally quantum-correlated beams, like the twin beams generated in our experiment, will result in $\sigma <1$, with a smaller $\sigma$ corresponding to a larger degree of quantum correlations (i.e., two-mode squeezing).
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{Sigma.pdf}
\caption{
\textcolor{black}{Temporal quantum noise reduction $\sigma$ as a function of absorption $\alpha$ for the intensity squeezed light (blue squares) and coherent light (red dots). Dashed blue line is the theoretical prediction with $\eta_p=0.61$, $\eta_c=0.63$ and $r=1.1$.}
\label{Sigma}}
\end{center}
\end{figure}
In Fig.~\ref{Sigma}, we plot $\sigma$ as a function of absorption $\alpha$ for the squeezed light together with coherent light. For each $\alpha$, we average 5 sets of 200 kinetic series and designate the error bar with one standard deviation. As expected, $\sigma < 1$ for the squeezed light (blue squares), while $\sigma \cong 1$ when the twin beams are replaced with two coherent beams (red dots). The notable degradation of the temporal quantum noise reduction measured by the EMCCD camera with respect to the one measured by balanced photodiodes in Fig.~\ref{Setup}(c) can be mainly attributed to
a much worse quantum efficiency of the EMCCD camera at 795~nm (merely 70~\% as opposed to at least 94~\% for photodiodes). We also repeated the experiment with different pulse duty cycles (i.e., $\delta t$ in Eq.~(\ref{sigma})), but they seemed to play an nonessential role on the quantum noise reduction as long as we were in the shot-noise-limited regime, i.e., $\sigma$ is still close to unity for coherent beams.
From Eqs.~(\ref{DEL}) and~(\ref{QD}) we can easily arrive at
\begin{equation}
\begin{aligned}
\begin{split}
\text{Quantum Advantage (dB)} = 10\times\text{log}_{10}\frac{\Delta \alpha_{\text{sqz}}}{\Delta \alpha_{\text{snl}}}\\
= 10\times\text{log}_{10}\sqrt{\frac{\langle\Delta^2 \hat{N_{\alpha}}\rangle_{\text{snl}}}{\langle\Delta^2 \hat{N_{\alpha}}\rangle_{\text{sqz}}}} = 10\times\text{log}_{10}\sqrt{\frac{1}{\sigma}}.
\end{split}
\label{QD2}
\end{aligned}
\end{equation}
We thus can use the same data depicted in Fig.~\ref{Sigma} to plot the quantum advantage versus absorption $\alpha$. The results are shown in Fig.~\ref{QuantumAdvantage}. Theoretical predictions for the temporal quantum noise reduction characterization $\sigma$ and the quantum advantage as a function of absorption $\alpha$ are plotted as dashed blue lines in Figs.~\ref{Sigma} and~\ref{QuantumAdvantage}, where excellent agreements between experiment and theory can be seen. At those faint absorption levels ($\alpha \leq 10$~\%) in Fig.~\ref{QuantumAdvantage}, the observed quantum advantage can be more than 1.2 dB, although \textcolor{black}{\textit{total}} optical losses \textcolor{black}{(including the transmission loss imposed by optics and imperfect detection quantum efficiency imposed by the EMCCD camera)} in the paths of the twin beams are significant - nearly $39$~\% in the probe path and nearly $37$~\% in the conjugate path. This is mainly due to a relatively low quantum efficiency of the EMCCD camera at 795~nm ($\sim 70~\%$) and imperfect transmission of the band pass filters ($\sim94\%$) mounted in front of the light-proof box. \textcolor{black}{If we were able to overcome this main obstacle of the experiment by employing a much more efficient camera, we would have a much higher quantum advantage approaching 3~dB as implied by the theoretical curves in Fig.~\ref{Theory}. We notice that a most recent work~\cite{doi:10.1063/5.0009681} has also demonstrated quantum advantage in absorption measurement using a single-mode amplitude squeezed light generated with an optical phase-sensitive amplifier. The reported advantage (according to Fig.~5 in Ref.~\cite{doi:10.1063/5.0009681}) is less than 0.5~dB.}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{QuantumAdvantage.pdf}
\caption{
\textcolor{black}{Quantum advantage as a function of absorption $\alpha$. Dashed blue line is the theoretical prediction with $\eta_p=0.61$, $\eta_c=0.63$ and $r=1.1$. The quantum advantage is only significant ($> 1$~dB) for small values of $\alpha$ ($< 20~\%$), and for $\alpha > 60~\%$ there is no quantum advantage.}
\label{QuantumAdvantage}}
\end{center}
\end{figure}
\textcolor{black}{It is worth mentioning that taking measurements using photodetectors would yield better results due to photodiodes' much higher quantum efficiency. However, the main drawback of using photodetectors is their much higher power requirement. For an EMCCD camera, a few nW input power is more than enough to yield a clear signal-to-noise ratio, however, for a photodetector to provide sufficient signal clearance from its electronic noise floor, the input power has to be in the range of tens of $\mu$W. For example in our experiment, in order to have a signal noise power that is 10 dB above the electronic noise floor, we have to shine a coherent beam of light of at least 50~$\mu$W to the photodetector (given our squeezing level of 6.5~dB, that implies a merely 3.5 dB clearance from the electronic noise floor for 50~$\mu$W squeezed light). One of the most important implementations of our experimental scheme is to characterize biological samples without imposing light-induced damages, a much higher input light power would hence defeat this purpose.}\\
\section{Discussion}
Overall, our experiment realizes a practical scheme that allows the SNL in the direct absorption measurement to be overcome. We demonstrate that by using the intensity squeezed light more than 1.2~dB quantum advantage is achieved for the measurement sensitivity at faint absorption levels ($\leq10\%$). We thus experimentally demonstrate the advantage of squeezed light for measurements on open systems. \textcolor{black}{We also theoretically demonstrate that more quantum advantage ($>$~3~dB) is very likely attainable by means of a proper optical loss management.} We use seeded FWM process in an atomic $^{85}$Rb vapor cell to generate the quantum-correlated twin beams of light. It is also the first experiment that uses quantum light generated with FWM instead of SPDC to demonstrate a sub-shot-noise absorption measurement. Major advantages of this FWM-based quantum light generation scheme include an ultra-high photon-pair flux up to~$10^{16}$ photons/s, which is a few orders of magnitude higher than the fluxes produced by SPDCs~\cite{Jechow:08,Villabona-Monsalve:2018fy,Varnavski:2017rp}, and narrow-band probe and conjugate beams ($\sim 20$~MHz)~\cite{Clark:2014vf,Glasser2012a}, which can be readily integrated into quantum networks through coupling with micro-resonators/cavities. \textcolor{black}{Also, although the small bandwidth feature of the twin beams is not used in the experiment, we do take advantage of it by making a `single-mode' approximation for the twin beams in the theoretical analysis. The fact that our experimental results agree very well with the theory based on the `single-mode' approximation confirms the importance of the narrow band feature of the twin beams.} Moreover, the seeded FWM process offers sufficient gains in a single-pass configuration producing bright quantum-correlated beams of light without a cavity, making it possible to preserve the multi-spatial-mode nature of the bright twin beams~\cite{PhysRevLett.109.043602,Corzo:11}. Our quantum light generation together with the direct absorption measurement scheme reported here can be therefore greatly beneficial to many applications involving characterizing chemical and biological samples, where sub-SNL absorption measurements are highly desirable~\cite{Genovese_2016,QuantumImaging}.
We gratefully acknowledge the support of Air Force Office of Scientific Research (Award No. FA-9550-18-1-0141), Office of Naval Research (Award No. N00014-20-1-2184), and the Robert A. Welch Foundation (Grants No. A-1261 \& A-1943). F.L. acknowledges support from the Herman F. Heep and Minnie Belle Heep Texas A\&M University Endowed Fund held/administered by the Texas A\&M Foundation.
|
2,877,628,090,821 | arxiv | \section{Introduction}
The Encke Gap is a 320-km-wide opening in the outer part of Saturn's A ring centered on the orbit of the small moon Pan. In addition to Pan itself, this gap contains several faint ringlets with spectral and photometric properties that indicate they are composed primarily of dust-sized grains less than 100 microns wide. These ringlets attracted interest when they were first observed by the Voyager spacecraft because they contained prominent ``clumps'' of bright material associated with distinct ``kinks'' in the ringlets' radial position \citep{Smith82, FB97}. However, it was difficult to investigate the structure and dynamics of these longitudinally-confined features due to the restricted amount of data obtained by the Voyager missions.
Now, thanks to the Cassini spacecraft, a much more extensive data set is available for investigations of the Encke Gap ringlets. In particular, the Encke Gap has now been imaged multiple times since Cassini arrived at Saturn in 2004, allowing the evolution and motion of this material to be tracked over timescales from weeks to years. Cassini data also provide information about other dusty ringlets in Saturn's rings \citep{Porco05, Horanyi09}, which can help clarify the dynamical processes operating in the Encke Gap. For example, a ringlet located within the Cassini Division's Laplace Gap demonstrates ``heliotropic'' behavior: its geometric center is displaced away from Saturn's center towards the Sun \citep{Hedman10}. This happens because the particles in this ringlet are sufficiently small that solar radiation pressure can induce significant orbital eccentricities. Since the spectral and photometric properties of the Encke gap ringlets indicate that they are also composed primarily of dust-sized particles \citep{Hedman11}, their structure should also be affected by such non-gravitational forces.
After a brief introduction to the Encke Gap's architecture (Section 2), this report will describe the Cassini imaging observations of the Encke Gap obtained between 2004 and 2011 that provide the best information about the structure and evolution of material in this region (Section 3). Section 4 documents the distribution and motion of bright clumps in the denser ringlets. This study reveals that the bright clumps do not follow the expected trajectories of test particles under the influence of the combined gravitational fields of Saturn and Pan. Section 5 discusses structures produced by Pan's perturbations on the nearby dusty material. Section 6 examines the orbital properties of the particles in the ringlets and demonstrates that non-gravitational forces like solar radiation pressure are indeed influencing the structure of these ringlets. Finally, Section 7 discusses some of the physical processes that could explain the longitudinal variations in the ringlets' orbital properties, the distribution of both the clumps along each ringlet and the radial locations of the ringlets within the gap. Note that these theoretical considerations only represent an initial examination of some of the dynamical phenomena that could be relevant to the Encke Gap ringlets' structure and evolution, and are not meant to provide an exhaustive
or complete picture of the ringlets' complex dynamics.
\section{Architecture of the Encke Gap}
\begin{figure}
\centerline{\resizebox{4in}{!}{\includegraphics{gapim_080912.pdf}}}
\caption{One of the highest resolution images of the Encke Gap obtained by
the Cassini spacecraft. This observation was made on day 183 of 2004 during Cassini's orbit insertion (N1467351325). The image has been heavily stretched to show the ringlets in the Encke Gap, causing the regions outside the gap to appear saturated. Labels mark the positions of the four ringlets observed in this region. The inner edge of the gap appears scalloped because Pan's gravity has excited radial motions in the nearby ring material \citep{Porco05}.}
\label{egapim}
\end{figure}
\begin{figure}
\centerline{\resizebox{4in}{!}{\includegraphics{egap_hrradplot_020411.pdf}}}
\caption{Profiles of average brightness versus radius through the gap derived
from the two observations of this gap with the best combination of resolution and signal-to-noise. Brightness is measured in terms of normal $I/F$, which is the observed $I/F$ values multiplied by the cosine of the emission angle (see Section 3). The upper profile is derived from the same image shown in Figure~\ref{egapim}, while the lower profile is derived from images taken on day 223 of 2009 during Saturn's equinox. Both profiles show the same basic features, including three narrow ringlets and a broad shelf at 133,680 km (for the names of these features, see Figure~\ref{egapim}). Note the differences in radial positions and relative brightnesses of the three narrow ringlets. These are due to the longitudinal variability of these structures.}
\label{egapprof}
\end{figure}
The basic architecture of the Encke Gap is best illustrated by Figures~\ref{egapim} and~\ref{egapprof}, which provide images and radial brightness profiles derived from the highest resolution and best signal-to-noise images of the Encke Gap obtained so far by Cassini (cf. Porco {\it et al.} 2005). These images and plots show that most of the faint material in this region is organized into three narrow ringlets and one broader feature. One narrow ringlet lies near the center of the gap, close to Pan's orbit at 133,584 km from Saturn's center. This feature is designated the ``Pan ringlet'' here, although it could just as well be called the ``central ringlet''. The two other narrow ringlets are situated on either side of the Pan ringlet. For want of a better terminology (thus far, no moon has been found within either of these ringlets), we will call the ringlet centered around 133,484 km the ``inner ringlet'' and the ringlet centered around 133,720 km the ``outer ringlet''. Note that the widths, peak brightnesses and locations of all three ringlets are different for the two profiles shown in Figure~\ref{egapprof}. This is an example of the longitudinal variability exhibited by all three of these ringlets. Closer inspection of these images and profiles reveals a broad shelf of material extending inward from the outer ringlet to an orbital radius of about 133,680 km. This shelf, which was called the ``fourth ringlet" by Porco {\it et al.} (2005), is considerably fainter than the other features in the Encke Gap and can only be seen with an appropriate combination of image resolution and viewing geometry. This broad feature also appears to be much more homogeneous than the three narrow ringlets. While wakes can be observed in this feature close to Pan (see Section 5 below), we have never observed anything like the clumps or kinks seen in the other three ringlets.
These ringlets all exist within a complex dynamical environment that is strongly influenced by the gravity of Saturn's small moon Pan \citep{Showalter91}. Pan travels in a nearly circular orbit (eccentricity $\sim 10^{-5}$) through the center of the gap with a semi-major axis $a_P=133,584$ km and an orbital period of 0.575 days \citep{Jacobson08}. Due to Keplerian shear, material within and surrounding the gap drifts in longitude relative to Pan and therefore periodically encounters the moon. Since the gap is so narrow, these relative motions are very slow and encounters with Pan are correspondingly infrequent. For example, particles at the edges of the gap (at orbital radii of 133,423 km and 133,745 km) will reach conjunction with Pan only once every 543 orbits, or roughly every 315 days.
Nevertheless, each time a particle has a close encounter with Pan, its orbital parameters will be perturbed by the moon's gravity. Indeed, Pan's influence is clearly visible in both the few-kilometer-high waves on the edges of the gap and the moonlet wakes found in the A-ring material on either side of the gap \citep{CS85, Showalter86, Horn96, Weiss09}. Based on the amplitudes of the waves Pan generates at the edge of the Encke Gap, the mass ratio of Pan to Saturn ($m_P/M_S$) has been estimated to be about $0.8*10^{-11}$, which corresponds to a mass $m_P \simeq 5*10^{15}$ kg \citep{Porco07, Weiss09}.
\begin{figure}
\resizebox{6in}{!}{\includegraphics{hill_illustration_081012x.pdf}}
\caption{Schematic representation of the expected particle trajectories relative to Pan, computed using Hill's equations \citep{MD99}. Units of Hill radii (indicated along the bottom and left axes) are converted into physical coordinates (indicated along the top and right axes), assuming Pan's Hill radius is 18 km and that Pan's semi-major axis $a_P=$133,584 km. Note that the trajectories are computed assuming particles approach Pan on initially circular orbits with a range of semi-major axes $a$. The particles approach Pan from the left when $a<a_P$ and from the right when $a>a_P$. Dark shaded bands at the top and bottom of the plot indicate the edges of the gap, and the lighter shaded bands indicate the locations of the inner, Pan and outer ringlets. }
\label{hillplot}
\end{figure}
Particles orbiting within the Encke Gap are even more strongly affected by Pan's gravity. Figure~\ref{hillplot} illustrates the expected trajectories of small particles within the Encke Gap, assuming that the only forces acting on the particles come from Pan's and Saturn's gravitational fields. These trajectories are computed using Hill's eqautions (cf. Murray and Dermott 1999), and the scale of structures in this diagram is set by Pan's Hill radius $R_H = a_P(m_P/3M_S)^{1/3} \simeq 18$ km. For example, while particles on orbits more than a few Hill radii from Pan's semi-major axis drift past the moon, particles orbiting close to $a_P$ are unable to drift past Pan, but will instead execute horseshoe or tadpole motion around the moon's L3, L4 and L5 Lagrange points (i.e. their orbital longitude relative to Pan will librate instead or circulate). The transition between these two regimes occurs at a critical distance from Pan's semi-major axis $\Delta a_{crit} \simeq 2.4 a_P(m_P/M_S)^{1/3} \simeq 65$ km \citep{DM81, MD99}. However, orbits with semi-major axes near $a_P\pm\Delta a_{crit}$ are actually highly unstable because they involve extremely close encounters with Pan \citep{DM81}. Such close encounters produce large changes in the particles' orbital semi-major axes and eccentricities, and cause the orbital parameters to undergo large stochastic variations \citep{DQT89}. Particles in this ``chaotic zone'' are likely to be lost either to collisions with the moon itself or with the gap edges. Numerical experiments and analytical theory suggest that the orbits of particles drifting past the moon will become chaotic when the semi-major axes are closer to Pan's orbit than $\Delta a_{d} \simeq 1.3a_P(m_P/M_S)^{2/7} \simeq 120$ km \citep{DQT89}. Similarly, particles on horseshoe orbits will become chaotic when their semi-major axes are greater than $\Delta a_{h} \simeq f_{h} a_P(m_P/M_S)^{1/3}$ from Pan's orbit, where $f_{h}$ is a numerical constant between 0.5 \citep{WW74, GT82} and 1.3 \citep{Dermott80}. Stable simple horseshoe orbits are therefore only found within 15 or 35 km of Pan's orbit.
The Pan ringlet always lies within $\Delta a_{crit}$ of Pan's orbit, and thus almost certainly consists of material moving in horseshoe and tadpole orbits around the moon's Lagrange points \citep{Showalter91}. By contrast, the inner, outer and fourth ringlets all are more than $\Delta a_{crit}$ from 133,584 km, and thus are likely composed of material that drifts continuously past Pan. The motions of the bright clumps in the inner and outer ringlets, as well as the presence of moonlet wakes in all these structures are consistent with this supposition (see below). However, note that both the inner and fourth ringlets may overlap the semi-major axis range where particle orbits should be chaotic (i.e., they lie within $\Delta a_d$ of Pan's orbit). This could imply that inter-particle interactions or some other process is affecting these particles' orbits and stabilizing these ringlets. Indeed, one might be tempted to regard the outer edge of the inner ringlet and the inner edge of the fourth ringlet as marking the edges of the chaotic zone.
\section{Observations and data reduction procedures}
This investigation of the Encke Gap structures will rely exclusively on pictures obtained by the Narrow Angle Camera (NAC) of the Imaging Science Subsystem onboard the Cassini spacecraft \citep{Porco04}. The observations that are most informative about the overall structure and dynamics of the Encke Gap ringlets include:
\begin{itemize}
\item Movie sequences obtained when the camera pointed at one place in the Encke Gap and watched material orbit through the field of view over a significant fraction of an orbital period. These observations provide snapshots of the longitudinal structure of the ringlets at particular times. The thirteen movies used in this analysis, which are the best in terms of longitudinal coverage, are listed in Table~\ref{moslist}.
\item The so-called SATELLORB observations designed to periodically observe various small moons in order to refine and track their orbits. A subset of these images targeted at Pan also capture nearby parts of the Encke Gap. Specifically, Table~\ref{suplist} lists 189 images where the ring opening angle was sufficiently high (more than $1^\circ$), the radial resolution was sufficiently good (better than 20 km/pixel) and a sufficiently broad range of longitudes were observable (at least 1$^\circ$). These images were obtained in between the more extensive movies, and thus provide additional information about the evolution and motion of certain clumps.
\item The PANORBIT observation made in 2007-143 during Rev 45. This is a sequence of 158 images (N1558590310- N155861997, emission angle 68$^\circ$, phase angle 79$^\circ$) targeted at Pan as it moved around the planet. These images also captured the part of the Encke Gap surrounding Pan, enabling us to observe how the structure of the central ringlet changes with true anomaly.
\end{itemize}
We also presented above some data from selected high-resolution, high signal-to-noise images of the Encke gap (N1467351325 and N1628681217-N16281691, see Figure~\ref{egapim} and ~\ref{egapprof}). However, this report will not include a thorough analysis of all the highest resolution images of the Encke Gap. While such images can provide very useful data regarding the fine-scale morphology of individual clumps, we will limit our scope here to the region's global behavior.
All the relevant images were calibrated using the standard CISSCAL routines \citep{Porco04} to remove instrumental backgrounds, apply flatfields and convert the raw data numbers to $I/F$, a standardized measure of reflectance that is
unity for a Lambertian surface at normal incidence and emission. The images were geometrically navigated using the appropriate SPICE kernels and this geometry was refined based on the position of sharp ring edges in the field of view. Whenever practical, this navigation used the outer edge of the Keeler Gap as a fiducial, but when the resolution of the images was either insufficient to resolve this gap or so high that the gap was not present in the field of view, the edges of the Encke Gap were used instead. While neither the Keeler Gap's outer edge nor the Encke Gap's edges are perfectly circular, the variations in the relevant edge positions are sufficiently small (only a few km) that they do not impact efforts to quantify and track the longitudinal positions of the clumps. However, these imperfections cannot be ignored in detailed studies of the ringlets' radial positions (see below).
For the high-resolution images described above, the rings are sufficiently homogeneous that we can reduce the geometrically-navigated data from each image into a single radial brightness profile by simply averaging over all longitudes. For the other observations, however, a single image can contain multiple clumps or kinks, so reducing the data to a single radial scan is not appropriate. Instead, the brightness measurements from each image are re-projected to produce ``maps'' of the Encke Gap on a uniform grid of radii and longitudes relative to Pan (derived from the appropriate SPICE kernels). For the SATELLORB and PANORBIT observations, these maps provide a useful basis for subsequent data analysis. However, for the movie sequences listed in Table~\ref{moslist}, which cover a broad range of co-rotating longitudes at a single time, individual images are less useful than the combined data set. Hence the relevant maps derived from individual images are interpolated onto a common radius and longitude scale and then assembled into a single mosaic spanning a large fraction of the Encke Gap (see Figure~\ref{mos0}). These mosaics can then be processed using the same basic procedures as the individual maps.
\begin{figure}[tb]
\resizebox{6in}{!}{\includegraphics{hiphamov30_pgg_forjab_062512.pdf}}
\caption{Example of part of a mosaic generated from Rev 030 HIPHAMOVD observation. This mosaic shows the brightness of the rings as a function of radius and longitude, and within this figure one can clearly see clumps in the Pan ringlet at a radius of 133,584 km and the Inner ringlet at 133,484 km. One can even see a few features in the outer ringlet just interior to the Gap's outer edge at 133,745 km.}
\label{mos0}
\end{figure}
Besides re-projecting the data into convenient maps and mosaics, the relevant geometric information is also used to compute the cosine of the emission angle $\mu$. By multiplying the observed brightness values by this quantity, the observed $I/F$ can be converted into an estimate of the ``normal $I/F$'', which for low optical depth features like the Encke Gap ringlets should be independent of emission angle.
Depending on the resolution and quality of the observation, different procedures were used to quantify the brightness and location of these ringlets. The finite resolution of the images influence both the peak brightness and radial width of the ringlets, so the brightness of the ringlet is instead quantified using the radially integrated normal $I/F$ of the ringlet, or ``normal equivalent width'' (abbreviated NEW in Figures~\ref{panb1},~\ref{inb1} and ~\ref{outb1} below). For low optical-depth features like the Encke Gap ringlets, this integrated quantity is independent of the image resolution. Profiles of normal equivalent width versus longitude derived from different observations can therefore be compared to one another relatively easily and reliably.
Whenever possible, the ringlet's radial brightness profile at each longitude was fit to a Lorentzian in order to obtain estimates of both the ringlets' radial position and its equivalent width. The fitting procedure for each ringlet is tuned to minimize contamination from the other ringlets and to cope with variations in the radial position of the ringlet with longitude and time.
For the Pan and inner ringlets, extrema in the derivative of the radial brightness profile are used to make a preliminary estimate of the location of the ringlet and to determine the radial range included in the fit. For the Pan (inner) ringlet, the point of maximum positive slope between 133,520 and 133,600 km (133,420 km and 133,500 km) provides an estimate of the ringlets' inner edge position $r_1$, while the point of largest negative slope between 133,560 and 133,630 km (133,470 and 133,530 km) yields an estimate for the ringlet's outer edge location $r_2$. The average of these two numbers therefore provides an estimate of the center of the ringlet, and a radial region centered on this location with a width that is the larger of 60 km and $2(r_2-r_1)$ is selected and fit to a Lorentzian plus linear background (the lower limit of 60 km ensures that the fitted region is broad enough to contain the entire ringlet, see Figure~\ref{egapprof}).
The outer ringlet is located closer to the edge of the gap than the other ringlets, and therefore required a somewhat more complex procedure that includes removing the background signal due to the nearby gap edge. This background was estimated by interpolating the brightness profile on either side of the ringlet, which requires a preliminary estimate of the ringlet's position and radial extent. The center of the ringlet is estimated as the location of the minimum in the second derivative of the brightness profile between 133,710 and 133,730 km. Preliminary estimates of the ringlet edge positions were obtained as the maximum of 20 km and 1.5 times the distance to the minimum slope within 20 km of the ringlet center (the lower limit of 20 km ensures that the fitted region is broad enough to contain the entire ringlet, see Figure~\ref{egapprof}). However, in order to obtain a sensible background level, the outer edge of the fit region is constrained to at least two radial bins short of the point of maximum slope on the gap edge. The background level under the ringlet is then obtained by a spline interpolation of the brightness data outside the selected region. The interpolation is actually applied to the logarithm of the brightness measurements because the abrupt change in slope near the edge of the gap made interpolation of the raw brightness measurements difficult. After removing the background, the remaining data are then fit to a Lorentzian plus constant offset.
For observations obtained at lower resolutions or at lower phase angles (where the ringlets are comparatively faint), the above fitting routines were not appropriate and so it was not possible to estimate the radial positions of the ringlet. However, the integrated brightness of the ringlet can still be computed. For the Pan ringlet we compute the integrated brightness within 50 km of 133,585 km. A background level based on the average brightness outside this region can be removed from these profiles if required. For the inner and outer ringlets, which lie closer to the edges of the gap, the radial region containing the ringlet and the appropriate background levels are computed using the same basic method as described in the previous paragraph. The edges of the ringlet region are determined based on extrema in the slopes, and the background in this region is determined by a cubic spline interpolation of the log-transformed data on either side of this region.
Mosaics where the peak-fitting procedures were successful are marked with P or R in Table~\ref{moslist}. By contrast, mosaics where only the integrated brightness could be computed are marked with an I. The data obtained from the SATELLORB observations (Table~\ref{suplist}) are entirely derived from simple integrations, and the PANORBIT observations are all processed with peak-fitting routines. Note that the different resolutions and processing techniques used on these different data sets could potentially complicate any effort to compare the absolute brightness of the ringlets derived from different observations, and hence we will not attempt such photometric comparisons here. Instead, this paper will focus exclusively on the structure and morphology of these ringlets, which are more robustly determined by these procedures.
Uncertainties in these relative brightness and position estimates are dominated by systematic errors in the fits and background removal rather than statistical noise, and thus are difficult to quantify {\it a priori}. Based on the lack of obvious long-wavelength drifts outside the clump-rich regions in the brightness profiles for the inner and Pan ringlets, systematic errors in the brightness structure of the clumps in these ringlets are expected to be negligible. The brightness variations outside the clumps are more substantial for the outer ringlet, but even here the morphology of the clumps are very repeatable between observations (see Figure~\ref{outb1} below), so systematic errors in the brightness of these clumps should also be small (probably less than 10\%). Finally, the repeatability of long-wavelength structure in the radial positions for these ringlets (see Section 6) implies that systematic errors in the radial positions of the inner and Pan ringlets are typically less than 1 km. However, these estimates are based on heuristic {\it a posteriori} arguments and not rigorous quantitative analyses. Hence in order to avoid giving a misleadingly precise impression of the relevant uncertainties, we will not plot error bars on the various longitudinal profiles presented in this paper.
\section{Brightness variations in the ringlets}
\begin{figure}
{\resizebox{3in}{!}{\includegraphics{mosaicdisp_sum_pg1_080912x.pdf}}}
{\resizebox{3in}{!}{\includegraphics{mosaicdisp_sum_pg2_080912x.pdf}}}
\caption{Images of the Encke Gap mosaics constructed from the observing sequences listed in Table~\ref{moslist}. The data from the SATSRCH observation are not shown here due to their low resolution. Each panel displays the ring brightness as a function of radius and longitude relative to Pan. Each image is individually stretched to best highlight the ringlets in the gap. Black regions in each map correspond to areas that were not observed during the observing sequence. Note the restricted longitude range of the clumps in the central Pan ringlet, and the steady movement of the clumps in the inner and outer ringlets relative to Pan.}
\label{mos1}
\end{figure}
Figure~\ref{mos0} illustrates the brightness variations that can be seen within the inner, outer and Pan ringlets. All three ringlets contain localized regions of enhanced brightness, which we interpret here as concentrations or ``clumps'' of material.\footnote{Alternative interpretations of the brightness variations as the result of vertical structures producing changes in the amount of material along certain lines of sight are much less plausible. If the bright regions were just the result of projection effects, then the distribution of these features would change radically with the observation geometry.
Instead, image sequences taken in very different observing geometries exhibit the same basic pattern of clumps (see Table~\ref{moslist} and Figures~\ref{panb1},~\ref{inb1} and \ref{outb1}), which is much more consistent with simple variations in the local particle density.} Figure~\ref{mos1} shows the full mosaics derived from most of the observations listed in Table~\ref{moslist} (the SATSRCH observation is not illustrated due to its lower resolution). These mosaics show that these clumps are not distributed randomly along each ringlet. In particular, the clumps in the Pan ringlet are always found between longitudes of $0^\circ$ and $ +60^\circ$ in a Pan-centered coordinate system, that is, between Pan and its leading Lagrange point. Studies of Voyager images of this ringlet taken around 1980 \citep{FB97} showed a similar pattern, indicating that such an asymmetric clump distribution is a persistent feature of this ringlet.
Next, consider the inner and outer ringlets. These features are located outside of Pan's horseshoe zone (see above), so this material should drift slowly relative to Pan. Indeed, the clumps in the inner ringlet can be observed to slip slowly ahead of Pan, while those in the outer ringlet move slowly backwards, as expected. However, within each ringlet, the distribution of clumps is again remarkably persistent. For the inner ringlet, the clumps cluster in a region between 110$^\circ$ and 160$^\circ$ wide. This is again consistent with the Voyager observations 25 years earlier \citep{FB97}, implying that something may be preventing these clumps from efficiently dispersing all around the ringlet. The clumps in the outer ringlet, by contrast, seem to be a bit more broadly distributed, with a dense cluster of clumps roughly 20$^\circ$ wide lagging 120$^\circ$ behind a more spread-out array of clumps (see top right panel of Figure~\ref{mos1}). Again, this basic pattern of clumps seems to persist for years. Note that all the clumps in both the inner and outer ringlets drifted past Pan multiple times during the course of these observations, so the distribution of the clumps in these ringlets appears to be moderately robust against perturbations from that moon.
The evolution of these clumps' morphology and spatial distribution between 2004 and 2011 can be more closely examined with the longitudinal brightness profiles shown in Figures~\ref{panb1},~\ref{inb1} and \ref{outb1}. These plots show the radially-integrated brightness of the ringlets as a function of longitude derived from the various mosaics listed in Table~\ref{moslist}. Also useful are the plots shown in Figures~\ref{clumptrackpan}-\ref{clumptrackdet2}, ~\ref{clumptrackin} and~\ref{clumptrackout}, which graph the positions of brightness maxima in these profiles as functions of time. In order to facilitate comparisons between observations taken at various times, a different co-rotating longitude system has been used to plot the data for each ringlet.
Identifying individual clumps and tracking their motions is challenging because clumps are not always isolated brightness peaks that drift relative to each other. Instead, regions of enhanced brightness have a range of morphologies, including tightly-packed clusters and looser archipelagos of brightness maxima that can split, merge or even drift as units.
This complicates any effort to quantify the motion or evolution of these structures, and consequently we will not attempt to generate a comprehensive catalog of these features. However, in all three ringlets, certain regions consisting of one or more bright clumps appear to be remarkably persistent across the various observations. Hence we can identify and track these broader-scale features over several years with some degree of confidence (cf. Showalter 2004), although we must admit that even some of these features could form or dissolve between observations taken years apart. In the following sections, we will examine the overall distribution of the brightness maxima and the detailed evolution of a few particular structures in each ringlet. \nocite{Showalter04}
\subsection{Pan ringlet}
\begin{figure}
\centerline{\resizebox{3.2in}{!}{\includegraphics{satsrchintplot_080912.pdf}}}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_pr_pg1_080912.pdf}}}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_pr_pg2_080912.pdf}}}
\caption{Plot of the {\bf Pan ringlet's} radially-integrated brightness (normal equivalent width) versus longitude from Pan based on the data from the observations listed in Table~\ref{moslist}. The 000/SATSRCH profile comes from radial integration of the brightness profile, while the other brightness profiles are all derived from Lorentzian fits to the ringlet. Fits with peak radii more than 30 km from 133,585 km are removed and the remaining data smoothed over 5 samples for the sake of clarity. Narrow spikes between 23$^\circ$ and 30$^\circ$ in the 00A/SPKMOVPER profile and around $60^\circ$ in the 044/FMOVIE profile are due to stars and cosmic rays, while the clumps all have a finite longitudinal width. }
\label{panb1}
\end{figure}
\begin{figure}
\centerline{\resizebox{5in}{!}{\includegraphics{clumptrack_ov_021711.pdf}}}
\caption{Plot showing the locations of brightness peaks in the {\bf Pan ringlet} as a function of longitude and time. The black plusses are measurements derived from the largely complete mosaics shown in Figure~\ref{panb1}, while the green diamonds are derived from the SATELLORB images listed in Table~\ref{suplist}. Note that the latter data only cover the region immediately in front of Pan.}
\label{clumptrackpan}
\end{figure}
\begin{figure}
\centerline{\resizebox{5in}{!}{\includegraphics{clumptrack_detdig_021711.pdf}}}
\caption{Longitudinal profiles of the Pan ringlet brightness obtained between days 140 and 250 of 2005. The profiles are stacked vertically with spacings proportional to their time separation, and the green diamonds mark the locations of brightness maxima at the times given on the right-hand vertical axis. Dotted lines tracing the motion of particular clumps are included to guide the eye. Note the clump that starts near 5.5$^\circ$ first drifts towards Pan, but then appears to reverse direction between days 200 and 220, such that it collides with the clump that had been following it between days 220 and 240.}
\label{clumptrackdet1}
\end{figure}
\begin{figure}
\centerline{\resizebox{5in}{!}{\includegraphics{clumptrack_detdig2_021811.pdf}}}
\caption{Longitudinal profiles of the Pan ringlet brightness obtained between days 330 of 2007 and 70 of 2008. The profiles are stacked vertically with spacings proportional to their time separation, and the green diamonds mark the locations of brightness maxima at the times given on the right-hand vertical axis. In this case, the motions of individual clumps are less obvious. However, the morphology of the clump around 5$^\circ$ in front of Pan changes in an interesting way. In 2007, this clump had an asymmetric profile with a single brightness peak. In 2008 a second peak appears and the two peaks begin to separate. Around day 50, each of those two peaks splits to produce a total of four peaks, which again move apart over time.}
\label{clumptrackdet2}
\end{figure}
First, let us consider the Pan ringlet data shown in Figures~\ref{panb1} and~\ref{clumptrackpan}. Note that the coordinate system used in these plots is simply longitude relative to Pan. When this region was first observed in 2004 the clumps were concentrated in three regions roughly 5$^\circ$, 20$^\circ$ and 50$^\circ$ in front of Pan. Over the next year, the clumps less than 30$^\circ$ in front of Pan seem to rapidly converge into a region roughly $5^\circ$ in front of Pan, while the clumps around 50$^\circ$ dispersed slightly. When these clumps were again seen in late 2006, the clumps could still be divided into two groups. The smaller group close to Pan appears to have spread over the region between 5$^\circ$ and 10$^\circ$, while the clumps $50^\circ$ in front of Pan had continued to disperse. In fact, this group appears to have split into two clusters, one centered around $35^\circ$ and one remaining around 45$^\circ$. Over the next year and a half, the cluster closest to Pan spread away from Pan, while the cluster around 35$^\circ$ drifted slowly towards Pan. During 2008-2009, one of the clumps appears to stay within 5$^\circ$ of Pan, while the remaining clumps from this region appear to have drifted outward so that they were seen a little beyond 10$^\circ$ in early 2009. At the same time, the clumps around 35$^\circ$ dispersed and the clumps around $45^\circ$ shifted a bit closer to Pan. The motions of these different clumps during the next year were modest, but during this time a new clump cluster seems to have formed roughly 17$^\circ$ in front of Pan. As can be seen in Figure~\ref{panb1}, this feature started as a broad hump in the Rev 109 LRHPENKMV data, then became a stronger peak with two maxima in the Rev 115 FMOVIEEQX data, which then moved apart to become a pair of clumps in subsequent observations. By the middle of 2010, clumps were distributed throughout much of the region between 0$^\circ$ and $60^\circ$ in front of Pan.
The fastest drift rates observed in these data are associated with the clumps that moved from just outside 20$^\circ$ to about 5$^\circ$ between mid-2004 and late 2006. These clumps moved at a rate of between 0.035$^\circ$/day and $0.040^\circ$/day relative to Pan. However, this drift rate appears to be unusual, and most of the other clump features only moved a few degrees per year, or less than 0.01$^\circ$/day relative to Pan. If these drift rates were due to the clump material having slightly different semi-major axes from Pan, then most of these clumps would be within 1.5 km of $a_P$, with the fast-moving clumps being only 5-6 km away. However, the actual trajectories of these clumps are not consistent with those expected for concentrations of material at such semimajor axes (cf Murray and Dermott 1999). Particles at these locations would be expected to execute horseshoe or tadpole motion around Pan's Lagrange points, where the particle approaches Pan at some speed, turns around, then recedes at the same speed until it is somewhere beyond 60$^\circ$ in front of Pan. The clump trajectories shown in Figure~\ref{clumptrackpan} do not match these expectations. For example, consider the most distant clump from Pan, which is a relatively isolated feature between 2005 and 2010 and thus can be tracked with confidence. It first emerges from the leading side of a large clump complex in 2005, when it is moving slowly away from Pan towards the leading Lagrange point at 60$^\circ$. However, in 2006-2008, this clump seems to have stalled at about $56^\circ$, and in 2009 and 2010 it is clearly moving towards Pan, away from the Lagrange point. This clump therefore accelerated away from Pan's Lagrange point between 2007 and 2009, which is inconsistent with any sort of horseshoe or tadpole orbit. This clump therefore is not moving like a simple test particle in the combined gravitational fields of Pan and Saturn.
Even more curious are the motions of the clumps found within 10$^\circ$ of Pan, which can be studied in greater detail thanks to the extensive SATELLORB observations of these regions in both 2005 and 2007-2008. Figures~\ref{clumptrackdet1} and~\ref{clumptrackdet2} illustrate how these clumps evolved over the course of these two time periods. During the 2005 observation sequence, the clump closest to Pan steadily drifts outwards at a rate of about 0.004$^\circ$/day, while the other clumps are initially drifting towards Pan at rates between 0.029$^\circ$/day and 0.035$^\circ$/day. (see Figure~\ref{clumptrackdet1}). If these approaching clumps were on horseshoe orbits, their semi-major axes would be $\delta a \sim$4-5 km exterior to Pan's. Such particles should be able to approach Pan until they reach a critical distance $y_{\rm min}$, where they will turn around on their horseshoe orbits. This minimum distance can be calculated from the semi-major axis separation \citep{DM81}:
\begin{equation}
y_{\rm min} = \frac{8}{3}\frac{m_p}{M_S}\left(\frac{a_P}{\delta a}\right)^2a_P.
\end{equation}
For such clumps, $y_{\rm min}$ corresponds to 1$^\circ$-1.5$^\circ$, but none of these approaching clumps ever gets that close to Pan. Instead, the closest of the approaching clumps seems to stop moving when it gets only 4$^\circ$ in front of Pan, and even starts moving away from Pan a bit before it appears to merge with the clump that had been following it. Looking at the profiles obtained between days 220 and 230 of 2005, it almost appears as if this clump was ``repelled'' by the slowly-moving clump at 2$^\circ$ (Note that additional peaks appeared in both clumps during this time). Yet this same clump then seems to have merged with the clump that had been following it just a few weeks later. Note the two profiles from around day 245 were both obtained at the same phase angle (about 60$^\circ$), so the sudden brightening at 4$^\circ$ could be the result of this merging event. In any case, these data demonstrate the interactions of these clumps can be quite complex.
By contrast, the clumps seen during late 2007-2008 do not appear to move very much (see Figure~\ref{clumptrackdet2}). Instead, we can observe the morphology of the clump around $5^\circ$ slowly change over time. In late 2007, this clump has a single obvious brightness maximum, but in early 2008 a second maximum appears and the two maxima begin to drift apart. Sometime around day 50 of 2008, each of these two maxima splits again to produce a total of four maxima, all separating from each other. This transformation of one clump into multiple clumps is similar to that seen in the region 17$^\circ$ in front of Pan during 2009 described above. But in addition to these morphological changes, what is remarkable is that the clump is not moving at all during this time, which is inconsistent with any of the drift rates seen in Figure~\ref{clumptrackdet1}. Indeed, looking at Figure~\ref{clumptrackpan}, we notice that the clump closest to Pan (if it can be interpreted as a persistent feature) has moved alternately closer and further from Pan between 2004 and 2010. Again, this indicates that the motions of these clumps cannot be easily described in terms of simple horseshoe motion, and we will re-consider this issue at the end of this report.
\subsection{Inner ringlet}
\begin{figure}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_ir_pg1_101212.pdf}}}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_ir_pg2_101212.pdf}}}
\caption{Plot of the {\bf inner ringlet's} radially-integrated brightness (normal equivalent width) versus co-moving longitude based on the data from the observations listed in Table~\ref{moslist}.
This longitude system drifts forward relative to Pan at 0.7060$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). These brightness profiles are all derived from Lorentzian fits to the ringlet, except for the 00A/SPKMOVPER observation, which is derived from direct radial integration. Fits with peak radii more than 20 km from 133,490 km or peak widths greater than 100 km are removed and the remaining data smoothed over 5 samples to improve the display. Note the region in front of the clumps in the 044/ FMOVIE data is noisy due to nearby data gaps.}
\label{inb1}
\end{figure}
\begin{figure}
\centerline{\resizebox{5in}{!}{\includegraphics{clumptrack_in_121411.pdf}}}
\caption{Plot showing the locations of brightness peaks in the {\bf inner ringlet} as functions of longitude and time. The longitude system drifts forward relative to Pan at 0.7060$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). The black plusses are measurements derived from the mostly complete mosaics shown in Figure~\ref{inb1}. Note that some clumps are missing at certain times owing to data gaps in the observations. The gray lines indicate Pan's longitude in this coordinate system.}
\label{clumptrackin}
\end{figure}
The inner ringlet data shown in Figures~\ref{inb1} and~\ref{clumptrackin} are plotted in a longitude system that drifts {\em forward} relative to Pan at a rate of 0.7060$^\circ$/day, and has its origin at Pan's location at an epoch time of 170000000 ET (2005-142T02:12:15 UTC). Assuming the \citet{Jacobson06} values for Saturn's gravitational field parameters, this rate corresponds to a semi-major axis of 133,484 km, which is consistent with the observed location of this ringlet (Figure~\ref{egapprof}). When the clumps in this ringlet were first seen in 2004-2005, they could also be divided into a few large groups. The largest cluster of clumps was located at co-rotating longitudes of about 90$^\circ$, while two smaller clusters were found at +10$^\circ$ and -10$^\circ$. Finally, an isolated clump could be seen around 60$^\circ$ co-rotating longitude. These clumps dispersed from a region 110$^\circ$ wide in 2004 to cover a region about 160$^\circ$ wide in 2010. This expansion is due to a combination of the steady backward drift of the most trailing set of clumps and the steady forward drift of the leading edge of the large clump cluster during this time. However, the trailing edge of the large clump cluster remains fixed around $80^\circ$ during the same time period, so this cluster actually disperses during this time. Indeed, this group of clumps seem to split in two, with a gap forming around 85$^\circ$. The clump cluster around $10^\circ$ also does not move much in this coordinate system, but it does seem to spread and grow in complexity as time goes on. Finally, the isolated feature that was at $60^\circ$ in 2004 initially drifts backward at a steady rate, but then seems to stall sometime in 2008 or 2009 at a longitude of about 30$^\circ$.
The fastest relative motions are between the two ends of the clump region, which drifted 0.025$^\circ$/day to 0.030$^\circ$/day relative to each other. This is comparable to the fastest drift rates observed in the Pan ringlet, indicating a basic similarity in the dynamics within these two regions. If these drift rates were simply due to differences in the particles' mean motions, this would imply that the clumps cover a semi-major axis range of about 4 km. However, as with the Pan ringlet, such an interpretation is questionable because the clumps do not always follow simple trajectories. For example, the clump initially at 60$^\circ$ went from drifting backwards at a rate of about 0.02$^\circ$/day to nearly motionless in this coordinate system, which would correspond to a semi-major axis shift of over 2 km if this clump were simply a test particle. While this clump did have conjunctions with Pan in early 2008 and 2009 (see Figure~\ref{clumptrackin}), these Pan encounters probably cannot explain the sudden deceleration of this clump. The expected semi-major axis shift experienced by a particle on a semi-major axis $a_P\pm\Delta a$ due to an encounter with Pan can be estimated by combining Equations 10.52 and 10.57 of \citet{MD99}:
\begin{equation}
{\delta a} \sim 3.3a\left(\frac{m_p}{M_S}\right)^2\left(\frac{a}{\Delta a}\right)^5.
\label{dap}
\end{equation}
For the inner ringlet, $\Delta a \simeq 100$ km, so $\delta a$ is only 0.1-0.2 km, much smaller than the shift required to explain the change in this clump's drift rate. Again, the unusual accelerations of this clump suggest that the motions of these clumps are more complex than those of isolated particles.
\subsection{Outer ringlet}
\begin{figure}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_0out_pg1_101212.pdf}}}
{\resizebox{3in}{!}{\includegraphics{mosaicfitploty_0out_pg2_101212.pdf}}}
\caption{Plot of the {\bf outer ringlet's} radially-integrated brightness (normal equivalent width) versus co-moving longitude based on the data from the observations listed in Table~\ref{moslist}.
This longitude system drifts backwards relative to Pan at 0.9581$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). These brightness profiles are all derived from Lorentzian fits to the ringlet, except for the 008/LPHRLFMOV observation, which is derived from direct integration. Fits with peak radii more than 20 km from 133,715 km or peak widths greater than 40 km or less than 10 km are removed and the remaining data smoothed over 5 samples for the sake of clarity.}
\label{outb1}
\end{figure}
\begin{figure}
\centerline{\resizebox{5in}{!}{\includegraphics{clumptrack_out_121511.pdf}}}
\caption{Plot showing the locations of brightness peaks in the {\bf outer ringlet} as functions of longitude and time. The longitude system drifts backward relative to Pan at 0.9590$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). The black plusses are measurements derived from the largely complete mosaics shown in Figure~\ref{outb1}. Note that some clumps are missing in certain time periods due to data gaps in the observations. The gray lines indicate the longitude of Pan in this coordinate system. Note the data from 2005 were noisy, so the few peaks between 80$^\circ$ and 110$^\circ$ are likely spurious.}
\label{clumptrackout}
\end{figure}
The outer ringlet data shown in Figures~\ref{outb1} and~\ref{clumptrackout} are plotted using a longitude system that drifts {\em backwards} relative to Pan at a rate of 0.9581$^\circ$/day and has its origin at Pan's longitude at an epoch time of 170000000 ET (2005-142T02:12:15 UTC). This corresponds to a semi-major axis of 133,720 km assuming \citet{Jacobson06} values for Saturn's gravity field. Again, this semi-major axis is consistent with the observed location of the ringlet. Since this ringlet lies just 30 km interior to the Encke gap's outer edge, only 10 of the mosaics yielded useful profiles. Still, these are enough to document that the clumps in this ringlet form two well-separated groups. One tight cluster of clumps is located at a co-rotating longitude of about 20$^\circ$, while a more dispersed archipelago of peaks extends between about 110$^\circ$ and 160$^\circ$, with a couple of outlying isolated clumps at 170$^\circ$ and 190$^\circ$.
Compared to the clumps in the Pan and inner ringlets, the clumps in the outer ringlet seem less time-variable. For example, the dense clump cluster always has a sharp isolated spike at about 12$^\circ$, a broader peak around 19$^\circ$, and a series of narrow spikes at larger longitudes. The pattern of narrow spikes between 110$^\circ$ and 200$^\circ$ is also remarkably repeatable across the observations. Indeed, the most obvious change in these clumps is a slight backwards drift of the material between 110$^\circ$ and 130$^\circ$ between 2007 and 2009. Even this drift is less than 0.01$^\circ$/day, so the relative drift rates in this ringlet are much less than those found in the other two ringlets. If we assume the drifts are due to different particle mean motions, then these clumps would have a semi-major axis spread of only about 1.5 km, as opposed to the 4-km widths of the other two ringlets. However, given the trajectories of the clumps in the other two ringlets are inconsistent with those of test particle orbits, we caution against taking these numbers too literally. Nevertheless, the outer ringlet does appear to have a narrower radial profile than either the inner and outer ringlets (see Figure~\ref{egapprof}), so the particles in this ringlet may be more tightly confined in semi-major axis than those in the other two.
\section{Pan's perturbations on the other ringlets}
\begin{figure}
\resizebox{6in}{!}{\includegraphics{mosaicdisp_sumxclose_sel_080912.pdf}}
\caption{Images of the region around Pan in the three highest signal-to-noise mosaics. Note the Pan-induced waves and wakes in the inner, outer and fourth ringlets (as well as the gap edges). Also note the differences in the inner-ringlet's wave morphology among the observations, which are likely due to differences in the ringlet-particles' true anomalies prior to their conjunctions with Pan.}
\label{wakedispdet}
\end{figure}
\begin{figure}
\resizebox{3in}{!}{\includegraphics{mosaicdisp_sumxclose_pg1_080912.pdf}}
\resizebox{3in}{!}{\includegraphics{mosaicdisp_sumxclose_pg2_080912.pdf}}
\caption{Images of the regions around Pan derived from most of the observations
listed in Table~\ref{moslist}. Note the waves in the inner ringlet generated by Pan's gravitational perturbations. Whenever the disturbed part of the ringlet is clump-free,
the wave damps within about 3$^\circ$. By contrast, the waves in the clumpy regions can persist over 10$^\circ$ downstream from the moon.}
\label{wakedispov}
\end{figure}
One way to probe the various ringlets' orbital properties is to examine how they respond to Pan's gravitational perturbations. These are most clearly seen in Figure~\ref{wakedispdet}, which shows close-ups of the region around Pan in the three highest signal-to-noise mosaics derived from the observations in Table~\ref{moslist}. In all these mosaics, the portion of the inner ringlet just in front of Pan exhibits periodic wiggles. Close inspection of these images reveals that the part of the outer ringlet immediately behind Pan also displays a series of wiggles, and a similarly periodic brightness variation can even be seen in the fourth ringlet. All of these periodic patterns are likely due to Pan's gravitational perturbations on this ring material.
Particles drifting past a massive object like Pan will have their orbits perturbed by the moon's gravity. If the particles were initially on circular orbits, then the moon's gravity throws the particles onto eccentric orbits with initially aligned pericenters (see Figure~\ref{hillplot}). These particles' organized epicyclic motion causes them to move in and out as they drift downstream of the moon, forming a series of ripples with a characteristic wavelength of $3\pi\Delta a$, where $\Delta a$ is the semi-major axis difference between the particles and the moon \citep{Dermott81, SB82}. The wavelengths of the ripples in both the inner and outer ringlets are consistent with this explanation.
In reality, the particles in these ringlets do not all have the same semi-major axis, so their epicyclic motions gradually slip out of phase, producing density variations like those seen in the fourth ringlet, and perhaps the inner ringlet as well. In dense rings, these density variations eventually lead to collisions that should cause any coherent pattern to dissipate. However, in these low optical depth ringlets, collisions are rare. Even so, as the epicyclic motions of the particles slip further and further out of phase, any coherent pattern should eventually dissipate. The distance these patterns extend beyond Pan therefore provides information about the range of semi-major axes present in these ringlets.
While the qualitative appearance of these structures is reasonable, a truly rigorous analysis of such structures would need to account for the fact that the particles do not approach Pan on circular orbits. For example, as we will discuss in more detail below, the inner ringlet possesses finite forced and free eccentricities. The orbital changes induced by Pan therefore depend not only on the particles' semi-major axis, but also their true anomalies during conjunction \citep{SB82, DQT89}. Indeed, if we compare the mosaics derived from the two LRHPENKMV observations from Rev 124, we can see some differences in the wave morphology in the inner ringlet that can be attributed to its finite eccentricity. In the earlier observation, the minima in radius appear to be sharper than the maxima, while in the later observation, which was obtained on the opposite side of the ring and thus viewed the same material half an orbital/epicyclic period later, the maxima appear to be sharper than the minima. Such patterns could be consistent with Pan's gravitational perturbations on an eccentric ringlet, but confirming this will require detailed simulations that are beyond the scope of this report.
While a rigorous analysis of these wavy patterns is not feasible here, we can use fairly simple arguments to obtain some useful insights into the semi-major axis dispersion in different regions of the inner ringlet. Consider Figure~\ref{wakedispov}, which shows close-ups of all the relevant mosaics. These reveal that the ripples in the inner ringlet extend different distances downstream from Pan depending on whether the disturbed region contains clumps or not. When there are no clumps in the disturbed region (the Rev 34 HIPHAMOVD, Rev 44 FMOVIE, Rev 124 LRHPENKMV and Rev 132 SHRTMOVIE observations), the ripples in the inner ringlet dissipate within a few degrees of Pan. By contrast, when the disturbed region does contain clumps, as in the Rev 008 LPHRLFMOV, Rev 030 HIPHAMOVE, Rev 51 LPMRDFMOV, Rev 053 LPHRDFMOV, Rev 109 LRHPENKMOV and Rev 115 FMOVIEEQX observations, the ripples can persist as far as 10$^\circ$ downstream of Pan. Since the distance the ripples extend downstream of Pan is set by the semi-major axis dispersion within the ringlet, this suggests that the clumps contain particles with a smaller range of semi-major axes than the rest of the ringlet.
We can make this qualitative observation a bit more quantitative if we assume the center of the ringlet is $\Delta a$ from $a_P$, and the ringlet consists of particles with a range of semi-major axes $\delta a$. In this case, we expect any coherent pattern produced by Pan to smear out when the epicyclic motions of particles at $\Delta a \pm \delta a$ are out of phase by 180$^\circ$. This will occur at a distance $x_d$ downstream from Pan where $x_d=3\pi(\Delta a + \delta a/2)(N-1/4)$ and $x_d=3\pi(\Delta a - \delta a/2)(N+1/4)$ for the same $N$. This condition is satisfied when $ N \simeq \Delta a/(2\delta a)$, or when $x_d \simeq (3\pi/2) \Delta a^2/\delta a$.
In the clump-free regions of the Pan ringlet, the wave seems to damp within $1^\circ-2^\circ$ of Pan, so $x_d$ is between 2500 and 5000 km, which corresponds to a semi-major axis spread $\delta a$ between 10 and 20 km. By contrast, in the clumpy regions the waves extend over $10^\circ-15^\circ$, implying damping lengths of order 30,000 km, and semi-major axis spreads of order 1-2 km. Both of these numbers are reasonable, given the overall width of the ringlet, the persistence of the clumps, and the slow drift rates of clumps relative to each other.
\section{Ringlet Orbital Parameters}
The above analysis of the distribution and evolution of the clumps in these various ringlets reveals some surprising patterns. In particular, the relative motions of these features are inconsistent with those expected for clumps of material moving in the combined gravity fields of Saturn and Pan. Thus, in order to better understand the dynamics of both these features and the ringlets as a whole, we will now use the apparent radial positions of these ringlets to investigate their orbital properties.
The following studies will focus exclusively on the Pan and inner ringlets because both these ringlets are sufficiently far from the Encke Gap edges that our fitting algorithms can yield reliable estimates of their radial positions. By contrast, for most of the observations considered here, the outer ringlet is only barely resolved from the outer gap edge. While our ringlet-fitting procedures can still provide useful information about the morphology and distribution of the clumps in the outer ringlet, the corresponding radial position estimates are more sensitive to the background signal from the nearby gap edge. Obtaining robust estimates of the outer ringlet's position is particularly difficult outside of the clumps, where the ringlet is comparatively faint. As will become clear below, detailed comparisons among multiple observations over a broad range of longitudes are needed to make sense of the radial positions of the inner and Pan ringlets. At present, the outer ringlet data are not sufficient to do these comparisons, so we will not examine the radial structure of the outer ringlet further here.
Determining the orbital properties of the clumpy inner and Pan ringlets is not as straightforward as measuring the shapes of such non-circular ring features like the dense Huygens ringlet or even the dusty ringlet in the outer Cassini Division. The shapes of the latter ring features can be determined by simply measuring their radial positions at multiple inertial longitudes, provided we assume that the ring particles' orbital properties are the same at all co-rotating longitudes. This, however, is clearly not a valid assumption for clumpy features like the Encke Gap ringlets. Instead, we can only obtain useful information about the Encke Gap ringlets' orbital properties by comparing observations of the same co-rotating longitudes $\lambda_c$ at different inertial longitudes $\lambda_i$. This obviously complicates the analysis, and forces us to focus our attention on a few particularly informative data sets. Furthermore, many of the relevant observations can only provide sensible orbital information if the ringlets are assumed to exhibit ``heliotropic'' behavior similar to that previously identified in a dusty ringlet in the Cassini Division \citep{Hedman10}. While this was not unexpected, given that both this Cassini Division ringlet and the Encke Gap ringlets are made out of comparably small particles \citep{Hedman11}, it does further complicate the analysis of the ringlets' radial structure.
After summarizing the theory and formalism for describing heliotropic ring features, we first consider the Rev 045 PANORBIT data, which yield complete orbit information for a small part of the Pan ringlet in the vicinity of the moon at one time. Then we examine the Rev 124 LRHPENKMV data, where multiple clumps in both the Pan and inner ringlets were observed at two very different inertial longitudes. These observations clarify that the kinks associated with the clumps in both ringlets are due to variations in the particles' orbital eccentricites. Finally, we use the mosaics illustrated in Figure~\ref{mos1} to study the large-scale variations in these ringlets' orbital properties.
\subsection{Properties of heliotropic ringlets}
\label{helio}
\citet{Hedman10} provide a detailed discussion of the dynamics of narrow heliotropic ringlets, based on observations of the dusty ``charming ringlet'' in the Cassini Division's Laplace Gap. That ringlet exhibits systematic variations in its observed radial position in a coordinate system fixed relative to the Sun, such that the geometric center of that ringlet was displaced away from Saturn's center towards the Sun. This unusual behavior is due to solar radiation pressure producing a forced eccentricity $e_f$ in the orbits of the tiny grains that form this ringlet \citep{BHS01}. However, the shape of this ringlet also varied with time. These variations could be modeled by assuming the ringlet traced out the orbit of a particle with both a forced eccentricity generated by solar radiation pressure and a free eccentricity precessing around the planet at the local rate. While it remains unclear what process coordinates the particles' motions within the ringlet so as to maintain this free eccentricity, this model still provides a useful way to parameterize the ringlet's morphology. As we will demonstrate below, the dusty Encke Gap ringlets also exhibit time-variable eccentricities that can be modeled as a forced component aligned with the Sun and a freely-processing component. We will therefore employ this decomposition to describe the shape of the Encke-Gap ringlets.
None of the observations to date indicates that the Encke-Gap ringlets have any detectable inclination, so (for the sake of simplicity) these ringlets will be assumed to lie exactly in Saturn's equatorial plane, In that case, the radial position of a heliotropic ringlet as a function of inertial longitude $\lambda_i$ can be expressed as:
\begin{equation}
r(\lambda_i, t)=a-ae(t)\cos[\lambda_i-\varpi(t)],
\end{equation}
where the eccentricity $e$ and pericenter $\varpi$ are slowly-varying functions of time. These quantities are given by:
\begin{equation}
e\cos(\varpi-\lambda_\Sun)=-e_f+e_l\cos(\varpi_l+\dot{\varpi}_lt)
\end{equation}
\begin{equation}
e\sin(\varpi-\lambda_\Sun)=e_l\sin(\varpi_l+\dot{\varpi}_lt),
\end{equation}
where $\lambda_\sun$ is the Sun's inertial longitude, $e_f$ is the forced eccentricity induced by solar radiation pressure, and $e_l$, $\varpi_l$ and $\dot{\varpi}_l$ parametrize the magnitude, orientation and precession rate of the free component of the eccentricity, respectively. Note that since the alignments of the free and forced eccentricities have different time-dependencies, these two components of the total eccentricity can be separated from one another by comparing measurements made at different times. For the purposes of this analysis, we will assume that the free eccentricity's precession rate $\dot{\varpi}_l$ is basically the precession due to Saturn's finite oblateness, $\dot{\varpi}_0$, which is $3.2^\circ$/day in the Encke Gap. Thus the orbital properties of the ringlet are specified by the parameters $a$, $e_f$, $e_l$ and $\varpi_l$, which for the Encke Gap ringlets may be functions of co-rotating longitude $\lambda_c$.
\subsection{Orbital elements of the Pan ringlet near Pan}
\begin{figure}
\centerline{\resizebox{4.5in}{!}{\includegraphics{panorbit_plot_121411.pdf}}}
\caption{Orbital elements of the Pan ringlet derived from the PANORBIT observation. The top two panels show the integrated brightness and radial position of the Pan ringlet derived from two images, one taken close to the sub-solar longitude, and the other taken near Saturn's shadow. Note that the ringlet is found displaced outward from Pan's orbit on the sunward side of the rings, and inwards on the side near Saturn's shadow. The bottom two panels show the ringlet's semi-major axis, eccentricity and pericenter longitude derived from all the useful images in this sequence. Statistical error bars are not plotted for reasons of clarity, but are consistent with the scatter in the estimates (i.e. they are around 0.5 km in $a$ and $ae$ and 5$^\circ$ in the pericenter in front of Pan, and 1-2 km in $a$ and $ae$ and 10-20$^\circ$ in the pericenter behind Pan). In front of Pan, the ringlet has a semi-major axis close to that of Pan, a finite eccentricity, and a pericenter anti-aligned with the Sun. Note that the eccentricity is reduced in the vicinity of the bright clumps between 5$^\circ$ and 6$^\circ$. Behind Pan, where the ringlet is fainter, the semi-major axis is systematically outside the orbit of Pan and the pericenter deviates from exactly 180$^\circ$. }
\label{panorbit}
\end{figure}
The PANORBIT observation from Rev 045 is a useful starting point for investigations of the ringlet's orbital properties because it consists of 158 images of Pan and the surrounding rings as the moon moved around the planet. The resulting images cover roughly 210$^\circ$ in true anomaly, with some gaps where the planet appeared behind the rings or when the rings themselves were in Saturn's shadow. These images were all re-projected onto a common scale in radius and longitude relative to Pan (sampling distances of 5 km and 0.02$^\circ$ respectively), and then the radial brightness profile at each longitude in each scan was fit to a Lorentzian in order to estimate the integrated brightness and radial position of the Pan ringlet. However, due to the changing viewing geometry and resolution of the images over the course of the observation, the radial position estimates had to be refined based on measurements of the position of the Encke-Gap's edges in each image.
For each longitude in each image, the locations of both gap edges were estimated as the points of maximum slope in the radial brightness profile, which were found by fitting peaks to the {\em derivative} of the brightness profile. The edge waves generated by Pan cause the radial positions of both edges to vary by a few kilometers within each image, so we did not individually adjust each estimate of the ringlet's radial position. Instead, we simply computed a single offset for each image based on the median deviation of both edges from their nominal positions at 133,423 km and 133,745 km. The resulting offsets varied over a range of about 6 km with an $m=2$ pattern. Such a pattern would not be confused with the $m=1$ pattern due to a real eccentricity, but removing these offsets still improves the reliability of the subsequent analysis.
The top pair of panels of Figure~\ref{panorbit} show two representative profiles of the Pan ringlets' brightness and radial position derived from two images in the PANORBIT sequence. One of these images (N1558598811) was obtained when Pan was only 12$^\circ$ from the sub-solar longitude, while the other (N1558615821) was obtained when Pan was over $130^\circ$ from the sub-solar longitude, and thus closer to Saturn's shadow. The integrated brightness profiles derived from these two images are very similar, up to an overall normalization that can probably be attributed to slight differences in the phase angles of the two observations (83$^\circ$ versus 76$^\circ$) and small uncertainties in the background subtraction. However, the radial position of the ringlet in the two images show clear systematic differences. The observation taken when Pan was near the sub-solar longitude shows the ringlet displaced exterior to Pan's semi-major axis at 133584 km, while the data taken closer to Saturn's shadow are shifted towards smaller radii. These variations in the apparent radial position of the ringlet around Pan can be most easily explained if the ringlet particles are on eccentric orbits with aligned pericenters. Furthermore, the directions of these displacements are consistent with the ringlet being heliotropic, with a forced eccentricity that tends to place the particles' orbital pericenters 180$^\circ$ from the Sun. At the same time, it is also apparent that the orbital properties of the ringlet depend upon the co-rotating longitude relative to Pan. The most obvious example of this is the distinct ``kink'' in the ringlet's radial position associated with the bright clumps around 5$^\circ$ in front of Pan.
Images from a single observing sequence (i.e. taken at a single time) do not provide sufficient information to determine all the parameters in a heliotropic model: $a$, $e_f$, $e_l$ and $\varpi_l$. However, we can derive estimates of the instantaneous values of $a$, $e$ and $\varpi$ at each co-rotating longitude by fitting the observed radial positions $r$ from all the relevant images to the function:
\begin{equation}
r=a-ae\cos(\lambda_i-\varpi).
\end{equation}
Note that due to variations in the viewing geometry, the range of $\lambda_i$ observed depends somewhat on $\lambda_c$. Also note that images obtained when the ring was in shadow, backlit by the planet, or yielded radial positions more than 50 km from 133584 km were excluded prior to performing these fits. Based on the residuals to these fits, we estimate the statistical uncertainties on these parameters are around 0.5 km in $a$ and $ae$ and 5$^\circ$ in $\varpi$ for longitudes in front of Pan (where the signal is stronger), and 1-2 km in $a$ and $ae$ and 10-20$^\circ$ in $\varpi$ for longitudes behind Pan
The bottom two panels of Figure~\ref{panorbit} show the estimated values of $a$, $ae$ and $\varpi$ as functions of co-rotating longitude relative to Pan. These plots indicate that for the portion of the ringlet in front of Pan, $a$ is close to Pan's semi-major axis, $ae$ is around 12 km, and the orbital pericenter is almost exactly 180$^\circ$ from the Sun. On the other hand, the part of the ringlet falling behind Pan displays a slightly lower eccentricity, a pericenter that gets as far as $80^\circ$ from the anti-Sun direction, and a semi-major axis that is displaced by about 3 km exterior to $a_P$.
No single observation can prove that this ringlet is heliotropic, but $\varpi$ always being almost exactly 180$^\circ$ from the Sun at all longitudes in front of Pan is certainly consistent with what one would expect for a heliotropic ringlet with $e_f>>e_l$. However, since the pericenter does deviate from $\lambda_\sun+180^\circ$ behind Pan, the entire ringlet cannot just have eccentricities forced by solar radiation pressure. Both these results are consistent with the analysis of the mosaics described at the end of this section, which provides separate estimates of $e_f$ and $e_l$.
While these data do not provide strong constraints on the origin of the ringlet's eccentricity, they do clearly demonstrate that the kink in the ringlet's radial position at $5^\circ$ corresponds to a region of reduced eccentricity. By contrast, neither $a$ nor $\varpi$ vary noticeably within this region.
\subsection{Orbital element variations associated with clumps}
\begin{figure}
\resizebox{6in}{!}{\includegraphics{mosaicfitplot124_pr_121511.pdf}}
\caption{The integrated brightness and radial position of the clumps in the Pan ringlet obtained from the Rev 124 LRHPENKMV observations. These profiles were derived from Lorentzian fits to the radial brightness profiles whose radial scales were refined using the position of the Encke-Gap's outer edge. Fits with peak radii more than 20 km from 133,584 km are removed and the remaining data smoothed over 5 samples for the sake of clarity. These two observations imaged the same ring region at two different longitudes, one close to the sub-solar point and one close to Saturn's shadow. Note that the variations in the radial position of the ringlet are reversed at the two locations, suggesting that the observed kinks in the ringlet are due primarily to eccentricity variations.}
\label{mov124pan}
\end{figure}
\begin{figure}
\resizebox{6in}{!}{\includegraphics{mosaicfitplot124_irc_080912.pdf}}
\caption{Brightness and radial position profiles of the clumps in the inner ringlet obtained from the Rev 124 LRHPENKMV observations. These profiles were derived from Lorentzian fits to the relevant brightness profiles whose radial scales were refined based on the observed positions of the Encke Gap's outer edge. Fits with peak radii more than 10 km from 133,484 km, widths greater than 100 km or less than 10 km, or peak brightnesses greater than 0.02 are removed and the remaining data smoothed over 5 samples for the sake of clarity. The longitude system used here drifts forward relative to Pan at 0.7060$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). These two observations imaged the same region in the ring at two different longitudes, one close to the sub-solar point and one close to Saturn's shadow. Note that the variations in the radial position of the ringlet are reversed at the two locations, suggesting that the observed kinks in the ringlet are due primarily to eccentricity variations.}
\label{mov124in}
\end{figure}
The LRHPENKMV observation sequence from Rev 124 was deliberately designed to investigate the orbital properties of the kinks in the Encke Gap ringlets. During this observation, the camera first stared at a point in the Encke Gap near the sub-solar longitude, then it looked at a point on the opposite side of the rings, near Saturn's shadow. The timing of these two pointings was chosen so that the same co-rotating longitudes would be observed at both locations.
Figures~\ref{mov124pan} and~\ref{mov124in} show the integrated brightness and radial position profiles for both Pan and inner ringlets derived from these observations. Again, the radial position estimates were refined based on the observed positions of the Encke Gap edges in the observed mosaics. Since we are looking at regions immediately in front of Pan, only the less-disturbed outer edge of the gap was used for this purpose. This edge position was measured by fitting a peak to the derivative of the radial brightness profiles. The edge positions were low-pass filtered using a 2$^\circ$ wide boxcar to remove fine-scale structure associated with the wavy edges, and then used to compute a correction that would place the smoothed edge at 133,745 km at all co-rotating longitudes. These corrections remove some broad-scale ripples in the ringlets' radial positions, but do not affect the fine-scale variations seen in Figures~\ref{mov124pan} and~\ref{mov124in}.
For both ringlets, the two brightness profiles are essentially the same, up to an overall normalization factor due to the slight phase-angle difference between the two observations. However, the radial positions at the two locations are quite different. Since these two data sets were obtained on opposite sides of the planet, the average of the two radial positions corresponds to the semi-major axis of the ringlet, while the difference between them is proportional to $ae$ (the constant of proportionality depending on the pericenter location).
As with the PANORBIT observations, the Pan ringlet is displaced outwards from Pan's orbit when viewed near the sub-solar longitude and is displaced inwards when viewed near Saturn's shadow. This coincidence strongly suggests that this ringlet exhibits heliotropic behavior. The PANORBIT and LRHPENKMV observations were obtained 960 days apart, and the expected apsidal precession rate of this ringlet is 3.2$^\circ$/day, so any freely-precessing eccentricity would place the pericenter on opposite sides of the planet during the two observations. Thus the ring's pericenter can only be on the anti-solar side of the planet in both observations if the eccentricity is forced by the Sun.
On the other hand, the observed part of the inner ringlet is actually found closer to the planet on the sunward side of the rings. Thus this material does not exhibit the same consistently heliotropic behavior as the clumps in the Pan ringlet, and it must have a finite free eccentricity. However, just as the PANORBIT observation alone could not provide solid proof that the Pan ringlet was heliotropic, these data alone cannot be used to argue that the inner ringlet has zero forced eccentricity due to solar radiation pressure. Indeed, examinations of the data from all the mosaics indicate that the inner ringlet does have a finite forced heliotropic eccentricity (see Section 6.4).
For both ringlets, there is a strong anti-correlation between the radial position variations observed at the sub-solar longitude and those seen at the anti-solar longitude. This implies that the kinks in both ringlets are primarily due to variations in the particles' orbital eccentricities, which is consistent with the analysis of the PANORBIT images described above. Furthermore, the kinks are clearly associated with the clumps in the brightness profile. In the Pan ringlet, all the locations where the separation between the two radial position curves reaches a minimum correspond to a peak in the brightness profiles. Similarly, whenever the radial position of the inner ringlet reaches a local minimum on the sunward side of the rings (and a local maximum on the anti-solar side), there is a corresponding peak in the ringlet's brightness. This implies that these brightness maxima correspond to regions with anomalous eccentricities. However, there are also multiple brightness maxima in both ringlets that do not correspond to obvious extrema in the radial position curves. This was also the case in the PANORBIT data, where the clump closest to Pan is not associated with an obvious kink.
Variations in the particles' semi-major axes can also be detected in these observations. For example, in the Pan ringlet the two position profiles are roughly symmetric about $a_P=133,584$ km along most of the region within $50^\circ$ of Pan, which requires a semi-major axis close to $a_P$. However, beyond $50^\circ$, both curves shift outwards, suggesting that the semi-major axis here is exterior to $a_P$. However, these semi-major axis variations appear to be on a broader scale than the eccentricity variations responsible from the sharp kinks in these profiles. These broad-scale trends can be clarified by comparing these data to those derived from the other mosaics.
\subsection{Large-scale orbital element variations}
\begin{figure}
\resizebox{3in}{!}{\includegraphics{mosaicfitplotx_pr_edge_pg1_080912.pdf}}
\resizebox{3in}{!}{\includegraphics{mosaicfitplotx_pr_edge_pg2_080912.pdf}}
\caption{Plots showing the edge-corrected radial positions of the Pan ringlet as a function of co-rotating longitude. For clarity, fits with peak radii more than 30 km from 133,585 km are removed and the remaining data are smoothed over 5 samples. Still some narrow spikes corresponding to misfits can be seen in many of the profiles. The sawtooth pattern in the Rev 034 HIPHAMOVD observation is an artifact that may be associated with the finite eccentricity of this ringlet and the finite longitudinal span of the images. Also, while the Rev 132 SHRTMOVIE data are shown here, they are not used in later fits to the orbital elements due to the restricted longitudinal coverage of this data set. Nevertheless, it is clear that in all the profiles the radial position of the ringlet shifts outwards between 50$^\circ$ and 70$^\circ$ in front of Pan.}
\label{rpan}
\end{figure}
\begin{figure}
\resizebox{3in}{!}{\includegraphics{mosaicfitplotx_ir_edge_pg1_101212.pdf}}
\resizebox{3in}{!}{\includegraphics{mosaicfitplotx_ir_edge_pg2_101212.pdf}}
\caption{Plots showing the edge-corrected radial positions of the inner ringlet as a function of co-rotating longitude. This longitude system drifts forward relative to Pan at 0.7060$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). These brightness profiles are all derived from Lorentzian fits to the ringlet. Fits with peak radii more than 20 km from 133,490 km or peak widths greater than 100 km are removed, and the remaining data are smoothed over 5 samples for the sake of clarity. The Rev 008 SPKMOVPER data are not shown here because of their low quality (the panel is kept just for ease of comparison to Figure~\ref{rpan}, and the Rev 132 SHRTMOVIE data are not included in subsequent orbital fits because of their limited longitudinal extent. In many of these profiles, there appears to be an increase in the fit radius at longitudes between 110$^\circ$ and 130$^\circ$, just in front of the clump-rich region.}
\label{rin}
\end{figure}
Both the PANORBIT and LRHPENKMV observations provide detailed but restricted information about the variations in the ringlets' orbital properties. In order to place these observations in context, and to better understand these ringlets' global structure, we now turn our attention back to the large-scale mosaics. Figures~\ref{rpan} and~\ref{rin} show the edge-corrected radial positions of the ringlets as functions of co-rotating longitudes derived from the mosaics listed in Table~\ref{moslist} with sufficient resolution to obtain sensible estimates of the ringlets' radial positions. As above, these radial positions have been corrected based on the positions of the edges within the mosaic, which were measured at each longitude by fitting a peak to the derivative of the radial brightness profiles. Since we are only looking at broad-scale trends in these plots, filtering out the edge waves was not necessary in this case. However, we avoid using either edge when it is observed between 0$^\circ$ and 40$^\circ$ downstream of Pan, due to large-scale variations in the edge position in these highly disturbed regions.
If we first consider the Pan ringlet data, we can note that the overall radial position of the ringlet depends on the observed inertial longitude relative to the Sun. The sequences taken near the sub-solar longitude (Rev 008 LPHRLFMOV, Rev 051 LPMRDFMOV, Rev 053 LPHRDFMOV, Rev 115 FMOVIEEQX, the second LRHPENKMV in Rev 124 and Rev 132 SHRTMOVIE) all show the ringlet displaced exterior to Pan's orbit, while those taken further from the sub-solar point (Rev 030 HIPHAMOVE, Rev 034 HIPHMOVD, Rev 109 LRHPENKMV, and the first LRHPENKMV in Rev 124) show the ringlet either near to, or displaced inwards from, Pan's orbit. While this suggests that this ringlet is heliotropic, there is also evidence that this ringlet's radial position is not strictly controlled by the Sun. For example, compare the Rev 008 LPHRLFMOV to the Rev 115 FMOVIEEQX data. The latter was obtained closer to the sub-solar point, but the former shows a more extreme outward radial offset, indicating that this ringlet also has a finite free eccentricity independent of the forced heliotropic eccentricity. Furthermore, we can detect common trends among all these profiles, such as an outward shift between $50^\circ$ and $70^\circ$ in front of Pan, that could be attributed to variations in the ringlet's semi-major axis.
The inner ringlet profiles, by contrast, do not provide clear evidence for heliotropic behavior (The ringlets' average radial position is not obviously correlated the observed longitude relative to the Sun). Still, clear systematic variations in the ringlet's mean radial position can be found among these observations, indicating that this ringlet does have a finite eccentricity. Also, we can detect an outward shift in the region between 110$^\circ$ and 130$^\circ$ in most of the profiles. This occurs immediately in front of the clump-rich region, suggesting a change in the ringlet's semi-major axis at this location, similar to that found in the Pan ringlet.
The nature of these broad-scale variations and trends can be clarified by fitting the radial position data at each co-rotating longitude to the heliotropic model described in Section~\ref{helio} above. This model has a small number of free parameters $a$, $e_f$, $e_l$, $\varpi_l$ and possibly $\dot{\varpi}_l$; and at most co-rotating longitudes there are sufficient radial position measurements to determine this many parameters.
However, in order to keep outliers from corrupting the fits, we first down-sample the edge-corrected radial position-estimates by averaging over 1$^\circ$ wide bins in co-rotating longitude. Uncertainties is these estimates were conservatively estimated as the standard deviations of the relevant estimates, which are typically around 1 km. Furthermore, we only use a sub-set of the mosaics, which are marked with an $R$ in Table~\ref{moslist}. Specifically, we exclude the Rev 00A SPKMOVPER data (and the Rev 008 LPHRLFMOV data for the inner ringlet) due to the low spatial resolution of these images. We also exclude the Rev 044 FMOVIE data because the gaps around the inner edge corrupt the edge corrections, and the Rev 132 SHRTMOVIE data because they only cover a small range of longitudes and at most longitudes the inner edge data are insufficient to correct the ringlets' radial positions. This leaves nine profiles for the Pan ringlet and eight profiles for the inner ringlet, which should still be enough to fit all the model parameters. However, many of these profiles do not cover all co-rotating longitudes, so at some locations the model cannot be adequately constrained.
\begin{figure}
\resizebox{6in}{!}{\includegraphics{mosaicfit_raddigplot_pr_prate318_080912.pdf}}
\caption{Plots of the Pan ringlet's orbital elements as functions of co-rotating longitude derived from the mosaics marked with an $R$ in Table~\ref{moslist}. The semi-major axis is measured from the Encke Gap center at 133,584 km. These fits assume the free precession rate was 3.21$^\circ$/day (3.18$^\circ$/day relative to the Sun), using an epoch time of 2008-001T00:00:00. Statistical error bars on these estimates are not shown for reasons of clarity, but are between 0.5 km and 1 km for $a$ and $ae$, and about 5$^\circ$ for the pericenter location.}
\label{fitpan}
\end{figure}
\begin{figure}
\resizebox{6in}{!}{\includegraphics{mosaicfit_raddigplot_ir_prate318_080912.pdf}}
\caption{Plots of the inner ringlet's orbital elements as functions of co-rotating longitude derived from the mosaics marked with $R$ in Table~\ref{moslist}. The semi-major axis is measured from the Encke Gap center at 133,584 km. The co-rotating longitude system drifts forward relative to Pan at 0.7060$^\circ$/day with an epoch time of 170000000 ET (2005-142T02:12:15 UTC). These fits assume the free precession rate was 3.21$^\circ$/day (3.18$^\circ$/day relative to the Sun), using an epoch time of 2008-001T00:00:00. Statistical error bars on these estimates arenot shown for reasons of clarity, but are between 0.5 km and 1 km for $a$ and $ae$, and about 5$^\circ$ for the pericenter location.}
\label{fitin}
\end{figure}
Figures~\ref{fitpan} and~\ref{fitin} show the heliotropic parameters $a$, $e_f$, $e_l$ and $\varpi_l$ as functions of co-rotating longitude in both the Pan and inner ringlets. Note that because we are mostly interested in large-scale trends, we do not attempt to account for the motions of clumps or for the waves generated by Pan in the inner ringlet in these calculations. Furthermore, in order to reduce the number of free parameters in these fits, the free precession rate was held fixed at 3.21$^\circ/$day (3.18$^\circ$/day relative to the Sun). Allowing the precession rate to float did not change the overall trends, but gave rise to increased scatter in the parameters, especially $\varpi_l$. Varying the assumed precession rate also did not affect the trends in the fit parameters significantly. Fitted parameters are only plotted at co-rotating longitudes with more than four radial position measurements. The statistical uncertainties on these parameters are between 0.5 and 1 km for $a$, $ae_f$ and $ae_l$, and around 5$^\circ$ for $\varpi_l$. Thus the large-scale trends seen in these plots are highly significant, however we caution that smaller-scale fluctuations might reflect systematic errors in individual observations.
First, consider the fit parameters for the Pan ringlet shown in Figure~\ref{fitpan}. These parameters generally show nice, smooth trends, except in the region between 0$^\circ$ and 60$^\circ$ in front of Pan. the excess scatter in this region arises because this analysis does not account for clumps drifting through this region. Despite this, the mean orbital elements in this region are consistent with those derived from the Rev 045 PANORBIT observation. In particular, the semi-major axis scatters around $a_P$, and the forced eccentricity is much larger than the free eccentricity. Thus neglecting the motions of the clumps does not appear to prevent us from obtaining sensible orbital elements.
Outside the clumpy region, we find that the values of $e_f$, $e_l$ and $\varpi_l$ do not vary much with co-rotating longitude. Furthermore, the forced and free components of the eccentricity are comparable to each other. These particles' orbits therefore periodically become nearly circular, and since $\varpi_l$ varies by less than 90$^\circ$ around the ring, the eccentricity variations in the entire ringlet are synchronized somehow. This behavior is very similar to that previously observed in the dusty Cassini Division ringlet \citep{Hedman10}.
By contrast, the ringlets' semi-major axes vary systematically with co-rotating longitude outside the clump-rich region. Behind Pan, the semi-major axis seems to increase linearly with distance from Pan. This trend seems to saturate when the radial displacement reaches 8 km exterior to Pan. In front of the clump-rich region, the semi-major axis rises rapidly from $a_P$ to ($a_P+8$ km) within a space of 60$^\circ$. The latter semi-major axis shift is responsible for the radial position shift visible in all the profiles in Figure~\ref{rpan}.
Turning to the inner ringlet's parameters illustrated in Figure~\ref{fitin}, many of the same trends are apparent, but there are some important differences as well. In this case, the clumps extend between co-rotating longitudes of -30$^\circ$ and 120$^\circ$, but are not common outside the regions centered around $0^\circ$ and $100^\circ$. The clump-rich region has the lowest semi-major axes of $a_P-100$ km, which corresponds to the semi-major axis required to match the clumps' mean motion. Beyond the clump-rich region, the semi-major axis is displaced outwards, following trends very similar to those seen in the Pan ringlet. Also, in the regions far from the clumps, $e_f$, $e_l$ and $\varpi_l$ are all roughly constant, and $e_f\simeq e_l$, just like for the Pan ringlet. However, unlike the Pan ringlet, the free eccentricity is close to, or even higher than, the forced eccentricity across the entire region covered by the clumps. This is consistent with the lack of an obvious heliotropic signature in the Rev 124 LRHPENKMV data described above (see Section 6.3).
\section{Discussion}
The above observations reveal that the fine material in the Encke Gap is sculpted by multiple processes. The overall architecture of the dusty material and the disturbances found near Pan demonstrate that Pan's gravity does influence the motions of particles in this region. Meanwhile, the heliotropic forced eccentricities indicate that non-gravitational forces also affect the distribution of particles within the gap. The anomalous motions of the bright clumps in the narrow ringlets suggest that interactions among the dust grains themselves probably also play a role in sculpting this material. The dynamics of the dust in the Encke gap are therefore quite complex, and a detailed theoretical analysis of this system is beyond the scope of this report. Still, we can provide some initial speculations and calculations that can provide a basis for such future modeling efforts that will be the subject of a future paper.
First, we use the magnitude of the heliotropic forced eccentricities to estimate the typical particle sizes in the ringlets and confirm that these are broadly consistent with previous estimates based on the ringlets' light-scattering properties. Then we examine the apparent variations in the inner and Pan ringlets' semi-major axes with co-rotating longitude and explore how these could be explained by radial transport of small particles. Next, we consider the role of particle collisions and argue that they may be responsible for some of the observed longitudinal variations in these ringlets' semi-major axes, as well as the formation of bright clumps. Finally, we suggest that the locations of the clump-rich regions in the Pan and inner ringlets may be determined by the competition between non-gravitational azimuthal drag forces and Pan's gravitational perturbations.
It is important to keep in mind that the following discussions focus primarily on dynamical phenomena that could explain some of the better documented trends in the currently-available data, and additional processes not considered below may well be important in sculpting the dusty material in the Encke Gap. For example, we are still unable to ascertain what could be exciting the ``free'' components of the ringlet's eccentricities. Also, since we have not yet been able to determine the outer ringlet's orbital properties, we cannot explore its dynamics in detail at present. Furthermore, the wide variety of processes considered in these discussions may interact and interfere with one another in very complex ways, and some of these still-unexplained features of these ringlets could reflect dynamical phenomena that will require some of the interpretations given below to be reconsidered and/or revised.
\subsection{Heliotropic behavior and particle sizes}
Away from the bright clumps, the Pan ringlet and the inner ringlet exhibit similar combinations of forced and free eccentricities, with $ae_f \simeq ae_l \simeq 5$ km. The similar magnitudes of $e_f$ and $e_l$ imply that these particles' orbits periodically become exactly circular. One possible explanation for this is that the particles were launched from source bodies on nearly circular orbits. In this case, even though solar radiation pressure imparts a forced eccentricity to these particles' orbits, the condition that they began on circular orbits would require that $e_f \simeq e_l$ and that the particles' orbits periodically return to a circular state. However, this simple explanation is complicated by the observation that $\varpi_l$ doesn't vary much with longitude in either ringlet. This means that the orbits of all the particles in each ringlet become nearly circular at the same time, which would not naturally occur if all these particles moved independently from each other and were produced at different times. Similarly coordinated motions have been observed previously in the so-called ``charming ringlet'' in the Laplace Gap in the outer Cassini Division \citep{Hedman10}, so this synchronization of free pericenters appears to be a common feature of narrow dusty ringlets.
As discussed in \citet{Hedman10}, collisions among a ringlets' particles will naturally tend to align the particles' orbital pericenters. Such inter-particle collisions could therefore produce the observed coordinated motions if the collisions are sufficiently frequent and if the particles can maintain finite free orbital eccentricities. Even outside the clumps, the Encke Gap ringlets' optical depths are about an order of magnitude higher than that of the ``charming ringlet'' (see Hedman {\it et al.} 2011), so collisions are more likely to be sufficiently frequent to align pericenters in the Encke Gap. Maintaining a finite free eccentricity is a bigger challenge, since collisions among the ring particles would also tend to dissipate $e_l$. \citet{Hedman10} explores what sorts of terms in the particles' equations of motion could support the free eccentricity of the dusty Cassini Division ringlet. For the Encke Gap ringlets, we have the additional constraint that $e_f \simeq e_l$, which could help clarify the origin of $e_l$ in these ringlets. For example, perhaps it becomes easier for particles with different orbital semi-major axes to maintain their aligned pericenters against differential precession when all the particles' orbits periodically become circular. A full exploration of such ideas will likely require numerical simulations of these ringlets.
Despite this lingering uncertainty regarding the free component of the ringlets' eccentricity, the magnitude of the forced eccentricities can still provide a useful estimate of the typical particle sizes in these ringlets because the value of $e_f$ can be computed using orbital perturbation theory \citep{Hedman10}:
\begin{equation}
e_f \simeq \frac{n}{\dot{\varpi}_0}\left[\frac{3}{2}(1-\epsilon+\sin(2\pi\epsilon)/6\pi)\frac{F_\sun}{F_G}\cos B_\sun\right],
\end{equation}
where $n$ is the particles' mean motion, $\dot{\varpi}_0$ is the apsidal precession rate, $F_\Sun/F_G$ is the ratio of the solar radiation force acting on the particle to Saturn's gravitational force, $\epsilon$ is the fraction of the particles' orbit that is in shadow, and $B_\sun$ is the solar elevation angle. For particles in the Encke Gap, $n=626^\circ$/day, $\dot{\varpi}_0=3.2^\circ$/day and $F_\Sun/F_G \simeq 1.6*10^{-5} Q_{pr}/(r_g/1\mu m)$, where $Q_{pr}$ is an efficiency factor dependent on the particle properties \citep{BLS79}, and $r_g$ is the particle's physical radius.
For the Encke Gap ringlets, $\epsilon < 0.15$, and for the images considered here, $|B_\Sun| < 25^\circ$, so $1-\epsilon+\sin(2\pi\epsilon)/6\pi$ and $\cos B_\sun$ can both only range between 0.9 and 1. Thus the heliotropic forced eccentricity can be expressed as a function of particle size:
\begin{equation}
e_f\simeq 0.0042 \frac{Q_{pr}}{r_g/1 \mu{\rm m}}.
\label{efpred}
\end{equation}
Strictly speaking, this calculation applies to individual ring particles, and the observed radial displacements of the ringlet represent the average motions of all the particles within the ringlet. Thus the measured heliotropic components of the ringlets' eccentricities provide estimates of an effective mean particle size in these ringlets.
For both the inner and Pan ringlets, $ae_f \sim 5$ km, implying that the particles in both ringlets have effective mean radii around 100$Q_{pr}$ microns. This estimate is plausible given previous studies of these and other dusty, heliotropic rings. For example, the ``charming ringlet" exhibits larger heliotropic radial excursions than the Encke Gap ringlets, indicating that the typical particle size is around 20$Q_{pr}$ microns \citep{Hedman10}, or a few times smaller than the particles in the Encke Gap. This is consistent with studies of the transmission spectra of all these ringlets, which contain a narrow dip that can be attributed to particles in the 10-50 micron size range \citep{Hedman11}. This spectral feature is weaker in the Encke Gap ringlets than it is in the ``charming ringlet'', implying that the Encke Gap ringlets contain a bigger fraction of larger particles.
\subsection{Radial transport in the Encke Gap}
Turning from eccentricities to semi-major axes, the longitudinal variations in the mean radial position of the inner and Pan ringlets outside of the clump-rich regions suggest that the semi-major axes of the ringlets' particles are drifting towards and away from Saturn. Since the particles in the clumps have the smallest semi-major axes, they should also have the shortest orbital periods and fastest orbital speeds. Hence we may also reasonably infer that the particles outside the clump-rich regions are drifting backwards in longitude relative to the clumps, and thus there is a steady stream of material flowing out from the trailing edge of the clump-rich region in each ringlet. If this is correct, then the observed trends in both ringlets' positions imply that the particles outside the clumps initially move outwards away from Saturn, but then reverse course and move back inwards when they approach the leading edge of the clump-rich regions.
More quantitatively, the observed trends in the ringlets' positions can be translated into estimates of the particles' radial migration rate. Say that at a given location in a ringlet, the particles' average semi-major axis drift rate $da/dt=v_a$. Furthermore, say the average semi-major axis of these particles $a$ is different from that of Pan or the clumps $a_0$. In that case, the particles will also drift longitudinally in a co-rotating system fixed to Pan or the clumps at a speed $v_\lambda=-1.5n(a-a_0)$, where $n$ is the mean motion of the clumps. The trajectory of these particles in the co-rotating frame therefore has the following slope:
\begin{equation}
\theta =\frac{1}{a_0}\frac{da}{d\lambda_c}=\frac{v_{a}}{v_{\lambda }}=-\frac{2}{3}\frac{v_a}{n}\frac{1}{a-a_0}.
\label{slope}
\end{equation}
Hence an observed slope $\theta$ in the ringlet implies a radial migration rate $v_a=-1.5n(a-a_0)\theta$.
Such migration rates may be compared with the rates that could be generated by various perturbation forces. Changing a particle's orbital semi-major axis also changes its orbital energy, so the most efficient way to generate a nonzero $v_a$ is to accelerate the particle along its direction of motion with an azimuthal force. If the average azimuthal force applied to the ring particle over one orbit is $F_\lambda$, then the particle's semi-major axis will drift at the following rate \citep{Burns76}:
\begin{equation}
v_{a} \simeq 2an\frac{F_\lambda}{F_G},
\label{drag1}
\end{equation}
where $F_G$ is Saturn's central gravitational force on the particle. Note the above equation assumes the particle's orbital eccentricity is small, which is reasonable for the Encke Gap ringlets. Combined with Equation~\ref{slope}, this expression can be used to estimate the forces required to produce an observed trend in a given ringlet.
The following subsections will explore what processes might be responsible for the various trends observed in the ringlets. First, we examine the apparent outwards motion behind the clumps and investigate whether this can be ascribed to interactions with the magnetospheric plasma. Then we consider the inwards motion just in front of the clumps and suggest that this may be due to collisions among different populations of ring particles.
\subsection{Outwards migration due to drag forces}
In both the inner and Pan ringlets, the semi-major axis drops steadily by about 7 km between $-180^\circ$ and $0^\circ$ in the co-rotating frame, which implies that: $\theta \simeq -1.7\times10^{-5}$. Hence, Equation~\ref{slope} implies that the particles in this particular region are drifting outwards at the following rate:
\begin{equation}
v_{aD} \sim +3\times10^{-5} {\rm m/s} \frac{(a-a_0)}{10 {\rm km}}.
\label{vadest}
\end{equation}
Similarly, Equation~\ref{drag1} implies that the magnitudes of the azimuthal force in these regions are:
\begin{equation}
\frac{F_\lambda }{F_G} \simeq 10^{-9} \frac{(a-a_0)}{10 {\rm km}}.
\label{fdfg}
\end{equation}
Note that both the migration rate and the perturbing force must increase with distance from the clump's semi-major axis in order to maintain the observed nearly constant slope.
One possible explanation for these radial motions is an interaction with the magnetospheric plasma. The ions in the plasma co-rotate with Saturn's magnetic field and thus move around the planet faster than particles orbiting at the Keplerian rate inside the Encke Gap. Thus, when these ions collide with the charged dust grains, the resulting momentum exchange accelerates the ring particles and causes them to slowly spiral outwards, as desired. Furthermore, the variations in the migration rate with distance from $a_0$ could be explained if the moon and/or dense clumps in these ringlets absorbed the plasma in their vicinity, sharply reducing the plasma density around the clumps' semi-major axis.
Unfortunately, it is not yet clear whether these sorts of interactions with plasma ions are sufficient to produce the observed trends in the ringlets' radial positions. The simplest expression for the azimuthal force experienced by a particle of radius $r_g$ due to these interactions is $F_D=\pi r_g^2\rho_i w^2$, where $\rho_i$ is the plasma ion mass density, $w=a(n-\Omega_S)$ is the azimuthal speed of the plasma ions relative to the ring particles, and $\Omega_S \simeq 810^\circ/$day is Saturn's rotation rate. Note that this is a highly over-simplified expression for the plasma interaction force, but it is a reasonable approximation for the tenuous plasma expected to exist within the rings \citep{Grun84}. Meanwhile, Saturn's gravitational pull on the particle $F_G$ can be written as $n^2a m$, where $n$ and $a$ are the particle's orbital mean motion and semi-major axis, and $m$ is the particle's mass, which can in turn be expressed in terms of the particle's radius $r_g$ and mass density $\rho_g$. The ratio of these two forces then becomes:
\begin{equation}
\frac{F_D}{F_G} \simeq \frac{3}{4}\frac{\rho_i}{\rho_g} \frac{a}{r_g} \left(1-\Omega_S/n\right)^2.
\end{equation}
For the particles in the Encke gap, $a \simeq 133,500$ km and $n \simeq 626^\circ/$day. Also, since these ringlets are composed primarily of water ice, we may assume that $\rho_g \simeq 1$ g/cm$^3$. Furthermore the magnitude of the ringlets' heliotropic forced eccentricities implies that $r_g \simeq 100 \mu$m (see above). Finally, the mass density of the plasma in the Encke gap can be estimated from data obtained by Cassini when it flew over the A ring during Saturn orbit insertion. Measurements made by various instruments demonstrate that the plasma surrounding the rings consists primarily of O$^+$ and O$^+_2$ \citep{Tokar05, Waite05, Young05}, so the mass per ion should be between 16 and 32 amu. Unfortunately, the number density of ions within the Encke Gap $n_i$ is not so well determined. During its passage over the rings, Cassini encountered ion densities above the rings between 0.1/cm$^3$ and 1.0/cm$^3$ \citep{Tokar05, Waite05}, but numerical models suggest that the ion number density at the ringplane could be as high as 10-100/cm$^3$\citep{Tseng10, Tseng11}. Taking $n_i=10/$cm$^3$ as a fiducial number, and assuming an equal mix of O$^+$ and O$^+_2$ in the ring's ionosphere, we can then estimate the above force ratio as:
\begin{equation}
\frac{F_D}{F_G} \simeq 3\times10^{-11}\left(\frac{n_i}{10{\rm/cm}^3}\right)\left(\frac{100 \mu{\rm m}}{r_g}\right)
\end{equation}
This is an order of magnitude less than the force required to produce the observed trends, and so simple plasma drag may be insufficient to produce the required outwards migration. However, the above calculation is very rough, and the force would be larger if the ion density in the Encke Gap is higher than 10/cm$^3$, the particles are less massive than assumed here, or the coupling between the plasma and the ring particles has been significantly underestimated by neglecting the Coloumb scattering between the charged grains and plasma ions (cf. Gr\"un {\it et al.} 1984). More detailed simulations of the plasma environment within the Encke Gap will therefore be needed in order to determine whether plasma drag could be responsible for the outward motions of these small grains.
Thus far, we have not been able to identify any other plausible physical process that could produce the observed outward trends in the ringlets' radial positions. However, whatever is causing these motions does not appear to be a localized phenomenon. Given that the radial positions of both the inner and Pan ringlets drift steadily outwards for over 180$^\circ$ in co-rotating longitude, some process is likely causing particles to accelerate azimuthally throughout the inner and central parts of the Encke Gap (the situation in the outer part of the gap is less clear). This perturbation therefore could have some relevance to other aspects of the ringlets' structure, even if we cannot yet identify how it is generated. In the following discussions, we use the generic term ``drag force'' to describe this as-yet unidentified azimuthal acceleration.
\subsection{Inwards migration from collisions and clump formation from instabilities}
While steady azimuthal forces can potentially explain both ringlets' outward displacement with increasing distance behind the clump-rich regions, it does not explain the opposite trend found just in front of these regions. This trend would require some process that transports material back inwards towards the planet and towards the clumps' semi-major axis. We propose that collisions among the particles in each ringlet are responsible for this inward motion. Furthermore, we suggest that the clumps themselves arise from an instability associated with such inter-particle collisions.
Whatever their origin, the drag forces discussed in the previous section cause the particles to spiral away from the planet, and to drift further and further outwards and backwards relative to the clump-rich part of the ringlet. Eventually, these ``drifters'' will move sufficiently far backwards that they will pass by the clump-rich regions. Extrapolating from the observed trends, these drifters will have semi-major axes that are only about 10-15 km exterior to the clump particles. If all the drifting particles had the same semi-major axes and were on perfectly circular orbits, they could just pass by the clumps and continue to spiral outwards. However, these particles are not all on simple circular orbits. Besides the mean forced and free components of the eccentricity discussed above, the finite widths of these ringlets suggest that their particles possess a finite range of eccentricities and semi-major axes. The radial widths of both the inner and central ringlets are greater than 10 km (see Figure~\ref{egapprof}), so the drifters can actually pass through the clumps and collide with that material. Furthermore, the relative velocities of the drifters and the clumps is small, so there are many opportunities for particles to collide before they drift past the clumps.
Since the drifting particles' semi-major axes are larger than those of the typical clump particles, the drifters are most likely to experience collisions with clump material near the periapses of their own orbits, when they will be moving faster than most of the clump material. Such collisions will therefore tend to knock the drifters backwards, slowing their orbital motion and causing their semi-major axes to decay inwards towards Saturn and the clump. The rate at which the drifting particles migrate towards the clumps due to such collisions is just the product of the semi-major axis shift induced by each collision and the collision frequency. To first order, the semi-major-axis shift per collision will be of order the semi-major axis difference between the drifter and the clumps, while the collision rate for a drifter will be the particle's mean motion times the clumps' optical depth. Hence the relevant radial drift rate should be of order:
\begin{equation}
v_{ac} \sim -\tau_c n (a-a_c),
\end{equation}
where $a_{c}$ is the semi-major axis of the clump particles, and $\tau_c$ is the clump optical depth. When the drifting particles initially encounter the clumps, they will have $a-a_c \simeq 10$ km, and the typical clump optical depth $\tau_c \simeq 0.1$ \citep{Hedman11}, so $v_{ac} \simeq -0.1$ m/s. By comparison the outward migration rate due to the drag forces is only $v_{aD} \sim 3\times10^{-5}$ m/s (see Equation~\ref{vadest}). Hence, collisions with the clump particles should be an efficient way to halt and reverse the outward migration of the drifting material.
It is important to note that these collisions not only affect the radial migration of particles, but also their longitudinal motion. By forcing the particles' semi-major axes to converge towards that of the clump, these interactions reduce the rate at which these particles drift past the clumps. Thus particles initially drifting past the clumps could get stuck in the clumps, raising the clump's density and increasing the likelihood that additional drifting particles will slow down in the clump's vicinity. This instability could potentially also explain the unusual motions of the clumps. In this scenario, the clumps would not represent a fixed set of particles. Instead, particles would be constantly entering and leaving the clump. Hence the apparent motion of the clump is controlled by how quickly particles get trapped or escape from this region, which does not necessarily correspond to the trajectory of any individual ring particle. Furthermore, as particles with different orbital elements converge on these dense regions, gradual variations in orbital eccentricities could transform into sharp features like the kinks. The dynamics of these clumps are quite complex and numerical simulations along the lines of those done by \citet{Lewis11} will likely be needed to evaluate whether the accelerations and orbital characteristics of the observed clumps are consistent with the above hypotheses. Such simulations will also probably be needed to determine whether inter-particle collisions can cause the radial position of the ringlet to begin to fall $\sim30^\circ$ in front of the clump-rich regions.
\subsection{Pan's gravity, the distribution of clumps and the location of the ringlets}
\begin{figure}[tbp]
\resizebox{6in}{!}{\includegraphics{horseshoedrag.pdf}}
\caption{Schematic representation of the asymmetric trajectories of the particles in the Pan ringlet due to the combined action of drag forces and Pan's gravity in a reference frame that co-rotates with Pan. Note radius increases upwards in this diagram, longitude increases to the right, and Pan's orbit is displayed as the dashed line. Also note that in this cartoon the radial (vertical) scale is highly exaggerated relative to the longitudinal (horizontal) scale. The particles are assumed to remain on nearly circular orbits in this cartoon, and initially have a range of longitudes along Pan's semi-major axis. On the right side of the figure, the particles are drifting outwards due to drag forces, while on the left they are undergoing horseshoe motion due to Pan's gravitational perturbations. Due to the intrinsic asymmetry of these motions, these particles are more likely to be found just in front of Pan, which is also where the clumps are located.}
\label{horseshoedrag}
\end{figure}
In the previous subsection, we proposed that collisions among the ringlet particles could keep ringlet material from drifting too far away from the semi-major axes of the relevant clumps. However, we still need to find a way to anchor the clumps at particular semi-major axes and prevent them from slowly drifting outwards under the influence of the relevant drag forces. It turns out that for both the Pan and the inner ringlets, the gravitational perturbations from Pan are likely responsible for maintaining the clumps at nearly constant semi-major axes.
For the Pan ringlet, the importance of Pan's gravity is not surprising. As discussed above, the entire Pan ringlet occupies the horseshoe zone surrounding Pan's orbit. As demonstrated by \citet{Murray94}, particles can be trapped in this region even in the presence of drag forces, so long as the latter do not allow a particle to escape the horseshoe region before it has a close encounter with the moon (see also Murray and Dermott 1999). In this case, we can estimate that the outwardly-drifting particles would have semi-major axes around 15 km exterior to Pan if they avoided collisions with any clump material. This lies comfortably within $\Delta a_h$ for Pan, so Pan's gravity should be able to keep the particles in such a ringlet from dispersing.
Furthermore, the combination of Pan's gravity and the outward migration induced by the drag forces could naturally produce the asymmetric distribution of clumps in the Pan ringlet (see Figure~\ref{horseshoedrag}). Imagine we launch fine debris on circular orbits at a range of longitudes relative to Pan, and for the sake of simplicity, let us neglect eccentricities driven by solar radiation pressure. These particles will then remain on circular orbits but they will all migrate outwards and drift backwards relative to Pan under the influence of the drag forces. These particles will encounter Pan at various positive values of $\delta a_{\rm before}=a-a_P$, and Pan's gravity will force all of them onto orbits with $\delta a_{\rm after}=-\delta a_{\rm before}$, so that they will begin to move forward relative to Pan. After the encounter, the steady outward migration will resume, and barring any collisions among the ring particles, the trajectories of the particles will form closed loops with one end at their start location and the other on the leading side of Pan. The average semi-major axis of all these particles therefore equals $a_P$, and the density of particles is highest in the region just in front of Pan. Since material naturally collects in front of Pan, collisions among the ringlet particles will favor the formation of clumps in this region, consistent with the observations. (Recall that because the clumps might not follow the trajectory of any individual particles, the clumps themselves would not necessarily follow trajectories like those shown in Figure~\ref{horseshoedrag}.)
Since no comparably massive moon has been identified in the inner ringlet, the clumps here cannot be similarly anchored by such horseshoe motion. Instead, we argue that the material in the inner ringlet is maintained by a balance between drag forces pulling particles outwards and Pan's gravitational perturbations pushing them inwards. As discussed above, some process is causing the particles far from the clumps to drift outwards at a rate of $v_{aD} \sim +3\times10^{-5} {\rm m/s}[(a-a_0)/10 {\rm km}]. $
On the other hand, each time the particles in the inner ringlet pass by Pan, their semi-major axes will be shifted inwards by the amount stipulated in Equation~\ref{dap}. The frequency of such encounters is $\Delta n=1.5n\Delta a/a$, so these perturbations will cause the particles to migrate inward at a rate:
\begin{equation}
v_{aP} \sim -5an\left(\frac{m_p}{M_S}\right)^2\left(\frac{a}{\Delta a}\right)^4.
\end{equation}
For the inner ringlet, $\Delta a /a \sim 0.0007$, which together with the current estimate of Pan's mass $m_p/M_S \sim 0.8*10^{-11}$ \citep{Porco07, Weiss09} yields $v_{aP} \sim -2\times10^{-5}$ m/s, which is remarkably close to the above value for $v_{aD}$. Hence the inner ringlet may well be situated in a region where the torques from drag forces and Pan's gravity balance, halting the radial motion of material. Indeed, material dispersed within the inner half of the gap will naturally collect at this location, as material closer to the planet is pushed outwards by drag forces and material closer to Pan is driven inwards. These competing forces, coupled with collisions among the particles, could then lead to the formation of a narrow ringlet.
A similar balancing of forces could potentially explain the distribution of material in the outer part of the Encke Gap (i.e., the narrow outer ringlet and the broader ``fourth ringlet''). However, since Pan's gravitational perturbations should always cause material to move away from Pan's orbit, such a balancing act would require some process that caused material in the outer part of Encke Gap to migrate inwards. One way this could occur is if the processes that accelerate particles in the inner and Pan ringlets decelerate the particles in the outer part of the Encke Gap, and thus cause particles to move away from both edges of the gap. Unfortunately, the data considered here do not have sufficient resolution to provide secure information about the orbital properties of the outer ringlet. Hence we cannot evaluate such possibilities at present. Future studies using higher-resolution observations should clarify the orbital properties of this ringlet, and thus provide additional insights into the dynamics of dust within the Encke Gap. For example, any trends in the semi-major axis could reveal whether particles in the outer half of the gap are migrating radially in the same way as the other two ringlets.
\section{Summary}
The Cassini observations of the dusty ringlets in the Encke Gap reveal a number of interesting dynamical phenomena:
\begin{itemize}
\item The bright clumps in the central Pan ringlet are confined to a longitudinal region roughly 60$^\circ$ wide just in front of Pan.
\item The bright clumps in the inner and outer ringlets cover less than 180$^\circ$ in co-rotating longitude, and the distribution of clumps is not obviously disrupted by conjunctions with Pan.
\item Within the inner and Pan ringlets, clumps drift relative to each other at rates of up 0.04$^\circ$/day, while the largest relative drift rates observed in the outer ringlet are near 0.01$^\circ$/day.
\item Clumps in the Pan and inner ringlets are observed to merge and split. They also accelerate in surprising ways and follow trajectories that are inconsistent with those expected for isolated particles moving in the combined gravitational fields of Saturn and Pan.
\item The orbital elements of the particles in both the inner and Pan ringlets vary systematically with co-rotating longitude.
\item Both the inner and Pan ringlets exhibit some heliotropic behavior, and outside the clumps, the free eccentricity is approximately equal to the forced eccentricity that is induced by solar radiation pressure.
\item ``Kinks" in the Pan and inner ringlets associated with the clumps appear to correspond to variations in the ring-particle's eccentricities. In the Pan ringlet, these kinks
seem to be locations where the heliotropic forced eccentricity is reduced.
\item The semi-major axes of both the inner and Pan ringlets vary with co-rotating longitude. They reach a minimum within the clump-rich regions and are up to 10 km larger outside of this region.
\end{itemize}
\section*{Acknowledgements}
We acknowledge the support of the Cassini Imaging Team, the Cassini Project and NASA. This work was funded by NASA Cassini Data Analysis Program Grants
NNX09AE74G and NNX12AC29G. We also wish to thank S. Charnoz and an anonymous reviewer for their helpful comments.
\begin{table}
\caption{Movie sequences used to construct mosaics}
\label{moslist}
\resizebox{6.5in}{!}{\begin{tabular}{|c|c|l|l|c|c|c|c|c|c|c|c|c|}\hline
Rev & Sequence & Date & Images &
Em. & Phase & Solar & Obs. & Mosaic & \multicolumn{3}{c|}{Quality Flags$^c$} \\
& & & & Angle & Angle & Long.$^a$ & Long.$^a$ & Res.$^b$ & Pan & Inner & Outer \\ \hline
000 & SATSRCH & 2004-173 & N1466448221-N1466504861 (119) &
106$^\circ$ & 67$^\circ$ & 159$^\circ$ & 178$^\circ$ & 20 km/pix & I & X & X\\
00A & SPKMOVPER & 2004-320 & N1479201492-N1479254052 (74) &
102$^\circ$ & 84$^\circ$ & 165$^\circ$ & 156$^\circ$ & 14 km/pix & P & I & X\\
008 & LPHRLFMOV & 2005-138 & N1495091875-N1495139739 (194) &
109$^\circ$ & 42$^\circ$ & 172$^\circ$ & 216$^\circ$ & 5 km/pix & R & P & I \\
030 & HIPHAMOVE & 2006-279 & N1538861755-N1538900050 (70) &
77$^\circ$ & 159$^\circ$ & 191$^\circ$ & 302$^\circ$ & 6 km/pix & R & R & P\\
034 & HIPHAMOVD & 2006-331& N1543346569-N1543387061 (46) &
70$^\circ$ & 158$^\circ$ & 193$^\circ$ & 305$^\circ$ & 5 km/pix & R & R & P\\
044 & FMOVIE & 2007-125 & N1557020880-N1557071468 (134) &
61$^\circ$ & 81$^\circ$ & 198$^\circ$ & 180$^\circ$ & 6 km/pix & P & P & P\\
051 & LPMRDFMOV & 2007-291 & N1571435192-N1571475337 (260) &
86$^\circ$ & 56$^\circ$ & 204$^\circ$ & 170$^\circ$ & 7 km/pix & R& R & X\\
053 & LPHRDFMOV & 2007-334 & N1575141899-N1575189603 (134) &
80$^\circ$ & 52$^\circ$ & 205$^\circ$ & 165$^\circ$ & 5 km/pix & R & R & P\\
109 & LRHPENKMV & 2009-107 & N1618663507-N1618688110 (60) &
47$^\circ$ & 117$^\circ$ & 221$^\circ$ & 302$^\circ$ & 4 km/pix & R & R & P\\
115 & FMOVIEEQX & 2009-211& N1637609661-N1627655251 (149) &
62$^\circ$ & 100$^\circ$ & 224$^\circ$ & 237$^\circ$ & 5 km/pix & R & R & P\\
124 & LRHPENKMV & 2010-007 & N1641576230-N1641603998 (104) &
106$^\circ$ & 118$^\circ$ & 229$^\circ$ & 81$^\circ$ & 5 km/pix & R & R& P\\
124 & LRHRENKMV & 2010-008 & N1641604730-N1641631010 (91) &
107$^\circ$ & 129$^\circ$ & 229$^\circ$ & 268$^\circ$ & 5 km/pix & R & R& P\\
132 & SHRTMOVIE & 2010-153 & N165413619- N1654175167 (240) &
78$^\circ$ & 141$^\circ$ & 233$^\circ$ & 289$^\circ$ & 2 km/pix & P & P & P\\
\hline
\end{tabular}}
$^a$ Longitudes measured relative to ring's ascending node on the J2000 coordinate system.
$^b$ Resolution of mosaics generated from the images, which oversample
the original pixels by roughly a factor of 2.
$^c$ X=no attempt to derive brightness profiles. I=Brightness profiles derived by integration over a radial range. P=Brightness profiles derived using a peak-fitting routine. R=Radial locations derived from peak-fitting routine suitable to
determining ringlet orbital elements.
\end{table}
\begin{table}
\caption{Supplementary images containing the region around Pan}
\label{suplist}
\resizebox{6.5in}{!}{\begin{tabular}{|c c|c c|c c|c c|c c|}\hline
Image & Date & Image & Date & Image & Date & Image & Date & Image & Date \\ \hline
N1492024160 & 2005-102 & N1552731154 & 2007-075 & N1575012478 & 2007-333 & N1583628328 & 2008-068 & N1603375318 & 2008-296 \\
N1492759120 & 2005-111 & N1552731197 & 2007-075 & N1575012511 & 2007-333 & N1583758349 & 2008-069 & N1603375361 & 2008-296 \\
N1493446920 & 2005-119 & N1553898401 & 2007-088 & N1575055318 & 2007-333 & N1583758382 & 2008-069 & N1603721360 & 2008-300 \\
N1493544975 & 2005-120 & N1553898444 & 2007-088 & N1575055351 & 2007-333 & N1586079511 & 2008-096 & N1603721403 & 2008-300 \\
N1495641779 & 2005-144 & N1553936876 & 2007-089 & N1575629792 & 2007-340 & N1586079554 & 2008-096 & N1604570501 & 2008-310 \\
N1495713539 & 2005-145 & N1553936919 & 2007-089 & N1575629835 & 2007-340 & N1586106286 & 2008-096 & N1604570544 & 2008-310 \\
N1495770990 & 2005-146 & N1554110742 & 2007-091 & N1575676367 & 2007-340 & N1586106329 & 2008-096 & N1606481890 & 2008-332 \\
N1495814115 & 2005-146 & N1554110785 & 2007-091 & N1575676410 & 2007-340 & N1586166616 & 2008-097 & N1607328286 & 2008-342 \\
N1496700636 & 2005-156 & N1555229824 & 2007-104 & N1575800823 & 2007-342 & N1586166659 & 2008-097 & N1607328329 & 2008-342 \\
N1497235299 & 2005-163 & N1555229867 & 2007-104 & N1575800866 & 2007-342 & N1587821608 & 2008-116 & N1610355419 & 2009-011 \\
N1497276055 & 2005-163 & N1555508391 & 2007-107 & N1576171776 & 2007-346 & N1587821651 & 2008-116 & N1610355462 & 2009-011 \\
N1498058015 & 2005-172 & N1555508434 & 2007-107 & N1576171819 & 2007-346 & N1588751210 & 2008-127 & N1610899512 & 2009-017 \\
N1498825460 & 2005-181 & N1555556437 & 2007-108 & N1577141652 & 2007-357 & N1588751253 & 2008-127 & N1610899555 & 2009-017 \\
N1499520329 & 2005-189 & N1555556480 & 2007-108 & N1577141695 & 2007-357 & N1590835414 & 2008-151 & N1612537044 & 2009-036 \\
N1499726971 & 2005-191 & N1555615492 & 2007-108 & N1577512965 & 2007-362 & N1591525824 & 2008-159 & N1612537087 & 2009-036 \\
N1500341195 & 2005-199 & N1555615535 & 2007-108 & N1577513008 & 2007-362 & N1591525867 & 2008-159 & N1616991490 & 2009-088 \\
N1500516231 & 2005-201 & N1555708703 & 2007-109 & N1578630743 & 2008-010 & N1591997427 & 2008-164 & N1616991533 & 2009-088 \\
N1501156540 & 2005-208 & N1555708746 & 2007-109 & N1578630786 & 2008-010 & N1591997460 & 2008-164 & N1619963567 & 2009-122 \\
N1502133340 & 2005-219 & N1556520958 & 2007-119 & N1579656750 & 2008-022 & N1592072518 & 2008-165 & N1619963610 & 2009-122 \\
N1502133373 & 2005-219 & N1556520991 & 2007-119 & N1579656793 & 2008-022 & N1592072551 & 2008-165 & N1622382064 & 2009-150 \\
N1502581803 & 2005-224 & N1558417179 & 2007-141 & N1579750261 & 2008-023 & N1596292933 & 2008-214 & N1622382097 & 2009-150 \\
N1502581836 & 2005-224 & N1558417222 & 2007-141 & N1579750304 & 2008-023 & N1596292976 & 2008-214 & N1622592755 & 2009-152 \\
N1502650783 & 2005-225 & N1558547905 & 2007-142 & N1580528781 & 2008-032 & N1596720406 & 2008-219 & N1622592788 & 2009-152 \\
N1502650816 & 2005-225 & N1558547948 & 2007-142 & N1580528824 & 2008-032 & N1596720449 & 2008-219 & N1623652033 & 2009-165 \\
N1503573529 & 2005-236 & N1559285595 & 2007-151 & N1580566252 & 2008-032 & N1597462656 & 2008-228 & N1623652076 & 2009-165 \\
N1503573562 & 2005-236 & N1559285638 & 2007-151 & N1580566295 & 2008-032 & N1597462699 & 2008-228 & N1623757093 & 2009-166 \\
N1504218268 & 2005-243 & N1559710457 & 2007-156 & N1580614147 & 2008-033 & N1597488396 & 2008-228 & N1623757136 & 2009-166 \\
N1504341929 & 2005-245 & N1559710500 & 2007-156 & N1580614190 & 2008-033 & N1597488439 & 2008-228 & N1623822254 & 2009-167 \\
N1549374582 & 2007-036 & N1559841843 & 2007-157 & N1580653027 & 2008-033 & N1600167160 & 2008-259 & N1623822297 & 2009-167 \\
N1549374625 & 2007-036 & N1559841886 & 2007-157 & N1580653070 & 2008-033 & N1600167203 & 2008-259 & N1625116703 & 2009-182 \\
N1552517897 & 2007-072 & N1559885869 & 2007-158 & N1580766488 & 2008-034 & N1601291283 & 2008-272 & N1625116736 & 2009-182 \\
N1552517940 & 2007-072 & N1559885912 & 2007-158 & N1580766531 & 2008-034 & N1601291316 & 2008-272 & N1627546060 & 2009-210 \\
N1552606713 & 2007-073 & N1560054860 & 2007-160 & N1581513703 & 2008-043 & N1602109066 & 2008-281 & N1627546103 & 2009-210 \\
N1552606756 & 2007-073 & N1560054903 & 2007-160 & N1581513746 & 2008-043 & N1602109109 & 2008-281 & N1628912570 & 2009-226 \\
N1552645698 & 2007-074 & N1573672968 & 2007-317 & N1582637241 & 2008-056 & N1602501762 & 2008-286 & N1628912603 & 2009-226 \\
N1552645741 & 2007-074 & N1573673011 & 2007-317 & N1582637274 & 2008-056 & N1602501805 & 2008-286 & N1633029034 & 2009-273 \\
N1552688328 & 2007-074 & N1574856717 & 2007-331 & N1583401346 & 2008-065 & N1603175686 & 2008-294 & N1633029067 & 2009-273 \\
N1552688371 & 2007-074 & N1574856760 & 2007-331 & N1583401389 & 2008-065 & N1603175729 & 2008-294 & & \\
\hline
\end{tabular}}
\end{table}
|
2,877,628,090,822 | arxiv | \section{Introduction}
{\it Dynamical triangulations} (DT) were invented as a nonperturbative
regularization of bosonic string theory and thus also of two-dimensional
quantum gravity coupled to conformal matter.
This program was both a failure --
in showing that even in a nonperturbative
setting no bosonic string theory exists in dimension two or larger --
and an amazing success, in providing a versatile
regularization of 2d quantum gravity coupled to conformal
matter with central charge $c \leq 1$, i.e. noncritical string theory.
Surprisingly, in many ways the regularized theory turned out to be easier
to solve analytically than the corresponding continuum theory.
Encouraged by this, DT was generalized to provide a regularization of
quantum gravity in three \cite{av,am3,andre}
and four dimensions \cite{aj,am4}. The na\"ive
expectation was that {\it if} a
`stand-alone' four-dimensional theory of quantum gravity existed,
the regularized theory should have a second-order phase transition,
which could be used to define a continuum theory
of quantum gravity. Second-order transitions are usually characterized
by a divergent correlation length associated with propagating field degree(s) of
freedom, in the case at hand presumably of a gravitonic nature.
If a given phase transition point was a UV fixed point, one
could also attempt to make contact with Weinberg's asymptotic-safety scenario,
for which plenty of corroborating evidence has been found recently
(see \cite{reuter-review} for reviews).
A phase transition point was indeed located in 4d DT, and at first
believed to be of second order \cite{aj,fractal4d,more4d}.
However, analyzing larger lattice systems changed the verdict to a
first-order phase transition where no obvious continuum limit
could be defined, at least not when using the Regge version of the
Einstein-Hilbert action \cite{Bialas:1996wu,deBakker:1996zx}.
Neither did one find convincing
evidence of a good classical behaviour of large-scale geometry away from the
phase transition.
Partly triggered by this impasse, a modified lattice model
in terms of {\it Causal Dynamical Triangulations} (CDT) was proposed, and
subsequently shown to have
long-distance properties in agreement with (semi-)classical gravity \cite{cdt}.
It still uses the Regge-Einstein-Hilbert action, but
assumes the existence of a global time-foliation, and
has a more complicated phase diagram than the simplest DT
model.\footnote{The notable similarities with `Ho\v rava-Lifshitz gravity'
\cite{horava} are being explored, see e.g. \cite{Ambjorn:2010hu}.}
In what follows, we will provide strong, new
evidence that -- unlike its Euclidean counterpart -- 4d CDT quantum gravity
possesses a second-order phase transition line.
\section{Causal Dynamical Triangulations}
We begin with a brief account of CDT, focusing on several important aspects
(see
\cite{ Ambjorn:2005qt,Ambjorn:2001cv,Ambjorn:2007jv,Ambjorn:2008wc}
for technical details and \cite{reviews} for reviews).
CDT can be characterized as a nonperturbative path integral which
is as close as possible to a canonical quantization: spacetime histories
share a global foliation, where each leaf is a spatial hypersurface, given in terms of
a three-dimensional triangulation of fixed topology $\mathcal{T}$,
built from equilateral
tetrahedra with link length $a_s$, and labeled by a discrete proper time $t_n$.
Adjacent hypersurfaces are connected by four-simplices, resulting in
spacetime histories of the form of four-dimensional
triangulations of topology $\mathcal{T}\times[0,1]$.
We use $\mathcal{T}=S^3$ and impose
periodic boundary conditions in time, such that the spacetime
topology is $S^3\times S^1$.
The geometry of each spacetime is fixed by how the four-simplices
are `glued together' to
form a simplicial manifold, and by the lengths of its links, which come in two types:
{\it spacelike} links which lie entirely within a given hypersurface, and {\it timelike}
links whose endpoints lie on adjacent hypersurfaces, and which have a
(squared) edge length $a_t^2=\alpha a_s^2$, for some fixed relative scaling
parameter $\alpha<0$.
The CDT gravitational path integral is a sum over all geometrically inequivalent
triangulations of this type with a fixed number of time steps,
with amplitudes depending on the
above-mentioned Regge-Einstein-Hilbert action \cite{Regge:1961px}.
Since in four dimensions analytical methods are mostly unavailable
we use Monte Carlo simulations to extract physical results.
To do this we must convert the path integral to a
statistical partition function by applying a Wick rotation, which
due to the foliated structure exists globally \cite{Ambjorn:2001cv}.
It can be implemented
by rotating
$\alpha\rightarrow -\alpha$ in the lower-half complex plane, leading to
the {\it Euclidean} Regge action
\begin{eqnarray}
S_E =\frac{1}{G}\int\sqrt g(-R+2\Lambda)
\rightarrow -(\kappa_0+6\Delta)N_0+\kappa_4 N_4+
\Delta(N_4+N_4^{(4,1)}) \label{reggeaction},
\end{eqnarray}
where $N_0$ and $N_4$ denote the total number of vertices and four-simplices and
$N_4^{(4,1)}$ counts the subset of four-simplices which have four vertices on one
hypersurface and the fifth one on a neighbouring one.
The couplings $\kappa_0$, $\kappa_4$ and $\Delta$ are
linearly related to the bare inverse gravitational coupling,
the bare cosmological coupling and the parameter $\alpha$
introduced above. Redefining $\tilde{\kappa}_4=\kappa_4+\Delta$,
we obtain the Euclidean Regge action
implemented in the computer simulations, namely,
\begin{equation}
S_{Regge} = -(\kappa_0+6\Delta)N_0+
\tilde{\kappa}_4 N_4+\Delta N_4^{(4,1)}=:
-\kappa_0 N_0 +\tilde{\kappa}_4 N_4+\Delta\, \mathrm{conj}(\Delta),
\label{softwareaction}
\end{equation}
where for later convenience we have introduced the quantity
$\mathrm{conj}(\Delta)=N_4^{(4,1)}-6 N_0$
conjugate to $\Delta$. This turns the gravitational
path integral into a statistical
partition function with Boltzmann weights $\exp (-S_{Regge})$.
We will use the freedom to switch to a different ensemble, obtained
by keeping $N_4$ (measuring the four-volume of the system) fixed, instead of its
conjugate $\kappa_4$.
In this way we can treat $N_4$ as a finite-size scaling parameter
which does not appear in the phase
diagram of the putative continuum theory. The remaining
couplings $\kappa_0$ and $\Delta$ span the phase diagram,
which we will go on to explore in the next section.
\section{The phase diagram of CDT}
\label{section:phasediagram}
A qualitative description of the CDT phase diagram first appeared
in \cite{Ambjorn:2005qt}, with a quantitative plot presented later in
\cite{Ambjorn:2010hu}.
The diagram exhibits three phases,
labeled A, B and C in Fig.\ \ref{fig:pd_plot}.
A geometric characterization of the phases can be given in terms of
their distinct spatial volume profiles $N_3(t)$, measuring
the three-volume in lattice units as a function of proper time $t$.
As described in detail elsewhere \cite{Ambjorn:2008wc,Ambjorn:2007jv},
the average large-scale geometry found in phase C shows
the scaling behaviour of a genuine four-dimensional universe.
The average volume profile matches beautifully that of a Euclidean
de Sitter spacetime. Even the quantum fluctuations
around this {\it emergent background geometry} are described well by a
cosmological minisuperspace action \cite{Ambjorn:2008wc,semi}.
The situation in the other phases is completely different.
The typical volume profile
of a configuration in phase A shows an almost uncorrelated sequence of
spatial slices, while the configurations in phase B are characterized
by an almost vanishing time extension.
\begin{figure}[t]
\centerline{\scalebox{1.3}{\includegraphics{phasediagram.eps}}}
\caption{The phase diagram of CDT. The large crosses represent actual
measurements.}
\label{fig:pd_plot}
\end{figure}
A preliminary analysis in \cite{Ambjorn:2010hu} suggested that the
A-C transition is of first order, similar in nature
to the first-order transition observed in the DT formalism mentioned
above. However, as also pointed out in \cite{agjlgt}, to nail down this result
the numerical evidence still needs to be strengthened.
By contrast, our interest in the present letter is in the order of the B-C transition,
about which considerable doubt remained in \cite{Ambjorn:2010hu}.
We will present strong evidence below that it is a second-order transition line.
Before doing so, it is instructive to analyze how the transitions change as we
move along the respective phase transition lines, while holding the system size
fixed. The A-C transition line is characterized by a jump in
$N_0$, the variable conjugate to the coupling constant $\kappa_0$,
when we cross the line by changing $\kappa_0$. We see no
appreciable change in this signal when we move along the A-C
transition line, although we have not examined closely the triple point
where all three phases meet.
The situation is very different for the
B-C transition. When changing the coupling constant $\Delta$ and
crossing the transition line, we observe a jump in the variable
$\mathrm{conj}(\Delta)$.
However, moving to smaller values of $\kappa_0$ on the left, the jump
decreases. Around $\kappa_0=1.0$, no signature of a
phase transition remains. We conclude that the
B-C transition has an endpoint, which for $N_4=80k$ is located
around $\kappa_0=1.0$. Moving to the right the jump increases, and
around $\kappa_0 = 2.3$ the transition is so strong that we
get stuck in metastable states. The dashed part of the B-C line
in Fig.\ \ref{fig:pd_plot} marks the region where
conventional methods are
insufficient to measure the location of the phase transition with acceptable
accuracy. We are analyzing currently whether
the use of multicanonical Monte Carlo simulations
can help in resolving this issue.
\section{The order of the B-C phase transition}
\label{sec:ptorder}
Measuring the order of phase transitions requires some care.
To confirm that a phase transition is {\it not} a first-, but a
second-order transition, one can try to measure
various so-called critical exponents.
One such exponent measures the shift of a transition point with system size.
Recall first how this works for a conventional system such as the
Ising model with volume $V=L^d$, where $d$ is the dimension of the
system \cite{Newman:1999a}. Considering the Ising model's
temperature-driven phase
transition and using the location of the maximum of
the magnetic susceptibility to define a transition point $\beta^c(V)$,
one finds the power-law behaviour
\begin{equation}
|\beta^c(\infty)-\beta^c(V)|\propto V^{-1/\nu d}
\label{eq:ptshift}
\end{equation}
for sufficiently large system sizes.
The exponent $\nu$ governs the increase of the correlation length in
a second-order transition as one moves towards the
critical point $\beta^c(\infty)$ on an infinite lattice.
For first-order transitions
there is no correlation length and one expects a
scaling like (\ref{eq:ptshift}), with $\nu d$ replaced by
an exponent $\tilde{\nu}$ where $\tilde{\nu}=1$ \cite{MeyerOrtmanns:1996ea}.
A sufficiently strong violation of $\tilde{\nu}=1$ therefore
signals the presence of a second-order transition.
Another quantity of interest is the Binder cumulant
\begin{equation}
B_{\cal O}=\frac{1}{3}\left(1-\frac{\left<{\cal O}^4\right>}
{\left<{\cal O}^2\right>^2}\right)
\label{binder}
\end{equation}
associated with an observable $\cal O$ \cite{MeyerOrtmanns:1996ea},
which is always nonpositive and zero if the histogram of
$\cal O$ is Gaussian. Evaluating $B_{\cal O}$ as a function of the couplings,
its local minima lie at transition points. We can measure these
minima for different system sizes and by extrapolation determine
$B_{\cal O}^{\min}(1/N_4=0)$. At a second-order transition the
histogram of $\cal O$ should converge to a single Gaussian distribution
with the Binder cumulant going to zero. At a first-order transition it
will go to a nonzero constant, however, its value at a weak first-order
transition can be small.
\begin{figure}[t]
\centerline{\scalebox{1.3}{\includegraphics{shiftexpBC.eps}}}
\caption{Measuring the location $\Delta^c$ of B-C transition points at
$\kappa_0=2.2$ for different system sizes $N_4$
to determine the shift exponent $\tilde{\nu}$.}
\label{fig:shiftexpBC}
\end{figure}
To analyze the B-C transition, we fixed $\kappa_0=2.2$ and used systems
of size 40, 50, 60, 80, 100, 120, 140 and 160$k$. The number of
sweeps used was approximately $2.5\cdot 10^6$, with
one sweep consisting of one million attempted Monte Carlo moves.
We have measured the shift exponent $\tilde\nu$ for the asymmetry
parameter $\Delta$ using
\begin{equation}
\Delta^c(N_4)=\Delta^c(\infty)-C N_4^{-1/\tilde{\nu}},
\label{eq:ptshiftBC}
\end{equation}
where $C$ is a proportionality factor, and $\Delta^c$ has been defined
using the location of the maximum of the susceptibility
$\chi_{\mathrm{conj}(\Delta)}\! =\!
\left<\mathrm{conj}(\Delta)^2\right>\! -\!\left<\mathrm{conj}(\Delta)\right>^2$.
Fig.\ \ref{fig:shiftexpBC} shows the measured data points (error bars
too small to be included) and the best fit through all of them,
yielding $\tilde{\nu}=2.39(3)$. To judge whether
our range of system sizes lies inside the scaling region we have
performed a sequence of fits by successively removing the data points
of lowest four-volume, leading to $\tilde\nu$-values
$2.51(3)$, $2.49(3)$ and
$2.51(5)$, where the last fit was performed with all but five data
points removed.
This suggests that the data point with the lowest four-volume lies
outside the scaling region. Removing it from the fit we obtain
\begin{equation}
\tilde{\nu}=2.51(3).
\end{equation}
This result makes a strong case for a second-order transition,
since the prediction $\tilde{\nu}=1$ for a first-order transition is
clearly violated. (By contrast, for the A-C transition one finds
$\tilde{\nu} \approx 1$
as will be reported elsewhere \cite{to-appear}.)
\begin{figure}[t]
\centerline{\scalebox{1.3}{\includegraphics{bincumBC.eps}}}
\caption{Dependence of the minimum of the
Binder cumulant $B_{\mathrm{conj}(\Delta)}$ on the (inverse)
system size at the B-C
transition. At a second-order transition, $B^{\min}\!\rightarrow\! 0$ in
the infinite-volume limit. (Fit excludes the two points on the right.)}
\label{fig:bincumBC}
\end{figure}
Lastly, we have investigated how the minimum of the Binder cumulant
(\ref{binder}) depends on the system size. Fig.\ \ref{fig:bincumBC}
shows $B_{\mathrm{conj}(\Delta)}^{\min}$ as a function of
\emph{inverse} system size (errors
approximately equal to the dot radii).
Inside the scaling region the minimum of the cumulant is expected
to have a power-law behaviour. To understand which data
points lie inside the scaling region, we have again performed
a sequence of fits by successively removing the points
of lowest four-volume. This has led to the exclusion of
the data points for $N_4=40k$ and
$N_4=50k$ from the fit shown in Fig.\ \ref{fig:bincumBC}.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|r|}
\hline
observable $\cal O$ & $B_{\cal O}^{\min}(N_4\rightarrow\infty)$ \\
\hline
\hline
$\mathrm{conj}(\Delta)$ & $-0.003(4)$ \\
\hline
$N_4^{(4,1)}$ & $-0.001(3)$ \\
\hline
$N_2$ & $-0.0000001(3)$ \\
\hline
$N_1$ & $-0.000003(7)$ \\
\hline
$N_0$ & $0.0000(3)$ \\
\hline
\end{tabular}
\end{center}
\caption{Measurements of $B_{\cal O}^{\min}(N_4\!\rightarrow\!\infty)$
for various observables $\cal O$, where $N_k$ denotes the number
of $k$-dimensional (sub-)simplices in the triangulation.}
\label{tab:bincumBC}
\end{table}
Table \ref{tab:bincumBC} collects the extrapolations
$B_{\cal O}^{\min}(N_4\!\rightarrow\!\infty)$ for several observables $\cal O$.
As indicated earlier, it is always difficult to make a strong case
for a second-order transition based on Binder cumulant measurements alone,
because weak first-order transitions may show a convergence to a
nonzero value close to zero.
Nevertheless, in the case at hand all our measurements are mutually
consistent and in excellent agreement with the limiting value 0,
further corroborating our claim of the second-order nature of the transition.
\section{Discussion}
We have succeeded in our goal of determining the order of the B-C
transition in CDT quantum gravity by applying two distinct methods, namely,
measuring the shift exponent and analyzing Binder cumulants.
The measured shift exponent
$\tilde{\nu}=2.51(3)$ represents a strong violation of the
prediction $\tilde{\nu}=1$ for a first-order transition.
Also the results of the Binder cumulant
analysis are clearly and unambiguously consistent with the
second-order nature of the transition.
From this we conclude that there is strong evidence
that the B-C transition is of second order, making four-dimensional
CDT quantum gravity the first known instance of a dynamically triangulated
model (without matter coupling) in any signature and dimension which
displays such a transition. This result is
potentially very attractive. It opens the door to studying
critical phenomena in CDT and
defining a continuum limit where the lattice spacing (the UV cut-off)
is taken to zero, just as one does in standard lattice quantum field theories
with nondynamical geometry.
\vspace{.5cm}
\noindent {\bf Acknowledgements.} SJ thanks Prof.\ G.T.\ Barkema
for fruitful discussions and valuable advice on
the numerical aspects of this work. JA thanks the ITP, Utrecht for hospitality.
RL acknowledges support by the Netherlands
Organisation for Scientific Research (NWO) under their VICI
program. The contributions by SJ and RL are
part of the research programme of the
Foundation for Fundamental Research on Matter (FOM), financially
supported by NWO.
|
2,877,628,090,823 | arxiv | \section{WHY CLOUD?}
The unprecedented growth of the global smartphone market over the last decade has been mirrored by the more recent emergence and rapid expansion of enterprise Cloud Computing Platforms (CCP). CCP provide on-demand computing, storage and software accessible over the internet, allowing for the remote offloading of process-intensive tasks. This approach to server technology is an increasingly common long-term strategy for replacing the traditional manually maintained client-server hardware set-up \cite{labati2020cloud}.
A clear advantage of CCPs are that they allow users of mobile devices to gain access to significant processing power, well beyond the means of any existing mobile device. This approach allows for patients to use even very dated mobile hardware to access the latest advances in automated medical image analysis. This essentially means that continual advances in this field are not tied to the computing capability of mobile devices, as such devices are simply consuming services from CCPs. Additionally, scalability becomes easier to manage, given the virtualised nature of cloud services. There is a growing trend in the use of ensemble CNNs in medical image analysis, whereby multiple CNNs are used to form a final prediction. Distributing mobile apps that use multiple models is not practical or possible given the limited permissible size of apps when distributed via online app stores. There is also the issue of intellectual property protection. Android apps are particularly easy to reverse engineer, so having the CNNs run on the server instead of the user's mobile device means that trained models are never publicly exposed.
\section{SYSTEM ARCHITECTURE}
The two major components created for the evaluation were (1) a cross-platform mobile app, and (2) a cloud-based deep learning framework that performed inference on foot photographs sent from mobile clients. A cross-platform framework was chosen for the development of the mobile client since the ultimate goal of this research is to provide patients with a means of remotely monitoring and diagnosing DFU using their own smart-phones, which primarily comprise of Android or iOS devices. An overview of the system physical architecture is shown in Fig. \ref{fig:PhysArchitecture}. The following sections describe how these components were utilised in the creation of our proposed framework.
\begin{figure*}
\centering
\includegraphics[scale=0.56]{PhysicalArchitecture.pdf}
\caption{An overview of the physical architecture showing the major structural components involved in the implementation.}
\label{fig:PhysArchitecture}
\end{figure*}
\subsection{Mobile App}
Cross-platform development can help to reduce the time and costs associated with developing apps for multiple mobile platforms. The mobile app developed for our evaluation was created using Ionic, a cross-platform framework based on the earlier Cordova framework. Screens within Ionic apps are rendered onto a standard WebView, in the same way that web pages are rendered in web browsers. There are also native elements within the framework however, including the ability to interface with the device's hardware features, such as sensors and cameras. Fig. \ref{fig:MobileAppScreens} shows the main data capture screens within the mobile app.
The primary objective of our initial proof of concept evaluation was to determine the usability and reliability of our cross-platform mobile client and cloud-based framework in real-world settings. Ease of use was a primary motivating factor behind the design of the mobile app. Screens within the app display context-sensitive information in the form of an information bar at the top of each screen that was used to guide the user through the process of acquiring and uploading foot photographs. The UI and validation were designed so that it was not possible for the user to take the wrong action. Examples of this include:
\begin{itemize}
\item It was not possible to retake a photograph for the current foot if one had already been taken and uploaded.
\item The user could not upload a photograph for any foot more than once.
\item It was not possible to change left foot ``checked" tickbox if the left foot photo had been uploaded.
\item It was not possible to change right foot ``checked" tickbox if the right foot photo had been uploaded.
\end{itemize}
Ionic utilises a Model View Controller (MVC) architecture, implemented using Angualr.js, which separates data, presentation of data and business logic. App data, including application state, is stored in a local SQLite database.
\begin{figure*}
\centering
\includegraphics[scale=0.37]{footsnap_6.png}
\caption{UI screenshots from the proposed cross-platform mobile client. From left to right, (top-left) patient QR code is scanned, (top-middle) clinician enters details of each foot being examined, (top-right) clinician enters number of visible ulcers, (bottom-left) photo acquisition of foot, (bottom-middle) inference result returned by cloud service, (bottom-right) examination complete.}
\label{fig:MobileAppScreens}
\end{figure*}
\subsection{Oracle Mobile Cloud Service Software Development Kit (SDK)}
Oracle provides a SDK for several mobile development platforms, including Ionic, which enables mobile clients to interface with Oracle Mobile Hub (OMH). The Oracle Mobile Cloud Service SDK is a HyperText Transfer Protocol Secure client layer, through which requests can be made to OMH and associated services using JavaScript Object Notation (JSON) via REpresentational State Transfer (REST) to transfer data between clients and the cloud service.
\subsection{Cloud Platform}
The cloud platform services developed for our proof-of-concept clinical evaluation were implemented using Oracle Cloud Infrastructure (OCI). OCI is an online enterprise scale cloud service offering Infrastructure as a Service (IaaS), Platform as a Service (PaaS) and Software as a Service (SaaS). A breakdown of these service models are described in the following sub-sections.
\subsubsection{Platform as a Service}
Oracle Mobile Hub (OMH) and the Autonomous Transaction Processing Instance (ATPI) represent the PaaS elements used in the evaluation. OMH provides a gateway for mobile clients to access other internal cloud services, and includes features such as identity management, analytics, and Application Programming Interface (API) management.
The ATPI hosts the Oracle 18c database, which is used for storing all data relating to the evaluation, including foot details entered by clinicians, photographs taken during patient appointments, inference results returned from the deep learning model and clinician confirmation of agreement with inference results. ATPI offers multiple deployment options that automatically configures the database depending on its targeted use case. For our evaluation, the ATPI was configured to use the Autonomous Transaction Processing workload type, which optimises the database with a bias towards processing high volumes of random data access.
\subsubsection{Infrastructure as a Service}
Oracle Compute represents the IaaS component of the project, consisting of a Virtual Machine (VM). Virtualisation software allows multiple guest systems to run on a single physical platform, where isolated environments can be created by multiplexing host's computing cycles and virtualising hardware resources. The VM hosts the core of the business logic, together with the frozen inference graph used for inference. For our proof-of-concept clinical evaluation, the operating system used was Ubuntu 16.04.6 LTS (xenial) with Nvidia GPU Cloud Machine Image shape, which defines the hardware configurations that are available to the VM instance. Hardware available on this shape included an Intel Xeon Gold 5120 2.20GHz CPU, and an Nvidia Tesla P100 SXM2 16GB GPU. We created two python programs to run on the VM, which were responsible for processing network, database, and image inference operations.
The first of the two Python programs, ServerPy, handled incoming requests from mobile clients via OMH over REST. All incoming requests are handled by Flask - a Python web framework that allows for the routing of incoming REST requests to Python classes. Requests are made to either add new data to the database, update existing data, or to retrieve data from the database. Adding data to the database includes adding new patient foot data, patient foot photographs, and clinician agreement confirmation with inference results. Sending data from the database to requesting clients takes the form of server status codes, app version checks to ensure the user is using the correct version of the mobile app, and the results of completed inference requests. Additionally, when new photographs are received by ServerPy, the details are added to a jobs table in the ATPI database. The second Python program, AnnotatePy, was responsible for periodically reading the jobs table and retrieving the oldest incomplete job. The job is then processed, using TensorFlow for inference. Once inference is completed, the results are added to the database, and the job is marked as complete. This process operates as a queue, using a first in first out principle.
\subsection{Deep Learning Framework}
The DFU localisation model trained by Goyal et al. \cite{goyal2018robust} was selected for use during our proof-of-concept clinical evaluation. This single classifier model showed the highest mAP (91.8) in a comparison of supervised deep learning models trained and evaluated with DFU. The model was trained using 1,775 DFU images, with ground truth labelling provided by clinical diabetic foot experts at Lancashire Teaching Hospitals NHS Foundation Trust. It implements Faster R-CNN as the object localisation network to process feature extraction, with Inception-ResNetV2 used to classify the extracted feature maps. This model was trained using two-tier (partial and full) transfer learning using the MS COCO dataset, implements three distinct steps to perform localisation:
\begin{enumerate}
\item Feature extraction using Inception V2 which serves as input for later stages (proposals and classifier).
\item Generation of proposals and refinement.
\item Region of interest classifier and bounding box regressor to fine-tune bounding box accuracy.
\end{enumerate}
The model was trained using a heterogeneous dataset consisting of non-standardised images of DFU. Aspects such as orientation, distance from foot, capture device type, resolution, focal length, exposure time, ISO speed ratings, variances in the amount of the foot visible in the image, and lighting conditions resulted in a high level of variability in image characteristics. It could be argued that a non-standardised dataset is more desirable in the training of such a model, since this would increase the viability of its use in real world settings where a system would need to be able to take into account numerous uncontrolled environment variables.
\begin{table*}
\caption{Summary of Results for Individual Question Responses, reported in mean $\pm$ SD (standard deviation).}
\label{tab:questions}
\scalebox{1.0}{
\begin{tabular}{|l|l|l|}
\hline
& Question & mean$\pm$SD \\ \hline
Q1 & The app was easy to use & 6.50$\pm$0.55\\ \hline
Q2 & It was easy for me to learn to use the app & 6.83$\pm$0.41\\ \hline
Q3 & The navigation was consistent when moving between screens & 5.83$\pm$0.98\\ \hline
Q4 & The interface of the app allowed me to use all the functions offered by the app & 6.00$\pm$0.89\\ \hline
Q5 & Whenever I made a mistake using the app, I could recover easily and quickly & 4.75$\pm$2.22\\ \hline
Q6 & I like the interface of the app & 5.83$\pm$0.98\\ \hline
Q7 & The information in the app was well organised, so I could easily find the information I needed & 6.00$\pm$1.27\\ \hline
Q8 & The app adequately acknowledged and provided information to let me know the progress of my action & 5.00$\pm$2.35\\ \hline
Q9 & Overall, I am satisfied with this app & 6.00$\pm$1.27\\ \hline
Q10 & This app has all the functions and capabilities I expected it to have & 4.80$\pm$2.28\\ \hline
\end{tabular}
}
\end{table*}
\section{FINDINGS}
Usability is a key factor in the adoption of mobile health apps, especially in cases where users are not within the typical age range of mobile users \cite{zapata2015mhealth}. Therefore, at the end of our six-month proof-of-concept evaluation, users of the system, clinicians, were asked to complete a usability questionnaire. The University of Pittsburgh Usability Questionnaire (Standalone Mobile Health App for Health Care Providers template) \cite{pitt2019usability} was used to obtain the qualitative measures for our evaluation. Zhou et al. validated the PITT Usability Questionnaire, and it was shown to have high internal consistency reliability \cite{zhou2019usability}. Questions that were not relevant to the app in its current state, such as questions 9, 15 and 18 were excluded as they were considered not to be relevant to the use of the app in its current prototype stage. We also added an additional free-text section at the end of the questionnaire asking clinicians to provide any other details of their experience using the app, and any recommendations for improvement. Six participating clinicians completed the questionnaire, with questions scored between 1 and 7; 1 being disagree and 7 being agree. They were also able to select a not applicable option if they believed that the statement did not relate to themselves.
\subsection{Quantitative Analysis}
Table \ref{tab:questions} shows a summary of mean and standard deviation for the ratings of each question. The questionnaire results indicate that all participating clinicians report high levels of satisfaction when using the app, with most of the mean scores being above 5, of a maximum score of 7. Questions 1 (m = 6.50; sd = 0.55) and 2 (m = 6.83; sd = 0.41), which relate to ease of use, provide the highest scores, which we regard as a good indicator that the app would be easy for patients to use in home settings, and meets one of the main criteria when taking into account the design of the app. The lowest scoring questions were Question 5 (m = 4.75; sd = 2.22) and Question 10 (m = 4.80; sd = 2.28), which related to how the app responds to user mistakes and expected app functionality respectively. This would indicate that the app design might benefit from further adjustments to enable users to more easily correct their mistakes. However, these issues would be negated in a patient-focused app since it would not contain any of the data entry elements currently present in the clinician-focused app prototype.
\subsection{Qualitative Analysis}
The free text responses provided by participating clinicians showed varying results that were not obvious to gauge from the answers to the Likert scale questions. Most participating clinicians were in agreement that the app was easy to use and functioned as expected. However, some clinicians experienced connectivity issues with the app due to the restrictive nature of free hospital WiFi, which resulted in occasionally slow upload of foot photographs. Such restrictions may mean that connected devices are automatically disconnected after a period of inactivity, with the only way to reconnect being via the device web browser, a process that has to be completed manually by the user.
Clinicians also agreed that when the app detected an ulcer, the ulcer localisation results were generally highly accurate. Other responses noted the number of false positive detections, with clinicians indicating that such detections would occur around callouses or extravasation areas around the wound. However, one response noted that extravasations detected as ulcers would at least direct patients to the clinic for assessment. One response noted that the app would be less useful for clinicians in its current state as they already knew how to recognise the presence of an ulcer. However, other responses disagreed with this statement, highlighting the importance of regular photographic capture of DFUs for screening and remote serial analysis. A device that allows patients to self-screen at home could encourage diabetic patients to check their feet more regularly, and would enable clinicians to check patients' feet without the need for hospital visits. It was also noted that older patients might have difficulty using the app without assistance. This could be addressed with the help of a partner, family member or carer. Another solution would be to use of a selfie-stick attachment for the mobile device.
\section{Recommendations and Future Work}
The two Python programs that run on the VM, ServerPy and AnnotatePy, might be better deployed as Linux services, so that they are able to automatically recover from system crashes and other system malfunctions. For the mobile app, in the case of a larger scale patient-focused evaluation, it might be preferable to implement push notifications so that users do not have to keep the app open while waiting for the cloud service to return inference results. A form of data protection might also be employed to ensure that all data and photographs are encrypted on the device. Given that an internet connection is required for the app to function, encryption of all local data could be achieved by requesting encryption keys from the cloud service as and when needed, but without ever actually storing the keys on the device itself.
A larger evaluation might require a more robust server-side solution, where multiple GPUs could be employed to complete a larger volume of jobs simultaneously. This could be achieved by sending each job to a single GPU, or distributing the workload of multiple jobs to multiple GPUs simultaneously. Further investigation is required to determine the most performant solution in this scenario.
Our framework has been designed to encourage frequent patient self-monitoring, supporting early detection of DFU that will lead to earlier signposting to treatment and therefore improved ulcer healing. Early intervention is an important factor in improving healing rates. Tools and education programmes to give patients the knowledge and motivation to manage their condition are essential in the goal of reducing the negative effects of diabetes and DFU \cite{boodoo2017mhealth}.
Many people diagnosed with DFU are older adults, therefore it will be important to ensure that any future apps created for use with our framework are extremely simple and easy to use. They should require the absolute minimum input from the user, and results should be presented in a form that are easy to understand. Usability will be the primary defining objective for a patient-focused version of the app. Minimal complexity will ensure the greatest adoption and impact of the system.
In the current system implementation, foot photographs are uploaded and stored in an Oracle 18c database as Binary Large Objects, together with all the other data captured during the evaluation. In the context of an initial evaluation, this approach was deemed appropriate, given that the average photograph file size was 60KB. However, in the context of a larger study, it may be beneficial to separate photographs and database data. Excessive file storage within the database may impact its performance over time. Photographs could instead be saved to the VM file system. However, this may require careful design to ensure that photographs are archived and easily retrievable, which would likely involve managing directories. A more desirable alternative might be to utilise a cloud storage layer, where photographs are given unique IDs and are sent to the storage layer which removes the need to manage file system elements. Photographs can be easily retrieved from the storage layer using an API and the photograph UID. Further ahead, the amount of data collected by our system could increase exponentially, especially if used internationally. Such datasets would classify as Big Data, whereby a NoSQL database may prove to be a superior fit for such large volumes of data processing.
During the analysis phase of the project, we explored the possibility of using a serverless solution, whereby the setup of a VM to host the Python applications could be bypassed. Instead, packages would be uploaded to a server space where application methods can be triggered by events received via REST requests. However, this approach to cloud computing is still in the early stages, with most providers not exposing access to GPU resources using this method.
Following the positive results in user acceptance from our proof-of-concept evaluation, we plan for a larger scale study to be undertaken. This follow-up study will be patient-focused, where the app will be simplified and distributed to a larger number of users. In this study, the app will be used by patients, their friends, family, or carers, instead of clinicians. This next stage will provide confirmation of whether the app and associated technologies are suitable for large-scale real-world use. The technologies developed will form the basis of a platform to support future research into areas such as:
\begin{itemize}
\item Automated early detection of DFU, including the detection of signs of pre-ulceration.
\item Automated classification and segmentation of DFU types: (1) DFU with no infection and no ischemia, (2) DFU with infection and no ischemia, (3) DFU with ischemia and no infection, (4) DFU with infection and ischemia.
\item Automated segmentation of DFU tissue types determined by colour and texture features, including necrotic, epithelial, granulation and slough.
\item Automated non-contact methods of monitoring DFU healing status over time.
\item Automated non-contact methods of monitoring the periwound (surrounding tissue of a wound) as a potential indicator of wound healing.
\item Automated non-contact methods of monitoring DFU edemia - a condition characterised by the swelling of all or parts of the foot.
\item Automated non-contact methods used for DFU pathophysiology.
\end{itemize}
\section{CONCLUSION}
In this work, we developed a cross-platform mobile app and a cloud-based deep learning framework for the automatic detection of DFU. The system was assessed for usability via qualitative methods, which showed that the system scored highly for system usability when used by clinicians in clinical settings. This work will provide the basis for a more extensive patient-focused evaluation of the system to determine its effectiveness when used by patients and their carers. The dataset obtained over the six-month evaluation period will be used to retrain the existing deep learning model to improve its effectiveness in detecting DFU at various stages of development. The longitudinal data will be used to form the basis of a refined model that will be used to detect the early signs of DFU.
To the best of our knowledge, the framework created for this research is the first of its kind, where DFU can be automatically detected and localised by a fully integrated framework of state-of-the-art technologies with an easy to use app, producing high confidence scores, where inference is performed in the cloud. This could lead to the eventual expansion of our system for use as a tool, not just for patients to self-monitor, but also as a diagnosis tool for medical experts. Our framework can now be used as a platform for further research, including early detection of DFU, and monitoring of DFU healing status over time. Further ahead, our platform could be expanded into other areas of research and automatic medical wound analysis, including other pathologies, on any part of the human body.
\section{ACKNOWLEDGMENT}
The authors would like to thank Oracle Research for providing the IaaS and PaaS technologies that enabled our clinical evaluation to take place. Gratitude is also extended to Salford Royal NHS Foundation Trust, Lancashire Teaching Hospitals NHS Foundation Trust and Manchester University NHS Foundation Trust for their extensive support during the usability study. This project is funded by The Manchester Metropolitan Strategic Operation Fund and Oracle Innovator Accelerator Programme.
|
2,877,628,090,824 | arxiv | \section{#1}\setcounter{equation}{0}}
\newcommand{\subsection}{\subsection}
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\newcommand{{i.e.}}{{i.e.}}
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\newcommand{-\penalty10000\zhs}{-\penalty10000\hskip 0pt plus 0pt}
\newcommand{--\penalty10000\zhs}{--\penalty10000\hskip 0pt plus 0pt}
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\newcommand{\abs}[1]{\left\vert#1\right\vert}
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\newcommand{\mbox{${1\over 2}$}}{\mbox{${1\over 2}$}}
\newcommand{\mbox{${1\over 4}$}}{\mbox{${1\over 4}$}}
\newcommand{\mbox{${1\over 8}$}}{\mbox{${1\over 8}$}}
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\newcommand{{\textstyle {\mbox{{\sevenrm i}}\over 16}}}{{\textstyle {\mbox{{\sevenrm i}}\over 16}}}
\newcommand{\mbox{\sevenrm can}}{\mbox{\sevenrm can}}
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\begin{document}
\font\csc=cmcsc10 scaled \magstep2
\font\sevenrm=cmr7 scaled \magstep1
\font\twelverm=cmr12
\title{\hfill {\twelverm SI--95--4\\
\hfill hep-th/9506162}\\
\ifn y\vrule height 2.5cm width 0cm
\else\vrule height 0.7cm width 0cm
\fi
{\bf Topological Yang--Mills Theory\\ with Two Fermionic Charges.\\
A Superfield Approach\\ on K\"ahler Manifolds}}
\author{{\csc
H.\ D.\ Dahmen,
S.\ Marculescu\thanks{E-mail:\quad{\tt
marculescu@hrz.uni-siegen.d400.de}}} \ and {\csc
T.\ Portmann\thanks{E-mail:\quad{\tt
portmann@sicip1.physik.uni-siegen.de}}}\and
{\em Universit\"at-\penalty10000\zhs Gesamthochschule-\penalty10000\zhs Siegen,\/}\and
{\em D--57068 Siegen,
Germany\/}}
\date{(to be published in Nucl.\ Phys.\ B)}
\maketitle
\begin{abstract}
The four-\penalty10000\zhs dimensional topological Yang--\penalty10000\zhs Mills theory with two
anticommuting charges is naturally formulated on K\"{a}hler manifolds. By
using a superspace approach we clarify the structure of the Faddeev--\penalty10000\zhs Popov
sector and determine the total action. This enables us to perform perturbation
theory around any given instanton configuration by manifestly maintaining all
the symmetries of the topological theory. The superspace formulation is very
useful for recognizing a trivial observable ({i.e.}\ having vanishing correlation
functions only) as the highest component of a gauge invariant superfield. As
an example of non-\penalty10000\zhs trivial observables we construct the complete solution to
the simultaneous cohomology problem of both fermionic charges. We also show
how this solution has to be used in order to make Donaldson's interpretation
possible.
\end{abstract}
\newpage
\renewcommand{\arraystretch}{1.4}
\sect{Introduction}
\label{intro}
There is a renewed interest in topological Yang--\penalty10000\zhs Mills (TYM) theory \cite{I}
over the past two years. While an immediate physical meaning is still a matter
of debate (see however \cite{II}, \cite{an}), TYM certainly offers powerful methods
\cite{III}, \cite{Mo} for extracting non-\penalty10000\zhs perturbative information
\cite{IV}, \cite{V} about supersymmetric chromodynamics.
Before presenting the content of our paper we would like to give a short
review of TYM. For more details one should consult the excellent articles
\cite{VII}, \cite{VIII}.
\subsection{Review of Topological Yang--\penalty10000\zhs Mills Theory}
\label{review}
The basic property of TYM is that its action can be written as the
variation of some gauge invariant expression. The variation itself acts on
the gauge field as a shift and is nilpotent. The former property allows one to
formulate TYM on curved manifolds.
There are essentially two ways of deriving the TYM action, each of them having
its own merit. One can, for instance, construct an action incorporating the
following set of subsidiary conditions: self-\penalty10000\zhs duality (instanton), fixing of
the topological shift and the BRS gauge fixing \cite{IX}, \cite{X}. Being
directly related to instanton calculus it can be conveniently used for explicit
calculations \cite{XI}, \cite{Anse}.
Another way \cite{I} to obtain TYM is by twisting the Euclidean $N=2$
supersymmetric gauge theory and by coupling thereafter to external gravity.
While the last step breaks down the original $N=2$ supersymmetry, some
supersymmetries may exist on the curved background as global symmetries. We
call them fermionic symmetries.
A single fermionic symmetry is preserved on arbitrary Riemannian manifolds. One
can show~\cite{XIII} that two or four symmetries remain unbroken if the
manifold is K\"{a}hler or hyper-\penalty10000\zhs K\"{a}hler, respectively, because each
K\"ahler structure is equivalent to a corresponding Killing spinor.
For the rest of this short review the TYM has a single fermionic symmetry
$\mbox{\rm q}$, hence it is formulated on a Riemannian four-\penalty10000\zhs manifold. The action
produced by twisting differs by $\mbox{\rm q}$-\penalty10000\zhs exact
terms from that obtained via subsidiary conditions. However, the correlation
functions are the same \cite{XI}.
The TYM obtained by twisting gains in clarity when formulated in superspace
\cite{XIV} despite the consequence that BRS symmetry has to be introduced in
terms of superfields.
TYM imposes very strong restrictions on the possible observables. Only those
objects which are not highest components of gauge invariant superfields can
have non-\penalty10000\zhs trivial correlations. The proper mathematical background for
constructing such observables is equivariant cohomology \cite{Cartan}. Its use
in TYM has been initiated by \cite{Ouvry} and further developped in
\cite{Kalkman}, \cite{Stora}.
The only known examples of TYM observables are the Donaldson polynomials
\cite{Don}, (for a review see \cite{DK}).
They can be obtained \cite{Brooks}, \cite{Dah} by twisting some components of
the $N=2$
superconformal anomaly \cite{Sohn}. By an appropriate change of renormalization
prescriptions one can obtain the Donaldson polynomials in the semiclassical
approximation. However, this approximation turns out to be sufficient as a
consequence of the non-\penalty10000\zhs renormalization theorem for chiral fields
\cite{Gris}, \cite{Dunbar}.
Further restrictions on the observables come through the path integral
representation. For TYM there is a canonical functional measure \cite{I} $[\mbox{\rm d}
m][\mbox{\rm d}\widetilde m]$, where $m$ denotes the moduli and ${\widetilde m}$
their $\mbox{\rm q}$-\penalty10000\zhs transformation. The above measure is supplemented with simple
prescriptions for integrating out all non-\penalty10000\zhs zero modes \cite{I} (as well as
some
zero modes \cite{Anse}). The integration over the fermionic zero modes amounts
to replacing ${\widetilde m}$ everywhere by $\mbox{\rm d} m$, such that the correlation
function becomes the integral of a certain top-\penalty10000\zhs form over the moduli space
\cite{I}.
Another consequence of the fermionic zero modes ${\widetilde m}$ is the
vanishing of the partition function. This means that the action of TYM has an
Abelian global invariance---hereafter called ghost number symmetry---that is
not shared by the path integral. The amount ${\Delta}U$ of violation of the
ghost number symmetry is obtained by integrating the twisted $\mbox{\rm I}\!\mbox{\rm R}$-\penalty10000\zhs anomaly
\cite{Dah}, \cite{Wu} over the manifold. One can choose the ghost number $U$
such that its
variation ${\Delta}U$ exactly compensates the dimension of the moduli space of
self-\penalty10000\zhs dual instantons \cite{Atiy}. This provides us with the following
selection rule:
The total ghost number of the observables in a non-\penalty10000\zhs vanishing correlation
function equals the dimension of the instanton moduli space.
\subsection{Content of the Paper}
\label{content}
In the present paper we consider TYM with two anticommutig symmetries along
with the items described previously. The twisting appropriate for this case has
been found for the first time in \cite{XIII}. Due to the existence of a Killing
spinor which is necessary to preserve the second supersymmetry the Riemannian
manifold has an integrable complex structure whose K\"ahler form is closed,
thus being in fact a K\"{a}hler manifold.
We use this formulation in section~\ref{super} to set up a superspace
approach. By extending the method of \cite{XIV} to two Grassmann variables we
encounter constraints in superspace. They are solved in terms of a
prepotential, a pair of chiral-\penalty10000\zhs antichiral connections and a chiral-\penalty10000\zhs
antichiral
pair of antisymmetric tensors. As a consequence, the gauge symmetry is
replaced by the local chiral symmetry. Furthermore, the gauge symmetry is
completely fixed if one is postulating subsidiary conditions for the chiral
connections. The gauge and Faddeev--\penalty10000\zhs Popov terms are then easily constructed.
Section~\ref{symme} deals with the symmetries of the theory. On K\"{a}hler
manifolds each conservation law involves two contragredient (with respect to
conservation indices) but inequivalent tensors. They belong to different
superfields and have different positions within each multiplet. Most prominent
examples are the energy-\penalty10000\zhs momentum tensor and the pair of fermionic symmetry
currents. These objects represent (up to improvement terms which leave the
conservation law unchanged) the highest components of gauge invariant
superfields.
The `classical' theory under discussion is invariant under two global Abelian
symmetries, whereas one encounters anomalies at the quantum level. In
section~\ref{pertu} we use this opportunity to check our superspace formalism in
perturbation theory. We evaluate the gravitational contribution to both
Abelian anomalies and found that they are equal. Moreover, they are equal to
the dimension of the instanton moduli space. The result coincides with the one
obtained for a Riemannian manifold admitting a K\"{a}hler structure.
In section~\ref{donal} we give the solution to the simultaneous
cohomology problem of both fermionic charges.
In contrast to TYM with only one fermionic symmetry, this solution consists of
local observables which may
depend on a larger number of fields, including antisymmetric fields.
We can, however, show that the correlation functions of the integrated
observables can be interpreted as integrals of top-forms over the moduli
space of selfdual instantons.
A different treatment of TYM with two fermionic charges has been given in
\cite{Park} and \cite{Holo}. A short comparision with our work is provided
at the end of section~\ref{donal}.
\sect{Superspace Approach}
\label{super}
Consider a superspace with two Grassmann coordinates $\theta$,
$\bar\theta$ related by complex conjugation $\mbox{\rm i}\bar\theta =
\overline{\mbox{\rm i}\theta}$. The commuting coordinates $z^m$ and $z^{\bar m}$
with $m = 1,2$, $\bar m = \bar 1, \bar 2 $ represent local holomorphic and
antiholomorphic coordinates, respectively on a K\"{a}hler manifold
${\cal K}$ without boundary and endowed with the metric $(1,1)$-\penalty10000\zhs form
\begin{equation}\label{i}
\gamma = g_{m \bar n}\mbox{\rm d} z^m \mbox{\rm d} z^{\bar n} = \partial\bar\partial h \; .
\end{equation}
Here, $h$ is the K\"{a}hler potential of the metric.
To these coordinates one associates the superconnections
$A_{m}$, $A_{\bar m} = - (A_{m})^{\cal y}$, $A_{\theta}$ and
$A_{\bar{\theta}} = - (A_{\theta})^{\cal y}$, in short $A_M$ with $M = m, \bar m,
\theta, \bar{\theta}$. Under an infinitesimal gauge transformation they
change as
\begin{equation}\label{ii}
\delta A_M = \mbox{\rm D}_M K = {\nabla}_M + [ A_M, K ] \;
\end{equation}
where $K = - K^{\cal y}$. We use ${\nabla}_m$, ${\nabla}_{\bar m}$ to represent
K\"{a}hler derivatives, while $\theta$, $\bar{\theta}$ directions are flat,
{i.e.}
${\nabla}_{\theta} = {\partial}_{\theta}$, ${\nabla}_{\bar{\theta}} =
{\partial}_{\bar{\theta}}$.
We also introduce the following covariant superfields: the anti-\penalty10000\zhs Hermitean
scalar $\Lambda = - {\Lambda}^{\cal y}$ and the pair of complex conjugate,
antisymmetric and anticommuting tensors $X_{mn}$,
$X_{{\bar m}{\bar n}} = - (X_{mn})^{\cal y}$. They transform as
\begin{equation}\label{iii}
\delta X = [ K, X ] \;
\end{equation}
where $X = \Lambda, X_{mn}$ or $X_{\bar m \bar n}$.
All superfields are taken in an irreducible representation of the compact and
semi-\penalty10000\zhs simple group
$\mbox{\rm G}$ with anti-\penalty10000\zhs Hermitean generators $t_a = - t^{\cal y}_a$ where $a =
1,\ldots, n=\dim \mbox{\rm G}$ and with totally antisymmetric structure constants
$c_{abc}$ defined
through $[ t_a, t_b ] = c_{abc} t_c$. The generators $t_a$ are normalized by
$\mbox{\rm Tr} (t_a t_b) = - {\delta}_{ab}$.
From superconnections one can construct the superfield strengths $F_{mn}$,
$F_{m{\theta}}$, $F_{m{\bar{\theta}}}$,
$F_{\theta \theta}$, their complex conjugates, $F_{m{\bar n}}$ and
$F_{\theta {\bar{\theta}}}$. They are defined by
\begin{equation}\label{iv}
F_{MN} = {\nabla}_M A_N + A_M A_N - (-)^{|M||N|}\left( M \leftrightarrow N
\right) \;
\end{equation}
where $\abs{M}$ is the grading of $M$, {i.e.}\ $\abs{M} = 0$ for $M = m, {\bar m}$
and $\abs{M} = 1$ for $M = \theta, \bar{\theta}$. The superfield strengths
transform covariantly, {i.e.}\ as~(\ref{iii}).
\subsection{Constraints in Superspace}
\label{constra}
In order to obtain the desired TYM one imposes the constraints
\begin{equation}\label{v}
F_{\theta \theta} = F_{\bar{\theta} \bar{\theta}} =
F_{m \bar{\theta}} = F_{\bar{m} \theta} = 0 \; ;
\end{equation}
\begin{equation}\label{vi}
\mbox{\rm D}_{\bar{\theta}}X_{mn} + i F_{mn} = 0 \; ; \qquad
\mbox{\rm D}_{\theta}X_{\bar m \bar n} + i F_{\bar m \bar n} = 0 \; .
\end{equation}
Notice the similarity of~(\ref{v}) with the constraints in superspace for the
$N=1$ supersymmetric gauge multiplet \cite{Wess}. The role of~(\ref{v}) is to
ensure that TYM has a single gauge field.
The fermionic symmetries are represented by the Grassmann derivatives
$\mbox{\rm q}=\partial_\theta$ and $\bar{\q}=\partial_{\bar\theta}$. They are nilpotent and
anticommute: $\mbox{\rm q}^2=\bar{\q}^2=\{\mbox{\rm q},\bar{\q}\}=0$.
\subsection{Wess--\penalty10000\zhs Zumino Gauge}
\label{wesszu}
We define the fields of TYM through covariant superfields and their Grassmann
covariant derivatives:
\begin{equation}\label{vii}
\begin{array}{rclrcl}
\psi_m &=& -\lwc{F_{m\theta}}\; ;\qquad &
\psi_{\bar m} &=& -\lwc{F_{\bar m\bar\theta}}\; ;\\
\varphi &=& -\mbox{\rm i}\lwc{F_{\theta\bar\theta}}\; ;\qquad &
\lambda &=& \lwc{\Lambda}\; ;\\
g_+ &=& 2\lwc{\mbox{\rm D}_\theta \Lambda}\; ;\qquad &
g_- &=& 2\lwc{\mbox{\rm D}_{\bar\theta}\Lambda}\; ;\\
k &=& \lwc{\left[\mbox{\rm D}_\theta,\mbox{\rm D}_{\bar\theta}\right] \Lambda}\; ;\qquad &
f_{m\bar n} &=& \lwc{F_{m\bar n}}\; ;\\
\chi_{mn} &=& \lwc{X_{mn}}\; ;\qquad &
\chi_{\bar m\bar n} &=& \lwc{X_{\bar m\bar n}}\; ;\\
b_{mn} &=& \lwc{\mbox{\rm i} \mbox{\rm D}_\theta X_{mn}}\; ;\qquad &
b_{\bar m\bar n} &=& \lwc{\mbox{\rm i} \mbox{\rm D}_{\bar\theta} X_{\bar m\bar n}}\; ;\\
f_{mn} &=& \lwc{\mbox{\rm i} \mbox{\rm D}_{\bar\theta} X_{mn}}\; ;\qquad &
f_{\bar m\bar n} &=& \lwc{\mbox{\rm i} \mbox{\rm D}_\theta X_{\bar m\bar n}}\; .\\
\end{array}
\end{equation}
The vertical bar means the lowest component of the superfield, {i.e.}\ at
$\theta=\bar\theta=0$.
Further components of these superfields can be obtained from~(\ref{vii}) with
the help of Bianchi identities and constraints~(\ref{v}), (\ref{vi}). They
are related to~(\ref{vii}) through the gauge covariant derivatives
$\mbox{\rm D}_m$, $\mbox{\rm D}_{\bar m}$ or vanish.
The transformations of the fields defined by~(\ref{vii}) are generated by the
Grassmann covariant derivatives
\begin{equation}\label{viii}
\delta\left(\lwc{X}\right)
= \mbox{\rm i} \left( \zeta + \mbox{\rm i} \xi \right) \lwc{\mbox{\rm D}_\theta X}
+ \mbox{\rm i} \left( \zeta - \mbox{\rm i} \xi \right) \lwc{\mbox{\rm D}_{\bar\theta} X} \; .
\end{equation}
Here $X$ represents any of the covariant superfields on the rhs.\
of~(\ref{vii}); $\zeta$ and $\xi$ are real, anticommuting parameters.
From~(\ref{viii}) one can see that the fermionic transformation
$\mbox{\rm q}=\partial_\theta$ (or $\bar{\q}=\partial_{\bar\theta}$ resp.) is always accompanied by
a gauge transformation $\lwc{\left[A_\theta,X\right]}$ (or
$\lwc{\left[A_{\bar\theta},X\right]}$ resp.) restoring the gauge covariance.
From the transformation of the various field strength $f_{mn}$, $f_{\bar
m\bar
n}$, and $f_{m\bar n}$ one can deduce how the gauge fields are transforming,
albeit up to a gauge transformation. Usually, in the Wess--\penalty10000\zhs Zumino (WZ)
gauge the last gauge transformation is suppressed. A simple calculation leads
to the following transformation rules:
\begin{equation}\label{ix}
\vcenter{\hbox{%
$\begin{array}{rclrcl}
\delta a_m &=& \mbox{\rm i}(\zeta+\mbox{\rm i}\xi)\,\psi_m\; ;\;&
\delta a_{\bar m} &=& \mbox{\rm i}(\zeta-\mbox{\rm i}\xi)\,\psi_{\bar m}\; ;\\
\delta \psi_m &=& (\zeta-\mbox{\rm i}\xi)\,\mbox{\rm D}_m\varphi\; ;\;&
\delta \psi_{\bar m} &=& (\zeta+\mbox{\rm i}\xi)\,\mbox{\rm D}_{\bar m}\varphi\; ;\\
\delta \varphi &=& 0\; ;\\
\delta \lambda &=& {\textstyle {\mbox{{\sevenrm i}}\over 2}}(\zeta+\mbox{\rm i}\xi)\,g_+ + {\textstyle {\mbox{{\sevenrm i}}\over 2}}(\zeta-\mbox{\rm i}\xi)\,g_-;\\
\delta g_+ &=& -\mbox{\rm i}(\zeta-\mbox{\rm i}\xi)\,(k-\mbox{\rm i}[\varphi,\lambda])\; ;\;&
\delta g_- &=& -\mbox{\rm i}(\zeta+\mbox{\rm i}\xi)\,(k+\mbox{\rm i}[\varphi,\lambda]);\\
\end{array}$}\hbox to 125mm {\hfill
$\begin{array}{rcl}
\delta k &=& \frac{1}{2}(\zeta+\mbox{\rm i}\xi)\,[\varphi,g_+]
- \frac{1}{2}(\zeta-\mbox{\rm i}\xi)\,[\varphi,g_-]\; ;\\
\delta \chi_{mn} &=& (\zeta+\mbox{\rm i}\xi)\,b_{mn}
+ (\zeta-\mbox{\rm i}\xi)\,f_{mn}\; ;\\
\delta \chi_{\bar m\bar n} &=& (\zeta+\mbox{\rm i}\xi)\,f_{\bar m\bar n}
+ (\zeta-\mbox{\rm i}\xi)\,b_{\bar m\bar n}\; ;\\
\delta b_{mn}
&=& -\mbox{\rm i}(\zeta-\mbox{\rm i}\xi)\,(\mbox{\rm D}_m\psi_n-\mbox{\rm D}_n\psi_m+[\varphi,\chi_{mn}])\;
;\\
\delta b_{\bar m\bar n}
&=& -\mbox{\rm i}(\zeta+\mbox{\rm i}\xi)\,(\mbox{\rm D}_{\bar m}\psi_{\bar n}-\mbox{\rm D}_{\bar n}\psi_{\bar
m}+[\varphi,\chi_{\bar m\bar n}])\; .\\
\end{array}$\hfill}}
\end{equation}
This choice of the field components enables one to declare them (K\"ahler)
metric independent. It follows that fermionic symmetries and the variation with
respect to the metric always commute.
The transformations~(\ref{ix}) close into $\varphi$-\penalty10000\zhs field dependent gauge
transformations.
In section~\ref{donal} we shall give the fermionic transformations generated
by the nilpotent, anticommutative operations $\mbox{\rm q}$ and $\bar{\q}$.
\subsection{Action}
\label{action}
The action in superspace is given by
\begin{equation}\label{x}
{\cal S} = \mbox{${1\over 4}$} \int_{\cal K} \mbox{\rm d}^2 z \mbox{\rm d}^2 {\bar z} \; g \;
\partial_\theta\partial_{\bar\theta} \; \mbox{\rm Tr} \left\{ -
\mbox{${1\over 4}$} X_{mn}X^{mn} + \Lambda\left(\mbox{\rm i} F -
[\mbox{\rm D}_\theta,\mbox{\rm D}_{\bar\theta}]\Lambda\right)\right\}\;
\end{equation}
where $g =\det g_{m\bar n}$ and $F = g^{\bar nm}F_{m\bar n}$. Indices
are raised by the inverse K\"{a}hler metric $g^{\bar nm}$ of the K\"ahler
manifold ${\cal K}$.
The equations of motion in superspace are
\begin{equation}\label{xi}
\begin{array}{rcl}
\mbox{\rm D}_\theta X_{mn} &=& 0\; ;\qquad\qquad \mbox{\rm D}_{\bar\theta} X_{\bar m\bar n} = 0\; ;\\
\mbox{${1\over 2}$}\,\mbox{\rm D}^n X_{mn} + \mbox{\rm D}_\theta \mbox{\rm D}_m \Lambda &=& 0\; ;\qquad
\mbox{${1\over 2}$}\,\mbox{\rm D}^{\bar n} X_{\bar m\bar n} + \mbox{\rm D}_\theta \mbox{\rm D}_{\bar m} \Lambda = 0\; ;\\
F+2\mbox{\rm i}[\mbox{\rm D}_\theta,\mbox{\rm D}_{\bar\theta}]\Lambda &=& 0\; ;\\
\{\mbox{\rm D}_m,\mbox{\rm D}^m\}\Lambda + {\textstyle {\mbox{{\sevenrm i}}\over 2}}\{X_{mn},X^{mn}\}
&+& 4\mbox{\rm i}\{\mbox{\rm D}_\theta \Lambda,\mbox{\rm D}_{\bar\theta} \Lambda\}
- 2\mbox{\rm i}\left[\Lambda,[\Lambda,F_{\theta\bar\theta}]\right]=0\; .
\end{array}
\end{equation}
From~(\ref{x}) one can get the action in component fields:
\begin{equation}\label{xii}
\begin{array}{rcl}
{\cal S} &=& \mbox{${1\over 8}$} \vrule height 8mm width 0pt \displaystyle\int_{\cal K} \mbox{\rm d}^2 z \mbox{\rm d}^2 {\bar z} \; g \;
\mbox{\rm Tr} \left\{\lambda\{\mbox{\rm D}_m,\mbox{\rm D}^m\}\varphi + \mbox{\rm i} fk -k^2
+ \mbox{${1\over 2}$}\left(f_{mn}f^{mn}\right.\right.\\
&& \left. - b_{mn}b^{mn}\right) + \mbox{\rm i}\left(g_+\mbox{\rm D}_m\psi^m + g_-\mbox{\rm D}^m\psi_m -
\chi_{mn}\mbox{\rm D}^m\psi^n - \chi^{mn}\mbox{\rm D}_m\psi_n\right)\\
&& \left. + 2\mbox{\rm i}\lambda\{\psi_m,\psi^m\}
+ \varphi\left(\mbox{${1\over 2}$}\{\chi_{mn},\chi^{mn}\} + \{g_+,g_-\}\right) -
[\varphi,\lambda]^2
\right\}\; .
\end{array}
\end{equation}
It coincides (up to some field redefinition) with that obtained in \cite{XIII}
by
twisting $N=2$ supersymmetric Yang--\penalty10000\zhs Mills theory (SYM).
\subsection{Solution of the Constraints}
\label{solut}
One cannot solve the constraints~(\ref{vi}) in superspace. Instead one can
take them into account by means of Lagrange multipliers. For this purpose one
introduces a new action
\begin{equation}\label{xiii}
\begin{array}{rcl}
{\widetilde{\cal S}} &=& {\cal S} + \mbox{${1\over 16}$}\vrule height 8mm width 0pt \displaystyle\int_{\cal K}\mbox{\rm d}^2 z \mbox{\rm d}^2
{\bar z}\, g\,\left(\partial_\theta \mbox{\rm Tr} \left\{L^{mn}\left(\mbox{\rm D}_{\bar\theta}X_{mn}
+ \mbox{\rm i} F_{mn}\right)\right\}\right.\\
&& + \left.\partial_{\bar\theta} \mbox{\rm Tr}
\left\{L_{mn}\left(\mbox{\rm D}_\theta X^{mn} + \mbox{\rm i} F^{mn}\right)\right\}\right) \;
\end{array}
\end{equation}
where $L_{mn}$ and $L_{{\bar m}{\bar n}} = - (L_{mn})^{\cal y}$ are a pair of
complex conjugate, anticommuting, and antisymmetric superfields satisfying
\begin{equation}\label{xiv}
\mbox{\rm D}_\theta L_{mn} = \mbox{\rm D}_{\bar\theta} L_{\bar m\bar n}
= 0 \; .
\end{equation}
The (covariant) superfields entering~(\ref{xiii}) are subject to the
constraints~(\ref{v}) and~(\ref{xiv}).
The solution of these constraints can be given in terms of
\begin{itemize}
\item
a Hermitean prepotential $V=V^{\cal y}$,
\item
a chiral-\penalty10000\zhs antichiral pair of superconnections $\phi_m$, $\phi_{\bar m} =
-(\phi_m)^{\cal y}$ depending on a single Grassmann variable
$\partial_{\bar\theta}\phi_m = \partial_\theta \phi_{\bar m} = 0$, and
\item
a pair of chiral-\penalty10000\zhs antichiral, anticommuting, and antisymmetric superfields
$M_{mn}$, $M_{\bar m\bar n} = -(M_{mn})^{\cal y}$ obeying $\partial_\theta M_{mn} =
\partial_{\bar\theta} M_{\bar m\bar n} = 0$.
\end{itemize}
It can be presented in the form
\begin{equation}\label{xv}
\begin{array}{rclrcl}
A_\theta &=& \mbox{\rm e}^{-\frac{V}{2}}\partial_\theta \mbox{\rm e}^{\frac{V}{2}} \; ;\qquad &
A_{\bar\theta} &=& \mbox{\rm e}^{\frac{V}{2}}\partial_{\bar\theta} \mbox{\rm e}^{-\frac{V}{2}} \; ;\\
A_m &=& \mbox{\rm e}^{\frac{V}{2}}\left(\phi_m + \nabla_m\right) \mbox{\rm e}^{-\frac{V}{2}} \;
;\qquad &
A_{\bar m} &=& \mbox{\rm e}^{-\frac{V}{2}}\left(\phi_{\bar m} + \nabla_{\bar m}\right)
\mbox{\rm e}^{\frac{V}{2}} \; ;\\
L_{mn} &=& \mbox{\rm e}^{-\frac{V}{2}} M_{mn} \mbox{\rm e}^{\frac{V}{2}} \; ;\qquad &
L_{\bar m\bar n} &=& \mbox{\rm e}^{\frac{V}{2}} M_{\bar m\bar n} \mbox{\rm e}^{-\frac{V}{2}} \; .\\
\end{array}
\end{equation}
The constraint superfields $V$, $\phi_m$, $\phi_{\bar m}$, $M_{mn}$, and
$M_{\bar m\bar n}$ are determined up to local chiral transformations.
\subsection{BRS Symmetry in Superspace}
\label{brst}
It is convenient to describe the chiral transformations by a BRS (nilpotent)
operation. For this purpose one introduces a pair of chiral-\penalty10000\zhs antichiral,
anticommuting superfields $C$, $C^{\cal y}$ with $\partial_{\bar\theta} C = \partial_\theta
C^{\cal y} = 0$. The unconstrained prepotentials transform as follows:
\begin{equation}\label{xvi}
\begin{array}{rclrcl}
\mbox{\rm s} \mbox{\rm e}^V &=& \mbox{\rm e}^V C + C^{\cal y} \mbox{\rm e}^V \; ;\\
\mbox{\rm s} \phi_m &=& {\cal D}_m C \; ;\qquad &
\mbox{\rm s} \phi_{\bar m} &=& -{\cal D}_{\bar m} C^{\cal y} \; ;\\
\mbox{\rm s} M_{mn} &=& \{ C^{\cal y}, M_{mn} \} \; ;\qquad &
\mbox{\rm s} M_{\bar m\bar n} &=& -\{ C, M_{\bar m\bar n} \} \;\\
\end{array}
\end{equation}
where ${\cal D}_m$ (or ${\cal D}_{\bar m}$ resp.) is the gauge covariant
derivative constructed with $\phi_m$ (or $\phi_{\bar m}$ resp.). The gauge
symmetry is represented by an anti-\penalty10000\zhs Hermitean Faddeev--\penalty10000\zhs Popov ghost
superfield
\begin{equation}\label{xvii}
K = \frac{C - C^{\cal y}}{2} + \tanh\pounds_{\frac{V}{4}}\left(
\frac{C + C^{\cal y}}{2} \right) \;
\end{equation}
where $\pounds_X = [X,\;]$ denotes the Lie bracket of the superfield $X$. The
`matter' transforms as
\begin{equation}\label{xviii}
\begin{array}{c}
\mbox{\rm s} \Lambda = - [ K, \Lambda ] \; ;\\
\mbox{\rm s} X_{mn} = - \{ K, X_{mn} \} \; ;\qquad
\mbox{\rm s} X_{\bar m\bar n} = - \{ K, X_{\bar m\bar n}\} \; .
\end{array}
\end{equation}
The local chiral symmetry is fixed if only the corresponding connections
$\phi_m$, $\phi_{\bar m}$ obey subsidiary conditions, {e.g.}\ $\nabla^m\phi_m =
\nabla_m\phi^m = 0$. In order to find the gauge fixing and Faddeev--\penalty10000\zhs Popov-\penalty10000\zhs
terms one introduces a pair of chiral-\penalty10000\zhs antichiral (commuting) superfields
$D$, $D^{\cal y}$ with $\partial_{\bar\theta} D = \partial_\theta D^{\cal y} = 0$, as well as
their BRS variations $B = \mbox{\rm s} D$ and $B^{\cal y} = \mbox{\rm s} D^{\cal y}$. The superfields $B$,
$B^{\cal y}$ are anticommuting and form a chiral-\penalty10000\zhs antichiral pair. They serve as
Lagrange multipliers for the gauge fixing conditions characterized by the real
parameter $\alpha$. The BRS (trivial) term of the action is
\begin{equation}\label{xix}
\begin{array}{rcl}
{\cal S}' &=& -\mbox{${1\over 4}$}\,\mbox{\rm s}\vrule height 8mm width 0pt \displaystyle\int_{\cal K}\mbox{\rm d}^2 z \mbox{\rm d}^2{\bar
z}\,g\,\Bigl(\partial_\theta\mbox{\rm Tr}\left\{D \left(\nabla^m \phi_m
- \alpha\partial_\theta B^{\mbox{\rm \vphantom{{\cal y}}}}
\right)\right\}\Bigr.\\
&& + \Bigl.\partial_{\bar\theta}\mbox{\rm Tr}\left\{D^{\cal y} \left(\nabla_m \phi^m
- \alpha\partial_\theta B^{\cal y} \right)\right\}\Bigr) \; .
\end{array}
\end{equation}
The total action is $\widetilde S + S'$ and represents the starting point of
all perturbative or non-\penalty10000\zhs perturbative considerations.
\subsection{Correlation Functions}
\label{correl}
We would like to study correlation functions of the form
\begin{equation}\label{xx}
\vev{\prod_i {\cal O}_i} = \int [\mbox{\rm d}\widetilde{\mu}]\,\prod_i {\cal O}_i\,
\exp\left\{-\frac{1}{e^2}\left(\widetilde S + S'\right)
\right\}\;
\end{equation}
where ${\cal O}_i$ are gauge invariant ($\mbox{\rm s} {\cal O}_i = 0$) and metric
independent polynomials in the fields, $[\mbox{\rm d}\widetilde{\mu}]$ denotes the path
integral
measure of all unconstrained superfields, {i.e.}\ $V$, $\phi_m$, $\phi_{\bar m}$,
$M_{mn}$, $M_{\bar m\bar n}$, $X_{mn}$, $X_{\bar m\bar n}$, $\Lambda$, $C$,
$C^{\cal y}$, $D$, $D^{\cal y}$, $B$, and $B^{\cal y}$; $e$ is the gauge coupling constant.
If one assumes that $\mbox{\rm q} {\cal O}_i = \bar{\q} {\cal O}_i = 0$, the correlation
functions are independent of the coupling constant. The reason for this
property is the form of the total action
\begin{equation}\label{xxi}
\widetilde S + S' = \mbox{\rm q}\bar{\q}{\cal V} + \mbox{\rm q}{\cal W} - \bar{\q}\bar{\cal W}
\end{equation}
where ${\cal V}$ is a Hermitean superfield and ${\cal W}$, $\bar{\cal W}$ is a
pair of complex conjugate chiral-\penalty10000\zhs antichiral anticommuting superfields, {i.e.}\
$\bar{\q}{\cal W} = \mbox{\rm q}\bar{\cal W} = 0$.
Many observables are constructed from the (total) action by operations which
commute with both $\mbox{\rm q}$ and $\bar{\q}$ and therefore have the form~(\ref{xxi}). They
are highest components of BRS invariant superfields. Of course, they are $\mbox{\rm q}$
and $\bar{\q}$ invariant. Observables which are highest components can be shown
to have vanishing correlation functions only.
\sect{Symmetries}
\label{symme}
Most of the symmetries of supersymmetric field theories are encoded in the
supercurrent \cite{Ferr}, a multiplet containing the energy-\penalty10000\zhs momentum tensor,
the
supersymmetry and the $\mbox{\rm I}\!\mbox{\rm R}$-\penalty10000\zhs symmetry currents. For $N=2$ supersymmetry
an additional isovector current \cite{Sohn} corresponding to the automorphism
$SU(2)$
symmetry group belongs to the supermultiplet. (Of course, the number of
supersymmetry currents is doubled.)
The twisting procedure enables one to get the above currents for the TYM and
to recast the result into appropriate superfields \cite{Dah}, \cite{Marc}. The
method has
the advantage to be easily applicable \cite{Alva} to any $N=2$ supersymmetric
gauge
theory, {e.g.}\ to super-\penalty10000\zhs Yang--\penalty10000\zhs Mills coupled to relaxed hypermultiplet
\cite{Howe}.
The approach we shall use below is adapted to the
relative simple structure of the TYM obtained from pure $N=2$ supersymmetric
Yang--\penalty10000\zhs Mills theory.
\subsection{Energy-\penalty10000\zhs Momentum Tensor}
\label{emt}
Consider a variation of the metric in the action~(\ref{xii}). The canonical
energy-\penalty10000\zhs momentum tensor defined by
\begin{equation}\label{iii_i}
\delta_g {\cal S} = - \mbox{${1\over 8}$}\int_{\cal K}\mbox{\rm d}^2z\mbox{\rm d}^2\bar z\,g\,
\delta g^{\bar nm} \vartheta_{m\bar n}
\end{equation}
has the form
\begin{equation}\label{iii_ii}
\begin{array}{rcl}
\vartheta_{m\bar n} &=& \mbox{\rm Tr}\{-\mbox{\rm i} k f_{m\bar n}
+ \mbox{\rm D}_m\lambda \mbox{\rm D}_{\bar n}\varphi + \mbox{\rm D}_m\varphi \mbox{\rm D}_{\bar n}\lambda
-\mbox{\rm i}\left(\psi_m \mbox{\rm D}_{\bar n} g_- + \psi_{\bar n} \mbox{\rm D}_m g_+\right)\\
&& -2\mbox{\rm i}\lambda\{\psi_m,\psi_{\bar n}\} - g_{m\bar n}[k^2-\mbox{\rm i} kf
+ \mbox{\rm D}_p\lambda \mbox{\rm D}^p\varphi + \mbox{\rm D}_p\varphi \mbox{\rm D}^p\lambda\\
&& -\mbox{\rm i}\left(\psi_p \mbox{\rm D}^p g_- + \psi^p \mbox{\rm D}_p g_+\right)
-2\mbox{\rm i}\lambda\{\psi_p,\psi^p\} - \mbox{\rm i}\varphi\{g_+,g_-\}\\
&& + [\varphi,\lambda]^2]\} \; .
\end{array}
\end{equation}
It is the last component of the superfield
$-2\mbox{\rm i}\mbox{\rm Tr}\left\{\Lambda\left[F_{m\bar n} - g_{m\bar n}\left(F +
\mbox{\rm i}[\mbox{\rm D}_\theta, \mbox{\rm D}_{\bar\theta}]\Lambda\right)\right]\right\}$. The on-\penalty10000\zhs
shell version of the latter reads
\begin{equation}\label{iii_iii}
Q_{m\bar n} = -2\mbox{\rm i}\mbox{\rm Tr}\left\{\Lambda\left(F_{m\bar n} - \mbox{${1\over 2}$} g_{m\bar n} F
\right)\right\}
\end{equation}
and obeys the conservation laws
\begin{equation}\label{iii_iv}
\begin{array}{rcl}
\partial_\theta\nabla^{\bar n} Q_{m\bar n} + \nabla^n J_{mn} &=& 0\; ;\\
\partial_{\bar\theta}\nabla^n Q_{n\bar m} + \nabla^{\bar n} J_{\bar m\bar n}
&=& 0
\end{array}
\end{equation}
where
\begin{equation}\label{iii_v}
\begin{array}{rcl}
J_{mn} &=& \mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F_{(m}{}^p X_{n)p} + 2 F_{m\theta} \mbox{\rm D}_n \Lambda
- \Lambda \mbox{\rm D}_{[m} F_{n]\theta}\right\}\; ;\\
J_{\bar m\bar n} &=& \mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F^{\bar p}{}_{(\bar m}
X_{\bar n)\bar p} - 2 F_{\bar m\bar\theta} \mbox{\rm D}_{\bar n} \Lambda
+ \Lambda \mbox{\rm D}_{[\bar m} F_{\bar n]\bar\theta}\right\}\; .\\
\end{array}
\end{equation}
In deriving these expressions we used the equations of motion~(\ref{xi});
hence $J_{mn}$ and its complex conjugate $J_{\bar m\bar n}$ have no definite
off-\penalty10000\zhs shell continuation like $Q_{m\bar n}$.
The energy-\penalty10000\zhs momentum tensor $\vartheta_{m\bar n}$ obeys the conservation
laws
\begin{equation}\label{iii_vi}
\nabla^n \vartheta_{mn} + \nabla^{\bar n} \vartheta_{m\bar n} = 0 \; ;\qquad
\nabla^n \vartheta_{n\bar m} + \nabla^{\bar n} \vartheta_{\bar m\bar n}
= 0
\end{equation}
where
\begin{equation}\label{iii_vii}
\begin{array}{rcl}
\vartheta_{mn} &=& \mbox{\rm Tr}\left\{-\mbox{${1\over 2}$} f_{(m}{}^p f_{n)p}
+ \mbox{\rm D}_{(m} \lambda \mbox{\rm D}_{n)} \varphi + \mbox{\rm i}\chi_{np} \mbox{\rm D}_m \psi^p
- \mbox{\rm i}\psi_n \mbox{\rm D}_m g_-\right\}\; ;\\
\vartheta_{\bar m\bar n} &=& \mbox{\rm Tr}\left\{\mbox{${1\over 2}$} f^{\bar p}{}_{(\bar m}
f_{\bar n)\bar p} + \mbox{\rm D}_{(\bar m} \lambda \mbox{\rm D}_{\bar n)} \varphi
+ \mbox{\rm i}\chi_{\bar n\bar p} \mbox{\rm D}_{\bar m} \psi^{\bar p}
- \mbox{\rm i}\psi_{\bar n} \mbox{\rm D}_{\bar m} g_+\right\}\; .\\
\end{array}
\end{equation}
The conservation partners $\vartheta_{mn}$, $\vartheta_{\bar m\bar n}$ of the
energy-\penalty10000\zhs momentum tensor are higher components of the superfields $J_{mn}$
and $J_{\bar m\bar n}$ but obviously not highest component like
$\vartheta_{m\bar n}$.
The correlation functions of $\vartheta_{m\bar n}$ are vanishing. From the
special structure of~(\ref{iii_iv}) it follows that the correlation functions
of all the components of $\nabla^n J_{mn}$, $\nabla^{\bar n} J_{\bar m\bar n}$
(and in particular of $\nabla^n \vartheta_{mn}$, $\nabla^{\bar n}
\vartheta_{\bar m\bar n}$) vanish.
Of course, radiative corrections may alter some of the above conclusions.
For Riemannian manifolds one can however show that the energy-\penalty10000\zhs momentum
tensor remains highest component. In particular, there is no contribution to
the energy-\penalty10000\zhs momentum trace coming from the (Riemannian) manifold
\cite{Dah_Lett}. We expect a similar property for TYM on K\"{a}hler
manifolds.
\subsection{Fermionic Symmetries}
\label{fermi}
Consider the fermionic transformations~(\ref{ix}) with $z,\bar z$-\penalty10000\zhs
dependent parameters $\zeta$, $\xi$ and pick up terms proportional to
(K\"ahler) derivatives of these parameters. The fermionic symmetry currents
defined by
\begin{equation}\label{iii_ii_v}
\begin{array}{rcl}
\delta {\cal S} &=& {\textstyle {\mbox{{\sevenrm i}}\over 16}}\vrule height 8mm width 0pt \displaystyle\int_{\cal K}\mbox{\rm d}^2 z\mbox{\rm d}^2\bar z\,g\,
\Bigl[\nabla^m(\zeta + \mbox{\rm i}\xi)\,s^{-\mbox{\sevenrm can}}_m
+ \nabla^{\bar m}(\zeta + \mbox{\rm i}\xi)\,s^{-\mbox{\sevenrm can}}_{\bar m}\Bigr. \\
&& + \Bigl.\nabla^m(\zeta - \mbox{\rm i}\xi)\,s^{+\mbox{\sevenrm can}}_m
+ \nabla^{\bar m}(\zeta - \mbox{\rm i}\xi)\,s^{+\mbox{\sevenrm can}}_{\bar m}\Bigr]\\
\end{array}
\end{equation}
are
\begin{equation}\label{iii_ii_vi}
\begin{array}{rcl}
s^{+\mbox{\sevenrm can}}_m &=& 2\mbox{\rm Tr}\left\{f_{mn} \psi^n - g_- \mbox{\rm D}_m \varphi\right\} \; ;\\
s^{+\mbox{\sevenrm can}}_{\bar m} &=& 2\mbox{\rm Tr}\left\{\chi_{\bar m\bar n} \mbox{\rm D}^{\bar n} \varphi
+ \mbox{\rm i} k\psi_{\bar m} + \varphi[\lambda,\psi_{\bar m}]\right\} \; ;\\
s^{-\mbox{\sevenrm can}}_m &=& 2\mbox{\rm Tr}\left\{\chi_{mn} \mbox{\rm D}^n \varphi
- \mbox{\rm i} k\psi_m + \varphi[\lambda,\psi_m]\right\} \; ;\\
s^{-\mbox{\sevenrm can}}_{\bar m} &=& 2\mbox{\rm Tr}\left\{f_{\bar m\bar n} \psi^{\bar n}
- g_+ \mbox{\rm D}_{\bar m} \varphi\right\} \; .\\
\end{array}
\end{equation}
Due to the equations of motion~(\ref{xi}) they obey the conservation
rules
\begin{equation}\label{iii_ii_vii}
\nabla^m s^{\pm\mbox{\sevenrm can}}_m + \nabla^{\bar m} s^{\pm\mbox{\sevenrm can}}_{\bar m} = 0\; .
\end{equation}
In order to find the superfields whose components are related to fermionic
symmetry
currents we investigate at first the global symmetries of the action.
\subsection{Global Symmetries}
\label{global}
The action~(\ref{x}) is invariant under a global $SU(2)$
\begin{equation}\label{iii_iii_viii}
\begin{array}{rclrcl}
\delta \psi_m &=& -v\psi_m - u_{mn}\psi^n\; ;\qquad &
\delta \psi_{\bar m} &=& v\psi_{\bar m} - u_{\bar m\bar n}\psi^{\bar n}\; ;\\
\delta \chi_{mn} &=& -v\chi_{mn} - u_{mn}g_-\; ;\qquad &
\delta \chi_{\bar m\bar n} &=&-v\chi_{\bar m\bar n}-u_{\bar m\bar n}g_+\; ;\\
\delta g_+ &=& -vg_+ + \mbox{${1\over 2}$} u_{mn}\chi^{mn}\; ;\qquad &
\delta g_- &=& vg_- + \mbox{${1\over 2}$} u_{\bar m\bar n}\chi^{\bar m\bar n}\;
\end{array}
\end{equation}
and under a global Abelian group:
\begin{equation}\label{iii_iii_ix}
\begin{array}{rclrcl}
\delta \psi_m &=& u \psi_m\; ;\qquad &
\delta \psi_{\bar m} &=& u \psi_{\bar m}\; ;\\
\delta \varphi &=& 2u\varphi\; ;\qquad &
\delta \lambda &=& -2u\lambda\; ;\\
\delta \chi_{mn} &=& -u\chi_{mn}\; ;\qquad &
\delta \chi_{\bar m\bar n} &=& -u\chi_{\bar m\bar n}\; ;\\
\delta g_+ &=& -u g_+\; ;\qquad &
\delta g_- &=& -u g_-\; .
\end{array}
\end{equation}
The parameters $u$, $v$ are real and $u_{mn}$, $u_{\bar m\bar n}$ are
antisymmetric and complex conjugate to each other, {i.e.}\ $u_{\bar m\bar n} = -
\overline{u_{mn}}$. The corresponding currents are given by
\begin{equation}\label{iii_iii_x}
\begin{array}{rcl}
\delta {\cal S} &=& \mbox{${1\over 8}$}\vrule height 8mm width 0pt \displaystyle\int_{\cal K}\mbox{\rm d}^2 z\mbox{\rm d}^2\bar z\,g\,
\Bigl[\mbox{${1\over 8}$}\left(\nabla^p u^{mn} b_{mnp} + \nabla^{\bar p} u^{mn}
b_{mn\bar p}\right.\Bigr.\\
&& \left. + \nabla^p u^{\bar m\bar n} b_{\bar m\bar np} + \nabla^{\bar p}
u^{\bar m\bar n} b_{\bar m\bar n\bar p}\right)\\
&& \Bigl. -\mbox{\rm i} \left(\nabla^m u b_m + \nabla^{\bar m} u b_{\bar m}\right)
- \left(\nabla^m v b'_m + \nabla^{\bar m} v b'_{\bar m}\right)\Bigr]\; .
\end{array}
\end{equation}
The conserved combinations $b_m \pm b'_m$ and $b_{\bar m} \pm b'_{\bar m}$ are
responsible for two global Abelian symmetry groups $\mbox{\rm I}\!\mbox{\rm R}_\pm$. Through their
charges they associate to each field two additive (real) quantum numbers
$r_\pm$.
One can still modify the above combinations of currents without disturbing
their conservation. A convenient choice is to improve $b_{\bar m} + b'_{\bar
m}$ to a $\mbox{\rm q}$ variation and $b_m - b'_m$ to a $\bar{\q}$ variation. The improved
currents, denoted by $b^+_{\bar m}$ and $b^-_m$ are defined by
\begin{equation}\label{iii_iii_xi}
b^+_{\bar m} = -4\mbox{\rm i}\mbox{\rm q}\mbox{\rm Tr}\{\lambda\psi_{\bar m}\}\; ;\qquad
b^-_m = -4\mbox{\rm i}\bar{\q}\mbox{\rm Tr}\{\lambda\psi_m\}\; .
\end{equation}
The last step is to construct the conservation partners $b^+_m$ and $b^-_{\bar
m}$. The superfields are then easily guessed to be
\begin{equation}\label{iii_iii_xii}
\begin{array}{rcl}
B^+_m &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F^n{}_{\bar\theta}
X_{mn} - F_{\theta\bar\theta} \mbox{\rm D}_m\Lambda\right\}\; ;\\
B^+_{\bar m} &=& 4\mbox{\rm i}\partial_\theta\mbox{\rm Tr}\{\Lambda F_{\bar m\bar\theta}\}
\end{array}
\end{equation}
and
\begin{equation}\label{iii_iii_xiii}
\begin{array}{rcl}
B^-_m &=& 4\mbox{\rm i}\partial_{\bar\theta}\mbox{\rm Tr}\{\Lambda F_{m\theta}\}\; ;\\
B^-_{\bar m} &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F^{\bar n}{}_\theta
X_{\bar m\bar n} - F_{\theta\bar\theta} \mbox{\rm D}_{\bar m}\Lambda\right\}\; .
\end{array}
\end{equation}
They obey the conservation rule
\begin{equation}\label{iii_iii_xiv}
\nabla^m B^\pm_m + \nabla^{\bar m} B^\pm_{\bar m} = 0\; .
\end{equation}
The first components of $B^\pm_m$, $B^\pm_{\bar m}$ are the $\mbox{\rm I}\!\mbox{\rm R}_\pm$-\penalty10000\zhs
currents discussed above; none of them is observable since none of them is
simultaneously annihilated by both $\mbox{\rm q}$ and $\bar{\q}$. The next components are the
improved currents of the fermionic symmetries:
\begin{equation}\label{iii_iii_xv}
\begin{array}{rcl}
s^+_m &=& 2\mbox{\rm Tr}\left\{f_{mn}\psi^n + \varphi \mbox{\rm D}_m g_-\right\}\; ;\\
s^+_{\bar m} &=& -\mbox{\rm Tr}\left\{f \psi_{\bar m} + 2g_- \mbox{\rm D}_{\bar m} \varphi +
2\varphi[\lambda,\psi_{\bar m}]\right\}\; ;\\
s^-_m &=& -\mbox{\rm Tr}\left\{-f \psi_m + 2g_+ \mbox{\rm D}_m \varphi + 2\varphi[\lambda,
\psi_m]\right\}\; ;\\
s^-_{\bar m} &=& 2\mbox{\rm Tr}\left\{f_{\bar m\bar n}\psi^{\bar n} + \varphi \mbox{\rm D}_{\bar m}
g_+\right\}\; .
\end{array}
\end{equation}
One can easily check that eqs.~(\ref{iii_iii_xv}) differ
from~(\ref{iii_ii_vi}) by terms which do not violate the conservation law.
Of the list~(\ref{iii_iii_xv}) only $s^+_{\bar m}$ and $s^-_m$ are
local observables since they are both $\mbox{\rm q}$ and $\bar{\q}$ invariant. However, being
the highest component of the superfields $4\mbox{\rm i}\mbox{\rm Tr}\{\Lambda F_{\bar m\bar
\theta}\}$ and $4\mbox{\rm i}\mbox{\rm Tr}\{\Lambda F_{m\theta}\}$ respectively, all their
correlation functions vanish.
There are two more conserved currents which correspond to commuting parameters
$u_{mn}$, $u_{\bar m\bar n}$ \cite{Dah}, \cite{Dem}. They give rise to the
antichiral
superfields
\begin{equation}\label{iii_iii_xvi}
\begin{array}{rcl}
B_{mnp} &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{X_{mn} F_{p\theta}\right\}\; ;\\
B_{mn\bar p} &=& 8\mbox{\rm i} g_{[m\bar p} \partial_\theta \mbox{\rm Tr}\left\{\Lambda F_{n]\theta}
\right\}
\end{array}
\end{equation}
and to their complex conjugates (chiral ones)
\begin{equation}\label{iii_iii_xvii}
\begin{array}{rcl}
B_{\bar m\bar np} &=& 8\mbox{\rm i} g_{p[\bar m} \partial_{\bar\theta} \mbox{\rm Tr}\left\{\Lambda
F_{\bar n]\bar\theta}\right\}\; ;\\
B_{\bar m\bar n\bar p} &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{X_{\bar m\bar n} F_{\bar
p\bar\theta}\right\}\; .
\end{array}
\end{equation}
The conservation rules read
\begin{equation}\label{iii_iii_xviii}
\begin{array}{rcl}
\nabla^p B_{mnp} + \nabla^{\bar p} B_{mn\bar p} &=& 0\; ;\\
\nabla^p B_{\bar m\bar np} + \nabla^{\bar p} B_{\bar m\bar n\bar p} &=& 0\; .
\end{array}
\end{equation}
Two new local observables emerge:
\begin{equation}\label{iii_iii_xix}
\begin{array}{rcccl}
s_{mnp} &=& \lwc{\partial_{\bar\theta} B_{mnp}} &=& 4\mbox{\rm Tr}\left\{\chi_{mn} \mbox{\rm D}_p
\varphi
- f_{mn} \psi_p \right\}\; ;\\
s_{\bar m\bar n\bar p} &=& \lwc{\partial_\theta B_{\bar m\bar n\bar p}} &=&
4\mbox{\rm Tr}\left\{\chi_{\bar m\bar n} \mbox{\rm D}_{\bar p} \varphi - f_{\bar m\bar n}
\psi_{\bar
p} \right\}\; .
\end{array}
\end{equation}
They are highest components of an antichiral or chiral superfield,
respectively,
and therefore have vanishing correlation functions.
\subsection{Taking BRS into Account}
\label{BRS}
Also if BRS symmetry is taken into account, the trivial observables discussed
previously remain higher components of gauge invariant superfields. The
starting point is the total action $\widetilde {\cal S}+{\cal S}'$, which
includes
also the Lagrange multipliers $L_{mn}$, $L_{\bar m\bar n}$, the Faddeev--\penalty10000\zhs
Popov ghosts $C$, $C^{\cal y}$, and their antighosts $D$, $D^{\cal y}$.
The dynamics of the total action is somewhat intricate---the superfields
$X_{mn}$, $X_{\bar m\bar n}$ are set equal to the corresponding Lagrange
multipliers
\begin{equation}\label{iii_iv_xxi}
X_{mn} = L_{mn}\; ;\qquad X_{\bar m\bar n} = L_{\bar m\bar n}\; ,
\end{equation}
therefore becoming covariantly chiral.
Some equations
of motion (see~(\ref{xi})) have to be modified:
\begin{equation}\label{iii_iv_xx}
\begin{array}{rcl}
\mbox{\rm D}^n L_{mn} + 2\mbox{\rm D}_\theta \mbox{\rm D}_m \Lambda + 2\mbox{\rm i}\mbox{\rm e}^{-\frac{V}{2}}
\left(\nabla_m B^{\cal y} - \left[C^{\cal y},\nabla_m D^{\cal y}\right]\right)
\mbox{\rm e}^{\frac{V}{2}} &=& 0\; ;\\
\mbox{\rm D}^{\bar n} L_{\bar m\bar n} + 2\mbox{\rm D}_{\bar\theta} \mbox{\rm D}_{\bar m} \Lambda
+ 2\mbox{\rm i}\mbox{\rm e}^{\frac{V}{2}}
\left(\nabla_{\bar m} B + \left[C,\nabla_{\bar m} D\right]\right)
\mbox{\rm e}^{-\frac{V}{2}} &=& 0\; .
\end{array}
\end{equation}
The gauge is fixed by
\begin{equation}\label{iii_iv_xxii}
\nabla^m \phi_m = 2\alpha\partial_\theta B\; ;\qquad \nabla^{\bar m} \phi_{\bar m}
=
2\alpha\partial_{\bar\theta} B^{\cal y}\; .
\end{equation}
Finally, ghosts and antighosts obey the equations
\begin{equation}\label{iii_iv_xxiii}
\begin{array}{rclrcl}
\nabla^m {\cal D}_m C &=& 0\; ;\qquad &
\nabla^{\bar m} {\cal D}_{\bar m} C^{\cal y} &=& 0\; ;\\
{\cal D}_m \nabla^m D &=& 0\; ;\qquad &
{\cal D}_{\bar m} \nabla^{\bar m} D^{\cal y} &=& 0\; .
\end{array}
\end{equation}
After some calculation one gets the following system of BRS modified current
superfields corresponding to the fermionic symmetries:
\begin{equation}\label{iii_iv_xxiv}
\begin{array}{rcl}
\widetilde B^+_m &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F^n{}_{\bar\theta}
X_{mn} - F_{\theta\bar\theta} \mbox{\rm D}_m \Lambda + \mbox{\rm i}\mbox{\rm s}\left(
\nabla_m D^{\cal y}\mbox{\rm e}^V\partial_{\bar\theta}\mbox{\rm e}^{-V} \right) \right\} \; ;\\
\widetilde B^+_{\bar m} &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\partial_\theta\left(\Lambda
F_{\bar m\bar\theta}\right) - \mbox{\rm i}\mbox{\rm s}\left(D^{\cal y} \partial_{\bar\theta}
\phi_{\bar m}\right) \right\} \; ; \\
\widetilde B^-_m &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\partial_{\bar\theta}\left(\Lambda
F_{m\theta}\right) - \mbox{\rm i}\mbox{\rm s}\left(D \partial_\theta \phi_m\right)
\right\} \; ; \\
\widetilde B^-_{\bar m} &=& 4\mbox{\rm i}\mbox{\rm Tr}\left\{\mbox{${1\over 2}$} F^{\bar n}{}_\theta
X_{\bar m\bar n} - F_{\theta\bar\theta} \mbox{\rm D}_{\bar m} \Lambda + \mbox{\rm i}\mbox{\rm s}\left(
\nabla_{\bar m} D\mbox{\rm e}^{-V}\partial_\theta\mbox{\rm e}^V \right) \right\} \;.
\end{array}
\end{equation}
The observable currents are $\widetilde s^+_{\bar m}$ and $\widetilde s^-_m$.
The differences $\widetilde s^+_{\bar m} - s^+_{\bar m}$ and $\widetilde s^-_m
- s^-_m$ are gauge variations of the second component of the chiral
superfields $-4\mbox{\rm Tr}\{D^{\cal y}
\partial_{\bar\theta} \phi_{\bar m}\}$ and $-4\mbox{\rm Tr}\{D \partial_\theta \phi_m\}$,
respectively.
\sect{Perturbative Check}
\label{pertu}
One would like to have some confirmation about the correctness of the field
theoretic description of TYM on K\"{a}hler manifolds. In the present section
we shall compute in perturbation theory the gravitational contribution to
$\mbox{\rm I}\!\mbox{\rm R}_{\pm}$-\penalty10000\zhs anomalies. The latter can be considered as finite radiative
corrections
to the conservation law for the Abelian currents. It will turn out that the
anomalies are equal.
When integrated over the K\"{a}hler manifold they both yield the (formal)
dimension
of the instanton moduli space. The resulting number is well known in the
mathematical literature and has a standard representation by local
polynomials in the curvature.
On the other hand $\mbox{\rm I}\!\mbox{\rm R}_{\pm}$-\penalty10000\zhs anomalies get perturbative contributions
from all the (super)fields of the model. Hence, one can in fact check the
proposed superspace description.
Since our calculation is limited to one-\penalty10000\zhs loop approximation, we shall
regularize the $\mbox{\rm I}\!\mbox{\rm R}_{\pm}$-\penalty10000\zhs currents by point-\penalty10000\zhs splitting. This way all
the symmetries which are supposed to be preserved at the quantum level will be
automatically taken into account.
\subsection{Green Functions}
\label{green}
The Green functions can be obtained from the linearized equations of motion
with sources. The source term that is added to the total action has the form
\begin{equation}\label{iv_i}
\begin{array}{c}
\mbox{${1\over 4}$}\vrule height 8mm width 0pt \displaystyle\int_{\cal K} \mbox{\rm d}^2 z \mbox{\rm d}^2{\bar z}g \left\{
{\partial}_{\theta}
\mbox{\rm Tr} \left( \mbox{${1\over 2}$} K_{mn}M^{mn} + J^m{\phi}_m + J_{B}B + J_{C}C + J_{D}D
\right) \right. \; \\
+ {\partial}_{\bar{\theta}} \mbox{\rm Tr} \left(
\mbox{${1\over 2}$} K^{mn}M_{mn} + J_m M^m + J_{B^{\cal y}} B^{\cal y} + J_{C^{\cal y}} + J_{D^{\cal y}}
D^{\cal y} \right) \; \\
+ {\partial}_{\theta}
{\partial}_{\bar{\theta}} \mbox{\rm Tr} \left. \left( \mbox{${1\over 2}$} J^{mn}X_{mn} +
\mbox{${1\over 2}$} J_{mn}X^{mn} + J_{\Lambda}{\Lambda} + J V \right) \right\}
\end{array}
\end{equation}
where the sources $K_{mn}$, $K_{\bar m \bar n}$, $J_m$, $J_{\bar m}$, $J_B$,
$J_{B^{\cal y}}$, $J_C$, $J_{C^{\cal y}}$, $J_D$, $J_{D^{\cal y}}$, $J_{mn}$,
$J_{\bar m \bar n}$, $J_{\Lambda}$, $J$ and the corresponding superfields
$M_{\bar m \bar n}$, $M_{mn}$, ${\phi}_{\bar m}$, ${\phi}_m$, $B$, $B^{\cal y}$,
$C$, $C^{\cal y}$, $D$, $D^{\cal y}$, $X_{\bar m \bar n}$, $X_{mn}$, $\Lambda$, $V$ have
the same symmetry and reality (or chirality) properties.
The linearized equations of motion are
\[
\begin{array}{rcl}
\mbox{${1\over 2}$}{\nabla}^n M_{mn} + {\nabla}_m \left( B^{\cal y} - \mbox{\rm i}
{\partial}_{\theta}\Lambda \right) & = & J_m \; ; \\
\mbox{${1\over 2}$}{\nabla}^{\bar n} M_{\bar m \bar n} + {\nabla}_{\bar m} \left( B -
\mbox{\rm i} {\partial}_{\bar{\theta}}\Lambda \right) & = & J_{\bar m} \; ;
\end{array}
\]
\begin{equation}\label{iv_ii}
\begin{array}{rclrcl}
\mbox{\rm i} {\nabla}_{[ m}{\phi}_{n ]} + {\partial}_{\bar{\theta}}X_{mn} &=& 2
K_{mn} \; ;\; &
\mbox{\rm i} {\nabla}_{[ \bar m}
{\phi}_{\bar n ]} + {\partial}_{\theta}X_{\bar m \bar n} &=& 2 K_{\bar m \bar n}
\; ; \\
X_{mn} &=& M_{mn} + 2 J_{mn} \; ;\; &
X_{\bar m \bar n} &=& M_{\bar m \bar n} + 2 J_{\bar m \bar n} \; ; \\
{\nabla}^m{\phi}_m &=& 2 \alpha {\partial}_{\theta}B + J_{B} \; ; \; &
{\nabla}_m {\phi}^m
&=& 2 \alpha {\partial}_{\bar{\theta}}B^{\cal y} + J_{B^{\cal y}} \; ;\\
{\nabla}_m {\nabla}^m C &=& J_{D} \; ;\; &
{\nabla}^m{\nabla}_m
C^{\cal y} &=& J_{D^{\cal y}} \; ;\\
{\nabla}_m{\nabla}^m D &=& J_{C} \; ;\; &
{\nabla}^m{\nabla}_m D^{\cal y} &=& J_{C^{\cal y}} \; ;
\end{array}
\end{equation}
\[
\begin{array}{rcl}
{\nabla}_m {\nabla}^m V + {\nabla}_m {\phi}^m -
{\nabla}^m {\phi}_m + 4\mbox{\rm i} {\partial}_{\theta}
{\partial}_{\bar{\theta}} \Lambda + \mbox{\rm i} J_{\Lambda} &=& 0 \; ;\\
{\nabla}_m {\nabla}^m \Lambda + \mbox{\rm i} J & = & 0 \; .
\end{array}
\]
Eqs.~(\ref{iv_ii}) can be converted into differential equations for the Green
functions (GF). The solution can be expressed in terms of two basic GF, the
scalar GF $G(z, z')$ and the vector one $G_{m {\bar n}'}(z, z')$ the
definition of which is given by
\begin{equation}\label{iv_iii}
\begin{array}{rcl}
{\nabla}_{p}{\nabla}^{p}G(z, z') & = & - \delta (z, z') \; ; \\
{\nabla}_{p}{\nabla}^{p}G_{m {\bar n}'}(z, z') & = & - g_{m {\bar n}'}( z, z' )
\delta (z, z')
\end{array}
\end{equation}
where
\begin{equation}\label{iv_iv}
\delta (z, z') = {g}^{-1}{\delta}^2 (z - z'){\delta}^2({\bar z}-
{\bar z}') \; .
\end{equation}
The factor $g_{m {\bar n}'}( z, z' )$ is explained below in
section~\ref{short}.
By taking ${\cal K}$ a compact K\"ahler manifold and by observing that the
Laplacian ${\nabla}_{p}{\nabla}^{p}$ is an elliptic operator one can assume
that eqs.~(\ref{iv_iii}) have unique solutions.
We choose to work in the gauge $\alpha = 1$. Below we list only those Green
functions which are necessary for the computation of anomalies:
\begin{eqnarray}\label{iv_v}
\vev{X_{mna} {\phi}_{{\bar p}'b}} & = & 2 \mbox{\rm i} {\delta}_{ab} (
\bar{\theta} -
{\bar{\theta}}' ) {\nabla}_{[ m} G_{n ] {\bar p}'}( z, z' ) \; ; \nonumber \\
\vev{X_{\bar m \bar n a} {\phi}_{p' b}} & = & 2 \mbox{\rm i} {\delta}_{ab} ( \theta -
{\theta}' ) {\nabla}_{[ \bar m} G_{p' \bar n ]} ( z, z' ) \; ; \nonumber \\
\vev{{\Lambda}_a {V'}_b } & = & \mbox{\rm i} {\delta}_{ab} ( \theta -
{\theta}' ) ( \bar{\theta} - {\bar{\theta}}' ) G( z, z' ) \; ; \nonumber \\
\vev{{\phi}_{ma} {B'}_b } & = & - {\delta}_{ab} ( \theta -
{\theta}' ) {\nabla}_m G( z, z' ) \; ; \nonumber \\
\vev{{\phi}_{\bar m a} {{\bar B}'}_b } & = & - {\delta}_{ab} ( \bar{\theta} -
{\bar{\theta}}' ) {\nabla}_{\bar m} G( z, z' ) \; ; \\
\vev{V_a {B'}_b } & = & - {\delta}_{ab} ( \theta -
{\theta}' ) G( z, z' ) \; ; \nonumber \\
\vev{V_a {{\bar B}'}_b } & = & - {\delta}_{ab} ( \bar{\theta} -
{\bar{\theta}}' ) G( z, z' ) \; ; \nonumber \\
\vev{C_a {D'}_b } & = & {\delta}_{ab} ( \theta -
{\theta}' ) G( z, z' ) \; ; \nonumber \\
\vev{{\bar C}_a {{\bar D}'}_b } & = & {\delta}_{ab} ( \bar{\theta} -
{\bar{\theta}}' ) G( z, z' ) \; .\nonumber
\end{eqnarray}
The brackets on the left hand side of~(\ref{iv_vi}) indicate that the two-\penalty10000\zhs
point function has to be calculated with the formula
\begin{equation}\label{iv_vi}
\vev{\ldots } = \int [\mbox{\rm d}\widetilde{\mu}]\,\ldots\,\exp \left\{ -\frac{1}{e^2}
{\cal S}_{\mbox{\sevenrm lin}} \right\} \; .
\end{equation}
The linearized action ${\cal S}_{\mbox{\sevenrm lin}}$ leads to the
equations~(\ref{iv_ii}), albeit with the source superfields set equal to zero.
\subsection{Short Distance Behaviour}
\label{short}
GF have singularities in the scalar variable $\sigma$, {i.e.}\ one fourth of
the geodetic interval squared between the points $( z^m , z^{\bar m} )$ and
$( z^{m'} , z^{{\bar m}'} )$. The variable satisfies
\begin{equation}\label{iv_vii}
\sigma = g^{\bar n m} {\sigma}_m {\sigma}_{\bar n} \;
\end{equation}
where
\begin{equation}\label{iv_viii}
{\sigma}_m = {\nabla}_m \sigma \; ; \qquad \qquad {\sigma}_{\bar m} =
{\nabla}_{\bar m} \sigma \; .
\end{equation}
The residue of the singularity involves the so called parallel
displacement matrix
$g_{m {\bar n}'}( z, z' )$ \cite{DWitt} defined by
\begin{equation}\label{iv_ix}
\left( {\sigma}^p {\nabla}_p + {\sigma}^{\bar p} {\nabla}_p \right)
g_{m {\bar n}'}( z, z' ) = 0 \; ; \qquad \left[ g_{m {\bar n}'} \right]
= g_{m \bar n} \; .
\end{equation}
The square bracket used in the boundary condition means coinciding
arguments, {i.e.}\ $z^{m'} = z^m$ and $z^{{\bar m}'} = z^{\bar m}$.
The scalar Green function has the following short distance behaviour
\cite{DWitt}:
\begin{equation}\label{iv_x}
G( z, z') = - \frac{\Gamma ( z, z' )}{64{\pi}^2} \left\{
\frac{1}{\sigma} + [v_0]
\ln \sigma + {\cal O}( {\sigma}_m \ln \sigma ) \right\} \;
\end{equation}
valid up to terms of at least first order in ${\sigma}_m \ln \sigma$ or
${\sigma}_{\bar m} \ln \sigma$. The functions $\Gamma ( z, z' )$ and
$v_0 ( z, z' )$ can be determined from differential equations with boundary
conditions that are similar to~(\ref{iv_ix}). The only relation we shall need
in the following is:
\begin{equation}\label{iv_xi}
\left( {\nabla}_m + g_{m {\bar n}'} {\nabla}^{{\bar n}'} \right) \ln \Gamma
( z, z' ) = {\cal O}_2 ( {\sigma}_m , {\sigma}_{\bar m} ) \; ;
\quad [\Gamma] = 1 \;
\end{equation}
where the symbol ${\cal O}_2$ on the right hand side of the first
equation~(\ref{iv_xi}) means terms at least bilinear in ${\sigma}_m$ and
${\sigma}_{\bar m}$.
Instead of the vector GF it proves convenient to introduce \cite{Chri} two
bilocal
unprimed tensors
\begin{equation}\label{iv_xii}
\begin{array}{rcl}
{\overline{G}}_{m \bar n} ( z, z' ) & = & g^{{\bar p}'}_{\bar n} ( z, z' )
G_{m {\bar p}'} ( z, z' ) \; ; \\
{\widetilde{G}}_{m \bar n} ( z, z' ) & = & g^{p'}_m ( z, z' ) G_{{p}' \bar n}
( z, z' ) \; .
\end{array}
\end{equation}
The short distance expansion of the vector GF reads now
\begin{equation}\label{iv_xiii}
{\overline{G}}_{m \bar n} ( z, z' ) = g_{m \bar n} G ( z, z' ) -
\frac{\Gamma ( z, z' )}{128{\pi}^2} R_{m \bar n} \ln \sigma +
{\cal O} ({\sigma}_m \ln \sigma ) \;
\end{equation}
where $R_{m \bar n}$ is the Ricci tensor of the K\"{a}hler manifold.
\subsection{Regularization of Superfield Currents}
\label{regul}
We assume that the fermionic symmetries are preserved in perturbation theory.
The corresponding currents $s^{\pm}_m$, $s^{\pm}_{\bar m}$ receive radiative
corrections, can be, however, redefined as to remain conserved.
As a consequence one
can use the superspace approach for quantum computations. However,
the conservation law~(\ref{iii_iii_xiv}) of the superfield currents
$B^{\pm}_m$, $B^{\pm}_{\bar m}$ is violated in perturbation theory. The
breakdown of current conservation gives rise to $\mbox{\rm I}\!\mbox{\rm R}_{\pm}$-\penalty10000\zhs anomalies.
For evaluating the gravitational contribution it is sufficient to compute the
vacuum expectation value of the left hand side of the conservation
law~(\ref{iii_iii_xiv}). Here we consider explicitly only the $\mbox{\rm I}\!\mbox{\rm R}_{+}$-\penalty10000\zhs
anomaly. The appropriate superfield currents were given
in~(\ref{iii_iv_xxiv}). In the one-\penalty10000\zhs loop approximation one uses the
linearized expressions
\begin{equation}\label{iv_xiv}
\begin{array}{rcl}
{\widetilde{B}}^{+}_{m} & = & 4\mbox{\rm i}\mbox{\rm Tr}\Bigl\{ \frac{1}{2}g^{\bar p n}
X_{mn}{\partial}_{\bar{\theta}}\left( {\nabla}_{\bar p}V + {\phi}_{\bar n}
\right) + {\partial}_{\theta}{\partial}_{\bar{\theta}}V{\nabla}_m \Lambda
\Bigr. \; \\
& & + \mbox{\rm i}\Bigl. \left( {\nabla}_m D^{\cal y}
{\partial}_{\bar{\theta}}C^{\cal y} - {\nabla}_m B^{\cal y}{\partial}_{\bar{\theta}}V
\right) \Bigr\} \; ;
\end{array}
\end{equation}
\begin{equation}\label{iv_xv}
\begin{array}{rcl}
{\widetilde{B}}^{+}_{\bar m} & = & 4 \mbox{\rm i}\mbox{\rm Tr}\Bigl\{ {\partial}_{\bar{\theta}}
\left( {\nabla}_{\bar m}V + {\phi}_{\bar m} \right) {\partial}_{\theta}\Lambda
- \Lambda{\partial}_{\theta}{\partial}_{\bar{\theta}}{\nabla}_{\bar m}V
\Bigr. \; \\
& & - \mbox{\rm i}\Bigl. \left( D^{\cal y}
{\partial}_{\bar{\theta}}{\nabla}_{\bar m}C^{\cal y} + B^{\cal y}
{\partial}_{\bar{\theta}}{\phi}_{\bar m} \right) \Bigr\} \; .
\end{array}
\end{equation}
The linearized currents are regularized by symmetric point-\penalty10000\zhs splitting being
summarized in the following rules:
\begin{enumerate}
\item
For each term of~(\ref{iv_xiv}) and~(\ref{iv_xv}) one takes into account both
factor orderings with equal weight.
\item
Each primed index, {i.e.}\ corresponding to the coordinates $( z^{m'},
z^{{\bar m}'} )$ is accompanied by a parallel displacement matrix
element $g_{m{\bar n}'}(z, z')$ or by its inverse $g^{{\bar n}'m}(z', z )$.
\end{enumerate}
As an example let us write down the regularized version of~(\ref{iv_xiv})
\begin{equation}\label{iv_xvi}
\begin{array}{rcl}
{\widetilde{B}}^{+}_m ( z, z' ) & = & 2 \mbox{\rm i}\mbox{\rm Tr}\Bigl\{\mbox{${1\over 2}$}\left(
g^{{\bar p}' n}X_{mn}{\partial}_{{\bar{\theta}}'}{\phi}_{{\bar p}'} -
g^{n'}_m g^{{\bar p} r'}
{\partial}_{\bar{\theta}}{\phi}_{\bar p} X_{n' r'} \right) \Bigr. \;
\\
& & + {\nabla}_m \Lambda
{\partial}_{{\theta}'} {\partial}_{{\bar{\theta}}'} V' + g^{n'}_m
{\partial}_{\theta} {\partial}_{\bar{\theta}} V {\nabla}_{n'} {\Lambda}' \;
\\
& & + \mbox{\rm i}\left({\nabla}_m D^{\cal y}
{\partial}_{{\bar{\theta}}'}C^{\prime{\cal y}} + g^{n'}_m {\partial}_{\bar{\theta}} C^{\cal y}
{\nabla}_{n'} B^{\prime{\cal y}} \right) \; \\
& & - \mbox{\rm i}\Bigl.\left( {\nabla}_m
B^{\cal y} {\partial}_{{\bar{\theta}}'} V' - g^{n'}_m {\partial}_{\bar{\theta}} V
{\nabla}_{n'} B^{\prime{\cal y}} \right) \Bigr\} \; .
\end{array}
\end{equation}
One finds a similar expression for ${\widetilde{B}}^{+}_{\bar m} ( z, z' )$.
The vacuum expectation values of the regularized currents are
\begin{equation}\label{iv_xvii}
\begin{array}{rcl}
\vev{{\widetilde{B}}^{+}_m (z, z')} & = & 2 n \Bigl\{ g^{{\bar p} n'}
{\nabla}_{[ m} G_{n ]{\bar p}'} - {\nabla}_m G \Bigr. \; \\
& & + \Bigl. g^{n'}_m \left(
g^{{\bar p} r'}{\nabla}_{[ n'}G_{r' ] \bar p} - {\nabla}_{n'} G
\right) \Bigr\} \; ; \\
\vev{{\widetilde{B}}^{+}_{\bar m} (z, z')} & = & 0 \; .
\end{array}
\end{equation}
Recalling the
meaning of the square bracket in eqs.~(\ref{iv_ix}) and~(\ref{iv_x}) the
gravitational contribution to the $\mbox{\rm I}\!\mbox{\rm R}_{+}$-\penalty10000\zhs anomaly becomes
\begin{equation}\label{iv_xviii}
B^{(+)} (z) = \frac{1}{2} {\nabla}^m \left[ \vev{ {\widetilde{B}}^{+}_m (z, z')
} \right] \;
\end{equation}
can be evaluated with the help of Synge's theorem \cite{Chri}
\begin{equation}\label{iv_xix}
B^{(+)} (z) = \frac{1}{2} \left[ \vev{ \left( {\nabla}^m + g^{{\bar n}' m}
{\nabla}_{{\bar n}'} \right){\widetilde{B}}^{+}_m (z, z') } \right]
\; .
\end{equation}
If one inserts~(\ref{iv_xvii}) into~(\ref{iv_xix}) and if one tries to exhibit
the combinations~(\ref{iv_xii}) in place of the vector GF, one gets
\begin{equation}\label{iv_xx}
\begin{array}{rcl}
B^{(+)} (z) & = & n \left[ \left\{ \left( g^{\bar n p'} {\nabla}_{[ \bar s}
g_{p' \bar u ]} - g_{p' [ \bar s} g_{r' \bar u ]} {\nabla}^{p'} g^{\bar n r'}
\right) \left( {\nabla}^{\bar s}\, {\overline{G}}^{\bar u}_{\bar n} +
{\nabla}^{\bar s} {\widetilde{G}}^{\bar u}_{\bar n} \right. \right. \right. \;
\\
& & + \left. g^{{\bar t}'}_{\bar n} {\nabla}^{\bar s}
g^{\bar v}_{{\bar t}'} {\overline{G}}^{\bar u}_{\bar v} - g^{\bar s t'}
g^{\bar u v'} {\nabla}_{t'} g^w_{v'} {\widetilde{G}}_{w \bar n} \right) \;
\\
& & + \left( {\nabla}_{{\bar m}'} g^{{\bar m}'}_{\bar s}
- g_{n' \bar s} {\nabla}_{\bar m} g^{\bar m n'} \right) \left( 2
g^{\bar s u'} {\nabla}_{u'} G + {\nabla}^{[ \bar s}
{\widetilde{G}}^{\bar u ]}_{ \bar u} \right. \; \\
& & - \left. \left. \left. g^{[ \bar s t'}
g^{\bar u ] v'} {\nabla}_{t'} g^p_{v'} {\widetilde{G}}_{p \bar u}
\right) \right\} \right] \; .
\end{array}
\end{equation}
Now one uses the short distance expansions~(\ref{iv_x}) and~(\ref{iv_xiii})
in~(\ref{iv_xx}). One realizes immediately that only singularities of the form
$ {\sigma}^{-2}$ contribute to~(\ref{iv_xx}). The residues can be evaluated by
means of the formulae:
\begin{equation}\label{iv_xxi}
\begin{array}{l}
g^{p'}_n {\nabla}_{[ \bar s} g_{p' \bar u ]} -
g_{p' [ \bar s} g_{r' \bar u ]} {\nabla}^{p'} g^{r'}_n \\
\qquad\qquad = \vrule height 8mm width 0pt \displaystyle\frac{1}{12}
{\sigma}^a {\sigma}^b {\sigma}^{\bar c} \left( {R^r}_{ab[ \bar s}
R_{\bar u ] rn \bar c} - {R^r}_{an [ \bar s} R_{\bar u ] rb \bar c} \right)
+ \ldots \\
{\nabla}_{{\bar m}'} g^{{\bar m}'}_{\bar s} -
g_{n' \bar s} {\nabla}_{\bar m} g^{\bar m n'} \\
\qquad\qquad =\vrule height 8mm width 0pt \displaystyle - \frac{1}{6}
{\sigma}^a {\sigma}^b {\sigma}^{\bar c} \left( \frac{1}{2} {R^p}_{ra \bar s}
{R^r}_{pb \bar c} + \frac{1}{4} {R^p}_{ab \bar c} R_{p \bar s} + {R^p}_a
R_{\bar c pb \bar s} \right) + \ldots
\end{array}
\end{equation}
where the dots collect all the terms which are unimportant for the present
calculation. The result reads
\begin{equation}\label{iv_xxii}
\begin{array}{rcl}
B^{(+)} (z) & = &\vrule height 8mm width 0pt \displaystyle \frac{n}{128{\pi}^2} \Biggl[
\frac{{\sigma}^a{\sigma}^{\bar b}}{\sigma}
\Biggl\{ \frac{{\sigma}^c{\sigma}^{\bar d}}{3 \sigma} \left(
{R^p}_{ac \bar b} R_{p \bar d} + {R^p}_a R_{\bar b pc \bar d} \right. \Biggr.
\Biggr.\\
& & - \Biggl. \Biggl. \left. {R^p}_{ra \bar b}
{R^r}_{pc \bar d}
- {{R^p}_{ac}}^r R_{\bar b pr \bar d} \right) + {R^p}_a R_{p \bar b} -
R R_{a \bar b} \Biggr\} \Biggr] \; .
\end{array}
\end{equation}
The average over the directions of ${\sigma}_m$, ${\sigma}_{\bar m}$
amounts to the following replacements:
\begin{equation}\label{iv_xxiii}
{\sigma}^a{\sigma}^{\bar b} {\rightarrow} g^{\bar b a} \frac{\sigma}{2} \;
; \qquad
{\sigma}^a {\sigma}^b {\sigma}^{\bar c}{\sigma}^{\bar d} {\rightarrow} \left(
g^{\bar c a} g^{\bar d b} + g^{\bar d a} g^{\bar c b} \right)
\frac{{\sigma}^2}{6} \; .
\end{equation}
One can avoid the average if one performs the subtraction of certain
direction dependent terms \cite{Niel} in the regularized current.
As a result the gravitational contribution to the $\mbox{\rm I}\!\mbox{\rm R}_{+}$-\penalty10000\zhs anomaly
assumes
the manifestly local form
\begin{equation}\label{iv_xxiv}
B^{(+)} = \frac{n}{64{\pi}^2} \left( \frac{1}{3} {R^a}_b {R^b}_a - \frac{1}{4}
R^2 - \frac{1}{12} {{R^a}_{bc}}^d {{R^b}_{ad}}^c \right) \; .
\end{equation}
It turns out that the contribution $B^{(-)}$ of the external gravity to the
$\mbox{\rm I}\!\mbox{\rm R}_{-}$-\penalty10000\zhs anomaly has the same expression.
Before discussing the result~(\ref{iv_xxiv}) let us comment on its
derivation. The contribution of the Faddeev--\penalty10000\zhs Popov ghosts cancels
in~(\ref{iv_xxiv}), hence the same result is obtained if one keeps only the
first three terms in~(\ref{iv_xiv}) and~(\ref{iv_xv}). The cancellation is due
to the special interplay between BRS and fermionic symmetries.
suggests that the calculation could be performed
without Faddeev--\penalty10000\zhs Popov ghosts, but with the ghosts
introduced in section~\ref{descent}.
Eq.~(\ref{iv_xxiv}) can be written in the more familiar form
\begin{equation}\label{iv_xxv}
B^{(\pm)} = n ( H - E ) \; .
\end{equation}
Here $E$ and $H$ are the K\"{a}hler analogs of the Euler and Hirzebruch
(signature) densities:
\begin{equation}\label{iv_xxvi}
\begin{array}{rcl}
E & = &\vrule height 8mm width 0pt \displaystyle \frac{1}{128{\pi}^2} \left( {{R^m}_{pr}}^n {{R^p}_{mn}}^r -
2 {R^m}_n {R^n}_m + R^2 \right) \; ; \\
H & = &\vrule height 8mm width 0pt \displaystyle \frac{1}{192{\pi}^2} \left( {{R^m}_{pr}}^n {{R^p}_{mn}}^r -
{R^m}_n {R^n}_m \right) \; .
\end{array}
\end{equation}
Recall that on a Riemannian manifold without boundary, the
densities~(\ref{iv_xxvi}) have the expressions \cite{Eguc}
\begin{equation}\label{iv_xxvii}
E = \frac{1}{32{\pi}^2} {\widetilde{R}}_{\lambda \mu \nu \rho}
{\widetilde{R}}^{\nu \rho \lambda \mu} \; ; \qquad
H = \frac{1}{48{\pi}^2} {\widetilde{R}}_{\lambda \mu \nu \rho}
R^{\nu \rho \lambda \mu} \;
\end{equation}
where $R_{\lambda \mu \nu \rho}$ is the curvature tensor and
${\widetilde{R}}_{\lambda \mu \nu \rho}$ its dual
\begin{equation}\label{iv_xxviii}
{\widetilde{R}}_{\lambda \mu \nu \rho} = \frac{1}{2}
\sqrt{g}\,{\epsilon}_{\lambda \mu \alpha \beta}
{{R}_{\nu \rho}}^{\alpha \beta} \; .
\end{equation}
One can write the curvature tensor $R_{\lambda \mu \nu \rho}$ in spinorial
form and decompose it into irreducible spinors \cite{Wess}, \cite{Wald}. In
passing to
K\"{a}hler four-\penalty10000\zhs manifolds one realizes that the curvature tensor has the
structure $R_{\bar m np \bar r}$, symmetric in $\bar m$, $\bar r$ and $n$,
$p$ respectively. In terms of irreducible spinors it has the form
\begin{equation}\label{iv_xxix}
\begin{array}{rcl}
R_{\bar m pr \bar n} & = & 4 \left\{ 2 \left( g_{p \bar m} g_{r \bar n} +
g_{r \bar m} g_{p \bar n} \right) U - e^{\alpha}_{\bar m} e^{\beta}_p
e^{\gamma}_r e^{\delta}_{\bar n} U_{\alpha \beta \gamma \delta}
\right. \; \\
& & + \left. \left( e^{\alpha}_{\bar m}
e^{\beta}_p g_{r \bar n} + e^{\alpha}_{\bar n} e^{\beta}_r g_{p \bar m} \right)
U_{\alpha \beta} \right\} \; .
\end{array}
\end{equation}
Here $U$, $U_{\alpha \beta}$ and $U_{\alpha \beta \gamma \delta}$ are
irreducible spinors, completely symmetric in their indices. The zweibeins
$e^{\alpha}_m$, $e^{\beta}_{\bar n}$ convert spinor indices into holomorphic
and anti-\penalty10000\zhs holomorphic ones.
The densities~(\ref{iv_xxvi}) can also be written in terms of irreducible
spinors. For K\"{a}hler manifolds one finds:
\begin{equation}\label{iv_xxx}
\begin{array}{rcl}
E & = &\vrule height 8mm width 0pt \displaystyle \frac{1}{8{\pi}^2} \left( U_{\alpha \beta \gamma \delta}
U^{\alpha \beta \gamma \delta} + 4 U_{\alpha \beta} U^{\alpha \beta}
+ 48 U^2 \right) \; ; \\
H & = &\vrule height 8mm width 0pt \displaystyle \frac{1}{12{\pi}^2} \left( U_{\alpha \beta \gamma \delta}
U^{\alpha \beta \gamma \delta} - 24 U^2 \right) \; .
\end{array}
\end{equation}
By using~(\ref{iv_xxix}) it is possible to express~(\ref{iv_xxx}) through the
curvature tensor over the K\"{a}hler manifold. The result is~(\ref{iv_xxvi}).
\sect{Donaldson Cohomology}
\label{donal}
Let $({\cal K}, \gamma)$ be a compact K\"{a}hler four-\penalty10000\zhs manifold with
K\"{a}hler form $\gamma$ given by~(\ref{i}). Let $\mbox{\rm E}$ be a complex vector
bundle with structure group $\mbox{\rm G}$ assumed Lie, compact and
semi-\penalty10000\zhs simple. The connection on $\mbox{\rm E}$ splits into a $(1,0)$ part $a =
a_{m}\mbox{\rm d} z^m$ and a $(0,1)$ part ${\bar a} = a_{\bar m}\mbox{\rm d} z^{\bar m}$. The
curvature two-\penalty10000\zhs form can be decomposed into its $(2,0)$, $(0,2)$ and $(1,1)$
parts as follows:
\begin{equation}\label{v_i}
\begin{array}{c}
f^{(2,0)} = {\partial}a + a^2 \; ; \qquad f^{(0,2)} =
{\bar{\partial}}{\bar a} + {\bar a}^2 \; ; \\
f^{(1,1)} = {\partial}{\bar a} + {\bar{\partial}}a + \{ a, \bar a
\} \; .
\end{array}
\end{equation}
They obey the Bianchi identities
\begin{equation}\label{v_ii}
\begin{array}{c}
\mbox{\rm D} f^{(1,1)} + {\bar{\mbox{\rm D}}} f^{(2,0)} = 0 \; ; \qquad {\bar{\mbox{\rm D}}} f^{(1,1)} +
\mbox{\rm D} f^{(0,2)} = 0 \; ; \\
\mbox{\rm D} f^{(2,0)} = \bar{\mbox{\rm D}} f^{(0,2)} = 0 \; .
\end{array}
\end{equation}
By using~(\ref{v_ii}) one derives the basic identities (For simplicity we limit
ourselves to invariant polynomials quadratic in the curvature)
\begin{equation}\label{v_iii}
\begin{array}{rcl}
{\partial}\; \mbox{\rm Tr} \left(\mbox{${1\over 2}$} {f^{(1,1)}}^2 + f^{(2,0)}
f^{(0,2)} \right) + {\bar{\partial}}\; \mbox{\rm Tr} f^{(1,1)} f^{(2,0)} & = & 0
\; ; \\
{\bar{\partial}}\; \mbox{\rm Tr} \left(\mbox{${1\over 2}$} {f^{(1,1)}}^2 + f^{(2,0)}
f^{(0,2)} \right) + {\partial}\; \mbox{\rm Tr} f^{(1,1)} f^{(0,2)} & = & 0 \; .
\end{array}
\end{equation}
Obviously, the last terms in both eqs.~(\ref{v_iii}) vanish, rendering the
invariant $(2,2)$-\penalty10000\zhs form $\mbox{\rm Tr} \left( \frac{1}{2} f^{(1,1) \; 2} +
f^{(2,0)}f^{(0,2)} \right)$ closed with respect to both $\partial$ and
$\bar{\partial}$.
Locally, the closed form can be represented as
\begin{equation}\label{v_iv}
\mbox{\rm Tr} \left(\mbox{${1\over 2}$} {f^{(1,1)}}^2 + f^{(2,0)}f^{(0,2)} \right) =
\bar{\partial}K + \partial{\bar K}
\end{equation}
where
\begin{equation}\label{v_v}
\begin{array}{rcl}
K & = & \mbox{${1\over 2}$} \mbox{\rm Tr} \left( {\bar a}\partial a + a f^{(1,1)}
\right) \; ; \\
\bar K & = & \mbox{${1\over 2}$} \mbox{\rm Tr} \left( a \bar{\partial}\bar a + \bar a
f^{(1,1)} \right) \; .
\end{array}
\end{equation}
One can easily check that $\partial \bar{\partial}K = \partial \bar{\partial}
\bar K = 0$. While eqs.~(\ref{v_v}) render the cohomology of $\partial$, $
\bar{\partial}$ trivial, they are not gauge invariant.
\subsection{Descent Equations and Their Solution}
\label{descent}
The fermionic symmetries $\mbox{\rm q}$, $\bar{\q}$ act as follows
\begin{equation}\label{v_vi}
\begin{array}{rclrcl}
\mbox{\rm q} a &=& \psi - \mbox{\rm D} \omega \; ; & {\bar{\q}} a & =& - \mbox{\rm D} {\bar{\omega}}
\; ; \\
\mbox{\rm q}{\bar a} &=& - {\bar{\mbox{\rm D}}} \omega \; ; & {\bar{\q}}{\bar a} & =&
{\bar{\psi}} - {\bar{\mbox{\rm D}}}{\bar{\omega}} \; ; \\
\mbox{\rm q}{\psi} &=& [ \psi , \omega ] \; ; & {\bar{\q}}{\psi} & =& - \mbox{\rm i} \mbox{\rm D} {\varphi} +
[ \psi , {\bar{\omega}} ] \; ; \\
\mbox{\rm q} {\bar{\psi}} &=& - \mbox{\rm i} {\bar{\mbox{\rm D}}}{\varphi} + [ {\bar{\psi}} , \omega ] \; ; &
{\bar{\q}}{\bar{\psi}} &=& [ {\bar{\psi}} , {\bar{\omega}} ] \; ; \\
\mbox{\rm q} \omega &=& - {\omega}^2 \; ; & {\bar{\q}}{\bar{\omega}} &=& -
{\bar{\omega}}^2 \; ;\\
\mbox{\rm q}{\varphi} &=& [ \varphi , \omega ] \; ; & {\bar{\q}}{\varphi} &=& [ \varphi ,
{\bar{\omega}} ]\; ; \\
\mbox{\rm i} {\varphi} &=& \mbox{\rm q}{\bar{\omega}} + \bar{\q}\omega + \{ \omega ,
{\bar{\omega}} \} \; .
\end{array}
\end{equation}
The ghosts $\omega$, $\bar{\omega}$ are the first components of the
Grassmann superconnections $A_{\theta}$ and $A_{\bar{\theta}}$ respectively.
They occur as supergauge transformations in superspace (see
section~\ref{wesszu}) and ensure the nilpotency and anticommutativity of the
fermionic symmetries
\begin{equation}\label{v_vii}
\mbox{\rm q}^2 = \bar{\q}^2 = \mbox{\rm q}\bar{\q} + \bar{\q}\mbox{\rm q} = 0 \; .
\end{equation}
Here a comment is in order, since apparently one cannot separate the action
of $\mbox{\rm q}$ on $\bar{\omega}$ from that of $\bar{\q}$ on $\omega$. In fact, the
ghosts $\omega$, $\bar{\omega}$ can be expressed through the same prepotential
$V$, as discussed in section~\ref{solut}. Of course, $V$ is determined up to
chiral gauge transformations, {i.e.}\ up to local parameters which are annihilated
either by $\mbox{\rm q}$ or by $\bar{\q}$.
The procedure we shall now describe is an extension of the
construction \cite{AtiyU}, \cite{Stor}, \cite{Kanno} to complex manifolds.
Let ${\cal A}$ be
the space of all connections on the complex vector bundle $\mbox{\rm E}$ and ${\cal G}$
be the group of gauge transformations. The quotient ${\cal B} = {\cal A}
{\setminus}{\cal G}$ is the set of all gauge equivalent connections. Replace
$\partial$ by
$\Delta = \partial + {\bar{\q}}$ and $\bar{\partial}$ by $\bar{\Delta} = \bar{\partial} + \mbox{\rm q}$.
The derivations $\Delta$, $\bar{\Delta}$ act over the product space
${\cal K} \times {\cal B}$ and satisfy
\begin{equation}\label{v_viii}
{\Delta}^2 = {\bar{\Delta}}^2 = {\Delta}{\bar{\Delta}} + {\bar{\Delta}}
{\Delta} = 0 \; .
\end{equation}
Furthermore, one defines an adjoint bundle ${\cal E}$ over ${\cal K} \times
{\cal B}$. Let ${\cal A} = a + {\bar{\omega}}$ and ${\bar{\cal A}} = {\bar a}
+ {\omega}$ be the connections on ${\cal E}$. One can construct generalized
forms for the curvature
\begin{equation}\label{v_ix}
\begin{array}{c}
{\cal F}^{(2,0)} = {\Delta}{\cal A} + {\cal A}^2 \; ; \qquad
{\cal F}^{(0,2)} = {\bar{\Delta}}{\bar{\cal A}} + {\bar{\cal A}}^2
\; ; \\
{\cal F}^{(1,1)} = {\Delta}{\bar{\cal A}} + {\bar{\Delta}}{\cal A}
+ \{ {\cal A} , {\bar{\cal A}} \} \; .
\end{array}
\end{equation}
The quantities~(\ref{v_ix}) satisfy Bianchi identities similar to~(\ref{v_ii})
\begin{equation}\label{v_x}
\begin{array}{c}
\Delta{\cal F}^{(2,0)} + [ {\cal A}, {\cal F}^{(2,0)} ] = 0 \; ; \qquad
\bar{\Delta}{\cal F}^{(0,2)} + [ \bar{\cal A}, {\cal F}^{(0,2)} ] = 0 \; ;
\\
\Delta{\cal F}^{(1,1)} + \bar{\Delta}{\cal F}^{(2,0)} + [ {\cal A},
{\cal F}^{(1,1)} ] + [ \bar{\cal A}, {\cal F}^{(2,0)} ] = 0 \; ;
\\
\Delta{\cal F}^{(0,2)} + \bar{\Delta}{\cal F}^{(1,1)} + [ {\cal A},
{\cal F}^{(0,2)} ] + [ \bar{\cal A}, {\cal F}^{(1,1)} ] = 0 \; .
\end{array}
\end{equation}
Also basic identities look similar to~(\ref{v_iii}):
\begin{equation}\label{v_xi}
\begin{array}{rcl}
{\Delta}\; \mbox{\rm Tr} \left( \frac{1}{2} {\cal F}^{(1,1) \; 2} + {\cal F}^{(2,0)}
{\cal F}^{(0,2)} \right) + {\bar{\Delta}}\; \mbox{\rm Tr} \; {\cal F}^{(1,1)}
{\cal F}^{(2,0)} & = & 0 \; ; \\
{\bar{\Delta}}\; \mbox{\rm Tr} \left( \frac{1}{2} {\cal F}^{(1,1) \; 2} +
{\cal F}^{(2,0)}{\cal F}^{(0,2)} \right) +
{\Delta}\; \mbox{\rm Tr} \; {\cal F}^{(1,1)}{\cal F}^{(0,2)} & = & 0 \; .
\end{array}
\end{equation}
However, the last terms of~(\ref{v_x}) do not vanish, since
\begin{equation}\label{v_xii}
\mbox{\rm Tr} \; {\cal F}^{(1,1)} {\cal F}^{(2,0)} = \mbox{\rm Tr} f^{(2,0)} \left(
{\bar{\psi}} + \mbox{\rm i} {\varphi} \right) \; ; \; \; \mbox{\rm Tr} \; {\cal F}^{(1,1)}
{\cal F}^{(0,2)} = \mbox{\rm Tr} f^{(0,2)} \left( {\psi} + \mbox{\rm i} {\varphi} \right) \; .
\end{equation}
Due to Bianchi identities~(\ref{v_x}) the expressions~(\ref{v_xi}) are
$\Delta$-\penalty10000\zhs close and ${\bar{\Delta}}$-\penalty10000\zhs close respectively
\begin{equation}\label{v_xiii}
\Delta \mbox{\rm Tr}{\cal F}^{(1,1)}{\cal F}^{(2,0)} = 0 \; ; \qquad \bar{\Delta}
\mbox{\rm Tr}{\cal F}^{(1,1)}{\cal F}^{(0,2)} = 0 \; .
\end{equation}
By enlarging the field
manifold one can make them $\Delta$ and ${\bar{\Delta}}$ exact. This feature
makes the theory of Donaldson polynomials somewhat different from that of
TYM with a single fermionic symmetry.
Let us introduce the forms $\chi$, $b$ and $\bar{\chi}$, $\bar b$ of $(2,0)$
and $(0,2)$ type, respectively. They obey
\begin{equation}\label{v_xiv}
\begin{array}{rclrcl}
\mbox{\rm q}{\chi} & =& -\mbox{\rm i} b - \{ \omega , \chi \} \; ; & {\bar{\q}}{\chi} & =& -\mbox{\rm i}
f^{(2,0)} - \{ \bar{\omega} , {\chi} \} \; ; \\
\mbox{\rm q}{\bar{\chi}} & =& - \mbox{\rm i} f^{(0,2)} - \{ \omega , {\bar{\chi}} \} \; ; &
{\bar{\q}}{\bar{\chi}} & =& - \mbox{\rm i} {\bar b} - \{ \bar{\omega}, {\bar{\chi}} \}
\; ; \\
\mbox{\rm q} b & =& [ b , \omega ] \; ; & {\bar{\q}} b & =& \mbox{\rm D} {\psi} -
[ \varphi , {\chi} ] + [ b , \bar{\omega} ] \; ; \\
\mbox{\rm q}{\bar b} & =& {\bar{\mbox{\rm D}}}{\bar{\psi}} - [ \varphi , {\bar{\chi}} ] +
[ {\bar b} , \omega ] \; ; & {\bar{\q}}{\bar b} & =& [ {\bar b} , \bar{\omega} ]
\; .
\end{array}
\end{equation}
One can check that
\begin{equation}\label{v_xv}
\begin{array}{rccccl}
\mbox{\rm Tr} \; {\cal F}^{(1,1)}{\cal F}^{(2,0)} & =& \mbox{\rm i} {\bar{\q}}\; \mbox{\rm Tr}
{\chi} \left( \bar{\psi} + \mbox{\rm i} {\varphi} \right) & =& \mbox{\rm i} {\Delta}\; \mbox{\rm Tr}
{\chi} \left( \bar{\psi} + \mbox{\rm i} {\varphi} \right) & \; ; \\
\mbox{\rm Tr} \; {\cal F}^{(1,1)}{\cal F}^{(0,2)} & =& \mbox{\rm i} \mbox{\rm q} \; \mbox{\rm Tr} {\bar{\chi}}
\left( {\psi} + \mbox{\rm i} {\varphi} \right) & =& \mbox{\rm i} {\bar{\Delta}}\;
\mbox{\rm Tr} {\bar{\chi}} \left( {\psi} + \mbox{\rm i} {\varphi} \right) & \; .
\end{array}
\end{equation}
By inserting~(\ref{v_xiv}) into~(\ref{v_x}) one gets
\begin{equation}\label{v_xvi}
\begin{array}{rcl}
\Delta \mbox{\rm Tr} \left(\mbox{${1\over 2}$} {{\cal F}^{(1,1)}}^2 + {\cal F}^{(2,0)}
{\cal F}^{(0,2)} \right) - \mbox{\rm i} \bar{\Delta} \mbox{\rm Tr} \chi \left( \bar{\psi} +
\mbox{\rm i} \varphi \right) & = & 0 \; ; \\
\bar{\Delta} \mbox{\rm Tr} \left(\mbox{${1\over 2}$} {{\cal F}^{(1,1)}}^2 + {\cal F}^{(2,0)}
{\cal F}^{(0,2)} \right) - \mbox{\rm i} \Delta \mbox{\rm Tr} \bar{\chi} \left( \psi + \mbox{\rm i}
\varphi
\right) & = & 0 \; .
\end{array}
\end{equation}
Let us now define
\begin{equation}\label{v_xvii}
\begin{array}{rcl}
{\cal W} & = & \vrule height 8mm width 0pt \displaystyle - \frac{c}{2{\pi}^2} \left\{ \mbox{\rm Tr} \left(
\mbox{${1\over 2}$}{\cal F}^{(1,1) \; 2} + {\cal F}^{(2,0)}{\cal F}^{(0,2)} \right)
\right. \; \\
& & \vrule height 8mm width 0pt \displaystyle - \mbox{\rm i} {\Delta} \left. \;
\mbox{\rm Tr} {\bar{\chi}} \left( \psi + \mbox{\rm i} {\varphi} \right) - \mbox{\rm i} {\bar{\Delta}}\;
\mbox{\rm Tr} {\chi} \left( {\bar{\psi}} + \mbox{\rm i} {\varphi} \right) \right\} \; .
\end{array}
\end{equation}
Here the factor in front is conventional, and $c$ is the second Casimir
invariant of the adjoint representation of the ${\cal G}$, {i.e.}\ $c_{acd}c_{bcd}
= c {\delta}_{ab}$. Eqs.~(\ref{v_xv}) take the form
\begin{equation}\label{v_xviii}
\Delta{\cal W} = 0 \; ; \qquad \bar{\Delta}{\cal W} = 0 \; .
\end{equation}
By expanding ${\cal W}$ in the ghost numbers associated to $\mbox{\rm I}\!\mbox{\rm R}_{\pm}$
(lower bracket) one gets the descent equations:
\begin{equation}\label{v_xix}
\begin{array}{rclrcl}
\mbox{\rm q} W^{(2,2)}_{(0,0)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial}W^{(2,1)}_{(1,0)} \; ; & \qquad
\bar{\q} W^{(2,2)}_{(0,0)} & =& - \partial W^{(1,2)}_{(0,1)} \; ; \\
\mbox{\rm q} W^{(2,1)}_{(1,0)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial}W^{(2,0)}_{(2,0)} \; ; & \qquad
\bar{\q} W^{(2,1)}_{(1,0)} & =& - \partial W^{(1,1)}_{(1,1)} \; ; \\
\mbox{\rm q} W^{(1,2)}_{(0,1)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial}W^{(1,1)}_{(1,1)} \; ; & \qquad
\bar{\q} W^{(1,2)}_{(0,1)} & =& - \partial W^{(0,2)}_{(0,2)} \; ; \\
\mbox{\rm q} W^{(1,1)}_{(1,1)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial}W^{(1,0)}_{(2,1)} \; ; & \qquad
\bar{\q} W^{(1,1)}_{(1,1)} & =& - \partial W^{(0,1)}_{(2,1)} \; ; \\
\mbox{\rm q} W^{(2,0)}_{(2,0)} &\vrule height 8mm width 0pt \displaystyle =& 0 \; ; & \qquad
\bar{\q} W^{(2,0)}_{(2,0)} & =& - \partial W^{(1,0)}_{(2,1)} \; ; \\
\mbox{\rm q} W^{(0,2)}_{(0,2)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial} W^{(0,1)}_{(1,2)} \; ; & \qquad
\bar{\q} W^{(0,2)}_{(0,2)} & =& 0 \; ; \\
\mbox{\rm q} W^{(1,0)}_{(1,2)} &\vrule height 8mm width 0pt \displaystyle =& 0 \; ; & \qquad
\bar{\q} W^{(1,0)}_{(2,1)} & =& - \partial W^{(0,0)}_{(2,2)} \; ; \\
\mbox{\rm q} W^{(0,1)}_{(1,2)} &\vrule height 8mm width 0pt \displaystyle =& - \bar{\partial} W^{(0,0)}_{(2,2)} \; ; & \qquad
\bar{\q} W^{(0,1)}_{(1,2)} & =& 0 \; ; \\
\mbox{\rm q} W^{(0,0)}_{(0,0)} &\vrule height 8mm width 0pt \displaystyle =& 0 \; ; & \qquad
\bar{\q} W^{(0,0)}_{(2,2)} & =& 0 \; .
\end{array}\end{equation}
The upper bracket indicates the type of form on K\"{a}hler manifold.
In order to obtain the solution of~(\ref{v_xix}) we write~(\ref{v_xvii}) in
the form
\begin{equation}\label{v_xx}
\begin{array}{rcl}
{\cal W} & = & \vrule height 8mm width 0pt \displaystyle - \frac{c}{4{\pi}^2}\; \mbox{\rm Tr} \; { \left( f^{(1,1)} +
f^{(2,0)} + f^{(0,2)} + {\psi} + {\bar{\psi}} + \mbox{\rm i} {\varphi} \right) }^2 \;
\\
& & \vrule height 8mm width 0pt \displaystyle + \frac{\mbox{\rm i} c}{2{\pi}^2} \left\{ {\Delta}\;
\mbox{\rm Tr} {\bar{\chi}} \left( \psi + \mbox{\rm i} {\varphi} \right) + {\bar{\Delta}}\;
\mbox{\rm Tr} {\chi} \left( {\bar{\psi}} + \mbox{\rm i} {\varphi} \right) \right\} \; .
\end{array}
\end{equation}
By expanding ${\cal W}$ according to $r_{\pm}$ numbers one gets
\begin{equation}\label{v_xxi}
\vcenter{\hbox{%
$\begin{array}{rcl}
W^{(2,2)}_{(0,0)} & = &\vrule height 8mm width 0pt \displaystyle \frac{c}{4{\pi}^2} \mbox{\rm Tr} \left\{\mbox{${1\over 2}$} {f^{(1,1)}}^2
+ f^{(2,0)}f^{(0,2)} - \mbox{\rm i} \partial \left( \bar{\chi}\psi \right) -
\mbox{\rm i} \bar{\partial}\left( \chi \bar{\psi}\right)
\right\} \; ; \\
W^{(2,1)}_{(1,0)} & = &\vrule height 8mm width 0pt \displaystyle \frac{c}{4{\pi}^2} \mbox{\rm Tr} \left( \varphi {\bar{\mbox{\rm D}}}\chi
- f^{(1,1)}\psi - b \bar{\psi} \right)\; ; \\
W^{(1,2)}_{(0,1)} & = &\vrule height 8mm width 0pt \displaystyle \frac{c}{4{\pi}^2} \mbox{\rm Tr} \left( \varphi \mbox{\rm D} \bar{\chi}
- f^{(1,1)}\bar{\psi} - \bar{b}\psi \right) \; ; \\
\end{array}$}\hbox{%
$\begin{array}{rclrcl}
W^{(2,0)}_{(2,0)} & = &\vrule height 8mm width 0pt \displaystyle \frac{c}{4{\pi}^2} \mbox{\rm Tr} \left(\mbox{${1\over 2}$}\psi^2 - \mbox{\rm i}
\varphi b \right) \; ;\;& W^{(0,2)}_{(0,2)} &=& \vrule height 8mm width 0pt \displaystyle\frac{c}{4{\pi}^2}
\mbox{\rm Tr} \left(\mbox{${1\over 2}$}\bar{\psi}^2 - \mbox{\rm i} \varphi \bar{b} \right) \; ;
\\
W^{(1,1)}_{(1,1)} & = &\vrule height 8mm width 0pt \displaystyle \frac{c}{4{\pi}^2} \mbox{\rm Tr} \left( \mbox{\rm i} \varphi f^{(1,1)}
+
\psi \bar{\psi} \right) \; ;\;& W^{(1,0)}_{(2,1)} &=&\vrule height 8mm width 0pt \displaystyle
\frac{\mbox{\rm i} c}{4{\pi}^2} \mbox{\rm Tr} \varphi \psi \; ; \\
W^{(0,1)}_{(1,2)} & = &\vrule height 8mm width 0pt \displaystyle \frac{\mbox{\rm i} c}{4{\pi}^2} \mbox{\rm Tr} \varphi \bar{\psi} \;
;\;&
W^{(0,0)}_{(2,2)} &=& \vrule height 8mm width 0pt \displaystyle - \frac{c}{8{\pi}^2} \mbox{\rm Tr} {\varphi}^2 \; .
\end{array}$}}
\end{equation}
The numbers in brackets can be checked by using their additivity as well as
table~1.
\begin{table}\label{tab}
$$
\begin{array} {||l|c|c|c|c|r|r|r|r|r|r|r|r|c||} \hline
& a & \bar{a} & \psi &\vphantom{\bar{\bar{\psi}}}\bar{\psi} & \chi &
\bar{\chi} & b & \bar{b} & \varphi & \lambda & g_{+} & g_{-} & k \\ \hline
r_+ & 0 & 0 & 1 & 0 & 0 & -1 & 1 & -1
&
1 & -1 & 0 & -1 & 0 \\ \hline
r_- & 0 & 0 & 0 & 1 & -1 & 0 & -1 & 1
&
1 & -1 & -1 & 0 & 0 \\ \hline
p & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 0 &
0 & 0 & 0 & 0 & 0 \\ \hline
q & 0 & 1 & 0 & 1 & 0 & 2 & 0 & 2 &
0 & 0 & 0 & 0 & 0 \\ \hline
d & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 &
0 & 2 & 2 & 2 & 2 \\ \hline
\end{array}
$$
\caption{Quantum numbers and form degrees of various fields.}
\end{table}
The meaning of the letters in the first column is the following: $r_+$, $r_-$
are the quantum numbers of the global Abelian symmetries,
$(p, q)$ is the complex form degree and $d$ the canonical dimension of the
field.
One can show further that
\begin{equation}\label{v_xxii}
W^{(0,0)}_{(2,2)} = \frac{c}{8{\pi}^2} \left\{ \mbox{\rm q} \mbox{\rm Tr} \left( \mbox{\rm i} \varphi -
\omega \bar{\omega} \right) \bar{\omega} + \bar{\q} \mbox{\rm Tr} \left( \mbox{\rm i}
\varphi + \bar{\omega} \omega \right) \omega \right\} \; .
\end{equation}
Nevertheless, $W^{(0,0)}_{(2,2)}$ is a nontrivial element of the (equivariant)
cohomology of $\mbox{\rm q}$ and $\bar{\q}$, since it does not depend on $\omega$ or
$\bar{\omega}$. In other words, both $\mbox{\rm Tr} ( \mbox{\rm i} \varphi - \omega
\bar{\omega} ) \bar{\omega}$ and $\mbox{\rm Tr} ( \mbox{\rm i} \varphi + \bar{\omega}\omega
) \omega$ are not gauge invariant.
Hence, $W^{(p,q)}_{(r_+,r_-)}$ are local
observables whose correlation functions might be nonvanishing.
Of course, the $W^{(p,q)}_{(r_+,r_-)}$ are gauge invariant, {i.e.}\ $\mbox{\rm s}
W^{(p,q)}_{(r_+,r_-)} = 0$. Notice that $\mbox{\rm s}$, $\mbox{\rm q}$ and $\bar{\q}$ anticommute
with each other.
\subsection{Cohomology Classes}
\label{cohom}
Let us consider the equivalence classes of $(p,q)$-\penalty10000\zhs forms which are both
$\partial$ and $\bar{\partial}$ closed but not exact. A $(p,q)$-\penalty10000\zhs form is
exact if either ${\omega}_{(p,q)} = \partial \bar{\partial}{\phi}_{(p-1,q-1)}$
or ${\omega}_{(p,0)} = \partial {\phi}_{p-1}(z)$ or else ${\omega}_{(0,q)} =
\bar{\partial}{\bar{\phi}}_{q-1}(\bar z)$, respectively; here $p, q {\geq}1$.
The equivalence classes make up a vector space known as the Dolbeault
cohomology group ${\cal H}^{(p,q)}({\cal K}; \partial, \bar{\partial} )$.
(The composition law is the additive group structure of the vector space.)
Let us define
\begin{equation}\label{v_xxiii}
{\Omega}_{(r_+,r_-)} = \int_{{\cal K}}W^{(2-p,2-q)}_{(r_+,r_-)}{\omega}_{(p,q)}
\end{equation}
where ${\omega}_{(p,q)}$ is a $\partial$ and $\bar{\partial}$ closed $(p,q)$-\penalty10000\zhs
form independent of the fields. One can check that~(\ref{v_xxiii}) is
annihilated by both fermionic charges $\mbox{\rm q}$, $\bar{\q}$. Furthermore, if
${\omega}_{(p,q)}$ is exact, then ${\Omega}_{(r_+=q,r_-=p)}$ is highest
component, {i.e.}\ it can be written in one of the following ways: $\mbox{\rm q} \bar{\q}
{\Phi}_{(q-1,p-1)}$, $\mbox{\rm q}{\Phi}_{(q-1,0)}$ or $\bar{\q} {\Phi}_{(0,p-1)}$.
We shall see in the next subsection that $\mbox{\rm q}$, $\bar{\q}$ can be interpreted
as complex exterior derivatives on the instanton moduli space ${\cal M}$.
Since ${\Omega}_{(q,p)}$ is a $(q,p)$-\penalty10000\zhs form closed with respect to both $\mbox{\rm q}$
and $\bar{\q}$, it belongs to the Dolbeault group ${\cal H}^{(q,p)}({\cal M}; \mbox{\rm q},
\bar{\q} )$. The Donaldson map between the Dolbeault cohomology groups relates
${\omega}_{(p,q)}\in {\cal H}^{(p,q)}({\cal K}; \partial, \bar{\partial} )$
to ${\Omega}_{(q,p)}\in {\cal H}^{(q,p)}({\cal M}; \mbox{\rm q}, \bar{\q} )$.
\subsection{Donaldson Invariants}
\label{doninv}
For computing invariant correlation functions one needs the integrated
observables~(\ref{v_xxiii}). It is convenient to express them in the form:
\begin{eqnarray}\label{v_xxiv}
{\Omega}_{(0,0)} & = & \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left( \mbox{${1\over 2}$}
{f^{(1,1)}}^2 + f^{(2,0)}f^{(0,2)} \right) \; ; \nonumber \\
{\Omega}_{(1,0)} & = & - \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left\{
f^{(1,1)}\psi + \mbox{\rm i} \mbox{\rm q} \left( \chi \bar{\psi} \right) \right\}
{\omega}_{(0,1)}
\; ; \nonumber \\
{\Omega}_{(0,1)} & = & - \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left\{
f^{(1,1)}\bar{\psi} + \mbox{\rm i} \bar{\q} \left( \bar{\chi} \psi \right) \right\}
{\omega}_{(1,0)} \; ; \nonumber \\
{\Omega}_{(2,0)} & = & \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left\{
\mbox{${1\over 2}$} {\psi}^2 + \mbox{\rm q} \left( \chi \varphi \right) \right\} {\omega}_{(0,2)}
\; ; \nonumber \\
{\Omega}_{(0,2)} & = & \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left\{
\mbox{${1\over 2}$}{\bar{\psi}}^2 + \bar{\q} \left( \bar{\chi} \varphi \right) \right\}
{\omega}_{(2,0)} \; ; \\
{\Omega}_{(1,1)} & = & \frac{c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \left( \mbox{\rm i} \varphi
f^{(1,1)} + \psi \bar{\psi} \right) {\omega}_{(1,1)} \; ; \nonumber \\
{\Omega}_{(2,1)} & = & \frac{\mbox{\rm i} c}{4{\pi}^2} \int_{\cal K} \mbox{\rm Tr} \varphi \psi
{\omega}_{(1,2)} \; ; \qquad {\Omega}_{(1,2)} = \frac{\mbox{\rm i} c}{4{\pi}^2}
\int_{\cal K} \mbox{\rm Tr} \varphi \bar{\psi} {\omega}_{(2,1)} \; ; \nonumber \\
{\Omega}_{(2,2)} & = & - \frac{c}{8{\pi}^2} \int_{\cal K} \mbox{\rm Tr} {\varphi}^2
{\omega}_{(2,2)} \; . \nonumber
\end{eqnarray}
The correlation functions of the observables~(\ref{v_xxiv}) are the well-known
Donaldson invariants and have the form
\begin{equation}\label{v_xxv}
\vev{\prod_{i} {\Omega}_{(p_i,q_i)}} = \int [\mbox{\rm d}\mu] \prod_{i}
{\Omega}_{(p_i,q_i)} \,
\exp \left\{ - \frac{1}{e^2} {\cal S} \right\} \; .
\end{equation}
where the functional integral is performed upon the fields $\mu$ figuring
in table~1.
Since the observables ${\Omega}_{(p,0)}$, ${\Omega}_{(0,q)}$ depend on
$\chi$, $\bar{\chi}$, a direct integration of the non-\penalty10000\zhs zero modes in the
path integral is not possible.
Nevertheless, one can show that the correlation
functions~(\ref{v_xxv}) remain unchanged if the equations~(\ref{v_xxiv})
are replaced by
another system of observables depending
only upon the gauge fields and their various topological ghosts. The new
observables -- denoted by ${\widetilde{\Omega}}_{(p,q)}$ -- are obtained from
${\Omega}_{(p,q)}$ by interchanging $\mbox{\rm q}$ and $\bar{\q}$. A simple calculation
leads to
\begin{eqnarray}\label{v_xxvi}
{\widetilde{\Omega}}_{(1,0)} & = & - \frac{c}{4{\pi}^2} \int_{\cal K}
\mbox{\rm Tr} \left\{
f^{(1,1)}\psi + f^{(2,0)} \bar{\psi} \right\}{\omega}_{(0,1)}
\; ; \nonumber \\
{\widetilde{\Omega}}_{(0,1)} & = & - \frac{c}{4{\pi}^2} \int_{\cal K}
\mbox{\rm Tr} \left\{
f^{(1,1)}\bar{\psi} + f^{(0,2)} \psi \right\}
{\omega}_{(1,0)} \; ; \\
{\widetilde{\Omega}}_{(2,0)} & = & \frac{c}{4{\pi}^2} \int_{\cal K}
\mbox{\rm Tr} \left\{
\mbox{${1\over 2}$} {\psi}^2 - \mbox{\rm i} f^{(2,0)} \varphi \right\} {\omega}_{(0,2)}
\; ; \nonumber \\
{\widetilde{\Omega}}_{(0,2)} & = & \frac{c}{4{\pi}^2} \int_{\cal K}
\mbox{\rm Tr} \left\{
\mbox{${1\over 2}$}{\bar{\psi}}^2 - \mbox{\rm i} f^{(0,2)} \varphi \right\}
{\omega}_{(2,0)} \; , \nonumber
\end{eqnarray}
while the other observables obviously are not affected.
For the proof let us write the generating functional of~(\ref{v_xxv}) in the
form (${\alpha}_q$, ${\beta}_p$ are arbitrary numbers)
\begin{equation}\label{v_xxvii}
\vev{ \exp \left\{ \sum_q {\alpha}_q \left( A_q + \bar{\q} B_q \right) +
\sum_p {\beta}_p \left( C_p + \mbox{\rm q} D_p \right) \right\} } \;
\end{equation}
where $A_q + \bar{\q} B_q$,
$C_p + \mbox{\rm q} D_p$ are the Donaldson integrated observables~(\ref{v_xxiv}). Being
invariant under both $\mbox{\rm q}$ and $\bar{\q}$ they obey
\begin{equation}\label{v_xxviii}
\bar{\q} A_q = \mbox{\rm q} C_p = 0 \; ; \quad \mbox{\rm q} A_q = \bar{\q} \mbox{\rm q} B_q \; ; \quad \bar{\q}
C_p = \mbox{\rm q} \bar{\q} D_p \; .
\end{equation}
Of course, for some $q$ or $p$ one can have $B_q = 0$ or $D_p = 0$.
Let us deform~(\ref{v_xxvii}) continuosly into the generating functional of
the new observables ${\widetilde{\Omega}}_{(p,q)}$ by means of
\begin{equation}\label{v_xxix}
\vev{ \exp \left\{ \sum_q {\alpha}_q \left[ A_q + \left( (1 - u)\bar{\q} + u \mbox{\rm q}
\right) B_q \right] +
\sum_p {\beta}_p \left[ C_p + \left( (1 - u)\mbox{\rm q} + u \bar{\q} \right) D_p \right]
\right\} } \;
\end{equation}
where $u$ is a real parameter in the interval $[ 0, 1 ]$. By differentiating
with respect to $u$ and by using the conditions~(\ref{v_xxviii}) one can show
that~(\ref{v_xxix}) does not depend on $u$. The generating functional of the
new observables is obtained from~(\ref{v_xxix}) by taking $u = 1$. Due to the
$u$ independence it coincides, however, with~(\ref{v_xxvii}). The integrand of
any correlation function has been transformed into an expression depending
only on the variables $a$, $\bar a$, their topological ghosts $\psi$,
$\bar{\psi}$ and the ghost for ghosts $\varphi$. Hence we proved
\begin{equation}\label{v_xxx}
\vev{\prod_{i} {\Omega}_{(p_i,q_i)}} = \vev{\prod_{i}
{\widetilde{\Omega}}_{(p_i,q_i)}} \; .
\end{equation}
The evaluation of the correlation function proceeds along the same lines as
for TYM with a single fermionic symmetry \cite{I}, by taking advantage of
working with the action
\begin{equation}\label{v_xxxi}
\frac{1}{8} \mbox{\rm q} \bar{\q} \int_{\cal K} \mbox{\rm Tr} \left( \chi \bar{\chi} - \mbox{\rm i} {\gamma}^2
\lambda f \right) \;
\end{equation}
which differs from (2.10) by a $\mbox{\rm q} \bar{\q} $-\penalty10000\zhs term.
One can now integrate out all non-\penalty10000\zhs zero modes. It is usually assumed that
there are no zero modes in the variables $\chi$, $\bar{\chi}$, $b$ and $\bar
b$.
There is a
special prescription for handling the ghost for ghosts: The field $\varphi$ has
to be replaced by the solution $\vev{\varphi}$ of the differential equation
\cite{Park}
\begin{equation}\label{v_xxxii}
g^{\bar n m} \left( \{ \mbox{\rm D}_m , \mbox{\rm D}_{\bar n} \} \vev{\varphi} + 2 \mbox{\rm i} \{ {\psi}_m
, {\psi}_{\bar n} \} \right) = 0 \;
\end{equation}
where ${\psi}_m$, ${\psi}_{\bar m}$ are the zero modes of the topological
ghosts.
In solving eq.~(\ref{v_xxxii}) one can meet zero modes of $\varphi$,
for which the procedure of ref.~\cite{Anse} should be extended to the
K\"{a}hler case. For simplicity we shall assume in the following that
also such zero modes are absent.
The path integral measure takes its canonical form $[\mbox{\rm d} a] [\mbox{\rm d} \bar a] [\mbox{\rm d}
\psi] [\mbox{\rm d}\bar{\psi}]$ where $a$ and $\bar a$ are solutions of the self-\penalty10000\zhs
duality conditions:
\begin{equation}\label{v_xxxiii}
f_{mn} = f_{\bar m \bar n} = g^{\bar n m} f_{m \bar n} = 0 \; .
\end{equation}
Since $\psi$ and $\bar{\psi}$ are the zero modes of the topological ghosts,
they obey the following equations of motion:
\begin{equation}\label{v_xxxiv}
\mbox{\rm D}_{[m} {\psi}_{n]} = \mbox{\rm D}_{[ \bar m } {\psi}_{\bar n ]} = \mbox{\rm D}_m {\psi}^m = \mbox{\rm D}^m
{\psi}_m = 0 \; .
\end{equation}
One can show \cite{I}, \cite{Park} that instanton deformations orthogonal to
purely
gauge transformations obey identical equations.
Hence $\psi$ and $\bar{\psi}$ are tangent vectors to the instanton moduli
space ${\cal M}$. Since $\mbox{\rm q}$, $\bar{\q}$ relate $a$, $\bar a$ to $\psi$,
$\bar{\psi}$, they play the role of exterior derivatives on ${\cal M}$.
The integration of $\psi$, $\bar{\psi}$ is straightforward and transforms the
integrand into a wedge product of $(p_i,q_i )$-\penalty10000\zhs forms over the moduli space
\begin{equation}\label{v_xxxv}
\vev{\prod_{i} {\Omega}_{(p_i,q_i)} } = \int_{\cal M} \prod_{i}
{\Phi}_{(p_i,q_i)} \; .
\end{equation}
In writing down eq.~(\ref{v_xxxv}) we assumed that ${\cal M}$ can be
considered a finite dimensional K\"{a}hler manifold \cite{Koba}.
One can now establish a selection rule for the correlation functions as given
by~(\ref{v_xxxv}). The action ${\cal S}$ needed
for computing the left hand side is invariant under the global Abelian symmetry
$\mbox{\rm I}\!\mbox{\rm R}_{+}\otimes \mbox{\rm I}\!\mbox{\rm R}_{-}$. In contrast the integration measure transforms
under $\mbox{\rm I}\!\mbox{\rm R}_{\pm}$ with certain weights that are equal and exactly compensate
the dimension of ${\cal M}$. Therefore ${\Omega}_{(p_i,q_i)}$ should provide
the compensating total weights
\begin{equation}\label{v_xxxvi}
\sum_{i} p_i = \sum_{i} q_i = \dim {\cal M} \; .
\end{equation}
This means that the integrand of~(\ref{v_xxxv}) is a top-\penalty10000\zhs form, {i.e.}\ a $(
\dim {\cal M}, \dim {\cal M} )$-\penalty10000\zhs form over ${\cal M}$.
The careful reader may have noticed that the correlation functions were defined
by using the action (2.10) in which the BRS gauge fixing has been neglected.
The importance of the BRS gauge fixing conditions both for interpreting and
computing Donaldson invariants has been emphasized for TYM with a single
fermionic charge in \cite{XI}.
In our case the BRS gauge fixing appears in the total action $\widetilde{\cal
S} + {\cal S}'$ suggesting its use in defining the correlation functions.
Since the eqs.~(\ref{v_xiv}) relating $\chi$, $\bar{\chi}$ to $f^{(2,0)}$ and
$f^{(0,2)}$ respectively, now become equations of motion, we cannot start from
the old observables ${\Omega}_{(p,q)}$, but rather from the new ones
${\widetilde{\Omega}}_{(p,q)}$. After performing the functional integration
over the chiral superfields $M_{mn}$, $M_{\bar m \bar n}$ one recovers the
full system of eqs.~(\ref{v_xiv}) and one can infer that~(\ref{v_xxx}) still
holds (albeit with a gauge fixed action). This is important in order to make
sure that we are discussing the correlation functions of the solution to the
cohomology problem of $\mbox{\rm q}$ and $\bar{\q}$.
It is possible to develop an analysis for TYM with two fermionic charges
similar to that performed in \cite{XI} in order to show that the BRS gauge
fixing in superspace is equivalent to Witten's method of computing the
correlation functions. Let us write the gauge-\penalty10000\zhs fixing action in the form
\begin{equation}\label{v_xxxvii}
-\mbox{${1\over 4}$}\,\mbox{\rm s}\int_{\cal K}\gamma\left\{\mbox{\rm q}\,\mbox{\rm Tr}\, d\bar\partial u
- \bar{\q}\,\mbox{\rm Tr}\, d^{\cal y}\partial\bar u\right\}\;.
\end{equation}
Here $d$ and $d^{\cal y}$ are the first components of the chiral superfields $D$
and $D^{\cal y}$. The $(1,0)$ and $(0,1)$ forms
\begin{equation}\label{v_xxxviii}
u = u_m \mbox{\rm d} z^m\;;\qquad \bar u = u_{\bar m}\mbox{\rm d} z^{\bar m}
\end{equation}
are constructed from the chiral connection superfields
\begin{equation}\label{v_xxxix}
\phi_m = u_m + \theta\pi_m\;;\qquad \phi_{\bar m} = u_{\bar m} +
\bar{\theta}{\pi}_{\bar m} \: .
\end{equation}
In view of the above gauge fixing term one can start from the following path
integral for the correlation functions
\begin{equation}\label{v_xl}
\vev{{\cal O}} = \int [\mbox{\rm d} \hat\mu][\mbox{\rm d} V]\,{\cal O}(\hat\mu,V)\,
\exp{\left\{-\frac{1}{e^2}{\cal S}[\hat\mu,V]\right\}}
\delta(\nabla^m\phi_m)
\delta(\nabla^{\bar m}\phi_{\bar m})
\hat\Delta(\phi,\bar\phi)\;
\end{equation}
where $\hat\mu$ represents the collection of superfields $\phi_m$, $\phi_{\bar
m}$, $\Lambda$, $X_{mn}$, $X_{\bar m\bar n}$; ${\cal O}(\hat\mu,V)$ denotes a
gauge invariant function (a product of Donaldson polynomials) and
\begin{equation}\label{v_xli}
\hat\Delta^{-1}(\phi,\bar\phi)
= \int_{\cal G}[\mbox{\rm d} g]\,\delta(\nabla^m\phi_m^g)
\delta(\nabla^{\bar m}\phi_{\bar m}^g)
\end{equation}
is the Faddeev--\penalty10000\zhs Popov (super)determinant.
From now on we will express all superfields by components. We would like to
consider the gauge group ${\cal G}$ consisting of chiral transformations with
the parameters $\eta$ and $\eta^{\cal y}$ acting on the field components as follows:
\begin{equation}\label{v_xlii}
\begin{array}{rclrcl}
u'_m &=& \mbox{\rm e}^{-\eta} (u_m + \nabla_m) \mbox{\rm e}^\eta\;;\quad&
u'_{\bar m} &=& \mbox{\rm e}^{\eta^{\cal y}} (u_{\bar m} + \nabla_{\bar m})
\mbox{\rm e}^{-\eta^{\cal y}}\;;\\
\pi'_m &=& \mbox{\rm e}^{-\eta} \pi_m \mbox{\rm e}^\eta\;;\quad&
\pi'_{\bar m} &=& \mbox{\rm e}^{\eta^{\cal y}} \pi_{\bar m}
\mbox{\rm e}^{-\eta^{\cal y}}\;;\\
\mbox{\rm e}^{v'} &=& \mbox{\rm e}^{\eta{\cal y}} \mbox{\rm e}^v \mbox{\rm e}^\eta\;
\end{array}
\end{equation}
where $v$ is the first component of the superfield $V$. The components of
non-\penalty10000\zhs chiral gauge superfields transform according to unitary transformations
generated by $h^{\cal y}=-h$
\begin{equation}\label{v_xliii}
\begin{array}{rclrcl}
a'_m &=& \mbox{\rm e}^{-h} (a_m + \nabla_m) \mbox{\rm e}^h\;;\quad&
a'_{\bar m} &=& \mbox{\rm e}^{-h} (a_{\bar m} + \nabla_{\bar m})
\mbox{\rm e}^h\;;\\
\pi'_m &=& \mbox{\rm e}^{-h} \pi_m \mbox{\rm e}^h\;;\quad&
\pi'_{\bar m} &=& \mbox{\rm e}^{-h} \pi_{\bar m}
\mbox{\rm e}^h\;;\\
\omega' &=& \mbox{\rm e}^{-h} \omega \mbox{\rm e}^h\;;\quad&
\bar\omega' &=& \mbox{\rm e}^{-h} \bar\omega \mbox{\rm e}^h\;;\\
\varphi' &=& \mbox{\rm e}^{-h}\varphi\mbox{\rm e}^h\;.
\end{array}
\end{equation}
(As can be seen from (2.17) $h$ depends highly non-\penalty10000\zhs trivially upon $\eta$,
$\eta^{\cal y}$, and $v$. The components of $\Lambda$, $X_{mn}$, and
$X_{\bar m\bar n}$ transform similarly, but they are not of interest for us
here.)
Finally, the matrix
$\mbox{\rm e}^{\frac{v}{2}}$ transforms in one of the following equivalent ways
\begin{equation}\label{v_xliv}
\mbox{\rm e}^{\frac{v'}{2}} = \mbox{\rm e}^{\eta^{\cal y}}\mbox{\rm e}^{\frac{v}{2}}\mbox{\rm e}^h
= \mbox{\rm e}^{-h}\mbox{\rm e}^{\frac{v}{2}}\mbox{\rm e}^{\eta}
\end{equation}
and serves to relate components of chiral and of gauge superfields
\begin{equation}\label{v_xlv}
\begin{array}{rclrcl}
u_m &=& \mbox{\rm e}^{-\frac{v}{2}} (a_m + \nabla_m) \mbox{\rm e}^{\frac{v}{2}}\;;\quad&
u_{\bar m} &=& \mbox{\rm e}^{\frac{v}{2}} (a_{\bar m} + \nabla_{\bar m})
\mbox{\rm e}^{-\frac{v}{2}}\;;\\
\pi_m &=& {\cal D}_m\omega
+ \mbox{\rm e}^{-\frac{v}{2}}\left(\psi_m - \mbox{\rm D}_m\omega\right)\mbox{\rm e}^{\frac{v}{2}}\;;\quad&
\pi_{\bar m} &=& {\cal D}_{\bar m}\bar\omega
+ \mbox{\rm e}^{\frac{v}{2}}\left(\psi_{\bar m} - \mbox{\rm D}_{\bar m}\bar\omega\right)
\mbox{\rm e}^{-\frac{v}{2}}\;.
\end{array}
\end{equation}
Let us now perform a chiral transformation on the path integral measure and on
the integrand of~(\ref{v_xl}). Everything but the $\delta$-\penalty10000\zhs function is
invariant under such a transformation.
If one chooses the chiral transformation such that $v'=0$, one gets from
~(\ref{v_xlii})--(\ref{v_xliv}) the relations
\begin{equation}\label{v_xlvi}
\begin{array}{rclrcl}
\mbox{\rm e}^\eta &=& \mbox{\rm e}^{-\frac{v}{2}}\mbox{\rm e}^h\;;\quad&
\mbox{\rm e}^{\eta^{\cal y}} &=& \mbox{\rm e}^{-h}\mbox{\rm e}^{-\frac{v}{2}}\;;\\
u'_m &=& a'_m\;;\quad&
u'_{\bar m} &=& a'_{\bar m}\;;\\
\pi'_m &=& \psi'_m\;;\quad&
\pi'_{\bar m} &=& \psi'_{\bar m}\;.
\end{array}
\end{equation}
One can show that
\begin{eqnarray}\label{v_lii}
V & = & 2 \theta {\omega}' - 2 \bar{\theta}{\bar{\omega}}' + \theta
\bar{\theta} \left( \mbox{\rm i} {\varphi}' - 2 \{ {\omega}',
{\bar{\omega}}' \} \right) \; ; \nonumber \\
{\phi}_m & = & {a'}_m + \theta{{\psi}'}_m \; ; \qquad {\phi}_{\bar m} =
{a'}_{\bar m} + \bar{\theta}{{\psi}'}_{\bar m} \;
\end{eqnarray}
is the supersymmetry gauge in which $\mbox{\rm q} = {\partial}_{\theta}$ and $\bar{\q} =
{\partial}_{\bar{\theta}}$ have the action given by~(\ref{v_vi}).
Now we perform the change of field variables
\begin{equation}\label{v_xlvii}
\begin{array}{rclrcl}
u_m &\to& a_m\;;\quad&
u_{\bar m} &\to& a_{\bar m}\;;\\
\pi_m &\to& \psi_m\;;\quad&
\pi_{\bar m} &\to& \psi_{\bar m}\;,
\end{array}
\end{equation}
so that any dependence of $v$, $\omega$, and
$\bar\omega$ in ${\cal S}$ and ${\cal O}$ disappears and moreover the
corresponding Jacobians are equal to one. The system of variables
$\hat\mu$ is replaced by $\mu$. One can easily check that
\begin{equation}\label{v_xlviii}
\begin{array}{rl}
\vrule height 8mm width 0pt \displaystyle\vev{{\cal O}} = \int [\mbox{\rm d}\mu] & \vrule height 8mm width 0pt \displaystyle {\cal O}(\mu)\,\exp{\left\{-
\frac{1}{e^2}{\cal S}[\mu]\right\}} \\
&\vrule height 8mm width 0pt \displaystyle \times \delta(\nabla^m a'_m)
\delta(\nabla^{\bar m} a'_{\bar m})
\delta(\nabla^m \psi'_m)
\delta(\nabla^{\bar m} \psi'_{\bar m})
\hat{\Delta}(a,\psi;\bar a,\bar\psi)\;
\end{array}
\end{equation}
where the new Faddeev--\penalty10000\zhs Popov (super)determinant is obtained by integrating
over the unitary subgroup generated by $h$:
\begin{equation}\label{v_xlix}
\begin{array}{rcl}
\vrule height 8mm width 0pt \displaystyle\hat{\Delta}^{-1}(a,\psi;\bar a,\bar\psi) &=& \vrule height 8mm width 0pt \displaystyle \int [\mbox{\rm d} h]\,
\delta(\nabla^m a'_m)
\delta(\nabla^{\bar m} a'_{\bar m}) \\
&&\vrule height 8mm width 0pt \displaystyle \times \delta(\nabla^m \psi'_m)
\delta(\nabla^{\bar m} \psi'_{\bar m}) \; .
\end{array}
\end{equation}
The above considerations lead to
the following path integral for the correlation functions:
\begin{equation}\label{v_l}
\vev{{\cal O}} = \int [\mbox{\rm d}\mu][\mbox{\rm d}\nu]\, {\cal O}(\mu)\,
\exp{\left\{-\frac{1}{e^2}{\cal S}[\mu,\nu]\right\}}\;
\end{equation}
where
\begin{equation}\label{v_li}
\begin{array}{rcl}
{\cal S}[\mu,\nu] &=&\vrule height 8mm width 0pt \displaystyle\mbox{${1\over 4}$}\int_{\cal K}\Bigl\{\mbox{${1\over 2}$}\mbox{\rm q}\bar{\q}\,\mbox{\rm Tr} \left(
\chi \bar{\chi} - \mbox{\rm i} {\gamma}^2 \lambda f \right)\Bigr.\\
&&\vrule height 8mm width 0pt \displaystyle\Bigl. -\,\mbox{\rm s} \gamma \left[\mbox{\rm q}\,\mbox{\rm Tr}\, d\, \bar{\mbox{\rm D}}_0(a-a_0) - \bar{\q}\,\mbox{\rm Tr}\,
d^{\cal y}\,\mbox{\rm D}_0(\bar a - \bar a_0)\right]\Bigr\}\;
\end{array}
\end{equation}
and $\nu$ stands for the fields $d$, $d^{\cal y}$ as well as all the fields obtained
from them by applying $\mbox{\rm s}$, $\mbox{\rm q}$, and $\bar{\q}$. We also introduced the background
gauge field forms $a_0$, $\bar a_0$ and the corresponding covariant
differentials $\mbox{\rm D}_0$, $\bar{\mbox{\rm D}}_0$ upon which $\mbox{\rm s}$, $\mbox{\rm q}$, and $\bar{\q}$ act trivially.
The similarity of~(\ref{v_l}) with the expression used in~\cite{XI} shows that
the prescription to evaluate the Donaldson invariants can be derived from the
standard (with gauge fixed action) path integral also for TYM with two
fermionic charges. Of course, the prescription coincides with that obtained by
neglecting the gauge fixing term.
A first systematic attempt to compute Donaldson invariants of smooth, oriented,
compact four-\penalty10000\zhs manifolds has been given by Kronheimer and Mrowka \cite{KM}.
They showed that the Donaldson invariants of the so-\penalty10000\zhs called manifols of
simple type
exhibit universal relations. It has been conjectured \cite{KM}, \cite{Mo} that
all simply-\penalty10000\zhs connected four-\penalty10000\zhs manifolds with $b_2^{+}$ ( $b_2^{+}$
is the number of independent self-\penalty10000\zhs dual forms ) are of simple type.
Subsequently, almost all $SU(2)$
and $SO(3)$ Donaldson invariants for K\"ahler four-\penalty10000\zhs manifolds of
simple type with
$b_2^{+} = \dim {\cal H}^{(1,1)}({\cal K};\partial ,\bar{\partial}) > 1$
have been calculated by Witten \cite{III}, making use of the known infrared
behaviour of $N=1$ supersymmetric gauge theories. On the other hand,
precise formulas relating the Donaldson invariants to Seiberg--\penalty10000\zhs Witten
invariants ( for a review see \cite{mm} ) have been conjectured in \cite{Mo}.
In sharp contrast to Donaldson
invariants, which are defined on the moduli space of instantons, Seiberg--\penalty10000\zhs
Witten invariants are associated to moduli spaces of abelian monopoles.
The Seiberg--\penalty10000\zhs Witten theory is a powerful method which allows
the calculation of all Donaldson invariants in case of K\"ahler manifolds of
simple type as mentioned above.
In order to make contact with
the present work we point out that the manifolds of simple type can only
have correlation functions of the observables ${\Omega}_{(1,1)}$ and
${\Omega}_{(2,2)}$.
A different approach based on the holomorphic Yang--\penalty10000\zhs Mills theory \cite{Holo}
has been proposed in \cite{Hyun} and used for computing correlation
functions of the product ${\Omega}_{(2,0)}{\Omega}_{(0,2)}$.
Concerning the mathematical literature we refer to \cite{FS} where
the Donaldson invariants are obtained by means of the so-called blowup
formula. A first step in proving the formulas conjectured by Witten \cite{Mo}
has been made in \cite{PT}.
Finally let us mention two papers \cite{HP}, \cite{Sd} where the question of
computing Donaldson invariants for four-\penalty10000\zhs manifolds with $b_2^{+} \leq 1$
is raised.
Some results of this section have been obtained in refs.\ \cite{Park},
\cite{Holo}. They concern Donaldson observables with equal ghost numbers
${\Omega}_{(p,p)}$. Here we included the off-\penalty10000\zhs diagonal Donaldson
observables, thereby completing the interpretation of the fermionic charges as
complex derivations on the instanton moduli space.
\sect{Conclusions}
In the present paper we formulated TYM theory with two fermionic charges on the
superspace consisting of a K\"{a}hler four-\penalty10000\zhs manifold and two Grassmann
variables. In contrast to TYM theory with a single fermionic charge, we had to
impose certain constraints in superspace. We solved the constraints and showed
that the gauge transformations were replaced by local chiral transformations.
Then we elucidated the structure of the Faddeev--\penalty10000\zhs Popov ghost sector and
determined
the total action.
Furthermore, we used the action for perturbatively computing
the (K\"{a}hler) gravitational contribution to the dimension of the instanton
moduli space. In performing the calculation we showed how the covariant
point-\penalty10000\zhs splitting technique can be extended to K\"{a}hler manifolds.
Insisting on
the specific form taken by the local conservation law on K\"{a}hler manifolds
we discussed in some detail the global symmetries of the action.
We showed that the associated currents representing locally these symmetries
(en\-er\-gy-\penalty10000\zhs momentum tensor, fermionic and antisymmetry currents)
are highest components of gauge invariant superfields. BRS symmetry does not
alter this property, while the irreducibility of the multiplets is
sometime lost. In any case, all their correlation functions vanish.
In addition we also determined the non-\penalty10000\zhs trivial observables.
They are cohomology classes
of both fermionic symmetry operations. Some of the classes involve additional
fields, absent in TYM with a single fermionic charge. Nevertheless, we could
show that the correlation functions of all non-\penalty10000\zhs trivial observables can be
represented as integrals of top-\penalty10000\zhs forms over the instanton moduli space.
\section*{Acknowledgements}
We would like to thank Friedemann Brandt for his comments on the
algebraic cohomology problem. One of us (S.\ M.) would like to acknowledge
useful discussions with Oleg Ogievetsky and Raymond Stora.
|
2,877,628,090,825 | arxiv | \section{Introduction}
Let $x, y \in \mathbb{C}^k$ be $k$-dimensional complex-valued vectors.
We denote their inner product as
\[
\inner{x}{y} = \sum_{i=1}^k x_i^* y_i \, .
\]
Now suppose we have a directed graph $G=(V,E)$. Let us associate a vector $x_v \in \mathbb{C}^k$ with each vertex $v$, and consider the product over all edges $(u,v)$ of the inner products of the corresponding vectors:
\begin{equation}
\label{eq:prod}
\prod_{(u,v) \in E} \inner{x_u}{x_v} \, .
\end{equation}
For instance, for the graph in Figure~\ref{fig:example} this product is
\begin{equation}
\label{eq:prod-example}
\prod_{(u,v) \in E} \inner{x_u}{x_v}
= \inner{x_1}{x_2} \inner{x_2}{x_3} \inner{x_3}{x_1} \abs{\inner{x_3}{x_4}}^2 \, .
\end{equation}
The expectation of this product, where each $x_v$ is chosen independently and uniformly from the set of vectors in $\mathbb{C}^k$ of norm $1$, is a type of moment, where each $x_v$ appears with order $d^{\rm in}_v+d^{\rm out}_v$. It is a function of the graph $G$ and the dimension $k$, which we denote as follows:
\[
q(G;k) = \mathop\mathrm{Exp}_{\{x_v\}} \prod_{(u,v) \in E} \inner{x_u}{x_v} \, .
\]
\begin{figure}
\begin{center}
\includegraphics[width=1.8in]{example}
\end{center}
\caption{A little directed graph. We use the edge labels in the proof of Theorem~\ref{thm:main}.}
\label{fig:example}
\end{figure}
A simple observation is that $q(G;k)$ is zero unless $G$ is Eulerian---that is, unless $d^{\rm in}_v = d^{\rm out}_v$ for each vertex $v$. Since $x_v$ appears in the product $d^{\rm in}_v$ times unconjugated and $d^{\rm out}_v$ times conjugated, multiplying $x_v$ by ${\rm e}^{i\theta}$ multiplies $q(G;k)$ by ${\rm e}^{i\theta (d^{\rm in}_v-d^{\rm out}_v)}$. But multiplying by a phase preserves the uniform measure, so the expectation is zero if $d^{\rm in}_v \ne d^{\rm out}_v$ for any $v$.
So, let us suppose that $G$ is Eulerian. In that case, what is $q(G;k)$? Does it have a combinatorial interpretation? And how difficult is it to calculate?
Our main result is this:
\begin{theorem}
\label{thm:main}
For any $k \ge 2$, computing $q(G;k)$, given $G$ as input, is $\#P$-hard under Turing reductions.
\end{theorem}
\noindent
If we extend $\#P$ to rational functions in the natural way, then we can replace $\#P$-hardness in this theorem with $\#P$-completeness.
Our proof is very simple; we show that $q(G;k)$ is essentially identical to an existing graph polynomial, which is known to be $\#P$-hard to compute. Along the way, we will meet some nice combinatorics, and glancingly employ the representation theory of the unitary and orthogonal groups.
\section{The circuit partition polynomial}
A \emph{circuit partition} of $G$ is a partition of $G$'s edges into circuits. Let $r_t$ denote the number of circuit partitions containing $t$ circuits; for instance, $r_1$ is the number of Eulerian circuits. The \emph{circuit partition polynomial} $j(G;z)$ is the generating function
\begin{equation}
\label{eq:j}
j(G;z) = \sum_{t=1}^\infty r_t z^t \, .
\end{equation}
For instance, for the graph in Figure~\ref{fig:example} we have $j(G;z) = z+z^2$. This polynomial was first studied by Martin~\cite{martin}, with a slightly different parametrization; see also~\cite{arratia,bollobas,bouchet,ellis,jaeger,vergnas79,vergnas88}.
Now consider the following theorem.
\begin{theorem}
\label{thm:q-j}
For any Eulerian directed graph $G=(V,E)$,
\begin{equation}
\label{eq:q-j}
q(G;k) =
\left( \prod_{v \in V} \frac{(k-1)!}{(k+d_v-1)!} \right)
j(G;k) \, ,
\end{equation}
where $d_v$ denotes $d^{\rm in}_v = d^{\rm out}_v$.
\end{theorem}
\begin{proof}
Given a vector $x \in \mathbb{C}^k$ and an integer $d$, the outer product of $x^{\otimes d} = \underbrace{x \otimes \cdots \otimes x}_{d\ {\rm times}}$ with itself is a tensor of rank $2d$, or equivalently a linear operator on $(\mathbb{C}^k)^{\otimes d}$:
\[
\ket{x^{\otimes d}} \bra{x^{\otimes d}} = \big( \ket{x} \bra{x} \big)^{\otimes d} \, .
\]
In terms of indices, we can write
\[
\ket{x^{\otimes d}} \bra{x^{\otimes d}}^{\alpha_1 \alpha_2 \cdots \alpha_d}_{\beta_1 \beta_2 \cdots \beta_d}
= \prod_{\ell=1}^d x_{\alpha_\ell} x_{\beta_\ell}^* \, .
\]
Then $\prod_{(u,v) \in E} \inner{x_u}{x_v}$ is a contraction of the product of these tensors, where upper and lower indices correspond to incoming and outgoing edges respectively. For instance, for the graph in Figure~\ref{fig:example} we can rewrite the product~\ref{eq:prod-example} as
\[
\prod_{(u,v) \in E} \inner{x_u}{x_v}
= \ket{x_1} \bra{x_1}^\gamma_\alpha
\; \ket{x_2} \bra{x_2}^\alpha_\beta
\; \ket{x_3 \otimes x_3} \bra{x_3 \otimes x_3}^{\beta \eta}_{\gamma \delta}
\; \ket{x_4} \bra{x_4}^\delta_\eta \, .
\]
Here we use the Einstein summation convention, where any index which appears once above and once below is automatically summed from $1$ to $k$. Now, since the $x_v$ are independent for different $v$, we can compute $q(G;k)$ by taking the expectation over each $x_v$ separately. This gives a contraction of the tensors
\begin{equation}
\label{eq:xd-def}
X_d = \mathop\mathrm{Exp}_{x} \ket{x^{\otimes d}} \bra{x^{\otimes d}} \, ,
\end{equation}
where $d=d_v$, over all $v$.
In order to calculate $X_d$, we introduce some notation. Let $S_d$ denote the symmetric group on $d$ elements. We identify a permutation $\pi \in S_d$ with the linear operator on $(\mathbb{C}^k)^{\otimes d}$ which permutes the $d$ factors in the tensor product. That is,
\[
\pi \left( x_1 \otimes x_2 \otimes \cdots \otimes x_d \right) = x_{\pi(1)} \otimes x_{\pi(2)} \otimes \cdots \otimes x_{\pi(d)} \, ,
\]
or, using indices,
\[
\pi^{\alpha_1 \alpha_2 \cdots \alpha_d}_{\beta_1 \beta_2 \cdots \beta_d}
= \prod_{\ell=1}^d \delta^{\alpha_{\pi(\ell)}}_{\beta_\ell} \, ,
\]
where $\delta^i_j$ is the Kronecker delta operator, $\delta^i_j=1$ if $i=j$ and $0$ if $i \ne j$. Diagrammatically, $\pi$ is a gadget with $d$ incoming edges and $d$ outgoing edges, wired to each other according to the permutation $\pi$.
We have the following lemma:
\begin{lemma}
\label{lem:xd}
With $X_d$ defined as in~\eqref{eq:xd-def}, if $x$ is uniform in the set of vectors in $\mathbb{C}^k$ of norm $1$, then
\begin{equation}
\label{eq:xd}
X_d
= \frac{(k-1)!}{(k+d-1)!} \sum_{\pi \in S_d} \pi \, .
\end{equation}
\end{lemma}
\begin{proof}
First, $X_d$ is a member of the commutant of the group $U(k)$ of $k \times k$ unitary matrices, since these preserve the uniform measure. That is $X_d$ commutes with $U^{\otimes d}$ for any $U \in U(k)$. By Schur duality, the commutant is a quotient of the group algebra $\mathbb{C}[S_d]$; namely, the image of $\mathbb{C}[S_d]$ under the identification above. Thus $X_d$ is a superposition of permutations, $\sum_{\pi \in S_d} a_\pi \pi$.
We also have $X_d \pi = \pi X_d = X_d$ for any $\pi$. Thus $X_d$ is proportional to the uniform superposition on $S_d$, or equivalently the projection operator $\Pi_{\rm sym} = (1/d!) \sum_\pi \pi$ onto the totally symmetric subspace $V_{\rm sym}$ of $(\mathbb{C}^k)^{\otimes d}$. Since $\tr X_d = \mathop\mathrm{Exp}_x \abs{x}^{2d} = 1$ while $\tr \Pi_{\rm sym} = \dim V_{\rm sym}$, we have $X_d = (1/\dim V_{\rm sym}) \,\Pi_{\rm sym}$.
Finally, $\dim V_{\rm sym}$ is the number of ways to label the $d$ factors of the tensor product with basis vectors $\{e_1,\ldots,e_k\}$ in nondecreasing order---or, for aficionados, the number of semistandard tableaux with one row of length $d$ and content ranging from $1$ to $k$. This gives $\dim V_{\rm sym} = {k+d-1 \choose d}$.
To illustrate some ideas that will recur in the next section, we give an alternate proof. First, note that $\tr \pi$ is the number of ways to label each of $\pi$'s cycles with a basis vector ranging from $1$ to $k$, or $k^{c(\pi)}$ where $c(\pi)$ denotes the number of cycles (including fixed points). Thus
\begin{equation}
\label{eq:sd-genfunc}
\tr \sum_{\pi \in S_d} \pi
= \sum_{\pi \in S_d} k^{c(\pi)} \, .
\end{equation}
To compute this generating function, we use the fact that each permutation $\pi \in S_d$ appears once in the following product, where $1$ denotes the identity permutation, and $\tau_{ij}$ denotes the transposition of the $i$th and $j$th object:
\begin{equation}
\label{eq:sd-sum}
\sum_{\pi \in S_d} \pi = 1 (1 + \tau_{1,2}) (1+\tau_{1,3}+\tau_{2,3}) \cdots (1+\tau_{1,d}+\tau_{2,d}+\cdots+\tau_{d-1,d}) \, .
\end{equation}
This product works by describing a permutation $\pi_t$ of $t$ objects inductively as a permutation $\pi_{t-1}$ of the first $t-1$ objects, composed either with the identity or with a transposition swapping the $t$th object with one of the previous $t-1$. If we apply the identity, then the $t$th object is a fixed point, and $c(\pi_t) = c(\pi_{t-1})+1$, gaining a factor of $k$ in~\eqref{eq:sd-genfunc}; but if we apply a transposition, then $c(\pi_t)=c(\pi_{t-1})$. Thus~\eqref{eq:sd-sum} becomes
\[
\sum_{\pi \in S_d} k^{c(\pi)} = k (k+1) (k+2) \cdots (k+d-1) = \frac{(k+d-1)!}{(k-1)!} \, .
\]
Comparing traces again gives~\eqref{eq:xd}.
\end{proof}
All that remains is to interpret the operators $X_{d_v}$, and their contraction, diagrammatically. Lemma~\ref{lem:xd} tells us that, for each vertex $v$ of $G$, taking the expectation over $x_v$ converts it to a sum over all $d_v!$ ways to wire the incoming edges to the outgoing edges. But doing this at each vertex gives us a sum over all cycle partitions of $G$. Contracting these tensors gives the number of ways to label each cycle in a each partition with a basis vector ranging from $1$ to $k$, so each cycle contributes a factor of $k$. Along with the scaling factor in~\eqref{eq:xd}, this completes the proof.
\end{proof}
Next we show that the cycle partition polynomial is $\#P$-hard. To our knowledge, the following theorem first appeared in~\cite{ellis-sarmiento}; we prove it here for completeness.
\begin{theorem}
\label{thm:nump}
For any fixed $z > 1$, computing $j(G;z)$ from $G$ is $\#P$-hard under Turing reductions.
\end{theorem}
\begin{proof}
Recall that the \emph{Tutte polynomial} of an undirected graph $G=(V,E)$ can be written as a sum over all subsets $S$ of $E$,
\begin{equation}
\label{eq:tutte}
T(G;x,y) = \sum_{S \subseteq E} (x-1)^{c(S)-c(G)} \,(y-1)^{c(S)+\abs{S}-n} \, .
\end{equation}
Here $c(G)$ denotes the number of connected components in $G$. Similarly, $c(S)$ denotes the number of connected components in the spanning subgraph $(V,S)$, including isolated vertices. When $x=y$, we have
\begin{equation}
\label{eq:tutte2}
T(G;x,x) = \sum_{S \subseteq E} (x-1)^{c(S)+\ell(S)-c(G)} \, ,
\end{equation}
where $\ell(S) = c(s)+\abs{S}-n$ is the total excess of the components of $S$, i.e., the number of edges that would have to be removed to make each one a tree.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{medial}
\end{center}
\caption{A planar graph $G$ (black) and its oriented medial graph $G_m$ (gray).}
\label{fig:medial}
\end{figure}
If $G$ is planar, then we can define a directed medial graph $G_m$ as in Figure~\ref{fig:medial}. Each vertex of $G_m$ corresponds to an edge of $G$, edges of $G_m$ correspond to shared vertices in $G$, and we orient the edges of $G_m$ so that they go counterclockwise around the faces of $G$. Each vertex of $G_m$ has $d^{\rm in} = d^{\rm out} = 2$, so $G_m$ is Eulerian.
The following identity is due to Martin~\cite{martin}; see also~\cite{vergnas79}, or~\cite{austin} for a review.
\begin{equation}
\label{eq:martin}
j(G_m;z) = z^{c(G)} \,T(G;z+1,z+1) \, .
\end{equation}
We prove this using a one-to-one correspondence between subsets $S \subseteq E$ and circuit partitions of $G_m$. Let $v$ be a vertex of $G_m$, corresponding to an edge $e$ of $G$. Then the circuit partition connects each of $v$'s incoming edge to the outgoing edge on the same side of $e$ if $e \in S$, and crosses over to the other side if $e \notin S$. It is easy to prove by induction that the number of circuits is then $c(S)+\ell(S)$, in which case~\eqref{eq:tutte2} yields~\eqref{eq:martin}.
The theorem then follows from the fact, proven by Vertigan~\cite{vertigan}, that the Tutte polynomial for planar graphs is $\#P$-hard under Turing reductions, except on the hyperbolas $(x-1)(y-1) \in \{1, 2\}$ or when $(x,y) \in \{(1,1), (-1,-1), (\omega,\omega^*), (\omega^*,\omega)\}$ where $\omega = {\rm e}^{2\pi i /3}$. Thus computing $j(G;z)$ for any $z > 1$ is $\#P$-hard, even in the special case where $G$ is planar and where every vertex has $d^{\rm in}=d^{\rm out}=2$.
\end{proof}
\section{Real-valued vectors}
We can also consider the case where the $x_v$ are real-valued, and are chosen uniformly from the set of vectors in $\mathbb{R}^k$ of norm $1$. In this case, the inner product $\inner{x_u}{x_v}$ becomes symmetric, so the graph $G$ becomes undirected. We might then expect $q(G;k)$ to be related to the circuit partition polynomial for undirected circuits, and indeed this is the case.
We again wish to compute the tensor $X_d = \mathop\mathrm{Exp}_x \ket{x^{\otimes d}} \bra{x^{\otimes d}}$. First, let $M_d$ denote the set of perfect matchings of $2d$ objects; note that
\[
\abs{M_d} = (2d-1)!! = (2d-1)(2d-3) \cdots 5 \cdot 3 \cdot 1 = \frac{(2d)!}{2^d d!} \, .
\]
We can identify each matching $\mu \in M_d$ with a linear operator on $(\mathbb{R}^k)^{\otimes d}$, where the first $d$ objects correspond to upper indices, and the last $d$ correspond to lower indices. However, in addition to permutations that wire upper indices to lower ones with a bipartite matching, we now also have ``cups'' and ``caps'' that wire two upper indices, or two lower indices, to each other. For instance, if $d=2$ then $M_d$ includes three operators, corresponding to the three perfect matchings of $4$ objects:
\begin{equation}
\label{eq:m2}
\delta^{\alpha_1}_{\beta_1} \delta^{\alpha_2}_{\beta_2} = \raisebox{-7pt}{\includegraphics[height=18pt]{identity}} \, , \quad
\delta^{\alpha_1}_{\beta_2} \delta^{\alpha_2}_{\beta_1} = \raisebox{-7pt}{\includegraphics[height=18pt]{exchange}} \, , \quad
\mbox{and} \quad
\delta^{\alpha_1,\alpha_2} \delta_{\beta_1,\beta_2} = \raisebox{-7pt}{\includegraphics[height=18pt]{cupcap}} \, .
\end{equation}
The first two of these operators correspond to the identity permutation and the transposition $\tau_{1,2}$ respectively, as in the previous section. The third one is a \emph{cupcap}; it is the outer product of the vector $\sum_{i=1}^k e_i \otimes e_i$ with itself, where $e_i$ denotes the $i$th basis vector in $\mathbb{R}^k$. We denote it $\gamma_{1,2}$, and more generally $\gamma_{ij} = \delta^{\alpha_i,\alpha_j} \delta_{\beta_i,\beta_j}$.
Now, in the real-valued case, Lemma~\ref{lem:xd} becomes the following:
\begin{lemma}
\label{lem:xd-real}
If $x$ is uniform in the set of vectors in $\mathbb{R}^k$ of norm $1$, then
\begin{equation}
\label{eq:xd-real}
X_d
= \frac{1}{k(k+2)(k+4)\cdots(k+2d-2)} \sum_{\mu \in M_d} \mu
= \frac{(k-2)!!}{(k+2d-2)!!} \sum_{\mu \in M_d} \mu \, ,
\end{equation}
where $n!! = n(n-2)(n-4) \cdots 6 \cdot 4 \cdot 2$ if $n$ is even, and $n(n-2)(n-4) \cdots 5 \cdot 3 \cdot 1$ if $n$ is odd.
\end{lemma}
\begin{proof}
Analogous to the complex case, $X_d$ is a member of the commutant of the group $O(k)$ of $k \times k$ orthogonal matrices, since these preserve the uniform measure. That is, $X_d$ commutes with $O^{\otimes d}$ for any $O \in O(k)$. The commutant of $O(k)$ is the \emph{Brauer algebra}; namely, the algebra consisting of linear combinations of the operators $\mu \in M_d$. Thus $X_d$ is of the form $\sum_{\mu \in M_d} a_\mu \mu$.
In addition to being fixed under permutations as in the complex case, $X_d$ is also fixed under partial transposes, which switch some upper indices with some lower ones. Thus $X_d$ is proportional to the uniform superposition $\sum_{\mu \in M_d} \mu$. To find the constant of proportionality, we again compare traces.
As in the case of permutations, the trace of an operator $\mu \in M_d$ is $k^{c(\mu)}$, where $c(\mu)$ is the number of loops in the diagram resulting from joining the upper indices to the lower ones. For instance, for the operators in~\eqref{eq:m2}, we have
$\tr 1 = k^2$, $\tr \tau_{1,2} = k$, and $\tr \gamma_{1,2} = k$. Thus we wish to calculate
\begin{equation}
\label{eq:md-genfunc}
\tr \sum_{\mu \in M_d} \mu
= \sum_{\mu \in M_d} k^{c(\mu)} \, .
\end{equation}
We can write $\sum_{\mu \in M_d}$ as a product, analogous to~\eqref{eq:sd-sum}:
\[
\sum_{\pi \in S_d} \pi = 1 (1 + \tau_{1,2} + \gamma_{1,2}) (1+\tau_{1,3}+\gamma_{1,3}+\tau_{2,3}+\gamma_{2,3})
\cdots (1+\tau_{1,d}+\gamma_{1,d}+\cdots+\tau_{d-1,d}+\gamma_{d-1,d}) \, .
\]
This product describes a matching $\mu_t$ of $2t$ objects inductively as a matching $\mu_{t-1}$ of the first $2(t-1)$ objects, composed either with the identity, or with a transposition or cupcap connecting the $t$th upper object with the $i$th lower one and the $t$th lower object with the $i$th upper one, or vice versa. If we apply the identity, then the $t$th upper object is matched to the $t$th lower one, and $c(\mu_t) = c(\mu_{t-1})+1$, gaining a factor of $k$ in~\eqref{eq:md-genfunc}; but if we apply a transposition or cupcap, then $c(\mu_t)=c(\mu_{t-1})$. Thus~\eqref{eq:md-genfunc} becomes
\[
\sum_{\pi \in S_d} k^{c(\pi)} = k (k+2) (k+4) \cdots (k+2d-2) = \frac{(k+2d-2)!!}{(k-2)!!} \, .
\]
We again have $\tr X_d = \mathop\mathrm{Exp}_x \abs{x}^{2d} = 1$, and comparing traces gives~\eqref{eq:xd-real}.
\end{proof}
As before, $q(G;k)$ is a contraction of the tensors $X_d$. However, now $G$ is undirected, with no distinction between incoming and outgoing edges, so at each vertex of degree $d_v$ the appropriate tensor is $X_{d_v/2}$. Applying Lemma~\ref{lem:xd-real} to each $v$ sums over all the ways to match $v$'s edges with each other, and hence sums over all possible partitions of $G$'s edges into undirected cycles. The trace of the resulting diagram is again the number of ways to label each cycle with a basis vector. So, if define a polynomial $j_{\rm undirected}(G;z)$ as $\sum_{t=1}^\infty r_t z^t$, where $r_t$ is the number of partitions with $t$ cycles, then Theorem~\ref{thm:q-j} becomes
\begin{theorem}
\label{thm:q-j-real}
For any undirected graph $G=(V,E)$ where every vertex has even degree, if we define $q(G;k)$ by selecting the $x_v$ independently and uniformly from the set of vectors in $\mathbb{R}^k$ with norm $1$, then
\begin{equation}
\label{eq:q-j-real}
q(G;k) =
\left( \prod_{v \in V} \frac{(k-2)!!}{(k+d_v-2)!!} \right)
j_{\rm undirected}(G;k) \, .
\end{equation}
\end{theorem}
\noindent
To our knowledge, the computational complexity of $j_{\rm undirected}(G;z)$ is open, although it seems likely that it is also $\#P$-hard.
\section{The Gaussian distribution}
Our results above assume that each $x_v$ is chosen uniformly from the set of vectors in $\mathbb{C}^k$ or $\mathbb{R}^k$ of norm $1$. Another natural measure would be to choose each component of $x_v$ independently from the Gaussian distribution with variance $1/k$, so that $\mathop\mathrm{Exp}[\norm{x_v}^2]=1$.
For vectors in $\mathbb{C}^k$, the probability density of the norm $\norm{x}$ is then
\begin{equation}
\label{eq:pr}
p\left( \norm{x} \right) = \frac{2k^{k+1}}{k!} \norm{x}^{2k-1} {\rm e}^{-k \norm{x}^2} \, ,
\end{equation}
Compared to the case where $\norm{x_v}=1$, each $x_v$ contributes scaling factor of $\norm{x_v}^{2d}$ to the product~\eqref{eq:prod}. The even moments of~\eqref{eq:pr} are
\[
\mathop\mathrm{Exp}\left[ \norm{x}^{2d} \right] = \frac{(d+k-1)!}{k^d (k-1)!} \, ,
\]
so in the Gaussian distribution~\eqref{eq:q-j} becomes
\begin{equation}
\label{eq:q-j-gaussian}
q(G;k) =
\left( \prod_{v \in V} \frac{1}{k^{d_v}} \right)
j(G;k)
= \frac{1}{k^m} \,j(G;k) \, ,
\end{equation}
where $m$ denotes the number of edges.
We could also have derived this directly from the Gaussian analog of Lemma~\ref{lem:xd}. If $x$ is chosen according to the Gaussian distribution on $\mathbb{C}^k$, and we again let $X_d$ denote $\mathop\mathrm{Exp}_x \ket{x^{\otimes d}} \bra{x^{\otimes d}}$, then
\begin{equation}
\label{eq:xd-gaussian}
X_d
= \frac{1}{k^d} \,\sum_{\pi \in S_d} \pi \, .
\end{equation}
Similarly, in the real-valued case, if we choose each component of $x \in \mathbb{R}^k$ from the Gaussian distribution on $\mathbb{R}$ with variance $1/k$, then~\eqref{eq:q-j-real} becomes
\begin{equation}
\label{eq:q-j-real-gaussian}
q(G;k)
= \frac{1}{k^m} \,j_{\rm undirected}(G;k) \, ,
\end{equation}
since
\begin{equation}
\label{eq:xd-real-gaussian}
X_d
= \frac{1}{k^d} \,\sum_{\mu \in M_d} \mu \, .
\end{equation}
Both~\eqref{eq:xd-gaussian} and~\eqref{eq:xd-real-gaussian} are consequences of Wick's Theorem~\cite{isserlis,wick}, that if $x_1,\ldots,x_{2t}$ obey a multivariate Gaussian distribution with mean zero, then
\[
\mathop\mathrm{Exp}\left[ \prod_{i=1}^{2t} x_i \right] = \sum_{\mu \in M_t} \prod_{(i,j) \in \mu} \mathop\mathrm{Exp}[x_i x_j] \, .
\]
\paragraph{Acknowledgments.} We are grateful to Piotr \'Sniady for teaching us the sum~\eqref{eq:sd-sum}, and to Jon Yard for introducing us to the Brauer algebra. This work was supported by the NSF under grant CCF-0829931, and by the DTO under contract W911NF-04-R-0009.
|
2,877,628,090,826 | arxiv |
\section*{Abstract}
Code initialization---the step of loading code, executing \emph{static} code, filling caches, and forming re-used connections---tends to dominate cold-start time in serverless compute systems such as \aws \awslambda. Post-initialization memory snapshots, cloned and restored on start, have emerged as a viable solution to this problem, with incremental snapshot and fast restore support in VMMs like \firecracker.
Saving memory introduces the challenge of managing high-value memory contents, such as cryptographic secrets. Cloning introduces the challenge of restoring the uniqueness of the VMs, to allow them to do unique things like generate UUIDs, secrets, and nonces. This paper examines solutions to these problems in the \emph{every microsecond counts} context of serverless cold-start, and discusses the state-of-the-art of available solutions. We present two new interfaces aimed at solving this problem---\texttt{MADV\_WIPEONSUSPEND} and VmGenId---and compare them to alternative solutions.
\section{Benchmarks}\label{sec:benchmarks}
In order to understand the steady-state and restore-time performance impact of these mechanisms, we ran a number of benchmarks on typical x86 (EC2 m5.12xlarge) server. We first considered the impact of reseeding: getting entropy from the kernel (\texttt{/dev/urandom}) or hardware (\texttt{RDRAND} or equivalent), and passing it into the userspace PRNG (using openssl 1.1's \texttt{RAND\_seed}).
This basic reseeding is, as expected, very fast. Reseeding 32 bytes from \texttt{/dev/urandom} (including \texttt{open}, \texttt{read}, and \texttt{close}) took a mean of $11\mu s$, with insignificant deviation. Reseeding 32 bytes from \texttt{RDRAND} or \texttt{RDSEED} took $0.6\mu s$ per run with insignificant deviation (consistent with Intel's advertised performance for RDRAND and RDSEED~\cite{intelrng2018}).
Checking the guard page (either a page mapped with \texttt{MADV\_WIPEONSUSPEND} or the \texttt{mmap}ed VmGenId page) is, again, very fast. For the simple case of generating a sequential nonce (just a 128-bit increment operation, in other words), enabling the check reduces throughput by 13x. Doing any kind of meaningful work inside the check, on the other hand, causes its cost to amortize to near-zero. For example, running OpenSSL's default \texttt{md\_rand} generator with or without the guard page check showed no statistically significant difference. Absolute differences will vary from platform to platform, but we expect these relative results to be durable across platforms.
\section{Conclusion}\label{sec:conclusion}
Snapshot, clone and restore are useful and powerful primitives for reduced cold start times in serverless compute platforms. Using these primitives introduces several challenges: breaking common network protocols, moving in-memory state onto disk, and losing the ability for short-lived clones to make unique decisions. We described two new kernel interfaces that we proposed for tackling this problem in context of Linux and \aws \awslambda, one of which has been contributed to the mainline Linux kernel. This mechanism, a Linux flavor of VmGenId, provides a flexible way to approach these problems, with little performance overhead, relatively easy adoption, and wide applicability.
\section{Userspace Interfaces for Uniqueness}\label{sec:interfaces}
Solving the uniqueness problem strongly enough for cryptographic purposes requires a mechanism which can deterministically reseed userspace PRNGs with new entropy at restore time.
This mechanism must also support the high-throughput and low-latency use-cases that led programmers to pick a userspace PRNG in the first place (so simply reverting to \texttt{getrandom} is not acceptable); be usable by both application code and libraries; allow transparent retrofitting behind existing popular PRNG interfaces without changing application code; it must be efficient, especially on restore; and be simple enough for wide adoption.
Efficiency is a particularly important concern. The default PRNGs in many languages, including C, Java, and Python, are not cryptographically secure, a choice typically driven by performance. This frequently leads to programmers using these weak random number generators for cryptographic purposes. Over 350 CVEs have been allocated in 2020 alone for security issues introduced by this class of bugs. Reducing the efficiency of CSPRNGs further would increase adoption of non-CS PRNGs, leading to more bugs and insecure cryptography implementations.
\subsection{Fork and \texttt{MADV\_WIPEONFORK}}
A similar problem with userspace PRNGs is introduced by \texttt{fork}, as processes memory is duplicated in both the parent and child processes. A new \texttt{madvise} flag, \texttt{MADV\_WIPEONFORK}, was introduced in Linux 4.14 in 2017.
Memory pages marked \texttt{MADV\_WIPEONFORK} are set to all zeros in the child process after the call to \texttt{fork}, \texttt{clone} and related calls.
PRNGs which mark their internal state this way, or put a guard variable in a page marked this way, can deterministically detect they have been forked, and reseed from kernel or hardware randomness.
This does not require additional system calls, and adds only the overhead of a single memory access per call to the random number generation library.
BSDs support a similar approach, using the \texttt{MAP\_INHERIT\_ZERO} flag to the \texttt{minherit} system call.
The same interface is useful in other ways too. A process which handles cryptographic keys or other high-value material can mark them as \texttt{MADV\_WIPEONFORK} to ensure that they are not needlessly copied into child processes. It can also be combined with \texttt{mlock} and \texttt{MADV\_DONTDUMP} to ensure that secrets aren't saved onto disk.
\texttt{MADV\_WIPEONFORK} is superficially similar to \texttt{pthread\_atfork}, but is more robust because it is not thread-specific, and works across all ways to clone a process. \texttt{pthread\_atfork} can be bypassed, intentionally or unintentionally, by using the \texttt{clone} syscall directly, and by some other interfaces like \texttt{posix\_spawn}. This makes \texttt{pthread\_atfork} insufficiently robust for use in widely-used libraries.
\subsection{Suspend and \texttt{MADV\_WIPEONSUSPEND}}
By analogy to \texttt{MADV\_WIPEONFORK}, we have contributed a new \texttt{MADV\_WIPEONSUSPEND} flag to the Linux kernel\footnote{Our contributions here focus on Linux because we use it as the guest OS in \aws \awslambda's microVMs, but are widely applicable to other guest operating systems, unikernels, and library OSs}, which marks pages to be wiped when a VM is suspended (a precursor to snapshotting). We expect that PRNGs and cryptographic libraries mark their state or guard variables as both \texttt{MADV\_WIPEONFORK} and \texttt{MADV\_WIPEONSUSPEND} and use the same reseeding logic to handle both the fork and VM clone cases.
Wiping on suspend, rather than restore, has two benefits. One is performance: in our serverless use-case, restore is on the latency-critical cold-start path, so handling memory wiping at suspend time optimizes restore latency. Second is security: applications which have high-value secrets they don't want to include in the snapshot, or intentionally want to retrieve directly from a key management service or hardware security module (HSM) on restore, can mark these secrets to be excluded from the snapshot.
\texttt{MADV\_WIPEONSUSPEND} has the benefits of being simple, having very little performance overhead, and being easy to use both in libraries and application code. The performance overhead of \texttt{MADV\_WIPEONFORK} and \texttt{MADV\_WIPEONSUSPEND} is small, about 600ns per random generation. In an implementation of NIST's CTR\_DRBG, performing the necessary checks reduced per-core throughput by just TODO\%.
\subsection{System Generation ID}
The Windows Server 2012 version of the Hyper-V hypervisor introduced the VM Generation ID (VmGenId). The guest OS detects that significant changes have occurred, including cloning and restore, by a change in the VmGenId. Microsoft's VmGenId implementation~\cite{VmGenId2012} exposes a 128-bit unique number, a UUID, in a memory location that is discovered using a special ACPI device name. While interacting with the low-level implementation is not convenient for applications and libraries, and they are unlikely to have permissions to do so, the VmGenId is used in the Windows Cryptography API to avoid problems with cloning. Other hypervisors, including VMWare and QEMU, have since added support for VmGenID.
We have contributed support for a similar mechanism to the Linux kernel, called System Generation ID (SysGenId). The Linux version implements a device driver which exposes a read-only device \texttt{/dev/SysGenId} to userspace, which contains a monotonically increasing generation counter. Libraries and applications are expected to \texttt{open()} the device, and then call \texttt{read} which blocks until the SysGenId changes. Following an update, \texttt{read()} calls no longer block until the application acknowledges the new VmGenId by \texttt{write}ing it back to the device. Non-blocking \texttt{read} calls return \texttt{EWOULDBLOCK} when their is no new SysGenId available. Alternatively, libraries can \texttt{mmap} the device to get a single shared page which contains the latest VmGenId at offset 0.
Linux SysGenId also supports a notification mechanism exposed as two \texttt{ioctls} on the device. \texttt{SysGenId\_GET\_OUTDATED\_WATCHERS} immediately returns the number of file handles that were open during the last SysGenId change but have not yet acknowledged the new id. \texttt{SysGenId\_WAIT\_WATCHERS} blocks until there are no open file handles on the device which haven't acknowledged the new id. These two interfaces are intended for serverless and container control planes, which want to confirm that all application code has detected and reacted to the new SysGenId before sending an invoke to the newly-restored sandbox. This notification mechanism, unlike signals, allows SysGenId to be used inside libraries without requiring changes to application code.
The Linux SysGenId implementation also supports a VmGenId, as defined by Microsoft. SysGenId is the frontend driver exposing the 32-bit generation ID, which depends on a backend driver to detect when the system's identity changes. VmGenId is one such backend. when the VmGenId changes, the SysGenId is increased. Other backends to SysGenId can be defined, making it more general, and allowing support for cases like userspace checkpoint restore (e.g. CRIU~\cite{Criu2021}), and Linux containers. In this sense, SysGenId is a generalization of VmGenId.
\subsection{Comparing Solutions}
The three solutions \texttt{MADV\_WIPEONSUSPEND}, Microsoft's VmGenId, and Linux SysGenId) to handling uniqueness during VM cloning are compared in Table \ref{table:comparison}. In \texttt{MADV\_WIPEONSUSPEND}'s favor is its compatibility with \texttt{MADV\_WIPEONFORK}, its additional use for excluding secrets from snapshots, and the fact that it can be used in containers and sandboxes that don't have the ability to open or mmap files. The popularity of container technology, and sandboxing approaches like \texttt{seccomp-bpf} make this property especially interesting. Many sandboxes and containers want to prevent processes from opening files and devices entirely. \texttt{open}, \texttt{read} and \texttt{write} give a potential attacker a lot more power than just \texttt{madvise}.
In SysGenId's favor is its flexibility, including the ability to monitor for when processes and libraries have fully caught up to the latest version of a container. On Linux, both solutions have little memory overhead (involving reading from a special page), and can be used by non-root processes. This same flexibility comes with some risk. By enabling libraries and applications to do blocking work on the restore path (while the system blocks on \texttt{SYSGENID\_WAIT\_WATCHERS}), they can re-introduce a form of cold-start latency. Application and library authors must be encouraged to use SysGenId for lightweight purposes only.
\begin{table}[ht]
\begin{tabular}{ |l|lll| }
\hline
& MADV & VmGenId & SysGenId \\
\hline
Mechanism & Guard Page & UUID & Inc. Id \\
Works for fork & Yes & No & No \\
Secret hiding & Yes & No & No \\
In-memory & Yes & Yes & Yes \\
Notification & No & No & Yes \\
Non-root & Yes & No & Yes \\
Min-privilege & Yes & No & No \\
Entropy & No & Yes & No \\
Containers & No & No & Yes \\
\hline
\end{tabular}
\caption{Comparison of features of \texttt{MADV\_WIPEONSUSPEND}, Microsoft's VmGenId, and Linux SysGenId}
\label{table:comparison}
\end{table}
In Microsoft's solution's favor is its use of a high-quality UUID VmGenId, which can be used directly as a node identifier for distributed protocols, for logging, and tracing. The UUID, which Microsoft~\cite{VmGenId2012} describe as \emph{"a 128-bit, cryptographically random integer value identifier"} can also be used directly as a high-entropy seed for PRNGs, removing the need to separately reseed from the kernel or hardware. It can also be used directly for the deterministic construction of IVs (see NIST SP800-38D, section 8.2.1). Microsoft's VmGenId has existed for longer than either of the Linux solutions, and is built into the Windows Cryptography API, meaning that it already has extensive userspace support.
\subsection{When is a VM the same VM?}
While \texttt{MADV\_WIPEONSUSPEND} has fairly obvious semantics, SysGenId raises the question of under which circumstances the ID should be changed. Microsoft change their VmGenId on restore, copy, clone, recovery from backup, and (in some cases) on failover. The VmGenId does not change on reboot, pause, resume, live migration, or even on host reboot. While most of these seem like reasonable decisions, it is not clear that they are a good match for a serverless environment.
Consider the case of restoring from a snapshot without cloning. The resulting microVM is, for practical purposes, the same microVM as the one that was snapshotted, as long as the system can reason that it can never be clones. This means that SysGenId is just that---a generation id---and not the id of the microVM itself. Figuring out the right ID for a microVM is a separate system-level concern.
\subsection{Alternative approaches}
One alternative to approaching this problem at the system level is to approach it at the level of cryptographic algorithms. Rogaway and Shrimpton proposed~\cite{rogaway2007, rogaway2007siv} authenticated encryption modes that do not depend on a random nonce for their strongest security properties. Gueron et al proposed~\cite{gueron2015} and standardized~\cite{rfc8452} AES-GCM-SIV provides a nonce misuse-resistant mode, which keeps it's confidentiality and authentication properties in case of IV re-use (exposing only message equality) with only small performance overhead. Ristenpart~\cite{ristenpart2010} proposed changes to the TLS protocol to resist nonce-reuse. Despite these advances, and many others, many of the cryptographic protocols and modes widely deployed in production remain vulnerable to nonce reuse. We do not expect this situation to change quickly.
Another alternative is to move towards deprecating userspace CSPRNGs, and encourage implementors to use random numbers provided by the kernel or hardware. Discussing the merits of this approach is beyond the scope of this paper, but it is clear than making such a change is not possible without changing the performance of many libraries and applications, and changing many pieces of software to adopt new RNG interfaces. A system-level approach allows us to approach this change incrementally, which is essential given the size of the installed base and the goal of serverless platforms to work with existing application software.
A similarly major architectural change is to build applications which rely on cryptographic services provided by hardware security modules (HSMs), key management services (such as \kms), trusted execution environments (TEEs), and secure enclaves (such as \enclaves). In many cases this would be a good choice, but again isn't always applicable due to application-specific requirements, data plane performance requirements, and large bodies of existing code.
\subsection{Atomicity and TOCTOU}
Both the \texttt{MADV\_WIPEONSUSPEND} and SysGenId options provide sufficient mechanism for a RNG library to ensure that it does not generate duplicate random numbers in multiple cloned VMs\footnote{TLA+ specifications demonstrating this property are available at http://redacted.example.com/}. In the SysGenId case, this requires that the SysGenId update happens before (or atomically with) the restore operation. It is up to the VMM and guest kernel to ensure that this ordering is not violated.
Serverless systems (like \aws \awslambda) explicitly track manage the requests in-flight in a running VM, and ensure that VMs cannot perform any actions outside the context of handling a request. These systems can therefore fully quiesce a VM and ensure that it does not handle any requests until the notification process is complete. This is sufficient to ensure that random numbers are never re-used across multiple requests, provided that application code does not cache values between requests.
In general, however, none of these mechanisms are sufficient to strongly prevent re-use, due to \emph{time of creation time of use} (TOCTOU) issues. In any program which does not atomically generate, use, and discard the generated value, a clone operation between generation and use, or between use and discard, can cause re-use. Atomicitiy of such complex operations is not generally supported. Some external fence provided by the environment is needed to ensure that reseeding is complete before connections are made or used, storage is modified, or the system has any other side-effect. Load balancer and container orchestrator health checks are likely good places to tie in this logic.
\section{Introduction}\label{sec:introduction}
When \aws \awslambda receives a request to invoke a serverless function, the system attempts to match the request to an already-running microVM containing a copy of the function code. At scale, this \emph{warm start} succeeds with high probability--both because of the averaging effects of scale, and because serverless systems attempt to predict the number of incoming requests using techniques such as reinforcement learning~\cite{Balaji2020}, LSTMs~\cite{gunasekaran2020}, and ARIMA models~\cite{shahrad2020}. When warm start does not succeed, the system falls back on \emph{cold start}, which involves starting a \firecracker~\cite{agache2020} microVM (or taking one from a pool), loading the function code, and initializing the code to get it ready to handle the request. The last step of the the cold start, initialization, is typically the one that takes longest. Du et al~\cite{Du2020} found this initialization step to take between 25\% and 95\% of total start-up time, mirroring our own experiences. As an additional challenge for cloud providers, this initialization step is almost entirely within the customer's control, leaving little opportunity to optimize or improve it on behalf of the customer (beyond simply allocating additional resources to its execution).
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\columnwidth]{images/timeline}
\caption{Time line of the life cycle of a microVM in \aws \awslambda}
\label{fig:timeline}
\end{figure}
Figure \ref{fig:timeline} shows a typical time line of the life cycle of a microVM in \awslambda. The microVM boot, booting a minimal Linux kernel in \firecracker, with a minimal set of userspace daemons, and network setup, takes approximately 200ms. This is hidden from the customer through the use of a small pool of pre-booted microVMs. The cold start portion, including downloading the function code into the microVM and initialization, frequently takes up to 60s but can be as little as 20ms for small static binaries. The download is typically short: at 25Gb/s, downloading \awslambda's maximum package size of 250MB takes only 80ms. Finally, each invoke experiences a warm start delay caused by routing, typically less than 10ms. The same microVM is then re-used, for future invokes of the same version of the same function, to amortize the cold start costs.
The time and resources taken by initialization depend strongly on the customer's choice of language and framework, and on additional work the customer has chosen to do at process start time. Of the popular languages, Java typically exhibits the longest cold-start times. This is driven by three main factors: the start-up time of the JVM itself, the decompression and loading of class code, and the execution of \emph{static} code in the loaded classes. It is typically the last of these which drives long cold-start times, because Java allows for the execution of effectively arbitrary code at static initialization time. While Java code initialization and static execution is multithreaded, the specification~\cite{jls11} limits parallelism by defining a strict recursive order on class initialization. This problem is not only limited to Java. The same outcome is common, but less ubiquitous, in workloads written in Javascript, Python, and C\#. Workloads written in native languages like C, C++, Rust and Go tend to initialize quickly, largely because it's a less-common pattern to do extensive work at startup in these languages.
Post-initialization snapshots have been proposed as a solution to this initialization time problem~\cite{Du2020, Cadden2020}, to the general problem of cloud scaling~\cite{bryant2011}, to fault-tolerance~\cite{lorch2015}, and even for accelerating hypervisor fuzzing~\cite{schumilo2021}. Simply, initialization is complete, the virtual machine is snapshotted, and when a cold start is needed the snapshot is cloned and restored. This approach is attractive, because microVM restore can be made very fast. In our own experiments, \firecracker snapshot restore can take as little as 4ms. Catalyzer~\cite{Du2020} reports latencies reliably below 10ms for a broad range of application types.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{images/timeline_snap}
\caption{Time line of the life cycle of a microVM in \aws \awslambda with snapshot and clone}
\label{fig:timelineclone}
\end{figure}
Figure \ref{fig:timelineclone} shows the timeline for a microVM with snapshot and clone. The entire boot and cold start portion can be hidden from the customer by performing it once at function creation time. The blocking portion then requires only a snapshot restore, a much faster operation for which the entire latency is under the control of \aws.
\subsection{Challenges}
Snapshotting and clone-on-restore introduce several system-level challenges. First, by their nature snapshots turn data in RAM into data on storage. This data can include cryptographic secrets and other high-value tokens which the programmer wished to control carefully, irrespective of the security properties of the snapshot storage and distribution layer. Second, cloning duplicates memory state, which is a challenge for applications which want to generate unique data such as request IDs, UUIDs, and cryptographic nonces. Third, cloning is not compatible with existing session-layer network protocols such as TCP and TLS, which assume that the client is a single unique entity with a single identity. Re-establishing these connections can add significantly to restore latency~\cite{hunhoff2020, hunhoff2020poster}, and can even cause correctness issues in some protocols (a type of replay attack). This paper focuses on the second challenge---uniqueness---but some of the solutions we analyze also help address the challenges posed by high-value secrets and network protocols.
\section{The Value of Uniqueness}\label{sec:uniqueness}
Most modern distributed systems depend on the ability of nodes to generate unique or random values. RFC4122~\cite{rfc4122} version 4 UUIDs are widely used as unique database keys, and as request IDs used for log correlation and distributed tracing. In this context, duplicate IDs will lead to correctness issues, logical data corruption, incorrect traces, and confusing or misleading logs. UUIDs are also used as tokens for API idempotency, where multiple calls with the same token behave as a single call. Here, duplicate IDs can lead to availability or correctness issues. Many common distributed systems protocols, including consensus protocols like Paxos~\cite{Lamport2001}, and ordering protocols like vector clocks, rely on the fact that participants can uniquely identify themselves. Cloning without changing identity could be seen as an unintentional Sybil~\cite{douceur2002} attack, which most distributed systems protocols don't tolerate.
Jitter, the intentional addition of pseudorandomness to values like retry intervals, sampling intervals, timeouts and resource limits are widely used to improve the resilience of distributed systems. For example, adding jitter to the retry intervals used by clients of an overloaded system reduce the clustering of retries in time, helping systems recover faster from transient overload. Adding jitter to periodic jobs, like \texttt{cron}, helps avoid seasonalities like start-of-day and top-of-hour spikes. Systems that aren't unique will tend to cluster, and clustering leads to correlated load, which leads to lower system utilization.
Cryptography is the most critical application of unique data. Any predictability in the data used to generate cryptographic keys---whether long-lived keys for applications like storage encryption or ephemeral keys for protocols like TLS---fundamentally compromise the confidentiality and authentication properties offered by cryptography. A potentially less-obvious vulnerability relates to initialization vectors (IVs) and nonces used in common block cipher modes like counter mode (CTR) and Galois counter mode (GCM). Using the same IV with the same key multiple times breaks the security of these modes. In many protocols, the IV is translated in the clear, making it easy for an attacker to tell if an IV has been re-used. Dworkin in NIST SP800-38D~\cite{dworkin2007} says:
\begin{quote}
The probability that the authenticated encryption function ever will be invoked with the same IV and the same key on two (or more) distinct sets of input data shall be no greater than $2^{-32}$.
\ldots
In practice, this requirement is almost as important as the secrecy of the key.
\end{quote}
These properties---the need for unique IVs and keys with both low predictability and very small probability duplication---require that even cloned MicroVMs have access to high-quality entropy, and the means to use it. In some cases, such as AES-GCM, these values need only be unique and not random, so access to a unique per-VM identifier is sufficient. Solutions to the entropy problem are well-understood, and widely deployed in virtualized environments. Typically, entropy is injected into the guest, where it can be accessed using devices (like \texttt{/dev/urandom}) or system calls (like \texttt{getrandom}). Hardware RNGs, such as Intel's \texttt{RDSEED} and \texttt{RDRAND}~\cite{inteldev2013} (itself the output of a DRBG seeded by a hardware RNG), can also provide high-quality entropy inside cloned MicroVMs. We aren't the first to notice these kinds of cryptographic vulnerabilities. For example, Ristenpart and Yilek~\cite{ristenpart2010} describe successful attacks against TLS 1.0 relying on the-reuse of VM snapshots.
Many applications don't obtain their entropy directly from the kernel or hardware, and instead deploy userspace pseudo-random number generators (PRNGs) or deterministic random bit generators (DRBGs). Widely-deployed cryptographically-secure PRNGs include NIST's CTR\_DRBG~\cite{barker2015}, Java's \texttt{SecureRandom}, and Javascript's \texttt{getRandomValues}. Most of these PRNGs are deterministic algorithms that get their seeds from the kernel or hardware RNGs, either at startup or periodically, and \emph{stretch} them into an unpredictable sequence of random bits.
This presents a challenge to the serverless compute provider: even if they follow best practices such as reseeding kernel randomness, and customers follow best practices like using a cryptographically secure PRNG (CSPRNG), the combination is not secure. For platforms like \aws \awslambda which allow customers to run arbitrary code, even arbitrary binaries, addressing this challenge is not trivial.
|
2,877,628,090,827 | arxiv | \section{Introduction}
\label{sec:intro}
Jets, collimated showers of energetic final-state particles, and the physical observable related to jets
are not only most closely related to the perturbative QCD (pQCD) dynamics, but also among the
most intensively studied objects in high-energy physics~\cite{Salam:2009jx,EKS}. Jet productions in elementary
collisions can be accurately calculated and compared to a wealth of experimental
data on jet observables, measured in $e^+ e^-$, deep inelastic scattering and hadron-hadron reactions.
Jets provide not only a laboratory to test and advance pQCD but are also related to important
signatures of new physics (beyond the Standard Model).
QCD theory predicts that in relativistic heavy-ion collisions (HIC) a new kind of matter,
the quark-gluon plasma (QGP) consisting of deconfined quarks and gluons, may be created.
Theoretical studies and experimental measurements have
shown that fast partons traversing the QGP medium will undergo multiple scattering and
lose energy due to collisional and radiative processes~\cite{GVWZ}.
This parton energy loss and the corresponding suppression of hadron cross sections in nucleus-nucleus (A+A)
reactions have been dubbed {\it jet quenching}. Jet quenching effects
in the hot QCD medium will surely alter the production of reconstructed jets, which contain one or several
energetic partons prior to hadronization~\cite{Vitev:2008rz}. The concurrent advance in understanding jet production
and parton energy loss in hot QCD medium has spawned a new direction of research: the physics of jets
in high-energy nucleus-nucleus collisions at RHIC and at the LHC.
In this manuscript we discuss the modification of
jet production in relativistic heavy-ion reactions. We include selected results for inclusive jet
productions~ \cite{Vitev:2009rd,He:2011pd}, $Z^0/\gamma^*$-tagged jet
cross sections~\cite{Neufeld:2010fj,Neufeld:2012df} and the dijet transverse momentum
imbalance~\cite{He:2011pd,He:2011sg,HNVZ-2012} in high-energy nuclear collisions.
Our theoretical simulations will be compared with experimental data on jets when available, and predictions for future measurement are also
presented.
\section{Inclusive jet production in HIC}
\label{sec:dijet}
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.8in,height=2.5in,angle=0]{jet_LHC.png}
\hspace{0.6cm}
\includegraphics[width=2.8in,height=2.5in,angle=0]{jet_LHC_R.png} \vspace*{-.1in}
\caption{ Nuclear modification factors for inclusive jets in Pb+Pb
collisions at the LHC (left panel). Numerical simulation of ratios of inclusive jet cross sections
at different jet size parameter $R$ with and without non-perturbative effect at
Pb+Pb at $\sqrt{s_{NN}}=2.76$~TeV (right panel). }
\label{fig:1-jet}
\end{center}
\end{figure}
Within the pQCD collinear factorization approach, jet cross sections at $ {\cal O}( \alpha_s^3 ) $
in hadron-hadron collisions can generally be expressed as follows~\cite{EKS,Vitev:2009rd,He:2011pd}:
\begin{eqnarray}
\!\!\!\! \!\!\! \frac{d\sigma}{dV_{\rm n}}= \frac{1}{2!}\int dV_{\rm 2}
\frac{d\sigma(2\rightarrow2)}{dy_1dy_2d^2E_{T\, 1}d^2E_{T\,2} } \, S_2(p_1^{\mu},p_2^{\mu})
+\frac{1}{3!}\int dV_{\rm 3} \frac{d\sigma(2\rightarrow3)}
{dy_1dy_2dy_3d^2E_{T\, 1}d^2E_{T\,2} d^2E_{T\,3} } \,S_3(p_1^{\mu},p_2^{\mu},p_3^{\mu}) \,\, .
\label{di-pt}
\end{eqnarray}
Here, $V_{\rm n} = dy_1\cdots dy_nd^2E_{T\,1}\cdots d^2E_{T\,n}$ stands for the multi-parton phase space.
Up to $ {\cal O}( \alpha_s^3 ) $ two
contributions should be taken into account: one
from $2 \rightarrow 2$ processes at leading order (LO) and the higher-order virtual corrections (given by the first term
in above equation). The second one is from $2 \rightarrow 3$ processes, represented by the second term.
In relativistic heavy-ion collisions, energetic partons or jets produced from the hard scattering
will pass through the hot/dense QCD medium and the jet-medium interactions will degrade the
energy of jets and alter the jet shapes. In our theoretical simulations of jet production in the
nuclear environment we utilize the formalism of in-medium parton splitting~\cite{Ovanesyan:2011kn}, which
in the soft gluon approximation naturally gives the fully differential distribution of the
medium-induced energy loss of fast partons in the QCD medium~\cite{GLV}.
Fig.~\ref{fig:1-jet} illustrates the dependence of the
nuclear modification factor for single jet production $R_{AA}^{\rm 1-jet}$
in central Pb+Pb collisions on the jet size $R$ with an acceptance cut without ($p_T^{\min}=0$~GeV)
and with ($p_T^{\min}>0$~GeV) collisional energy loss~\cite{He:2011pd}.
The left panel of Fig.~\ref{fig:1-jet} presents $R_{AA}^{\rm 1-jet}$ due to both initial-state and final-state effects
for different jet sizes $R$ and different coupling constant $g_{\rm med}$. The influence of cold nuclear matter effects
is elucidated. The nuclear modification factor $R_{AA}^{\rm 1-jet}$ shows a clear dependence on jet size
because for larger $R$ more medium-induced gluon radiation will be recaptured within the jet~\cite{Vitev:2009rd}.
However, our results in Fig.~\ref{fig:1-jet} only include radiative energy loss. If collisional energy loss is included,
the dependence of $R_{AA}^{\rm 1-jet}$ on the jet size $R$ will almost disappear~\cite{He:2011pd}.
The right panel shows the ratio of jet cross sections at two different jet radii at three sets of jet-medium coupling
constant $g_{\rm med}$. Non-perturbative effects may affect this ratio, as shown in the figure.
\section{Tagged jet production in HIC}
\label{sec:tagged-jet}
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.6in,height=2.2in,angle=0]{Z_jet.png}
\hspace{0.6cm}
\includegraphics[width=2.6in,height=2.2in,angle=0]{Z_jet_AJ.png} \vspace*{-.1in}
\caption{ Numerical simulations of $Z^0/\gamma^*$ tagged jet transverse momentum spectra in p+p and Pb+Pb
reactions at $\sqrt{s_{NN}}=4.0$~TeV (left panel).
The $Z^0/\gamma^*$ tagged jet transverse momentum asymmetry for p+p and Pb+Pb collisions at
$\sqrt{s} = 2.76$~TeV for two different $R=0.2, 0.4$ (right panel). }
\label{fig:Z-jet}
\end{center}
\end{figure}
To investigate jet production in the nuclear environment and infer the properties of
the QGP, it is important to place constraints on the magnitude of the parton shower energy
in the medium~\cite{Zhang:2011ak}. Electroweak vector bosons, such as $Z^0, W^{\pm}$ and $\gamma$,
produced in conjunction with jets can provide such constraints on average~\cite{Neufeld:2010fj,Neufeld:2012df,Dai:2012am}.
For example, in $Z^0$-tagged jet production at leading-order obeys
the relation \begin{math} p_{T_{\rm jet}} \end{math} = \begin{math} p_{T,Z^0} \end{math}. Next-to-leading order
corrections are very important, as we will see below.
Because $Z^0$ and isolated photons do not interact strongly,
they can escape from the hot medium undisturbed. By measuring the difference between $p_{T_{\rm jet}}$ and $p_{T_,Z^0}$
in the final state, one can deduce the average energy loss of the jet in QCD matter.
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in,height=1.9in,angle=0]{dijet_Raa_Mjj.png}
\hspace{0.6cm}
\includegraphics[width=2.5in,height=1.9in,angle=0]{dijet_Raa_AJ.png} \vspace*{-.1in}
\caption{Nuclear modification factors for the dijet invariant mass spectra (left panel) and the dijet transverse energy asymmetry distributions (right panel) in Pb+Pb collisions at the LHC with different nPDFs. }
\label{fig:di-1}
\end{center}
\end{figure}
Recently the first ${\cal O}(G_F\alpha_s^2)$\ calculations on $Z^0/\gamma^*$-tagged jet production
and event asymmetry in HIC with final-state
parton energy loss effect have been done~\cite{Neufeld:2010fj,Neufeld:2012df}. This approach
has been extended to study isolated-photon tagged jets in A+A collisions
at ${\cal O}(\alpha_{\rm em} \alpha_s^2)$~\cite{Dai:2012am}. The left panel of Fig.~\ref{fig:Z-jet} shows
$Z^0/\gamma^*$ tagged jet transverse momentum spectra in p+p and Pb+Pb at
$\sqrt{s} = 4,0$~TeV. Due to parton energy loss effects, the tagged jet spectra are shifted
to smaller $ p_{T_{jet}}$ values.
If we plot the ratio of the cross section in Pb+Pb reaction (per binary collision) to the one in p+p, as shown in the
insert of the left panel figure,
we observe a transition from the enhancement at small momentum to suppression at the large momemtum
~\cite{Neufeld:2010fj}.
The right panel of Fig.~\ref{fig:Z-jet} presents the event asymmetry, where we define
$A_J = ({p_T}_1 - {p_T}_2)/({p_T}_1 + {p_T}_2) $
whith ${p_T}_1$ and ${p_T}_2$ being the transverse momenta of the leading and subleading jets. For the case of
$Z^0/\gamma^*$ tagged jet production ${p_T}_1=p_{T,Z^0}$ and ${p_T}_2=p_{T_{\rm jet}} $. We observe that
with jet-medium interaction the tagged jet $A_J$ distributions in Pb+Pb reactions are significantly broader
and shifted to $A_J > 0$. The underlying physics mechanism is medium-induced parton splitting:
energy is lost due to large-angle radiation out of the jet reconstruction parameter
$R$. Consequently, the smaller the jet radius, the larger
the width and the average asymmetry of these distributions will be~\cite{Neufeld:2012df} .
\section{Dijet asymmetry in high-energy nucleus-nucleus collisions}
\label{sec:dijet}
Recently, the ATLAS and CMS collaborations at the LHC~\cite{Aad:2010bu,Chatrchyan:2011sx} published
results on the enhancement of the transverse energy imbalance
for dijet in Pb+Pb collisions. This was the first evidence for the quenching of reconstructed jets at the LHC.
In the framework of pQCD, we investigate dijet productions to $ {\cal O}( \alpha_s^3 ) $ at RHIC and the LHC
by including both initial- and final-state nuclear matter effects~\cite{He:2011pd,He:2011sg}.
The dijet invariant mass $M_{JJ}$ is defined as the invariant mass of all particles in the final state, $[(\sum p^{\mu}_n)^2]^{1/2}$. Cold nuclear matter(CNM) effects on $M_{JJ}$ and $A_J$ distributions in Pb+Pb collisions at $\sqrt{s}=2.76$~TeV~ are
displayed in Fig.~\ref{fig:di-1} by implementing four parametrization sets of nuclear parton distributions (nPDFs). The left
panel shows dijet invariant mass spectra are enhanced in a wide region of $M_{JJ}$ due to CNM effects, which is opposite to the suppression resulting from the final-state QGP effects. The right panel of Fig.~\ref{fig:di-1} shows that the dijet asymmetry
distribution is insensitive to the CNM effects and, thus, provides a robust observable to probe the final-state effects in the QGP.
The energy fraction $z=E_{T2}/E_{T1}$, which represents the transverse energy imbalance of dijet production,
plays a similar role to the asymmetry $A_J$. The normalized dijet asymmetry
distributions in central Pb+Pb collisions at $\sqrt{s}=2.76$~TeV and imbalance distribution in central Au+Au
collisions at $\sqrt{s}=200$~GeV are presented in Fig.~\ref{fig:di-2}. We see that in A+A collisions
these distributions are enhanced due to the energy loss of jets propagating the QGP.
For radiative energy losses, there is distinct dependence on the jet cone radius R.
When the collisional energy dissipation of the art on shower is also considered, the dependence of
imbalance distributions on jet sizes is very weak,
but the sensitivity to the coupling strength of jet to medium is amplified~\cite{He:2011pd}. Furthermore,
jet-medium interactions shift the $z$ distribution to the left and,
thus, give smaller averaged $\langle z\rangle $ in Au+Au collisions~\cite{HNVZ-2012}, see the right
panel of Fig.~\ref{fig:di-2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in,height=2.0in,angle=0]{dijet_AJ_LHC.png}
\hspace{0.6cm}
\includegraphics[width=2.5in,height=2.0in,angle=0]{dijet_z_RHIC.png} \vspace*{-.1in}
\caption{ The dependence of the normalized dijet asymmetry distributions $A_J$ on
the jet radii
in Pb+Pb at the LHC (left panel). Similar dependencies on coupling strength are studied and for momentum imbalance
$z$ distribution in Au+Au reactions at RHIC (right panel). }
\label{fig:di-2}
\end{center}
\end{figure}
|
2,877,628,090,828 | arxiv | \section{Introduction}
Charting the complex interface of biological membranes is essential in determining how localized interactions contribute to overall cellular function. Containing an elaborate mesh of membrane proteins, assorted lipids and domains of many types, regional lipid orientation within membranes can give us great insight on their interactions with neighboring molecules. Before approaching the complexities of a true cellular membrane less sophisticated model membranes have been characterized using techniques ranging from NMR to various x-ray methods \cite{phtr8,Sampcharct6,Dppcret2}. These traditional methods accurately predict structural properties of lipid membrane multi-stacks and give great insight in the physical behavior of these systems. One drawback is that they do not yield information on low curvature/single membrane systems naturally found in biological cellular membranes. Those techniques also are limited in the lateral information they are able to obtain across the lipid bilayer. To address these limitations a high resolution technique that has an elevated degree of sensitivity to structural changes throughout planar membrane systems was implemented.
In addition, many have chosen methods that mainly focus on exploiting the anisotropic nature of lipids to obtain structural information of the lipid molecules oriented in the membrane \cite{ Dppcret6, Dppcret8, Dppcret7, Dppcret4,Dppcret2}. In particular, various studies have been used to measure the effective retardance of the membrane systems on 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) bilayers in the $L_{\beta^{\prime}}$ gel state \cite{intro10, Dppcret4, blip8}. Under these conditions an average constant molecular tilt, $\phi\, \sim \,32^\circ$ \cite{Dppcret5}, across the structure can be assumed. Knowing the perpendicular component of the refractive index $n_{\bot}$ of the acyl chains, the optical orientation $\theta $ and parallel refractive index $n_{\|}$ of the acyl chains can be determined. All mentioned techniques were used in studying both bulk and planar lipid membrane systems, but give limited lateral information. To address this issue Lee \textit{et al.} \cite{intro10} were able to explore these structural effects by measuring the effective retardance by using a Near-Field Scanning Optical Microscope (NSOM). Doing so they were able to obtain a lateral resolution on the order of $\sim$ 100 nm across a planar membrane system. While obtaining a high lateral sensitivity with NSOM, the measurement was limited in its sensitivity of the retardance, leaving room for improvement on the detectability of the acyl chain tilt variations across planar membrane systems.
In studying lipid systems it is also common to investigate their thermodynamic properties to better understand the localized interactions that take place between molecules. Including temperature controlled NMR and x-ray experiments \cite{NMR1,NMR2,Saxs1}, techniques like differential scanning calorimetry \cite{PTran2} have been used to study the first order phase transitions in lipid systems. In this paper an attempt to address these limitations by introducing a polarization modulating technique \cite{NSOMref7} in combination with a temperature controlled Near-Field Scanning Optical Microscope is introduced. Doing so a high degree of sensitivity of the acyl chain tilt, with a lateral resolution $\sim$ 100 nm has been obtained. As a consequence of the enhanced a more accurate difference in index of refraction on the membrane's plane is readily available. The combined use of these techniques with temperature control allow to study the structural variations involved in first order phase transitions planar bilayer systems.
\vfill
\section{Experimental Details}
\subsection{Sample Preparation}
\begin{figure}[b]
\centering
\subfigure[] {\includegraphics[width=.9\linewidth,height=.6\linewidth]{Fig1a.png}}
\hspace{12pt}
\subfigure[] {\includegraphics[width=.9\linewidth,height=.6\linewidth,trim=40 10 40 40, clip = true]{Fig1b.png}}\\
\caption{AFM image of supported DPPC bilayers near an edge. (a) 2.6 $\mu$m $\times$ 2.6 $\mu$m topographical contrast image. The low height regions represent the underlying glass slide (b) Plot of the height data taken from a line cut of the image in a), showing a height difference of $\approx$ 5.2 nm. }
\label{afmchar}
\end{figure}
Hydrophilic glass substrates were prepared by sonicating Fisher brand glass cover slips in detergent and deionized (DI) water separately for $\sim$ 15$-$20 minutes each. This was followed with a wet-chemical oxidation process using a piranha solution (H$_2$SO$_4$~:~H$_2$O$_2$ = 3:1) for $\approx$ 5 min \cite{Sampprep2, Sampprep4}, to create a flat hydrophobic surface. The cover slips were then rinsed and sonicated for $\sim$ 30 min in DI water and finally thermally dried under normal atmospheric conditions.
Supported lipid bilayers were formed on the prepared hydrophilic glass substrates by vesicle fusion \cite{Sampprep1, Sampprep4}. 1.0 - 0.5 mg/ml of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) in $100 \text{ mM NaCl}~:~30 \text{ mM NaH}_2\text{PO}_4$ was sonicated at $\approx 60^{\circ} \text{C}$ until clear to create small unilamellar vesicles (SUV’s). 100 $\mu$L of solution was placed on the glass substrates and left to equilibrate at room temperature for $\approx $ 30 mins. The samples were then baked at $60^{\circ} \text{C}$ for 45 to 60 min. After rinsing the samples with DI water, they were either re-hydrated and characterized with an AFM under a fluid cell (see figure \ref{afmchar}), or kept in a chamber at 100\% relative humidity until imaged with the PM-NSOM system.
\subsection{Experimental Setup and Supporting Details}
The total PM-NSOM setup is schematically shown in figure \ref{PMNSOMset}. All measurements were conducted with NSOM probes coated with $\sim$ 5 nm of Cr and $\approx$ 150-200 nm of Al, with diameters ranging from $\approx$ 80 to 100 nm (verified with a scanning electron microscope). Monochromatic light from the (5-6 mW) helium neon laser ($\lambda \,=$~632.8~nm) was mechanically chopped and passed through a linear polarizer. All other optical elements are positioned with respect to this direction defined to be 0$^{\circ}$. After the linear polarizer, light passed through a photo elastic modulator PEM (45$^{\circ}$, frequency 41.9 kHz) and QWP (0$^{\circ}$ ). The light was then coupled into a single mode optical fiber, which was connected to an universal fiber polarizer, used to control the polarization going into the NSOM probe \cite{NSOMref7}. The NSOM tip-to-sample distance ($\sim$ 10 nm) was controlled utilizing a shear force feedback system \cite{NSOMref14,NSOMref15}. The light from the NSOM probe and sample was collected with an objective lens, and sent through a QWP (0$^{\circ}$), an analyzer (-45$^{\circ}$) and finally collected by a Si PIN diode detector. The current from the diode was converted to a voltage signal by means of a transimpedance amplifier and measured by three lock-in amplifiers (LIA). The three LIA's were locked at the frequency of the mechanical chopper (proportional to the intensity of the laser, DC value), the first and second harmonics of the PEMs frequency respectively.
In the absence of a sample, it was initially assumed that the universal polarizer negates any polarizing effects from the fiber and NSOM probe. Under these circumstances, a Jones matrix formalism was applied and it was determined that the signal at the detector is proportional to $ \frac{1}{2}-\frac{1}{2}\sin (A)\sin (S) \cos (2 \theta ) +\frac{1}{2} \cos (A) \sin (S)\sin (2 \theta )$, where $A = B \cos \left(\omega t\right)$, $B$ is the magnitude of the retardance and $\omega $ is the angular frequency set by the PEM (The PEM was calibrated such that the $J_{0}(B)=0$ ). The lock-in amplifiers are used to obtain a signal normalized by the DC term at the first harmonic ($I_{\omega}$) and the second harmonic ($I_{2\omega}$),
\begin{equation}
I_{\omega} = \gamma J_1(B) \sin (S) \cos (2 \theta)
\label{intens1}
\end{equation}
\begin{equation}
I_{2\omega} = \gamma J_2(B) \sin (S) \sin (2 \theta)
\label{intens2}
\end{equation}
\noindent Hence,
\begin{equation}
\theta =\frac{1}{2} \arctan \left(\frac{- I_{2\omega}\text{ }J_1(B)}{I_{\omega} J_2(B)}\right)
\end{equation}
\noindent and the retaradance $S$ is,
\begin{eqnarray}
S &=& \arcsin \left(\frac{ I_{\omega}}{ \gamma J_1(B)\cos (2 \theta )}\right)\nonumber \\
&=&\arcsin \left(\frac{ - I_{2\omega}}{ \gamma J_2(B)\sin (2 \theta )}\right)
\end{eqnarray}
\noindent where $\gamma$ is a the term associated with the lock-in measurement (i.e. $\gamma =\,\frac{\pi}{2}$ due to the DC component being a square wave for our study). During the alignment process in the absence of a sample, both $I_{\omega}$ and $I_{2\omega}$ in equations \ref{intens1} and \ref{intens2} are minimized, which implies that the retardances associated with the NSOM probe and fiber approach zero.
Moving the sample in and out of the optical path while aligning the NSOM system is not ideal due to the user's risk of damaging the probe and the cumbersome process of attaching another. It would be advantageous to have a method where the alignment could be made over the sample of unknown retardance. To understand this effect, the sample and probe were modeled as two independent retarding elements, with retardances $S$ and $\delta$ inserted between the two QWP in the previously described setup. Following the same Jones matrix formalism the expected intensities of the two retarding objects $S$ and $\delta$ oriented at different angles were calculated. During the alignment process, where $I_{\omega}$ and $I_{2\omega}$ are minimized, the we can assume the orientation of the NSOM probe/fiber and sample are made equivalent. Since all retardances are small, the intensities,
\begin{equation}
I_{\omega} = \gamma J_1(B) \sin (S+\delta) \cos (2 \theta)
\end{equation}
and
\begin{equation}
I_{2\omega} = \gamma J_2(B) \sin (S+\delta) \sin (2 \theta)
\end{equation}
\noindent are obtained.
From these expression, it is seen that instead of an absolute value of $S$, one can align the system over the sample and obtain a deviation in the retardance $\Delta S$ and a relative orientation $\theta$ across the sample.
To ensure the correctness of our calibration approach and the viability of our previous calculations, PM-NSOM measurements were conducted on cleaved stepped muscovite (mica) substrates. The mica substrate showed steps of different heights corresponding to different number of layers of the crystal. Both far-field and near-field measurements were made of the retardance $S$ on mica, the far field measurements used to make sure NSOM determinations were accurate. Both aforementioned approaches for measuring the retardance and orientation using the NSOM apparatus were performed. It was observed that on average the overall difference was $\Delta S \sim$ 0.3 mrad between the two methods. Using the height topography from the NSOM measurement the birefringence of the muscovite crystal was determined to be $\sim$ 0.0025, which falls within accepted values \cite{mica1, mica2}. It was noted that the images taken with the system aligned over the sample, yielded more detail in $\Delta S$ than just measuring $S$. We suspect that this increase in sensitivity to sample variations is a consequence of the measurements being done from a minimum instead of an offset signal, highlighting an additional advantage of aligning the system over the sample.
All experiments were conducted at 100$\%$ relative humidity, to ensure membrane structure contained the characteristic acyl chain tilt of $\approx$ 32$^{\circ}$ in the $L_{\beta^{\prime}}$ phase at room temperature. The measurements were taken with scan times ranging from 100 ms to 450 ms per point, where $I_{1\omega}$, $I_{2\omega}$, $\Delta S$, $\theta $ and topography data were collected using a data acquisition board and LabView computer program. Trace and retrace images were collected to confirm measurement reliability.
Temperature controlled PM-NSOM measurements were performed similarly to those conducted at room temperature but including the following procedures. A cylindrical aluminum chamber with a small slit in one side to accommodate for the NSOM probe holder was placed around the NSOM probe (figure \ref{PMNSOMset}) to minimize the lateral heat gradients across the sample. The chamber was thermally isolated with an insulating foam casing and the temperature was controlled using a Peltier thermoelectric cooler (TEC) plate to move heat in and out of the chamber. The temperature of the sample was measured using a sensor embedded into the aluminum sample holder. The sample holder rested on a Delrin stand with a stainless steel bottom to improve mechanical stability. With this design the temperature stability was within $\pm \textbf{ } 0.07 ^{ \circ}$C. A water trough surrounded the sample holder, which preserved the required 100$\%$ relative humidity.
During a typical run the chamber's temperature was first heated beyond $T_m$ to ensure the lipid membrane was well into the $L_{\alpha}$ state. The NSOM probe was then engaged with the sample. The chamber was cooled below $T_m$ and heated back to the initial temperature. This was proven to be the most reliable method to counter the probe crashing events that occurred due to the thermal expansion of the Delrin separator.
\begin{figure}[!]
\begin{center}
\subfigure[]{\label{MNSOMset1}}{\includegraphics[width=.95\linewidth,height=.7\linewidth]{Fig2a.png}}\\
\subfigure[]{\label{MNSOMset2}}{\includegraphics[width=.95\linewidth,height=.8\linewidth,trim = 0 0 0 0, clip]{Fig2b.png}}
\subfigure[]{\label{MNSOMset3}}{\includegraphics[width=.49\linewidth,height=.6\linewidth,trim = 0 0 0 10, clip]{Fig2c.png}}
\subfigure[]{\label{MNSOMset4}}{\includegraphics[width=.49\linewidth,height=.6\linewidth,trim = 0 0 0 40,clip]{Fig2d.png}}
\caption{Diagram of PM-NSOM setup. (a) PM-NSOM setup highlighted with components for room temperature studies (b) Diagram of the temperature controlled chamber with water trough to maintain 100 \% relative humidity. (c,d) Photographs of the temperature controlled chamber.}
\label{PMNSOMset}
\end{center}
\end{figure}
\section{Results}
The anisotropic nature of lipids yields differences in the index of refraction (or birefringence ($n_e - n_o$)) along orthogonal directions in the membrane's plane. Because the polarizability of the lipid molecule is asymmetric the refractive indices along the length of the acyl chains ($n_{\parallel}$) and perpendicular ($n_{\perp}$) are not equal. In this study, the principal optical axis is assumed to lie parallel to the length of the acyl chains $\phi \sim$ 32$^{\circ}$ with respect to the membranes' normal for DPPC. Polarized light of wavelength $\lambda$ propagating in the $z$-direction, parallel to the membranes' normal, experiences a retardance between the $x$ and $y$ components of the electric field according to \cite{intro10}
\begin{eqnarray}
\Delta S &=& \frac{2\pi (n_e - n_o)t}{\lambda} \nonumber \\
&=& \frac{2\pi t}{\lambda} \frac{1}{\sin \left( \arctan \left(\frac{n_{\|}{}^2}{n_{\bot }{}^2}\cot (\phi )\right)+\phi \right)} \nonumber \\ && \cdot \frac{n_{\bot }n_{\|}} {\sqrt{n_{\bot }{}^2\sin^2 (\phi )+n_{\|}{}^2\cos^2 (\phi )}}-n_{\bot }.
\label{biref2}
\end{eqnarray}
\noindent The extraordinary ray corresponds to the electric field parallel to the optical axis, where as for the ordinary ray the electric field is perpendicular to it. The two rays were indistinguishable for the collection optics used, due to the overall thickness of the membrane.
Topographic measurements were made across the sample until a discontinuity in $S$ and $\theta$ were found. From the image of $S$ in figure \ref{DPPCMT2S1}, a measured $\Delta S $ of $\approx$ 3.9 $\pm$ 0.4 mrad was obtained by taking the average $S$ inside the hole and subtracting from a region away from the determined edge. A birefringence $(n_e \,-\, n_o)\,=\, \frac{\Delta S \lambda}{2\pi \textit{t}} \approx$ 0.073 $\pm$ 0.008, using $\lambda$ = 632.8 nm and \textit{t} $=$ 5.2 nm $\pm$ 0.4 \cite{Dppcret3} was obtained. These results agree well with previous measurements with a significant reduction in error \cite{intro10}.
Furthermore an increase in $\Delta S$ to $\approx\,$ 0.0075 mrad was obtained at the edge of the hole. It is expected that $\phi$ is greater where the membrane forms a boundary. Since the size of the hole is smaller than the probe's diameter, it did not show a significant change in height, see figure \ref{DPPCMT2S4}.
\begin{figure}[!]
\begin{center}
\subfigure{\label{DPPCMT2S1}\includegraphics[width=.95\linewidth,height=.7\linewidth, trim = 0 0 0 0, clip]{Fig3a.png}}
\hspace{24 pt}
\hspace{24 pt}
\subfigure[]{\label{DPPCMT2S4}\includegraphics[width=.95\linewidth,height=.6\linewidth,clip=true,trim= 270 0 10 160,clip]{Fig3b.png}}
\caption{PM-NSOM picture of DPPC supported on glass, (a) $S$ as a function of position. (b) Topography obtained over the same area.}
\label{Dppcretard1}
\end{center}
\end{figure}
\begin{figure}[!]
\begin{center}
\subfigure{\label{DPCTmpS}\includegraphics[width=.95\linewidth,height=.7\linewidth, trim = 0 10 0 0, clip]{Fig4.png}}\\
\hspace{36pt}
\caption{Temperature controlled PM-NSOM 512 nm $\times $ 512 nm images of DPPC supported on glass. An image of the measured $S$, showing a $\Delta $ $S$ of $\approx $ 3.8 $\pm$ 0.3 mrad at the $T_m$, which is highlighted by the white dotted line. This change $S$ indicates a change of $\phi $ of the acyl chains.}
\label{temp1}
\end{center}
\end{figure}
A series of temperature PM-NSOM experiments probed the main phase transition temperature $T_m \sim 41^{\circ}$C of DPPC across a 512 nm $\times$ 512 nm area. The 16 $\times$ 16 pixel images were taken over $\approx$ 3.5 min at a rate of $\sim$ 0.06 $^{\circ}$C$/$min. The series started at $T \approx 38^\circ$ and finishing at $T \approx 42^\circ$. Figure \ref{DPCTmpS}, highlights the main transition occurring over on image time scale. In the figure, $S$ is observed to remain constant at low $T$ and then it shows a jump of $\Delta S \approx \text{(}3.84 \pm 0.20 \text{) } \text{mrad}$ at $\approx$ 41.1$^{\circ}$C. This $\Delta S$ is interpreted as the average position of the acyl chains change from their characteristic $\langle \phi \rangle$ of $\approx 32^{\circ}$ in the $L_{\beta^\prime}$ to the $\langle \phi \rangle$ $\rightarrow$ 0 in the $L_{\alpha}$ state. A change of $\approx 21^{\circ}$ at $T_m$ was observed in $\theta$ at the same position highlighted in figure \ref{DppcRetversustmp}. This change in $\theta$ corresponds to the sample in the $L_{\beta^\prime }$ state with optical system having one $\theta$, and changing to a different value, characteristic of the optical system, when $T > T_m$.
\begin{figure}[!]
\begin{center}
\subfigure[]{\label{DPRetGrallH}\includegraphics[width=.95\linewidth,height=.7\linewidth,trim= 20 0 90 50,clip]{Fig5a.png}}\\
\subfigure[]{\label{DPRetGrH}\includegraphics[width=.95\linewidth,height=.7\linewidth, angle = 0,trim= 0 0 90 50,clip]{Fig5b.png}}\\
\caption{$ \Delta S$ values from a temperature controlled PM-NSOM measurement of DPPC supported on glass. (a) Graph of $\Delta S_{avg} $ versus temperature as the system was heated, taken from (512 nm)$^2$ images showing a change in $S$ across $T_m$ of $\approx $ 4.0 $\pm$ 0.4 mrad. (b) Cooling data graph of $\Delta S_{avg} $ versus temperature, taken from $\sim$ (100 nm)$^2$ images.}
\label{DppcRetversustmp}
\end{center}
\end{figure}
The next series of temperature controlled experiments decreased the image acquisition time to $\approx $ 50 s, and the temperature change per image was $\sim$ 0.05$^{\circ}$C at a rate $\sim$ 3 $^{\circ}$C/min. This was done to explore the hysteresis effects of the 1$^{st}$ order phase transition of the planar membrane system. Images taken were identical in size to the previous described temperature controlled experiments. The results for $S$ are seen in figure \ref{DppcRetversustmp}, where a $\Delta S$ of $\approx$ 3.5 mrad is observed across $T_m$. Figure \ref{DPRetGrallH} Shows that $S \approx 4.7$~mrad for $T < T_m$ and $\approx $ 1 mrad when $T > T_m$. This translates to the acyl chains' $\langle \phi \rangle$ transitioning from $\sim$ 32$^{\circ}$ to zero, calculated using equation \ref{biref2}. This is what we expect to occur as the lipid bilayer goes from the $L_{\beta^\prime }$ into the L$_\alpha$ phase.
\section{Discussion}
With the knowledge obtained from the PM-NSOM experiments at room temperature on supported DPPC in the $L_{\beta^\prime }$ phase, physical parameters about the lipid molecules in the the membrane can be extracted. Birefringence yields information on the polarizability of the lipid molecules. Following the model of rigid cylinders used by Salamon \textit{et al}. \cite{Dppcret4} equation \ref{biref2} and the relationship $ n_i{}^2= \frac{\alpha _i\epsilon _o}{\left(V -\alpha _iL_i\right)}+ \epsilon _o,$ (where $V$ is volume and $L_i$ is the shape factor for a cylinder \cite{AnisoMat1}) can be used to determine the transverse ($\alpha_{t}$) and longitudinal ($\alpha_{l}$) polarizabilites. The values for two acyl chains was determined to be, $\alpha_{t} \, = \, 44.2 \text{\AA}^3$ and $\alpha_{l}\, =\, 94.4 \text{\AA}^3$, assuming the area per lipid ($A$) and acyl chain length ($l$) to be 47.9 $\text{\AA}^2$ and $17.2\,\text{\AA}$ respectively \cite{Dppcret5}. These values are very close to the theoretically calculated polarizabilities $\alpha_{t}$ = 25.14 ${\text{\AA}}^3$ and $\alpha_{l}$ = 45.8 $\text{\AA}^3$ \cite{Dppcret4, AnisoMat1} of a single palmitic acid C$_{16}$ .
Beyond extracting physical parameters, the model previously presented can be used to create a three dimensional representation of the average direction of the orientation of the acyl chains of the lipid molecules within the membrane. Using the values for $n_\bot$ and $n_\|$, the average orientaion $\phi$ of the acyl chains across the NSOM probe's aperture with respect to the membrane's normal. Figure \ref{DPPC512chainmap} utilized the data taken from figure \ref{temp1}, representing the molecules average orientation as blue rods. As shown, when the membrane is below $T_m$, $\langle \phi \rangle $ was $\approx $ 32$^{\circ}$ while when it is heated above the phase transition temperature it was in the $L_{\alpha}$ phase, where $\langle \phi \rangle $ is zero.
\begin{figure}[!]
\begin{center}
\subfigure{{\includegraphics[width=.95\linewidth,height=.7\linewidth, angle = 0]{Fig6a.png}}}\\
\subfigure{{\includegraphics[width=.95\linewidth,height=.5\linewidth, scale = .1,trim= 120 0 120 0,clip]{Fig6b.png}}}
\caption{ 3D representation of the average acyl chain orientation throughout the membrane. The imag was reconstructed from the data in figure \ref{DPCTmpS}) Each tube portrays the average position of many lipid molecules contained in a $\sim$ (100 nm)$^2$ area.}
\label{DPPC512chainmap}
\end{center}
\end{figure}
Through further analysis from the data taken in figure \ref{DppcRetversustmp}, we observed that $\langle S \rangle $ varies as $T_m$ was approached. This observation could derive from varying orientation that exist in the $P_{{\beta}^\prime }$ phase. A characteristic that cannot be confirmed with PM-NSOM. Previous experiments on on multi-layered bilayer systems have measured the "ripples" and their periodicity, which are $<<$ than the diameter of the NSOM aperture \cite{blip9,blip10}.
The hysteresis observed under the applied heating rates is comparable to what has previously been reported \cite{PTran1, PTran2}. What was not initially evident was that as $T_m$ was approached, the overall variability in $S$ increased, displayed in the graph of the variance ($\Delta S^2$) versus $T$ in figure \ref{fluct2}. Within a Landau-Ginzburg picture of the first-order phase transition in nematic-to-isotropic systems \cite{phtr12} the observations can be understood. It is expected that as the system crosses the melting temperature ($T_m$) and approach the supper-cooled or super-heated temperature ($T^*$) the shape of the free energy curve changes, as shown in figure \ref{fluct1}, where the free energy is depicted as a function of the order parameter $\Gamma$. In the process of heating, as $T$ increases past $T_m$, the shape of the free energy functional and the width of the metastable minimum slightly increases, allowing for an increase in the fluctuations of $\Gamma$ and consequently of any thermodynamic quantity. This increase of deviations from the mean in $\Gamma$ is then correlated to the subtle increase in $\langle \Delta S^2\rangle$ as the system approached $T^*$. The increase in the variance in $S$ correlates with an increase in state variability from figure \ref{fluct1}. It can be clearly seen that before the transition $\langle \Delta S^2\rangle$ is smaller in magnitude than when the system approaches $T^*$ ($\sim 42 ^{\circ}$C). This may suggest that this observation highlights these fluctuations in the metastable region predicted in the Landau-Ginzburg model, which will be investigated further in future work.
\begin{figure}[!]
\centering
\subfigure[]{\label{fluct1}}\includegraphics[width=.95\linewidth,height=.7\linewidth, angle = 0,trim= 0 30 20 10,clip]{Fig7a.png}
\subfigure[]{\label{fluct2}}\includegraphics[width=.95\linewidth,height=.7\linewidth, angle = 0,trim= 0 10 90 20,clip]{Fig7b.png}
\caption{(a) Graphical depiction of the free energy of a system as a function of the order parameter($\Gamma$). (b) Graph of $\langle \Delta S^2 \rangle$ versus $T$, extracted by analyzing the various of the image in the heating data presented in figure \ref{DPRetGrallH}.}
\label{retvar1}
\end{figure}
\section{Conclusions}
PM-NSOM was utilized in determining the anisotropic structural properties in supported lipid bilayers. The technique was shown to improve previous measurements on $S$ by increasing the sensitivity and independently determining the direction of the projection of the acyl chains onto the membrane's surface. With that increased sensitivity, the longitudinal and transverse polarizability can be more accurately determined yielding results comparable to their theoretically calculated values. It was also shown that one can obtain accurate lateral high resolution $\Delta S$ and orientation images with the PM-NSOM system while aligning the system over the sample. This proved to be imperative in conducting the temperature controlled experiments and allowed one to measure relative values of $S$ and $\theta$.
By adding temperature control to PM-NSOM the main phase transition from the gel state to the liquid disorder phase was observed. The melting temperature of a single DPPC lipid bilayer was found to be in good agreement with previously reported work. The information from $S$ and $\theta$ allowed to create a three dimensional model of the average orientation of the lipid molecules within the membrane. The system's sensitivity and control allowed for the observations of increased fluctuations as the metastable region of the phase diagram of the lipid membrane was probed.
Future work could extend into membrane systems containing lipid mixtures or even protein/lipid complexes, where the properties of domains with different phases or orientations may be exploited. With temperature control, the lateral dynamics of phase separations or even changes in the orientation of the lipid bilayer through regions of interest could be observed. Overall it was shown how PM-NSOM can be utilized as a highly sensitive/non-invasive technique to study single lipid bilayer systems and achieve structural information beyond the ability of conventional optical techniques.
\begin{acknowledgments}
\section{acknowledgments}
We would like to acknowledge the use of the atomic force microscope associated with the Integrated Nanosystems Development Institute (INDI) and Nanoscale Imaging Center (NIC) at Indiana University Purdue University Indianapolis.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
|
2,877,628,090,829 | arxiv | \section{Introduction}
It is well known that Einstein's theory of General Relativity is not straightforward to quantize. This is easily seen from the fact that GR is not perturbatively renormalizable. Simply put, one can attempt to quantize GR as an ordinary spin-two field in flat Minkowski spacetime, in the following way (for a nice review see \cite{Hamber}). Starting from the usual Einstein-Hilbert action
$$
S_{EH} = \int d^4x\, \sqrt{-g} R,
$$
one rewrites the metric tensor $g_{\mu\nu}$ as the flat Minkowski metric $\eta_{\mu\nu}$ and the spin-two field $h_{\mu\nu}$ as
$$
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},
$$
and substitutes it into the action, rewriting it in terms of the new variable $h_{\mu\nu}$. Thereby one obtains
$$
\begin{array}{r}
\displaystyle S_{EH} = \int d^4x\, h_{\mu\nu} \square h^{\mu\nu} + (\text{gauge fixing terms}) + \\
+ (\text{self-interaction terms}). \\
\end{array}
$$
The D'Alambertian operator is defined in the usual way, in flat Minkowski space, $\square \equiv \eta^{\mu\nu} \partial_{\mu} \partial_{\nu}$. From here one can proceed to perform the standard field theory quantization in the naive way --- first formulate the free quantum field theory, and then introduce interactions perturbatively.
However, very soon one is bound to face the difficulty of nonrenormalizability of this theory. The tree-level Feynman diagrams are finite, the one-loop divergences can be removed by wavefunction renormalization, but at the two-loop level a Lagrangian counterterm of the form
$$
{\cal L}_2 = \frac{const}{\varepsilon^2} R^{\alpha\beta}{}_{\mu\nu} R^{\mu\nu}{}_{\rho\sigma} R^{\rho\sigma}{}_{\alpha\beta} \qquad (\varepsilon\to 0)
$$
appears \cite{GoroffSagnotti}, which is nonzero on-shell. Here $\varepsilon = 4-D$ is the cutoff parameter from dimensional regularization scheme. At higher loop levels similar terms involving $R^4$, $R^5$, etc. terms are also expected to appear, rendering the theory perturbatively nonrenormalizable. This means that in order to remove all divergences one needs to introduce at least one additional coupling constant for each loop level. The infinite number of these coupling constants implies the loss of predictive power of the theory, since all experiments doable in principle can only ever fix a finite number of coupling constants.
This property of General Relativity has been known for quite some time, and there are various research directions which attempt to address this issue. They can be broadly separated into two classes, by the methodology.
The first class of approaches considers modifying or substituting GR by another theory, which should preferably be renormalizable. Such attempts have evolved into vast research directions such as supergravity, string field theory, noncommutative geometry, and so on. The goal of each proposed model is to have a renormalizable theory that looks like GR at least on the length scales which can be tested experimentally, while at the same time have only a finite number of coupling constants. These coupling constants could then in principle be used to predict the values of the infinite set of coupling constants appearing in the perturbative quantum gravity approach.
The second class of approaches is based on the point of view that abandons the renormalization paradigm, and essentially gives physical meaning to the cutoff parameters of some particular regularization scheme. In other words, the assumption is that at some scale (typically expected to be near the Planck scale) expectation values of the physical observables will start to depend explicitly on cutoff parameters. This dependence is assumed to be measurable (in principle), rather than being removed by renormalization. These attempts have also evolved into vast research directions such as loop quantum gravity, causal dynamical triangulations, causal set theory, etc. The goal of all proposed models is exactly the same as before --- predict some definite values for the infinite number of coupling constants present in the perturbative quantum gravity.
All these research directions have had limited success, and in the absence of any experimental data relevant at the Planck scale, none of these directions can be preferred over the others.
In what follows, we shall be mainly concerned with the approach of loop quantum gravity (for a review see \cite{Rovelli}), more specifically spin foam models, and we shall propose one novel particular model that addresses some serious issues present in all other spin foam models so far.
In section \ref{LQGSection} we shall give a short overview of the status of LQG in general and spin foam models in particular. We will argue that the main drawbacks of all 4D spin foam models stem from the fact that tetrad fields are not basic variables of the theory. Section \ref{TwoGroupSection} deals with the categorical generalization of the Poincar\' e group, called the Poincar\' e $2$-group. This will give us the necessary mathematical tools to reformulate the GR action in a convenient way which includes tetrad fields as basic variables. The analysis of this new action is then given in section \ref{SpincubeSection}, with a sketch of a quantization procedure giving rise to the so-called spincube model. Section \ref{DiscussionSection} contains conclusions and discussion of the results.
\section{\label{LQGSection}Loop Quantum Gravity and Spin Foam Mo\-dels}
A detailed review of the Loop Quantum Gravity approach can be found in \cite{Rovelli}. Here we just give some basic properties at an informal level.
The basic idea of LQG is to choose diffeomorphism-invariant quantities as basic degrees of freedom for the gravitational field, and then perform a canonical nonperturbative quantization of gravity in terms of these quantities. The natural candidates for basic variables turned out to be Wilson loops, and subsequently their generalizations called spin networks. This choice of variables introduces a natural diffeomorphism-invariant cutoff at the Planck length scale $l_P$, thereby rendering the theory UV-finite. The quantization is performed in the Schr\" odinger picture, and provides one with a mathematically well-defined constructions of the kinematical Hilbert space for the theory and some basic operators for geometric observables such as lengths, areas and volumes of space. Evolution in time is embodied in the Hamiltonian constraint, corresponding to the Wheeler-de Witt equation in the LQG setting.
The main features of such canonical approach to quantization are as follows. The theory represents a nonperturbative quantization of GR, and can in principle be applied to the study of physical systems where gravity is the dominant factor at short distances --- such systems include the black hole and cosmological singularities. It gives one a mathematical handle on a well-defined Hilbert space of states for the gravitational field, thereby giving some insight into the quantum mechanical features of gravity. The natural basis for the Hilbert space is the set of the {\em spin network states}, combinatorial graphs colored by the irreducible representations of the $SU(2)$ group, and corresponding intertwiners. Finally, the study of the geometric observables --- the length, area and volume operators --- reveals that each of them has a discrete spectrum, giving rise to the geometric interpretation of the gravitational field wavefunctional, as well as the discrete character of space.
The theory also has some drawbacks. First, the Hamiltonian constraint is not uniquely defined, due to the usual ordering problems present in quantum mechanics. Second, even if one chooses some particular ordering, the Hamiltonian constraint is extremely complicated and impossible to solve in practice. This severely limits the possibility for any practical calculations and the study of the dynamics of the theory. As the main obstacle, the proof of the correct semiclassical limit of the theory is still missing, as well as any attempt to predict the coupling constants from the perturbative gravity approach.
A way to resolve these drawbacks has been found in the spin foam approach \cite{RovelliSpinFoams}. The idea is to give up canonical quantization, but instead attempt a covariant, path-integral quantization of the theory. Building on the results of the canonical approach, one wants to define the gravitational path-integral
$$
Z = \int {\cal D} g_{\mu\nu} \exp \left( i S_{EH}[g_{\mu\nu}] \right)
$$
in some way, in order to be able to calculate expectation values of observables, both in deep quantum regime and the semiclassical regime. This approach tends to give one a good handle on the dynamics of the theory, in addition to all features of the canonical approach.
The basic procedure of defining $Z$ goes as follows. One starts from the Plebanski action for General Relativity,
$$
S = \int B_{ab}\wedge R^{ab} + \phi^{abcd} B_{ab} \wedge B_{cd}.
$$
The first part of this action represents the topological $BF$ theory for the $SO(3,1)$ group. The $R^{ab}$ is the curvature $2$-form, a field strength ``$F$'' for the $SO(3,1)$ connection $1$-form $\omega^{ab}$. The $B_{ab}$ is the Lagrange multiplier $2$-form. The second part of the action is the Plebanski constraint, featuring $B_{ab}$ and the $0$-form Lagrange multiplier $\phi^{abcd}$. The purpose of the constraint is to enforce the $B_{ab}$ to be a simple $2$-form (i.e. an exterior product of two $1$-forms). This constraint is therefore called ``simplicity constraint'', and it can be shown that the simplicity requirement of the $B_{ab}$ field is enough to convert the topological $BF$ theory into General Relativity. The fact that $B_{ab}$ is simple gives rise to nontrivial degrees of freedom in the theory, reducing the equation of motion for $\omega^{ab}$ from Riemann-flat to Ricci-flat.
The second step is the quantization of the topological $BF$ theory. This can be done in a rigorous way by employing the methods of topological quantum field theory. One first discretizes spacetime into $4$-simplices, motivated by the structure of space in the canonical LQG, and rewrites the $BF$ action in the form
$$
\int B_{ab} \wedge R^{ab} \stackrel{\text{discr.}}{\longrightarrow} \sum_{\triangle} B_{\triangle} R_{\triangle},
$$
where the sum goes over all triangles in the triangulation. Then one defines a topological invariant
$$
\begin{array}{lcl}
Z & \equiv & \displaystyle \int {\cal D}\omega \int {\cal D} B \exp \Big( i\sum_{\triangle} B_{\triangle} R_{\triangle} \Big) = \\
& = & \displaystyle \sum_{\Lambda} \prod_f A_2(\Lambda_f) \prod_v A_4(\Lambda_v). \\
\end{array}
$$
Here $\Lambda$ are the irreducible representations of $SO(3,1)$, labeling the faces $f$, edges $e$ and vertices $v$ of the Poincar\' e dual lattice corresponding to the triangulation. The colored $2$-complex dual to the spacetime triangulation is called a {\em spin foam}. The amplitudes $A_2(\Lambda)$ and $A_4(\Lambda)$ are determined such that $Z$ is in fact a topological invariant --- the total expression must not depend on the particular choice of the spacetime triangulation. In that way one arrives at the TQFT corresponding to the $BF$ theory for the $SO(3,1)$ group, commonly called the {\em Ooguri spin foam model}. Of course, the invariant $Z$ may be (and actually is) badly divergent, but that is not important at this stage, since we are only interested in the structure of the path integral.
The last step in the quantization procedure is to enforce the simplicity constraint on the $BF$ path integral at the quantum level. The exact technique for this is quite involved \cite{EPRL,FK}, but the bottom-line is that one projects the $SO(3,1)$ irreducible representations $\Lambda$ to the $SU(2)$ representations present in the canonical LQG formalism, in order to obtain the same structure of the Hilbert space on the spin foam boundary. The resulting theory is not topologically invariant, but represents one possible rigorous definition for the theory of quantum gravity. The most advanced spin foam model in this respect is the EPRL/FK model, developed independently by two research groups \cite{EPRL,FK}.
The main feature of spin foam models is that they correct some drawbacks of the canonical theory, primarily the dynamical sector is more under control. In addition, there remains a certain ambiguity in the choice of the amplitudes $A_2$ and $A_4$. This can be conveniently utilized to redefine the model such that it becomes IR-finite and to have a correct semiclassical limit \cite{MVefact,MVfiniteness}. One can also employ standard QFT methods to calculate the effective action for the model in the semiclassical limit, which opens a possibility to explicitly determine the coupling constants from perturbative quantum gravity.
Unfortunately, the spin foam models introduce their own set of problems. Aside from the ``unusual'' properties like fuzziness of geometry at the Planck scale, all spin foam models suffer from two major handicaps. The first is related to the fact that, in addition to the good semiclassical limit, all models have {\em additional semiclassical limits}, which do not give rise to the standard GR, but to the so-called {\em area-Regge geometry}. Since these different classical limits are not observed in experiments, one needs some additional mechanism to suppress such solutions. However, so far no mechanism could be constructed to deal with this problem.
The second handicap is related to the inability of the spin foam models to couple matter fields to gravity. Namely, the basic geometric variables which are employed in description of spacetime geometry are areas and volumes of space, but not lengths. This situation makes it extremely complicated (and in the case of massive fermionic matter even impossible) to incorporate matter fields into the spin foam model. Even if doable (see \cite{RovelliFermions} for the massless fermion coupling), the resulting theory would be too complicated to be useful for any calculation.
As it turns out, both of these handicaps have a common origin --- the edge lengths in the triangulation are not well-defined at the quantum level. This is itself a consequence of the choice of spin network states as basic degrees of freedom in the canonical LQG --- the choice which emphasizes the spin connection $\omega^{ab}$, while entirely ignoring the tetrad fields $e^a$. At the level of spin foam models, it is easy to see that the Plebanski constraint was purposefully designed to require the simplicity of $B_{ab}$, while avoiding to explicitly state that (the dual of) $B_{ab}$ is the product of two tetrad $1$-forms. The reason for this is that the tetrad fields do not appear as variables in the topological $BF$ sector of the theory, which is being used for the definition of the path integral.
In the remainder of this paper we will present a novel way to address this main difficulty, and to introduce tetrad fields explicitly in the topological sector of the theory. However, in order to do this, it is important to introduce some mathematical concepts which provide the background formalism for the new model.
\section{\label{TwoGroupSection}Poincar\' e 2-group}
We begin by giving a very brief review of the so-called {\em categorification ladder}, an important and active research topic in category theory. We shall not attempt at any rigor, leaving out most of the details, which can be found for example in \cite{BaezHuerta} and references therein.
In the branch of mathematics called {\em category theory}, one defines a structure called a {\em category} as a set of {\em objects} and a set of {\em morphisms} between those objects, satisfying some basic axioms. Such a structure is fairly general and does not have many interesting properties itself. However, this generality allows one to use it for all sorts of purposes. For example, one can define the usual structure of a {\em group} as a category which has only one object, while all morphisms (mapping the object onto itself) are invertible. The composition rules for the morphisms can be chosen to be the group multiplication, thereby providing an isomorphism between a given group and the corresponding category with one element.
The first step in the categorification ladder is to introduce the concept of a {\em 2-category}. A $2$-category consists of a set of objects, a set of morphisms and a set of {\em 2-morphisms}, maps between morphisms. Intuitively, if a category can be represented by a linear graph of dots (objects) and arrows connecting them (morphisms), a $2$-category can be represented by a planar graph, consisting of dots (objects), arrows connecting them (morphisms) and ``surface arrows'' mapping one arrow into another (see \cite{BaezHuerta} for details and pictures). The main point is that the dimensionality of the graph has been raised by one. The categorification ladder can continue by introducing a $3$-category (or in general an $n$-category) by a similar process, leading to $3$-dimensional (in general $n$-dimensional) graphs.
In analogy with a group, one can then define a {2-group}, as a $2$-category which has only one element, while all morphisms and $2$-morphisms are invertible. A $2$-group is a categorical generalization of a group, and is not a group itself. One can prove that any $2$-group is equivalent to a {\em crossed module}, a structure that has been studied independently by mathematicians before the idea of the categorification ladder has even been introduced. A crossed module is a quadruple $(G,H, \partial,\triangleright)$. This is a pair of groups $G$ and $H$, such that $\partial: H \to G$ is a homomorphism and $\triangleright : G\times H \to H$ is an action of $G$ on $H$ such that certain axioms are satisfied, which turn out to be directly related to the structure of a $2$-category, see \cite{BaezHuerta}. The elements of $G$ represent the $1$-morphisms, while the elements of the semidirect product $G\ltimes H$ represent the $2$-morphisms. The canonical example of a $2$-group relevant for physics is the Poincar\'e $2$-group, where $G = SO(3,1)$, $H={\ensuremath{\mathbb{R}}}^4$, $\partial$ is a trivial homomorphism and $\triangleright$ is the usual action of the Lorentz transformations on the ${\ensuremath{\mathbb{R}}}^4$ space. The Lorentz group is the group of morphisms, while the usual Poincar\'e group is the group of $2$-morphisms.
The main feature of the whole $2$-group formalism is that one can generalize the concept of a {\em holonomy} along a line to its two-dimensional analog --- a {\em surface holonomy}. The initial interest in this came from string theory. A point-particle travels along a world line in spacetime, and one is naturally led to the concept of a parallel transport along a given line. String theory promotes the point particle into a one-dimensional object --- a string --- which then travels along a world surface in spacetime. Thus one would like to have a concept of a {\em parallel transport along a given surface}. One of the main aims of the $2$-category and $2$-group formalism is to introduce and formalize this concept.
Given the strong categorical relationship between groups and $2$-groups, one can construct a gauge theory on a $4$-manifold ${\cal M}$ based on a crossed module $(G,H, \partial,\triangleright)$ of Lie groups by using $1$-forms $A$, which take values in the Lie algebra $\mathfrak g$ of $G$, and $2$-forms $\beta$, which take values in the Lie algebra $\mathfrak h$ of $H$ \cite{GPP,fmm}. The forms $A$ and $\beta$ transform under the usual gauge transformations $g: {\cal M} \to G$ as
$$
A \to g^{-1} A g \,+\,g^{-1} d g \,,\quad \beta\to g^{-1} \triangleright \beta \,,
$$
while the gauge transformations generated by $H$ are given by
$$
A \to A\,+\,\partial\eta\,,\quad\beta\to \beta +d \eta + A \wedge^\triangleright \eta + \eta\wedge \eta \, ,
$$
where $\eta$ is a one-form taking values in $\mathfrak h$, see \cite{fmm}. When the group $H$ is Abelian, which happens in the Poincar\'e 2-group case, then the $\eta\wedge\eta$ term vanishes, and one obtains the gauge transformations given in \cite{GPP}.
The pair $(A,\beta)$ represents a $2$-connection on a $2$-fiber bundle associated to the $2$-Lie group $(G,H)$ and the manifold ${\cal M}$. The corresponding curvature forms are given by
$$
{\cal F} = dA + A\wedge A -\partial\beta \,,\quad
{\cal G} = d \beta + A \wedge^\triangleright \beta \,,
$$
and they transform as
$$
{\cal F} \to g^{-1} {\cal F} g\,, \quad {\cal G} \to g^{-1} \triangleright {\cal G} \,,
$$
under the usual gauge transformations, while
$$
{\cal F} \to {\cal F}\,,\quad {\cal G} \to {\cal G} +{\cal F} \wedge^\triangleright \eta\,,
$$
under the $H$-gauge transformations.
One can introduce a natural topological gauge theory determined by the vanishing of the 2-curvature
$$
{\cal F} = 0 \,,\quad {\cal G} =0 \,.
$$
These equations can be obtained from the action
$$
S =\int \langle B \wedge {\cal F} \rangle_{\mathfrak g} + \langle C \wedge {\cal G} \rangle_{\mathfrak h} \,,
$$
where $B$ is a Lagrange multiplier $2$-form taking values in $\mathfrak g$, $C$ is a Lagrange multiplier $1$-form taking values in $\mathfrak h$, $\langle\;\;,\;\;\rangle_{\mathfrak g}$ is a $G$-invariant nondegenerate bilinear form in $\mathfrak g$ and $\langle\;\;,\;\;\rangle_{\mathfrak h}$ is a $G$-invariant nondegenerate bilinear form in $\bf h$. This action is called $BFCG$ action, in analogy with the $BF$ theory action. The gauge transformations of the Lagrange multiplier fields are given by
$$
B \to g^{-1} B g \,,\quad C \mapsto g^{-1} \triangleright C \,,
$$
for the usual gauge transformations, while
$$
B \to B - [C,\eta] \,,\quad C \mapsto C \,,
$$
for the $H$-gauge transformations.
Let us now examine the case of the Poincar\'e 2-group. In this case $A=\omega^{ab}J_{ab}$, $\beta = \beta^a P_a$, where $a,b\in\{0,1,2,3\}$,
$J_{ab}$ are the generators of the Lorentz group while $P_a$ are the generators of the translation group ${\ensuremath{\mathbb{R}}}^4$. Consequently
$$
\begin{array}{cclcl}
{\cal F} & = & \displaystyle (d\omega^{ab} + \omega^a{}_c\wedge\omega^{cb} )J_{ab} & = & R^{ab}J_{ab} , \\
{\cal G} & = & \displaystyle \left(d\beta^a + \omega^a{}_b \wedge \beta^b \right) P_a & = & \left(\nabla\beta^a \right) P_a. \\
\end{array}
$$
The $G$-gauge transformations are the local Lorentz rotations
$$
\omega \to g^{-1} \omega g + g^{-1} dg \,,\quad \beta \to g^{-1} \triangleright \beta \,,
$$
while the $H$-gauge transformations are the local translations
$$
\delta_\varepsilon \omega^{ab} =0 \,,\quad \delta_\varepsilon \beta^a = d\varepsilon^a + \omega^a{}_b \wedge \varepsilon^b \,,
$$
where $\eta = \varepsilon^a P_a$.
The $BFCG$ action then becomes
$$
S = \int_{{\cal M}} \left( B^{ab}\wedge R_{ab} + C_a \wedge \nabla\beta^a \right)\,,
$$
where
$$
\delta_\varepsilon B = 0 \,,\quad \delta_\varepsilon C = 0 \,.
$$
At this point a very important observation is in order. The transformation properties of the $1$-form $C^a$ are the same as the transformation properties of the tetrad $1$-form $e^a$ under the local Lorentz and the diffeomorphism transformations. In addition, the equation of motion for $C^a$ is $\nabla C^a =0$, just like the no-torsion equation for the tetrad, $\nabla e^a =0$. Based on this, {\em we identify the Lagrange multiplier $C^a$ with the tetrad field $e^a$}, and write the action in the form
$$
S = \int_{{\cal M}} \left( B^{ab}\wedge R_{ab} + e^a \wedge \nabla \beta_a \right) \,. \label{tga}
$$
In this way one can construct a categorical generalization of the topological $BF$ action. The new action is again topological, but more rich in structure, since the tetrad fields are explicitly present. In addition, the $2$-group formalism provides a framework to construct a topological quantum field theory from this action, in analogy with the $BF$ case. This provides us with the necessary tools to construct a categorical generalization of a spin foam model, based on the $BFCG$ action instead of the $BF$ action. The explicit presence of the tetrads should help us resolve the two handicaps of spin foam models discussed in section \ref{LQGSection}.
\section{\label{SpincubeSection}The Spincube Model}
The first step in the construction of the new model is to write the action for General Relativity, starting from the $BFCG$ action. In order to do this, all we need is the simplicity constraint,
$$
B_{ab} = \varepsilon_{abcd}\, e^c\wedge e^d\,,
$$
which can now be added into the action as it stands, as opposed to the $BF$ case where the Plebanski constraint had to be introduced due to the absence of the tetrads $e^a$ in the $BF$ action. Therefore, one can write the {\em constrained $BFCG$ action} in the form
\begin{equation} \label{2pgr}
S = \int_{{\cal M}} \Big[ B^{ab}\wedge R_{ab} + e^a \wedge \nabla \beta_a -
\phi_{ab}\wedge \left( B^{ab} - \varepsilon^{abcd}e_c \wedge e_d \right) \Big]\,,
\end{equation}
where $\phi_{ab}$ is an additional Lagrange multiplier $2$-form field, introduced in order to enforce the simplicity constraint.
The equations of motion are obtained by varying $S$ with respect to $B$, $e$, $\omega$, $\beta$ and $\phi$, respectively, to give:
$$
\begin{array}{l}
R_{ab} - \phi_{ab} = 0\,,\\
\nabla\beta_a + 2\varepsilon_{abcd} \phi^{bc}\wedge e^d = 0\,, \\
\nabla B_{ab} - e_{[a} \wedge \beta_{b]} = 0\,, \\
\nabla e_a = 0\,, \\
B_{ab} - \varepsilon_{abcd} e^c \wedge e^d = 0\, .\\
\end{array}
$$
With the usual assumption that the tetrad fields are nondegenerate, these equations can be reworked into an equivalent form:
$$
\phi^{ab} = R^{ab}, \qquad B_{ab} = \varepsilon_{abcd} e^c \wedge e^d, \qquad \beta^a = 0,
$$
$$
\nabla e^a = 0\,, \qquad \varepsilon_{abcd} R^{bc}\wedge e^d = 0\,.
$$
The first three equations determine $\beta^a$ and the multipliers $B_{ab}$ and $\phi_{ab}$ in terms of $e^a$ and $\omega^{ab}$. The fourth equation is the no-torsion equation, which determines the connection $\omega^{ab}$ to be the Levi-Civita connection (a function of the tetrads $e^a$). The last equation is nothing but the Einstein field equation for the only remaining field $e^a$. Thus we see that the action (\ref{2pgr}) is classically equivalent to General Relativity. More precisely, it is equivalent to the Einstein-Cartan theory,
$$
S_{EC} = \int_{{\cal M}} \varepsilon_{abcd} e^a \wedge e^b \wedge R^{cd} \,,
$$
since the torsion is equal to zero as an equation of motion rather than by definition.
Given the new action for General Relativity, we can proceed with the covariant quantization in analogy with the spin foam models. The action has the topological term and the constraint term, so as a first step we construct a topological quantum field theory by defining the path integral for the $BFCG$ part of the action. In the second step, we enforce the constraint term by requiring a suitable restriction in the path integral of the topological theory.
One begins by triangulating spacetime into $4$-simplices, and rewriting the topological part of the action in the form
$$
\sum_{\triangle} B_{\triangle} R_{\triangle} + \sum_{l} e_l ( \nabla\beta )_l,
$$
where the first sum goes over all triangles and the second goes over all edges in the triangulation of the spacetime manifold. Then one constructs a topologically invariant path integral in the form (see \cite{MVtwoPoincare} for the details of the construction)
\begin{equation} \label{StateSuma}
\begin{array}{lcl}
Z & \equiv & \displaystyle \int {\cal D}\omega \int {\cal D} B \int {\cal D} e \int {\cal D} \beta \,
\exp \Big( i\sum_{\triangle} B_{\triangle} R_{\triangle} + i \sum_{l} e_l ( \nabla\beta )_l \Big) = \\
\vphantom{\displaystyle\int} & = & \displaystyle \sum_{\Lambda} \prod_p A_1(\Lambda_p) \prod_f A_2(\Lambda_f) \prod_v A_4(\Lambda_v). \\
\end{array}
\end{equation}
The labels $\Lambda = (L_p,m_f)$, where $L_p \in\ensuremath{\mathbb{R}}_0^+$ and $m_f\in\ensuremath{\mathbb{Z}}$, are now irreducible representations of the Poincar\' e $2$-group, and in addition to vertices $v$ and faces $f$ of the Poincar\' e dual lattice, we also take the product over all the polyhedra $p$, since they are dual to the edges of the triangulation and naturally appear in the construction due to the presence of the $e\wedge \nabla\beta$ term in the $BFCG$ action. The amplitudes $A_1(\Lambda)$, $A_2(\Lambda)$ and $A_4(\Lambda)$ are chosen so that $Z$ does not change under the action of the Pachner moves, which guarantees its independence of the triangulation. The polyhedra are colored with $L_p$, which have the interpretation as lengths of triangulation edges, while faces are colored with $m_f$, which have the interpretation as areas of the triangles in the triangulation. In the topological theory, edge lengths and triangle areas are independent of each other.
Note that the path integral is not defined over a colored $2$-complex (the spinfoam), but rather over a colored $3$-complex (called {\em spincube}).
Finally, we can impose the simplicity constraint, in order to turn the topological path integral into a realistic model for quantum gravity. Based on the geometric interpretation of the variables, the constraint actually says that a very natural requirement should be enforced --- the triangle areas must be compatible with the corresponding edge lengths. This can be formalized in the requirement
$$
|m_f|l_P^2 = A_f (L) , \quad \forall f
$$
where $A_f(L)$ is the Heron formula for the triangle area in terms of its edges. The Planck length appears naturally in order to balance the dimensions of the two sides of the equation. As a last step, one redefines the amplitudes $A_1$, $A_2$ and $A_4$ in order to render the theory IR-finite, as well as to enforce the correct semiclassical limit, in a way similar to the spinfoam models.
Note that imposing this constraint leaves only edge lengths as independent variables in the theory, so that the ``area-Regge'' problem present in spinfoam models is resolved automatically. In addition, the edge length variables allow for a completely straightforward coupling of matter fields to the spincube model. Namely, at the level of the classical theory, one can introduce fermions via the action
\begin{equation} \label{DejstvoSaFermionom}
\begin{array}{ccl}
S & = & \displaystyle \int \Big[ B^{ab}\wedge R_{ab} + e^a \wedge \nabla \beta_a - \phi_{ab}\wedge \left( B^{ab} - \varepsilon^{abcd}e_c \wedge e_d \right) \Big] + \\
& & \displaystyle \hphantom{mm} + i \kappa_1 \int \varepsilon_{abcd} \, e^a \wedge e^b \wedge e^c \wedge \bar{\psi} \left[ \gamma^d \stackrel{\leftrightarrow}{d}{} + \{ \omega , \gamma^d \} + \frac{im}{2}\,e^d \right] \psi + \\
& & \displaystyle \hphantom{mmmmmmm} + i\kappa_2 \int \varepsilon_{abcd} e^a \wedge e^b \wedge \beta^c \, \bar{\psi}\gamma_5\gamma^d \psi \, , \\
\end{array}
\end{equation}
where $\omega = \omega_{ab} [\gamma^a , \gamma^b]/8$, $\kappa_1 = 8\pi l_P^2 /3$ and $\kappa_2 = - 2 \pi l_P^2 $. The first term is the constrained $BFCG$ action, while the second and third terms introduce fermion coupling which results in the same equations of motion as in the ordinary Einstein-Cartan theory with fermions.
The quantization procedure of the action (\ref{DejstvoSaFermionom}) is essentially the same as the one without fermions. The only difference is in the fact that the vertex amplitude $A_4$ will change to reflect the presence of the fermionic matter, as
$$
A_4 \to A_4 \exp\left[ iS_R^{(\rm ferm)}(L,\psi) \right] \, ,
$$
where $S_R^{(\rm ferm)}$ is the Regge discretized action of a fermion field $\psi$ coupled to gravity. The expressions which appear in $S_R^{(\rm ferm)}$ can be easily obtained, in contrast to the EPRL/FK model case, where the expression for the $4$-simplex volume is impossible to define uniquely in terms of the spin foam variables \cite{RovelliFermions}.
Similarly to (\ref{DejstvoSaFermionom}), one can also couple other matter fields to (\ref{2pgr}) in a completely straightforward way, including gauge and scalar fields, the cosmological constant, the Holst term, and so on.
\section{\label{DiscussionSection}Conclusions}
The proposed 2-group reformulation of GR can be used to obtain a categorical ladder generalization of Loop Quantum Gravity. The advantage of this generalization is that the edge lengths of a triangulation become the basic dynamical variables. This will facilitate the construction of the path integral such that the classical limit of the corresponding quantum theory is GR and the coupling of matter will be much easier to accomplish.
The categorical nature of the theory implies that the edge labels of a spacetime triangulation should be the 2-group irreducible representations on a $2$-Hilbert space. Note that this is not unique, since one can also use the category of chain complexes of vector spaces in order to define the representations, see \cite{fmm,cfm}. The structure of the chain-complex representations is different from the 2-Hilbert space representations, which means that chain-complex representation theory defines an alternative quantization of GR. Hence it would be interesting to develop the chain-complex representation theory of the Poincar\' e 2-group.
The physical significance of 2-Hilbert space representations could be better understood by performing a canonical quantization of the action (\ref{2pgr}).
As far as the construction of 4-manifold invariants based on the $BFCG$ state sum is concerned, one would have to regularize the topological state sum/integral based on the amplitude (\ref{StateSuma}) such that the triangulation independence is preserved. One way to do it is to try to implement a gauge-fixing procedure, see \cite{bafr}. Another way is to find a quantum group regularization, since there are strong indications that categorified quantum groups and their representations will be important for the construction of $4$-manifold invariants \cite{crfr}. Hence one can try to find a crossed module of Hopf algebras which is a deformation of the Poincar\' e 2-group, and then try to find an appropriate $2$-category of representations which will give a finite topological state sum.
\ack
This work has been partially supported by FCT project PTDC/MAT/099880/2008. MV was also supported by the FCT grant SFRH/BPD/46376/2008.
\section*{References}
|
2,877,628,090,830 | arxiv | \section{Introduction} In this paper, our aim is to count the
number of rational points of bounded height on the surface $S
\subset \mathbb{P}^6$ given by
\begin{align}
x_{3}^{2} + x_{0} x_{5} + x_{1} x_{6} & = x_{2} x_{3} - x_{0}
x_{6} =x_{1} x_{2} + x_{0} x_{3} + x_{0} x_{4}= 0,\nonumber \\
x_{3} x_{5} + x_{4} x_{5} + x_{6}^{2} & = x_{2} x_{5} - x_{4} x_{6}
= x_{1} x_{5} - x_{3} x_{6}= 0,\label{equations}\\
x_{4}^{2} + x_{0} x_{5} + x_{2} x_{6} & = x_{3} x_{4} - x_{0} x_{5}
= x_{1} x_{4} - x_{0} x_{6}= 0.\nonumber
\end{align}
This surface is an example of a singular del Pezzo surface of degree
$6$. A priori, it might not be clear why this is a natural
diophantine problem. However in 1989, Manin and his collaborators
\cite{FMT89} formulated a general conjecture on the number of
rational points of bounded height on Fano varieties. There is a
programme (see \cite{BB07} or \cite{DT07} for example) to try to
prove this conjecture for Fano surfaces, namely \emph{del Pezzo
surfaces} and their singular counterparts. Such surfaces have a
well-known classification in terms of their singularity type and
degree. See \cite{Man86} and \cite{CT88} for more information on
smooth and singular del Pezzo surfaces respectively, and
\cite{Bro07} for a general overview of Manin's conjecture for del
Pezzo surfaces.
The surface $S$ has one singularity of type $\mathbf{A}_2$, which we
can resolve using blow-ups to create two exceptional curves on the
minimal desingularisation $\widetilde{S}$ of $S$. The set of
equations $(\ref{equations})$ correspond to the embedding induced by
a divisor in the \emph{anticanonical divisor class}. Since $S$ is
singular normal with only rational double points, by \cite[Prop.
0.1]{CT88} an anticanonical divisor of $S$ can be taken to be any
divisor on $S$ which pulls back to an anticanonical divisor on the
minimal desingularisation $\widetilde{S}$. The anticanonical
embedding is a natural choice, for example in this embedding the
lines are exactly the $(-1)$-curves and Manin's conjecture takes a
simpler form. The height function associated to the chosen embedding
is the usual height on projective space, namely given $x \in
S(\mathbb{Q})$, we have $H(x)=\max_{0\leq i \leq 6}|x_i|$, where
$(x_0,\ldots,x_6)$ is a primitive integer vector in the affine cone
above $x$. Further details about the geometry of $S$ can be found in
Lemma~\ref{lem:geom}.
Now, $S$ contains the two lines
\begin{align*}
&L_1: x_1=x_3=x_4=x_5=x_6=0, \\
&L_2: x_2=x_3=x_4=x_5=x_6=0,
\end{align*}
which both contain ``many" rational points whose contribution will
dominate the counting problem. Hence, it is natural to let
$U=S\setminus \{L_1 \cup L_2\}$ and take
$$N_{U,H}(B)=\#\{ x \in U(\mathbb{Q}) : H(x) \leq B\}$$
to be the associated counting function. In this context, Manin's
conjecture predicts an asymptotic formula of the shape
$$N_{U,H}(B) \sim c_{\widetilde{S},H}B (\log B)^ {\rho-1}$$
as $B \to \infty$, where $\rho=\rank (\Pic(\widetilde{S}))=4$ and
$c_{\widetilde{S},H}$ is some constant. In this paper, we establish
a significantly sharper version of this estimate.
\begin{thm}\label{thm:asym}
Let $\varepsilon >0$. Then there is a monic cubic polynomial $P \in
\mathbb{R}[x]$ such that
$$N_{U,H}(B) = c_{\widetilde{S},H} B P(\log B) +
O_\varepsilon(B^{7/8+\varepsilon})$$
where
$c_{\widetilde{S},H}=\alpha(\widetilde{S}) \tau_\infty(\widetilde{S}) \prod_p \tau_p(\widetilde{S})$ and
\begin{align*}
\alpha(\widetilde{S}) &= 1/432,\quad\tau_p(\widetilde{S})=
\left(1-\frac{1}{p}\right)^4\left(1 + \frac{4}{p} + \frac{1}{p^2}\right), \\
\tau_\infty(\widetilde{S}) & =
6\int_{\{t,v,u \in \mathbb{R}:0<|t(ut+v^2)|,|uvt|,
|uvt+v^3|,|u^2t|,|u^2t+uv^2|,u^3,u^2v\leq1\}} \mathrm{d}u\mathrm{d}v\mathrm{d}t.
\end{align*}
\end{thm}
The leading constant in this expression agrees with the prediction
of Peyre \cite{Pey95}, which we shall verify in Section
\ref{subsec:constant}. The calculation of the real density
$\tau_\infty(\widetilde{S})$ poses something of a challenge, since
in our case $S$ is not given by a complete intersection, so standard
methods for calculating this constant do not apply. In Section
\ref{subsec:constant} we also prove a general result which assists
in the calculation of the $p$-adic densities $\tau_p(\widetilde{S})$
(See Lemma~\ref{lem:densities}).
The second theorem of this paper is intimately related to the above
asymptotic formula. We give an explicit expression and meromorphic
continuation of the associated \emph{height zeta function}
\begin{equation}
Z_{U,H}(s)= \sum_{x \in U(\mathbb{Q})}
\frac{1}{H(x)^{s}}. \label{HZF}
\end{equation}
To state the result, let $\re(s)
> 0$ and define
\begin{equation}\label{E12}
\begin{split}
E_1(s+1)&=\zeta(4s+1)\zeta(3s+1)^2\zeta(2s+1), \\
E_2(s+1)&=\frac{\zeta(7s + 3)^4\zeta(8s + 3)^2}
{\zeta(4s + 2)^3\zeta(5s + 2)^2\zeta(6s +
2)\zeta(10s+4)}.
\end{split}
\end{equation}
It is clear that $E_1(s)$ and $E_2(s)$ have a meromorphic
continuation to the whole complex plane. Also $E_1(s)$ has a single
pole of order $4$ at $s=1$ and $E_2(s)$ is holomorphic on $\re(s) >
3/4$. We then prove the following.
\begin{thm}\label{thm:HZF}
Let $\varepsilon >0$, then
$$Z_{U,H}(s)=E_1(s)E_2(s)G_1(s) + \frac{12/\pi^2 +
2\lambda}{s-1} + G_2(s).$$
Here, $\lambda \in \mathbb{R}$ is a constant and $G_1(s)$ and $G_2(s)$ are
complex functions that are holomorphic on $\re(s) > 5/6$ and
$\re(s) \geq 3/4 + \varepsilon$
respectively and satisfy $G_1(s) \ll_{\varepsilon} 1$ and $G_2(s) \ll_{\varepsilon} (1 +
|\im(s)|)$ on these half-planes.
In particular, $(s-1)^4Z_{U,H}(s)$ has a holomorphic continuation to
the half-plane $\re(s) >5/6$.
\end{thm}
Expressions for $G_1(s)$ and $G_2(s)$ can be found in
(\ref{G2}),(\ref{G11}),(\ref{G1}) and Lemma~\ref{lem:G12}. Here
$E_1(s)E_2(s)G_1(s)$ and $G_2(s)$ correspond to the main term and
error term in the counting argument respectively and $12/\pi^2$
corresponds to an isolated conic in the surface. We only prove the
existence of $\lambda$, however a keen reader can build an explicit
(and complicated) expression for it using the work in Section
\ref{subsec:error}. We shall only say that $\lambda$ arises
naturally in the proof as an error term created by approximating a
sum by an integral and has appeared in some form in other works
(e.g. \cite{BB07}), however it is currently severely lacking in
geometric interpretation.
We will show in Lemma~\ref{lem:geom} that the surface $S$ is an
equivariant compactification of $\mathbb{G}_a^2$, so that the work
of Chambert-Loir and Tschinkel \cite{CT02} applies, where they have
already achieved an analytic continuation of the associated height
zeta function and an asymptotic formula for the counting problem.
However, our results are stronger for a number of reasons. Firstly,
we do not use the fact that $S$ is an equivariant compactification
of $\mathbb{G}_a^2$, so our methods seem applicable to more general
situations. We also get an explicit expression for the height zeta
function in terms of the Riemann zeta function, which gives a better
insight into how these zeta functions look and behave for a concrete
example. Furthermore, whereas \cite{CT02} only gives a holomorphic
continuation of $(s-1)^4Z_{U,H}(s)$ to an unspecified half-plane
$\re(s)>1-\delta$, we are able to show that $\delta=1/6$ is
acceptable, and that $\delta=1/4$ appears to be a natural boundary
under the assumption of the Riemann hypothesis. As a consequence, we
get an explicit (and stronger) error term in our asymptotic formula.
The first important step in the proof of Theorem~\ref{thm:HZF} is to
relate the counting problem on $S$ to that of counting integral
points on the associated \emph{universal torsor}. Universal torsors
were introduced by Colliot-Th\'{e}l\`{e}ne and Sansuc in
\cite{CTS87} to aid the study of the Hasse principle and weak
approximation. However, Salberger \cite{Sal98} showed that they
could be a valuable tool in counting problems on varieties. In
general a variety may have more than one universal torsor, however
in our case there is only one universal torsor (see Section
\ref{subsec:torsor} for further details). It can be visualised as a
certain open subset $\mathcal{T}$ of the affine variety in
$\mathbb{A}^7$ given by the following equation
$$\eta_2\alpha_1^2+ \eta_3\alpha_2 +\eta_4\alpha_3 = 0.$$
For our purposes, the universal torsor is a variety with a
surjective morphism $\pi:\mathcal{T} \to S$ defined over
$\mathbb{Q}$, and an action of $\mathbb{G}_m^4$ on $\mathcal{T}$
which preserves the fibres of $\pi$ and acts freely and transitively
on them. Exact definitions can be found in the above references, and
a concrete realisation of the universal torsor can be found in Lemma
\ref{lem:torsor}.
To relate the two counting problems we find a suitable set-theoretic
section of the map $\pi$, which corresponds to requiring that we
count certain integral points satisfying the universal torsor
equation and certain coprimality conditions. Previous methods for
achieving this in similar problems have been the ``elementary
method" \cite[Section 4]{BB07} and the ``blow-up method"
\cite[Section 4]{DT07}. The first method involves looking for
divisibility relations given by the equations of the surface, and
then performing a lengthy chain of substitutions to pull out any
highest common factors among the variables. The second method
involves knowing which exact points of $\mathbb{P}^2$ are blown-up
to create your surface, and using these to guide you through various
algebraic manipulations.
Here we present a new method, which uses the action of
$\mathbb{G}_m^4$ on the universal torsor. Essentially, we use this
action to ``rescale" each point in each fibre to a unique point.
Since the universal torsor (if it exists) of a more general variety
always has a free and transitive group action on its fibres, this
method is more likely to generalise to other situations than the
previously two mentioned methods. See Lemma~\ref{lem:section} for
more details.
\textbf{Notation}: To simplify notation, throughout this paper
$\varepsilon$ is any positive real number which all implied
constants are allowed to depend upon. We use the common practice
that $\varepsilon$ can take different values at different points of
the argument.
\textbf{Acknowledgments}: The author is funded by an EPSRC student
scholarship and is grateful for the help and support of Tim
Browning, and for useful conversations with Per Salberger, Emmanuel
Peyre, Ulrich Derenthal, Tomer Schlank, Tony Scholl and R\'{e}gis de
la Bret\`{e}che. We are also indebted to the referee for their
careful reading of the preliminary manuscript and many useful
comments.
\section{Preliminary Steps}
\subsection{Some Geometry}
The underlying geometry of the surface $S$ is well understood, and
we gather some facts about it in the following lemma, which also
helps to fix some notation.
\begin{lem}\label{lem:geom}
Let $S$ be given by (\ref{equations}). Then the following holds.
\begin{itemize}
\item $S$ is a split singular del Pezzo surface of degree $6$ given by
its anticanonical embedding.
\item It contains the singular point $(1:0:0:0:0:0:0)$ of type
$\mathbf{A}_2$.
\item The only lines in $S$ are given by
\begin{align*}
&L_1: x_1=x_3=x_4=x_5=x_6=0, \\
&L_2: x_2=x_3=x_4=x_5=x_6=0.
\end{align*}
In particular $U=S\setminus\{L_1 \cup
L_2\}=S\setminus\{x_5=0\}$.
\item $S$ is the closure of $\mathbb{P}^2$ under the rational
map $\varphi:\mathbb{P}^2 \dashrightarrow S$ given by
\begin{align*}
&\varphi(x_3:x_5:x_6) =(\varphi_0(x_3,x_5,x_6):\cdots:\varphi_6(x_3,x_5,x_6))= \\
&(-x_3^2x_5-x_3 x_6^2:x_3x_5x_6:-x_3x_5x_6-x_6^3:
x_3x_5^2:-x_3x_5^2-x_5x_6^2:x_5^3:x_5^2x_6),
\end{align*}
where
$\Gamma(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(1))=\langle x_3,x_5,x_6 \rangle$.
\item The group law on $\varphi(\mathbb{G}_a^2)= U$
extends to an action on $S$ by translation. i.e. $S$ is
an equivariant compactification of $\mathbb{G}_a^2$.
\end{itemize}
\end{lem}
\begin{proof}
First, it is clear that $\varphi$ defines
an isomorphism $U \cong \mathbb{G}_a^2$. Hence the divisor class
group of $S$ is generated by the $L_1$ and $L_2$, as $\Pic(\mathbb{G}_a^2)=0$.
It is simple enough to check that the induced group
law on $U$ extends to an action on all of $S$. However as mentioned in the introduction
we will not use this fact in this paper, so the proof is
omitted and can be found in \cite{DL10}.
Resolving the singularity explicitly via blow-ups
creates two exceptional curves $E_1$ and $E_2$ on the minimal desingularisation $\widetilde{S}$.
The singularity is of type $\mathbf{A}_2$ and
$\Pic(\widetilde{S}) = \langle E_1,E_2,E_3,E_4 \rangle
\cong \mathbb{Z}^4$, where $E_3$ and $E_4$
are the strict transforms of $L_1$ and $L_2$ respectively.
Now, one can use the adjunction formula \cite[Ch. V, Prop. 1.5]{Har77} to show that
$-K_{\widetilde{S}}=4E_1 + 2E_2 + 3E_3 +
3E_4$, which proves that $K_{\widetilde{S}}^2=6$. Also,
one can show that the pull back of the hyperplane section on $S$
is $-K_{\widetilde{S}}$, thus proving that $S$ is a singular del
Pezzo surface of degree $6$ given by its anticanonical
embedding.
Finally, we note that the $\mathbf{A}_2$ singular del Pezzo surface of degree $6$
contains only two lines by the classification of singular del Pezzo
surfaces \cite[Prop. 8.3]{CT88}. These are both defined over $\mathbb{Q}$,
so the surface is indeed split.
\end{proof}
We also include the extended Dynkin diagram of $\widetilde{S}$ in
Figure \ref{Dyn:Diag}, which records the intersection behaviour of
relevant curves on $\widetilde{S}$. This can be derived from the
proof of Lemma~\ref{lem:geom}, or found in \cite[Sec. 5]{Der06}.
Here $E_1,E_2,E_3$ and $E_4$ are as in the proof of Lemma
\ref{lem:geom} and
\begin{align*}
&A_1: S \cap \{x_1=x_2=x_6=0\} ,
\quad A_2: S \cap\{x_0=x_1=x_3=0\},\\
&A_3: S\cap\{x_0=x_2=x_4=0\}.
\end{align*}
These rational curves correspond to generators of the nef cone and
will be needed in our work in section \ref{subsec:torsor}.
\begin{figure}[hbt]
\centerline{
\xymatrix{A_2 \ar@{-}[rr] \ar@{-}[dr] \ar@{-}[dd]&& E_3 \ar@{-}[dr] \\
& A_1 \ar@{-}[dl] \ar@{-}[r] & E_2 \ar@{-}[r] & E_1\\
A_3 \ar@{-}[rr] && E_4 \ar@{-}[ur]}}
\caption{The extended Dynkin diagram for $\widetilde{S}$.}
\label{Dyn:Diag}
\end{figure}
\subsection{Calculating Peyre's Constant}\label{subsec:constant}
In this section we shall verify that the constant achieved in the
asymptotic formula for Theorem~\ref{thm:asym} is in agreement with
the conjectural expression as formulated by Peyre \cite[Sec.
2]{Pey95}. Since our surface is split, it is birational to
$\mathbb{P}^2$ over $\mathbb{Q}$. So the constant is equal to the
following three factors multiplied together:
\begin{itemize}
\item The volume $\alpha(\widetilde{S})$ of a certain polytope
in the cone of effective divisors,
\item The real density $\tau_\infty(\widetilde{S})$,
\item The $p$-adic densities $\prod_{p } \tau_p(\widetilde{S})$.
\end{itemize}
By the work of \cite[Table 3]{Der07} we know that
$$\alpha(\widetilde{S}) = \frac{1}{432},$$
which is in agreement with the constant $\alpha(\widetilde{S})$ in
Theorem~\ref{thm:asym}.
We shall now calculate the real density, which corresponds to the
measure of some region, where we consider
$\widetilde{S}(\mathbb{R})$ as a real analytic manifold. Since
removing a codimension one subset does not change this volume, we
may consider the measure of the coordinate chart $U=S\setminus \{x_5
= 0\}$, with local coordinates $x_3$ and $x_6$. By Lemma
\ref{lem:geom}, this is just a reflection of the fact that our
surface is a compactification of $\mathbb{A}^2$ with $\varphi$ as a
local homeomorphism. Since $S$ is given by its anticanonical
embedding, we have by \cite[Section 2.2.1]{Pey95}
\begin{align*}
\tau_\infty(\widetilde{S})
& =\int_{\mathbb{R}^2}\frac{\mathrm{d}x_3 \mathrm{d}x_6}
{\max(|x_3^2 +x_3 x_6^2|,|x_3x_6|,|x_3x_6+x_6^3|,
|x_3|,|x_3+x_6^2|,1,|x_6|)} \\
& =\int_{\mathbb{R}^2}
\int_{x_5\geq \{\max(|x_3^2 +x_3 x_6^2|,|x_3x_6|,|x_3x_6+x_6^3|,
|x_3|,|x_3+x_6^2|,1,|x_6|)}\frac{\mathrm{d}x_3 \mathrm{d}x_5 \mathrm{d}x_6}{x_5^2}
\\
& =3\int_{\{t,v,u \in \mathbb{R}:0<|t(ut+v^2)|,|uvt|,
|uvt+v^3|,|u^2t|,|u^2t+uv^2|,u^3,|u^2v|\leq1\}} \mathrm{d}u\mathrm{d}v\mathrm{d}t,
\end{align*}
where we have used the change of variables
$$x_3=t/u,x_5=u^{-3},x_6=v/u.$$
Then noticing that we have the obvious automorphism $v \mapsto -v$
in the above integral, this gives the required expression for the
constant in Theorem~\ref{thm:asym}. We note that more generally, the
real density of any anticanonically embedded del Pezzo surface can
be calculated similarly by knowing which linear system of cubics in
$\mathbb{P}^2$ determines the given embedding.
The calculation of the $p$-adic densities for similar problems (see
\cite{BB07} for example) have normally involved a ``hands-on''
approach to point counting modulo $p$ for each prime $p$. Here we
opt for a more general method, which applies to any surface that is
the blow-up of $\mathbb{P}^2$ at a sequence of (possibly infinitely
near) rational points. First we recall some definitions.
\begin{df}
Let $V$ be a non-singular projective variety defined over $\mathbb{Q}$. A \emph{model}
for $V$ over $\mathbb{Z}$ is a projective morphism of schemes $\mathcal{V} \to
\Spec{\mathbb{Z}}$,
whose generic fibre is isomorphic to $V$. For each prime $p$, we denote by
$\mathcal{V}_p=\mathcal{V}\times_{\Spec\mathbb{Z}}\Spec{\mathbb{F}_p}$ the
reduction of $\mathcal{V}$ modulo $p$.
We say that $V$ has \emph{everywhere good reduction} if there exists a
model whose structure morphism is a \emph{smooth morphism}
(i.e. $\mathcal{V}_p$ is a non-singular variety for each prime $p$).
\end{df}
\begin{lem}\label{lem:reduction}
Let $S$ be a surface over $\mathbb{Q}$ with everywhere good reduction, and
$\pi:\widetilde{S} \to S$ the blow-up of $S$ at a rational point
$P$. Then $\widetilde{S}$ also has everywhere good reduction.
\end{lem}
\begin{proof}
Let $\mathcal{S}$ be the model of $S$ with everywhere good reduction. Since $\mathcal{S}$ is projective, the rational point $P$
extends uniquely to an integral point $\mathcal{P}$ of $\mathcal{S}$.
Then the scheme $\widetilde{\mathcal{S}}$, which is defined to be
the blow-up of $\mathcal{S}$ at $\mathcal{P}$, is a model for
$\widetilde{S}$. For every prime $p$ it is clear that $\widetilde{\mathcal{S}}_p$ is simply
the blow-up of $\mathcal{S}_p$ at a smooth $\mathbb{F}_p$-point,
so $\widetilde{\mathcal{S}}$ also has everywhere good
reduction.
\end{proof}
Now let $S,\mathcal{S},\widetilde{S}$ and $\widetilde{\mathcal{S}}$
be as in Lemma~\ref{lem:reduction}. Then it is clear that for every
prime $p$ we have $\#\widetilde{\mathcal{S}}_p(\mathbb{F}_p)
=\#\mathcal{S}_p(\mathbb{F}_p) + p$, since blowing up a smooth
$\mathbb{F}_p$-point replaces one $\mathbb{F}_p$-point by a copy of
$\mathbb{P}^1_{\mathbb{F}_p}$, which has $p+1$
$\mathbb{F}_p$-points. We can use this simple fact to prove the
following.
\begin{lem}\label{lem:densities}
Let $S$ be a surface over $\mathbb{Q}$ which is the blow-up of $\mathbb{P}^2$ at
$r$ (possibly infinitely near) rational points. Then for every
prime $p$ the local density at $p$ is
$$\tau_p(S)= \left(1 - \frac{1}{p}\right)^{r+1}\left(1 +
\frac{r+1}{p} + \frac{1}{p^2}\right).$$
\end{lem}
\begin{proof}
We begin by noting that the definition of $\tau_p(S)$ is independent of the choice
of model, as pointed out in \cite[Def. 2.2]{Pey95}. Since $\mathbb{P}^2$ has everywhere
good reduction, then so does $S$ by Lemma~\ref{lem:reduction}.
Let $\mathcal{S}$ be the corresponding model, then
$\#\mathcal{S}_p(\mathbb{F}_p)=1 + (r+1)p + p^2$ since $\#\mathbb{P}^2(\mathbb{F}_p)=1 + p + p^2$. It is also
clear that $\Pic(\mathcal{S}_p) \cong \mathbb{Z}^{r+1}$ with
trivial galois action, hence the associated Artin L-function is
$\zeta(s)^{r+1}$. This gives the correct ``convergence factors"
and the result follows.
\end{proof}
Applying Lemma~\ref{lem:densities} to $\widetilde{S}$ (which is
split by Lemma~\ref{lem:geom}) with $r=3$, we deduce the result.
\section{The Proof}
\subsection{Passage to the Universal Torsor}\label{subsec:torsor}
As mentioned in the introduction, the first step in the proof is
transferring the problem of counting rational points on the surface
$S$, to counting integral points on the corresponding universal
torsor $\mathcal{T}$.
A variety may in general have more than one universal torsor,
however in our case there is only one. Indeed if a smooth projective
variety $V$ over a field $k$ has a universal torsor, then the set of
isomorphism classes of universal torsors is a principal homogeneous
space under $H^1(k,T)$, where $T=\Hom(\Pic(V),\mathbb{G}_m)$ is the
N\'{e}ron-Severi torus. However in our case $T=\mathbb{G}_m^4$ since
$\Pic(\widetilde{S})\cong\mathbb{Z}^4$ with trivial galois action,
and also $H^1(\mathbb{Q},\mathbb{G}_m^4)=0$ by Hilbert's theorem
$90$. Hence $\widetilde{S}$ can have at most one universal torsor.
However, the existence of a rational point on $\widetilde{S}$
implies the existence of a universal torsor. These facts (and more)
can be found in \cite[Sec. 2.3]{Sko01}. The following lemma gives us
a concrete description of the universal torsor.
\begin{lem} \label{lem:torsor}
Let
$$\Cox(\widetilde{S})=\bigoplus_{(n_1,n_2,n_3,n_4) \in \mathbb{Z}^4}
H^0(\widetilde{S},\mathcal{O}(E_1)^{\otimes n_1}\otimes \cdots \otimes
\mathcal{O}(E_4)^{\otimes n_4})$$
be the Cox ring of $\widetilde{S}$. Then
\begin{itemize}
\item $\Cox(\widetilde{S}) \cong
\mathbb{Q}[\alpha_1,\alpha_2,\alpha_3,\eta_1,\eta_2,\eta_3,\eta_4]
/(\eta_2\alpha_1^2 + \eta_3\alpha_2 + \eta_4\alpha_3).$
\item The universal torsor $\mathcal{T}$ of $\widetilde{S}$ is an
open subset of $\Spec(\Cox(\widetilde{S}))$.
\item We have a commutative diagram
$$\xymatrix{\mathcal{T} \ar[r]^{\widetilde{\pi}} \ar[dr]_{\pi} & \widetilde{S} \ar[d] \\ & S }$$
where $\pi$ is the map
\begin{equation}
\begin{split} \label{pi}
\pi(\bfeta,\bfalpha) \mapsto &(
\alpha_2 \alpha_3: \eta_1\eta_2\eta_3\alpha_1\alpha_2 :
\eta_1\eta_2\eta_4\alpha_1\alpha_3:
\eta_1^2\eta_2\eta_3^2\eta_4\alpha_2 \\
&:\eta_1^2\eta_2\eta_3\eta_4^2\alpha_3 :
\eta_1^4\eta_2^2\eta_3^3\eta_4^3:
\eta_1^3\eta_2^2\eta_3^2\eta_4^2\alpha_1 ).
\end{split}
\end{equation}
\item The action of a point $(k_1,k_2,k_3,k_4) \in \mathbb{G}_m^4$ on the universal torsor
is $\eta_i \mapsto k_i \eta_i$ for $i=1,2,3,4$, and
\begin{align*}
&\alpha_1 \mapsto
k_1k_3k_4\alpha_1, \quad
\alpha_2 \mapsto k_1^2k_2k_3k_4^2\alpha_2,\quad
\alpha_3 \mapsto k_1^2k_2k_3^2k_4\alpha_3.
\end{align*}
\end{itemize}
\end{lem}
\begin{proof}
The calculation of the Cox ring, the map $\pi$ and the action of the
N\'{e}ron-Severi torus on the Cox ring can be found in
\cite{Der06}. That the universal torsor is an open subset of
$\Spec(\Cox(\widetilde{S}))$ is well-known, see \cite[Cor. 2.16,
Prop. 2.9]{HK00} for example.
\end{proof}
In fact, everything we need to know about the universal torsor can
be deduced from first principles. Firstly, it is not actually
necessary for us to calculate explicitly which open subset of
$\Spec(\Cox(\widetilde{S}))$ the universal torsor corresponds to.
However, it is easy to check that the action given in Lemma
\ref{lem:torsor} is well-defined and that it preserves the fibres
of $\pi$. Moreover, $\pi$ is surjective on its domain of definition
since $\varphi^{-1} \circ\pi$ is surjective onto $U$, where
$\varphi$ and $U$ are as in Lemma~\ref{lem:geom}. And also, it is
easy enough to see that $\pi$ hits every point on $S\setminus U$ as
well, hence it is surjective.
In particular, when we consider the universal torsor as being over
$U$, it is simple to see that we get a free and transitive action on
the fibres of $\pi$ on the corresponding open subset where
$\eta_1\eta_2\eta_3\eta_4\neq0$. That is, it is clear that
$\Spec(\Cox(\widetilde{S}))\setminus\{\eta_1\eta_2\eta_3\eta_4=0\}$
is a $U$-torsor under $\mathbb{G}_m^4$.
Now to find a suitable section of the morphism $\pi$. Bearing in
mind that we are counting points on $U$ where $x_5 \neq 0$, we see
that the rational points which have some coordinate equal to zero
lie in the image of the points on the torsor where
$\alpha_1\alpha_2\alpha_3=0$. These are exactly the curves $A_1,A_2$
and $A_3$ given in Figure \ref{Dyn:Diag}. They are rational curves,
and it is easy enough to show that the corresponding counting
functions satisfy
$$N_{A_1}(B)=\frac{12}{\pi^2}B + O(B^{1/2}), \quad
N_{A_2}(B)=N_{A_3}(B)=O(B^{2/3}).$$
Since these have been taken into account, we can now assume that
each coordinate of each rational point is non-zero.
\begin{lem} \label{lem:section}
Above each rational point $x \in U(\mathbb{Q})$ with non-zero
coordinates, there is a unique integral point $(\bfalpha,\bfeta)$ on
the universal torsor satisfying
\begin{align*}
&(\alpha_1,\eta_1\eta_3\eta_4)=(\alpha_2,\eta_1\eta_2\eta_4)=(\alpha_3,\eta_1\eta_2\eta_3)=1, \\
&(\eta_2,\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1, \\
&\eta_1,\eta_2,\eta_3,\eta_4 > 0, \alpha_1\alpha_2\alpha_3
\neq0.
\end{align*}
\end{lem}
\begin{proof}
We should note that we are guided to the above coprimality conditions
by Figure \ref{Dyn:Diag}, whereby two
variables are coprime if and only if the corresponding curves do
not intersect each other.
First let $(\bfalpha,\bfeta)$ be an integral point on the
universal torsor lying above a rational point with non-zero
coordinates. Suppose that there is a prime $p \mid(\eta_1,\alpha_1)$. Then
using the torsor action in Lemma~\ref{lem:torsor} with $k_1=1/p,k_2=p^3,k_3=k_4=1$, we
map
\begin{align*}
\eta_1 & \mapsto \eta_1/p, \quad \eta_2 \mapsto p^3\eta_2, \\
\alpha_1 &\mapsto \alpha_1/p, \quad\alpha_2 \mapsto p\alpha_2,
\quad\alpha_3 \mapsto p\alpha_3.
\end{align*}
So we have successfully managed to divide $\eta_1$ and
$\alpha_1$ by $p$, and left the other variables as integers, meaning that if they have any common factor
we can remove it. A very similar argument
works for $\eta_3$ and $\eta_4$, so we can assume
$$(\alpha_1,\eta_1\eta_3\eta_4)=1.$$
We now fix our choice of $\alpha_1$ modulo $\{\pm1\}$, meaning that from now on we
impose the condition $|k_1k_3k_4|=1$. This simplifies the action
on $\alpha_2$ and $\alpha_3$ to
$$\alpha_2 \mapsto \frac{k_2}{k_3}\alpha_2, \quad \alpha_3 \mapsto
\frac{k_2}{k_4}\alpha_3.$$
Carrying on with the same procedure, if $p \mid
(\alpha_2,\eta_1)$, take $k_1=1/p,k_2=k_4=1,k_3=p$ to get
$(\alpha_2,\eta_1)=1$ and for $p \mid (\alpha_2,\eta_4)$ take
$k_1=k_2=1, k_3=p,k_4=1/p$ to get $(\alpha_2,\eta_4)=1$.
We have now come to interesting part of the proof, since so far we have not
used the equation of the universal torsor, but now we are driven to
use it since it encodes divisibility conditions. Namely, if $p \mid
(\alpha_2,\eta_2)$, then $p$ must also divide $\eta_4$ or $\alpha_3$. But
$(\alpha_2,\eta_4) =1$, so we are safe to choose
$k_1=k_3=k_4=1,k_2=1/p$ and keep $\alpha_3$ as an integer. So
we have successfully shown that we can choose
$$(\alpha_2,\eta_1\eta_2\eta_4)=1.$$
We fix this choice of $\alpha_2$ modulo $\{\pm1\}$, which is equivalent to
requiring $|k_2|=|k_3|$.
The reader should now be familiar with the method and can check
that we can assume $(\alpha_3,\eta_1\eta_2\eta_3)=1$ after
performing the following
\begin{itemize}
\item If $p \mid (\alpha_3,\eta_1)$, choose
$k_1=1/p,k_2=k_3=1,k_4=p$,
\item If $p \mid (\alpha_3,\eta_3)$, choose
$k_1=p,k_2=k_3=1/p,k_4=1$,
\item If $p \mid (\alpha_3,\eta_2)$, contradiction since
$(\alpha_3,\eta_3)=(\alpha_2,\eta_2)=1$.
\end{itemize}
So fixing $\alpha_3$ modulo $\{\pm1\}$, we are restricted to
$$|k_2|=|k_3|=|k_4|.$$
But if $p \mid (\eta_2,\eta_3,\eta_4)$, choosing
$k_1=p^2,k_2=k_3=k_4=1/p$ then gives $(\eta_2,\eta_3,\eta_4)=1$, and moreover
the torsor equation implies they must also be pairwise coprime.
Finally, by choosing the $\eta_i$ to be positive, we have used all
degrees of freedom in the torsor action and so the choice of
integral point is unique.
\end{proof}
Using this lemma, we see that counting those points $x \in
U(\mathbb{Q})$ satisfying the height bound $H(x) \leq B$, is
equivalent to counting the unique integral points above them on the
universal torsor which satisfy the bound
$H(\pi(\bfalpha,\bfeta))\leq B$. Naively, this corresponds to $7$
separate height conditions. However, using the map $\varphi$ from
Lemma~\ref{lem:geom}, we know that we actually have 3 degrees of
freedom. With this in mind, we define
\begin{align} \nonumber
X_3 &= \left(\frac{\eta_1^2\eta_2\eta_3^2\eta_4}{X_5^2B}\right) =
\left(B\eta_1^2\eta_2\eta_4^3\right)^{-1/3}, \nonumber\\
X_5 &= \left(\frac{\eta_1^4\eta_2^2\eta_3^3\eta_4^3}{B}\right)^{1/3},\label{height1}\\
X_6 &= \left(\frac{\eta_1^3\eta_2^2\eta_3^2\eta_4^2}{X_5^2B}\right) =
\left(\frac{\eta_1\eta_2^2}{B}\right)^{1/3}, \nonumber\\ \nonumber
\end{align}
and let $\overline{\varphi_i}(\alpha_1,\alpha_2) =\varphi_i(\alpha_2
X_3,X_5,\alpha_1 X_6)$ for $i=0,1,2,4,$ and
$\overline{\varphi_3}(\alpha_2) =\varphi_3(\alpha_2 X_3,X_5,1)
,\overline{\varphi_6}(\alpha_1) =\varphi_6(1,X_5,\alpha_1 X_6)$.
Then it is clear that the height condition
$H(\pi(\bfalpha,\bfeta))\leq B$ is equivalent to the condition
\begin{align}
&|\overline{\varphi_i}(\alpha_1,\alpha_2)|,|\overline{\varphi_3}(\alpha_2)|\leq 1, i =0,1,2,4, \label{height2}\\
&X_5,\overline{\varphi_6}(\alpha_1) \leq 1 \label{height3}.
\end{align}
Finally, on noticing we have the obvious automorphism $\alpha_1
\mapsto -\alpha_1$ on the torsor, we have shown the following.
\begin{lem} \label{problem1}
The counting function for $U$ satisfies
$$N_U(B) = 2T(B) + \frac{12}{\pi^2}B + O(B^{2/3})$$
where
$$ T(B) = \#\left\{
\begin{array}{ll}
(\bfalpha,\bfeta) \in \mathbb{Z}^7
\end{array}:
\begin{array}{ll}
\eta_2\alpha_1^2+ \eta_3\alpha_2 +\eta_4\alpha_3 = 0, (\ref{height2}),(\ref{height3}), \\
(\alpha_1,\eta_1\eta_3\eta_4)=(\alpha_2,\eta_1\eta_2\eta_4)= (\eta_2,\eta_3)=1,\\
(\alpha_3,\eta_1\eta_2\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1, \\
\alpha_1,\eta_1,\eta_2,\eta_3,\eta_4 > 0, \alpha_2\alpha_3
\neq0.
\end{array} \right\} .$$
\end{lem}
We note that we have the natural upper bound $\alpha_1 \leq 1/X_5^2
X_6$ given by $\overline{\varphi_6}$. However, we can actually do
better than this, which will be quite important to improving our
error term later on. Notice that $\overline{\varphi_4}$ and
$\overline{\varphi_3}$ imply
$$ -\frac{1}{X_3X_5^2} \leq \alpha_2 \leq \frac{1}{X_3X_5^2}\left(1 -
\alpha_1^2X_5X_6^2\right).$$
Rearranging this in terms of $\alpha_1$, we deduce the stronger
bound
\begin{equation}
\alpha_1\leq \frac{\sqrt{2}}{X_6\sqrt{X_5}}. \label{bound:alpha1}
\end{equation}
\subsection{\Mob Inversion}
Now we shall use \Mob inversion to remove the coprimality conditions
on the $\alpha_i$'s. Recalling the counting problem in Lemma
\ref{problem1} and the height conditions (\ref{height3}), it makes
sense to define
\begin{align}
\mathcal{N}=&
\left\{\bfeta \in \mathbb{Z}^4 :
\begin{array}{ll}
\eta_1,\eta_2,\eta_3,\eta_4 > 0, X_5 \leq 1,\\
(\eta_2,\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1.
\end{array}
\right\} .\label{bigeta}
\end{align}
Then it is clear that
$$T(B)=\sum_{\bfeta \in \mathcal{N}} \sum_{\substack{\alpha_1 > 0 \\
(\alpha_1,\eta_1\eta_3\eta_4)=1 \\ \overline{\varphi_6}(\alpha_1)
\leq 1}} S $$ where
$$S = \#\left\{\alpha_2,\alpha_3 \in \mathbb{Z} :
\begin{array}{ll}
\alpha_2\alpha_3 \neq 0, (\ref{height2}) \mbox{ holds},\\
(\alpha_2,\eta_1\eta_2\eta_4)=(\alpha_3,\eta_1\eta_2\eta_3)=1, \\
\eta_2\alpha_1^2+ \eta_3\alpha_2 +\eta_4\alpha_3 = 0.
\end{array}
\right\}.$$
Now using \Mob inversion on $(\alpha_3,\eta_1\eta_2\eta_3)=1$ gives
us
$$S = \sum_{k_3\mid\eta_1\eta_2\eta_3}\mu(k_3) S_{k_3}$$
where
$$S_{k_3} = \#\left\{\alpha_2,\alpha_3 \in \mathbb{Z} :
\begin{array}{ll}
\alpha_2\alpha_3 \neq 0,(\ref{height2}) \mbox{ holds}, \\
(\alpha_2,\eta_1\eta_2\eta_4)=1, \\
\eta_2\alpha_1^2+ \eta_3\alpha_2 +k_3\eta_4\alpha_3 = 0.
\end{array}
\right\}.$$
However $S_{k_3} \neq 0$ if and only if $(k_3,\eta_2\eta_3)=1$, so
$$S = \sum_{\substack{k_3\mid\eta_1 \\(k_3,\eta_2\eta_3)=1}}\mu(k_3) S_{k_3}.$$
A similar argument yields
\begin{equation}
T(B)=\sum_{\bfeta \in \mathcal{N}}
\sum_{\substack{k_3\mid\eta_1 \\ (k_3,\eta_2\eta_3)=1}}\mu(k_3)
\sum_{\substack{k_2\mid\eta_1\eta_2 \\ (k_2,k_3\eta_4)=1}} \mu(k_2)
\sum_{\substack{\alpha_1 > 0 \\ (\alpha_1,\eta_1\eta_3\eta_4)=1 \\ \overline{\varphi_6}(\alpha_1) \leq 1}}
S_{k_2,k_3} \label{problem2}
\end{equation}
where $$S_{k_2,k_3} = \#\left\{\alpha_2,\alpha_3 \in \mathbb{Z} :
\begin{array}{ll}
\alpha_2\alpha_3 \neq 0, k_2\eta_3\alpha_2 + \eta_2\alpha_1^2 +k_3\eta_4\alpha_3=0,\\
|\overline{\varphi_i}(\alpha_1,k_2\alpha_2)|,|\overline{\varphi_4}(k_2\alpha_2)|\leq 1, i
=0,1,2,3.
\end{array}
\right\}.$$
\subsection{Sum over $\alpha_2$ and $\alpha_3$ via Congruences}
In this section, we shall perform the summation over $\alpha_2$. We
note that there are no conditions on $\alpha_3$ other than the
equation of the universal torsor, so we find that
$$S_{k_2,k_3} = \#\left\{\alpha_2 \in \mathbb{Z} :
\begin{array}{ll}
\alpha_2 \neq 0, k_2\eta_3\alpha_2 \equiv -\eta_2\alpha_1^2 \pmod{k_3\eta_4},\\
|\overline{\varphi_i}(\alpha_1,k_2\alpha_2)|,|\overline{\varphi_4}(k_2\alpha_2)|\leq 1, i
=0,1,2,3.
\end{array}
\right\}.$$
However since $(k_2\eta_3,k_3\eta_4)=1$, $\alpha_2$ is uniquely
determined modulo $k_3\eta_4$. For any integers $q,n_0,a,b$ with
$a<b$, we have the simple estimate
$$\#\{n \in \mathbb{Z} \cap [a,b]: n \equiv n_0 \pmod q\} =
\frac{b-a}{q} + O(1).$$
Using this and the change of variables $t \mapsto k_2tX_3$, we see
that
\begin{equation}
S_{k_2,k_3} = \frac{1}{k_2k_3\eta_4X_3}F_1(X_5,\alpha_1X_6)+
O(1) \label{def:Sk2k3}
\end{equation}
where $F_1(u,v)$ is defined by the following result.
\begin{lem}\label{lem:F1}
Let $$F_1(u,v)
=\int_{\{t\in
\mathbb{R}:,0<|t(ut+v^2)|,|uvt|,|uvt+v^3|,|u^2t|,|u^2t+uv^2|\leq1\}}
\mathrm{d}t$$ for $u,v\geq0$ and $(u,v) \neq (0,0)$. Then
\begin{enumerate}
\item[(a)] For $u \neq 0$, $$F_1(u,v) \leq \frac{2}{\sqrt{u}}.$$
\item[(b)] $F_1(u,v)$ is piecewise differentiable with respect to $u$ and $v$.
\end{enumerate}
\end{lem}
\begin{proof}
The differentiability condition is clear, so it remains to prove the inequality.
First let $M(u,v)=\vol\{t\in \mathbb{R}:|t(ut+v^2)|
\leq 1\}$, then we have
$M(u,v) = \vol\{t\in \mathbb{R}: v^4/4u^2-1/u \leq t^2 \leq 1/u + v^4/4u^2\}$
after completing the square. If $v^4/4u^2 \geq 1/u$, then using
the simple fact that $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$ for
all non-negative real numbers $a$ and $b$, we deduce that
$$M(u,v) = \sqrt{v^4/4u^2 + 1/u} - \sqrt{v^4/4u^2- 1/u} \leq
\sqrt{2/u}.$$
Similarly, if $v^4/4u^2 \leq 1/u$ then $M(u,v) = \sqrt{v^4/4u^2 +
1/u} \leq 2/\sqrt{u}$.
\end{proof}
We now have our first error term in the counting problem
(\ref{problem2}). First recall that $\sum_{k\mid n} |\mu(k)| =
2^{\omega(n)}$ where $\omega(n)$ is the number of prime divisors of
$n$, that we have the stronger bound on $\alpha_1$ given by
(\ref{bound:alpha1}), and the definition (\ref{bigeta}) of
$\mathcal{N}$. Using these, we see that the overall contribution to
the error term from (\ref{def:Sk2k3}) is
\begin{align*}
& \ll \sum_{\eta_1^4\eta_2^2\eta_3^3\eta_4^3\leq B}
\sum_{k_3\mid\eta_1} |\mu(k_3)|
\sum_{k_2\mid\eta_1\eta_2} |\mu(k_2)|
\sum_{|\alpha_1|\leq \frac{\sqrt{2}}{X_6\sqrt{X_5}}} 1 \\
& \ll B^{1/2} \sum_{\eta_1^4\eta_2^2\eta_3^3\eta_4^3\leq B}
\frac{2^{\omega(\eta_1)} 2^{\omega(\eta_1\eta_2)}
}{\eta_1\eta_2\eta_3^{1/2}\eta_4^{1/2}} \\
& \ll B^{1/2} \sum_{\eta_1^4\eta_2^2\eta_3^3\leq B}
\frac{4^{\omega(\eta_1)} 2^{\omega(\eta_2)}
}{\eta_1\eta_2\eta_3^{1/2}} \cdot\frac{B^{1/6}}{\eta_1^{2/3}\eta_2^{1/3}\eta_3^{1/2}}
\ll B^{2/3+\varepsilon}
\end{align*}
since $2^{\omega(n)} \leq d(n) \ll n^\varepsilon$, where $d(n)$ is
the usual divisor function. This error term is clearly satisfactory
for Theorem~\ref{thm:asym}.
\subsection{Sum over $\alpha_1$}
Recall that the main term in our counting problem is given by
(\ref{problem2}) and (\ref{def:Sk2k3}). Applying \Mob inversion to
remove the coprimality condition in the sum over $\alpha_1$ gives
$$\sum_{\substack{\alpha_1 > 0 \\ (\alpha_1,\eta_1\eta_3\eta_4)=1 \\ \overline{\varphi_6}(\alpha_1) \leq 1}}
F_1(X_5,\alpha_1X_6) =
\sum_{k_1\mid\eta_1\eta_3\eta_4}\mu(k_1) \sum_{0<\alpha_1 \leq 1/k_1X_5^2X_6} F_1(X_5,\alpha_1k_1X_6).$$
A natural step is to now apply Euler-Maclaurin summation. To
simplify our notation in what follows, we shall use Stieltjes
integral notation, and also use $\{\cdot\}$ to denote the fractional
part of a real number.
\begin{lem}\label{lem:F2E}
We have
$$\sum_{0<\alpha_1 \leq 1/k_1X_5^2X_6} F_1(X_5,\alpha_1k_1X_6) =
\frac{1}{k_1X_6}F_2(X_5) + E(\bfeta,k_1,B),$$
where for $u>0$ we have
\begin{align*}
F_2(u)&=\int_0^{\frac{1}{u^2}}F_1\left(u,v\right)\mathrm{d}v ,\\
&=\int_{\{t,v \in \mathbb{R}:0<|t(ut+v^2)|,|uvt|,|uvt+v^3|,|u^2t|,|u^2t+uv^2|,u^2v\leq1\}}
\mathrm{d}v\mathrm{d}t ,
\end{align*}
and
$$E(\bfeta,k_1,B) =\int_0^{1}\left\{\frac{v}{k_1X_5^2X_6}\right\}\mathrm{d}F_1\left(X_5,\frac{v}{X_5^2}\right)
-\left\{\frac{1}{k_1X_5^2X_6}\right\}F_1\left(X_5,\frac{1}{X_5^2}\right).$$
We also have the bounds
\begin{equation}
|E(\bfeta,k_1,B)| \leq \frac{6}{\sqrt{X_5}}, \qquad F_2(u) \leq
\frac{4}{\sqrt{u}}. \label{bounds:F2E}
\end{equation}
\end{lem}
\begin{proof}
Euler-Maclaurin summation gives
\begin{align*}
&\sum_{0<\alpha_1 \leq 1/k_1X_5^2X_6} F_1(X_5,\alpha_1k_1X_6) \\
& =\int_0^{\frac{1}{k_1X_5^2X_6}}F_1\left(X_5,vk_1X_6\right)\mathrm{d}v -
\int_0^{\frac{1}{k_1X_5^2X_6}}F_1\left(X_5,vk_1X_6\right)\mathrm{d}\{v\}.
\end{align*}
Changing variables and applying integration by parts gives
the first part of the lemma. As for the first upper bound, recall the properties of $F_1$ given in
Lemma~\ref{lem:F1}. Then we have
$$ \left|E(\bfeta,k_1,B)\right|
\leq 2F_1\left(X_5,\frac{1}{X_5^2}\right) + F_1\left(X_5,0\right)
\leq \frac{6}{\sqrt{X_5}}. $$
For the second upper bound, note that $|uvt|\leq 1$ and $|uvt +
v^3|\leq 1$ imply that $v \leq 2^{1/3}$, hence
$$F_2(u) \leq \int_0^{2^{1/3}}F_1\left(u,v\right)\mathrm{d}v \leq
\frac{4}{\sqrt{u}}.$$
\end{proof}
\subsection{Making a Lower Order Term Explicit}\label{subsec:error}
The counting problem (\ref{problem2}) now stands as
\begin{align*}
T(B)=& \sum_{\bfeta \in \mathcal{N}}
\frac{F_2(X_5)}{\eta_4X_3X_6}
\sum_{\substack{k_3\mid\eta_1 \\
(k_3,\eta_2\eta_3)=1}}\frac{\mu(k_3)}{k_3}
\sum_{\substack{k_2\mid\eta_1\eta_2 \\ (k_2,k_3\eta_4)=1}}
\frac{\mu(k_2)}{k_2}
\sum_{k_1\mid\eta_1\eta_3\eta_4}
\frac{\mu(k_1)}{k_1} \\
&+T_1(B)
\end{align*}
where $T_1(B)$ denotes the same expression, but with $
F_2(X_5)/k_1X_6$ replaced by $E(\bfeta,k_1,B)$. It turns out that
there is a term of order $B$ in $T_1$, which we shall handle by
performing the sum over $\eta_2$ explicitly. Taking out the factors
which depend on $\eta_2$ and recalling the definition of
$\mathcal{N}$ in (\ref{bigeta}) and the height conditions
(\ref{height1}), we see that
\begin{equation}
T_1(B)=B^{1/3}\sum_{\substack{\eta_1^4\eta_3^3\eta_4^3 \leq B \\
(\eta_3,\eta_4)=1 }}
\eta_1^{2/3} \sum_{k_1\mid\eta_1\eta_3\eta_4}\mu(k_1)
T_2\left(\eta_1,\eta_3,\eta_4,k_1, \widetilde{X_5}\right)
\label{def:T1}
\end{equation}
where we define
\begin{equation}
\widetilde{X_5}=\eta_2/X_5^{3/2}=\sqrt{B/(\eta_1^4\eta_3^3\eta_4^3)}
\label{X5:tilde}
\end{equation}
and
\begin{align*}
T_2(\eta_1,\eta_3,\eta_4,k_1,\widetilde{X_5})&= \\
\sum_{\substack{\eta_2 \leq \widetilde{X_5} \\
(\eta_2,\eta_3\eta_4)=1}} &\eta_2^{1/3}E(\bfeta,k_1,B)
\sum_{\substack{k_3\mid\eta_1 \\
(k_3,\eta_2\eta_3)=1}}\frac{\mu(k_3)}{k_3}
\sum_{\substack{k_2\mid\eta_1\eta_2 \\ (k_2,k_3\eta_4)=1}}
\frac{\mu(k_2)}{k_2} .
\end{align*}
This is essentially a sum involving an arithmetic function and a
real valued function, so partial summation is the natural method to
use. However first we need to unravel this arithmetic function to
get a multiplicative function in $\eta_2$. To simplify our notation,
let
\begin{equation}
\phi^*(a_1,\ldots,a_n)=\prod_{p\mid(a_1,\ldots,a_n)}\left(1 -
\frac{1}{p}\right), \label{phi^*}
\end{equation}
and we use the shorthand $\phi^*(a)=\phi^*(a,a)$. There will also be
unfortunate $2$-adic conditions we shall need to take care of, so we
define
$$
\mathcal{N}_0 = \{(\eta_1,\eta_3,\eta_4) \in \mathbb{N}^3:
2\nmid\eta_1 \mbox{ or } 2 \mid \eta_3\eta_4\},\quad
\mathcal{N}_1 = \mathbb{N}^3 \setminus \mathcal{N}_0.
$$
\begin{lem} \label{lem:nu}
We have
$$T_2(\eta_1,\eta_3,\eta_4,k_1,\widetilde{X_5})= \psi(\eta_1,\eta_3,\eta_4)
\sum_{\eta_2 \leq \widetilde{X_5}} \nu_{\eta_1,\eta_3,\eta_4}(\eta_2) \eta_2^{1/3}E(\bfeta,k_1,B),$$
where
\begin{align*}
\psi(\eta_1,\eta_3,\eta_4)& = \phi^*(\eta_1,\eta_3\eta_4)
\prod_{\substack{p\mid \eta_1 \\ p \nmid \eta_3\eta_4 \\ p\neq2}}
\left(1-\frac{2}{p}\right), \\
\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(\eta_2)& =
\left\{
\begin{array}{ll}
\phi^*(\eta_2) \prod_{\substack{p\mid \eta_1,\eta_2 \\ p \neq 2}} \left(1-\frac{2}{p}\right)^{-1}
,&(\eta_2,\eta_3\eta_4)=1,\\
0,& \mbox{otherwise},
\end{array}
\right.
\end{align*}
and if $(\eta_1,\eta_3,\eta_4) \in \mathcal{N}_i$, then
$$\nu_{\eta_1,\eta_3,\eta_4}(\eta_2)=\left\{
\begin{array}{ll}
\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(\eta_2), &2^{i}\mid\eta_2,\\
0,& \mbox{otherwise}.
\end{array}\right.$$
\end{lem}
\begin{proof}
One can verify the following expression
\begin{align*}
\sum_{\substack{k_3\mid\eta_1 \\
(k_3,\eta_2\eta_3)=1}}\frac{\mu(k_3)}{k_3}
\sum_{\substack{k_2\mid\eta_1\eta_2 \\ (k_2,k_3\eta_4)=1}}
\frac{\mu(k_2)}{k_2}
=\phi^*(\eta_1,\eta_3\eta_4)\phi^*(\eta_2)
\prod_{\substack{p\mid \eta_1 \\ p \nmid \eta_2\eta_3\eta_4 }}
\left(1-\frac{2}{p}\right),
\end{align*}
by checking its value at prime powers and recalling that
$(\eta_2,\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1$.
We want this to be written as a
multiplication function of $\eta_2$ times some other arithmetic function independent of $\eta_2$. In
order to do this, we need to split up the product over primes, but we
can only safely do this if it is non-zero, i.e. if $2\nmid \eta_1$ or
$2\mid\eta_2\eta_3\eta_4$. So we have defined $\nu_{\eta_1,\eta_3,\eta_4}$ be zero exactly when
$2\mid\eta_1,2\nmid \eta_2\eta_3\eta_4$ and the coprimality conditions are not satisfied,
and simplified its definition in the remaining cases.
\end{proof}
Note that $\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}$ is a
multiplicative function of $\eta_2$, but
$\nu_{\eta_1,\eta_3,\eta_4}$ is not. The next natural step is to
find the average order of $\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}$.
However to simplify our notation and argument, from now on we shall
assume that $(\eta_1,\eta_3,\eta_4) \in \mathcal{N}_0.$ The other
case is almost exactly the same, the only difference being the
condition that $\eta_2$ must be even, and it will still contribute a
power of $B$ to the main term and give the same error term. With
this in mind, we have the following.
\begin{lem}\label{lem:nu_2}
Let $V(s)$ be the Dirichlet series associated to $\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}$ and
$\widetilde{V}(s)=V(s)/\zeta(s)$. Then $\widetilde{V}(s)$ is
a holomorphic and bounded function on $\re(s) > 0$ satisfying $0 \leq \widetilde{V}(1)\ll
2^{\omega(\eta_1)}$ and
$$\sum_{n \leq X} \widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(n) = \widetilde{V}(1)X + O(2^{\omega(\eta_1)}X^\varepsilon).$$
\end{lem}
\begin{proof}
For $p\neq2$, it is easy to see that
\begin{align*}
\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(p^k)& = \left\{
\begin{array}{ll}
\left(1- \frac{1}{p}\right) , &\quad p\nmid \eta_1\eta_3\eta_4, \\
\left(\frac{1- 1/p}{1-2/p}\right) , &\quad p \mid
\eta_1, p \nmid \eta_3\eta_4, \\
0 , &\quad \mbox{otherwise}.
\end{array}\right.
\end{align*}
Then by considering Euler products, one can check that $V(s)$ is
equal to
\begin{align*}
\frac{\zeta(s)V'(s)}{\zeta(s+1)}
\prod_{ p \mid \eta_1\eta_3\eta_4}
\left(1 + \frac{1-1/p}{p^s-1}\right)^{-1}
\prod_{\substack{p \mid \eta_1 , p \neq 2\\ p \nmid \eta_3\eta_4}}
\left(1 + \frac{1-1/p}{(1-2/p)(p^s-1)}\right)
\end{align*}
where $V'(s)$ is some function corresponding to the Euler factor at the prime $2$.
So $\widetilde{V}(s)$ has the properties stated in the lemma.
Ignoring convergence issues for now, we have
\begin{align*}
\sum_{n \leq X} \widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(n) & = \sum_{n \leq X}
((\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)*1)(n)\\
&= \sum_{n \leq X} \sum_{d\mid n}(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)(d) \\
& = X\sum_{d =1}^\infty \frac{(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)(d)}{d}
+O\left(X^{\varepsilon}\sum_{d =1}^\infty \frac{|(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)(d)|}{d^\varepsilon}\right) \\
\end{align*}
where we have used the trivial bound $[x]=x + O(x^\varepsilon)$. To
make this rigorous, first note that
$$\lim_{s\to1}\sum_{d =1}^\infty \frac{(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)(d)}{d^s}=
\lim_{s\to1}V(s)\zeta(s)^{-1}=\widetilde{V}(1).$$
Next, we need to find an expression for the Dirichlet series
$V^+(s)$ of $|(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)|$. It is easy to verify that
$$(\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}*\mu)(p^k) = \left\{
\begin{array}{ll}
\widetilde{\nu}_{\eta_1,\eta_3,\eta_4}(p) - 1, &\quad k=1,\\
0,&\quad k>1.
\end{array}\right.$$
By considering Euler products, one can check that $V^+(s)$ is a holomorphic and
bounded function of $s$ on $\re (s)>0$ and satisfies
\begin{align*}
\widetilde{V}^+(\varepsilon)
& \ll
\prod_{\substack{p \mid \eta_1 \\ p \nmid \eta_3\eta_4}}\left(1 +
\frac{1}{(p-2)p^\varepsilon}\right) \ll 2^{\omega(\eta_1)}
\end{align*}
on this domain. Thus we are done.
\end{proof}
We shall now perform the summation over $\eta_2$, and to do this we
will need a slight abuse of notation. Namely, we define
$$\widetilde{E}(t) = E(\eta_1,t,\eta_3,\eta_4,k_1,B),$$ where $E$ is
given in Lemma~\ref{lem:F2E}, and for this we also need to think of
$X_5$ and $X_6$ as being functions of $\eta_2$. Recalling the
expression we had for $T_2$ as given in Lemma~\ref{lem:nu} and using
Lemma~\ref{lem:nu_2}, by partial summation we have
\begin{align*}
&\sum_{ \eta_2 \leq \widetilde{X_5}} \nu_{\eta_1,\eta_3,\eta_4}(\eta_2) \eta_2^{1/3}\widetilde{E}(\eta_2) \\
& = \widetilde{X_5}^{1/3}\widetilde{E}(\widetilde{X_5})
\sum_{\eta_2 \leq \widetilde{X_5}} \nu_{\eta_1,\eta_3,\eta_4}(\eta_2)
- \int_0^{\widetilde{X_5}} \sum_{\eta_2 \leq t} \nu_{\eta_1,\eta_3,\eta_4}(\eta_2) \mathrm{d}\left(t^{1/3}\widetilde{E}(t)\right)\\
& = \widetilde{V}(1)\int_0^{\widetilde{X_5}}t^{1/3}\widetilde{E}(t)\mathrm{d}t +
O\left(2^{\omega(\eta_1)} |\widetilde{E}(\widetilde{X_5})|
\widetilde{X_5}^{1/3+\varepsilon}\right) \\
&=\widetilde{V}(1)\widetilde{X_5}^{4/3}\int_0^1u^{1/3}\widetilde{E}(u\widetilde{X_5})\mathrm{d}u
+ O\left(B^\varepsilon |\widetilde{E}(\widetilde{X_5})|
\widetilde{X_5}^{1/3+\varepsilon}\right).
\end{align*}
We now note an interesting feature, namely that
$\widetilde{E}(\widetilde{X_5})$ is actually independent of $B$.
Indeed, viewing $X_5$ and $X_6$ as functions of $\eta_2$, we find
that $X_5(u\widetilde{X_5})=u^{2/3}$ and
$X_6(u\widetilde{X_5})=\eta_1\eta_3\eta_4/u^{2/3}$. Hence
$\widetilde{E}(u\widetilde{X_5})$ is independent of $B$ and moreover
by (\ref{bounds:F2E}) we deduce that
$$ \widetilde{E}(u\widetilde{X_5}) \leq \frac{6}{u^{1/3}}.$$
Hence referring back to (\ref{def:T1}), the overall error term
contribution to $T_1(B)$ in this case is
\begin{align*}
&\ll B^{1/3+\varepsilon}\sum_{\eta_1^4\eta_3^3\eta_4^3 \leq B}
\eta_1^{2/3} 2^{\omega(\eta_1\eta_3\eta_4)} \widetilde{X_5}^{1/3+\varepsilon} \\
&\ll B^{1/2+\varepsilon}\sum_{\eta_1^4\eta_3^3\eta_4^3 \leq B}
\frac{1}{\eta_3^{1/2}\eta_4^{1/2}} \ll B^{3/4 + \varepsilon}
\end{align*}
which is satisfactory. Now we can finally make the main term of
$T_1$ explicit, which in the case $ (\eta_1,\eta_3,\eta_4) \in
\mathcal{N}_0$ is
\begin{align*}
B\sum_{{\eta_1,\eta_3,\eta_4 \in \mathcal{N}_0}}
\widetilde{V}(1)\frac{\psi(\eta_1,\eta_3,\eta_4)}{\eta_1^2\eta_3^2\eta_4^2}
\sum_{k_1\mid\eta_1\eta_3\eta_4}\mu(k_1)
\int_0^1u^{1/3}\widetilde{E}(u\widetilde{X_5})\mathrm{d}u.
\end{align*}
We know that $\widetilde{E}(u\widetilde{X_5})\leq 6/u^{1/3}$ is
actually independent of $B$, so letting the sum over the $\eta_i$ go
to infinity, we get a main term of the form $\lambda'B$ where
$\lambda' \in \mathbb{R}$ is some constant and an error term of the
order
\begin{align*}
&\ll B^{1+\varepsilon}\sum_{\eta_1^4\eta_3^3\eta_4^3 > B}
\frac{1}{\eta_1^2\eta_3^2\eta_4^2}
\ll B^{3/4+\varepsilon}
\end{align*}
which is satisfactory. This was only for the case
$(\eta_1,\eta_3,\eta_4) \in \mathcal{N}_0$, however it is clear that
the sum over the case where $(\eta_1,\eta_3,\eta_4) \in
\mathcal{N}_1$ is almost exactly the same and hence it is omitted.
So returning to the original problem (\ref{def:T1}), we have shown
that there exists a constant $\lambda \in \mathbb{R}$ such that
$$T_1(B)=\lambda B + O(B^{3/4+\varepsilon}).$$
\subsection{Summation over the $\eta_i$}
We now know that
\begin{align*}T(B)=&\sum_{\bfeta \in \mathcal{N}}
\frac{\vartheta(\bfeta) F_2(X_5)}{\eta_4X_3X_6} + \lambda B + O(B^{3/4+\varepsilon}),
\end{align*}
where $X_3,X_5,X_6$ and $\mathcal{N}$ are given by (\ref{height1})
and (\ref{bigeta}), $F_2$ is as in Lemma~\ref{lem:F2E}, and we
define
$$\vartheta(\bfeta)=\sum_{\substack{k_3\mid\eta_1 \\
(k_3,\eta_2\eta_3)=1}}\frac{\mu(k_3)}{k_3}
\sum_{\substack{k_2\mid\eta_1\eta_2 \\ (k_2,k_3\eta_4)=1}}
\frac{\mu(k_2)}{k_2}
\sum_{k_1\mid\eta_1\eta_3\eta_4}\frac{\mu(k_1)}{k_1}$$ when
$(\eta_2,\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1$ and
$\vartheta(\bfeta) = 0$ otherwise. We have already simplified a very
similar sum in Lemma~\ref{lem:nu}, and using a similar method one
can check that
\begin{align}
\vartheta(\bfeta) & =\phi^*(\eta_1)\phi^*(\eta_2)\phi^*(\eta_3)\phi^*(\eta_4)
\prod_{\substack{p\mid \eta_1 \\ p \nmid \eta_2\eta_3\eta_4}}
\left(1-\frac{2}{p}\right) \label{vartheta}
\end{align}
when $(\eta_2,\eta_3)=(\eta_2,\eta_4)=(\eta_3,\eta_4)=1$ and
$\vartheta(\bfeta) = 0$ otherwise. Recalling the height conditions
(\ref{height1}) it follows that
\begin{align*}
T(B)&
=B^{2/3}\sum_{n \leq B}\Delta(n) F_2\left(\left(\frac{n}{B}\right)^{1/3}\right)
+ \lambda B + O(B^{3/4+\varepsilon})
\end{align*}
where
\begin{equation}
\Delta(n)=\sum_{\eta_1^4\eta_2^2\eta_3^3\eta_4^3=n}
\vartheta(\bfeta) \left(\frac{\eta_1}{\eta_2}\right)^{1/3}.
\label{Delta}
\end{equation}
Hence we have the expression
\begin{equation}\label{N(B)}
N_{U,H}(B)=2B^{2/3}\sum_{n \leq B}\Delta(n)
F_2\left(\left(\frac{n}{B}\right)^{1/3}\right)
+\left(\frac{12}{\pi^2} + 2\lambda\right)B + O(B^{3/4+\varepsilon})
\end{equation}
for the counting function.
\subsection{The Height Zeta Function}
In this section we shall prove Theorem~\ref{thm:HZF} on the height
zeta function $Z_{U,H}(s)$ as defined in (\ref{HZF}). A standard
application of Perron's formula \cite[Lemma 3.12]{Tit86} gives us an
expression for the counting function $N_{U,H}(B)$ in terms of the
zeta function via an inverse Mellin transform. Then performing the
corresponding Mellin transform tells us that for $\re (s)\gg1$ we
have
\begin{equation}
Z_{U,H}(s)=s\int_1^\infty u^{-s-1}N_{U,H}(u)\mathrm{d}u \label{zeta
Mellin}
\end{equation}
where $s=\sigma + it$ is a complex variable. Recalling (\ref{N(B)}),
we have $Z_{U,H}(s) = Z_1(s) + Z_2(s)$ where
\begin{align}
Z_1(s) &= 2s\int_1^\infty u^{-s-1/3} \sum_{n \leq u}\Delta(n)
F_2\left(\left(\frac{n}{u}\right)^{1/3}\right) \mathrm{d}u, \nonumber\\
Z_2(s) &= \frac{12/\pi^2 + 2\lambda}{s-1} + G_2(s),\label{G2}\\
G_2(s) &= s\int_1^\infty u^{-s-1}R(u)\mathrm{d}u,\nonumber
\end{align}
and $R(u)$ is some function such that $R(u)\ll u^{3/4+\varepsilon}$
for all $\varepsilon>0$. From this it follows that $G_2(s)$ is
holomorphic on the half-plane $\re(s)\geq 3/4+\varepsilon$, and
moreover
\begin{align*}
G_2(s)& \ll|s|\int_1^\infty u^{-\sigma -1}u^{3/4+\varepsilon}\mathrm{d}u
\ll\frac{|1 + i\frac{t}{\sigma}|}{|\frac{3}{4\sigma}-1|}
\ll 1 + |t|
\end{align*}
on this domain (note that here we use the common abuse of notation
that $\varepsilon$ is allowed to take different values
simultaneously). In particular $Z_2(s)$ has a meromorphic
continuation to the same half-plane with a simple pole at $s=1$ of
residue $12/\pi^2 + 2\lambda$.
Now that $Z_2(s)$ is under control, let us turn our attention to
$Z_1(s)$. Define $\Delta$'s Dirichlet series by
$D(s)=\sum_{n=1}^\infty \Delta(n) n^{-s}$. Then by choosing a
suitable $s$ to make sure that change of sum and integral are valid,
we can simplify $Z_1$ by
\begin{align}
Z_1(s) &= 2s\sum_{n=1}^\infty\Delta(n)\int_n^\infty u^{-s-1/3}
F_2\left(\left(\frac{n}{u}\right)^{1/3}\right) \mathrm{d}u \nonumber \\
&=2sD\left(s-\frac{2}{3}\right)\int_1^\infty u^{-s-1/3}
F_2\left(\left(\frac{1}{u}\right)^{1/3}\right) \mathrm{d}u \nonumber \\
&=D\left(s-\frac{2}{3}\right)G_{1,1}(s), \label{Z1}
\end{align}
where
\begin{align}
G_{1,1}(s) & = 6s\int_0^1 u^{3(s-1)}F_2(u)\mathrm{d}u\label{G11}.
\end{align}
A standard application of \cite[Lemma 4.3]{Tit86} combined with
(\ref{bounds:F2E}) now tells us that $G_{1,1}(s)$ is a bounded and
holomorphic function on the half-plane $\re(s) > 5/6$. Recalling the
definition of $\Delta$ in (\ref{Delta}), we find that
\begin{align*}
D(s + 1/3)&=
\sum_{\eta_1,\eta_2,\eta_3,\eta_4=1}^\infty\frac{\vartheta(\eta_1,\eta_2,\eta_3,\eta_4)}
{\eta_1^{4s+1}\eta_2^{2s+1}\eta_3^{3s+1}\eta_4^{3s+1}}\\
&= \prod_p \sum_{k_i=0}^\infty
\frac{\vartheta(p^{k_1},p^{k_2},p^{k_3},p^{k_4})}{p^{(4s+1)k_1 +
(2s+1)k_2 + (3s+1)(k_3+k_4)}}.
\end{align*}
After recalling the expression for $\vartheta(\bfeta)$ in
(\ref{vartheta}) and using the fact that $\vartheta(\bfeta) \neq 0$
if and only if $(n_2,n_3)=(n_2,n_4)=(n_3,n_4)=1$, this sum greatly
simplifies and it is easy to see that $D(s + 1/3) = \prod_p D_p(s +
1/3)$ where
\begin{align*}
D_p(s + 1/3) =1 +
&\left(1 - \frac{1}{p} \right)
\left(\frac{1}{p^{2s+1}-1} +
\frac{2}{p^{3s+1}-1} +
\frac{1-2/p}{p^{4s+1}-1}
\right) \\
+&
\left(1-\frac{1}{p}\right)^2
\left(\frac{1}{p^{4s+1}-1}\right)
\left(\frac{1}{p^{2s+1}-1} +
\frac{2}{p^{3s+1}-1}
\right).
\end{align*}
Recalling the definition of $E_1(s)$ and $E_2(s)$ given by
(\ref{E12}), we can prove the following.
\begin{lem} \label{lem:G12}
We have
$$D(s+1/3) = E_1(s+1)E_2(s+1)G_{1,2}(s+1)$$
where $G_{1,2}(s+1)$ is holomorphic and bounded on the half-plane
$\mathcal{H}=\{s \in \mathbb{C} : \re(s)\geq-1/3 + \varepsilon\}$.
\end{lem}
\begin{proof}
Defining $G_{1,2}(s+1)=D(s+1/3)/(E_1(s+1)E_2(s+1))$, it is clear that it will be
enough to show that $G_{1,2}(s+1) = \prod_p(1 + O(1/p^{1+\varepsilon}))$
on $\mathcal{H}$. A routine calculation tells us that
\begin{align*}
D_p(s+1/3)&\left(1 - \frac{1}{p^{4s+1}}\right) =
1 - \frac{3}{p^{4s+2}} + \frac{2}{p^{4s+3}} \\
&+\left(1-\frac{1}{p}\right)\left(1 - \frac{1}{p^{4s+2}}\right)\left(\frac{1}{p^{2s+1}-1} + \frac{2}{p^{3s+1}-1}
\right).
\end{align*}
Now on $\mathcal{H}$ we have the following estimates
\begin{align*}
\frac{1}{p^{4s+2}} & =
O\left(\frac{1}{p^{2/3+\varepsilon}}\right) , \quad
\frac{1}{p^{2s+1}-1} =
O\left(\frac{1}{p^{1/3+\varepsilon}}\right), \\
\frac{1}{p^{4s+3}} & =
O\left(\frac{1}{p^{5/3+\varepsilon}}\right), \quad
\frac{1}{p^{3s+1}-1} =
O\left(\frac{1}{p^{\varepsilon}}\right).
\end{align*}
So on $\mathcal{H}$ we have
\begin{align*}
D_p(s+1/3)\left(1 - \frac{1}{p^{4s+1}}\right) &=
1 - \frac{3}{p^{4s+2}} + \frac{1}{p^{2s+1}-1} \\
&+\frac{2}{p^{3s+1}-1}\left(1 - \frac{1}{p^{4s+2}}\right) +
O\left(\frac{1}{p^{1+\varepsilon}}\right).
\end{align*}
And finally an easy calculation gives us
\begin{align*}
\frac{D_p(s+1/3)}{E_{1,p}(s+1)}= &1 - \frac{3}{p^{4s+2}}
- \frac{2}{p^{5s+2}} - \frac{1}{p^{6s+2}} +
\frac{4}{p^{7s+3}} \\
& \quad+ \frac{2}{p^{8s+3}} - \frac{1}{p^{10s+4}}
+ O\left(\frac{1}{p^{1+\varepsilon}}\right)
\end{align*}
where $E_{1,p}(s+1)$ is the corresponding Euler factor of
$E_1(s+1)$, thus proving the claim.
\end{proof}
Thus letting
\begin{align}
G_1(s)=G_{1,1}(s)G_{1,2}(s) \label{G1}
\end{align}
and combining (\ref{G2}),(\ref{Z1}) and (\ref{G11}) with Lemma
\ref{lem:G12}, we have proved Theorem~\ref{thm:HZF}.
\subsection{The Asymptotic Formula}
In this section we shall prove Theorem~\ref{thm:asym}. Our starting
point is the expression for the counting function given by
(\ref{N(B)}), which we shall simplify using partial summation and
the properties of the Dirichlet series $D(s)$ deduced in
Lemma~\ref{lem:G12}. In what follows let $M(B) = \sum_{n \leq B}
\Delta(B)$.
\begin{lem}\label{lem:sum_Delta}
We have
$$M(B) = \frac{E_2(1)G_{1,2}(1)}{144}B^{1/3}Q(\log
B) + O(B^{7/8-2/3 + \varepsilon})$$
where $Q \in \mathbb{R}[x]$ is some monic cubic polynomial.
\end{lem}
\begin{proof}
Letting $T \in [1,B]$, Perron's formula \cite[Theorem 3.12]{Tit86} tells us that for
non-integral $B$ we have
$$M(B) = \frac{1}{2\pi
i}\int_{1/3 + \varepsilon-iT}^{1/3 +
\varepsilon+iT}D(s)\frac{B^s}{s}ds + O\left(\frac{B^{1+\varepsilon}}{T}\right).$$
Changing variables and using Lemma~\ref{lem:G12} we deduce that
$$M(B) = \frac{1}{2\pi
iB^{2/3}}\int_{1 + \varepsilon-iT}^{1 +
\varepsilon+iT}E_1(s)E_2(s)G_{1,2}(s)\frac{B^s}{s-2/3}ds +
O\left(\frac{B^{1+\varepsilon}}{T}\right).$$
Now let $a \in [7/8,1)$ and let $\Gamma$ be the rectangular contour
through the points $a-iT,a+iT,1+\varepsilon - iT,1+\varepsilon +
iT$. Then, as we have already shown, $E_2(s)$ and $G_{1,2}(s)$ are
holomorphic and bounded inside this contour, and $E_1(s)$ has a pole
of order $4$ at $s=1$. Recalling that $\zeta(s)$ has a simple pole
of order $1$ at $s=1$ with residue $1$, we have
$\lim_{s\to1}E_1(s)(s-1)^4=4\cdot3^2\cdot2=72$. Also we have the
following Taylor series
$$B^s=B\sum_{n=1}^\infty \frac{(\log B)^n (s-1)^n}{n!}$$
which gives us the residue
$$\mbox{Res}_{s=1}\left\{E_1(s)E_2(s)G_{1,2}(s)\frac{B^s}{s-2/3}\right\}
= \frac{E_2(1)G_{1,2}(1)}{144}BQ(\log B)$$ where $Q \in \mathbb{R}[x]$
is some monic cubic polynomial. So letting $$\mathcal{E}(s)
=\sum_{n \leq B} \Delta(n) - \frac{E_2(1)G_{1,2}(1)}{144}B^{1/3}Q(\log
B)$$ and applying Cauchy's
residue theorem to the contour $\Gamma$, we deduce that
\begin{align*}
\mathcal{E}(s) &\ll B^{-2/3}\left(\int_{a-iT}^{a+iT} +
\int_{a-iT}^{1+\varepsilon-iT} +
\int_{1+\varepsilon+iT}^{a+iT}\right)
\left|E_1(s)\frac{B^s}{s}\right|ds + \frac{B^{1+\varepsilon}}{T}.
\end{align*}
From \cite[Ch. II.3.4, Theorem 6]{Ten95} we have the bound
$$\zeta(\sigma + it) \ll
|t|^{(1-\sigma)/3 + \varepsilon}, \quad \mbox{ if } \sigma\in [1/2,1]. $$
Note that our choice of $a$ implies that in the strip $a < \re(s) <
1$, we have $4\sigma-3, 3\sigma-2, 2\sigma-1
>1/2$, so $|E_1(s)| \ll |t|^{4(1-\sigma)+\varepsilon}.$ Then the
contribution from the first horizontal contour is
\begin{align*}
\int_{a-iT}^{1+\varepsilon-iT}\left|E_1(s)\frac{B^s}{s}\right|ds
& \ll \int_{a}^{1+\varepsilon}T^{3-4\sigma+\varepsilon}B^{\sigma}d\sigma
\\
&\ll \frac{B^{1+\varepsilon}T^{\varepsilon}}{T} +
B^aT^{3-4a+\varepsilon},
\end{align*}
and the same bound is obtained for the other horizontal contour. For
the vertical contour we will use well-known estimates for the fourth
moment of the zeta function. First note that
$$\int_{a-iT}^{a+iT} \left|E_1(s)\frac{B^s}{s}\right|ds
\ll B^a\int_{-T}^T \frac{|E_1(a+it)|}{1+|t|}\mathrm{d}t.$$ Now let
$0 < U \ll T$ and consider the following dyadic interval
$$\int_{U}^{2U} \frac{|E_1(a+it)|}{1+|t|}\mathrm{d}t
\ll \frac{1}{U}\int_{U}^{2U} |E_1(a+it)|\mathrm{d}t =
\frac{J(U)}{U},$$ say. H\"{o}lder's inequality now tells us that
$$J(U) \leq J_4(U)^{1/4} J_3(U)^{1/2} J_2(U)^{1/4}$$
where $J_k(U)=\int_{U}^{2U}|\zeta(k(a -1)+1+kit)|^4\mathrm{d}t$. Now
by convexity \cite[Ch. VII.8]{Tit86} and the fact that we have
$\int_0^T|\zeta(1/2+it)|^4 \ll T\log^4T$ by \cite[Th. 1]{HB79}, we
see that for $\sigma\in [1/2,1]$ we have
$$\int_U^{2U}|\zeta(\sigma+it)|^4\mathrm{d}t \ll U^{1+\varepsilon}.$$
Hence we deduce that $J(U) \ll U^{1+\varepsilon}$. Now summing over
these dyadic intervals we find
$$\int_0^T \frac{|E_1(a+it)|}{1+|t|}\mathrm{d}t \ll
T^{\varepsilon}.$$ The same estimate holds over the interval
$[-T,0]$, and so putting everything together we find an overall
error of $$\mathcal{E}(s) \ll \frac{B^{1+\varepsilon}}{T} + B^{a -2/3 +
\varepsilon}.$$ Taking $T=B,a=7/8+\varepsilon$, the
error we obtain is satisfactory for the
lemma.
\end{proof}
Using this lemma we can deduce the following.
\begin{lem}\label{lem:sum_Delta_F2}
We have
\begin{align*}
&\sum_{n \leq B}\Delta(n) F_2\left(\left(\frac{n}{B}\right)^{1/3}\right) \\
& =\frac{E_2(1)G_{1,2}(1)}{144}\left(\int_0^1F_2(u)\mathrm{d}u\right) B^{1/3}P(\log B) +
O(B^{7/8-2/3 + \varepsilon})
\end{align*}
where $P \in \mathbb{R}[x]$ is some monic cubic polynomial.
\end{lem}
\begin{proof}
For ease of notation let $C=E_2(1)G_{1,2}(1)/144$. Applying partial
summation, using (\ref{bounds:F2E}) and Lemma~\ref{lem:sum_Delta} we
deduce that
\begin{align*}
&\sum_{n \leq B}\Delta(n)
F_2\left(\left(\frac{n}{B}\right)^{1/3}\right) \\
& =F_2(1)M(B) - \int_1^B M(t)
\mathrm{d}F_2\left(\left(\frac{t}{B}\right)^{1/3}\right) \\
& =C\int_1^B F_2\left(\left(\frac{t}{B}\right)^{1/3}\right)
\mathrm{d}\left(t^{1/3}Q(\log t)\right) +
O\left(B^{7/8-2/3+\varepsilon}\right)\\
\end{align*}
It remains to simplify the main term. In what follows we focus on
the leading term of the polynomial $Q$, the lower order terms being
dealt with similarly. After changing variables we deduce that it
equals
\begin{align*}
& CB^{1/3}\int_{1/B^{1/3}}^1 F_2(u)
\mathrm{d}\left(u (\log u^3B)^3\right) \\
& =CB^{1/3}(\log B)^3\int_{B^{-1/3}}^1 F_2(u)
\mathrm{d}u + \cdots,
\end{align*}
where all the implied lower order terms are easily seen to be of the
order $O(B^{1/3}(\log B)^2\int_{B^{-1/3}}^1 F_2(u) u^{\varepsilon}
\mathrm{d}u)$. On using (\ref{bounds:F2E}) to deduce that
$$\int_0^{B^{-1/3}} F_2(u)u^\varepsilon\mathrm{d}u \ll B^{-1/6 + \varepsilon},$$
the result follows.
\end{proof}
Hence, combing Lemma~\ref{lem:sum_Delta_F2} with (\ref{N(B)}), we
deduce the asymptotic formula given in Theorem~\ref{thm:asym}. One
can also verify the leading constant, after noticing that
$\tau_{\infty}(\widetilde{S})=6\int_0^1 F_2(u) \mathrm{d}u$ and
using Lemma~\ref{lem:G12} to deduce that
$\prod_p\tau_p(\widetilde{S}) = E_2(1)G_{1,2}(1)$.
|
2,877,628,090,831 | arxiv | \section{Introduction}
Open systems that exchange matter with the environment represent a major challenge for theoreticians and simulators. In fact, the variation of the number of particles during the evolution of the system corresponds to a sudden change of the total amount of microscopic information that one must process and analyze within a consistent system-environment physical framework. In a previous work co-authored with Matej Praprotnik, \cite{physrep} an overview of theoretical principles and (mostly classical) simulation techniques for systems with open boundaries available in the literature has been discussed. In \cite{physrep} it was concluded {that when modeling matter classically, there is a satisfactory understanding of how to treat the case of varying the number of particles on a numerical level as well as on a conceptual level, with the corresponding dimensional change of the phase space. On the other hand, departing from a classical representation and instead modeling this process in the quantum case becomes far more complex.} In standard situations, e.g. in equilibrium, a classical particle arriving from the external environment into a system needs only to accommodate locally and does not modify abruptly the overall state of the system. The same concept does not hold in the case of quantum particles where the change in the number of particles completely redefines the quantum state of the system. Quantum particles are characterized by quantum correlations or better, by their entanglement, which implies that the most accurate knowledge of a system does not imply the most accurate knowledge of its parts \cite{kais,markusreiher,tecmer,ijqc}, thus the gain/loss of information for particles entering/leaving a subsystem (of specific interest) must be treated and interpreted with special care. In this work I will focus on the treatment of one specific, albeit relevant, class of quantum systems, that is many-electron systems that exchange particles, {i.e. electrons, or atoms/molecules (electrons + classical nuclei)}, with a large environment. The description of the environment and its coupling to the (sub)system are the two main ingredients for the construction of a computational procedure that can simulate the process of exchange. The environment can be considered with its full electronic structure, that is, if rigorously treated, the system of interest is merely a subsystem of a fully resolved quantum large system. In reality, pragmatic approximations are used to fully resolve the environment at a reasonable computational price. In general, the conceptual advantage of this model is that the electronic coupling (system-environment) is explicitly taken into consideration, and as a consequence the electronic correlations between the subsystem and the environment are, with some degree of accuracy, explicitly considered into the electronic properties of the subsystem. The computational disadvantage is that one needs to treat either large systems, which would be in most of the cases prohibitive for current computational resources, or a small environment which is likely to suffer from artifacts due to the reduced size. Alternatively, the environment can be considered as an ideal statistical reservoir which is assumed to provide or adsorb particles according to the electronic chemical potential. It follows that the system is described as a quantum Grand Canonical ensemble without the need of explicitly calculating the electronic properties of the environment. The advantage in such a case is that one can focus only on the (sub-)system of interest, but the obvious disadvantage is that information regarding quantum correlations with the environment cannot be derived in any manner.
From the computational point of view, the two categories outlined above can be differentiated in terms of calculations at a fixed number of electrons (energy minimization of the system+environment) and calculations at a fixed chemical potential and a varying number of electrons for the system of interest.
The general conceptual framework outlined above leads us to the backbone of the paper. I have taken as guiding examples the theoretical and computational procedures employed in two relevant classes of applications: (i) in nanotechnology and (ii) in chemical and biological physics. More specifically, to the first category belong the subjects of nanoelectronics and electrochemistry, where the varying number of particles corresponds to the transport of electrons between a (molecular) system and the environment acting as an electrode. To the second category belongs the field of solvation chemistry, where a molecular subsystem exchanges molecules with the large thermodynamic bath in which the subsystem is embedded, for example biomolecules in water. Here the exchange of electrons occurs through the exchange of molecules (i.e. electrons and nuclei) and {from a computational point of view, in order to make calculations more feasible, the environment shall be treated classically \cite{qmmmad2,qmmmad,cpcluigi}. In this case, the combination of quantum mechanics of electrons and the thermodynamics/statistical mechanics of the classical scale becomes particularly delicate \cite{physrep}.} Regarding the methods of calculation, among the electronic structure techniques, {Density Functional Theory (DFT) \cite{dft}, due to its intrinsically statistical mechanical structure, facilitates the extension of its principles to the treatments of electrons in various ensembles and at the same time it allows for the use of functionals beyond the total energy. The previous statement can be verified by consulting the seminal book of Parr and Yang \cite{bookdft} where DFT is derived in terms of density matrix and corresponding statistical ensembles}. As a consequence, DFT represents the most flexible and popular approach to the Grand Canonical treatment of electrons; a large amount of studies of chemical and physical systems where the DFT Grand Canonical approach has been used are present in the literature (see e.g. \cite{lozovoi,sprik,anatole,auer,arias,ayers1} and references therein). In general, all of the applications reported above use DFT, in various forms and in combination with other techniques, as the electronic structure method of preference. Beyond DFT, among the most advanced (wavefunction-based) electronic structure techniques, one finds high level quantum chemical methods in Fock space. In fact, this is a natural approach to create and destroy electrons in a system \cite{bochum}. In addition, Quantum Monte Carlo (QMC), given its stochastic/statistical nature \cite{qmc-generic}, is flexible enough to treat the case of a varying number of particles and thus, treat the Grand Canonical ensemble \cite{qmc-size-a,qmc-size-b,qmc-size1,qmc-size2}. I will discuss the principles of the abovementioned QMC for one specific example, although it must be underlined that such an approach has not been explicitly used to treat the physical exchange of electrons, but has been employed mostly as a numerical trick that minimizes the size effects of the calculations{, i.e. it improves the convergence of calculated quantities to their corresponding value in the thermodynamic limit}. Nevertheless, as an electronic structure Grand Canonical technique, the QMC approach may be embedded in multiscale methodologies of the near future that will treat the physical process of varying the number of electrons and molecules. Finally, the paper is concluded with a discussion where the current ideas and related techniques are put in perspective. Possible combinations and modifications are suggested, which may be likely to optimize the methodology and hopefully push it beyond the current frontiers of applicability to physical systems.
\section{Electron flow: nanoelectronics and electrochemistry}
The passage of electrons from a molecule to its environment that acts as an electrode (and vice versa) is the prototypical situation occurring in nanoelectronics and electrochemistry. In nanoelectronics, the most popular case treated is that of a molecule that acts as a junction/bridge between two surfaces (metals or semiconductors) and allows the passage of a current of electrons from one surface to the other \cite{nanoel1,nanoelnic,nanoelrefpap,nanoel-advts}. In electrochemistry, a typical example is the flow of electrons between the reactants and a catalytic surface as a chemical reaction proceeds \cite{46ofarias}. In the next two sections, I will analyze how these two situations are treated at methodological level.
\subsection{Electron Transport in Molecular Junctions}
Figure \ref{junction} depicts the typical structure of a simple molecular junction attached to a right and left electrode.
\begin{figure}[htbp]
\centering
\includegraphics[clip=true,trim=0.1cm 0cm 0cm 0.1cm,width=9cm]{junction.jpg}
\caption{Schematic illustration of a junction. Left and right lead are part of the environment (electrode), while the molecule is the system of interest through which the electrons flow. For numerical convenience, often the system of interest is the molecule and some of the metal atoms of the electrode, this approach goes under the name of ``extended molecule''.}
\label{junction}
\end{figure}
{Differently from the other examples treated in this work (systems in equilibrium), the theoretical description of the transport of electrons from one electrode to the other is particularly challenging. In fact in such a case one deals with the more complex situation of non-equilibrium for a quantum many-body system.} In the analysis of the methodological aspects relevant to our current discussion, I will follow the recent perspective paper of Thoss and Evers \cite{nanoelrefpap}, which exhaustively traces the current state of the art in the field (see also references in the special issue about frontiers in molecular electronics \cite{spcnano}). They make a major distinction between {\it weakly correlated systems}, that is, systems where the energy transfer between charge carriers and the molecule can be neglected, and {\it strongly correlated systems}, where electron-electron or electron-phonon correlation have a significant impact. The weakly correlated regime, for its methodological aspects related to the system-environment treatment, is of particular interest to our specific analysis and will be discussed in the next section. For the strongly correlated regime I will not discuss the specific aspects of the strong correlations and its physical implications, instead I will restrict the attention to its treatment in the framework of a dynamic model of quantum open systems that exchange {electrons} with a reservoir.
\subsubsection{Weakly correlated regime}
The weakly correlated regime allows the description of a molecular junction within the approximation of quasi-static scattering potential and is valid in the regime of linear response, that is, the transport is treated as an elastic process. Within this picture, the electronic properties of interest can be calculated by Kohn-Sham Density Functional theory (KS DFT)\cite{dft}. The current-voltage characteristic, following the Landauer formula is \cite{nanoelnic,landauer,ratner}:
\begin{equation}
I(V)=\frac{2e}{h}\int_{-\infty}^{\infty}T(E)[f_{L}(E)-f_{R}(E)]dE
\end{equation}
where $V$ is the applied voltage and $E$ the energy levels, $f_{L,R}(E)=f(E-\mu_{L,R})$ is the Fermi distribution function of the lead at the corresponding chemical potentials $\mu_{L,R}$. In this context, the key quantity is the (molecular) transmission function, $T(E)$ which can be written as (see also \cite{koentopp}):
\begin{equation}
T=tr {\bf \Gamma_{L}\mathcal{G}_{V}\Gamma_{R}
\mathcal{G}^{\dagger}_{V}}.
\end{equation}
The quantities involved in the formula are: ${\bf \mathcal{G}}^{-1}_{V}={\bf \mathcal{G}}^{-1}_{0V}-{\bf \Sigma}_{L}-{\bf \Sigma}_{R}$, where the resolvent matrix is:
\begin{equation}
{\bf \mathcal{G}}^{-1}_{0V}({\bf x},{\bf x}^{'},E)=\sum_{n}\frac{\phi_{n}({\bf x})\phi^{*}_{n}({\bf x}^{'})}{E-\epsilon_{n}+i\eta/2}
\end{equation}
the sum is over the KS energies and orbitals, $\epsilon_{n}$ and $\phi_{n}$ for the uncoupled molecule calculated with a standard DFT approach. ${\bf \Sigma}_{L,R}$ are the self-energies, so that in the approximation of non-interacting particles, ${\bf \Gamma_{L,R}}=i({\bf \Sigma}_{L,R}-{\bf \Sigma}^{\dagger}_{L,R})$. The self-energy is the coupling term between molecule and leads, and can be formulated in terms of a hopping matrix: ${\bf \Sigma}_{L,R}={\bf t}_{L,R}{\bf g}_{L,R}{\bf t}^{\dagger}_{L,R}$. The elements of the hopping matrix are defined as: $t_{L,R}({\bf X}N, {\bf x}n)$, describing the hopping process of an electron in an orbital $N$ of an atom at position ${\bf X}$ of the molecule to an orbital $n$ of an atom at position {\bf x}, in the lead. These elements are approximated by their bulk values and are calculated using independent DFT simulations of the lead (i.e. usually a large metal cluster without the molecule) and the molecule. Finally, the surface Green's function, $<{\bf x}n|{\bf g}_{L,R}|{\bf x}^{'}n^{'}>$, that is the effect of broken translational symmetry on the density of states of the metal occurring at the interface plane perpendicular to the lead-molecule direction, is also calculated in {the same DFT simulation of the metal cluster used to find the elements of the hopping matrix.} This description reports the very essential aspects to give an idea of the methodological procedure used in the field; there are of course technical improvements (e.g. ``the extended molecule'' model, that is the molecule and some atoms/layers of the leads) and further conceptual developments of this basic picture. Advantages, limitations and open problems are well described and discussed in Reference \cite{nanoelrefpap}. For the current focus the relevant methodological aspects are: (i) the model of environment/lead, is fully resolved in its electronic structure, although in the pragmatic approximation of large clusters, (ii) its coupling to the system/molecule is done through the hopping process by which electrons are exchanged. This latter aspect requires the construction of a matrix obtained by (iii) modeling the system of interest/junction as an ``isolated molecule'' treated, as for the bulk of the metal, with standard DFT approaches at a constant number of electrons. The approach underlined in this section makes use of a static picture where the dynamics of electron exchange between molecule and lead is expressed through the probabilistic event of the hopping. An interesting methodological variation for the calculation of $T(E)$ where the leads are characterized only by their chemical potentials (given as an input), without the need of knowing their electronic structure, has been proposed by Arnold, Weigend and Evers \cite{quasi-gc}. They develop a general scheme, which despite requiring standard quantum chemistry calculations for the bridging molecule (or extended molecule), in practice gives as a result electronic properties calculated at fixed chemical potentials (of the left and right lead) and a variable number of electrons. This is an effective and computationally robust way to realize the Grand Canonical Ensemble in an implicit manner from a series of (properly looped) standard fixed-particle quantum calculations. Its potential application can go far beyond the case of electron transport and can certainly be used for systems in equilibrium that will be described later on. Beyond the static approaches discussed so far, there exists a class of methods that explicitly considers the electron dynamics and can treat also the {\it strongly correlated} regime. They are based on the Liouville equation for the density matrix of the subsystem of interest (reduced density matrix of the junction). In essence, such an equation describes the dynamical behaviour of electrons in a molecule considered as an open quantum system; such an approach is described in the next section.
\subsubsection{Electron Dynamics: Liouville Equation for the Density Matrix and related methods}
In an interesting work of Emch and Sewell \cite{emsew}, the Liouville equation for the time evolution of a quantum subsystem (S) embedded in an ideal bath (R) is rigorously derived in terms of the evolution of the density matrix of the subsystem. The interesting aspects for the focus of this work is that the authors suggest a coupling term between the S and R also for the case in which the bath is a reservoir of particles (in addition to energy). The essence of the Emch-Sewell model is the filtering of the microscopic degrees of freedom of R through the Zwanzig projector \cite{zwanzig}. The process of filtering leads to effective actions of such microscopic degrees of freedom in terms of actions of statistically averaged macroscopic quantities.
The standard Liouville-Neumann equation: $\frac{d}{dt}\rho=-iL\rho$, with $\rho$ the density matrix of the system and $L=[H,*]$ the Liouville operator with Hamiltonian, $H$, is transformed through the projector operator $\mathcal{P}$ into the equivalent master equation for $\mathcal{P}\rho$:
\begin{equation}
\frac{d}{dt}\mathcal{P}\rho(t)+i\mathcal{P}L\mathcal{P}\rho(t)+\int_{0}^{t}dt^{'}\mathcal{P}L(I-\mathcal{P})\mathcal{U}(t-t^{'})(I-\mathcal{P})L\mathcal{P}\rho(t^{'})=0
\label{eq1}
\end{equation}
$I$ is the identity operator and $\mathcal{U}(t)=exp[-i(I-\mathcal{P})L(I-\mathcal{P})t]$. S and R are initially uncorrelated/independent $\rho(0)=\rho_{R}(0)\otimes \rho_{S}(0)$. The initial state of R is given by the measurement of the set of macroscopic variables, thus $\mathcal{P}\rho(0)=\rho(0)$ and since the average/macroscopic quantities of the reservoir are time independent, then $\mathcal{P}\rho(t)=\rho_{R}(0)\otimes \rho_{S}(t)$ thus Eq.\ref{eq1} can now be written as a self-contained equation for S:
\begin{equation}
\rho_{R}(0)\otimes\left(\frac{d}{dt}+i L_{eff}^{S}\right)\rho_{S}(t)=-\int_{0}^{t}dt^{'}\mathcal{K}(t-t^{'})\rho_{R}(0)\otimes\rho_{S}(t^{'})
\label{eq2}
\end{equation}
considering the trace with respect to R on both sides one obtains the master equation for $\rho_{S}(t)$:
\begin{equation}
\left(\frac{d}{dt}+i L_{eff}^{S}\right)\rho_{S}(t)=-\int_{0}^{t}dt^{'}\mathcal{K}^{S}(t-t^{'})\rho_{S}(t^{'}).
\label{eq3}
\end{equation}
$\mathcal{K}^{S}(t)\rho_{S}(t)=Tr_{R}\{\mathcal{K}(t)\rho_{R}(0)\otimes \rho_{S}(t)\}$. As anticipated above, the interesting point for our focus is that $ L^{S}_{eff}$ and $\mathcal{K}(t)$ can be used to define the exchange of particles between R and S. The Hamiltonian of the system can be divided in three parts: $H=H_{R}+H_{S}+H_{I}$, the relevant part is the latter term, that is the Hamiltonian of interaction between R and S. Correspondingly, one has the related Liouville operators $L_{R}, L_{S}, L_{I}$.
The authors define $H_{I}=\int_{\Omega_{R}}dx\int_{\Omega_{S}}dy V(x,y) J_{R}(x)\otimes J_{S}(y)$, with $x,y$ configuration coordinates of R and S respectively, $\Omega_{R}, \Omega_{S}$ the volumes occupied and $J_{R}(x),J_{S}(y)$ operators acting on the Hilbert subspace of R and S respectively. Such operators represents intensive variables, for example particle number, which could be function of the creation and annihilation operators for particles in R and in S. The intensity of their action is regulated by $V(x,y)$, a potential of (direct) interaction between R and S (e.g Voltage in a junction-lead system).
Furthermore, $L^{S}_{eff}=L_{S}+L^{S}_{I}$, with $L^{S}_{I}\rho=[V_{S},\rho]$ and $V_{S}=\int_{\Omega_{S}}dy\langle V(y)\rangle_{0} J_{S}(y)$. The kernel is defined as follows:
\begin{equation}
\mathcal{K}(t)=\mathcal{P}\mathcal{U}_{S}(t)L_{I}(t)(I-P)\mathcal{U}^{'}(t) L_{I}\mathcal{P}
\end{equation}
with
\begin{equation}
\mathcal{U}_{S}(t)=exp\{-i L_{S}t\}$, $L_{I}(t)=exp[i(L_{R}+L_{S})t] L_{I} exp[-i(L_{R}+L_{S})t]
\end{equation}
and
\begin{equation}
\mathcal{U}^{'}(t)=exp\{-\int_{0}^{t}dt^{'} (I-\mathcal{P}L_{I}(t^{'})(I-\mathcal{P})\}.
\end{equation}
{The treatment is (in principle) exact, but the memory kernel , $\mathcal{K}_{S}(t-t^{'})$, cannot be analytically determined thus it must be approximated. As suggested in \cite{nanoelrefpap}, recent research has brought substantial advances in the field \cite{memory1,memory2}.}
A similar formulation in which the Liouville-Neumann equation considers a stochastic coupling to a bath is the so-called Kossakowski-Lindblad equation \cite{linda,kosslinda}:
\begin{equation}
\dot{\rho(t)}=L(\rho)=i[H,\rho]+\frac{1}{2}\sum_{j}([\mathbb{L}_{j}\rho,\mathbb{L}^{+}_{j}]+[\mathbb{L}_{j},\mathbb{L}^{+}_{j}\rho])
\label{lineq}
\end{equation}
with $H$ the system Hamiltonian, $\mathbb{L}_{j}$,$\mathbb{L}_{j}^{+}$ operators that carry the interaction of the system with a reservoir (Lindblad operators). The term $\sum_{j}([\mathbb{L}_{j}\rho,\mathbb{L}^{+}_{j}]+[\mathbb{L}_{j},\mathbb{L}^{+}_{j}\rho])$, gives Eq.\ref{lineq} the form of a rate equation (quantum jumps of the system under the action of the environment), $[\mathbb{L}_{j}\rho,\mathbb{L}^{+}_{j}]$ and $[\mathbb{L}_{j},\mathbb{L}^{+}_{j}\rho]$ can be interpreted as transition rates between two events (e.g. particles exchange with a reservoir and transition from $N$ to $N^{'}$). The use of the creation and annihilation operators inevitably leads to the treatment of the problem in Fock space, within standard one-particle orbitals of quantum chemistry, or Bloch states and plane waves \cite{kosov,neuhauser}, further routes to describe the electron transport originating from the master equation and references for specific determinations of the memory kernel are discussed in Ref.\cite{nanoelrefpap,tamar}.
It must be reported that methods based on the direct calculation of the time evolution of the wavefunction of the system, rather than its density matrix, are also present in the literature, however in such a case, the wavefunction must be calculated for the whole system (molecule+leads). In order for this calculation to be computationally feasible, the leads must be approximated by a finite system \cite{thosswf}. The most popular approach in such a context is the time-dependent Hartree method (MCTDH) \cite{lubic,vanleuven}. For electron transport, the exchange of electrons between system (molecule) and environment (leads) occurs through the change of orbital occupations (from lead to molecule and molecule to lead) \cite{thosswf2}. In such case correlations are explicitly taken into account, however, the computational effort may be sizable \cite{nanoelrefpap}. Beyond electron transport, within the same wavefunction approach, attempts to restrict the treatment only to a subsystem and add the rest (environment) as e.g. a sink of electrons (i.e. for treating ionization processes), led to different technical and conceptual problems; the most relevant is the fact that the modified Hamiltonian, with a sink: $H+i\Gamma$ with $\Gamma$ a one body potential vanishing on the domain of the molecule, is a non-Hermitian operator \cite{kvaal}. Overcoming such a limitation implies the fulfillment of the requirement that the quantum dynamical semigroup for the time evolution must be trace preserving, Markovian and strictly positive. In practice, one follows the application of the Gorini-Lindblad theorem \cite{gor,linda}, and thus arrives at the Kossakowski-Lindblad equation discussed before. The previous statement implies that the density matrix, when the number of electrons is varying in time, is the proper quantity to consider rather than the direct wavefunction of the system. The use of the Liouville equation discussed so far implies two distinct computational strategies for the partitioning and definition of system and environment. In the approach of the projector operator, the density matrix should, in principle, be defined for the whole space and the physics of the subsystem is obtained by the action of Zwanzig projector operator. It is clear that while the accuracy of the results would be very high, its computational costs are likely to be prohibitive for systems of reasonable size (see also comments about Fock space), and of course reasonable approximations are possible and needed (see e.g. \cite{nanoelrefpap} and references therein). The Liouville equation in the approach of Kossakowski-Lindblad, requires instead only the definition of a transition probability without the necessity of explicit calculations of the electronic properties of the reservoir. Though, of course, its accuracy is directly linked to the construction of a physically well funded stochastic term, for example from transition rates determined experimentally.
In general, as underlined before, the treatment of a problem in Fock space is the natural approach to describe the passage of electrons from one system to another, but in practice the computation may be difficult. In fact, it requires the construction of the density matrix through the treatment of a large number of states to which {electrons} can be located according to the action of the creation and annihilation operators. This means that one needs a predefined reasonable ``active space'' \cite{bochum}. If the explicit time dependence of the electron flow is not of primary relevance, an alternative to the use of Fock space is given by DFT treated in the Grand Canonical ensemble. Calculations for electrochemistry are often done following this approach, and this subject will be treated in the next section.
\subsection{Grand Canonical Density Functional Theory and its application to Electrochemistry}
Electrochemical systems can typically be schematized as an electrolyte, usually consisting of ions in a solvent, and a surface, acting as an electrode, which promotes an electrochemical reaction \cite{46ofarias,cat1}. Electrons are absorbed from or injected into the solution as the reaction proceeds (see also Figure \ref{electro} for a pictorial example).
\begin{figure}[htbp]
\centering
\includegraphics[clip=true,trim=0.1cm 0cm 0cm 0.1cm,width=10cm]{electr.jpg}
\caption{Schematic illustration of an electrochemical process at a metal surface (reduction of formic acid). Panel (a), $H_{2}$ and $CO_{2}$ molecules solvated in water react, through the supply of electrons from the surface, and produce formic acid. Next electrons are released and adsorbed by the surface, Panel (b). In this case the system of interest are the reactants, $H_{2}$ and $CO_{2}$ molecules and the product, the formic acid molecules, while the environment consists of the solvating bath of water and the metal surface which acts as a reservoir of electrons.}
\label{electro}
\end{figure}
From the point of view of simulation, modeling such systems are extremely challenging: the full quantum treatment of the system is computationally prohibitive, while a pragmatic partitioning of the total system as subsystem of interest (where the chemical reaction occurs), and the environment (electrolyte and electrode), allows multiscale simulation techniques to be applied with success \cite{arias}. Later on I will describe a specific model/example that treats the electrolyte, but for the moment, the interesting aspect is the treatment of the electrode as a Grand Canonical reservoir of electrons for the (sub)system of interest. In fact, such a view can be generalized to the problem of a charged surface or a surface in an external electric field which can be treated as a system at constant chemical potential, $\mu$ and varying number of electrons $N$ (for methodologically-oriented work see e.g. \cite{ali,thomas,auer,arias,ayers1}). The use of DFT calculations with constant $\mu$ and varying $N$ leads to the introduction of the (exact) Helmholtz free energy, $A$, for interacting electrons at finite temperature in an external potential $V({\bf r})$. $A$ satisfies the so-called Hohenberg-Kohn-Mermin variational theorem \cite{dft,mermin} (see also Reference \cite{arias}, of which I will follow the formalism):
\begin{equation}
A=\min_{\{n({\bf r})\}}\left(A_{HKM}[n({\bf r})]+\int V({\bf r}) n({\bf r}) d{\bf r}\right)
\label{A1}
\end{equation}
with $A_{HKM}[n]$ the universal free energy functional of the electron density $n({\bf r})$, for any external potential $V({\bf r})$ (in atomic units).
As for the universal energy functional of Hohenberg and Kohn, $A_{HKM}[n({\bf r})]$ is not known and can be only approximated:
\begin{equation}
A_{HKM}[n]=A_{ni}[n]+E_{H}[n({\bf r})]+E_{XC}[n({\bf r})]
\label{A2}
\end{equation}
where $A_{ni}[n]$ is the non-interacting free energy, $E_{H}[n]$ is the Hartree term and $E_{XC}[n]$ is the exchange and correlation energy. The relevant quantity here (compared to standard DFT) is:
\begin{equation}
A_{ni}[n]=\min_{\{(\psi_{i}({\bf r}),f_{i})\to n({\bf r})\}}\sum_{i}\left(\frac{f_{i}}{2}\int |\nabla\psi_{i}({\bf r})|^{2}d{\bf r}- T S(f_{i})\right)
\label{A3}
\end{equation}
that is, the minimum over all the single-particle orbitals $\psi_{i}({\bf r})$ and the corresponding occupation factors $f_{i}\in [0,1]$ which leads to the electron density: $n({\bf r})=\sum_{i}f_i|\psi_{i}({\bf r})|^{2}$. $T$ is the temperature and $S(f_{i})$ is the single-particle entropy:
\begin{equation}
S(f)=-f\ln f-(1-f)\ln(1-f).
\end{equation}
The minimization leads to the single-particle KS equations:
\begin{equation}
-\frac{\nabla^{2}}{2}\psi_{i}({\bf r})+V_{KS}({\bf r})\psi_{i}({\bf r})=\epsilon_{i}\psi_{i}({\bf r})
\end{equation}
for the stationarity w.r.t. $\psi_{i}({\bf r})$ and to the Fermi Occupation condition:
\begin{equation}
f_{i}=\frac{1}{1+e^{\frac{\epsilon_{i}-\mu}{T}}}
\end{equation}
for the stationarity w.r.t. $f_{i}$, with $V_{KS}({\bf r})=V({\bf r})+\frac{\delta}{\delta n({\bf r})}(E_{H}[n]+E_{XC}[n])$. For standard DFT, $\mu$ is a Lagrange multiplier and the fixed number of {electrons} is assured by the condition: $\sum_{i}f_{i}=N$. In a Grand Canonical ensemble instead, the free energy to minimize is not $A$ but the Grand Free Energy:
\begin{equation}
\Phi=A-\mu N
\end{equation}
In such a case, for the optimization procedure applied above, one modification is required, that is the Lagrange multiplier term $-\mu(\sum_{i}f_{i}-N)$ is replaced by $-\mu\sum_{i}f_{i}$. Such modification removes the constraint on fixed $N$ and implements the Legendre transformation from $A$ to $\Phi$; $\mu$ is now given as an input while $N$ is variable.
The grand potential of the electrons can be expressed also in a different formalism, for example following Alavi {\it et al.} \cite{ali}:
\begin{equation}
\Phi[n({\bf r})]=-\frac{2}{\beta}\ln det\left(1+e^{-\beta(\mathcal{H}-\mu)}\right)-\int d{\bf r} n({\bf r})\left(\frac{\phi({\bf r})}{2}+\frac{\delta\Phi_{xc}}{\delta n({\bf r})}\right)+\Phi_{xc}
\end{equation}
with $\Phi_{xc}$ the finite-temperature exchange-correlation grand potential, $\phi({\bf r})$ the Hartree potential, $\beta=\frac{1}{k_{B}T_{e}}$ the electronic temperature parameter, $\mathcal{H}=-\frac{1}{2}\nabla^{2}+V({\bf r})$ the one-electron Hamiltonian, and the effective density-dependent potential, $V({\bf r})=V_{ext}({\bf r})+\phi({\bf r})+\frac{\delta\Phi_{xc}}{\delta n({\bf r})}$, with $V_{ext}({\bf r})$ the external potential. The exponential form can then be efficiently evaluated through the Trotter approximation as a product of $P$ high temperature matrices: $e^{-\beta\mathcal{H}}=\left(e^{-\beta\mathcal{H}/P}\right)^{P}$, with $P$ a large integer, so that $\epsilon=\beta/P$ is small and one can write: $e^{-\epsilon(K+V)}=e^{-\epsilon V/2}e^{-\epsilon K}e^{-\epsilon V/2}+\mathcal{O}(\epsilon^{3})$; based on such an idea, it was possible to devise efficient linear system-size scaling schemes \cite{parr1,parr2,thomas}.
In general, compared to fixed $N$ calculations, those at constant $\mu$ carry several technical problems. For example, the large fluctuations of $N$ (and $n({\bf r})$) at the initial stage, the need of compensating charges because of the finiteness of the slabs representing the surface and periodic boundary conditions \cite{lozovoi,auer,arias}, or the fact that $E_{XC}[n]$ are defined for integer numbers of electrons \cite{perdew,vuilleumier,weitao}. However, technical solutions were made available and thus the method can be routinely used nowadays for applications. The approach outlined above makes possible the introduction of a reservoir of electrons that adds or removes {such} particles as, e.g., a chemical reaction on a surface/electrode proceeds, so that our system of interest consists of the reactants while the surface represents the (active) environment. However, as underlined before, the effect of the solvent often plays a key role and its quantum treatment would be computationally prohibitive. The most popular approximation consists of employing continuum solvation models \cite{cont1,cont2}, however an interesting proposal based on the so-called joint density-functional theory (JDFT) \cite{jdft1,jdft2} has been elaborated in the context of Grand Canonical DFT reported above \cite{arias}. JDFT consists of a variational theorem similar to the Hohenberg-Kohn theorem which allows for the description of the free energy of a solvated system in terms of the electron density $n({\bf r})$ of the solute and of nuclear densities for the solvent, $N_{\alpha}({\bf r})$, where $\alpha$ indicates a nuclear species:
\begin{equation}
A=\min_{\{n({\bf r}), N_{\alpha}({\bf r})\}}\left(A_{JDFT}[n({\bf r}),N_{\alpha}({\bf r})]+\int V({\bf r})n({\bf r})d{\bf r}+\sum_{\alpha}V_{\alpha}({\bf r})N_{\alpha}({\bf r}) d{\bf r}\right)
\label{jdft1}
\end{equation}
similarly to the Hohenberg-Kohn-Mermin functional, $V({\bf r})$ is the external electron potential, $V_{\alpha}({\bf r})$ is the external potential for the nuclei of the solvent and $A_{JDFT}[n({\bf r}),N_{\alpha}({\bf r})]$ is a universal functional independent of the two potentials and can be separated as:
\begin{equation}
A_{JDFT}[n,N_{\alpha}]=A_{HKM}[n]+A_{diel}[n,N_{\alpha}].
\end{equation}
The solvent is described in terms of average density and not in terms of individual atomistic configurations which would imply expensive sampling computations with e.g. molecular dynamics or Monte Carlo, while the system of interest is treated at full quantum (Grand Canonical) accuracy. However, often the solvent and the corresponding configuration space that it can access are of crucial importance for the chemical or physical event of interest and thus the continuum approximation is far too drastic and it requires an explicit molecular treatment of the solvent \cite{bulo-matter}. In general, the region of interest is usually localized in space, e.g. the solvation region of a solute, thus the quantum accuracy may be needed only in a small portion of the system while the rest of the system can be treated at a coarser (classical) level. This is a typical scenario for, e.g. biophysical systems such as proteins and membranes in water, whose relevance for the current academic research as well as for technology has stimulated the development of multiscale computational methodologies, such as Adaptive Quantum Mechanics/Molecular Mechanics (A-QM/MM) \cite{qmmmad} and Adaptive Resolutions Simulation (AdResS) \cite{physrep}. In such cases, the environment is a classical system and provides/takes molecules from the quantum region, thus the exchange of electrons is strictly related to the exchange of entire molecules. This kind of approach is the subject of the next section.
\section{Molecules in Solution: Exchange of Molecules between system and reservoir}
Many interesting systems in chemical physics and in particular in biochemistry are characterized by events that happen locally, e.g. in solution. In such a context, the interesting event takes place in a specific subregion of the system (where electrons play a key role) embedded in a larger thermodynamic bath of, e.g., solvating molecules. The proper (full quantum/electronic treatment) in space and time, of such a class of systems is in most of the cases prohibitive; however, a decisive step forward was taken following the idea of space partitioning of Warshel and Karplus \cite{warschkarplus} and of Birge, Sullivan and Kohler \cite{birge}, that is, each region is treated at the most convenient resolution, i.e. quantum if electrons play a major role or coarser otherwise. Later on, the idea of partitioning was extended further by Warshel and Levitt \cite{warshlev}, who described the enzymatic reaction of lysozyme; their seminal work is nowadays recognized as the first example of the so-called Quantum Mechanics/Molecular Mechanics (QM/MM) method. This method has represented a technical revolution in the field because it allows for the treatment of systems that, up to that moment, were thought to be intractable at quantum level. For this reason, the pioneers of the technique, Martin Karplus, Michael Levitt and Arieh Warschel were awarded the Nobel Prize in Chemistry in 2013. Today, QM/MM represents a very precious computational tool for tackling problems in various field of chemistry and physics with applications in a broad range of subjects and disciplines (see e.g. \cite{roth,thiel}). However, in the standard QM/MM method, the quantum region and the classical region are fixed, that is, there is exchange of energy, but not of matter. The physical implication is that the QM region represents an artificial Canonical ensemble whose accuracy increases by increasing the size of the QM region and checking that the effect of the environment is negligible. The most recent development in the field attempts to technically remove the constraints of a fixed number of molecules in the QM region and thus, allow the exchange of molecules between the quantum and the classical region (see e.g. \cite{qmmmad}).
I have already mentioned such an approach, referring to it as: ``Adaptive Quantum Mechanics/Molecular Mechanics (A-QM/MM)''; its advantages and limitations will be discussed in the next sections. In parallel, within the context of classical multiscale simulation, (molecular) adaptive resolution methods for classical systems have been developed \cite{physrep}. Typically such methods concurrently couple regions at (classical) atomistic resolution with regions at coarse-grained resolution in such a way that when a molecule passes from one region to the other, it slowly changes resolution without creating sizable physical artifacts.
The next frontier for classical adaptive resolution methods is the extension to molecules with electrons. In this context, a computational framework that I have recently proposed, electronic Quantum Adaptive Resolution Simulation (el-QM-AdResS) \cite{cpcluigi}, where the classical adaptive resolution technique is combined with the electronic scale, is discussed in the following sections. Finally, a discussion of the similarities, differences and possible synergies of A-QM/MM and el-QM-AdResS concludes the chapter.
\subsection{A-QM/MM: Partitioning the system on-the-fly}
In the following discussion, I will take as a main reference the exhaustive review by Hai Lin and collaborators \cite{qmmmad} where the state of the art of the field is critically analyzed (for an additional complementary view see also Ref.\cite{qmmmad2}). A-QM/MM, as anticipated above, is an extension of the standard QM/MM method. The essence of A-QM/MM consists of an on-the-fly reclassification of QM and MM atoms/molecules during the simulation so that the molecules included in the QM or in the MM region can be dynamically updated according to the evolution of the system in space. A straightforward approach consists of an ``abrupt'' scheme of reclassification (see Figure \ref{abrupt-a-qmmm}): classical molecules that at time $t_{0}$ lie at the border of the QM region may directly enter into the QM region at time $t_{1}=t_{0}+\Delta t$, ($\Delta t$ time step of the simulation) and be reclassified as QM molecules for the calculations of the next time step; equivalently, QM molecules lying at the border with the MM region and moving at time $t_{1}$ into the MM region are reclassified as classical for the calculations of the next time step.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{abrupt-a-qmmm.jpg}
\caption{Schematic representation of the A-QM/MM method. Molecules that at time $t$ are in the QM region are treated at quantum mechanical level, molecules outside, in the MM region, are treated with standard classical force-field methods. {The example shown here involves water molecules: the quantum molecules are schematized as nuclei around which the electron density is distributed and the classical molecules are schematized as standard atoms using a classical MM force field.}}
\label{abrupt-a-qmmm}
\end{figure}
It was found out that such a sudden change implies the hopping between different energy surfaces whose discontinuity causes numerical instabilities and artificial results \cite{heydenqmmm}. For such a reason, the latest generation of A-QM/MM methods are centered around more involved computational schemes based on the introduction of a buffer region. At each time-step of the simulation,
the buffer region is partitioned in different subsets and the standard
QM/MM interactions are defined for all possible subsets (see Figure \ref{M-part}). Next, for each partition, the molecules of the corresponding subset of the buffer are included in the QM region.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{M-partitioning.jpg}
\caption{Example of the role of the buffer and the corresponding partitioning schemes for $M=4$, for a system of water molecules. As in the previous figure the quantum mechanical character is indicated by the electron density distributed around the nuclei.}
\label{M-part}
\end{figure}
Finally, for each of these ``extended'' QM regions a
standard QM/MM calculation is done. {The total potential is then defined as a
weighted average of the individual potentials from each individual simulation}:
\begin{equation}
U({\bf r})=\sum_{i}^{M}f_{i}({\bf r})U_{i}({\bf r}).
\label{qmmmint}
\end{equation}
Here, each $U_{i}({\bf r})$ corresponds to one of the $M$ partitions of the system in a group of QM
molecules and a group of MM molecules and $f_{i}({\bf r})$ is a switching function depending on the coordinates of the single molecules. The switching function is constructed with the intuitive but well justified idea that the quantum energy of molecules in the buffer at larger distances from the center of the region of interest (active site) contributes less than the energy of molecules which are closer to the active site. This is the essential technical and conceptual point of the method. In fact, the dynamical evolution produced by the energy surface, can now be updated and it
creates a new configuration on which, in turn, the partitioning step and the corresponding weighted potential (or forces) are applied for the time evolution of the next time step. From the technical point of view, the empirical approach to the smoothing of the coupling at the interface (either via potential or via inter-molecular forces) turned out to be very powerful and, in my view, can be considered a very relevant step forward in the development of truly multiscale simulation technologies for condensed matter and chemical physics, as testified by a large number of successful applications \cite{bulos1,nielsen,csanys,watanabe,lins1,lins2} (see also additional references in the topical reviews \cite{qmmmad2,qmmmad}). Yet, Lin and coworkers point out a series of technical problems, but above all conclude that despite the large number of encouraging results, A-QM/MM cannot be considered a truly predictive tool since its results always require a case-by-case validation, i.e. reproduction of some reference results (from experiment or larger QM/MM calculations). {In the next section, I will discuss conceptual aspects that are mandatory for the construction of a A-QM/MM approach with automatic {\it a priori} physical control criteria that can transform A-QM/MM into a predictive tool without the need for external validation.}
\subsection{A-QM/MM: Physical validation and the necessity of a Grand Canonical view}
In Reference \cite{qmmmad}, it is explicitly stated that, in principle, the QM region can be made as small as possible. This statement is certainly true from a technical point of view, however it carries the first conceptual bug of the current QM/MM methods (in general). In fact, if one is interested in a realistic quantum description of a specific region of interest, then the coupling energy between the QM region and the MM region must be negligible compared to the energy of the QM region, otherwise the electronic spectrum is essentially determined by the classical part. It is obvious that in QM/MM the electronic wavefunction and corresponding energy spectrum of the QM region cannot be the same as if the region was in a full QM environment. In good approximation, this situation holds only in the case of localized systems/properties, that is, when there is a negligible coupling energy with the environment (i.e. a reasonably separable Hamiltonian). This would be the first internal criterion of control for the validity of a QM/MM calculation. If we now take a step forward and move to the A-QM/MM approach, there are further conceptual problems. A very relevant one concerns the physical validity of the approach used in the partitioning scheme. Specifically: as said before, at quantum mechanical level, in terms of wavefunctions and electronic spectra, an open QM region embedded in a MM reservoir, in general, cannot be equivalent to a QM region embedded in a full QM reservoir. The equivalence can only hold from the quantum statistical mechanics point of view, if (and only if) the QM region of the A-QM/MM, is interpreted as a Grand Canonical region at the same macroscopic (thermodynamic) and electronic chemical potential of a QM region embedded in a full quantum environment.
{In this respect, the concept of {\it ``Nearsightedness of electronic Matter''} introduced by W.Kohn \cite{nearkohn, nearkohn2}, provides a clear physical principle for the definition of a subsystem embedded in a larger environment. In fact it states that, at fixed chemical potential, local electronic properties depends on the effective external potential (i.e. the environment) only at nearby points. The effects of the external potential beyond a certain distance are negligible for the local properties. Furthermore, in a recent work of Fias, Heidar-Zadeh, Geerlings, and Ayers, it has been shown that the response kernel to the environment is local at constant chemical potential, thus in any partitioning scheme one should use the concept of constant chemical potential \cite{ayers-gerl}.}
To my knowledge, the current partitioning schemes of A-QM/MM do not use {any of the criteria of control listed above}.
Moreover, in a recent work by Miranda-Quintana and Ayers about electronic systems with a varying number of {electrons} in DFT, it is discussed the possibility that interpolations of property-values between electron numbers is not consistent with a physical ensemble average \cite{quintana-ayers}.\\
Their finding may be extended to the current discussion and would imply that any average property obtained by averaging over the different $M$ partitions of the A-QM/MM system, where each calculation is done minimizing the energy $E_{i}$ at a given (fixed) number of electrons $N_{i}$, is inconsistent with statistical mechanics.\\
An extension of the A-QM/MM method that includes the internal criteria of control described in this section, has been proposed by myself and is based on the inclusion of molecules with electrons within the classical atomistic/coarse-grained Adaptive Resolution scheme (AdResS) in its Grand Canonical version (GC-AdResS) \cite{cpcluigi}. The essence of the idea is reported in the next section.
\subsection{el-QM-AdResS: A Grand Canonical electronic system embedded in a classical reservoir}
The Adaptive Resolution Simulation technique (AdResS) \cite{jcp,annurev} for classical systems allows molecules to transform their resolution according to the region in space where they are instantaneously located. The root model of the computational algorithm consists of a space dependent interpolation for the force between two molecules, $\alpha,\beta$ (see Fig.\ref{cartoon-adress}):
\begin{equation}
F_{\alpha \beta} = w(X_{\alpha})w(X_{\alpha})F_{\alpha\beta}^{AT} + [1 - w(X_{\alpha})w(X_{\alpha})]F_{\alpha\beta}^{CG}
\end{equation}
where $F_{\alpha\beta}^{AT}$ is the atomistic force and $F_{\alpha\beta}^{CG}$ is the coarse-grained force.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{adress-cl.jpg}
\caption{Schematic representation of the AdResS simulation set up. AT indicates the atomistic region; $\Delta$ is the region where molecules change resolution according to $w(x)$, and CG is the coarse-graining region. {Adapted with permission from Figure 1 of Ref.\cite{cpcluigi}. License number 4355330763152, Copyright 2018, Elsevier}}
\label{cartoon-adress}
\end{figure}
$w(x)$ ($x$ is the coordinate of the center of mass of a molecule) smoothly goes from 0 to 1 in a transition region $\Delta$. The system is embedded in a thermostat that assures a target thermodynamic equilibrium. The conceptual solidity of the original AdResS method is then enhanced by the addition of a force, ${\bf F}_{th}(x)$ (thermodynamic force), which acts on the center of mass of the molecule in $\Delta$. Such a force assures that, in situations of equilibrium, the effective chemical potential of the whole system corresponds to that of the (target) atomistic resolution \cite{jcpsimon,prx,njp}. ${\bf F}_{th}(x)$, has been derived within a rigorous statistical mechanics model for molecular systems with open boundaries \cite{prl12,jctchan,prx,mujcp,njp,pre16,physrep} (thus AdResS became Grand Canonical (GC-) AdResS) and it was found out that a numerically convenient way to calculate ${\bf F}_{th}(x)$ during a simulation (equilibration run), consists of expressing it as the gradient of the number density of the molecules in an iterative form \cite{prl12}: $F_{k+1}^{th}(x)=F_{k}^{th}(x) - \frac{M_{\alpha}}{[\rho_{ref}]^2\kappa}\nabla\rho_{k}(x)$,
$M_{\alpha}$ is the mass of the molecule, $\kappa$ a (conveniently) tunable constant, $\rho_{k}(x)$ is the molecular density as a function of the position in $\Delta$ at the $k$-th iteration and $\rho_{ref}$ is the density of reference, decided {\it a priori} according to the thermodynamic state point at which we wish to do the simulation.
The convergence criterion depends on the accuracy required for the simulation
but, based upon experience, $|\rho_{final}-\rho_{ref}|$ should always be below $10\%$ in $\Delta$. The accuracy of this method has been proven over the last ten years over a large range of systems and problems, see e.g. \cite{physrep} for an overview, and Refs.\cite{jcpil,advts} for the most recent applications. In this context, it is important to notice that the coupling scheme allows one to interface any classical molecular representations (e.g. two different atomistic models, atomistic and path integral representation of molecules \cite{lujcppi,cpcanim}, to mention a few). There also exists a version based on a global Hamiltonian \cite{raff1,raff2}, which is, in essence, technically equivalent to the method outlined above (although conceptually confusing, see discussion in \cite{physrep}). Given the technical equivalence, it is not a surprise that calculations repeated with the Hamiltonian scheme give the same results obtained years before with the (GC-)AdResS method (compare \cite{lujcppi,cpcanim}, with \cite{raffpi1,raffpi2}; \cite{matdna} with \cite{raffdna}; and \cite{mujcp} with \cite{raffmu}).\\
In order to include electrons, the theoretical framework of el-QM-AdResS, proposed in Reference \cite{cpcluigi}, is based on two simple concepts (see also pictorial representation in Figure \ref{el-qm-ad-scheme}):
\begin{itemize}
\item Use the classical GC-AdResS scheme as a computational interface for the quantum region (i.e. equivalent to the MM region of a QM/MM scheme).\\
GC-AdResS will take care of providing and removing the (classical) nuclei (i.e. the skeleton of the molecule) in the QM region according to the macroscopic (thermodynamic) chemical potential.
\item The electrons of the QM region are treated as a system in contact with an ideal (Grand Canonical) reservoir of electrons whose exchange of particles is regulated by a properly chosen (see explanation later) electronic chemical potential.
\end{itemize}
\begin{figure}[h!]
\centering
\includegraphics[width=0.80\textwidth]{el-qm-adress.jpg}
\caption{Pictorial representation of the el-QM-AdResS. ${\mu}_{macro}^{QM}$ is the macroscopic chemical potential assured by the thermodynamic force in the MM and the $\Delta$ region. In essence, as in the $\Delta$ region of a purely classical GC-AdResS, an iterative procedure is applied so that the average number density of molecules in the MM region (and of course in the $\Delta$ and CG regions of el-QM-AdResS) is equal to the molecule number density of reference ({at which the QM region automatically is}). ${\mu}_{el}^{QM}=\mu_{el}^{ref}$ is the electronic chemical potential of the QM region (which should be the same as that of a corresponding full quantum calculation of the bulk/environment).}
\label{el-qm-ad-scheme}
\end{figure}
The scheme follows three automatic criteria of control for assuring physical consistency of the calculation:
\begin{itemize}
\item The coupling between the QM region and the classical reservoir must be such that: $\langle H_{QM-Ad}\rangle << \langle H_{QM}\rangle$ at any time, with $ H_{QM-Ad}$ the coupling Hamiltonian of the QM region with the GC-AdResS interface (as an example of application of this criterion in GC-AdResS, see \cite{njp,lujcppi}).
\item The macroscopic chemical potential must be such that $\mu_{macro}^{QM}=\mu_{macro}^{GC-AdResS}$, i.e. the whole system is at the same macroscopic chemical potential at any time.
\item The electronic chemical potential in the QM region must satisfy the condition: $\mu_{el}^{QM}=\mu_{el}^{ref}$, where $\mu_{el}^{ref}$ is the electronic chemical potential the QM region would have if the whole system was treated at quantum resolution. For example, for a molecule solvated in water, the QM region would be composed of the solute and the water molecules of the first solvation shells embedded in bulk water, thus the electronic chemical potential of reference would be that of bulk water.
\end{itemize}
The first condition is a simple check that assures the choice of a physically consistent size of the QM region. The second condition is fulfilled by the thermodynamic force, calculated with the same formula used in standard GC-AdResS for the region $\Delta$, but now calculated for (and applied in) the MM region at the interface with the QM region. Finally, the third condition can be implemented by performing electronic structure calculations for the QM region at constant $\mu_{el}$, with the number of electrons being the variable in the (Grand Canonical) energy functional minimization (in the same spirit of the Grand Canonical DFT calculations of the previous sections). The electronic chemical potential can be determined from separate calculations of prototype systems that represent a reasonable bulk/environment (see example of liquid water before). The main advantage of the el-QM-AdResS scheme consists in the assurance regarding the statistical mechanical consistency of the results; however, {it carries with it open questions regarding the integration of its conceptual aspects and its technical implementation.} In fact, from the technical point of view, one major challenge is the abrupt interface between the QM and the GC-AdResS region. It is possible that the thermodynamic force, acting in the MM region directly adjacent to the QM region which, in essence, equalizes the molecular number density in an iterative process (equilibration run), may not converge as expected. It is encouraging though that recent tests have proved that this may not be a problem; in fact, an abrupt interface can be turned into an advantage \cite{prlyn}. Another challenge is the fact that $N$, the number of electrons in the QM region, may not be integer, which implies that some fractional charge is delocalized in the classical region and it must be taken into account. In the meantime, some technical solutions have been proposed \cite{cpcluigi} and probably the optimal development of this idea is to combine it with the already well developed technical A-QM/MM schemes. A brief discussion about this possibility is considered in the next section.
\subsection{el-QM-AdResS + A-QM/MM: A path to a technically powerful and conceptually self-validating innovative scheme}
el-QM-AdResS is, as a matter of fact, technically similar to A-QM/MM, and some work in merging AdResS with A-QM/MM has been already made in the group of Rosa Bulo \cite{bulo-adr}. A proposed way to proceed further would be to extend the criteria of physical validity used in el-QM-AdResS to A-QM/MM and, from the other side, use a buffer region and the corresponding partitioning scheme of A-QM/MM in el-QM-AdResS. The resulting method would be a new A-QM/MM where each extended QM region, corresponding to each partition, is treated within the (same) $\mu_{el}$ constant scheme (and implicitly, through the thermodynamic force, at the same $\mu_{macro}$). As a consequence, the average potential (or forces) calculated over all partitioning schemes in the buffer region would correspond to an average over different, but equivalent (i.e. same $\mu_{el}$ and same $\mu_{macro}$), Grand Canonical systems, and thus, it would have a solid statistical mechanics justification. At the same time the technical problem of the abrupt interface in el-QM-AdResS could be solved by the smoothing technique of the buffer region, where the thermodynamic force can also be applied and smoothly calculated. Moreover, GC-AdResS allows a further coupling of the atomistic resolution to coarse-grained models and beyond (continuum models) \cite{matej1,matej2}, thus, the fully-realized A-QM/MM approach would represent a truly multiscale method.\\
Most of the QM calculations are done within the DFT scheme, however, one can, in principle, go beyond DFT and aim for a higher level of accuracy. For example, Hofer and H\"{u}nenberger have recently explored the idea of going beyond DFT and using the resolution-of-identity second-order M{\o}ller-Plesset perturbation (RIMP2), which is a sort of first level beyond DFT for describing electron correlations \cite{hunen}. In this perspective, it is important to also consider other electronic structure methods that can accurately describe electron correlations but, given the context of this work, also assure exchange of matter, and thus calculations involving a varying number of electrons. In the previous chapters, quantum chemistry approaches in Fock space have already been discussed, however, one may also extend the discussion to other methods such as Quantum Monte Carlo (QMC). As previously anticipated, QMC techniques in the Grand Canonical ensemble are developed and used mostly for technical reasons (alleviating size effects), but in any case, in this context, they may be included in a scheme such as el-QM-AdResS or A-QM/MM. A discussion of QMC methods where the number of electrons can vary is reported in the next section.
\section{Quantum Monte Carlo {and a} varying number of electrons}
Quantum Monte Carlo methods can be classified among the most accurate techniques for describing electron correlations. There are multiple QMC approaches, each with its own positive and negative aspects; however, in general they are all closely related. {Here, I will not provide an overview of specific QMC techniques (for an updated discussion see Reference \cite{qmc-generic} and references therein) but I will instead focus on some generic features that can be applied to different QMC approaches and involve the possibility of carrying out QMC calculations in a Grand Canonical ensemble}. The work of reference which I will follow is the paper by Lin, Zong and Ceperley who introduced the so-called ``twist average Grand Canonical ensemble'' (TA-GCE) \cite{qmc-size1}.\\
TA-GCE has been developed in order to reproduce results of the thermodynamic limit with calculations involving a (relatively small) finite number of electrons, thus circumventing finite size effects. The starting point of this idea is the observation that in QMC the phase factor of the many-electron wavefunction is usually assumed to return to the same value if {electrons} cross the periodic boundary and return to the same position. The technical consequence of this assumption is that, for delocalized electrons, the convergence of properties to their thermodynamic limit is very slow. In Reference \cite{qmc-size1}, Lin, Zong and Ceperley proposed an alternative that allows {electrons} to {take up a phase angle} when they cross the periodic boundary. This is the basic concept of twist averaged boundary conditions (TABC); the pre-existing literature background they refer to is a work by Gros \cite{qmc-size-a} who has shown that TABC, applied to the Hubbard model, gives exact results in the Grand Canonical ensemble for non-interacting particles. Thus, for the focus of this paper, this is a procedure of interest for the treatment of a varying number of electrons. In the following, I will report the basic ingredients of the idea of Lin, Zong and Ceperley. In non-interacting homogeneous systems with periodic boundary conditions, plane waves describe single particle states. For simplicity of, exposition the spinless case is considered, so that the single state of the system in a, e.g., cubic box of linear size $L$, can be written as: $\phi({\bf r}, {\bf k})\propto e^{i{\bf k}{\bf r}}$. In order to satisfy the TABC one must have: ${\bf k}_{\bf n}=(2\pi+\theta)/L$, with ${\bf n}$ an integer vector. In the Canonical ensemble, the $N$ lowest energy states form the ground state and a Slater determinant of such states gives the ground state wavefunction of the system. In the Grand Canonical ensemble one has that: $E_{\alpha,N}=E_{\alpha,N}(\theta)$, where $(\alpha,N)$ labels the quantum states. The probability of a given state is: $P(\alpha,N)\propto exp\{-\beta[E_{\alpha,N}(\theta)-N\mu]\}$, with $\mu$ the chemical potential of the system. In the ground state, that is for $\beta\to \infty$, the wavefunction must not optimize the energy $E_{\alpha,N}$, as in the Canonical ensemble, but the quantity: $E_{\alpha,N}(\theta)-\mu N$. Next, they show that the occupation number is exactly what one gets if the calculation were to be done in the thermodynamic limit, and thus TABC (now renamed TA-GCE) carries no size effects. The procedure implies that the number of {electrons} varies, since, for a given $\theta$ and Fermi wave vector, the number of occupied states will vary. The fluctuations in the particle number is then derived and it is shown to be $\propto N^{1/4}$ in the limit for $N\to \infty$. The treatment discussed so far concerns non-interacting systems; the extension to interacting systems is based on the argument that the theory of Fermi Liquids states that low lying excited states of a non-interacting system are in a one-to-one correspondence with non-interacting states, thus TA-GCE is likely to reduce finite size effects for interacting systems, as well. There are technical difficulties, though, that must be reported. One is that the wavefunction must be optimized at each value of $\theta$, which implies additional computational resources. Another problem is the treatment of net charges that are not balanced, but one can show that the average
charge of the supercell (averaged over the BC) is exactly given by the density; so, in many cases, this is sufficient to get rid of the problem of
a charged supercell. In general, reasonable solutions to such difficulties have been found, and the method has been applied to calculate several properties of interacting electrons \cite{qmc-size2,markus1,markus2,markus3}. In the schematic picture of {\it ``system/environment''}, with which I have analyzed methods and applications so far, the TA-GCE approach can be visualized as the construction of a generic ``infinite'' environment (reservoir of the Grand Canonical ensemble) via the introduction of the phase factor of the wavefunction that breaks the symmetry of the periodicity. This characteristic suggests that in principle, TA-GCE, in the not too distant future, may find use beyond its current technical utility of circumventing finite size effects. In this perspective, TA-GCE could be employed, for example, to describe the (small) QM part of A-QM/MM or el-QM-AdResS treated in the previous section, or for the QM description of the electrode in electrochemistry. Besides TA-GCE, other routes have been explored within the QMC scheme. An interesting recent example concerns the possibility of considering systems with a variable number of electrons for studying photoemission and inverse photoemission phenomena \cite{photo}. The method used is the Full Configuration Interaction QMC method (FCIQMC) developed in the group of Ali Alavi \cite{fciqmc1,fciqmc2,fciqmc3}. In essence, the method uses a Monte Carlo sampling of Slater determinants and calculates the real-time propagation of the wave function of the system as an
electron is added or removed. Differently from TA-GCE, electrons are not introduced in a Grand Canonical fashion, but the method is specifically designed to compute electronic spectra upon sudden addition or
removal of an electron (as in photoemission). In a qualitative explanation, this is equivalent to saying that the removed electron has been put in a stationary, non-interacting state somewhere at infinity. It must be added that nowadays it is even possible to run small QMC calculations on desktop {computers; however, the computational costs of any QMC approach for large-scale systems routinely treated by, e.g., DFT, requires high-performance resources that may be beyond the means of research groups of a moderate size. Nevertheless, or, actually, because of such limitations, there is the need to significantly intensify the amount of theoretical work being done in this field}.
\section{Conclusions and Perspectives}
The overall analysis of the various situations discussed in this paper leads me to the conclusion that {what lies at the core of the problem that is simulating open, electronic systems,} is not the lack of accurate theoretical tools, but rather the difficulty of merging the corresponding simulation techniques, specific to multiple aspects of the problem, in a physically consistent way. A proper, non-empirical merging is a necessary step for building simulation tools with predictive power, without the mandatory need for a case-by-case external validation. In the last decade we have witnessed the exponential increase of multiscale simulation techniques, mostly based on pragmatic empiricism for solving specific problems in the short term. The challenge of the next decade, in my view, should be the construction of multiscale techniques where the scale-coupling interfaces are based only on solid physical principles with corresponding well defined formulas for possible errors induced by approximations. Such a project requires methodological and theoretical work with, as much as possible, mathematical rigor, thus strengthening even further the concept that methodology (across fields and disciplines) is a self-standing, relevant field. On a broader scale, the project would require a combined effort of applied mathematicians, theoretical physicists and simulators (including physicists, chemists and materials scientists) with constructive exchanges is every direction.
In this perspective, the aim of this paper is to offer to the molecular simulation and related communities a bird's-eye view (of a theoretical physicist) on recent progress in the computational treatment of many-electron systems embedded in a fully-open larger environment. This paper shall not be intended as a comprehensive review but rather as a stimulating discussion for future developments, taking, as possible starting points, the methods, ideas and applications discussed in the various sections. These examples discussed have not been chosen because they are necessarily more relevant than others, but because together they sample the field in a rather uniform way. Their overall analysis leads to the conclusion that, as a QM technique, DFT clearly plays a key role, however quantum chemical methods (beyond Hartree-Fock) and QMC methods have been developed, at least at a conceptual level, up to the point of allowing calculations at a varying number of electrons that mimic the exchange of matter with the environment. Yet, their computational price is still prohibitive for treating systems such as molecules in solution or the metal surfaces associated with electrochemistry. The hope is that with the rapid evolution of the technology related to computational capabilities, the application of such methods to the above-mentioned systems may become possible. An intermediate step, that I can foresee, may be represented by the development of methods similar to the QM/MM approach with open boundaries, but with different levels of QM resolution (for a similar idea within the density matrix embedding approach see \cite{knizia}). {An example could be Grand Canonical QMC embedded in a quantum reservoir treated at the (Grand Canonical) DFT level, itself embedded in a reservoir of classical molecules. Analogous to the coupling criteria of el-QM-AdResS, one would require that the electronic and macroscopic chemical potentials are equal at the various interfaces.} In summary, the future challenge lies in the construction of a solid theory of boundary conditions, at the interface of varying resolutions, in order to achieve a global physical consistency. A selection of starting points has been reported here and these are now ready to be developed even further.
\section*{Acknowledgments}{This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through the grant CRC 1114: ``Scaling Cascades in Complex Systems'', project C01. I would like to thank Paul Ayers, Ferdinand Evers and Christian Krekeler for useful suggestions that have clarified relevant concepts discussed in this work. I am grateful to Ali Alavi, David Ceperley, Markus Holzmann and Ravishankar Sundararaman, for a critical reading of the sections of the manuscript related to their field of expertise. {I wish to thank John Whittaker for a critical reading of the manuscript and for his precious suggestions.}}
|
2,877,628,090,832 | arxiv | \section{Introduction}
The Sydney-AAO Multi-object Integral Field Spectrograph (SAMI) is a
prototype wide-field system at the Anglo-Australian Telescope (AAT)
deploying $13 \times 61$-core imaging fibre bundles (hexabundles) over
a 1-degree field of view \citep{2012clb+}. The hexabundles, together
with ancillary sky and calibration fibres, are mounted on a plug plate
located at the prime focus of the telescope. Each plate is a 24~cm
diameter and 3~mm thick steel disc, pre-drilled with holes
corresponding to the on-sky positions of targets for several distinct
pointings. Typically this is 3 fields for science plates, and 8 fields
in the case of plates used for the set-up and calibration of the
instrument.
A science field consists of a guide-star located at the centre of the
plate, a flux calibration star, 26 blank-sky positions and 12 science
targets located in the 1-degree field-of-view. A calibration field
contains either 2 visual alignment stars, used in the initial coarse
plate rotation alignment process, or 13 stars used for distortion
calculations and fine plate rotation alignment.
\section{Configuring the Plate}
The process of determining the positions of the science plate holes
involves defining 3 stacked observing fields composed of a common
guide-star hole at the centre of the plate, and choosing the flux star
and science targets for each field taking into account separation
constraints between targets in the same field and in the two other
fields (simplistically, plug-holes should not overlap). The 26
blank-sky positions are allocated to each field with the additional
constraint that only 26 sky holes are to be drilled in the plate, so
the sky positions for each of the 3 fields must map to the same
physical holes (see Figure~\ref{p048_FigStackedFields}).
Configuration of the calibration plate holes is simpler, as there are
no sky positions to find which are common to all fields, but star
plates contain a greater number of stacked fields with one common
guide-star hole: 4 visual alignment fields plus 4 fields for the
astrometric distortion model and fine rotation alignment.
\begin{figure}[!t]
\centering
\includegraphics[width=0.7\linewidth]{stackedFullPlate3dWithLabels.eps}
\caption {Representation of a science plate composed of three fields. The target
positions are unique to a given field, whereas the sky positions are
valid for each of the three field but use a single common hole on the plate.}
\label {p048_FigStackedFields}
\end{figure}
\noindent The steps involved in configuring a plate are as follows:
\begin{enumerate}
\item {\bf Field Creation:} From a target catalogue and a catalogue of
suitable calibration and guide stars, a database of candidate
observing fields is constructed, based on several positional
constraints set by the physical limits imposed by the fibre
connector assembly, the position of the guide camera and the need to
physically place and remove connectors between one observation and
the next. \newline
- Separation between a target and the guide star: $7.6$\ arcmin
$<$~separation $<0.5$~degrees\\
- Separation between two targets in the same field $> 3.8$\ arcmin
\item{\bf Field Stacking:} Taking into account inter-field exclusion
rules, where the separation of two targets in different fields
$>$3\ arcmin, multiple fields are stacked to form a single plate.
Each field is taken from a different RA range, so that a night’s
observation of 3 fields can be performed without the need to stop to
change plates.
\item{\bf Sky Fibre Positioning:} A new method is being implemented
for determining the position of the 26 sky fibres
(Figure~\ref{p048_FigSkyPos}). First a grid is placed over the field
with cell size = hole exclusion size / 2. Any cells which fall within the
exclusion zone of any star or target are then removed from the grid.
The most isolated remaining cell is then located (i.e. the cell
furthest from its nearest neighbour) and tested for suitability as a
sky position. This involves checking that, in each of the 3 fields
comprising the plate there are no sources above the magnitude limit
within the sky fibre's field-of-view. If the position is suitable,
the grid cell is marked as full and the nearest-neighbour distances
are recalculated. This process is iterated until all 26 sky
positions have been obtained.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\linewidth]{plateWithGrid3WithLabels.eps}
\caption {Finding suitable positions for sky fibres to ensure
uniform coverage over the plate. This example shows the invalid
(i.e. occupied) grid cells as gray and the order in which the sky
positions have been determined (1 - 4).}
\label {p048_FigSkyPos}
\end{figure}
\item{\bf Transformation:} Once the plate positions are defined,
instrument and sky compensation transforms are applied. These
include the optical distortion model and the atmospheric
differential refraction models, based on the expected time of
observation of each field
\item{\bf Generation:} The configuration process creates two products:
the first is a CSV file containing the hole positions for the plate. A final
adjustment is made for the estimated dome temperature at observing time
and ambient temperature at fabrication (when the holes are drilled)
and these data are sent to the plate manufacturer. The second
product is a set of observing files, which are used to drive the Telescope
Control System at observing time.
\item{\bf Verification and Visualisation:} Visualisation is done using
the Aladin application \citep{2000bfb+}. The various components
making up a plate --- the fields-of-view of the telescope,
hexabundles, sky fibres and guide camera, the hole information and
exclusion zones are separately layered onto an image of the
field (Figure~\ref{p048_FigAladin}). This is done using the Aladin FoV mechanism, sending the
relative positions from the centre of the field in a VOTable
format \citep{2011owd+}. The survey target positions are also plotted as a separate
absolute position $(\alpha, \delta)$ layer. This facilitates
verification by eye of several items, including:
1) The target position is within the field-of-view of the fibre
hexabundle and that both of these coincide with the DSS galaxy;
2) The field-of-view of the sky fibres is empty, as expected;
3) The distribution of sky fibres is uniform across the plate; and
4) There are no bright stars or extended galaxies sufficiently
close to a target position so as to pose a contamination risk.
\begin{figure}[ht]
\centering
\includegraphics[width=0.73\linewidth]{aladinView.eps}
\caption{Aladin screenshot showing SAMI's $1^\circ$ field-of-view (white),
guide camera (red), hexabundle (yellow)
and sky fibre (blue) positions, superimposed on a DSS image.}
\label{p048_FigAladin}
\end{figure}
\end{enumerate}
\section{Automation}
As part of a prototype project this configuration process has until
now been a painstaking task involving the use of several software
packages, scripts, stand-alone code and a lot of manual configuration
and checking. As SAMI evolves from a technology demonstrator to a
survey instrument with an expected observing catalogue of several
thousand targets, this approach will no longer be feasible, for
reasons of both efficiency and the increased likelihood of error.
Consequently, the process for configuring SAMI plates is now in the
process of being automated. This consists of a C++ layer which carries
out the optimisation of target and sky positions for each field and
plate, and applies the required atmospheric, telescope and thermal
models to convert between sky and plate positions. This process is
controlled by a Java layer which also provides visualisation of the
process to the user by means of Aladin, using
VOTables as the data transport mechanism. The aim is to take away the
tedium of plate configuration, whilst giving the user control over the
process, by presenting them with a way of easily checking the validity
of an automatically generated plate and allowing them to drive
subsequently finer configuration cycles until a satisfactory plate
configuration is achieved.
|
2,877,628,090,833 | arxiv | \section{SUPPLEMENTAL MATERIAL}
\end{widetext}
|
2,877,628,090,834 | arxiv | \section{Introduction}
\label{intro}
\textbf{\emph{Scientific motivations.}}
When searching for the signals of new particles, or when aiming to detect new astronomical objects, a common difficulty arising in the analysis of the data collected by the detectors is the impossibility of correctly specifying the background distribution. In physics and astronomy, we typically refer with ``background'' or ``noise'' to the signal of all the astrophysical sources which are not those we aim to discover.
Unfortunately, since many sources contribute to the background, its distribution is particularly difficult to model \citep[e.g.,][]{priel, dauncey, algeriBANFF}.
Moreover, if the model postulated by the scientists is rejected, it is often difficult to identify the invalidating causes. As a result, this aspect is typically addressed by conducting a suite of exploratory analyses to adequately constrain the parameters involved, followed by a validity check for a newly postulated distribution. Given the complexity of the models investigated through physics experiments, however, this may result in a substantial investment of resources \cite[e.g.,][]{GAMBIT1}; even more so when having to choose between tens of plausible theoretical models \citep[e.g.,][]{pat}.
\textbf{\emph{Statistical formulation of the problem.}} In statistical terms, these difficulties translate into two main questions arising in the statistical analysis of multivariate data. Specifically, given a random vector $\bm{X}=(X_1,\dots,X_p)$, we may wonder:
\begin{itemize}
\item[Q1.] \emph{is the distribution of $\bm{X}$ correctly specified and, if not, in what way the true data distribution diverges from that
hypothesized under the null hypothesis?}
\item[Q2.] \emph{How can we improve our postulated model?} \emph{Or in other words, can we provide a data-driven correction of for it?}
\end{itemize}
As noted by \citet{pearson1938}, smooth tests, originally introduced by \citet{neyman37}, naturally allow us to capture and model the departure of $f$ from $g$ and thus, they offer the framework to directly address Q1 and Q2.
In order to provide a high level overview on smooth tests, let $f$ be the true (unknown) probability density function (pdf) of a random variable $X\in\Real$, $g$ is the hypothesized density and $G$ the respective cumulative density function (cdf). For example, in the above-mentioned problem of background mismodeling, $f$ represents the true background distribution and $g$ is the background model postulated by the scientists. A smooth model for the true probability law $f$ can be specified as
\begin{equation}
\label{skewG}
f(x)= g(x)d\bigl(G(x)\bigl)=g(x)\biggl\{1+\sum_{j\geq 1}\theta_j T_j\bigl[G(x)\bigl]\biggl\},
\end{equation}
where $d(G(x))=\frac{f(x)}{g(x)}$ is the likelihood ratio and the term in the curly brackets is an orthonormal expansion for $d$.
A smooth test \citep[e.g.,][]{neyman37,barton53, ledwina} consists of testing if any of the coefficients $\theta_j$ in \eqref{skewG} is different from zero. Finally, by estimating $d\bigl(G(x)\bigl)$ and constructing adequate confidence bands, it is possible to visualize the nature of the departure of $f$ from $g$.
Despite their usefulness, smooth tests are mainly limited to the univariate setting. In light of this, the main methodological task of this work is to extend this framework to allow for the analysis of multivariate data.
\textbf{\emph{Main results and organization.}}
The theoretical framework is presented in Section \ref{modelling}. There, we define a suitable expansion of the likelihood ratio through orthonormal functions on the unit cube.
As shown in Sections \ref{estimation} and \ref{inference}, such representation substantially simplifies the subsequent stages of estimation, model selection and (post-selection) inference.
In Section \ref{diagnostics}, we discuss a simple ANOVA-like testing strategy to identify possible sources of mismodeling. Power studies are conducted via simulations in both Sections \ref{inference} and \ref{diagnostics}.
As noted above, this work finds its main motivations in the context of astrophysical searches. Therefore, in Section \ref{bkg} we illustrate how iGOF can be used to address the problem of mismodeling of the cosmic background considering a realistic simulation from the Fermi Large Area Telescope \citep{atwood}. Despite this article mainly focuses on the analysis of continuous data, extensions to the discrete setting are discussed in Section \ref{discrete}.
Section \ref{conclusions} collects a summary of the results and a discussion of the limitations of iGOF. Technical proofs and codes are provided in the Supplementary Material.
A summary of the main notation used throughout the paper is available in the Appendix.
\section{Theoretical framework }
\label{modelling}
\subsection{Transformations of the likelihood ratio on the unit cube}
\label{Rosenblatt}
Suppose $F$ is the true distribution function of a random vector $\textbf{X}\in \mathcal{X}\subseteq \Real^p$ and denote with $G$ its hypothesized distribution. $F$ and $G$ are assumed to be continuous with densities $f$ and $g$. Furthermore, assume that $f(\bm{x})=0$ whenever $g(\bm{x})=0$. For every $\bm{x}=(x_1,\dots,x_p)\in \mathcal{X}$, the hypothesized density $g$ is such that
\[g(\bm{x})=\prod_{d=1}^p g_d(x_d|{\bm{x}}_{<d}),\]
where $\bm{x}_{<d}=(x_1,\dots,x_{d-1})$ and $g_1,\dots,g_p$ are suitable densities with associated cdfs and quantile functions $G_d$ and $Q_{d}$, for all $d=1,\dots,p$. The likelihood ratio between $F$ and $G$ can be specified as
\begin{equation}
\label{jcd}
d(\bm{u})=\frac{f\bigl(\bm{Q}(\bm{u})\bigl)}{g\bigl(\bm{Q}(\bm{u})\bigl)},\quad\bm{u}\in[0,1]^p
\end{equation}
where $\bm{u}=(u_1,\dots,u_p)=\bigl(G_1(x_1),\dots,G_p(x_p|{\bm{x}}_{<p})\bigl)=\bm{G}_R(\bm{x})$ is the Rosenblatt transformation \citep{rosenblatt}\footnote{Notice that, in general, $\bm{G}_R(\bm{x}) \not\equiv G(\bm{x})$ as the Rosenblatt's transform $\bm{G}_R(\bm{x})\in[0,1]^d$ whereas the cdf $G(\bm{x})\in [0,1]$.}. Whereas,
$\bm{Q}(\bm{u})=\bigl(Q_{1}(u_1),\dots,Q_{p}(u_p)\bigl)$, for all $d=1,\dots,p$.
In the bivariate setting, for instance, let $G_1\equiv G_{X_1}$ and $G_2\equiv G_{X_2|X_1}$, i.e., the hypothesized marginal cdf of $X_1$ and the hypothesized conditional cdf of $X_2|X_1$, respectively. Hence, \eqref{jcd} specifies as
\[d(u_1,u_2)=\frac{f_{X_1X_2}\bigl(Q_{1}(u_1),Q_{2}(u_2)\bigl)}{g_{X_1X_2}\bigl(Q_{1}(u_1),Q_{2}(u_2)\bigl)}.\]
\begin{remark}
\label{independence}
As a plausible alternative to Rosenblatt's transform, one could choose each $G_d\equiv G_{X_d}$, which corresponds to assuming independence among the components of $\bm{X}$. In this setting, \eqref{jcd} is the copula density \citep[e.g.,][]{nelsen} of $\bm{X}$ under $G$. Despite this choice could simplify substantially the computations, it would not allow to test models where the components of $\bm{X}$ are assumed to be dependent. Moreover, it is worth pointing out that there are situations where such transformation cannot be specified (e.g., Section \ref{bkg}).
\end{remark}
To provide a sufficiently detailed representation of the substructures characterizing the distribution of $\bm{X}$ (see Q1 in Section \ref{intro}), a natural approach is that of expressing \eqref{jcd} by means of a suitable orthonormal basis in $L^2[0,1]^p$.
For instance, let $T_{j_d}(u_d)$ be the $j_d^{th}$ normalized shifted Legendre polynomial evaluated at $u_d=G_d(x_d|\bm{x}_{<d})$, with $T_0(u_d)=1$, $T_1(u_d)=\sqrt{12}(u_d-0.5)$, etc. (e.g., Section \ref{Ex1app}, Supplementary Material). Each $\{T_{j_d}(u_d)\}_{j_d\geq 0}$ forms a basis in $L^2[0,1]$. Hence, we can exploit a well known result in Hilbert space theory \citep[e.g., Proposition 2][p.50]{reedbook} which asserts that given two orthonormal bases $\{\psi_{j}\}$, $\{\phi_{k}\}$ for the Hilbert spaces $\mathcal{H}_1$, $\mathcal{H}_2$, then $\{\psi_{j}\otimes \phi_{k}\}$ is an orthonormal basis for $\mathcal{H}_1\otimes\mathcal{H}_2$. It follows that the tensor product basis $\{T_{j_1,\dots,j_p}(\bm{u})\}_{j_1\dots j_p\geq 0}$ of functions
\begin{equation}
\label{Sjs}
T_{j_1\dots j_p}(\bm{u})=\prod_{d=1}^p T_{j_d}(u_d)
\end{equation}
forms an orthonormal basis on $L^2[0,1]^p$, the Hilbert space of square integrable function over the $p$-dimensional unit cube.
Notice that despite any orthonormal basis in $[0,1]$ could be used to construct a tensor product basis in $[0,1]^p$, here we focus on the normalized shifted Legendre polynomials. This choice is justified by the fact that the latter are special cases of the so called LP-score functions \citep[e.g.,]{LPksamples}. As discussed in Section \ref{discrete}, the LP score functions allow for extensions to the discrete setting.
Finally, under the assumption $d(\bm{u})\in L^2[0,1]^p$, we can write
\begin{equation}
\label{jcd_rep}
d(\bm{u})=\sum_{j_1\geq 0,\dots,j_p\geq 0}\theta_{j_1\dots j_p}T_{j_1\dots j_p}(\bm{u}),\qquad \text{ $\bm{u}\in [0,1]^p$}
\end{equation}
with $\theta_{j_1\dots j_p}=\int_{[0,1]^p}T_{j_1\dots j_p}(\bm{u})d(\bm{u})\text{d}\bm{u}$. The expansion in \eqref{jcd_rep} follows from Theorem II.6 in \citet[][]{reedbook} and it is equivalent to say that the sum on the right-hand side converges to $d(\bm{u})$ in $L^2[0,1]^p$.
\begin{remark}
\label{whyU}
An anonymous referee correctly pointed out that the likelihood ratio can also be expanded through an orthonormal expansion on the original domain $\mathcal{X}$, bypassing the need of Rosenblatt's transform. In our context, however, the Rosenblatt transform is particularly useful for two main reasons. First of all, one can show (Proposition \ref{moments_prop} to follow) that $d(\bm{u})$ corresponds to the density of $\bm{U}=\bm{G}_R(\bm{X})$.
This fact will simplify substantially the estimation process (Section \ref{estimation}). Second, by transforming the data on the compact compact domain $[0,1]^p$ we will be able to exploit exploit results from random fields theory to perform inference (Section \ref{inference}).
\end{remark}
\section{Estimation}
\label{estimation}
The summations in \eqref{jcd_rep} are taken up to infinity. However, to make the expansion operational, it is necessary to truncate the series in \eqref{jcd_rep} at integers values $m_1,\dots,m_p$. That is because, effectively, the coefficients $\theta_{j_1\dots j_p}$ need to be estimated and, consequently, the more terms are included in \eqref{jcd_rep}, the larger the variance of the resulting estimator of $d(\bm{u})$ (see Section \ref{selection} for a more detailed discussion on model selection).
For the sake of simplifying the notation in this section and those to follow, denote with $\mathcal{K}$ the set
\begin{equation}
\label{kappa}
\mathcal{K}:=\biggl\{\{j_1\dots j_p\},\text{ with } j_d=0,\dots,m_d, \text{ for all } d=1,\dots,p, \text{ and } \sum_{d=1}^p j_d\neq 0 \biggl\}
\end{equation}
of cardinality $|\mathcal{K}|=M=\prod_{d=1}^p(m_d+1)-1$. That is, $\mathcal{K}$ contains all the $p-$tuples $\{j_1\dots j_p\}$ of indexes $j_d=0,\dots,m_d$, $d=1,\dots,p$ apart from the $p-$tuple $\{0\dots 0\}$.
Let $\bm{\theta}$ be the $M\times 1$ vector of components ${\theta}_{k}$, with $k\in \mathcal{K}$.
Similarly, denote with $\bm{T}(\bm{u})$ the $M\times 1$ vector of elements $T_{k}(\bm{u})$, $k\in \mathcal{K}$.
Consider $\bm{x}_1,\dots,\bm{x}_n$, a sample of $n$ i.i.d. observations from $\bm{X}$, and let $\bm{U}=\bm{G}_R(\bm{X})$ be the respective Rosenblatt transformation.
Denote with $\bm{u}_1,\dots,\bm{u}_n$ the sample of elements $\bm{u}_i=\bm{G}_R(\bm{x}_i).$
The parameter ${\bm{\theta}}$ can be estimated by means of the vector $\widehat{\bm{\theta}}$ of components
\begin{equation}
\label{theta_est}
\widehat{\theta}_{k}=\frac{1}{n}\sum_{i=1}^n T_{k}(\bm{u}_i)\quad\text{ for all $k\in \mathcal{K}$ },
\end{equation}
The mean and covariance matrix of $\widehat{\bm{\theta}}$ and an estimator of $d(\bm{u})$ are given in Proposition \ref{moments_prop}.
\begin{proposition}
\label{moments_prop}
The likelihood ratio $d\bigl(\bm{u}\bigl)$ is the density of the random vector $\bm{U}$ and
\begin{equation}
\label{moments}
E[\widehat{\bm{\theta}}]=\bm{\theta}\quad\text{and}\quad \text{Cov}(\widehat{\bm{\theta}})=\bm{\Sigma}
\end{equation}
where $\bm{\Sigma}$ has diagonal elements $\frac{\sigma^2_{k}}{n}=\frac{1}{n}V\bigl[T_{k}(\bm{U})\bigl]$ and non-diagonal elements
$\frac{\sigma_{k,h}}{n}=\frac{1}{n}\text{Cov}\bigl[T_{k}(\bm{U}),T_{h}(\bm{U})\bigl]$, with $k,h\in \mathcal{K}$.
Furthermore, if $F\equiv G$, the equalities in \eqref{moments} reduce to
\begin{equation}
\label{moments0}
E[\widehat{\bm{\theta}}]=\bm{0} \quad\text{and}\quad \text{Cov}(\widehat{\bm{\theta}})=\frac{1}{n}\bm{I}_M,
\end{equation}
where $\bm{0}$ is the $M\times1$ zero vector and $\bm{I}_M$ is the $M\times M$ identity matrix.
Finally, an estimator of $d(\bm{u})$ is
\begin{align}
\label{jcd_est}
\widehat{d}(\bm{u})&=1+\widehat{\bm{\theta}}'\bm{T}(\bm{u}),
\end{align}
and has variance $V\Bigl[\widehat{d}(\bm{u})\Bigl]=\bm{T}(\bm{u})'\bm{\Sigma}\bm{T}(\bm{u})$.
\end{proposition}
It is worth pointing out that, when the likelihood ratio is formulated as a density, it is often referred to as \emph{comparison density} \citep[e.g.,][]{parzen2004}.
\begin{figure*}[!htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=70mm ]{Fig1a} & \hspace{-1cm} \includegraphics[width=71mm ]{Fig1b}\\
\end{tabular*}
\caption{True (left panel) and estimated (right panel) likelihood ratio for Example I. The estimate on the right has been obtained via \eqref{jcd_est} with $m_1=4$ and $m_2=3$. The components of the $\widehat{\bm{\theta}}$ vector have been selected via the AIC criterion in \eqref{BIC}.
}
\label{fig1}
\end{figure*}
Combining \eqref{skewG}, \eqref{jcd} and \eqref{jcd_est} an estimate of $f$ is
\begin{align}
\label{skewG_est}
\widehat{f}(\bm{x})=g(\bm{x})\widehat{d}\bigl(\bm{G}_R(\bm{x})\bigl)=g(\bm{x})\bigl[1+\widehat{\bm{\theta}}'\bm{T}\bigl( \bm{G}_R(\bm{x})\bigl)\bigl]
\end{align}
Notice that the estimator $\widehat{f}$ incorporates the information carried by the hypothesized model $g$; whereas, the estimator in the square brackets provides a data-driven correction for it. Furthermore, define the integrated squared bias (ISB) of $\widehat{d}(\bm{u})$ to be
\begin{equation}
\label{ISBdef}
ISB=\int_{[0,1]^p}\Bigl(E[\widehat{d}(\bm{u})]-d(\bm{u})\Bigl)^2\text{d}\bm{u}.
\end{equation}
From Proposition \ref{ISB_prop} it follows that the closer $g$ is to $f$ in terms of squared normalized distance the lower the ISB of $\widehat{d}(\bm{u})$.
\begin{proposition}
\label{ISB_prop}
The integrated squared bias of the estimator in \eqref{jcd_est} is
\begin{equation}
\label{ISB}
\bigintsss_{[0,1]^p}\biggl(\frac{f\bigl(Q(\bm{u})\bigl)- g\bigl(Q(\bm{u})\bigl)}{g\bigl(Q(\bm{u})\bigl)}\biggl)^2\text{d} \bm{u}-\bm{\theta}'\bm{1},
\end{equation}
where $\bm{1}$ is the $M\times 1$ unit vector.
\end{proposition}
The estimate in \eqref{skewG_est} is essentially that of a smooth model \citep[e.g.,][]{rayner90}, that is, a smoothed version of the true underlying probability function. Similarly to the smooth model proposed by \citet{barton53} in the univariate setting, the estimator in \eqref{skewG_est} may lead to estimate that are not \emph{bona-fide}, i.e, they may be negative and/or they may not integrate/sum up to one. In this manuscript we focus on \eqref{skewG_est} mostly for the sake of mathematical convenience in deriving the inferential results of Section \ref{inference}. Nonetheless, bona-fide estimators can be constructed similarly to the univariate case as described in \citet{algeri_zhang}.
\emph{\textbf{Example I.}} In direct searches for dark matter, the dominant background sources are neutron recoils which may produce signals mimicking those expected from dark matter candidates \citep[e.g.,][]{westerdale}. As a toy example, suppose we are interested in assessing the validity of a given distribution for the nuclear recoil background specified over the energy region $\mathcal{X}=[5,20]KeVnr\times[0,17]KeVnr$. Each observations in $\mathcal{X}$ corresponds to the scintillation of photons ($X_1$) and ionization electrons ($X_2$) \citep[e.g.,][]{aprile}. The hypothesized background distribution, $G_{X_1X_1}$, is that of a truncated bivariate normal with mean vector $(12,8)$, variances $8$ and $12$ and covariance $2$. Moreover, suppose that one additional background source is present. The latter is also a bivariate normal with the same mean vector, variances $4$ and $20$ and covariance $5$. Thus, the true model, $F_{X_1X_1}$, involves a mixture of two, overlapping truncated bivariate Gaussians with mixture parameter $0.15$. In order to estimate the likelihood ratio, set $G_1=G_{X_1}(x_1)$ and $G_2=G_{X_2|X_1}$. The estimated likelihood ratio, obtained over a sample of $n=5,000$, is shown in the right panel of Figure \ref{fig1}, whereas the left panel shows the true likelihood ratio. A closed form expression for the estimate shown on the right panel is given in equation \ref{LR1} in the Supplementary Material.
While the estimate obtained recovers the main departures from uniformity, the contours highlight that the estimator is rather noisy. Therefore, it is important to investigate the properties of \eqref{jcd_est} to assess the significance of the deviations observed.
\section{Inference and model selection}
\label{inference}
\subsection{Pre-selection inference}
A smooth test for $H_0:G\equiv F$ versus $H_1:G\not\equiv F$ consists in reformulating the problem as a test for uniformity of $\bm{U}$.
Specifically, \eqref{jcd} implies that $F\equiv G$ whenever $d(\bm{u})=1$, and thus
\begin{equation}
\label{test0}
\begin{split}
H_0: d(\bm{u})=1 \quad\text{$\forall \bm{u}\in [0,1]^p$} \quad\text{versus} \quad H_1 : \exists \bm{u}\in [0,1]^p \text{ s.t. } d(\bm{u})\neq 1.
\end{split}
\end{equation}
It is easy to see that $d(\bm{u})=1$ for all $\bm{u}\in [0,1]^p$, when all ${\theta}_{k}$, $k\in\mathcal{K}$, are identically equal to zero. Hence, in practice, we test
\begin{equation}
\label{test}
H_0: \bm{\theta}= \bm{0} \qquad\text{vs}\qquad H_1 : \bm{\theta}\neq \bm{0}.
\end{equation}
Notice that $H_0$ in \eqref{test0} implies $H_0$ in \eqref{test}, but the opposite is not true in general. Whereas, $H_1$ in \eqref{test} does imply $H_1$ in \eqref{test0}.
With a little
abuse of nomenclature, in this section and those to follow, we will refer to $G$ as the ``null model''. Furthermore, we will refer to $H_0$ in \eqref{test0} when generically saying ``under $ H_0$''. However, most of the results presented here, only require validity of the milder $H_0$ in \eqref{test}
To conduct our inference, we consider the so-called \emph{deviance} test statistics, i.e.,
\begin{equation}
\label{deviance}
D=n\widehat{\bm{\theta}}'\widehat{\bm{\theta}}.
\end{equation}
Its asymptotic null distribution is given in Theorem \ref{normality}.
\begin{theorem}
\label{normality}
If $H_0$ is true, then
\vspace{-0.2cm}
\begin{equation}
\label{thetaH0}
\sqrt{n}\widehat{\bm{\theta}}\xrightarrow d N(\bm{0},\bm{I}), \quad\text{as $n\rightarrow\infty$}
\end{equation}
\vspace{-0.2cm}
where $N(\bm{0},\bm{I})$ denotes a standard multivariate normal distribution. Furthermore,
\begin{equation}
\label{DH0}
D \xrightarrow d \chi^2_{M}, \quad\text{as $n\rightarrow\infty$},
\end{equation}
where $M$ is the length of $\widehat{\bm{\theta}}$.
\end{theorem}
Corollary \ref{dhat_cor} follows directly from Theorem \ref{normality}.
\begin{corollary}
\label{dhat_cor}
Denote with $\{\widehat{d}(\bm{u})\}$ the random field indexed by $\bm{u}\in [0,1]^p$ with components as in \eqref{jcd_est}. Moreover, assume that $\widehat{\theta}_k=o(n^{-1/2})$ for all $k\not\in \mathcal{K}$. If $H_0$ is true,
\begin{equation}
\label{rf}
\Biggl\{\frac{\widehat{d}(\bm{u})-1}{\sqrt{\frac{1}{n}\bm{T}(\bm{u})'\bm{T}(\bm{u})}}\Biggl\}\xrightarrow d \bm{Z}(\bm{u}), \quad\text{as $n\rightarrow\infty$,}
\end{equation}
where $\bm{Z}(\bm{u})$ denotes a Gaussian random field with mean zero, unit variance and covariance function
$\text{Cov}\Bigl( \bm{Z}(\bm{u}), \bm{Z}(\bm{u}^\dag)\Bigl)=\frac{\bm{T}(\bm{u})'\bm{T}(\bm{u}^\dag)}{\sqrt{\bm{T}(\bm{u})'\bm{T}(\bm{u})\bm{T}(\bm{u}^\dag)'\bm{T}(\bm{u}^\dag)}}.$
\end{corollary}
At this stage, constructing inference on the basis of Theorem \ref{normality} and Corollary \ref{dhat_cor} would be tempting. However, to guarantee the validity of our results we must take into account that, when estimating the likelihood ratio in \eqref{jcd_est}, a model selection procedure is likely to be implemented. Unfortunately, when a model is selected by a pool of possibilities, such process introduces an additional source of variability and thus the resulting inference is automatically affected \citep[e.g.,][]{berk}. Section \ref{selection} addresses this aspect directly.
\subsection{Post-selection inference}
\label{selection}
The estimate of the likelihood ratio considered so far involves up to $M$ functions $T_{k}(\bm{u})$. Nonetheless, it is possible that not all of these $M$ terms are needed to capture the departures of $G$ from $F$ and indeed, it is often convenient to remove some of them in order to avoid unnecessary sources of noise. Various criteria have been proposed in literature for density estimation and smooth models \citep[e.g.,][]{LPmode, algeri20} and which can be easily extended to the multivariate setting. Here, we focus on the approach of \citet{LPmode} and which specifies as follows.
Let $\widehat{\theta}_{(k)}$ be the $k^{th}$ largest $\widehat{\theta}_{k}$ estimate in order of magnitude, i.e., $\widehat{\theta}_{(1)}^2\geq \widehat{\theta}_{(2)}^2\geq\dots\geq \widehat{\theta}_{(M)}^2$. Select the $K$ largest coefficients which maximize either
\begin{equation}
\label{BIC}
\text{BIC}(K) = \sum_{(k)=1}^K\widehat{\theta}^2_{(k)} - \frac{K \log n }{n}\quad\text{or}\quad \text{AIC}(K) = \sum_{(k)=1}^R\widehat{\theta}^2_{(k)} - \frac{2K}{n}.
\end{equation}
Notice that, as defined in \eqref{kappa}, each $k$ is a $p-$tuple of indexes $j_1\dots j_p$, whereas $(k)$ is the integer value corresponding to the order of magnitude of the respective coefficient $\widehat{\theta}_{k}$. Hence, the summations in \eqref{BIC} and those to follow are taken over $(k)=1,\dots,K$, that is the $K$ $p-$tuples of indexes $j_1\dots j_p$ with the $K^{th}$ largest estimates $\widehat{\theta}_{k}$.
An estimate of $d(\bm{u})$, is then selected via \eqref{BIC} from the family of estimators
\begin{equation}
\label{estAIC}
\widehat{d}_{(K)}(\bm{u})=1+\sum_{(k)=1}^{K}\widehat{\theta}^2_{(k)}T_{(k)}(\bm{u}),\quad\text{K=1,\dots,M}
\end{equation}
where the subscript $(K)$ is used to emphasize that the estimator in \eqref{estAIC} includes only the $K^{th}$ largest $\widehat{\theta}_k$ estimated coefficients.
Clearly, the choice of BIC or AIC is arbitrary and, from a practical standpoint, the BIC tends to lead to smoother estimates than the AIC.
The selection rules in \eqref{BIC} compare $M$ possible models assuming that each $m_d$, for $d=1,\dots,D$ was fixed before the researcher looked at the data. Valid post-selection inference can then be constructed as in Theorem \ref{normality} and Corollary \ref{naive}. The respective proofs are provided in the Supplementary material.
\begin{corollary}
\label{naive}
Denote with $\widehat{d}_{(K*)}$ the estimator of $d(\bm{u})$ selected via \eqref{BIC}, and let $D_{(K^*)}=\sum_{(k)=1}^{K^*}\widehat{\theta}^2_{(k)}$ be the respective deviance statistics.
As $n\rightarrow \infty$, a valid post-selection bound for the p-value to test \eqref{test0} is
\begin{equation}
\label{pvalue}
\text{p-value}_{adj}=P(\chi^2_{M }>D_\text{obs}),
\end{equation}
where $D_\text{obs}$ is the value of $D_{(K^*)}$ observed.
\end{corollary}
Where the bound in \eqref{pvalue}, follows from the fact that the estimators in \eqref{estAIC} are nested, for all $K=1,\dots,M-1$, and thus each $D_{(K)}$ is stochastically lower or equal than $D_{(M)}$. Hence, for all $K=1,\dots,M-1$, $P(D_{(K)}>D_\text{obs})$ is smaller than $P(D_{(M)}>D_\text{obs})$.
In order to grasp further insights on the deviations of $G$ from $F$, it is worth constructing adequate confidence bands. This can be done, while accounting for post-selection adjustments, as in Corollary \ref{naive2}.
\begin{corollary}
\label{naive2}
Denote with $\widehat{d}_{(K*)}$ the estimator of $d(\bm{u})$ selected via \eqref{BIC}, and denote with
$SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]$ its standard error under $H_0$.
Valid (post-selection adjusted) $(1-\alpha)\%$ confidence regions, under $H_0$ in \eqref{test0}, are
\begin{equation}
\label{CIband}
\biggl[1-c_{\alpha/2} SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl],1+c_{\alpha/2}SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]\biggl]\qquad \text{for all $\bm{u}\in [0,1]^p$}
\end{equation}
with $c_{\alpha/2}$ such that
\begin{equation}
\label{calpha}
P\Biggl(\sup_{\bm{u}}\biggl\{\frac{\widehat{d}_{(M)}(\bm{u})-1}{SE_0\bigl[\widehat{d}_{(M)}(\bm{u})\bigl]}\biggl\}>c_{\alpha/2} \Biggl|H_0\Biggl)=\frac{\alpha}{2}.
\end{equation}
\end{corollary}
Intuitively, $\widehat{d}_{(M)}$ is the least smooth among all the estimators considered; hence, we expect that the random field resulting from $\widehat{d}_{(M)}$ has the largest probability of crossing the fixed level $c_{\alpha/2}$.
From a theoretical perspective, a highly non-trivial aspect in the construction of \eqref{CIband} is the estimation of the quantile $c_{\alpha/2}$. Probabilities such \eqref{calpha} are known in literature as \emph{excursion probabilities} \citep[e.g.,][]{adler2000} and which cannot be expressed in closed form. A possible solution for constructing the confidence bands in \eqref{CIband}, is that of proceeding by estimating $SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]$ and $c_{\alpha/2}$ via Monte Carlo simulations \citep[see][Algorithm 1]{algeri_zhang}. Unfortunately, in the most crucial (astro)physical searches the level of significance required to claim a new discovery is typically in the order of $\alpha=10^{-7}$ \citep[e.g.,][]{lyons2015}, and thus Monte Carlo simulations may be computationally prohibitive. This is further aggravated when dealing with complex models for which even a single Monte Carlo replicate can be highly expensive in terms of both computational and time resources.
As a valid alternative, for continuous $F$ and $G$, accurate approximations for \eqref{calpha} under mild smoothness conditions exist \citep[e.g.,][]{taylor2008}.
In our setting, smoothness follows from the fact that the random field in \eqref{rf} and the respective limit can be written as a linear combination of the functions $\frac{T_k(\bm{u})}{\sqrt{\sum_{k\in\mathcal{K}}T^2_k(\bm{u}) }}$
(see proof of Corollary \ref{dhat_cor} in the Supplementary Material) which are composition of Legendre polynomials, and thus, admit infinite partial derivatives.
An approximation for the left-hand side of \eqref{calpha} is
\begin{equation}
\label{LK}
\bigl(1-\Phi(c_{\alpha/2})\bigl)+\mathcal{L}_1\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\pi}+\mathcal{L}_2\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\sqrt{2}\pi^{3/2}}+O\biggl(\exp\Bigl(-\frac{\gamma c_{\alpha/2}^2}{2}\Bigl)\biggl),\quad\text{as $n\rightarrow\infty$,}
\end{equation}
\noindent for some $\gamma>1$ \citep{takemura}. In \eqref{LK}, $\mathcal{L}_1$ and $\mathcal{L}_2$ are constant known as Lipischitz-Killing curvatures and are typically estimated numerically \citep[e.g.,][]{TOHM}. Notice that the error rate in \eqref{LK} decreases exponentially fast, as $\alpha\rightarrow \infty$. Therefore, this solution is particularly amenable to overcome the issues arising when dealing with stringent significance requirements.
\begin{figure*}[htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=72mm ]{Fig2a}&\includegraphics[width=72mm ]{Fig2b} \\
\end{tabular*}
\caption{Simulated and approximated confidence regions for Example I. The left panel corresponds to the (post-selection) confidence regions and deviance p-value obtained via a simulation of size $B=10,000$. The right panel shows to the (post-selection adjusted) confidence regions and deviance p-value computed as in \eqref{pvalue} and \eqref{CIband}. Darker shades correspond to significant deviations of the estimated likelihood ratio above one. Lighter shades correspond to significant deviations below one.
}
\label{fig2}
\end{figure*}
\begin{table}[!h]
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|cccccc|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
& $n=500$ & $n=1000$ & n=2000 & $ n=5000$ & $ n=7000$ & $ n=10,000$ \\
\hline
Type I error &0.0540 & 0.0500 & 0.0499 & 0.0482 & 0.0508 &0.04930\\
($\pm$ SE) & ($\pm$ 0.0023) & ($\pm$ 0.0022) &($\pm$ 0.0022) & ($\pm$ 0.0021) & ($\pm$ 0.0022) & ($\pm$ 0.0022) \\
\hline
Power& 0.2157 &0.4456 &0.8063 & 0.9995 &1.0000& 1.0000 \\
($\pm$ SE)& ($\pm$ 0.0041) &($\pm$ 0.0050) &($\pm$ 0.0040) & ($\pm$ 0.0002) &($\pm$ 0.0000) & ($\pm$ 0.0000)\\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{Simulated probability of type I error and power for Example I considering different sample sizes. The nominal level is chosen to be $\alpha=0.05$. Each simulation involves $B=10,000$ replicates.}
\label{time}}
\end{table}
As one may expect, the simplicity of the post-selection adjustments in \eqref{pvalue} and \eqref{CIband} comes with a price. Specifically, they can be rather conservative for increasing values of $M$.
However, as shown below for Example I and in the sections to follow, \eqref{pvalue} still leads to high power even if the sample size is only moderately large. Similarly, \eqref{CIband} can be quite accurate and match closely the confidence regions obtained by simulating directly the distribution of \eqref{rf}, while repeating the selection process at each replicate.
\emph{\textbf{Example I (continued).}} The estimate of the likelihood ratio in the right panel of Figure \ref{fig1} has been obtained by setting $m_1=4$ and $m_2=3$ and selecting the terms of the respective tensor basis via the AIC rule in \eqref{BIC}.
The AIC procedure selects $9$ terms out of $M=19$.
The post-selection adjusted p-value and $95\%$ confidence regions are shown in the right panel of Figure \ref{fig2}.
The confidence contours are constructed by setting equal to one all the values of $\widehat{d}$ contained within the bands in \eqref{CIband}. Whereas the quantile $c_\alpha$ has been calculated by solving
\begin{equation}
\label{solver}
\bigl(1-\Phi(c_{\alpha/2})\bigl)+\mathcal{L}_1\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\pi}+\mathcal{L}_2\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\sqrt{2}\pi^{3/2}}-\frac{\alpha}{2}=0
\end{equation}
and estimating $\mathcal{L}_1$ and $\mathcal{L}_2$ by means of the \texttt{R} package \texttt{TOHM} \citep{TOHMpkg} as described in \citet{TOHM}. This approach led to $c_{\alpha}=3.5568$.
The confidence contours suggest that the most prominent deviations occur in correspondence of the regions $[10,12]\times[0,5]$ and $[12,15]\times[12,17]$. Here, the estimator of $d(\bm{u})$ shows significant deviations above one and thus we conclude that the postulated model underestimates the truth over these areas. The presence of significant departures of $G_{X_1X_2}$ from $F_{X_1X_2}$ are confirmed by the deviance test (adjusted p-value $\sim 3.97 \cdot 10^{-11}$).
The left panel of Figure \ref{fig2} shows the confidence regions and deviance p-value obtained by means of a Monte Carlo simulation involving $B=10,000$ replicates. The selection procedure has been implemented at each replicate. While more conservative, the confidence regions computed via \eqref{CIband} and \eqref{solver}, approximate reasonably well those obtained via simulation.
Finally, we investigate the probability of type I error and the power of the deviance test based on \eqref{pvalue}. Table \ref{time} reports the results obtained considering a suite of five simulations, each of size $B=10,000$, conducted using five different sample sizes. For all $n$ considered, the probability of type I error observed is approximately the same than the nominal level $\alpha=0.05$. Whereas, the power increases rapidly with $n$. For the smallest samples sizes considered, i.e., $n=500$ and $n=1000$, the power is rather low ($\sim 22\%$ and $\sim 45\%$, respectively). However, it has to be noted that, in our example, the mixture parameter is $0.15$; therefore the deviations from the postulated model effectively account for only $\sim 75$ and $\sim 150$ data points when $n=500$ and $n=1000$, respectively.
In principle the plots in Figures 1, 2, and those to follow, could also be visualized in the quantile domain. The latter is to be preferred when working with long tailed distribution where one may expect that only a few observations have been detected over large regions of the $\bm{X}$ domain. In those situations, the quantile representation would allow to magnify the differences observed over the the most ``data-abundant'' regions. An more detailed discussion of this aspect, and adequate graphical comparisons can be found in \citep{algeri20}.
\section{iGOF-diagnostic analysis}
\label{diagnostics}
The constructs introduced so far allow us to assess the validity of the postulated model, obtain an estimate of the likelihood ratio test to visualize where and how departures of $g$ from $f$ occur, and construct a data driven correction for the initial model $g$ (equation \ref{skewG_est}). Unfortunately, however, a visual inspection is only possible when $p\leq 3$. Nevertheless, when $p>3$, more insights on the sources of mismodeling affecting $G$ can be obtained by conducting an ANOVA-like analysis where random sub-vectors of $\bm{X}$ are tested individually, from the largest to the smallest.
Without loss of generality, let $\bm{X}_q=(X_1,\dots,X_q)$ be the random collecting the first $q<p$ components of $\bm{X}$. Denote with $F_q$ the true cdf of $\bm{X}_q$ and let $G_q$ be its postulated cdf. Moreover, assume that the density of $G_q$ can be specified as
\begin{equation}
\label{C1}
g_{q}\bigl(\bm{x}_q\bigl)=\prod_{d=1}^q g_d\bigl(x_d|\bm{x}_{<d}\bigl)\quad\text{for all $d=1,\dots,q$.}
\end{equation}
Similarly to \eqref{jcd} and \eqref{jcd_rep}, we can then express the likelihood ratio of $\bm{X}_q$ on the $q$ dimensional unit cube via
\begin{equation}
\label{LRq}
d(\bm{u}_q)=\frac{f_q\bigl(\bm{Q}_q(\bm{u}_q)\bigl)}{g_q\bigl(\bm{Q}_q(\bm{u}_q)\bigl)}=\sum_{j_1\geq 0,\dots,j_q\geq 0}\theta_{j_1\dots j_q}T_{j_1\dots j_q}(\bm{u}_q),\quad\bm{u_q}\in[0,1]^q
\end{equation}
where $\bm{u}_q=\bigl(G_1(x_1),\dots,G_q(x_q|\bm{x}_{<q})\bigl)$, and thus $\bm{u}_q$ is a sub-vector of $\bm{u}=\bm{G}_R(\bm{x})$.
Whereas, similarly to \eqref{Sjs}, one can write the tensor basis functions $T_{j_1\dots j_q}$ as
\begin{equation}
\label{Sj2}
T_{j_1\dots j_q}(\bm{u}_q)=\prod_{d=1}^q T_{j_d}(u_d)=\prod_{d=1}^p T_{j_d}(u_d) \quad \text{ with } j_d=0, \text{ for all } d=q+1.
\end{equation}
The last equality follows from the fact that $T_0\bigl(G(x_d|\bm{x}_{<d})\bigl)=1$ for all $d=1,\dots,p$, and thus each $T_{j_1\dots j_q}(\bm{u}_q)=T_{j_1\dots j_q0\dots 0}(\bm{u})$. Consequently, the $\theta_{j_1\dots j_q}$ coefficients are equal to the theta $\theta_{j_1\dots j_p}$ whenever $j_d=0$, for all $d=q+1$.
As a result, we can easily perform inference for $\bm{X}_q$ by means of the estimators $\widehat{\theta}_k$ in \eqref{theta_est}, without the need of an entirely new estimation procedure.
Specifically, denote with $\mathcal{K}_q$ and $\mathcal{K}^*$ the subsets of $\mathcal{K}$ in \eqref{kappa}
\begin{align}
\label{kappaq}
\mathcal{K}_q&:=\biggl\{k=\{j_1\dots j_p\}\in \mathcal{K} \text{ with } j_d=0, \text{ for all } d=q+1,\dots, p\biggl\}\\
\label{kappaK}
\mathcal{K}^*&:=\biggl\{k=\{j_1\dots j_p\}\in \mathcal{K} \text{ with } (k)\leq K^*\biggl\}
\end{align}
of cardinality $|\mathcal{K}_q|= M_q=\prod_{d=1}^q(m_d+1)-1$ and $|\mathcal{K}^*|=K^*$. Recall that $K^*$ is the value minimizing either the AIC or BIC in \eqref{BIC}, and thus,
$\mathcal{K}^*$ collects all the $p-$tuple of indexes in $\mathcal{K}$ which have been ultimately selected when constructing the estimator $\widehat{d}_{(K^*)}$ and the deviance statistics $D_{(K^*)}$ in Corollaries \ref{naive} and \ref{naive2}. To test
\begin{equation}
\label{testQ}
H_0: G_q=F_q\quad\text{versus} \quad H_1: G_{q}\neq F_{q}
\end{equation}
we may consider the test statistics $D_q=n\sum_{k\in \mathcal{K}_q }\widehat{\theta}^2_k$ and proceed as in Theorem \ref{normality}.
Whereas, valid post-selection inference can be obtained as in Theorem \ref{anova_theo}.
\begin{theorem}
\label{anova_theo}
As $n\rightarrow \infty$, a valid post-selection bound for the p-value to test \eqref{testQ} is
\begin{equation}
\label{test3}
\text{p-value}_{q,adj}=P(\chi^2_{M_{q}}>D_{obs}),
\end{equation}
where $D_{obs}$ being the value of the test statistics
\begin{equation}
\label{Dq}
D^*_q=n\sum_{k\in \mathcal{K}_q\cap\mathcal{K}^* }\widehat{\theta}^2_k
\end{equation}
observed, $\mathcal{K}_q$ and $\mathcal{K}^*$ as in \eqref{kappaq} and \eqref{kappaK} and $\widehat{\theta}_k$ as in \eqref{theta_est}.
\end{theorem}
Theorem \ref{anova_theo} follows directly from \eqref{LRq} and \eqref{Sj2} and, in virtue of the orthogonality of the $T_k$ functions and condition \eqref{C1}.
\begin{remark}
Because of condition \eqref{C1}, Theorem \ref{anova_theo} holds only for random sub-vectors of $\bm{X}$ whose Rosenblatt transform $u_q$ includes all the conditioning, from the higher to the lower, necessary to recover $g_{q}\bigl(\bm{x}_q\bigl)$. To some extent, this condition can be seen as the iGOF counterpart of the marginality principle advocated by \citet{nelder} in the context of ANOVA, and which consists in taking into account of the hierarchy of the main effects and interactions in a given model.
\end{remark}
Similarly to the ANOVA, Theorem \ref{anova_theo} allows us to construct an iGOF-diagnostic table to identify the source of mismodeling for a given random vector $\bm{X}$ and its components. Below we show how this can be done in practice for the case of a $7$-dimensional random vector.
\emph{\textbf{Example II.}} We consider a sample of $n=5000$ observations from a random vector $\bm{X}=(X_1,\dots,X_7)$ with components distributed as summarized in the second column of Table \ref{mismodeling}. Table \ref{anova} collects the results obtained by applying Theorem \ref{anova_theo} to test the validity of the models specified for different sub-vectors of $\bm{X}$. The overall deviance test is reported in the first row and correctly reject the null model.
\begin{center}
\begin{table}[!h]
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|c|c|c|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\vspace{-0.1cm}
& & & \\
\textbf{Variable} & \textbf{True} ($F$) & \textbf{Hypothesized} ($G$) & \textbf{Correct} \\
\vspace{-0.1cm}
& & & \\
\hline
\vspace{-0.1cm}
& & & \\
$X_6|X_1,X_2,X_5$& Laplace$\Bigl[e^{0.03x_1+0.02x_2+0.01x_2^2+0.02x_5},1\Bigl]$& Laplace$\Big[e^{0.03x_1+0.02x_2+0.02x_5},1\Bigl]$& No \\
\vspace{-0.1cm}
&&&\\
$X_1,X_2,X_5$& $N\left[\left(\begin{array}{c}
10\\
15\\
11
\end{array}\right),\left(\begin{array}{ccc}
4 & 0.5 & 0\\
0.5 & 3 & 1\\
0 & 1 & 5
\end{array}\right)\right]$ & $N\left[\left(\begin{array}{c}
10\\
15\\
11
\end{array}\right),\left(\begin{array}{ccc}
4 & 0.5 & 0\\
0.5 & 3 & 1\\
0 & 1 & 5
\end{array}\right)\right]$ & Yes \\
\vspace{-0.1cm}
&&&\\
$X_4|X_3$& Exponential$\Bigl(\frac{1}{x_3}\Bigl)$ & Exponential$\Bigl(\frac{1}{x_3}\Bigl)$ & Yes \\
\vspace{-0.1cm}
&&&\\
$X_3$& Exponential$(1)$ & Exponential$(0.9)$ & No \\
\vspace{-0.1cm}
&&&\\
$X_7$& $T_3$ & Cauchy$(0,1)$ & No \\
&&&\\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{True and postulated model for Example II. The last column highlights where mismodeling occurs. }
\label{mismodeling}}
\end{table}
\end{center}
Similarly, the test in the second row, rejects the hypotheses that the vector $(X_1,X_2,X_5,X_6)$ is modelled correctly, and fails to rejects the model for $(X_1,X_2,X_5)$. This aspect is particularly important as it highlights that the mismodeling occurs only with respect to the conditional distribution of $X_6|X_1,X_2,X_5$. The tests in the fourth and fifth row show that the vector $(X_3,X_4)$ has been mismodeled and one source of mismodeling is the marginal of $X_3$. Ultimately, the test for $X_7$ also correctly rejects the null hypothesis of Cauchy distribution.
Table \ref{simul} collects the results of a simulation obtained by repeating the diagnostic analysis in Table \ref{anova} through a simulation of $B=10,000$ replicates, while considering different sample sizes. Even when the sample size considered is only $500$, the most prominent deviations are captured with probability one, whereas, the model for $(X_1,X_2,X_5)$ is never rejected. More issues arise in diagnosing mismodeling of $X_3$ and, consequently, $(X_3,X_4)$ for smaller samples. For instance, even when $n=1000$ the power of the procedure in detecting departures of $G_{X_3}$ from $F_{X_3}$ is only $\sim56\%$ and $\sim3\%$ for $(X_3,X_4)$. It has to be noted, however, that detecting mismodeling of $X_3$ is a particularly challenging task. As shown in Figure \ref{figexp}, the postulated and the true pdf of $X_3$ are very close one-another; this minor differences are further ``diluted'' when considering the joint distribution of $(X_3,X_4)$, since $X_4|X_3$ is correctly specified. Nevertheless, such minor deviations are detected with high power for larger sample sizes.
\begin{minipage}{\textwidth}
\begin{minipage}[b]{0.49\textwidth}
\centering
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|c|c|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
$\bm{X}_q$ & \textbf{df}& \textbf{(Adjusted)} \\
&&\textbf{ p-value } \\
\hline
$\bm{X}$& 16383 & $<10^{-130}$ \\
$(X_1,X_2,X_5,X_6)$& 256 & $<10^{-130}$ \\
$(X_1,X_2,X_5)$&63 &$1$ \\
$(X_3,X_4)$& 15 & $8.467\cdot 10^{-08}$ \\
$X_3$&3 &$1.525\cdot 10^{-13}$ \\
$X_7$& 3 & $2.599\cdot 10^{-122}$ \\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\captionof{table}{iGOF-diagnostic table. The third column reports the (post-selection adjusted) deviance p-values in \eqref{test3} with $\bm{X}_q$ specified as in the first column. The second column corresponds to the degrees of freedom used in the calculation of the p-value, namely, $M_{q}$. }
\label{anova}}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\textwidth}
\centering
\centering
\includegraphics[width=55mm ]{Exponential} \\
\captionof{figure}{Comparing the postulated (red dashed line) and the true model (green solid line) of $X_3$. }
\label{figexp}
\end{minipage}
\end{minipage}\\
\vspace{0.5cm}
\begin{table}[!h]
\fontsize{8}{8}\selectfont{
\centering
\begin{tabular}{|c|cccccc|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
& & & & & & \\
\multirow{2}{*}{$\bm{X}_q$} &\multicolumn{6}{c|}{\textbf{Sample size ($n$)}} \\
& & & & & & \\
&$500$ & $1000$ &$2000$&$3000$& $5000$ & $10,000$ \\
\hline
& & & & & & \\
$\bm{X}$& 1 & 1 & 1 & 1 & 1 & 1 \\
$(X_1,X_2,X_5,X_6)$& 1 & 1 & 1& 1 & 1 & 1 \\
$(X_1,X_2,X_5)$& 0 & 0 & 0 & 0 & 0 & 0 \\
$(X_3,X_4)$& 0.0069 & 0.0342 & 0.2384 & 0.5939& 0.9615 &1 \\
& ($\pm$0.0008) & ($\pm$0.0018)& ($\pm$0.0049) & ($\pm$0.0049) & ($\pm$ 0.0019) & \\
$X_3$ & 0.2360 &0.5560 & 0.9153 &0.9868 &0.9999 &1 \\
& ($\pm$0.0043)& ($\pm$0.0050) & ($\pm$0.0028) & ($\pm$0.0011)& ($\pm$ 0.0001) & \\
$X_7$& 1 &1 &1& 1& 1 &1 \\
& & & & & & \\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{Performance of the iGOF-diagnostic analysis for different sample sizes. For values different from zero and one the Monte Carlo errors $(\pm SE)$ are also reported. The significance level considered is $\alpha=0.05$.}
\label{simul}}
\end{table}
\begin{figure*}[htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=72mm ]{Fig3a}&\includegraphics[width=72mm ]{Fig3b} \\
\end{tabular*}
\caption{Simulated and approximated confidence regions for the Fermi LAT simulation. The left panel corresponds to the (post-selection) confidence regions and deviance p-value obtained via a simulation of size $B=10,000$. The right panel shows to the (post-selection adjusted) confidence regions and deviance p-value computed as in \eqref{pvalue} and \eqref{CIband}. Darker shades correspond to significant deviations of the estimated likelihood ratio above one. Lighter shades correspond to significant deviations below one.
}
\label{Fermi}
\end{figure*}
\section{A diagnosis of background mismodeling}
\label{bkg}
When conducting searches for new phenomena, mismodeling of the background distribution can dramatically compromise
the sensitivity of the experiment. Specifically, overestimating the background can increase the chances of false negatives. Whereas, underestimating the background may lead to claiming false discoveries. To illustrate how iGOF can be used to understand if and how the postulated background model have been misspecified, we consider a simulated observation by the Fermi Large Area Telescope (LAT) \cite{atwood} obtained with the \emph{gtobssim} package\footnote{\url{http://fermi.gsfc.nasa.gov/ssc/data/analysis/software}} and previously published in \citet{TOHM}. The simulation includes a
realistic representations of the instrumental noise of the detector and present backgrounds.
The region of interest corresponds to a disc in the sky of $30^\circ$ radius and centered at ($195$ RA, $28$ DEC), where RA and DEC are the coordinates in the sky. Here we assume that, despite the cosmic background is known to follow a uniform distribution over the search area, it is unclear if the instrumental error is effectively negligible, or if it has a prominent effect on the underlying distribution. Therefore, we set $G_{X_1X_2}$ to be the cdf of a uniform distribution with support $\mathcal{X}_1\times\mathcal{X}_2=[165,195]\times[28-\sqrt{30^2-(x-195)^2}, 28+\sqrt{30^2-(x-195)^2}]$ and we proceed by estimating the likelihood ratio via \eqref{jcd_est} over a sample of $n=68658$ observations. Specifically, we set $m_1=m_2=4$ and we select the components of $\widehat{\bm{\theta}}$ via the BIC criterion in \eqref{BIC}. The resulting estimate is
\begin{equation}
\label{Fermi_est}
\widehat{d}\bigl(G_{1}(x_1),G_{2}(x_2|x_1)\bigl)=1+0.022T_{1}\bigl(G_1(x_1)\bigl)-0.043T_{1}\bigl(G_2(x_2|x_1)\bigl)+0.041T_{2}\bigl(G_2(x_2|x_1)\bigl).
\end{equation}
In order to assess the significance of the deviations captured by \eqref{Fermi_est}, we compute both the confidence regions and deviance p-values via \eqref{CIband} and \eqref{pvalue}. The results are reported in the right panel of Figure \ref{Fermi}, whereas the left panel shows the confidence regions and deviance p-value obtained via simulation. Similarly to what we have observed for Example I (see Figure \ref{fig2}), despite the approximate confidence bands are more conservative, they still allow to capture the main departures from uniformity. Indeed, in both cases, we can see that the prominent deviations of the true underlying model from the postulated uniform distribution occur in proximity of low values of $X_2$. Whereas, at the center-left of the search area, the uniform model significantly underestimates the model inclusive of the instrumental error. Finally, it follows from \eqref{skewG_est} that an updated model background distribution which accounts for these deviations can be constructed as in \eqref{skewG_est} by simply multiplying the uniform pdf by the estimated likelihood ratio in \eqref{Fermi_est}.
\section{Extensions to the discrete case}
\label{discrete}
The methods discussed so far focus on the case where $F$ and $G$ are continuous. However, extensions to the discrete setting can be derived by rewriting the expansion in \eqref{jcd_rep} through an orthonormal set of functions suitable to model discrete data. This can be done, for instance, by means of the so-called ``LP\footnote{In the \emph{LP} acronym, the letter \emph{L} typically denotes nonparametric methods based on quantiles, whereas \emph{P} stands for polynomials \cite[Supp S1]{LPksamples}.} score functions'', recently introduced \citep[e.g.,][]{LPksamples} and which can be seen as a generalization of the Legendre polynomials valid in both the continuous and discrete setting.
Specifically, when $p=1$, a complete orthonormal basis of LP score functions in $L^2(G)$ can be specified by letting the first component to be $T_0\bigl[G(x)\bigl]=1$. Subsequent components $\{T_j\bigl[G(x)\bigl]\}_{j>0}$ are obtained by Gram-Schimidt orthonormalization of powers of
\begin{equation}
\label{T1}
T_1\bigl[G(x)\bigl]=\frac{G_{\text{mid}}(x)-E[G_{\text{mid}}(x)]}{\sqrt{V(G_{\text{mid}}(x))}}=\frac{G(x)-0.5p_{G}(x)-0.5}{\sqrt{[1-\sum_{x\in \mathcal{X}}p_{G}^3(x)]/12}},
\end{equation}
where $G_{\text{mid}}(x)=G(x)-0.5p_{G}(x)$ is the \emph{mid-distribution function}, which it has been shown in \citet{parzen2004} to have mean $0.5$ and variance $[1-\sum_{x\in \mathcal{X}}p_{G}^3(x)]/12$, with $\mathcal{X}$ being the set of distinct points in the support of $X$ and $p_{G}(x)=P(X=x)$ if $X\sim G$.
Therefore, $T_1\bigl[G(x)\bigl]$ is the standardized mid-distribution and orthonormality of the $T_j\bigl[G(x)]$ functions in $L^2(G)$ follows by the first equality in \eqref{T1} and by Gram-Schmidt process.
Notice that, for continuous $X$, $G_{\text{mid}}(x)=G(x)$ and $\sum_{x\in \mathcal{X}}p_{G}^3(x)=0$, consequently, the LP score functions reduce to normalized shifted Legendre polynomials. The latter are effectively the result of a Gram-Schmidt orthonormalization applied to powers of $G(x)$.
Whereas, the LP score functions are obtained by orthonormalizing powers of the standardized mid-distribution function with respect to the measure $G$.
Recall that, in our context, the cdfs $G_d$, $d=1,\dots,p$ are the conditional and marginal distribution functions specified in the Rosenblatt's transform $\bm{G}_R(\bm{x})$.
Hence, an orthonormal basis in $L^2\bigl(G_d)$ is $\Bigl\{T_{j_d}\bigl[G_d(\cdot|\cdot)\bigl]\Bigl\}_{j_d\geq 0}$ with $T_{0}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]=1$
and subsequent components
\begin{align}
\label{T0etc}
\quad T_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]&=\frac{\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]}{\bigl|\bigl|\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl|\bigl|_{G_d}}, \quad \text{for all $j_p\geq 1$, where}\\
\label{Tjd}
\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]&=T^{j_d}_{1}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\\
&-\sum_{k=1}^{{j_p}-1}\bigl<T^{{j_d}}_1\bigl[G_d(x_d|\bm{x}_{<d})\bigl],T_k\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl>_{G_d}T_k\bigl[G_d(x_d|\bm{x}_{<d})\bigl],\\
\bigl<T_1^{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl],&T_{k_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl>_{G_d}=\\&\int T_1^{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]T_{k_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\text{d}G_d(x_d|\bm{x}_{<d})
\end{align}
and $||\cdot||_{G_d}=\sqrt{<\cdot,\cdot>_{G_d}}$.
When $p>1$ a suitable tensor basis in $L^2(G)$ can then be constructed as in Proposition \ref{Sjs}.
$\{T_{j_1,\dots,j_p}(\bm{u})\}_{j_1\dots j_p\geq 0}$. Orthonormality of the $T_{j_1,\dots,j_p}(\bm{u})$ score functions can be verified directly as shown in Section \ref{LPbasis} of the Supplementary Material.
\section{Discussion}
\label{conclusions}
This work proposes an informative approach to goodness-of-fit which connects exploratory and confirmatory data analysis to study multivariate distributions. By transforming the likelihood ratio on the unit cube, confidence regions can be constructed as in Corollary \ref{naive2} to identify regions of the supportwhere significant deviations occurs. While this approach is practical only for problems in at most three dimensions, in more dimensions a detailed diagnosis of mismodeling can be achieved by means of the iGOF-diagnostic analysis proposed in Section \ref{diagnostics}. These tools can be used to directly address Q1 in Section \ref{intro}.
For instance, given the panacea of theories available on the nature of dark matter, experimentalists aiming to detect it often face the dilemma of selecting which of the tens of theoretical models (mainly non-nested) available should be tested \citep[e.g.,][]{pat}. If one was to test it using the procedure discussed in this paper, even when a given model is rejected, it is possible to gain further insight on the shape of the departure of the true data distribution and the null model and ultimately use such information to ``rule out'' other models which would be inconsistent with such deviation.
Moreover, as we aimed for when formulating Q2 in Section \ref{intro}, the true probability function of the data can be estimated semi-parametrically via \eqref{skewG_est}, while assessing the validity of the model postulated by the scientists. Interestingly, the resulting estimate incorporates the knowledge carried by the hypothesized model and thus, it provides a data-driven update for it in the direction of the true distribution of the data.
Despite the usefulness of the methods presented here in applied settings, and in the physical sciences in particular (e.g., Section \ref{bkg}), they are not exempt from limitations. For instance, several problems in physics and astronomy, often involve no more than 8 or 10 dimensions and/or can be reduced to 2D planes \citep[e.g.,][]{aprile}. In this context, choosing $m_d$ equal to $3$ or $4$ for all $d=1,\dots,p$, is often sufficient to avoid overfitting and, eventually, lack of power by implementing adequate model selection strategies and for sufficiently large samples (see Sections \ref{inference} and \ref{diagnostics}). In more dimensions, however, the method suffers from the curse of dimensionality \citep[e.g.][]{friedman}, as the size of the LP tensor basis increases exponentially fast with $p$. In this context, a regularized solution could be particularly valuable \citep[see for instance][]{signoretto} when analyzing, for instance, data coming from large astronomical surveys such as the Large Synoptic Survey Telescope (LSST) survey \citep[e.g.,][]{tyson}.
Furthermore, the unitary representation of the likelihood ratio in \eqref{jcd} relies on the Rosenblatt transform and which can lead to different configurations of $\bm{U}$ and, potentially, different estimators. Despite this aspect would require adequate treatment on its own, it is worth noting that this problem is essentially the same arising in the context of vine copulas \citep[e.g.,][]{Nagler} and for which adequate model selection procedures exists \citep[e.g.,][]{panagiotelis,dissmann}.
Finally, the inferential procedures presented here extend classical smooth tests to the multivariate setting and allow us to visualize graphically the departure of $F$ from $G$ and study their substructures. Despite this article focuses on simple null hypothesis, that is, the postulated model is assumed to be fully specified,
classical results on smooth tests \citep[e.g.,][Sec 4.2.2.3 and 5.2.2.3 ]{thas} can be used to show to derive asymptotic tests in the parametric setting. Unfortunately, however, the asymptotic approximations are known to be rather slow in the parametric case. Therefore, in practical applications, when $G$ depends on unknown parameters it is recommended to perform inference by means of the parametric bootstrap and which has been shown by \citet{babu} to be consistent also in the multivariate setting.
\section{Introduction}
\label{intro}
\textbf{\emph{Scientific motivations.}}
When searching for the signals of new particles, or when aiming to detect new astronomical objects, a common difficulty arising in the analysis of the data collected by the detectors is the impossibility of correctly specifying the background distribution. In physics and astronomy, we typically refer with ``background'' or ``noise'' to the signal of all the astrophysical sources which are not those we aim to discover.
Unfortunately, since many sources contribute to the background, its distribution is particularly difficult to model \citep[e.g.,][]{priel, dauncey, algeriBANFF}.
Moreover, if the model postulated by the scientists is rejected, it is often difficult to identify the invalidating causes. As a result, this aspect is typically addressed by conducting a suite of exploratory analyses to adequately constrain the parameters involved, followed by a validity check for a newly postulated distribution. Given the complexity of the models investigated through physics experiments, however, this may result in a substantial investment of resources \cite[e.g.,][]{GAMBIT1}; even more so when having to choose between tens of plausible theoretical models \citep[e.g.,][]{pat}.
\textbf{\emph{Statistical formulation of the problem.}} In statistical terms, these difficulties translate into two main questions arising in the statistical analysis of multivariate data. Specifically, given a random vector $\bm{X}=(X_1,\dots,X_p)$, we may wonder:
\begin{itemize}
\item[Q1.] \emph{is the distribution of $\bm{X}$ correctly specified and, if not, in what way the true data distribution diverges from that
hypothesized under the null hypothesis?}
\item[Q2.] \emph{How can we improve our postulated model?} \emph{Or in other words, can we provide a data-driven correction of for it?}
\end{itemize}
As noted by \citet{pearson1938}, smooth tests, originally introduced by \citet{neyman37}, naturally allow us to capture and model the departure of $f$ from $g$ and thus, they offer the framework to directly address Q1 and Q2.
In order to provide a high level overview on smooth tests, let $f$ be the true (unknown) probability density function (pdf) of a random variable $X\in\Real$, $g$ is the hypothesized density and $G$ the respective cumulative density function (cdf). For example, in the above-mentioned problem of background mismodeling, $f$ represents the true background distribution and $g$ is the background model postulated by the scientists. A smooth model for the true probability law $f$ can be specified as
\begin{equation}
\label{skewG}
f(x)= g(x)d\bigl(G(x)\bigl)=g(x)\biggl\{1+\sum_{j\geq 1}\theta_j T_j\bigl[G(x)\bigl]\biggl\},
\end{equation}
where $d(G(x))=\frac{f(x)}{g(x)}$ is the likelihood ratio and the term in the curly brackets is an orthonormal expansion for $d$.
A smooth test \citep[e.g.,][]{neyman37,barton53, ledwina} consists of testing if any of the coefficients $\theta_j$ in \eqref{skewG} is different from zero. Finally, by estimating $d\bigl(G(x)\bigl)$ and constructing adequate confidence bands, it is possible to visualize the nature of the departure of $f$ from $g$.
Despite their usefulness, smooth tests are mainly limited to the univariate setting. In light of this, the main methodological task of this work is to extend this framework to allow for the analysis of multivariate data.
\textbf{\emph{Main results and organization.}}
The theoretical framework is presented in Section \ref{modelling}. There, we define a suitable expansion of the likelihood ratio through orthonormal functions on the unit cube.
As shown in Sections \ref{estimation} and \ref{inference}, such representation substantially simplifies the subsequent stages of estimation, model selection and (post-selection) inference.
In Section \ref{diagnostics}, we discuss a simple ANOVA-like testing strategy to identify possible sources of mismodeling. Power studies are conducted via simulations in both Sections \ref{inference} and \ref{diagnostics}.
As noted above, this work finds its main motivations in the context of astrophysical searches. Therefore, in Section \ref{bkg} we illustrate how iGOF can be used to address the problem of mismodeling of the cosmic background considering a realistic simulation from the Fermi Large Area Telescope \citep{atwood}. Despite this article mainly focuses on the analysis of continuous data, extensions to the discrete setting are discussed in Section \ref{discrete}.
Section \ref{conclusions} collects a summary of the results and a discussion of the limitations of iGOF. Technical proofs and codes are provided in the Supplementary Material.
A summary of the main notation used throughout the paper is available in the Appendix.
\section{Theoretical framework }
\label{modelling}
\subsection{Transformations of the likelihood ratio on the unit cube}
\label{Rosenblatt}
Suppose $F$ is the true distribution function of a random vector $\textbf{X}\in \mathcal{X}\subseteq \Real^p$ and denote with $G$ its hypothesized distribution. $F$ and $G$ are assumed to be continuous with densities $f$ and $g$. Furthermore, assume that $f(\bm{x})=0$ whenever $g(\bm{x})=0$. For every $\bm{x}=(x_1,\dots,x_p)\in \mathcal{X}$, the hypothesized density $g$ is such that
\[g(\bm{x})=\prod_{d=1}^p g_d(x_d|{\bm{x}}_{<d}),\]
where $\bm{x}_{<d}=(x_1,\dots,x_{d-1})$ and $g_1,\dots,g_p$ are suitable densities with associated cdfs and quantile functions $G_d$ and $Q_{d}$, for all $d=1,\dots,p$. The likelihood ratio between $F$ and $G$ can be specified as
\begin{equation}
\label{jcd}
d(\bm{u})=\frac{f\bigl(\bm{Q}(\bm{u})\bigl)}{g\bigl(\bm{Q}(\bm{u})\bigl)},\quad\bm{u}\in[0,1]^p
\end{equation}
where $\bm{u}=(u_1,\dots,u_p)=\bigl(G_1(x_1),\dots,G_p(x_p|{\bm{x}}_{<p})\bigl)=\bm{G}_R(\bm{x})$ is the Rosenblatt transformation \citep{rosenblatt}\footnote{Notice that, in general, $\bm{G}_R(\bm{x}) \not\equiv G(\bm{x})$ as the Rosenblatt's transform $\bm{G}_R(\bm{x})\in[0,1]^d$ whereas the cdf $G(\bm{x})\in [0,1]$.}. Whereas,
$\bm{Q}(\bm{u})=\bigl(Q_{1}(u_1),\dots,Q_{p}(u_p)\bigl)$, for all $d=1,\dots,p$.
In the bivariate setting, for instance, let $G_1\equiv G_{X_1}$ and $G_2\equiv G_{X_2|X_1}$, i.e., the hypothesized marginal cdf of $X_1$ and the hypothesized conditional cdf of $X_2|X_1$, respectively. Hence, \eqref{jcd} specifies as
\[d(u_1,u_2)=\frac{f_{X_1X_2}\bigl(Q_{1}(u_1),Q_{2}(u_2)\bigl)}{g_{X_1X_2}\bigl(Q_{1}(u_1),Q_{2}(u_2)\bigl)}.\]
\begin{remark}
\label{independence}
As a plausible alternative to Rosenblatt's transform, one could choose each $G_d\equiv G_{X_d}$, which corresponds to assuming independence among the components of $\bm{X}$. In this setting, \eqref{jcd} is the copula density \citep[e.g.,][]{nelsen} of $\bm{X}$ under $G$. Despite this choice could simplify substantially the computations, it would not allow to test models where the components of $\bm{X}$ are assumed to be dependent. Moreover, it is worth pointing out that there are situations where such transformation cannot be specified (e.g., Section \ref{bkg}).
\end{remark}
To provide a sufficiently detailed representation of the substructures characterizing the distribution of $\bm{X}$ (see Q1 in Section \ref{intro}), a natural approach is that of expressing \eqref{jcd} by means of a suitable orthonormal basis in $L^2[0,1]^p$.
For instance, let $T_{j_d}(u_d)$ be the $j_d^{th}$ normalized shifted Legendre polynomial evaluated at $u_d=G_d(x_d|\bm{x}_{<d})$, with $T_0(u_d)=1$, $T_1(u_d)=\sqrt{12}(u_d-0.5)$, etc. (e.g., Section \ref{Ex1app}, Supplementary Material). Each $\{T_{j_d}(u_d)\}_{j_d\geq 0}$ forms a basis in $L^2[0,1]$. Hence, we can exploit a well known result in Hilbert space theory \citep[e.g., Proposition 2][p.50]{reedbook} which asserts that given two orthonormal bases $\{\psi_{j}\}$, $\{\phi_{k}\}$ for the Hilbert spaces $\mathcal{H}_1$, $\mathcal{H}_2$, then $\{\psi_{j}\otimes \phi_{k}\}$ is an orthonormal basis for $\mathcal{H}_1\otimes\mathcal{H}_2$. It follows that the tensor product basis $\{T_{j_1,\dots,j_p}(\bm{u})\}_{j_1\dots j_p\geq 0}$ of functions
\begin{equation}
\label{Sjs}
T_{j_1\dots j_p}(\bm{u})=\prod_{d=1}^p T_{j_d}(u_d)
\end{equation}
forms an orthonormal basis on $L^2[0,1]^p$, the Hilbert space of square integrable function over the $p$-dimensional unit cube.
Notice that despite any orthonormal basis in $[0,1]$ could be used to construct a tensor product basis in $[0,1]^p$, here we focus on the normalized shifted Legendre polynomials. This choice is justified by the fact that the latter are special cases of the so called LP-score functions \citep[e.g.,]{LPksamples}. As discussed in Section \ref{discrete}, the LP score functions allow for extensions to the discrete setting.
Finally, under the assumption $d(\bm{u})\in L^2[0,1]^p$, we can write
\begin{equation}
\label{jcd_rep}
d(\bm{u})=\sum_{j_1\geq 0,\dots,j_p\geq 0}\theta_{j_1\dots j_p}T_{j_1\dots j_p}(\bm{u}),\qquad \text{ $\bm{u}\in [0,1]^p$}
\end{equation}
with $\theta_{j_1\dots j_p}=\int_{[0,1]^p}T_{j_1\dots j_p}(\bm{u})d(\bm{u})\text{d}\bm{u}$. The expansion in \eqref{jcd_rep} follows from Theorem II.6 in \citet[][]{reedbook} and it is equivalent to say that the sum on the right-hand side converges to $d(\bm{u})$ in $L^2[0,1]^p$.
\begin{remark}
\label{whyU}
An anonymous referee correctly pointed out that the likelihood ratio can also be expanded through an orthonormal expansion on the original domain $\mathcal{X}$, bypassing the need of Rosenblatt's transform. In our context, however, the Rosenblatt transform is particularly useful for two main reasons. First of all, one can show (Proposition \ref{moments_prop} to follow) that $d(\bm{u})$ corresponds to the density of $\bm{U}=\bm{G}_R(\bm{X})$.
This fact will simplify substantially the estimation process (Section \ref{estimation}). Second, by transforming the data on the compact compact domain $[0,1]^p$ we will be able to exploit exploit results from random fields theory to perform inference (Section \ref{inference}).
\end{remark}
\section{Estimation}
\label{estimation}
The summations in \eqref{jcd_rep} are taken up to infinity. However, to make the expansion operational, it is necessary to truncate the series in \eqref{jcd_rep} at integers values $m_1,\dots,m_p$. That is because, effectively, the coefficients $\theta_{j_1\dots j_p}$ need to be estimated and, consequently, the more terms are included in \eqref{jcd_rep}, the larger the variance of the resulting estimator of $d(\bm{u})$ (see Section \ref{selection} for a more detailed discussion on model selection).
For the sake of simplifying the notation in this section and those to follow, denote with $\mathcal{K}$ the set
\begin{equation}
\label{kappa}
\mathcal{K}:=\biggl\{\{j_1\dots j_p\},\text{ with } j_d=0,\dots,m_d, \text{ for all } d=1,\dots,p, \text{ and } \sum_{d=1}^p j_d\neq 0 \biggl\}
\end{equation}
of cardinality $|\mathcal{K}|=M=\prod_{d=1}^p(m_d+1)-1$. That is, $\mathcal{K}$ contains all the $p-$tuples $\{j_1\dots j_p\}$ of indexes $j_d=0,\dots,m_d$, $d=1,\dots,p$ apart from the $p-$tuple $\{0\dots 0\}$.
Let $\bm{\theta}$ be the $M\times 1$ vector of components ${\theta}_{k}$, with $k\in \mathcal{K}$.
Similarly, denote with $\bm{T}(\bm{u})$ the $M\times 1$ vector of elements $T_{k}(\bm{u})$, $k\in \mathcal{K}$.
Consider $\bm{x}_1,\dots,\bm{x}_n$, a sample of $n$ i.i.d. observations from $\bm{X}$, and let $\bm{U}=\bm{G}_R(\bm{X})$ be the respective Rosenblatt transformation.
Denote with $\bm{u}_1,\dots,\bm{u}_n$ the sample of elements $\bm{u}_i=\bm{G}_R(\bm{x}_i).$
The parameter ${\bm{\theta}}$ can be estimated by means of the vector $\widehat{\bm{\theta}}$ of components
\begin{equation}
\label{theta_est}
\widehat{\theta}_{k}=\frac{1}{n}\sum_{i=1}^n T_{k}(\bm{u}_i)\quad\text{ for all $k\in \mathcal{K}$ },
\end{equation}
The mean and covariance matrix of $\widehat{\bm{\theta}}$ and an estimator of $d(\bm{u})$ are given in Proposition \ref{moments_prop}.
\begin{proposition}
\label{moments_prop}
The likelihood ratio $d\bigl(\bm{u}\bigl)$ is the density of the random vector $\bm{U}$ and
\begin{equation}
\label{moments}
E[\widehat{\bm{\theta}}]=\bm{\theta}\quad\text{and}\quad \text{Cov}(\widehat{\bm{\theta}})=\bm{\Sigma}
\end{equation}
where $\bm{\Sigma}$ has diagonal elements $\frac{\sigma^2_{k}}{n}=\frac{1}{n}V\bigl[T_{k}(\bm{U})\bigl]$ and non-diagonal elements
$\frac{\sigma_{k,h}}{n}=\frac{1}{n}\text{Cov}\bigl[T_{k}(\bm{U}),T_{h}(\bm{U})\bigl]$, with $k,h\in \mathcal{K}$.
Furthermore, if $F\equiv G$, the equalities in \eqref{moments} reduce to
\begin{equation}
\label{moments0}
E[\widehat{\bm{\theta}}]=\bm{0} \quad\text{and}\quad \text{Cov}(\widehat{\bm{\theta}})=\frac{1}{n}\bm{I}_M,
\end{equation}
where $\bm{0}$ is the $M\times1$ zero vector and $\bm{I}_M$ is the $M\times M$ identity matrix.
Finally, an estimator of $d(\bm{u})$ is
\begin{align}
\label{jcd_est}
\widehat{d}(\bm{u})&=1+\widehat{\bm{\theta}}'\bm{T}(\bm{u}),
\end{align}
and has variance $V\Bigl[\widehat{d}(\bm{u})\Bigl]=\bm{T}(\bm{u})'\bm{\Sigma}\bm{T}(\bm{u})$.
\end{proposition}
It is worth pointing out that, when the likelihood ratio is formulated as a density, it is often referred to as \emph{comparison density} \citep[e.g.,][]{parzen2004}.
\begin{figure*}[!htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=70mm ]{Fig1a} & \hspace{-1cm} \includegraphics[width=71mm ]{Fig1b}\\
\end{tabular*}
\caption{True (left panel) and estimated (right panel) likelihood ratio for Example I. The estimate on the right has been obtained via \eqref{jcd_est} with $m_1=4$ and $m_2=3$. The components of the $\widehat{\bm{\theta}}$ vector have been selected via the AIC criterion in \eqref{BIC}.
}
\label{fig1}
\end{figure*}
Combining \eqref{skewG}, \eqref{jcd} and \eqref{jcd_est} an estimate of $f$ is
\begin{align}
\label{skewG_est}
\widehat{f}(\bm{x})=g(\bm{x})\widehat{d}\bigl(\bm{G}_R(\bm{x})\bigl)=g(\bm{x})\bigl[1+\widehat{\bm{\theta}}'\bm{T}\bigl( \bm{G}_R(\bm{x})\bigl)\bigl]
\end{align}
Notice that the estimator $\widehat{f}$ incorporates the information carried by the hypothesized model $g$; whereas, the estimator in the square brackets provides a data-driven correction for it. Furthermore, define the integrated squared bias (ISB) of $\widehat{d}(\bm{u})$ to be
\begin{equation}
\label{ISBdef}
ISB=\int_{[0,1]^p}\Bigl(E[\widehat{d}(\bm{u})]-d(\bm{u})\Bigl)^2\text{d}\bm{u}.
\end{equation}
From Proposition \ref{ISB_prop} it follows that the closer $g$ is to $f$ in terms of squared normalized distance the lower the ISB of $\widehat{d}(\bm{u})$.
\begin{proposition}
\label{ISB_prop}
The integrated squared bias of the estimator in \eqref{jcd_est} is
\begin{equation}
\label{ISB}
\bigintsss_{[0,1]^p}\biggl(\frac{f\bigl(Q(\bm{u})\bigl)- g\bigl(Q(\bm{u})\bigl)}{g\bigl(Q(\bm{u})\bigl)}\biggl)^2\text{d} \bm{u}-\bm{\theta}'\bm{1},
\end{equation}
where $\bm{1}$ is the $M\times 1$ unit vector.
\end{proposition}
The estimate in \eqref{skewG_est} is essentially that of a smooth model \citep[e.g.,][]{rayner90}, that is, a smoothed version of the true underlying probability function. Similarly to the smooth model proposed by \citet{barton53} in the univariate setting, the estimator in \eqref{skewG_est} may lead to estimate that are not \emph{bona-fide}, i.e, they may be negative and/or they may not integrate/sum up to one. In this manuscript we focus on \eqref{skewG_est} mostly for the sake of mathematical convenience in deriving the inferential results of Section \ref{inference}. Nonetheless, bona-fide estimators can be constructed similarly to the univariate case as described in \citet{algeri_zhang}.
\emph{\textbf{Example I.}} In direct searches for dark matter, the dominant background sources are neutron recoils which may produce signals mimicking those expected from dark matter candidates \citep[e.g.,][]{westerdale}. As a toy example, suppose we are interested in assessing the validity of a given distribution for the nuclear recoil background specified over the energy region $\mathcal{X}=[5,20]KeVnr\times[0,17]KeVnr$. Each observations in $\mathcal{X}$ corresponds to the scintillation of photons ($X_1$) and ionization electrons ($X_2$) \citep[e.g.,][]{aprile}. The hypothesized background distribution, $G_{X_1X_1}$, is that of a truncated bivariate normal with mean vector $(12,8)$, variances $8$ and $12$ and covariance $2$. Moreover, suppose that one additional background source is present. The latter is also a bivariate normal with the same mean vector, variances $4$ and $20$ and covariance $5$. Thus, the true model, $F_{X_1X_1}$, involves a mixture of two, overlapping truncated bivariate Gaussians with mixture parameter $0.15$. In order to estimate the likelihood ratio, set $G_1=G_{X_1}(x_1)$ and $G_2=G_{X_2|X_1}$. The estimated likelihood ratio, obtained over a sample of $n=5,000$, is shown in the right panel of Figure \ref{fig1}, whereas the left panel shows the true likelihood ratio. A closed form expression for the estimate shown on the right panel is given in equation \ref{LR1} in the Supplementary Material.
While the estimate obtained recovers the main departures from uniformity, the contours highlight that the estimator is rather noisy. Therefore, it is important to investigate the properties of \eqref{jcd_est} to assess the significance of the deviations observed.
\section{Inference and model selection}
\label{inference}
\subsection{Pre-selection inference}
A smooth test for $H_0:G\equiv F$ versus $H_1:G\not\equiv F$ consists in reformulating the problem as a test for uniformity of $\bm{U}$.
Specifically, \eqref{jcd} implies that $F\equiv G$ whenever $d(\bm{u})=1$, and thus
\begin{equation}
\label{test0}
\begin{split}
H_0: d(\bm{u})=1 \quad\text{$\forall \bm{u}\in [0,1]^p$} \quad\text{versus} \quad H_1 : \exists \bm{u}\in [0,1]^p \text{ s.t. } d(\bm{u})\neq 1.
\end{split}
\end{equation}
It is easy to see that $d(\bm{u})=1$ for all $\bm{u}\in [0,1]^p$, when all ${\theta}_{k}$, $k\in\mathcal{K}$, are identically equal to zero. Hence, in practice, we test
\begin{equation}
\label{test}
H_0: \bm{\theta}= \bm{0} \qquad\text{vs}\qquad H_1 : \bm{\theta}\neq \bm{0}.
\end{equation}
Notice that $H_0$ in \eqref{test0} implies $H_0$ in \eqref{test}, but the opposite is not true in general. Whereas, $H_1$ in \eqref{test} does imply $H_1$ in \eqref{test0}.
With a little
abuse of nomenclature, in this section and those to follow, we will refer to $G$ as the ``null model''. Furthermore, we will refer to $H_0$ in \eqref{test0} when generically saying ``under $ H_0$''. However, most of the results presented here, only require validity of the milder $H_0$ in \eqref{test}
To conduct our inference, we consider the so-called \emph{deviance} test statistics, i.e.,
\begin{equation}
\label{deviance}
D=n\widehat{\bm{\theta}}'\widehat{\bm{\theta}}.
\end{equation}
Its asymptotic null distribution is given in Theorem \ref{normality}.
\begin{theorem}
\label{normality}
If $H_0$ is true, then
\vspace{-0.2cm}
\begin{equation}
\label{thetaH0}
\sqrt{n}\widehat{\bm{\theta}}\xrightarrow d N(\bm{0},\bm{I}), \quad\text{as $n\rightarrow\infty$}
\end{equation}
\vspace{-0.2cm}
where $N(\bm{0},\bm{I})$ denotes a standard multivariate normal distribution. Furthermore,
\begin{equation}
\label{DH0}
D \xrightarrow d \chi^2_{M}, \quad\text{as $n\rightarrow\infty$},
\end{equation}
where $M$ is the length of $\widehat{\bm{\theta}}$.
\end{theorem}
Corollary \ref{dhat_cor} follows directly from Theorem \ref{normality}.
\begin{corollary}
\label{dhat_cor}
Denote with $\{\widehat{d}(\bm{u})\}$ the random field indexed by $\bm{u}\in [0,1]^p$ with components as in \eqref{jcd_est}. Moreover, assume that $\widehat{\theta}_k=o(n^{-1/2})$ for all $k\not\in \mathcal{K}$. If $H_0$ is true,
\begin{equation}
\label{rf}
\Biggl\{\frac{\widehat{d}(\bm{u})-1}{\sqrt{\frac{1}{n}\bm{T}(\bm{u})'\bm{T}(\bm{u})}}\Biggl\}\xrightarrow d \bm{Z}(\bm{u}), \quad\text{as $n\rightarrow\infty$,}
\end{equation}
where $\bm{Z}(\bm{u})$ denotes a Gaussian random field with mean zero, unit variance and covariance function
$\text{Cov}\Bigl( \bm{Z}(\bm{u}), \bm{Z}(\bm{u}^\dag)\Bigl)=\frac{\bm{T}(\bm{u})'\bm{T}(\bm{u}^\dag)}{\sqrt{\bm{T}(\bm{u})'\bm{T}(\bm{u})\bm{T}(\bm{u}^\dag)'\bm{T}(\bm{u}^\dag)}}.$
\end{corollary}
At this stage, constructing inference on the basis of Theorem \ref{normality} and Corollary \ref{dhat_cor} would be tempting. However, to guarantee the validity of our results we must take into account that, when estimating the likelihood ratio in \eqref{jcd_est}, a model selection procedure is likely to be implemented. Unfortunately, when a model is selected by a pool of possibilities, such process introduces an additional source of variability and thus the resulting inference is automatically affected \citep[e.g.,][]{berk}. Section \ref{selection} addresses this aspect directly.
\subsection{Post-selection inference}
\label{selection}
The estimate of the likelihood ratio considered so far involves up to $M$ functions $T_{k}(\bm{u})$. Nonetheless, it is possible that not all of these $M$ terms are needed to capture the departures of $G$ from $F$ and indeed, it is often convenient to remove some of them in order to avoid unnecessary sources of noise. Various criteria have been proposed in literature for density estimation and smooth models \citep[e.g.,][]{LPmode, algeri20} and which can be easily extended to the multivariate setting. Here, we focus on the approach of \citet{LPmode} and which specifies as follows.
Let $\widehat{\theta}_{(k)}$ be the $k^{th}$ largest $\widehat{\theta}_{k}$ estimate in order of magnitude, i.e., $\widehat{\theta}_{(1)}^2\geq \widehat{\theta}_{(2)}^2\geq\dots\geq \widehat{\theta}_{(M)}^2$. Select the $K$ largest coefficients which maximize either
\begin{equation}
\label{BIC}
\text{BIC}(K) = \sum_{(k)=1}^K\widehat{\theta}^2_{(k)} - \frac{K \log n }{n}\quad\text{or}\quad \text{AIC}(K) = \sum_{(k)=1}^R\widehat{\theta}^2_{(k)} - \frac{2K}{n}.
\end{equation}
Notice that, as defined in \eqref{kappa}, each $k$ is a $p-$tuple of indexes $j_1\dots j_p$, whereas $(k)$ is the integer value corresponding to the order of magnitude of the respective coefficient $\widehat{\theta}_{k}$. Hence, the summations in \eqref{BIC} and those to follow are taken over $(k)=1,\dots,K$, that is the $K$ $p-$tuples of indexes $j_1\dots j_p$ with the $K^{th}$ largest estimates $\widehat{\theta}_{k}$.
An estimate of $d(\bm{u})$, is then selected via \eqref{BIC} from the family of estimators
\begin{equation}
\label{estAIC}
\widehat{d}_{(K)}(\bm{u})=1+\sum_{(k)=1}^{K}\widehat{\theta}^2_{(k)}T_{(k)}(\bm{u}),\quad\text{K=1,\dots,M}
\end{equation}
where the subscript $(K)$ is used to emphasize that the estimator in \eqref{estAIC} includes only the $K^{th}$ largest $\widehat{\theta}_k$ estimated coefficients.
Clearly, the choice of BIC or AIC is arbitrary and, from a practical standpoint, the BIC tends to lead to smoother estimates than the AIC.
The selection rules in \eqref{BIC} compare $M$ possible models assuming that each $m_d$, for $d=1,\dots,D$ was fixed before the researcher looked at the data. Valid post-selection inference can then be constructed as in Theorem \ref{normality} and Corollary \ref{naive}. The respective proofs are provided in the Supplementary material.
\begin{corollary}
\label{naive}
Denote with $\widehat{d}_{(K*)}$ the estimator of $d(\bm{u})$ selected via \eqref{BIC}, and let $D_{(K^*)}=\sum_{(k)=1}^{K^*}\widehat{\theta}^2_{(k)}$ be the respective deviance statistics.
As $n\rightarrow \infty$, a valid post-selection bound for the p-value to test \eqref{test0} is
\begin{equation}
\label{pvalue}
\text{p-value}_{adj}=P(\chi^2_{M }>D_\text{obs}),
\end{equation}
where $D_\text{obs}$ is the value of $D_{(K^*)}$ observed.
\end{corollary}
Where the bound in \eqref{pvalue}, follows from the fact that the estimators in \eqref{estAIC} are nested, for all $K=1,\dots,M-1$, and thus each $D_{(K)}$ is stochastically lower or equal than $D_{(M)}$. Hence, for all $K=1,\dots,M-1$, $P(D_{(K)}>D_\text{obs})$ is smaller than $P(D_{(M)}>D_\text{obs})$.
In order to grasp further insights on the deviations of $G$ from $F$, it is worth constructing adequate confidence bands. This can be done, while accounting for post-selection adjustments, as in Corollary \ref{naive2}.
\begin{corollary}
\label{naive2}
Denote with $\widehat{d}_{(K*)}$ the estimator of $d(\bm{u})$ selected via \eqref{BIC}, and denote with
$SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]$ its standard error under $H_0$.
Valid (post-selection adjusted) $(1-\alpha)\%$ confidence regions, under $H_0$ in \eqref{test0}, are
\begin{equation}
\label{CIband}
\biggl[1-c_{\alpha/2} SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl],1+c_{\alpha/2}SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]\biggl]\qquad \text{for all $\bm{u}\in [0,1]^p$}
\end{equation}
with $c_{\alpha/2}$ such that
\begin{equation}
\label{calpha}
P\Biggl(\sup_{\bm{u}}\biggl\{\frac{\widehat{d}_{(M)}(\bm{u})-1}{SE_0\bigl[\widehat{d}_{(M)}(\bm{u})\bigl]}\biggl\}>c_{\alpha/2} \Biggl|H_0\Biggl)=\frac{\alpha}{2}.
\end{equation}
\end{corollary}
Intuitively, $\widehat{d}_{(M)}$ is the least smooth among all the estimators considered; hence, we expect that the random field resulting from $\widehat{d}_{(M)}$ has the largest probability of crossing the fixed level $c_{\alpha/2}$.
From a theoretical perspective, a highly non-trivial aspect in the construction of \eqref{CIband} is the estimation of the quantile $c_{\alpha/2}$. Probabilities such \eqref{calpha} are known in literature as \emph{excursion probabilities} \citep[e.g.,][]{adler2000} and which cannot be expressed in closed form. A possible solution for constructing the confidence bands in \eqref{CIband}, is that of proceeding by estimating $SE_0\bigl[\widehat{d}_{(K^*)}(\bm{u})\bigl]$ and $c_{\alpha/2}$ via Monte Carlo simulations \citep[see][Algorithm 1]{algeri_zhang}. Unfortunately, in the most crucial (astro)physical searches the level of significance required to claim a new discovery is typically in the order of $\alpha=10^{-7}$ \citep[e.g.,][]{lyons2015}, and thus Monte Carlo simulations may be computationally prohibitive. This is further aggravated when dealing with complex models for which even a single Monte Carlo replicate can be highly expensive in terms of both computational and time resources.
As a valid alternative, for continuous $F$ and $G$, accurate approximations for \eqref{calpha} under mild smoothness conditions exist \citep[e.g.,][]{taylor2008}.
In our setting, smoothness follows from the fact that the random field in \eqref{rf} and the respective limit can be written as a linear combination of the functions $\frac{T_k(\bm{u})}{\sqrt{\sum_{k\in\mathcal{K}}T^2_k(\bm{u}) }}$
(see proof of Corollary \ref{dhat_cor} in the Supplementary Material) which are composition of Legendre polynomials, and thus, admit infinite partial derivatives.
An approximation for the left-hand side of \eqref{calpha} is
\begin{equation}
\label{LK}
\bigl(1-\Phi(c_{\alpha/2})\bigl)+\mathcal{L}_1\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\pi}+\mathcal{L}_2\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\sqrt{2}\pi^{3/2}}+O\biggl(\exp\Bigl(-\frac{\gamma c_{\alpha/2}^2}{2}\Bigl)\biggl),\quad\text{as $n\rightarrow\infty$,}
\end{equation}
\noindent for some $\gamma>1$ \citep{takemura}. In \eqref{LK}, $\mathcal{L}_1$ and $\mathcal{L}_2$ are constant known as Lipischitz-Killing curvatures and are typically estimated numerically \citep[e.g.,][]{TOHM}. Notice that the error rate in \eqref{LK} decreases exponentially fast, as $\alpha\rightarrow \infty$. Therefore, this solution is particularly amenable to overcome the issues arising when dealing with stringent significance requirements.
\begin{figure*}[htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=72mm ]{Fig2a}&\includegraphics[width=72mm ]{Fig2b} \\
\end{tabular*}
\caption{Simulated and approximated confidence regions for Example I. The left panel corresponds to the (post-selection) confidence regions and deviance p-value obtained via a simulation of size $B=10,000$. The right panel shows to the (post-selection adjusted) confidence regions and deviance p-value computed as in \eqref{pvalue} and \eqref{CIband}. Darker shades correspond to significant deviations of the estimated likelihood ratio above one. Lighter shades correspond to significant deviations below one.
}
\label{fig2}
\end{figure*}
\begin{table}[!h]
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|cccccc|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
& $n=500$ & $n=1000$ & n=2000 & $ n=5000$ & $ n=7000$ & $ n=10,000$ \\
\hline
Type I error &0.0540 & 0.0500 & 0.0499 & 0.0482 & 0.0508 &0.04930\\
($\pm$ SE) & ($\pm$ 0.0023) & ($\pm$ 0.0022) &($\pm$ 0.0022) & ($\pm$ 0.0021) & ($\pm$ 0.0022) & ($\pm$ 0.0022) \\
\hline
Power& 0.2157 &0.4456 &0.8063 & 0.9995 &1.0000& 1.0000 \\
($\pm$ SE)& ($\pm$ 0.0041) &($\pm$ 0.0050) &($\pm$ 0.0040) & ($\pm$ 0.0002) &($\pm$ 0.0000) & ($\pm$ 0.0000)\\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{Simulated probability of type I error and power for Example I considering different sample sizes. The nominal level is chosen to be $\alpha=0.05$. Each simulation involves $B=10,000$ replicates.}
\label{time}}
\end{table}
As one may expect, the simplicity of the post-selection adjustments in \eqref{pvalue} and \eqref{CIband} comes with a price. Specifically, they can be rather conservative for increasing values of $M$.
However, as shown below for Example I and in the sections to follow, \eqref{pvalue} still leads to high power even if the sample size is only moderately large. Similarly, \eqref{CIband} can be quite accurate and match closely the confidence regions obtained by simulating directly the distribution of \eqref{rf}, while repeating the selection process at each replicate.
\emph{\textbf{Example I (continued).}} The estimate of the likelihood ratio in the right panel of Figure \ref{fig1} has been obtained by setting $m_1=4$ and $m_2=3$ and selecting the terms of the respective tensor basis via the AIC rule in \eqref{BIC}.
The AIC procedure selects $9$ terms out of $M=19$.
The post-selection adjusted p-value and $95\%$ confidence regions are shown in the right panel of Figure \ref{fig2}.
The confidence contours are constructed by setting equal to one all the values of $\widehat{d}$ contained within the bands in \eqref{CIband}. Whereas the quantile $c_\alpha$ has been calculated by solving
\begin{equation}
\label{solver}
\bigl(1-\Phi(c_{\alpha/2})\bigl)+\mathcal{L}_1\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\pi}+\mathcal{L}_2\frac{e^{-\frac{c_{\alpha/2}^2}{2}}}{\sqrt{2}\pi^{3/2}}-\frac{\alpha}{2}=0
\end{equation}
and estimating $\mathcal{L}_1$ and $\mathcal{L}_2$ by means of the \texttt{R} package \texttt{TOHM} \citep{TOHMpkg} as described in \citet{TOHM}. This approach led to $c_{\alpha}=3.5568$.
The confidence contours suggest that the most prominent deviations occur in correspondence of the regions $[10,12]\times[0,5]$ and $[12,15]\times[12,17]$. Here, the estimator of $d(\bm{u})$ shows significant deviations above one and thus we conclude that the postulated model underestimates the truth over these areas. The presence of significant departures of $G_{X_1X_2}$ from $F_{X_1X_2}$ are confirmed by the deviance test (adjusted p-value $\sim 3.97 \cdot 10^{-11}$).
The left panel of Figure \ref{fig2} shows the confidence regions and deviance p-value obtained by means of a Monte Carlo simulation involving $B=10,000$ replicates. The selection procedure has been implemented at each replicate. While more conservative, the confidence regions computed via \eqref{CIband} and \eqref{solver}, approximate reasonably well those obtained via simulation.
Finally, we investigate the probability of type I error and the power of the deviance test based on \eqref{pvalue}. Table \ref{time} reports the results obtained considering a suite of five simulations, each of size $B=10,000$, conducted using five different sample sizes. For all $n$ considered, the probability of type I error observed is approximately the same than the nominal level $\alpha=0.05$. Whereas, the power increases rapidly with $n$. For the smallest samples sizes considered, i.e., $n=500$ and $n=1000$, the power is rather low ($\sim 22\%$ and $\sim 45\%$, respectively). However, it has to be noted that, in our example, the mixture parameter is $0.15$; therefore the deviations from the postulated model effectively account for only $\sim 75$ and $\sim 150$ data points when $n=500$ and $n=1000$, respectively.
In principle the plots in Figures 1, 2, and those to follow, could also be visualized in the quantile domain. The latter is to be preferred when working with long tailed distribution where one may expect that only a few observations have been detected over large regions of the $\bm{X}$ domain. In those situations, the quantile representation would allow to magnify the differences observed over the the most ``data-abundant'' regions. An more detailed discussion of this aspect, and adequate graphical comparisons can be found in \citep{algeri20}.
\section{iGOF-diagnostic analysis}
\label{diagnostics}
The constructs introduced so far allow us to assess the validity of the postulated model, obtain an estimate of the likelihood ratio test to visualize where and how departures of $g$ from $f$ occur, and construct a data driven correction for the initial model $g$ (equation \ref{skewG_est}). Unfortunately, however, a visual inspection is only possible when $p\leq 3$. Nevertheless, when $p>3$, more insights on the sources of mismodeling affecting $G$ can be obtained by conducting an ANOVA-like analysis where random sub-vectors of $\bm{X}$ are tested individually, from the largest to the smallest.
Without loss of generality, let $\bm{X}_q=(X_1,\dots,X_q)$ be the random collecting the first $q<p$ components of $\bm{X}$. Denote with $F_q$ the true cdf of $\bm{X}_q$ and let $G_q$ be its postulated cdf. Moreover, assume that the density of $G_q$ can be specified as
\begin{equation}
\label{C1}
g_{q}\bigl(\bm{x}_q\bigl)=\prod_{d=1}^q g_d\bigl(x_d|\bm{x}_{<d}\bigl)\quad\text{for all $d=1,\dots,q$.}
\end{equation}
Similarly to \eqref{jcd} and \eqref{jcd_rep}, we can then express the likelihood ratio of $\bm{X}_q$ on the $q$ dimensional unit cube via
\begin{equation}
\label{LRq}
d(\bm{u}_q)=\frac{f_q\bigl(\bm{Q}_q(\bm{u}_q)\bigl)}{g_q\bigl(\bm{Q}_q(\bm{u}_q)\bigl)}=\sum_{j_1\geq 0,\dots,j_q\geq 0}\theta_{j_1\dots j_q}T_{j_1\dots j_q}(\bm{u}_q),\quad\bm{u_q}\in[0,1]^q
\end{equation}
where $\bm{u}_q=\bigl(G_1(x_1),\dots,G_q(x_q|\bm{x}_{<q})\bigl)$, and thus $\bm{u}_q$ is a sub-vector of $\bm{u}=\bm{G}_R(\bm{x})$.
Whereas, similarly to \eqref{Sjs}, one can write the tensor basis functions $T_{j_1\dots j_q}$ as
\begin{equation}
\label{Sj2}
T_{j_1\dots j_q}(\bm{u}_q)=\prod_{d=1}^q T_{j_d}(u_d)=\prod_{d=1}^p T_{j_d}(u_d) \quad \text{ with } j_d=0, \text{ for all } d=q+1.
\end{equation}
The last equality follows from the fact that $T_0\bigl(G(x_d|\bm{x}_{<d})\bigl)=1$ for all $d=1,\dots,p$, and thus each $T_{j_1\dots j_q}(\bm{u}_q)=T_{j_1\dots j_q0\dots 0}(\bm{u})$. Consequently, the $\theta_{j_1\dots j_q}$ coefficients are equal to the theta $\theta_{j_1\dots j_p}$ whenever $j_d=0$, for all $d=q+1$.
As a result, we can easily perform inference for $\bm{X}_q$ by means of the estimators $\widehat{\theta}_k$ in \eqref{theta_est}, without the need of an entirely new estimation procedure.
Specifically, denote with $\mathcal{K}_q$ and $\mathcal{K}^*$ the subsets of $\mathcal{K}$ in \eqref{kappa}
\begin{align}
\label{kappaq}
\mathcal{K}_q&:=\biggl\{k=\{j_1\dots j_p\}\in \mathcal{K} \text{ with } j_d=0, \text{ for all } d=q+1,\dots, p\biggl\}\\
\label{kappaK}
\mathcal{K}^*&:=\biggl\{k=\{j_1\dots j_p\}\in \mathcal{K} \text{ with } (k)\leq K^*\biggl\}
\end{align}
of cardinality $|\mathcal{K}_q|= M_q=\prod_{d=1}^q(m_d+1)-1$ and $|\mathcal{K}^*|=K^*$. Recall that $K^*$ is the value minimizing either the AIC or BIC in \eqref{BIC}, and thus,
$\mathcal{K}^*$ collects all the $p-$tuple of indexes in $\mathcal{K}$ which have been ultimately selected when constructing the estimator $\widehat{d}_{(K^*)}$ and the deviance statistics $D_{(K^*)}$ in Corollaries \ref{naive} and \ref{naive2}. To test
\begin{equation}
\label{testQ}
H_0: G_q=F_q\quad\text{versus} \quad H_1: G_{q}\neq F_{q}
\end{equation}
we may consider the test statistics $D_q=n\sum_{k\in \mathcal{K}_q }\widehat{\theta}^2_k$ and proceed as in Theorem \ref{normality}.
Whereas, valid post-selection inference can be obtained as in Theorem \ref{anova_theo}.
\begin{theorem}
\label{anova_theo}
As $n\rightarrow \infty$, a valid post-selection bound for the p-value to test \eqref{testQ} is
\begin{equation}
\label{test3}
\text{p-value}_{q,adj}=P(\chi^2_{M_{q}}>D_{obs}),
\end{equation}
where $D_{obs}$ being the value of the test statistics
\begin{equation}
\label{Dq}
D^*_q=n\sum_{k\in \mathcal{K}_q\cap\mathcal{K}^* }\widehat{\theta}^2_k
\end{equation}
observed, $\mathcal{K}_q$ and $\mathcal{K}^*$ as in \eqref{kappaq} and \eqref{kappaK} and $\widehat{\theta}_k$ as in \eqref{theta_est}.
\end{theorem}
Theorem \ref{anova_theo} follows directly from \eqref{LRq} and \eqref{Sj2} and, in virtue of the orthogonality of the $T_k$ functions and condition \eqref{C1}.
\begin{remark}
Because of condition \eqref{C1}, Theorem \ref{anova_theo} holds only for random sub-vectors of $\bm{X}$ whose Rosenblatt transform $u_q$ includes all the conditioning, from the higher to the lower, necessary to recover $g_{q}\bigl(\bm{x}_q\bigl)$. To some extent, this condition can be seen as the iGOF counterpart of the marginality principle advocated by \citet{nelder} in the context of ANOVA, and which consists in taking into account of the hierarchy of the main effects and interactions in a given model.
\end{remark}
Similarly to the ANOVA, Theorem \ref{anova_theo} allows us to construct an iGOF-diagnostic table to identify the source of mismodeling for a given random vector $\bm{X}$ and its components. Below we show how this can be done in practice for the case of a $7$-dimensional random vector.
\emph{\textbf{Example II.}} We consider a sample of $n=5000$ observations from a random vector $\bm{X}=(X_1,\dots,X_7)$ with components distributed as summarized in the second column of Table \ref{mismodeling}. Table \ref{anova} collects the results obtained by applying Theorem \ref{anova_theo} to test the validity of the models specified for different sub-vectors of $\bm{X}$. The overall deviance test is reported in the first row and correctly reject the null model.
\begin{center}
\begin{table}[!h]
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|c|c|c|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\vspace{-0.1cm}
& & & \\
\textbf{Variable} & \textbf{True} ($F$) & \textbf{Hypothesized} ($G$) & \textbf{Correct} \\
\vspace{-0.1cm}
& & & \\
\hline
\vspace{-0.1cm}
& & & \\
$X_6|X_1,X_2,X_5$& Laplace$\Bigl[e^{0.03x_1+0.02x_2+0.01x_2^2+0.02x_5},1\Bigl]$& Laplace$\Big[e^{0.03x_1+0.02x_2+0.02x_5},1\Bigl]$& No \\
\vspace{-0.1cm}
&&&\\
$X_1,X_2,X_5$& $N\left[\left(\begin{array}{c}
10\\
15\\
11
\end{array}\right),\left(\begin{array}{ccc}
4 & 0.5 & 0\\
0.5 & 3 & 1\\
0 & 1 & 5
\end{array}\right)\right]$ & $N\left[\left(\begin{array}{c}
10\\
15\\
11
\end{array}\right),\left(\begin{array}{ccc}
4 & 0.5 & 0\\
0.5 & 3 & 1\\
0 & 1 & 5
\end{array}\right)\right]$ & Yes \\
\vspace{-0.1cm}
&&&\\
$X_4|X_3$& Exponential$\Bigl(\frac{1}{x_3}\Bigl)$ & Exponential$\Bigl(\frac{1}{x_3}\Bigl)$ & Yes \\
\vspace{-0.1cm}
&&&\\
$X_3$& Exponential$(1)$ & Exponential$(0.9)$ & No \\
\vspace{-0.1cm}
&&&\\
$X_7$& $T_3$ & Cauchy$(0,1)$ & No \\
&&&\\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{True and postulated model for Example II. The last column highlights where mismodeling occurs. }
\label{mismodeling}}
\end{table}
\end{center}
Similarly, the test in the second row, rejects the hypotheses that the vector $(X_1,X_2,X_5,X_6)$ is modelled correctly, and fails to rejects the model for $(X_1,X_2,X_5)$. This aspect is particularly important as it highlights that the mismodeling occurs only with respect to the conditional distribution of $X_6|X_1,X_2,X_5$. The tests in the fourth and fifth row show that the vector $(X_3,X_4)$ has been mismodeled and one source of mismodeling is the marginal of $X_3$. Ultimately, the test for $X_7$ also correctly rejects the null hypothesis of Cauchy distribution.
Table \ref{simul} collects the results of a simulation obtained by repeating the diagnostic analysis in Table \ref{anova} through a simulation of $B=10,000$ replicates, while considering different sample sizes. Even when the sample size considered is only $500$, the most prominent deviations are captured with probability one, whereas, the model for $(X_1,X_2,X_5)$ is never rejected. More issues arise in diagnosing mismodeling of $X_3$ and, consequently, $(X_3,X_4)$ for smaller samples. For instance, even when $n=1000$ the power of the procedure in detecting departures of $G_{X_3}$ from $F_{X_3}$ is only $\sim56\%$ and $\sim3\%$ for $(X_3,X_4)$. It has to be noted, however, that detecting mismodeling of $X_3$ is a particularly challenging task. As shown in Figure \ref{figexp}, the postulated and the true pdf of $X_3$ are very close one-another; this minor differences are further ``diluted'' when considering the joint distribution of $(X_3,X_4)$, since $X_4|X_3$ is correctly specified. Nevertheless, such minor deviations are detected with high power for larger sample sizes.
\begin{minipage}{\textwidth}
\begin{minipage}[b]{0.49\textwidth}
\centering
\fontsize{9}{9}\selectfont{
\centering
\begin{tabular}{|c|c|c|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
$\bm{X}_q$ & \textbf{df}& \textbf{(Adjusted)} \\
&&\textbf{ p-value } \\
\hline
$\bm{X}$& 16383 & $<10^{-130}$ \\
$(X_1,X_2,X_5,X_6)$& 256 & $<10^{-130}$ \\
$(X_1,X_2,X_5)$&63 &$1$ \\
$(X_3,X_4)$& 15 & $8.467\cdot 10^{-08}$ \\
$X_3$&3 &$1.525\cdot 10^{-13}$ \\
$X_7$& 3 & $2.599\cdot 10^{-122}$ \\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\captionof{table}{iGOF-diagnostic table. The third column reports the (post-selection adjusted) deviance p-values in \eqref{test3} with $\bm{X}_q$ specified as in the first column. The second column corresponds to the degrees of freedom used in the calculation of the p-value, namely, $M_{q}$. }
\label{anova}}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\textwidth}
\centering
\centering
\includegraphics[width=55mm ]{Exponential} \\
\captionof{figure}{Comparing the postulated (red dashed line) and the true model (green solid line) of $X_3$. }
\label{figexp}
\end{minipage}
\end{minipage}\\
\vspace{0.5cm}
\begin{table}[!h]
\fontsize{8}{8}\selectfont{
\centering
\begin{tabular}{|c|cccccc|}
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
& & & & & & \\
\multirow{2}{*}{$\bm{X}_q$} &\multicolumn{6}{c|}{\textbf{Sample size ($n$)}} \\
& & & & & & \\
&$500$ & $1000$ &$2000$&$3000$& $5000$ & $10,000$ \\
\hline
& & & & & & \\
$\bm{X}$& 1 & 1 & 1 & 1 & 1 & 1 \\
$(X_1,X_2,X_5,X_6)$& 1 & 1 & 1& 1 & 1 & 1 \\
$(X_1,X_2,X_5)$& 0 & 0 & 0 & 0 & 0 & 0 \\
$(X_3,X_4)$& 0.0069 & 0.0342 & 0.2384 & 0.5939& 0.9615 &1 \\
& ($\pm$0.0008) & ($\pm$0.0018)& ($\pm$0.0049) & ($\pm$0.0049) & ($\pm$ 0.0019) & \\
$X_3$ & 0.2360 &0.5560 & 0.9153 &0.9868 &0.9999 &1 \\
& ($\pm$0.0043)& ($\pm$0.0050) & ($\pm$0.0028) & ($\pm$0.0011)& ($\pm$ 0.0001) & \\
$X_7$& 1 &1 &1& 1& 1 &1 \\
& & & & & & \\
\noalign{\global\arrayrulewidth0.05cm}
\hline
\noalign{\global\arrayrulewidth0.05pt}
\end{tabular}
\caption{Performance of the iGOF-diagnostic analysis for different sample sizes. For values different from zero and one the Monte Carlo errors $(\pm SE)$ are also reported. The significance level considered is $\alpha=0.05$.}
\label{simul}}
\end{table}
\begin{figure*}[htb]
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{}c@{}c@{}}
\includegraphics[width=72mm ]{Fig3a}&\includegraphics[width=72mm ]{Fig3b} \\
\end{tabular*}
\caption{Simulated and approximated confidence regions for the Fermi LAT simulation. The left panel corresponds to the (post-selection) confidence regions and deviance p-value obtained via a simulation of size $B=10,000$. The right panel shows to the (post-selection adjusted) confidence regions and deviance p-value computed as in \eqref{pvalue} and \eqref{CIband}. Darker shades correspond to significant deviations of the estimated likelihood ratio above one. Lighter shades correspond to significant deviations below one.
}
\label{Fermi}
\end{figure*}
\section{A diagnosis of background mismodeling}
\label{bkg}
When conducting searches for new phenomena, mismodeling of the background distribution can dramatically compromise
the sensitivity of the experiment. Specifically, overestimating the background can increase the chances of false negatives. Whereas, underestimating the background may lead to claiming false discoveries. To illustrate how iGOF can be used to understand if and how the postulated background model have been misspecified, we consider a simulated observation by the Fermi Large Area Telescope (LAT) \cite{atwood} obtained with the \emph{gtobssim} package\footnote{\url{http://fermi.gsfc.nasa.gov/ssc/data/analysis/software}} and previously published in \citet{TOHM}. The simulation includes a
realistic representations of the instrumental noise of the detector and present backgrounds.
The region of interest corresponds to a disc in the sky of $30^\circ$ radius and centered at ($195$ RA, $28$ DEC), where RA and DEC are the coordinates in the sky. Here we assume that, despite the cosmic background is known to follow a uniform distribution over the search area, it is unclear if the instrumental error is effectively negligible, or if it has a prominent effect on the underlying distribution. Therefore, we set $G_{X_1X_2}$ to be the cdf of a uniform distribution with support $\mathcal{X}_1\times\mathcal{X}_2=[165,195]\times[28-\sqrt{30^2-(x-195)^2}, 28+\sqrt{30^2-(x-195)^2}]$ and we proceed by estimating the likelihood ratio via \eqref{jcd_est} over a sample of $n=68658$ observations. Specifically, we set $m_1=m_2=4$ and we select the components of $\widehat{\bm{\theta}}$ via the BIC criterion in \eqref{BIC}. The resulting estimate is
\begin{equation}
\label{Fermi_est}
\widehat{d}\bigl(G_{1}(x_1),G_{2}(x_2|x_1)\bigl)=1+0.022T_{1}\bigl(G_1(x_1)\bigl)-0.043T_{1}\bigl(G_2(x_2|x_1)\bigl)+0.041T_{2}\bigl(G_2(x_2|x_1)\bigl).
\end{equation}
In order to assess the significance of the deviations captured by \eqref{Fermi_est}, we compute both the confidence regions and deviance p-values via \eqref{CIband} and \eqref{pvalue}. The results are reported in the right panel of Figure \ref{Fermi}, whereas the left panel shows the confidence regions and deviance p-value obtained via simulation. Similarly to what we have observed for Example I (see Figure \ref{fig2}), despite the approximate confidence bands are more conservative, they still allow to capture the main departures from uniformity. Indeed, in both cases, we can see that the prominent deviations of the true underlying model from the postulated uniform distribution occur in proximity of low values of $X_2$. Whereas, at the center-left of the search area, the uniform model significantly underestimates the model inclusive of the instrumental error. Finally, it follows from \eqref{skewG_est} that an updated model background distribution which accounts for these deviations can be constructed as in \eqref{skewG_est} by simply multiplying the uniform pdf by the estimated likelihood ratio in \eqref{Fermi_est}.
\section{Extensions to the discrete case}
\label{discrete}
The methods discussed so far focus on the case where $F$ and $G$ are continuous. However, extensions to the discrete setting can be derived by rewriting the expansion in \eqref{jcd_rep} through an orthonormal set of functions suitable to model discrete data. This can be done, for instance, by means of the so-called ``LP\footnote{In the \emph{LP} acronym, the letter \emph{L} typically denotes nonparametric methods based on quantiles, whereas \emph{P} stands for polynomials \cite[Supp S1]{LPksamples}.} score functions'', recently introduced \citep[e.g.,][]{LPksamples} and which can be seen as a generalization of the Legendre polynomials valid in both the continuous and discrete setting.
Specifically, when $p=1$, a complete orthonormal basis of LP score functions in $L^2(G)$ can be specified by letting the first component to be $T_0\bigl[G(x)\bigl]=1$. Subsequent components $\{T_j\bigl[G(x)\bigl]\}_{j>0}$ are obtained by Gram-Schimidt orthonormalization of powers of
\begin{equation}
\label{T1}
T_1\bigl[G(x)\bigl]=\frac{G_{\text{mid}}(x)-E[G_{\text{mid}}(x)]}{\sqrt{V(G_{\text{mid}}(x))}}=\frac{G(x)-0.5p_{G}(x)-0.5}{\sqrt{[1-\sum_{x\in \mathcal{X}}p_{G}^3(x)]/12}},
\end{equation}
where $G_{\text{mid}}(x)=G(x)-0.5p_{G}(x)$ is the \emph{mid-distribution function}, which it has been shown in \citet{parzen2004} to have mean $0.5$ and variance $[1-\sum_{x\in \mathcal{X}}p_{G}^3(x)]/12$, with $\mathcal{X}$ being the set of distinct points in the support of $X$ and $p_{G}(x)=P(X=x)$ if $X\sim G$.
Therefore, $T_1\bigl[G(x)\bigl]$ is the standardized mid-distribution and orthonormality of the $T_j\bigl[G(x)]$ functions in $L^2(G)$ follows by the first equality in \eqref{T1} and by Gram-Schmidt process.
Notice that, for continuous $X$, $G_{\text{mid}}(x)=G(x)$ and $\sum_{x\in \mathcal{X}}p_{G}^3(x)=0$, consequently, the LP score functions reduce to normalized shifted Legendre polynomials. The latter are effectively the result of a Gram-Schmidt orthonormalization applied to powers of $G(x)$.
Whereas, the LP score functions are obtained by orthonormalizing powers of the standardized mid-distribution function with respect to the measure $G$.
Recall that, in our context, the cdfs $G_d$, $d=1,\dots,p$ are the conditional and marginal distribution functions specified in the Rosenblatt's transform $\bm{G}_R(\bm{x})$.
Hence, an orthonormal basis in $L^2\bigl(G_d)$ is $\Bigl\{T_{j_d}\bigl[G_d(\cdot|\cdot)\bigl]\Bigl\}_{j_d\geq 0}$ with $T_{0}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]=1$
and subsequent components
\begin{align}
\label{T0etc}
\quad T_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]&=\frac{\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]}{\bigl|\bigl|\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl|\bigl|_{G_d}}, \quad \text{for all $j_p\geq 1$, where}\\
\label{Tjd}
\text{\r{T}}_{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]&=T^{j_d}_{1}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\\
&-\sum_{k=1}^{{j_p}-1}\bigl<T^{{j_d}}_1\bigl[G_d(x_d|\bm{x}_{<d})\bigl],T_k\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl>_{G_d}T_k\bigl[G_d(x_d|\bm{x}_{<d})\bigl],\\
\bigl<T_1^{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl],&T_{k_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\bigl>_{G_d}=\\&\int T_1^{j_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]T_{k_d}\bigl[G_d(x_d|\bm{x}_{<d})\bigl]\text{d}G_d(x_d|\bm{x}_{<d})
\end{align}
and $||\cdot||_{G_d}=\sqrt{<\cdot,\cdot>_{G_d}}$.
When $p>1$ a suitable tensor basis in $L^2(G)$ can then be constructed as in Proposition \ref{Sjs}.
$\{T_{j_1,\dots,j_p}(\bm{u})\}_{j_1\dots j_p\geq 0}$. Orthonormality of the $T_{j_1,\dots,j_p}(\bm{u})$ score functions can be verified directly as shown in Section \ref{LPbasis} of the Supplementary Material.
\section{Discussion}
\label{conclusions}
This work proposes an informative approach to goodness-of-fit which connects exploratory and confirmatory data analysis to study multivariate distributions. By transforming the likelihood ratio on the unit cube, confidence regions can be constructed as in Corollary \ref{naive2} to identify regions of the supportwhere significant deviations occurs. While this approach is practical only for problems in at most three dimensions, in more dimensions a detailed diagnosis of mismodeling can be achieved by means of the iGOF-diagnostic analysis proposed in Section \ref{diagnostics}. These tools can be used to directly address Q1 in Section \ref{intro}.
For instance, given the panacea of theories available on the nature of dark matter, experimentalists aiming to detect it often face the dilemma of selecting which of the tens of theoretical models (mainly non-nested) available should be tested \citep[e.g.,][]{pat}. If one was to test it using the procedure discussed in this paper, even when a given model is rejected, it is possible to gain further insight on the shape of the departure of the true data distribution and the null model and ultimately use such information to ``rule out'' other models which would be inconsistent with such deviation.
Moreover, as we aimed for when formulating Q2 in Section \ref{intro}, the true probability function of the data can be estimated semi-parametrically via \eqref{skewG_est}, while assessing the validity of the model postulated by the scientists. Interestingly, the resulting estimate incorporates the knowledge carried by the hypothesized model and thus, it provides a data-driven update for it in the direction of the true distribution of the data.
Despite the usefulness of the methods presented here in applied settings, and in the physical sciences in particular (e.g., Section \ref{bkg}), they are not exempt from limitations. For instance, several problems in physics and astronomy, often involve no more than 8 or 10 dimensions and/or can be reduced to 2D planes \citep[e.g.,][]{aprile}. In this context, choosing $m_d$ equal to $3$ or $4$ for all $d=1,\dots,p$, is often sufficient to avoid overfitting and, eventually, lack of power by implementing adequate model selection strategies and for sufficiently large samples (see Sections \ref{inference} and \ref{diagnostics}). In more dimensions, however, the method suffers from the curse of dimensionality \citep[e.g.][]{friedman}, as the size of the LP tensor basis increases exponentially fast with $p$. In this context, a regularized solution could be particularly valuable \citep[see for instance][]{signoretto} when analyzing, for instance, data coming from large astronomical surveys such as the Large Synoptic Survey Telescope (LSST) survey \citep[e.g.,][]{tyson}.
Furthermore, the unitary representation of the likelihood ratio in \eqref{jcd} relies on the Rosenblatt transform and which can lead to different configurations of $\bm{U}$ and, potentially, different estimators. Despite this aspect would require adequate treatment on its own, it is worth noting that this problem is essentially the same arising in the context of vine copulas \citep[e.g.,][]{Nagler} and for which adequate model selection procedures exists \citep[e.g.,][]{panagiotelis,dissmann}.
Finally, the inferential procedures presented here extend classical smooth tests to the multivariate setting and allow us to visualize graphically the departure of $F$ from $G$ and study their substructures. Despite this article focuses on simple null hypothesis, that is, the postulated model is assumed to be fully specified,
classical results on smooth tests \citep[e.g.,][Sec 4.2.2.3 and 5.2.2.3 ]{thas} can be used to show to derive asymptotic tests in the parametric setting. Unfortunately, however, the asymptotic approximations are known to be rather slow in the parametric case. Therefore, in practical applications, when $G$ depends on unknown parameters it is recommended to perform inference by means of the parametric bootstrap and which has been shown by \citet{babu} to be consistent also in the multivariate setting.
|
2,877,628,090,835 | arxiv | \section{Introduction}
\label{sec:intro}
\subsection{Background}
\label{ssec:bgnd}
Sometimes constraints are maintained effortlessly, an example being $\nabla\cdot\mathbf{B}=0$ in electrodynamics which if initially true remains true, while alternatively in most cases dynamical equations must be modified to maintain constraints, an example being $\nabla\cdot\mathbf{v}=0$ in fluid mechanics. The need to apply constraints arises in a variety of contexts, ranging from gauge field theories \citep[e.g.][]{sundermeyer} to optimization and control \citep[e.g.][]{bloch}. A very common approach is to use the method of Lagrange multipliers, which is taught in standard physics curricula for imposing holonomic constraints in mechanical systems. Alternatively, \citet{dirac50}, in pursuit of his goal of quantizing gauge field theories, introduced a method that uses the Poisson bracket.
The purpose of the present article is to explore different methods for imposing the compressibility constraint in ideal (dissipation-free) fluid mechanics and its extension to magnetohydrodynamics (MHD), classical field theories intended for classical purposes. This endeavor is richer than might be expected because the different methods of constraint can be applied to the fluid in either the Lagrangian (material) description or the Eulerian (spatial) description, and the constraint methods have different manifestations in the Lagrangian (action principle) and Hamiltonian formulations. Although Lagrange's multiplier is widely appreciated, it is not so well known that he used it long ago for imposing the incompressibility constraint for a fluid in the Lagrangian variable description \citep{lagrange}. More recently, Dirac's method was applied for imposing incompressibility within the Eulerian variable description, first in \citet{turski99,turski01} and followed up in several works \citep{pjmLB09,pjmTC09,pjmCT12,pjmCT14,pjmCGBT13}. Given that a Lagrangian conservation law is not equivalent to an Eulerian conservation law, it remains to elucidate the interplay between the methods of constraint and the variables used for the description of the fluid. Thus we have three dichotomies: the Lagrangian vs.\ Eulerian fluid descriptions, Lagrange multiplier vs.\ Dirac constraint methods, and Lagrangian vs.\ Hamiltonian formalisms. It is the elucidation of the interplay between these, along with generalizing previous results, that is the present goal.
It is well known that a free particle with holonomic constraints, imposed by the method of Lagrange multipliers, is a geodesic flow. Indeed, Lagrange essentially observed this in \citet{lagrange} for the ideal fluid when he imposed the incompressibility constraint by his method. Lagrange did this in the Lagrangian description by imposing the constraint that the map from initial positions of fluid elements to their positions at time $t$ preserves volume, and he did this by the method of Lagrange multipliers. It is worth noting that Lagrange knew the Lagrange multiplier turns out to be the pressure, but he had trouble solving for it. Lagrange's calculation was placed in a geometrical setting by \citet{arnold-diffeo} (see also Appendix 2 of \cite{arnold-book}), where the constrained maps from the initial conditions were first referred to as volume preserving diffeomorphisms in this context. Given this background, in our investigation of the three dichotomies described above we emphasize geodesic flow.
For later use we record here the incompressible Euler equations for the case with constant density and the case where density is advected. The equations of motion, allowing for density advection, are given by
\bal
\frac{\partial \mathbf{v}}{\partial t} &= -\mathbf{v}\cdot\nabla\mathbf{v} -\frac1{\rho}\nabla p\,,
\label{momentum}
\\
& \nabla \cdot \mathbf{v}=0\,,
\label{solenoidal}
\\
\frac{\partial \rho}{\partial t} &= -\mathbf{v}\cdot\nabla\rho\,,
\eal
where $\mathbf{v}(\mathbf{x},t)$ is the velocity field, $\rho(\mathbf{x},t)$ is the mass density, $p(\mathbf{x},t)$ is the pressure, and $\mathbf{x}\in D$, the region occupied by the fluid. These equations are generally subject to the free-slip boundary condition $\mathbf{n}\cdot \mathbf{v}|_{\partial D}=0$, where $\mathbf{n}$ is normal to the boundary of $D$. The pressure field that enforces the constraint \eqref{solenoidal} is obtained by setting $\partial (\nabla\cdot \mathbf{v})\partial t=0$,
which implies
\begin{equation}
\Delta_\rho p:=\nabla\cdot\left(\frac1{\rho}\nabla p\right)=-\nabla \cdot (\mathbf{v}\cdot\nabla\bf v)\,.
\label{elliptic}
\end{equation}
For reasonable assumptions on $\rho$ and boundary conditions, \eqref{elliptic} is a well-posed elliptic equation \citep[see e.g.][] {evans}, so we can write
\begin{equation}
p= -\Delta_\rho^{-1}\nabla\cdot(\mathbf{v}\cdot\nabla\bf v)\,.
\label{presrho}
\end{equation}
For the case where $\rho$ is constant we have the usual Green's function expression
\begin{equation}
p(\mathbf{x},t)=- \, \rho \int \! d^3x' \, G(\mathbf{x},\mathbf{x}')\, \nabla'\! \cdot (\mathbf{v}'\cdot\nabla'\bf v')\,,
\label{pG}
\end{equation}
where $G$ is consistent with Neumann boundary conditions \citep{orszag86} and $\mathbf{v}'=\mathbf{v}(\mathbf{x}',t)$. Insertion of \eqref{pG} into \eqref{momentum} gives
\begin{equation}
\frac{\partial \mathbf{v}}{\partial t} = -\mathbf{v}\cdot\nabla\mathbf{v} +\nabla\int\! d^3x' \, G(\mathbf{x},\mathbf{x}')\, \nabla'\!\cdot (\mathbf{v}'\cdot\nabla'\bf v')\,,
\label{momentumCl}
\end{equation}
which is a closed system for $\mathbf{v}(\mathbf{x},t)$.
For MHD, equation \eqref{momentum} has the additional term $(\nabla\times\mathbf{B})\times \mathbf{B}/\rho$ added to the righthand side. Consequently for this model, the source of \eqref{presrho} is modified by the addition of this term to $-\mathbf{v}\cdot\nabla\bf v$.
\subsection{Overview}
\label{ssec:over}
Section \ref{sec:constraints} contains material that serves as a guide for navigating the more complicated calculations to follow. We first consider the various approaches to constraints in the finite-dimensional context in Sections \ref{ssec:lagrange}--\ref{ssec:holoD}. Section \ref{ssec:lagrange} briefly covers conventional material about holonomic constraints by Lagrange multipliers -- here the reader is reminded how the free particle with holonomic constraints amounts to geodesic flow. Section \ref{ssec:dirac} begins with the phase space action principle, whence the Dirac bracket for constraints is obtained by Lagrange's multiplier method, but with phase space constraints as opposed to the usual holonomic configuration space constraints used in conjunction with Hamilton's principle of mechanics, as described in Section \ref{ssec:lagrange}. Next, in Section \ref{ssec:holoD}, we compare the results of Sections \ref{ssec:lagrange} and \ref{ssec:holoD} and show how conventional holonomic constraints can be enforced by Dirac's method. Contrary to Lagrange's method, here we obtain explicit expressions, ones that do not appear in conventional treatments, for the Christoffel symbol and the normal force entirely in terms of the original Euclidean coordinates and constraints. Section \ref{sec:constraints} is completed with Section \ref{ssec:d+1}, where the previous ideas are revisited in the $d+1$ field theory context in {preparation} for the fluid and MHD calculations. Holonomic constraints, Dirac brackets, with local or nonlocal constraints, and geodesic flow are treated.
In Section \ref{sec:unconstrained} we first consider the compressible (unconstrained) fluid and MHD versions of Hamilton's variational principle, the principle of least action, with Lagrange's Lagrangian in the Lagrangian description. From this we obtain in Section \ref{ssec:HamDesc} the canonical Hamiltonian field theoretic form in the Lagrangian variable description, which is transformed in Section \ref{ssec:LtoE}, via the mapping from Lagrangian to Eulerian variables, to the noncanonical Eulerian form. Section \ref{sec:unconstrained} is completed by an in depth comparison of constants of motion in the Eulerian and Lagrangian descriptions, which surprisingly does not seem to appear in fluid mechanics or plasma physics textbooks. The material of this section is necessary for understanding the different manifestations of constraints in our dichotomies.
Section \ref{sec:dirac} begins with Section \ref{ssec:LagCon} that reviews Lagrange's original calculations. Because the incompressibility constraint he imposes is holonomic and there are no additional forces, his equations describe infinite-dimensional geodesics flow on volume preserving maps. The remaining portion of this section contains the most substantial calculations of the paper. In Section \ref{ssec:LDconTh} for the first time Dirac's theory is applied to enforce incompressibility in the Lagrangian variable description. This results in a new Dirac bracket that generates volume preserving flows. As in Section \ref{sssec:HCD}, which serves as a guide, the equations of motion generated by the bracket are explicit and contain only the constraints and original variables. Next, in Section \ref{ssec:EDcon}, a reduction from Lagrangian to Eulerian variables is made, resulting in a new Eulerian variable Poisson bracket that allows for density advection while preserving incompressibility. This was an heretofore unsolved problem. Section \ref{ssec:comparison} ties together the results of Sections \ref{ssec:LDconTh}, \ref{ssec:EDcon}, and \ref{ssec:CoM}. Here both the Eulerian and Lagrangian Dirac constraint theories are compared after they are evaluated on their respective constraints, simplifying their equations of motion. Because Lagrangian and Eulerian conservation laws are not identical, we see that there are differences. Section \ref{sec:dirac} concludes in Section \ref{ssec:AoI} with a discussion of the full algebra of invariants, that of the ten parameter Galilean group, for both the Lagrangian and Eulerian descriptions. In addition the Casimir invariants of the theories are discussed.
The paper concludes with Section \ref{sec:conclusion}, where we briefly summarize our results and speculate about future possibilities.
\section{Constraint methods}
\label{sec:constraints}
\subsection{Holonomic constraints by Lagrange's multiplier method}
\label{ssec:lagrange}
Of interest are systems with Lagrangians of the form $L(\dot{q}, q)$ where the overdot denotes time differentiation and $q=(q^1,q^2,\dots, q^N)$. Because nonautonomous systems could be included by appending an additional degree of freedom, explicit time dependence is not included in $L$.
Given the Lagrangian, the equations of motion are obtained according to Hamilton's principle by variation of the action
\begin{equation}
S[q]=\int_{t_0}^{t_1}\!\!dt\, L(\dot{q}, q)\,;
\label{HamPrin}
\end{equation}
i.e.
\begin{equation}
\delta S[q;\delta q]:= \frac{d}{d\epsilon} S[q +\epsilon \delta q]\Big|_{\epsilon=0}
=\int_{t_0}^{t_1}\!\!dt\left( \frac{d }{dt} \frac{\partial L}{\partial \dot{q}^i} - \frac{\partial L}{\partial {q}^i}\right)\delta q^i
=\int_{t_0}^{t_1}\!\!dt\, \frac{\delta S[q]}{\delta q^i(t)}\, \delta q^i= 0\,,
\end{equation}
for all variations $\delta q(t)$ satisfying $\delta q(t_0)=\delta q(t_1)=0$, implies Lagrange's equations of motion, i.e.
\begin{equation}
\frac{\delta S[q]}{\delta q^{i}(t)} = 0\quad \Rightarrow\quad \frac{d }{dt} \frac{\partial L}{\partial \dot{q}^i} - \frac{\partial L}{\partial {q}^i}=0\,, \qquad i=1,2,\dots, N\,.
\label{LagEOM}
\end{equation}
Holonomic constraints are real-valued functions of the form $C^A(q)$, $A=1,2,\dots, M$, which are desired to be constant on trajectories. Lagrange's method for implementing such constraints is to add them to the action and vary as follows:
\begin{equation}
\delta S_\lambda :=\delta (S + \lambda_A C^A)=0\,,
\label{Lagconstraint}
\end{equation}
yielding the equations of motion
\begin{equation}
\frac{\delta S_{\lambda}[q]}{\delta q^{i}(t)} = 0\quad \Rightarrow\quad
\frac{d }{dt} \frac{\partial L}{\partial \dot{q}^i} - \frac{\partial L}{\partial {q}^i}=\lambda_A\frac{\partial C^A}{\partial q^i}\,, \qquad i=1,2,\dots, N\,,
\label{LagForce}
\end{equation}
with the forces of constraint residing on the right-hand side of \eqref{LagForce}. Observe in \eqref{Lagconstraint} and \eqref{LagForce} repeated sum notation is implied for the index $A$. The $N$ equations of \eqref{LagForce} with the $M$ numerical values of the constraints $C^A(q)=C_0^A$, determine the $N+M$ unknowns $\{q^i\}$ and $\{\lambda_A\}$. In practice, because solving for the Lagrange multipliers can be difficult an alternative procedure, a example of which we describe in Section \ref{sssec:HG}, is used.
We will see in Section \ref{ssec:LagCon} that the field theoretic version of this method is how Lagrange implemented the incompressiblity constraint for fluid flow. For the purpose of illustration and in preparation for later development, we consider a finite-dimensional analog of Lagrange's treatment.
\subsubsection{Holonomic constraints and geodesic flow via Lagrange}
\label{sssec:HG}
Consider $N$ noninteracting bodies each of mass $m_i$ in the Eucledian configuration space $\mathbb{E}^{3N}$ with cartesian coordinates $\mathbf{q}_i=(q_{xi},q_{yi},q_{zi})$, {where as in Section \ref{ssec:lagrange} $i=1,2,\dots,N$, but our configuration space has dimension $3N$.} The Lagrangian for this system is given by the usual kinetic energy,
\begin{equation}
L= \sum_{i=1}^N \frac{m_i}{2} \, \dot{\mathbf{q}}_i\cdot \dot{\mathbf{q}}_i\,,
\label{fpl}
\end{equation}
with the usual ``dot" product. The Euler-Lagrange equations for this system, equations \eqref{LagEOM}, are the uninteresting system of $N$ free particles. As in Section \ref{ssec:lagrange} we constrain this system by adding constraints $C^A(\mathbf{q}_1,\mathbf{q}_2,\dots,\mathbf{q}_N)$, where again $A=1,2,\dots, M$, leading to the equations
\begin{equation}
m_i\ddot{\mathbf{q}}_i = \lambda_A \frac{\partial C^A}{\partial \mathbf{q}_i}\,.
\label{ELgeod}
\end{equation}
Instead of solving the $3N$ equations of \eqref{ELgeod} together with the $M$ numerical values of the constraints, in order to determine the unknowns $\mathbf{q}_i$ and $\lambda_A$, we recall the alternative procedure, which dates back to Lagrange (see e.g.\ Sec. IV of \citet{lagrange}) and has been taught to physics students for generations \citep[see e.g.][] {whittaker,corben}. With the alternative procedure one introduces generalized coordinates that account for the constraints, yielding a smaller system on the constraint manifold, one with the Lagrangian
\begin{equation}
L=\frac12 g_{\mu\nu}(q) \, \dot{q}^\mu \dot{q}^\nu\,, \hspace{1cm} \mu,\nu= 1,2\dots, 3N-M\,,
\label{laggeo}
\end{equation}
where
\begin{equation}
g_{\mu\nu}= \sum_{i=1}^Nm_i \frac{\partial \mathbf{q}_i}{\partial q^{\mu}}\cdot\frac{\partial \mathbf{q}_i}{\partial q^{\nu}}
\,.
\label{gdef}
\end{equation}
Then Lagrange's equations \eqref{LagEOM} for the Lagrangian \eqref{laggeo} are the usual equations for geodesic flow
\begin{equation}
\ddot{q}^\mu+\Gamma^\mu_{\alpha\beta}\, \dot{q}^\alpha\dot{q}^\beta=0\,,
\label{geoflow}
\end{equation}
where the Christoffel symbol is as usual
\begin{equation}
\Gamma^\mu_{\alpha\beta}=\frac12 g^{\mu\nu}\left(
\frac{g_{\nu\alpha}}{\partial q^\beta} + \frac{g_{\nu\beta}}{\partial q^\alpha} - \frac{g_{\alpha\beta}}{\partial q^\nu}
\right)
\,.
\label{Csymb}
\end{equation}
If the constraints had time dependence, then the procedure would have produced the Coriolis and centripetal forces, as is usually done in textbooks.
Thus, we arrive at the conclusion that free particle dynamics with time-independent holonomic constraints is geodesic flow.
\subsection{Dirac's bracket method}
\label{ssec:dirac}
So, a natural question to ask is ``How does one implement constraints in the Hamiltonian setting, where phase space constraints depend on both the configuration space coordinate $q$ and the canonical momentum $p$"? {(See e.g.\ \cite{sundermeyer,AKN} for a general treatment and \cite{moncrief} for a treatment in the context of the ideal fluid and relativity and a selection of earlier references.) } To this end we begin with the phase space action principle
\begin{equation}
S[q,p]=\int_{t_0}^{t_1}\!\!dt\, \left[ p_i \dot{q}^i - H(q,p)\right]\,,
\end{equation}
where again repeated sum notation is used for $i=1,2,\dots, N$. Independent variation of $S[q,p]$ with respect to $q$ and $p$, with $\delta q(t_0)=\delta q(t_1)=0$ and no conditions on $\delta p$, yields Hamilton's equations,
\begin{equation}
\dot{p}_i =-\frac{\partial H}{\partial q^i} \qquad \mathrm{and}\qquad
\dot{q}^i = \frac{\partial H}{\partial p_i} \,,
\label{finite-ham}
\end{equation}
or equivalently
\begin{equation}
\dot{z}^{\alpha}=\{z^\alpha,H\} \,,
\end{equation}
which is a rewrite of \eqref{finite-ham} in terms of the Poisson bracket on phase space functions $f$ and $g$,
\begin{equation}
\{ f,g\} =\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}- \frac{\partial g}{\partial q^i}\frac{\partial f}{\partial p_i}
= \frac{\partial f}{\partial z^\alpha}\mathbb{J}_c^{\alpha\beta}\frac{\partial g}{\partial z^\beta}\,,
\label{FDcanbkt}
\end{equation}
where in the second equality we have used $z=(q,p)$, so $\alpha,\beta=1,2,\dots, 2N$ and the cosymplectic form (Poisson matrix) is
\begin{equation}
\mathbb{J}_c = \left( \begin{array}{cc}
\mathbb{O}_N & \mathbb{I}_N
\\
-\mathbb{I}_N & \mathbb{O}_N \end{array} \right) \,,
\label{Jc}
\end{equation}
with $\mathbb{O}_N$ being an $N\times N$ block of zeros and $\mathbb{I}_N$ being the $N\times N$ identity.
Proceeding as in Section \ref{ssec:lagrange}, albeit with phase space constraints $D^a(q,p)$, $a=1,2,\dots, 2M< 2N$, we vary
\begin{equation}
S_\lambda[q,p]=\int_{t_0}^{t_1}\!\!dt\, \left[ p_i \dot{q}^i -H(q,p) + \lambda_a
D^a\right]\,,
\end{equation}
and obtain
\begin{equation}
\dot{p}_i =- \frac{\partial H}{\partial q^i}
+ \lambda_a \frac{\partial D^a}{\partial q^i}
\qquad \mathrm{and}\qquad
\dot{q}^i =\frac{\partial H}{\partial p_i}
- \lambda_a \frac{\partial D^a}{\partial p_i}\,.
\label{condyn}
\end{equation}
Next, enforcing $\dot D^a=0$ for all $a$, will ensure that the constraints stay put. Whence, differentiating the $D^a$ and using \eqref{condyn} yields
\begin{eqnarray}
\dot{D}^a&=& \frac{\partial D^a}{\partial q^i}{\dot q}^i +
\frac{\partial D^a}{\partial p_i}{\dot p}_i \nonumber\\
&=&\{D^a,H\} - \lambda_b \{D^a,D^b\} \equiv
0\,.
\label{Cdyn}
\end{eqnarray}
We assume $\mathbb{D}^{ab}:=\{D^a,D^b\}$ has an inverse, $\mathbb{D}^{-1}_{ab}$, which requires there be an even number of constraints, $a,b=1,2,\dots, 2M$, because odd antisymmetric matrices have zero determinant. Then upon solving
(\ref{Cdyn}) for $\lambda_b$ and inserting the result into \eqref{condyn} gives
\begin{equation}
\dot{z}^{\alpha}=\{z^\alpha,H\} - \mathbb{D}^{-1}_{ab}\{z^\alpha,D^{a}\}\{D^b,H\}\,.
\label{czdyn}
\end{equation}
From (\ref{czdyn}), we obtain a generalization of the Poisson bracket, the Dirac bracket,
\begin{equation}
\{f,g\}_*=\{f,g\} - \mathbb{D}^{-1}_{ab}\{f,D^{a}\}\{D^b,g\}\,.
\label{DB}
\end{equation}
which has the degeneracy property
\begin{equation}
\{f,D^a\}_*\equiv 0\ \,.
\label{DCas}
\end{equation}
for all functions $f$ and indices $a=1,2,\dots, 2M$.
The generation of the equations of motion via a Dirac bracket, i.e.
\begin{equation}
\dot{z}^{\alpha}=\{z^\alpha,H\}_*\,,
\label{evolv}
\end{equation}
which is equivalent to \eqref{czdyn}, has the advantage that the Lagrange multipliers $\lambda_A$ have been eliminated from the theory.
Note, although the above construction of the Dirac bracket is based on the canonical bracket of \eqref{FDcanbkt}, his construction results in a valid Poisson bracket if one starts from any valid Poisson bracket (cf.\ \eqref{eqn:PBgene} of Section \ref{ssec:d+1} and Section \ref{ssec:LtoE}), which need not have a Poisson matrix of the form of \eqref{Jc} \citep[see e.g.] []{pjmLB09}. We will use such a bracket in Section \ref{ssec:EDcon} when we apply constraints by Dirac's method in the Eulerian variable picture. Also note, for our purposes it is not necessary to describe primary vs.\ secondary constraints (although we use the latter), and the notions of weak vs.\ strong equality. We refer the reader to \citet{dirac50,sudarshan,HRT,sundermeyer} for treatment of these concepts.
\subsection{Holonomic constraints by Dirac's bracket method}
\label{ssec:holoD}
A connection between the approaches of Lagrange and Dirac can be made. From a set of Lagrangian constraints $C^A(q)$, where $A=1,2,\dots,M$, one can construct an additional $M$ constraints by differentiation,
\begin{equation}
\dot{C}^A=\frac{\partial C^A}{\partial q^i} \dot{q}^i= \frac{\partial C^A}{\partial q^i}\frac{\partial H}{\partial p_i}\,,
\end{equation}
where the second equality is possible if \eqref{LagEOM} possesses the Legendre transformation to the Hamiltonian form. In this way we obtain an even number of constraints
\begin{equation}
D^a(q,p)= \left(C^A(q)\,, \dot{C}^{A'}(q,p)\right)\,,
\label{LDcon}
\end{equation}
where $A= 1,2,\dots, M$, $A'= M+1, M+2,\dots, 2M$, and $a=1,2,\dots, 2M.$
With the constraints of \eqref{LDcon} the bracket $\mathbb{D}^{ab}= \{D^a,D^b\}$ needed to construct \eqref{DB} is easily obtained,
\begin{equation}
\mathbb{D}
= \left( \begin{array}{cc}
\mathbb{O}_M & \{C^A, \dot{C}^{B'}\}
\\
\{\dot{C}^{A'},C^{B}\} & \{\dot{C}^{A'},\dot{C}^{B'}\} \end{array} \right) =:
\left( \begin{array}{cc}
\,\mathbb{O}_M & \mathbb{S}
\\
\!-\mathbb{S} & \mathbb{A} \end{array} \right)\,,
\label{bbD}
\end{equation}
where $\mathbb{O}_M$ is an $M\times M$ block of zeros and $ \mathbb{S}$ is the following $M\times M$ symmetric matrix with elements
\begin{equation}
\mathbb{S}^{AB}:= \{C^A, \dot{C}^{B}\} =
\frac{\partial^2 H}{\partial p_i \partial p_j}\, \frac{\partial C^A}{\partial q^i} \frac{\partial C^B}{\partial q^j}
\,,
\label{DAB}
\end{equation}
and $\mathbb{A}$ is the following $M\times M$ antisymmetric matrix with elements
\bal
\mathbb{A}^{AB}&:= \{\dot{C}^{A'},\dot{C}^{B'}\}
\label{D2D2}\\
&=\frac{\partial^2 H}{\partial p_i \partial p_k}\left[
\frac{\partial^2 H}{\partial q^i \partial p_j}
\left(
\frac{\partial C^A}{\partial q^j} \frac{\partial C^B}{\partial q^k} - \frac{\partial C^B}{\partial q^j} \frac{\partial C^A}{\partial q^k}
\right)
+
\frac{\partial H}{\partial p_j}
\left(
\frac{\partial^2 C^A}{\partial q^i \partial q^j} \frac{\partial C^B}{\partial q^k} - \frac{\partial^2 C^B}{\partial q^i \partial q^j} \frac{\partial C^A}{\partial q^k}
\right)
\right]\,.
\nonumber
\eal
Assuming the existence of $\mathbb{D}^{-1}$, the $2M\times 2M $ inverse of \eqref{bbD}, the Dirac bracket of \eqref{DB} can be constructed. A necessary and sufficient condition for the existence of this inverse is that det\,$\mathbb{S}\neq 0$, and when this is the case the inverse is given by
\begin{equation}
\mathbb{D}^{-1}
=
\left( \begin{array}{cc}
\mathbb{S}^{-1}\! \mathbb{A}\mathbb{S}^{-1} & - \,\mathbb{S}^{-1}
\\
\\
\, \mathbb{S}^{-1} & \mathbb{O}_M \end{array} \right)\,.
\label{bbDI}
\end{equation}
Because of the block diagonal structure of \eqref{bbDI}, the Dirac bracket \eqref{DB} becomes
\begin{equation}
\{f,g\}_*=\{f,g\} + \mathbb{S}^{-1}_{AB}\left(
\{f,C^{A}\}\{\dot{C}^{B},g\} -\{g,C^{A}\}\{\dot{C}^{B},f\}
\right) + \mathbb{S}^{-1}_{AC}\, \mathbb{A}^{CD}\, \mathbb{S}^{-1}_{DB}\, \{f,{C}^{A}\}\{{C}^{B},g\}\,,
\label{DBbb}
\end{equation}
which has the form
\bal
\{f,g\}_*&=\{f,g\} - (\mathbb{P}_\perp)^i_j\left( \frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_j} - \frac{\partial g}{\partial q^i}\frac{\partial f}{\partial p_j}\right)
+ \mathbb{Q}^{ij} \, \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial p_j}
\nonumber\\
&= \frac{\partial f}{\partial q^i} \, \mathbb{P}^i_j\frac{\partial g}{\partial p_j} - \frac{\partial g}{\partial q^i}\, \mathbb{P}^i_j\frac{\partial f}{\partial p_j}
+ \mathbb{Q}^{ij}\, \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial p_j}
\,,
\label{DBbbP}
\eal
where the matrices $\mathbb{P}=I-\mathbb{P}_{\perp}$, with
\begin{equation}
(\mathbb{P}_{\perp})_j^i = \mathbb{S}^{-1}_{AB} \, \frac{\partial^2 H}{\partial p_i \partial p_k}
\frac{\partial C^A}{\partial q^j} \frac{\partial C^B}{\partial q^k} \,,
\end{equation}
and $\mathbb{Q}$, a complicated expression that we will not record, are crafted using the constraints and Hamiltonian so as to make $\{f,g\}_*$ preserve the constraints.
The equations of motion that follow from \eqref{DBbb} are
\bal
\dot{q}^{\ell}&=\{q^{\ell},H\}_*=\frac{\partial H}{\partial p_\ell}+ \mathbb{S}^{-1}_{AB}\left(
\{q^{\ell},C^{A}\}\{\dot{C}^{B},H\} -\{H,C^{A}\}\{\dot{C}^{B},q^{\ell}\}
\right)
\nonumber\\
&\hspace{3cm} + \mathbb{S}^{-1}_{AC}\, \mathbb{A}^{CD}\, \mathbb{S}^{-1}_{DB}\, \{q^{\ell},{C}^{A}\}\{{C}^{B},H\}\,,
\label{eomq}\\
\dot{p}_{\ell}&=\{p_{\ell},H\}_*=-\frac{\partial H}{\partial q^\ell}+ \mathbb{S}^{-1}_{AB}\left(
\{p_{\ell},C^{A}\}\{\dot{C}^{B},H\} -\{H,C^{A}\}\{\dot{C}^{B},p_{\ell}\}
\right)
\nonumber\\
&\hspace{3cm} + \mathbb{S}^{-1}_{AC}\, \mathbb{A}^{CD}\,\mathbb{S}^{-1}_{DB}\, \{p_{\ell},{C}^{A}\}\{{C}^{B},H\}\,.
\label{eomp}
\eal
Given the Dirac bracket associated with the $\mathbb{D}$ of \eqref{DAB}, dynamics that enforces the constraints takes the form of \eqref{evolv}. Any system generated by this bracket will enforce Lagrange's holonomic constraints; however, only initial conditions compatible with
\begin{equation}
D^a\equiv 0\,, \qquad \forall \qquad a=M+1, M+2, \dots, 2M\,,
\end{equation}
or equivalently
\begin{equation}
\dot{C}^A= \frac{\partial C^A}{\partial q^i}\frac{\partial H}{\partial p_i}=\{C^A,H\}\equiv 0\,, \qquad \forall \qquad A=1, 2, \dots, M\,,
\label{dotcon}
\end{equation}
will correspond to the system with holonomic constraints. Using \eqref{dotcon} and $\{q^{\ell},C^{A}\}\equiv 0$, \eqref{eomq} and \eqref{eomp} reduce to
\bal
\dot{q}^{\ell}&=\{q^{\ell},H\}_*=\frac{\partial H}{\partial p_\ell}\,,
\label{eomqr}\\
\dot{p}_{\ell}&=\{p_{\ell},H\}_*=-\frac{\partial H}{\partial q^\ell}+ \mathbb{S}^{-1}_{AB}
\{p_{\ell},C^{A}\}\{\dot{C}^{B},H\} \,,
\label{eompr}
\eal
where
\begin{equation}
\{\dot{C}^{B},H\}=\left(\frac{\partial^2 H}{\partial q^i\partial p_j}\frac{\partial C^B}{\partial q^j} + \frac{\partial H}{\partial p_j}\frac{\partial^2 C^B}{\partial q^iq^j}\right) \frac{\partial H}{\partial p_i}
- \frac{\partial C^B}{\partial q^i} \frac{\partial^2 H}{\partial p_i\partial p_j} \frac{\partial H}{\partial q^j}\,.
\label{dotCH}
\end{equation}
Thus the Dirac bracket approach gives a relatively simple system for enforcing holonomic constraints. It can be shown directly that if initially $\dot{C}^A$ vanishes, then the system of \eqref{eomqr} and \eqref{eompr} will keep it so for all time.
\subsubsection{Holonomic constraints and geodesic flow via Dirac}
\label{sssec:HCD}
Let us now consider again the geodesic flow problem of Section \ref{sssec:HG}: the $N$ degree-of-freedom free particle system with holonomic constraints, but this time within the framework of Dirac bracket theory. For this problem the unconstrained configuration space is the Euclidean space $\mathbb{E}^{3N}$ and we will denote by $\mathcal{Q}$ the constraint manifold within $\mathbb{E}^{3N}$ defined by the constancy of the constraints $C^A$.
The Lagrangian of \eqref{fpl} is easily Legendre transformed to the free particle Hamiltonian
\begin{equation}
H= \sum_{i=1}^N \frac1{2m_i} \, {\mathbf{p}}_i\cdot {\mathbf{p}}_i\,,
\label{fph}
\end{equation}
where $\mathbf{p}_i=m_i\dot{\mathbf{q}}_i$. For this example the constraints of \eqref{LDcon} take the form
\begin{equation}
D^a= \left(C^A(\mathbf{q}_1,\mathbf{q}_2,\dots,\mathbf{q}_N),
\frac{\partial C^{A'}(\mathbf{q}_1,\mathbf{q}_2,\dots,\mathbf{q}_N)}{\partial \mathbf{q}_i}
\cdot \frac{\mathbf{p}_i}{m_i}
\right)\,,
\end{equation}
the $M\times M$ matrix $\mathbb{S}$ has elements
\begin{equation}
\mathbb{S}^{AB} = \sum_{i=1}^N\, \frac1{m_i}
\frac{\partial C^A}{\partial \mathbf{q}_i} \cdot \frac{\partial C^B}{\partial \mathbf{q}_i}
\,,
\label{fpDAB}
\end{equation}
and the $M\times M$ matrix $\mathbb{A}$ is
\begin{equation}
\mathbb{A}^{AB} = \sum_{i,j=1}^N\, \frac1{m_i m_j} \,\mathbf{p}_i \cdot
\left[
\frac{\partial^2 C^A}{\partial \mathbf{q}_i \partial \mathbf{q}_j}\cdot \frac{\partial C^B}{\partial \mathbf{q}_j}
- \frac{\partial^2 C^B}{\partial \mathbf{q}_i \partial \mathbf{q}_j}\cdot \frac{\partial C^A}{\partial \mathbf{q}_j}
\right]\,.
\end{equation}
The Dirac bracket analogous to \eqref{DBbbP} is
\begin{equation}
\{f,g\}_*= \sum^N_{ij=1}\,\left[ \frac{\partial f}{\partial \mathbf{q}_i} \cdot \overset\leftrightarrow{\mathbb{P}}_{ij} \cdot \frac{\partial g}{\partial \mathbf{p}_j}
- \frac{\partial g}{\partial \mathbf{q}_i} \cdot \overset\leftrightarrow{\mathbb{P}}_{ij} \cdot \frac{\partial f}{\partial \mathbf{p}_j}
+ \frac{\partial f}{\partial \mathbf{p}_i}\cdot \overset\leftrightarrow{\mathbb{Q}}_{ij} \cdot \frac{\partial g}{\partial \mathbf{p}_j}
\right]\,,
\label{geoBkt}
\end{equation}
where
$\overset\leftrightarrow{\mathbb{P}}_{ij}
= \overset\leftrightarrow{\mathbb{I}}_{ij} - \overset\leftrightarrow{\mathbb{P}}_{\perp\, ij}$ with the tensors
\bal
\overset\leftrightarrow{\mathbb{P}}_{\perp\, ij}&:=
\sum^N_{k=1}\,\mathbb{S}^{-1}_{AB}\, \frac{\partial^2 H}{\partial \mathbf{p}_i \partial \mathbf{p}_k} \cdot \frac{\partial C^B}{\partial \mathbf{q}_k}
\frac{\partial C^A}{\partial \mathbf{q}_j} = \mathbb{S}^{-1}_{AB}\,\frac1{m_i} \frac{\partial C^B}{\partial \mathbf{q}_i} \frac{\partial C^A}{\partial \mathbf{q}_j}\,,
\\
\overset\leftrightarrow{\mathbb{Q}}_{\, ij}&:= \sum^N_{k=1}\,\mathbb{S}^{-1}_{AB}\,
\left[
\frac{\partial C^A}{\partial \mathbf{q}_j}\, \frac{\mathbf{p}_k}{m_k} \cdot \frac{\partial^2 C^B}{\partial \mathbf{q}_k \partial \mathbf{q}_i}
- \frac{\partial C^A}{\partial \mathbf{q}_i} \, \frac{\mathbf{p}_k}{m_k} \cdot \frac{\partial^2 C^B}{\partial \mathbf{q}_k \partial \mathbf{q}_j}
\right]
+ \mathbb{S}^{-1}_{AC}\, \mathbb{A}^{CD}\, \mathbb{S}^{-1}_{DB}\, \frac{\partial C^{A}}{\partial \mathbf{q}_i} \frac{\partial C^B}{\partial \mathbf{q}_j}
\\
&=: \overset\leftrightarrow{\mathbb{T}}_{ij} - \overset\leftrightarrow{\mathbb{T}}_{ji}
+\overset\leftrightarrow{\mathbb{A}}_{ij} \,,
\label{TA}
\eal
where $\overset\leftrightarrow{\mathbb{A}}_{ij}$ is the term with
$\mathbb{S}^{-1}_{AC} \mathbb{A}^{CD} \mathbb{S}^{-1}_{DB}$.
Observe $\overset\leftrightarrow{\mathbb{A}}_{ij}= -\overset\leftrightarrow{\mathbb{A}}_{ji}$ because $\mathbb{A}^{CD}=- \mathbb{A}^{DC}$ and
\bal
\sum_{k=1}^N\, \overset\leftrightarrow{\mathbb{P}}_{\perp\, ik} \cdot \overset\leftrightarrow{\mathbb{P}}_{\perp\, kj}
&= \sum_{k=1}^N\,\left(
\mathbb{S}^{-1}_{AB}\,\frac1{m_i} \frac{\partial C^B}{\partial \mathbf{q}_i} \frac{\partial C^A}{\partial \mathbf{q}_k}
\right)
\cdot
\left(
\mathbb{S}^{-1}_{A'B'}\,
\frac1{m_k} \frac{\partial C^{B'}}{\partial \mathbf{q}_k} \frac{\partial C^{A'}}{\partial \mathbf{q}_j}
\right)
\nonumber\\
&= \left(
\mathbb{S}^{-1}_{AB}\,\frac1{m_i} \frac{\partial C^B}{\partial \mathbf{q}_i}\right) \mathbb{S}^{-1}_{A'B'}\, \left( \sum_{k=1}^N\, \frac{\partial C^A}{\partial \mathbf{q}_k}
\cdot
\frac1{m_k} \frac{\partial C^{B'}}{\partial \mathbf{q}_k}\right)
\frac{\partial C^{A'}}{\partial \mathbf{q}_j}
\nonumber\\
&= \left(
\mathbb{S}^{-1}_{AB}\,\frac1{m_i} \frac{\partial C^B}{\partial \mathbf{q}_i}\right) \mathbb{S}^{-1}_{A'B'}\, \mathbb{S}^{A B'}
\frac{\partial C^{A'}}{\partial \mathbf{q}_j} = \overset\leftrightarrow{\mathbb{P}}_{\perp\, ij} \,.
\eal
Also observe for the Hamiltonian of \eqref{fph}
\bal
\sum^N_{j=1}\, \overset\leftrightarrow{\mathbb{P}}_{\perp\, ij} \cdot \frac{\partial H}{\partial \mathbf{p}_j}&=
\sum^N_{j=1}\, \overset\leftrightarrow{\mathbb{P}}_{\perp\, ij} \cdot\frac{\mathbf{p}_j}{m_j}
\equiv 0
\label{finP}\,,
\\
\sum^N_{j=1}\, \frac{\partial H}{\partial \mathbf{p}_j}\cdot \overset\leftrightarrow{\mathbb{T}}_{ij} &=
\sum^N_{j=1}\, \frac{\mathbf{p}_j}{m_j} \cdot \overset\leftrightarrow{\mathbb{T}}_{ij}
\equiv 0 \,,
\label{finT}
\\
\sum^N_{j=1}\, \overset\leftrightarrow{\mathbb{A}}_ {ij} \cdot \frac{\partial H}{\partial \mathbf{p}_j}
&= -\sum^N_{j=1}\, \overset\leftrightarrow{\mathbb{A}}_ {ji} \cdot \frac{\partial H}{\partial \mathbf{p}_j} \equiv 0\,,
\label{finA}
\eal
when evaluated on the constraint $\dot{C}^{A,B}=0$, while
\begin{equation}
\sum^N_{j=1}\, \frac{\partial H}{\partial \mathbf{p}_j}\cdot \overset\leftrightarrow{\mathbb{P}}_{\perp\, ji} \neq 0
\quad \mathrm{and} \quad
\sum^N_{i=1}\, \frac{\partial H}{\partial \mathbf{p}_i}\cdot \overset\leftrightarrow{\mathbb{T}}_{ij} \neq 0\,,
\label{finPTneq}
\end{equation}
when evaluated on the constraint $\dot{C}^{A,B}=0$.
Thus, the bracket of \eqref{geoBkt} yields the equations of motion
\bal
\dot{\mathbf{q}}_i&=\{\mathbf{q}_i,H\}_*=\frac{\partial H}{\partial \mathbf{p}_i}=\frac{\mathbf{p}_i}{m_i}\,,
\label{eomqrg}\\
\dot{\mathbf{p}}_{i}&
= - \frac{\partial C^A}{\partial \mathbf{q}_i} \mathbb{S}^{-1}_{AB}
\sum_{j,k=1}^N \frac{\mathbf{p}_j}{m_j}\cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot\frac{\mathbf{p}_k}{m_k} \,,
\label{eomprg}
\eal
or
\begin{equation}
\ddot{\mathbf{q}}_i=- \frac1{m_i} \frac{\partial C^A}{\partial \mathbf{q}_i}
\mathbb{S}^{-1}_{AB}
\sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k \,
= - \sum_{j,k=1}^N\dot{\mathbf{q}}_j\cdot \widehat{\boldsymbol{\Gamma}}_{i,jk}\cdot \dot{\mathbf{q}}_k \,,
\label{finalD}
\end{equation}
where
\begin{equation}
\widehat{\boldsymbol{\Gamma}}_{i,jk}:=\frac1{m_i}\frac{\partial C^A}{\partial \mathbf{q}_i}\otimes
\mathbb{S}^{-1}_{AB} \,
\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \,,
\label{PCS}
\end{equation}
is used to represent the normal force.
Observe, \eqref{finalD} has two essential features: as noted, its righthand side is a normal force that projects to the constraint manifold defined by the constraints $C^A$ and within the constraint manifold it describes a geodesic flow, all done in terms of the original Euclidean space coordinates where the initial conditions place the flow on $\mathcal{Q}$ by setting the values $C^A$ for all $A=1,2,\dots, M$. We will show this explicitly.
First, because the components of vectors normal to $\mathcal{Q}$ are given by $\partial C^A/\partial \mathbf{q}_i$ for $A=1,2,\dots, M$, this prefactor on the righthand side of \eqref{finalD} projects as expected. Upon comparing \eqref{finalD} with \eqref{ELgeod} we conclude that the coefficient of this prefactor must be the Lagrange multipliers, i.e.,
\begin{equation}
\lambda_A= - \mathbb{S}^{-1}_{AB}
\sum_{k,j=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k
\,.
\end{equation}
Thus, we see that Dirac's procedure explicitly solves for the Lagrange multiplier.
Second, to uncover the geodesic flow we can proceed as usual by projecting explicitly onto $\mathcal{Q}$. To this end
we consider the transformation between the Euclidean configuration space $\mathbb{E}^{3N}$ coordinates
\begin{equation}
(\mathbf{q}_1, \mathbf{q}_2, \dots,\mathbf{q}_i,\dots, \mathbf{q}_N) \,, \qquad\mathrm{where} \qquad i=1,2,\dots, N
\end{equation}
and another set of coordinates
\begin{equation}
(q^1,q^2, \dots,q^a,\dots,q^{3N}) \,, \qquad\mathrm{where} \qquad a=1,2,\dots, 3N\,,
\label{untailored}
\end{equation}
which we tailor as follows:
\begin{equation}
(q^1,q^2, \dots,q^\alpha\dots, q^n, C^1,C^2,\dots, C^A,\dots, C^{M})\,,
\end{equation}
where $\alpha=1,2,\dots, n$, $A=1,2,\dots, M$, and $n= 3N-M$. Here we have chosen $q^{n+A}=C^A$ and $n$ is the actual number of degrees of freedom on the constraint surface $\mathcal{Q}$. We can freely transform back and forth between the two coordinates, i.e.
\begin{equation}
(\mathbf{q}_1, \mathbf{q}_2, \dots,\mathbf{q}_i,\dots, \mathbf{q}_N) \longleftrightarrow (q^1,q^2, \dots,q^a,\dots,q^{3N}) \,.
\end{equation}
Note, the choice $q^{n+A}=C^A$ could be replaced by $q^{n+A}=f^A(C^1,C^2, \dots, C^M)$ for arbitrary independent $f^A$, but we assume the original $C^A$ are optimal. Because $q^\alpha$ are coordinates within $\mathcal{Q}$, tangent vectors to $\mathcal{Q}$ have the components $\partial \mathbf{q}_i/\partial q^\alpha$, and there is one for each $\alpha =1,2,\dots, n$. The pairing of the normals with tangents is expressed by
\begin{equation}
\sum_{i=1}^N \frac{\partial \mathbf{q}_i}{\partial q^\alpha}\cdot \frac{\partial C^A}{\partial \mathbf{q}_i} =0\,,
\qquad \alpha=1,2,\dots,n;\ A=1,2,\dots, M\,.
\end{equation}
Let us now consider an alternative procedure that the Dirac constraint method provides. Proceeding directly we calculate
\begin{equation}
\dot{q}^a= \sum_{i=1}^N \frac{\partial q^a}{\partial \mathbf{q}_i}\cdot \dot{\mathbf{q}}_i\,.
\end{equation}
Observe that on $\mathbb{E}^{3N}$ the matrix ${\partial q^a}/{\partial \mathbf{q}_i}$ is invertible and the full metric tensor and its inverse in the new coordinates are given as follows:
\begin{equation}
g^{ab}=\sum_{i=1}^N\frac1{m_i} \frac{\partial q^a}{\partial \mathbf{q}_i}\cdot\frac{\partial q^b}{\partial \mathbf{q}_i}
\qquad\mathrm{and}\qquad
g_{ab}=\sum_{i=1}^N {m_i} \frac{\partial \mathbf{q}_i}{\partial q^a}\cdot \frac{\partial \mathbf{q}_i}{\partial q^b}\,.
\label{totalg}
\end{equation}
The metric tensor on $\mathcal{Q}$ of \eqref{gdef} is obtained by restricting $g_{ab}$ to $a,b\leq n$ and $g^{\alpha\beta}$ is obtained by inverting $g_{\alpha\beta}$ and not by restricting $g_{ab}$.
Proceeding by differentiating again we obtain
\begin{equation}
\ddot{q}^a= \sum_{i=1}^N \frac{\partial q^a}{\partial \mathbf{q}_i}\cdot \ddot{\mathbf{q}}_i +
\sum_{i,j=1}^N \dot{\mathbf{q}}_i\cdot \frac{\partial^2 q^a}{\partial \mathbf{q}_i \partial \mathbf{q}_j}\cdot \dot{\mathbf{q}}_j\,, \qquad a=1,2,\dots, 3N.
\label{ddqa}
\end{equation}
Now inserting \eqref{finalD} into \eqref{ddqa} gives
\begin{equation}
\ddot{q}^a= - \sum_{i=1}^N \frac1{m_i}\frac{\partial q^a}{\partial \mathbf{q}_i}\cdot \frac{\partial C^A}{\partial \mathbf{q}_i}
\, g_{AB}
\sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k
+
\sum_{i,j=1}^N \dot{\mathbf{q}}_j\cdot \frac{\partial^2 q^a}{\partial \mathbf{q}_i \partial \mathbf{q}_j}\cdot \dot{\mathbf{q}}_i\,,
\label{ddqaa}
\end{equation}
where we have recognized that
\begin{equation}
g^{AB}= \mathbb{S}^{AB} =
\sum_{i=1}^N\, \frac1{m_i}
\frac{\partial C^A}{\partial \mathbf{q}_i} \cdot \frac{\partial C^B}{\partial \mathbf{q}_i}
\end{equation}
and, as was necessary for the workability of the Dirac bracket constraint theory, $g_{AB}= \mathbb{S}^{-1}_{AB}$ must exist. This quantity is obtained by inverting $\mathbb{S}^{AB}$ and not by restricting $g^{ab}$.
Equation \eqref{ddqaa} is an expression for the full system on $\mathbb{E}^{3N}$. However, for $a>n$, we know
$\ddot{q}^a=\ddot{C}^A=0$, so the two terms of \eqref{ddqaa} should cancel. To see this, in the first term of \eqref{ddqaa} we set $q^a=C^C$ and observe that this first term becomes
\bal
- \sum_{i=1}^N \frac1{m_i}\frac{\partial C^C}{\partial \mathbf{q}_i}\cdot \frac{\partial C^A}{\partial \mathbf{q}_i}
\, g_{AB}
\sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k
&= - g^{CA}\, g_{AB}
\sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k
\nonumber\\
&= - \sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^C}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k\,.
\eal
Now, for $a\leq n$, say $\alpha$, the righthand side gives a Christoffel symbol expression for the geodesic flow; viz.
\bal
\ddot{q}^\alpha&= - \sum_{i=1}^N \frac1{m_i}\frac{\partial q^\alpha}{\partial \mathbf{q}_i}\cdot \frac{\partial C^A}{\partial \mathbf{q}_i}
\, g_{AB}
\sum_{j,k=1}^N \dot{\mathbf{q}}_j \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \dot{\mathbf{q}}_k
+
\sum_{j,k=1}^N \dot{\mathbf{q}}_j\cdot \frac{\partial^2 q^\alpha}{\partial \mathbf{q}_j \partial \mathbf{q}_k}\cdot \dot{\mathbf{q}}_k
\nonumber \\
&=-\Gamma^\alpha_{\mu\nu}\, \dot{q}^\mu \dot{q}^\nu
\,,
\label{ddqaa2}
\eal
where
\begin{equation}
\Gamma^\alpha_{\mu\nu} =
\sum_{i=1}^N \frac1{m_i}\frac{\partial q^\alpha}{\partial \mathbf{q}_i}\cdot \frac{\partial C^A}{\partial \mathbf{q}_i}
\, \mathbb{S}^{-1}_{AB}
\sum_{j,k=1}^N \frac{\partial {\mathbf{q}}_j}{\partial q^\mu} \cdot\frac{\partial^2 C^B}{\partial \mathbf{q}_j\mathbf{q}_k} \cdot \frac{\partial {\mathbf{q}}_k}{\partial q^\nu}
+
\sum_{j,k=1}^N \frac{\partial {\mathbf{q}}_j}{\partial q^\mu}
\cdot \frac{\partial^2 q^\alpha}{\partial \mathbf{q}_j \partial \mathbf{q}_k}\cdot \frac{\partial {\mathbf{q}}_k}{\partial q^\nu}
\label{newGA}
\end{equation}
is an expression for the Christoffel symbol in terms of the original Euclidean coordinates, the constraints, and the choice of coordinates on $\mathcal{Q}$.
Using \eqref{newGA} one can calculate an analogous expression for the Riemann curvature tensor on $\mathcal{Q}$ from the usual expression
\begin{equation}
R^\alpha_{\beta\gamma\delta} = \frac{\partial \Gamma^\alpha_{\delta\beta}}{\partial q^\gamma} - \frac{\partial \Gamma^\alpha_{\gamma\beta}}{\partial q^\delta}
+ \Gamma^\alpha_{\gamma \lambda} \Gamma^\lambda_{\delta \beta} - \Gamma^\alpha_{\delta \lambda} \Gamma^\lambda_{\gamma\beta} \,,
\end{equation}
using $\partial /\partial q^\gamma= \sum_i ( \partial \mathbf{q}_i/\partial q^\gamma)\cdot \partial /\partial \mathbf{q}_i$. This gives the curvature written in terms of the original Euclidean coordinates, the constraints, and the chosen coordinates on $\mathcal{Q}$.
\subsection{$d+1$ field theory}
\label{ssec:d+1}
The techniques of Sections \ref{ssec:lagrange}, \ref{ssec:dirac}, and \ref{ssec:holoD} have natural extensions to field theory.
Given independent field variables $\Psi^\mathcal{A}(\mu,t)$, indexed by $\mathcal{A}= 1, 2,\dots, \ell$, where the independent variable $\mu =(\mu^1,\mu^2,\dots, \mu^d)$. The field theoretic version of Hamilton's principle of \eqref{HamPrin} is embodied in the action
\begin{equation}
S[\Psi]=\int_{t_0}^{t_1}\!\!dt\, L[\Psi,\dot\Psi] \,,
\qquad \mathrm{with}\qquad
L[\Psi,\dot\Psi]=\int \! d^d\!\mu\, \mathcal{L}(\Psi,\dot\Psi,\partial \Psi)\,,
\label{fieldLag}
\end{equation}
where we leave the domain of $\mu$ and the boundary conditions unspecified, but freely drop surface terms obtained upon integration by parts. The Lagrangian density $\mathcal{L}$ is assumed to depend on the field components $\Psi$ and $\partial\Psi$, which is used to indicate all possible partial derivatives with respect of the components of $\mu$. Hamilton's principle with \eqref{fieldLag} gives the Euler-Lagrange equations,
\begin{equation}
\frac{\delta S[\Psi]}{\delta\Psi^\mathcal{A}(\mu,t)} = 0\quad \Rightarrow\quad \frac{d }{dt} \frac{\partial L}{\partial \dot{\Psi}^\mathcal{A}} + \frac{\partial }{\partial \mu} \frac{\partial L}{\partial \partial{\Psi}^\mathcal{A}} - \frac{\partial L}{\partial \Psi^\mathcal{A}}=0\,,
\label{FLagEOM}
\end{equation}
where the overdot implies differentiation at constant $\mu$. Local holonomic constraints $C^A(\Psi,\partial \Psi)$ are enforced by Lagrange's method by amending the Lagrangian
\begin{equation}
L_\lambda[\Psi,\dot\Psi]=\int \! d^d\!\mu\, \big( \mathcal{L}(\Psi,\dot\Psi,\partial \Psi) + \lambda_A C^A(\Psi,\partial \Psi)\big) \,,
\label{CfieldLag}
\end{equation}
with again $A=1,2,\dots, M$ and proceeding as in the finite-dimensional case.
In the Hamiltonian field theoretic setting, we could introduce the conjugate momentum densities $\pi_\mathcal{A}$, $\mathcal{A}= 1, 2,\dots, \ell$, with the phase space action
\begin{equation}
S_{\!\lambda}[\Psi,\pi]=\int_{t_0}^{t_1}\!\!dt\!\!\int \! \!d^d\!\mu\,
\left[ \pi_\mathcal{A} \dot{\Psi}^\mathcal{A} -\mathcal{H} + \lambda_A
D^A\right]\,, \end{equation}
with Hamiltonian density $\mathcal{H}$ and local constraints $D^a$ depending on the values of the fields and their conjugates. Instead of following this route we will jump directly to a generalization of the field theoretic Dirac bracket formalism that would result.
Consider a Poisson algebra composed of functionals of field variables ${\chi}^\mathcal{A}({\mu},t)$ with a Poisson bracket of the form
\begin{equation}
\label{eqn:PBgene}
\{F,G\}=\int\! \!d^d \mu \, F_{\chi} \cdot \mathbb{J} ({\chi}) \cdot G_{\chi}\,,
\end{equation}
where $F_{\chi}$ is a shorthand for the functional derivative of a functional $F$ with respect to the field $\chi$ \citep[see e.g.][]{pjm98} and $F_{\chi} \cdot {\mathbb J} \cdot G_{\chi}= F_{\chi^\mathcal{A}} \,{\mathbb J}^{\mathcal{A}\mathcal{B}} \, G_{\chi^\mathcal{B}}$, again with repeated indices summed. Observe the fields
${\chi}^\mathcal{A}({\mu},t)$ need not separate into coordinates and momenta, but if they do the Poisson operator $ \mathbb{J} $ has a form akin to that of \eqref{Jc}. By a Poisson algebra we mean a Lie algebra realization on functionals, meaning the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity and that there is an associative product of functionals that satisfies the Leibniz law. From the Poisson bracket the equations of motion are given by $\dot{\chi}=\{{\chi},H\}$, for some Hamiltonian functional $H[{\chi}]$.
Dirac's constraint theory is generally implemented in terms of canonical Poisson brackets \citep[see e.g.][]{dirac50,sudarshan,sundermeyer}, but it is not difficult to show that his procedure also works for noncanonical Poisson brackets \citep[see e.g.\ an Appendix of][] {pjmLB09}.
We impose an even number of local constraints which we write as $D^a(\mu)=$ constant, a shorthand for $D^a[\chi(\mu)]$, with the index $a=1,2,\dots, 2M$, bearing in mind that they depend on the fields $\chi$ and their derivatives. As in the finite-dimensional case, the Dirac bracket is obtained from the matrix $\mathbb{D}$ obtained from the bracket of the constraints,
\begin{equation}
\mathbb{D}^{ab}(\mu,\mu')=\{D^a(\mu),D^b(\mu')\}\,,
\nonumber
\end{equation}
where we note that $\mathbb{D}^{ab}(\mu,\mu')=-\mathbb{D}^{ba}(\mu',\mu)$.
If $\mathbb{D}$ has an inverse, then the Dirac bracket is defined as follows:
\begin{equation}
\label{eq:dbkt}
\{F,G\}_*=\{F,G\}-\int \!\!d^d\mu\!\! \int\!\! d^d\mu'\, \{F,D^a(\mu)\}\mathbb{D}^{-1}_{ab}(\mu,\mu')\{D^b(\mu'),G\}\,,
\end{equation}
where the coefficients $\mathbb{D}^{-1}_{ab}(\mu,\mu')$ satisfy
$$
\int \!\!d^d\mu' \, \mathbb{D}^{-1}_{ab}(\mu,\mu')\mathbb{D}^{bc}(\mu',\mu'')=\int \!\! d^3\mu' \,
\mathbb{D}^{cb}(\mu,\mu')\mathbb{D}^{-1}_{ba}(\mu',\mu'')
=\delta_{a}^{c}\delta(\mu-\mu'')\,,
$$
consistent with $\mathbb{D}^{-1}_{ab}(\mu,\mu')=-\mathbb{D}^{-1}_{ba}(\mu',\mu)$.
We note, the procedure is effective only when the coefficients $\mathbb{D}^{-1}_{a b}(\mu,\mu')$ can be found. If $\mathbb{D}$ is not invertible, then one needs, in general, secondary constraints to determine the Dirac bracket.
\subsubsection{Field theoretic geodesic flow}
In light of Section \ref{sssec:HG}, any field theory with a Lagrangian density of the form
\begin{equation}
\mathcal{L} = \frac12 \dot{\Psi}^\mathcal{A}(\mu,t)\, \eta_{\mathcal{A} \mathcal{B}} \, \dot{\Psi}^\mathcal{B}(\mu,t) \,,
\label{FTGlag}
\end{equation}
with the metric $\eta_{\mathcal{A} \mathcal{B}} =\delta_{\mathcal{A} \mathcal{B}}$ being the Kronecker delta, subject to time-independent holonomic constraints can be viewed as geodesic flow on the constraint surface. This is a natural infinite-dimensional generalization of the idea of Section \ref{sssec:HG}.
\section{Unconstrained Hamiltonian and action for fluid}
\label{sec:unconstrained}
\subsection{Fluid action in Lagrangian variable description}
\label{ssec:fluidaction}
The Lagrangian variable description of a fluid is described in Lagrange's famous work \citep{lagrange}, while historic and additional material can be found in \citet{serrin59,newcomb62,VKF67,pjm98}. Because the Lagrangian description treats a fluid as a continuum of particles, it naturally is amenable to the Hamiltonian form. The {Lagrangian variable} is a coordinate that gives the position of a fluid element or parcel, as it is sometimes called, at time $t$. We denote this variable by $\mathbf{q}=\mathbf{q}(\mathbf{a},t)=(q^1,q^2,q^3)$, which is measured relative to some cartesian coordinate system.
Here $\mathbf{a}=(a^1,a^2,a^3)$ denotes the {fluid element label}, which is often defined to be the position of the fluid element at the initial time, $\mathbf{a}=\mathbf{q}(\mathbf{a},0)$, but this need not always be the case. The label $\mathbf{a}$ is a continuum analog of the discrete index that labels a generalized coordinate in a finite degree-of-freedom system. If $D$ is a domain that is fully occupied by the fluid, then at each fixed time $t$, $\mathbf{q}\colon D\rightarrow D$ is assumed to be 1-1 and onto. Not much is really known about the mathematical properties of this function, but we will assume that it is as smooth as it needs to be for the operations performed. Also, we will assume we can freely integrate by parts dropping surface terms and drop reference to D in our integrals.
When discussing the ideal fluid and MHD we will use repeated sum notation with upper and lower indices even though we are working in cartesian coordinates. And, unlike in Section 2, latin indices, $i,j,k,\ell$ etc.\ will be summed over 1,2, and 3, the cartesian components, rather than to $N$. This is done to avoid further proliferation of indices and we trust confusion will not arise because of context.
Important quantities are the deformation matrix, $\partial q^i/\partial a^j$ and its Jacobian determinant $\mathcal{J}:= \det(\partial q^i/\partial a^j)$, which is given by
\begin{equation}
\mathcal{J}= \frac1{6}\epsilon_{kj\ell}\epsilon^{imn} \frac{\partial q^k}{\partial a^i} \frac{\partial q^j}{\partial a^m} \frac{\partial q^\ell}{\partial a^n} \,,
\label{J3}
\end{equation}
where $\epsilon_{ijk}=\epsilon^{ijk}$ is the purely antisymmetric (Levi-Civita) tensor density.
We assume a fluid element is uniquely determined by its label for all time. Thus, $\mathcal{J}\neq 0$ and we can invert $\mathbf{q}=\mathbf{q}(\mathbf{a},t)$ to obtain the label associated with the fluid element at position
$\mathbf{x}$ at time $t$, $\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)$.
For coordinate transformations $\mathbf{q}=\mathbf{q}(\mathbf{a},t)$ we have
\begin{equation}
\frac{\partial q^{k}}{\partial a^j}\, \frac{A^i_k}{\mathcal{J}} = \delta^i_j\,,
\label{inverseAJ}
\end{equation}
where $A^i_k$ is the cofactor of $\partial q^k/\partial a^i$ , which can be written as follows:
\begin{equation}
A^i_k= \frac12\epsilon_{kj\ell}\epsilon^{imn} \frac{\partial q^{j}}{\partial a^m}\frac{\partial q^{\ell}}{\partial a^n}\,.
\label{co3}
\end{equation}
Using $\mathbf{q}(\mathbf{a},t)$ or its inverse $\mathbf{q}^{-1}(\mathbf{x},t)$, various quantities can be written either as a function of $\mathbf{x}$ or $\mathbf{a}$. For convenience we list additional formulas below for latter use:
\bal
\mathcal{J} & = \frac{1}{3}A_{\ell}^{k}\frac{\partial q^{\ell}}{\partial a^{k}}\,,
\\
A^j_i &= \frac{\partial \mathcal{J}}{\partial (\partial q^i/\partial a^j)} \,,
\\
\frac{\partial (A_{i}^{k}f)}{\partial a^{k}}&=A_{i}^{k}\,\frac{\partial f}{\partial a^{k}}\,,
\label{dAda}
\\
\delta \mathcal{J} &= A_{i}^{k}\frac{\partial \delta{q}^{i}}{\partial a^{k}} \hspace{1.33cm} \mathrm{or}\qquad \dot{\mathcal{J}}=A_{i}^{k}\frac{\partial \dot{q}^{i}}{\partial a^{k}}\,,
\label{dedet}
\\
\delta\left(\frac{A_{\ell}^{k}}{\mathcal{J}}\right)\frac{\partial q^{\ell}}{\partial a^{u}}&=-\frac{A_{i}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{u}}\delta q^{i}
\qquad \mathrm{or}\qquad
\delta\left(\frac{A_{\ell}^{k}}{\mathcal{J}}\right)=-\frac{A_{i}^{k}A_{\ell}^{u}}{\mathcal{J}^{2}}\frac{\partial}{\partial a^{u}}\delta q^{i}\,,
\label{eq:AJvar}
\\
{{A}_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial f}{\partial a^{k}} \right]}
& {=A_{i}^{k}\frac{\partial}{\partial a^{k}}\left[\frac{A_{\ell}^{u}}{\mathcal{J}}\frac{\partial f}{\partial a^{u}}\right]\,,\forall\, f}
\,,
\label{IDen}
\eal
which follow from the standard rule for differentiation of determinants and the expression for the cofactor matrix. {For example, the commutator expression of \eqref{IDen} follows easily from \eqref{eq:AJvar}, which in turn follows upon differentiating \eqref{inverseAJ}. These formulas are all of classical origin, e.g., the second equation of \eqref{dedet} is the Lagrangian variable version of a formula due to Euler \citep[see e.g.][] {serrin59}. }
Now we are in a position to recreate and generalize Lagrange's Lagrangian for the ideal fluid action principle. On physical grounds we expect our fluid to possess kinetic and internal energies, and if magnetized, a magnetic energy. The total kinetic energy functional of the fluid is naturally given by
\begin{equation}
T[\dot{\mathbf{q}}]:=\frac1{2} \int \!d^3a\, \rho_0(\mathbf{a})\, |\dot \mathbf{q}|^2\,,
\label{Tq}
\end{equation}
where $\rho_0$ is the mass density attached to the fluid element labeled by $\mathbf{a}$ and $\dot\mathbf{q}$ denotes time differentiation of $\mathbf{q}$ at fixed label $\mathbf{a}$. Note, in \eqref{Tq}
$|\dot \mathbf{q}|^2=\dot{q}_i\dot{q}^i$, where in general $\dot{q}_i=g_{ij}\,\dot{q}^i$, but we will only consider the cartesian metric where $g_{ij}=\delta_{ij}=\eta_{ij}$.
Fluids are assumed to be in local thermodynamic equilibrium and thus can be described by a function $U(\rho,s)$, an internal energy per unit mass that depends on the specific volume $\rho^{-1}$ and specific entropy $s$. If a magnetic field $\mathbf{B}(\mathbf{x},t)$ were present, then we could add dependence on $|\mathbf{B}|$ as in \citet{pjm82} to account for pressure anisotropy. { \citep[See also][where this appears in the context of gyroviscosity.]{pjmLA14,pjmLW20}} The internal energy is written in terms of the Eulerian density and entropy (see Section \ref{ssec:LtoE}) since we expect the fluid at each Eulerian observation point to be in thermal equilibrium. From $U$ we compute the temperature and pressure according to the usual differentiations, $T=\partial U/\partial s$ and $p=\rho^2\partial U/\partial {\rho}$. For MHD, the magnetic energy $H_B=\int\! d^3x\, |\mathbf{B}|^2/2$ in Lagrangian variables would be added. For the ideal fluid, the total internal energy functional is
\begin{equation}
V[\mathbf{q}]:= \int \!d^3a\, \rho_0 \, U\left({\rho_0}/{\mathcal{J}}, s_0\right)
\,.
\label{Vq}
\end{equation}
Here we have used the fact that a fluid element carries a specfic entropy $s=s_0(\mathbf{a})$ and a mass determined by $\rho=\rho_0(\mathbf{a})/\mathcal{J}$. In Section \ref{ssec:LtoE} we will describe in detail the map from Lagrangian to Eulerian variables.
Thus, the special case of the action principle of \eqref{fieldLag} for the ideal fluid has Lagrange's Lagrangian $L[\mathbf{q},\dot\mathbf{q}]=T - V$. Variation of this action gives the Lagrangian equation of motion for the fluid
\begin{equation}
\rho_0 \ddot{q}_i=- A^j_i\, \frac{\partial}{\partial a^j}
\left(
\frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho}
\right)\,,
\label{lagEOM}
\end{equation}
with an additional term that describes the $\mathbf{J}\times\mathbf{B}$ force in Lagrangian variables for MHD. See, e.g., \citet{newcomb62,pjm98,pjm09} for details of this calculation and the MHD extension.
\subsection{Hamiltonian formalism in Lagrangian description}
\label{ssec:HamDesc}
Upon defining the momentum density as usual by
\begin{equation}
\pi_i=\frac{\delta L}{\delta \dot{q}^i}=\rho_0 \,\dot{q}_i\,,
\label{piDef}
\end{equation}
we can obtain the Hamiltonian by Legendre transformation, yielding
\begin{equation}
H[{\boldsymbol{\pi}},\mathbf{q}]=T + V= \int \!d^3a\,\left( \frac{|{\boldsymbol{\pi}}|^2}{2 \rho_0 }\,
+ \rho_0 U\left({\rho_0}/{\mathcal{J}}, s_0\right)\right)\,,
\label{LagHam}
\end{equation}
where $|\boldsymbol{\pi}|^2= \pi^i\pi_i= \pi_i \eta^{ij} \pi_j$.
This Hamiltonian with the canonical Poisson bracket,
\begin{equation}
\left\{F,G\right\} =\int\! d^{3}a\,\left(\frac{\delta F}{\delta q^{i}}\frac{\delta G}{\delta\pi_{i}}-\frac{\delta G}{\delta q^{i}}\frac{\delta F}{\delta\pi_{i}}\right)\,,
\label{cbkt}
\end{equation}
yields
\bal
\dot{q}^i&= \{q^i, H\}= \pi^i\!/\rho_0 \,,
\label{qdot}
\\
\dot{\pi}_i&= \{\pi_i, H\}= -
A^j_i\, \frac{\partial}{\partial a^j}
\left(
\frac{\rho_0^2}{\mathcal{J}^2} \frac{\partial U}{\partial \rho}
\right)\,.
\label{pidot1}
\eal
Equations \eqref{qdot} and \eqref{pidot1} are equivalent to \eqref{lagEOM}. For MHD a term $H_B$ is added to \eqref{LagHam} \citep[see][] {newcomb62,pjm09}. We will give this explicitly in the constraint context in Section \ref{sssec:LDholo} after discussing the Lagrange to Euler map.
\subsection{Hamiltonian formalism in Eulerian description via the Lagrange to Euler map}
\label{ssec:LtoE}
In order to understand how constraints in terms of the Lagrangian variable description relate to those in terms of the Eulerian description, in particular $\nabla\cdot \mathbf{v}=0$, it is necessary to understand the mapping from Lagrangian to Eulerian variables.
Thus, we record here the relationship between the two unconstrained descriptions, i.e., how the noncanonical Hamiltonian structure of the compressible Euler's equations relates to the Hamiltonian structure described in Section \ref{ssec:HamDesc}.
For the ideal fluid, the set of Eulerian variables can be taken to be $\{\mathbf{v},\rho, s\}$, where $\mathbf{v}(\mathbf{x},t)$ is the velocity field at the Eulerian observation point, $\mathbf{x}=(x,y,z)=(x^1,x^2,x^3)$ at time $t$ and, as as noted in Section \ref{ssec:fluidaction}, $\rho(\mathbf{x},t)$ is the mass density and $s(\mathbf{x},t)$ is the specific entropy.
In order to describe magnetofluids the magnetic field $\mathbf{B}(\mathbf{x},t)$ would be appended to this set. It is most important to distinguish between the Lagrangian fluid element position and label variables,
$\mathbf{q}$ and $\mathbf{a}$, and the Eulerian observation point $\mathbf{x}$, the latter two being independent variables. Confusion exists in the literature because some authors use the same symbol for the Lagrangian coordinate $\mathbf{q}$ and the Eulerian observation point $\mathbf{x}$.
The Lagrangian and Eulerian descriptions must clearly be related and, indeed, knowing $\mathbf{q}(\mathbf{a},t)$ we can obtain $\mathbf{v}(\mathbf{x},t)$. If one were to insert a velocity probe into a fluid at $(\mathbf{x},t)$ then one would measure the velocity of the fluid element that happened to be at that position at that time. Thus it is clear that $\dot{\mathbf{q}}(\mathbf{a},t)=\mathbf{v}(\mathbf{x},t)$, where recall the overdot means the time derivative at constant $\mathbf{a}$. But, which fluid element will be at $\mathbf{x}$ at time $t$? Evidently $\mathbf{x}=\mathbf{q}(\mathbf{a},t)$, which upon inversion yields the label of that element that will be measured,
$\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)$. Thus, the Eulerian velocity field is given by
\begin{equation}
\mathbf{v}(\mathbf{x},t) =\left.\dot{\mathbf{q}}(\mathbf{a},t)\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}=\dot{\mathbf{q}}\circ\mathbf{q}^{-1}(\mathbf{x},t)\,.
\end{equation}
Properties can be attached to fluid elements, just as a given mass is identified with a given particle in mechanics. For a continuum system it is natural to attach a mass density, $\rho_0(\mathbf{a})$, to the element labeled by $\mathbf{a}$. Whence the element of mass in a given volume is given by $\rho_0d^3a$ and this amount of mass is preserved by the flow, i.e.\ $\rho(\mathbf{x},t)d^3x=\rho_0(\mathbf{a})d^3a$. Because the locus of points of material surfaces move with the fluid are determined by $\mathbf{q}$, an initial volume element $d^3a$ maps into a volume element $d^3x$ at time $t$ according to
\begin{equation}
d^3x=\mathcal{J} d^3a\,,
\label{vol}
\end{equation}
Thus, using \eqref{vol} we obtain $\rho_0=\rho \mathcal{J}$ as used in Section \ref{ssec:fluidaction}.
Other quantities could be attached to a fluid element; for the ideal fluid, entropy per unit mass, $s(\mathbf{x},t)$, is such a quantity. The assumption that each fluid element is isentropic then amounts to $s=s_0$. Similarly, for MHD a magentic field, $B_0(\mathbf{a})$, can be attached, and then the frozen flux assumption yields $B\cdot d^2x=B_0\cdot d^2a$. An initial area element $d^2a$ maps into an area element $d^2x$ at time $t$ according to
\begin{equation}
(d^2x)_i= A^j_i \, (d^2a)_j\,.
\label{area}
\end{equation}
Using (\ref{area}) we obtain $\mathcal{J} B^i=B_0^j\, \partial q^i/\partial a^j$.
Sometimes it is convenient to use another set of Eulerian density variables: $\{\mathbf{M},\rho, \sigma,\mathbf{B}\}$, where $\sigma=\rho s$ is the entropy per unit volume, and $\mathbf{M}=\rho \mathbf{v}$ is the momentum density. These Eulerian variables can be represented by using the Dirac delta function to `pluck out' the fluid element that happens to be at the Eulerian observation point $\mathbf{x}$ at time $t$. For example, the mass density $\rho(\mathbf{x},t)$ is obtained by
\begin{equation}
\rho({\mathbf{x}},t)=\int \!d^3a
\, \rho_0(\mathbf{a}) \, \delta\left({\mathbf{x}}-{\mathbf{q}}\left(\mathbf{a},t\right)\right)
=\left. \frac{\rho_0}{\mathcal{J}}\right|_{\mathbf{a}=\mathbf{q}^{-1}({\mathbf{x}},t)}\,.
\label{rhoEu}
\end{equation}
The density one observes at $\mathbf{x}$ at time $t$ will be the one attached to the fluid element that happens to be there then, and this fluid element has a label given by solving $\mathbf{x}=\mathbf{q}(\mathbf{a},t)$. The second equality of (\ref{rhoEu}) is obtained by using the three-dimensional version of the delta function identity $\delta(f(x))=\sum_i \delta (x-x_i)/|f'(x_i)|$, where $f(x_i)=0$. Similarly, the entropy per unit volume is given by
\begin{equation}
\sigma({\mathbf{x}},t)=\int\! d^3a\,
\sigma_0(\mathbf{a}) \, \delta\left({\mathbf{x}}-{\mathbf{q}}\left(\mathbf{a},t\right)\right)
=\left. \frac{\sigma_0}{\mathcal{J}}\right|_{\mathbf{a}=\mathbf{q}^{-1}({\mathbf{x}},t)}\,,
\label{siEu}
\end{equation}
which is consistent with $\sigma_0(\mathbf{a})=\rho_0(\mathbf{a}) s_0(\mathbf{a})$ and $s(\mathbf{x},t)=\left. s_0(\mathbf{a})\right|_{\mathbf{a}=\mathbf{q}^{-1}({\mathbf{x}},t)}$, where the latter means $s$ is constant along a Lagrangian orbit. Proceeding, the momentum density, $\mathbf{M}=(M_1,M_2,M_3)$, is related to the Lagrangian canonical momentum (defined in Section \ref{ssec:HamDesc}) by
\begin{equation}
\mathbf{M}(\mathbf{x},t)=\int \!d^3a \,
\boldsymbol{\pi}(\mathbf{a},t) \, \delta\left({\mathbf{x}}-{\mathbf{q}}(\mathbf{a},t)\right)
= \left.
\frac{{\boldsymbol{\pi}}(\mathbf{a},t)}{\mathcal{J}}
\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}\,,
\label{MEu}
\end{equation}
where for the ideal fluid and MHD, \ ${\boldsymbol{\pi}}(\mathbf{a},t)=(\pi_1,\pi_2,\pi_3)=\rho_0 \dot{\mathbf{q}}$. Lastly,
\begin{equation}
B^i(\mathbf{x},t)=\int \!d^3a \, \frac{\partial q^i(\mathbf{a},t)}{\partial a^j} B_0^j(\mathbf{a}) \,
\delta\left({\mathbf{x}}-{\mathbf{q}}(\mathbf{a},t)\right)
= \left.
\frac{\partial q^i(\mathbf{a},t)}{\partial a^j}\frac{B_0^j(\mathbf{a})}{\mathcal{J}}
\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}\,,
\end{equation}
for the components of the magnetic field. It may be unfamiliar to view the magnetic field as density, but in MHD it obeys a conservation law. {Geometrically, however, it naturally satisfies the equations of a vector density associated with a differential 2-form as was observed in \citet{pjm82} and \citet{TY93}. }
To obtain the noncanonical Eulerian Poisson bracket we consider functionals $F[\mathbf{q},\mathbf{\pi}]$ that are restricted so as to obtain their dependence on $\mathbf{q}$ and $\mathbf{\pi}$ only through the Eulerian variables. Upon setting $F[\mathbf{q},\boldsymbol{\pi}]=\bar{F}[\mathbf{v},\rho,\sigma]$, equating variations of both sides,
\begin{equation}
\delta F = \int d^3a \left[\frac{\delta F}{\delta \mathbf{q}}\cdot\delta \mathbf{q} +
\frac{\delta F}{\delta \mathbf{\pi}}\cdot\delta \mathbf{\pi}\right]
= \int d^3x \,\left[\frac{\delta\bar F}{\delta\rho}\delta\rho +
\frac{\delta\bar F}{\delta \sigma}\delta \sigma +
\frac{\delta \bar F}{\delta \mathbf{M}}
\cdot\delta \mathbf{M}\right]=\delta\bar{F}\,,
\label{deltaF}
\end{equation}
varying the expressions \eqref{rhoEu}, \eqref{siEu}, and \eqref{MEu}, substituting the result into
\eqref{deltaF}, and equating the independent coefficients of $\delta\mathbf{q}$ and $\delta{\boldsymbol{\pi}}$, we obtain
\bal
\frac{\delta F}{\delta \mathbf{q}}& = \int d^3x\,\left[\rho_ 0\,\nabla
\frac{\delta\bar F}{\delta\rho} + \sigma_0\,\nabla
\frac{\delta\bar F}{\delta\sigma}
+ \pi_{i} \nabla \frac{\delta\bar F}{\delta M_{i}} \right]\,
\delta(\mathbf{x}-\mathbf{q}) \,,
\label{deFq}
\\
\frac{\delta F}{\delta {\boldsymbol{\pi}}} &= \int d^3x\, \frac{\delta\bar F}{\delta \mathbf{M}} \,
\delta (\mathbf{x}-\mathbf{q})\,.
\label{deFpi}
\eal
(See \cite{pjm98} and \cite{pjmG80} for details.)
Upon substitution of \eqref{deFq} and \eqref{deFpi}, expressions of the functional chain rule that relate Lagrangian functional derivatives to the Eulerian functional derivates, into \eqref{cbkt}
yields the following bracket expressed entirely in terms of the Eulerian fields $\{\mathbf{M},\rho,\sigma\}$:
\begin{eqnarray}
\{F,G\} &=& -\int \!d^3 x\, \Bigg[
M_i \Bigg(\frac{\delta F}{\delta M_j} \frac{\partial}{\partial x^j}
\frac{\delta G}{\delta M_i} - \frac{\delta G}{\delta M_j}
\frac{\partial}{\partial x^j} \frac{\delta F}{\delta M_i} \Bigg)
\nonumber\\
&+& \rho \, \Bigg(\frac{\delta F}{\delta \mathbf{M}} \cdot
\nabla \frac{\delta G}{\delta \rho}
- \frac{\delta G}{\delta \mathbf{M}} \cdot \nabla
\frac{\delta F}{\delta \rho}\Bigg)
+ \sigma\, \Bigg(\frac{\delta F}{\delta \mathbf{M}} \cdot
\nabla \frac{\delta G}{\delta \sigma} -
\frac{\delta G}{\delta \mathbf{M}} \cdot \nabla
\frac{\delta F}{\delta \sigma}\Bigg)
\Bigg] \,.
\label{Mbkt}
\end{eqnarray}
In \eqref{Mbkt} we have dropped the overbars on the Eulerian functional derivatives.
The bracket for MHD is the above with the addition of the following term, which is obtained by adding a $\mathbf{B}$ contribution to \eqref{deltaF}:
\bal
\{F,G\}_B &=-\int \!d^3 x\, \Bigg[ \mathbf{B}\cdot
\left(
\frac{\delta F}{\delta \mathbf{M}} \cdot\nabla \frac{\delta G}{\delta \mathbf{B}}
- \frac{\delta G}{\delta \mathbf{M}} \cdot\nabla \frac{\delta F}{\delta \mathbf{B}}
\right)
\nonumber\\
&\hspace{1cm} + \mathbf{B}\cdot
\left(
\nabla \left( \frac{\delta F}{\delta \mathbf{M}} \right ) \cdot \frac{\delta G}{\delta \mathbf{B}}
- \nabla \left( \frac{\delta G}{\delta \mathbf{M}} \right ) \cdot \frac{\delta F}{\delta \mathbf{B}}
\right)
\Bigg]\,,
\label{Bbkt}
\eal
where dyadic notation is used; for example, $\mathbf{B}\cdot[\nabla (\mathbf{D}) \cdot \mathbf{C}]=\sum_{i,j} B_i C_j \partial D_j/\partial x_i$, for vectors $\mathbf{B},\mathbf{D}$, and $\mathbf{C}$. Alternatively, the bracket in terms of
$\{\mathbf{v},\rho,s,\mathbf{B}\}$ is obtained using chain rule expressions, e.g.,
\begin{equation}
\frac{\delta F}{\delta \rho}\Bigg|_{\mathbf{v},s} = \frac{\delta F}{\delta
\rho}\Bigg|_{\mathbf{M},s}
+\frac{\mathbf{M}}{\rho}\cdot \frac{\delta F}{\delta \mathbf{M}} + \frac{\sigma}{\rho}
\frac{\delta F}{\delta \sigma}\,,
\label{chn1}
\end{equation}
yielding
\bal
\{F,G\} &= -\int\!d^3 x \, \Bigg[\Bigg(
\frac{\delta F}{\delta \rho}\nabla\cdot\frac{\delta G}{\delta \mathbf{v}}
- \frac{\delta G}{\delta \rho}\nabla\cdot\frac{\delta F}{\delta \mathbf{v}}\Bigg)
+ \Bigg(\frac{\nabla \times \mathbf{v}}{\rho}\cdot\frac{\delta G}{\delta
\mathbf{v}}\times\frac{\delta F}{\delta \mathbf{v}}\Bigg)
\nonumber\\
&\hspace{2cm} + \frac{\nabla s}{\rho}\cdot\Bigg(\frac{\delta F}{\delta s}
\frac{\delta G}{\delta \mathbf{v}} -
\frac{\delta G}{\delta s}\frac{\delta F}{\delta \mathbf{v}}\Bigg)
\Bigg]
\,,
\label{vbkt}
\eal
and
\bal
\{F,G\}_B & = -\int\!d^3 x \, \Bigg[
\mathbf{B}\cdot
\left(
\frac{1}{\rho} \frac{\delta F}{\delta \mathbf{v}} \cdot\nabla \frac{\delta G}{\delta \mathbf{B}}
- \frac{1}{\rho} \frac{\delta G}{\delta \mathbf{v}} \cdot\nabla \frac{\delta F}{\delta \mathbf{B}}
\right)
\nonumber\\
&\hspace{1cm}
+\mathbf{B}\cdot
\left(
\nabla \left(\frac{1}{\rho} \frac{\delta F}{\delta \mathbf{v}} \right ) \cdot \frac{\delta G}{\delta \mathbf{B}}
- \nabla \left(\frac{1}{\rho} \frac{\delta G}{\delta \mathbf{v}} \right ) \cdot \frac{\delta F}{\delta \mathbf{B}}
\right)
\Bigg]\,.
\label{vBbkt}
\eal
The bracket of \eqref{vbkt} plus that of \eqref{vBbkt} with the Hamiltonian
\begin{equation}
H[\rho, s,\mathbf{v},\mathbf{B}]=\int\! d^3x\left(\frac12 \rho |\mathbf{v}|^2 +\rho U(\rho,s) +\frac12 |\mathbf{B}|^2\right)
\end{equation}
gives the Eulerian version of MHD in Hamiltonian form, $\partial \mathbf{v}/\partial t=\{\mathbf{v},H\}$, etc., and similarly using \eqref{Mbkt} plus \eqref{Bbkt} with the Hamiltonian expressed in terms of $(\mathbf{M},\rho, \sigma,\mathbf{B})$. Ideal fluid follows upon neglecting the $\mathbf{B}$ terms.
\subsection{Constants of motion: Eulerian vs.\ Lagrangian}
\label{ssec:CoM}
In oder to compare the imposition of constraints in the Lagrangian and Eulerian descriptions, it is necessary to compare Lagrangian and Eulerian conservations laws. This is because constraints, when enforced, are conserved quantities. The comparison is not trivial because time independent quantities in the Eulerian description can be time dependent in the Lagrangian description.
Consider a Lagrangian function $f(\mathbf{a},t)$, typical of the Lagrangian variable description, and
the relation $\mathbf{x}=\mathbf{q}(\mathbf{a},t)$, which relates an Eulerian observation point $\mathbf{x}$ to a corresponding fluid element trajectory value. The function $f$ can be written in either picture by composition, as follows:
\begin{equation}
f(\mathbf{a},t)=\tilde{f}(\mathbf{x},t)= \tilde{f}(\mathbf{q}(\mathbf{a},t),t)\,,
\end{equation}
where we will use a tilde to indicated the Eulerian form of a Lagrangian function. Application of the chain rule gives
\begin{equation}
\left. \frac{A^i_k}{\mathcal{J}} \, \frac{\partial f}{\partial a^i}\,\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)} =\frac{\partial \tilde{f}}{\partial x^k}
\qquad \mathrm{and}\qquad
\left. \frac{A_{\ell}^{k}}{\mathcal{J}}\,\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\mathcal{J}}\right)\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)} = \nabla\cdot \mathbf{v} \,,
\label{ELgrad}
\end{equation}
with the second equality of \eqref{ELgrad} being a special case of the first. Similarly,
\begin{equation}
\left.\dot f \, \right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}= \frac{\partial \tilde{f}}{\partial t} + \left. \dot{q}^i(\mathbf{a},t)\, \frac{\partial \tilde{f}}{\partial x^i}\,\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}=
\frac{\partial \tilde{f}}{\partial t} + \mathbf{v}\cdot \nabla \tilde{f}(\mathbf{x},t)\,,
\label{lagDet}
\end{equation}
where recall an overdot denotes the time derivative at constant $\mathbf{a}$, $\partial/\partial t$ denotes the time derivative at constant $\mathbf{x}$, and $\nabla$ is the Eulerian gradient with components $\partial/\partial x^i$ as used in \eqref{ELgrad}. Because the Jacobian determinant $\mathcal{J}$ is composed of derivatives of $\mathbf{q}$, we have $\mathcal{J}(\mathbf{a},t)|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)} =\tilde{\mathcal{J}}(\mathbf{x},t)$, whence one obtains a formula due to Euler \citep[see e.g.][] {serrin59},
\begin{equation}
\frac{\partial \tilde\mathcal{J}}{\partial t} + \mathbf{v}\cdot \nabla \tilde \mathcal{J}= \tilde\mathcal{J} \, \nabla\cdot \mathbf{v}\,,
\label{Jeuler}
\end{equation}
{which can be compared to its Lagrangian version of \eqref{dedet}.}
Now, consider a conservation law in the Lagrangian variable description,
\begin{equation}
\dot \mathcal{D}_L + \frac{\partial \Gamma_{\mathcal{D}_L}^i}{\partial a^{i}}=0\,,
\label{Lcon}
\end{equation}
where the density $\mathcal{D}_L(\mathbf{a},t)$ has the associated flux $\boldsymbol{\Ga}_{\mathcal{D}_L}$. Then, the associated conserved quantity is
\begin{equation}
\mathcal{I}_{\mathcal{D}_L}=\int \!d^3a \, \mathcal{D}_L\,,
\label{ILcon}
\end{equation}
which satisfies $d\mathcal{I}_{\mathcal{D}_L}/dt =0$ provided surface terms vanish. Similarly, an Eulerian conservation law with density $\mathcal{D}_E$ and flux ${\boldsymbol{\Ga}}_{\mathcal{D}_E}$ is
\begin{equation}
\frac{\partial\mathcal{D}_E}{\partial t} + \frac{\partial \Gamma_{\mathcal{D}_E}^i}{\partial x^{i}}=0
\label{Econ}
\end{equation}
and the following is similarly constant in time:
\begin{equation}
\mathcal{I}_{\mathcal{D}_E}=\int \!d^3x \, \mathcal{D}_E\,.
\end{equation}
The relationship between the two conservation laws (\ref{Lcon}) and (\ref{Econ}) can be obtained by defining
\begin{equation}
\tilde\mathcal{D}_L={\mathcal{J}}\mathcal{D}_E\,, \qquad \tilde\Gamma^i_{\mathcal{D}_L}= A^i_k\, \bar\Gamma_{\mathcal{D}_E}^k\,,
\qquad {\rm and}\qquad \mathbf{\Gamma}_{\mathcal{D}_E}= \mathbf{ \bar\Gamma}_{\mathcal{D}_E} + \mathbf{v}\, \mathcal{D}_E\,,
\label{ELcon}
\end{equation}
and their equivalence follows from (\ref{dedet}), (\ref{lagDet}), and (\ref{Jeuler}). Given a Lagrangian conservation law, one can use \eqref{ELcon} to obtain a corresponding Eulerian conservation law. The density $\mathcal{D}_E$ is obtained from the first equation of \eqref{ELcon}, a piece of the Eulerian flux $\bar{\boldsymbol{\Ga}}_{\mathcal{D}_E}$ from the second, which then can be substitued into the third equation of \eqref{ELcon} to obtain the complete Eulerian flux $\boldsymbol{\Ga}_{\mathcal{D}_E}$. An Eulerian conservation law is most useful when one can write $\mathcal{D}_E$ and $\boldsymbol{\Ga}_{\mathcal{D}_E}$ entirely in terms of the Eulerian variables of the fluid.
The simplest case occurs when $\mathcal{D}_L$ only depends on $\mathbf{a}$, in which case the corresponding flux is zero and
$\partial \mathcal{D}_L/\partial t=0$ and $d\mathcal{I}_{\mathcal{D}_L}/dt=0$ follow directly because (\ref{ILcon}) has no time dependence whatsoever. Any attribute attached to a fluid element only depends on the label $\mathbf{a}$ and this has a trivial conservation law of this form. However, such trivial Lagrangian conservation laws yield nontrivial Eulerian conservation laws. Observe, even thought
$\mathbf{\bar\Gamma}_{\mathcal{D}_E}\equiv 0$ by \eqref{ELcon}, $\boldsymbol{\Ga}_{\mathcal{D}_E}= \mathbf{v} \mathcal{D}_E\neq 0$. Consider the case of the entropy where $\mathcal{D}_L=s_0(\mathbf{a})$, whence $s(\mathbf{x},t)=s_0(\mathbf{a}(\mathbf{x},t))$ and by \eqref{lagDet},
\begin{equation}
\frac{\partial s}{\partial t} +\mathbf{v}\cdot \nabla s=0\,,
\end{equation}
with the quantity $s=s_0/\mathcal{J}$ being according to \eqref{ELcon} the Eulerian conserved density, as can be verified using \eqref{Jeuler}. But, as it stands, this density cannot be written in terms of Eulerian fluid variables. However, $\sigma_0=\rho_0s_0$ is also a trivial Lagrangian conserved density and according to (\ref{ELcon}) we have the Eulerian density $\rho_0s_0/\mathcal{J}=\rho s=\sigma$ that satisfies
\begin{equation}
\frac{\partial \sigma}{\partial t} + \nabla \cdot(\mathbf{v}\sigma)=0\,.
\end{equation}
Thus, it follows that any advected scalar has an associated conserved quantity obtained by multiplication by $\rho$.
As another example, consider the quantity $B_0^i \partial q^j/\partial a^i$. This quantity is the limit displacement between two nearby fluid elements, i.e., $\mathbf{q}(\mathbf{a},t) -\mathbf{q}(\mathbf{a} +\delta\mathbf{a},t)$ along the initial magnetic field as $\delta \mathbf{a} \rightarrow 0$. Evidently,
\begin{equation}
\dot{\left(B_0^i\frac{ \partial q^j}{\partial a^i}\right)}= B_0^i\frac{ \partial \dot{q}^j}{\partial a^i}
= \frac{\partial}{\partial a^i} \left(B_0^i \dot{q}^j\right)\,,
\label{Bpara}
\end{equation}
where the second equality follows if the initial magnetic field is divergence free. This is of course another trivial conservation law, for the time derivative of a density that is a divergence will always be a divergence. However, let us see what this becomes in the Eulerian description. According to \eqref{ELcon} the corresponding Eulerian density is $\mathcal{D}_E=\mathcal{D}_L/\mathcal{J}$; so, the density associated with this trivial conservation law \eqref{Bpara} is
\begin{equation}
B^j(\mathbf{x},t) = \left.\frac{B_0^i}{\mathcal{J}}\frac{ \partial q^j}{\partial a^i}\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}\,.
\end{equation}
which as we saw in Section \ref{ssec:LtoE} is the expression one gets for the MHD magnetic field because of flux conservation. That the divergence-free magnetic field satisfies a conservation law is clear from
\begin{equation}
\frac{\partial \mathbf{B}}{\partial t}= -\mathbf{v}\cdot \nabla \mathbf{B}+ \mathbf{B}\cdot \nabla \mathbf{v} - \mathbf{B}\, \nabla \cdot \mathbf{v}=\nabla \cdot \overset\leftrightarrow{{T}}\,,
\label{Bcon}
\end{equation}
where the tensor $\overset\leftrightarrow{{T}}$ of the last equality is
\begin{equation}
\overset\leftrightarrow{{T}} = \mathbf{B} \otimes\mathbf{v}-\mathbf{v} \otimes \mathbf{B}\,.
\end{equation}
Thus we have another instance where a trivial Lagrangian conservation law leads to a nontrivial Eulerian one.
Although $B_0^i { \partial q^j}/{\partial a^i}$ does not map into an expression entirely in terms of our set of Eulerian variables, we can divide it by $\rho_0$, a quantity that only depends on the label $\mathbf{a}$, and obtain
\begin{equation}
\left.\frac{B_0^i}{\rho_0}\frac{ \partial q^j}{\partial a^i}\right|_{\mathbf{a}=\mathbf{q}^{-1}(\mathbf{x},t)}= \frac{B^j}{\rho}\,.
\end{equation}
Eulerianizing the counterpart of \eqref{Bpara} for this expression gives
\begin{equation}
\frac{\partial }{\partial t} \left(\frac{\mathbf{B}}{\rho}\right) + \mathbf{v}\cdot \nabla \left(\frac{\mathbf{B}}{\rho}\right)
= \frac{\mathbf{B}}{\rho} \cdot \nabla \mathbf{v}\,,
\label{Brho}
\end{equation}
which is not an Eulerian conservation law. This is to be expected because, unlike what we did to get \eqref{Bcon}, we have Eulerianized without using \eqref{ELcon}. In light of its relationship to $\mathbf{q}(\mathbf{a},t) -\mathbf{q}(\mathbf{a} +\delta\mathbf{a},t)$, the quantity $\mathbf{B}/\rho$ has been described as a measure of the distance of points on a magnetic field line \citep[see e.g.][] {vkampen}. This was predated by analogous arguments for vorticity \citep[see e.g.][] {serrin59}.
\section{Constraint theories for the incompressible ideal fluid}
\label{sec:dirac}
\subsection{The incompressible fluid in Lagrangian variables}
\label{ssec:LagCon}
In order to enforce incompressibility, Lagrange added to his Lagrangian the constraint $\mathcal{J}=1$ with the Lagrange multiplier $\lambda(\mathbf{a},t)$,
\begin{equation}
L_\lambda[\mathbf{q},\dot{\mathbf{q}}]= T[\dot{\mathbf{q}}] + \lambda \, \mathcal{J}\,,
\label{lagla}
\end{equation}
with $T$ given \eqref{Tq}. Here we have dropped $V$ because incompressible fluids contain no internal energy. Upon insertion of \eqref{lagla} into the action of Hamilton's principle it is discovered that $\lambda$ corresponds to the pressure. The essence of this procedure was known to Lagrange. (See \citet{serrin59} for historical details and \citet{sommerfeld} for an elementary exposition.) This procedure yields
\begin{equation}
\rho_0\ddot q^i=-A^i_j \, \frac{\partial \lambda}{\partial a^j}\,,
\label{lageomla}
\end{equation}
where use has been made of \eqref{dedet}. The Eulerian form of \eqref{lageomla} is clearly
$\rho(\partial \mathbf{v}/\partial t+\mathbf{v}\cdot\nabla \mathbf{v})=-\nabla \lambda$, whence it is clear that $\lambda$ is the pressure. Although Lagrange knew the Lagrange multiplier was the pressure, he could only solve for it in special cases. The general procedure of Section \ref{ssec:bgnd} was not available because Green's function techniques and the theory of elliptic equations were not at his disposal.
\subsubsection{Lagrangian volume preserving geodesic flow}
If the constraint is dropped from \eqref{lagla}, we obtain free particle motion for an infinite-dimensional system, the ideal fluid case of \eqref{FTGlag} of Section \ref{ssec:d+1}, which is analogous to the finite-dimensional case of Section \ref{sssec:HG}. Because the constraint
$\mathcal{J}=1$ only depends on the derivatives of $\mathbf{q}$, it is a configuration space constraint; thus, it is an holonomic constraint. As is well-known and reviewed in Section \ref{sssec:HG}, free particle motion with holonomic constaints is geodesic flow. Thus, following Lagrange, it is immediate that the ideal incompressible fluid is an infinite-dimensional version of geodesic flow.
Lagrange's calculation was placed in a geometric/group theoretic setting in \cite{arnold-diffeo} (see also Appendix 2 of \cite{arnold-book} and \cite{khesin}). Given that the transformation $\mathbf{a}\leftrightarrow \mathbf{q}$, at any time, is assumed to be a smooth invertible coordinate change, it is a Lie group, one referred to as the diffeomorphism group. With the additional assumption that these transformations are volume preserving, Lagrange's constraint $\mathcal{J}=1$, the transformations form a subgroup, the group of volume preserving diffeomorphisms. Thus, Lagrange's work can be viewed as geodesic flow on the group of volume preserving diffeomorphisms.
Although Arnold's assumptions of smoothness etc.\ are mathematically dramatic, his description of Lagrange's calculations in these terms has spawned a considerable body of research. Associated with a geodesic flow is a metric, and whence one can calculate a curvature. In his original work, Arnold added the novel calculation of the curvature in the mathematically more forgiving case of two-dimensional flow with periodic boundary conditions.
\subsection{Lagrangian-Dirac constraint theory}
\label{ssec:LDconTh}
More recently there have been several works \citep{pjmTC09,pjmCGBT13,pjmLB09},
following \citet{turski99,turski01}, that treat the enforcement of
the incompressibility constraint of hydrodynamics by Dirac's method
of constraints \citep{dirac50}. In these works the compressibility
constraint was enforced in the Eulerian variable description
of the fluid using the noncanonical Poisson bracket of Section \ref{ssec:LtoE}
as the base bracket of a generalization of Dirac's constraint theory. We will return to this approach in Section \ref{ssec:LtoE} where we revisit and extend Dirac's constraint results for the fluid in the Eulerian variable description. Here, apparently for the first time, we consider the incompressibility constraint in the Lagrangian variable description, where the canonical Poisson bracket of \eqref{cbkt} is the base for the construction of a Dirac bracket.
We adapt \eqref{eq:dbkt} for the fluid case at hand with the supposition of only two local constraints,
which we write as
\begin{equation}
D^a(\mathbf{a}') =\int d^3a \,D^a(\mathbf{a})\, \delta(\mathbf{a}-\mathbf{a}')\,,
\label{deexp}
\end{equation}
where $a=1,2$ and $D^a(\mathbf{a})$ is a shorthand for a function of $\mathbf{q}(\mathbf{a}, t)$ and $\boldsymbol{\pi}(\mathbf{a},t)$ and their derivatives with respect to $\mathbf{a}$. Then the matrix $\mathbb{D}$ is a $2\times 2$ matrix with the components
\begin{equation}
\mathbb{D}^{ab}(\mathbf{a},\mathbf{a}')=\{D^a(\mathbf{a}),D^b(\mathbf{a}')\}\,,
\nonumber
\end{equation}
using the canonical bracket of \eqref{cbkt}.
To construct the Dirac bracket
\begin{equation}
\label{eqn:DBF}
\{F,G\}_*=\{F,G\}-\int \!\!d^3a\!\! \int\!\! d^3a'\, \{F,D^a(\mathbf{a})\}\mathbb{D}^{-1}_{ab}(\mathbf{a},\mathbf{a}')\{D^b(\mathbf{a}'),G\},
\end{equation}
we require the inverse, which satisfies
\begin{equation}
\int\!d^{3}a\,\,\mathbb{D}^{ac}(\mathbf{a}^{\prime},\mathbf{a})\,\mathbb{D}^{-1}_{cb}(\mathbf{a},\mathbf{a}^{\prime\prime})=\delta^{a}_{b}\,\delta(\mathbf{a}^{\prime}-\mathbf{a}^{\prime\prime})\,.
\label{inverse}
\end{equation}
Rather than continuing with the general case, which is unwieldy, we proceed to the special case for the incompressible fluid, an infinite-dimensional version of the holonomic constraints discussed in Section \ref{sssec:HCD}.
\subsubsection{Lagrangian-Dirac incompressibility holonomic constraint}
\label{sssec:LDholo}
Evidently we will want our holonomic incompressibility constraint to be $\mathcal{J}$. However, it is convenient to express this by choosing
\begin{equation}
D^1=\ln\left(\frac{\mathcal{J}}{\rho_{0}}\right)\,.
\label{D1}
\end{equation}
This amounts to the same constraint as $\mathcal{J}=1$ with the value $D^1=-\ln(\rho_0)$. The scaling of $\mathcal{J}$ in \eqref{D1} by $\rho_0(\mathbf{a})$ is immaterial because it is a time-independent quantity.
To obtain the second constraint we follow suit and set
\begin{equation}
D^{2}=\dot{D}^1=\frac{A_{\ell}^{k}}{\mathcal{J}}\,\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)
= \eta^{\ell j}\,\frac{A_{\ell}^{k}}{\mathcal{J}}\,\frac{\partial}{\partial a^{k}}\left(\frac{\pi_{j}}{\rho_{0}}\right)\,,
\label{constraints-1-1}
\end{equation}
where recall we assume $\eta^{\ell j}=\delta^{\ell j}$ and $\pi_j$ is given by \eqref{piDef}. That the constraint $D^2$ is the time derivative of $D^1$ requires the definition of $\pi_j$ of \eqref{piDef} that uses the Hamiltonian $ \int \!d^3a\ {|{\boldsymbol{\pi}}|^2}/{(2\rho_0)}$.
Observe, that constraints $D^1$ and $D^2$ are local constraints in that they are
enforced pointwise \citep[see e.g.][] {pjmF11}, i.e., they are enforced
on each fluid element labeled by $\mathbf{a}$. Equation (\ref{D1})
corresponds in the Eulerian picture to $-\ln(\rho)$, while
the second constraint
of (\ref{constraints-1-1}), the Lagrangian time derivative of the
first constraint, corresponds in the Eulerian picture to $\nabla\cdot\mathbf{v}$,
which can be easily verified using the second equation of \eqref{ELgrad}.
Note, the particular values of these constraints of interest are, of course,
$\mathcal{J}=1$ and $\nabla\cdot\mathbf{v}= 0$, but the dynamics
the Dirac bracket generates will preserve any values of these constraints.
For example, we could set $\mathcal{J}=f\left(\mathbf{a}\right)$
where the arbitrary function $f$ is less than unity for some $\mathbf{a}$
and greater for others, corresponding to regions of fluid elements
that experience contraction and expansion. Also note, because we have used $\boldsymbol{\pi}$ with the up index in \eqref{constraints-1-1}; thus as seen in the second equality it depends on the metric. This was done to make it have the Eulerian form
$\nabla\cdot\mathbf{v}$.
For the constraints \eqref{D1} and \eqref{constraints-1-1}, $\mathbb{D}$ only depends on two quantities because $D^{1}$ does not depend on $\boldsymbol{\pi}$, i.e.\ $\{D^{1},D^{1}\}= 0$ and $\{D^{1},D^{2}\}=-\{D^{2},D^{1}\}$.
Thus the inverse has the form
\begin{equation}
\mathbb{D}^{-1}=\left(\begin{matrix}\mathbb{D}^{-1}_{11} & \mathbb{D}^{-1}_{12}\\
\mathbb{D}^{-1}_{21} & 0
\end{matrix}\right)\,,
\end{equation}
giving rise to the conditions
\begin{equation}
\mathbb{D}^{-1}_{12}\cdot \mathbb{D}^{21}=\mathcal{I}=\mathbb{D}^{-1}_{21}\cdot \mathbb{D}^{12}\, \qquad
\mathrm{and}
\qquad
\mathbb{D}^{-1}_{11}\cdot \mathbb{D}^{12}+ \mathbb{D}^{-1}_{12}\cdot \mathbb{D}^{22}=0
\,,
\label{condition}
\end{equation}
where $\mathcal{I}$ is the identity. Thus, the inverse is easily tractable if the inverse of $\mathbb{D}^{12}$
exists; whence,
\begin{equation}
\mathbb{D}^{-1}_{11}=-\mathbb{D}^{-1}_{12}\cdot \mathbb{D}^{22}\cdot \mathbb{D}^{-1}_{21}\,.
\label{condition2}
\end{equation}
In the above the symbol `$\,\cdot\,$' is used to denote the product
with the sum in infinite dimensions, i.e., integration over $d^3a$ as in \eqref{inverse}.
Equation (\ref{condition2}) can be rewritten in an abbreviated form with implied integrals on repeated arguments as
\begin{equation}
\mathbb{D}^{-1}_{11}(\mathbf{a}^{\prime},\mathbf{a}^{\prime\prime})=\mathbb{D}^{-1}_{21}(\mathbf{a}^{\prime},\hat{\mathbf{a}})\cdot \mathbb{D}^{22}(\hat{\mathbf{a}},\check{\mathbf{a}})
\cdot\mathbb{D}^{-1}_{21}(\check{\mathbf{a}},\mathbf{a}^{\prime\prime})\,.
\label{condition2-1}
\end{equation}
In order to obtain $\mathbb{D}$ and its inverse, we need the functional derivatives of $D^{1}$ and $D^2$. These are obtained directly by writing these local constraints as in \eqref{inverse}, yielding
\bal
\frac{\delta D^{1}(\mathbf{a}^{\prime})}{\delta q^{i}(\mathbf{a})} & = -A_{i}^{k}\frac{\partial}{\partial a^{k}}\frac{\delta(\mathbf{a}-\mathbf{a}^{\prime})}{\mathcal{J}} \,,
\label{D1q}\\
\frac{\delta D^{1}(\mathbf{a}^{\prime})}{\delta\pi_{i}(\mathbf{a})} & = 0
\label{D1pi}
\,,
\eal
where use has been made of \eqref{dedet}, and
\bal
\frac{\delta D^{2}(\mathbf{a}^{\prime})}{\delta q^{i}(\mathbf{a})} & = \frac{\partial}{\partial a^{u}}\left(\frac{A_{i}^{k}A_{\ell}^{u}}{\mathcal{J}^{2}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\delta(\mathbf{a}-\mathbf{a}^{\prime})\right) \,,
\label{D2pi}\\
\frac{\delta D^{2}\left(\mathbf{a}^{\prime}\right)}{\delta\pi_{i}(\mathbf{a})} & = -\frac{\, \eta^{ij}}{\rho_{0}}\frac{\partial}{\partial a^{m}}\left(\frac{A_{j}^{m}}{\mathcal{J}}\delta\left(\mathbf{a}-\mathbf{a}^{\prime}\right)\right)\,,
\label{D2q}
\eal
where use has been made of (\ref{eq:AJvar}) and recalling we have \eqref{dAda} at our disposal.
Let us now insert \eqref{D1q}, \eqref{D1pi}, \eqref{D2pi}, and \eqref{D2q} into the canonical
Poisson bracket \eqref{cbkt}, to obtain
\bal
\mathbb{D}^{12}(\mathbf{a},\mathbf{a}^{\prime}) &= \{D^{1}(\mathbf{a}),D^{2}(\mathbf{a}^{\prime})\}
\nonumber\\
&= -\frac{A_{i}^{\ell}}{\mathcal{J}}\frac{\partial}{\partial a^{\ell}}\left(\frac{\eta^{ij}}{\rho_{0}}
{A_{j}^{k}} \frac{\partial}{\partial a^{k}}\left(\frac{\delta(\mathbf{a}-\mathbf{a}^{\prime})}{\mathcal{J}}\right)\right) \,,
\label{D12}
\eal
which corresponds to the symmetric matrix $\mathbb{S}$ of \eqref{DAB} and \eqref{fpDAB} and
\bal
\mathbb{D}^{22}({\mathbf{a}},\mathbf{a}') &=
\{D^{2}({\mathbf{a}}),D^{2}(\mathbf{a}')\}
\nonumber
\\
& = \frac{A_{i}^{k}A_{\ell}^{u}}{\mathcal{J}^{2}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\frac{\partial}{\partial a^{u}}\left[\frac{ \eta^{ij}}{\rho_{0}}\frac{\partial}{\partial a^{m}}\left(\frac{A_{j}^{m}}{\mathcal{J}}\delta\left(\mathbf{a}-\mathbf{a}'\right)\right)\right]
\nonumber\\
& \hspace{1.5cm} - \frac{A_{i}^{m}}{\mathcal{J}}\frac{\partial}{\partial a^{m}}
\left[\frac{\eta^{ij}}{\rho_{0}}\frac{\partial}{\partial a^{u}}\left(\frac{A_{j}^{k}A_{\ell}^{u}}{\mathcal{J}^{2}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\delta(\mathbf{a}-\mathbf{a}')
\right)\right] \,,
\label{eq:C22-2}
\eal
which corresponds to the antisymmetric matrix $\mathbb{A}$ of \eqref{D2D2}. Observe the symmetries corresponding to the matrices $\mathbb{S}$ and $\mathbb{A}$, respectively, are here
\bal
\int \! d^3a' \, \mathbb{D}^{12}(\mathbf{a},\mathbf{a}')\, \phi (\mathbf{a}')&=\int d^3a' \mathbb{D}^{12}(\mathbf{a}',\mathbf{a})\, \phi (\mathbf{a}') \,,
\nonumber\\
\int \! d^3a' \, \mathbb{D}^{22}(\mathbf{a},\mathbf{a}')\, \phi (\mathbf{a}')&=-\int d^3a' \mathbb{D}^{22}(\mathbf{a}',\mathbf{a})\, \phi (\mathbf{a}')\,,
\nonumber
\eal
for all functions $\phi$. The first follows from integration by parts, while the second is obvious from its definition.
Using (\ref{inverse}), the first condition of \eqref{condition} is
\begin{equation}
\int\!d^{3}a^{\prime\prime}\,\,\mathbb{D}^{12}(\mathbf{a}^{\prime},\mathbf{a^{\prime\prime}})\,\mathbb{D}^{-1}_{21}(\mathbf{a^{\prime\prime}},\hat{\mathbf{a}})=\delta(\mathbf{a}^{\prime}-\hat{\mathbf{a}})\,,
\label{inverseA}
\end{equation}
which upon substitution of \eqref{D12} and integration gives
\begin{equation}
-\frac{A_{i}^{\ell}}{\mathcal{J}} \frac{\partial}{\partial a^{\ell}}
\left[ \frac{\eta^{ij}}{\rho_{0}} A_{j}^{k}\, \frac{\partial}{\partial a^{k}}\left(\frac{\mathbb{D}_{21}^{-1}(\mathbf{a},\mathbf{a}^{\prime\prime})}{\mathcal{J}}\right)\right]=\delta(\mathbf{a}-\mathbf{a}^{\prime\prime})\,.
\label{eq:ddDinv21}
\end{equation}
We introduce the formally self-adjoint operator (cf.\ \eqref{dAda})
\begin{equation}
\Delta_{\rho_{0}}f:=\frac{A_{i}^{\ell} }{\mathcal{J}}\frac{\partial}{\partial a^{\ell}}\left[ \frac{\eta^{ij}}{\rho_{0}} A_{j}^{k} \frac{\partial}{\partial a^{k}}\left(\frac{f}{\mathcal{J}}\right)\right]\,,
\end{equation}
i.e., an operator that satisfies
\begin{equation}
\int\!d^3a\, f(\mathbf{a}) \, \Delta_{\rho_0} g(\mathbf{a}) =\int\!d^3a \, g(\mathbf{a}) \, \Delta_{\rho_0} f(\mathbf{a}) \,,
\end{equation}
a property inherited by its inverse $\Delta_{\rho_{0}}^{-1}$. Thus we can rewrite equation (\ref{eq:ddDinv21}) as
\begin{equation}
\mathbb{D}_{21}^{-1}(\mathbf{a},\mathbf{a}^{\prime\prime})
=-G_{0}\left(\mathbf{a},\mathbf{a}^{\prime\prime}\right)=-\Delta_{\rho_{0}}^{-1}\delta(\mathbf{a}-\mathbf{a}^{\prime\prime})\,,
\label{eq:Dinv21-1}
\end{equation}
where $G_{0}$ represents the Green function associated with (\ref{eq:ddDinv21}).
In order to obtain $\mathbb{D}^{-1}_{21}$, we find it convenient to transform \eqref{eq:Dinv21-1} to Eulerian variables. Using $\mathbf{x}=\mathbf{q}(\mathbf{a},t)$ we find
\begin{equation}
\frac{\mathbb{D}^{-1}_{21}(\mathbf{a},\mathbf{a}^{\prime})}{\mathcal{J}}=-G(\mathbf{x},\mathbf{x}^{\prime})=-G(\mathbf{q}(\mathbf{a}),\mathbf{q}(\mathbf{a}^{\prime}))\,,
\end{equation}
where $G$ satisfies
\begin{equation}
\nabla\cdot\left(\frac{1}{\rho}\nabla G\right)=-\Delta_{\rho}\frac{\mathbb{D}^{-1}_{21}(\mathbf{a},\mathbf{a}^{\prime})}{\mathcal{J}}=\mathcal{J}\delta(\mathbf{x}-\mathbf{x}^{\prime}).
\end{equation}
Here use has been made of identities \eqref{dAda} and \eqref{ELgrad}.
As noted in Section \ref{ssec:bgnd}, under physically reasonable conditions, the operator
\begin{equation}
\Delta_{\rho}f=\Delta_{\rho_{0}}\left(\mathcal{J}f\right)=\nabla\cdot\left(\frac1{\rho} \nabla f\right)
\label{nabrho}
\end{equation}
has an inverse. Thus we write
\begin{equation}
\mathbb{D}^{-1}_{21}(\mathbf{a},\mathbf{a}^{\prime})=-\mathcal{J} \Delta_{\rho}^{-1}
\Big(\mathcal{J}\,
\delta\big(\mathbf{q}(\mathbf{a},t)-\mathbf{q}(\mathbf{a}',t)\big)
\Big)\,.
\label{eq:Dinv21}
\end{equation}
Now, using $\mathbb{D}^{-1}_{21}=-\mathbb{D}^{-1}_{12}$, the element $\mathbb{D}^{-1}_{11}$ follows directly from (\ref{condition2}).
For convenience we write the Dirac bracket of \eqref{eqn:DBF} as follows:
\begin{equation}
\left\{ F,G\right\} _{*}=\left\{ F,G\right\} -\left[F,G\right]^{D} \,,
\label{eq:dbkt1}
\end{equation}
where
\begin{align}
\left[F,G\right]^{D}& :=\sum_{a,b=1}^{2}\left[ F,G\right]^D_{ab}=\int\!d^{3}a\,\int\!d^{3}a^{\prime}\,\,\left\{ F,D^{a}\left(\mathbf{a}\right)\right\} \mathbb{D}_{ab}^{-1}(\mathbf{a},\mathbf{a}^{\prime})\left\{ D^{b}\left(\mathbf{a}^{\prime}\right),G\right\} \,.
\label{eq:FG}
\end{align}
Because $\mathbb{D}_{22}^{-1}=0$ and $\left[ F,G\right]^D _{12}=-[ G,F]^D_{21}$, we only need to calculate $[F,G]^D_{11}$
and $[F,G]^D_{21}$.
As above, we substitute \eqref{D1q}, \eqref{D1pi}, \eqref{D2pi}, and \eqref{D2q} into the bracket (\ref{cbkt})
and obtain
\begin{align}
\left\{ F,D^{1}\left(\mathbf{a}\right)\right\} & =-\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\delta F}{\delta\pi_{i}}\right),\label{eq:F-D1}\\
\left\{ F,D^{2}\left(\mathbf{a}\right)\right\} & = \frac{A_{\ell}^{k}}{\mathcal{J}}
\frac{\partial}{\partial a^{k}}\left(\frac{\eta^{i\ell } }{\rho_{0}}\frac{\delta F}{\delta q^{i}}\right)
+\frac{A_{i}^{k}}{\mathcal{J}} \frac{A_{\ell}^{u}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\frac{\partial}{\partial a^{u}}\left(\frac{\delta F}{\delta\pi_{i}}\right)\,.
\label{eq:F-D2}
\end{align}
Then, exploiting the antisymmetry of the Poisson bracket, it is straightforward
to calculate analogous expressions for the terms $\left\{ D^{1,2},G\right\}$.
We first analyze the operator
\begin{align}
[F,G]^D_{11} & =\!\int\!d^{3}a\!\!\int\!\!d^{3}a^{\prime}\!\!\int\!d^{3}\hat{a}\!\!\int\!d^{3}\check{a} \left\{ F,D^{1}\!\left(\mathbf{a}\right)\right\} \mathbb{D}_{21}^{-1}(\mathbf{a},\hat{\mathbf{a}})\,\mathbb{D}^{22}(\hat{\mathbf{a}},\check{\mathbf{a}})\,\mathbb{D}_{21}^{-1}(\check{\mathbf{a}},\mathbf{a}^{\prime})\left\{ D^{1}\!\left(\mathbf{a}^{\prime}\right),G\right\} ,
\end{align}
where we used the second condition of \eqref{condition} to replace $\mathbb{D}_{11}^{-1}$.
Upon inserting \eqref{eq:Dinv21-1} and \eqref{eq:F-D1}, this equation can
be rewritten as
\begin{align}
[F,G]^D_{11} & =-\int\!d^{3}a\,\int\!d^{3}a^{\prime}\,\int\!d^{3}\hat{a}\,\,\int\!d^{3}\check{a}
\nonumber \\
&\hspace{1cm} \left[\frac{A^h_{j}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\left(\frac{\delta F}{\delta\pi_{j}}\right)\Delta_{\rho_{0}}^{-1}\delta(\mathbf{a}-\hat{\mathbf{a}})\right]_{\mathbf{a}=\mathbf{a}}\!\!\!
\mathbb{D}^{22}(\hat{\mathbf{a}},\check{\mathbf{a}})\,\left[\frac{A_{r}^{s}}{\mathcal{J}}\frac{\partial}{\partial a^{s}}\left(\frac{\delta G}{\delta\pi_{r}}\right)\Delta_{\rho_{0}}^{-1}\delta(\check{\mathbf{a}}-\mathbf{a})\right]_{\mathbf{a}=\mathbf{a}^{\prime}} ,
\end{align}
where the subscripts on the right delimiters indicate that $\mathbf{a}$
is to be replaced after the derivative operations, including those that occur in $\mathcal{J}$
and $A_{i}^{j}$.
Integrating this expression by parts
with respect to $\mathbf{a}$ and $\mathbf{a}^{\prime}$ yields
\begin{align}
[F,G]^D_{11} & =-\int\!d^{3}\hat{a}\,\,\int\!d^{3}\check{a}\,\,\left[\Delta_{\rho_{0}}^{-1}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\left(\frac{\delta F}{\delta\pi_{j}}\right)\right)\right]_{\mathbf{a}=\hat{\mathbf{a}}}\!\!\!\mathbb{D}^{22}(\hat{\mathbf{a}},\check{\mathbf{a}})\,\left[\Delta_{\rho_{0}}^{-1}\left(\frac{A_{r}^{s}}{\mathcal{J}}\frac{\partial}{\partial a^{s}}\left(\frac{\delta G}{\delta\pi_{r}}\right)\right)\right]_{\mathbf{a}=\check{\mathbf{a}}} ,
\label{eq:FG11}
\end{align}
and then substituting \eqref{eq:C22-2} gives
\begin{align}
[F,G]^D_{11} & =-\!\int\!d^{3}\hat{a}\!\!\int\!d^{3}\check{a}
\left[\Delta_{\rho_{0}}^{-1}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\left(\frac{\delta F}{\delta\pi_{j}}\right)\right)\right]_{\mathbf{a}=\hat{\mathbf{a}}}\!
\left\{\frac{A_{i}^{k} A_{\ell}^{u}}{\mathcal{J}^2}\frac{\partial}{\partial a^{k}}
\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\frac{\partial}{\partial a^{u}}\left[\frac{\eta^{in}}{\rho_{0}}\frac{\partial}{\partial a^{m}}\left(\frac{A_{n}^{m}}{\mathcal{J}}\delta\left(\mathbf{a}-\check{\mathbf{a}}\right)\right)\right]\right.\nonumber \\
& \hspace{.25cm} \left.- \frac{A_{i}^{m}}{\mathcal{J}} \frac{\partial}{\partial a^{m}}\left[\frac{\eta^{in}}{\rho_{0}}\frac{\partial}{\partial a^{u}}\left(\frac{A_{n}^{k}}{\mathcal{J}}\frac{A_{\ell}^{u}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\delta(\mathbf{a}-\check{\mathbf{a}})\right)\right]
\right\} _{\mathbf{a}=\hat{\mathbf{a}}}\,
\left[\Delta_{\rho_{0}}^{-1}\left(\frac{A_{r}^{s}}{\mathcal{J}} \frac{\partial}{\partial a^{s}}\left(\frac{\delta G}{\delta\pi_{r}}\right)\right)\right]_{\mathbf{a}=\check{\mathbf{a}}}.
\label{eq:FG11-bis}
\end{align}
Then, by means of integrations by parts we can remove the derivatives
from the term $\delta(\mathbf{a}-\check{\mathbf{a}})$ and perform
the integral. After relabeling the integration variable as $\mathbf{a}$
to simplify the notation, (\ref{eq:FG11-bis}) becomes
\begin{align}
[F,G]^D_{11} & =\int\!d^{3}a\,\, \mathlarger{\Bigg\{}
\rho_{0}\, \eta^{ui}\frac{A_{u}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)
\left(\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{\ell}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{i}
- \left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{i}\left.
\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{\ell}\right)
\nonumber \\
&\hspace{1cm} + \, \eta^{ni} {A_{\ell}^{u}} \frac{\partial}{\partial a^{u}}\left[\frac{A_{n}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\right]\left[\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{i}\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}\left(\frac{\delta F}{\delta\pi_{j}}\right)\right)\right.
\nonumber \\
& \hspace{5cm}\left.-\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{i}\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\left(\frac{\delta G}{\delta\pi_{j}}\right)\right)\right]
\mathlarger{\Bigg\}}\,,
\label{eq:FG11-L}
\end{align}
where we introduced the projection operator
\begin{equation}
\left(\mathbb{P}_{\rho_0\perp}\right)^{i}_j\, {z}^j=
\frac{\eta^{i\ell}}{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\, z^{j}\right)\right]
=: \mathbb{P}_{\rho_0\perp}\mathbf{z}\,\big|^{i}\,,
\label{eq:P0perp}
\end{equation}
where in the last equality we defined a shorthand for convenience; {thus,
\begin{equation}
\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta \boldsymbol{\pi}}\,\right|_{\ell}:= \frac1{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\,
\frac{\delta F}{\delta \pi_{j}}\right)\right]\,.
\end{equation}
}
It is straightforward to prove that $\mathbb{P}_{\rho_0\perp}$ represents
a projection, i.e.\ $\mathbb{P}_{\rho_0\perp}\left(\mathbb{P}_{\rho_0\perp}\mathbf{z}\right)=\mathbb{P}_{\rho_0\perp}\mathbf{z}$ for each $\mathbf{z}$, which in terms of indices would have an $i$th component given by $(\mathbb{P}_{\rho_0\perp})^i_j (\mathbb{P}_{\rho_0\perp})^j_k \,z^k = (\mathbb{P}_{\rho_0\perp})^i_ k\, z^k$.
Also, $\mathbb{P}_{\rho_0\perp}$ is formally self-adjoint with respect to the following weighted inner product:
\begin{equation}
\int d^3a \, \rho_0 \, w_i \, (\mathbb{P}_{\rho_0\perp})^i_j \, z^j = \int d^3a \, \rho_0 \, z_i \, (\mathbb{P}_{\rho_0\perp})^i_j \, w^j \,.
\label{PperpSA}
\end{equation}
The projection operator complementary to $\mathbb{P}_{\rho_0\perp}$ is given by
\begin{equation}
\mathbb{P}_{\rho_0}= I - \mathbb{P}_{\rho_0\perp}\,,
\label{P}
\end{equation}
where $I$ is the identity.
Now let us return to our evaluation of $[ F,G]_D$ and analyze the contribution
\begin{align}
[F,G]^D_{21} & =\int\!d^{3}a\,\int\!d^{3}a^{\prime}\,\,\left\{ F,D^{2}\left(\mathbf{a}\right)\right\} \mathbb{D}_{21}^{-1}(\mathbf{a},\mathbf{a}^{\prime})\left\{ D^{1}\left(\mathbf{a}^{\prime}\right),G\right\} .
\end{align}
Using \eqref{eq:Dinv21}, \eqref{eq:F-D1}, and \eqref{eq:F-D2}, this
equation can be rewritten as
\begin{align}
[F,G]^D_{21} & =-\int\!d^{3}a\,\int\!d^{3}a^{\prime}\,\,\left[
\frac{A_{i}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\frac{\eta^{in}}{\rho_{0}}\frac{\delta F}{\delta q^{n}}\right)+\frac{A_{i}^{k}}{\mathcal{J}} \frac{A_{\ell}^{u}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\frac{\partial}{\partial a^{u}}\left(\frac{\delta F}{\delta\pi_{i}}\right)\right]\nonumber \\
&\hspace{3.5cm} \times \Delta_{\rho_{0}}^{-1}\delta(\mathbf{a}-\mathbf{a}^{\prime})\left[\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\left(\frac{\delta G}{\delta\pi_{j}}\right)\right]_{\mathbf{a}=\mathbf{a}^{\prime}}
\end{align}
and, integrating by parts to simplify the $\delta(\mathbf{a}-\mathbf{a}^{\prime})$
term, results in
\begin{align}
[F,G]^D_{21} & =\int\!d^{3}a\,\, \mathlarger{\Bigg\{}
\frac{\delta F}{\delta q^{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}
+\rho_{0}\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\frac{\delta F}{\delta\pi_{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{\ell}\nonumber \\
&\hspace{1cm} + A_{\ell}^{u}\frac{\partial}{\partial a^{u}}\left[\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\right]\,\frac{\delta F}{\delta\pi_{i}}\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}\left(\frac{\delta G}{\delta\pi_{j}}\right)\right)
\mathlarger{\Bigg\}}
\,.
\label{eq:FG21-L}
\end{align}
We can now combine the operators $[F,G]^D_{11}$,
$[F,G]^D_{21}$, and $[F,G]^D_{12}=-[G,F]^D_{21}$,
given by (\ref{eq:FG11-L}) and (\ref{eq:FG21-L}), to calculate the
Dirac bracket (\ref{eq:dbkt1}).
First, we rewrite (\ref{eq:FG}) as
\begin{align}
[F,G]^D & =\int\!d^{3}a\,\, \mathlarger{\Bigg\{}
\frac{\delta F}{\delta q^{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}-\frac{\delta G}{\delta q^{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}
\nonumber \\
& +\rho_{0} \frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,
\left(\left.\mathbb{P}_{\rho_0} \frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{\ell}-\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{\ell}\right)
\nonumber \\
& + A_{\ell}^{u}\frac{\partial}{\partial a^{u}}\left[\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\right]\,\left[\left.\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}\left(\frac{\delta G}{\delta\pi_{j}}\right)\right)\right.
\nonumber \\
& \hspace{4cm}\left.-\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}\left(\frac{\delta F}{\delta\pi_{j}}\right)\right)\right]
\mathlarger{\Bigg\}}
\,.
\label{eq:FG-mid}
\end{align}
Using the identity of \eqref{IDen} with $z^\ell$ set to $\pi^\ell/\rho_0$,
\begin{equation}
A_{\ell}^{u}\frac{\partial}{\partial a^{u}}\left[\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\right]= A_{i}^{k}\frac{\partial}{\partial a^{k}}\left[\frac{A_{\ell}^{u}}{\mathcal{J}}\frac{\partial}{\partial a^{u}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\right]\,,
\end{equation}
and integrating by parts, (\ref{eq:FG-mid}) becomes
\begin{align}
[F,G]^D & =\int\!d^{3}a\,\, \mathlarger{\Bigg\{}
\frac{\delta F}{\delta q^{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}-\frac{\delta G}{\delta q^{i}}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}
\nonumber \\
& +\rho_{0}\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\left(\left.
\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{\ell}
- \left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{\ell}\right)
\nonumber \\
& -\rho_{0} \frac{A_{\ell}^{u}}{\mathcal{J}}\frac{\partial}{\partial a^{u}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\left(\left.
\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{i}
-\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{i}\right)
\mathlarger{\Bigg\}}\,,
\label{eq:FG-final}
\end{align}
where we used
\begin{equation}
A_{i}^{k}\frac{\partial}{\partial a^{k}}\mathbb{P}_{\rho_0}\mathbf{z}\,\big|^{i}=A_{i}^{k}\frac{\partial}{\partial a^{k}} \left(\mathbb{P}_{\rho_0}\right)^i_j z^j= 0\,,\qquad\mathrm{for\ all}\ \ \mathbf{z}\,,
\label{Ldiv}
\end{equation}
which follows from the definitions \eqref{eq:P0perp}, viz.
\begin{equation}
\frac{A_{i}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\mathbb{P}_{\rho_0\perp}\right)^i_j z^{j}=
\frac{A_{i}^{k}}{\mathcal{J}}\frac{\partial z^{i}}{\partial a^{k}} \,,
\label{divPperp}
\end{equation}
and \eqref{P}. Also, upon inserting $\mathbb{P}_{\rho_0\perp}= I - \mathbb{P}_{\rho_0}$ in the last line of \eqref{eq:FG-final}, symmetry implies we can drop the $\mathbb{P}_{\rho_0\perp}$.
Finally, upon substituting (\ref{eq:FG-final}) into (\ref{eq:dbkt1}), we obtain
\begin{align}
\left\{ F,G\right\} _{*} & =\int\!d^{3}a\,\, \mathlarger{\Bigg\{}
\frac{\delta F}{\delta q^{i}}\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}-\frac{\delta G}{\delta q^{i}}\left.\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}
\nonumber \\
& - \rho_{0}\frac{A_{i}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\left(\left.
\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|_{\ell}
- \left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|_{\ell}\right)
\nonumber \\
& +\rho_{0} \frac{A_{\ell}^{k}}{\mathcal{J}}\frac{\partial}{\partial a^{k}}\left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,\left(\left.
\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}
\frac{\delta G}{\delta{\pi^i}}
-\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}
\frac{\delta F}{\delta \pi^i} \right)
\mathlarger{\Bigg\}}\,.
\label{eq:dbkt2}
\end{align}
Once more inserting $\mathbb{P}_{\rho_0\perp}= I- \mathbb{P}_{\rho_0}$, rearranging, and reindexing gives
\begin{align}
\left\{ F,G\right\} _{*} & =- \int\!d^{3}a\,\rho_0 \, \mathlarger{\Bigg\{}
\frac1{\rho_0}\frac{\delta G}{\delta q^{i}}\left.\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{i}
- \frac1{\rho_0} \frac{\delta F}{\delta q^{i}}\left.\mathbb{P}_{\rho_0}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{i}
+ \mathcal{A}_{mn}
\left.\mathbb{P}_{\rho_0\perp} \frac{\delta F}{\delta\boldsymbol{\pi}}\right|^m
\left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{n}
\nonumber \\
&
\hspace{ 3cm} + \mathcal{T}_{mn}\,
\left(
\frac{\delta F}{\delta{\pi_m}} \left.\mathbb{P}_{\rho_0\perp}\frac{\delta G}{\delta\boldsymbol{\pi}}\right|^{n}
- \frac{\delta G}{\delta \pi_m}
\left.\mathbb{P}_{\rho_0\perp}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^{n}
\right)
\mathlarger{\Bigg\}}\,,
\label{eq:dbkt-2}
\end{align}
where
\begin{equation}
\mathcal{A}_{nm}:= \eta_{\ell m} D^\ell_n- \eta_{\ell n} D^\ell_m
\qquad
\mathrm{and}
\qquad
\mathcal{T}_{mn}:= \eta_{\ell n} D^\ell_m + \eta_{mn} D^2 \,,
\label{CB}
\end{equation}
with
\begin{equation}
D_m^{\ell}= \frac{A_{m}^{k}}{\mathcal{J}} \frac{\partial}{\partial a^{k}} \left(\frac{\pi^{\ell}}{\rho_{0}}\right)\,.
\end{equation}
Note the trace $D^\ell_\ell= D^2$, which we will eventually set to zero. Equation \eqref{eq:dbkt-2} gives the Dirac bracket for the incompressibility holonomic constraint. This bracket with the Hamiltonian
\begin{equation}
H=\int \! d^3a \, \frac{|\boldsymbol{\pi}|^2}{2\rho_0} =\int \! d^3a \, \eta^{mn} \frac{\pi_m \pi_n}{2\rho_0} \,,
\label{ham}
\end{equation}
produces dynamics that fixes $\mathcal{J}$ and thus enforces incompressibility provided the constraint $D^2=0$ is used as an initial condition.
For MHD we add to $H$ the following:
\begin{equation}
H_B=\int \! d^3a \, \eta_{mn} \frac{B_0^j B_0^k }{2\mathcal{J}}\,\frac{\partial q^m}{\partial a^j} \frac{\partial q^n}{\partial a^k} \,.
\end{equation}
We note, any Hamiltonian that is consistent with \eqref{constraints-1-1} can be used to define a constrained flow.
Proceeding to the equations of motion, we first calculate $\dot{q}^i$,
\bal
\dot{q}^i&= \{q^i,H\}_{*}
= \left(\mathbb{P}_{\rho_0}\right)^i_j \frac{\delta H}{\delta \pi_j}
= \frac{\delta H}{\delta \pi_i} - \frac{\eta^{i\ell}}{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}} \frac{\delta H}{\delta \pi_j}\right)\right]
\nonumber \\
& = \frac{\pi^i}{\rho_0} - \frac{\eta^{i\ell}}{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}} \frac{\pi^j}{\rho_0}\right)\right]\,.
\label{LDeomQ}
\eal
The equation for $\dot{\pi}_i$ is more involved. Using the adjoint property of \eqref{PperpSA}, which is valid for both $\mathbb{P}_{\rho_0\perp}$ and
$\mathbb{P}_{\rho_0}$, we obtain
\bal
\dot{\pi_i}&= \{\pi_i,H\}_{*} =- \rho_0 \left( \mathbb{P}_{\rho_0}\right)^j_i \frac1{\rho_0}
\frac{\delta H}{\delta q^j}
- \rho_{0}\, \left(\mathbb{P}_{\rho_0\perp}\right)^m_i
\left(
\mathcal{A}_{mn}
\left(\mathbb{P}_{\rho_0\perp} \right)^n_k
\frac{\delta H}{\delta \pi_k}
\right)
%
\nonumber\\
&\hspace{2cm} +\rho_{0}\, \left(\mathbb{P}_{\rho_0\perp} \right)^n_i
\left(
\mathcal{T}_{mn}
\frac{\delta H}{\delta \pi_m}
\right)
- \rho_{0}\, \mathcal{T}_{in} \left(\mathbb{P}_{\rho_0\perp} \right)^n_k
\frac{\delta H}{\delta \pi_k}
\nonumber\\
& =
- \rho_{0}\, \left(\mathbb{P}_{\rho_0\perp}\right)^m_i
\left(
\mathcal{A}_{mn}
\left(\mathbb{P}_{\rho_0\perp} \right)^n_k
\frac{\pi^k}{\rho_0}
\right)
%
+ \rho_{0}\, \left(\mathbb{P}_{\rho_0\perp} \right)^n_i
\left(
\mathcal{T}_{mn}
\frac{\pi^m}{\rho_0}
\right)
- \rho_{0}\, \mathcal{T}_{in} \left(\mathbb{P}_{\rho_0\perp} \right)^n_k
\frac{\pi^k}{\rho_0}\,,
\label{LDeomP}
\eal
which upon substitution of the definitions of $\mathbb{P}_{\rho_0}$, $\mathcal{A}_{mn}$, and $\mathcal{T}_{mn}$ of \eqref{eq:P0perp} and \eqref{CB} yields a complicated nonlinear equation.
Equations \eqref{LDeomQ} and \eqref{LDeomP} are infinite-dimensional versions of the finite-dimensional systems of \eqref{eomq} and \eqref{eomp} considered in Section \ref{sssec:HCD}. There, equations \eqref{eomq} and \eqref{eomp} were reduced to \eqref{eomqr} and \eqref{eompr} upon enforcing the holomomic constraint by requiring that initially $D^2=0$. Similarly we can enforce the vanishing of $D^2$ of \eqref{constraints-1-1}, which is compatible with the Hamiltonian \eqref{ham}. Instead of addressing this evaluation now, we find the meaning of various terms is much more transparent when written in terms of Eulerian variables, which we do in Section \ref{ssec:EDcon}. We then return to these Lagrangian equations in Section \ref{ssec:comparison} and make comparisons. Nevertheless, the solution of equations \eqref{LDeomQ} and \eqref{LDeomP}, $\mathbf{q}(\mathbf{a},t)$, with the initial conditions $D^1=-\ln \rho_0$ and $D^2=0$, is a volume preserving transformation at any time $t$.
\subsection{Eulerian-Dirac constraint theory}
\label{ssec:EDcon}
Because we chose the form of constraints $D^{1,2}$ of \eqref{D1} and \eqref{constraints-1-1} to be Eulerianizable, it follows that we can transform easily the results of Section \ref{sssec:LDholo} into Eulerian form. This we do in Section \ref{sssec:LDEP}. Alternatively, we can proceed as in \citet{turski99,turski01,pjmTC09,pjmCGBT13,pjmLB09}, starting from the Eulerian noncanonical theory of Section \ref{ssec:LtoE} and directly construct a Dirac bracket with Eulerian constraints. This is a valid procedure because Dirac's construction works for noncanonical Poisson brackets, as shown, e.g., in \citet{pjmLB09}, but it does not readily allow for advected density. This direct method with uniform density is reviewed in Section \ref{sssec:EDD}, where it is contrasted with the results of Section \ref{sssec:LDEP}.
\subsubsection{Lagrangian-Dirac constraint theory in the Eulerian picture}
\label{sssec:LDEP}
In a manner similar to that used to obtain \eqref{deFq} and \eqref{deFpi}, we find the functional derivatives transform as
\begin{equation}
\frac{\delta F}{\delta\pi_{i}}=\frac{1}{\rho}\frac{\delta \bar{F}}{\delta \varv_{i}},\quad\frac{1}{\rho_{0}}\frac{\delta F}{\delta q^{i}}=\frac{\partial}{\partial x^{i}}\frac{\delta \bar{F}}{\delta\rho}-\frac{1}{\rho}\frac{\delta \bar{F}}{\delta s}\frac{\partial s}{\partial x^{i}}-\frac{1}{\rho}\frac{\delta \bar{F}}{\delta \varv_{\ell}}\frac{\partial \varv_{\ell}}{\partial x^{i}} \,,
\label{chain}
\end{equation}
where the expressions on the left of each equality are clearly Lagrangian variable quantities, while on the right they are Eulerian quantities represented in terms of Lagrangian variables.
Substituting these expressions into (\ref{eq:dbkt}) and dropping the bar on $F$ and $G$ gives the following bracket in terms of the Eulerian variables:
\begin{align}
\left\{ F,G\right\} _{*} & =-\int\!d^{3}x\,\, \mathlarger{\Bigg\{}
\left(\left.
\mathcal{P}_{\rho}\frac{\delta F}{\delta\mathbf{v}}\right|^{i} \frac{\partial}{\partial x^{i}}\frac{\delta G}{\delta\rho}
- \left.\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}\right|^{i}\frac{\partial}{\partial x^{i}}\frac{\delta F}{\delta\rho}\right)
\nonumber \\
& +\frac{1}{\rho}\frac{\partial s}{\partial x^{i}}
\left(\frac{\delta F}{\delta s}\left.\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}\right|^{i}
- \frac{\delta G}{\delta s}\left.\mathcal{P}_{\rho}\frac{\delta F}{\delta\mathbf{v}}\right|^{i}\right)
\nonumber \\
& +\frac{1}{\rho}\frac{\partial \varv^{\ell}}{\partial x^{i}}\left(\left.\mathcal{P}_{\rho}\frac{\delta F}{\delta\mathbf{v}}\right|_{\ell}\left.\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}\right|^{i}
-\left.\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}\right|_{\ell}\left.\mathcal{P}_{\rho}\frac{\delta F}{\delta\mathbf{v}}\right|^{i}\right)
\nonumber \\
& +\frac{1}{\rho}\frac{\partial \varv^{\ell}}{\partial x^{\ell}}\,\left(\left.\mathcal{P}_{\rho}\frac{\delta F}{\delta\mathbf{v}}\right|^{i}\frac{\delta G}{\delta \varv_{i}}
-\left.\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}\right|^{i}\frac{\delta F}{\delta \varv_{i}}\right)
\mathlarger{\Bigg\}}\,,
\label{eq:dbkt-3}
\end{align}
where we used the relations \eqref{vol} and \eqref{ELgrad}
and we introduced the Eulerian projection operator
\begin{equation}
\left.\mathcal{P}_{\rho} \frac{\delta F}{\delta \mathbf{v}}\right|^i = (\mathcal{P}_{\rho})^i_j\frac{\delta F}{\delta \varv_j}
= \left. \rho\mathbb{P}_{\rho_0}\frac{\delta F}{\delta\boldsymbol{\pi}}\right|^i
\qquad
\mathrm{and}
\qquad
\left.\mathcal{P}_{\rho} \frac{\delta F}{\delta \mathbf{v}}\right|_i =\eta_{ij}\left.\mathcal{P}_{\rho} \frac{\delta F}{\delta \mathbf{v}}\right|^j \,,
\end{equation}
with
\begin{equation}
\left(\mathcal{P}_{\rho}\right)^i_j z^j=\delta^i_j - \eta^{ik} \frac{\partial}{\partial x^k} \left[\Delta_\rho^{-1} \frac{\partial }{\partial x^j}
\left(\frac{z^j}{\rho}\right)\right]\,,
\end{equation}
which is easily seen to satisfy $(\mathcal{P}_{\rho})^i_j (\mathcal{P}_{\rho})^j_k=(\mathcal{P}_{\rho})^i_k$.
Observe, like its Lagrangian counterpart, $\mathcal{P}_{\rho}$ is formally self-adjoint; however, this time we found it convenient to define the projection in such a way that the self-adjointness is with respect to a different weighted inner product, viz.
\begin{equation}
\int \frac{d^3x}{\rho} \, w_i \, (\mathcal{P}_{\rho })^i_j \, z^j = \int \frac{d^3x}{\rho} \, z_i \, (\mathcal{P}_{\rho})^i_j \, w^j \,.
\end{equation}
In terms of usual cartesian vector notation
\begin{equation}
\mathcal{P}_{\rho}\frac{\delta G}{\delta\mathbf{v}}=\frac{\delta G}{\delta\mathbf{v}}-\nabla\Delta_{\rho}^{-1}\nabla\cdot\left(\frac{1}{\rho}\frac{\delta G}{\delta\mathbf{v}}\right) \,.
\label{eq:Peul}
\end{equation}
Upon writing $\mathcal{P}_{\rho}=I-\mathcal{P}_{\rho \perp}$ and decomposing an arbitrary vector field as
\[
\mathbf{z}=-\nabla\Phi+\rho\nabla\times\mathbf{A},
\]
this projection operator yields the component $\mathcal{P}_{\rho}\mathbf{z}=\rho\nabla\times\mathbf{A}$.
{Therefore, if $\nabla\rho\times\mathbf{A}=0$, then this operator projects into the space of incompressible vector fields.} For convenience we introduce the associated projector
\begin{equation}
\mathbb{P}_\rho \mathbf{v} := \mathbf{v} -\frac1{\rho} \nabla\Delta_{\rho}^{-1}\nabla\cdot \mathbf{v}
= \frac1{\rho} {\mathcal{P}_{\rho}}(\rho \mathbf{v}) \,,
\label{Pmap}
\end{equation}
which has the desirable property
\begin{equation}
\nabla\cdot (\mathbb{P}_\rho \mathbf{v})=0 \quad \forall\ \mathbf{v}
\qquad\mathrm{compared\ \ to}\qquad
\nabla\cdot \left(\frac1{\rho} {\mathcal{P}_{\rho}} \mathbf{w}\right)=0 \quad \forall\ \mathbf{w}\,.
\label{divP}
\end{equation}
Upon writing $\mathbb{P}_{\rho}=I-\mathbb{P}_{\rho \perp}$ and decomposing an arbitrary vector field $\mathbf{v}$ as
\[
\mathbf{v}=-\frac1{\rho}\, \nabla\Phi+ \nabla\times\mathbf{A},
\]
this projection operator yields the component $\mathbb{P}_{\rho}\mathbf{v}= \nabla\times\mathbf{A}$, while $\mathbb{P}_{\rho \perp}\mathbf{v} = \nabla \Phi/\rho$.
Note, $\mathbb{P}_\rho$ is the Eulerianization of
$\mathbb{P}_{\rho_0}$ and it is not difficult to write \eqref{eq:dbkt-4} in terms of this quantity.
Upon adopting this usual vector notation, the bracket (\ref{eq:dbkt-3})
can also be written as
\begin{eqnarray}
\left\{ F,G\right\} _{*} & = & -\int\,d^{3}x\,\left[\nabla\frac{\delta G}{\delta\rho}\cdot\mathcal{P}_{\rho} \frac{\delta F}{\delta\mathbf{v}}-\nabla\frac{\delta F}{\delta\rho}\cdot\mathcal{P}_{\rho} \frac{\delta G}{\delta\mathbf{v}}\right.
\nonumber \\
& & +\frac{\nabla s}{\rho}\cdot\left(\frac{\delta F}{\delta s}\mathcal{P}_{\rho} \frac{\delta G}{\delta\mathbf{v}}-\frac{\delta G}{\delta s}\mathcal{P}_{\rho} \frac{\delta F}{\delta\mathbf{v}}\right)
\nonumber \\
& & +\frac{\nabla\times\mathbf{v}}{\rho}\cdot\left(\mathcal{P}_{\rho} \frac{\delta G}{\delta\mathbf{v}}\times\mathcal{P}_{\rho} \frac{\delta F}{\delta\mathbf{v}}\right)
\nonumber \\
& & +\left.\frac{\nabla\cdot\mathbf{v}}{\rho}\left(\frac{\delta F}{\delta\mathbf{v}}\cdot
\mathcal{P}_{\rho} \frac{\delta G}{\delta\mathbf{v}}
-\frac{\delta G}{\delta\mathbf{v}}\cdot\mathcal{P}_{\rho} \frac{\delta F}{\delta\mathbf{v}}\right)\right]\,.
\label{eq:dbkt-4}
\end{eqnarray}
For MHD there is a magnetic field contribution to \eqref{chain} and following the steps that lead to \eqref{eq:dbkt-4} we obtain
\bal
\{F,G\}_{*B} & = -\int \!d^3 x \, \Bigg[
\mathbf{B}\cdot
\left(
\frac{1}{\rho} \mathcal{P}_{\rho}\frac{\delta F}{\delta \mathbf{v}} \cdot\nabla \frac{\delta G}{\delta \mathbf{B}}
- \frac{1}{\rho} \mathcal{P}_{\rho}\frac{\delta G}{\delta \mathbf{v}} \cdot\nabla \frac{\delta F}{\delta \mathbf{B}}
\right)
\nonumber\\
&\hspace{1cm}
+\mathbf{B}\cdot
\left(
\nabla \left(\frac{1}{\rho}\mathcal{P}_{\rho} \frac{\delta F}{\delta \mathbf{v}} \right ) \cdot \frac{\delta G}{\delta \mathbf{B}}
- \nabla \left(\frac{1}{\rho}\mathcal{P}_{\rho} \frac{\delta G}{\delta \mathbf{v}} \right ) \cdot \frac{\delta F}{\delta \mathbf{B}}
\right)
\Bigg]\,.
\label{PvBbkt}
\eal
With the exception of the last term of \eqref{eq:dbkt-4} proportional to $\nabla\cdot \mathbf{v}$ and the presence of the Eulerian projection operator $\mathcal{P}_\rho$, \eqref{eq:dbkt-4} added to \eqref{PvBbkt} is identical to the noncanonical Poisson bracket for the ideal fluid and MHD as given in \citet{pjmG80}.
By construction, we know that \eqref{eq:dbkt-4} satisfies the Jacobi identity -- this follows because it was obtained by Eulerianizing the canonical Dirac bracket in terms of Lagrangian variables.
Guessing the bracket and proving Jacobi for \eqref{eq:dbkt-4} directly would be a difficult chore, giving credence to the path we have followed in obtaining it.
To summarize, the bracket of \eqref{eq:dbkt-4} together with the Hamiltonian
\begin{equation}
H=\frac12\int d^3x\, \rho \,|\mathbf{v}|^2\,,
\label{HamAgain}
\end{equation}
the Eulerian counterpart of \eqref{ham}, generates dynamics that can preserve the constraint $\nabla\cdot\mathbf{v}=0$. If we add $H_B=\int d^3x\, |\mathbf{B}|^2/2$ to \eqref{HamAgain} and add \eqref{PvBbkt} to \eqref{eq:dbkt-4}, then we obtain incompressible MHD. The fluid case is the Eulerian counterpart of the volume preserving geodesic flow, described originally by Lagrange in Lagrange variables. Upon performing a series of straightforward manipulations, we obtain the following equations of motion for the flow:
\bal
\frac{\partial \rho}{\partial t}&= \left\{ \rho,H\right\} _{*} = -\nabla \cdot\mathcal{P}_{\rho}\frac{\delta H}{\delta\mathbf{v}} = -\nabla \rho \cdot \mathbb{P}_\rho \mathbf{v}
\,,
\label{Pden}\\
\frac{\partial s}{\partial t}&= \left\{s,H\right\} _{*} = -\frac{\nabla s}{\rho}\cdot\mathcal{P}_{\rho}\frac{\delta H}{\delta\mathbf{v}}
= - \nabla s \cdot \mathbb{P}_\rho \mathbf{v}\,,
\label{Pent}\\
\frac{\partial \mathbf{v}}{\partial t}&= \left\{ \mathbf{v},H\right\} _{*} = -\frac1{\rho} \mathcal{P}_{\rho} \left( \rho \nabla \frac{\delta H}{\delta \rho}\right)
+ \frac1{\rho} \mathcal{P}_{\rho}\left( {\nabla s}\frac{\delta H}{\delta s}\right)
-\frac1{\rho} \mathcal{P}_{\rho}
\left((\nabla\times\mathbf{v}) \times \mathcal{P}_{\rho} \frac{\delta H}{\delta\mathbf{v}}
\right)
\nonumber\\
& - \frac{\nabla\cdot \mathbf{v}}{\rho} \mathcal{P}_{\rho} \frac{\delta H}{\delta\mathbf{v}}
+ \frac1{\rho} \mathcal{P}_{\rho} \left( {\nabla\cdot \mathbf{v}} \, \frac{\delta H}{\delta\mathbf{v}}
\right)
\nonumber\\
&= - \mathbb{P}_\rho \nabla \frac{|\mathbf{v}|^2}{2} - \mathbb{P}_\rho \left( (\nabla\times \mathbf{v})\times \mathbb{P}_\rho \mathbf{v}\right) - (\nabla\cdot \mathbf{v})\, \mathbb{P}_\rho \mathbf{v}
+ \, \mathbb{P}_\rho \left(\mathbf{v}\, \nabla\cdot \mathbf{v}\right)\,.
\label{pvt}
\eal
If we include $H_B$ we obtain additional terms to \eqref{pvt} generated by \eqref{PvBbkt} for the projected $\mathbf{J}\times\mathbf{B}$ force.
Observe, equation \eqref{pvt} is not yet evaluated on the constraint $D^2=0$, which in Eulerian variables is $\nabla\cdot\mathbf{v}=0$.
As noted at the end of Section \ref{ssec:LDconTh}, we turn to this task in Section \ref{ssec:comparison}.
\subsubsection{Eulerian-Dirac constraint theory direct with uniform density}
\label{sssec:EDD}
For completness we recall the simpler case where the Eulerian density $\rho$ is uniformly constant, which without loss of generality can be scaled to unity. This case was considered in \citet{turski99,turski01,pjmCT12,pjmCGBT13} (although a trick of using entropy as density was employed in \citet{pjmCGBT13} to treat density advection). In these works the Dirac constraints were chosen to be the pointwise Eulerian quantities
\begin{equation}
\mathcal{D}^1=\rho \qquad \mathrm{and} \qquad \mathcal{D}^2=\nabla\cdot \mathbf{v}
\,,
\label{Edirac}
\end{equation}
and the Dirac procedure was effected on the purely Eulerian level.
This led to the projector
\begin{equation}
\mathbb{P}:=\mathbb{P}_{\rho=1}=1 -\nabla \Delta^{-1}\nabla\,\cdot \ \,,
\label{eq:P}
\end{equation}
where $\Delta=\Delta_{\rho=1}$, and the following Dirac bracket:
\begin{eqnarray}
\{F,G\}_*&=&- \int d^3x\, \Bigg[
\frac{\nabla s}{ \rho} \cdot \left( \frac{\delta F}{\delta s} \mathbb{P}\frac{\delta G}{\delta \bf v}
- \frac{\delta G}{\delta s} \mathbb{P}\frac{\delta F}{\delta \bf v} \right)
\nonumber\\
&& \hspace{1cm} - \frac{\nabla\times {\bf v}}{\rho}\cdot \left( \mathbb{P}\frac{\delta F}{\delta \bf v}\times \mathbb{P}\frac{\delta G}{\delta \bf v}\right) \Bigg]\,.
\label{eqn:PBD}
\end{eqnarray}
Incompressible MHD with constant density is generated by adding the following to \eqref{eqn:PBD}
\begin{eqnarray}
\{F,G\}_{*B}&=&- \int d^3x\, \Bigg[\frac{\bf B}{\rho} \cdot \left(\mathbb{P}\frac{\delta F}{\delta \bf v} \cdot \nabla \frac{\delta G}{\delta \bf B}
-\mathbb{P}\frac{\delta G}{\delta \bf v} \cdot \nabla \frac{\delta F}{\delta \bf B} \right)
\nonumber \\
&& \hspace{1cm} + {\bf B}\cdot \left(\nabla \left(\frac1{\rho}\mathbb{P}\frac{\delta F}{\delta \bf v}\right) \cdot \frac{\delta G}{\delta \bf B}- \nabla \left(\frac1{\rho} \mathbb{P}\frac{\delta G}{\delta \bf v}\right) \cdot \frac{\delta F}{\delta \bf B}\right)\Bigg]\,,
\label{eqn:BPBD}
\end{eqnarray}
and adding $|\mathbf{B}|^2/2$ to the integrand of \eqref{HamAgain}.
The bracket of \eqref{eqn:PBD} differs from that of \eqref{eq:dbkt-4} in two ways: the projector $\mathcal{P}_{\rho}$ is replaced by the simpler projector $\mathbb{P}$ and it is missing the term proportional to $\nabla\cdot \mathbf{v}$. Given that $\nabla\cdot \mathbf{v}$ cannot be set to zero until after the equations of motion are obtained, this term gives rise to a significant differences between the constant and nonconstant density Poisson brackets and incompressible dynamics.
\subsection{Comparison of the Eulerian-Dirac and Lagrangian-Dirac constrained theories}
\label{ssec:comparison}
Let us now discuss equations \eqref{Pden}, \eqref{Pent} and \eqref{pvt}. Given that $\nabla\cdot \mathbb{P}_{\rho}\mathbf{v}=0$ (cf.\ \eqref{divP})
it is clear that the density and entropy are advected by the incompressible velocity field $\mathbb{P}_{\rho}\mathbf{v}$, as expected. However, the meaning of \eqref{pvt} remains to be clarified. To this end we take the divergence of \eqref{pvt} and again use \eqref{divP} to obtain
\begin{equation}
\frac{\partial(\nabla\cdot \mathbf{v})}{\partial t} = -\nabla\cdot\left(\nabla\cdot \mathbf{v} \, \mathbb{P}_\rho \mathbf{v}\right)
= -\left(\mathbb{P}_\rho \mathbf{v}\right) \cdot \!\nabla\, (\nabla \cdot \mathbf{v})\,.
\label{divvt}
\end{equation}
Thus $\nabla\cdot\mathbf{v}$ itself is advected by an incompressible velocity field. As with any advection equation, if initially
$\nabla\cdot\mathbf{v} = 0$ , it will remain uniformly zero. After setting $\nabla\cdot \mathbf{v} =0$ in equation \eqref{pvt} it collapses down to
\begin{equation}
\frac{\partial \mathbf{v}}{\partial t}= - \mathbb{P}_\rho \left(\mathbf{v}\cdot \nabla \mathbf{v}\right)\,;
\label{Eeom}
\end{equation}
this is the anticipated equation of motion, the momentum equation of \eqref{momentum} with the insertion of the pressure given by \eqref{presrho}.
Given the discussion of Lagrangian vs.\ Eulerian constants of motion of Section \eqref{ssec:CoM}, that $\nabla\cdot \mathbf{v}$ is advected rather than pointwise conserved is to be expected. Our development began with the constraints $D^{1,2}$ of \eqref{D1} and \eqref{constraints-1-1} both of which are pointwise conserved by the Dirac procedure, i.e.\ $\dot\mathcal{D}_L\equiv 0$. This means their corresponding fluxes are identically zero, i.e., in \eqref{Lcon} we have $\mathbf{\Gamma}_{\mathcal{D}_L}\equiv 0$ for each. Thus the flux component $\mathbf{ \bar\Gamma}_{\mathcal{D}_E}$ of \eqref{ELcon} vanishes and the Eulerian flux for both $D^1$ and $D^2$ have the form $\mathbf{v} {\mathcal{D}_E}$. Because $D^1$ and $D^2$ Eulerianize to $-\ln(\rho)$ and $\nabla\cdot\mathbf{v}$, respectively, we expected equations of the from of \eqref{divvt} for both. We will see in Section \ref{ssec:AoI} that the equation for $D^1$ in fact follows also because the constraints are Casimir invariants.
Let us return to \eqref{LDeomP} and compare with the results of Section \ref{sssec:HCD}. Because the incompressibility condition is an holonomic constraint and Section \ref{sssec:HCD} concerns holonomic constraints for the uncoupled $N$-body problem, both results are geodesic flows. In fact, one can think of the fluid case as a continuum version of that of Section \ref{sssec:HCD} with an infinity of holonomic constraints-- thus we expect similarities between these results. However, because the incompressibility constraints are pointwise constraints, the comparison is not as straightforward as it would be for global constraints of the fluid.
To make the comparison we first observe that the term $\mathcal{A}_{mn}$ of \eqref{eq:dbkt-2} must correspond to the term $\overset\leftrightarrow{\mathbb{A}}_{ij}$ of \eqref{TA}, since their origin follows an analogous path in the derivation, both are antisymmetric, and both project from both the left and the right.
The analog of \eqref{finP} according to \eqref{eq:P0perp} is
\begin{equation}
\left(\mathbb{P}_{\rho_0\perp}\right)^{i}_j\, \frac{\pi^j}{\rho_0}=
\frac{\eta^{i\ell}}{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(\frac{A_{j}^{h}}{\mathcal{J}}\frac{\partial}{\partial a^{h}}\, \frac{\pi^j}{\rho_0}\right)\right]
= \frac{\eta^{i\ell}}{\rho_{0}} A_{\ell}^{u} \frac{\partial}{\partial a^{u}}\left[\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}\left(D^2\right)\right]
\equiv 0 \,,
\label{infinP}
\end{equation}
when evaluated on $D^2=0$. Unlike \eqref{finP} a sum, which would here be an integral over $d^3a$, does not occur because the constraint $D^2$ is a pointwise constraint as opposed to a global constraint. Also, because the constraints are pointwise, the $\overset\leftrightarrow{\mathbb{T}}_{ij}$ is analogous to the terms with $\mathcal{T}_{mn}$ that also have a factor of the projector $\mathbb{P}_{\rho_0\perp}$, giving the results analogous to \eqref{finT}.
Just as in Section \ref{sssec:HCD}, we obtain $\pi^i=\rho_0\dot{q}^i$ from \eqref{LDeomQ} when evaluated on the constraint $D^2=0$ and only a single term involving the $\mathcal{T}_{mn}$ contributes to the momentum equation of motion \eqref{LDeomP}. {We obtain
\bal
\dot{\pi}_i&= \rho_0 \eta_{in} \left(\mathbb{P}_{\rho_0\perp}\right)^{n}_r\,
\Big(
\dot{q}^m\, \eta^{rs} \mathcal{T}_{ms}
\Big)
\nonumber\\
&= A^u_i\frac{\partial }{\partial a^u}
\left\{\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}} \left[\frac{A_{\ell}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}
\left( \dot{q}^m\,
\frac{A^k_m}{\mathcal{J}} \frac{\partial \dot{q}^\ell}{\partial a^k}
\right) \right]\right\}
\nonumber\\
&= A^u_i\frac{\partial }{\partial a^u}
\left\{\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}} \left[\frac{A_{\ell}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}}
\left(
\frac{A^k_m}{\mathcal{J}} \frac{\partial (\dot{q}^m\dot{q}^\ell)}{\partial a^k}
\right) \right]\right\} \,,
\label{pidot}
\eal
where the second equality follows upon substitution of
\[
\mathcal{T}_{ms} \rightarrow \eta_{\ell s} \frac{A^k_m}{\mathcal{J}} \frac{\partial \dot{q}^\ell}{\partial a^k}\,,
\quad \mathrm{for} \quad D^2=0\,,
\]
which follows from \eqref{CB}, while the third follows again from $D^2=0$ according to \eqref{constraints-1-1}. Thus,
\bal
\ddot{q}^i&= \eta^{i\ell} \frac{A^u_\ell}{\rho_0} \frac{\partial }{\partial a^u}
\left\{\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}
\left[\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}} \left(\frac{A^f_k}{\mathcal{J}}
\frac{\partial (\dot{q}^j\, \dot{q}^k) }{\partial a^f}
\right) \right]\right\}
\nonumber\\
&
= \left(\mathbb{P}_{\rho_0\perp} \right)^{i}_j\,
\left(\frac{A^f_k}{\mathcal{J}}
\frac{\partial (\dot{q}^j\, \dot{q}^k) }{\partial a^f}
\right)
=: -\, \widehat{\Gamma}^{\,i}_{jk}(\dot{q}^j, \dot{q}^k)\,,
\label{infPCS}
\eal
where in \eqref{infPCS} we have defined $ \widehat{\Gamma}^{\,i}_{jk}(\dot{q}^j, \dot{q}^k)$, the normal force operator for geodesic flow, analogous to that of \eqref{finalD}.}
{As was the case for the $ \widehat{\boldsymbol{\Gamma}}_{i,jk}$ of \eqref{PCS}, $ \widehat{\Gamma}^{\,i}_{jk}$ possesses symmetry: given arbitrary vector fields $\mathbf{V}$ and $\mathbf{W}$
\begin{equation}
\widehat{\Gamma}^{\,i}_{jk}(V^j, W^k):= -\eta^{i\ell} \frac{A^u_\ell}{\rho_0} \frac{\partial }{\partial a^u}
\left\{\frac{\Delta_{\rho_{0}}^{-1}}{\mathcal{J}}
\left[\frac{A_{j}^{h}}{\mathcal{J}} \frac{\partial}{\partial a^{h}} \left(\frac{A^f_k}{\mathcal{J}}
\frac{\partial (V^j\, W^k) }{\partial a^f}
\right) \right]\right\} = \widehat{\Gamma}^{\,i}_{jk}(V^k, W^j)\,.
\label{GA}
\end{equation}
where the second equality follows from the commutation relation of \eqref{IDen}.}
Equation \eqref{infPCS} defines geodesic flow on the group of volume preserving diffeomorphisms, as was the case in Section \ref{sssec:HCD}, it does so in terms of the original coordinates, i.e., without specifically transforming to normal coordinates on the constraint surfaces which here are infinite-dimensional.
Now we are in position to close the circle by writing \eqref{infPCS} in Eulerian form. We will do this for the ideal fluid, but MHD follows similarly. {As usual the term $\ddot{q}^i$ becomes the advective derivative $\partial \mathbf{v}/\partial t +\mathbf{v}\cdot\nabla\mathbf{v}$, the projector $\mathbb{P}_{\rho_0\perp}$ becomes $\mathbb{P}_{\rho\perp}$ (using $\Delta_{\rho_0}^{-1}=\mathcal{J} \Delta_{\rho}^{-1}$) when Eulerianized, and
}
the $ \widehat{\Gamma}^i_{jk}$ term becomes $\mathbb{P}_{\rho \perp}\left(\nabla\cdot (\mathbf{v}\otimes \mathbf{v})\right)$. Thus \eqref{infPCS} is precisely the Lagrangian form of \eqref{Eeom}, written as follows:
\begin{equation}
\frac{\partial \mathbf{v}}{\partial t}= - \mathbb{P}_\rho \big(\nabla\cdot (\mathbf{v}\otimes \mathbf{v})\big)= - \mathbb{P}_\rho \left(\mathbf{v}\cdot \nabla \mathbf{v}\right)\,.
\end{equation}
Similarly, the Lagrangian version of \eqref{divvt} follows easily from \eqref{infPCS}. To see this we operate with the counterpart of taking the Eulerian divergence on the first line of \eqref{pidot} and make use of \eqref{divPperp},
\bal
\frac{A^h_n}{\mathcal{J}} \frac{ \partial}{\partial a^h} \frac{\dot{\pi}^n}{\rho_0}
&= \frac{A^h_n}{\mathcal{J}} \frac{ \partial}{\partial a^h}
\left(\mathbb{P}_{\rho_0\perp}\right)^{n}_r\,
\Big(
\dot{q}^m\, \eta^{rs} \mathcal{T}_{ms}
\Big)
=
\frac{A^h_n}{\mathcal{J}} \frac{ \partial}{\partial a^h}
\Big(
\dot{q}^m\, \eta^{ns} \mathcal{T}_{ms}
\Big)
\nonumber\\
&= \delta^n_\ell \frac{A^h_n}{\mathcal{J}} \frac{ \partial}{\partial a^h}
\left(
\dot{q}^m \frac{A^k_m}{\mathcal{J}} \frac{\partial \dot{q}^\ell}{\partial a^k}
\right) \,,
\eal
which in Eulerian variables becomes
\begin{equation}
\nabla\cdot\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v} \right)
= \nabla\cdot\left(\mathbf{v}\cdot\nabla\mathbf{v} \right)\qquad \mathrm{or} \qquad
\nabla\cdot\frac{\partial \mathbf{v}}{\partial t} =\frac{\partial\, }{\partial t} \nabla\cdot\mathbf{v} =0\,.
\end{equation}
In Lagrangian variables we have the trivial conservation laws
\begin{equation}
\dot{\rho_0}= 0 \qquad\mathrm{and} \qquad \dot{s_0}= 0 \,,
\end{equation}
where the corresponding fluxes are identically zero. However, as is evident from \eqref{Pden} and \eqref{Pent} we obtain nontrivial conservation laws for $\rho$ and $s$ with nonzero fluxes. Thus we see again, consistent with Section \ref{ssec:CoM}, how Lagrangian and Eulerian conservation laws are not equivalent.
For the special case where $\rho_0=\mathcal{J}=1$ one could proceed directly from \eqref{momentumCl}, write it in terms of the Lagrangian variables, and obtain \eqref{infPCS}. However, without the constraint theory, one would not immediately see it is Hamiltonian and in fact geodesic flow on an infinite-dimensional manifold.
\subsection{Incompressible algebra of invariants}
\label{ssec:AoI}
In closing this section, we examine the constants of motion for the constrained system.
The Poisson bracket together with the set of functionals that commute with the Hamiltonian, i.e., that satisfy $\{H, I_a\}=0$ for $a = 1, 2,\dots, d$, constitute the $d$-dimensional algebra of invariants, a subalgebra of the infinite-dimensional Poisson bracket realization on all functionals. This subalgebra is a Lie algebra realization associated with a symmetry group of the dynamical system, and the Poisson bracket with $\{I_a, \cdot\, \}$ yields the infinitesimal generators of the symmetries, i.e., the differential operator realization of the algebra. This was shown for compressible MHD in \cite{pjm82}, where the associated Lie algebra realization of the 10 parameter Galilean group on functionals was described. This algebra is homomorphic to usual representations of the Galilean group, with the Casimir invariants being in the center of the algebra composed of elements that have vanishing Poisson bracket with all other elements.
A natural question to ask is what happens to this algebra when incompressibility is enforced by our Dirac constraint procedure. Obviously the Hamiltonian is in the subalgebra and $\{H, \cdot\, \}_*$ clearly generates time translation, and this will be true for any Hamiltonian, but here we use Hamiltonian of \eqref{HamAgain}.
Inserting the momentum
\begin{equation}
\mathbf{P}=\int \!d^3x \, \rho \mathbf{v}
\end{equation}
into \eqref{eq:dbkt-4} with the Hamiltonian \eqref{HamAgain} gives
\begin{equation}
\{\mathbf{P}, H\}_* = 0
\label{PH}
\end{equation}
without assuming $\nabla\cdot \mathbf{v}=0$. To see this, we use \eqref{eq:Peul} to obtain
\begin{equation}
\mathcal{P}_{\rho}\frac{\delta H}{\delta\mathbf{v}}= \rho \mathbf{v}
-\nabla\Delta_{\rho}^{-1}\nabla\cdot \mathbf{v}
\qquad\mathrm{and}\qquad
\mathcal{P}_{\rho}\frac{\delta P_i}{\delta \varv_j}= \rho \, \delta_{ij} \,,
\label{dePH}
\end{equation}
which when inserted into \eqref{eq:dbkt-4} gives
\begin{eqnarray}
\left\{ P_i,H\right\} _{*} & = & -\int\! d^{3}x\,\bigg[\frac{\rho}{2} \frac{\partial |\mathbf{v}|^2}{\partial x_i}
+ \varv_i \nabla \cdot
\left(
\rho \mathbf{v} - \nabla\Delta_{\rho}^{-1}\nabla\cdot \mathbf{v}
\right)
\nonumber \\
& & +\left[
(\nabla\times\mathbf{v})\times
\left( \rho \mathbf{v} - \nabla\Delta_{\rho}^{-1}\nabla\cdot \mathbf{v} \right)
\right]_i
\nonumber \\
& & + (\nabla\cdot\mathbf{v})
\left[ \left( \rho \mathbf{v} - \nabla\Delta_{\rho}^{-1}\nabla\cdot \mathbf{v} \right)_i
-\rho \varv_i\right]
\bigg]=0
\,,
\label{PHbkt}
\end{eqnarray}
as expected. The result of \eqref{PHbkt} follows upon using standard vector identities, integration by parts, and the self-adjointness of $\Delta_{\rho}^{-1}$.
The associated generator of space translations that satisfies the constraints is given by the operator $\{\mathbf{P}, \ \cdot\ \}_*$, which can be shown directly. And, it follows that
\begin{equation}
\{P_i,P_j\}_*= 0\,, \qquad\ \forall\ i,j=1,2,3\,.
\label{PPbkt}
\end{equation}
Because the momentum contains no $s$ dependence the the second line of \eqref{eq:dbkt-4} vanishes and using $\mathcal{P}_{\rho}{\delta P_i}/{\delta \varv_j}= \rho \, \delta_{ij}$ of \eqref{dePH} it is clear the last line involving $\nabla\cdot \mathbf{v}$ of \eqref{eq:dbkt-4} also vanishes. The result of \eqref{PPbkt} is obtained because the first and third lines cancel.
Next, consider the angular momentum
\begin{equation}
\mathbf{L}=\int\! d^3x \, \rho\, \mathbf{x}\times\mathbf{v}\,.
\label{angMom}
\end{equation}
We will show
\begin{equation}
\left\{ L_i,H\right\} _{*} =0\,.
\label{LH}
\end{equation}
Using $\mathcal{P}_{\rho} {\delta L_i}/{\delta \mathbf{v}}={\delta L_i}/{\delta \mathbf{v}}$, {which follows from \eqref{eq:Peul} with $\partial(\epsilon_{ik\ell} x_\ell)/\partial x^\ell=0$, }
the fact that $\{L_i, H\}=0$ for the compressible fluid, and $ \mathcal{P}_{\rho}= I- \mathcal{P}_{\rho\perp}$, we obtain
\begin{eqnarray}
\left\{ L_i, H\right\} _{*} & = & \int\! d^{3}x\,\Bigg[
- \nabla\frac{\delta L_i}{\delta\rho}\cdot\mathcal{P}_{\rho\perp}(\rho\mathbf{v})
\nonumber \\
& & +\left(
\frac{\delta L_i}{\delta\mathbf{v}}\times \frac{\nabla\times\mathbf{v}}{\rho}
+ \frac{\nabla\cdot \mathbf{v}}{\rho}\, \frac{\delta L_i}{\delta\mathbf{v}}
\right)
\cdot \mathcal{P}_{\rho\perp} (\rho\mathbf{v})
\Bigg]\,.
\label{LH1}
\end{eqnarray}
Next, recognizing that $\mathcal{P}_{\rho\perp}(\rho\mathbf{v})= \nabla \Delta_\rho^{-1} \nabla\cdot\mathbf{v}$ and integrating by parts, we obtain
\begin{equation}
\left\{ L_i, H\right\} _{*} = \int\!d^{3}x\, \Delta_\rho^{-1}( \nabla\cdot\mathbf{v})\, \Bigg[
\nabla^2\frac{\delta L_i}{\delta\rho}
- \nabla\cdot \left(
\frac{\delta L_i}{\delta\mathbf{v}}\times \frac{\nabla\times\mathbf{v}}{\rho}
+ \frac{\delta L_i}{\delta\mathbf{v}} \, \frac{\nabla\cdot \mathbf{v}}{\rho}
\right)
\Bigg]\,.
\label{LH2}
\end{equation}
Then upon inserting
\begin{equation}
\frac{\delta L_i}{\delta \rho}= \epsilon_{ijk}x_j\varv_k
\qquad\mathrm{and}\qquad \frac{\delta L_i}{\delta \varv_j}= \rho \, x_k\epsilon_{ikj} \,,
\label{deLH}
\end{equation}
and using standard vector analysis we obtain \eqref{LH}.
Because $\mathcal{P}_{\rho} {\delta L_i}/{\delta \mathbf{v}}={\delta L_i}/{\delta \mathbf{v}}$, the first and third lines of \eqref{eq:dbkt-4} produce
\begin{equation}
\{L_i,L_j\}_*= \epsilon_{ijk} L_k \,,
\end{equation}
just as they do for the compressible fluid (and MHD), while the fourth line manifestly vanishes. Similarly, it follows that that
$\{\mathbf{L},\cdot\, \}_*$ is the generator for rotations.
To obtain the full algebra of invariants we need $\{L_i,P_j\}_*$. However because $\mathcal{P}_{\rho} {\delta P_i}/{\delta \mathbf{v}}={\delta P_i}/{\delta \mathbf{v}}$ and $\mathcal{P}_{\rho} {\delta L_i}/{\delta \mathbf{v}}={\delta L_i}/{\delta \mathbf{v}}$, it follows as for the compressible fluid that $\{L_i, P_j\}_*= \epsilon_{ijk} P_k$.
Finally, consider the following measure of the position of the center of mass, the generator of Galilean boosts,
\begin{equation}
\mathbf{G}= \int\! d^3x\, \rho\, (\mathbf{x} - \mathbf{v} t)\,.
\end{equation}
Calculations akin to those above reveal
\begin{equation}
\{G_i, G_j\}_* = 0\,,\quad \{G_i, P_j\}_* = 0\,, \quad \{G_i, H\}_* = P_i\,, \quad \{L_i, G_j\}_* =\epsilon_{ijk} G_k\,.
\end{equation}
Thus the bracket \eqref{eqn:PBD} with the set of ten invariants $\{H, \mathbf{P},\mathbf{L}, \mathbf{G}\}$ is at once a closed subalgebra of Poisson bracket realization on all functionals and produces an operator realization of the Galilean group \citep[see e.g.][] {sudarshan} that is homomorphic to the operator algebra of $\{L_i, \ \cdot\ \}_*$, $\{P_i , \ \cdot\ \}_*$, etc.\ with operator commutation relations. This remains true for MHD with the only change being the addition of $H_B$ to the Hamiltonian.
Thus, the Galilean symmetry properties of the ideal fluid and MHD are not affected by the compressibility constraint. However, based on past experience with advected quantities, we do expect a new Casimir invariant of the form
\begin{equation}
\hat C[\rho,s]=\int\! d^3x \,\hat\mathcal{C}(\rho,s)\,.
\label{hatCas}
\end{equation}
To see that $\{\hat C, F\}_*=0$ for any functional $F$, where $\hat\mathcal{C}(\rho,s)$ is an arbitrary function of its arguments, we calculate
\begin{equation}
\{F, \hat C\}_*= -\int\!d^{3}x\,\frac1{\rho}
\left[
\rho\, \nabla \frac{\partial \hat \mathcal{C}}{\partial \rho} - \frac{\partial \hat \mathcal{C}}{\partial s}\, \nabla s
\right]
\cdot \mathcal{P}_\rho \frac{\delta F}{\delta\mathbf{v}}\,,
\label{hatCas2}
\end{equation}
and since $\nabla \times (\rho\, \nabla {\partial \hat\mathcal{C}}/{\partial \rho} - {\partial \hat\mathcal{C}}/{\partial s}\, \nabla s)=0$ we write it as $\nabla p$, giving for \eqref{hatCas2}
\begin{equation}
\{F, \hat C\}_*= -\int\! d^{3}x\,\frac1{\rho}
\nabla p
\cdot \mathcal{P}_\rho \frac{\delta F}{\delta\mathbf{v}}\,.
\label{hatCas3}
\end{equation}
Thus, integration by parts and use of \eqref{divP} imply $\{F, \hat C\}_*=0$ for all functionals $F$. Note, without loss of generality we can write $\hat \mathcal{C}(\rho,s)= \rho U(\rho, s)$, in which case $p=\rho^2 \partial U/ \partial \rho$. Thus, it is immaterial whether or not one retains the internal energy term $ \int\! d^3x\, \rho U(\rho,s)$ in the Hamiltonian.
Now, \eqref{hatCas} is not the most general Casimir. Because both $\rho$ and $\nabla\cdot\mathbf{v}$ are Lagrangian pointwise Dirac constraints, we expect the following to be an Eulerian Casimir
\begin{equation}
\hat C[\rho,s,\nabla\cdot\mathbf{v}]=\int\! d^3x \, \mathcal{C}(\rho,s,\nabla\cdot\mathbf{v})\,,
\label{newCas}
\end{equation}
where $\mathcal{C}$ is an arbitrary function of its arguments. To see that $\{ C, F\}_*=0$ for any functional $F$, we first observe that
\begin{equation}
\frac{\delta C}{\delta \mathbf{v}}= -\nabla \frac{\partial \mathcal{C}}{\partial \nabla\cdot \mathbf{v}}
\end{equation}
and, as is evident from \eqref{eq:Peul}, that $\nabla\cdot (\mathcal{P}_\rho\nabla\Phi)=0$ for all $\Phi$; hence, all the ${\delta C}/{\delta \mathbf{v}}$ terms vanish except the first term of the last line of \eqref{eq:dbkt-4}. This term combines with the others to cancel, just as for the calculation of $\hat\mathcal{C}$.
For constant density, entropy, and magnetic field, the bracket of \eqref{eqn:PBD} reduces to
\begin{equation}
\{F,G\}_*= -\int \!d^3x\, \frac{\nabla\times {\bf v}}{\rho}\cdot \left( \mathbb{P}\frac{\delta F}{\delta \bf v}\times \mathbb{P}\frac{\delta G}{\delta \bf v}\right)\,,
\label{eqn:RPBD}
\end{equation}
whence it is easily seen that the helicity
\begin{equation}
C_{\varv\cdot\nabla\times \varv}=\int\! d^3x\, \mathbf{v}\cdot\nabla\times \mathbf{v}
\end{equation}
is a Casimir invariant because $\mathbb{P}\, (\nabla\times\mathbf{v})= \nabla\times\mathbf{v}$.
This Casimir is lost when entropy and density are allowed to be advected, for it is no longer a Casimir invariant of \eqref{eq:dbkt-4}.
Now, let us consider invariants in the Lagrangian description. Without the incompressibility constraints, the Hamiltonian has a standard kinetic energy term and the internal energy depends on $\partial q/\partial a$, an infinitesimal version of the two-body interaction, if follows that just like the $N$-body problem the system has Galilean symmetry, and because the Poisson bracket in the Lagrangian description \eqref{cbkt} is canonical there are no Casimir invariants. With the incompressibility constraint, the generators of the algebra now respect the constraints, with Dirac constraints being Casimirs and the algebra of constraints now having a nontrivial center. Because the Casimirs are pointwise invariants, we expect the situation to be like that for the Maxwell Vlasov equation \cite{pjm82}, where the following is a Casimir
\begin{equation}
C_{\nabla\cdot B}[\mathbf{B}]=\int \!d^3x\, \mathcal{C}(\nabla\cdot \mathbf{B}, \mathbf{x})\,,
\label{divB}
\end{equation}
with $\mathcal{C}$ being an arbitrary function of its arguments. {Because both nabla $\nabla \cdot \mathbf{B}$ and $\mathcal{J}$ are pointwise constraints, analogous to \eqref{divB} we expect the following Casimir:}
\begin{equation}
\hat C[\mathcal{J}]=\int \!d^3a\, \hat\mathcal{C}(\mathcal{J}, \mathbf{a})\,.
\end{equation}
Indeed, only the first term of \ \eqref{eq:dbkt-2} contributes when we calculate $\{\hat C, G\}_*$ and this term vanishes by \eqref{Ldiv} because
\begin{equation}
\frac{\delta \hat C}{\delta q^i}=-\frac{\partial }{\partial a^\ell} \left( A^\ell_i\frac{\partial \hat\mathcal{C}}{\partial \mathcal{J}}\right)\,,
\end{equation}
{which follows upon making use of \eqref{dedet}.} Similarly, it can be shown that the full Casimir is
\begin{equation}
\hat C[D^1,D^2]=\int \!d^3a\, \hat\mathcal{C}(D^1,D^2,\mathbf{a})\,,
\end{equation}
a Lagrangian Casimir consistent with \eqref{newCas}.
For MHD, the magnetic helicity,
\begin{equation}
C_{A\cdot B}= \int \!d^3 x \, \mathbf{A}\cdot \mathbf{B}\, ,
\end{equation}
where $\mathbf{B}=\nabla\times \mathbf{A}$ is easily seen to be preserved and a Casimir up to the usual issues regarding gauge conditions and boundary terms \citep[see][] {finn}.
We know that the cross helicity
\begin{equation}
C_{\varv\cdot B}= \int\!d^3 x \, \mathbf{v}\cdot \mathbf{B}\, ,
\end{equation}
is a Casimir of the compressible barotropic MHD equations, and it is easy to verify that it is also a Casimir of \eqref{eqn:PBD} added to \eqref{eqn:BPBD}, that is for uniform density. However, it is not a Casimir for the case with advected density, i.e., for the bracket of \eqref{eq:dbkt-4} added to \eqref{PvBbkt}.
\section{Conclusions}
\label{sec:conclusion}
In this paper we have substantially investigated constraints, particularly incompressibility for the ideal fluid and MHD, for the three dichotomies described in Section \ref{ssec:bgnd}: the Lagrangian vs.\ Eulerian fluid descriptions, Lagrange multiplier vs.\ Dirac constraint methods, and Lagrangian vs.\ Hamiltonian formalisms. An in depth description of the interplay between the various fluid and MHD descriptions was given, with an emphasis on Dirac's constraint method. Although we mainly considered geodesic flow for simplicity, the Dirac's Poisson bracket method can be used to find other forces of constraint in a variety of fluid and plasma contexts.
Based on our results, many avenues for future research are presented. We mention a few. Since the Hamiltonian structure of extended and relativistic MHD are now at hand \citep{pjmKLWW14,AKY15,pjmDP15,pjmLM16,pjmDL16,pjmKT20} calculations analogous to those presented here can be done for a variety of magnetofluid models. {Another valuable class of models that could be studied, ones that are known to have Lagrangian and Hamiltonian structure, are those with various finite-Larmor-radius effects \citep[e.g.][]{pjmTWG08,izacard11,tassi14,tassi19} }
Another avenue for future research would be to address stability with constraints. In a previous series of papers \citep{pjmAP10,pjmAP12,pjmAP13,pjmAPE15,pjmAP16} we have investigated Hamiltonian based stability, generalizations of the MHD energy principle or the ideal fluid Rayleigh criterion, within the Lagrangian, energy-Casimir, and dynamically accessible frameworks. Because Dirac's method adds Casimirs, the Dirac constraints, one gets a richer set of equilibria from the energy-Casimir variational principle and these can be tested for Lyapunov stability. Similarly, the method of dynamical accessibility \citep[see][]{pjm98} based on constrained variations induced by the Poisson operator will enlarge the set of stable equilibria.
{Recently there has been consider research in the development of structure preserving computational algorithms. \citep[See, e.g.,][for review.]{pjm17} These are algorithms that preserve various geometric, Hamiltonian, variational, and other structure of fluid, kinetic, and other physical models. In the plasma community, in particular, we mention \citet{evstati13,hong16,pjmXQLYZH16,pjmKKS17}, but there is a large body of additional work by these and other authors. Given how the finite-dimensional material of Section \ref{sec:constraints} so strongly parallels the infinite-dimensional material of Section \ref{sec:dirac}, notably the structure of geodesic flow, a natural avenue for future research would be to develop numerical algorithms that preserve this structure.}
Lastly, we mention that there is considerable geometric structure behind our calculations {that could be further developed. Our results can be restated }in geometric/Lie group language \citep[see e.g.][]{bloch}. Also, Arnold's program for obtaining the Riemann curvature for geodesic flow on the group of volume preserving diffeomorphisms can be explored beginning from our results of Section \ref{sec:dirac}. We did not feel this special issue would be the appropriate place to explore these ideas.
\section*{Acknowledgment}
\noindent PJM was supported by U.S. Dept.\ of Energy under contract \#DE-FG02-04ER-54742. He would also like to acknowledge support from the Humboldt Foundation and the hospitality of the Numerical Plasma Physics Division of the IPP, Max Planck, Garching. FP would like to acknowledge the hospitality of the Institute for Fusion Studies of the University of Texas at Austin.
\bibliographystyle{jpp}
|
2,877,628,090,836 | arxiv | \section{Introduction \& Related Work} \label{sec:intro}
Exploration is one of the central problems of RL and has been studied primarily assuming access to an online environment \citep{sutton2018}. Offline RL, the problem of learning a policy from a previously collected experience history, has gained a lot of attention in the recent years \citep{offlineRLtutorial, fu2020d4rl}. Yet the combination of the two, i.e., reasoning about exploration and agent's learning process given just an offline dataset, though a problem of tremendous potential value, has not been covered extensively by the RL research community. \textit{Offline policy evaluation} (OPE), a subproblem of offline RL, focuses on evaluating performance of a fixed policy using an offline dataset (\cref{fig:OfflineSim}).
In many real-world scenarios, including recommender systems, personalizable web services, robots required to adapt to new tasks, etc., instead of having fixed policies, we would like the agent to continue learning after deployment. This requires the agent to explore and react to its experience in the environment by adapting its policy. OPE ignores these factors and is not the right framework to assess such agents. In this work we propose \textit{offline learner simulation} (OLS) as a way to evaluate non-stationary agents given just an offline dataset.
\vspace{-0.2em}
A natural approach to simulate learners can be to leverage model-based RL (also used successfully as an OPE method - see \citet{fu2021benchmarks}) - by learning a world model on the offline dataset, and then using that model to generate new rollouts. While simple to use, the learned model incurs bias that may be hard to measure and reason about. On the other side of the spectrum, we have non-parametric approaches replaying data in certain ways to match the true environment's distribution: \citet{john2011} discussed this approach in the contextual bandit setting, and \citet{mandel2016offline} extended the idea to Markov decision processes (MDPs). These methods are provably unbiased and allow for simulating a learning process with provable absolute accuracy, but become inefficient for all but the simplest toy problems. More realistic environments, with rich observations and stochastic transitions, would lead to simulation terminating after few steps, deeming these methods impractical.
\vspace{-0.2em}
In this work, we incorporate recent advances in latent state discovery \citep{du2019provably, misra2020kinematic, acstate2022}, which allow one to recover the unobserved latent states from potentially high-dimensional rich observations, with the model-free data-driven approaches proposed in \cite{mandel2016offline}, to improve their efficiency and practicality. In the preliminary experiments, we evaluate the methods by comparing the obtained simulations to the true online learners in terms of fidelity and efficiency. We show that newly proposed methods are able to simulate a learning process with high fidelity, and are capable of producing longer simulations than fully non-parametric approaches.
\section{Problem Setup}
We consider reinforcement learning (RL) in block Markov decision processes (Block MDPs), defined by a large (possibly infinite) observation space $\Xcal$, a finite unobservable state space $\Scal$, a finite action space $\Acal$, transition function $p: \Scal \times \Acal \to \Delta(\Scal)$, emission function $q: \Scal \to \Delta(\Xcal)$, reward function $r: \Xcal \times \Acal \times \Xcal \to \Delta(\RR)$, initial state distribution $\mu_0 \in \Delta(\Scal)$, and discount factor $\gamma \in [0,1]$. A policy $\pi: \Xcal \to \Delta(\Acal)$ specifies a distribution over actions for each observation. In RL, a \textit{learner} (often realized by executing a learning algorithm), denoted by $\AA$, defines a mapping from some history of interactions of arbitrary length $\tau\in \mathcal{H}$ to a policy $\pi_{\tau}: \Xcal \to \Delta(\Acal)$. Here, a history of interactions of length $t$ is an ordered sequence of transition tuples $\tau_{1:t} = [(x_{t'}, a_{t'}, r_{t'}, x'_{t'})]_{t'=1}^{t} \in \Hcal_{t}$, and $\Hcal = \Hcal_0 \cup \Hcal_1 \cup \cdots $ is the set of all histories. Consider the interaction cycle between the learner and the environment: at step $t$, the learner has seen its interactions with the environment during steps $t'=1\dots t$ and makes use of the history so far $\tau_{1:t}$ to define its policy $\pi_t = \pi_{\tau_{1:t}}$ for subsequent interaction(s). Overall, the learner follows a non-stationary policy where, importantly, the sequence of policies $\{\pi_1, \dots, \pi_t\}$ that constitute this non-stationary policy is not known in advance.
\begin{wrapfigure}[12]{L}{0.33\textwidth}
\centering
\includegraphics[scale=0.55, trim=70 10 25 25]{fig/OfflineSim.drawio.pdf}
\renewcommand\figurename{Fig.}
\vspace{-0.5cm}
\captionof{figure}{OPE vs OLS}
\label{fig:OfflineSim}
\end{wrapfigure}
In this work, we consider the problem of offline learner simulation (OLS), which is used to gain an understanding of what a learner would ``perform'' in the real environment; we might be interested in how the learner would gather data and explore the environment, or how quickly the learner would converge to the optimal policy. Given a logged dataset of past interactions $\Dcal = \{x_i, a_i, r_i, x_i'\}_{i=1}^{n}$, in order to simulate a (black-box) learner up to step $T$, it is necessary to provide the learner with a history $\tau_{1:T} = [(x_{t}, a_{t}, r_{t}, x'_{t})]_{t=1}^{T} \sim p^{\textsf{sim}}_{\Hcal}$ that is drawn from a distribution identical to the one observed if the learner interacted with the real environment $p^{\textsf{real}}_{\Hcal}$. This is in contrast to OPE, where it is usually sufficient to obtain a value function estimate (\cref{fig:OfflineSim}).
\begin{minipage}{0.52\textwidth}\setlength{\parskip}{1ex}
\subsection{Evaluation Protocol}
A ``good'' offline simulation should run as accurately as possible for as long as possible. Therefore, we propose to quantify the success of offline simulations via two aspects - efficiency and fidelity.
Efficiency can be measured by the length of histories generated by the simulation before it terminates. While some simulation approaches allow the learner to run indefinitely, this often comes at the cost of large biases. Therefore, we also consider simulation approaches that have the option to ``terminate''. In \cref{fig:metrics-example}, \textsf{sim2} is more efficient than \textsf{sim1} because it terminates after more simulation steps.
\end{minipage}
\hfill
\begin{minipage}{0.46\textwidth}
\centering
\setlength{\abovecaptionskip}{0pt}
\includegraphics[scale=0.65]{fig/metrics.pdf}
\captionof{figure}{Example of the ground-truth learning curve as well as two offline simulations (details of this experiment are in \cref{appx:metrics-details}).}
\label{fig:metrics-example}
\end{minipage}
To measure fidelity, in theory, we want to compare the distribution of histories generated by the simulation $p^{\textsf{sim}}_{\Hcal}$ to the real distribution $p^{\textsf{real}}_{\Hcal}$. Since these distributions may be difficult to represent analytically, in practice, we instead use aggregate scalar statistics $g(p_{\Hcal})$ as proxy measures.\footnote{We caution that $g(p^{\textsf{sim}}_{\Hcal})$ being close to $g(p^{\textsf{real}}_{\Hcal})$ is only a necessary condition for the $p^{\textsf{sim}}_{\Hcal}$ being close to $p^{\textsf{real}}_{\Hcal}$, but not a sufficient condition.} For example, we may have $g(p_{\Hcal_t}) = \EE_{\tau_{1:t} \sim \Hcal_t}[V(\AA(\tau_{1:t}))]$ - the expected policy performance after the learner has received a length-$t$ history $\tau_{1:t} \sim p_{\Hcal_t}$.
Note that the specific choice of $g$ is dependent on the learner and the environment; some alternative choices include: state visitation distribution in the history $\tau_{1:t}$, or the learner's model parameters. The expectation over distributions of histories can be approximated empirically using averaged results from multiple simulation runs. Suppose we are interested in the fidelity of a simulation from training step $1$ to $T$. As shown in \cref{fig:metrics-example}, we can visualize $g(p_{\Hcal_t})$ for $t = 1 \cdots T$ as learning curves, and measure the error in simulation as the RMSE between the learning curves from the simulation vs the ground-truth over all steps (alternatively, one may use the mean/max absolute error). For $T=50000$, \textsf{sim1} is a more ``accurate'' simulation than \textsf{sim2}, even though \textsf{sim2} eventually converged to the correct policy after $T=50000$ whereas \textsf{sim1} is not able to run till convergence. Comparisons of fidelity are only meaningful for the same $T$.
Both efficiency and fidelity are important for offline simulation, yet it is not straightforward to define a single metric that captures both. Since there is usually a trade-off between the two (similar to the bias-variance trade-off), in our experiments below, we consider these two aspects separately.
\section{Non-parametric \& Semi-parametric Offline Simulation}
\citet{mandel2016offline} proposed several non-parametric approaches for OLS in RL including the queue-based evaluator (QBE) and per-state rejection sampling (PSRS). We focus on PSRS because it was shown to be more efficient than QBE. In PSRS, each transition in the logged dataset is only considered once, and is either accepted and given to the learner, or rejected and discarded. This ensures the unbiasedness of the overall simulation. In \cref{appx:alg-1}, we restate these two algorithms with a few modifications to allow for more general settings such as episodic problems with multiple initial states and non-stationary logging policies.
A key step in PSRS (as the name suggests) is line 3 ({\crefname{algorithm}{Alg.}{Algs.}{\cref{alg:psrs}}}) which groups transitions in the logged dataset into queues based on the from-state, and the simulation terminates whenever it hits an empty queue. This works reasonably well for tabular MDPs. For block MDPs, a naive approach is to treat the observations as the keys to group transitions by. Since we often have an infinite observation space, every observation is only seen in the logged dataset once, making this approach very inefficient and impractical because the queues would be empty before we are able to simulate long enough. Fortunately, recent advances in latent state discovery \citep{du2019provably,misra2020kinematic,acstate2022} allow one to recover the unobserved latent states from potentially high-dimensional rich observations. For simulating such block MDPs, we propose to first learn a latent state encoder, preprocess the logged dataset into latent states, and perform PSRS while grouping transitions using the latent states (high-level pseudo code shown in \cref{alg:psrs-latent-high-level}; for more details, see \cref{appx:alg-2}). Importantly, the simulation interfaces with learners using raw observations and is thus compatible with the learners that would be used with the original block MDP.
\begin{algorithm}[h]
\caption{OLS using Latent Per-State Rejection Sampling: high-level pseudo code}
\label{alg:psrs-latent-high-level}
\begin{algorithmic}[1]
\State \textbf{Input:} Logged dataset $\Dcal = \{(x_i, a_i, r_i, x_i')\}$ recorded by policy $\pi_b$, initial observation $x_0$
\State \textbf{Input:} Learner $\AA$ that maps from history $\tau$ to policy $\pi$
\State {\textbf{Input:} Encoder $f$ that maps from observation $x$ to latent state $z$}
\State {\textbf{Preprocess:} Calculate $z_i = f(x_i)$ for all $x_i$}
\State \textbf{Initialize} queues[$z$], $\forall z$: group transitions from $\Dcal$ by latent state $z$, into randomized queues
\Statex // {\color{gray}Start simulation}
\State \textbf{Initialize} $\tau = $ [\ ]
\State $x = x_0$ \Comment{{\color{gray}initial observation}}
\For{step $t = 1$ to $\infty$}
\State $\pi = \AA(\tau)$ \Comment{{\color{gray}update learner with the history}}
\State {$z = f(x)$} \Comment{{\color{gray}encode observation}}
\While{no transition has been accepted for this step}
\If{queues[$z$] is empty} terminate \EndIf
\State Sample a transition from queues[$z$]
\State Perform rejection sampling \Comment{\parbox[t]{.55\linewidth}{\textcolor{gray}{accept or reject the transition tuple based on similarity between action distributions of the current policy $\pi$ and behavior policy $\pi_b$, given observation $x$}}}
\EndWhile
\State $\tau$.append($x,a,r,x'$) \Comment{{\color{gray}update history with new transition}}
\If{episode ends}
\State $x = x_0$ \Comment{{\color{gray}start new episode}}
\Else
\State $x=x'$
\EndIf
\EndFor
\State \textbf{Output:} History $\tau$
\end{algorithmic}
\end{algorithm}
\section{Proof of Concept Experiments} \label{sec:exp}
In this section, we conducted empirical evaluations of offline simulation for simple block MDPs. First, in a problem with known latent states, we show that using the latent states for offline simulation is more efficient than using the observations, without sacrificing simulation fidelity. Building on this result, we then consider a more challenging scenario where the latent states are not known and must be discovered from data, and demonstrate the effectiveness of our semi-parametric simulation approach even when the learned latent state encoder is not perfect. As the building block for our experiments, we introduce a $5 \times 5$ grid world navigation task (\cref{fig:grid-noise}-left) modified from \citet{zintgraf2019varibad}, further described in \cref{appx:exp}.
\begin{figure}
\centering
\vspace{-2.7ex}
\setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{ccc}
\makecell{\includegraphics[scale=0.35,align=c]{fig/gridworld.pdf} \\ \\ $\Acal = \{\circ, \uparrow, \rightarrow, \downarrow, \leftarrow\}$} &
\includegraphics[scale=0.5,align=c]{fig/v824_noise4bit_ql_learning_curves.pdf} &
\includegraphics[scale=0.55,align=c]{fig/v824_noise4bit_ql_combined.pdf} \\[-2.5ex]
\end{tabular}
\caption{\small Grid-world with discrete observations. \textbf{Left}: latent state space and action space. Observations consist of the latent state concatenated with 4 random bits. \textbf{Middle}: estimated value of starting state for real Q-learning and two simulations (using observations and using latent states directly). \textbf{Right}: fidelity and efficiency of the simulations. Both simulations have perfect fidelity, but using latent states allows for simulating longer. }
\label{fig:grid-noise}
\end{figure}
\noindent\textbf{Grid-World with Discrete Observations.} In this setting, we use a discrete observation space $\Xcal = \Scal \times \{0,1\}^k$ induced by the emission function $x = q(s) = s + (b_1 + \dots + b_{k}2^{k-1})|\Scal|$, where an observation is made up of the underlying state and $k$ bits $b_1 \dots b_k$ that are stochastically sampled at every time step. We first collected 1,000 episodes following a uniformly random policy, and then used PSRS on the logged dataset to simulate tabular Q-learning. We compared PSRS with the observations $x$ vs PSRS with the underlying states $s$, and in each case, we repeated the simulation procedure for 100 runs with different random seeds. We show results for $k=4$, where the observation space is 16 times larger than the state space. \cref{fig:grid-noise}-middle shows the learning curves for all runs, where we track the estimated value of the initial state as the $g$ function, since we are in the tabular setting and have transparency over the learner's internal parameters. In \cref{fig:grid-noise}-right we show the aggregate result for fidelity and efficiency. Both are accurate simulations as their learning curves are both overlapping exactly with real Q-learning, but using latent states is more efficient than using raw observations and led to simulations about twice as long for this problem. Additional variations of this experiment are explored further in \cref{appx:discrete-obs}.
\begin{figure}
\centering
\setlength{\abovecaptionskip}{-12pt}
\begin{tabular}{cc}
(a)
\includegraphics[scale=0.28, align=t, trim=0 0 0 0mm]{fig/grid_state_visitation.png}
(b)
\includegraphics[scale=0.28, align=t, trim=0 0 0 0mm]{fig/grid_state_latent.png} &
\scalebox{0.9}{
\begin{tabular}[t]{c|cc}
\toprule
\textbf{Simulation} & \textbf{Efficiency} ($\uparrow$) & \textbf{Fidelity} ($\downarrow$) \\
\midrule
Real & $\infty$ & 0 \\
\midrule
PSRS-oracle & 17 & 0.173 \\
PSRS-encoder & 16 & 0.203 \\
PSRS-obs-only & 50 & 1.148 \\
PSRS-act-only & 50 & 1.221 \\
PSRS-random & 50 & 1.514\\
\bottomrule
\end{tabular}}
\end{tabular}
\vspace*{-2mm}
\caption{Grid world with continuous observations: experiment results. \textbf{Left}: state visitation distribution and learned latent states. \textbf{Right}: quantitative results comparing efficiency (median simulation length in epochs, higher is better) and fidelity (RMSE of validation performance curves compared to PPO in a real environment, lower is better).}
\label{fig:grid-continuous}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{fig/new_fig/grid_ppo_FINAL_learning_2.pdf}
\includegraphics[scale=0.5]{fig/new_fig/grid_ppo_FINAL_validation.pdf}
\caption{Learning curves of a PPO agent in grid world with continuous observations: ``Real'' and various PSRS simulations. \textbf{Left}: learning performance, i.e., average episode return \emph{within} each training epoch. \textbf{Right}: validation performance, i.e., average episode return as measured in a real validation environment, \emph{after} each training epoch. All results are averaged over 10 runs. OLS using latent PSRS, denoted here as ``PSRS-encoder'' faithfully reconstructs learning curves of the real, online PPO agent.}
\label{fig:grid-ppo-curves}
\end{figure}
\noindent\textbf{Grid-World with Continuous Observations.} Next, we consider a more complex observation space, where the observation $x$ is a randomly sampled 2D coordinate within the grid cell of state $s$ (normalized to be between 0 and 1). We similarly collected the logged dataset using a uniformly random policy, leading to the state visitation distribution shown in \cref{fig:grid-continuous}a. Note that since the observation space is now continuous -- essentially no two $x$'s are the same -- PSRS on the observation space is no longer practical. Therefore, we trained a neural network encoder for discrete latent states for kinematic inseparability abstraction, using the contrastive estimation objective similar to \citet{misra2020kinematic}. While we used a latent dimension of 50, it ended up learning only 20 discrete latent states as shown in \cref{fig:grid-continuous}b. We subsequently used the learned state encoder in offline simulation of a PPO agent using PSRS and compared to several baselines. We visualized the learning curves (\cref{fig:grid-ppo-curves}) by tracking policy performance within each training epoch (\cref{fig:grid-ppo-curves}-left) and in a validation environment at the end of each epoch, obtained via averaging the returns from 10 Monte-Carlo rollouts in the true environment (\cref{fig:grid-ppo-curves}-right). Summarized numerical results on efficiency and fidelity of the validation performance are shown in \cref{fig:grid-continuous}-right. Despite the errors in the latent states from the learned encoder, it performs close to the oracle encoder, and both are close to the online PPO in the real environment. All other baselines, despite being efficient, are far from accurate simulations and have non-negligible error. Overall, this experiment shows promise that for block MDPs, we can learn to encode the observations into latent states and then do offline simulations in an efficient and accurate manner, completely using offline data. Further experimental details, including explanation of the used baselines, are in \cref{appx:continuous-obs}.
\section{Conclusion}
In this work, we studied offline learner simulation (OLS), which allows one to evaluate an agent's learning process using offline datasets. We formally described the evaluation protocol for OLS in terms of efficiency and fidelity, and proposed semi-parametric simulation approaches to handle block MDPs with rich observations by extending existing non-parametric approaches and leveraging recent advances in latent state discovery. Through preliminary experiments, we show the advantage of this approach even when the learned latent states are not perfectly correct. Code to reproduce experimental results will be released publicly upon publication of this paper.
Besides applications in recommender systems and robotics, OLS may be especially useful for multi-task and meta-learning settings, where simulation on a subset of tasks may inform us about future adaptive performance on other tasks. It may prove to be a crucial component to succeed in offline meta-RL \citep{OfflineMetaRL, mitchell2021offline, offlineMetaRLWithOnlineSelfSupervision}. Future work should also consider removing the assumption on discrete latent topology and accounting for exogenous processes (further discussed in \cref{appx:exoMDP}) to handle a wider class of problems.
\newpage
|
2,877,628,090,837 | arxiv | \section*{Introduction}
The vision of the semantic web is to build a massive network of distributed, interconnected, machine-readable data \cite{1}\cite{2}. The goal is not only for software programs to be able to access and query the data itself, but also to make automated inferences based on the meaning that is encoded therein. The core components of the semantic web have now been established by the W3C: we have RDF \cite{3}, a language for describing data; OWL \cite{4}, a language for defining ontologies; and SPARQL \cite{5}, a language for querying RDF. In addition, several OWL reasoners \cite{6}\cite{7}\cite{8} have been implemented which are capable of classifying data when given an ontology and a set of instance data.
Unfortunately, crucial infrastructure for querying and reasoning across distributed datasets is still missing. Current SPARQL implementations handle remote data sets by downloading them to the site of the query engine in their entirety \cite{9}, and reasoners are likewise dependent on a single, centralized dataset. In the realm of bioinformatics, a distributed framework for querying and reasoning would be particularly valuable. There are now more than a thousand biological databases on the web \cite{10}, containing distinct but fundamentally interrelated information about DNA sequences, protein structures, networks of metabolic reactions, chemical properties of molecules, and so on. The need for a simple and effective means of integrating these databases is evidenced by the numerous publications \cite{11}--\cite{14}, data warehouses \cite{15}--\cite{18}, and software systems \cite{19}--\cite{25} that have been inspired by the problem. One such system is BioMOBY; the SHARE project described here upgrades and extends BioMOBY, creating a general purpose architecture for querying and reasoning over the semantic web.
\section*{Past Work: BioMoby}
BioMoby\footnote{Moby is not an acronym, it's just a name. The name comes from the conference where the idea was conceived: MOBY-DIC (Model Organism Bring Your Own Database Interface Conference).} is a simple framework for defining and discovering interoperable web services. Although Moby is a generic solution which can be applied to any type of service, bioinformatics is the area in which it is currently being used. Under Moby, services communicate according to a shared messaging format, and all inputs and outputs of services are specified in terms of a centralized Moby datatype ontology. This ontology defines both syntax and semantics for a large number bioinformatics datatypes such as DNA sequences, Gene Ontology \cite{26} terms, Single Nucleotide Polymorphisms (SNPs), and so on. For example, the object for representing a protein sequence is called \textbf{AminoAcidSequence} and has two member values: an integer for storing the length of the sequence, and a string for storing the sequence itself. Each datatype specifies its own serialization into XML, and new datatypes may be introduced by any user of the system. The precise specification of datatypes allows services to be easily chained into \emph{workflows}, in which the output of one service becomes the input of the next.
In addition to a datatype ontology, Moby also maintains a large working registry of services. The registry now holds approximately 1500 web services which perform a wide variety of tasks such as database retrieval, alignment of sequences, identification of protein domains, prediction of subcellular localization, etc. The most important feature of the Moby registry is the ability to query for services by input or output datatype. This enables the stepwise, interactive construction of workflows which perform complex analyses. Moby workflows may be constructed in a GUI environment such as Taverna \cite{27}, or executed immediately as they are traversed, by means of a client such as GBrowse Moby \cite{28}.
The Moby architecture is depicted in Figure \ref{fig:mobyframework}.
\begin{figure}
\centering
\includegraphics[width=5.5in,natwidth=3600,natheight=2901]{Moby_Framework_Enlarged.jpg}
\caption{Typical usage of the BioMOBY framework. (1) The user begins with data that matches a certain Moby datatype. Usually this data is a bare identifier, which corresponds to the default Moby datatype \textbf{Object}. (2) The user queries the registry for services that consume her identifier as input. (3) The registry returns a list of such services. (4) The user chooses a service from the list, based on the desired type of analysis. (5) The user's data is sent to the chosen service, in this case \textbf{getAminoAcidSequence}, and the service is executed. (6) The service returns its output, in this case a data object of type \textbf{AminoAcidSequence}. (7) The user repeats steps 1-6, until the desired analysis of the data is complete. The reader may try steps 1-6 using the GBrowse Moby client at http://moby.ucalgary.ca/gbrowse\_moby.}
\label{fig:mobyframework}
\end{figure}
\section*{Recent Work: SPARQL Queries Resolved By Web Services}
One of the main limitations of BioMoby is its reliance on a custom XML format, making it difficult for Moby services to be used within other frameworks. Unfortunately, the invention of an extensible data syntax was necessary as BioMOBY predates the advent of RDF. SHARE is a major revision of the MOBY framework which corrects this shortcoming and establishes a completely generic, open framework based on semantic web standards. At the same time, SHARE introduces higher-level querying and reasoning functionality.
The SHARE system is based on the following key observation: whenever a web service computes a result, it is in effect generating an RDF triple. The subject of this triple is the input, the object is the output, and the predicate is the relationship that is established between the input and the output by the service call. In other words, the predicate is defined by the behaviour of the service. For example, a service that retrieves a list of GO (Gene Ontology) annotations for a protein generates triples of the form ``$<$protein ID$>$ hasGOTerm $<$GO term ID$>$'', as shown in Figure \ref{fig:servicetriple}. It is logical then, to annotate the service itself with the predicate \textbf{hasGOTerm}.\footnote{More accurately, a predicate annotation connects one input and one output of a service. A Moby service may have arbitrarily many inputs and outputs, with differing datatypes.}
\begin{figure}
\centering
\includegraphics[width=5.5in, natwidth=3300,natheight=1523]{Web_Service_Generates_a_Triple_Enlarged.jpg}
\caption{The key observation behind the SHARE framework: a web service invocation generates an implicit RDF triple. The subject of this triple is the input, the object is the output, and the predicate is the relationship established between the input and output, as determined by the behaviour of the service. In this case, the service consumes a GI (Genbank Identifier) for a protein, and returns one or more GO terms which annotate the protein. The implicit relationship is \textbf{hasGOTerm}.}
\label{fig:servicetriple}
\end{figure}
The system provides a specialized SPARQL engine which utilizes these predicate annotations to retrieve data ``on demand'' from web services. The syntax of a SHARE query is identical to that of a standard SPARQL query, with the only difference being the resolution behaviour. A query is resolved by: (1) identifying any predicates that can be matched to services, (2) retrieving data from these services, and (3) allowing the query to be resolved as usual on the local triple store. Figure \ref{fig:querypark} shows an example query which asks: ``What transcription factors have been implicated in Parkinson's Disease?''.
\begin{figure}
{\tt
SELECT ?transcriptionFactor\\
WHERE\\
\{\\
?transcriptionFactor SHARE:hasGOTerm GO:0006351 .\\
?transcriptionFactor SHARE:associatedWithDisease OMIM:168600 .\\
\}
}
\caption{A hypothetical SHARE query, which finds transcription factors implicated in Parkinson's Disease. Supposing both \textbf{hasGOTerm} and \textbf{associatedWithDisease} have been assigned to services, the proteins with the specified predicate values can be retrieved dynamically via web service invocations.}
\label{fig:querypark}
\end{figure}
SHARE depends on access to a large central registry of services which are annotated with appropriate predicates. This registry is provided by the existing BioMoby framework and community. Services participating in the SHARE system are required to follow two simple rules: (1) All inputs and outputs of services must be RDF documents, and (2) All inputs and outputs must be specified in terms of OWL classes. A ``seed'' ontology of OWL classes will be provided based on existing BioMoby datatypes, but the system will be completely open to expansion; service providers may specify their interfaces in terms of any OWL classes they choose. The use of OWL to specify interfaces, rather than WSDL \cite{29}, will enable description of both the syntax \emph{and the meaning} of service arguments, thus allowing for a community of truly interoperable services. In addition, service providers will be encouraged to supply predicate annotations for their services. However, as it does no harm to assign multiple predicates to the same service, any users of the system will be able to assign predicates as well.
An early prototype of SHARE, with example queries, is accessible at \\ http://cardioshare.icapture.ubc.ca/cardioSHARE/query.
The system represents a valuable enhancement to standard query systems, as it offers a straightforward mechanism for querying across any number of data sources. In effect, the target of a SHARE query is an enormous \emph{virtual graph}, consisting of all triples that can generated by the complete set of participating services.\footnote{This includes the full set of $\sim$1500 BioMoby services already in the system.} Beyond providing a large, integrated dataset, the system has several additional advantages. As a web service based framework, participating services need not be simple retrieval mechanisms for data; they are capable of performing any calculation that can be accomplished by software. SHARE is therefore not only a framework for integrating databases, but also a framework for integrating analytical programs. A further advantage of the system is that new services may be added by anyone, and the responsibility for maintaining these services is distributed to their creators.
Intuition might suggest that SHARE queries, because they must retrieve data from many remote sources, are vastly slower than equivalent queries on a data warehouse. This is not necessarily the case. For example, one important optimization trick for speeding up query resolution is the use of \emph{inverse services}. Considering the example query in Figure \ref{fig:querypark}, the system might naively find proteins that are associated with Parkinsons (OMIM:168600) by feeding every known protein into a web service that returns OMIM codes. However, it is equally possible that there is a service which accepts OMIM codes as input and return associated proteins.\footnote{In fact, there is such a service in the BioMoby registry, and it is called \textbf{MOBYSHoundGiFromOMIM}.} In the latter case, the question can be answered with a single service invocation.
\section*{Current Work: DL Reasoning Resolved By Web Services}
In a similar fashion, the SHARE framework will extend an OWL reasoner to use predicate annotations on services. When determining instances of a class, the reasoner will have the ability to test property restrictions by means of web service invocations. For example, we could define an OWL class called \textbf{ParkinsonTranscriptionFactor} with the restrictions (hasGOTerm hasValue GO:0006351) and (associatedWithDisease hasValue OMIM:168600). We could then answer the question posed in the previous section, by finding instances of this class. This is completely equivalent to the SPARQL query posed in Figure \ref{fig:querypark}.
The SPARQL and DL reasoning aspects of SHARE will be tied together by allowing an OWL class to be referenced within a SPARQL query. This facility will allow users to formulate complex queries in simple, abstract language. For instance, the original query in Figure \ref{fig:querypark} could be extended to find transcription factors which are both implicated in Parkinson's disease and also have experimentally solved 3D structures (Figure \ref{fig:querypark2}).
\begin{figure}
{\tt
SELECT ?transcriptionFactor\\
WHERE\\
\{\\
?transcriptionFactor rdf:type SHARE:ParkinsonTranscriptionFactor .\\
?transcriptionFactor SHARE:hasSolved3DStructure ?structure .\\
\}
}
\caption{A hypothetical SHARE query, which finds transcription factors that are both implicated in Parkinson's Disease and have at least one experimentally solved 3D structure. The \textbf{rdf:type} triple tells the query engine to match \textbf{?transcriptionFactor} to instances of the OWL class \textbf{ParkinsonTranscriptionFactor}. The system retrieves these instances by invoking web services corresponding to the predicates (\textbf{hasGOTerm} and \textbf{associatedWithDisease}) that have been used to define the class. Each of the instances is then sent to one or more web services that have been annotated with \textbf{hasSolved3DStructure}, in order to retrieve any solved 3D structures that are available.}
\label{fig:querypark2}
\end{figure}
It is reasonable to ask what purpose the reasoner extension serves if classification is exactly like querying, but with the additional restrictions imposed by OWL-DL. The advantage of the reasoner approach can be seen if one imagines defining classes in terms of other classes. If instead of being defined by specific URI values for properties, \textbf{ParkinsonTranscriptionFactor} was defined by the intersection of \textbf{ParkinsonAssociatedProtein} and \textbf{TranscriptionFactor} classes, each having a long list of property restrictions, the equivalent SPARQL query would likely be quite complex. The use of OWL classes provides modularity, reusability, and simplicity when formulating queries.
In addition to enabling reasoning across distributed data sources, the SHARE reasoner will enable classification over large-scale datasets without the need to make changes to existing reasoning algorithms. This is possible for the same reason that large-scale SPARQL queries are possible; the use of inverse services (as explained above) filters out large amounts of irrelevant data that would otherwise have to be processed by the query engine or reasoner.
From a bioinformatics perspective, one of the most interesting applications of the SHARE reasoner will be its ability to automatically ``lift'' raw data into an ontology. If a user wants to gather a complete list of instances for each class in an ontology, all they will have to do is assign the properties of the ontology to available web services, and then run the reasoner. This is interesting because the majority of data annotation in bioinformatics is still done manually with controlled vocabularies such as the Gene Ontology.
The first application of SHARE will be in the analysis of clinical data relating to heart disease. This will entail the development of a SHARE ontology to encode expert knowledge about cardiovascular disease. The research environment provided by this ontology, together with the SHARE framework, will be called CardioSHARE.
\section*{Conclusion}
Currently there are no widely accepted systems for querying or reasoning across distributed data sources. The SHARE framework provides these capabilities, by means of simple extensions to existing query engines and reasoners. At the same time, SHARE allows these tools to operate on vastly larger datasets than would otherwise be possible. The price that must be paid for achieving these improvements is typical of data integration projects in general. First, the system must gain widespread community support in order to have any true value for its users. Fortunately, we already have access to a large community of service providers and users, through the legacy of the BioMoby system. Secondly, service providers must play by a shared set of rules. In the case of SHARE, the rules are simple: the inputs and outputs of services must be RDF documents that are described by OWL classes.
\section*{Acknowledgements}
The development of SHARE and CardioSHARE is made possible by the support of the Heart and Stroke Foundation of British Columbia and Yukon. MDW is funded for CardioSHARE through an operating grant from the Canadian Institutes of Health Research. BioMOBY was developed under support from the Genome Canada/Genome Alberta bioinformatics Platform. Hardware for both projects has been provided by Sun Microsystems and IBM. Core laboratory funding is provided by the Natural Sciences and Engineering Research Council of Canada (NSERC).
|
2,877,628,090,838 | arxiv | \section{Background}
In 2004 Hegarty \cite{hegarty2004permutations} introduced the notion of permutations that destroy arithmetic progressions in finite cyclic groups.
\begin{defn}
Given a permutation $\pi:\mathbb Z/n\mathbb Z\to \mathbb Z/n\mathbb Z$, a three term arithmetic progression $(a, a+r, a+2r)$, with not all terms equal, is called \textit{preserved} in $\pi$ if $\pi(a+2r)-2\pi(a+r)+\pi(a)=0$. A permutation is said to \textit{destroy} all arithmetic progressions if it has no preserved arithmetic progressions.
\end{defn}
For the sake of simplicity, a three-term arithmetic progression will be denoted an AP and a permutation that destroys all APs will be called AP-Destroying. This notion can be extended to permutations which destroy $k$-term arithmetic progressions and Hegarty \cite{hegarty2004permutations} demonstrated that for $n\neq 3,4$ there exists a permutation of $\mathbb Z/n\mathbb Z$ that destroys all $k$-term arithmetic progressions for all $k\ge 4$. However, classifying which cyclic groups have an AP-Destroying permutation has been resistant to proof. In particular Hegarty \cite{hegarty2004permutations} gave the following conjecture regarding AP-Destroying permutations based on computational evidence.
\begin{conj}
For $n\not\in\left\{2,3,5,7\right\}$, there exists an AP-Destroying permutation $\pi:\mathbb Z/n\mathbb Z\to \mathbb Z/n\mathbb Z$.
\end{conj}
This conjecture was proved for sufficiently large $n$ by Hegarty and Martinsson \cite{hegarty2015permutations} in 2015.
\begin{thm}\label{current}
For $n\ge(9\times11\times16\times17\times19\times23)^2$, there exists a AP-Destroying permutation $\pi:\mathbb Z/n\mathbb Z\to \mathbb Z/n\mathbb Z$.
\end{thm}
However given that $(9\times11\times16\times17\times19\times23)^2\approx 1.4\times 10^{14}$, any purely computational approach is out of reach in order to establish Hegarty's original conjecture. We instead base our construction on that of Elkies and Swaminathan \cite{Ashvin}, who proved the following result.
\begin{thm}
Let $p$ be a prime with $p\ge 11$. Then there exists an AP-Destroying permutation $\pi:\mathbb Z/p\mathbb Z\to \mathbb Z/p\mathbb Z$.
\end{thm}
Following this approach we establish the original conjecture of Hegarty \cite{hegarty2004permutations}.
\begin{thm}\label{main}
For $n\not\in\left\{2,3,5,7\right\}$, there exists a AP-Destroying permutation $\pi:\mathbb Z/n\mathbb Z\to \mathbb Z/n\mathbb Z$.
\end{thm}
Note that Hegarty \cite{hegarty2004permutations} computationally checked that each of the values $n\in\left\{2,3,5,7\right\}$ does not have an AP-Destroying permutation, so it suffices to prove that the remaining values do have an AP-Destroying permutation.
\section{Preliminary Reductions}
The starting point for our proof is a theorem from Hegarty \cite{hegarty2004permutations} that can be used to simplify the general case to five infinite classes of integers and a finite exceptional set. (Note that the theorem given by Hegarty \cite{hegarty2004permutations} applies more generally for abelian groups.)
\begin{thm}\label{Product}
If there exists an AP-Destroying permutation for $\mathbb Z/m\mathbb Z$ and $\mathbb Z/n\mathbb Z$, there exists a AP-Destroying permutation for $\mathbb Z/mn\mathbb Z$. Note that $m$ and $n$ are not necessarily coprime.
\end{thm}
Given this theorem it is possible to reduce the set of integers necessary to prove the desired result. This reduction is given without proof in Hegarty \cite{hegarty2004permutations}. (There appears to be a slight error in the version given by Hegarty \cite{hegarty2004permutations} as it excludes the case when $n=343$.)
\begin{thm}
In order to prove Theorem \ref{main}, it suffices to prove the cases $\{p, 2p, 3p, 5p, 7p~|~p$ prime and $p\ge 11\}$ and the integers $\left\{p_1p_2,p_1p_2p_3\right\}$ with $p_i\in\left\{2,3,5,7\right\}$, not necessarily distinct.
\end{thm}
\begin{proof}
Suppose that $n=2^{a_1}3^{a_2}5^{a_3}7^{a_4}p_1^{b_1}\ldots p_k^{b_k}$. If $2^{a_1}3^{a_2}5^{a_3}7^{a_4}\not\in\left\{1, 2,3,5,7\right\}$ then find a AP-Destroying permutation for each $p_i^{b_i}$ and $2^{a_1}3^{a_2}5^{a_3}7^{a_4}$ and the result follows from the previous lemma. The last integer can be constructed as $a_1+a_2+a_3+a_4\ge 2$ so we can represent $a_1+a_2+a_3+a_4$ as a sum of $2$'s and $3$'s and using this we can construct $2^{a_1}3^{a_2}5^{a_3}7^{a_4}$ as a product of products of $2$ or $3$ primes in $\left\{2,3,5,7\right\}$. Otherwise we take the $2^{a_1}3^{a_2}5^{a_3}7^{a_4}p_1$ and $\frac{n}{2^{a_1}3^{a_2}5^{a_3}7^{a_4}p_1}$ in order to represent $n$ an find a permutation for each of the integers independently.
\end{proof}
To prove the result for the cases $\{2p, 3p, 5p, 7p~|~p$ prime and $p\ge 11\}$ we model our construction based on the one used by Elkies and Swaminathan \cite{Ashvin} to demonstrate Theorem 4. The key similarity is the following lemma of Elkies and Swaminathan from \cite{Ashvin}, which we will rely heavily on as well. Note that in the statement below, and elsewhere in this paper we will not distinguish between an arithmetic progression and its reverse.
\begin{lemma}\label{hi}
Suppose that $\pi: \mathbb Z/p\mathbb Z\to \mathbb Z/p\mathbb Z$ is the following permutation with $t\neq 0$:
\[\pi:=\begin{cases}
t & x=0
\\ 0 & x=1
\\ \frac{t}{x} & x\notin{0, 1}
\end{cases}.\]
Then the only APs preserved by $\pi$ are $(0, \frac{3}{2}, 3)$, $(\frac{1}{3}, \frac{2}{3}, 1)$.
\end{lemma}
Elkies and Swaminathan \cite{Ashvin} then performed two transpositions in order to eliminate these preserved APs and this demonstrated the case when $n$ is prime. In the case $n=2p$ we will ``glue" together two such permutations in a careful manner so that there is exactly one preserved AP, and then using a single transposition we eliminate preserved AP. In the remaining cases however we are able to significantly simplify this approach by directly giving an AP that has no arithmetic progression, avoiding the need for a transposition. In each of these cases however we will not simply be able to show $2p, 3p, 5p, 7p$ for all primes $p$ directly; instead, certain character estimates will show it for $p$ sufficiently large. Thus we show the conjecture to be true for all $n\le2500$ using computational techniques, and this will be a starting point for the analysis in the remaining cases. Note that the $5p$ case, where $p>500$ is assumed, is the limiting case here. All mentioned computational files can be found on the corresponding arXiv submission.
One piece of machinery that is used multiple times in this paper is the Hasse-Weil bound. (Elkies and Swaminathan \cite{Ashvin} similarly require such character estimates, but they can make do with the elementary Hasse bound.) Note that the version we are using is equivalent to counting the number of points on the hyperelliptic curve $y^2=g(x)
$ over a finite field and the bound we are using was proven for curves by Weil in \cite{weil1949numbers}.
\begin{thm}\label{hasse-weil}
Let $\mathbb{F}_p$ be the field with $p$ elements, $p$ being prime,
and let $(\frac{\cdot}{p})$ be the Legendre symbol.
If $f \in \mathbb{F}_p[x]$ is a polynomial of degree $2g+1$ or
$2g+2$ such that $g$ is not a constant times a perfect square in $\mathbb{F}_p[x]$, then
\[
\Bigl|\sum_{y \in \mathbb Z/p\mathbb Z} (\frac{f(y)}{p})\Bigr| \leq 2g\sqrt{p}+1.
\]
\end{thm}
\section{AP-Destroying Permutations for $\mathbb Z/2p\mathbb Z$}
Our initial construction is the following permutation $\pi_2: \mathbb Z/2\mathbb Z\times \mathbb Z/p\mathbb Z\to \mathbb Z/2\mathbb Z\times \mathbb Z/p\mathbb Z$, for a parameter $t\notin\left\{0, 1\right\}$ to be chosen later:
\[\pi_2 := \setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.2}
\left\{\begin{array}{l @{\quad} l}
(0, 0)\to (1, t) & (1, 0)\to (0, 1)
\\(0, 1)\to (1, 0) & (1, 1)\to (0,0)
\\(0, x)\to (0, \frac{1}{x}), x\notin\left\{0, 1\right\} & (1, x)\to (1, \frac{t}{x}), x\notin\left\{0, 1\right\}
\end{array}\right.\]
\begin{lemma}\label{Lemma2pPartA}
Suppose that
\[t\not\in \left\{0,1,\frac{1}{4},4,\frac{1}{9},9\right\} \]
and furthermore
\[(\frac{1-\frac{1}{t}}{p})=(\frac{1-t}{p})=-1.\]
Then the only three term arithmetic progressions preserved by $\pi_2$ are $\left\{(0, 1), (1,1), (0, 1)\right\} $ and $\left\{(1, 1), (0, 1), (1, 1)\right\}$.
\end{lemma}
\begin{proof}
We proceed via contradiction. Suppose that $t$ satisfies the above properties, and that some other three term arithmetic progression $T$ is preserved. Let $U$ be the image of $T$. Furthermore denote by $T_2$ and $T_p$ the$\mod 2$ and$\mod p$ components of $T$, respectively, and define $U_2$ and $U_p$ similarly. We separate cases based on the numbers of elements of $T_p$ which are in $\{0,1\}$.
\begin{enumerate}[\text{Case }1.]
\item$T_p$ is of the form $(a-r, a, a+r)$ with $\left\{a-r, a, a+r\right\}\cap \left\{0, 1\right\}=\emptyset$. We take cases which exhaust the possible values of $T_2$.
\begin{enumerate}[\text{Case 1.}a.]
\item $T_2=(0, 0, 0)$ or $(1,1,1)$. Then $U_p=(\frac{1}{a-r}, \frac{1}{a}, \frac{1}{a+r})$ or $U_p=(\frac{t}{a-r}, \frac{t}{a}, \frac{t}{a+r})$ depending on $T_2$. In either case, since $t\nequiv 0$, $U_p$ being an AP is equivalent to $\frac{2}{a}\equiv \frac{1}{a-r}+\frac{1}{a+r}\mod p$, which is equivalent to $r^2\equiv 0$. However, this is impossible as $T$ would then be a degenerate AP.
\item $T_2=(0, 1, 0).$ Then $U_p=(\frac{1}{a-r}, \frac{t}{a}, \frac{1}{a+r})$ and $U_p$ being an AP is equivalent to $\frac{2t}{a}\equiv \frac{1}{a-r}+\frac{1}{a+r}\mod p$. This is equivalent to$(\frac{r}{a})^2\equiv 1-\frac{1}{t}$, which is impossible as $(\frac{1-\frac{1}{t}}{p})=-1$.
\item $T_2=(1, 0, 1).$ Then $U_p=(\frac{t}{a-r}, \frac{1}{a}, \frac{t}{a+r})$. Hence $\frac{2}{a}\equiv \frac{t}{a-r}+\frac{t}{a+r}\mod p$, which is equivalent with $(\frac{r}{a})^2\equiv1-t$. However, this is impossible as $(\frac{1-t}{p})=-1$.
\end{enumerate}
\item We now consider the case where $|T_p\cap \{0,1\}|=1$. It therefore follows, reversing the AP if necessary, that either $T_p=(1,1+r,1+2r)$, $T_p=(1-r,1,1+r)$, $T_p=(0,r,2r)$, or $T_p=(-r,0,r)$.
\begin{enumerate}[\text{Case 2.}a.]
\item $T_p=(1,1+r,1+2r).$ Note that if $T_2=(0,w_1,w_2)$ then $U_2=(1,w_1,w_2)$ or vice versa and both of these can not be APs.
\item $T_p=(1-r,1,1+r).$ There are now four possible cases of $T_2$. If $T_2=(0,0,0)$ or $(0,1,0)$ then $U_p=(\frac{1}{1-r},0,\frac{1}{1+r})$. This being an AP is equivalent to $\frac{1}{1-r}+\frac{1}{1+r}\equiv 0$. Simplifying, this is equivalent to $\frac{2}{1-r^2}\equiv 0$ which is impossible. If $T_2=(1,0,1)$ or $(1,1,1)$ then $U_p=(\frac{t}{1-r},0,\frac{t}{1+r})$. This being an AP is equivalent to $\frac{t}{1-r}+\frac{t}{1+r}\equiv 0$. Simplifying, this is equivalent to $\frac{2t}{1-r^2}\equiv 0$ which is impossible as $t\nequiv 0$.
\item $T_p=(0,r,2r).$ There are now four possible cases of $T_2$. If $T_2=(0,0,0)$ then $U_2=(1,0,0)$ which is not an AP modulo $2$. If $T_2=(1,1,1)$ then $U_2=(0,1,1)$ which is not an AP modulo $2$. If $T_2=(0,1,0)$ then $U_2=(1,1,0)$ which is not an AP modulo $2$. Finally if $T_2=(1,0,1)$ then $U_2=(0,0,1)$ which is not an AP modulo $2$.
\item $T_p=(-r,0,r).$ There are now four possible cases of $T_2$. If $T_2=(0,0,0)$ then $U_p=(\frac{-1}{r},t,\frac{1}{r})$ which is not an AP as $t\nequiv 0$. If $T_2=(0,1,0)$ then $U_p=(\frac{-1}{r},1,\frac{1}{r})$ which is not an AP as $1\nequiv 0$. If $T_2=(1,1,1)$ then $U_p=(\frac{-t}{r},1,\frac{t}{r})$ which is not an AP as $1\nequiv 0$. If $T_2=(1,0,1)$ then $U_p=(\frac{-t}{r},t,\frac{t}{r})$ which is not an AP as $t\nequiv 0$.
\end{enumerate}
\item In the final case we have that at least two elements of $T_p$ are in $\{0,1\}.$ Reversing the AP if necessary, this gives the cases $T_p=(0,0,0)$, $(1,1,1)$, $(0,\frac{1}{2},1)$, $(0,1,2)$, or $(-1,0,1)$. The second case gives exactly the APs mentioned in the statement of the lemma and therefore it suffices to study the other four cases.
\begin{enumerate}[\text{Case 3.}a.]
\item $T_p=(0,0,0)$. In order for $T$ to not be a trivial progression, $T_2=(0,1,0)$ or $(1,0,1).$ In the first case, $U_p=(t,1,t)$ which is not an AP as $t\nequiv 1$. In the second case, $U_p=(1,t,1)$ which is not an AP as $t\nequiv 1$.
\item $T_p=(0,\frac{1}{2},1).$ If $T_2=(0,0,0)$ then $U_p=(t,2,0)$ but $t\nequiv 4$. If $T_2=(1,1,1)$ then $U_p=(1,2t,0)$ but $t\nequiv \frac{1}{4}$. If $T_2=(0,1,0)$ then $U_p=(t,2t,0)$ but $t\nequiv 0$. Finally if $T_2=(1,0,1)$ then $U_p=(1,2,0)$ which is never an AP.
\item $T_p=(0,1,2).$ If $T_2=(0,0,0)$ then $U_2=(1,1,0)$. If $T_2=(1,1,1)$ then $U_2=(0,0,1)$. If $T_2=(0,1,0)$ then $U_2=(1,0,0)$. Finally if $T_2=(1,0,1)$ then $U_2=(0,1,1)$. In none of these cases is $U_2$ an AP.
\item $T_p=(-1,0,1).$ If $T_2=(0,0,0)$ then $U_2=(0,1,1)$. If $T_2=(1,1,1)$ then $U_2=(1,0,0)$. If $T_2=(0,1,0)$ then $U_2=(0,0,1)$. Finally if $T_2=(1,0,1)$ then $U_2=(1,1,0)$. In none of these cases is $U_2$ an AP.
\end{enumerate}
\end{enumerate}
\end{proof}
Now we claim that a $t$ with the conditions of the previous lemma exists for every prime $p\ge 31$.
\begin{lemma}\label{Lemma2pPart2}
For $p\ge 31$, there exists a $t$ such that
\[t\not\in \left\{0,1,\frac{1}{4},4,\frac{1}{9},9\right\} \]
and
\[(\frac{1-\frac{1}{t}}{p})=(\frac{1-t}{p})=-1.\]
\end{lemma}
\begin{proof}
First note that $(\frac{1-\frac{1}{t}}{p})=(\frac{t(t-1)}{p})$ for $t\nequiv 0$. Then note that
\begin{align*}
\sum_{t\in \mathbb Z/p\mathbb Z}(1-(\frac{1-t}{p}))(1-(\frac{t(t-1)}{p}))
&= \sum_{t\in \mathbb Z/p\mathbb Z}1-(\frac{1-t}{p})-(\frac{t(t-1)}{p})+(\frac{-t(t-1)^2}{p})
\\& \ge p-4
\end{align*}
where we have used that $(\frac{(t-1)^2}{p})=1$ for $t\neq 1$ and the Hasse-Weil bound. Therefore the number of $t\in \mathbb Z/p\mathbb Z$ which satisfy $(\frac{t(t-1)}{p})=(\frac{1-t}{p})=-1$ is at least $\frac{p-5}{4}$ as $t=0, 1$ together contribute exactly 1 in total to the sum. For $p\ge31$, we have $\frac{p-5}{4}>6$ so for such $p$ there exists a $t$ outside of those in the set $\{0,1,\frac{1}{4},4,\frac{1}{9},9\}$ as required.
\end{proof}
Now choose any such fixed $t$ satisfying the above conditions. Consider the following adjustment of $\pi_2$:
\[\pi_2^y := \setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.2}
\left\{\begin{array}{l @{\quad} l}
(0, 0)\to (1, t) & (1, 0)\to (0, 1)
\\(0, 1)\to (0, \frac{1}{y}) & (1, 1)\to (0,0)
\\(0, y)\to (1, 0) & (1, x)\to (1, \frac{t}{x}), x\notin\left\{0, 1\right\}
\\ (0, x)\to (0, \frac{1}{x}), x\notin\left\{0, 1,y\right\}.
\end{array}\right.\]
We claim that there exists a $y$ for which $\pi_2^y$, which is $\pi_2$ with the values of $(0, y)$ and $(0, 1)$ exchanged, is AP-Destroying permutation for some choice of $y$. In particular, we claim the following.
\begin{lemma}
Suppose that
\[y\not\in\left\{0,1,-1,2, \frac{1}{2},\frac{1}{3}, 4, \frac{4}{t}, \frac{1}{2t+1}\right\},\]
\[(\frac{1-ty}{p})=(\frac{1-y}{p})=(\frac{(4t-1)^2y^2-2(4t+1)y+1}{p})=-1,\] and that \[(\frac{1-9y}{p})=1.\]
Then $\pi_2^y$ is AP.
\end{lemma}
\begin{proof}
Note that the only difference between $\pi_2$ and $\pi_2^y$ is the exchange of $(0,1)$ and $(0,y)$. Note that this transposition destroys the APs $\{(1,0),(0,0),(1,0)\}$ and $\{(0,0),(1,0),(0,0)\}$, and it suffices to demonstrate that we created no new APs. Due to Lemma \ref{Lemma2pPartA} these APs must contain $(0,1)$ or $(0,y)$. We have four cases.
\begin{enumerate}[\text{Case }1.]
\item $T$ contains $(0,y)$ and $T_p=(y, y+r, y+2r)$. We take two cases based on the possibilities for $T_2$.
\begin{enumerate}[\text{Case 1.}a.]
\item $T_2=(0, 0, 0)$. First note that $r\nequiv 0$, as otherwise $T$ is a trivial AP. Then if $y+2r\nequiv 0$ it follows that $U_2=(1,\cdot,0)$ which is never an AP. If $y+2r\equiv 0$ then $T_p=(y,\frac{y}{2},0).$ Since $y\neq 2$, $U_p=(0,\frac{2}{y},t)$ but $y\nequiv \frac{4}{t}$ so this is not an AP.
\item $T_2=(0, 1, 0)$. If $r\equiv 0$, then $U_p=(0, \frac{t}{y}, 0)$, which is never an AP since $t\nequiv 0$. Otherwise, if $y+2r\nequiv0$ then $U_2=(1,\cdot,0)$ which is not an AP. If $y+2r\equiv 0$ then $T_p=(y,\frac{y}{2},0).$ Since $y\nequiv 2$ then $U_p=(0,\frac{2t}{y},t)$ but $y\nequiv 4$ so this is not an AP.
\end{enumerate}
\item $T$ contains $(0,y)$ and $T_p=(y-r, y, y+r)$. We take two cases based on the possibilities for $T_2$.
\begin{enumerate}[\text{Case 2.}a.]
\item $T_2=(0, 0, 0)$. First note that $r\nequiv 0$, as otherwise $T$ is a trivial AP. If $\left\{y+r, y-r\right\}\cap \left\{0, 1\right\}=\emptyset$, then $U_p=(\frac{1}{y-r}, 0, \frac{1}{y+r})$, which is not an AP as $y\nequiv 0$. By symmetry, it suffices to check the cases $y-r\equiv 0, 1$. If $y-r\equiv0$ then $y+r\equiv2y\nequiv 1$ as $y\nequiv \frac{1}{2}$. Therefore we have $U_2=(1, 1, 0)$, which is not an AP. In the case $y-r\equiv1$, we have $y+r\equiv2y-1\notin\left\{1, y\right\}$. Now $2y-1\nequiv 0$, as $y\nequiv \frac{1}{2}$. Therefore, $U_p=(\frac{1}{y},0, \frac{1}{2y-1})$, which is not an AP as $y\nequiv \frac{1}{3}$.
\item $T_2=(1, 0, 1)$. If $r\equiv 0$, $U_p=(\frac{t}{y},0,\frac{t}{y})$ which is never an AP as $t\neq 0$ and thus $r\nequiv 0$ suffices. If $\left\{y+r, y-r\right\}\cap \left\{0, 1\right\}=\emptyset$, then $U_p=(\frac{t}{y-r}, 0, \frac{t}{y+r})$, which is not an AP as $yt\nequiv 0$. If $y-r\equiv 0$, then $y+r\nequiv 1$ as $y\nequiv \frac{1}{2}$. Furthermore since $y+r\nequiv 0$ it follows that $U_2=(0,1,1)$ which is not an AP. If $y-r\equiv 1$, then $y+r\nequiv 0$ as $y\nequiv\frac{1}{2}$. Since $y+r\nequiv 1$ it follows that $U_2=(0,1,1)$ which is never an AP.
\end{enumerate}
Note that in the following two cases, we may assume that $T$ does not contain $(0, y)$ as these have been handled.
\item $T$ contains $(0,1)$ and $T_p=(1, 1+r, 1+2r)$. We take two cases based on the possibilities for $T_2$.
\begin{enumerate}[\text{Case 3.}a.]
\item $T_2=(0, 0, 0)$. First note that $r\nequiv 0$, as otherwise $T$ is a trivial AP. If $1+2r\in \left\{0,y\right\}$, then $U_2=(1,\cdot,0)$ is not an AP. If $1+r\equiv 0$, then $U_p=(\frac{1}{y}, t, -1)$, which is impossible as $y\nequiv \frac{1}{2t+1}$. Similarly, $1+r\equiv y$ yields $U_p=(\frac{1}{y}, 0, \frac{1}{2y-1})$, which is not an AP since $y\nequiv \frac{1}{3}$. Therefore it suffices to study the general case where $U_p=(\frac{1}{y}, \frac{1}{1+r}, \frac{1}{1+2r})$. The condition for this being an AP is a quadratic in $r$ and has discriminant $(9y-1)(y-1)$. This is not a perfect square as $(\frac{1-9y}{p})=1$ and $(\frac{1-y}{p})=-1$ by assumption.
\item $T_2=(0, 1, 0)$. If $r\equiv 0$, then $U_p=(\frac{1}{y}, 0, \frac{1}{y})$, which is not an AP. If $1+2r\in\left\{0, y\right\}$, then $U_2=(0,\cdot,1)$ is not an AP. If $1+r\equiv 0$, then since $y\nequiv -1$, we have $U_p=(\frac{1}{y}, 1, -1)$, which is not an AP as $y\nequiv \frac{1}{3}$. Finally, in the general case we have $U_p=(\frac{1}{y}, \frac{t}{1+r}, \frac{1}{1+2r})$. The condition for this sequence being an AP is a quadratic in $r$, and its discriminant is $(1-4t)^2y^2-2(4t+1)y+1$. However this is not a perfect square by assumption.
\end{enumerate}
\item $T$ contains $(0,1)$ and $T_p=(1-r, 1, 1+r)$. We take two cases based on the possibilities for $T_2$.
\begin{enumerate}[\text{Case 4.}a.]
\item $T_2=(0, 0, 0)$. First note that $r\nequiv 0$, as otherwise $T$ is a trivial AP. If $1-r\equiv 0$, then $2\equiv 1+r\notequiv y$.
It follows that $U_2=(1,0,0)$, which is not an AP. Since we can assume $1-r\nequiv y$ due to previous cases and we can reverse the AP as necessary, it suffices to consider $\{1-r,1+r\}\cap \{0,1,y\}=\emptyset$. In the remaining cases it follows that $U_p=(\frac{1}{1-r}, \frac{1}{y}, \frac{1}{1+r})$, which implies $y\equiv 1-r^2$. But this is impossible since $(\frac{1-y}{p})=-1$.
\item $T_2=(1, 0, 1)$. If $r\equiv 0$, then $U_p=(0, \frac{1}{y}, 0)$, which is never an AP. If $1-r\equiv 0$, then $1+r\equiv 2$, so $U_2=(0,0,1)$ which is not an AP. Finally, in the general case $U_p=(\frac{t}{1-r}, \frac{1}{y}, \frac{t}{1+r})$. The condition for this being an AP is equivalent to $r^2\equiv1-ty$, which is impossible as $(\frac{1-ty}{p})=-1$.
\end{enumerate}
\end{enumerate}
This exhausts all possible cases, so the proof is complete.
\end{proof}
Having shown this, we finally proceed to showing the existence of $y$ which satisfies the hypotheses of Lemma $12$.
\begin{lemma}\label{Lemma2pPart3}
For $p>500$ and a fixed $t$ which satisfies the hypotheses of Lemma 10, there exists a $y$ which satisfies the hypotheses of Lemma 12.
\end{lemma}
\begin{proof}
Let $f_1=y^2(4t-1)^2-2(4t+1)y+1$. We consider
\[\sum_{y\in \mathbb Z/p\mathbb Z}(1-(\frac{1-ty}{p}))(1-(\frac{1-y}{p}))(1-(\frac{f_1}{p}))(1+(\frac{(1-9y)}{p})).\]
Expanding this product yields the $p$ plus $15$ terms of the form $\sum_{y=0}^{p-1}\pm(\frac{\pm g(y)}{p})$ where $g(y)$ is the product of some terms in the set
\[\{1-y, 9y-1, 1-ty, f_1\}.\]
We claim that none of the $g(y)$ which arise are perfect squares. To see this we instead prove the stronger claim that no two terms share a root and thus it suffices to show that the discriminant of the product is nonzero. In particular \[\Delta((y-1)(9y-1)(ty-1)f_1)=2^{28} t^3 (t-9)^2 (t-4)^2 (t-1)^8 (9 t-1)^2\] and all roots of the discriminant are in the set of excluded $t$. Hence each of the $15$ sums is at most $4\sqrt{p}+1$ in absolute value using the Hasse-Weil bound, so the entire sum is at least $p-60\sqrt{p}-15$. When $p>10000$, we have $\frac{p-60\sqrt{p}-15}{16}\ge \frac{40\sqrt{p}-15}{16}>9$. So, more than $9$ values of $y$ contribute a nonzero term to the above sum, which means that some $y$ outside of the required exceptional set satisfies
\[(\frac{1-ty}{p})=(\frac{1-y}{p})=(\frac{(1-4t)^2y^2-2(4t+1)y+1}{p})=-1\] and \[(\frac{1-9y}{p})=1\]
as required. Hence there exists an AP-Destroying permutation for $n=2p, p>10000$. In the cases $500<p<10000$, the existence of $y$ is verified in LegrendeSymbol2p.java.
\end{proof}
\section{AP-Destroying Permutations for $\mathbb Z/3p\mathbb Z$}
For each constant $t\in \mathbb Z/p\mathbb Z, t\notin\{0, 1\}$, we can define the following permutation:
\[\pi_3 := \setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.2}
\left\{\begin{array}{l @{\quad} l @{\quad}l}
(0, 0)\to (1, 0) & (0, 1)\to (1, 1) & (0, x)\to (0, \frac{1}{x}), x\notin \{0, 1\}
\\ (1, 0)\to (2, t) & (1, 1)\to (2, 0) & (1, x)\to (1, \frac{1}{x}), x\notin \{0, 1\}
\\ (2, 0)\to (0, 1) & (2, 1)\to (0, 0) & (2, x)\to (2, \frac{t}{x}), x\notin \{0, 1\}
\end{array}\right.\]
\begin{lemma}
Suppose that $t\in \mathbb Z/p\mathbb Z$ such that
\[t\notin \left\{-1, 0, 1, \frac{1}{2}, 2, 9\right\}\]
and
\[(\frac{t(t-1)}{p})=(\frac{(t-1)(t-9)}{p})=-1.\]
Then $\pi_3$ is AP.
\end{lemma}
\begin{proof}
Suppose for sake of contradiction that some arithmetic progression $T$ is preserved, and let $U$ be its image. Denote by $T_3, T_p, U_3, U_p$ the projections of $T$ and $U$ modulo $3$ and $p$ respectively. We take three cases:
\begin{enumerate}[\text{Case }1.]
\item Three elements of $T_p$ are in $\{0, 1\}$. Then since $T_p$ is an AP and $p>2$, this implies $T_p=(0, 0, 0)$ or $T_p=(1, 1, 1)$. In the former case, $U_p$ is a permutation of $(0, 1, t)$, which is not an AP as $t\notin\{-1, \frac{1}{2}, 2\}$. In the latter case, $U_p$ is a permutation of $(1, 0, 0)$, which is not an AP. Hence case $1$ is impossible.
\item One or two elements of $T_p$ are in $\{0, 1\}$. Consider the triple $T_3'$ obtained by incrementing each of the three elements in $T_3$. Note that $\pi_3$ increments the mod $3$ value of its input if that input is $0$ or $1$ mod $p$, and otherwise the mod $3$ value stays the same. It follows that if one element of $T_p$ is in $\{0, 1\}$, then $U_3$ differs from $T_3$ in exactly one element, and if two elements of $T_p$ are in $\{0, 1\}$, then $U_3$ differs from $T_3'$ in exactly one element. In both cases, $U_3$ cannot be an AP.
\item None of the elements of $T_p$ are in $\{0, 1\}$. Let $T_p=(a-r, a, a+r)$. Then we take four cases based on the possible values of $T_3$.
\begin{enumerate}[\text{Case 3.}a.]
\item $T_3=(0, 0, 0)$ or $(1, 1, 1)$. Then $U_p=(\frac{1}{a-r}, \frac{1}{a}, \frac{1}{a+r})$. It follows that $\frac{1}{a-r}+\frac{1}{a+r}=\frac{2}{a}$, so $r\equiv 0$, which is impossible.
\item $T_3=(2, 2, 2)$. Then $U_p=(\frac{t}{a-r}, \frac{t}{a}, \frac{t}{a+r})$. Since $t\nequiv 0$, this reduces to the previous case.
\item $T_3=(0, 1, 2), (1, 0, 2), (2, 0, 1),$ or $(2, 1, 0)$. By symmetry, we may suppose $T_3$ is of one of the first two triplets. Then $U_p=(\frac{1}{a-r}, \frac{1}{a}, \frac{t}{a+r})$. Solving the AP condition as a quadratic in $r$, we obtain a discriminant $(t-1)(t-9)$. This, however, is not a perfect square mod $p$ by assumption.
\item $T_3=(0, 2, 1)$ or $(1, 2, 0)$. Then $U_p=(\frac{1}{a-r}, \frac{t}{a}, \frac{1}{a+r})$. Solving the AP condition as a quadratic in $r$, we obtain a discriminant $16t(t-1)$. This, however, is not a perfect square mod $p$ by assumption.
\end{enumerate}
\end{enumerate}
\end{proof}
Now we prove that for $p\ge 31$, some prime satisfying the conditions of Lemma 8 exists.
\begin{lemma}
For $p\ge 31$, there exists a $t\in \mathbb Z/p\mathbb Z$ such that
\[t\notin \{-1, 0, 1, \frac{1}{2}, 2, 9\}\]
and
\[(\frac{t(t-1)}{p})=(\frac{(t-1)(t-9)}{p})=-1.\]
\end{lemma}
\begin{proof}
We may calculate
\[
\sum_{t\in\mathbb Z/p\mathbb Z}(1-(\frac{(t-1)(t-9)}{p}))(1-(\frac{t(t-1)}{p}))\]\[= p+\sum_{t\in\mathbb Z/p\mathbb Z}((\frac{t(t-9)(t-1)^2}{p})-(\frac{t(t-1)}{p})-(\frac{(t-1)(t-9)}{p}))\ge p-4
\]
where we have used the Hasse-Weil Bound and that $(\frac{(t-1)^2}{p})=1$ for $t\nequiv 1$.
It follows that the number of solutions to $(\frac{t(t-1)}{p})=(\frac{(t-1)(t-9)}{p})=-1$ over $t\in\mathbb Z/p\mathbb Z$ is at least $\frac{p-5}{4}>6$, so that there is in particular some $t$ outside of the exceptional set satisfying these conditions. For this value of $t$, $\pi_3$ is an AP-Destroying permutation, as desired.
\end{proof}
\section{AP-Destroying Permutations for $\mathbb Z/5p\mathbb Z$}
For each constant $t\in \mathbb Z/p\mathbb Z, t\notin\{-1, 0, 1\}$, we can define the following permutation:\[\pi_5 := \setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.2}
\left\{\begin{array}{l @{\quad} l @{\quad}l}
(0, 0)\to (3, 1) & (0, 1)\to (3, 0) & (0, x)\to (0, \frac{t}{x}), x\notin\{0, 1\}
\\ (1, 0)\to (2, 0) & (1, 1)\to (2, t) & (1, x)\to (1, \frac{t+1}{x}), x\notin\{0, 1\}
\\ (2, 0)\to (1, t+1) & (2, 1)\to (1, 0) & (2, x)\to (2, \frac{t}{x}), x\notin\{0, 1\}
\\ (3, 0)\to (4, 1) & (3, 1)\to (4, 0) & (3, x)\to (3, \frac{1}{x}), x\notin\{0, 1\}
\\ (4, 0)\to (0, t) & (4, 1)\to (0, 0) & (4, x)\to (4, \frac{1}{x}), x\notin\{0, 1\}.
\end{array}\right.\]
We first note two properties of the permutation $\sigma: \mathbb Z/5\mathbb Z\to \mathbb Z/5\mathbb Z$ defined by $\sigma(0)=3, \sigma(1)=2, \sigma(2)=1, \sigma(3)=4, \sigma(4)=0$. The first is that $\sigma(i)\neq i$ for each $i$, so that in particular no AP with exactly two elements in the rightmost column can be preserved. Also, the only APs preserved by $\sigma$ are $(3, 1, 4), (0, 1, 2)$, and their reverses. In particular, every AP preserved by $\sigma$ contains $1$.
\begin{lemma}
Suppose that $t\in \mathbb Z/p\mathbb Z$ such that
\[t\notin \{-3,-2, -1, 0, 1, 2, 3, 4, -\frac{3}{2}, -\frac{4}{3}, -\frac{3}{4}, -\frac{2}{3}, -\frac{1}{2},-\frac{1}{3}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2},\frac{2}{3}, \frac{3}{4}, \frac{3}{2}\}\]
and
\[(\frac{9t-16}{p})=(\frac{9-16t}{p})=(\frac{t+1}{p})=(\frac{(t-1)(t-9)}{p})=(\frac{(t-1)(9t-1)}{p})=-1, (\frac{t}{p})=1.\]
Then $\pi_5$ is AP.
\end{lemma}
\begin{proof}
Suppose for sake of contradiction that some arithmetic progression $T$ is preserved, and let $U$ be its image. Denote by $T_5, T_p, U_5, U_p$ the projections of $T$ and $U$ modulo $5$ and $p$ respectively. We take four cases:
\begin{enumerate}[\text{Case }1.]
\item Three of the elements of $T_p$ are in $\{0, 1\}$. Then since $T_p$ is an $AP$, we must have either $T_p=(0, 0, 0)$ or $T_p=(1, 1, 1)$. In the first case, $U_p$ is an AP formed with elements in $\{0, 1, t, t+1\}$ not all equal. But this is impossible as $t\notin\{-2, -1, 0, 1, 2, \frac{1}{2}, -\frac{1}{2}\}$. The second case is also impossible since the only APs preserved by $\sigma$ contain $1$.
\item Two of the elements of $T_p$ are in $\{0, 1\}$. Then there are three possible values of $T_p$ up to symmetry.
\begin{enumerate}[\text{Case 2.}a.]
\item $T_p=(0, \frac{1}{2}, 1)$. Then the first element of $U_p$ is in $\{0, 1, t, t+1\}$, the middle element is in $\{2, 2t, 2(t+1)\}$, and the last element is in $\{0, t\}$. Checking the $24$ potential combinations, there are no APs for $t$ outside of the set
\[\{-2, -1, 0, 1, 2, 3, 4, -\frac{3}{2}, -\frac{4}{3}, -\frac{3}{4}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{3}{2}\}\]
\item $T_p=(-1, 0, 1)$. Then the first element of $U_p$ is in $\{-t-1, -t, -1\}$, the middle element is in $\{1, 0, t, t+1\}$, and the last element is in $\{0, t\}$. One of the $24$ possible combinations is $(-t, 0, t)$. However, this can only be the case if $T_5=(0, 1, 1)$ or $(2, 1, 1)$, neither of which are APs. Checking the remaining $23$ potential combinations, there are no APs for $t$ outside of the set
\[\{-3, -2, -1, 0, 1, 3, -\frac{3}{2}, -\frac{2}{3}, -\frac{1}{2}, -\frac{1}{3}\} \]
\item $T_p=(0, 1, 2)$. Then the first element of $U_p$ is in $\{0, 1, t, t+1\}$, the second element is in $\{0, t\}$, and the third is in $\{\frac{1}{2}, \frac{t}{2}, \frac{t+1}{2}\}$. Checking the $24$ potential combinations, there are no APs for $t$ outside of the set
\[\{-3, -2, -1, 0, 1, 2, 3, -\frac{3}{2}, -\frac{2}{3}, -\frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}\}\]
\end{enumerate}
\item One of the elements of $T_p$ is in $\{0, 1\}$. Then since $\sigma(i)\neq i$ for $0\le i\le 4$, it follows that $U_5$ cannot be an AP.
\item None of the elements of $T_p$ are in $\{0, 1\}$. Let $T_p=(a-r, a, a+r)$. If all coordinates of $T_5$ are equal, then since $t\notin\{0, -1\}$ we would have $\frac{1}{a-r}+\frac{1}{a+r}\equiv \frac{2}{a}$. But this implies $r=0$, which is impossible. Then there are seven remaining cases based on the possible values of $T_5$, up to reverses.
\begin{enumerate}[\text{Case 4.}a.]
\item $T_5=(0, 1, 2)$. Then $U_p=(\frac{t}{a-r}, \frac{t+1}{a}, \frac{t}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $t+1$, which isn't a perfect square by assumption.
\item $T_5=(0, 2, 4)$ or $(2, 0, 3)$. Then $U_p=(\frac{t}{a-r}, \frac{t}{a}, \frac{1}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $(9t-1)(t-1)$, which isn't a perfect square by assumption.
\item $T_5=(0, 3, 1)$ or $(2, 4, 1)$. Then $U_p=(\frac{t}{a-r}, \frac{1}{a}, \frac{t+1}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $9-16t$, which isn't a perfect square by assumption.
\item $T_5=(0, 4, 3)$ or $(2, 3, 4)$. Then $U_p=(\frac{t}{a-r}, \frac{1}{a}, \frac{1}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $(t-1)(t-9)$, which isn't a perfect square by assumption.
\item $T_5=(3, 1, 4)$. Then $U_p=(\frac{1}{a-r}, \frac{t+1}{a}, \frac{1}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $t(t+1)$, which isn't a perfect square by assumption.
\item $T_5=(1, 0, 4)$ or $(1, 2, 3)$. Then $U_p=(\frac{t+1}{a-r}, \frac{t}{a}, \frac{1}{a+r})$. The condition that this is an AP is a quadratic in $r$ with discriminant $t(9t-16)$, which isn't a perfect square by assumption.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{lemma}
For $p>500$ there exists a $t$ such that
\[t\notin \{-3,-2, -1, 0, 1, 2, 3, 4, -\frac{3}{2}, -\frac{4}{3}, -\frac{3}{4}, -\frac{2}{3}, -\frac{1}{2},-\frac{1}{3},\frac{1}{3}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2},\frac{2}{3}, \frac{3}{4}, \frac{3}{2}\}\]
and
\[(\frac{9t-16}{p})=(\frac{9-16t}{p})=(\frac{t+1}{p})=(\frac{(t-1)(t-9)}{p})=(\frac{(t-1)(9t-1)}{p})=-1, (\frac{t}{p})=1.\]
\end{lemma}
\begin{proof}
We consider
\[\sum_{t\in \mathbb Z/p\mathbb Z}(1-(\frac{9t-16}{p}))(1-(\frac{9-16t}{p}))(1-(\frac{t+1}{p}))\]\[(1-(\frac{(t-1)(t-9)}{p}))(1-(\frac{(9t-1)(t-1)}{p}))(1+(\frac{t}{p})).\]
Expanding this product yields the $p$ plus $63$ terms of the form $\sum_{t\in\mathbb Z/p\mathbb Z}\pm(\frac{\pm f(t)}{p})$ where $f(t)$ is the product of some terms in the set
\[\{9t-16,9-t,1+t,(t-1)(t-9),(t-1)(9t-1),t\}.\]
We claim that none of the $f(y)$ which arise are perfect squares. To see this it suffices note that the roots $\left\{\frac{16}{9},\frac{9}{16},0,1,-1,9,\frac{1}{9}\right\}$ are all distinct for $p>500$ and no terms involving both $(t-1)(t-9)$ and $(9t-1)(t-1)$ give perfect squares. Upon expanding it can be verified that we get $4$ terms of degree $1$, $9$ terms of degree $2$, $16$ terms of degree $3$, $19$ terms of degree $4$, $12$ terms of degree $5$, and $3$ terms of degree $6$. Using the Hasse Weil bound it follows that
\[\sum_{y\in \mathbb Z/p\mathbb Z}(1-(\frac{9t-16}{p}))(1-(\frac{9-16t}{p}))(1-(\frac{t+1}{p}))\]\[(1-(\frac{(t-1)(t-9)}{p}))(1-(\frac{(9t-1)(t-1)}{p}))(1+(\frac{t}{p}))\]
\[\ge p-13-35(2\sqrt{p}+1)-15(4\sqrt{p}+1)\] while the sum over the excluded $t$ is at most $22(64)=1408$ and the sum over the roots not in the excluded set is at most $4(64)=256$. It follows that for $p>21000$ that the sum in question is greater than $1408+256$ so such a $t$ exists and for $500<p\le 21000$ the existence of such $t$ is verified in LegrendeSymbol5p.java.
\end{proof}
\section{AP-Destroying Permutations for $\mathbb Z/7p\mathbb Z$}
For each constant $t\in \mathbb Z/p\mathbb Z, t\notin\{0, 1\}$, we can define the following permutation:\[\pi_7 := \setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.2}
\left\{\begin{array}{l @{\quad} l @{\quad}l}
(0, 0)\to (0, 1) & (0, 1)\to (0, 0) & (0, x)\to (6, \frac{t}{x}), x\notin\{0, 1\}
\\ (1, 0)\to (1, 1) & (1, 1)\to (1, 0) & (1, x)\to (0, \frac{1}{x}), x\notin\{0, 1\}
\\ (2, 0)\to (2, 0) & (2, 1)\to (2, t) & (2, x)\to (4, \frac{1}{x}), x\notin\{0, 1\}
\\ (3, 0)\to (3, 1) & (3, 1)\to (3, 0) & (3, x)\to (2, \frac{t}{x}), x\notin\{0, 1\}
\\ (4, 0)\to (5, 1) & (4, 1)\to (5, 0) & (4, x)\to (3, \frac{1}{x}), x\notin\{0, 1\}
\\ (5, 0)\to (6, t) & (5, 1)\to (6, 0) & (5, x)\to (5, \frac{1}{x}), x\notin\{0, 1\}
\\ (6, 0)\to (4, 1) & (6, 1)\to (4, 0) & (6, x)\to (1, \frac{1}{x}), x\notin\{0, 1\}.
\end{array}\right.\]
We first note several properties of the permutations $\sigma_1, \sigma_2: \mathbb Z/7\mathbb Z\to \mathbb Z/7\mathbb Z$ defined by
\[\sigma_1(0)=0, \sigma_1(1)=1, \sigma_1(2)=2, \sigma_1(3)=3, \sigma_1(4)=5, \sigma_1(5)=6, \sigma_1(6)=4\]
\[\sigma_2(0)=6, \sigma_2(1)=0, \sigma_2(2)=4, \sigma_2(3)=2, \sigma_2(4)=3, \sigma_2(5)=5, \sigma_2(6)=1\]
The first is that both $\sigma_1$ and $\sigma_2$ are almost AP-Destroying; that is, they each only preserve two APs up to reversals. Namely, $\sigma_1$ preserves $(0, 1, 2)$ and $(1, 2, 3)$ while $\sigma_2$ preserves $(1, 4, 0)$ and $(4, 0, 3)$. Furthermore, for any AP $(a, b, c)$ mod $7$, the images $(\sigma_1(a), \sigma_2(b), \sigma_2(c))$ and $(\sigma_2(a), \sigma_1(b), \sigma_2(c))$ are not APs.
\begin{lemma}
Suppose that $t\in \mathbb Z/p\mathbb Z$ such that
\[t\notin \{-2, -1, 0, 1, 2, 3, 4, -\frac{1}{2}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2},\frac{2}{3}, \frac{3}{4}\}\]
and
\[(\frac{(t-1)(t-9)}{p})=(\frac{(9t-1)(t-1)}{p})=-1.\]
Then $\pi_7$ is AP.
\end{lemma}
\begin{proof}
Suppose for sake of contradiction that some arithmetic progression $T$ is preserved, and let $U$ be its image. Denote by $T_7, T_p, U_7, U_p$ the projections of $T$ and $U$ modulo $7$ and $p$ respectively. We take four cases:
\begin{enumerate}[\text{Case }1.]
\item Three of the elements of $T_p$ are in $\{0, 1\}$. Then since $T_p$ is an AP, it must be equal to $(0, 0, 0)$ or $(1, 1, 1)$. Furthermore, $\sigma_1$ only preserves the APs $(0, 1, 2)$ and $(1, 2, 3)$. In both cases, neither these nor their reverses yield APs for $U_7$.
\item Two of the elements of $T_p$ are in $\{0, 1\}$. Then there are three cases up to symmetry according to the possible values of $T_p$.
\begin{enumerate}[\text{Case 2.}a.]
\item $T_p=(0, \frac{1}{2}, 1)$. Consider $U_p$. The possible values of the first coordinate are $\{0, 1, t\}$, the possible values of the second coordinate are $\{2, 2t\}$, and the possible values of the third coordinate are $\{0, t\}$. Considering the $12$ possible combinations, there are no APs for
\[t\notin\{0, 2, 3, 4, \frac{1}{4}, \frac{1}{3}\}.\]
\item $T_p=(0, 1, 2)$. Consider $U_p$. The possible values of the first coordinate are $\{0, 1, t\}$, the possible values of the second coordinate are $\{0, t\}$, and the possible values of the third coordinate are $\{\frac{1}{2}, \frac{t}{2}\}$. Considering the $12$ possible combinations, there are no APs for
\[t\notin\{-2, 0, -\frac{1}{2}, \frac{1}{4}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}\}\]
\item $T_p=(-1, 0, 1)$. Consider $U_p$. The possible values of the first coordinate are $\{-1, -t\}$, the possible values of the second coordinate are $\{0, 1, t\}$, and the possible values of the third coordinate are $\{0, t\}$. The AP $(-t, 0, t)$ never occurs since it forces the second and third coordinates of $T_7$ to be $2$ and the first to be in $\{0, 3\}$. Considering the other $11$ possible combinations, there are no APs for
\[t\notin\{-2, -1, 0, 3, -\frac{1}{2}\}\]
\end{enumerate}
\item One of the elements of $T_p$ is in $\{0, 1\}$. Then due to the mentioned properties of $\sigma_1$ and $\sigma_2$, it follows that $U_7$ is not an AP.
\item None of the elements of $T_p$ are in $\{0, 1\}$. Let $T_p=(a-r, a, a+r)$. Note that the $T_7$ coordinates cannot be equal, since that would force $\frac{1}{a-r}+\frac{1}{a+r}\equiv \frac{2}{a}$ or $r\equiv 0$, which is impossible. Then since $\sigma_2$ only preserves $(1, 4, 0)$ and $(4, 0, 3)$, we have two cases up to symmetry:
\begin{enumerate}[\text{Case 4.}a.]
\item $T_7=(1, 4, 0)$. Then $U_p=(\frac{1}{a-r}, \frac{1}{a}, \frac{t}{a+r})$. Solving the AP condition for $r$ yields a quadratic with discriminant $(t-1)(t-9)$, which is not a perfect square by assumption.
\item $T_7=(4, 0, 3)$. Then $U_p=(\frac{1}{a-r}, \frac{t}{a}, \frac{t}{a+r})$. Solving the AP condition for $r$ yields a quadratic with discriminant $(t-1)(9t-1)$, which is not a perfect square by assumption.
\end{enumerate}
\end{enumerate}
\end{proof}
\begin{lemma}
For $p\ge 66$ there exists a $t$ such that $t\notin \{-2, -1, 0, 1, 2, 3, 4, -\frac{1}{2}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2},\frac{2}{3}, \frac{3}{4}\}$ and $(\frac{(t-9)(t-1)}{p})=(\frac{(9t-1)(t-1)}{p})=-1$.
\end{lemma}
\begin{proof}
Since there are $13$ excluded elements and $2$ additional roots of $(t-9)(t-1)$ and $(9t-1)(t-1)$, it suffices to demonstrate that \[\sum_{t\in \mathbb Z/p\mathbb Z}(1-(\frac{(t-9)(t-1)}{p}))(1-(\frac{(9t-1)(t-1)}{p}))\ge 15(4)+1.\] However note that \[\sum_{t\in \mathbb Z/p\mathbb Z}(1-(\frac{(t-9)(t-1)}{p}))(1-(\frac{(9t-1)(t-1)}{p}))\]\[=p-\sum_{t\in \mathbb Z/p\mathbb Z}(\frac{(t-9)(t-1)}{p})+ (\frac{(9t-1)(t-1)}{p})-(\frac{(t-9)(9t-1)(t-1)^2}{p})\ge p-4\] where the Hasse Weil-Bound and that $(\frac{(t-1)^2}{p})=1$ for $t\nequiv 1$ is used. Since $p-4>61$ the result follows.
\end{proof}
\section{Computational Techniques}
In the previous sections, computational techniques are often required to ensure the existence of AP-Destroying permutations. For $n=2p, 3p, 5p, 7p$ corresponding to $n\le 2500$, we verified the existence of an AP-Destroying permutation via a descent algorithm; see DescentPermutation.java. In particular, we choose a random starting permutation, and only administer random transpositions if they decrease the total number of APs preserved. This condition can be checked in time linear in $n$ for each iteration. Empirically, the running time of this algorithm appeared to be roughly quadratic in $n$, which suggests that a random permutation descends to an AP-Destroying permutation with positive probability.
Whenever larger permutations were required, we calculated the necessary value of $t$ or $y,t$ directly, and this appears in many of the lemmas scattered throughout the proof. This was done instead of directly generating permutations due to the run time of this algorithm being empirically linear in $n$ versus quadratic for the above.
\section{Application to Finite Abelian Groups}
In the previous sections, we've classified which finite cyclic groups have AP-Destroying permutations. One particularly useful result is the following result of Hegarty which allows one to quotient out by subgroups with an AP-Destroying permutation.
\begin{thm}\label{Product}
If there exists an AP-Destroying permutation for $H$ and $G/H$, there exists an AP-Destroying permutation for $G$.
\end{thm}
Previously Elkies and Swaminathan \cite{Ashvin} demonstrated that all finite $p$-groups with odd order have an AP-Destroying permutation. We extend their result by classifying all finite abelian groups with odd order that have an AP destroying permutation.
\begin{thm}
Let $G$ be a finite abelian group with odd order greater than $7$. Then $G$ has an AP-Destroying permutation.
\end{thm}
\begin{proof}
We first claim that the result holds if $\Omega(|G|)\le 2$, where $\Omega(n)$ denotes the number of prime factors of $n$ with multiplicity. Indeed, if $G$ is of the form $\mathbb Z/p\mathbb Z\times \mathbb Z/q\mathbb Z$ for primes $p\neq q$ or $\mathbb Z/p\mathbb Z$ or $\mathbb Z/p^2\mathbb Z$ for a prime $p$, then the result follows from the main theorem. Finally the case $G=(\mathbb Z/p\mathbb Z)^2$ follows from the result of Elkies and Swaminathan \cite{Ashvin}.
\\
\\ Now we consider the case $\Omega(|G|)=3$. If $G$ is in the set below, then the direct verification of the existence of an AP-Destroying permutation is in FiniteAbelian.java.
\[\{(\mathbb Z/3\mathbb Z)^2\times \mathbb Z/5\mathbb Z, (\mathbb Z/3\mathbb Z)^2\times \mathbb Z/7\mathbb Z, (\mathbb Z/5\mathbb Z)^2\times \mathbb Z/3\mathbb Z, (\mathbb Z/5\mathbb Z)^2\times \mathbb Z/7\mathbb Z, (\mathbb Z/7\mathbb Z)^2\times \mathbb Z/3\mathbb Z\] \[(\mathbb Z/7\mathbb Z)^2\times \mathbb Z/5\mathbb Z,
\mathbb Z/9\mathbb Z\times \mathbb Z/3\mathbb Z, \mathbb Z/25\mathbb Z\times \mathbb Z/5\mathbb Z, \mathbb Z/49\mathbb Z\times \mathbb Z/7\mathbb Z\}\]
Other than the above set and cyclic groups, all other groups $G$ of odd order with $\Omega(|G|)=3$ have $\mathbb Z_p$ as a subgroup for some prime $p>11$ or are $G=(\mathbb Z/p\mathbb Z)^3$ for $p=3, 5, 7$. In the latter case the result follows from the result of Elkies and Swaminathan \cite{Ashvin} while in the former $G$ has an AP-Destroying permutation due to Theorem \ref{Product} along with the case $\Omega(|G|)\le 2$.
\\
\\ Finally, we prove the full result using strong induction on $\Omega(|G|)$, with base cases $\Omega(|G|)\in \{2, 3\}$ established. Suppose the result holds for $2\le \Omega(|G|)\le k$, an that $\Omega(|G|)=k+1$. Then there exists some product $pq$ of two possibly equal primes $p, q$ such that there is an order $pq$ subgroup $H$ of $G$. Then $H$ and $G/H$ both have an AP-Destroying permutation by the inductive hypothesis, so $G$ does as well by Theorem 20. This completes the induction.
\end{proof}
We remark that there are infinite families of even-order abelian groups which do not have an AP-Destroying permutation. For example, the following is true, which is mentioned in Remark 4.2 in \cite{hegarty2004permutations}
\begin{proposition}
Suppose that $H$ is an abelian group with $|H|<2^k$. Then $G=(\mathbb Z/2\mathbb Z)^k\times H$ has no AP-Destroying permutation.
\end{proposition}
\begin{proof}
Suppose otherwise, and let $\sigma:(\mathbb Z/2\mathbb Z)^k\times H\to (\mathbb Z/2\mathbb Z)^k\times H$ be such a permutation. Let $\pi_H: G\to H$ be the projection of $G$ onto the second coordinate. Then since $2^k>|H|$, there exist some $a\neq b\in (\mathbb Z/2\mathbb Z)^k$ such that $\pi_H\circ \sigma(a, 0)=\pi_H\circ \sigma(b, 0)$. But then $\{(a, 0), (b, 0), (a, 0)\}$ is an AP preserved by $\sigma$, a contradiction. So no such AP-Destroying permutation exists as required.
\end{proof}
In particular, if the largest odd number dividing a positive integer $n$ is less than $\sqrt{n}$, then there exists a finite abelian group of order $n$ which does not have an AP-Destroying permutation.
\section{Conclusion}
In this paper, we have resolved a conjecture of Hegarty. In particular, we proved that there exists an AP-Destroying permutation for all cyclic groups of order not in the set $\{2, 3, 5, 7\}$. However, as the last section demonstrates, this result does not immediately resolve the case for all finite abelian groups, and in fact for every positive integer $k$ there is a finite abelian group whose order is a multiple of $k$ which does not have any AP-Destroying permutation. In light of this, the following question is still open.
\begin{prob}
For which even order finite abelian groups do there exist AP-Destroying permutations?
\end{prob}
\section{Acknowledgements}
This research was conducted at the University of Minnesota Duluth REU and was supported by NSF grant 1659047. The authors would like to thank Joe Gallian and Ashvin Swaminathan for suggesting the topic of AP-Destroying permutations, and Aaron Berger, Evan O'Dorney, Colin Defant and Joe Gallian for reading over the manuscript. Finally we would like to thank the anonymous referee who pinpointed numerous errors throughout the manuscript and simplified the presentation of several cases in the above analysis.
\bibliographystyle{plain}
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2,877,628,090,839 | arxiv | \section{Introduction}
With the development of Internet, tons of videos are continuously generated in video content platforms, like YouTube, Netflix, Bilibili, etc. It shows that video understanding becomes more and more indispensable. Most videos are untrimmed in nature, thus temporal action detection is a fundamental task, which aims to detect start time, end time and semantic label of action instances.
Current mainstream approaches~\cite{lin2018bsn,qing2021temporal,xu2020g} are multi-step solutions which achieve good performance. They include proposal generation, action classification, ensemble results of classifiers and proposal post-processing. However, they fall short in efficiency and flexibility, especially for videos with diverse semantic labels. In recent years, there are also some works focused on single network\cite{lin2021learning,xu2017r}, but they fail to yield comparable results as those of multi-step approaches.
To simplify the pipeline of TAD, we propose a novel single network with remarkable performance, dubbed Faster-TAD. Inspired by Faster-RCNN\cite{ren2015faster}, we jointly learn temporal proposal generation, action classification, and proposal refinement with multi-task loss, sharing information for end-to-end update.
We observe many challenges in temporal action detection. Firstly, to tackle the extreme duration variation of action instances, ranging from second to minute, boundary-based mechanism\cite{lin2019bmn} is adopted to generate proposals instead of the traditional anchor-based method. Secondly, proposal context is helpful for the recognition of the proposal label. To enhance semantic information for action instances, we bring an innovative Context-Adaptive Proposal Module for classification, in which we propose a new Proximity-Category Proposal Block to obtain context, a Self-Attention Block to construct relationships among proposals, and a Cross-Attention Block to learn relevant context in raw videos for proposals. Thirdly, we propose a new Fake-Proposal Block to make refinement module to be trained with various offsets relative to ground truth boundary. Last but not least, many complex human activities have long duration and consist of atomic actions, so action recognition models such as CSN\cite{tran2019video} are often adopted to extract the features of video clips as input for subsequent localization task. Nevertheless, action recognition model trained with limited input frames and complex human activity labels lacks atomic action information which may improve the recognition of boundaries and classes. To address this issue, we utilize atomic features trained on an atomic action dataset SEAL\cite{chen2022SEAL} as auxiliary features.
To sum up, our contributions are as follow:
\begin{enumerate}
\item We propose a unified network Faster-TAD for temporal action detection with the architecture which is similar to Faster-RCNN.
\item In classification head, we propose a new Context-Adaptive Proposal Module, which consists of Proximity-Category Proposal Block, Self-Attention Block, and Cross-Attention Block. This Module greatly enhance semantic information for proposals.
\item In proposal regression refinement, we propose a new Fake-Proposal Generation Block, with which we obtain more valid and diverse proposals for refinement.
\item We find that feature representation is very important for this task, and demonstrate that atomic action features are helpful for complex activity detection.
\end{enumerate}
\section{Related Works}
\subsubsection{Temporal Action Proposal Generation.}
Temporal action detection task can be divided into temporal proposal generation and action classification. Approaches for action proposal generation can be grouped into two categories. The first method\cite{buch2017sst,gao2017turn,heilbron2016fast} is producing proposals with multi-scale anchors, which is inflexible and lack of boundary precision. The second method generates proposals via locally locating temporal boundaries and globally evaluating the probability of potential action. BSN\cite{lin2018bsn} is the representative work. They later improved this work to BMN\cite{lin2019bmn}, which utilizes a Boundary-Matching confidence map to evaluate the probability of proposal globally. Boundary-Matching confidence map enumerates all possible combination of temporal locations, bringing promotion in both efficiency and effectiveness. Boundary-Matching confidence map can be called as an anchor set in extreme form. Recently, TCA-Net\cite{qing2021temporal} is proposed for temporal proposal refinement. G-TAD\cite{xu2020g} is produced to find effective video context.
\vspace{-0.5cm}
\subsubsection{Single Model Method.}
There are some works focused on single network, like Faster-RCNN-like works, Transformer-based works. Most Faster-RCNN-like works adopt pre-defined anchors to generate proposals. R-C3D\cite{xu2017r} encodes the video streams using a three-dimensional fully convolutional network, utilizing a streamline pipeline of Faster-RCNN. TAL-Net \cite{chao2018rethinking} improved receptive field alignment using a multi-scale architecture. GTAN \cite{long2019gaussian} introduced Gaussian kernels to dynamically optimize temporal scale of each action proposal. These works alleviate the problem of inflexible proposal generation to some extent, but still lack of boundary precision compared with boundary-based method and not taking full of semantic features of proposals. Recently, some Transformer-based works \cite{liu2021end,tan2021relaxed} are introduced, which encode proposal with transformer block. However, the problem of insensitivity to boundaries has not been effectively solved. To tackle these challenges of temporal action detection, we propose a single network named as Faster-TAD.
\begin{figure}[t]
\centering
\includegraphics[height=5cm,width=12cm]{faster-tad-framwork.png}
\caption{Overview of our method. Given an untrimmed video, Faster-TAD can generate proposals and simultaneously (1) refine the boundary and (2) classify the proposal in a context-adaptive way. We construct our Faster-TAD with feature sequences extracted from raw video as inputs.}
\label{figure:faster-tad framework}
\end{figure}
\section{Our Approach}
\subsection{Overview of Framework}
As shown in Fig.~\ref{figure:faster-tad framework}, we propose a Faster-RCNN like network in temporal action detection, Faster-TAD. By jointing temporal proposal generation and action classification with multi-task loss and shared features, Faster-TAD simplifies the pipeline of TAD.
The input to our pipeline is a video sequence $X=\{x_t\}_{t=1}^T$, Following recent temporal action proposal generation methods \cite{buch2017sst,escorcia2016daps,lin2019bmn,xu2020g,zhao2017temporal}, we construct our Faster-TAD with feature sequences extracted from raw video as inputs by SwinTransformer\cite{liu2021swin} Extractor. The features of every $\tau$ consecutive frames are averaged and each set of the $\tau$ are named as a clip. In this way, input can be represented by $X\in\mathbb{R}^{C\times L}$, where $C$ is the feature dimension of each clip, and $L$ is the number of clips.
We first process the feature sequences with a base module to extract shared features, which consists of a CNN Layer, a Relu Layer, and a GCNeXt\cite{xu2020g} Block. We then exert a Proposal Generation Mechanism to obtain most credible $K$ coarse proposals, where $K$ is 120. Proposals and shared features are further utilized to get more accurate boundaries by Boundary Regression Refinement Module\cite{qing2021temporal}. At the same time, shared features and proposals are employed to get the semantic labels of action instances with Context-Adaptive Proposal Module. We make some improvements to tackle the challenges in temporal action detection.
\subsection{Proposal Generation Mechanism}
In anchor-based object detection tasks, anchors are generated in advance. The RPN Network\cite{ren2015faster} has two branches. Position Regression branch gets more accurate proposals, by regressing offset between anchors and ground truth regions. The Classification branch predicts positive score of anchors. Compared with object detection, a challenge of temporal action detection is the extreme duration variation of action instances. It is difficult to generate anchors to cover the receptive filed of all actions. So, we adopt Confidence-Matching mechanism\cite{lin2019bmn} to generate proposals instead of the traditional anchor-based method.
Proposal Generation Mechanism contains two branches, Temporal Evaluation Module(TEM) and Proposal Evaluation Module(PEM). We first adopt a Transformer Layer~\cite{vaswani2017attention} to get semantic information. Temporal Evaluation Module aims to evaluate the starting and ending probabilities for all temporal locations in untrimmed video. In Proposal Evaluation Module, we adopt SGAlign\cite{xu2020g} Block to generate Boundary-Matching (BM) confidence map, which aims to evaluate the probability of proposal globally. Boundary-Matching confidence map enumerates all possible combination of temporal locations, which can be called as an anchor set in extreme form. We use boundary probability sequences and BM confidence map to generate proposals during post processing.
\subsection{Context-Adaptive Proposal Module}
In object detection task, there is limited relation between objects in different spaces. However, in temporal action detection task, action instance is closely related to other actions in the same video. For example, if a semantic label of proposal is high jumping, the preceding action is likely to be running. The context greatly helps to classify the semantic label of proposals. We construct this module with Proximity-Category Proposal Block, Self-Attention Block, and Cross-Attention Block, learning useful context adaptively for each proposal.
\begin{figure}[t]
\centering
\includegraphics[scale=0.45]{classification_head2.PNG}
\caption{Context-Adaptive Proposal Module. Proposal features are generated from proposal generation outputs and the shared features by a ROI layer. Then, encoder layer is followed to further encode the proposal representation. Finally, Self and Cross Attention block is applied to model the proposal semantic features.}
\label{fig:classification_head}
\end{figure}
Our Context-adaptive Proposal Module is illustrated in Fig.~\ref{fig:classification_head}. Proposal features are extracted from shared features by the ROI\cite{he2017mask} layer with sampled proposals. Compared with shared features ($X\in\mathbb{R}^{C\times L}$), Proposal features can be represented by $F_p\in\mathbb{R}^{N\times C\times T}$, where $N$ is the number of coarse sampled proposals, $C$ is the feature dimension of each clip, and $T$ is temporal resolution processed by ROI layer. The method to sample proposals are described in {``Proximity-Category Proposal Block’’} subsection. Encoder consists of three encoder layers to obtain deeper semantic features for each proposal. We employ Residual Block\cite{he2016deep} as the encoder layer. After three encoder layers, the temporal dimension of proposal features turn into $1/8 \times T$. Proposal features are afterwards flattened along the last two dimensions to a feature sequence ($P\in\mathbb{R}^{N\times \frac{T}8C})$, where $T$ is set to 16.
\begin{figure}
\centering
\includegraphics[scale=0.23]{Auxiliary-Category.PNG}
\caption{Illustration of Proximity-Category proposals (PC proposals for short). The first row shows the ground truth segments. The second row is the output of our Proposal Generation Mechanism. The last row shows that the proposal with unsatisfied IoU will be assigned to a Proximity-Category according to its nearby ground truth segment. For example, proposal 2 has a label of {``using the rowing machine - proximity’’}.}
\label{proximity-category}
\end{figure}
\subsubsection{Proximity-Category Proposals Block}
Previous classification methods generally regard the proposal with IoU larger than a threshold as positive proposals, and other proposals as negative proposals. IoU is the matching score between the proposal and ground truth(GT).
In general, there are two methods to classify proposals\cite{liu2016ssd,redmon2016you,ren2015faster}. The first is to only reduce classification loss for positive proposals, the other is to add a negative category and assign all negative proposals to the negative category. However, negative proposals also contain many semantic information, and it isn't a sensible approach to classify all negative proposals into one category. As illustrated in Fig. \ref{proximity-category}, we propose a new proposal selection approach, called Proximity-Category Proposals Block. We define proposals with proximity category as Proximity-Category Proposals(PC Proposals). For example, in Fig. \ref{proximity-category}, proposal 2 is a PC Proposal of {``using the rowing machine - proximity’’}, and proposal 4 is a PC proposal of {``clean and jerk - proximity’’}. In this way, we expand the origin $M$ categories into $2M$ categories. We propose a high-low threshold method to set semantic labels for each proposal, according to the maximum IoU value between the proposal and ground true segments:
\begin{tiny}
\begin{equation}
\label{eq:sematic label}
{f_0}^{2M-1}(i)=\begin{cases}
f(i=idx)=1.0, f(i\neq idx)=0.0 &\text{if}\ IoU_{best}\geq{\tau}_1;\\[0.4cm]
\begin{split}
&f(i=idx)=\alpha IoU_{best},f(i=(idx+M))=1-\alpha IoU_{best}, \\
&f(i\neq {idx} \text{ and } i\neq {(idx+M)})=0.0
\end{split}
&\text{if}\ {\tau}_1>IoU_{best}\geq{{\tau}_2};\\[0.6cm]
f(i=(idy+M))=1.0,f(i\neq {(idy+M)})=0.0&\text{if}\ IoU_{best}<{\tau}_2;
\end{cases}
\end{equation}
\end{tiny}
\noindent where,
\begin{gather}
\label{eq:sematic label23}
idx=G(argmax(IoUs))\\
idy=G(argmin(Dists))
\end{gather}
As illustrated in Eq (\ref{eq:sematic label}), ${\tau}_1$ is the high threshold of $IoU$, ${\tau}_2$ is the low threshold. $M$ is the number of original categories, $i$ is ranging frame $0$ to $2M-1$ in Eq \ref{eq:sematic label}. ${IoUs}$ are the IoU values between proposal and the ground truth segments. ${IoU_{best}}$ is the max value of ${IoUs}$. $idx$ and $idy$ are the index of label for proposal, $(idx+M)$ and $(idy+M)$ are index of Proximity-Category label for proposal. $\alpha$ is a hyperparameter. As illustrated in Eq (\ref{eq:sematic label23}), $G$ is a function that maps ground truth position to ground truth label index. ${Dists}$ stand for the center point distance values between proposal and the ground truth segments.
In order to make the ratio of positive proposals and PC Proposals close to 1:1, we select positive proposals first, and then select PC Proposals with the highest scores until $N$ coarse proposals are sampled.
\subsubsection{Self-Attention Block.}
We utilize Self-Attention\cite{vaswani2017attention} Block to capture relationships between proposals. Our decoder consists of 2 decoder blocks. At each block $l$, Query/Key/Value sequences are computed for proposal features from the representation ${P}^l$ encoder by the preceding block:
\begin{gather}
{Q}^l={K}^l={V}^l={P}^l
\end{gather}
Where $l$ is the block index. The multi-head variant of the self-attention computation is popularly used because of jointly attention to information from different representation sub-spaces. Multi-Head Attention\cite{vaswani2017attention} uses scaled dot-product attention, which is defined as:
\begin{gather}
MultiHead-Attention(Q^l,K^l,V^l)=Concat({head_1}^l,...,{head_H}^l)W, \\
{head_h}^l=softmax(\frac{{Q_h}^l\times {K_h^{Tl}}}{\sqrt{d}})V_h
\end{gather}
Where $H$ is the number of heads, $h$ is the head index, $W$ stands for parameter matrices, $d$ is the dimensionality of the hidden representations.
We set the LayerNorm\cite{ba2016layer} after the Multi-Head Attention layer, followed with residual connection. We call this approach Middle-LN Transformer Layer, which is defined as:
\begin{equation}
{P_{out}}^l=LN(MultiHead-Attention({Q}^l,{K}^l,{V}^l))+{Q}^l
\end{equation}
\subsubsection{Cross-Attention Block.}
We permute the shared features at the dimension of 0 and 1, which is encoded from raw video features by base module, and set the permuted features as key sequences and value sequences. It is represented by Eq (\ref{eq:cross-attn}). That is to say, Cross-Attention Block attend to learn the relationship between proposals and every clips in raw video. This block greatly increases the semantic information of proposals, bringing a large performance improvement.
\begin{equation}
{K_{cross}}^l={V_{cross}}^l=permute(X),
{Q_{cross}}^l={P_{out}}^l \label{eq:cross-attn}
\end{equation}
where $X\in\mathbb{R}^{C\times L}$ is shared features, $K_{cross}\in\mathbb{R}^{L\times C}$ is key sequences, $V_{cross}\in\mathbb{R}^{L\times C}$ is value sequences, $C$ is the feature dimension of each clip, and $L$ is the number of clips.
Like Self-Attention Block, we utilize Middle-LN Transformer Layer to learn relationships, which is defined as:
\begin{equation}
{P_{cross}}^l=LN(MultiHead-Attention({Q_{cross}}^l,{K_{cross}}^l,{V_{cross}}^l)+{Q_{cross}}^l
\end{equation}
${P_{cross}}^l$ are further employed to get semantic features of proposals by a two-layer Feed-Forward Network(FFN)\cite{vaswani2017attention}, which is defined as:
\begin{equation}
P^{l+1}=FFN({P_{cross}}^l)
\end{equation}
\subsection{Fake-Proposal Block}
We propose a gt-based Fake-Proposal Block in Boundary Regression Refinement Module. We select the $\mu$ proposals with the highest confidence from $K$ coarse proposals as part of the proposals that will be refined. Then, we generate the rest $K-\mu$ fake proposals based on gt. As illustrated in Fig. \ref{Fake-Proposal}, we have $[0,\pm\frac{1}{8},\pm\frac{1}{6},\pm\frac{1}{4}]$ seven modes to change boundaries of GT. That is to say, each GT can generate 49 fake proposals. For each GT, we choose extension ways in order to generate fake proposals until $K-\mu$ fake proposals are generated, where $K$ and $\mu$ are set to 120 and 90.
The Fake-Proposal Block ensures that the refinement module can be trained with various boundary offsets relative to GT, which is beneficial for regression. On the other hand, this block feeds more valid proposals to the refinement module.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{Fake-gt.PNG}
\caption{Fake-Proposal Block. The red arrows indicate the different offsets based on the origin boundary of an ground truth segment. This block generates $[0,\pm\frac{1}{8},\pm\frac{1}{6},\pm\frac{1}{4}]$ seven modes to offset for each boundary of GT.}
\label{Fake-Proposal}
\end{figure}
\subsection{Auxiliary-Features Block \label{Auxiliary-Features Block}}
Many complex human activities have long duration and consist of atomic actions. Action recognition model Swin Transformer\cite{liu2021swin} is adopted to extract features of each clip as input for subsequent localization task. Nevertheless, action recognition model is trained with limited semantic information. To address this issue, we adopt atomic features as auxiliary features extracted by Slowfast\cite{feichtenhofer2019slowfast} trained on \textit{SEAL Clips}\cite{chen2022SEAL}. We designed a feature aggregation method named Auxiliary-Features Block to adapt to the two streams input. As shown in Fig. \ref{Auxiliary-feature}. Main and auxiliary features are combined in a simple way after going through two separate base modules. \textit{SEAL} \cite{chen2022SEAL} develop a novel large-scale video dataset of multi-grained spatio-temporally Action Localization. \textit{SEAL Clips} localizes atomic actions in space during a two-second clip.
\begin{figure}[t]
\centering
\includegraphics[scale=0.25]{auxiliary-feature.PNG}
\caption{Auxiliary-Feature Block. Two streams of features go through base module respectively. After that, they are combined along the temporal dimension. The rest of the network keeps the same.\label{Auxiliary-feature}}
\vspace{-0.5cm}
\end{figure}
\subsection{Training and Inference}
\subsubsection{Training.}
To jointly learn regressing temporal proposals and classifying action labels, an unified multi-task framework is exploited for optimization. The training details of Faster-TAD are introduced in this section. Once having both the coarse and refined prediction of temporal boundary, confidence score of positive proposal, and semantic label, we can optimize the model with the following objective function:
\begin{equation}
L= {L_{C}}+{L_{R}}+\lambda{L_2}(\Theta)
\end{equation}
where $\lambda$ is hyper-parameter, set to 0.0001, ${L_2}(\Theta)$ is $L_2$ regularization loss, ${L_{C}}$ is loss of generating coarse proposals, and ${L_{R}}$ is loss of refining proposals and classifying labels. ${L_{C}}$ and ${L_{R}}$ are defied as:
\begin{gather}
L_{C}= {L_{TEM}}^{BCls}+\lambda_1{L_{PEM}}^{BCls}+\lambda_2{L_{PEM}}^{Loc},\\
L_{R}={L_{R}}^{Loc} +\gamma{L_{R}}^{Cls}
\end{gather}
where weight term $\lambda_1$, $\lambda_2$, and $\gamma$ are set to 1, 10, and 0.5, ${L_{TBR}}^{BCls}$ and ${L_{PEM}}^{BCls}$ are weighted binary logistic regression loss $L_{bl}$, adopted from BMN\cite{lin2019bmn}. we adopt $L_2$ loss for regression loss ${L_{PEM}}^{Loc}$. ${L_{R}}^{Loc}$ is a regression refined loss, adopted from TCA-Net\cite{qing2021temporal}.
${L_{R}}^{Cls}$ is softmax focal loss $\ell_{focal}$\cite{lin2017focal} between classification prediction and ground truth labels $y$:
\begin{equation}
{L_R}^{Cls}=\frac{1}{2M}\sum_i^{2M} \ell_{focal}(\hat{y_i},{y_i})
\end{equation}
where $M$ is the number of original classification categories, $2M$ includes the number of original categories and proximity-categories.
\subsubsection{Inference.}
At inference time, top $K$ coarse proposals are proceeded to Context-Adaptive Proposal Module and Temporal Regression Refinement Module, predicting classification and regression score for each refined proposals. Following recent temporal action detection methods \cite{lin2019bmn,xu2020g,zhao2017temporal}, we construct predicted actions $\Psi=\left\{{{\psi}_k=((\hat{t}_{s,k},\hat{t}_{e,k}),\hat{c}_k,\hat{p}_k)}\right\}_{k=1}^K$, where $(\hat{t}_{s,k},\hat{t}_{e,k})$ refer to the predicted action boundaries, $\hat{c}_k$ is the predicted action category, and $\hat{p}_k$ is the fused confidence score of this prediction, $K$ is set to 120.
\section{Experiment}
\subsection{Datasets}
\vspace{-0.1cm}
\subsubsection{ActivityNet-1.3\cite{caba2015activitynet}}
is a large-scale action understanding dataset for action recognition, temporal detection, proposal generation and dense captioning tasks. It contains 19,994 temporally annotated untrimmed videos with 200 action categories, which are divided into training, validation and testing sets by the ratio of 2:1:1.
\vspace{-0.38cm}
\subsubsection{HACS Segments\cite{zhao2019hacs}}
is a recently introduced dataset for temporal localization of human actions collected from web videos, therefore the results of many early methods on this dataset are not available. HACS Segments contains 139K action segments densely annotated in 50K untrimmed videos spanning 200 action categories.
\vspace{-0.38cm}
\subsubsection{SoccerNet-Action Spotting\cite{giancola2018soccernet}}
is a large-scale dataset for soccer video understanding. The dataset consists of 400 videos from soccer broadcast games for training, 100 videos for validation, and 50 separate games for test(challenge). In SoccerNet-Action Spotting, validation set is defined as test, and test set is called challenge. In action spotting task, 17 classes are annotated with a single timestamp, making annotations quite scattered in long videos. To convert the annotation format to traditional temporal action detection style, we take the annotated timestamp as the start moment and delay the timestamp by 4 seconds as the end moment.
\subsection{Implementation Details}
\subsubsection{Features.}
For ActivityNet-1.3 and HACS Segments, we employ pre-extracted features as inputs. To fully demonstrate the effectiveness of our network, we use multiple features as input to conduct multiple experiments. The video features are extracted using Swin Transformer\cite{liu2021swin} trained on HACS Clips, which is called as Swin Feature in this paper. Auxiliary-Features are extracted by Slowfast\cite{feichtenhofer2019slowfast} trained on SEAL Clips\cite{chen2022SEAL}, which is defined as Slowfast-A Feature($A$ stands for atomic). We extract both features with windows of $size = 32$, $stride=32$. and we resize video features to 100 clips using linear interpolation. In addition, we also adopt TSP\cite{alwassel2021tsp} Feature trained on ActivityNet-1.3 as input. In training, we do not adopt any videos without actions. For SoccerNet Action Spotting, we train two feature extractors using Swin Transformer with 17 positive classes and 1 negative class. Two feature extractors have 3 and 6 second window sizes respectively to capture different context ranges. Features are combined using Auxiliary-Features Block mentioned in Chapter \ref{Auxiliary-Features Block}.
\subsubsection{Training and Inference.}
We train our model in a single network, with batch size of 64 on 8 gpus. The learning rate is $6\times{10}^{-4}$ for the first 3 epochs, and is reduced by 10 in epoch 3 and 7. We train the model with total 10 epochs. In inference, we apply Soft-NMS\cite{bodla2017soft} for post-processing, and select the top-M prediction for final evaluation. M is 120 for both ActivityNet-1.3 and Hacs Segments. We do not adjust hyper-parameters for HACS Segments, using the same hyper-parameters as those on ActivityNet-1.3.
\begin{table}[t]
\centering
\caption{Action detection results on validation set of ActivityNet-1.3, measured by mAP(\%) at different tIoU thresholds and the average mAP(\%). $S-N$ stands for Single-Network. $Ensemble \ of \ classifiers$ stands for ensemble video level classification results. $CUHK$ is from\cite{xiong2016cuhk}, $UntrimmedNet$ is from \cite{wang2017untrimmednets}. $Slowfast-A$ is extracted by Slowfast model trained on SEAL Clips\cite{chen2022SEAL}. {``-’’} indicates unknown results.}
\label{tab:tab1}
\scalebox{0.92}{
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Method & Feature & S-N & Ensemble of classifiers & 0.5 & 0.75 & 0.95 & Avg \\\hline
\multicolumn{8}{l}{Self-contained methods}\\\hline
R-C3D\cite{xu2017r} & C3D & $\checkmark$ & $\times$ & 26.8 & - & - & - \\
TAL-Net\cite{chao2018rethinking} & I3D & $\checkmark$ & $\times$ & 38.23 & 18.30 & 1.30 & 20.22 \\
P-GCN\cite{zeng2019graph} & I3D & $\times$ & $\times$ & 42.90 & 28.14 & 2.47 & 26.99 \\
TadTR\cite{liu2021end} & I3D & $\checkmark$ & $\times$ & 40.85 & 28.44 & 7.84 & 27.75 \\
ContextLoc\cite{zhu2021enriching} & I3D & $\checkmark$ & $\times$ & 51.24 & 31.40 & 2.83 & 30.59\\
Lin et al.\cite{lin2021learning} & I3D & \checkmark & $\times$ & 52.4 & 35.3 & 6.5 & \textbf{34.4} \\
\hline
\multicolumn{8}{l}{Ensemble Action-labels}\\\hline
P-GCN\cite{zeng2019graph} & I3D & $\times$ & UntrimmedNet & 48.26 & 33.16 & 3.27 & 31.11 \\
BMN\cite{lin2019bmn} & TS & $\times$ & CUHK & 50.07 & 34.78 & 8.29 & 33.85 \\
G-TAD\cite{xu2020g} & TS & $\times$ & CUHK & 50.36 & 34.60 & 9.02 & 34.09 \\
TSP\cite{alwassel2021tsp} & TSP & $\times$ & CUHK & 51.26 & 37.12 & 9.29 & 35.81 \\
BMN-CSA\cite{sridhar2021class} & TSP & $\times$ & CUHK & 52.64 & 37.75 & 7.94 & 36.25 \\
TCANet[BMN]\cite{qing2021temporal} & Slowfast &$\times$ & CUHK & 54.33 & 39.13 &8.41 & 37.56\\
PRN\cite{wang2021proposal} & CSN & $\times$ & PRN & 57.9 & - & - & \textbf{39.4} \\\hline
\multicolumn{8}{l}{Our method}\\\hline
Faster-TAD & TSP & \checkmark & $\times$ & 51.29 & 36.19 & 10.22 & 35.32\\
Faster-TAD & TSP+Slowfast-A & \checkmark & $\times$ & 52.20 & 36.97 & 10.10 & 35.98 \\
Faster-TAD & Swin & \checkmark & $\times$ & 57.39 & 39.97 & 10.48 & \textbf{39.09} \\
Faster-TAD & Swin+Slowfast-A & \checkmark & $\times$ & 58.30 & 40.77&11.28 & \textbf{40.01} \\\hline
\end{tabular}}
\end{table}
\subsection{Comparison with State-of-the-art Results}
\subsubsection{ActivityNet-1.3.}
Table \ref{tab:tab1} demonstrates the temporal action detection performance comparison on validation set of ActivityNet-1.3. Faster-TAD reports the highest average mAP results on this large-scale dataset. Our approach outperforms existing single-network detector by a large margin of 5.61\% mAP. Our single network outperforms these multi-step method by 0.61\% mAP, even multi-step detector using ensemble results of classifiers.
\vspace{-0.1cm}
\subsubsection{HACS Segments.}
Table \ref{tab:tab2} compares Faster-TAD with state-of-the-art detectors on HACS Segments. Our approach obtains remarkable performance of 38.39\% average mAP, and outperforms existing single-network detector by a large maigin of 7.56\% mAP.
\vspace{-0.1cm}
\subsubsection{SoccerNet-Action Spotting.}
\cite{zhou2021feature} is the winner of SoccerNet-Action Spotting 2021. As shown in Table \ref{tab:tab3}, with Faster-TAD network, we reached an mAP of 54.09\% in tight test set, bringing a gain of 8.77\% mAP in shown set.
\begin{table}[t]
\centering
\caption{Action detection results on validation set of HACS-Segments, measured by mAP(\%) at different tIoU thresholds and the average mAP(\%). $S-N$ stands for Single-Network. $Ensemble \ of \ classifiers$ stands for ensemble video level classification results. $Slowfast-A$ is extracted by Slowfast model trained on SEAL Clips\cite{chen2022SEAL}. {``-’’} indicates unknown results. Results of BMN are from \cite{qing2021temporal}.}
\label{tab:tab2}
\scalebox{0.92}{
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Method & Feature & S-N & Ensemble of classifiers & 0.5 & 0.75 & 0.95 & Avg \\\hline
TadTR\cite{liu2021end} & I3D & $\checkmark$ & $\times$ & 45.16 & 30.70 & 11.78 & 30.83 \\
G-TAD\cite{xu2020g}& I3D & $\times$ & - & 41.08 & 27.59 & 8.34 & 27.48 \\
BMN\cite{lin2019bmn} & Slowfast & $\times$ & - & 52.49 & 36.38 & 10.37 & 35.76 \\
TCANet[BMN]\cite{qing2021temporal} & Slowfast &$\times$ & - & 56.74 & 41.14 &12.15 & \textbf{39.77}\\
\hline
\multicolumn{8}{l}{Our method}\\\hline
Faster-TAD & Swin & \checkmark & $\times$ & 54.13 & 37.10 & 12.02 & 36.92 \\
Faster-TAD & Swin+Slowfast-A & \checkmark & $\times$ & 55.63 & 38.72 & 12.90 & \textbf{38.39} \\\hline
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{Action detection results on test set of SoccerNet-Action Spotting, measured by tight average-mAP introduced by \cite{giancola2018soccernet}. The test set is divided to shown part and unshown part according to the action visibility. We report the performance on Shown, Unshown and all test set. Validation set is defined as test, and test set is called challenge in SoccerNet.}\label{tab:tab3}
\scalebox{1.0}{
\begin{tabular}{c|c|c|c|c}
\hline
Method & Feature & Shown & Unshown & All \\\hline
Zhao et al.\cite{zhou2021feature}(2021 top1) & BaiDu & 52.33&25.63&47.05 \\
Faster-TAD(ours) & Swin(3s) & 56.91 & 24.38 & 50.34\\
Faster-TAD(ours) & Swin(3s+6s) & 61.10 & 25.50 & 54.09 \\\hline
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{Ablation study of Single Network(SN), Proximity-Category(PC), Self-Attention(SA), Cross-Attention(CA), Fake-Proposal(FP) and Auxiliary-Features(AF) on ActivityNet-1.3 in terms of average mAP(\%). Each experiment is repeated five times, and we report the mean, standard deviation(std), and max values.}\label{tab:tab4}
\begin{tabular}{c|c c c c c c |c|c|c}
\hline
Method & SN & PC & SA & CA & FP & AF& mAP max & mAP mean & mAP std \\\hline
Faster-TAD & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & 38.02&37.97&0.048 \\
Faster-TAD & \checkmark & $\times$ &$\times$&$\times$&$\times$&$\times$&38.21&38.12&0.064 \\
Faster-TAD & \checkmark & \checkmark &$\times$&$\times$&$\times$&$\times$&38.68&38.57&0.098 \\
Faster-TAD & \checkmark & \checkmark &$\checkmark$&$\times$&$\times$&$\times$&38.76&38.65&0.131 \\
Faster-TAD & \checkmark & \checkmark &$\checkmark$&$\checkmark$&$\times$&$\times$&38.95&38.85&0.094 \\
Faster-TAD & \checkmark & \checkmark &$\checkmark$&$\checkmark$&$\checkmark$&$\times$&39.09&38.99&0.068 \\
Faster-TAD & \checkmark & \checkmark &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&40.01&39.88&0.123 \\\hline
\end{tabular}
\end{table}
\subsection{Ablation Study}
As shown in Table \ref{tab:tab4}, in order to illustrate the effectiveness of each module, we performed five experiments for each configuration and obtained the maximum mAP, average mAP, and variance for each configuration. For the purpose of the stability of experiments, the average mAP is adopted below to prove the effectiveness of each module.
\subsubsection{Single-Network}
We use the video level classification results to replace the classification part of Faster-TAD, according to the general multi-step methods. Video level classification results are reached by using Swin Transformer model trained on HACS clips training set, obtaining 92.1\% top1 accuracy and 98.8\% top5 accuracy on HACS Clips validation set. The method is the same as video feature extraction of Faster-TAD. Table \ref{tab:tab3} shows that jointing temporal proposal generation and action classification with multi-task loss and shared feature can get better performance(+0.15\% mAP).
\subsubsection{Context-adaptive Proposal Module}
We optimize the model using the traditional method of reducing only the classification loss of positive proposals, and obtain 38.12\% mAP. On the other hand, we adopt the our Proximity-Category Proposal Module to reduce the classification loss, getting 38.57\% mAP with 0.45\% mAP improvement.
As shown in Table \ref{tab:tab4}, we adopt our Self-Attention Block to conduct relations among proposals, reaching 38.65\% mAP with 0.08\% gain.
By using our Cross-Attention Block to conduct relations between proposals and clips of raw video, we obtain a mAP gain of 0.2\%. It is demonstrated that our Context-adaptive Proposal Module effectively enhance semantic information for each proposal, bringing a gain of 0.73\% mAP.
\subsubsection{Fake-Proposal Block}
Table \ref{tab:tab4} reports the effectiveness of the Fake-Proposal Block, with which our network obtained 0.14\% mAP improvement.
\subsubsection{Auxiliary-Feature Block}
As shown in Table \ref{tab:tab4}, our Auxiliary-Feature has rich semantic information, with which our network obtained 0.89\% mAP improvement.
\section{Conclusions}
In this paper, We propose a novel network for temporal action detection in a single network, dubbed Faster-TAD. Faster-TAD includes Context-Adaptive Proposal Module to adaptively learn the semantic information of proposals by introducing attention mechanism across proposals to whole video and considering context as proximity-category proposals. Then the Fake Proposal based on the ground truth boundary with different offsets improves the Boundary Regression Module. Also, we found feature representation trained on atomic actions is very useful for complex activity detection. Our network can aggregate features with different semantic information and further improve the performance. Extensive experiments demonstrate that Faster-TAD outperforms existing single-network detector by a large margin on many benchmarks, obtaining state-of-the-art results on ActivityNet-1.3 and SoccerNet-Action Spotting.
\clearpage
\bibliographystyle{splncs04}
|
2,877,628,090,840 | arxiv | \section{Introduction}
Supersymmetry (SUSY) is widely viewed as the most likely theory beyond the Standard Model (SM). In SUSY extensions of the SM, the spectrum of sparticle masses is strongly dependent on the mechanism for SUSY breaking. Gauge- and gravity-mediated SUSY breaking are the most commonly considered mechanisms. In the case of gauge-mediated SUSY breaking (GMSB), the gravitino can be much lighter than the mass of the sparticles, making it a natural candidate for the lightest SUSY partner (LSP) and dark matter. For a complete cosmology of GMSB, we also need a mechanism for baryogenesis. In the MSSM, Affleck-Dine (AD) baryogenesis \cite{ad} is a particularly simple and effective way to generate the baryon asymmetry. A complex flat direction field gains a large expectation value, which later begins coherent oscillations in the real and imaginary directions. The interaction of the evolving field with B violating operators induces a net asymmetry in the flat-direction condensate.
It is well-known that AD condensates are unstable with respect to spatial perturbations \cite{ks,km1,km2,km3,kk1,kk2,kk3,fk}. These grow and fragment the condensate, producing Q-balls. The cosmology of Q-balls depends on the dimension of the B-violating operator and the nature of SUSY breaking. In gravity-mediated models the Q-balls are unstable and, if their charge is large, these may decay below the freeze-out temperature of neutralino dark matter \cite{km1,km2}. In this case dark matter and baryon number are simultaneously produced when the Q-balls decay, with $n_{DM} = 3 n_{B}$ by R-parity conservation. Therefore $\Omega_{B}/\Omega_{DM} = m_{n}/(3 m_{DM})$, where $m_{n}$ is the nucleon mass. For neutralino dark matter with mass $\gae 100 \GeV$, as expected in gravity-mediated SUSY breaking models, this produces a value much smaller than the observed baryon-to-dark matter ratio $\approx 1/6$, which requires $m_{DM} \approx 2 \GeV$. In this case the neutralino DM candidates must annihilate rapidly after production, pointing to Higgsino-like dark matter \cite{yamag}.
While this is an appealing mechanism for non-thermal dark matter, the possibility of directly producing both dark matter and the baryon asymmetry from Q-ball decay is lost. This is only possible with a dark matter candidate of mass $\approx$ 2 GeV. In \cite{rs} a light axino LSP in gravity-mediated SUSY breaking was proposed. Alternatively, in GMSB models a natural low-mass DM candidate is the gravitino.
However, Q-ball formation in GMSB AD baryogenesis is difficult to achieve without phenomenological problems \cite{kssw1,kssw2,shoeQ}. The flat-direction potential depends on the messenger mass. For field values much larger than the messenger mass, the potential becomes almost flat due to the suppression of gauge-mediation by the flat-direction field. If the magnitude of the flat-direction field when Q-balls form is much larger than the messenger mass, then the Q-balls will have an energy-per-charge, $E/Q$, which decreases as $Q^{-1/4}$ \cite{ks}. For large enough $Q$, $E/Q$ is less than the nucleon mass, implying that the Q-balls are absolutely stable if they carry baryon number. This leads to severe problems. A stable Q-ball captured by a neutron star will absorb baryon number, reducing its mass and eventually destabilizing the neutron star. This severely constrains AD baryogenesis in such GMSB models, ruling out all flat directions lifted by conventional B-conserving non-renormalizable potential terms \cite{kssw1}. More precisely, $E/Q \propto Q^{-1/4}$ is true if the AD field $\phi$ in the Q-ball is on the approximately constant part of the potential. (These are referred to as flat direction (FD) Q-balls in \cite{kssw1}.) For sufficiently large Q-ball charge, $\phi$ becomes large enough for the non-renormalizable terms to become important in the Q-ball solution. This happens once $Q > Q_{c}$, where for the $d = 6$ Q-balls we consider $Q_{c} \sim 10^{24}$ \cite{kssw1}. For larger charge, the Q-balls become thin-walled, with constant field magnitude $\phi_{c} \sim 10^{14} \GeV$ inside and radius increasing as $Q$ increases. (These are referred to as curved direction (CD) Q-balls in \cite{kssw1}.) If Q-balls have a density comparable to dark matter (as we would expect if the Q-balls are stable and the baryon asymmetry is due to Affleck-Dine baryogenesis), then they will be excluded by neutron star stability if the non-renormalizable lifting terms conserve baryon number. FD type Q-balls imply a neutron star lifetime $\sim 10^{10}$ years, whereas CD Q-balls imply a lifetime $\sim 1500$ years \cite{kssw1}. The initial charge of Q-balls from AD baryogenesis will typically be less than $Q_{c}$ and therefore they will initially be FD type Q-balls. However, since $Q_{c} \ll Q_{ns}$, where $Q_{ns} \sim 10^{57}$ is the baryon number of a neutron star, the charge of any FD Q-ball trapped by a neutron star will rapidly grow by baryon absorption to become larger than $Q_{c}$ (for a constant absorption rate, this will occur in a time $\sim Q_{c}/Q_{ns} \times 10^{10}$ years $\sim$ $10^{-15}$ s), transforming them in CD Q-balls and destabilizing the neutron star.
AD baryogenesis in GMSB could work if the messenger mass is sufficiently large that condensate fragmentation occurs when the amplitude of the flat-direction scalar is in the approximately quadratic part of the potential. In this case the resulting Q-balls will be unstable. It may then be possible to produce dark matter via late decay of Q-balls to out-of-equilibrium NLSPs which subsequently decay to gravitino with mass $m_{3/2} \approx 2 \GeV$.\footnote{An alternative mechanism for achieving unstable Q-balls in GMSB models is to produce Q-balls with a small baryonic charge, $10^{12} \lae Q \lae 10^{18}$. In this case the large suppression of the mass of the scalars forming the Q-ball does not occur and the Q-balls can decay directly to out-of-equilibrium gravitinos \cite{ksh1}.}
In the case of gravity-mediated SUSY breaking, it is known that $d = 6$ flat directions lifted by superpotential terms of the form $(u^{c}d^{c}d^{c})^2$ can produce Q-balls with large enough baryon number to decay after neutralino freeze-out but before nucleosynthesis \cite{km1,km2}. We may expect similar behaviour in the case of GMSB flat directions with sufficiently large messenger mass. Therefore in the following we will focus on the example of $d=6$ flat directions
\footnote{$d = 4$ directions are disfavoured for AD baryogenesis as the $d =4$ superpotential terms $QQQL$ and $u^{c}u^{c}d^{c}e^{c}$ cause too rapid proton decay, although this does not exclude AD leptogenesis along the $(H_{u}L)^2$ direction.
$d =4$ flat directions may also thermalize before AD baryogenesis can occur \cite{cmr}.
$d =5$ directions require superpotential terms which violate R-parity, but LSP dark matter assumes R-parity conservation.}. Our method will be to numerically study a phenomenological GMSB flat-direction potential which models the suppression of the SUSY-breaking masses at field values greater than the messenger scale.
Our paper is organized as follows. In Section 2 we introduce the phenomenological GMSB flat-direction potential. In Section 3 we consider AD baryogenesis and condensate fragmentation along a $d = 6$ MSSM flat-direction in a GMSB model with $m_{3/2} \approx 2 \GeV$. We solve the field equations to obtain the baryon density and so determine the reheat temperature. We then perform a semi-analytical study of condensate fragmentation using the phenomenological potential and estimate the time of formation and baryonic charge of the condensate fragments. In Section 4 we present our conclusions.
\section{Flat-direction potential in GMSB}
GMSB models are based on SUSY breaking in a hidden sector which is transmitted to the MSSM via vector pairs of messenger fields with SM gauge charges. The messenger superfield scalar components acquire SUSY breaking mass splittings from their interaction with the hidden sector. The messengers then induce masses for MSSM gauginos at 1-loop and soft SUSY breaking scalar mass squared terms at 2-loops.
This is true so long as the flat-direction field $\Phi$ does not give the masses to the gauge fields which are greater than the messenger mass $M_{m}$. Once $|\Phi| \gae M_{m}$, the transmission of SUSY breaking via gauge fields is suppressed\footnote{More accurately, $g |\Phi| \gae M_{m}$, where $g$ is the relevant gauge coupling. For the $(u^{c}d^{c}d^{c})^2$ direction of interest to us here, $g$ is the strong coupling. Since $g \approx 1$, for simplicity we will not include $g$ explicitly in the potential.}. In general, all SUSY breaking mass terms and A-terms are expected to be proportional to the order parameter of SUSY breaking, $<F_{S}>$, therefore once $|\Phi| \gae M_{m}$ all SUSY breaking mass parameters are expected to be proportional to $<F_{S}>/|\Phi|$ \cite{deg}. Thus we must include a $|\Phi|^{-1}$ suppression factor for the soft SUSY breaking mass terms at large $|\Phi|$. A 2-loop calculation shows that the flat-direction effective potential has in fact a squared logarithmic dependence on $|\Phi|$ \cite{deg}.
Therefore to model $d = 6$ AD baryogenesis in GMSB models, we consider the following phenomenological flat-direction potential
$$ V(\Phi) = m_{s}^2 M_{m}^2 \ln^{2} \left(1 + \frac{|\Phi|}{M_{m}}\right)\left(1 + K \ln \left( \frac{|\Phi|^2}{M_{m}^2} \right) \right) $$
$$+ m_{3/2}^2\left(1 + \hat{K} \ln \left( \frac{|\Phi|^2}{M_{m}^2} \right) \right)|\Phi|^2
- cH^2 |\Phi|^2 + $$
\be{e1}
(A W + h.c.) +
\left| \frac{\Phi^{5}}{5! \tM^{3}}\right|^2 ~,\ee
where
\be{e2} W = \frac{\Phi^6}{6! \tM^3} ~.\ee
We include the factor $6!$ in \eq{e2} so that the physical strength of the interactions is dimensionally of the order of $\tM$. The first term in \eq{e1} is due to GMSB with messenger mass $M_{m}$. The factor multiplying this takes into account 1-loop radiative corrections due to gaugino loops once $|\Phi| \lae M_{m}$, with $K \approx -(0.01-0.1)$ \cite{km1,km2,km3}. The second term is due to gravity-medated SUSY breaking including the 1-loop correction term $ \hat{K}$. (For simplicity we set $\hat{K} = K$.) We have also included a Hubble correction to the mass squared term. We do not include a Hubble correction to the A-term, as the interaction leading to such terms is typically proportional to the inflaton field and so averages to zero over its coherent oscillations \cite{kawaA}. The scale of the non-renormalizable terms, $\tM$, will depend on the origin of the superpotential term responsible for lifting the flat direction. This could be gravitational, $\tM \sim M_{p}$ (where $M_{p} = 2.4 \times 10^{18} \GeV$) or due to the new physics suggested by MSSM coupling constant unification, $\tM \sim M_{U} \approx 10^{16} \GeV$. In the following we will focus on the case where $\tM \sim M_{p}$.
For the A-term we consider
\be{e3} A = m_{3/2} + \frac{a_{o} m_{s}}{\left(1 +
\frac{|\Phi|^2}{M_{m}^{2}}\right)^{1/2} } ~.\ee
The first term in \eq{e3} represents the A-term due to gravity-mediated SUSY breaking. The second term models the A-term in gauge-mediated SUSY breaking at $|\Phi| \lae M_{m}$, which is generated at 1-loop from the gaugino masses and is therefore suppressed relative to the A-term in gravity-mediated models, with $a_{o} \sim 0.01$. The suppression factor $(1 + |\Phi|^2/M_{m}^{2})^{-1/2}$ models the $1/|\Phi|$ suppression of the GMSB A-term at $|\Phi| \gg M_{m}$.
As a reference model for GMSB we will consider a metastable
SUSY breaking sector \cite{mura}, although our results depend only on the SUSY breaking
contribution to the messenger scalar masses and therefore can easily be adapted to GMSB models more generally.
The superpotential of the SUSY breaking sector is
\be{e4} W_{SB} = - \mu^2 S + \kappa S f \overline{f} + M f \overline{f} ~,\ee
where $S$ is the SUSY breaking field and $f$, $\overline{f}$ are the messenger fields.
The K\"ahler potential is
\be{e5} K = |S|^2 - \frac{|S|^4}{4 \Lambda^2} + O\left( \frac{|S|^6}{\Lambda^4} \right) ~,\ee
which is the generic form of K\"ahler potential for GMSB in a metastable vacuum.
The potential is then
$$ V = |\mu^2 - \kappa f \overline{f} |^2 \left(1 + \frac{|S|^2}{\Lambda^2}
+ O\left(\frac{|S|^4}{\Lambda^4}\right) \right) $$
\be{e6} + | \kappa S f + M f|^2 + | \kappa S \overline{f} + M \overline{f}|^2
~.\ee
There is a local SUSY breaking minimum at $S = f = \overline{f} = 0$ if $M^2 > \kappa \mu^2$, in which case $F_{S} = \mu^2$. At the local SUSY breaking minimum the SUSY messenger mass and SUSY breaking mass squared splitting are
\be{e7} M_{m} = M \;\;\;\;\; ; \;\;\;\; F_{m} = \kappa F_{S} = \kappa \mu^2 ~.\ee
The masses squared of the scalar components of the messengers $f$, $\overline{f}$ are therefore $m^2_{f} = m^2_{\overline{f}} = M_{m}^2 \pm \kappa \mu^2$, while the fermion mass is $M_{m}$. Soft SUSY-breaking masses for gauginos and scalars in the MSSM sector of order $m_{s}$ are then generated by messenger loops
\be{e8} m_{s} \approx \frac{g^2}{16 \pi^2} \frac{F_{m}}{M_{m}} \equiv \frac{g^2}{16 \pi^2} \frac{\kappa \mu^2}{M_{m}} ~.\ee
The gravitino mass is
\be{e9} m_{3/2} = \frac{F_{S}}{\sqrt{3} M_{p}} = \frac{\mu^2}{\sqrt{3} M_{p}} ~.\ee
Therefore the messenger mass is related to $m_{s}$ and $m_{3/2}$ by
\be{e10} M_{m} \approx \frac{g^2}{16 \pi^2} \frac{\sqrt{3} \kappa m_{3/2} M_{p}}{m_{s}} ~.\ee
Thus
\be{e11} M_{m} \approx 5 \times 10^{13} g^2 \kappa \left( \frac{m_{3/2}}{2 \GeV} \right)
\left( \frac{100 \GeV}{m_{s}} \right) \GeV ~.\ee
From this we see that a large messenger mass scale is required to have $m_{3/2} \approx 2 \GeV$. The importance of this for $d = 6$ AD baryogenesis is that, for plausible values of $\tilde{M}$, the messenger mass can be comparable to the initial value of the flat direction scalar when
$\Phi$ begins to oscillate and the baryon asymmetry is generated.
To get a rough estimate of the value of $|\Phi|$ at the onset of oscillations, $|\Phi|_{osc}$, we compute the value of the minimum of $V(\Phi)$ for the case where only the $-c H^2 |\Phi|^2$ term and the SUSY non-renormalizable term are included. Setting $cH^2 = m_{s}^2$, the value when the potential is destablized by the mass squared term, then gives
$$ |\Phi|_{osc} \sim \left(\frac{5!^{2}}{5} \right)^{1/8} \left(m_{s}^2 \tM^{6} \right)^{1/8}$$
\be{e12} = 3 \times 10^{14} \GeV \times
\left( \frac{m_{s}}{100 \GeV} \right)^{1/4} \left( \frac{\tM}{10^{18} \GeV} \right)^{3/4} ~.\ee
The value of $\tM$ is a free parameter in AD baryogenesis models. A natural possibility is that the non-renormalizable terms are associated with the completion of the theory at the Planck scale. However, even with this assumption there are different possibilities for how the Planck scale enters. If one expects the strength of the interaction to be dimensionally determined by $M_{P}$, then the factorial term should be included in the superpotential and so $\tM \approx M_{P}$. But it is also often assumed that the Planck scale enters in the Lagrangian rather than the vertex, in which case $W = \Phi^6/M_{p}^3$ and so $\tM \approx M_{p}/(6!)^{1/3} = 2.7 \times 10^{17} \GeV$. We will therefore consider a range of $\tM$ from $
2 \times 10^{17}\GeV$ to $M_{p}$\footnote{Such scales may also arise in the case where $\tM$ is due to exchange of heavy particles of mass $M_{U} \sim 10^{16} \GeV$, once suppression of the non-renormalizable operators due to couplings is included.}.
For the case where $g^2 \kappa \approx 1$, $m_{s} \approx 100 \GeV$ and $\tM = 2 \times 10^{17} \GeV$, the messenger mass scale is $\approx 5 \times 10^{13} \GeV$ while $|\Phi|_{osc} \approx 8 \times 10^{13} \GeV$. Therefore it is possible that the messenger mass scale can be close to the value of $|\Phi|$ at the onset of oscillations. In addition, there will be a significant time delay between the onset of oscillations and the eventual fragmentation of the AD condensate and Q-ball formation. Therefore even if $|\Phi|$ is greater than $M_{m}$ when oscillations begin, it is possible that $|\Phi|$ will be less than $M_{m}$ when the condensate fragments. If Q-balls form when $|\Phi| \lae M_m$, then $E/Q$ will not be strongly suppressed, in contrast to the case of conventional GMSB Q-balls with $|\Phi| \gg M_{m}$.
Thus $d = 6$ flat directions, combined with $m_{3/2} \approx 2 \GeV$ and $\tM \approx 10^{17}-10^{18} \GeV$, may lead to condensate fragmentation in a region of the potential where unstable Q-balls can form. In order to better understand the evolution of the flat-direction condensate, in the next section we will numerically evolve $\Phi$ using the phenomenological GMSB potential given in \eq{e1}.
\section{Affleck-Dine Baryogenesis and Condensate Fragmentation}
\subsection{$d = 6$ AD Baryogenesis in GMSB with a large messenger mass}
The baryon asymmetry in the late-time coherent oscillations of the real and imaginary parts of $\Phi$ is induced by the B-violating A-term. The baryon number density is $n_{B} = n_{Q}/3$, where $n_{Q}$ is the global $U(1)$ charge density corresponding to the number asymmetry in $\Phi$ particles:
\be{e12a} n_{Q} = i\left( \dot{\Phi}^{\dagger} \Phi - \Phi^{\dagger} \dot{\Phi} \right) \equiv \phi_{1} \dot{\phi}_{2} - \dot{\phi}_{1} \phi_{2} ~.\ee
The factor $1/3$ accounts for the baryon number of $\Phi$ along the $(u^{c}d^{c}d^{c})^2$ direction.
In the following we will use subscript $b$ to denote quantities at the time when the baryon asymmetry in a comoving volume becomes fixed. The present baryon asymmetry to entropy is then
\be{e12b} \frac{n_{B}}{s} = \frac{n_{B\;b}T_{R}}{4 H_{b}^{2} M_{p}^2}
~.\ee
The observed baryon asymmetry is $(n_{B}/s)_{obs} = 1.8 \pm 0.1 \times 10^{-10}$. This allows us to fix the reheating temperature for a given
flat-direction potential by computing $H_{b}$ and $n_{B\;b}$.
In Fig.1 we show the growth of the baryon asymmetry per comoving volume as a function of $|\Phi|/M_{m}$ for the case $M_{m} = 5 \times 10^{13} \GeV$, $m_{s} = 100 \GeV$, $K = -0.1$ and $\tM = M_{p}$, where $|\Phi|$ is the maximum value of the magnitude of $\Phi$ on the ellipitical trajectory. (We assume $a_{o} = 0.01$ in all our calculations.) This shows the rapid increase in the asymmetry once the $\Phi$ oscillations begin, with the asymmetry becoming fixed once $|\Phi|/M_{m} \lae 3$. In Fig.2 we show the trajectory of $\Phi$ in the complex plane when $|\Phi|/M_{m} = 0.54$. A noteworthy feature of this is that the trajectory is not yet ellipitical but in fact precesses, even though the baryon asymmetry per comoving volume is constant. This indicates that the GMSB deviation from a $|\Phi|^2$ potential is still significant at this time, even though the A-terms have become negligible and the baryon number in a comoving volume is constant.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=figFP1.eps, width=0.3\textwidth, angle = -90}
\caption{Growth of the baryon asymmetry per comoving volume as a function of $|\Phi|/M_{m}$.}
\label{fig1}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\epsfig{file=figFP2.eps, width=0.3\textwidth, angle = -90}
\caption{Trajectory of $\Phi$ in the complex plane after formation of the baryon asymmetry. The ellipse precesses but the baryon asymmetry per comoving voume is constant.}
\label{fig2}
\end{center}
\end{figure}
In Table 1 we give the AD baryogenesis parameters and $T_{R}$ for a range of $\tM$ and $M_{m}$ for the case $m_{s} = 100 \GeV$. In Table 2 we show the case with $m_{s} = 500 \GeV$. From this we see that in most cases the baryon asymmetry forms when $|\Phi|/M_{m}$ is larger than or of the order of 1.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline $m_{s}$ & $\tM $ & $M_{m} $ & & $n_{B\;b}$ & $H_{b}$ &
$\frac{|\Phi|}{M_{m}}$ & $T_{R}$ \\
\hline 100 & $2.4 \times 10^{18}$ & $5 \times 10^{13}$ & & $1.4 \times 10^{28}$ & 6.8 & 2.5 & 14.4 \\
\hline 100 & $2.4 \times 10^{18}$ & $5 \times 10^{12}$ & & $4.3 \times 10^{27}$ & 1.6 & 22.6 & 2.4 \\
\hline 100 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & & $4.3 \times 10^{25}$ & 0.4 & 75.0 & 14.7 \\
\hline 100 & $2.0 \times 10^{17}$ & $5 \times 10^{13}$ & & $3.7 \times 10^{26}$ & 15.4 & 0.5 & 2701 \\
\hline 100 & $2.0 \times 10^{17}$ & $5 \times 10^{12}$ & & $2.9 \times 10^{26}$ & 5.7 & 14.1 & 464 \\
\hline 100 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & & $1.2 \times 10^{26}$ & 2.0 & 17.3 & 140 \\
\hline
\end{tabular}
\caption{\footnotesize{AD baryogenesis parameters for $m_{s} = 100 \GeV$. (Dimensionful quantities in GeV units.) }}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $m_{s} $ & $\tM $ & $M_{m} $ & $n_{B\;b} $ & $H_{b}$ &
$\frac{|\Phi|}{M_{m}}$ & $T_{R}$ \\
\hline 500 & $2.4 \times 10^{18}$ & $1 \times 10^{13}$ & $1.2 \times 10^{28}$ & 9.7 & 14.9 & 33.3 \\
\hline 500 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & $6.7 \times 10^{27}$ & 1.9 & 119.6 & 2.1 \\
\hline 500 & $2.0 \times 10^{17}$ & $1 \times 10^{13}$ & $9.3 \times 10^{26}$ & 33.0 & 3.2 & $5.1 \times 10^{3}$ \\
\hline 500 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & $1.2 \times 10^{26}$ & 6.8 & 19.7 & 845 \\
\hline
\end{tabular}
\caption{\footnotesize{AD baryogenesis parameters for $m_{s} = 500 \GeV$. (Dimensionful quantities in GeV units.) }}
\end{center}
\end{table}
\subsection{Perturbation growth and condensate fragmentation}
To study the growth of perturbations of the AD condensate we will use the approach of \cite{ks}. This assumes that the baryon asymmetry in the condensate is maximal i.e. the trajectory of $\Phi$ is circular in the absence of expansion ("Q-matter"). This is not true for GMSB models with $a_{o} \ll 1$ and $m_{3/2} \ll m_{s}$, which produce ellipitical trajectories. However, the growth rate of energy density perturbations for the ellipitical trajectory will be similar, since in this case the growth of perturbations will be determined by the negative pressure of the condensate averaged over oscillation cycles, which turns out to be essentially independent of the field trajectory for a given potential and maximum value of the field on the trajectory, $\phi_{max}$. The average pressure for the limiting case of a linear oscillating condensate (i.e. a real field) has been discussed in \cite{turner}, where the average pressure was shown to be
\be{rx1} \frac{<p>}{\rho} + 1 = 2 \frac{\int_{0}^{\phi_{max}} \left( 1 - \frac{V}{V_{max}} \right)^{1/2} d \phi }{\int_{0}^{\phi_{max}} \left( 1 - \frac{V}{V_{max}} \right)^{-1/2} d \phi} ~ ~.\ee
Perturbations of the energy density of wavenumber ${\bf k}$ then grow according to
\be{rx2} \delta \ddot{\rho}_{{\bf k}} = -<p> |{\bf k}|^2 \delta \rho_{{\bf k}} ~.\ee
In this we have neglected expansion since the growth rate is typically large compared with $H$.
In the case of the circular condensate, the pressure is constant and determined by the magnitude of the field and the potential,
\be{rx3} \frac{p}{\rho} = \frac{\frac{\phi V^{'}}{2} - V}{\frac{\phi V^{'}}{2} + V} ~.\ee
This expression allows us to derive
simple expressions for the pressure and growth rate of perturbations which depend only on the potential and its derivatives, unlike the linear oscillation case where
the average pressure must be calculated numerically via \eq{rx1}.
The key observation that allows us to also use \eq{rx3} for non-circular trajectories is that the pressures in a circular condensate and in a linear condensate for a given $\phi_{max}$ are essentially identical. For the case of a polynomial potential, the pressures can be calculated exactly in both cases and are exactly equal, $<p>/\rho = (n-2)/(n+2)$ for $V(\Phi) \propto |\Phi|^{n}$ \cite{turner}. For a non-polynomial potential there is no analytical
expression and the integrals in \eq{rx1} must be calculated numerically. (This is not straightforward as the integrand in the denominator of \eq{rx1} is divergent as
$V \rightarrow V_{max}$.) We have numerically integrated \eq{rx1} for the potential \eq{e1} and find that the negative pressure is equal to within 10$\%$ for a given $\phi_{max}$.
Therefore the growth rate of perturbations of the circular and linear condensate will be essentially the same. We can expect the same to be true for ellipitical trajectories which are between these limiting cases. This allows us to use the more convenient expression for the pressure \eq{rx3} to calculate the growth of perturbations of the ellipitical condensate and so the time when they become non-linear. The similarity of the growth of perturbations in the circular and linear condensates is physically reasonable, as the growth of perturbations is essentially due to an attractive interaction between the scalar particles, and the strength of the interaction is the same for real and complex scalars which have the same potential. This approach is also in agreement with the numerical results of \cite{kk3}, where it was found that the size of the condensate fragments in the circular and elliptical cases differs by a factor of at most 2.
We first consider condensate fragmentation for a circular condensate in the potential \eq{e1}.
The flat-direction field can be written as
\be{e13} \Phi = \frac{1}{\sqrt{2}} R(\bfx,t) e^{i \Omega(\bfx,t)} ~.\ee
The $\Phi$ field equations are then
\be{e14} \ddot{\Omega} + 3 H \dot{\Omega} - \frac{1}{a^2} \nabla^{2} \Omega + 2 \frac{\dot{R}}{R} \dot{\Omega} - \frac{2}{a^2 R} \nabla \Omega . \nabla R = 0 ~\ee
\be{e15} \ddot{R} + 3 H \dot{R} - \frac{1}{a^2} \nabla^2 R - \dot{\Omega}^2 R + \frac{1}{a^2} (\nabla \Omega)^2 R + V^{'} = 0 ~,\ee
where $V^{'} = dV/dR$.
In this it has been assumed that $V$ is a function of $R$ alone, so these equations can be used to study growth of perturbations only once the A-terms are no longer significant in $V$ i.e. once $H < H_{b}$. The growing mode solutions have the form
\be{e15a} \delta R \;\;,\; \delta \Omega \propto e^{S(t) - i \bfk.\bfx} ~.\ee
Substituting these into \eq{e13} and \eq{e14} and assuming that $\alpha = \dot{S} = constant$ gives \cite{ks}
$$ \left[ \alpha^2 + 3 H \alpha + \frac{\bfk^2}{a^2} + \frac{2 \dot{R}}{R} \alpha \right] \left[\alpha^2 + 3 H \alpha + \right.$$
\be{e16} \left. \frac{|\bfk|^2}{a^2}
-\dot{\Omega}^2 + U^{''}(R) \right] + 4 \dot{\Omega}^2 \left[ \alpha - \frac{\dot{R}}{R} \right] \alpha = 0 ~.\ee
$\alpha$ is assumed to be constant because the rate of growth of perturbations is large compared with the rate of expansion of the Universe, in which case the time dependence in \eq{e16}, which is entirely due to expansion in the case of circular trajectory, can be neglected. In this case $\dot{R}$ can also be set to zero, as this is non-zero only because of expansion. In this case the equation simplifies to
\be{e16a} \left[ \alpha^2 + \frac{|\bfk|^2}{a^2} \right] \left[\alpha^2 + \frac{|\bfk|^2}{a^2}
-\dot{\Omega}^2 + V^{''}(R) \right] + 4 \dot{\Omega}^2 \alpha^2 = 0 ~.\ee
This can be solved for $\alpha^2$
\be{e17} \alpha^2 = \frac{|\bfk|^2}{a^2} \frac{1}{ \left(
V^{''} + 3 \dot{\Omega}^2 \right) } \left( \dot{\Omega}^2 -
V^{''} - 16 \frac{|\bfk|^2}{a^2} \frac{ \dot{\Omega}^4 } {\left(V^{''} + 3 \dot{\Omega}^{2}\right)^{2} } \right) ~.\ee
From this the largest value of $|\bfk|$ for which growth is possible is
\be{e18} \left| \frac{\bfk_{max}}{a} \right|^2 \equiv K_{m}^2 \approx \left( \frac{4 \dot{\Omega}^2}{V^{''} + 3 \dot{\Omega}^2}\right) \times
\left(\dot{\Omega}^2 - V^{''}\right) ~,\ee
where we have defined $K_{m} = \left|\bfk_{max}/a\right|$.
For a homogeneous Q-matter background with $\dot{R} = \ddot{R} = 0$, \eq{e15} implies that $\dot{\Omega}^{2} = V^{'}/R$.
The solution for $\alpha$ in \eq{e17} gives the value of $\alpha(t)$ on a time-scale short compared to $H^{-1}$.
One can then re-introduce the $t$ dependence of $R$ and $\dot{\Omega}$ due to the expansion of the Universe. The growth factor $S(t)$ is then given by
\be{e19} S(t) = \int_{t_{*}}^{t} \alpha(t) dt ~,\ee
where $t_{*}$ is the time at which a mode of wavenumber $\bfk$ begin to grow.
For a given mode $\bfk$, the perturbation of the condensate will only begin to grow once $|\bfk/a| < K_m$. Suppose for a given mode this occurs at a time $t_{*}$. The subsequent growth of the perturbation follows from
\be{e20} S(t) = \int_{\alpha_{*}}^{\alpha} \frac{\alpha(t)}{a H} da ~,\ee
where
\be{e21} \alpha(\bfk,a)
\approx \left( \frac{|\bfk|^2}{a^2} \frac{ \left( \dot{\Omega}^2 - V^{''} \right) }{ \left( V^{''} + 3 \dot{\Omega}^2 \right)} \right)^{1/2} ~.\ee
With $H \propto a^{-3/2}$ during inflaton-domination, we find from \eq{e19}
\be{e22} S(\bfk,a) = 2 \alpha(\bfk,a) \left( 1 - \frac{a_{*}^{1/2}}{a^{1/2}} \right) H^{-1} ~.\ee
At a given value of $a$, the mode with the maximum growth is found by maximizing $S(\bfk,a)$ with respect to $|\bfk|$. Since $\alpha \propto |\bfk|$ and $ a_{*}(k) = K_{m}^{-1} |\bfk|$
we find
\be{e23} S(\bfk) \propto |\bfk| - \frac{|\bfk|^{3/2}}{a^{1/2} K_{m}^{1/2}} ~.\ee
This is maximized at
\be{e24} \frac{|\bfk|_{frag}}{a} = \frac{4}{9} K_{m} ~.\ee
Therefore at each value of $a$, the mode which has the largest growth will have wavenumber given by
\eq{e24}. The value of $S$ for this mode is
\be{e24a} S = \frac{2}{3} \alpha H^{-1} ~.\ee
The condition for the condensate to fragment is
\be{e25} \frac{\delta R}{R} = \frac{\delta R_{o}}{R_{o}} e^{S} \; \gae 1 \; ~,\ee
for the mode with wavenumber \eq{e24}. The diameter of the fragments is then given by
\be{e25} \lambda_{frag} \approx \frac{2 \pi }{\left(\left|\bfk\right|_{frag}/a\right)} ~,\ee
while their global $U(1)$ charge ($Q = 3B$) is
\be{e25a} Q \approx n_{B} \frac{4 \pi}{3} \left( \frac{\lambda_{frag}}{2} \right)^{3} ~.\ee
We expect the same value of $S$ to apply to the elliptical trajectory when $R = R_{max}$, the maximum value of $R$ for the ellipse. We will therefore use this to estimate the time of fragmentation and the value of $|\Phi|/M_{m}$ at fragmentation. Given the energy density and baryon density when the condensate fragments, we can then compute the energy and baryon number of the fragments.
To compute the time of fragmentation of the condensate, we need a spectrum of initial perturbations. These are related to the primordial density perturbations.
For the initial perturbation entering the horizon we expect $\delta R/R \approx \delta \Omega/\Omega \approx \delta \rho/\rho$, the usual primordial density perturbation. This is certainly true of the amplitude, since $R \propto H^{1/4}$ at the minimum of the potential
due to the Hubble mass correction. For the phase, once the perturbation enters the horizon we can expect field dynamics to
transfer the perturbation of $R$ into a correlated $\Omega$ perturbation of a similar magnitude. There could also be additional phase fluctuations independent of the amplitude which will result in isocurvature baryon perturbations, but these are model dependent and we will not consider them here.
With $\delta R_{o}/R_{o} \approx \delta \rho/\rho \approx 10^{-4}$, the condition for fragmentation becomes $S \gae 11$.
In Figure 3 we show the growth factor $S$ as a function of $|\Phi|/M_{m}$ for the case $m_{s} = 100 \GeV$, $K = -0.1$, $M_{m} = 5 \times 10^{13} \GeV$ and $\tM = M_{p}$. It can be seen that fragmentation occurs at $|\Phi|/M_{m} \approx 0.1$, even though the baryon asymmetry forms at a much larger value, $|\Phi|/M_{m} \approx 2.5$.
In Table 3 we show the condensate fragmentation parameters for the case $m_{s} = 100 \GeV$ and in Table 4 for the case where $m_{s} = 500 \GeV$. We see that fragmentation can easily occur with $|\Phi|/M_{m} \lae 1$, in particular if $\tM < M_{p}$ and $|K| < 0.1$.
The field trajectory at fragmentation is strongly ellipitical, indicating that the total energy density in the $\Phi$ field is large compared with the energy density in Q-matter of the same charge density. We can estimate this ratio for a strongly elliptical trajectory by
\be{e12c} r_{E} \equiv \frac{\rho}{\left(E/Q\right) n_{Q}} \approx \frac{V(|\Phi|)}{m_{s} n_{Q}} ~.\ee
This is correct for the case where $|\Phi|$ for the Q-matter configuration is sufficiently less than $M_{m}$ that it can be considered to be described by $|\Phi|^2$ potential with $E/Q \approx m_{s}$. Otherwise $E/Q$ will be less than $m_{s}$, so that \eq{e12c} will be an underestimate of $r_{E}$. $r_{E}$ also gives the ratio of the energy of the condensate fragments to the energy of a Q-ball of the same charge. For the cases being considered we find that $r_{E}$ is mostly in the range 10-60 for $m_{s} = 100 \GeV$ and 20-90 for $m_{s} = 500 \GeV$.
We next estimate the decay temperature of the Q-balls under the assumption that the condensate fragments evolve into gravity mediated-type Q-balls of the same charge, corresponding to Q-balls in a potential $V = m_{s}^{2} |\Phi|^2 (1 + K \ln (|\Phi|^{2})$.
The decay rate of a Q-ball has an upper bound given by \cite{Qdecay}
\be{e26} \Gamma_{d} \leq \frac{\omega^3 A}{192 \pi^2 Q} ~,\ee
where the Q-ball solution is proportional to ${\rm exp}(i \omega t)$ and $A$ is the area of the Q-ball. The upper limit is expected to be saturated for flat-direction Q-balls. The decay temperature is then
\be{e27} T_{d} = \left( \frac{\omega^3 R^2 M_{p}}{48 \pi k_{T} Q} \right)^{1/2} ~,\ee
where $k_{T} = (\pi^2 g(T)/90)^{1/2} \approx 4.7$ using $g(T) \approx 200$ for the MSSM.
Dimensionally we expect $\omega \sim m$ and $R \sim m^{-1}$, so we can parameterize $\omega^3 R^2$ as $\omega^3 R^2 = f_{Q} m$. In particular, for gravity-mediated Q-balls $\omega \approx m$ and $R^2 = 2/(|K|m^2)$, therefore $f_{Q} = 2/|K|$.
This assumes $\Phi$ decay to scalar pairs is kinematically excluded, otherwise the decay rate can be enhanced by a factor $f_{s} \sim 10^{3}$ since the decay is not Pauli-blocked within the volume \cite{km2}. The decay temperature can then be written as
\be{e28} T_{d} = 60 \MeV f_{Q}^{1/2} \left( \frac{m}{100 \GeV} \right)^{1/2} \left( \frac{10^{20}}{Q} \right)^{1/2} ~.\ee
In Table 5 we show the gravity-mediated type Q-ball properties for the case $m_{s} = 100 \GeV$ and in Table 6 for the case $m_{s} = 500 \GeV$. In most cases the Q-ball decay temperature is well above the temperature of the onset of nucleosynthesis $\approx 1 \MeV$ but well below the typical temperature of NLSP freeze-out $\approx M_{LSP}/20 \gae 5 \GeV$.
(The decay temperature will be somewhat higher if $f_{s} > 1$.) Therefore such Q-balls can be a source of non-thermal NLSPs which subsequently decay to gravitino dark matter.
The condensate lumps will evolve into gravity-mediated Q-balls if at the end of their evolution $|\Phi|/M_{m} \ll 1$,
so that the dynamics of the Q-ball are entirely due to the radiatively-corrected $|\Phi|^2$ potential. The initial
values of $|\Phi|/M_{m}$ from Tables 3 and 4 are not always very much less than 1. However, since the energy in the initial condensate fragments is larger than the final energy in the Q-ball by a factor $r_{E}$, the field in the final Q-ball configuration
will be smaller by a factor $\approx 1/\sqrt{r_{E}}$. In most cases this is enough to ensure $|\Phi|/M_{m} < 1$ for the Q-balls.
Note that if $|\Phi|/M_{m}$ is less than 1 but not very small, the Q-ball may correspond to a new type of Q-ball which is intermediate between the gravity- and gauge-mediated cases. We will study the structure of such intermediate Q-balls in a future discussion
\cite{dm2}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $m_{s}$ & $K$ & $\tM $ & $M_{m} $ & $\frac{|\Phi|}{M_{m}}$
& $K_{m}$ & $n_{B\;frag}$ & $r_{E}$ \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $5 \times 10^{13}$ & 0.12 & 54.4 & $5.0 \times 10^{25}$ & 19.1 \\
\hline 100 & -0.01 & $2.4 \times 10^{18}$ & $5 \times 10^{13}$ & 0.017 & 21.1 & $2.9 \times 10^{24}$ & 24.3 \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $5 \times 10^{12}$ & 3.2 & 24.9 & $3.0 \times 10^{26}$ & 5.8 \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & 12.3 & 9.6 & $4.3 \times 10^{26}$ & 1.6 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $5 \times 10^{13}$ & 0.006 & 45.1 & $1.8 \times 10^{23}$ & 43.3 \\
\hline 100 & -0.01 & $2.0 \times 10^{17}$ & $5 \times 10^{13}$ & 0.0009 & 14.6 & $1.1 \times 10^{21}$ & 57.9 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $5 \times 10^{12}$ & 0.26 & 55.1 & $2.7 \times 10^{24}$ & 16.3 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & 2.3 & 29.5 & $7.0 \times 10^{24}$ & 7.3 \\
\hline
\end{tabular}
\caption{\footnotesize{Condensate fragmentation parameters for $m_{s} = 100 \GeV$. (Dimensionful quantities in GeV units.)}}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $m_{s} $ & $K$ & $\tM$ & $M_{m}$ & $\frac{|\Phi|}{M_{m}}$
& $K_{m}$ & $n_{B\;frag}$ & $r_{E}$ \\
\hline 500 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{13}$ & 2.2 & 149.9 & $7.3 \times 10^{26}$ & 31.8 \\
\hline 500 & -0.01 & $2.4 \times 10^{18}$ & $1 \times 10^{13}$ & 2.2 & 153.3 & $7.3 \times 10^{26}$ & 31.8 \\
\hline 500 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & 21.2 & 30.6 & $8.3 \times 10^{26}$ & 1.9 \\
\hline 500 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{13}$ & 0.15 & 276 & $4.7 \times 10^{24}$ & 72.2 \\
\hline 500 & -0.01 & $2.0 \times 10^{17}$ & $1 \times 10^{13}$ & 0.03 & 118 & $1,2 \times 10^{23}$ & 85.3 \\
\hline 500 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & 3.8 & 111 & $2.1 \times 10^{25}$ & 20.0 \\
\hline
\end{tabular}
\caption{\footnotesize{Condensate fragmentation parameters for $m_{s} = 500 \GeV$. (Dimensionful quantities in GeV units.)}}
\end{center}
\end{table}
\begin{figure}[htbp]
\begin{center}
\epsfig{file=figFP3.eps, width=0.3\textwidth, angle = -90}
\caption{Growth factor S(\emph{t}) as a function of $|\Phi|/M_{m}$.}
\label{fig2}
\end{center}
\end{figure}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline $m_{s}$ & $K$ & $\tM$ & $M_{m}$ & $Q$
& $T_{d}(\MeV) $ \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $5 \times 10^{13}$ & $1.1 \times 10^{23}$ & 8.1 \\
\hline 100 & -0.01 & $2.4 \times 10^{18}$ & $5 \times 10^{13}$ & $4.0 \times 10^{22}$ & 42 \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $5 \times 10^{12}$ & $8.0 \times 10^{24}$ & 1.0 \\
\hline 100 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & $6.5 \times 10^{25}$ & 0.3 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $5 \times 10^{13}$ & $2.6 \times 10^{20}$ & 167
\\
\hline 100 & -0.01 & $2.0 \times 10^{17}$ & $5 \times 10^{13}$ & $1.3 \times 10^{20}$ & 236 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $5 \times 10^{12}$ & $6.5 \times 10^{21}$ & 33 \\
\hline 100 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & $9.2 \times 10^{25}$ & 0.3 \\
\hline
\end{tabular}
\caption{\footnotesize{Q-ball charge and decay temperature for $m_{s} = 100 \GeV$. (Dimensionful quantities in GeV units except $T_{d}$.)}}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline $m_{s}$ & $K$ & $\tM$ & $M_{m}$ & $Q$
& $T_{d} (\MeV)$ \\
\hline 500 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{13}$ & $7.8 \times 10^{22}$ & 22 \\
\hline 500 & -0.01 & $2.4 \times 10^{18}$ & $1 \times 10^{13}$ & $7.8 \times 10^{22}$ & 69 \\
\hline 500 & -0.1 & $2.4 \times 10^{18}$ & $1 \times 10^{12}$ & $1.1 \times 10^{25}$ & 1.8 \\
\hline 500 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{13}$ & $7.7 \times 10^{19}$ & 683 \\
\hline 500 & -0.01 & $2.0 \times 10^{17}$ & $1 \times 10^{13}$ & $2.7 \times 10^{19}$ & 3651 \\
\hline 500 & -0.1 & $2.0 \times 10^{17}$ & $1 \times 10^{12}$ & $5.7 \times 10^{21}$ & 79 \\
\hline
\end{tabular}
\caption{\footnotesize{Q-ball charge and decay temperature for $m_{s} = 500 \GeV$. (Dimensionful quantities in GeV units except $T_{d}$.)}}
\end{center}
\end{table}
\section{Conclusions}
In this paper we have revisited Affleck-Dine baryogenesis and Q-ball formation in GMSB models. We have been concerned with two questions: (i) whether unstable Q-balls can naturally form in GMSB AD baryogenesis models and (ii) whether such unstable Q-balls can be a source of gravitino dark matter. To answer these questions we have focused on a $d = 6$ flat direction with $m_{3/2} = 2 \GeV$, which is consistent with gravitino dark matter from Q-ball decay.
We find that it is possible for condensate fragmentation in $d = 6$ AD baryogenesis to occur when $|\Phi|/M_{m}$ is less than or of the order of 1. The importance of this is that fragmentation does not occur on the plateau of the GMSB potential and so does not lead to formation of stable Q-balls which would destabilize neutron stars. Instead the fragments form either in the intermediate region between the
$|\Phi|^2$ region and the GMSB plateau or well within the $|\Phi|^2$ dominated region, where we can expect the Q-balls to behave like unstable gravity-mediated Q-balls. If the condensate fragments subsequently evolve into Q-balls of the same charge, then the Q-balls should be of approximately gravity-mediated form and unstable, with decay temperatures typically in the range 10 MeV - 1 GeV. Decay of such Q-balls to NLSPs could then be a non-thermal source of gravitino dark matter. Even if $|\Phi|/M_{m} \sim 1$ when the condensate fragments, the final value of $|\Phi|/M_{m}$ in the Q-ball will typically be significantly less than 1 once the fragment has lost its excess energy, since the energy of the initial condensate fragment is much larger than the energy in its baryonic charge.
In some cases the value of $|\Phi|$ will be less than the messenger scale but not sufficiently small that the Q-ball will be a gravity-mediated type Q-ball. In this case we need to consider a new type of Q-ball, intermediate between gauge- and gravity-mediated Q-balls, whose structure will be determined by the region of the potential where the
corrections due to the messenger scale are still significant. We will study the structure of these intermediate Q-balls in a future analysis \cite{dm2}.
Due to the suppressed A-terms in GMSB models, the energy in the condensate fragments is typically larger than the energy in the baryon asymmetry by a factor of 10-100.
In our discussion we have assumed that the condensate fragments of charge Q subsequently evolve into Q-balls of the same charge.
One may question whether this assumption is reasonable. We have considered condensate fragments which initially form with only a small amount of energy in their baryonic charge relative to their mass. This is consistent with the most recent numerical simulation of condensate fragmentation, presented in \cite{fk}, where the initial condensate fragments are called "first generation Q-balls". Their subsequent evolution is found to depend on how ellipitical the condensate trajectory is. It was found that if $r_{E}$ (essentially equivalent to $\epsilon^{-1}$ in \cite{fk}) is less than about 10, most fragments are quasi-stable and will probably slowly evolve into Q-balls. However, if $r_{E}$ is 100 or more, then the initial oscillon-like fragments break up into Q-balls of positive and negative charge. Thus our assumption of evolution to Q-balls is consistent with the results of \cite{fk} if $r_{E} \lae 10$, but in most of our examples $r_{E}$ is between 10 and 100, where the evolution of the fragments is not clear.
The non-linear dynamics of oscillon-like objects requires very high resolution simulations to evaluate their stability \cite{oscillons}. The spatial size of the simulations of \cite{fk} is sufficiently large to observe many condensate fragments, but this necessarily means that the resolution of a single fragment is not optimized; a high-resolution simulation of just one fragment would be the best way to clearly establish the value of $r_{E}$ for which the fragments are quasi-stable. The emission of oppositely charged Q-ball pairs might be interpreted as a way for the condensate fragments to lose excess energy, but this appears to be contradicted by the observation of quasi-stable fragments with $r_{E} \approx 10$ in the simulation of \cite{fk}, which suggests complex non-linear dynamics favouring spherically symmetric oscillon-like states. This is an interesting question for future research.
Is Q-ball decay in GMSB AD baryogenesis a testable model for the origin of gravitino dark matter? The model predicts a large messenger scale, which will be reflected in the spectrum of SUSY particles \cite{strumia}. Moreover, the 2 GeV gravitino mass predicted by the model is sufficiently large that there should be a significant gravity-mediated contribution to the spectrum of SUSY particle masses. The model also predicts that decay of the thermal relic NLSP density must underproduce gravitino dark matter when $m_{3/2}$ is assumed to be 2 GeV, in order that the contribution from thermal NLSP decay is less than that from Q-ball decay. These features may be testable at the LHC; in particular, identification of the 2 GeV gravity-mediated contribution to SUSY breaking masses would provide strong evidence\footnote{The phenomenological advantages of an O(1) GeV gravitino with respect to flavour problems and dark matter have been discussed in the context of "'sweet spot" \cite{ss} and "Goldilocks" \cite{gold} supersymmetry.}. The GMSB flat direction might also produce correlated baryon and dark matter isocurvature perturbations, if the GMSB potential in the angular direction is sufficiently flat and $|\Phi|$ has the right magnitude during inflation.
In summary, we have shown that unstable Q-balls are a natural possibility in GMSB AD baryogenesis with a large messenger scale. Such unstable Q-balls are essential if AD baryogenesis in GMSB is to be consistent with astrophysical constraints from neutron star stability. The requirement of a large messenger scale is also consistent with the 2 GeV gravitino mass required if gravitino dark matter comes from Q-ball decay. This points to a striking consistency between the requirements for gravitino dark matter from Q-ball decay and successful Affleck-Dine baryogenesis in GMSB models.
|
2,877,628,090,841 | arxiv |
\section{Introduction}
With the rapid development of active safety and autonomous functions, modern automotive systems have become complex cyber-physical systems that involve close interactions between the cyber domain (i.e., automotive electronic systems) and the physical domain (i.e., mechanical components and surrounding physical environment). The design and validation of these systems span across multiple layers, as illustrated in Fig.~\ref{fig:automotivelayers}. The function layer defines various automotive system functionalities in sensing, control, computation, communication, etc., and captures their interaction with the physical environment. In this paper, as we consider connected vehicle applications, the function layer can be further divided into the vehicular network layer and the individual vehicle function layer. The architecture layer defines the platform on which the system functionalities are implemented. It could include multiple sub-layers such as the software layer and the hardware layer. It may also include the layers of mechanical and physical components (e.g., engines, brakes, wires), but those are beyond the scope of this paper. In the AUTOSAR (Automotive Open System Architecture) standard, the automotive software layer could be further divided into a layer of runnables and a layer of software tasks connected with signals, as shown in~\cite{2015_ICCPS_Deng}.
\begin{figure}[htbp]
\centering
\includegraphics[width=1\linewidth]{figure/automotive_layers_v2.pdf}
\caption{An illustration of the different layers for automotive system design, verification and validation. }
\label{fig:automotivelayers}
\end{figure}
Traditionally, the design of different automotive layers is often carried out in an isolated fashion. However, the fast growing complexity of system functionality and architecture, as well as the close interaction between the cyber and physical domains, has led to strong dependency between layers and made those isolated approaches ineffective. For instance, whether an advanced control function or a new security feature can be deployed in a vehicle often depends on the availability of computation and communication resource at the architecture layer; whether a connected vehicle application can meet the safety and performance requirements depends on the communication delay and reliability of vehicular ad-hoc network. Thus, it is critical to adopt a cross-layer design methodology to holistically address the multiple layers in automotive systems.
In this paper, we will first introduce our previous works in cross-layer design for individual vehicles (Section~\ref{sec:individual_vehicles}) and for connected vehicles (Section~\ref{sec:connected_vehicles}). We will then present our initial results in developing a new cross-layer methodology for systems that allow the weakly-hard constraints (Section~\ref{sec:ongoing_works}).
\section{Cross-Layer Design for Individual Vehicles}
\label{sec:individual_vehicles}
Vehicle design spans across multiple layers. As shown in Fig.~\ref{fig:automotivelayers}, various automotive functionalities (e.g., sensing and control algorithms) can be captured at the function layer with formal or semi-formal models. These models are implemented at the architecture layer, often as software tasks running on the hardware platform. Traditionally, the functionality design, the generation of tasks from function models (which may go through a runnable layer as in AUTOSAR), and the mapping of tasks onto the hardware platform, are often done in isolated steps. However, as automotive systems are time-critical and resource-limited, design choices at the higher layers (e.g., functionality design or runnable generation) have significant impact on whether efficient or even feasible designs can be found at the lower layers. This has motivated our work in cross-layer design for individual vehicles, as introduced below.
\subsection{Holistic Software Synthesis from Function to Architecture}
In~\cite{2015_ICCPS_Deng}, we propose a model-based software synthesis flow for AUTOSAR-based automotive systems. The cross-layer flow conducts runnable generation from the function model, task generation from the runnables, and task mapping onto a multicore platform in a holistic framework. Different from traditional approaches where runnable generation is performed merely from functional perspective and isolated from task generation and mapping, our approach explicitly addresses architectural properties in runnable generation, in particular regarding timing and schedulability.
In runnable generation, the functional blocks are mapped to runnables, as shown in Fig.~\ref{fig:automotivelayers}. Two algorithms are proposed in~\cite{2015_ICCPS_Deng} to explore different runnable generation options, while considering system modularity\footnote{As in~\cite{2015_ICCPS_Deng}, modularity is a metric that reflects the IP disclosure degree, and is measured by the number of runnables generated. A runnable generation can effectively hide the information of the internal block structure if the number of runnables (and their dependencies) is significantly smaller than the number of internal blocks. The optimal granularity is achieved when there is fewest number of generated runnables (when under certain constraints such as reusability and/or schedulability).}, reusability, code size, and schedulability: In a top-down method, a mixed integer linear programming (MILP) formulation is used to create the initial solution with the maximum modularity and reusability (i.e., no false input-output dependencies), and then a heuristic is used to decompose the runnables to improve schedulability. In a bottom-up method, another MILP formulation generates the initial solution with the maximum schedulability and a heuristic gradually merges runnables to optimize modularity.
To facilitate schedulability analysis, a formalism called Firing Execution Time Automaton (FETA) is developed, which can accurately capture the worst-case runnable timing behavior. In task generation and mapping, two algorithms are developed to group runnables into software tasks and map tasks onto Electronic Control Units (ECUs) on the hardware platform, while considering schedulability and memory cost for inter-task communication. For schedulability analysis, FETA is also applied at the task level to capture task timing behavior.
The experimental results in~\cite{2015_ICCPS_Deng} demonstrate that it is important to address timing and schedulability during the generation of runnables from function models, as the decisions at this stage already have significant impact on the eventual system feasibility. In particular, it is shown that there are strong trade-offs between modularity and schedulability. Previous methods that do not consider schedulability often lead to runnable generation solutions that have optimal modularity but are infeasible for task generation and mapping. Moreover, the proposed FETA formalism provides a holistic timing representation for functional blocks, runnables, and tasks, and is shown to be effective for schedulability analysis across these different layers.
\subsection{Cross-Layer Design for Automotive Security}
Security has become a pressing issue for automotive systems in recent years, especially with the increase of vehicle connectivity and complexity. Various security protection mechanisms, such as message authentication, encryption, and anomaly detection, have been proposed for automotive systems. However, the successful deployment of these techniques depends on the available resources and whether the additional overhead may lead to timing violations of existing functions.
It is thus important to take a cross-layer approach to address the design of security features together with other system objectives, including architecture layer properties such as timing and schedulability.
\subsubsection{Security-Aware Software Synthesis}
Traditional automotive software synthesis flow does not address security.
It is often difficult to add security mechanisms after the software synthesis process is completed (i.e., after task allocation and scheduling are decided), because of the tight timing and resource constraints.
On the other hand, fixing the design of security mechanisms before software synthesis could often result in infeasible systems. Thus, we propose to address the security (function layer) together with the software synthesis (architecture layer) in an integrated formulation.
In~\cite{2013_ICCAD_Lin, 2014_ICCAD_Lin}, we explore security mechanisms to protect communication messages against replay and masquerade attacks, which could happen when a malicious attacker compromises an ECU and then either replays legitimate messages on the communication bus or sends messages pretending as another ECU. Adopting message authentication codes (MACs) may prevent such attacks by authenticating each message with a key that only the message sender and receiver have, however they also incur overhead and could lead to timing and resource violations.
In~\cite{2013_ICCAD_Lin}, we consider adding MACs to Controller Area Networks (CAN) bus messages. Longer MACs make it harder for brute-force attacks, but also increase CAN message sizes and could lead to infeasible designs.
To address these trade-offs, we develop an MILP-based algorithm to quantitatively explore the security design, including the messages to authenticate, MAC lengths and sharing strategies, together with the software synthesis options. Experimental results demonstrate that such holistic consideration can significantly increase the chance to find designs that satisfy both security and schedulability constraints -- although in some cases feasible solutions still cannot be found, given the limited bandwidth and message size of CAN.
In~\cite{2014_ICCAD_Lin}, we explore adding MACs to future automotive bus protocols that are based on TDMA communication (e.g., FlexRay, Time-Triggered Protocol, Time-Triggered Ethernet), which provide much higher bandwidths and larger message sizes. However, applying security mechanisms still requires careful analysis and design to avoid violations on other design constraints. In this work, we leverage the time-division property and adopt a key sharing mechanism that is based on time-delayed release of keys.
This mechanism protects against masquerade attacks, however may lead to long message transmission delays. We quantitatively model the worst-case transmission delays under time-delayed release of keys, and develop a simulated annealing based method to holistically optimize task allocation and scheduling, TDMA-based network scheduling, and the key-release interval to meet both timing and security constraints.
\subsubsection{Security-Performance Trade-offs under Architectural Constraints}
With limited resources, improving automotive system security may require sacrificing other objectives, and such trade-off should be addressed in a quantitative and holistic manner. In~\cite{2016_TCAD_Zheng}, we explore the trade-off between security and control performance for a CAN-based system, while meeting schedulability constraints.
We consider a system model where multiple control tasks share the same ECU and communicate with sensors and actuators. Malicious attackers may eavesdrop on the messages between sensors and control tasks, and try to reconstruct the system state. This not only results in a loss of privacy, but could further be used as the basis for other attacks. Applying encryption techniques may prevent such attacks, however also introduces computation and communication overhead. For each encrypted message, a decryption task needs to be added on the ECU, which may force the control tasks to increase their activation periods to maintain schedulable.
In~\cite{2016_TCAD_Zheng}, we present a cross-layer formulation to address system security level, control performance, and schedulability. The security level is the difficulty for attackers to restore the system states, measured through either Observability Gramian or analysis based on Kalman filter. We quantitatively model how the security level depends on the number of encrypted sensing channels (messages). On the other hand, for each control task, we model the relation between its performance and its period as an exponentially decaying function. Intuitively, having more encrypted messages increases system security level, but may lead to the increase of control task periods and thus worse control performance.
We then develop a simulated annealing based algorithm to explore the choices of message encryption and control periods, under schedulability constraints. The experimental results demonstrate the clear trade-off between security and control performance, and show the importance of holistically considering these cross-layer properties.
\section{Cross-Layer Design for Connected Vehicles}
\label{sec:connected_vehicles}
\input{cross_layer_connected_vehicle.tex}
\input{ongoing.tex}
\input{Weakly_hard_control_performance.tex}
\section{Conclusion}
This paper presents several cross-layer methods for the design of automotive systems, including our prior works on systems with hard deadlines and our new results on weakly-hard systems. We believe that the strong dependencies between different function and architecture layers (even more so in weakly-hard systems) make it critical to take a cross-layer approach for addressing the design of automotive systems, and similar methodology might be applicable to other cyber-physical systems such as airplanes, robots, and various Internet-of-Things systems.
\section*{Acknowledgment}
The authors gratefully acknowledge the support from the National Science Foundation grants 1834701, 1834324, and 1839511, and the Office of Naval Research grant N00014-19-1-2496.
\bibliographystyle{IEEEtran}
\section{Cross-Layer Design with Weakly-Hard Paradigm}
\label{sec:ongoing_works}
Traditionally, the timing behavior of automotive functions has been specified based on hard constraints, where every instance of a task (or message) has to complete its execution (or transmission) by a pre-defined deadline. This is also the assumption in our previous works introduced in Section~\ref{sec:individual_vehicles} and~\ref{sec:connected_vehicles}. While such timing model facilitates worst-case analysis of system behavior, it is often over-pessimistic and rigid, resulting in infeasible or over-conservative designs.
Many practical functions can in fact tolerate certain degrees of deadline misses, and their timing behavior can be described with the so-called \emph{weakly-hard} constraints, where bounded deadline misses are allowed. A common example is the $(m,\text{K})$ constraints, which specify that among any K consecutive instances of a task, at most $m$ of them can violate their execution deadlines~\cite{Bernat_TC_01}. Leveraging such weakly-hard constraints could more accurately define system timing requirements, significantly increase feasible design space under the typically-tight resource constraints in automotive systems, and improve design flexibility with additional timing slacks.
We believe that cross-layer design is particularly important for weakly-hard systems. To properly set the weakly-hard constraints (e.g., choose the values of $m$ and K) and effectively leverage their potential, it is essential to address the following two issues in a holistic manner: 1) at the function layer, ensure that system safety, stability, security, and other functional requirements can still be met under deadline misses allowed by weakly-hard constraints; 2) at the architecture layer, explore the design space under weakly-hard constraints to satisfy various system adaptation and retrofitting goals.
In our recent work~\cite{huang2019formal}, we consider the first issue, and develop an approach for analyzing system functional properties under given degree of deadline misses. This work is different from works that focus on feedback controller synthesis for stability, such as~\cite{linsenmayer2017stabilization}.
More specifically, our approach can determine whether a system is safe from an initial state under given $(m,\text{K})$ weakly-hard constraints. Previous verification methods could not be directly applied due to the lack of mechanism to capture and model the $(m,\text{K})$ specification at architecture level. To address this problem, our approach first carries out a series of transformations to abstract the $(m,\text{K})$ constraints, and then uses over-approximation based techniques to verify the safety of a new system model that combines the functional model and the abstraction. This approach is shown to be sound and effective in verifying system safety under weakly-hard constraints.
In another of our recent work~\cite{2019_ICCD_Liang}, we consider both issues, and develop a codesign approach to explore the addition of new security monitoring tasks by leveraging weakly-hard constraints for control tasks. The work studies the trade-off between control performance and system security level, when different degrees of deadline misses occur to the control tasks.
\subsection{System model}
We consider a set of tasks $\{\tau_i\}$ running on a single ECU. All tasks are periodically activated. Each task $\tau_i$ is modeled by its period $T_i$, deadline $D_i$, and the execution time $C_i$. The system is scheduled by the static-priority preemptive policy.
We consider a controller task $\tau_c$. The continuous-time dynamic of this linear time-invariant (LTI) system is:
\begin{equation}\label{equ:continuous}
\dot{\textbf{x}}(t)=A_c\textbf{x}(t) + B_c\textbf{u}(t)
\end{equation}
where $\textbf{x}(t)\in\mathbb{R}^{n}$ and $\textbf{u}(t)\in\mathbb{R}^{m}$.
We assume that $\tau_c$ is running under the Logical Execution Time (LET) paradigm~\cite{2003_IEEE_Henzinger} where it receives the system state from sensors at the beginning of each sampling period and applies the control input to the actuators at the deadline. If a deadline miss occurs, the controller will apply the last calculated control input (from previous periods) at the deadline. For simplification, we assume that the deadline of this controller is the same as its period, i.e. $D_c=T_c$. And the discrete-time system dynamic is:
\begin{equation}
\textbf{x}[k+1]=A\textbf{x}[k] + B\textbf{u}[k-p_k] \quad p_k=1,2,3,\ldots
\end{equation}
where
\begin{equation*}
A=e^{A_c T_c}, \quad B = \int_0^{T_c} e^{A_c t} B_c dt
\end{equation*}
Here $\textbf{u}[k-p_k]$ is the latest control input at time $t=T_ck$, and $p_k$ is the related delay factor. For instance, $p_k=1$ if the deadline at $t=T_ck$ is not missed.
The control law is derived by solving the discrete-time linear–quadratic regulator (LQR) problem. Assume such control law is designed without considering any deadline misses.
The LET paradigm eases the control design as there will be a constant sensing-actuating delay.
By introducing the augmented state vector $\textbf{z}[k]=\left[\textbf{x}^\top[k], \textbf{u}^\top[k-1]\right]^\top$, the system dynamic used for solving the LQR is:
\begin{equation}
\textbf{z}[k+1]=
\begin{bmatrix}
A & B\\
\textbf{0} & \textbf{0}
\end{bmatrix}
\textbf{z}[k] +
\begin{bmatrix}
\textbf{0}\\
\textbf{I}
\end{bmatrix}
\textbf{u}[k]
=A_z\textbf{z}[k] + B_z\textbf{u}[k]
\end{equation}
The control law $\textbf{u}[k]=-F\textbf{z}[k]$ is derived by minimizing the quadratic cost function:
\begin{equation}\label{equ:lqr_z}
J=\sum_{k=1}^\infty \left(\textbf{z}^\top[k]Q\textbf{z}[k]+\textbf{u}^\top[k]R\textbf{u}[k] \right)
\end{equation}
where $Q$ and $R$ are both positive semi-definite matrices.
\subsection{Control stability under deadline misses}
Let $p_k$ denote the control input delay at time step $k$. Based on task schedulability analysis, we can deduce that $p_k$ is bounded by a maximum delay $\hat{p}=\left\lceil\frac{R_c}{T_c}\right\rceil$, where $R_c$ represents the task worst-case response time.
Then, we consider a new augmented state vector: $$\xi[k]=\left[\textbf{x}^\top[k],\textbf{u}^\top[k-1],\ldots,\textbf{u}^\top[k-\hat{p}]\right]^\top$$
we can have the system dynamic as:
\begin{equation}
\xi[k+1] = A_\xi[k] \xi[k] + B_\xi u[k]
\end{equation}
\begin{equation}
A_\xi[k]=
\begin{bmatrix}
A & B_1 & \ldots & B_{\hat{p}-1} & B_{\hat{p}}\\
\textbf{0} & \textbf{0} & \ldots & \textbf{0} & \textbf{0}\\
\textbf{0} & \textbf{I} & \ldots & \textbf{0} & \textbf{0}\\
& & \ddots & & \\
\textbf{0} & \textbf{0} & \ldots & \textbf{I} & \textbf{0}
\end{bmatrix}
,\
B_\xi=
\begin{bmatrix}
\textbf{0}\\
\textbf{I}\\
\textbf{0}\\
\vdots \\
\textbf{0}
\end{bmatrix}
\end{equation}
where $B_{p_k}=B$ and $B_i=\textbf{0},\forall i\neq {p_k}$. As the control law $\textbf{u}[k]=-F\textbf{z}[k]$ is derived from the LQR problem~\eqref{equ:lqr_z}, we can rewrite the system dynamic as:
\begin{equation}\label{equ:wh_aug_dynamic}
\xi[k+1] = (A_\xi[k] - B_\xi F_\xi)\xi[k]=\phi[k]\xi[k]
\end{equation}
where $F_\xi = \left[F,\textbf{0}\right],\ \textbf{0}\in \mathbb{R}^{(\hat{p}-1)m}$.
As $B_\xi$ and $F_\xi$ are constant matrices and $A_\xi[k]$ is only related to $p_k$, there are $\hat{p}$ different $\phi[k]$, denoted as $\phi_1,\ldots,\phi_{\hat{p}}$. And we have $\phi[k]=\phi_{p_k}$.
The hyper-period $H_c$ of the controller task $\tau_c$ is the least common multiple of the periods of $\tau_c$ and its higher priority tasks. The execution pattern in each hyper-period is the same. As introduced in~\cite{2019_ICCD_Liang}, a weakly-hard schedulability analysis approach based on event-based simulation can derive the deadline miss pattern of the task in its hyper-period.
Moreover, during the simulation, the latest finished job can be recorded at each deadline, i.e. the delay factor $p_k$ will be recorded at $t=T_c*k$. Based on the consistency property of the hyper-period, we have $p_{k+iN_c} = p_k, \forall i\in \mathbb{Z}^+$, where $N_c = {H_c/T_c}$ is the number of jobs of $\tau_c$ in each hyper-period.
From the weakly-hard schedulability analysis, the delay factors $p_k$ in the hyper-period $k\in[0,N_c)$ are known. Thus, we have:
\begin{equation}
\begin{aligned}
\xi[k+N_c] &= \phi[k+N_c-1]\cdots\phi[k+1]\phi[k]\xi[k] \\
&= \prod^0_{i=k+N_c-1}\phi[i] \prod^k_{j=N_c-1}\phi[j]\xi[k]\\
&= \prod^0_{i=k+N_c-1}\phi_{p_i} \prod^k_{j=N_c-1}\phi_{p_j}\xi[k]\\
&= \Phi_k\xi[k]
\end{aligned}
\end{equation}
The following Theorem~\ref{thm:1} can then be used to check the control stability with deadline misses.
\begin{theorem}\label{thm:1}
The weakly-hard LTI system \eqref{equ:wh_aug_dynamic} is asymptotic stable if all eigenvalues of $\Phi_k$ are within the unit circle for all $k$:
\begin{equation}
|\lambda^i_k| < 1,\ \forall \lambda^i_k\in eig(\Phi_k),\ \forall k\in[0,N_c)
\end{equation}
\end{theorem}
\begin{proof}
As all eigenvalues of $\Phi_k$ are in the unit circle, the sub-series $\xi'[l] = \xi[k+lN_c] = \Phi_k^l\xi[k],\ l=0,1,2,3,\dots$ is asymptotic stable, which can be expressed as:
\begin{equation} \label{equ:k_stable}
\forall \varepsilon,\ \exists L_k(\varepsilon,\xi[k]),\ s.t.\ ||\xi[k+l N_c]|| \leq \varepsilon,\ \forall l \geq L_k
\end{equation}
As~\eqref{equ:k_stable} is satisfied by all $k\in [0,N_c)$, we have:
\begin{equation}
\forall \varepsilon,\ \exists L(\varepsilon),\ s.t.\ ||\xi[k]|| \leq \varepsilon,\ \forall k \geq L
\end{equation}
where $L(\varepsilon) = \max\{k+N_c L_k(\varepsilon,\xi[k])|\forall k\in [0,N_c) \}$. Thus, the weakly-hard LTI system \eqref{equ:wh_aug_dynamic} is asymptotic stable.
\end{proof}
\subsection{Experiments}
In our experiments, we use a Furuta inverted pendulum as the example control plant to analyze the weakly-hard control functionality. The modeling of the Furuta pendulum is introduced in~\cite{2018_ECRTS_Pazzaglia}.
The motor of the pendulum controls an arm that rotates in the horizontal plane. A pendulum is jointed to the arm and is free to rotate in the vertical plane. The system state is $\textbf{x}(t)=[\theta_r, \theta_p, \dot{\theta}_r, \dot{\theta}_p]^\top$, which are the angles of the arm and the pendulum, and their angular velocities. The control input $u(t)$ is the voltage applied to the motor.
The continuous-time dynamic~\eqref{equ:continuous} of this Furuta pendulum in the numerical form is:
\begin{equation}
A_c =
\begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1.6907 & -2.9968 & -0.0048\\
0 & 21.9176 & -3.0831 & 0.0626
\end{bmatrix}
, \
B_c = \begin{bmatrix}
0 \\
0 \\
3.8998 \\
4.0122
\end{bmatrix}
\end{equation}
Besides this pendulum control task, there are 7 regular periodic tasks that share the ECU. The periods of these tasks are varied from 60$ms$ to 300$ms$.
The total utilization of these 7 regular tasks is $72\%$, where the utilization is $\sum C_i/T_i$.
The execution time of the controller task $\tau_c$ is $C_c=15\ ms$. We assume that all regular tasks do not allow any deadline misses, and they are scheduled with the rate-monotonic policy.
The priority of $\tau_c$ is chosen to be the highest allowed one such that no low-priority regular tasks have deadline misses.
Under each period $T_c$, the control law $u[k]=-F\textbf{z}[k]$ is designed by solving the LQR problem \eqref{equ:lqr_z}, where $Q=diag(\alpha,\alpha,0,0,0)$ and $R = \beta$. $\alpha\in[0.1,10]$ and $\beta\in[0.1,1000]$ are the weights for states and control input in the quadratic cost. As the ratio $\alpha/\beta$ decreases, the control input intensity will reduce, while the convergence rate will be slower. We assume that the control input (i.e. the voltage apply to the motor) has an upper bound: $||F|| \leq 35$. During the control law design, it will find the highest $\alpha/\beta$ ratio that satisfies the $||F|| \leq 35$ constraint.
\begin{figure}[htbp]
\centering
\includegraphics[width=1\linewidth]{figure/controlPerf_WHconst_K35_2020.pdf}
\caption{The black line (with star points) shows the smallest $m$ that makes the weakly-hard constraint $(m, \text{K})$ schedulable for $\text{K}=20$, under different sampling periods. The blue line (with round points and red crosses) shows the control stability and control performance $J_c$ for corresponding sampling period. The control input constraint is $||F|| \leq 35$. The results show that considering weakly-hard constraints can significantly expand the feasible design space and improve performance.}
\label{fig:controlPerf_WHconst}
\end{figure}
We evaluate the system schedulability and control functionality for the sampling period $T_c$ from 55$ms$ to 300$ms$.
Besides control stability, we also evaluate the performance of the controller under different sampling period via simulation. Specifically, we generate 100 random initial states with random arrival time $t_0$ to represent disturbance and evaluate the cost among different sampling period. The cost $J_c$ is defined as the integral of the distance to the equilibrium:
\begin{equation}
J_c = \int_{t_0}^{t_0 + \Delta t} \theta_r^2(t) + \theta_p^2(t) dt
\end{equation}
The control performance of each sampling period is the average cost among these $100$ cases.
Fig.~\ref{fig:controlPerf_WHconst} shows the smallest $m$ that makes the weakly-hard constraint $(m, \text{K})$ schedulable for each $T_c$ and the corresponding control stability and performance. We can see that the controller is stable for period $T_c\in[90,225]$. There is no deadline miss for period larger than $210ms$, i.e, the feasible design space under traditional hard deadlines is only $[210,225]$. With weakly-hard constraints, the space is expanded to at least $[90,225]$, with many period choices between $[60,90]$ feasible as well.
For sampling period from about $130ms$ to $170ms$, the control performance is close to the best performance, while the feasible weakly-hard constraint varies from $(5,20)$ to $(1,20)$. The best performance is achieved when $T_c$ is at $132ms$, with weakly-hard constraint $(5, 20)$. When the sampling period gets larger than $225ms$, the performance deteriorates and eventually control becomes unstable. When the sampling periods gets smaller than $120ms$, there are more deadline misses and the control becomes worse as well.
We also evaluate this system with different control input constraints, e.g., from $||F|| \leq 25$ to $\leq 50$. The results demonstrate that a lower input constraint will lead to shorter sampling period for better performance. For instance, if $||F|| \leq 30$, the system is unstable when $T_c\geq 200\ ms$ (i.e., when no deadline misses). It is stable and reaches the optimal performance when $T_c\in [120,140]$, even though some deadlines are missed.
This case study shows that leveraging weakly-hard constraints can expand the feasible design space of controller sampling periods (from [210,225] to at least [90, 225]) and achieve better performance (best performance achieved at $132ms$).
Moreover, to the focus of this paper, the results show that it is critical to address weakly-hard systems with a cross-layer approach that considers both function and architecture layers. It confirms a major motivation for using weakly-hard constraints, i.e., to leverage the robustness at function layer (e.g., the robustness of control functions with respect to occasional deadline misses) for expanding the design/adaptation flexibility at architecture layer (e.g., the flexibility to explore more sampling periods or add more security monitoring tasks). |
2,877,628,090,842 | arxiv | \section{Introduction}
\subsection{Nonparametric Bayesian Shrinkage}
Bayesian analysis has long been a major methodological vehicle for
implementation of shrinkage ideas
in complex scenarios. There are two primary ways in which such
shrinkage is implemented. The
first is through use of prior distributions which shrink the unknowns
in some fashion---to prespecified locations or prespecified subspaces, depending
on the problem and type of prior. Thus an unknown normal mean could be
shrunk toward a specified prior mean;
a collection of unknown normal means could be shrunk toward the
hyperplane in which
the means are equal; and an unknown real function could be shrunk
toward the subspace of monotonic functions.
This is the Bayesian version of the type of shrinkage originating with
\citet{Stei56} and \citet{JaSt61}.
The second major Bayesian vehicle for shrinkage is Bayesian variable
selection, which sets some of the unknown parameters to zero. This is
often an overly drastic shrinkage, but is certainly not so in the
context of model selection, or in the context of
nonparametric function
estimation. In the latter setting, the unknown parameters that are set
to zero are typically coefficients of basis elements from a basis
representation of the function, and sparsity considerations strongly
encourage such shrinkage.
\begin{figure*}
\includegraphics{384f01.eps}
\caption{The radial velocity data (the $\times$'s)
for T Mon, and their fit to a fifth-order trigonometric polynomial.}
\label{TMon}
\end{figure*}
Both of these shrinkage concepts are herein utilized in nonparametric
function estimation with
dependent wavelets. The motivating application is to Cepheid variable
stars and is described
in the next subsection; the functions to be estimated
can have arbitrary shapes, but are quite smooth. It is to induce
sufficient smoothness
that will utilize both ty\-pes of shrinkage discussed above.
\subsection{The Astronomical Problem}
There is a class of
stars, called Cepheid variables, that pulsate with a regular and
distinctive periodic signature.
The stars actually grow larger and then smaller, and as a result their
luminosities vary
periodically along with their colors. Since there is a~physical
relationship between
the star's linear diameter, its luminosity, and its color, there are
actually two independent
periodically varying quantities.
A very interesting and useful property of these stars is that their
mean luminosities
are highly correlated with their pulsation period, in that the
shorter-period stars are less luminous than the longer-period ones. This
is very well approximated as a linear relation between the log of the period
and the log of the luminosity. As a consequence, if one knows the slope and
intercept of this relationship, and measures the period of
a Cepheid (which is trivial), one can infer the luminosity with quite
high precision.
This makes these stars very useful as ``standard candles,'' because
knowledge of
a star's luminosity as well as its observed brightness allows us to compute
the distance from the inverse square law. Knowing the distance to the individual
Cepheid also gives us the distance to the galaxy or cluster of stars in which
it is embedded. Thus, these stars are fundamental in setting the
distance scale of the
universe.
The most challenging feature of the problem statistically is that the
key photometry and radial velocity curves
for a star are unknown, and have no simple structure.
In \citet{barnesal03}, Fourier polynomials of finite (but unknown)
degree were used to represent
these two curves. For instance, Figure \ref{TMon} presents the data
concerning the radial velocity of the surface
of the star T Moncerotis, at various phases of the star's period
(the actual data are indicated by the $\times$'s) together with a fifth-order
trigonometric polynomial fit
to the data.
Because of the possibility of quite arbitrary shapes for the photometry
and velocity curves for Cepheid variable stars,
we instead desired to model the curves via much more flexible wavelet
decompositions.
\subsection{Computational Implementation}
Posterior inference in this setup
is formally equivalent to variable selection in a normal linear
regression problem with massively many candidate covariates.
Posterior simulation requires averaging and/or selection across
alternative models defined by the set of basis functions (wavelets)
which are included in the model.
In the context of normal-linear regression, common approaches are
guided search in the model space using the Occam's Window principle
(\citep{madiganraftery94}; \citep{rafteryal97});
Markov chain Monte Carlo simulation across the model space
(\citep{georgemcculloch97}; \citep{smithkohn96});
and importance sampling or Gibbs sampling based on analytic
approximations to the marginal posterior distribution on the model
indicator
(\citep{clydedesiparm96};
\citep{clydeparmvida98}).
See, for example,
\citet{clyde99}, Hoe\-ting et~al. (\citeyear{hoetingal99})
and \citet{clydegeorge04}
for reviews.
In this paper we introduce a scheme for fast posterior simulation
across the model space, marginalizing over the wavelet coefficients.
We use a computational strategy similar to that used by
\citet{georgemcculloch97} and \citet{smithkohn96}
to allow fast computation of marginal model probabilities when
considering models differing by only one wavelet basis function.
\section{Wavelet Representation}
Wavelet decomposition allows representation of any square integrable
function $f(x)$ as
\begin{equation}
\quad f(x) = \sum_{k\in Z} c_{J_0k} \phi_{J_0k}(x) +
\sum_{j \geq J_0} \sum_{k\in Z} d_{jk} \psi_{jk}(x).
\label{eqdwt}
\end{equation}
Here $\psi_{jk}(x) = 2^{j/2} \psi(2^j x -k)$
and $\phi_{jk}(x) = 2^{j/2} \cdot \phi(2^j x -k)$
are wavelets and scaling functions at level of detail $j$ and shift
$k$.
In the context of statistical modeling, (\ref{eqdwt}) allows for
inference about random functions by defining a
probability model for the coefficients
$\theta=(c_{J_0k},d_{jk},$ $j \geq J_0;~k \in\ZZ)$,
that is, (\ref{eqdwt}) provides a parameterization of a random function
$f$ in terms of the wavelet coefficients $\theta$.
See, for example,
\citet{vidamuel99} or
Ferreira and Lee (\citeyear{ferreiralee07}), Chapter~5,
for a review of wavelet representations
relevant for statistical modeling.\looseness=1
Perhaps the most common application of (\ref{eqdwt}) in
sta\-tistical modeling is to nonlinear regression where~$f(x)$
represents the unknown mean response $E(y|x)$ for an observation $y$ with
covariate $x$.
\citet{chipal97},
\citet{clydeparmvida98},
\citet{vida98},
\citet{semadenial04},
\citet{mahletal05},
\citet{WangWood06},
\citet{terBraak06} and
Abra\-movich, Angelini and De~Canditiis (\citeyear{abramovichal07}),
among many others,
discuss Bayesian inference in such models assuming equally spaced data,
that is,
covariate values $x_i$ are on a regular grid.
For equally spaced data the discrete wavelet transformation is
orthogonal. Together with assuming independent measurement errors and
a priori independent wavelet coefficients this leads to
posterior independence of the $d_{jk}$. Thus the problem essentially
reduces to a sequence of univariate problems, one for each wavelet
coefficient.
See, for example,
\citet{yaukohn99}
for a review.
Generalizations of wavelet techniques to non-equidistant (NES)
design impose additional con\-ceptual and computational burdens.
A reasonable approximation is to bin observations in equally spaced
bins and proceed as in the equally spaced case. If only few
observations are missing to complete an equally spaced grid, treating
these few as missing data leads to efficient implementations
(An\-toniadis,~Gr{\'e}goire and McKeague (\citeyear{antoniadisal94}); \citep{caibrown97}).
We propose instead an approach\break which does not depend on
posterior independence. Our approach includes informative dependent
priors with positive prior probabilities for vanishing wavelet
coefficients.\looseness=1
\vspace*{2pt}
\section{\texorpdfstring{Shrinkage of \textit{\lowercase{f\textup{(}x\textup{)}}}}{Shrinkage of $f(x)$}}
\vspace*{2pt}
\subsection{Shrinkage Toward a Smooth Subspace}
\label{secsubspace}
Because of the wavelet representation that will be used, a function space
prior can be defined by considering the function at the discrete points
$\{i/n, i=1,\ldots,n\}$, where $n=2^J$. Letting $f_i = f(i/n)$, consider
the difference process $d_i = f_i-f_{i-1}$.
A function space prior that ``shrinks toward\break smoothness'' can be defined by
imposing positive~cor\-relations on the $d_i$.
Specifically, let
$d=(d_1,\ldots,d_n)$, and define the prior to be $p(d) = N(0,\Delta)$ with
$\Delta_{ij} = \lambda\exp(-\beta|i-j|)$; that is, we assume a
multivariate normal prior with scale parameter $\lambda$ and log
correlations proportional to distance.
\begin{figure*}
\includegraphics{384f02.eps}
\caption{For $\beta=0.1$, the left panel plots simulations from the prior
process on the
unknown function conditioning on \textit{all} wavelet coefficients
included; the right panel shows for comparison prior simulations
conditional on setting those coefficients equal to zero which are
excluded by the universal wavelet thresholding rule with $\sqrt
{2n}\hat
\sig$ of \textit{Donoho and Johnstone} (\protect\citeyear{donohoejohnstone94}).}
\label{figprior}
\end{figure*}
Let $\Delta_{(11)}$ denote the left upper $(n-1)\times(n-1)$ submatrix
of $\Delta$ and partition $\Delta$ into
\[
\Delta= \left[
\matrix{
\Delta_{(11)} & \Delta_{(12)} \vspace*{2pt}\cr
\Delta_{(21)} & \Delta_{(22)}}
\right].
\]
Let $v=\operatorname{Var}(\sum_{i=1}^n d_i) =
\lambda\sum_{i=1}^n \sum_{j=1}^n \exp(-\beta|i-j|)$.
Assuming $f_0 \sim N(0,\lambda\sigma^2_0)$ we find
\[
p(f_0,\ldots,f_{n-1} | f_0=f_n) = N(0,\lambda V) ,
\]
with $V = A H_0 A'$,
\begin{eqnarray*}
A &=& \left[
\matrix{
1 & 0 & \cdots& 0\vspace*{2pt}\cr
1 & 1 & \cdots& 0\vspace*{2pt}\cr
\ldots\vspace*{2pt}\cr
1 & 1 & \cdots& 1}
\right],\\
H_0 &= &\left[
\matrix{
\sigma^2_0 & 0 \vspace*{2pt}\cr
0 & H}
\right]
\quad\mbox{and}\\
H &=& \Delta_{(11)}-\Delta_{(12)} \Delta_{(12)}'/v .
\end{eqnarray*}
In view of the normalization property, $\Vert\phi_{jk}\Vert=1$, scaling
coefficients at the highest level of detail $J$ are approximately
proportional to the represented function,
$c_{Jk} \approx2^{-J/2} f_k$.
Therefore the multivariate normal prior on $(f_0,\ldots,f_{n-1})$
implies
$
p(c_J) = N(0,\allowbreak r_J\cdot\lambda V)
$
where $r_J = 2^{-J}$. Following common practice in the use of wavelet
decomposition, we will ignore\vadjust{\goodbreak} the proportionality constant $r_J$ and
assume
\[
p(c_J) = N(0,\lambda V).
\]
As long as we also drop $r_J$ in the reconstruction of $f(x)$, ignoring
the proportionality constant will leave the final inference unchanged.
The prior $p(c_J) = N(0,\lambda V)$ implies a dependent multivariate normal
prior for the vector of all wavelet coefficients $d=(c_{J_0k},d_{jk},
j=J_0,\ldots,J, k=0,\ldots,\break 2^j-1)$
\begin{equation}
p(d | \gamma=1) = N(0,\lambda\Lambda).
\label{eq1}
\end{equation}
In principle $\Lambda$ can be found by explicitly computing the linear
operator of the wavelet decomposition. But from a computational point
of view this is unnecessary and undesirable. Instead
\citet{vannuccicorradi99} show how $\Lambda$ can be derived from $V$
as a bivariate wavelet decomposition of $V$.
\subsection{Shrinkage Through Wavelet Sparsity}
\label{secsparsity}
One of the important advantages of wavelet bases over alternative bases for
$L^2$ functions is the parsimony property of wavelet representations.
Reasonably regular functions are well approximated with only few
nonzero wavelet coefficients. Therefore\break ``shrinkage toward smoothness''
can also be
induced by setting many of
the wavelet coefficients to be zero. We thus assume positive prior
probability for vanishing wavelet coefficients.
Let $\gamma=(\gamma_1,\ldots,\gamma_l)$
denote the vector of indices of nonzero wavelet coefficients,
that is, $d_{jk}=0$ iff $(jk) \notin\gamma$.
We define a prior distribution on $\gamma$ with\vadjust{\goodbreak} geometrically
decreasing probability for nonzero wave\-let coefficients in higher
levels of detail $j$:
\[
\operatorname{Pr}(d_{jk}=0) = 1-\alpha^{j+1}.
\]
See, for example, \citet{abramovichal98} for a discussion of the
choice of $\alpha$.
We write
$\theta_\gamma$ for the subvector of nonzero wavelet coefficients
$d_{jk}$,
and we use $\gamma=1$ for the full model which includes all
coefficients $\gamma=((jk), j=J_0,\ldots,J$ and $k=0,\ldots,2^j-1)$.
The prior $p(\theta_\gamma\vert\gamma)$ for the wave\-let
coefficients under
model $\gamma$ is implied from~(\ref{eq1}) by conditioning the
multivariate normal on $\theta_h=0$, \mbox{$h \notin\gamma$}.
Let $\Omega=V^{-1}$ and write $\Omega_{(\gamma)}$ for the submatrix
with rows and columns $(\gamma_1,\ldots,\gamma_l)$.
Then
\begin{equation}
p(\theta_\gamma\vert \gamma) = N\bigl(0,\lambda\Omega_{(\gamma
)}^{-1}\bigr) =
N(0,\lambda\Lambda) .
\label{eqpr-g}
\end{equation}
We use $\Lambda$ to generically denote $\Omega_{(\gamma)}^{-1}$,
suppressing the dependence on $\gamma$ to simplify notation.
\subsection{Illustration of the Shrinkage Effects}
Figures \ref{figprior} and \ref{figprior9} demonstrate the ``shrinkage
toward smoothness'' behavior of the priors in Sections~\ref{secsubspace}
and~\ref{secsparsity}. The figures give realizations from the priors
specified in
the two subsections.
Figure~\ref{figprior} utilizes $\beta=0.1$ from the prior in Section~\ref{secsubspace}
and Figure~\ref{figprior9} utilizes $\beta=0.9$. The smaller $\beta$
induces much more
dependence, clearly resulting in smoother functions.
\begin{figure*}
\includegraphics{384f03.eps}
\caption{
Prior simulations as in Figure
\protect\ref{figprior}, but using $\beta=0.9$ (very little dependence).}
\label{figprior9}
\end{figure*}
The left panel of each figure is generated from use of only the prior
in Section \ref{secsubspace},
that is, all the wavelet coefficients are kept. In contrast, the right
panels of each figure show
what happens when many of the wavelet coefficients are set to zero.
(For simplicity, these were
produced using a standard wavelet thresholding rule.) Clearly, setting
many wavelet
coefficients to zero does seem to result in considerable additional
shrinkage toward smoothness.
\vspace*{3pt}\section{Posterior Simulation}\vspace*{3pt}
We implement posterior inference using Markov chain Monte Carlo
simulation.
Marginalizing over~$\theta_\gamma$, we use the posterior
probabilities $p(\gamma\vert y)$ to define a~Metropolis--Hastings scheme
which proposes moves in the model space by adding or deleting
one wavelet basis function at a time.
The computational effort of the proposed scheme is comparable to that of
\citet{georgemcculloch97} and
\citet{smithkohn96}, who suggest schemes based
on algorithms by
\citet{chambers71}
and
(\citeyear{dongarraal79})
which allow fast updating of a Choleski
decomposition of the cross-product matrix $X'X$.
The algorithms proposed by
\citet{georgemcculloch97}
and
\citet{smithkohn96}
allow computation of marginal posterior probabilities with $O(q^2)$
basic operations, whe\-re~$q$ is the number of covariates (basis
functions) included in the model.
We describe a similar efficient updating algorithm in a form suitable
for the wavelet regression problem.\looseness=1
\textit{Notation}.
Let $A_{ij}$ be the element in the $i$th row and $j$th
column of a matrix $A$, with $A_i$ being its $i$th column vector.
For a vector $\gamma=(\gamma_1,\ldots,\gamma_l)$ we denote with
$A_\gamma$ the submatrix consisting of \textit{columns}
$(\gamma_1,\ldots,\gamma_l)$, with $A_{(\gamma)}$ the submatrix
consisting of \textit{columns and rows} $(\gamma_1,\ldots,\gamma
_l)$, and
with $A_{(-\gamma)}$ the submatrix with rows and columns
$\gamma=(\gamma_1,\ldots,\gamma_l)$ removed.
Let $x_i,y_i$, $i=1,\ldots,N$, denote the observed data.
Let $h=1,\ldots,2^J$ index the wavelet coefficients
$d=(c_{J_0k},d_{jk})$ and
let $X$ denote the design matrix
\[
X_{ih} =
\cases{
\psi_{jk}(x_i) & $\mbox{for } h=2^{J_0}+1,\ldots,n,$\vspace*{3pt}\cr
\phi_{J_0k}(x_i) & $\mbox{for } h=1,\ldots,2^{J_0} ,$}
\]
where $(jk)$ are the wavelet indices corresponding to the $h$th
element in the vector $d$ of wavelet coefficients.
\textit{Likelihood.}
For a given model $\gamma$ the wavelet decomposition of the unknown
velocity curve $f$ implies a likelihood
\begin{equation}
y_i \vert \theta,\gamma\iid N(X_\gamma\theta_\gamma, S),\quad
i=1,\ldots,N ,
\label{eql}
\end{equation}
where $S = \operatorname{diag}(\sigma_i^2)$ with known variances $\sigma^2_i$,
$i=1,\ldots,N$.
\textit{Posterior}.
Together with prior (\ref{eqpr-g}) the likelihood implies a
multivariate normal posterior
$p(\theta_\gamma\vert y,\gamma) = N(\mu,\Sigma)$ with
\begin{eqnarray*}
\Sigma^{-1}& =&
\underbrace{(X_\gamma)'S^{-1}X_\gamma}_{Q^\gamma} +
1/\lambda \Omega_{(\gamma)}
\quad\mbox{and}\\[3pt]
\mu&=& \Sigma\cdot\underbrace{(X^\gamma)'S^{-1}y}_{v^\gamma}.
\end{eqnarray*}
Again, to simplify notation we suppress the dependence on $\gamma$ in
$\mu$ and $\Sigma$.
\subsection{Down Move}\vspace*{1pt}
$\!\!$Assume $\gamma\,{=}\,(\gamma_1,\ldots,\gamma_l)$ and consider a move
``down'' to the submodel $\gamma^*=(\gamma_1,\ldots,\gamma_{l-1})$.
Partition $\Sigma$ into
\[
\Sigma=
\left[
\matrix{
\Sigma_{(-l)} & \tS_l \vspace*{2pt}\cr
\tS_l' & \Sigma_{ll}}
\right]
\]
and similarly $\mu=(\mu_{(-l)},\mu_l)$.
Then
\[
p(\theta_{\gamma^*} \vert y,\gamma^*) = N(\mu^*,\Sigma^*) ,
\]
with $\Sigma^* = \Sigma_{(-l)} - \tS_l \Sigma_{ll}^{-1} \tS_l'$
and $\mu^* = \mu_{(-l)} +\break \tS\Sigma_{ll}^{-1}(-\mu_l)$.
Similarly,
$\Lambda^* = \Lambda_{(-l)} - \tL\Lambda_{ll}^{-1}\tL_l'$.
The corresponding ratio of marginal probabilities is
\[
\frac{p(y \vert\gamma^*)}{p(y \vert\gamma)} =
\biggl(\frac{\lambda \Lambda_{ll}}{\Sigma_{ll}}\biggr)^{1/2}
e^{-(1/2) \mu_l^2/\Sigma_{ll}} .
\]
This expression is easily verified using the candidate formula
$p(y \vert\gamma) =
p(\theta_\gamma\vert\gamma) p(y \vert\theta_\gamma,\gamma)/
p(\theta_\gamma\vert y,\gamma)$ and\break substituting $\theta_\gamma=0$.
\begin{figure*}[t!]
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{384f04a.eps}
& \includegraphics{384f04b.eps}\\
\footnotesize{(a) $\alpha=0.5$, $\beta=0.1$} & \footnotesize{(b) $\alpha=0.5$, $\beta=0.9$}\\[6pt]
\includegraphics{384f04c.eps}
& \includegraphics{384f04d.eps}\\
\footnotesize{(c) $\alpha=0.7$, $\beta=0.1$} & \footnotesize{(d) $\alpha=0.7$, $\beta=0.9$}
\end{tabular}
\caption{Posterior inference for T Moncerotis. In all four panels, the thick
smooth line shows the posterior mean
curve. The gray shaded margins show central 50\% (light gray) and central
90\% (dark gray) intervals. The points are the observed data points,
with little error bars showing 2 standard deviations for the
measurement error. Panel \textup{(a)} shows inference under
$\beta=0.1$ and $\alpha=0.5$.
Panels \textup{(b)} through \textup{(d)} show posterior inference using $\beta=0.9$ (\textup{b}
and \textup{d}) and $\alpha=0.7$ (\textup{c} and \textup{d}).
Fixing $\beta=0.9$ essentially assumes independence of the $d_i$ and
implies less
smoothing; setting $\alpha=0.7$ greatly decreases the number of
wavelet coefficients set to zero.}
\label{fig4}
\end{figure*}
\vspace*{1pt}\subsection{Up Move}\vspace*{1pt}
Consider a move from $\gamma$ to $\gamma^*=(\gamma^*_1,\gamma)$.
Denote with $(\mu,\Sigma)$ and $\Lambda$ the posterior and prior
moments under the (current) model $\gamma$:
\[
p(\theta_\gamma\vert \gamma,y) = N(\mu,\Sigma) \quad\mbox{and}\quad
p(\theta_\gamma\vert \gamma) = N(0,\lambda\Lambda).
\]
Similarly, let $(\mu^*,\Sigma^*)$ and $\Lambda^*$ denote the posterior
and prior moments under the (proposed) model $\gamma^*$:
\begin{eqnarray*}
p(\theta_{\gamma^*} \vert \gamma^*,y) &=& N(\mu^*,\Sigma^*)\quad
\mbox{and}\\
p(\theta_{\gamma^*} \vert \gamma^*) &=& N(0,\lambda\Lambda^*).
\end{eqnarray*}
For posterior simulation we use a lower triangular Choleski
decomposition of the posterior variance/\break covariance matrix,
$T T'=\Sigma$ and $T^{*\prime} T^* = \Sigma^*$.
The new moments $\mu^*,\Sigma^*$ and $\Lambda^*$ and the Choleski
decomposition $T^*$ are computed using the
following expressions.
Let
$Q^*=(X^{\gamma^*})'S^{-1}X^{\gamma^*},
\Omega^*=\Omega_{(\gamma^*)},
Q=(X^{\gamma})'\cdot\break S^{-1} X^{\gamma}$ and
$\Omega=\Omega_{(\gamma)}$
and partition
\[
Q^* =
\left[
\matrix{
Q^*_{11} & \tQ^{*\prime}_1 \vspace*{2pt}\cr
\tQ^{*}_1 & Q}
\right]
\quad\mbox{and}\quad
\Omega^* =
\left[
\matrix{
\Omega^*_{11} & \tO_1^{*\prime} \vspace*{2pt}\cr
\tO_1^* & \Omega}
\right].
\]
Let $b=\tQ^*_1 + 1/\lambda\tO_1^*$,
$h=\Sigma b$,
$c=\tQ_{11}^* + 1/\lambda \Omega^*_{11}$,
$b_0=\tO_1^*$,
$h_0 = \Lambda\tO_1^*$ and
$c_0 = \Omega^*_{11}$.
Then
\begin{eqnarray*}
\Sigma^*& =&
\left[
\matrix{
0 & 0\vspace*{2pt}\cr
0 & \Sigma}
\right]
+ \frac1{c-b'h}
\left[
\matrix{
1 & -h'\vspace*{2pt}\cr
-h & h h'}
\right]
\quad\mbox{and}
\end{eqnarray*}
\begin{eqnarray*}
\hspace{-8pt}
\Lambda^* &=&
\left[
\matrix{
0 & 0\vspace*{2pt}\cr
0 & \Lambda}\right]
+ \frac1{c_0-b_0'h_0}
\left[
\matrix{
1 & -h_0'\vspace*{2pt}\cr
-h_0 & h_0 h_0'}
\right],
\\[3pt]
\hspace{-8pt}
\mu^* &=&
\pmatrix{
0\vspace*{2pt}\cr \mu}
+
(c-b'h) \Sigma^*_1 \Sigma^{*\prime}_1 v^{(\gamma^*)},
\end{eqnarray*}
and
$T^*$ is obtained by augmenting $T$ with a new first column
$w=\Sigma_1^*/\sqrt{\Sigma_{11}^*}$ to
\[
T^* =
\left[
\matrix{
& 0\vspace*{2pt}\cr
w & T
}
\right].
\]
The corresponding ratio of marginal probabilities is, by symmetry to
the down move,
\[
\frac{p(y \vert\gamma)}{p(y \vert\gamma^*)} =
\biggl( \frac{\lambda\Lambda^*_{11}}{\Sigma^*_{11}}\biggr)^{1/2}
e^{-(1/2) \mu_1^{*2}/\Sigma^*_{11}} .
\]
\section{Example}
We apply the above methodology to the data for the star T Moncerotis,
as shown
in Figure \ref{TMon}, for the choices $\beta=0.1$ (strong dependence of the $d_i$)
and $\alpha=0.5$ (inducing a moderate level of sparsity). The resulting
nonparametric posterior is difficult
to summarize; some features of this posterior are presented in Figure
\ref{fig4}(a).
It is, of course, one of the strengths of the Bayesian approach to shrinkage
that uncertainty in the shrinkage estimate [the posterior mean of
$f(x)$, given by the
thick center line in Figure \ref{fig4}(a)] can also be given. This is
crucial in
characterizing the (considerable) uncertainty in the eventual estimate of
distance to the star (see \citep{barnesal03}).
Figure \ref{fig4} also indicates the effect on the T Moncerotis data
of each of the shrinkage priors in Sections \ref{secsubspace} and~\ref{secsparsity}.
Panel (b) shows the effect of the prior
in Section \ref{secsubspace}; setting $\beta=0.9$ effectively
makes the $d_i$ independent. Panel (c) shows the effect of the
prior in Section \ref{secsparsity}; setting $\alpha=0.7$ greatly decreases the
number of
wavelet coefficients set to zero. In both cases, the
posterior functions appear to be unreasonably rough
and the uncertainty in the shrinkage estimate appears
to be unreasonably large. Pa\-nel~(d), which effectively uses neither of the
shrinkage techniques, is especially unsatisfactory.
\section*{Acknowledgments}
This research was supported in part by NSF Grants DMS-01-03265,
DMS-06-35449 and
DMS-07-57549-001. We are grateful to Thomas Barnes for providing us
with the data analyzed herein.
|
2,877,628,090,843 | arxiv | \section{Introduction}
\label{sec:intro}
Speech-to-text translation (ST) has been traditionally addressed by pipeline approaches
involving several components \cite{iwsltprec_2019}.
The most important blocks are the speech recognition (ASR),
which converts the input audio into its transcript,
and the neural machine translation (NMT),
which translates the transcript into the target language.
Direct ST models \cite{berard_2016,weiss2017sequence} recently gained attention as an alternative approach, thanks to their appealing promises to overcome some of the pipeline systems' problems, such as error propagation and loss of information present in the audio (prosody in particular).
Both pipeline and direct solutions, however, can be significantly affected by mismatches in the segmentation of the input between training and test data.
On one side, the two solutions involve the use of training data segmented at sentence level. This, for instance, holds for the parallel corpora normally used to train the NMT component of the pipeline approach, as well as for all the available ST corpora used for
direct ST
training.
On the other side, at inference time both solutions will be exposed to data segmented according to criteria that look at properties of the audio input rather than at linguistic notions like sentence well-formedness. The most widespread approach consists in fact in using voice activity detection (VAD) to split the audio stream into chunks,
which are input to the ST system.
In particular, VAD systems determine
whether a given short (usually 10-30 ms) audio segment
actually contains speech,
and this information is used in the context of ST for two purposes:
\textit{i)} dividing the audio stream into segments containing uninterrupted speech;
\textit{ii)} filtering out audio
segments containing other sounds.
Since VAD is solely based on the alternation between human voice, silences
and other sounds, the resulting splits might not correspond to well-formed sentences but to fragments of one or more sentences.
The impact of feeding an ST model trained on ``clean'' data with sub-optimal, not linguistically-motivated segmentations varies according to the characteristics of the VAD employed and its settings. Very aggressive settings reduce the generation of long (cross-sentential) segments,
which are difficult to handle by neural models that are typically very sensitive to input length.
On the downside, they produce
short (sub-sentential) segments that might not provide enough context for
proper translation.
To address this problem, pipeline systems include an additional component that
re-segments the ASR output to
provide the NMT
with well-formed sentences \cite{matusov_segm, oda-etal-2014-optimizing, Cho2017}.
Since this solution is not possible for direct
ST,
where
the two steps are not decoupled,
researchers have worked on
alternative audio segmentation techniques.
In the 2019 IWSLT offline ST task \cite{iwsltprec_2019}, for instance,
the best direct ST system \cite{potapczyk_tomasz_2019} had one of its key features in the
segmentation method.
Instead of working on the segmentation algorithm, in this paper we aim to make our direct ST models more robust to VAD-segmented data.
To train them on a data distribution more similar to the one fed at inference time, we generate an artificial dataset by
randomly
re-segmenting clean (i.e. sentence-based) ST data.
Then, we experiment with two approaches: \textit{i)}
fine-tuning on
the new dataset;
\textit{ii)}
improving our direct ST model with the capability to look back and attend to the preceding segment as contextual information.
Our experiments
show that the proposed context-based solution
effectively handles the segmentation of different VAD systems and configurations,
reducing the drop in translation quality caused by
segmentation mismatches in the training and test data by up to 55\%.
\section{Context-aware ST}
The idea of exploiting contextual information to improve translation
has been successfully applied in NMT
\pp{\cite{wang-etal-2017-exploiting-cross,zhang-etal-2018-improving,bawden-etal-2018-evaluating,kim-etal-2019-document}}.
In our use case, unlike \cite{wang-etal-2017-exploiting-cross},
we are interested only in modeling short-range cross-segment dependencies
\mn{to cope with the sub-optimal}
breaks introduced by
VAD
segmentation.
We hence consider as context only the segment
immediately preceding the one to be translated, leaving
out of our study hierarchical
approaches modeling the whole document as context.
Moreover, while in document-level NMT the best approach is to use the source side of the sentence(s) as contextual information,
in the ST scenario it is not trivial to understand which side is best.
\pp{On one hand,}
audio source avoids the error propagation and exposure bias introduced by using as context the translations generated at inference time.
\pp{On the other,} these problems are balanced by the easiness of extracting information from text rather than from audio \cite{instance_based}.
In this work, we study both options.
To integrate context information into the model, we explore the two solutions that gave the best results for NMT \cite{kim-etal-2019-document}.
\mg{They} respectively use sequential \cite{zhang-etal-2018-improving} and parallel \cite{bawden-etal-2018-evaluating} decoders.
We also experimented with the integration of
context information in the encoder \cite{zhang-etal-2018-improving}, but
the trainings were either very unstable (when using audio as context)
or ineffective, \pp{eventually} leading to worse results. For this reason, we do not consider
this type of integration in the rest of the paper.
Finally, supported by previous findings \cite{kim-etal-2019-document},
we
\pp{neither}
investigate the concatenation of the context with the current input \cite{agrawal_ctx}, \pp{nor}
the combination of encoded representations of
\mn{the two}
\cite{voita-etal-2018-context}.
\pp{Our base model is an adaptation of Transformer \cite{transformer}:
\mg{its} encoder is enhanced
to take into account the characteristics of speech input
by means of two 2D convolutional layers and a logarithmic distance penalty in its self-attention
layers \mnn{\cite{digangi:interspeech19}}.}
Both the sequential and the parallel decoder use a multi-encoder approach,
with an additional encoder dedicated to the context information.
\mg{However,} they differ in the way this information is integrated into the base model.
The context encoder is composed of Transformer encoder layers,
but its input depends on the modality of the segment used as context, i.e. text or audio.
When we use the generated translations as context,
its tokens are converted into vectors with
\textit{word embeddings} (namely, we re-use the decoder embeddings), summed with \textit{positional encoding} and then provided to the encoder Transformer layers.
When we use the audio as context, the input audio features are first processed by the encoder of the base model and then
passed to the context encoder \cite{instance_based}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.25\textwidth]{decoder_seq.jpeg}
\caption{Sequential context integration.}
\label{fig:seq_arch}
\end{figure}
\noindent \textbf{Sequential} (Figure \ref{fig:seq_arch}). In each decoder Transformer layer,
an additional multi-head cross-attention sub-layer is introduced.
It queries the output $C_{out}$ of the context encoder
using the output $H_i$ of the $i$-th encoder cross-attention sub-layer.
The result $S_i$ of this operation is combined with $H_i$ using a position-wise gating mechanism, before being fed to the feed-forward network $\text{FFN}_i$.
Hence, the output of the $i$-th decoder layer $D_i$ is:
\begin{equation}
\lambda_i = \sigma (W_{hi} H_i + W_{si} S_i)
\label{gating_lambgda_eq}
\end{equation}
\begin{equation}
D_i = \text{FFN}_i (\lambda_i H_i + (1 - \lambda_i) S_i)
\label{decoder_gating_eq}
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{decoder_par.jpeg}
\caption{Parallel context integration.}
\label{fig:par_arch}
\end{figure}
\noindent \textbf{Parallel} (Figure \ref{fig:par_arch}). In each decoder Transformer layer, the output of the self-attention sub-layer is used as
query for both the encoder cross-attention and the context cross-attention defined in the same way as in the previous case.
The outputs of these two sub-layers are then combined using the
position-wise gating mechanism described in Eq.(\ref{decoder_gating_eq}).
To
avoid over-relying on
the context,
we add a regularization on the context gate.
Our regularization is slightly different from the one proposed by \cite{li2019regularized}:
we always penalize the context information,
so that the model will
use it only when it is strictly needed.
With the regularization factor, the resulting loss is:
\begin{equation}
\mathcal{L'} = \mathcal{L} + \alpha \sum_{i=0}^{N_d}(1 - \lambda_i)
\label{regularization_gate_eq}
\end{equation}
\section{\pp{Experimental settings}}
\subsection{\pp{Clean and artificial data}}
\pp{Our base models are trained and evaluated on English-German data drawn from the MuST-C corpus, the largest ST dataset currently available \cite{mustc}. MuST-C comprises 234K samples (corresponding to about 408 hours of speech) divided into training (229K), validation (1.5K) and test sets (3.5K).}
\label{sec_data}
\pp{To cope with segmentation mismatches between the clean data used for training and the VAD-processed ones handled at inference time, we generate an automatic re-segmentation of the MuST-C training and validation set.}
The re-segmentation starts by picking a random (with uniform distribution) \textit{split word} for each sample in the original English
\pp{transcripts.} Each fragment spanning from a \textit{split word} to the word before the next \textit{split word} becomes a segment of the new training set and the preceding fragment becomes its context. We extract the audio corresponding to each resulting transcript by leveraging word alignments computed with Gentle.\footnote{https://github.com/lowerquality/gentle/} Then, we retrieve the corresponding translations using word alignments generated with fast\_align \cite{dyer-etal-2013-simple}. In case of missing alignments (either with the audio or with the translation), the sample is discarded. The resulting training dataset contains 225K samples (4K less than the original), while
the validation set size is almost unchanged.
A manual check on a sample of the produced aligned segments revealed that about 96\% of them are acceptable. The most frequently observed issue is that some translations contain 1-2 words more than the optimum, mostly due to the lack of some word alignments and to word-reordering. This leads to the presence of overlapping words between the context and the target German references in 25\% of the samples. In early experiments, this caused model instability at inference time because models learnt to copy the final context words, up to producing nonsensical sequences of repeated tokens. We solved the issue by filtering out the overlapping words from the context.
\subsection{VAD and segmentation}
As we want our systems to be robust to different VAD
outputs, we test our models
on two different open source VAD tools:
LIUM \cite{meigner2010lium}
and WebRTC's VAD.\footnote{https://webrtc.org/. We use the open-source Python interface https://github.com/wiseman/py-webrtcvad}
For WebRTC we tested all the possible configurations, varying the \textit{frame size}
(allowed values are $10$ms, $20$ms and $30$ms) and the \textit{aggressiveness}
(ranging from $0$ to $3$, extremes included).
\mg{We discarded those}
producing either too long ($>60$s) or too many segments
($>5,100$, i.e. twice the segments of the original sentence-based segmentation \pp{of the MuST-C test set}).
In this way,
we ended up with three configurations,
whose characteristics are described in Table \ref{tab:vad_summary}.
Overall, the segments produced by WebRTC have much higher variance in their length (ranging from $0.40$s to $58.62$s) compared to LIUM (from $2.50$s to $18.63$s) and are significantly more ($>3,500$ vs $2,725$).
As anticipated in $\S\ref{sec:intro}$, this can affect the final performance of neural ST models, for which handling very long/short segments is
difficult.
However, from a qualitative standpoint,
a manual inspection of 50 samples showed that the split times selected by LIUM are less accurate than those selected by WebRTC: while the former
often splits fluent speech, the latter always selects
\mn{positions in which}
the speaker is silent.
\begin{table}[htbp]
\caption{Statistics for different segmentations of the MuST-C test set. ``Man.'' refers to the original sentence-based segmentation.}
\label{tab:vad_summary}
\centering
\begin{tabular}{l|c|c|ccc}
\toprule
\textbf{System} & Man. & LIUM & \multicolumn{3}{c}{WebRTC} \\
\textbf{Frame size} & ~~~ & ~~~ & 30ms & 20ms & 20ms \\
\textbf{Aggress.} & ~~~ & ~~~ & 3 & 2 & 3 \\
\midrule
\textbf{\% filt. audio} & 14.66 & 0.00 & 11.27 & 9.53 & 15.58 \\
\textbf{Num. segm.} & 2,574 & 2,725 & 3,714 & 3,506 & 5,005 \\
\textbf{Max len. (s)} & 51.97 & 18.63 & 48.84 & 58.62 & 46.76 \\
\textbf{Min len. (s)} & 0.05 & 2.50 & 0.60 & 0.40 & 0.40 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Training settings}
\pp{All our models are optimized}
with label smoothed cross entropy \cite{szegedy2016rethinking}
using the Adam optimizer \cite{adam} with a learning rate starting
from
$3\cdot10^{-4}$,
increasing linearly up to $5\cdot10^{-4}$ in the first $5,000$ steps and then decaying
with inverse square root policy.
The overall batch size was $512$ \textit{(audio, translation)} pairs.
We used the \textit{BIG} configuration from \mg{\cite{digangi:interspeech19}}
regarding all layers' hidden sizes. The number of context encoder layers $N_c$ is set to $1$, as \cite{zhang-etal-2018-improving} shows that this leads to the best results. Since~\cite{kim-etal-2019-document} has demonstrated that poorly regularized systems can
lead to ambiguous results when
\mn{integrating context}, we used $0.2$ dropout
and \textit{SpecAugment} \cite{Park_2019} to prevent this issue.
We performed preliminary experiments on a baseline model
(\textit{BASE\_MUSTC})
with $8$ encoder layers $N_e$ and $6$ decoder layers $N_d$ trained on the MuST-C En-De training set.
Since models using the generated translations as context are affected
by exposure bias,
we wanted to test our solution also in more realistic conditions,
with a stronger baseline model trained in rich data conditions.
This model
(\textit{BASE\_ALL})
was trained with $N_e$ set to $11$ and $N_d$ to $4$, on all the data available for the IWSLT 2020 evaluation campaign,\footnote{http://iwslt.org/doku.php?id=offline\_speech\_translation}
with knowledge distillation from an MT model and synthetic data generated translating the transcripts of ASR corpora.
Its training involves a pre-training on the synthetic data,
a fine-tuning on the data having ground-truth translations and
a second fine-tuning using label-smoothed cross entropy instead of knowledge distillation \mg{\cite{gaido-etal-2020-end}}.
All the context-aware models are initialized with the corresponding baseline model trained on sentence-segmented data.
We experimented with freezing all the pre-trained parameters as in~\cite{zhang-etal-2018-improving},
but freezing the decoder weights turned out to be harmful.
If freezed, decoder's layers are not able to adapt to the new inputs \mn{(with different segmentation)} and this slows down convergence
and leads to worse results.
\mg{We hence freeze only the encoder.}
\mg{Our code is based on fairseq \cite{ott-etal-2019-fairseq} and is available at https://github.com/mgaido91/FBK-fairseq-ST.}
\mn{Textual data were}
pre-processed with tokenization and punctuation normalization
\mn{performed using}
Moses \cite{koehn-etal-2007-moses}, and
\mn{were}
segmented with $8,000$ BPE merge rules \cite{sennrich2015neural}.
\mn{For the audio,}
we applied $40$ Mel filters
with window size of $25$ms and stride of $10$ms, \mn{performing} speaker normalization with XNMT~\cite{neubig-etal-2018-xnmt}.
To avoid out-of-memory errors, we excluded from the training set the audio segments longer than $20$ seconds.
\mn{In all cases, evaluation is performed on the best model according to the loss on the validation set.}
\mn{The metrics used are BLEU \cite{papineni2002bleu} and TER \cite{Snover:06}}\pp{, computed against the reference translations in the MuST-C En-De test set}.
\section{Results}
We performed preliminary experiments
\mn{with}
\textit{BASE\_MUSTC}
\mn{(scoring 21.08 BLEU on the original MuST-C En-De test set)}
to compare the context integration techniques and select the most suitable \mn{one} for ST.
We then compared the fine-tuning with the context-aware models using the stronger baseline model \textit{BASE\_ALL} (scoring 27.55 BLEU on the original test set).
\begin{table}[htbp]
\caption{\pp{Evaluation results on the VAD-segmented
test set. Notes: SRC=audio as context; TGT=generated translation as context; SEQ=sequential; PAR=parallel.}}
\label{tab:results_prelim}
\centering
\begin{tabular}{l|c|ccc}
\toprule
~~~ & \textbf{LIUM} & \multicolumn{3}{c}{\textbf{WebRTC}} \\
~~~ & ~~~ & \textbf{3, 30ms} & \textbf{2, 20ms} & \textbf{3, 20ms} \\
\midrule
BASE\_MUSTC & 17.32 & 17.82 & 17.75 & 16.31\\\hline
SRC SEQ & 19.08 & 18.81 & 18.00 & 17.42 \\
SRC PAR & 19.25 & 18.90 & 18.25 & 17.30 \\
TGT SEQ & 19.57 & \textbf{19.21} & 18.81 & \textbf{17.60} \\
TGT PAR & \textbf{20.01} & 18.98 & \textbf{18.82} & 17.32 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[t]
\caption{Comparison between base model, fine-tuning and context-aware models.}
\label{tab:results}
\centering
\begin{tabular}{l|cc|cccccc}
\toprule
~~~ & \multicolumn{2}{c}{\textbf{LIUM}} & \multicolumn{6}{c}{\textbf{WebRTC}} \\
~~~ & ~~~ & ~~~ & \multicolumn{2}{c}{\textbf{AGG=3, FS=30ms}} & \multicolumn{2}{c}{\textbf{AGG=2, FS=20ms}} & \multicolumn{2}{c}{\textbf{AGG=3, FS=20ms}} \\
\midrule
~~~ & BLEU ($\uparrow$) & TER ($\downarrow$) & BLEU ($\uparrow$) & TER ($\downarrow$) & BLEU ($\uparrow$) & TER ($\downarrow$) & BLEU ($\uparrow$) & TER ($\downarrow$) \\
\midrule
BASE\_ALL & 19.66 & 76.57 & 22.07 & 67.08 & 21.98 & 66.83 & 19.59 & 72.62 \\
\midrule
FINE-TUNE & 22.48 & 64.21 & 23.48 & 60.03 & \textbf{23.40} & 61.54 & 21.35 & 63.90 \\
\midrule
TGT SEQ & 23.18 & \textbf{58.60} & 22.85 & \textbf{58.49} & 22.59 & \textbf{59.79} & 21.11 & \textbf{60.51} \\
\hspace{2mm} + REG & 23.88 & 58.81 & \textbf{23.61} & 58.57 & 23.15 & 60.36 & 21.88 & 60.97 \\
TGT PAR & 23.77 & 59.02 & 23.34 & 58.94 & 22.91 & 60.09 & 21.75 & 60.77 \\
\hspace{2mm} + REG & \textbf{23.91} & 58.95 & 23.51 & 58.64 & \textbf{23.40} & 59.95 & \textbf{22.03} & 60.83 \\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Context information and integration}
\label{sec_res_ctx_type}
\mg{Table \ref{tab:results_prelim} shows that}
all the tested approaches outperform the baseline
\mg{on VAD-segmented data}
with a margin that ranges from 0.25 to 2.69 BLEU points.
This
\mg{indicates} that the context is useful to mitigate the effect of
\mn{VAD-based}
segmentation.
On LIUM, our models
\pp{achieve}
the highest score (\textit{TGT PAR}, 20.01 BLEU) and \mn{the largest} gain over the baseline;
on WebRTC the improvements are significant
\mn{but smaller.}
We argue that the reason lies in the different characteristics of the two tools.
\mn{The split positions selected by LIUM do not always correspond to actual pauses in the audio, which prevents the baseline model from disposing of all the information necessary
for translation.
This information, instead, is available to the context-aware models as they can access the previous segment.}
WebRTC, instead, produces very long/short segments, whose effect on context-aware models is limited:
the contribution of adding the previous segment is low both in case of very long segments, as only the first part is influenced by it, and in case of very short ones, as having a short segment as context means adding little information.
We also experimented with including manually-segmented data,
but it was not beneficial for any
\mn{of our models}.
\mn{Looking at the context modality (text vs audio), we observe that supplying the previously generated translation (TGT*) yields higher BLEU scores than supplying its corresponding audio (SRC*) with both the integration types (*SEQ and *PAR).}
\mn{This suggests that the audio representation produced by current ST models is less suitable than text to extract useful content information to support traslation.
In light of these observations, we decided to proceed with \textit{TGT SEQ} and \textit{TGT PAR} in the following experiments with the stronger \textit{BASE\_ALL} model.}
\subsection{Context vs fine-tuning}
In this section, we compare the performance of the fine-tuning
and the context-aware solutions.
In this way, we can disentangle the benefits produced by the context and those due to
the \mn{use of}
artificial training data.
The results \mn{in Table \ref{tab:results}} show that: \mn{\textit{i)}}
fine-tuning on the artificial data produces significant gains over \textit{BASE\_ALL}
\mn{(respectively, 2.82 \mn{BLEU points} on LIUM
and from 1.41 to 1.76 on WebRTC)}\mn{,
and \textit{ii)}}
\textit{TGT~PAR} outperforms \textit{TGT SEQ} on all datasets (by 0.32 to 0.64).
\textit{TGT PAR} without regularization is superior to the fine-tuning when the VAD splits very aggressively
(21.75 vs 21.35 on WebRTC 3, 20ms) or in non-pause positions (23.77 vs 22.48 on LIUM).
On the other VAD configurations, the results are close, but inferior to the fine-tuning.
Our intuition is that this behavior is caused by the noise added by the context-attention
when the context is not needed.
This is confirmed by the results obtained adding the context-gate regularization presented in Eq. (\ref{regularization_gate_eq}) (\textit{TGT PAR+REG} and \textit{TGT SEQ+REG}).
The regularization allows our best context-aware model (\textit{TGT PAR+REG}) to outperform the fine-tuned model on
\pp{3 out of 4}
VAD configurations tested \pp{(in
\mnn{one}
case
BLEU is on par)}
and improves both integration types.
\textit{TGT SEQ} benefits more from it, closing the gap with \textit{TGT PAR}.
The value of the hyperparameter $\alpha$ was chosen among $0.01$, $0.02$, $0.04$ and $0.08$:
we set it to $0.04$ as it provided the best loss on the validation set.
The difference between context-aware models and
fine-tuning is even more evident if we consider the TER \mn{metric (the lower the better)}.
In this case, \textit{TGT SEQ}
\mn{obtains}
the best scores in
\mg{every setting},
but the results of all context-aware models are close and are $2$ to $6$ points
\mn{better}
than those obtained with
fine-tuning.
We also noticed that {1-},\nolinebreak 2-,3- and
\mn{4-gram}
BLEU scores are always
\mn{significantly} higher
for the context-aware solutions than for the fine-tuning,
even when the overall BLEU
\mn{scores are}
similar. The reason lies in the brevity penalty,
as the context-aware models produce shorter translations.
Interestingly, the best result
(23.91 BLEU)
is obtained
\mn{by exploiting}
the context in one of
\mn{the
\mg{hardest} segmentations}
for the base model (19.66 BLEU).
This is coherent with the behavior observed in $\S\ref{sec_res_ctx_type}$.
\section{Analysis}
We performed a manual analysis of the translations produced by the
baseline and by our best context-aware model (\textit{TGT PAR} + REG) on the LIUM-segmented test set. The goal was to check whether the gains are actually due to the use of contextual information and to understand how this information is exploited.
We noticed three main issues solved by
the context-aware approach.
\mg{They} are all related to the presence of sub-sentential fragments located at the beginning or
the end of a segment.
First,
these fragments are often ignored by the
baseline model. Being trained \pp{only} on well-formed sentences from the clean MuST-C corpus, this model seems unable to handle segments reflecting truncated sentences and, instead of returning partial translations, it opts for ignoring part of the input audio.
Second, the base model produces \textit{hallucinations} \cite{Lee2018HallucinationsIN}
\mg{trying} to translate a sub-sentential fragment into a well-formed target sentence.
Our models, instead, produce
the translation corresponding to the incomplete fragment.
Third, the
baseline model translates the sub-sentential fragment and the adjacent sentence in the same segment into one single output sentence, mixing them.
In contrast, our models are able to translate them separately.
\section{Conclusions}
\mn{We} studied how to make ST
\mn{models
trained on data segmented at sentence-level
robust to
VAD-segmented audio supplied at inference time.}
\mn{To this aim, we
explored
different approaches to integrate contextual information
provided by
the segment preceding the one to be translated.}
Our experiments show that
adopting
a context-aware architecture,
combined with
\mn{training on}
artificial data
generated with random segmentation,
\mn{is beneficial to improve final translation quality.}
\pp{We}
also demonstrate
that, compared to the best automatic segmentation (22.07 BLEU), context-aware models
achieve results that are similar in the worst case
\pp{(22.03)}
and significantly better in the best case
\pp{(23.91)}.
In this case, our context-based approach allows to reduce by 55\% the performance gap of the base model
\pp{(19.66)}
with respect to optimal (i.e. sentence-level) manual segmentation
\pp{(27.55).}
All in all,
\pp{this suggests}
that working on models' robustness \pp{to sub-optimal VAD segmentation} is at least as promising as improving
\pp{the segmentation itself.}
\section{Acknowledgements}
This work is part of the ``End-to-end Spoken Language Translation in Rich Data Conditions'' project,\footnote{https://ict.fbk.eu/units-hlt-mt-e2eslt/} which is financially supported by an Amazon AWS ML Grant.
\bibliographystyle{IEEEtran}
|
2,877,628,090,844 | arxiv | \section{Introduction.}
The discovery of neutrino oscillations can be considered as one of the greatest triumphs of modern physics.
It began with atmospheric neutrino oscillations \cite{SUPERKAMIOKANDE} interpreted as
$\nu_{\mu} \rightarrow \nu_{\tau}$ oscillations, as well as
$\nu_e$ disappearance in solar neutrinos \cite{SOLAROSC}. These
results have been recently confirmed by the KamLAND experiment \cite{KAMLAND},
which exhibits evidence for reactor antineutrino disappearance.
As a result of these experiments we have a pretty good idea of the neutrino
mixing matrix and the two independent quantities $\Delta m^2$, e.g $|m_2^2-m^2_1|$ and $|m^2_3-m^2_2|$.
Fortunately these
two $\Delta m^2$ values are vastly different \cite{MALTONI04}, see Eq. (\ref{matdat}) below.
This means that the relevant $L/E$ parameters are very different. Thus for a given energy the experimental results can approximately be described as two generation oscillations. For an accurate description of the data, however, a three generation analysis is necessary.
In all of these analyses the
oscillation length is much larger than the size of the detector. So one is able to see the effect, if the detector is
placed in the right distance from the source. It is, however, possible to design an experiment with an oscillation
length of the order of the size of the detector.
This is achieved, if one considers a neutrino source with as low as practical average neutrino energy,
such as a triton source with a maximum energy of $18.6$ keV. Thus the average oscillation length is $6.5$m, which is
smaller than the radius of $10$m of a spherical gaseous TPC detector \cite{NOSTOSA}. Such low energy events can be detected by measuring recoiling electrons with a low threshold spherical TPC detector.
In a typical supernova an energy of about $10^{53}$ ergs is released in the form of neutrinos
\cite{BEACFARVOG},\cite{SUPERNOVA}. These neutrinos
are emitted within an interval of about $10$ s after the explosion and they travel to Earth undistorted, except that,
on their way to Earth, they may oscillate into other flavors.
Thus for traditional detectors
relying on the charged current
interactions the precise event rate may depend critically on the specific properties of the neutrinos. The
time integrated spectra in the case of charged current detectors, like the SNO experiment,
depend on the neutrino oscillations \cite{TKBT03}.
Recently it has become feasible to detect neutrinos by exploiting the neutral current interaction \cite{DKLEIN} and measuring
the recoiling nucleus. One employs gaseous TPC detectors with low threshold energies.
A description of the NOSTOS project and details of the spherical TPC detector with sub-keV threshold
are given in \cite{NOSTOSA},\cite{NOSTOSB}.
The whole system looks stable, robust and easy to maintain.
The neutral current interaction, through its vector component, can lead
to coherence, i.e. an additive contribution of all neutrons in the nucleus (the vector contribution of the
protons is tiny).
Finally, in spite of the great process been made in understanding neutrinos mainly via neutrino oscillations,
some fundamental issues remain unsettled. First are the neutrinos Dirac or Majorana particles? What is the absolute
scale of the neutrino masses? The first question can practically be answered only by neutrinoless double beta
decay, see, e.g., earlier reports \cite{VERGADOS} and references therein . Furthermore neutrinoless double beta decay offers the best hope for answering the second question down to
a few meV.
\section {Coherent neutrino nucleus scattering}
The standard neutral current left handed weak interaction can be cast in the form:
\begin{equation}
{\cal L_q}=-\frac{G_F}{\sqrt{2}}
\left[ \bar\nu_\alpha \gamma^\mu (1-\gamma^5)\nu_\alpha \right]
\left[ \bar q\gamma_\mu (g_V(q)-g_A(q)\gamma^5) q \right]
\label{weak1}
\end{equation}
(diagonal in flavor space).
At the nucleon level we get:
\begin{equation}
{\cal L_q}=-\frac{G_F}{\sqrt{2}}
\left[ \bar\nu_\alpha \gamma^\mu (1-\gamma^5)\nu_\alpha \right]
\left[ \bar N\gamma_\mu (g_V(N)-g_A(N)\gamma^5) N \right]
\label{weak3}
\end{equation}
with
\begin{equation}
g_V(p)=\frac{1}{2}-2 \sin^2{\theta_W}\simeq 0.04~~,~~g_A(p)=1.27 \frac{1}{2}~;~
g_V(n)=-\frac{1}{2}~~,~~g_A(n)=-\frac{1.27}{2}
\label{weak4}
\end{equation}
Beyond the standard level one has further interactions which need not be diagonal in flavor
space \cite{VERGIOM}. We are not. however, going to discuss such issues here.
The cross section for elastic neutrino nucleon scattering has extensively been studied
\cite{BEACFARVOG},\cite{VogEng}.
The energy of the recoiling particle can be written in dimensionless form
as follows:
\beq
y=\frac{2\cos^2{\theta}}{(1+1/x_{\nu})^2-\cos^2{\theta}}~~,~~
y=\frac{T_{recoil}}{m_{recoil}},x_{\nu}=\frac{E_{\nu}}{m_{recoil}}
\label{recoilen}
\eeq
The maximum energy occurs when $\theta=0$, $y_{max}=\frac{2}{(1+1/x_{\nu})^2-1}$,
in agreement with Eq. (2.5) of earlier work. \cite{BEACFARVOG}.
One can invert Eq. \ref{recoilen} and get the neutrino energy associated with a given recoil energy and
scattering angle.
From the above expressions we see that the vector current contribution, which may lead to coherence, is negligible
in the case of the protons. Thus the coherent contribution may come from the neutrons and is expected to be
proportional to the square of the neutron number.
The neutrino-nucleus coherent cross section takes the form:
\begin{eqnarray}
\left(\frac{d\sigma}{dT_A}\right)_{weak}&=&\frac{G^2_F Am_N}{2 \pi}~(N^2/4)F_{coh}(A,T_A,E_{\nu}),
\nonumber\\
& &F_{coh}(A,T_A,E_{\nu})=F(T_A) {\huge [}
\left (1+\frac{A-1}{A}\frac{T_A}{E_{\nu}} \right )
+(1-\frac{T_A}{E_{\nu}})^2
\nonumber\\
& &\left (1-\frac{A-1}{A}\frac{T_A}{m_N}\frac{1}{E_{\nu}/T_A-1} \right )
-\frac{Am_NT_A}{E^2_{\nu}} ]
\label{elaswAV}
\end{eqnarray}
where $F(T_A)$ is the nuclear form factor.
\section{Supernova Neutrinos}
\label{sec.supernova}
The number of neutrino events for a given detector depends on the neutrino spectrum and the distance of the
source. We will consider a typical case of a source which is about $10$ kpc, l.e. $D=3.1 \times 10^{22}$ cm ( of the order of the radius of the galaxy) with
an energy output of $3 \times 10^{53}$ ergs with a duration of about $10$ s. Furthermore we will assume for simplicity that each neutrino flavor is characterized by a
Fermi-Dirac like distribution times its characteristic cross section and we will not consider here the more
realistic distributions, which have recently become available \cite{NUSPECTRA}.
This is adequate for our purposes. Thus:
\beq
\frac{dN}{dE_{\nu}}=\sigma(E_{\nu})\frac{E^2_{\nu}}{1+exp(E_{\nu}/T)}=\frac{\Lambda}{JT}\frac{x^4}{1+e^x}~~,
~~x=\frac{E_{\nu}}{T}
\label{nudistr}
\eeq
with
$J=\frac{31\pi^6}{252}$, $\Lambda$ a constant and
$T$ the temperature of the emitted neutrino flavor.
Each flavor is characterized by its
own temperature as follows:
$$T=8 \mbox { MeV for } \nu_{\mu},\nu_{\tau},\tilde{\nu}_{\mu}, \tilde{\nu}_{\tau}
\mbox{ and } T=5 ~(3.5)\mbox{ MeV for } \tilde{\nu}_e ~(\nu_e)$$
The constant $\Lambda$ is determined by the requirement that the distribution yields the total energy of each
neutrino species.
$$U_{\nu}=\frac{\Lambda T}{J}\int_0^{\infty } dx \frac{x^5}{1+e^x}\Rightarrow \Lambda=\frac{U_{\nu}}{T}$$
We will further assume that $U_{\nu}=0.5 \times 10^{53}$ ergs
per neutrino flavor. Thus one finds:
$$\Lambda=0.89\times 10^{58}~(\nu_e),~~0.63\times 10^{58}~(\tilde{\nu}_e)~,0.39\times 10^{58}
\mbox{ (all other flavors)}$$
The differential event rate (with respect to the recoil energy) is proportional to the quantity:
\beq
\frac{dR}{dT_A}=\frac{\lambda (T)}{J}\int_0^{\infty } dx
F_{coh}(A,T_A,xT) \frac{x^4}{1+e^x}
\label{dRdT}
\eeq
with $\lambda(T)=(0.89,0.63,0.39)$ for $\nu_e,\tilde{\nu}_e$ and all other flavors respectively.
This is shown in Figs. \ref{fig:difr131a} and \ref{fig:difr131b}.
\begin{figure}[h]
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{1.0cm} {$F_{coh}$}}
\includegraphics[width=18pc]{plotdifp131.eps}
\hspace*{6.0cm}\tiny{$T_A \rightarrow$ MeV}
\caption{\label{fig:difr131a}The differential event rate as a function of the recoil energy $T_A$, in arbitrary units, for
Xe without quenching.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{1.0cm} {$F_{coh}$}}
\includegraphics[width=18pc]{quplotdifp131.eps}
\hspace*{6.0cm}\tiny{$T_A \rightarrow$ MeV}
\caption{\label{fig:difr131b}The same as in \ref{fig:difr131a} with quenching included.}
\end{minipage}
\end{figure}
Summing over all the neutrino species we can write \cite{NOSTOSB}:
\beq
\mbox{No of events}=C_{\nu} \frac{K(A,(T_A)_{th})}{K(40,(T_A)_{th})}Qu(A)
\label{sumevents}
\eeq
with
\beq
C_{\nu}=153 \left ( \frac{N}{22} \right )^2 \frac{U_{\nu}}{0.5\times 10^{53}ergs}
\left ( \frac{10kpc}{D}\right )^2 \frac{P}{10Atm}
\left[ \frac{R}{4m}\right]^3 \frac{300}{T_0}
\label{C2}
\eeq
$K(A,(T_A)_{th})$
is the rate at a given threshold energy divided by that at zero threshold. It depends
on the threshold
energy, the assumed quenching factor and the nuclear mass number. It is unity at $(T_A)_{th})=0$.
From the above equation we find
that, ignoring quenching, the following expected number of events:
\beq
1.25,~31.6,~153,~614,~1880\mbox{ for He, Ne, Ar, Kr and Xe}
\label{allrates}
\eeq
respectively. For other possible targets the rates can be found by the above formulas or interpolation.\\
The quantity $Qu(A)$ is the quenching factor \cite{SIMON03}-\cite{LIDHART}, assuming a threshold energy
$(T_A)_{th}=100$eV.
The parameter $Qu(A)$ takes the values:
\beq
0.49,~0.38,~0.35,~0.31,~0.29\mbox{ for He, Ne, Ar, Kr and Xe}
\label{quench2}
\eeq
respectively. The effect of quenching is larger in the case of heavy targets, since the average energy of
the recoiling nucleus is smaller.
The effect of quenching is exhibited in Figs \ref{fig:Kqua} and \ref{fig:Kqub} for the two interesting targets
Ar and Xe.
\begin{figure}[h]
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{-0.0cm} {\tiny $K(A,(T_A)_{th})\rightarrow $}}
\includegraphics[width=18pc]{th40.eps}
\hspace*{6.0cm}\tiny{$T_A \rightarrow$ MeV}
\caption{\label{fig:Kqua}The function $K(A,(T_A)_{th})$ versus $(T_A)_{th}$ for the target Ar. The short and long dash correspond to no quenching and quenching factor respectively.
For a threshold energy of $100$ eV the rates are quenched by factors of $3$ (see Eq. \ref{quench2}).}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{1.0cm} {$F_{coh}$}}
\includegraphics[width=18pc]{th131.eps}
\hspace*{6.0cm}\tiny{$T_A \rightarrow$ MeV}
\caption{\label{fig:Kqub}The same as in \ref{fig:Kqua} in the Xe. The effect of quenching now is $3.5$ }
\end{minipage}
\end{figure}
We should mention that it is of paramount importance to experimentally measure the quenching factor. The
above estimates were based on the assumption of a pure gas.
Such an effect will lead to an increase in the quenching factor and needs be measured.\\
Anyway the number of expected events including quenching and $E_{th}$=0.1 keV becomes:
\beq
0.61,~ 12.0,~ 53.5,~ 190,~ 545 \mbox{ for He, Ne, Ar, Kr and Xe}
\eeq
The inclusion of the form factor is important only in the case of Xe, in which case the above number of events
becomes $415$.
\section{Neutrino oscillations}
The neutrino mixing can be
parametrized as follows:
\begin{equation}
\rm{U}=
\left(
\begin{array}{ccc}
1 & 0 & 0\\
0 & c_{23} & s_{23} \\
0 & -s_{23} & c_{23}
\end{array} \right)
\left(
\begin{array}{ccc}
\nonumber
c_{13} & 0 & s_{13}~e^{i\delta}\\
0 & 1 & 0 \\
-s_{13}~e^{i\delta} & 0 & c_{13}
\end{array} \right)
\left(
\begin{array}{ccc}
c_{12} & s_{12} & 0\\
-s_{12} & c_{12} & 0 \\
0 & 0 & 1
\end{array} \right)
\nonumber
\end{equation}
where $s_{ij}=\sin{\theta_{ij}}$ and $c_{ij}=\cos{\theta_{ij}}$.
The neutrino oscillation data can be summarized as follows \cite{MALTONI04}:
\beq
\left(
\begin{array}{cccc}
\mbox {parameter}& \mbox{best fit} & 2 \sigma &3 \sigma \\
&&&\\
\Delta m^2_{31}(10^{-3} \mbox {eV}^2) & 1.3 & 1.7-2.9&1.4-33\\
\Delta m^2_{21}(10^{-5} \mbox {eV}^2 )& 8.1 & 7.3-8.7&7.2-9.1 \\
\sin^2{\theta_{12}} & 0.3 & 0.25-0.34&0.23-0.38\\
\sin^2{\theta_{23}} & 0.5 & 0.38-0.64&0.38-0.68\\
\sin^2{\theta_{13}} & 0.00 &\preceq 0.028&\preceq 0.047
\end{array} \right)
\label{matdat}
\eeq
In a three generation model the electron neutrino disappearance probability is given by:
\barr
P(\nu_e \rightarrow \nu_e)&=1& { -\cos^2{\theta_{13}} \sin^2{2 \theta_{12}} \sin^2 {\Delta_{21}}}
\nonumber\\
&&{-\cos^2{\theta_{12}} \sin^2 {2 \theta_{13}} \sin^2{ (\Delta_{32}-\Delta_{21})}} { -
\sin^2{\theta_{12}} \sin^2{2 \theta_{13}} \sin^2{\Delta_{32}}}
\earr
with
\beq
{\Delta_{21}=\frac{\Delta m^2_{21} L}{2 E_{\nu}}~,~\Delta_{32}=\frac{\Delta m^2_{31} L}{2 E_{\nu}}}
\eeq
In the presence of neutrino mixing both oscillation lengths contribute to the electron neutrino
disappearance.
For $|\Delta_{21}|\ll |\Delta_{32}|$ and $\theta_{13}\ll 1 $ we get
\beq
P(\nu_e \rightarrow \nu_e)=1-\sin^2{2 \theta_{12}} \sin^2{\frac{\Delta m^2_{21} L}{2 E_{\nu}}}-
\sin^2{2 \theta_{13}} \sin{\frac{\Delta m^2_{32} L}{2 E_{\nu}}}
\eeq
\beq{ P(\nu_e \rightarrow \nu_e)\simeq 1} { -\sin^2{2 \theta_{12}} \sin^2{\pi \frac{L} {50 L_{32}}}} {-
\sin^2{2 \theta_{13}} \sin{\pi \frac{L}{L_{32}}} }
\eeq
with
$${ L_{32}=\frac{2 E_{\nu}}{\pi \Delta^2_{32}}}= \mbox{ { small oscillation length}}~,~
L_{12}\approx50L_{23}= \mbox{ large oscillation length}$$
These are shown in Figs \ref{triton}-\ref{combined}.
\begin{figure}[h]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{triton.eps}
\caption{\label{triton}The small oscillation length $\nu_e$ disappearance expected to be seen in a TPC
low energy electron detector.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{combined.eps}
\caption{\label{combined} On top of the large oscillation length seen in reactor experiments we show the small oscillation
length due to $\theta_{13}$.}
\end{minipage}
\end{figure}
\section{Neutrino mass Limits from astrophysics and triton decay}
The neutrino oscillation data alone cannot determine the absolute neutrino mass scale and the sign of $\Delta m^2_{32}$.
We thus distinguish the following cases
\begin{itemize}
\item { Normal Hierarchy}:
$${ \Delta m^2_{SUN}=m^2_2-m^2_1}~,~{ \Delta m^2_{ATM}=m^2_3-m^2_1}$$
$${ m_1}~,~m_2=\sqrt{{ \Delta m^2_{SUN}}+{ m_1^2}}~,~m_3=\sqrt{{\Delta m^2_{ATM}}+{ m_1^2}}$$
\item {Inverted Hierarchy}:
$${\Delta m^2_{SUN}=m^2_2-m^2_1}~,~{\Delta m^2_{ATM}=m^2_2-m^2_3}$$
$${m_3}~,~m_2=\sqrt{{ \Delta m^2_{ATM}}+{ m_3^2}}~~,
~~~m_1=\sqrt{{\Delta m^2_{ATM}-\Delta m^2_{SUN}}+{ m_3^2}}$$
\item The degenerate scenario.
$$m_1=m_2=m_3>>\sqrt{|\Delta m^2_{23}|}$$
\end{itemize}
These problems can be tackled from other experiments as follows:
\begin{enumerate}
\item The astrophysics limit as follows:
\begin{itemize}
\item { Normal Hierarchy}:
$${m_1}+\sqrt{{ \Delta m^2_{SUN}}+{ m_1^2}}+\sqrt{{\Delta m^2_{ATM}}+{ m_1^2}}\leq { m_{astro}}$$
\item { Inverted Hierarchy}:
$${ m_3}+\sqrt{{ \Delta m^2_{ATM}}+{ m_3^2}}+
\sqrt{{ \Delta m^2_{ATM}-\Delta m^2_{SUN}}+{m_3^2}}\leq { m_{astro}}$$
\end{itemize}
unfortunately at present this limit is not very stringent \cite{COSMO06} $ m_{astro}<0.71$eV
\item The triton decay limit
\begin{itemize}
\item { Normal Hierarchy}:
$$c_{12}^2 c^2_{13}{ m^2_1}+s_{12}^2 c^2_{13}({ \Delta m^2_{SUN}}+{ m_1^2})+s^2_{13}({\Delta m^2_{ATM}}+{ m_1^2})\leq { m^2_{decay}}$$
\item {Inverted Hierarchy}:
$$s^2_{13}{ m^2_3}+s_{12}^2c^2_{13}({ \Delta m^2_{ATM}}+{ m_3^2})++c^2_{12}c^2_{13}({ \Delta m^2_{ATM}-\Delta m^2_{SUN}}+{ m_3^2})
\leq { m^2_{decay}}$$
\end{itemize}
This limit is also not very stringent \cite{DECAY02} $m_{decay}<$2.2eV
\end{enumerate}
The only process which at present offers the best chance of reaching limits comparable to the neutrino oscillation
data is neutrinoless double beta decay.
\section{Neutrinoless Double Beta Decay.}
Double beta decay of a nucleus A(N,Z) occurs when the ordinary $\beta$ decay to the nucleus $A(N,Z\pm 1)$ is forbidden due to
energy conservation or angular momentum mismatch, while the decay to one of the nuclei $A(N,Z\pm 2)$ is allowed.
Ignoring the non exotic decays involving neutrinos, the following decays are possible:
$$N(A,Z) \rightarrow N(A,Z+2)+e^- +e^- ~ (0\nu ~\beta \beta ~\mbox {-decay})$$
(the corresponding two neutrino decay has already been observed in many systems).
Furthermore, omitting the non exotic processes accompanied by neutrinos, the following processes
are possible:
$$N(A,Z) \rightarrow N(A,Z-2)+e^+ +e^+
\mbox {(double ~positron~ emission)}$$
$$ e^-_b + N(A,Z)~\rightarrow N(A,Z-2)+e^+ \mbox
{(electron ~positron ~conversion)}$$
$$e^- _b + e^- _b +N(A,Z) \rightarrow \mbox {N(A,Z-2)+2 X-rays
(Double electron capture)}$$
We will limit our discussion on the first of these processes, which is the most important experimentally.
We will adopt the most popular view and assume that the process proceeds via intermediate neutrinos.
The relevant half life time is given by:
\beq
[T_{1/2}^{0\nu}]^{-1} = G^{0\nu}_{01} \left |\frac{<m_{\nu}>}{m_e} \Omega_{\nu} \right |^2
\eeq
where $G^{0\nu}_{01}$ is well understood kinematical factor, $\Omega_{\nu}$ is the nuclear matrix
element and $<m_{\nu}>$ is the average neutrino mass given by:
\beq
\left < m_{\nu} \right >=c_{12}^2 c_{13}^2 m_1 +s^2_{12}c_{13}^2 e^{i\alpha} m_2+s_{13}^2 e^{i\beta} m_3
\eeq
with $c_{ij}=\cos{\theta_{ij}}$, $s_{ij}=\sin{\theta_{ij}}$, $m_i$, $i=1,2~3$ are the neutrino masses and $\alpha$ and $\beta$ the two relevant Majorana phases.
Once the nuclear matrix elements are known $<m_{\nu}>$ can be extracted from the data. From this the lightest neutrino mass
can be inferred, if neutrino oscillation data are incorporated. This analysis has already been
done \cite{PETCOV06}, \cite{VALLE06}
and we are not going to elaborate on it here. The main conclusions are:
If the degenerate scenario holds double beta decay observation is
within the goals of the current experiments. If the inverted hierarchy holds, there exists a lower bound on
the value of $<m_{\nu}>$ which is within the reach of the currently planned future experiments. If, however, the
normal hierarchy scenario holds there is no such lower bound and the road towards measuring $<m_{\nu}>$ may be very long.
The above conclusions depend, however, on the assumption that the neutrino mass mechanism dominates in $0\nu$ $\beta\beta$ decay.
There exist many other mechanisms which may contribute to double beta decay. The most prominent are intermediate
heavy neutrinos
and R-parity, and consequently lepton violating, supersymmetry \cite{VERGADOS}.
We will extend our formalism to consider the case of right handed currents: The mixing matrix is a $6\times 6$ and takes the form:
\beq
U=\left ( \begin{array}{c}\nu^0_L\\
\nu^{0c}_{L}
\end{array} \right )
\left ( \begin{array}{cc}
U^{11}~~~&U^{12}\\
U^{21}~~~&U^{22}
\end{array} \right )
\left ( \begin{array}{c}\nu_L \\
N_L\end{array} \right )
\eeq
where
$$\nu^0_L=(\nu_{eL},\nu_{\mu L},\nu_{\tau L})~;~
\nu^{0c}_L=(\nu^c_{eL},\nu^c_{\mu L},\nu^c_{\tau L})~\Longleftrightarrow \mbox{ right~handed~neutrino}$$
$$\nu_L=(\nu_{1L},\nu_{2L},\nu_{3L}) \mbox { light}~,~N_L=(N_{1L},N_{2L},N_{3L})\mbox { heavy}$$
If the right handed neutrino does not exist:
{ $$U=U^{11}=U_{MNS}$$
}Quite generally the half-life takes the form \cite{VERGADOS}:
\begin{eqnarray}
\lefteqn{
[T_{1/2}^{0\nu}]^{-1} = G^{0\nu}_{01}
\left\{ |X_{L}|^2 + |X_{R}|^2 -
{\tilde C}_1^\prime X_{L} X_{R}+...
\right. } & & \nonumber \\
&&+ {\tilde C}_2 |\lambda| X_{L} cos \psi_1
+ {\tilde C}_3 |\eta| X_{L} cos \psi_2
+ {\tilde C}_4 |\lambda|^2 + {\tilde C}_5 |\eta|^2
\nonumber \\
&& \left. + {\tilde C}_6
|\lambda||\eta| cos (\psi_1 -\psi_2) + Re ({\tilde C}_2
\lambda X_{R} + {\tilde C}_3 \eta X_{R}) \right\},
\end{eqnarray}
Where the left handed contribution is:
\begin{eqnarray}
X_{L}^{} = \eta_{\nu} \Omega_{\nu}+\eta^L_N \Omega_N+\eta_{SUSY} \Omega_{SUSY}
\end{eqnarray}
with $\Omega_{\nu}~,~\Omega_N~,~\Omega_{SUSY}$ the nuclear matrix elements, associated with light neutrinos, heavy
neutrinos and SUSY contributions respectively, while
$\eta_{\nu}~,~\eta^L_N~,~\eta_{SUSY}$ lepton violating parameters:\\
$\eta_{\nu}=\frac{<m_\nu >}{m_e}$ , $<m_\nu > ~ = ~ \sum^{3}_k~ (U^{(11)}_{ek})^2 ~ e^{i \alpha_k} ~ m_k$
$ \eta^L_{_N} ~ = ~ \sum^{3}_k~ (U^{(12)}_{ek})^2 ~
~~ e^{i \Phi_k} ~ \frac{m_p}{M_k}$
with $\alpha_k$, $\Phi_k$ Majorana phases and
$m_p$ ($m_e$) being the proton (electron) mass.
We see now that we have two types of interference.
\begin{itemize}
\item The interference between the various left handed contributions.\\
It is thus possible that the light neutrino mass mechanism may be cancelled by the other contributions, so that
the experiments go below the inverted hierarchy and still do not see the $0\nu$ double beta decay. This, however,
cannot happen in all nuclear systems, since the nuclear matrix elements behave very differently. The light neutrino
operator is long range, but the one resulting from heavy particle exchange is short ranged.
To be more specific consider { two left handed mechanisms}, one light ($\nu$) and one heavy H. Then
\beq
\frac{m_e/\Omega_{\nu}}{\sqrt{T_{1/2}(0\nu)G_{01}^{0\nu}}}=<m_{\nu}>+\eta_H m_e \frac{\Omega_H}{\Omega_{\nu}}
\eeq
or
\beq
\eta_H=\frac{b-<m_{\nu}>}{m_e r}~~,~~r=\frac{\Omega_H}{\Omega_{\nu}}~~,~~b=\frac{m_e/\Omega_{\nu}}{\sqrt{T_{1/2}(0\nu)G_{01}^{0\nu}}}
\eeq
Now for two targets $A_1$ and $A_2$ we get
$${ <m_{\nu}>=\frac{b_1 r_2-b_2 r_1}{r_2-r_1}}$$
{ More than two targets overdetermine the system and provide tests.}\\
If only the light neutrino mechanism is operative\\
$\Leftrightarrow b_1=b_2=b$ we get the standard expression $ <m_{\nu}>$, which is independent of the target:
In an analogous fashion we can write:
\beq
<m_{\nu}> \rightarrow <m_{\nu}>\left [1+(\eta^L_N/\eta_{\nu}) )(\Omega_N (A) /\Omega_{\nu} (A) \right]
\eeq
Assuming that the cancellation in the target $A_0$ is almost complete (given by c, $c<<1$), we find
\beq
<m_{\nu}> \rightarrow <m_{\nu}>\left [1-c\frac{\Omega_{\nu}(A_0)}{\Omega_{\nu}(A)}\frac{\Omega_{N}(A)}{\Omega_{N}(A_0)} \right]
\eeq
Even though for the neutrino mass mechanism recent and more reliable nuclear matrix elements have
appeared \cite{RFSV03}, we
need the matrix elements for the other mechanisms within the same QRPA model. So we are going to use the set
provided by the earlier paper \cite{PSVF99} (without p-n pairing). The factor inside the square bracket is plotted as a function of A for $A_0=^{76}$Ge and $A_0=^{136}$Xe in
Figs \ref{fig:frGe} and \ref{fig:frXe}.
\begin{figure}[h]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{frGe.eps}
\caption{\label{fig:frGe} The apparent $<m_{\nu}>$ when there exists 100$\%$ (thick curve), 90$\%$ (fine curve)
and 80$\%$(dotted curve) cancellation
in the target $A_0=^{76}$Ge.
}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{frXe.eps}
\caption{\label{fig:frXe} The same as in \ref{fig:frGe} in the case of $A_0=^{136}$Xe.}
\end{minipage}
\end{figure}
It is clear that such a cancellation cannot occur in all nuclear systems.
\item Interference between left and right handed lepton currents.\\
One can have two amplitudes independent of the neutrino mass if the chirality of the neutrinos is opposite.
The associated lepton violating parameters are indicated by $\lambda$ and $\eta$ given by:
\beq
{\ \eta} ~ = ~{ \epsilon}~~ \eta_{RL}~~,~~{ \lambda} ~=~ {\kappa}~~
{ \eta_{RL}}
\eeq
\beq{ \eta_{RL}} ~=~
\sum^{3}_j~ (U^{(21)}_{ej}U^{(11)}_{ej}) e^{i \alpha_j}
\eeq
\beq
{ \kappa}=m^2_L/m^2_R~~,~~{ \epsilon}=tan \zeta.
\eeq
$$m_L,m_R= \mbox{ gauge boson masses}~
\zeta= \mbox{ the gauge boson mixing angle}.$$
In the experimental limit on $^{76}Ge$ after using the nuclear matrix elements of \cite{PSVF99} we
obtain the constraint on the $<m_{\nu},\lambda$ and $<m_{\nu},\eta$ plane shown in Figs \ref{fig:mlam}
and \ref{fig:meta} respectively.
\begin{figure}[h]
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{-0.0cm} {$\left <m_{\nu} \right >\rightarrow $ eV}}
\includegraphics[width=18pc]{lamnopn.eps}
\hspace*{5.0cm}$\lambda \rightarrow10^{-6}$
\caption{\label{fig:mlam} The constraints on $<m_{\nu}>$ and $\lambda$ for the target $^{76}Ge$ assuming that they are relatively real.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\rotatebox{90}{\hspace{-0.0cm} {$\left <m_{\nu} \right >\rightarrow $ eV}}
\includegraphics[width=18pc]{etanopn.eps}
\hspace*{5.0cm}$\eta \rightarrow10^{-8}$
\caption{\label{fig:meta} The constraints on $<m_{\nu}>$ and $\eta$ for the target $^{76}Ge$ assuming that they are relatively real.}
\end{minipage}
\end{figure}
\end{itemize}
\section{Conclusions}
In this brief review we considered some aspects of neutrino physics, which in the frontiers of research
after the discovery of neutrino oscillations.
We first considered neutrinos as probes to detect and study supernova explosions.
We have seen that it is quite simple to detect typical supernova
neutrinos in our galaxy, provided that such a supernova explosion takes place (one explosion every 30 years is estimated \cite{SOLBERG}). The idea is to employ a small size spherical TPC detector filled with a high
pressure noble gas and measure nuclear recoils. An enhancement of the neutral current component is achieved via the coherent
effect of all neutrons in the target. Thus employing, e.g., Xe at $10$ Atm, with a feasible threshold energy
of about $100$ eV in the detection of the recoiling nuclei,
one expects between $400$ and $1900$ events, depending on the quenching factor and the nuclear form factor.
We believe that networks of such dedicated detectors, made out of simple, robust and cheap technology,
can be simply managed by an international scientific consortium and operated by students. This network
comprises a system, which can be maintained
for several decades (or even centuries). This is is a key point towards being able to observe
few galactic supernova explosions.
We then examined processes which depend on the mixing and mass of the neutrinos. We have shown that
there are many advantages in using very low energy neutrinos in the few keV and detecting them via
measuring electron recoils using low threshold gaseous spherical TPC detectors. Then one can look
for oscillations induced by the small mixing angle $\theta_{13}$ and the the small oscillation length,
associated with the large $\Delta m^2_{23}$, of tens of meters.
Then the full oscillation takes place inside the detector. With the realistic goal of measuring the distance
up to $1\%$ we hope to measure or put stringent limits on the small mixing angle $\theta_{13}$. At the same
time such an experiment will put a limit on the neutrino magnetic moment at the level of $10^{-12}\mu_b$. It
can also measure the Weinberg angle down to essentially zero momentum transfer.
We have also seen that neutrinoless double beta decay is the only process to decide whether
neutrinos are Dirac or Majorana particles and the best process to settle the question of the
absolute scale of neutrino mass. In order to be able to do so reliable nuclear matrix elements
must be available. Furthermore, even though the neutrino mass mechanism is the most popular
scenario in
this post neutrino oscillation period, other mechanisms may contribute or even dominate
this process. Interference between such mechanisms may invalidate any conclusions we draw
about the neutrino mass scale. This indeed maybe a problem, if one analyzes the data of only one
target. We have seen, however, that such ambiguities may be resolved, if one analyzes data
from many targets or experiments with different signatures, e.g. tracking versus deposited
energy.
\ack The author is indebted to the organizers of the Third Symposium on Large TPC's for Low Energy Rare Event Detection, Paris, Dec. 11-12, 2006
and NUMMY07 ENTApP Network Meeting, Durham, Jan. 10-12, 2007 for their support in attending the meetings
where this work was presented.
\section{References}
|
2,877,628,090,845 | arxiv | \section{Introduction}
\begin{document}
\title{Quantum Coding Theorems for Arbitrary Sources, Channels and
Entanglement Resources}
\author{{Garry~Bowen~and~Nilanjana~Datta
\thanks{This work was supported by the EPSRC (Research Grant GR/S92816).}%
\thanks{G. Bowen is with the Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK (e-mail: gab30@damtp.cam.ac.uk).}
\thanks{N. Datta is with the Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA, UK (e-mail: N.Datta@statslab.cam.ac.uk).}%
}
\markboth{Submitted to IEEE Transactions on Information Theory}{Bowen \& Datta: Quantum Information Spectrum Stuff}
\maketitle
\begin{abstract}
The information spectrum approach gives general formulae for optimal rates
of various information theoretic protocols, under minimal assumptions
on the nature of the sources, channels and entanglement resources involved.
This paper
culminates in the derivation of the
dense coding capacity for a noiseless quantum
channel, assisted by arbitrary shared entanglement, using this approach.
We also review
the currently known coding theorems, and their converses,
for protocols such as data compression for arbitrary quantum sources and
transmission of classical information through arbitrary quantum channels. In addition,
we derive the optimal rate of data compression for a mixed source.
\end{abstract}
\begin{keywords}
Quantum information, dense coding capacity, quantum data compression, classical capacity, information spectrum.
\end{keywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\PARstart{Q}{uantum} information theory generalizes the ideas of coding and
communication to include the nature of the physical system in which information
is encoded. The information spectrum approach of Han \& Verdu
\cite{verdu94,han} gives general formulae for many operational schemes in
information theory. It replaces the idea of typical
events (generally called typical sequences) in information theory, with high
probability events. The power of this approach lies in the lack of
assumptions about the source, channel and entanglement resource.
The quantum information spectrum was defined in terms of quantum states by
Hayashi \& Nagaoka \cite{hayashi03}, initially in the context of hypothesis
testing, and was used to determine a general expression for the classical
capacity of arbitrary quantum channels. The quantum information spectrum
extends the idea of high probability events to high probability subspaces
of states in a Hilbert space. In the commutative case, the quantum
information spectrum simply reduces to its classical counterpart.
In this paper we present a review of coding theorems for quantum data
compression and transmission of classical information through a
quantum channel. The rate of
compression for a mixed source is explicitly derived. A number of new
results are also presented, including the dense coding capacity for a noiseless
quantum channel, assisted by arbitrary shared entanglement.
\section{Preliminaries}
Let ${\cal B}({\cal H})$ denote the algebra of linear operators acting on
a finite--dimensional Hilbert space ${\cal H}$ of dimension $d$. The von Neumann entropy of a state $\rho$, i.e. a positive operator of
unit trace in ${\cal B}({\cal H})$, is defined as $S(\rho) = -\rm Tr \rho \log \rho$. Throughout this paper, we choose the logarithm to base $e$. We could
equally well choose an arbitrary base for
the logarithm. This would simply scale the unit of information.
A quantum channel is given by a
completely positive trace--preserving (CPTP) map $\Phi: {\cal
B}({\cal K}) \to {\cal B}({\cal H})$, where ${\cal K}$ and ${\cal
H}$ are the input and output Hilbert spaces of the channel.
\subsection{Spectral Projections}
The quantum information spectrum approach requires the extensive use of
spectral operators. For a self-adjoint operator $A$ written in its spectral
decomposition $A = \sum_i \lambda_i |i\rangle \langle i|$ we define the
positive spectral projection on $A$ as
\begin{equation}
\{ A \geq 0 \} = \sum_{\lambda_i \geq 0} |i\rangle \langle i|
\end{equation}
the projector onto the eigenspace of positive eigenvalues of $A$.
Corresponding definitions apply for the other spectral projections $\{ A < 0
\}$, $\{ A > 0 \}$ and $\{ A \leq 0 \}$. For two operators $A$ and $B$, we can
then define $\{ A \geq B \}$ as $\{ A - B \geq 0 \}$, and similarly for the
other ordering relations.
\subsection{Two Important Lemmas}
The following key lemmas are used repeatedly in the paper. For their proofs
see \cite{BD1}.
\begin{lemma}
\label{lemma}
For self-adjoint operators $A$, $B$ and any positive operator $0 \leq P \leq I$
the inequality
\begin{equation}
\mathrm{Tr}\big[ P(A-B)\big] \leq \mathrm{Tr}\big[ \big\{ A \geq B \big\}
(A-B)\big]
\label{eqn:first_ineq}
\end{equation}
holds.
\end{lemma}
\begin{lemma}
\label{lemma2}
For self-adjoint operators $A$ and $B$, and any completely positive
trace-preserving (CPTP) map $\mathcal{T}$ the inequality
\begin{equation}
\mathrm{Tr}\big[ \{\mathcal{T}(A) \geq \mathcal{T}(B) \}\mathcal{T}(A-B)\big]
\leq \mathrm{Tr}\big[ \big\{ A \geq B \big\} (A-B)\big]
\label{eqn:second_ineq}
\end{equation}
holds.
\end{lemma}
We also make use of the following proposition
\begin{proposition}
\label{cor0}
Given a state $\rho_n$ and a self-adjoint
operator $\omega_n$, we have
\begin{equation}
\mathrm{Tr}\big[\{\rho_n \ge e^{n\gamma}\omega_n \} \omega_n \bigr]
\le e^{-n\gamma}.
\end{equation}
for any real $\gamma$.
\end{proposition}
\begin{proof}
We have
\begin{equation}
\mathrm{Tr}\big[\{\rho_n \ge e^{n\gamma}\omega_n \} (\rho_n - e^{n\gamma}\omega_n \bigr]
\ge 0
\end{equation}
and hence, by rearranging terms
\begin{equation}
\mathrm{Tr}\big[\{\rho_n \ge e^{n\gamma}\omega_n \} \omega \bigr]
\le e^{-n\gamma} \mathrm{Tr}\big[\{\rho_n \ge e^{n\gamma}\omega_n \} \rho_n \bigr]
\le e^{-n\gamma}.
\end{equation}
where $\mathrm{Tr}\big[\{\rho_n \ge e^{n\gamma}\omega_n \} \rho_n \bigr] \leq 1$.
\end{proof}
\subsection{Quantum Spectral Information Rates}
As a generalization of the relative entropy, the spectral divergence allows information theory to include arbitrary sources and channels.
\begin{definition}
Given the difference operator $\Pi_n(\gamma) = \rho_n - e^{n\gamma}\omega_n$, the quantum spectral sup-(inf-)divergence rates are defined as
\begin{align}
\overline{D}(\rho \| \omega) &= \inf \Big\{ \gamma : \limsup_{n\rightarrow \infty} \mathrm{Tr}\big[ \{ \Pi_n(\gamma) \geq 0 \} \Pi_n(\gamma) \big] = 0 \Big\} \label{supdiv} \\
\underline{D}(\rho \| \omega) &= \sup \Big\{ \gamma : \liminf_{n\rightarrow \infty} \mathrm{Tr}\big[ \{ \Pi_n(\gamma) \geq 0 \} \Pi_n(\gamma) \big] = 1 \Big\} \label{infdiv}
\end{align}
respectively.
\end{definition}
The spectral entropies, conditional spectral entropies, and spectral mutual
information rates may all be expressed as a divegrence rate with appropriate
substitutions for the sequence of operators $\omega = \{ \omega_n
\}_{n=1}^{\infty}$. These are
\begin{align}
\overline{S}(\rho) &= -\underline{D}(\rho| I) \label{supent} \\
\underline{S}(\rho) &= -\overline{D}(\rho| I)
\end{align}
and for sequences of bipartite state $\rho^{AB} = \{\rho^{AB}_n\}_{n=1}^\infty$,
\begin{align}
\overline{S}(A|B) &= -\underline{D}(\rho^{AB}| I^{A}\otimes \rho^B) \\
\underline{S}(A|B) &= -\overline{D}(\rho^{AB}| I^{A}\otimes \rho^B) \\
\overline{S}(A:B) &= \overline{D}(\rho^{AB}| \rho^{A}\otimes \rho^B)\\
\underline{S}(A:B) &= \underline{D}(\rho^{AB}| \rho^{A}\otimes \rho^B),
\end{align}
giving all the spectral sup(inf)-information rates. Various properties and
relationships of these quantities are explored in \cite{BD1}.
\section{Data Compression for Arbitrary Quantum Sources}
A general quantum source consists of a sequence of density
$\rho = \{\rho_n\}_{n=1}^\infty$ acting on a corresponding
sequence of Hilbert spaces ${\cal{H}} = \{{\cal{H}}_n\}_{n=1}^\infty$.
A compression scheme for such a source, $\rho$, consists of two families
of quantum operations ${\cal{C}}_n$ and ${\cal{D}}_n$. Here
${\cal{C}}_n$ denotes the compression
operation which takes states in the original Hilbert space ${\cal{H}}_n$
to states in a Hilbert space ${\widetilde{{\cal{H}}_n}}$ such that
${\hbox{dim }} {\widetilde{{\cal{H}}_n}} \leq {\hbox{dim }} {{{\cal{H}}_n}}$.
Hence, ${\widetilde{{\cal{H}}_n}}$ can be regarded as the compressed
Hilbert space.
The corresponding decompression operation, ${\cal{D}}_n$, takes states
in ${\widetilde{{\cal{H}}_n}}$ to states in the original Hilbert space
${\cal{H}}_n$.
The compression scheme given by the family of combined
compression decompression maps ${\cal{D}}_n \circ {\cal{C}}_n$ is said to be
\textit{reliable}
if the entanglement fidelity $F(\rho_n, {\cal{D}}_n \circ {\cal{C}}_n)$
tends to $1$ as $n \rightarrow \infty$. Let $P_n$ denote the orthogonal
projection onto ${\widetilde{{\cal{H}}_n}}$.
The \textit{rate} of the compression scheme is determined by
\begin{equation}
R = \limsup_{n\rightarrow \infty} \frac{1}{n} \log M_n,
\end{equation}
where $M_n := {\rm Tr} P_n = \dim {\widetilde{{\cal{H}}_n}}$.
The objective is thus to obtain the optimal rate of reliable compression for a
given source $\rho$. Defining the optimal rate $\mathcal{R}$ as the infimum of
all reliable rates, leads to the following theorem.
\begin{theorem}
\label{coding}
The quantum spectral sup-entropy rate is optimal. Hence,
\begin{equation}
\mathcal{R} = \overline{S}(\rho)
\end{equation}
for a given source $\rho$. Equivalently, $(i)$ if $R> \overline{S}(\rho)$
then there exists a reliable compression scheme of rate $R$, and
$(ii)$ there can be no reliable compression scheme of
rate $R$ for $R< \overline{S}(\rho)$.
\end{theorem}
\begin{proof}[Proof of (i) :]
Suppose $R> \overline{S}(\rho)$.
Consider the compression operation, ${\cal{C}}_n$, defined by its action on
any state $\sigma_n \in {\cal{B}}({\cal{H}}_n)$ as follows:
\begin{equation}
{\cal{C}}_n(\sigma_n):= P_n\sigma_nP_n + \sum_{k} A_k \sigma_n A_k^\dagger,
\end{equation}
where $(a)$ $P_n$, the compression projection, i.e. the orthogonal projection
onto
the compressed Hilbert
space ${\widetilde{{\cal{H}}_n}}$, is given by
\begin{equation}
P_n:= \{\rho_n \ge e^{-n\gamma} I_n \},
\end{equation}
and $(b)$ $A_k := |\chi_0\rangle \langle k|$, with $|\chi_0\rangle$
being a fixed pure state in ${\widetilde{{\cal{H}}_n}}$ and
$\{|k\rangle\}$ being an orthonormal basis for the orthocomplement of
${\widetilde{{\cal{H}}_n}}$. Equivalently,
\begin{equation}
{\cal{C}}_n(\sigma_n):= P_n\sigma_nP_n + {\rm Tr}\bigl((I_n - P_n)\sigma_n\bigr)
| \chi_0\rangle \langle\chi_0 |.
\end{equation}
The corresponding decoding operation ${\cal{D}}_n$ is defined to be the
identity
on ${\widetilde{{\cal{H}}_n}}$.
If $\{C_n^j\}$ and $\{D_n^k\}$ denote
finite sets of Kraus operators of the quantum
operations ${\cal{C}}_n$ and ${\cal{D}}_n$ respectively, then
\begin{equation}
F_n:= F(\rho_n, {\cal{D}}_n \circ {\cal{C}}_n) =
\sum_{jk} |{\rm Tr}\bigl(D_n^k C_n^j \rho_n)|^2.
\label{entfid}
\end{equation}
and hence the entanglement fidelity is given by
\begin{eqnarray}
F(\rho_n, {\cal{D}}_n \circ {\cal{C}}_n) &=&
| {\rm Tr}(P_n \rho_n )|^2 + \sum_k |{\rm Tr}( A_k \rho_n )|^2 \nonumber\\
&\ge & |{\rm Tr}(P_n \rho_n )|^2 \nonumber\\
&\ge & |{\rm Tr}\bigl[P_n (\rho_n - e^{-n\gamma} I_n)\bigr]|^2 \nonumber\\
&= &|{\rm Tr}\bigl[\{\rho_n \ge e^{-n\gamma} I_n \}(\rho_n - e^{-n\gamma}
I_n)\bigr]|^2\nonumber\\
&= &|{\rm Tr}\bigl[\{ \Pi_n(\gamma) \geq 0 \} \Pi_n(\gamma)\bigr]|^2,\nonumber\\
\label{fidlim}
\end{eqnarray}
where $\Pi_n(\gamma) = \rho_n - e^{n\gamma}I_n$.
From the definitions in (\ref{supent}) and (\ref{infdiv}) it follows that
the RHS of \reff{fidlim} tends to $1$ as $n \rightarrow \infty$, for any
$\gamma > \overline{S}(\rho)$.
Utilizing Proposition \ref{cor0}, the dimension of the compression projections
$P_n$ is bounded for each $n$
by
\begin{equation}
\mathrm{Tr} P_n = \mathrm{Tr} \big[ \{ \rho_n \geq e^{-n\gamma} I_n \} \big]
\leq
e^{n\gamma} = e^{n(\overline{S}(\rho) + \delta)}
\end{equation}
for $\delta > 0$. Since this is true for all $\delta > 0$ we have $\mathcal{R}
\leq
\overline{S}(\rho)$.
[Proof of (ii) (Weak Converse):] Suppose $R < \overline{S}(\rho)$.
Without loss of generality, assume that ${\cal{C}}_n$ maps states in
${\cal{H}}_n$ to states in an $M_n$-dimensional Hilbert space
${\widetilde{{\cal{H}}_n}}$, with $M_n = \lfloor e^{nR}\rfloor$. Hence, if
$P_n$ is the orthogonal projection onto ${\widetilde{{\cal{H}}_n}}$ then
${\rm Tr} [P_n] = M_n \le e^{nR}$.
Let $\{C_n^j\}$ and $\{D_n^k\}$ denote
finite sets of Kraus operators for the quantum
operations ${\cal{C}}_n$ and ${\cal{D}}_n$ respectively. Obviously,
$P_n C_n^j = C_n^j$. Further, let $Q_n^k$ be the orthogonal
projection onto the subspace to which
${\widetilde{{\cal{H}}_n}}$ is mapped to by $D_n^k$. Then
$D_n^kC_n^j = D_n^kP_n C_n^j
= Q_n^k D_n^kP_n C_n^j = Q_n^k D_n^k C_n^j$. Moreover,
${\rm Tr} [Q_n^k ] \le {\rm Tr} [P_n]$ since ${\cal{D}}_n$ is a CPTP map.
The entanglement fidelity can be
expressed as
\begin{align}
F_n &= \sum_{jk} |{\rm Tr}\bigl(D_n^k C_n^j \rho_n)|^2\nonumber\\
&= \sum_{jk} |{\rm Tr}\bigl(Q_n^k D_n^k C_n^j \rho_n)|^2 \nonumber\\
&= \sum_{jk} |{\rm Tr}\bigl[ (D_n^k C_n^j \sqrt{\rho_n})(\sqrt{\rho_n}Q_n^k)
\bigr]|^2\nonumber\\
&\le \sum_{jk} {\rm Tr} \bigl[Q^k_n{\rho_n}Q_n^k\bigr]\cdot {\rm Tr}\bigl[ D_n^k
C_n^j {\rho_n}C_n^{j\dag}D_n^{k\dag}\bigr] \label{long}\\
&\le {\rm Tr}\bigl[P_n \rho_n\bigr] \label{long2}\\
&\le {\rm Tr}\bigl[\{\rho_n \ge e^{-n\gamma}I_n\}(\rho_n - e^{-n\gamma}I_n)\bigr]
\nonumber \\
&\phantom{=}\;+ e^{-n\gamma}{\rm Tr} P_n
\label{long1}
\end{align}
To arrive at \reff{long}, we have made use of the Cauchy Schwarz inequality for
the
Hilbert-Schmidt inner product, specifically $|{\rm Tr}(A^\dagger B)|^2 \le
{\rm Tr}(A^\dagger A)
\cdot {\rm Tr}(B^\dagger B)$. The inequality in \reff{long2} uses the inequality
${\rm Tr} Q_n^k\le {\rm Tr} P_n$, and the fact that $\mathcal{C}_n$ and
$\mathcal{D}_n$ are trace preserving maps. The final inequality in \reff{long1}
follows from Lemma \ref{lemma}.
Using the fact
that
${\rm Tr} P_n \le e^{nR}$, we have
\begin{equation}
F_n \leq {\rm Tr}\bigl[\{\rho_n \ge e^{-n\gamma}I_n\}(\rho_n -
e^{-n\gamma}I_n)\bigr]
+ e^{-n(\gamma - R)}.
\label{last}
\end{equation}
Choosing a number $\gamma$ and $\delta > 0$ such that $R = \gamma + \delta <
\overline{S}(\rho)$, the second term on RHS of \reff{last} tends to zero as $n
\rightarrow \infty$. However, since $\gamma < {\overline{S}}(\rho)$
the first term on RHS of \reff{last} does not converge to $1$ as
$n \rightarrow \infty$. Hence, the fidelity does not converge to 1 in the limit
as $n \rightarrow \infty$ and the compression scheme is not reliable.
\end{proof}
The proof of the weak converse above shows that for $R < \overline{S}(\rho)$
the entanglement fidelity cannot approach unity, and hence any compression
scheme will give an error with non-zero probability. To determine the rate at
which the probability of error converges to 1 for any compression protocol we
can equivalently determine the supremum of the rates for which the asymptotic
limit of the entanglement fidelity goes to zero. Here we prove the result for
the \textit{strong converse} rate denoted by $\mathcal{R}^*$.
\begin{theorem}
Coding a source $\rho$ at a rate less than the quantum inf-spectral entropy
rate gives an error with probability equal to one. That is
\begin{equation}
R < \underline{S}(\rho) \implies \lim_{n\rightarrow \infty}F_n = 0.
\end{equation}
or, equivalently $\mathcal{R}^* = \underline{S}(\rho)$.
\end{theorem}
\begin{proof}
From \reff{long1} we can immediately see that for rates $R <
\underline{S}(\rho)$ choosing a $\gamma = R + \delta < \underline{S}(\rho)$ we
obtain
\begin{equation}
\lim_{n\rightarrow \infty} F_n = 0
\end{equation}
and the compression scheme fails with probability approaching 1 as $n
\rightarrow \infty$.
\end{proof}
\subsection{Relationship to the von Neumann Entropy}
For any quantum information source $\rho$, the quantum spectral sup- and inf-
information rates are related to the von Neumann entropy in the following
manner.
\begin{lemma}
The sup-information and inf-information rates are related to the von Neumann
entropy by
\begin{equation}
\underline{S}(\rho) \leq \liminf_{n\rightarrow \infty} \frac{1}{n}S(\rho_n)
\leq \limsup_{n\rightarrow \infty} \frac{1}{n}S(\rho_n) \leq \overline{S}(\rho)
\end{equation}
for any source $\rho$.
\end{lemma}
\begin{proof} Let $\{\lambda_n^i\}$ denote the set of eigenvalues of
state $\rho_n$.
For the first inequality we have
\begin{align}
\frac{1}{n}S(\rho_n) &= - \frac{1}{n} \mathrm{Tr} \big[ \rho_n \log \rho_n
\big] \nonumber \\
&= - \frac{1}{n} \sum_i \lambda^i_n \log \lambda^i_n \nonumber \\
&\geq - \frac{1}{n} \sum_{\lambda_n^i < e^{-n(\underline{S}(\rho) -
\delta)}} \lambda^i_n \log \lambda^i_n \nonumber \\
&\geq - \frac{1}{n} \mathrm{Tr}\big[ \{ \rho_n < e^{-n(\underline{S}(\rho) -
\delta)} \} \rho_n \big] \log e^{-n(\underline{S}(\rho) - \delta)} \nonumber \\
&= (\underline{S}(\rho) - \delta)\mathrm{Tr}\big[ \{ \rho_n <
e^{-n(\underline{S}(\rho) - \delta)} \} \rho_n \big]
\end{align}
and from the definition of $\underline{S}(\rho)$ we have $\lim_{n \rightarrow
\infty} \mathrm{Tr}\big[ \{ \rho_n \leq e^{-n(\underline{S}(\rho) - \delta)} \}
\rho_n \big] = 1$, and this is true for all $\delta > 0$, implying
\begin{equation}
\underline{S}(\rho) \leq \liminf_{n\rightarrow \infty} \frac{1}{n}S(\rho_n)
\end{equation}
Similarly, we have
\begin{align}
\frac{1}{n}S(\rho_n) &= - \frac{1}{n} \sum_{\lambda_n^i \geq
e^{-n(\overline{S}(\rho) + \delta)}} \lambda^i_n \log \lambda^i_n \nonumber \\
&\phantom{=}\qquad \quad - \frac{1}{n} \sum_{\lambda_n^i <
e^{-n(\overline{S}(\rho) + \delta)}} \lambda^i_n \log \lambda^i_n \nonumber \\
&\leq (\overline{S}(\rho) + \delta)\mathrm{Tr}\big[ \{ \rho_n
\geq e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] \nonumber \\
&- \frac{1}{n} \mathrm{Tr}\big[Q_n \rho_n \log \, \rho_n\bigr],
\label{eq11}
\end{align}
where $Q_n := \{ \rho_n < e^{-n(\overline{S}(\rho) + \delta)} \}$. Let
$W_n := Q_n \rho_n Q_n$ and define the normalized state ${\widehat{W}}_n := W_n/(
\mathrm{Tr} W_n)$. Hence,
\begin{align}
\frac{1}{n}S(\rho_n)&\leq (\overline{S}(\rho) + \delta)\mathrm{Tr}\big[ \{ \rho_n
\geq e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] \nonumber \\
&- \frac{1}{n} \mathrm{Tr} W_n \bigl(\log \, W_n + \log \mathrm{Tr} W_n
- \log \mathrm{Tr} W_n \bigr)\nonumber\\
&= (\overline{S}(\rho) + \delta)\mathrm{Tr}\big[ \{ \rho_n
\geq e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] \nonumber \\
&- \frac{1}{n} \mathrm{Tr} W_n S({\widehat{W}}_n) - \frac{1}{n} H(\mathrm{Tr} W_n),
\nonumber\\
&\leq (\overline{S}(\rho) + \delta)\mathrm{Tr}\big[ \{ \rho_n
\geq e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] \nonumber \\
&+ \frac{1}{n} \log d_n\, \mathrm{Tr} W_n - \frac{1}{n} H(\mathrm{Tr} W_n)
\end{align}
In the above, $H(\cdot)$ denotes the Shannon entropy.
and $d_n={\hbox{dim\,}}\mathcal{H}_n$.
Since $\lim_{n \rightarrow \infty} \mathrm{Tr} W_n = \lim_{n \rightarrow \infty} \mathrm{Tr}\big[ \{ \rho_n <
e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] = 0$, the last term
vanishes in this limit. The second term also vanishes under the
assumption that for all $n$
\begin{equation}
\frac{1}{n} \log d_n < \beta
\end{equation}
for some $\beta < +\infty$. Moreover, since
$\lim_{n\rightarrow \infty} \mathrm{Tr}\big[ \{ \rho_n \geq
e^{-n(\overline{S}(\rho) + \delta)} \} \rho_n \big] = 1$, we have
\begin{equation}
\limsup_{n\rightarrow \infty} \frac{1}{n}S(\rho_n) \leq \overline{S}(\rho).
\end{equation}
The remaining inequality follows from the definition of $\liminf$ and
$\limsup$.
\end{proof}
\subsection{Mixed Sources}
Given two sources $\sigma= \{\sigma_n\}_{n=1}^\infty$ and
$\omega=\{\omega_n\}_{n=1}^\infty$, we define the mixed source
$\rho=\{\rho_n\}_{n=1}^\infty$ to be the
source for which
\begin{equation}
\rho_n = t \sigma_n + (1-t) \omega_n
\end{equation}
for $t \in (0,1)$.
\begin{theorem}
For the mixed source $\rho$ the optimal rate $\mathcal{R}$ is given by
\begin{equation}
\mathcal{R} = \max \big[ \overline{S}(\sigma), \overline{S}(\omega)\big],
\end{equation}
the maximum of the rates for either source $\sigma$ or $\omega$.
\end{theorem}
\begin{proof}
Let $\mathrm{Tr}\big[\Pi_n(\gamma)\big] = \mathrm{Tr}\big[ \{ \rho_n \geq e^{-n\gamma}I_n \} (\rho_n - e^{-n\gamma}I_n)\big]$, then from the linearity of the trace operation, we have
\begin{align}
\mathrm{Tr}\big[\Pi_n(\gamma)\big] &= t\, \mathrm{Tr}\big[ \{ \rho_n \geq e^{-n\gamma}I_n \} (\omega_n - e^{-n\gamma}I_n)\big] \nonumber \\
&\phantom{=}\; + (1-t)\, \mathrm{Tr}\big[ \{ \rho_n \geq e^{-n\gamma}I_n \} (\sigma_n - e^{-n\gamma}I_n)\big] \nonumber \\
&\leq t \, \mathrm{Tr}\big[ \{ \omega_n \geq e^{-n\gamma}I_n \} (\omega_n - e^{-n\gamma}I_n)\big] \nonumber \\
&\phantom{=}\; + (1-t) \, \mathrm{Tr}\big[ \{ \sigma_n \geq e^{-n\gamma}I_n \} (\sigma_n - e^{-n\gamma}I_n)\big]
\label{mixed1}
\end{align}
where the inequality follows from Lemma \ref{lemma}. Hence for any $\gamma = \overline{S}(\rho) + \delta$, the limit of the LHS goes to one, and hence both of the traces on the RHS must also approach one in the limit. This then implies that
\begin{equation}
\overline{S}(\rho ) \geq \max \big[ \overline{S}(\sigma), \overline{S}(\omega)
\big]
\end{equation}
as $\delta$ is arbitrary.
To prove the reverse inequality we explicitly construct a sequence of
projection operators. For each $\alpha>0$ we utilize the projections
$P_n^0:=\{\sigma_n \geq e^{-n\alpha}I_n \}$ and $Q_n:=\{\omega_n \geq e^{-n\alpha}I_n
\}$. Let $Q_n$ have the spectral projection $Q_n = \sum_{i=1}^K |i\rangle
\langle i|$,
with $K= {\rm Tr} Q_n$. Starting with $P_n^0$, we define a sequence of projection
operators $P_n^i$, $i=1, \ldots, K$, iteratively, as follows.
For each $i$, if $|i\rangle$ lies in the subspace onto which $P_n^{i-1}$
projects,
then we set $P_n^i = P_n^{i-1}$. Otherwise, we take the component of
$|i\rangle$ orthogonal
to this subspace, say $|i^{\perp}\rangle$, and let $P_n^i = P_n^{i-1} \oplus
|i^{\perp}\rangle \langle i^{\perp}|$.
From Lemma 1 it then follows that
\begin{align}
\mathrm{Tr}\big[\Pi_n(\gamma)\big] &\geq \mathrm{Tr}\big[P^K_n (\rho_n - e^{-n\gamma}I_n)\big] \nonumber \\
&= t\, \mathrm{Tr}\big[ P^K_n (\omega_n - e^{-n\gamma}I_n)\big] \nonumber \\
&\phantom{=}\; + (1-t)\, \mathrm{Tr}\big[ P^K_n (\sigma_n - e^{-n\gamma}I_n)\big] \nonumber \\
&\geq t \, \mathrm{Tr}\big[ \{ \omega_n \geq e^{-n\alpha}I_n \} \omega_n \big] \nonumber \\
&\phantom{=}\; + (1-t) \, \mathrm{Tr}\big[ \{ \sigma_n \geq e^{-n\alpha}I_n \} \sigma_n \big] \nonumber \\
&\phantom{=}\;- e^{-n\gamma}\mathrm{Tr}\big[ P^K_n\big] \nonumber \\
&\geq t \, \mathrm{Tr}\big[ \{ \omega_n \geq e^{-n\alpha}I_n \} \omega_n \big] \nonumber \\
&\phantom{=}\; + (1-t) \, \mathrm{Tr}\big[ \{ \sigma_n \geq e^{-n\alpha}I_n \} \sigma_n \big] \nonumber \\
&\phantom{=}\;- 2e^{-n(\gamma - \alpha)} \label{mixed2}
\end{align}
where $\mathrm{Tr}\big[ P^K_n\big] \leq 2e^{n\alpha}$, as the rank of the projector cannot be greater than the sum of the ranks of the projectors $P^0_n$ and $Q_n$. For every $\delta >
0$ and $\alpha = \max \big[\overline{S}(\sigma), \overline{S}(\omega)\big] +
\delta$, the limit of the sum of first two terms on the RHS goes to 1. By choosing $\gamma = \alpha + \delta$ this implies both the RHS and LHS converge to 1 and hence that
$$ \overline{S}(\rho) \le \max \big[\overline{S}(\sigma),
\overline{S}(\omega)\big].
$$
as $\delta$ is arbitrary.
\end{proof}
\begin{corollary}
The strong converse is given by
\begin{equation}
\mathcal{R}^* = \min \big[\underline{S}(\sigma), \underline{S}(\omega)\big]
\end{equation}
for any mixed source $\rho_n = t \sigma_n + (1-t) \omega_n$, for $t \in (0,1)$.
\end{corollary}
\begin{proof}
Choosing $\gamma$ and $\alpha$ such that the RHS and LHS of \reff{mixed1} and \reff{mixed2} go to zero, respectively, gives the required inequalities.
\end{proof}
A source obeys the strong converse property only if
\begin{equation}
\underline{S}(\rho) = \lim_{n \rightarrow \infty} \frac{1}{n}S(\rho_n) =
\overline{S}(\rho)
\end{equation}
Note that mixed sources do not obey the strong converse property if $\max \big[
\overline{S}(\sigma), \overline{S}(\omega) \big] > \min \big[
\underline{S}(\sigma), \underline{S}(\omega) \big]$. This can easily be shown to hold
for mixtures of stationary memoryless sources with different entropies
$S(\sigma) > S(\omega)$.
\section{Classical Capacity of an Arbitrary Quantum Channel}
\label{classicalcap}
In this section we obtain the classical capacity of a sequence
of arbitrary quantum channels in terms of the inf-spectral mutual information rate of bipartite separable states shared through the channel.
Let $\{{\cal{K}}_Q^{(n)}\}_{n=1}^\infty$ and
$\{{\cal{H}}_Q^{(n)}\}_{n=1}^\infty$
be two sequences of Hilbert spaces, and
let $\Lambda = \{ \Lambda^Q_n \}_{n=1}^{\infty}$ be a sequence
of quantum channels such that, for each $n$,
$$\Lambda^Q_n : {\cal{B}}({\cal{K}}_Q^{(n)}) \mapsto
{\cal{B}}({\cal{H}}_Q^{(n)}).
$$
Here ${\cal{K}}_Q^{(n)}$ denotes the Hilbert space at the input of the
channel $\Lambda^Q_n$, whereas ${\cal{H}}_Q^{(n)}$ denotes the Hilbert space
at its output.
Consider the following scenario. Suppose Alice has a set of messages,
labelled by the elements of the set ${\cal{M}} = \{1,2, \ldots, M_n\},$
which she would like to communicate to Bob, using the
quantum channel $\Lambda^Q_n$.
To do this, she encodes
each message into a quantum state of a physical system with Hilbert space
${\cal{K}}_Q^{(n)}$ and
sends this state to Bob through the quantum channel.
In order to infer the message that Alice communicated to him, Bob makes
a measurement (described by POVM elements) on the state
that he receives. The encoding and decoding operations together define
a quantum error correcting code (QECC). More precisely, a code
${\cal{C}}^{(n)}$ of size $M_n$ is given by a sequence $\{\rho_n^i,
E_n^i\}_{n=1}^{M_n}$
where each $\rho_n^i$ is a state in ${\cal{B}}({\cal{K}}_Q^{(n)})$ and each
$E_n^i$ is a
positive operator acting in ${\cal{H}}_Q^{(n)}$, such that $\sum_{i=1}^{M_n}
E_n^i \le {{{I}}}_n$.
Defining $E_n^0 = I_n - \sum_{i=1}^{M_n} E_n^i$, yields a resolution of
identity
in ${\cal{H}}_Q^{(n)}$. Hence, $\{E_n^i\}_{i=0}^{M_n}$ defines a POVM. An
output $i\ge 1$ would
lead to the inference that the state $\rho_n^i$ was transmitted through the
channel $\Lambda^Q_n$,
whereas the output $0$ is interpreted as a failure of any inference. In other
words, a code ${\cal{C}}^{(n)}$ is given by a triple $(M_n, \phi_n, E_n)$,
where
$\phi_n$ is the encoder, i.e., $\phi_n(i) = \rho_n^i$ for $i \in \{1,2,
\ldots, 2^{nR}\}$, and $E_n = \{E_n^i\}_{i=1}^{M_n}$ is the decoder. The
rate of the code is given by $({1}/{n}){\log M_n}$.
The average probability of error for such a code ${\cal{C}}^{(n)}$ is given by
\begin{equation}
P_e({\cal{C}}^{(n)}):= \frac{1}{M_n} \sum_{i=1}^{M_n} \left(1 - {\rm Tr}(\sigma_n^i
E_n^i)\right),
\label{codeerr}
\end{equation}
$\sigma_n^i$ being the output of the channel when the input is the $i^{th}$
codeword $\rho_n^i$. A quantity $R \in {\bf{R}}$ is said to be an
{\em{achievable rate}} if there exists an
$N \in {\bf{N}}$ such that for all $n \ge N$, there exits a sequence
of codes $\{{\cal{C}}^{(n)}\}_{n=1}^\infty$ with $M_n \ge e^{nR}$,
and $P_e({\cal{C}}^{(n)})\rightarrow 0$ as $n \rightarrow \infty$.
\medskip
The capacity of ${\Lambda}$ is defined as
\begin{equation}
C(\Lambda) := \sup R,
\end{equation}
where $R$ is an achievable rate.
\begin{theorem}
\label{thm_cap}
The classical capacity of a sequence of channels $\Lambda =
\{ \Lambda^Q_n \}_{n=1}^{\infty}$ is given by
\begin{equation}
C(\Lambda) = \max_{\rho^{AQ} \in \mathcal{S}} \underline{S}(A:\Lambda Q)
\label{statethm}
\end{equation}
where $(i)$ $\mathcal{S}$ is the set of sequences of separable states in
${\cal{B}}({\cal{H}}_{AQ})$, with ${\cal{H}}_{AQ}$ being
a sequence of Hilbert spaces ${\cal{H}}_{AQ} :=\{{\cal{H}}_A^{(n)} \otimes
{\cal{K}}_Q^{(n)}\}_{n=1}^\infty$, and $(ii)$ $\underline{S}(A:\Lambda Q)$ is
the
{\em{inf-spectral mutual information rate}} of a sequence of separable density
matrices $\{\rho_n^{A\Lambda Q}\}_{n=1}^\infty$.
\end{theorem}
Consider an arbitrary set ${\cal{X}}^{(n)}$ of indices and define
a separable state
$$\rho_n^{AQ} := \sum_{x \in {\cal{X}}^{(n)}}p_n^x \rho_{n,x}^A \otimes
\rho_{n,x}^Q,$$
acting in a Hilbert space ${\cal{H}}_A^{(n)} \otimes{\cal{K}}_Q^{(n)}$.
The set of codewords that Alice uses, to transmit her messages to Bob,
is a finite subset of the set
$$\{\rho_{n,x}^Q\,:\, x \in {\cal{X}}^{(n)}\}.$$
The state $\rho_n^{AQ}$
can be purified to the state
$$\rho_n^{AA'Q} := \sum_{x \in {\cal{X}}^{(n)}}p_n^x |x\rangle \langle x|^{AA'}
\otimes \rho_{n,x}^Q$$
in ${\cal{B}}({\cal{H}}_A^{(n)}\otimes{\cal{H}}_{A'}^{(n)}
\otimes{\cal{K}}_Q^{(n)})$.
Let $B$ denote the bipartite system with Hilbert space
${\cal{H}}_A^{(n)}\otimes{\cal{H}}_{A'}^{(n)}$ (and thus replace
the superscript $AA'$ by $B$). Let $Q$ denote system
with Hilbert space ${\cal{K}}_Q^{(n)}$.
A state $\rho_n^{B Q}$ of the
form given by \reff{rhoq} is referred to as a
{\em{classical--quantum state}}\footnote{
According to the terminology of \cite{windev} it is the density matrix
which one can associate to a {\em{c-q resource}} given by the ensemble
$\{p_n^x, \rho_{n,x}^Q\}$.}. If $X$ is a random variable with
probability mass function $\{p_n^x : x \in {\cal{X}}^{(n)}\}$, then
the state of the quantum system $Q$ is correlated with the values taken by the
classical index $X$. The state $\rho_n^{BQ}$ therefore represents the
preparation of quantum states $\rho_{n,x}^{Q}$ corresponding to classical
indices
$x \in {\cal{X}}^{(n)}$, according to the apriori distribution $\{p_n^x \}$.
The action of the channel $\Lambda^Q_n$ on the system $Q$
yields the state
\begin{eqnarray}
\rho_n^{B\Lambda Q} &:=& \bigl({{\hbox{id}}}_{B} \otimes\Lambda^Q_n\bigr)
\bigl(\rho_n^{BQ} \bigr)\nonumber\\
&=& \sum_{x \in {\cal{X}}^{(n)}}p_n^x |x\rangle \langle x|^{B}
\otimes\Lambda^Q_n\bigl( \rho_{n,x}^Q
\bigr)\nonumber\\
&:=& \sum_{x \in {\cal{X}}^{(n)}}p_n^x |x\rangle \langle x|^{B} \otimes
\rho_{n,x}^{\Lambda Q}.
\label{rhoq}
\end{eqnarray}
Here the superscript $\Lambda Q$ is used to denote the system $Q$ after
the action of the channel on it.
For the sequence of classical-quantum states $\{\rho_n^{B\Lambda Q}\}$
the {\em{inf-spectral mutual information rate}} is given by
\begin{equation}
\underline{S}(B:\Lambda Q) = \sup \Big\{ \gamma : \lim_{n\rightarrow \infty}
\mathrm{Tr}\big[ \{ \Pi_n(\gamma) \ge 0\} \Pi_n(\gamma)\big] = 1 \Big\}
\label{def1}
\\
\end{equation}
where $\Pi_n(\gamma) := \rho_n^{B\Lambda Q}- \rho_n^B \otimes \rho_n^{\Lambda
Q}$, and $\rho_n^B$,
$\rho_n^{\Lambda Q}$ are the reduced density matrices
of the systems $B$ and $\Lambda Q$ respectively.
The proof of the Theorem \ref{thm_cap} relies on the following
lemma proved in \cite{hayashi03}.
\begin{lemma}
\label{lemma3HN}
For any $n \in {\bf{N}}$, $M \in{\bf{N}}$, and
$\gamma \in {\bf{R}}$, given a probability distribution
$\{p^{{{x}}}_n\}$
on ${\cal{X}}^{(n)}$, there exists
a code ${\cal{C}}^{(n)}$ of size $|{\cal{C}}^{(n)}| = M$, whose average
probability of error satisfies the following bound:
\begin{eqnarray}
P_e({\cal{C}}^{(n)})
&\le & 2\sum_{x \in {\cal{X}}^{(n)}} p^{x}_n
{{\rm Tr}} \bigl[\{ \rho_{n,x}^{\Lambda Q} - e^{n\gamma} {{\rho}}_n^{\Lambda Q}
\le 0\}
\rho_{n,x}^{\Lambda Q}\bigr]\nonumber\\
& & + 4e^{-n\gamma} M,
\label{lem3}
\end{eqnarray}
where $${{\rho}}^{\Lambda Q}_n:= \sum_{x \in {\cal{X}}^{(n)}} p^{x}_{n}
\rho_{n,x}^{\Lambda Q}.$$
\end{lemma}
\bigskip
{\em{Proof of Theorem \ref{thm_cap}}} We shall first prove that for any rate
$0 < R <\underline{S}(B:\Lambda Q)$, the average probability of error
$P_e({\cal{C}}^{(n)}$ vanishes asymptotically. Here $\underline{S}(B:\Lambda
Q)$
denotes the inf-spectral mutual information rate for a sequence of
{\em{classical-quantum states}} $\{\rho_n^{B \Lambda Q}\}_{n=1}^\infty$
and is given by \reff{def1}.
Computing the reduced density matrices of the bipartite state
$\rho_n^{B\Lambda Q}$
(defined by \reff{rhoq}) yields
\begin{equation}
\rho^{B}_{n}\otimes \rho^{\Lambda Q}_{n} = \bigl(\sum_{x } p^{x}_{n}
| x\rangle \langle x|^B \bigr) \otimes {{\rho}}_n^{\Lambda Q},
\label{red1}
\end{equation}
where ${{\rho}}_n^{\Lambda Q} := \sum_{x } p^{x}_{n} \rho_{n,x}^{\Lambda
Q}.$
The difference operator $ \Pi_n(\gamma)$ appearing in \reff{def1} is given by
\begin{equation}
\Pi_n(\gamma) = \sum_{x}p^{x}_n |x\rangle \langle x|^B \otimes (\rho^{\Lambda
Q}_{n,x}
- e^{n\gamma}{{\rho}}_n^{\Lambda Q}).
\end{equation}
Note that
\begin{eqnarray}
&&
\mathrm{Tr}\Big[ \{ \Pi_n(\gamma) \ge 0 \}\Pi_n(\gamma)\Big]\nonumber\\
&=&
\mathrm{Tr}\Big[ \{ \Pi_n(\gamma) \ge 0 \} \Bigl(
\sum_{x \in {\cal{X}}^{(n)}} p^{x}_n
| x\rangle \langle x|^B \otimes \bigl(\rho^{\Lambda Q}_{n,x} -
e^{n\gamma}{{\rho}}_n^{\Lambda Q}
\bigr)\Big]\nonumber\\
&& =
\sum_{x} p^{x}_n\mathrm{Tr}
\Bigl[\{\rho^{\Lambda Q}_{n,x} \ge e^{n\gamma} {{\rho}}_n^{\Lambda Q}
\}\bigl(\rho^{\Lambda Q}_{n,x}-e^{n\gamma}{{\rho}}_n^{\Lambda Q}
\bigr)\Bigr].
\end{eqnarray}
Hence, $\underline{S}(B:\Lambda Q)$ is equivalently given by
$$
\sup \Big\{ \gamma : \lim_{n\rightarrow \infty}
\sum_{x} p^{x}_n\mathrm{Tr}
\bigl[\{\rho^{\Lambda Q}_{n,x} \ge e^{n\gamma}{{\rho}}_n^{\Lambda Q}\}
\bigl(\rho^{\Lambda Q}_{n,x} - e^{n\gamma}{{\rho}}_n^{\Lambda Q}
\bigr)\bigr]=1\Big\}.
$$
This implies that for any $\gamma < \underline{S}(B:\Lambda Q)$,
\begin{equation}\lim_{n\rightarrow \infty}\sum_{x} p^{x}_n\mathrm{Tr}
\bigl[\{\rho^{\Lambda Q}_{n,x} < e^{n\gamma}{{\rho}}_n^{\Lambda Q}\}
\rho^{\Lambda Q}_{n,x}\bigr] = 0.
\label{eq2}
\end{equation}
For $M_n = \lceil e^{nR} \rceil$, Lemma \ref{lemma3HN}
ensures the existence of a sequence of codes
$\{{\cal{C}}^{(n)}\}_{n=1}^\infty$ of size $|{\cal{C}}^{(n)}| = \lceil e^{nR}
\rceil$, such that for each $n$
\begin{eqnarray}
P_e({\cal{C}}^{(n)})
&\le & 2\sum_{x \in {\cal{X}}^{(n)}} p^{x}_n
{{\rm Tr}} \bigl[\{ \rho_{n,x}^{\Lambda Q} - e^{n\gamma} {{\rho}}_n^{\Lambda Q}
\le 0\}
\rho_{n,x}^{\Lambda Q}\bigr]\nonumber\\
& & + 4e^{-n\gamma} \lceil e^{nR} \rceil,
\label{eq22}
\end{eqnarray}
for any $\gamma \in {\bf{R}}$ and $c >0$.
From (\ref{eq2}) it follows that for any $\gamma < \underline{S}(B:\Lambda Q)$,
the first term on the RHS of (\ref{eq22}) vanishes in the
limit $n \rightarrow \infty$.
For all $\delta >0$, there exists $n_0 \in {\bf{N}}$, such that for all $n \ge n_0$, $\lceil e^{nR} \rceil \le e^{n(R + \delta)}$. Hence,
$$ 4e^{-n\gamma} \lceil e^{nR} \rceil
\leq 4^{-n(\gamma- (R + \delta))},$$
which vanishes as $n \rightarrow \infty$ for $\gamma > R + \delta$. Since
$\delta$ is arbitrary, it follows that any rate $R < \gamma
<\underline{S}(B:\Lambda Q)$ is achievable. More generally, any rate
$0 < R <\underline{S}(B:\Lambda Q)$ is achievable.
To prove the (weak) converse we are required only to show that for any code with rate larger than the capacity, there exists a probability distribution on the codewords such that the average probability of error does not vanish asymptotically.
Define a family of codes of size $M_n$ by the average state of the codewords $\rho^Q_n$. Note that the family includes all possible sets of $M_n$ codewords with the same average state. Given the family $\{ M_n, \rho^Q_n \}_{n=1}^{\infty}$ we can extend $\rho^Q_n$ to any separable state $\rho^{AQ}$ on an enlarged Hilbert space. The outcome of any measurement on $A$ is thus classically correlated with a state on $Q$.
Explicitly, we can assign the message that has been sent with the outcome of the set of measurements on $A$, described by a POVM $\{E_{n,i}^A\}$, such that message $i \in \{ 1,2,..., M_n \}$ is generated with probability
\begin{equation}
p_i = \mathrm{Tr}\big[ (E_{n,i}^A \otimes I^Q_n) \rho^{AQ}_n \big].
\label{prob}
\end{equation}
and results in the codeword
\begin{equation}
\rho^Q_{n,i} = \mathrm{Tr}_A \big[ \sqrt{E_{n,i}^A \otimes I^Q_n} \rho^{AQ}_n \sqrt{E_{n,i}^A \otimes I^Q_n}\big]
\end{equation}
which is then sent throught the noisy channel.
\noindent
The average probability of error can thus be expressed as
\begin{align}
P_e({\cal{C}}^{(n)}) &= 1 - \sum_{i=1}^{M_n} p_i {\rm Tr}\big[ E^{Q}_{n,i}\Lambda^Q_n \rho^Q_{n,i})\big] \nonumber \\
&= 1 - \sum_{i=1}^{M_n} {\rm Tr}\big[ (E^A_{n,i}\otimes E^{Q}_{n,i}) \rho^{A\Lambda Q}_n\big],
\end{align}
where $\rho^{A\Lambda Q}_n = (I^A_n \otimes \Lambda^Q_n)\rho^{AQ}_n$.
From Lemma \ref{lemma} it then follows that
\begin{align}
P_e({\cal{C}}^{(n)}) &\geq 1 - \mathrm{Tr}\big[ \{ \Pi_n(\gamma) \geq 0 \} \Pi_n (\gamma) \big] \nonumber \\
&\phantom{=}\;- e^{n\gamma}\mathrm{Tr}\big[ \sum_{i=1}^{M_n} E^A_{n,i}\rho^{A}_n \otimes E^{Q}_{n,i} \rho^{\Lambda Q}_n\big] \nonumber \\
&= 1 - \mathrm{Tr}\big[ \{ \Pi_n(\gamma) \geq 0 \} \Pi_n (\gamma) \big] \nonumber \\
&\phantom{=}\;- e^{n\gamma}\sum_{i=1}^{M_n} p_i \mathrm{Tr}\big[ E^{Q}_{n,i} \rho^{\Lambda Q}_n\big]
\end{align}
with $\Pi_n (\gamma) = \rho^{A\Lambda Q}_n - e^{n\gamma}\rho^{A}_n \otimes \rho^{\Lambda Q}_n$, and where the probability $p_i$ is given by \reff{prob}.
Choosing only those POVMs such that
\begin{equation}
p_i = \mathrm{Tr}\big[ E_{n,i}^A \rho^{A}_n \big] = \frac{1}{M_n}
\end{equation}
is sufficient to show that any code of size $M_n$ is not reliable. In this case
\begin{equation}
P_e({\cal{C}}^{(n)}) \geq 1 - \mathrm{Tr}\big[ \{ \Pi_n(\gamma) \geq 0 \} \Pi_n (\gamma) \big] - \frac{e^{n\gamma}}{M_n}
\label{lem4a}
\end{equation}
and for any $\delta >0$, choose $M_n = \lceil e^{nR} \rceil$ where $R = \underline{S}(A:\Lambda Q) + 2\delta$,
and $\gamma = \underline{S}(A:\Lambda Q) + \delta$. Thus, the third term on the
RHS of \reff{lem4a} vanishes in the limit $n \rightarrow \infty$. However,
the difference of the first two terms does not vanish and we have
$\limsup_{n\rightarrow \infty} P_e({\cal{C}}^{(n)}) \ge \epsilon_0$ for some $\epsilon_0 >0$.
We thus conclude that the classical capacity of a
sequence of channels $\Lambda =
\{ \Lambda^Q_n \}_{n=1}^{\infty}$ is given by
\begin{equation}
C(\Lambda) = \max_{\rho^{BQ} \in \mathcal{Q}} \underline{S}(B:\Lambda Q)
\end{equation}
where $\mathcal{Q}$ denotes the set of sequences of classical--quantum
states in ${\cal{B}}({\cal{H}}_{BQ})$, with ${\cal{H}}_{BQ}$ being
a sequence of Hilbert spaces ${\cal{H}}_{BQ} :=\{{\cal{H}}_B^{(n)} \otimes
{\cal{K}}_Q^{(n)}\}_{n=1}^\infty$,
The monotonicity of
the inf-spectral mutual information rate under CPTP maps
(see \cite{BD1}) implies that
$\underline{S}(B:\Lambda Q) \equiv \underline{S}(AA':\Lambda Q)
\geq \underline{S}(A:\Lambda Q)$.
This ensures that optimization over classical-quantum states is equivalent
to optimization over separable states, thus yielding the statement
\ref{statethm} of Theorem \ref{thm_cap}.
\section{Dense Coding}
Dense coding is the protocol by which prior shared entanglement between
a sender (Alice) and a receiver (BOB) is exploited for sending classical
messages through a noiseless quantum channel. Let ${\rho_n^{AB}}\in
{\cal{H}}_A^{(n)} \otimes {\cal{H}}_B^{(n)} $ be an entangled mixed
state that Alice and Bob initially share. As in Section \ref{classicalcap},
Alice has a set of messages,
labelled by the elements of the set ${\cal{M}}_n = \{1,2, \ldots, M_n\},$
which she wishes to communicate to Bob. However, the quantum channel
that she uses is noiseless. She encodes her messages into her part, $A$, of
the bipartite system $AB$ which is in the state ${\rho_n^{AB}}$. The codewords
are given by
$$\phi_n(i):=\rho_{n,i}^{AB} = ({\cal{E}}_{n,i}^A \otimes
{\hbox{id}}^B)\rho_{n}^{AB},$$
for $i= {\cal{M}}_n$. Here $\phi_{n}$ denotes the encoding map
for a code of size $M_n$ as defined in terms of the CPTP maps
${{\cal{E}}}_{n,i}^A$, $i \in {\cal{M}}_n$.
Let Bob's measurement on the states $\rho_{n,i}^{AB}$
that he receives, be given by
$E_n^{AB} =\{E_{n,i}^{AB}\}_{i=1}^{M_n}$, with each $ E_{n,i}^{AB}\ge 0$ and
$\sum_{i=1}^{M_n} E_{n,i}^{AB} \le I^{AB}_n$. The average probability of
error of the code ${\cal{C}}^{(n)}=(M_n, \phi_n^A, E_n^{AB})$ is given by
\begin{equation}
P_e({\cal{C}}^{(n)}):= \frac{1}{M_n} \sum_{i=1}^{M_n} \left(1 -
{\rm Tr}(\rho_{n,i}^{AB}
E_{n,i}^{AB})\right),
\label{codeerr2}
\end{equation}
The dense coding capacity for a sequence of bipartite states
$\rho^{AB} = \{ \rho^{AB}_n \}_{n=1}^{\infty}$ is defined as
\begin{equation}
C_{DC} := \sup R,
\end{equation}
where $R$ is an achievable rate.
\medskip
\begin{theorem}
The dense coding capacity for a sequence of bipartite states $\rho^{AB} = \{
\rho^{AB}_n \}_{n=1}^{\infty}$ is given by
\begin{equation}
C_{DC} = \log d - \min_{\Lambda} \overline{S}(\Lambda A |B)
\label{cdc}
\end{equation}
where $\Lambda = \{ \Lambda_n^A \}_{n=1}^{\infty}$ is a sequence of CPTP maps
on $A$.
\end{theorem}
\begin{proof}[Converse]
For a code ${\cal{C}}^{(n)}$ of $M_n$ codewords
$\rho_{n,i}^{AB} = (\phi_{n,i}^A \otimes {\hbox{id}}^B)\rho_{n}^{AB}$, and
measurement operators $E_{n,i}^{AB}$, $i=1, \ldots, M_n$, the average
probability of error \reff{codeerr2} satisfies
\begin{align}
P_e({\cal{C}}^{(n)})
&\geq 1 - \frac{1}{M_n} \sum_i \mathrm{Tr}\big[E_{n,i}^{AB}
\rho_{n,i}^{AB} - e^{-n\gamma} I_n^A \otimes \rho^{B}_n\big]\nonumber\\
&-
\frac{e^{-n\gamma}}{M_n}\mathrm{Tr}\Big[ E_{n,i}^{AB}
(I_n^A \otimes \rho^{B}_n) \Big] \nonumber \\
&\geq 1 - \frac{1}{M_n} \sum_i \mathrm{Tr}\big[ \Pi_{n,i}(\gamma) \big] -
\frac{e^{-n\gamma}}{M_n}\mathrm{Tr}\,I_n^A \nonumber \\
&\geq
1 - \max_i \mathrm{Tr}\big[ \Pi_{n,i}(\gamma) \big] - \frac{e^{n(\log d -
\gamma)}}{M_n}
\end{align}
where $\Pi^i_n(\gamma) = \{ \rho_{n,i}^{AB}\geq e^{-n\gamma} I_n^A
\otimes \rho^{B}_n \} \big( \rho_{n,i}^{AB} - e^{-n\gamma} I_n^A \otimes
\rho^{B}_n \big)$. In the above we have used Lemma \ref{lemma} and the facts
that $\sum_i E_{n,i}^{AB} \le I_n^{AB}$ and ${{\rm Tr}\,{I^A_n}}= e^{n \log d}$.
If we then assume that $M_n \geq e^{nR} = \log d - \min_{\Lambda}
\overline{S}(\Lambda A|B) + 2\delta$ for some $\delta > 0$,
then we can choose $\gamma = \min_{\phi} \overline{S}(\phi A|B) - \delta$,
and we find
\begin{equation}
\limsup_{n\rightarrow \infty} P_e({\cal{C}}^{(n)})
\geq \epsilon_0 > 0
\end{equation}
implying $C_{DC} \leq \log d - \min_{\phi} \overline{S}(\phi A|B)$.
\end{proof}
\begin{proof}[Coding]
Lemma \ref{lemma3HN}, adapted to the case of dense coding, states that
for any $n \in {\bf{N}}$, $M \in{\bf{N}}$, and
$\gamma \in {\bf{R}}$, given a probability distribution
$\{p^{{{x}}}_n\}$
on ${\cal{X}}^{(n)}$, where ${\cal{X}}^{(n)}$ is a finite set of
indices, there exists
a code ${\cal{C}}^{(n)}$ of size $|{\cal{C}}^{(n)}| = M$, whose average
probability of error satisfies the bound
\begin{eqnarray}
P_e({\cal{C}}^{(n)})
&\le & 2\sum_{x \in {\cal{X}}^{(n)}} p^{x}_n
{{\rm Tr}} \bigl[\{ \rho_{n,x}^{AB} < e^{n\gamma} {{\rho}}_n^{AB} \}
\rho_{n,x}^{AB}\bigr]\nonumber\\
& & + 4e^{-n\gamma} M,
\label{lem33}
\end{eqnarray}
where $${{\rho}}_n^{AB}:= \sum_{x \in {\cal{X}}^{(n)}} p^x_n
\rho_{n,x}
^{AB}.$$
Choose ${\cal{X}}^{(n)}$ to be a set of size $N_n= d^{2n}$
and define a probability distribution $\{p_n^x\}$ on it,
with $p_n^x = 1/{N_n} = e^{-2n \log d}$ for each $x \in {\cal{X}}^{(n)}$.
Further, consider states $\rho_{n,x}^{AB}$ defined as follows:
$$\rho_{n,x}^{AB} := \bigl(\mathcal{U}_{n,x}^A {{\Lambda}}_{n}^A
\otimes {\hbox{id}}^B)\bigr)\rho_{n}^{AB}.$$
Here ${{\Lambda}}_{n}^A$ denote quantum operations for which the
sequence $\{{{\Lambda}}_{n}^A\}_{n=1}^\infty$ minimizes
$\overline{S}(\Lambda A|B)$, and $(ii)$ $\mathcal{U}_{n,x}^A,$ $x \in
{\cal{X}}^{(n)}$, denotes
unitary encodings with the shift-multiply operators
$U_{(p,q)}$, with $p,q \in \{0,1,\ldots ,(d^n -1) \}$,
which are defined as follows (\cite{hiroshima, bowen1}):
$$U_{(p,q)}|j\rangle = e^{\frac{2\pi p j}{d}}|j + q\, ( \textrm{mod}\, d)\rangle,$$
with $\{|j\rangle :j \in \{0,1,\ldots ,(d^n -1) \}$ being an orthonormal
basis in a $d^n$-dimensional Hilbert space.
Let $$\rho_{n}^{\Lambda AB} :=
({\Lambda}_{n}^A \otimes {\hbox{id}}^B)\rho_{n}^{AB}.$$
For the ensemble $\{p^x_n, \rho_{x,n}^{AB}\}$
\begin{eqnarray}
\sum_{x \in {\cal{X}}^{(n)}} p^x_n
\rho_{n,x}^{AB}
&=&\sum_{x \in {\cal{X}}^{(n)}} p^x_n \bigl(\mathcal{U}_{n,x}^A
\otimes {\hbox{id}}^B)\rho_{n}^{\Lambda AB}\nonumber\\
&=& \frac{I^A_n}{d^n} \otimes \rho^B_n,
\end{eqnarray}
where $\rho^B_n$ is the reduced density matrix of the state
$\rho_n^{\Lambda AB}$.
\medskip
For the ensemble $\{p^x_n, \rho_{x,n}^{AB}\}$ defined above, let
$$\alpha_n:= \sum_{x \in {\cal{X}}^{(n)}} p^{x}_n
{{\rm Tr}} \bigl[\{ \rho_{n,x}^{AB} \ge e^{n\gamma} {{\rho}}_n^{AB} \}
\rho_{n,x}^{AB}\bigr]$$
We have that
\begin{align}
\alpha_n &\geq \frac{1}{N_n}\sum_{x \in {\cal{X}}^{(n)}}
{{\rm Tr}} \bigl[\{ \rho_{n,x}^{AB} \ge e^{n\gamma} {{\rho}}_n^{AB} \} \nonumber\\
&\phantom{=}\; \times \bigl(\rho_{n,x}^{AB} - e^{n\gamma} {{\rho}}_n^{AB} \bigr)
\bigr]\nonumber\\
&= {{\rm Tr}} \bigl[\{ \rho_{n}^{\Lambda AB} \ge e^{-n(\log d -\gamma)}
I^A_n \otimes \rho^B_n \} \nonumber\\
&\phantom{=}\;\times \bigl(\rho_{n}^{\Lambda AB} - e^{-n(\log d - \gamma)} {I^A_n}\otimes \rho^B_n \bigr)
\bigr].\label{dc}
\end{align}
In the above we have made use of the fact that the trace remains invariant
under a unitary transformation.
If $\gamma = \log d -
\overline{S}({\Lambda} A|B) -
\delta$ for any $\delta > 0$, the RHS of \reff{dc} goes to one as
$n \rightarrow \infty$. Hence the RHS of \reff{lem33} vanishes asymptotically,
implying that a rate
$R = \log d - \min_{\Lambda} \overline{S}(\Lambda A|B) - \delta$ is achievable for any $\delta > 0$.
\end{proof}
\subsection{Reduction to the {i.i.d.} Case}
For entanglement resources which are tensor products of identical bipartite
states $\rho^{AB}_N = \varrho_{AB}^{\otimes N}$, with $\varrho_{AB}
\in {\cal{B}}({\cal{H}})$, the dense coding capacity was shown in \cite{horodecki} to be given by
\begin{equation}
C_{DC} = \log d + S(B) - \inf_N \inf_{\Lambda_A^{(N)}} \frac{1}{N}
S\big( (\Lambda_A^{(N)}\otimes {\hbox{id}}_B^{(N)})\varrho_{AB}^{\otimes N} \big).
\label{hor}
\end{equation}
Here $S(B)=S(\varrho_B)$, where $\varrho_B$ is the reduced density
matrix of the system $B$, corresponding to the state $\varrho_{AB}$.
For sequences of {i.i.d.} states
${{\omega}} = \{ \vartheta^{\otimes n} \}_{n=1}^{\infty}$ and
${\sigma} = \{ \varsigma^{\otimes n} \}_{n=1}^{\infty}$, Theorem 4 of
\cite{nagaoka02}
states that
\begin{equation}
\underline{D}(\omega\| \sigma) = D(\vartheta \| \varsigma ) = \overline{D}(\omega \| \sigma).
\label{thm4}
\end{equation}
For bipartite states $\vartheta= \vartheta_{AB}$ and $\varsigma=
\tfrac{1}{d}I_A \otimes \vartheta_B$, \reff{thm4} implies that
\begin{equation}
\overline{S}(A|B) = S(A|B) = \underline{S}(A|B),
\end{equation}
where $S(A|B) = S(\vartheta^{AB}) - S(\vartheta^B)$.
This is because $\log d - \overline{S}(A|B) = \underline{D}(\omega \| \sigma)
= D(\vartheta \| \varsigma ) = \log d - \frac{1}{n}
S\bigl(\vartheta_{AB}^{\otimes n}|\vartheta_{B}^{\otimes n}\bigr) = \log d - S(A|B)$, and similarly for
$\overline{D}(\omega \| \sigma)$. If instead, we choose
$\vartheta_{AB}$ and $\varsigma_{AB}$ to be states in ${\cal{B}}({\cal{H}}^{\otimes N})$, given by
$$
\vartheta_{AB} := \bigl(\Lambda_A^{(N)}\otimes {\hbox{id}}_B^{(N)} \bigr)
\varrho_{AB}^{\otimes N},$$
and
$$\varsigma_{AB} = \tfrac{1}{d^N}I_A^{(N)} \otimes \varrho_B^{\otimes N},$$
then \reff{thm4} yields the identity
\begin{equation}
\overline{S}(\Lambda^{(N)} A|B) = \frac{1}{N}S\Bigl((\Lambda_A^{(N)}\otimes
{\hbox{id}}_B^{N})\varrho_{AB}^{\otimes N}\Bigr) - S(\varrho_B).
\end{equation}
Hence, in this case our expression \reff{cdc} for the dense coding capacity
reduces to \reff{hor}.
|
2,877,628,090,846 | arxiv | \section{Introduction}\label{section1}
We work over an infinite field $\FF$ of arbitrary characteristic $p=\Char{\FF}\geq0$. All vector spaces, algebras, modules as well as tensor products are over $\FF$ and all algebras are associative with unity unless otherwise stated. All ideals are two-sided.
\subsection{Notations}
Given $n>1$ we consider $n\times n$ {\it generic} matrices $X_k=(x_{ij}(k))_{1\leq i,j\leq d}$ ($k\geq 1$) with entries from the following polynomial algebra
$$R=R_{n}=\FF[x_{ij}(k)\,|\,1\leq i,j\leq n,\, k\geq 1].$$
Denote coefficients in the characteristic polynomial
of an arbitrary $n\times n$ matrix $A$ by $\sigma_t(A)$, i.e.,
$$\det(\lambda E -A )=\sum_{t=0}^{n} (-1)^t\lambda^{n-t}\sigma_t(A).$$
So, $\sigma_0(A)=1$, $\sigma_1(A)=\tr(A)$ and $\sigma_n(A)=\det(A)$.
The algebra of {\it matrix $GL(n)$-invariants} $R^{GL(n)}$ is known to be generated by $\si_t(A)$, where $1\leq t\leq n$ and $A$ is a monomial in generic matrices. Moreover, we can assume that $A$ ranges over {\it primitive} monomials, i.e., $A\neq B^l$ for $l>1$ and a monomial $B$ in generic matrices. The mentioned generators of $R^{GL(n)}$ were found by Sibirskii~\cite{Sibirskii_1968} and Procesi~\cite{Procesi_1976} in characteristic zero case and by Donkin~\cite{Donkin_1992a} in the general case. The formal definition of $R^{GL(n)}$ together with some properties can be found, for example, in~\cite{DKZ_2002}.
The algebra of {\it $n\times n$ matrices with forms} (or, the algebra of {\it concomitants})
$$\C_n=\alg_{\FF}\{X_1,X_2,\ldots,fE\}$$
is generated by generic matrices and $fE$, where $f$ ranges over $R^{GL(n)}$ and $E$ stands for the identity $n\times n$ matrix. The ideal of identities for the algebra $M_n(\FF)$ of $n\times n$ matrices over $\FF$ coincides with the ideal of identities for $\alg_{\FF}\{X_1,X_2,\ldots\}\subset\C_n$. So a description of identities for $\C_n$ can be applied to the problem of investigation of identities for $M_n(\FF)$. Note that the identities for $M_n(\FF)$ are described only in the case of $n=2$ and $p\neq2$ (see~\cite{Razmyslov_1973},~\cite{Koshlukov_2001},~\cite{Koshlukov_2004}). In particular, it is shown that the T-ideal of identities for $M_n(\FF)$ is finitely based in the case of $n=2$ and $p\neq2$, but it is an open problem for $n=p=2$ as well as in the case of $n>2$ and $p>0$.
We define the following notions.
\begin{enumerate}
\item[$\bullet$] Let $\X$ be the semigroup (without unity) freely generated by {\it letters} $x_1,x_2,\ldots$ and $\X^{\#}=\X\sqcup\{1\}$.
\item[$\bullet$] Let $\FF\X$ and $\FF\X^{\#}$ be the vector spaces with the bases $\X$ and $\X^{\#}$, respectively. Note that elements of $\FF\X$ and $\FF\X^{\#}$ are {\it finite} linear combinations of monomials from $\X$ and $\X^{\#}$, respectively.
\item[$\bullet$] Define a homomorphism of algebras $\phi_n:\FF\X^{\#}\to \alg_{\FF}\{E,X_1,X_2,\ldots\}$ by $1\to E$ and $x_k\to X_k$ for all $k\geq1$.
\end{enumerate}
Consider a {\it free} algebra $\algF$ for $R^{GL(n)}$, i.e., $\algF$ is a free commutative $\FF$-algebra, equipped with a surjective homomorphism $\Phi_{\algF}:\algF\to R^{GL(n)}$, whose kernel is called the {\it ideal of relations} for $R^{GL(n)}$ with respect to $\algF$. Then the algebra $\algF\otimes \FF\X^{\#}$ is called a {\it free} algebra for $\C_n$ and the kernel of the surjective homomorphism
$$\Psi_{\algF}:\algF\otimes \FF\X^{\#} \to \C_n,\qquad f\otimes b \to \Phi_{\algF}(f)\, \phi_n(b)$$
is the {\it ideal of relations} for $\C_n$ with respect to $\algF\otimes \FF\X^{\#}$. There are several ways to introduce a free algebra $\algF$ for $R^{GL(n)}$ and, consequently, for $\C_n$. Below we consider
\begin{enumerate}
\item[$\bullet$] the {\it absolutely} free algebra $\si\X$ for $R^{GL(n)}$,
\item[$\bullet$] the {\it large} free algebra $\AlgLarge$ for $R^{GL(n)}$ with the ideal of relations $\KLarge{n}$,
\item[$\bullet$] the {\it small} free algebra $\AlgSmall{n}$ for $R^{GL(n)}$ with the ideal of relations $\KSmall{n}$,
\item[$\bullet$] the large and small free algebras $\AlgLarge \otimes \FF\X^{\#}$ and $\AlgSmall{n} \otimes \FF\X^{\#}$, respectively, for $\C_n$ with the ideals of relations $\TLarge{n}$ and $\TSmall{n}$, respectively.
\end{enumerate}
Our main results are the following ones:
\begin{enumerate}
\item[$\bullet$] the ideals of relations $\KSmall{n}$ and $\TSmall{n}$ are finitely based (see Theorem~\ref{theo_GL_main});
\item[$\bullet$] the ideals $\KLarge{n}$ and $\TLarge{n}$ are finitely based if and only if $p=0$ (see Lemma~\ref{lemma_GL_fb});
\item[$\bullet$] similar results are obtained in case $p\neq2$ for the ideal of identities with forms of the $\FF$-algebra generated by $n\times n$ generic and transpose generic matrices (see Theorem~\ref{theo_O_main} and Lemma~\ref{lemma_O_fb}).
\end{enumerate}
Let us determine these free algebras.
\begin{enumerate}
\item[$\bullet$] Introduce the natural lexicographical linear order on $\X$ by setting $x_1>x_2>\cdots$ and $ab>a$ for $a,b\in\X$. (Note that we can actually consider any other lexicographical linear order).
\item[$\bullet$] Let $\AlgSmall{n}$ ($\si\X$, respectively) be a ring with unity of commutative polynomials over $\FF$ freely generated by ``symbolic'' elements $\si_t(a)$, where $1\leq t\leq n$ ($t\geq1$, respectively) and $a$ ranges over polynomials from $\FF\X$ with coefficient $1$ in the highest term with respect to the introduced lexicographical order on $\X$. Define
$$\si_t(\al a)=\al^t\si_t(a)$$
for $\al\in\FF$ and denote $\si_0(a)=1$, $\tr(a)=\si_1(a)$. Note that $\si_t(0)=0$ and $\AlgSmall{n}\subset \si\X$.
\item[$\bullet$] We say that $a,b\in\X$ are {\it cyclic equivalent} and write $a\stackrel{c}{\sim} b$
if $a=a_1a_2$ and $b=a_2a_1$ for some $a_1,a_2\in\X^{\#}$.
\item[$\bullet$] Let $\EX\subset\X$ be a subset of maximal (with respect to the introduced lexicographical order on $\X$) representatives of $\stackrel{c}{\sim}$-equivalence classes of {\it primitive} elements, i.e., for $a\in\EX$ we have $a\neq b^l$ for all $b\in\X$ and $l>1$.
\item[$\bullet$] Assume that $\AlgLarge$ is a ring with unity of commutative polynomials over $\FF$ freely generated by ``symbolic'' elements $\si_t(a)$, where $t>0$ and $a\in\EX$.
\end{enumerate}
There are the following maps between the defined free algebras. By Lemma~\ref{lemma_GL_Donkin}, we have the surjective homomorphism $\piLarge:\si\X\to\AlgLarge$. Define a surjective homomorphism $\piSmall{n}:\si\X\to\AlgSmall{n}$ by
$$\piSmall{n}(\si_t(a))=\left\{
\begin{array}{rl}
\si_t(a),& 1\leq t\leq n\\
0,& t> n\\
\end{array}
\right..$$
Consider the surjective homomorphism
$$\PhiAbs{n}:\si\X\to R^{GL(n)}$$
such that $\si_t(a) \to \si_t(\phi_n(a))$ for $1\leq t\leq n$ and $\si_t(a) \to 0$ for $t>n$, where $a\in\FF\X$. Since
$$\si_t(\al A)=\al^t\si_t(A)$$
holds for an arbitrary $n\times n$ matrix $A$ over a commutative $\FF$-algebra and $1\leq t\leq n$, the homomorphism $\PhiAbs{n}$ is well-defined. Similarly, we define surjective homomorphisms
$$\PhiLarge{n}:\AlgLarge\to R^{GL(n)}\;\text{ and }\;\PhiSmall{n}:\AlgSmall{n}\to R^{GL(n)}.$$
Its kernels $\KLarge{n}$ and $\KSmall{n}$, respectively, are the ideals of relations for $R^{GL(n)}$ in the large and small free algebra, respectively. Then it is well-known that the following diagram is commutative. Namely, its left triangle is commutative by the definition and its right triangle is commutative by Remark~\ref{remark_GL_diagram} (see below).
$$
\begin{picture}(0,120)
\put(0,95)
\put(0,-2){\vector(0,-1){58}
\put(15,0){\vector(3,-2){35}
\put(-15,0){\vector(-3,-2){35}
\put(-11,5){$\si\X$
\put(-75,-33){$\AlgSmall{n}$
\put(50,-33){$\AlgLarge$
\put(50,-40){\vector(-3,-2){35}
\put(-50,-40){\vector(3,-2){35}
\put(110,0){\vector(-3,-2){35}
\put(-110,0){\vector(3,-2){35}
\put(-6,-75){$R^{GL(n)}$
\put(-125,5){$\KSmall{n}$
\put(115,5){$\KLarge{n}$
\put(3,-33){$\scriptstyle\PhiAbs{n}$
\put(-40,-8){$\scriptstyle\piSmall{n}$
\put(33,-8){$\scriptstyle\piLarge$
\put(-35,-48){$\scriptstyle\PhiSmall{n}$
\put(25,-48){$\scriptstyle\PhiLarge{n}$
\put(-20,-90){\text{Diagram 1.}
\end{picture}
$
\noindent{}The homomorphisms $\PhiLarge{n}$ and $\PhiSmall{n}$ induce surjective homomorphisms
$$\PsiLarge{n}=\PhiLarge{n}\otimes\phi_n:\AlgLarge \otimes \FF\X^{\#} \to \C_n\;\text{ and }\;
\PsiSmall{n}=\PhiSmall{n}\otimes\phi_n:\AlgSmall{n}\otimes \FF\X^{\#} \to \C_n,$$
respectively. Its kernels $\TLarge{n}$ and $\TSmall{n}$, respectively, are the ideals of relations for $\C_n$ in the corresponding free algebras. For short, we write $\si_t(a)b$ for $\si_t(a)\otimes b$. We can depict the introduced maps as follows:
$$
\begin{picture}(0,120)
\put(0,95)
\put(-100,-33){$\AlgSmall{n}\otimes \FF\X^{\#}$
\put(35,-33){$\AlgLarge\otimes \FF\X^{\#}$
\put(50,-40){\vector(-3,-2){35}
\put(-50,-40){\vector(3,-2){35}
\put(110,0){\vector(-3,-2){35}
\put(-110,0){\vector(3,-2){35}
\put(-6,-75){$\C_n$
\put(-125,5){$\TSmall{n}$
\put(115,5){$\TLarge{n}$
\put(-35,-48){$\scriptstyle\PsiSmall{n}$
\put(25,-48){$\scriptstyle\PsiLarge{n}$
\put(-20,-90){\text{Diagram 2.}
\end{picture}
$
We say that an ideal $J$ of $\AlgSmall{n} \otimes \FF\X^{\#}$ is a {\it $\Tidskew$-ideal} if it is stable with respect every endomorphism $\varphi$ preserving $\si_t$, i.e.,
$$\varphi(\si_t(a)b)=\varphi(\si_t(a))\varphi(b)\text{ and }\varphi(\si_t(a))=\si_t(\varphi(a))$$
for all $a,b\in\FF\X$. These endomorphisms are determined by substitutions $x_k\to a_k$, where $a_k\in \FF\X$, $k>0$, and we call them {\it substitution} endomorphisms. A $\Tid$-ideal $J$ is {\it finitely based} if it is generated by a finite set $f_1,\ldots,f_s$ as $\Tid$-ideal, i.e., the ideal $J$ is generated by $\varphi(f_1),\ldots,\varphi(f_s)$, where $\varphi$ ranges over substitution endomorphisms. Similarly, we define the notion of a $\Tid$-ideal for $\AlgLarge \otimes \FF\X^{\#}$, $\AlgLarge$, and $\AlgSmall{n}$. Obviously, $\KLarge{n}$, $\KSmall{n}$, $\TLarge{n}$, and $\TSmall{n}$ are $\Tid$-ideals.
\subsection{Results for $\C_n$}
In case $p=0$ Razmyslov~\cite{Razmyslov_1974} and Procesi~\cite{Procesi_1976} showed that the $\Tid$-ideal $\KLarge{n}$ is generated by a single identity. In particular, $\KSmall{n}$, $\TLarge{n}$, $\TSmall{n}$ are finitely based in characteristic zero case. In case $p>n$ results of Samoilov~\cite{Samoilov_2007} imply that $\KSmall{n}$ and $\TSmall{n}$ are finitely based. In the case of arbitrary characteristic Zubkov~\cite{Zubkov_1996} described an infinite generating set for the $\Tid$-ideal $\KLarge{n}$ (see Theorem~\ref{theo_Zubkov}) and, therefore, for the ideals $\KSmall{n}$, $\TLarge{n}$, $\TSmall{n}$.
In our main result we established a finite generating sets for the $\Tid$-ideals $\KSmall{n}$ and $\TSmall{n}$ (see Theorem~\ref{theo_GL_main} and Remark~\ref{remark_theo_GL_main}). In particular, $\KSmall{n}$ and $\TSmall{n}$ are finitely based. Necessary definitions are given in Section~\ref{section2}. To prove Theorem~\ref{theo_GL_main}, in Section~\ref{section3} we obtained an essentially smaller than in~\cite{Zubkov_1996} generating set for $\KLarge{n}$ (see Theorem~\ref{theo_GL} and Remark~\ref{remark_theo_GL}). We also showed that $\KLarge{n}$ and $\TLarge{n}$ are finitely based if and only if $p=0$ (see Lemma~\ref{lemma_GL_fb}). Applying Theorem~\ref{theo_GL}, we completed the proof of Theorem~\ref{theo_GL_main} in Section~\ref{section4}.
\subsection{Results for $\CY_n$}
Assume that $p\neq2$. In Section~\ref{section5} we consider identities with forms for the $\FF$-algebra generated by $n\times n$ generic and transpose generic matrices, or, equivalently, identities for the algebra $\CY_n$ generated $X_i$, $X_i^T$, $fE$, where $i>0$ and $f$ ranges over the algebra $R^{O(n)}$ of {\it matrix $O(n)$-invariants}. A description of identities for $\CY_n$ can be applied to the problem of investigation of identities with transpose involution for $M_n(\FF)$. Note that the identities with transpose involution for $M_n(\FF)$ are described only in the case of $n=2$ and $p\neq2$ (see~\cite{Koshlukov_2005}).
Similarly to $\AlgLarge$ we introduce {\it large} free algebra $\AlgLargeY$ for $R^{O(n)}$ with the ideal of relations $\KLargeY{n}$. And similarly to $\AlgSmall{n}$ we introduce {\it small} free algebra $\AlgSmallY{n}$ for $R^{O(n)}$ with the ideal of relations $\KSmallY{n}$. Finally, similarly to $\TLarge{n}$ and $\TSmall{n}$ we define ideals of relations $\TLargeY{n}$ and $\TSmallY{n}$ for $\CY_n$ in the large and small free algebras $\AlgLargeY\otimes \FF\Y^{\#}$ and $\AlgSmallY{n}\otimes \FF\Y^{\#}$, respectively.
In case $p=0$ Procesi~\cite{Procesi_1976} described a finite generating set for the $\Tid$-ideal $\KLargeY{n}$. In particular, $\KSmallY{n}$, $\TLargeY{n}$, $\TSmallY{n}$ are finitely based in characteristic zero case. In the case of arbitrary characteristic an infinite generating set for the $\Tid$-ideal $\KLargeY{n}$ was described in~\cite{Lopatin_free},~\cite{Lopatin_Orel} (see Theorem~\ref{theo_Lopatin}).
We established a finite generating sets for the $\Tid$-ideals $\KSmallY{n}$ and $\TSmallY{n}$ (see Theorem~\ref{theo_O_main}). In particular, $\KSmallY{n}$ and $\TSmallY{n}$ are finitely based. Necessary definitions are given in Section~\ref{section5}. To prove Theorem~\ref{theo_O_main}, in Section~\ref{section6} we obtained an essentially smaller than in~\cite{Lopatin_free} generating set for $\KLargeY{n}$ (see Theorem~\ref{theo_O} and Remark~\ref{remark_theo_O}). We also showed that $\KLargeY{n}$ and $\TLargeY{n}$ are finitely based if and only if $p=0$ (see Lemma~\ref{lemma_O_fb}). Applying Theorem~\ref{theo_O}, we completed the proof of Theorem~\ref{theo_O_main} in Section~\ref{section7}. Note that the proof of Theorem~\ref{theo_O_main} uses the same approach as the proof of Theorem~\ref{theo_GL_main}, but it is essentially more difficult. Namely, instead of core Lemmas~\ref{lemma_GL_sets}, \ref{lemma_GL_key} in case of Theorem~\ref{theo_GL_main} we need
Lemmas~\ref{lemma_O_sets1}, \ref{lemma_O_key1}, \ref{lemma_O_sets2}, \ref{lemma_O_key2} to prove Theorem~\ref{theo_O_main}.
\begin{remark}
The notion algebra of matrix $GL(n)$-invariants $R_{n,d}^{GL(n)}$ from papers~\cite{Donkin_1992a}, \cite{Procesi_1976}, \cite{Razmyslov_1974}, \cite{Sibirskii_1968}, \cite{Zubkov_1996} is slightly different from ours. Namely, the algebra $R_{n,d}^{GL(n)}$ from the mentioned papers is generated by $\si_t(A)$, where $1\leq t\leq n$ and $A$ is a monomial in $X_1,\ldots,X_d$. Since $R^{GL(n)}=\bigcup_{d>0} R_{n,d}^{GL(n)}$, part~2 of Theorem~\ref{theo_Zubkov} holds for $R^{GL(n)}$. Similar remark also holds for the algebra of matrix $O(n)$-invariants $R^{O(n)}$ from Section~\ref{section5}.
\end{remark}
\section{Relations}\label{section2}
Denote $\NN=\{1,2,\ldots\}$ and $\NN_0=\NN\sqcup \{0\}$. Given $\un{t}=(t_1,\ldots,t_u)\in\NN^u$, we write $|\un{t}|$ for $t_1+\cdots+t_u$ and $\#\un{t}$ for $u$. For short, we write $1^t$ for $(1,\ldots,1)$ ($t$ times).
Let $\algA=\bigoplus_{k\in\NN_0} \algA_k$ be a graded algebra with $\algA_0=\FF$, $f,h,h_1,\ldots,h_r\in\algA$, and $J\vartriangleleft\algA$ be an ideal. We say that the relation $f=h$ belongs to the ideal $J$ (or, equivalently, holds modulo $J$) if $f-h\in J$. We also say that the relation $f=0$ follows from relations $h_1=0,\ldots,h_r=0$ if $f$ belongs to the ideal generated by $h_1,\ldots,h_r$. The relation $f=0$ is said to belong to $J$ modulo relations $h_1=0,\ldots,h_r=0$ if $f$ belongs to the ideal generated by $J,h_1,\ldots,h_r$. If $f=\sum_{i=1}^r \al_i f_i h_i$, where $\al_i\in\FF$ and $f_i,h_i\in\algA$ are homogeneous elements of positive degree ($1\leq i\leq r$), then we write $f\equiv0$. If $f-\sum_{i=1}^r \al_i f_i h_i$ belongs to $J$, where $\al_i,f_i,h_i$ are the same as above, then we say that $f\equiv0$ holds modulo $J$.
For $f=\si_t(a)\in\si\X$ with $a\in\X$ we set $\deg(f)=t\deg(a)$ and $\deg_x(f)=t\deg_x(a)$, where $x$ is a letter and $\deg_x(a)$ stands for a degree of the monomial $a$ in the letter $x$. In the same way we define a degree for elements of $\AlgLarge$ and $\AlgSmall{n}$. Denote the multidegree of $a\in\X$ by $\mdeg(a)=(\de_1,\de_2,\ldots)$, where $\de_i=\deg_{x_i}(a)$. For short, we write $\mdeg(a)=(\de_1,\ldots,\de_d)$ in case $\de_i=0$ for all $i>d$.
We use notation
$\{\ldots\}_{m}$ for {\it multisets}, i.e., given an equivalence $=$ on a set $S$ and
$a_1,\ldots,a_r,b_1,\ldots,b_s\in S$, we write $\{a_1,\ldots,a_r\}_{m} = \{
b_1,\ldots,b_s\}_{m}$ if and only if $r=s$ and
$$\#\{1\leq j\leq r\,|\,a_j=a_i\}=\#\{1\leq j\leq r\,|\,b_j=a_i\}$$
for all $1\leq i\leq r$. We also refer to $\{a_1,\ldots,a_r\}_{m}$ as a {\it multisubset} of $S$.
Consider some relations for $\C_n$ and $R^{GL(n)}$. Given $\un{t}\in\NN_0^u$, we denote by $\Omega(\un{t})$ the set of multisets
$$\omega=\{\underbrace{e_1,\ldots,e_1}_{k_1},\ldots,\underbrace{e_q,\ldots,e_q}_{k_q}\}_{m}
$
such that
\begin{enumerate}
\item[$\bullet$] $e_1,\ldots,e_q \in\EX$ are pairwise different and $k_1,\ldots,k_q\in\NN$ ($q>0$);
\item[$\bullet$] $k_1\mdeg(e_1)+\cdots+k_q\mdeg(e_q)=\un{t}$.
\end{enumerate}
We set $\si(\omega)=(-1)^{k_1+\cdots+k_q} \si_{k_1}(e_1)\cdots\si_{k_q}(e_q)$.
For $\un{x}=(x_1,\ldots,x_u)$ we define $\si_{\un{t}}(\un{x})\in \si\X$ as follows:
\begin{eq}\label{eq1}
\si_{\un{t}}(\un{x})= (-1)^{|\un{t}|}\!\!\! \sum_{\omega\in \Omega(\un{t})} \si(\omega).
\end{eq
If $\Omega(\un{t})$ is empty, then we set $\si_{\un{t}}(\un{x})=1$. For $t>0$ denote
$$F_t(\un{x})=\sum\si_{\un{t}}(\un{x}),$$
where the sum is taken over all $\un{t}\in\NN_0^u$ with $|\un{t}|=t$. For $\un{a}=(a_1,\ldots,a_u)$ with $a_1,\ldots,a_u\in\FF\X$ we set that $\si_{\un{t}}(\un{a})$ and $F_t(\un{a})$ are the results of substitutions $x_1\to a_1,\ldots,x_u\to a_u$ in $\si_{\un{t}}(\un{x})$ and $F_t(\un{x})$, respectively. By Amitsur's formula~\cite{Amitsur_1980}, for $1\leq t\leq n$ we have that
\begin{eq}\label{eq_Amitsur}
\si_t(a_1+\cdots+a_u)=F_t(\un{a})
\end{eq
is a relation for $R^{GL(n)}$, i.e., belongs to the kernel of $\PhiAbs{n}$.
\begin{example}\label{ex_21} Taking the image of relation~(\ref{eq_Amitsur}) in $R^{GL(n)}$ we obtain that for an arbitrary $n\times n$ matrices $A,B$ over a commutative $\FF$-algebra the following equalities hold:
\begin{enumerate}
\item[$\bullet$] $\si_2(A+B)=\si_2(A)+\si_2(B)+\tr(A)\tr(B)-\tr(AB)$,
\item[$\bullet$] $\si_3(A+B)=\si_3(A)+\si_3(B)+\si_2(A)\tr(B)-\tr(AB)\tr(A)+\tr(A^2B)$
$\qquad\qquad\qquad\qquad\qquad\quad$ $+\,\si_2(B)\tr(A)-\tr(AB)\tr(B)+\tr(B^2A)$.
\end{enumerate}
\end{example}
\bigskip
For $t\geq1$, $l\geq2$, and an $n\times n$ matrix $A$ over a commutative $\FF$-algebra we have the following well-known formula:
\begin{eq}\label{eq_P}
\si_t(A^l)=\sum\limits_{i_1,\ldots,i_{t l}\geq0}\be^{(t,l)}_{i_1,\ldots,i_{t l}}
\si_1(A)^{i_1}\cdots\si_{t l}(A)^{i_{t l}}
\end{eq
where we assume that $n\geq tl$ is large enough.
Denote the right hand side of~(\ref{eq_P}) by $P_{t,l}(A)$.
In~(\ref{eq_P}) coefficients $\be^{(t,l)}_{i_1,\ldots,i_{rl}} \in \ZZ_p\simeq \ZZ/p\ZZ$
do not depend on $A$ and $n$.
If we take a diagonal matrix $A=\diag(\al_1,\ldots,\al_n)$, $\al_i\in\FF$, then $\si_t(A^l)$ is a symmetric polynomial in $\al_1,\ldots,\al_n$ and $\si_k(A)$ is the
$k^{\rm th}$ elementary symmetric polynomial in $\al_1,\ldots,\al_n$, where $1\leq k\leq n$. Thus the coefficients
$\be^{(t,l)}_{i_1,\ldots,i_{tl}}$ can easily be found. Some information about the polynomial $P_{t,l}(A)$ is given in Lemma~\ref{lemma_P} (see below).
\begin{example} We have the following partial cases of formula~(\ref{eq_P}):
\begin{enumerate}
\item[$\bullet$] $\tr(A^2)=\tr(A)^2-2\si_2(A)$,
\item[$\bullet$] $\tr(A^3) = \tr(A)^3 - 3\si_2(A)\tr(A) + 3\si_3(A)$,
\item[$\bullet$] $\tr(A^4) = \tr(A)^4 - 4\si_2(A)\tr(A)^2 + 2\si_2(A)^2 + 4\si_3(A)\tr(A) - 4\si_4(A)$,
\item[$\bullet$] $\si_2(A^2) = \si_2(A)^2 - 2\si_3(A)\tr(A) + 2\si_4(A)$.
\end{enumerate}
\end{example}
\begin{remark}\label{remark_GL_notations} Let $f\in\si\X$. Taking the image of $f$ with respect to $\piLarge$ ($\piSmall{n}$, respectively), we can consider $f$ as an element of $\AlgLarge$ ($\AlgSmall{n}$, respectively). As an example, let $f_k=\si_k(x+y)\in\si\X$ for $k\geq1$ and letters $x\neq y$. Then $f_2$ in $\AlgLarge$ is $\si_2(x)+\si_2(y)+\tr(x)\tr(y)-\tr(xy)$. On the other hand, $f_3$ in $\AlgSmall{n}$ is zero in case $n=2$ and $f_3$ in $\AlgSmall{n}$ is $\si_3(x+y)$ in case $n\geq 3$.
\end{remark}
\bigskip
The next remark follows from Lemma~\ref{lemma_GL_Donkin} (see below) and the definition of $\AlgLarge$.
\begin{remark}\label{remark_GL_OK1}
Assume that $\un{t}\in\NN^u$ and $t=|\un{t}|$. Then $\si_{\un{t}}(x_1,\ldots,x_u)\in\AlgLarge$ is a {\it partial linearization} of $\si_t(x_1)$, i.e., it is the coefficient of $\la_1^{t_1}\cdots \la_u^{t_u}$ in $\si_t(\la_1 x_1+\cdots+\la_u x_u)\in\AlgLarge$ considered as a polynomial in $\la_1,\ldots,\la_u\in\FF$.
Moreover, for $\un{k}\in\NN^l$ with $k=|\un{k}|$ we have that $\si_{\un{t},\un{k}}(x_1,\ldots,x_{u+l})\in\AlgLarge$ is a partial linearization of $\si_{(\un{t},k)}(x_1,\ldots,x_{u},x_{u+1})$.
\end{remark}
\bigskip
As we have mentioned in Section~\ref{section1}, we will usually omit $\otimes$ in the elements of $\si\X \otimes \FF\X^{\#}$. Given $a\in\FF\X$ and $t\geq0$, let $\chi_t(a)\in\si\X \otimes \FF\X^{\#}$ be the Cayley--Hamilton polynomial, i.e,
\begin{eq}\label{eq_chi}
\chi_t(a)=\sum_{i=0}^t (-1)^i \si_i(a) a^{t-i}.
\end{eq
Note that $\chi_{0}(a)=1$. As in Remark~\ref{remark_GL_notations}, we can consider $\chi_{t}(a)$ as an element of $\AlgLarge\otimes \FF\X^{\#}$ as well as of $\AlgSmall{n}\otimes \FF\X^{\#}$. The Cayley--Hamilton theorem implies that
$$\chi_n(a)=0$$
is a relation for $\C_n$, i.e., belongs to $\TLarge{n}$ and $\TSmall{n}$. The proof of the following Theorem~\ref{theo_GL_main} and Remark~\ref{remark_theo_GL_main} is given in Section~\ref{section4}.
\begin{theo}\label{theo_GL_main}
\begin{enumerate}
\item[1.] The ideal of relations $\TSmall{n}$ for $\C_n$ is generated by $\KSmall{n}\otimes 1$ and
$\chi_n(a)=0$ for $a\in\FF\X$.
\item[2.] The ideal of relations $\KSmall{n}$ for $R^{GL(n)}\simeq \AlgSmall{n} / \KSmall{n}$ is generated by
\begin{enumerate}
\item[(a)] $\si_t(a+b)=F_t(a,b)$ for $1\leq t\leq n$, where $a,b\in\FF\X$;
\item[(b)] $\si_t(a^l)=P^{\pplus}_{t,l}(a)$ for $1\leq t\leq n$, $1<l\leq n$, where $a\in\X$;
\item[(c)] $\si_t(ab)=\si_t(ba)$ for $1\leq t\leq n$, where $a,b\in\X$;
\item[(d)] $\si^{\pplus}_{\un{t}}(a_1,\ldots,a_u)=0$ for $n<|\un{t}|\leq 2n$, where $\un{t}\in\NN^u$, $u>1$, and $a_i\in\X$ for all $i$
\end{enumerate}
\end{enumerate}
In particular, ideals $\TSmall{n}$ and $\KSmall{n}$ are finitely based.
\end{theo}
\bigskip
Relations (a), (b), (c) from Theorem~\ref{theo_GL_main} are called {\it free} relations, because, being considered as elements of $\si\X$, they belong to the kernel of $\PhiAbs{n}$ for all $t\geq 1$, $l>1$ and do not depend on $n$.
\begin{remark}\label{remark_theo_GL_main}
In the formulation of Theorem~\ref{theo_GL_main} we can assume that $\un{t}\in\NN^u$ from relation~(d) satisfies the following conditions:
\begin{eq}\label{eq_cond1}
t_1\geq \cdots \geq t_u,
\end{eq}
\vspace{-0.5cm
\begin{eq}\label{eq_cond2}
t_1,\ldots,t_u\in\{1,p,p^2,p^3,\ldots\},
\end{eq}
\vspace{-0.5cm
\begin{eq}\label{eq_cond3}
\text{either } |\un{t}|=n+1, \text{ or } n+1<|\un{t}|\leq 2n \text{ and } |\un{t}|-\min\{t_i\}\leq n.
\end{eq
In particular, if the second case from~(\ref{eq_cond3}) holds, then $t_i\neq 1$ for all $i$. These conditions enable us to diminish the number of multidegrees $\un{t}$ from~(d) considerably. Namely, it is not difficult to see that conditions~(\ref{eq_cond1}), (\ref{eq_cond2}), (\ref{eq_cond3}) imply that
\begin{enumerate}
\item[$\bullet$] if $p=0$ or $p>n$, then $\un{t}=1^{n+1}$;
\item[$\bullet$] if $\frac{n}{2}<p\leq n$, then $\un{t}$ belongs to the following list: $1^{n+1}$, $(p,1^{n+1-p})$, $(p,p)$;
\item[$\bullet$] if $\frac{n}{3}<p\leq \frac{n}{2}$ and $p\neq2$, then $\un{t}$ belongs to the following list: $1^{n+1}$, $(p,1^{n+1-p})$, $(p,p,1^{n+1-2p})$, $(p,p,p)$.
\end{enumerate}
\end{remark}
\bigskip
Note that in the formulation of Theorem~\ref{theo_GL_main} we can not consider elements $\si_t(a)$ for $n<t\leq 2n$, $a\in\FF\X$ instead of relations (d), because images of these elements in $\AlgSmall{n}$ are zeros.
\section{Large free algebra of $GL(n)$-invariants}\label{section3}
We start this section with the known description of the ideal of relations $\KLarge{n}$.
\begin{theo}(Zubkov~\cite{Zubkov_1996})\label{theo_Zubkov}
\begin{enumerate}
\item[1.] The ideal of relations $\TLarge{n}$ for $\C_n$ is generated by $\KLarge{n}\otimes 1$ and $\chi_n(a)=0$ for $a\in\FF\X$.
\item[2.] The ideal of relations $\KLarge{n}$ for $R^{GL(n)}\simeq \AlgLarge / \KLarge{n}$ is generated by $\si_t(a)=0$ for $t>n$ and $a\in\FF\X$.
\end{enumerate}
\end{theo}
\bigskip
The next lemma describes the large free algebra $\AlgLarge$ as a quotient of the absolutely free algebra $\si\X$.
\begin{lemma}\label{lemma_GL_Donkin}(Donkin~\cite{Donkin_1993a}) We have $\AlgLarge\simeq \si\X/ L$ for the ideal $L$ generated by
\begin{enumerate}
\item[(a)] $\si_t(a_1+\cdots+a_u)=F_{t}(a_1,\ldots,a_u)$,
\item[(b)] $\si_t(a^l)=P_{t,l}(a)$,
\item[(c)] $\si_t(ab)=\si_t(ba)$,
\end{enumerate}
where $t>0$, $l,u>1$, $a_1,\ldots,a_u\in \FF\X$, and $a,b\in\X$.
\end{lemma}
\bigskip
In this section we prove the following theorem together with Remark~\ref{remark_theo_GL}.
\begin{theo}\label{theo_GL}
The ideal of relations $\KLarge{n}$ for $R^{GL(n)}\simeq \AlgLarge / \KLarge{n}$ is generated by
\begin{enumerate}
\item[$\bullet$] $\si_{t}(a)=0$, where $n<t\leq 2n$ and $a\in\FF\X$
\item[$\bullet$] $\si_t(b)=0$, where $t>2n$ and $b\in\EX$.
\end{enumerate}
\end{theo}
\begin{remark}\label{remark_theo_GL}
We can reformulate Theorem~\ref{theo_GL} as follows: the ideal $\KLarge{n}$ is generated by
\begin{enumerate}
\item[$\bullet$] $\si_{\un{t}}(a_1,\ldots,a_u)=0$, where $\un{t}\in\NN^u$ ($u>1$) satisfies conditions~(\ref{eq_cond1}), (\ref{eq_cond2}), (\ref{eq_cond3}) and $a_i\in\X$ for $1\leq i\leq u$;
\item[$\bullet$] $\si_t(b)=0$, where $t>n$ and $b\in\EX$.
\end{enumerate}
\end{remark}
\bigskip
We split the proof of Theorem~\ref{theo_GL} and Remark~\ref{remark_theo_GL} into several lemmas. Denote by $J_t$ the ideal of $\AlgLarge$ generated by $\si_t(a)$, $a\in\FF\X$. Since the field $\FF$ is infinite, Remark~\ref{remark_GL_OK1} implies that elements $\si_{\un{t}}(a_1,\ldots,a_u)$ generate the ideal $J_{t}$ for $t=|\un{t}|$, where $\un{t}\in\NN^u$ and $a_1,\ldots,a_u\in\X$. We write $J_t^{(p)}$ for the $\FF$-subspace of $\AlgLarge$ spanned by $\si_{\un{t}}(a_1,\ldots,a_u)$ for $\un{t}\in\NN^u$ satisfying $t=|\un{t}|$, $t_i\in\{1,p,p^2,\ldots\}$ and $a_i\in\X$ for all $i$.
The key idea of the proof of Theorem~\ref{theo_GL} is the fact that $\si_{(k,t)}(a,b)\in J_t$ (see Lemma~\ref{lemma_GL_key}). This fact together with Lemma~\ref{lemma_GL_key2} enables us to show that $\si_{\un{t}}(a_1,\ldots,a_u)$, where $|\un{t}|>n$, $u>1$, $a_1,\ldots,a_u\in\X$, belongs to the ideal of $\AlgLarge$, generated by elements from Theorem~\ref{theo_GL}.
\begin{lemma}\label{lemma_P}
For a letter $x$ we have that
\begin{enumerate}
\item[1)] every summand of $P_{t,l}(x)\in\si\X$ contains a multiple $\si_k(x)$ with $k\geq t$; in particular, if $t>n$, then the image of every relation (b) from Lemma~\ref{lemma_GL_Donkin} with respect to $\piSmall{n}$ is zero;
\item[2)] if $p>0$, $t=p^r$, $l=p^s$ for $r\geq0$, $s>0$, then $P_{t,l}(x)=\si_t(x)^l$.
\end{enumerate}
\end{lemma}
\begin{proof}
1) We can assume that $t>1$. Let $P_{t,l}(x)=P+Q$ in $\si\X$, where $P,Q$ are polynomials in $\si_i(x)$, $i>0$, and every summand of $P$ ($Q$, respectively) contains $\si_k(x)$ for some $k\geq t$ (does not contain $\si_k(x)$ for any $k\geq t$, respectively). Let $Q$ be a non-zero polynomial. We set $n=t-1$. By part~2 of Theorem~\ref{theo_Zubkov} and Lemma~\ref{lemma_GL_Donkin}, $\piLarge(Q)$ lies in $\KLarge{n}$. Acting by $\PhiLarge{n}$, we obtain a non-trivial relation between $\tr(X),\si_2(X),\ldots, \si_n(X)$, where $X$ is the generic $n\times n$ matrix corresponding to the letter $x$. But it is well-known that these elements are algebraically independent over $\FF$; a contradiction.
\medskip
2) It follows from $(\al_1+\cdots+\al_m)^p=\al_1^p+\cdots+\al_m^p$ for $\al_1,\ldots,\al_m\in\FF$, $m>0$ and the reasoning after formula~(\ref{eq_P}).
\end{proof}
\begin{remark}\label{remark_GL_diagram} The right triangle of Diagram~1 is commutative. To show this we use definitions of $\PhiAbs{n}$ and $\PhiLarge{n}$ together with the claim that $\PhiAbs{n}$ sends relations~(a), (b), (c) of Lemma~\ref{lemma_GL_Donkin} to zero. In case $t>n$ this claim follows from part~2 of Theorem~\ref{theo_Zubkov} and part~1 of Lemma~\ref{lemma_P}, and in case $1\leq t\leq n$ see Section~\ref{section2}.
\end{remark}
\bigskip
\begin{lemma}\label{lemma_GL_OK2}
Assume that $\un{t}\in\NN^u$ and $t=|\un{t}|$. Then $t_1\!!\si_{\un{t}}(x_1,\ldots,x_u)=\si_{\un{t}'}(\underbrace{x_1,\ldots,x_1}_{t_1},x_2,\ldots,x_u)$ in $\AlgLarge$, where $\un{t}'$ stands for $(1^{t_1}\!,t_2,\ldots,t_u)$.
\end{lemma}
\begin{proof} We work in $\AlgLarge$.
Remark~\ref{remark_GL_OK1} implies that $\si_{\un{t}'}(\underbrace{x_1,\ldots,x_1}_{t_1},x_2,\ldots,x_u)$ is equal to the coefficient of $\la_1\cdots \la_{t_1}$ in $\si_{\un{t}}(\la_1 x_1+\cdots+\la_{t_1} x_1, x_2, \ldots, x_u) = (\la_1+\cdots+\la_t)^{t_1} \si_{\un{t}}(x_1,\ldots,x_u)$, where $\la_1,\ldots,\la_{t_1}\in\FF$. The required is proven.
\end{proof}
\begin{lemma}\label{lemma_GL_sets}
Given pairwise different letters $x_0,x,e_1,e_2,\ldots$, consider an endomorphism $\varphi$ of $\X$ defined by
$$\varphi(a)=\left\{
\begin{array}{rl}
x_0^i x,& \text{ if }a=e_i\\
a,& \text{ otherwise }\\
\end{array}
\right.$$
for any letter $a$. Let $\Theta_{I}\subset\EX$ ($\Theta_{II}\subset\EX$, respectively) be the set of all monomials in $x_0,x$ (in $x,e_1,e_2,\ldots$, respectively). Then $\varphi$ induces the well-defined bijection $\ov{\varphi}:\ov{\Theta}_{II} \sqcup\{\ov{x_0}\}\to \ov{\Theta}_I$ of sets of $\stackrel{c}{\sim}$-equivalence classes.
\end{lemma}
\begin{proof}
For $a\in\Theta_{II}$ we have $a=x$ or $a\stackrel{c}{\sim} e_{i_1}x^{j_1}\cdots e_{i_s}x^{j_s}$ for some $i_1,\ldots,i_s>0$ and $j_1,\ldots,j_s\geq0$. Then $\ov{\varphi}(\ov{a})=\ov{b}$ for $b=x$ or $b\stackrel{c}{\sim} x_0^{i_1}x^{j_1+1}\cdots x_0^{i_s}x^{j_s+1}$, respectively. The following fact completes the proof: if $c_1,c_2\in\X$, $c_1$ is primitive, and $c_1\stackrel{c}{\sim} c_2$, then $c_2$ is also primitive.
\end{proof}
\begin{lemma}\label{lemma_GL_key}
If $\un{t}\in\NN^u$ with $u>1$, then $\si_{\un{t}}(x_1,\ldots,x_u)\in J_{|\un{t}|-t_1}$.
\end{lemma}
\begin{proof}
We work in $\AlgLarge$. Assume that $u=2$. It is convenient to denote $x_0=x_1$, $x=x_2$, $e_1=x_3$, $e_2=x_4$ and so on. For short, we set $\un{t}=(k,t)$. In what follows, we use notations from Lemma~\ref{lemma_GL_sets}. Let $\Upsilon_I$ be the set of finite multisubsets of $\Theta_I$ and $\Upsilon_{II}$ be the set of finite multisubsets of $\Theta_{II} \sqcup\{x_0\}$. We define the $\stackrel{c}{\sim}$-equivalence on $\Upsilon_I$ naturally and denote by $\ov{\Upsilon}_I$ the set of all $\stackrel{c}{\sim}$-equivalence classes. Similarly we define $\ov{\Upsilon}_{II}$. Then Lemma~\ref{lemma_GL_sets} implies that $\ov{\varphi}:\ov{\Upsilon}_{II}\to \ov{\Upsilon}_I$ is a bijection.
Let us recall that $\Omega(\un{t})$ was defined in Section~\ref{section2}. Assume that $\omega$ belongs to $\ov{\Upsilon}_{I}$ or $\ov{\Upsilon}_{II}$. Since we work in $\AlgLarge$, the element $\si(\omega)$ is well-defined. For short, we write $\mdeg(\omega)$ for $\mdeg(\si(\omega))$.
By the definition,
\begin{eq}\label{eq1_lemma_GL_key}
\si_{(k,t)}(x_0,x)=(-1)^{k+t} \sum_{\omega\in \ov{\Omega}_I} \si(\omega),
\end{eq
where $\ov{\Omega}_I=\ov{\Omega}(k,t)=\{\omega\in\ov{\Upsilon}_{I}\,|\,\mdeg(\omega)=(k,t)\}$. For $\ov{\Omega}_{II}=\{\omega\in\ov{\Upsilon}_{II}\,|\,\mdeg(\ov{\varphi}(\omega))=(k,t)\}$ an isomorphism of sets $\ov{\Omega}_{II}\simeq \ov{\Omega}_I$ is determined by the restriction of $\ov{\varphi}$.
Given $d\geq0$ and $\Delta=(\al_0,\al,\al_1,\ldots,\al_d)\in\NN_0^{d+2}$, where $\al_d>0$ in case $d>0$, we denote $\varphi(\Delta)=\mdeg(\varphi(x_0^{\al_0}x^{\al}e_1^{\al_1}\cdots e_d^{\al_d}))=(\al_0+\sum_{i=1}^d i\al_i,\al+\sum_{i=1}^d \al_i)$ and $\ov{\Omega}_{II}^{\Delta}=\{\omega\in\ov{\Upsilon}_{II}\,|\,\mdeg(\omega)=\Delta\}$. Thus
\begin{eq}\label{eq2_lemma_GL_key}
\ov{\Omega}_{II}=\bigsqcup \ov{\Omega}_{II}^{\Delta},
\end{eq
where the union ranges over $\Delta$ satisfying $\varphi(\Delta)=(k,t)$. Consequently applying formula~(\ref{eq1_lemma_GL_key}), the isomorphism $\ov{\Omega}_{II}\simeq \ov{\Omega}_I$, and formula~(\ref{eq2_lemma_GL_key}) we obtain
$$\si_{(k,t)}(x_0,x)=(-1)^{k+t}\!\!\!\! \sum_{\varphi(\Delta)=(k,t)}\; \sum_{\omega\in\ov{\Omega}_{II}^{\Delta} } \si(\ov{\varphi}(\omega)).$
Note that $\sum_{\omega\in\ov{\Omega}_{II}^{\Delta} } \si(\ov{\varphi}(\omega))=(-1)^{\al_0+|\De'|} \si_{\al_0}(x_0)\, \si_{\Delta'}(x,\varphi(e_1),\ldots,\varphi(e_d))$, where $\Delta'$ stands for $(\al,\al_1,\ldots,\al_d)$. Since the condition $\varphi(\Delta)=(k,t)$ implies $|\De'|=t$, we have
\begin{eq}\label{eq3_lemma_GL_key}
\si_{(k,t)}(x_0,x)=\sum (-1)^{\al_0+k}\si_{\al_0}(x_0)\, \si_{\Delta'}(x,x_0 x,x_0^2 x,\ldots,x_0^d x)
\end{eq
for $\Delta'=(\al,\al_1,\ldots,\al_d)$, where the sum ranges over $d\geq0$, $\al_0,\al,\al_1,\ldots,\al_d\geq0$ such that $\al_d>0$ in case $d>0$,
$$\al_0+\sum_{i=1}^d i\al_i = k \;\text{ and }\;\al + \sum_{i=1}^d \al_i = t.$$
Thus, $\si_{(k,t)}(x_0,x)\in J_t$ and the required is proven for $u=2$.
Let $u>2$. By the considered case of the lemma, $\si_{(t_1,t-t_1)}(x_1,x_2)\in J_{t-t_1}$ for $t=|\un{t}|$. Remark~\ref{remark_GL_OK1} implies that $\si_{\un{t}}(x_1,\ldots,x_u)$ is a partial linearization of $\si_{(t_1,t-t_1)}(x_1,x_2)$. The fact that the ideal $J_{t-t_1}$ is closed with respect to taking partial linearizations completes the proof.
\end{proof}
\begin{example}\label{ex_GL_1}
For letters $x_0$ and $x$ the following equalities hold in $\AlgLarge$:
\begin{enumerate}
\item[$\bullet$] $\si_{(1,1)}(x_0,x)= \tr(x_0)\tr(x) - \tr(x_0x)\in J_1$;
\item[$\bullet$] $\si_{(2,2)}(x_0,x) = \si_2(x_0)\si_2(x) - \tr(x_0)\si_{(1,1)}(x,x_0x) + \si_2(x_0x) + \si_{(1,1)}(x,x_0^2x) \in J_2$;
\item[$\bullet$] for $\un{t}=(1,t_2,\ldots,t_u)$ we have $\si_{\un{t}}(x_1,\ldots,x_u) = \tr(x_1) \si_{(t_2,\ldots,t_u)}(x_2,\ldots,x_u) - \sum_{i=2}^u \si_{\un{t}^{(i)}}(x_1x_i, x_2,\ldots,x_u)\in J_{|\un{t}|-1}$, where $\un{t}^{(i)}=(1,t_2,\ldots,t_i-1,\ldots,t_u)$.
\end{enumerate}
The first two equalities are partial cases of key formula~(\ref{eq3_lemma_GL_key}) from the proof of Lemma~\ref{lemma_GL_key}.
\end{example}
\bigskip
A statement similar to the following lemma was proved by Samoilov in~\cite{Samoilov_2008}.
\begin{lemma}\label{lemma_GL_key2}
For every $\un{t}\in\NN^u$ we have $\si_{\un{t}}(x_1,\ldots,x_u)\in J_{|\un{t}|}^{(p)}$.
In particular, $\si_t(x_1)\in\AlgLarge$ is a polynomial in $\si_{p^i}(x_1^j)$ for $i\geq0$ and $j\geq1$.
\end{lemma}
\begin{proof} We work in $\AlgLarge$. If $p=0$, then applying Lemma~\ref{lemma_GL_OK2} several times we obtain the first claim of the lemma.
Assume $p>0$. The positive integer $t_1$ can be written in a base $p$ expansion in the following form: $t_1=\sum_{i=1}^{k} l_i p^{\al_i}$ for $1\leq l_1,\ldots,l_k\leq p-1$, $0\leq \al_1<\cdots<\al_k$, and $k\geq1$. Denote $l=l_1+\cdots+l_k$,
$$\un{t}' = (\underbrace{p^{\al_1},\ldots,p^{\al_1}}_{l_1},\ldots, \underbrace{p^{\al_k},\ldots,p^{\al_k}}_{l_k},t_2,\ldots,t_u),
\text{ and }
x_1^{(i)}=(\underbrace{x_1,\ldots,x_1}_i).
$
Lemma~\ref{lemma_GL_OK2} implies that for $\al=(p^{\al_1}!)^{l_1}\cdots (p^{\al_k}!)^{l_k}$ we have
$$\al\, \si_{\un{t}'}(x_1^{(l)},x_2,\ldots,x_u)=\si_{(1^{t_1},t_2,\ldots,t_u)}(x_1^{(t_1)},x_2,\ldots,x_u)\text{ and }$$
$$t_1!\, \si_{\un{t}}(x_1,\ldots,x_u)=\si_{(1^{t_1},t_2,\ldots,t_u)}(x_1^{(t_1)},x_2,\ldots,x_u).$$
Hence
\begin{eq}\label{eq4}
\si_{\un{t}}(x_1,\ldots,x_u)=\be_{t_1} \si_{\un{t}'}(x_1^{(l)},x_2,\ldots,x_u)
\end{eq
over $\QQ$, where $\be_{t_1}=\al/t_1!$. We claim that
\begin{eq}\label{eq_statement_beta}
\be_{t_1}\neq0 \text{ is well-defined over an arbitrary field } \FF \text{ of characteristic } p.
\end{eq
Denote $[\be]=\max\{\ga\in\ZZ\,|\,\ga\leq \be\}$ for $\be\in\QQ$. Given $m=p^\ga q\in\NN$ with $\ga,q\in\NN_0$ such that $p$ is not a divisor of $q$, we write $\ga_m$ for $\ga$. Note that $\ga_{m!}=\sum_{j\geq 1} \#\{1\leq i\leq m\,|\,p^j \text{ is a divisor of } i\}=\sum_{j\geq 1}[m/p^j]$. Therefore,
$$\ga_{t_1!}=\sum_{j\geq 1}\sum_{i=1}^k l_i\, [p^{\al_i-j}]=\sum_{i=1}^k l_i \left(p^{\al_i-1}+p^{\al_i-2}+\cdots+1\right)=\ga_{\al}.$$
Statement~(\ref{eq_statement_beta}) is proven. Thus~(\ref{eq4}) holds over $\FF$. Repeating this procedure for $t_2,\ldots,t_u$, we obtain the first claim of the lemma.
The proven part of the lemma implies that $\si_t(x_1)$ belongs to $\FF$-span of $\si_{\un{t}}(x_1,\ldots,x_1)$, where $\un{t}\in\NN^u$, $|\un{t}|=t$, and $t_i\in\{1,p,p^2,\ldots\}$ for all $i$. If $u>1$, then $\si_{\un{t}}(x_1,\ldots,x_1)$ is a polynomial in $\si_k(x_1^j)$ for $1\leq k<t$ and $j>0$. So we can apply the above reasoning to $\si_k(x_1^j)$ and so on. Finally, we prove the second claim of the lemma.
\end{proof}
Now we can prove Theorem~\ref{theo_GL} and Remark~\ref{remark_theo_GL}:
\begin{proof}
We work in $\AlgLarge$. Since the field $\FF$ is infinite, part~2 of Theorem~\ref{theo_Zubkov} together with Remark~\ref{remark_GL_OK1} implies that $\KLarge{n}$ is generated by
\begin{enumerate}
\item[(a)] $\si_{\un{t}}(a_1,\ldots,a_u)=0$ for $|\un{t}|>n$, where $\un{t}\in\NN^u$, $u>1$, and $a_1,\ldots,a_u\in\X$;
\item[(b)] $\si_t(b)=0$ for $t>n$, where $b\in\X$.
\end{enumerate}
Consider relations~(a). For short, we set $t=|\un{t}|$ and $\un{a}=(a_1,\ldots,a_u)$. Since $\si_{\un{t}}(\un{a})=\si_{\un{t}^{\si}}(a_{\si(1)},\ldots,a_{\si(u)})$ for all $\si \in S_u$, where $\un{t}^{\si}$ stands for $(t_{\si(1)},\ldots,t_{\si(u)})$, we can always assume that $t_1\geq \cdots \geq t_u$. By Lemma~\ref{lemma_GL_key2} we can assume that $t_i\in\{1,p,p^2,\ldots\}$ for all $i$. If $t>n+1$ and $t_i=1$ for some $i$, then Lemma~\ref{lemma_GL_key} implies that $\si_{\un{t}}(\un{a})\in J_{t-1}$. Repeating this procedure several times we obtain that every relation from~(a) follows from relations~(b) and such relations~(a) that have $\un{t}$ satisfying one of the following conditions:
\begin{enumerate}
\item[1)] $t=n+1$ and $t_i\in\{1,p,p^2,\ldots\}$ for all $i$;
\item[2)] $t>n+1$ and $t_i\in\{p,p^2,\ldots\}$ for all $i$.
\end{enumerate}
Consider the second case. If $t-t_{u}>n$, then $\si_{\un{t}}(\un{a})\in J_{t-t_u} \subset \KLarge{n}$ by Lemma~\ref{lemma_GL_key} and part~2 of Theorem~\ref{theo_Zubkov}. Therefore, we can assume that $t-t_{u}\leq n$. If $p>n$, then $t-t_u=t_1+\cdots+t_{u-1}\geq p>n$; a contradiction. Thus in case $p>n$ we can assume that $\un{t}$ satisfies condition~1). If $p\leq n$ and $t> 2n$, then $n\geq t-t_u> 2n - t_u$; thus $t_1,\ldots,t_u>n$ and $t-t_u > n$; a contradiction. Thus in case $p\leq n$ we can assume that $t\leq 2n$. Hence, $\un{t}$ satisfies conditions~(\ref{eq_cond1}), (\ref{eq_cond2}), (\ref{eq_cond3}).
Consider relations~(b). If $b=c^l$ for $l>1$ and $c\in\EX$, then $\si_t(b)=P_{t,l}(c)$ in $\AlgLarge$ (see Lemma~\ref{lemma_GL_Donkin}). Part~1 of Lemma~\ref{lemma_P} implies that relations (b) follow from
\begin{enumerate}
\item[($\rm b'$)] $\si_t(b)=0$ for $t>n$, where $b\in\EX$.
\end{enumerate}
The required is proven.
\end{proof}
As we have already mentioned, the first part of the following lemma is a reformulation of the result from~\cite{Razmyslov_1974} and~\cite{Procesi_1976}. The second part is new.
\begin{lemma}\label{lemma_GL_fb}
If $p=0$, then the ideal $\KLarge{n}\vartriangleleft \AlgLarge$ is generated by $\si_{n+1}(a)=0$ for $a\in\FF\X$; in particular, $\KLarge{n}$ is finitely based.
If $p>0$, then the ideal $\KLarge{n}\vartriangleleft \AlgLarge$ is not finitely based.
\end{lemma}
\begin{proof}
We work in $\AlgLarge$. Assume $p=0$. For short, we write $x$ for $x_1$ and $I$ for the ideal generated by $\si_{n+1}(a)=0$ for all $a\in\FF\X$. Theorem~\ref{theo_GL} together with Remarks~\ref{remark_theo_GL} and~\ref{remark_GL_OK1} implies that $\KLarge{n}$ lies in the ideal, generated by $I$ and $\si_t(b)=0$ for $t>n$ and $b\in\EX$. By Lemma~\ref{lemma_GL_key2}, $\si_t(x)$ is equal to a polynomial $f_t$ in $\tr(x^j)$, where $1\leq j\leq t$. Given $t>n$, we make substitutions $\tr(x^j)\to-\sum_{i=1}^n (-1)^{i} f_i \tr(x^{j-i})$, where $j>n$, in $f_t$. Repeating this procedure several times we obtain a polynomial in $\tr(x),\ldots,\tr(x^n)$, which we denote by $h_t$. Since $\si_{(n,1)}(x,x^{j-n})=\sum_{i=0}^n (-1)^{n-i} \si_{i}(x) \tr(x^{j-i})$ belongs to $I$ for $j>n$, the element $h_t$ is equal to $f_t$ modulo the ideal $I$. If $h_t$ is not equal to zero in $\AlgLarge$ for $t>n$, then the equality $\PhiLarge{n}(h_t)=0$ implies that $\tr(X_1),\ldots,\tr(X_1^n)$ are not algebraically independent over $\FF$; a contradiction. Thus, for $t>n$ we have $h_t=0$ and $f_t\in I$. The required is proven.
Assume $p>0$. Let $\KLarge{n}$ be finitely based, i.e., $\KLarge{n}$ is generated by some elements $f_1,\ldots,f_r\in\AlgLarge$ as $\Tid$-ideal. Denote by $m$ the maximal $t>0$ such that $\si_t(a)$ is a multiple of a summand of $f_i$ for some $i$ and $a\in\EX$. Consider a letter $x$ and $k\in\NN$ satisfying $p^k>m$ and $p^k>n$. We claim that
\begin{eq}\label{eq_claim}
\!\!\!\!\!\!\!\!\si_{p^k}(x)\in \KLarge{n} \text{ does not belong to }\Tid\text-\text{ideal of } \AlgLarge \text{ generated by } f_1,\ldots,f_r.
\end{eq
If the claim does not hold, then
\begin{eq}\label{eq2}
\si_{p^k}(x)=\sum_{i=1}^r f_i(\un{a}_i) f'_i
\end{eq
in $\AlgLarge$, where $f'_i\in\AlgLarge$, $\un{a}_i=(a_{i1},\ldots,a_{ir_i})$ for $a_{ij}\in\FF\X$ for all $i,j$, and $f_i(\un{a}_i)$ stands for the result of substitutions $x_1\to a_{i1},\ldots, x_{r_i}\to a_{ir_i}$ in $f_i$. The right hand side of~(\ref{eq2}) is equal in $\AlgLarge$ to a polynomial in elements $\{\si_i(b)\,|\, i>0,\; b\in\EX\}$, which are algebraically independent in $\AlgLarge$ (see Lemma~\ref{lemma_GL_Donkin}). Thus, there exists an $i$ such that the polynomial $f_i(\un{a}_i)\in\AlgLarge$ contains a summand $\si_{p^k}(x)$. Therefore, there are $t<p^k$, $c_1,\ldots,c_s\in\X$, and $\ga_1,\ldots,\ga_s\in\FF$ such that $\si_t(\ga_1 c_1+\cdots+\ga_s c_s)\in\AlgLarge$ contains a summand $\si_{p^k}(x)$. To write down $\si_t(\ga_1 c_1+\cdots+\ga_s c_s)$ as a polynomial in $\{\si_i(b)\,|\, i>0,\;b\in\EX\}$ we apply relations~(a) from Lemma~\ref{lemma_GL_Donkin} and then apply relations~(b) from Lemma~\ref{lemma_GL_Donkin} to the result several times. Note that the representation of $\si_t(\ga_1 c_1+\cdots+\ga_s c_s)$ in the mentioned form is unique (see Lemma~\ref{lemma_GL_Donkin}). Since $F_{t}(\ga_1 c_1,\ldots,\ga_s c_s)\in\si\X$ does not contain a summand $\si_{p^k}(x)$, we conclude that there exist $j<p^k$, $c\in\X$, and $l>1$ such that $\si_j(c^l)=P_{j,l}(c)\in\AlgLarge$ contains a summand $\si_{p^k}(x)$.
It is not difficult to see that we can assume that $c=x$ and $jl=p^k$. Hence $j=p^{k_0}$ and $l=p^{k-k_0}$ for $0\leq k_0<k$. By part~2 of Lemma~\ref{lemma_P}, we obtain $P_{j,l}(c)=\si_j(x)^l\in\AlgLarge$ does not contain a summand $\si_{p^k}(x)$; a contradiction.
Thus, claim~(\ref{eq_claim}) does hold and $\KLarge{n}$ is not finitely based.
\end{proof}
\section{Proof of Theorem~\ref{theo_GL_main}}\label{section4}
We prove Theorem~\ref{theo_GL_main} together with Remark~\ref{remark_theo_GL_main} at the end of this section. In this section we assume that~(a), (b), (c) are relations from part~2 of Theorem~\ref{theo_GL_main}. Similarly,~(d) stands for those relations (d) from part~2 of Theorem~\ref{theo_GL_main} that satisfy conditions from Remark~\ref{remark_theo_GL_main}.
As above, we assume that $n>1$.
\begin{lemma}\label{lemma_GL_3}
The ideal $\KSmall{n}$ of relations for $R^{GL(n)}\simeq \AlgSmall{n} / \KSmall{n}$ is generated by relations~(a), (c), (d) and
\begin{enumerate}
\item[($b_{t,l}$)] $\si_t(a^l)=P_{t,l}^{\pplus}(a)$ for $1\leq t\leq n$, $l>1$, $a\in\X$.
\end{enumerate}
\end{lemma
\begin{proof}
Let $I$ be the ideal of $\AlgSmall{n}$ generated by the elements from the formulation of the lemma. Note that in Lemma~\ref{lemma_GL_Donkin} we can assume that relations~(a) satisfy $u=2$ in case $t\leq n$ and $a_i=\be_i b_i$ for $\be_i\in\FF$, $b_i\in\X$ in case $t>n$. Since the ideal $\KLarge{n}$ is described by part~2 of Theorem~\ref{theo_Zubkov} and $\Ker(\piLarge)=L$ is considered in Lemma~\ref{lemma_GL_Donkin}, we obtain that the ideal
$\Ker(\PhiAbs{n})=\piLarge^{-1}(\KLarge{n})$ is generated by
\begin{enumerate}
\item[(${\rm a}'$)] $\si_t(a+b)=F_t(a,b)$, where $1\leq t\leq n$, $a,b\in\FF\X$;
\item[(${\rm a}''$)] $\si_t(\al_1 a_1+\cdots+\al_u a_u)=F_t(\al_1 a_1,\ldots,\al_u a_u)$, where $u>1$, $t>n$, $\al_i\in\FF$, $a_i\in\X$ for all $i$;
\item[(${\rm b}'$)] $\si_t(a^l)=P_{t,l}(a)$, where $t>0$, $l>1$, $a\in\X$;
\item[(${\rm c}'$)] $\si_t(ab)=\si_t(ba)$, where $t>0$, $a,b\in\X$;
\item[(${\rm d}'$)] $\si_t(a)=0$, where $t>n$, $a\in\FF\X$.
\end{enumerate}
We have $\piSmall{n}(\Ker(\PhiAbs{n}))=\KSmall{n}$. Since elements $\si_{\un{t}}(a_1,\ldots,a_u)$, where $|\un{t}|>n$, $a_i\in\X$, belong to $\Ker(\PhiAbs{n})$, relations~(d) belong to $\KSmall{n}$. Note that $\piSmall{n}$ sends (${\rm d}'$) to zero. Hence the ideal $\KSmall{n}$ of $\AlgSmall{n}$ is generated by the ideal $I$ and the images of (${\rm a}''$), (${\rm b}'$) in $\AlgSmall{n}$.
By part~1 of Lemma~\ref{lemma_P}, in case $t>n$ the image of relations (${\rm b}'$) in $\AlgSmall{n}$ is equal to zero. Since the field $\FF$ is infinite, we can take elements
\begin{enumerate}
\item[(${\rm a}'''$)] $\si_{\un{t}}(a_1,\ldots,a_u)\in \AlgSmall{n}$, where $u>1$, $|\un{t}|>n$, $a_1,\ldots,a_u\in\X$,
\end{enumerate}
instead of the image of (${\rm a}''$) in $\AlgSmall{n}$. By Theorem~\ref{theo_GL} and Remark~\ref{remark_theo_GL}, an element
$\si_{\un{t}}(a_1,\ldots,a_u)$, where $|\un{t}|>n$, $a_i\in\X$, of $\si\X$ belongs to the ideal of $\si\X$, generated by relations (${\rm b}'$), (${\rm c}'$), (${\rm d}'$), and (d), considered as elements of $\si\X$. Therefore, relations (${\rm a}'''$) belong to $I$. The required is proven.
\end{proof}
To complete the proof of part~2 of Theorem~\ref{theo_GL_main} it is enough to show that in Lemma~\ref{lemma_GL_3} we can assume that $1< l\leq n$ in relations ($b_{t,l}$). We prove this fact in Lemma~\ref{lemma_GL_2} (see below). The definition of $\equiv$-equivalence was given in Section~\ref{section2}.
\begin{lemma}\label{lemma_GL_equiv}
Given letters $x,y$ and $1\leq t\leq n$, we have that $\si_t(x^n y)\equiv0$ in $\AlgSmall{n}$ follows from $\si_{(kn,k)}(a,b)=0$ and (c), where $1\leq k\leq t$ and $a,b\in\X$.
\end{lemma}
\begin{proof} We work in the quotient of $\AlgSmall{n}$ by the ideal generated by (c). Assume that $x\neq y$. The proof is by induction on $1\leq t\leq n$.
Let $t=1$. Since
$$(-1)^{n}\si_{(n,1)}(x,y)=\tr(x^ny) + \sum_{i=1}^n (-1)^i \tr(x^{n-i}y)\si_i(x),$$
we obtain the required.
Assume $t>1$. Denote by $J$ the ideal generated by $\si_{(kn,k)}(a,b)=0$, $1\leq k\leq t$, $a,b\in\X$. Let $\Theta_t$ be the set of monomials $c\in\EX$ in letters $x,y$ such that $\deg_x(c)=nt$ and $\deg_y(c)=t$. Since the relation $\si_{(tn,t)}(x,y)=0$ belongs to $J$, we have that
\begin{eq}\label{eq3}
\sum (-1)^{t(n+1)-r} \!\!\!\!\sum_{c\in \Theta_{t/r}} \si_r(c)\equiv0 \;\text{ holds modulo }J,
\end{eq
where the sum is taken over $1\leq r\leq t$ with $r|t$. Note that for every $c\in\Theta_{t/r}$ there exists a $c_0\in\X$ such that $c\stackrel{c}{\sim} x^n c_0$. Thus the induction hypothesis implies that for $1\leq r<t$ we have that $\si_r(c)\equiv0$ follows from $\si_{(kn,k)}(a,b)=0$, where $1\leq k\leq r<t$ and $a,b\in\X$. Thus, $\si_r(c)\equiv0$ holds modulo $J$. Since $\Theta_1=\{c\}$ for $c\stackrel{c}{\sim}x^n y$, formula~(\ref{eq3}) implies the required.
\end{proof}
Let $I_n$ be the ideal of $\AlgSmall{n}$ generated by relations~(a), (b), (c), (d).
\begin{remark}\label{rem1} The ideal $I_n$ is closed with respect to substitutions $x_i\to a_i$ for $i>0$ and $a_i\in\X$.
\end{remark}
\begin{lemma}\label{lemma_GL_2}
Relations $(b_{t,l})$ belong to $I_n$, where $l>1$ and $1\leq t\leq n$.
\end{lemma}
\begin{proof}
We work in the quotient of $\AlgSmall{n}$ by the ideal generated by (c). Assume that $x,y$ are different letters. We prove by induction on $r>1$ the claim that relations
$$\begin{array}{cl}
{\rm (b_{\it t,l})}\;\, & \text{for }tl=r,\\
{\rm (h_{\it k})}:& \si_{(kn,n)}(x,y)=0 \text{ for }k(n+1)=r
\end{array}
$$
belong to $I_n$ for all $1\leq t\leq n$, $l>1$, $k\geq1$.
Let $r=2$. Since $n\geq2$, we obtain that $\rm (b_{1,2})$ belongs to $I_n$ and the set of relations ${\rm (h_{\it k})}$ is empty.
Assume that $r>2$. Let $h$ be a relation from $\rm (h_{\it k})$, where $k(n+1)=r$. Note that we can not claim that $h$ is a relation from (d). Since $h$ lies in $\KSmall{n}$, Lemma~\ref{lemma_GL_3} implies that $h$ belongs to $I_n$ modulo some relations from $\rm (b_{\it i,j})$ for $i,j>0$ satisfying $ij\leq \max\{\deg_x(h),\deg_y(h)\}=kn<r$. By the induction hypothesis, the mentioned relations from $\rm (b_{\it i,j})$ belong to $I_n$. Thus, all relations from $\rm (h_{\it k})$ belong to $I_n$.
Consider $\rm (b_{\it t,l})$, where $tl=r$. If $l\leq n$, then relations $\rm (b_{\it t,l})$ belong to $I_n$ by the definition.
Let $l>n$. By Lemma~\ref{lemma_GL_equiv}, $\si_t(x^n y)\equiv0$ follows from $\si_{(in,i)}(a,b)=0$, where $1\leq i\leq t$ and $a,b\in\X$. Note that $i(n+1)\leq tl=r$. If $i(n+1)<r$, then the induction hypothesis implies that $\rm (h_{\it i})$ belongs to $I_n$. On the other hand, if $i(n+1)=r$, then the proven part of the claim implies that $\rm (h_{\it i})$ belongs to $I_n$. By Remark~\ref{rem1},
$$\si_t(x^n y)\equiv0 \text{ holds modulo }I_n.$
Since $l>n$, we obtain that modulo the ideal $I_n$ the element $\si_t(x^l)=\si_t(x^n x^{l-n})$ is a polynomial in $\si_i(x^j)$ for $ij<\deg{\si_t(x^l)}=r$ with $1\leq i\leq n$. Applying $(b_{\it i,j})$ to $\si_i(x^j)$ and using the induction hypothesis, we can see that there is a polynomial $P_{t,l}'(x)$ in $\si_1(x),\ldots,\si_n(x)$ such that $\si_t(x^l)=P_{t,l}'(x)$ modulo the ideal $I_n$. Let $X$ be the $n\times n$ generic matrix corresponding to the letter $x$. Since $I_n$ is a subset of $\KSmall{n}$, $P_{t,l}'(X)=\si_t(X^l)=P_{t,l}^{\pplus}(X)$ is the equality of polynomials in $\si_1(X),\ldots,\si_n(X)$. But it is well-known that the latter elements are algebraically independent over $\FF$. Therefore, $P_{t,l}'(x)=P_{t,l}^{\pplus}(x)$ and $\si_t(x^l)=P_{t,l}(x)$ holds modulo $I_n$. By Remark~\ref{rem1}, relations $\rm (b_{\it t,l})$ belong to $I_n$. The claim is proven.
\end{proof}
Now we can prove Theorem~\ref{theo_GL_main} and Remark~\ref{remark_theo_GL_main}.
\begin{proof}
By part~1 of Theorem~\ref{theo_Zubkov}, the ideal $\TLarge{n}$ is generated by $\KLarge{n}\otimes 1$ and $\chi_n(a)=0$ for $a\in\FF\X$. For the sake of completeness, we point out that the mentioned result is a partial case of Lemma~\ref{lemma_73} (see below). Consider the surjective map $\PsiAbs{n}=\PhiAbs{n}\otimes \phi_n:\si\X\otimes\FF\X^{\#} \to \C_n$. Then the following diagram is commutative:
$$
\begin{picture}(0,120)
\put(0,95)
\put(0,-2){\vector(0,-1){58}
\put(15,0){\vector(3,-2){33}
\put(-15,0){\vector(-3,-2){33}
\put(50,-40){\vector(-3,-2){35}
\put(-50,-40){\vector(3,-2){35}
\put(110,0){\vector(-3,-2){33}
\put(-110,0){\vector(3,-2){33}
\put(-29,5){$\si\X\otimes \FF\X^{\#}$
\put(-100,-33){$\AlgSmall{n}\otimes \FF\X^{\#}$
\put(35,-33){$\AlgLarge\otimes \FF\X^{\#}$
\put(-6,-75){$\C_n$
\put(-125,5){$\TSmall{n}$
\put(115,5){$\TLarge{n}$
\put(3,-33){$\scriptstyle\PsiAbs{n}$
\put(-54,-8){$\scriptstyle\piSmall{n}\otimes\idmap$
\put(33,-8){$\scriptstyle\piLarge\otimes\idmap$
\put(-35,-48){$\scriptstyle\PsiSmall{n}$
\put(25,-48){$\scriptstyle\PsiLarge{n}$
\put(-20,-90){\text{Diagram 3.}
\end{picture}
$
Here $\idmap$ stands for the identical map on $\FF\X^{\#}$. The kernel $\TAbs{n}$ of $\PsiAbs{n}$ is equal to $(\piLarge\otimes\idmap)^{-1}(\TLarge{n})$.
Thus, the ideal $\TAbs{n}$ is generated by $(\piLarge\otimes\idmap)^{-1}(\KLarge{n}\otimes 1)=\piLarge^{-1}(\KLarge{n})\otimes 1 + \Ker(\piLarge)\otimes \FF\X^{\#}$ and $(\piLarge\otimes\idmap)^{-1}(\chi_n(a))$, $a\in\FF\X$. Since $\piLarge^{-1}(\KLarge{n})=\Ker(\PhiAbs{n})$ and $\Ker(\piLarge\otimes\idmap)=\Ker(\piLarge)\otimes \FF\X^{\#} \subset \Ker(\PhiAbs{n})\otimes \FF\X^{\#}$, the ideal $\TAbs{n}$ is generated by $\Ker(\PhiAbs{n})\otimes 1$ and $\chi_n(a)$, $a\in\FF\X$.
Note that $\TSmall{n}=(\piSmall{n}\otimes\idmap)(\TAbs{n})$. Thus the ideal $\TSmall{n}$ is generated by $(\piSmall{n}\otimes\idmap)(\Ker(\PhiAbs{n})\otimes 1)=\KSmall{n}\otimes 1$ and $\chi_n(a)$, $a\in\FF\X$, considered as an element of $\AlgSmall{n}\otimes \FF\X^{\#}$. Hence part~1 of Theorem~\ref{theo_GL_main} is proven. Part~2 of Theorem~\ref{theo_GL_main} follows immediately from Lemmas~\ref{lemma_GL_3} and~\ref{lemma_GL_2}.
\end{proof}
\section{Relations for matrix $O(n)$-invariants}\label{section5}
In the rest of the paper we assume that $p\neq2$. In this section we consider identities with forms for the $\FF$-algebra generated by $n\times n$ generic and transpose generic matrices or, equivalently, identities for the algebra $\CY_n$.
The algebra of {\it matrix $O(n)$-invariants} $R^{O(n)}$ is known to be generated by $\si_t(A)$, where $1\leq t\leq n$ and $A$ is a monomial in generic and transpose generic matrices. The mentioned generators of $R^{O(n)}$ were found by Sibirskii~\cite{Sibirskii_1968} and Procesi~\cite{Procesi_1976} in characteristic zero case and by Zubkov~\cite{Zubkov_1999} in the general case. In Section~\ref{section1} we denoted by
$$\CY_n=\alg_{\FF}\{X_1,X_1^T,X_2,X_2^T,\ldots,fE\}$$
the algebra generated by generic matrices, transpose generic matrices and $fE$, where $f$ ranges over $R^{O(n)}$.
Similarly to Section~\ref{section1} we define the following notions.
\begin{enumerate}
\item[$\bullet$] Let $\Y$ be the semigroup (without unity) freely generated by {\it letters} $x_1,x_1^T,x_2,x_2^T,\ldots$ and $\Y^{\#}=\Y\sqcup\{1\}$.
\item[$\bullet$] Introduce a lexicographical linear order on $\Y$ by setting $x_1>x_1^T>x_2>x_2^T>\cdots$ and $ab>a$ for $a,b\in\Y$.
\item[$\bullet$] Introduce the involution ${}^T$ on $\Y$ as follows. We set $(x_k)^T=x_k^T$, $(x_k^T)^T=x_k$ for all $k$ and $(a_1\cdots a_s)^T=a_s^T\cdots a_1^T\in\Y$.
\item[$\bullet$] We say that $a,b\in\Y$ are {\it cyclic equivalent} and write $a\stackrel{c}{\sim} b$
if $a=a_1a_2$ and $b=a_2a_1$ for some $a_1,a_2\in\Y^{\#}$. If $a\stackrel{c}{\sim} b$ or $a\stackrel{c}{\sim} b^T$, then we say that $a$ and $b$ are {\it equivalent} and write $a\sim b$.
\item[$\bullet$] Using $\Y$ instead of $\X$ and $\sim$-equivalence instead of $\stackrel{c}{\sim}$-equivalence, we introduce $\FF\Y$, $\FF\Y^{\#}$, $\AlgSmallY{n}$, $\si\Y$, $\EY$, $\AlgLargeY$, respectively, similarly to $\FF\X$, $\FF\X^{\#}$, $\AlgSmall{n}$, $\si\X$, $\EX$, $\AlgLarge$, respectively. Note that $\AlgSmallY{n}\subset \si\Y$.
\item[$\bullet$] The algebra $\si\Y$ ($\AlgLargeY$, $\AlgSmallY{n}$, respectively) is called the {\it absolutely} ({\it large}, {\it small}, respectively) free algebra for $R^{O(n)}$.
\item[$\bullet$] Let $\phi'_n:\FF\Y^{\#}\to \alg_{\FF}\{E,X_1,X_1^T,X_2,X_2^T,\ldots\}$ be the homomorphism of algebras defined by $1\to E$ and $x_k\to X_k$, $x_k^T\to X_k^T$ for all $k\geq1$.
\end{enumerate
\noindent{}Given $\un{t}\in\NN^u$ and $\un{a}=(a_1,\ldots,a_u)$ for $a_1,\ldots,a_u\in\FF\Y$, we define $\si_{\un{t}}(\un{a})\in\si\Y$ as the result of substitutions $x_1\to a_1,\ldots,x_u\to a_u$ in $\si_{\un{t}}(x_1,\ldots,x_u)\in\si\X$. Similarly we define elements $F_t(\un{a})$ and $P_{t,l}(b)$ of $\si\Y$, where $b\in\FF\Y$.
By Lemma~\ref{lemma_O_Lopatin} (see below), we have the surjective homomorphism $\piLargeY:\si\Y\to\AlgLargeY$. Using $\phi'_n$ instead of $\phi_n$, we define the surjective homomorphisms $\PhiAbsY{n}$, $\PhiLargeY{n}$, $\PhiSmallY{n}$, $\piSmallY{n}$, respectively, in the same way as $\PhiAbs{n}$, $\PhiLarge{n}$, $\PhiSmall{n}$, $\piSmall{n}$ (see the diagram below for the details). Denote by $\KLargeY{n}$ and $\KSmallY{n}$ the kernels of $\PhiLargeY{n}$ and $\PhiSmallY{n}$, respectively. Then the following diagram is commutative. Namely, its left triangle is commutative by the definition and the commutability of its right triangle can be shown in the same way as in Remark~\ref{remark_GL_diagram}.
$$
\begin{picture}(0,120)
\put(0,95)
\put(0,-2){\vector(0,-1){58}
\put(15,0){\vector(3,-2){35}
\put(-15,0){\vector(-3,-2){35}
\put(-11,5){$\si\Y$
\put(-75,-33){$\AlgSmallY{n}$
\put(50,-33){$\AlgLargeY$
\put(50,-40){\vector(-3,-2){35}
\put(-50,-40){\vector(3,-2){35}
\put(110,0){\vector(-3,-2){35}
\put(-110,0){\vector(3,-2){35}
\put(-6,-75){$R^{O(n)}$
\put(-125,5){$\KSmallY{n}$
\put(115,5){$\KLargeY{n}$
\put(3,-33){$\scriptstyle\PhiAbsY{n}$
\put(-40,-8){$\scriptstyle\piSmallY{n}$
\put(33,-8){$\scriptstyle\piLargeY$
\put(-35,-48){$\scriptstyle\PhiSmallY{n}$
\put(25,-48){$\scriptstyle\PhiLargeY{n}$
\put(-20,-90){\text{Diagram 4.}
\end{picture}
$
\noindent{}Denote the kernels of surjective homomorphisms
$$\PhiLargeY{n}\otimes\phi'_n:\AlgLargeY \otimes \FF\Y^{\#} \to \CY_n\;\text{ and }\;
\PhiSmallY{n}\otimes\phi'_n:\AlgSmallY{n}\otimes \FF\Y^{\#} \to \CY_n$$
by $\TLargeY{n}$ and $\TSmallY{n}$, respectively. These ideals are ideals of relations for $\CY_n$ in the corresponding free algebras.
Let $\un{t}\in\NN_0^u$, $\un{r}\in\NN_0^v$, $\un{s}\in\NN_0^w$ ($u,v,w>0$) with $|\un{r}|=|\un{s}|$. We set $y_1=x_{u+1},\ldots,y_v=x_{u+v}$ and $z_1=x_{u+v+1},\ldots,z_w=x_{u+v+w}$. In order to define $\si_{\un{t};\un{r};\un{s}}(\un{x};\un{y};\un{z})\in \FF\Y$ for $\un{x}=(x_1,\ldots,x_u)$, $\un{y}=(y_1,\ldots,y_v)$, $\un{z}=(z_1,\ldots,z_w)$, we consider the quiver (i.e., the oriented graph) $\Q=\Q(\un{x};\un{y};\un{z})$:
$$
\loopR{0}{0}{x_1,\ldots,x_u}
\xymatrix@C=1cm@R=1cm{
\vtx{1}\ar@1@/^/@{<-}[rr]^{y_1,y_1^T,\ldots,y_v,y_v^T} &&\vtx{2}\ar@1@/^/@{<-}[ll]^{z_1,z_1^T,\ldots,z_w,z_w^T}\\
\loopL{0}{0}{x_1^T,\ldots,x_u^T}\qquad\qquad,
$$
where there are $2v$ ($2w$, respectively) arrows from vertex $2$ to vertex $1$ (from $1$ to $2$, respectively) and there are $u$ loops in each of two vertices. By abuse of notation arrows of $\Q$ are denoted by letters from $\Y$. For an arrow $a$ denote by $a'$ its head and by $a''$ its tail. A sequence of arrows $a_1\cdots a_s$ of $\Q$ is a {\it path} of $\Q$ if $a_i''=a_{i+1}'$ for all $1\leq i< s$. The head of the path $a$ is $a'=a_1'$ and the tail is $a''=a_s''$. A path $a$ is {\it closed} if $a'=a''$. We introduce the following notations:
\begin{enumerate}
\item[$\bullet$] $\path{\Q}$ is the set of all (non-empty) paths in $\Q$;
\item[$\bullet$] $\LA\Q\RA\subset \Y$ is the semigroup (without unity) freely generated by closed paths in $\Q$ and $\LA\Q\RA^{\#}=\LA\Q\RA \sqcup\{1\}$; note that $\LA\Q\RA$ is closed with respect to the $\sim$-equivalence;
\item[$\bullet$] $\LA\widetilde{\Q}\RA = \EY \cap \LA\Q\RA$ is the set of maximal representatives of $\sim$-equivalence classes of primitive elements from $\LA\Q\RA$;
\item[$\bullet$] $\LA\ov{\Q}\RA$ is the set of all $\sim$-equivalence classes of primitive elements from $\LA\Q\RA$.
\end{enumerate
Denote the multidegree of a monomial $a$ in arrows of $\Q$ by $\mdeg(a)=(\deg_{x_1}(a) + \deg_{x_1^T}(a),\ldots,\deg_{z_w}(a) + \deg_{z_w^T}(a))$.
\begin{remark}\label{remark_primitive}
If $a,b\in\LA\Q\RA$ and $a\sim b^l$ for $l>1$, then there exists a $c\in\LA\Q\RA$ such that $a=c^l$.
\end{remark}
\bigskip
Let $\Omega=\Omega(\un{t};\un{r};\un{s})$ be the set of multisets
$$\omega=\{\underbrace{e_1,\ldots,e_1}_{k_1},\ldots,\underbrace{e_q,\ldots,e_q}_{k_q}\}_{m}
$
such that
\begin{enumerate}
\item[$\bullet$] $e_1,\ldots,e_q \in\LA\widetilde{\Q}\RA$ are pairwise different and $k_1,\ldots,k_q\in\NN$ ($q>0$);
\item[$\bullet$] $k_1\mdeg(e_1)+\cdots+k_q\mdeg(e_q)=(\un{t},\un{r},\un{s})$.
\end{enumerate}
We set $\si(\omega)=(-1)^{\xi} \si_{k_1}(e_1)\cdots\si_{k_q}(e_q)$ for
$$\xi=\sum_{i=1}^q k_i\left(\sum_{j=1}^v\deg_{y_j}{e_i}+\sum_{j=1}^w\deg_{z_j}{e_i}+1\right).$$
Then we define the following element of $\FF\Y$:
\begin{eq}
\si_{\un{t};\un{r};\un{s}}(\un{x};\un{y};\un{z})= (-1)^{|\un{t}|}\!\!\! \sum_{\omega\in \Omega(\un{t};\un{r};\un{s})} \si(\omega).
\end{eq
For empty $\Omega$ we set $\si_{\un{t};\un{r};\un{s}}(\un{x};\un{y};\un{z})=1$.
Given $a_i,b_j,c_k\in\FF\Y$, we define $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})$ as the result of the corresponding substitutions in $\si_{\un{t};\un{r};\un{s}}(\un{x};\un{y};\un{z})$. Note that
\begin{enumerate}
\item[$\bullet$] for $\un{r}=\un{s}=(0)$ we have $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})=\si_{\un{t}}(\un{a})$;
\item[$\bullet$] for $\un{t}=(t)$ and $\un{r}=\un{s}=(r)$ we denote $\si_{t,r}(a,b,c)=\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})$, where $\un{a}=(a)$, $\un{b}=(b)$, $\un{c}=(c)$.
\end{enumerate}
The element $\si_{t,r}(a,b,c)$ was introduced by Zubkov~\cite{ZubkovII}. Note that the definition from~\cite{ZubkovII} is different from our definition and their equivalence was established in Lemma~7.14 of~\cite{Lopatin_Orel}. More details can be found in Section~1.3 of~\cite{Lopatin_Orel}.
As in Section~\ref{section2}, we define elements $F_t(\un{a})$, $P_{t,l}(b)$ of $\si\Y$, where $a_i,b\in\FF\Y$.
\begin{remark}\label{remark_O_notations}
Let $f\in\si\Y$. Taking the image of $f$ with respect to $\piLargeY$ ($\piSmallY{n}$, respectively), we can consider $f$ as an element of $\AlgLargeY$ ($\AlgSmallY{n}$, respectively). As an example, see formulations of Theorems~\ref{theo_O_main},~\ref{theo_Lopatin},~\ref{theo_O}.
\end{remark}
\begin{remark}\label{remark_O_OK1}
Definition~4.1 together with Lemma 4.2 from~\cite{Lopatin_Orel} implies that $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})\in\AlgLargeY$ is a {\it partial linearization} of $\si_{t,r}(x,y,z)$, i.e., it is the coefficient of $\la_1^{t_1}\cdots \la_u^{t_u} \mu_1^{r_1}\cdots \mu_v^{r_v}\nu_1^{s_1}\cdots \nu_w^{s_w} $ in $\si_{t,r}(\la_1 a_1+\cdots+\la_u a_u, \mu_1 b_1+\cdots+\mu_v b_v, \nu_1 c_1+\cdots+\nu_w c_w)\in\AlgLargeY$ considered as a polynomial in $\la_1,\ldots,\la_u,\mu_1,\ldots,\mu_v,\nu_1,\ldots,\nu_w\in\FF$.
\end{remark}
Note that $\si_{t,r}$ has certain symmetries. Namely,
for $a,a_i,b,b_j,c,c_k\in\Y$ and $(a_1,\ldots,a_u)^T=(a_1^T,\ldots,a_u^T)$ we have
\begin{eq}\label{eq_O_transpose}
\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c}) = \si_{\un{t};\un{r};\un{s}}(\un{a};\un{b}^T;\un{c}^T) = \si_{\un{t};\un{s};\un{r}}(\un{a}^T;\un{c};\un{b}) \;\text{ in }\; \AlgLargeY.
\end{eq}
Let $t,r\geq0$. To introduce $\chi_{t,r}$ and $\zeta_{t,r}$, analogues of the Cayley--Hamilton polynomial $\chi_t$ for the algebra $\CY_n$, we consider the quiver $\Q=\Q(x;y;z)$, where $x=x_1$, $y=x_2$, $z=x_3$, and
\begin{enumerate}
\item[$\bullet$] denote by $L_{i,j}$ the set of pairwise different elements $e_1,\ldots,e_q\in \LA \Q\RA^{\#}$ with $e_1'=\cdots=e_q'=e_2''=\cdots=e_2''=1$ that satisfy $\mdeg(e_1)=\cdots=\mdeg(e_q)=(i,j,j)$;
\item[$\bullet$] denote by $M_{i,j}$ the set of pairwise different paths $e_1,\ldots,e_q$ in $\Q$ with $e_1'=\cdots=e_q'=2$ and $e_1''=\cdots=e_q''=1$ that satisfy $\mdeg(e_1)=\cdots=\mdeg(e_q)=(i,j,j+1)$.
\end{enumerate}
Then we consider the following elements of $\si\Y\otimes \FF\Y^{\#}$:
\begin{eq}\label{eq_O_chi}
\chi_{t,r}(x,y,z)=\sum_{i=0}^t \sum_{j=0}^r \si_{i,j}(x,y,z) \left( \sum_{e\in L_{t-i,r-j}} (-1)^{\xi}e \right),
\end{eq}
\begin{eq}\label{eq_O_zeta}
\zeta_{t,r}(x,y,z)=\sum_{i=0}^t \sum_{j=0}^r \si_{i,j}(x,y,z) \left( \sum_{e\in M_{t-i,r-j}} (-1)^{\xi}e \right),
\end{eq
where $\xi = i + \deg_{y}(e) + \deg_{z}(e)$. For short, here we have omitted $\otimes$. Note that $\chi_{0,0}(x,y,z)=1$, $\chi_{t,0}(x,y,z)=\chi_t(x)$, and $\zeta_{0,0}(x,y,z)=z^T-z$. For $a,b,c\in\FF\Y$ we define $\chi_{t,r}(a,b,c)$ and $\zeta_{t,r}(a,b,c)$ as the results of the corresponding substitutions. As in Remark~\ref{remark_O_notations}, we can consider $\chi_{t,r}(a,b,c)$ and $\zeta_{t,r}(a,b,c)$ as elements of $\AlgLargeY\otimes \FF\Y^{\#}$ as well as of $\AlgSmallY{n}\otimes \FF\Y^{\#}$. Connections between $\si_{t,r}$, $\chi_{t,r}$, $\zeta_{t,r}$ are given in Lemma~\ref{lemma_O_transpose} (see below).
\begin{example}\label{ex_54}
For $a,b,c\in\FF\Y$ and $\ov{b}=b-b^T$ the following equalities hold in $\si\Y$ and $\si\Y\otimes\FF\Y^{\#}$:
\begin{enumerate}
\item[$\bullet$] $\si_{0,1}(a,b,c) = -\tr(b\ov{c})$;
\item[$\bullet$] $\si_{1,1}(a,b,c) = \tr(a\ov{b}\ov{c}) - \tr(a)\tr(b\ov{c})$;
\item[$\bullet$] $\si_{0,2}(a,b,c) = \si_2(bc) + \si_2(bc^T) + \tr(bcbc^T) + \tr(bcb^Tc) - \tr(bcb^Tc^T) - \tr(bc)\tr(bc^T)$;
\item[$\bullet$] $\chi_{0,1}(a,b,c) = \ov{b}\ov{c} - \tr(b\ov{c})$ and $\zeta_{1,0}(a,b,c) = - a^T\ov{c} - \ov{c}a + \tr(a)\ov{c}$;
\item[$\bullet$] $\chi_{1,1}(a,b,c) = a\ov{b}\ov{c} + \ov{b}a^T\ov{c} + \ov{b}\ov{c}a - \tr(a)\ov{b}\ov{c} - \tr(b\ov{c})a - \tr(a\ov{b}\ov{c}) + \tr(a)\tr(b\ov{c})$;
\item[$\bullet$] $\zeta_{2,0}(a,b,c) = - (a^T)^2\ov{c} - a^T\ov{c}a - \ov{c}a^2 + \tr(a)a^T\ov{c} + \tr(a)\ov{c}a - \si_2(a)\ov{c}$;
\item[$\bullet$] $\zeta_{0,1}(a,b,c) = - \ov{c}\ov{b}\ov{c} + \tr(b\ov{c})\ov{c} $.
\end{enumerate}
\end{example}
\bigskip
The proof of the following theorem is given in Section~\ref{section7}.
\begin{theo}\label{theo_O_main}
\begin{enumerate}
\item[1.] The ideal of relations $\TSmallY{n}$ for $\CY_n$ is generated by $\KSmallY{n}\otimes 1$ and
\begin{enumerate}
\item[$\bullet$] $\chi_{t,r}(a,b,c)=0$ for $t+2r=n$;
\item[$\bullet$] $\zeta_{t,r}(a,b,c)=0$ for $t+2r=n-1$;
\end{enumerate}
where $a,b,c\in\FF\Y$.
\item[2.] The ideal of relations $\KSmallY{n}$ for $R^{O(n)}\simeq \AlgSmallY{n} / \KSmallY{n}$ is generated by
\begin{enumerate}
\item[(a)] $\si_t(a+b)=F_t(a,b)$ for $1\leq t\leq n$, where $a,b\in\FF\Y$;
\item[(b)] $\si_t(a^l)=P^{\pplus}_{t,l}(a)$ for $1\leq t\leq n$, $1<l\leq n$, where $a\in\Y$;
\item[(c)] $\si_t(ab)=\si_t(ba)$ for $1\leq t\leq n$, where $a,b\in\Y$;
\item[(d)] $\si_t(a)=\si_t(a^T)$ for $1\leq t\leq n$, where $a\in\Y$;
\item[(e)] $\si^{\pplus}_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})=0$ for $n<|\un{t}|+2|\un{r}|\leq 2n$, where $\un{t}\in\NN_0^u$, $\un{r}\in\NN_0^v$, $\un{s}\in\NN_0^w$ ($u,v,w>0$) satisfy $|\un{r}|=|\un{s}|$ and $a_i,b_j,c_k\in\Y$ for all $i,j,k$
\end{enumerate}
\end{enumerate}
Moreover, we can assume that $\un{t}$, $\un{r}$, $\un{s}$ from relation~(e) satisfy condition~(\ref{eq_cond1}) and the vector $(\un{t},\un{r},\un{s})$ without zero entries satisfies conditions~(\ref{eq_cond2}) and~(\ref{eq_cond3}).
In particular, ideals $\TSmallY{n}$ and $\KSmallY{n}$ are finitely based.
\end{theo}
\bigskip
Relations (a), (b), (c), (d) from Theorem~\ref{theo_O_main} are called {\it free} relations, because, being considered as elements of $\si\Y$, they belong to the kernel of $\PhiAbsY{n}$ for all $t\geq 1$, $l>1$ and do not depend on $n$.
\section{Large free algebra of $O(n)$-invariants}\label{section6}
We start this section with the known description of the ideal of relations $\KLargeY{n}$. We completed the proof of the following theorem in~\cite{Lopatin_free}, using results from~\cite{Lopatin_Orel} and the approach described in~\cite{ZubkovII}.
\begin{theo}\label{theo_Lopatin} The ideal of relations $\KLargeY{n}$ for $R^{O(n)}\simeq \AlgLargeY / \KLargeY{n}$ is generated by $\si_{t,r}(a,b,c)=0$ for $t+2r>n$, $t,r\geq0$, and $a,b,c\in\FF\Y$.
\end{theo}
\bigskip
The next lemma describes the large free algebra $\AlgLargeY$ as a quotient of the absolutely free algebra $\si\Y$. Its proof follows immediately from the proof of Lemma 3.1 from~\cite{Lopatin_free}.
\begin{lemma}\label{lemma_O_Lopatin} We have $\AlgLargeY\simeq \si\Y/ L'$ for the ideal $L'$ generated by
\begin{enumerate}
\item[(a)] $\si_t(a_1+\cdots+a_u)=F_{t}(a_1,\ldots,a_u)$,
\item[(b)] $\si_t(a^l)=P_{t,l}(a)$,
\item[(c)] $\si_t(ab)=\si_t(ba)$,
\item[(d)] $\si_t(a)=\si_t(a^T)$,
\end{enumerate}
where $t>0$, $l,u>1$, $a_1,\ldots,a_u\in \FF\Y$, and $a,b\in \Y$.
\end{lemma}
\bigskip
In this section we prove the following theorem together with Remark~\ref{remark_theo_O}:
\begin{theo}\label{theo_O}
The ideal of relations $\KLargeY{n}$ for $R^{O(n)}\simeq \AlgLargeY / \KLargeY{n}$ is generated by
\begin{enumerate}
\item[$\bullet$] $\si_{t,r}(a,b,c)=0$, where $n<t+2r\leq 2n$, $t,r\geq0$, and $a,b,c\in\FF\Y$;
\item[$\bullet$] $\si_t(b)=0$, where $t>2n$ and $b\in\EY$.
\end{enumerate}
\end{theo}
\begin{remark}\label{remark_theo_O}
We can reformulate Theorem~\ref{theo_O} as follows: the ideal $\KLargeY{n}$ is generated by
\begin{enumerate}
\item[$\bullet$] $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})=0$ for $n<|\un{t}|+2|\un{r}|\leq 2n$, where $\un{t}\in\NN_0^u$, $\un{r}\in\NN_0^v$, $\un{s}\in\NN_0^w$ ($u,v,w>0$) satisfy condition~(\ref{eq_cond1}), $|\un{r}|=|\un{s}|$, and $a_i,b_j,c_k\in\Y$ for all $i,j,k$; moreover, the vector $(\un{t},\un{r},\un{s})$ without zero entries satisfies conditions~(\ref{eq_cond2}) and~(\ref{eq_cond3});
\item[$\bullet$] $\si_t(b)=0$, where $t>n$ and $b\in\EY$.
\end{enumerate}
\end{remark}
\bigskip
We split the proof of Theorem~\ref{theo_O} and Remark~\ref{remark_theo_O} into several lemmas. Given $l\geq0$, we denote by $J_{l}$ the ideal of $\AlgLargeY$ generated by $\si_{t,r}(a,b,c)$ satisfying $l=t+2r$ and $a,b,c\in\FF\Y$. Since the field $\FF$ is infinite, Remark~\ref{remark_O_OK1} implies that elements $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})$ generate the ideal $J_l$ for $l=|\un{t}|+2|\un{r}|$, where $|\un{r}|=|\un{s}|$ and $a_i,b_j,c_k\in\Y$ for all $i,j,k$. We write $J_{l}^{(p)}$ for the $\FF$-subspace of $\AlgLargeY$ spanned by $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})$ for $t_i,r_j,s_k\in\{1,p,p^2,\ldots\}$ and $a_i,b_j,c_k\in\Y$ satisfying $l=|\un{t}|+2|\un{r}|$ and $|\un{r}|=|\un{s}|$. Note that the ideal $J_l$ is closed with respect to partial linearizations.
\begin{remark}\label{remark_O_OK2}
Remark~\ref{remark_O_OK1}, the definition of $\AlgLargeY$ and Lemma~\ref{lemma_O_Lopatin} imply that for $\un{k}\in\NN^{l}$ ($l>0$), $k=|\un{k}|$, $\un{e}=(e_1,\ldots,e_l)$ with $e_i\in\Y$ for all $i$, and a letter $x$ we have
\begin{enumerate}
\item[$\bullet$] $\si_{\un{t},\un{k};\un{r};\un{s}}(\un{a},\un{e};\un{b};\un{c})\in\AlgLargeY$ is a partial linearization of $\si_{\un{t},k;\un{r};\un{s}}(\un{a},x;\un{b};\un{c})$, where $|\un{r}|=|\un{s}|$;
\item[$\bullet$] $\si_{\un{t};\un{r},\un{k};\un{s}}(\un{a};\un{b},\un{e};\un{c})\in\AlgLargeY$ is a partial linearization of $\si_{\un{t};\un{r},k;\un{s}}(\un{a};\un{b},x;\un{c})$, where $|\un{s}|=|\un{r}|+k$;
\item[$\bullet$] $\si_{\un{t};\un{r};\un{s},\un{k}}(\un{a};\un{b};\un{c},\un{e})\in\AlgLargeY$ is a partial linearization of $\si_{\un{t};\un{r};\un{s},k}(\un{a};\un{b};\un{c},x)$, where $|\un{r}|=|\un{s}|+k$.
\end{enumerate}
\end{remark}
\bigskip
We will use the following Lemma~\ref{lemma_O_45} together with Remark~\ref{remark_O_45}:
\begin{lemma}\label{lemma_O_45}
Given $\un{t}=(t_1,\ldots,t_u)$ and $\un{a}=(a_1,\ldots,a_u)$, we write $\un{t}'$ for $(1^{t_1},t_2,\ldots,t_u)$ and $\un{a}^{(i)}$ for $(\underbrace{a_1,\ldots,a_1}_{i},a_2,\ldots,a_u)$. Then in case $\FF=\QQ$ we have the following equalities
$$\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})=
\frac{1}{t_1!} \si_{\un{t}';\un{r};\un{s}}(\un{a}^{(t_1)};\un{b};\un{c}) =
\frac{1}{r_1!} \si_{\un{t};\un{r}';\un{s}}(\un{a};\un{b}^{(r_1)};\un{c}) =
\frac{1}{s_1!} \si_{\un{t};\un{r};\un{s}'}(\un{a};\un{b};\un{c}^{(s_1)})$$
in $\AlgLargeY$.
\end{lemma}
\begin{proof}
Applying Remark~\ref{remark_O_OK2} instead of Remark~\ref{remark_GL_OK1}, we obtain the claim in the same way as we proved Lemma~\ref{lemma_GL_OK2}.
\end{proof}
\begin{remark}\label{remark_O_45}
Let $\algA_{\FF}=\AlgLargeY$. We write $\algA_{\ZZ}$ for the set of all $f\in\algA_{\QQ}$ with integer coefficients.
The natural surjective map $\ZZ\to \ZZ_p\subset \FF$ induces the well-defined homomorphism of rings $\algA_{\ZZ}\to \algA_{\FF}$. Here in case $p=0$ we assume that $\ZZ_p=\ZZ$.
\end{remark}
\bigskip
Similarly to a finite quiver $\Q(\un{x};\un{y};\un{z})$ from Section~\ref{section5} we can consider a quiver $\Q(a_1,a_2,\ldots;b_1,b_2,\ldots;c_1,c_2,\ldots)$ with infinitely many arrows, where $a_i,b_j,c_k\in\{x_1,x_2,\ldots\}$ are pairwise different letters for all $i,j,k>0$. Note that by path in a quiver we always mean a finite path.
Consider a quiver $\Q_{I}$ and a quiver $\Q_{II}$ with infinitely many arrows:
$$\Q_{I}=\Q(x_0,x;y;z) \;\text{ and }\; \Q_{II}=\Q(x,e_i; y, u_i, v_i, w_{ij}; z)_{i,j>0}.$$
\begin{lemma}\label{lemma_O_sets1}
Define a homomorphism $\varphi:\LA \Q_{II}\RA \to \LA \Q_{I}\RA$ of semigroups as follows:
$$\varphi(a^T)=\varphi(a)^T
\;\text{ and }\;
\varphi(a)=\left\{
\begin{array}{cl}
a,& \text{ if }a=x \text{ or } a=y \text{ or } a=z\\
x_0^ix ,& \text{ if } a=e_i\\
x_0^iy ,& \text{ if } a=u_i\\
x_0^iy^T ,& \text{ if } a=v_i^T\\
x_0^iy (x_0^T)^j ,& \text{ if } a=w_{ij}\\
\end{array}
\right.$
for an arrow $a$ of $\Q_{II}$ and $i,j>0$. We extend $\varphi$ to the map $\LA\Q_{II}\RA \sqcup\{x_0\} \to \LA\Q_{I}\RA$ by setting $\varphi(x_0)=x_0$. Then $\varphi$ induces the well-defined bijection $\ov{\varphi}:\LA \ov{\Q}_{II}\RA\sqcup\{\ov{x_0}\} \to \LA \ov{\Q}_{I}\RA$ of sets of $\sim$-equivalence classes of primitive elements.
\end{lemma}
\begin{proof}We split the proof into several statements. It is not difficult to see that $\varphi:\LA \Q_{II}\RA \to \LA \Q_{I}\RA$ is well-defined. We can extend $\varphi$ to the homomorphism $\path{\Q_{II}}\to\path{\Q_{I}}$, which we also denote by $\varphi$.
Denote by $\Omega_I$ the set of such elements $az,az^T\in\LA\Q_{I}\RA$ that a path $a$ satisfies $\deg_{z}(a)+\deg_{z^T}(a)=0$. In the same way we define $\Omega_{II}$. We claim that
\begin{eq}\label{eq_statement1}
\varphi:\Omega_{II}\to\Omega_I \text{ is a bijection}.
\end{eq
An arbitrary element of $\Omega_I$ can be written as $ab^{\de}\!cz^{\eta}$, where $\de,\eta\in\{1,T\}$ and
\begin{enumerate}
\item[$\bullet$] $a=x^{j_0} x_0^{i_1} x^{j_1}\cdots x_0^{i_r} x^{j_r}$ for $j_0\geq0$, $i_1,j_1,\ldots,i_r,j_r>0$, $r\geq0$; in particular, $a=1$ in case $j_0=r=0$;
\item[$\bullet$] $b=x_0^i y (x_0^T)^j$ for $i,j\geq0$;
\item[$\bullet$] $c=(x^T)^{k_1} (x_0^T)^{l_1}\cdots (x^T)^{k_s} (x_0^T)^{l_s}(x^T)^{k_{s+1}}$ for $k_1,l_1,\ldots,k_{s},l_{s}>0$, $k_{s+1}\geq0$, $s\geq0$; in particular, $c=1$ in case $s=k_1=0$.
\end{enumerate
Elements $a,b,c$ have unique preimages with respect to $\varphi$, namely,
$$\varphi^{-1}(b)=\left\{
\begin{array}{cl}
y,& \text{ if }i=j=0\\
u_i ,& \text{ if } i>0,\, j=0\\
v_j ,& \text{ if } i=0,\, j>0\\
w_{ij} ,& \text{ if } i>0,\, j>0\\
\end{array}
\right.,$
$\varphi^{-1}(a) = x^{j_0} e_{i_1} x^{j_1-1}\cdots e_{i_r} x^{j_r-1}$,
$\varphi^{-1}(c) = (x^T)^{k_1-1} e_{l_1}^T \cdots (x^T)^{k_s-1} e_{l_s}^T (x^T)^{k_{s+1}}$ for $a\neq1$ and $c\neq1$, respectively. Thus it is not difficult to see that $ab^{\de}\!cz^{\eta}$ has a unique preimage $\varphi^{-1}(ab^{\de}\!cz^{\eta})=\varphi^{-1}(a)(\varphi^{-1}(b))^{\de}\varphi^{-1}(c)z^{\eta}$. Statement~(\ref{eq_statement1}) is proven.
Consider paths $a,b\in\LA\Q_{II}\RA$. Then we claim that
\begin{eq}\label{eq_statement2}
a\sim b\text{ if and only if }\varphi(a)\sim\varphi(b).
\end{eq
Since $\varphi$ is a homomorphism, then $\varphi(a)\sim\varphi(b)$ follows from $a\sim b$.
Let $\varphi(a)\sim\varphi(b)$. Denote $\deg_z(a)+\deg_{z^T}(a)=\deg_z(b)+\deg_{z^T}(b)=r$. Note that if $r=0$, then Lemma~\ref{lemma_GL_sets} implies $a\sim b$. So we assume that $r>0$. Then $a\stackrel{c}{\sim} a_1\cdots a_r$ and $b\stackrel{c}{\sim} b_1\cdots b_r$ for $a_i,b_i\in\Omega_{II}$ ($1\leq i\leq r$).
Assume $\varphi(a)\stackrel{c}{\sim}\varphi(b)$. Since $\varphi(a_i),\varphi(b_i)\in\Omega_I$ for all $i$, there is a cyclic permutation $\pi=(1,2,\ldots,r)^l\in S_r$ for some $l>0$ such that $\varphi(a_i)=\varphi(b_{\pi(i)})$. Statement~(\ref{eq_statement1}) implies $a_i=b_{\pi(i)}$. Therefore, $a\stackrel{c}{\sim}b$. If $\varphi(a)\stackrel{c}{\sim}\varphi(b)^T$, then $\varphi(a)\stackrel{c}{\sim}\varphi(b^T)$ and $a\sim b$ follows from the proven part of statement~(\ref{eq_statement2}).
Let $b\in\LA\Q_{I}\RA$ and $b\neq x_0^l$ for all $l>0$. Then we claim that
\begin{eq}\label{eq_statement4}
\text{there exists an } a\in\LA\Q_{II}\RA \text{ satisfying }\varphi(a)\sim b.
\end{eq
Denote $\deg_z(b)+\deg_{z^T}(b)=r$. Note that if $r=0$, then Lemma~\ref{lemma_GL_sets} implies that $\varphi(a)=b$ for some $a\in\LA \Q_{II}\RA$. So we assume that $r>0$. Then $b\stackrel{c}{\sim} b_1\cdots b_r$ for $b_i\in\Omega_{I}$ ($1\leq i\leq r$). By statement~(\ref{eq_statement1}), there are $a_1,\ldots,a_r\in\LA \Q_{II}\RA$ such that $\varphi(a_i)=b_i$. Hence $\varphi(a_1\cdots a_r)\stackrel{c}{\sim} b$ and statement~(\ref{eq_statement4}) is proven.
Consider $a\in\LA\Q_{II}\RA$. We claim that
\begin{eq}\label{eq_statement5}
a \text{ is primitive if and only if } \varphi(a) \text{ is primitive}.
\end{eq
Let $\varphi(a)=b^l$ for $b\in\LA\Q_{I}\RA$ and $l>1$. If $\deg_z(a)+\deg_{z^T}(a)=0$, then the claim follows from Lemma~\ref{lemma_GL_sets}. Otherwise, $a\stackrel{c}{\sim} a_1\cdots a_r$ for $a_i\in\Omega_{II}$ and $r>0$. Hence, we obtain $b^l\stackrel{c}{\sim} \varphi(a_1)\cdots\varphi(a_r)$. By Remark~\ref{remark_primitive}, $\varphi(a_1)\cdots\varphi(a_r)=c^l$ for some $c\in\LA\Q_{I}\RA$. Since the last letter of $\varphi(a_r)$ is $z$ or $z^T$, the last letter of $c$ is also $z$ or $z^T$. Using the definition of $\Omega_{I}$ and the fact that $\varphi(a_i)\in\Omega_I$ for all $i$, we obtain that $c=\varphi(a_1)\cdots \varphi(a_s)$ for some $1\leq s<r$. Hence $\varphi(a_1)\cdots \varphi(a_r)=(\varphi(a_1)\cdots \varphi(a_s))^l$. In other words, $\varphi(a_i)=\varphi(a_{i+s})=\cdots=\varphi(a_{i+(l-1)s})$ for $1\leq i\leq s$. Statement~(\ref{eq_statement1}) implies $a_i=a_{i+s}=\cdots=a_{i+(l-1)s}$ for $1\leq i\leq s$, i.e., $a_1\cdots a_r=(a_1\cdots a_s)^l$. By Remark~\ref{remark_primitive}, $a$ is not primitive. The converse claim is trivial.
Now we can complete the proof of the theorem. Note that for $a\in\LA\Q_{II}\RA\sqcup\{x_0\}$ we have $\varphi(a)=x_0^l$ if and only if $l=1$ and $a=x_0$. By statements~(\ref{eq_statement2}) and (\ref{eq_statement5}), $\ov{\varphi}$ is a well-defined injective map. Statement~(\ref{eq_statement4}) implies that $\ov{\varphi}$ is a surjective map.
\end{proof}
\begin{lemma}\label{lemma_O_key1}
Given $x_0,x,y,z\in\Y$ and $k,t,r\geq0$, we have $\si_{k,t;r;r}(x_0,x;y;z)\in J_{t+2r}$.
\end{lemma}
\begin{proof}
We work in $\AlgLargeY$. Without loss of generality we can assume that $x_0,x,y,z\in\{x_1,x_2,\ldots\}$ are pairwise different letters. Assume that $e_i, y, u_i, v_i, w_{ij}, z$ ($i,j>0$) are pairwise different letters from $\{x_1,x_2,\ldots\}\backslash \{x_0,x,y,z\}$. In what follows, we use notations from Lemma~\ref{lemma_O_sets1}. Let $\Upsilon_I$ be the set of finite multisubsets of $\LA\Q_I\RA$ and $\Upsilon_{II}$ be the set of finite multisubsets of $\LA\Q_{II}\RA \sqcup\{x_0\}$. We define the $\sim$-equivalence on $\Upsilon_I$ naturally and denote by $\ov{\Upsilon}_I$ the set of all $\sim$-equivalence classes. Similarly we define $\ov{\Upsilon}_{II}$. Then Lemma~\ref{lemma_O_sets1} implies that $\ov{\varphi}:\ov{\Upsilon}_{II}\to \ov{\Upsilon}_I$ is a bijection.
Let use recall that the definition of $\Omega(\un{t};\un{r};\un{s})$ was given in Section~\ref{section5}. Assume that $\omega$ belongs to $\ov{\Upsilon}_I$ or $\ov{\Upsilon}_{II}$. Since we work in $\AlgLargeY$, the element $\si(\omega)$ is well-defined. For short, we write $\mdeg(\omega)$ for $\mdeg(\si(\omega))$. We refer to the entries of $\Delta=\mdeg(\omega)$ as follows:
\begin{enumerate}
\item[$\bullet$] $\De=(\al_0,\al; \be;\ga)$ for $\omega\in\ov{\Upsilon}_I$, where $\De=\mdeg(x_0^{\al_0}x^{\al} y^{\be} z^{\ga})$;
\item[$\bullet$] $\De=(\al_0,\al,\al_i; \be,\la_i,\mu_i,\nu_{ij};\ga)_{i,j>0}$ for $\omega\in\ov{\Upsilon}_{II}$, where $\De$ is equal to
$$\mdeg(x_0^{\al_0}x^{\al} y^{\be} z^{\ga} \prod_i x_i^{\al_i} u_i^{\la_i} v_i^{\mu_i} \prod_{j} w_{ij}^{\nu_{ij}} ).$$
Here we assume that only finitely many elements from $\{\al_i,\la_i,\mu_i,\nu_{ij}\}_{i,j>0}$ are non-zero.
\end{enumerate
In the first case (the second case, respectively) we say that $\De$ is a multidegree of type I (type II, respectively). By the definition,
\begin{eq}\label{eq1_lemma_O_key1}
\si_{k,t;r;r}(x_0,x;y;z)=(-1)^{k+t} \sum_{\omega\in \ov{\Omega}_I} \si(\omega),
\end{eq
where $\ov{\Omega}_I=\ov{\Omega}(k,t;r;r)=\{\omega\in\ov{\Upsilon}_{I}\,|\,\mdeg(\omega)=(k,t;r;r)\}$. For $\ov{\Omega}_{II}=\{\omega\in\ov{\Upsilon}_{II}\,|\,\mdeg(\varphi(\omega))=(k,t;r;r)\}$ an isomorphism of sets $\ov{\Omega}_{II}\simeq \ov{\Omega}_I$ is determined by the restriction of $\ov{\varphi}$.
Given a multidegree $\De$ of type II, we denote $\ov{\Omega}_{II}^{\De}=\{\omega\in\ov{\Upsilon}_{II}\,|\,\mdeg(\omega)=\De\}$ and $\varphi(\De)=\mdeg(\varphi(x_0^{\al_0}x^{\al} y^{\be} z^{\ga} \prod_i x_i^{\al_i} u_i^{\la_i} v_i^{\mu_i} \prod_{j} w_{ij}^{\nu_{ij}}))$. Thus
\begin{eq}\label{eq2_lemma_O_key1}
\ov{\Omega}_{II}=\bigsqcup \ov{\Omega}_{II}^{\Delta},
\end{eq
where the union ranges over $\Delta$ of type II satisfying $\varphi(\Delta)=(k,t;r;r)$. Consequently applying formula~(\ref{eq1_lemma_O_key1}), the isomorphism $\ov{\Omega}_{II}\simeq \ov{\Omega}_I$, and formula~(\ref{eq2_lemma_O_key1}) we obtain
$$\si_{k,t;r;r}(x_0,x;y;z)=(-1)^{k+t}\!\!\!\! \sum_{\varphi(\Delta)=(k,t;r;r)}\; \sum_{\omega\in\ov{\Omega}_{II}^{\Delta} } \si(\varphi(\omega)).$
Since
$$\deg_y(\varphi(c))+\deg_z(\varphi(c)) = \deg_y(c) + \sum_i\deg_{u_i}(c) + \sum_i\deg_{v_i}(c) + \sum_{ij}\deg_{w_{ij}}(c) + \deg_z(c)$
for all $c\in\LA \Q_{II}\RA$, we have $\si(\varphi(\omega))=\varphi(\si(\omega))$ for $\omega\in\ov{\Omega}_{II}^{\Delta}$. Therefore,
$$\sum_{\omega\in\ov{\Omega}_{II}^{\Delta} } \si(\varphi(\omega))=(-1)^{\al_0+|\Delta'|} \si_{\al_0}(x_0)\, \varphi( \si_{\Delta'}(x,e_i; y, u_i, v_i, w_{ij}; z)_{i,j>0}),$
where $\Delta'$ stands for $(\al,\al_i; \be,\la_i,\mu_i,\nu_{ij};\ga)_{i,j>0}$.
The condition $\varphi(\Delta)=(k,t;r;r)$ implies $|\Delta'|=t+2r$. Thus,
\begin{eq}\label{eq3_lemma_O_key1}
\begin{array}{c}
\si_{k,t;r;r}(x_0,x;y;z) \\
=\sum\limits(-1)^{\al_0+k}\si_{\al_0}(x_0)\, \si_{\Delta'}(x,x_0^i x; y, x_0^i y, y (x_0^T)^i, x_0^i y (x_0^T)^j; z)_{i,j>0},\\
\end{array}
\end{eq
where the sum ranges over $\Delta$ of type II satisfying $\varphi(\Delta)=(k,t;r;r)$.
The required is proven.
\end{proof}
\begin{example}\label{ex_O1}
For $x_0,x,y,z\in\Y$ and $t,r>0$ the following equalities of $\AlgLargeY$ are partial cases of key formula~(\ref{eq3_lemma_O_key1}) from the proof of Lemma~\ref{lemma_O_key1}:
\begin{enumerate}
\item[$\bullet$] $\si_{1,t;r;r}(x_0,x;y;z) =
\tr(x_0)\si_{t;r;r}(x;y;z)
- \si_{t-1,1;r;r}(x,x_0x;y;z)$
$- \si_{t;r-1,1;r}(x;y,x_0y;z)
- \si_{t;r-1,1;r}(x;y,yx_0^T;z) \in J_{t+2r}$;
\item[$\bullet$] $\si_{(2,0;2;2)}(x_0,x;y;z) =
\si_2(x_0)\si_{0;2;2}(x;y;z)$
$- \tr(x_0)\si_{0;1,1;2}(x;y,x_0y;z) - \tr(x_0)\si_{0;1,1;2}(x;y,yx_0^T;z)$
$+ \si_{0;2;2}(x;x_0y;z) + \si_{0;2;2}(x;yx_0^T;z)$
$+ \si_{0;1,1;2}(x;y,x_0^2y;z) + \si_{0;1,1;2}(x;y,y(x_0^T)^2;z)$
$+ \si_{0;1,1;2}(x;x_0y,yx_0^T;z)
+ \si_{0;1,1;2}(x;y,x_0yx_0^T;z) \in J_4$.
\end{enumerate}
\end{example}
\bigskip
Consider quivers $\Q_{III}=\Q(x;y_0,y;z)$ and $\Q_{IV}=\Q(x,e_1,e_2; y, y_1; z)$.
\begin{lemma}\label{lemma_O_sets2}
Define a homomorphism $\varphi:\LA \Q_{IV}\RA \to \LA \Q_{III}\RA$ of semigroups as follows:
$$\varphi(a^T)=\varphi(a)^T
\;\text{ and }\;
\varphi(a)=\left\{
\begin{array}{cl}
a,& \text{ if }a=x \text{ or } a=y \text{ or } a=z\\
y_0z ,& \text{ if } a=e_1\\
y_0z^T ,& \text{ if } a=e_2\\
y_0x^T ,& \text{ if } a=y_1\\
\end{array}
\right.$
for an arrow $a$ of $\Q_{IV}$. Then $\varphi$ induces the well-defined bijection $\ov{\varphi}:\LA \ov{\Q}_{IV}\RA \to \LA \ov{\Q}_{III}\RA$ of sets of $\sim$-equivalence classes of primitive elements.
\end{lemma}
\begin{proof}It is not difficult to see that $\varphi:\LA \Q_{IV}\RA \to \LA \Q_{III}\RA$ is well-defined. Denote by $\Omega_{III}$ the set of elements $a\in\LA\Q_{III}\RA$ satisfying $\deg_z(a)+\deg_{z^T}(a)=1$ and denote by $\Omega_{IV}$ the set of elements $a\in\LA\Q_{IV}\RA$ satisfying
$$\deg_z(a)+\deg_{z^T}(a)+\sum_{i=1}^2(\deg_{e_i}(a)+\deg_{e_i^T}(a))=1.$
We claim that
\begin{eq}\label{eq_statement6}
\ov{\varphi}: \ov{\Omega}_{IV}\to\ov{\Omega}_{III} \text{ is a bijection of $\sim$-equivalence classes}.
\end{eq
An arbitrary element of $\Omega_{III}$ is $\sim$-equivalent to $a=x^i b (x^T)^j z^{\de}$ for some $i,j\geq0$, $\de\in\{1,T\}$, and $b\in\{y_0,y\}$. If $\deg_{y_0}(a)=0$, then $a$ has a unique preimage $\varphi^{-1}(a)=a$. Otherwise, $b=y_0$ and $a$ also has a unique preimage, namely,
$$\varphi^{-1}(a)=\left\{
\begin{array}{cl}
x^i e_1 ,& \text{ if } j=0 \text{ and }\de=1\\
x^i e_2 ,& \text{ if } j=0 \text{ and }\de=T\\
x^i y_1 (x^T)^{j-1} z^{\de} ,& \text{ if } j>0\\
\end{array}
\right..$
Statement~(\ref{eq_statement6}) is proven.
Applying statement~(\ref{eq_statement6}) instead of statement~(\ref{eq_statement1}), we complete the proof of the lemma in a similar way as the proof of Lemma~\ref{lemma_O_sets1}.
\end{proof}
\begin{lemma}\label{lemma_O_key2}
Given $x,y_0,y,z\in\Y$ and $t,r,s\geq0$, we have $\si_{t;r,s;r+s}(x;y_0,y;z)\in J_{t+r+2s}$.
\end{lemma}
\begin{proof}
We work in $\AlgLargeY$. Without loss of generality we can assume that $x,y_0,y,z\in\{x_1,x_2,\ldots\}$ are pairwise different letters. Assume that $e_1,e_2, y_1$ are pairwise different letters from $\{x_1,x_2,\ldots\}\backslash \{x,y_0,y,z\}$. In what follows, we use notations from Lemma~\ref{lemma_O_sets2}. Let $\Upsilon_{III}$ be the set of finite multisubsets of $\LA\Q_{III}\RA$ and $\Upsilon_{IV}$ be the set of finite multisubsets of $\LA\Q_{IV}\RA$. Then Lemma~\ref{lemma_O_sets2} implies that $\ov{\varphi}:\ov{\Upsilon}_{IV}\to \ov{\Upsilon}_{III}$ is a bijection of sets of multisets of $\sim$-equivalence classes.
Assume that $\omega$ belongs to $\ov{\Upsilon}_{III}$ or $\ov{\Upsilon}_{IV}$. Since we work in $\AlgLargeY$, the element $\si(\omega)$ is well-defined. For short, we write $\mdeg(\omega)$ for $\mdeg(\si(\omega))$. We refer to the entries of $\Delta=\mdeg(\omega)$ as follows:
\begin{enumerate}
\item[$\bullet$] $\De=(\al; \be_0, \be;\ga)$ for $\omega\in\ov{\Upsilon}_{III}$, where $\De=\mdeg(x^{\al} y_0^{\be_0} y^{\be} z^{\ga})$;
\item[$\bullet$] $\De=(\al,\al_1,\al_2; \be,\be_1;\ga)$ for $\omega\in\ov{\Upsilon}_{IV}$, where $\De=\mdeg(x^{\al} x_1^{\al_1} x_2^{\al_2} y^{\be} y_1^{\be_1} z^{\ga})$.
\end{enumerate
In the first case (the second case, respectively) we say that $\De$ is a multidegree of type III (type IV, respectively). By the definition,
\begin{eq}\label{eq1_lemma_O_key2}
\si_{t;r,s;r+s}(x;y_0,y;z)=(-1)^{t}\!\! \sum_{\omega\in \ov{\Omega}_{III}} \si(\omega),
\end{eq
where $\ov{\Omega}_{III}=\ov{\Omega}(t;r,s;r+s)=\{\omega\in\ov{\Upsilon}_{III}\,|\,\mdeg(\omega) = (t;r,s;r+s)\}$. For $\ov{\Omega}_{IV} = \{\omega\in\ov{\Upsilon}_{IV}\,|\,\mdeg(\varphi(\omega))=(t;r,s;r+s)\}$ an isomorphism of sets $\ov{\Omega}_{IV}\simeq \ov{\Omega}_{III}$ is determined by the restriction of $\ov{\varphi}$.
Given a multidegree $\De$ of type IV, we denote $\ov{\Omega}_{IV}^{\De}=\{\omega\in\ov{\Upsilon}_{IV}\,|\,\mdeg(\omega)=\De\}$ and $\varphi(\De)=\mdeg(\varphi(x^{\al} x_1^{\al_1} x_2^{\al_2} y^{\be} y_1^{\be_1} z^{\ga}))$. Thus
\begin{eq}\label{eq2_lemma_O_key2}
\ov{\Omega}_{IV}=\bigsqcup \ov{\Omega}_{IV}^{\Delta},
\end{eq
where the union ranges over $\Delta$ of type IV satisfying $\varphi(\Delta)=(t;r,s;r+s)$. Consequently applying formula~(\ref{eq1_lemma_O_key2}), the isomorphism $\ov{\Omega}_{IV}\simeq \ov{\Omega}_{III}$, and formula~(\ref{eq2_lemma_O_key2}) we obtain
$$\si_{t;r,s;r+s}(x;y_0,y;z)=(-1)^{t}\!\!\!\! \sum_{\varphi(\Delta)=(t;r,s;r+s)}\; \sum_{\omega\in\ov{\Omega}_{IV}^{\Delta} } \si(\varphi(\omega)).$
Since $\deg_{y_0}(\varphi(c))+\deg_y(\varphi(c))+\deg_z(\varphi(c))$ is equal to
$$\left(\deg_y(c) + \deg_{y_1}(c) + \deg_z(c)\right) +
\left( 2\deg_{e_1}(c) + \deg_{e_2}(c) + \deg_{e_2^T}(c)\right)
$
for all $c\in\LA\Q_{IV}\RA$, we have $\si(\varphi(\omega))=(-1)^{\al_2}\varphi(\si(\omega))$ for $\omega\in\ov{\Omega}_{IV}^{\Delta}$. Therefore,
$$\sum_{\omega\in\ov{\Omega}_{IV}^{\Delta} } \si(\varphi(\omega))=(-1)^{|\Delta|+\al_2} \varphi(\si_{\Delta}(x,e_1, e_2; y, y_1; z)).$
A multidegree $\Delta$ of type IV satisfies the equality $\varphi(\Delta)=(t;r,s;r+s)$ if and only if
$$\begin{array}{rcl}
\al+\be_1 &=& t\\
\al_1+\al_2+\be_1 &=& r\\
\be&=&s\\
\ga+\al_1+\al_2&=&r+s\\
\end{array};
$
in particular, $|\Delta|=t+r+2s$. Thus,
\begin{eq}\label{eq3_lemma_O_key2}
\si_{t;r,s;r+s}(x;y_0,y;z) = \sum\limits (-1)^{\al_2+r} \si_{\Delta}(x, y_0 z, y_0 z^T; y, y_0x^T; z),
\end{eq
where the sum ranges over $\Delta$ of type IV satisfying $\varphi(\Delta)=(t;r,s;r+s)$. The required is proven.
\end{proof}
\begin{example}\label{ex_O2}
For $x,y_0,y,z\in\Y$ and $t>0$, $s\geq0$ the following equality of $\AlgLargeY$ is a partial case of key formula~(\ref{eq3_lemma_O_key2}) from the proof of Lemma~\ref{lemma_O_key2}:
$$\begin{array}{c}
\si_{t;1,s;s+1}(x;y_0,y;z) = -\si_{t,1;s;s}(x,y_0z;y;z) + \si_{t,1;s;s}(x,y_0z^T;y;z)\\
- \si_{t-1;s,1;s+1}(x;y,y_0x^T;z) \in J_{t+2s+1}.\\
\end{array}
$$
\end{example}
\begin{lemma}\label{lemma_O_key3}
Given $\un{t}\in\NN^u_0$, $\un{r}\in\NN^v_0$, $\un{s}\in\NN^w_0$ with $t=|\un{t}|$ and $r=|\un{r}|=|\un{s}|$ ($u,v,w>0$). Then for $f=\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})\in \AlgLargeY$, where $a_i,b_j,c_k\in\Y$, we have
\begin{enumerate}
\item[$\bullet$] $f\in J_{t+2r-t_1}$;
\item[$\bullet$] $f\in J_{t+2r-r_1}$ and $f\in J_{t+2r-s_1}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume $t_1>0$. By Remark~\ref{remark_O_OK2}, $f$ is equal to the coefficient of $\la_2^{t_2}\cdots\la_u^{t_u} \mu_1^{r_1}\cdots\mu_v^{r_v} \nu_1^{s_1}\cdots\nu_w^{s_w}$ in $h=\si_{\un{k};r;r}(a_1,\la_2 a_2+\cdots+\la_u a_u; \mu_1 b_1+\cdots+\mu_v b_v; \nu_1 c_1+\cdots+\nu_w c_w)$, where $\un{k}=(t_1,t-t_1)$ and $\la_2,\ldots,\nu_w\in\FF$. Lemma~\ref{lemma_O_key1} implies that $h$ belongs to $J_{t+2r-t_1}$. Since the ideal $J_{t+2r-t_1}$ is closed with respect to partial linearizations, we obtain $f\in J_{t+2r-t_1}$.
Let $r_1>0$. Applying Lemma~\ref{lemma_O_key2} instead of Lemma~\ref{lemma_O_key1}, we obtain $f\in J_{t+2r-r_1}$ by the same reasoning as above. Formula~(\ref{eq_O_transpose}) concludes the proof.
\end{proof}
\begin{lemma}\label{lemma_O_key4}
Given $\un{t}\in\NN_0^u$, $\un{r}\in\NN_0^v$, $\un{s}\in\NN_0^w$ with $t=|\un{t}|$ and $r=|\un{r}|=|\un{s}|$. Then $\si_{\un{t};\un{r};\un{s}}(\un{a};\un{b};\un{c})$ belongs to $J_{t+2r}^{(p)}$ for $a_i,b_j,c_k\in\Y$.
\end{lemma}
\begin{proof} We repeat the proof of Lemma~\ref{lemma_GL_key2}, applying Lemma~\ref{lemma_O_45} together with Remark~\ref{remark_O_45} instead of Lemma~\ref{lemma_GL_OK2}.
\end{proof}
Now we can prove Theorem~\ref{theo_O} and Remark~\ref{remark_theo_O}:
\begin{proof}
We work in $\AlgLargeY$. Since the field $\FF$ is infinite, Theorem~\ref{theo_Lopatin} together with Remark~\ref{remark_O_OK1} implies that $\KLargeY{n}$ is generated by
\begin{enumerate}
\item[(a)] $\si_{\un{t},\un{r},\un{s}}(\un{a},\un{b},\un{c})=0$ for $|\un{t}|+2|\un{r}|>n$, where $\un{t}\in\NN_0^u$, $\un{r}\in\NN_0^v$, $\un{s}\in\NN_0^w$ ($u,v,w>0$) satisfy $|\un{r}|=|\un{s}|$ and $a_i,b_j,c_k\in\Y$;
\item[(b)] $\si_t(b)=0$ for $t>n$, where $b\in\Y$.
\end{enumerate}
Applying Lemmas~\ref{lemma_O_key3} and~\ref{lemma_O_key4} instead of Lemmas~\ref{lemma_GL_key} and~\ref{lemma_GL_key2}, respectively, we complete the proof in the same way as we proved Theorem~\ref{theo_GL} at the end of Section~\ref{section3}.
\end{proof}
\begin{lemma}\label{lemma_O_fb}
If $p=0$, then the ideal $\KLargeY{n}\vartriangleleft \AlgLargeY$ is generated by $\si_{t,r}(a,b,c)=0$ for $t+2r=n+1$, $t,r\geq0$, and $a,b,c\in\FF\Y$; in particular, $\KLarge{n}$ is finitely based.
If $p>0$, then the ideal $\KLargeY{n}\vartriangleleft \AlgLargeY$ is not finitely based.
\end{lemma}
\begin{proof}
Using Theorem~\ref{theo_O}, Remarks~\ref{remark_theo_O},~\ref{remark_O_OK1} and Lemma~\ref{lemma_O_Lopatin}, instead of Theorem~\ref{theo_GL}, Remarks~\ref{remark_theo_GL},~\ref{remark_GL_OK1} and Lemma~\ref{lemma_GL_Donkin}, respectively, we prove this lemma in the same way as we proved Lemma~\ref{lemma_GL_fb}.
\end{proof}
\section{Proof of Theorem~\ref{theo_O_main}}\label{section7}
Assume that
\begin{eq}\label{eq5}
f=\sum_i \al_i f_i a_i\in \AlgLargeY\otimes \FF\Y^{\#}
\end{eq
for $\al_i\in\FF$, $f_i=\si_{t_{i1}}(b_{i1})\cdots \si_{t_{ir_i}}(b_{ir_i})\in \AlgLargeY$, $b_{ij}\in\Y$, $a_i\in\Y^{\#}$.
If $a_i\in\Y$ for all $i$, then we write $\tr(f)$ for $\sum_i \al_i f_i \tr(a_i)\in \AlgLargeY\otimes \FF\Y^{\#}$. Note that we do not define $\tr(f)$ for $f\in\AlgLargeY\otimes 1\subset \AlgLargeY\otimes \FF\Y^{\#}$. We say that $f$ does not contain a letter $x$ if $\deg_x(f_i)=0$ and $\deg_x(a_i)=0$ for all $i$.
\begin{lemma}\label{lemma_71}
Let $f\in \AlgLargeY\otimes \FF\Y^{\#}$ do not contain letters $x$ and $x^T$. Then
\begin{enumerate}
\item[1)] $f\in \TLargeY{n}$ if and only if $\tr(fx)\in \KLargeY{n}$;
\item[2)] $f=0$ if and only if $\tr(fx)=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
1) Note that $\tr(fx)\in\KLargeY{n}$ if and only if $\tr(\PsiLargeY{n}(f)X)=0$ for the generic $n\times n$ matrix $X$ corresponding to $x$. Since the trace bilinear form $\tr:M_n(\FF)\times M_n(\FF)\to \FF$ is nondegenerate, the last condition is equivalent to the fact that $\PsiLargeY{n}(f)=0$. The required is proven.
\smallskip
2) Let $f$ be given by formula~(\ref{eq5}). Since the equality $\al f_i\tr(a_i x) = f_j\tr(a_j x)$ in $\AlgLargeY\otimes \FF\Y^{\#}$ for some $\al\in\FF$ implies $\al=1$, $f_i=f_j$, and $a_i=a_j$, we obtain that the equality $f=0$ follows from
$\tr(fx)=0$.
\end{proof}
Analogues of formula~(\ref{eq_O_transpose}) hold for $\chi_{t,r}$ and $\zeta_{t,r}$.
\begin{lemma}\label{lemma_O_transpose}
For $x,a,b,c\in\Y$ we have
\begin{enumerate}
\item[1)] $\si_{t,1;r;r}(a,x;b;c) = (-1)^{t}\tr(\chi_{t,r}(a,b,c)x)\;$ and $\;\si_{t;r,1;r+1}(a;b,x;c) = (-1)^{t} \tr(\zeta_{t,r}(a,b,c)x)$ in $\AlgLargeY$;
\item[2)] $\chi_{t,r}(a,b,c)^T=\chi_{t,r}(a^T,c,b)=\chi_{t,r}(a,b^T,c^T)^T$ in $\AlgLargeY\otimes \FF\Y^{\#}$;
\item[3)] $\zeta_{t,r}(a,b,c)^T=\zeta_{t,r}(a,b^T,c^T)$ in $\AlgLargeY\otimes \FF\Y^{\#}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part~1 follows from the definitions and parts~2,~3 are consequences of formula~(\ref{eq_O_transpose}), part~1 of the lemma, and part~2 of Lemma~\ref{lemma_71}.
\end{proof}
\begin{lemma}\label{lemma_73}
The ideal of relations $\TLargeY{n}$ for $\CY_n$ is generated by $\KLargeY{n}\otimes 1$ and
\begin{enumerate}
\item[$\bullet$] $\chi_{t,r}(a,b,c)=0$ for $t+2r=n$;
\item[$\bullet$] $\zeta_{t,r}(a,b,c)=0$ for $t+2r=n-1$;
\end{enumerate}
where $a,b,c\in\FF\Y$.
\end{lemma}
\begin{proof}
By part~1 of Lemma~\ref{lemma_71}, an element $f\in\AlgLargeY\otimes \FF\Y^{\#}$ belongs to $\TLargeY{n}$ if and only if $\tr(fx)\in\KLargeY{n}$ for such a letter $x$ that neither $x$ nor $x^T$ is not contained in $f$. Since $\deg_{x}\tr(fx)+\deg_{x^T}\tr(fx)=1$, Theorem~\ref{theo_O} together with Remarks~\ref{remark_theo_O},~\ref{remark_O_OK2} implies that the last condition holds if and only if $\tr(fx)$ belongs to the ideal of $\AlgLargeY$, generated by
\begin{enumerate}
\item[(a)] $\si_{t,1;r;r}(a,e;b;c)=0$ for $t+2r=n$;
\item[(b)] $\si_{t;r,1;r+1}(a;b,e;c)=0$ and $\si_{t;r+1;r,1}(a;b;c,e)=0$ for $t+2r=n-1$;
\item[(c)] $h\tr(e)$ for $h\in\KLargeY{n}$;
\end{enumerate}
where $a,b,c\in\FF\Y$, $e=e_1 x^{\de} e_2$ for $e_1,e_2\in\Y^{\#}$ and $\de\in\{1,T\}$. By formula~(\ref{eq_O_transpose}), $\si_{t;r+1;r,1}(a;b;c,e)=\si_{t;r,1;r+1}(a^T;c^T,e^T;b^T)$ in $\AlgLargeY$. Thus, parts~1,~2,~3 of Lemma~\ref{lemma_O_transpose} imply that elements~(a) and~(b) of $\AlgLargeY$ coincide with elements
\begin{enumerate}
\item[$\bullet$] $\pm\tr(\chi_{t,r}(a,b,c)e_1xe_2)=0$ for $t+2r=n$;
\item[$\bullet$] $\pm\tr(\zeta_{t,r}(a,b,c)e_1xe_2)=0$ for $t+2r=n-1$;
\end{enumerate}
where $a,b,c\in\FF\Y$ and $e_1,e_2\in\Y^{\#}$. Finally, part~2 of Lemma~\ref{lemma_71} completes the proof.
\end{proof}
Now we can complete the proof of Theorem~\ref{theo_O_main}.
\begin{proof}
Applying Lemma~\ref{lemma_73}, we prove part~1 of Theorem~\ref{theo_O_main} exactly in the same way as we proved part~1 of Theorem~\ref{theo_O_main} at the end of Section~\ref{section4}. Using Lemma~\ref{lemma_O_Lopatin}, Theorem~\ref{theo_O}, Remark~\ref{remark_theo_O} instead of Lemma~\ref{lemma_GL_Donkin}, Theorem~\ref{theo_GL}, Remark~\ref{remark_theo_GL}, respectively, we repeat the reasoning from Section~\ref{section4} to prove part~2 of Theorem~\ref{theo_O_main}.
\end{proof}
\section*{Acknowledgements}
This paper was written during author's visit to Bielefeld University, sponsored by CRC 701 ``Spectral Structures and Topological Methods in Mathematics''. The author is grateful for this support. The author is also grateful to Professor Claus Michael Ringel for hospitality. This paper has also been partially supported by grants of Ministry of Education and Science of Russia \textnumero 14.B37.21.0359 and \textnumero 0859.
|
2,877,628,090,847 | arxiv | \section{Introduction}
Incompressible fluid dynamics underlies the vast majority of natural phenomena.
It is described by famous Navier-Stokes equation
\begin{equation}
\dot{v}_{\alpha} = \nu \partial_{\beta}^2 v_{\alpha} - v_{\beta}
\partial_{\beta} v_{\alpha} - \partial_{\alpha} p \\;\;
\partial_{\alpha}v_{\alpha} = 0 \label{eq1}
\end{equation}
which is nonlinear, and therefore hard to solve.
This nonlinearity makes life more interesting, though, as it leads to
turbulence. Solving this equation with appropriate initial and boundary
conditions we expect to obtain the chaotic behavior of velocity field.
The simplest boundary conditions correspond to infinite space with
vanishing velocity at infinity. We are looking for the translation
invariant probability distribution for velocity field, with infinite range of
the wavelengths. In order to compensate for the energy dissipation, we add the
usual random force to the Navier-Stokes equations, with the short wavelength support,
corresponding to large scale energy pumping.
One may attempt to describe this probability distribution by the
Hopf generating functional (the angular bracket denote time averaging, or
ensemble averaging over realizations of the random forces)
\begin{equation}
Z[J] = \left \langle \exp \left( \int d^3 r
J_{\alpha}(r)v_{\alpha}(r)\right)
\right \rangle \label{eq2}
\end{equation}
which is known to satisfy linear functional differential equation
\begin{equation}
\dot{Z} = H\left[J,\frac{\delta}{\delta J} \right] Z
\label{eq3}
\end{equation}
similar to the Schr\"odinger equation for Quantum Field Theory, and
equally hard to solve. Nobody managed to go beyond the Taylor
expansion in source $ J $ , which corresponds to the obvious chain of equations
for the equal time correlation functions of velocity
field in various points in space. The same equations could be obtained
directly from Navier-Stokes equations, so the Hopf equation looks
useless.
In this work\footnote{see also \ct{Loop} where this approach was initiated and
\ct{cells} where its relation with the generalized Hamiltonian dynamics and the
Gibbs-Boltzmann statistics was established} we argue, that one could
significantly simplify the Hopf functional without loosing information about
correlation functions.
This simplified functional depends upon the set of 3 periodic
functions of one variable
\begin{equation}
C : r_{\alpha} = C_{\alpha}(\theta)\\;\; 0< \theta< 2\pi
\end{equation}
which set describes the closed loop in coordinate space. The
correlation functions reduce to certain functional derivatives of our loop
functional with respect to $ C(\theta)$ at vanishing loop $ C \rightarrow 0 $.
The properties of the loop functional at large loop $ C $ also have
physical significance. Like the Wilson loops in Gauge Theory, they
describe the statistics of large scale structures of vorticity field, which is
analogous to the gauge field strength.
In Appendix A we recover the expansion in inverse powers of viscosity by direct
iterations of the loop equation.
In Appendix B we study the matrix formulation of the Navier-Stokes equation, which may
serve as a basis of the random matrix description of turbulence.
In Appendix C we study the reduced dynamics, corresponding to the
functional Fourier transform of the loop functional. We argue, that
instead of 3D Navier-Stokes equations one can use the 1D equations for the Fourier loop
$P_{\alpha}(\theta,t)$.
In Appendix D we discuss the relation between the initial data for
velocity field and the $P$ field, and we find particular realisation for these
initial data in terms of the gaussian random variables.
In Appendix E we introduce the generating functional for the scalar
products $ P_{\alpha}(\theta)P_{\alpha}(\theta') $. The advantage of this
functional over the original $\Psi[C]$ functional is the smoother continuum
limit.
In Appendix F we discuss the possible numerical implementations
of the reduced loop dynamics.
In Appendix G we show uniqueness of the tensor area law within certain class of
functionals.
In Appendix H we present the modern view at the old problem of the minimal
surface.
In Appendix I we show that the triple Kolmogorov correlation function
corresponds to a vanishing correlation of vorticity with two velocity fields.
\section{The Loop Calculus}
We suggest to use in turbulence the following version of the Hopf functional
\begin{equation}
\Psi \left[C \right] = \left \langle \exp
\left(
\frac{\imath\, }{\nu} \oint dC_{\alpha}(\theta)
v_{\alpha}\left(C(\theta)\right) \right) \right \rangle \label{eq4}
\end{equation}
which we call the loop functional or the loop field.
It is implied that all angular variable $\theta$ run from $ 0 $ to $ 2\pi$and
that all the functions of this variable are $ 2\pi$
periodic.\footnote{This parametrization of the loop is a matter of
convention, as the loop functional is parametric invariant.} The viscosity $
\nu $ was inserted in denominator in exponential, as the only parameter of
proper dimension. As we shall see below, it plays the role, similar to the
Planck's constant in Quantum mechanics, the turbulencecorresponding to the WKB
limit $ \nu \rightarrow 0 $. \footnote{One could also insert any numerical
parameter in exponential, but this factor could be eliminated by space- and/or
time rescaling.}
As for the imaginary unit $\imath\,$, there are two reasons to insert it in the
exponential. First, it makes the motion compact: the phase factor goes around
the unit circle, when the velocity field fluctuates. So, at large times one
may expect the ergodicity, with well defined average functional bounded by $1$
by absolute value. Second, with this factor of $\imath\,$, the irreversibility of
the problem is manifest. The time reversal corresponds to the complex
conjugation of $\Psi$, so that imaginary part of the asymptotic value of
$\Psi$ at $t \rightarrow \infty$ measures the effects of dissipation.
The loop orientation reversal $ C(\theta) \rightarrow C(2\pi - \theta) $ also leads to
the complex conjugation, so it is equivalent to the time reversal. This
symmetry implies, that any correlator of odd/even number of velocities should
be integrated odd/even number of times over the loop, and it must enter with
an imaginary/real factor. Later, we shall use this property in the area law.
We shall often use the field theory notations for the loop integrals,
\begin{equation}
\Psi \left[C \right] = \left \langle \exp
\left(
\frac{\imath\, }{\nu} \oint_C dr_{\alpha}v_{\alpha}
\right) \right \rangle \label{eq4'}
\end{equation}
This loop integral can be reduced to the surface
integral of vorticity field
\begin{equation}
\omega_{\mu\nu} = \partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu}
\end{equation}
by the Stokes theorem
\begin{equation}
\Gamma_C[v] \equiv \oint_C dr_{\alpha}v_{\alpha}= \int_{S} d
\sigma_{\mu\nu} \omega_{\mu\nu} \\;\; \partial S = C
\end{equation}
This is the well-known velocity circulation, which measures the net
strength of the vortex lines, passing through the loop $ C $. Would we fix
initial loop $ C $ and let it move with the flow, the loop field would be
conserved by the Euler equation, so that only the viscosity effects would be
responsible for its time evolution. However, this is not what we are trying to
do. We take the Euler rather than Lagrange dynamics, so that the loop is fixed
in space, and hence $\Psi$ is time dependent already in the Euler equations.
The difference between Euler and Navier-Stokes equations is the time irreversibility,
which leads to complex average $\Psi$ in Navier-Stokes dynamics.
It is implied that this field $\Psi\left[C\right]$ is invariant under
translations of the loop $ C(\theta) \rightarrow C(\theta)+ const $. The
asymptotic behavior at large time with proper random forcing reaches certain
fixed point, governed by the translation- and scale invariant equations, which
we derive in this paper.
The general Hopf functional (\ref{eq2}) reduces for the loop field for the
following imaginary singular source
\begin{equation}
J_{\alpha}(r) = \frac{\imath\, }{ \nu} \oint_C dr'_{\alpha} \delta^3
\left(r'-r \right) \label{eq8}
\end{equation}
The $\Psi$ functional involves connected correlation functions of the powers
of circulation at equal times.
\begin{equation}
\Psi[C] = \EXP{\sum_{n=2}^{\infty}\frac{\imath\,^{n}}{n!\,\nu^{n}}\,\VEV{\VEV{
\Gamma_C^n[v]}}}
\end{equation}
This expansion goes in powers of effective Reynolds number, so it diverges in
turbulent region. There, the opposite WKB approximation will be used.
Let us come back to the general case of the arbitrary Reynolds number. What
could be the use of such restricted Hopf functional? At first glance it seems
that we lost most of information, described by the Hopf functional, as the
general Hopf source $J$ depends upon 3 variables $ x,y,z $ whereas the loop $C$
depends of only one parameter $ \theta $. Still, this information can be
recovered by taking the loops of the singular shape, such as two infinitesimal
loops $R_1, R_2 $, connected by a couple of wires
\pct{Fig1}
The loop field in this case reduces to
\begin{equation}
\Psi \left[C \right] \rightarrow \left \langle \exp
\left(
\frac{\imath\,}{ 2\nu} \Sigma_{\mu\nu}^{R_1}\omega_{\mu\nu}(r_1)
+\frac{\imath\,}{ 2\nu} \Sigma_{\mu\nu}^{R_2 } \omega_{\mu\nu}(r_2)
\right) \right \rangle
\end{equation}
where
\begin{equation}
\Sigma_{\mu\nu}^R = \oint_R d r_{\nu}r_{\mu}
\end{equation}
is the tensor area inside the loop $R$. Taking functional derivatives with
respect to the shape of $R_1$ and $R_2$ prior to shrinking them to points, we
can bring down the product of vorticities at $r_1$ and $r_2$. Namely, the
variations yield
\begin{equation}
\delta\Sigma_{\mu\nu}^R= \oint _R\left(d r_{\nu}\delta r_{\mu}+ r_{\mu}d
\delta r_{\nu} \right) = \oint_R \left( d r_{\nu}\delta r_{\mu} -d
r_{\mu}\delta r_{\nu} \right)
\end{equation}
where integration by parts was used in the second term.
One may introduce the area derivative $\fbyf{}{\sigma_{\mu\nu}(r)}$, which
brings down the vorticity at the given point $ r $ at the loop.
\begin{equation}
-\nu^2 \frac{\delta^2 \Psi \left[C \right]}
{\delta \sigma_{\mu\nu}(r_1)\delta \sigma_{\lambda \rho}(r_2)}
\rightarrow \left \langle \omega_{\mu\nu}(r_1)
\omega_{\lambda \rho}(r_2) \right \rangle
\end{equation}
The careful definition of these area derivatives are or paramount importance
to us. The corresponding loop calculus was developed in\cite{Mig83} in the
context of the gauge theory. Here we rephrase
and further refine the definitions and relations established in that
work.
The basic element of the loop calculus is what we suggest to call the spike
derivative, namely the operator which adds the infinitesimal $
\Lambda $ shaped spike to the loop
\begin{equation}
D_{\alpha}(\theta,\epsilon) = \int_{\theta}^{\theta+2\epsilon}d \phi
\left(
1-\frac{\left|\theta +\epsilon - \phi\right|}{\epsilon }
\right)
\frac{\delta}{\delta C_{\alpha}(\phi)}
\end{equation}
The finite spike operator
\begin{equation}
\Lambda(r,\theta,\epsilon) =
\exp \left( r_{\alpha} D_{\alpha}(\theta,\epsilon) \right)
\end{equation}
adds the spike of the height $r$. This is the straight line from $
C(\theta) $ to $ C(\theta + \epsilon) + r$, followed by another
straight line from $ C(\theta+\epsilon)+r $ to $ C(\theta+2
\epsilon)$,
\pct{Fig2}
Note, that the loop remains
closed, and the slopes remain finite, only the second derivatives
diverge. The continuity and closure of the loop eliminates the
potential part of velocity; as we shall see below,
this is necessary to obtain the loop equation.
In the limit $ \epsilon \rightarrow 0 $ these spikes are invisible, at
least for the smooth vorticity field, as one can see from the Stokes
theorem (the area inside the spike goes to zero as $ \epsilon $).
However, taking certain derivatives prior to the limit $ \epsilon
\rightarrow 0 $ we can obtain the finite contribution.
Let us consider the operator
\begin{equation}
\Pi \left(r,r',\theta ,\epsilon \right) =
\Lambda \left(r, \theta,\frac{1}{2} \epsilon \right) \Lambda
\left(r',\theta,\epsilon \right)
\end{equation}
By construction it inserts the smaller spike on top of a bigger one,
in such a way, that a polygon appears
\pct{Fig3}
Taking the derivatives with respect to the vertices of
this polygon $ r, r' $ , setting $r=r'=0$ and
antisymmetrising, we find the tensor operator
\begin{equation}
\Omega_{\alpha\beta}(\theta,\epsilon) =
-\imath\, \nu D_{\alpha}\left(\theta,\frac{1}{2} \epsilon \right)
D_{\beta}\left(\theta,\epsilon \right) - \{\alpha \leftrightarrow\beta\}
\label{OM}
\end{equation}
which brings down the vorticity, when applied to the loop field
\begin{equation}
\Omega_{\alpha\beta}(\theta,\epsilon) \Psi \left[C \right]
\stackrel{\epsilon \rightarrow 0}{\longrightarrow}
\omega_{\alpha\beta}\left(C(\theta)\right)\Psi \left[C \right] \label{eqom}
\end{equation}
The quick way to check these formulas is to use formal functional
derivatives
\begin{equation}
\frac{\delta \Psi \left[C \right]}{\delta C_{\alpha}(\theta)} =
C'_{\beta}(\theta) \fbyf{\Psi \left[C
\right]}{\sigma_{\alpha\beta}\left(C(\theta)\right)}
\end{equation}
Taking one more functional derivative derivative we find the term with
vorticity times first derivative of the $ \delta $ function, coming from
the variation of $ C'(\theta) $
\begin{equation}
\frac{\delta^2 \Psi [C ]}{\delta C_{\alpha}(\theta) \delta
C_{\beta}(\theta')} = \delta'(\theta-\theta')\fbyf{\Psi \left[C
\right]}{\sigma_{\alpha\beta}\left(C(\theta)\right)} +
C'_{\gamma}(\theta) C'_{\lambda}(\theta')\frac{\delta^2\Psi \left[C
\right]}{\delta \sigma_{\alpha\gamma}\left(C(\theta)\right) \delta
\sigma_{\beta\lambda}\left(C(\theta')\right)}
\end{equation}
This term is the only one, which survives the limit $ \epsilon
\rightarrow 0 $ in our relation (\ref{eqom}).
So, the area derivative can be defined from the antisymmetric tensor part
of the second functional derivative as the coefficient in front of $
\delta'(\theta-\theta') $ . Still, it has all the properties of the first
functional derivative, as it can also be defined from the above first
variation.
The advantage of dealing with spikes is the control over the limit $\epsilon
\rightarrow 0$ , which might be quite singular in applications.
So far we managed to insert the vorticity at the loop $ C $ by
variations of the loop field. Later we shall need the vorticity off
the loop, in arbitrary point in space. This can be achieved by the
following combination of the spike operators
\begin{equation}
\Lambda \left(r,\theta,\epsilon \right) \Pi
\left(r_1,r_2,\theta+\epsilon,\delta \right) \\;\; \delta \ll \epsilon
\end{equation}
This operator inserts the $ \Pi $ shaped little loop at the top of the
bigger spike, in other words, this little loop is translated by a
distance $r$ by the big spike.
Taking derivatives, we find the operator of finite translation of the
vorticity
\begin{equation}
\Lambda \left(r,\theta,\epsilon \right)
\Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)
\end{equation}
and the corresponding infinitesimal translation operator
\begin{equation}
D_{\mu}(\theta,\epsilon)\Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)
\end{equation}
which inserts $ \partial_{\mu} \omega_{\alpha \beta} \left( C(\theta)
\right) $ when applied to the loop field.
Coming back to the correlation function, we are going now to construct
the operator, which would insert two vorticities separated by a distance.
Let us note that the global $ \Lambda $ spike
\begin{equation}
\Lambda \left(r,0,\pi \right) = \exp
\left(
r_{\alpha}\int_{0}^{2\pi}d
\phi \left(1- \frac{ \left|\phi-\pi \right|}{\pi} \right)
\frac{\delta}{\delta C_{\alpha}(\phi)}\right)
\end{equation}
when applied to a shrunk loop $ C(\phi) = 0 $ does nothing but
the backtracking from $0$ to $r$
\pct{Fig4}
This means that the operator
\begin{equation}
\Omega_{\alpha\beta}(0 ,\delta)\Omega_{\lambda \rho}(\pi ,\delta)
\Lambda \left(r,0,\pi \right)
\end{equation}
when applied to the loop field for a shrunk loop yields the vorticity
correlation function
\begin{equation}
\Omega_{\alpha\beta}(0 ,\delta)\Omega_{\lambda \rho}(\pi ,\delta)
\Lambda \left(r,0,\pi \right) \Psi [0] = \left \langle \omega_{\alpha
\beta}(0) \omega_{\lambda \rho}(r) \right \rangle
\end{equation}
The higher correlation functions of vorticities could be constructed in a
similar fashion, using the spike operators. As for the velocity, one
should solve the Poisson equation
\begin{equation}
\partial_{\mu}^2 v_{\alpha}(r) = \partial_{\beta} \omega_{\beta \alpha}(r)
\end{equation}
with the proper boundary conditions , say, $ v=0 $ at infinity.
Formally,
\begin{equation}
v_{\alpha}(r) =
\frac{1}{\partial_{\mu}^{2}}\partial_{\beta} \omega_{\beta \alpha}(r)
\end{equation}
This suggests the following formal definition of the velocity
operator
\begin{equation}
V_{\alpha}(\theta,\epsilon,\delta) = \frac{1}{D_{\mu}^2(\theta,\epsilon)}
D_{\beta}(\theta,\epsilon) \Omega_{\beta \alpha}(\theta,\delta)\\;\;
\delta \ll \epsilon
\label{VOM}
\end{equation}
\begin{equation}
V_{\alpha}(\theta,\epsilon,\delta)\Psi[C] \stackrel{\delta,\epsilon
\rightarrow 0}{\longrightarrow} v_{\alpha} \left(C(\theta) \right) \Psi[C]
\end{equation}
Another version of this formula is the following integral
\begin{equation}
V_{\alpha}(\theta,\epsilon,\delta)= \int d^3 \rho
\frac{\rho_{\beta}}{4 \pi |\rho|^3}\Lambda \left(\rho,\theta,\epsilon
\right)
\Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)
\end{equation}
where the $ \Lambda $ operator shifts the $ \Omega $ by a distance $
\rho $ off the original loop at the point $ r = C(\theta + \epsilon)
$
\pct{Fig5}
\section{Loop Equation}
Let us now derive exact equation for the loop functional.
Taking the time derivative of the original definition, and using the
Navier-Stokes equation we get in front of exponential
\begin{equation}
\oint_C d r_{\alpha} \frac{\imath\,}{ \nu}
\left(
\nu \partial_{\beta}^2 v_{\alpha} - v_{\beta}
\partial_{\beta} v_{\alpha} - \partial_{\alpha} p \right)
\end{equation}
The term with the pressure gradient yields zero after integration over
the closed loop, and the velocity gradients in the first two terms
could be expressed in terms of vorticity up to irrelevant gradient
terms, so that we find
\begin{equation}
\oint_C d r_{\alpha} \frac{\imath\,}{ \nu}
\left(
\nu \partial_{\beta} \omega_{\beta \alpha} - v_{\beta}
\omega_{\beta \alpha}
\right) \label{Orig}
\end{equation}
Replacing the vorticity and velocity by the operators discussed in the
previous Section we find the following loop equation (in explicit
notations)
\begin{eqnarray} &&
-\imath\,\dot{\Psi}[C] = \\ \nonumber &&
\oint d C_{\alpha}(\theta)
\left(
D_{\beta}(\theta,\epsilon) \Omega_{\beta \alpha}(\theta,\epsilon) +
\frac{1}{ \nu}
\int d^3 \rho
\frac{\rho_{\gamma}}{4 \pi |\rho|^3}\Lambda \left(\rho,\theta,\epsilon
\right)
\Omega_{\gamma\beta}(\theta+ \epsilon ,\delta)\Omega_{\beta
\alpha}(\theta,\delta) \right) \Psi[C]
\label{PsiC}
\end{eqnarray}
The more compact form of this equation, using the notations of
\cite{Mig83}, reads
\begin{eqnarray} &&
\imath\,\,\nu\dot{\Psi}[C] = {\cal H}_C\Psi \\ \nonumber &&
{\cal H}_C \equiv \nu^2\oint_{C} dr_{\alpha}
\left(
\imath\,\partial_{\beta} \frac{\delta }{\delta \sigma_{\beta \alpha}(r)}+
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta^2}{\delta \sigma_{\beta \alpha}(r)
\delta \sigma_{\beta \gamma}(r')}
\right)
\label{OLD}
\end{eqnarray}
Now we observe that viscosity $ \nu $ appears in front of time and
spatial derivatives, like the Planck constant $\hbar$ in Quantum
mechanics. Our loop hamiltonian ${\cal H}_C$ is not hermitean, due to
dissipation. It contains the second loop derivatives, so it represents a
(nonlocal!) kinetic term in loop space.
So far, we considered so called decaying turbulence, without external
energy source. The energy
\begin{equation}
E = \int d^3 r \frac{1}{2} \, v_{\alpha}^2
\end{equation}
would eventually all dissipate, so that the fluid would stop. In this case
the loop wave function $\Psi$ would asymptotically approach $1$
\begin{equation}
\Psi[C] \stackrel{t \rightarrow \infty}\longrightarrow 1
\end{equation}
In order to reach the steady state, we add to the right side of the Navier-Stokes
equation the usual gaussian random forces $f_{\alpha}(r,t)$ with the space
dependent correlation function
\begin{equation}
\VEV{f_{\alpha}(r,t)f_{\beta}(r',t')} = \delta_{\alpha\beta}\delta(t-t')F(r-r')
\end{equation}
concentrated at at small wavelengths, i.e. slowly varying with $r-r'$.
Using the identity
\begin{equation}
\VEV{f_{\alpha}(r,t) \Phi[v(.)]} = \int d^3 r' F(r-r')
\fbyf{\Phi[v(.)]}{v_{\alpha}(r')}
\end{equation}
which is valid for arbitrary functional $\Phi$ we find the following
imaginary potential term in the loop hamiltonian
\begin{equation}
\delta{\cal H}_C \equiv \imath\,\,U[C]= \frac{\imath\,}{\nu}\,\oint_{C}
dr_{\alpha}\oint_{C} dr'_{\alpha} F(r-r')
\end{equation}
Note, that orientation reversal together with complex conjugation changes
the sign of the loop hamiltonian, as it should. The potential part
involves two loop integrations times imaginary constant. The first term in
the kinetic part has one loop integration, one loop derivative times
imaginary constant. The second kinetic term has one loop integration, two
loop derivatives and real constant. The left side of the loop equation has
no loop integrations, no loop derivatives, but has a factor of $\imath\,$.
The relation between the potential and kinetic parts of the loop
hamiltonian depends of viscosity, or, better to say, it depends upon the
Reynolds number, which is the ratio of the typical circulation to
viscosity. In the viscous limit, when the Reynolds number is small, the
loop wave function is close to $1$. The perturbation expansion in $
\inv{\nu}$ goes in powers of the potential, in the same way, as in Quantum
mechanics. The second (nonlocal) term in kinetic part of the hamiltonian
also serves as a small perturbation (it corresponds to nonlinear term in
the Navier-Stokes equation).
The first term of this perturbation expansion is just
\begin{equation}
\Psi[C] \rightarrow 1 - \int \frac{d^3
k}{(2\pi)^3}\frac{\tilde{F}(k)}{2\nu^3\,k^2} \left|\oint_C d r_{\alpha} e^{\imath\, k
r}\right|^2
\end{equation}
with $\tilde{F}(k)$ being the Fourier transform of $F(r)$.
This term is real, as it corresponds to the two-velocity correlation. The
next term comes from the triple correlation of velocity, and this term is
purely imaginary, so that the dissipation shows up.
This expansion can be derived by direct iterations in the loop space as
in \cite{Mig83}, inverting the operator in the local part of the kinetic
term in the hamiltonian. This expansion is discussed in Appendix A. The
results agree with the straightforward iterations of the Navier-Stokes equations in
powers of the random force, starting from zero velocity.
So, we have the familiar situation, like in QCD, where the perturbation
theory breaks because of the infrared divergencies. For arbitrarily small
force, in a large system, the region of small $k$ would yield large
contribution to the terms of the perturbation expansion. Therefore, one
should take the opposite WKB limit $\nu \rightarrow 0$.
In this limit, the wave function should behave as the usual WKB wave
function, i.e. as an exponential
\begin{equation}
\Psi[C] \rightarrow \EXP{\frac{\imath\,\,S[C]}{\nu}}
\end{equation}
The effective loop Action $S[C]$ satisfies the loop space Hamilton-Jacobi
equation
\begin{equation}
\dot{S}[C] =-\imath\, U[C] + \oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta S}{\delta \sigma_{\beta \alpha}(r)}
\frac{\delta S}{\delta \sigma_{\beta \gamma}(r')}
\label{SC}
\end{equation}
The imaginary part of $S[C]$ comes from imaginary potential $U[C]$, which
distinguishes our theory from the reversible Quantum mechanics. The sign
of $\Im S$ must be positive definite, since $ |\Psi| <1$. As for the
real part of $S[C]$, it changes the sign under the loop orientation
reversal $C(\theta) \rightarrow C(2\pi-\theta) $.
At finite viscosity there would be an additional term
\begin{equation}
-\nu\oint_{C} dr_{\alpha}\partial_{\beta} \frac{\delta S[C]}{\delta
\sigma_{\beta\alpha}(r)}
-\imath\,\nu \oint_{C} dr_{\alpha}
\int d^3 r' \frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta^2 S[C]}{\delta \sigma_{\beta \alpha}(r) \delta \sigma_{\beta
\gamma}(r')}
\end{equation}
on the right of \rf{SC}. As for the term
\begin{equation}
-\oint_{C} dr_{\alpha}
\imath\,\left(\partial_{\beta}S[C]\right) \frac{\delta S[C]}{\delta
\sigma_{\beta \alpha}(r)}
\end{equation}
which formally arises in the loop equation, this term vanishes, since
$\partial_{\beta}S[C]=0$. This operator inserts backtracking at some point
at the loop without first applying the loop derivative at this point. As
it was discussed in the previous Section, such backtracking does not
change the loop functional. This issue was discussed at length in
\ct{Mig83}, where the Leibnitz rule for the operator $ \partial_{\alpha}
\fbyf{}{\sigma_{\beta\gamma}} $ was established
\begin{equation}
\partial_{\alpha} \fbyf{f(g[C])}{\sigma_{\beta\gamma}(r)} = f'(g[C])\partial_{\alpha}
\fbyf{g[C]}{\sigma_{\beta\gamma}(r)}
\end{equation}
In other words, this operator acts as a first order derivative on the loop
functional with finite area derivative (so called Stokes type functional).
Then, the above term does not appear.
The Action functional $ S[C] $ describes the distribution of the large
scale vorticity structures, and hence it should not depend of viscosity.
In terms of the above connected correlation functions of the circulation
this corresponds to the limit, when the effective Reynolds number
$\frac{\Gamma_C[v]}{\nu}$ goes to infinity, but the sum of the divergent
series tends to the finite limit. According to the standard picture of
turbulence, the large scale vorticity structures depend upon the energy
pumping, rather than the energy dissipation.
It is understood that both time $ t $ and the loop
size\footnote{As a measure of the loop size one may take the square root
of the minimal area inside the
loop.} $ |C| $ should be greater then the viscous scales
\begin{equation}
t \gg t_0 = \nu^{\frac{1}{2}}{\cal E}^{-\frac{1}{2}} \\;\;
|C| \gg r_0 = \nu^{\frac{3}{4}} {\cal E}^{-\frac{1}{4}}
\end{equation}
where $ {\cal E } $ is the energy dissipation rate.
It is defined from the energy balance equation
\begin{equation}
0 = \partial_t\VEV{\frac{1}{2}\,v_{\alpha}^2}= \nu \VEV{v_{\alpha}\partial^2 v_{\alpha}} +\VEV{f_{\alpha}v_{\alpha}}
\end{equation}
which can be transformed to
\begin{equation}
\frac{1}{4} \nu \VEV{\omega_{\alpha\beta}^2} = 3 F(0)
\end{equation}
The left side represents the energy, dissipated at small scale due to
viscosity, and the right side - the energy pumped in from the large scales
due to the random forces. Their common value is ${\cal E}$.
We see, that constant $F(r-r')$, i.e., $\tilde{F}(k)\propto \delta(k)$ is
sufficient to provide the necessary energy pumping. However, such forcing
does not produce vorticity, which we readily see in our equation. The
contribution from this constant part to the potential in our loop equation
drops out (this is a closed loop integral of total derivative). This is
important, because this term would have the wrong order of magnitude in
the turbulent limit - it would grow as the Reynolds number.
Dropping this term, we arrive at remarkably simple and universal
functional equation
\begin{equation}
\dot{S}[C] = \oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta S}{\delta \sigma_{\beta \alpha}(r)}
\frac{\delta S}{\delta \sigma_{\beta \gamma}(r')}
\label{KIN}
\end{equation}
The stationary solution of this equation describes the steady
distribution of the circulation in the strong turbulence. Note, that the
stationary solutions come in pairs $ \pm S$. The sign should be chosen
so, that $ \Im S > 0 $, to provide the inequality $ |\Psi| <1$.
\section{Scaling law}
The `Hamilton-Jacobi' equation without the potential term (\ref{KIN})
allows the family of the scaling solutions
\begin{equation}
S[C] = t^{2 \kappa -1}\phi \left[\frac{C}{t^{\kappa}} \right]
\end{equation}
with arbitrary index $ \kappa $. The scaling function satisfies the
equation
\begin{equation}
(2 \kappa -1 ) \phi[C] - \kappa \oint_{C} dr_{\alpha}
\frac{\delta \phi[C]}{\delta \sigma_{\beta \alpha}(r)}r_{\beta} =
\oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta \phi[C]}{\delta \sigma_{\beta \alpha}(r)}
\frac{\delta \phi[C]}{\delta \sigma_{\beta \gamma}(r')}
\end{equation}
The left side here was computed, using the chain rule differentiation of
functional.
Asymptotically, at large time, we expect the fixed point, which is the
homogeneous functional
\begin{equation}
S_{\infty}[C] = |C|^{2- \frac{1}{\kappa}} f \left[\frac{C}{|C|} \right]
\end{equation}
zeroing the right side of our `kinetic' functional equation
\begin{equation}
0=\oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta S_{\infty}[C]}{\delta \sigma_{\beta \alpha}(r)}
\frac{\delta S_{\infty}[C]}{\delta \sigma_{\beta \gamma}(r')}
\end{equation}
The Kolmogorov scaling \cite{Kolm41} would correspond to
\begin{equation}
\kappa = \frac{3}{2}
\end{equation}
in which case one can express the $ S $ functional in terms of $ {
\cal E } $
\begin{equation}
S[C] = {\cal E} t^2 \phi \left[\frac{C}{\sqrt{{\cal E}t^3}} \right]
\end{equation}
One can easily rephrase the Kolmogorov arguments in the loop space.
The relation between the energy dissipation rate and the velocity
correlator reads
\begin{equation}
{\cal E } = \left \langle v_{\alpha}(r_0) v_{\beta}(0) \partial_{\beta}
v_{\alpha}(0) \right \rangle
\end{equation}
where the point splitting at the viscous scale $r_0$ is introduced. Such
splitting is necessary to avoid the viscosity effects; without the
splitting the average would formally reduce to the total derivative
and vanish.
Instead of the point splitting one may introduce the finite loop of
the viscous scale $ |C| \sim r_0 $, and compute this correlator in
presence of such
loop. This reduces to the WKB estimates
\begin{equation}
\omega_{\alpha \beta}(r) \rightarrow
\frac{\delta S[C]}{\delta \sigma_{\alpha \beta}(r)} \\;\;
v_{\alpha}(r) = \int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\omega_{\alpha \gamma}(r')
\end{equation}
Using the generic scaling law for $ S $ we find
\begin{equation}
\omega \sim r_0^{- \frac{1}{\kappa}}\\;\;
v \sim r_0^{1- \frac{1}{ \kappa}}\\;\;
{\cal E} \sim r_0^{2 - \frac{3}{\kappa}}
\end{equation}
We see, that the energy dissipation rate would stay finite in the limit of
the vanishing viscous scale only for the Kolmogorov value of the index.
This argument looks rather cheap, but I think it is basically
correct. The constant value of the energy dissipation rate in the limit
of vanishing viscosity arises as the quantum anomaly in the field
theory, through the finite limit of the point splitting in the
correspondent energy current.\footnote{I am grateful to A.~Polyakov and
E.~Siggia for inspiring comments on this subject.}
There is another version of this argument, which I like better. The
dynamics of Euler fluid in infinite system would not exist, for the
non-Kolmogorov scaling. The extra powers of loop size would have to enter
with the size $L$ of the whole system, like
$\left(\frac{|C|}{L}\right)^{\epsilon} $. So, in the regime with finite energy
pumping rate ${\cal E}$ the infinite Euler system can exist only for the
Kolmogorov index. This must be the essence of the original Kolmogorov
reasoning \ct{Kolm41}.
The problem is that nobody proved that such limit exists, though. Within
the usual framework, based on the velocity correlation functions, one has
to prove, that the infrared divergencies, caused by the sweep, all cancel
for the observables. Within our framework these problems disappear, as we
shall see later.
As for the correlation functions in inertial range, unfortunately
those cannot be computed in the WKB approximation, since they involve the
contour shrinking to a double line, with vanishing area inside. Still,
most of the physics can be understood in loop terms, without these
correlation functions. The large scale behavior of the loop functional
reflects the statistics of the large vorticity structures, encircled by
the loop.
\section{Loop Equation for the Circulation PDF}
The loop field could serve as the generating function for the PDF $P_C(\Gamma)
$ for the circulation. The Fourier integral
\begin{equation}
P_C(\Gamma) = \int_{-\infty}^{\infty} \frac{d g}{2 \pi \nu}
\EXP{ \frac{\imath\, g }{\nu} \left(\oint_C d r_{\alpha} v_{\alpha}(r) - \Gamma \right)}
\end{equation}
can be analyzed in the same way as the loop field before. The only difference
is that the factors of $ g $ appear in front of various terms. These factors
can be replaced by
\begin{equation}
g \rightarrow \imath\, \nu \pp{\Gamma}
\end{equation}
acting on $P_C(\Gamma) $.
As a result we find
\begin{eqnarray} &&
\pp{\Gamma}\dot{P}_C(\Gamma) = -\oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta^2 P_C(\Gamma)}{\delta \sigma_{\beta \alpha}(r)
\delta \sigma_{\beta \gamma}(r')} \\ \nonumber &&
+\nu \pp{\Gamma}
\oint_{C} dr_{\alpha}\partial_{\beta}
\frac{\delta P_C(\Gamma)}{\delta \sigma_{\beta \alpha}(r)}
- U[C] \frac{\partial^3 P_C(\Gamma)}{\partial \Gamma^3}
\label{PDF}
\end{eqnarray}
All the imaginary units disappear, as they should. As for the viscosity and
forcing, these terms can be neglected in inertial range in the same way as
before. The only new thing is that one has to assume that $ \Gamma \gg \nu $ in
inertial range in addition to above assumptions about the size of the loop.
In absence of these terms there are no dimensional parameters so that the
following scaling laws hold (with the same index $ \kappa $ as before)
\begin{equation}
P_C(\Gamma) = t^{2 \kappa-1} F\left[\frac{C}{t^{\kappa}},\frac{\Gamma}{t^{2
\kappa-1}} \right]
\end{equation}
The factor $ t^{2 \kappa-1} $ came from the normalization of probability
density. Note, that this is more general law than before. Here we do not have
to use the WKB approximation for the PDF. In other words, the whole PDF rather
than just its decay at large $ \Gamma $ satisfies the scaling law.
The steady distribution would have the form of
\begin{equation}
P_C(\Gamma) \rightarrow \inv{\Gamma}\Phi\left[\frac{C}{\Gamma^{\frac{\kappa}{2
\kappa-1}}} \right]
\end{equation}
where the scaling functional $ \Phi $ satisfies the homogeneous equation
\begin{equation}
\oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta^2 \Phi[C]}{\delta \sigma_{\beta \alpha}(r)
\delta \sigma_{\beta \gamma}(r')} =0
\end{equation}
with the normalization condition
\begin{equation}
1=\int_{-\infty}^{\infty} \frac{d \Gamma}{ \Gamma}
\Phi\left[\frac{C}{\Gamma^{\frac{\kappa}{2 \kappa-1}}} \right]
\end{equation}
In principle, there could be different scaling functions for positive and
negative $ \Gamma $, rather than just absolute value $ |\Gamma| $ prescription.
This would correspond to above mentioned violation of the time reversal
symmetry.
However, as we mentioned above, there is no exact relation which would
eliminate the symmetric solution.
The Kolmogorov triple correlation function vanishes for vorticities (see
Appendix I), so that there is no restriction on the asymmetric part of the
circulation PDF. Nevertheless, the Kolmogorov scaling $ \kappa = \frac{3}{2} $
seems to me the most likely possibility, by the reasons discussed in the
previous section.
The homogeneous loop equation requires some boundary conditions at large loops,
to provide a meaningful solution. The asymptotic decrease of PDF
\begin{equation}
P_C(\Gamma) \sim \EXP{- Q\left[\frac{C}{\Gamma^{\frac{\kappa}{2 \kappa-1}}}
\right]}, Q \rightarrow \infty
\end{equation}
would lead to the same WKB equation as before
\begin{equation}
\oint_{C} dr_{\alpha}
\int d^3 r'\frac{r'_{\gamma}-r_{\gamma}}{4 \pi |r-r'|^3}
\frac{\delta Q[C]}{\delta \sigma_{\beta \alpha}(r)}
\frac{\delta Q[C]}{\delta \sigma_{\beta \gamma}(r')} =0
\end{equation}
We are studying this equation in the next section.
\section{Tensor Area law}
The Wilson loop in QCD decreases as exponential of the minimal area,
encircled by the loop, leading to the quark confinement. What is the
similar asymptotic law in turbulence? The physical mechanisms leading to
the area law in QCD are absent here. Moreover, there is no guarantee, that
$\Psi[C]$ always decreases with the size of the loop.
This makes it possible to look for the simple Anzatz, which was not
acceptable in QCD, namely
\begin{equation}
S[C] = s\left(\Sigma_{\mu\nu}^C\right)
\end{equation}
where
\begin{equation}
\Sigma_{\mu\nu}^C= \oint_C r_{\mu} d r_{\nu}
\end{equation}
is the tensor area encircled by the loop $C$. The difference between this
area and the scalar area is the positivity property. The scalar area
vanishes only for the loop which can be contracted to a point by removal
of all the backtracking. As for the tensor area, it vanishes, for example,
for the $8$ shaped loop, with opposite orientation of petals.
Thus, there are some large contours with vanishing tensor area, for which
there would be no decrease of the $\Psi$ functional.
In QCD the Wilson loops must always decrease at large distances, due to
the finite mass gap. Here, the large scale correlations are known to
exist, and play the central role in the turbulent flow. So, I see no
convincing arguments to reject the tensor area Anzatz.
This Anzatz in QCD not only was unphysical, it failed to reproduce the
correct short-distance singularities in the loop equation. In turbulence,
there are no such singularities. Instead, there are the large-distance
singularities, which all should cancel in the loop equation.
It turns out, that for this Anzatz the (turbulent limit of the) loop
equation is satisfied automatically, without any further restrictions.
Let us verify this important property. The first area derivative yields
\begin{equation}
\omega_{\mu\nu}^C(r)=\fbyf{S}{\sigma_{\mu\nu}(r)} = 2\pbyp{s}{
\Sigma_{\mu\nu}^C}
\end{equation}
The factor of $2$ comes from the second term in the variation
\begin{equation}
\fbyf{\Sigma^C_{\alpha\beta}}{\sigma_{\mu\nu}(r)}=
\delta_{\alpha\mu}\delta_{\beta\nu}-\delta_{\alpha\nu}\delta_{\beta\mu}
\end{equation}
Note, that the right side does not depend on $r$. Moreover, you can shift
$r$ aside from the base loop $C$, with proper wires inserted. The area
derivative would not change, as the contribution of wires drops.
This implies, that the corresponding vorticity $\omega_{\mu\nu}^C(r) $ is
space independent, it only depends upon the loop itself. The velocity can
be reconstructed from vorticity up to irrelevant constant sterms
\begin{equation}
v_{\beta}^C(r) = \frac{1}{2}\,r_{\alpha}\,\omega_{\alpha\beta}^C
\end{equation}
This can be formally obtained from the above integral representation
\begin{equation}
v_{\beta}^C(r) =\int d^3 r'\frac{r_{\alpha}-r'_{\alpha}}{4 \pi
|r-r'|^3}\omega_{\alpha\beta}^C
\label{INTG}
\end{equation}
as a residue from the infinite sphere $ R = |r'| \rightarrow \infty$. One may insert
the regularizing factor $ |r'|^{-\epsilon}$ in $\omega$, compute the
convolution integral in Fourier space and check that in the limit $
\epsilon \rightarrow 0^+$ the above linear term arises. So, one can use the above
form of the loop equation, with the analytic regularization prescription.
Now, the $v\,\omega$ term in the loop equation reads
\begin{equation}
\oint _C d r_{\gamma} \,v_{\beta}^C(r)\,\omega_{\beta\gamma}^C \propto
\Sigma_{\gamma\alpha}^C\,\omega_{\alpha\beta}^C\,\omega_{\beta\gamma}^C
\end{equation}
This tensor trace vanishes, because the first tensor is antisymmetric, and
the product of the last two antisymmetric tensors is symmetric with
respect to $\alpha\gamma$.
So, the positive and negative terms cancel each other in our loop
equation, like the "income" and "outcome" terms in the usual kinetic
equation. We see, that there is an equilibrium in our loop space kinetics.
{}From the point of view of the notorious infrared divergencies in
turbulence, the above calculation explicitly demonstrates how they cancel.
By naive dimensional counting these terms were linearly divergent. The
space isotropy lowered this to logarithmic divergency in \rf{INTG}, which
reduced to finite terms at closer inspection. Then, the explicit form of
these terms was such, that they all cancelled.
This cancellation originates from the angular momentum conservation in
fluid mechanics. The large loop $C$ creates the macroscopic eddy with
constant vorticity $\omega_{\alpha\beta}^C$ and linear velocity $ v^C(r)
\propto r$. This is a well known static solution of the Navier-Stokes equation. The
eddy is conserved due to the angular momentum conservation.The only
nontrivial thing is the functional dependence of the eddy vorticity upon
the shape and size of the loop $C$. This is a function of the tensor area
$\Sigma_{\mu\nu}^C$, rather than a general functional of the loop.
Combining this Anzatz with the space isotropy and the Kolmogorov scaling
law, we arrive at the tensor area law
\begin{equation}
\Psi[C] \propto \EXP{-
B\,\left(\frac{{\cal E}}{\nu^3}\left(\Sigma^C_{\alpha\beta}\right)^2\right)^{\frac{1}{3}} }
\label{AREA}
\end{equation}
The universal constant $B$ here must be real, in virtue of the loop
orientation symmetry. When the orientation is reversed $C(\theta) \rightarrow
C(2\pi-\theta)$, the loop integral changes sign, but its square, which
enters here, stays invariant. Therefore, the constant in front must be
real. The time reversal tells the same, since {\em both} viscosity $\nu$
and the energy dissipation rate ${\cal E}$ are time-odd. Therefore, the ratio
$\frac{{\cal E}}{\nu^3}$ is time-even, hence it must enter $\Psi[C]$ with the
real coefficient. Clearly, this coefficient $B$ must be positive, since $
\left|\Psi[C] \right|<1$.
Note, however, that we did not prove this law. The absence of decay for large
twisted loops with zero tensor area is suspicious. Also, the physics seems to
be different from what we expect in turbulence. The uniform vorticity, even a
random one, as in this solution, contrasts the observed intermittent
distribution. Besides, there clearly must be corrections to the asymptotic law,
whereas the tensor area law is {\em exact}. This is far too simple. We
discussed this unphysical solution mostly as a test of the loop technology.
\section{Scalar Area law}
Let us now study the scalar area law, which is a valid Anzatz for the
asymptotic decay of the circulation PDF.
The set of equations for the minimal surface is summarized in Appendix A.
All we need here is the following representation
\begin{equation}
A \rightarrow \inv{2L^2_{\Gamma}}\,
\int \int d \sigma_{\mu\nu}(x) d \sigma_{\mu\nu}(y)
\EXP{-\pi\frac{(x-y)^2}{L^2_{\Gamma}}}
\end{equation}
where $ L_{\Gamma} =|\Gamma|^{\frac{3}{4}} {\cal E}^{-\frac{1}{4}} $.
The distance $ (x-y)^2$ is measured in 3-space and integration goes
along the minimal surface. It is implied that its size is much larger than
$ L_{\Gamma} $.
In this limit the integration over, say, $ y $ can be performed along the
local tangent plane at $ x $ in small vicinity $ y-x \sim L_{\Gamma} $ ,
after which the factors of $ L_{\Gamma} $ cancel. We are left then with the
ordinary scalar area integral
\begin{equation}
A \rightarrow \frac{1}{2} \int d\sigma_{\mu\nu}(x)
d \sigma_{\mu\nu}(y) \delta^2(x-y) \rightarrow \int d^2x \sqrt{g}
\end{equation}
In the previous, regularized form the area represents so called Stokes
functional\ct{Mig83}, which can be substituted into the loop equation.
The area derivative of the area reads
\begin{equation}
\fbyf{A}{\sigma_{\mu\nu}(x)} =
\inv{L^2_{\Gamma}}\, \int d \sigma_{\mu\nu}(y)
\EXP{-\pi\frac{(x-y)^2}{L^2_{\Gamma}}}
\end{equation}
In the local limit this reduces to the tangent tensor
\begin{equation}
\fbyf{A}{\sigma_{\mu\nu}(x)} \rightarrow \int d \sigma_{\mu\nu}(y) \delta^2(x-y)
= t_{\mu\nu}(x)
\end{equation}
It is implied that the point $x$ approaches the contour from inside the
surface, so that the tangent tensor is well defined
\begin{equation}
t_{\mu\nu}(x) = t_{\mu}n_{\nu} - t_{\nu} n_{\mu}
\end{equation}
Here $ t_{\mu}$ is the local tangent vector of the loop, and $ n_{\nu} $
is the inside normal to the loop along the surface.
The second area derivative of the regularized area in this limit is just
the exponential
\begin{equation}
\frac{\delta^2
A}{\delta\sigma_{\alpha\beta}(x)\delta\sigma_{\gamma\delta}(y)}=
\inv{L^2_{\Gamma}}\, \EXP{-\pi\frac{(x-y)^2}{L^2_{\Gamma}}}
\end{equation}
Should we look for the higher terms of the asymptotic expansion at large
area we would have to take into account the shape of the minimal
surface, but in the thermodynamical limit we could neglect the curvature of
the loop and use the planar disk.
Let us use the general WKB form of PDF
\begin{equation}
P_C(\Gamma) = \inv{\Gamma}
\EXP{-Q\left(\frac{A}{t^{2\kappa}},\frac{\Gamma}{t^{2 \kappa-1}} \right)}
\end{equation}
We shall skip the arguments of effective action $ Q $.
We find on the left side of the loop equation
\begin{equation}
\partial_t Q \partial_{\Gamma} Q - \partial_t \partial_{\Gamma} Q
\end{equation}
On the right side we find the following integrand
\begin{equation}
\left(\left(\partial_{A}Q\right)^2-\partial^2_{A}Q\right)
\,\fbyf{A}{\sigma_{\alpha\beta}(r)}
\fbyf{A}{\sigma_{\gamma\beta}(r')} -
\partial_{A}Q \,\frac{\delta^2 A}{\delta\sigma_{\alpha\beta}(r)
\delta\sigma_{\gamma\beta}(r')}
\end{equation}
The last term drops after the $r' $ integration in virtue of symmetry.
The leading terms in the WKB approximation on both sides are those with
the first derivatives. We find
\begin{equation}
\partial_t Q \partial_{\Gamma}Q =
\left(\partial_{A}Q\right)^2 \oint_C d r_{\alpha} \fbyf{A}{\sigma_{\alpha\beta}(r)}
\int d^3 r' \frac{r_{\gamma}-r'_{\gamma}}{4\pi|r-r'|^3}\,
\fbyf{A}{\sigma_{\gamma\beta}(r')}
\end{equation}
In the last integral we substitute above explicit form of the
area derivatives and perform the $ d^3r' $ integration first. In the
thermodynamical limit only the small vicinity $ r'-y \sim L_{\Gamma} $
contributes, and we find
\begin{equation}
\int d^3 r' \frac{r_{\gamma}-r'_{\gamma}}{4\pi|r-r'|^3}\,
\fbyf{A}{\sigma_{\gamma\beta}(r')} \rightarrow
L_{\Gamma}^2 \,\int d \sigma_{\gamma\beta}(y)
\frac{r_{\gamma}-y_{\gamma}}{4\pi|r-y|^3}
\end{equation}
This integral logarithmically diverges. We compute it with
the logarithmic accuracy with the following result
\begin{equation}
\int d \sigma_{\gamma\beta}(y)
\frac{r_{\gamma}-y_{\gamma}}{4\pi|r-y|^3}
\propto \frac{t_{\beta}}{ \pi} \ln \frac{L^2_{\Gamma}}{A}
\end{equation}
The meaning of this integral is the average velocity in the WKB
approximation. This velocity is tangent to the loop, up to the next
correction terms at large area.
Now, the emerging loop integral vanishes due to symmetry
\begin{equation}
\oint_C d r_{\alpha} t_{\beta} t_{\alpha\beta} =0
\end{equation}
as the line element $ d r_{\alpha} $ is directed along the tangent vector $
t_{\alpha} $, and the tangent tensor $ t_{\alpha\beta} $ is antisymmetric.
Similar mechanism was used in the tensor area solution, only there the
cancellations emerged at the global level, after the closed loop
integration. Here the right side of the loop equation vanishes locally,
at every point of the loop. Anyway, we see, that the scalar area indeed
represents the steady solution of the loop equation in the leading WKB
approximation.
It might be instructive to compare this solution with another known exact
solution of the Euler dynamics, namely the Gibbs solution
\begin{equation}
P[v] = \EXP{-\beta \int d^3 r \frac{1}{2} v_{\alpha}^2 }
\end{equation}
For the loop functional it reads
\begin{equation}
\Psi_C(\gamma) = \EXP{-\frac{\gamma^2}{2\beta}\oint_C d r_{\alpha} \oint_C d r'_{\beta} \delta^3(r-r')}
\end{equation}
The integral diverges, and it corresponds to the perimeter law
\begin{equation}
\oint_C d r_{\alpha} \oint_C d r'_{\beta} \delta^3(r-r') \rightarrow r_0^{-2} \oint_C |dr|
\end{equation}
where $ r_0 $ is a small distance cutoff. For the PDF it yields
\begin{equation}
P(\Gamma) \propto \EXP{-\frac{\Gamma^2\beta r_0^2}{2 \oint_C |dr|}}
\end{equation}
When the Gibbs solution is substituted into the loop equation, we observe
the same thing. Average velocity is tangent to the loop, which leads to
vanishing integrand in the loop equation. The difference is that in our
case this is true only asymptotically, there are next corrections.
The shape of the function $ Q $ is not fixed by this equation in the leading
WKB approximation. In a scale invariant theory it is natural to expect the
power law
\begin{equation}
Q\left(\frac{A}{t^{2\kappa}},\frac{\Gamma}{t^{2 \kappa-1}} \right) \rightarrow
\mbox{const } \left(\Gamma^{2\kappa} A^{1-2\kappa}\right)^{\mu}
\label{MuLaw}
\end{equation}
There is one more arbitrary index $ \mu$ involved. Even for the Kolmogorov law
$ \kappa = \frac{3}{2} $ the $ \Gamma $ dependence remains unknown.
\section{Discussion}
So, we found two asymptotic solutions of the loop equation in the
turbulent limit, not counting the Gibbs solution. It remains to be seen, which
one (if any) is realized in turbulent flows. The tensor area solution is
mathematically cleaner, but its physical meaning contradicts the intermittency
paradigm. It corresponds to the uniform vorticity with random magnitude and
random direction, rather that the regions of high vorticity interlaced with
regions of low vorticity, observed in the turbulent flows.
The recent numerical experiments\cite{Umeki} favor the scalar area rather than
the tensor one. Also, the Kolmogorov scaling was observed in these experiments.
The Reynolds number was only $ \sim 100 $ which was too small to make any
conclusions. We have to wait for the experiments ( real or numerical ) with the
Reynolds numbers few orders of magnitude larger.
The scalar area is less trivial than the tensor one. The minimal area as a
functional of the loop cannot be represented as any explicit contour integral
of the Stokes type, therefore it corresponds to infinite number of higher
correlation functions present. Moreover, there could be several minimal
surfaces for the same loop, as the equations for the minimal surface are
nonlinear. Clearly, the one with the least area should be taken.
The natural generalization of this solution is the string Anzatz where the sum
over all surfaces bounded by the loop is taken
\begin{equation}
P_C(\Gamma) = \sum_{S: \partial
S=C}\EXP{-Q\left(\frac{A}{t^{2\kappa}},\frac{\Gamma}{t^{2 \kappa-1}} \right)}
\end{equation}
At large loop the minimal $ Q $ terms will remain. The extremum condition
\begin{equation}
\delta Q = \frac{\partial Q}{\partial A } \delta A =0
\end{equation}
will be satisfied for the minimal surface.
However, the sum over random surfaces is not well defined. The recent
studies\ct{QG} indicate that the typical closed surfaces degenerate to branched
polymers. For the surface bounded by a fixed loop this cannot happen, of
course. Still nobody knows how to compute such sums. The loop equation in
principle allows to systematically compute the corrections to the area law as
the WKB expansion.
The WKB solution is incomplete so far. The leading term in the loop equation is
annihilated by arbitrary function of the area (scalar or tensor). The similar
ambiguity was present in the Gibbs solution, where arbitrary function of the
Hamiltonian satisfied the Liouville equation for the velocity PDF. In that
case the ambiguity was removed by extra requirement of thermodynamic limit:
only the exponential of the hamiltonian would agree with the factorization of
the PDF for two remote parts of the system.
What could be a similar requirement here? The area of the minimal surface
represents the effective volume of the system at large loop. The circulation
can be written as a surface integral of vorticity, which makes the circulation
an extensive variable at this surface. The average vorticity
\begin{equation}
\bar{\omega} = \frac{\Gamma}{A}
\end{equation}
represents an intensive quantity. The thermodynamic limit would then
correspond to
\begin{equation}
Q = A q(\bar{\omega})
\end{equation}
Comparing this with the previous formula for $ Q $ \rf{MuLaw} we conclude that
$ \mu=1$. In this case
\begin{equation}
Q =\mbox{const }A \bar{\omega}^{2\kappa}
\end{equation}
In principle, there could be two different laws for positive and negative $
\Gamma $, due to violation of the time reversal invariance
\begin{equation}
Q \rightarrow q_{\pm} A |\bar{\omega}|^{2\kappa}
\end{equation}
Another line of argument might start with an assumption of decorrelated
average vorticity $ \omega_i $ at various parts $ S_i $ of the area $ A_0 $ of
the minimal surface. The net circulation, adding up from the large number $ n
\sim \frac{A}{A_0} \gg 1 $ of independent random terms $ \omega_i A_0 $ would
be a gaussian variable as a consequence of the law of large numbers. We would
have then
\begin{equation}
Q \sim \frac{\Gamma^2}{n A_0^2 \omega_i^2} = \frac{\Gamma^2 }{A A_0\omega_i^2}
\end{equation}
This would agree with the previous estimate at
\begin{equation}
\mu = \inv{\kappa}, A_0\omega_i^2\sim A^{1- \inv{\kappa}}
\end{equation}
so that
\begin{equation}
Q \rightarrow \mbox{const } \Gamma^2 A^{\inv{\kappa}-2}
\end{equation}
The natural assumption here would be that the vorticity variance $ \omega_i$
does not scale with the area $ A $, so that
\begin{equation}
A_0 \sim A^{1- \inv{\kappa}}
\end{equation}
The Gaussian behavior (with $ \kappa = \frac{3}{2} $ ) was observed in
numerical experiments \ct{Umeki}, but the Reynolds number was too low to make
conclusions at this point. There could be a scaling function
\begin{equation}
Q = q\left(\Gamma^{2} A^{\inv{\kappa}-2}\right)
\end{equation}
which starts linearly and then grows as a power, say, $ q(x) = (1+ a
\,x)^{\kappa} $. I suggest that this function should be studied in real and
numerical experiments. This would teach us something new about turbulence.
\section{Acknowledgments}
I am grateful to V.~Borue, I.~Goldhirsh, D.~McLaughlin, A.~Polyakov and
V.~Yakhot for stimulating discussions .
\newpage
|
2,877,628,090,848 | arxiv | \section{Introduction}
In physics, there are relations connecting different thermodynamic quantities in equilibrium, such as $P=(2/3)E$ for both noninteracting fermions~\cite{Walecka} and unitary Fermi gases~\cite{HoPRL04}. Here $P$ is the pressure and $E$ is the energy density, and a unitary Fermi gas consists of two-component attractive fermions on the verge of forming two-body bound states. There are also relations connecting different transport coefficients, such as the Wiedemann-Franz law showing the ratio of the thermal conductivity and the electric conductivity of normal electrons is proportional to the temperature~\cite{AshcroftBook}. There are, however, a third type of relations connecting thermodynamic quantities and transport coefficients. An example is the Einstein relation $D=\Gamma/(\partial n/\partial \mu)$ of Brownian motion~\cite{ChaikinBook}, where $D$ is the diffusion constant and is a transport coefficient, $\partial n/\partial \mu$ is the density susceptibility (which is related to the compressibility) and is a thermodynamic quantity, and $\Gamma$ is a dissipative coefficient associated with the relaxation rate of the system.
It was shown~\cite{EnssPRA12} that the shear viscosity $\eta$ is proportional to the pressure $P$ in a normal scale-invariant fluid, implying a relation $\eta=P\tau$ with $\tau$ being the relaxation time. By using a large-$N$ expansion and kinetic theory, the relation is shown to apply to normal unitary Fermi gases, where the divergent two-body $s$-wave scattering length renders the system scale invariant. The relation has been implemented in later studies of unitary Fermi gases~\cite{SchaferPRA15}. A particular feature of this relation is that $P$ is not a susceptibility, so it does not have the structure suggested by the fluctuation-dissipation theorem that leads to the Einstein relation~\cite{ChaikinBook}. The relaxation time $\tau$ naturally comes out of linear response or kinetic theories \cite{AshcroftBook,HaoPRL11,EnssPRA12}, but its measurements in cold-atoms can be challenging and past studies used the lifetime of quasi-particles as an estimation of $\tau$ \cite{HaoPRL11}. The relation $\eta=P\tau$ provides a more direct means for determining $\tau$ when $\eta$ and $P$ are measured experimentally. Since the ground state of an unitary Fermi gas is a superfluid, it is imperative to investigate how the relation is modified in the suprfluid phase.
There has been broad interest in transport and thermodynamic properties of ultracold atomic Fermi gases, which provide a clean testbed for analyzing strongly interacting systems and connect to other interacting quantum systems~\cite{KinastPRL05,BruunPRA07,SchaferPRA07,ourlongpaper,ThomasJLTP08,Nascimbene10,ThomasPRL14,ThomasScience11,PethickPRL11,HaoPRL11,ZwierleinNature11,HaoNJP11,EnssPRA12,SchaferPRA15}. The unitary Fermi gas is also closely related to the perfect quantum fluid~\cite{Turlapov08,Cao11} showing a minimal ratio between the shear viscosity and entropy. Although harmonic traps are frequently used in cold-atom experiments, recent progresses on realizing and measuring homogeneous Fermi gases~\cite{Mkherjee17,Horikoshi17} offer more direct check of many-body theories without complications from inhomogeneity.
Here we present a relation of superfluid unitary Fermi gases connecting the shear viscosity, pressure, superfluid density, chemical potential, and anomalous shear viscosity due to the order parameter. For the mean-field BCS-Leggett theory \cite{Leggett}, the relation is a consequence of the equations of state and a gauge-invariant linear response theory named the consistent fluctuations of order parameter (CFOP) theory \cite{OurPRD12,OurJLTP13}, which may also be constructed by a path integral formalism~\cite{HLYAP16}. Since strongly attractive interactions can lead to preformed pairs in unitary Fermi gases~\cite{Ourreview,OurAnnPhys}, one should distinguish condensed and noncondensed Cooper pairs. Following a pairing fluctuation theory consistent with the BCS-Leggett ground state~\cite{Ourreview,OurAnnPhys}, the noncondensed pair contributions to the thermodynamic quantities and transport coefficients in the relation are evaluated. A modified relation resembling the one in the absence of pairing fluctuations is found, and numerical calculations show that the modified relation represents a reasonable approximation.
The paper is organized as follows. Section~\ref{sec:quantities} introduces the thermodynamics quantities and transport coefficients involved in the relation of unitary Fermi superfluids. The exact relation for the BCS-Leggett theory and an approximate relation in the presence of pairing fluctuations are presented in Section~\ref{sec:relations}. The anomalous shear viscosity characterizing the momentum transfer through Cooper pairs will also be discussed there. Section~\ref{sec:conclusion} concludes our work. The theoretical framework, details, and derivations of the relations and associated quantities are summarized in the Appendix.
\section{Thermodynamic quantities and transport coefficients from BCS-Leggett theory}\label{sec:quantities}
\subsection{Mean-field theory}
The relation connecting thermodynamic and transport quantities of unitary Fermi superfluids can be derived from the mean-field BCS-Leggett theory.
The equations of states, which are usually called the gap and number equations, are given by \cite{Leggett}
$\frac{1}{g}=\sum_{\mathbf{k}}\frac{1}{2\epsilon_{\mathbf{k}}}-\frac{m}{4\pi a}=\sum_{\mathbf{k}}\frac{1-2f(E_{\mathbf{k}})}{2E_{\mathbf{k}}}$ and
$n=\sum_{\mathbf{k}}\Big[1-\frac{\xi_{\mathbf{k}}}{E_{\mathbf{k}}}+2\frac{\xi_{\mathbf{k}}}{E_{\mathbf{k}}}f(E_{\mathbf{k}})\Big]$,
where $m$ is the fermion mass, $g$ is the attractive coupling constant modeling the contact interaction between atoms, and $a$ is the two-body $s$-wave scattering length.
Here $f(x)=[\exp(x/k_B T)+1]^{-1}$ is the Fermi distribution function, $\xi_{\mathbf{k}}=\epsilon_\mathbf{k}-\mu=\frac{\hbar^2 k^2}{2m}-\mu$, and $E_{\mathbf{k}}=\sqrt{\xi^2_{\mathbf{k}}+\Delta^2}$ with $\Delta$ the order parameter or the energy gap. The chemical potential $\mu$ has to be self-consistently determined. Throughout the paper we will set $\hbar=1$ and $k_B=1$. The unitary point corresponds to $1/(k_Fa)=0$, where $k_F$ the Fermi momentum of a noninteracting Fermi gas with the same density. Via thermodynamic arguments~\cite{HoPRL04}, it has been shown that $P=\frac{2}{3}E$ for unitary Fermi gases. In the Leggett-BCS theory, the following expressions satisfy the relation $P=\frac{2}{3}E$ in the superfluid phase (with a proof in the Appendix):
\begin{eqnarray}
P&=&-\sum_{\mathbf{k}}(\xi_{\mathbf{k}}-E_{\mathbf{k}})-\frac{\Delta^2}{g}+2\sum_{\mathbf{k}}T\ln(1+e^{-\frac{E_{\mathbf{k}}}{T}}), \nonumber \\
E&=&\sum_{\mathbf{k}}(\xi_{\mathbf{k}}-E_{\mathbf{k}})+\frac{\Delta^2}{g}+2\sum_{\mathbf{k}}E_{\mathbf{k}}f(E_{\mathbf{k}})+\mu n.
\end{eqnarray}
The shear viscosity can be calculated by the Kubo formalism~\cite{Kadanoff61,BruunPRA07}:
\begin{eqnarray}\label{eta0}
\eta&=&-\lim_{\omega\rightarrow0}\lim_{q\rightarrow0}\textrm{Im}\frac{\Xi(\omega,\mathbf{q})}{\omega}=-m^2\lim_{\omega\rightarrow0}\lim_{q\rightarrow0}\frac{\omega}{q^2}\textrm{Im}\chi_{\textrm{T}}(\omega,\mathbf{q}) \nonumber \\
&=&\frac{1}{15\pi^2m^2}\int_{0}^{+\infty}dkk^6\frac{\xi^2_{\mathbf{k}}}{E^2_{\mathbf{k}}}\Big(-\frac{\partial f(E_{\mathbf{k}})}{\partial E_{\mathbf{k}}}\Big)\tau.
\end{eqnarray}
Here $\Xi(\omega,\mathbf{q})$ is the Fourier transform of $\Xi(\bar{\tau}-\bar{\tau}',\mathbf{q})=-i\theta(\bar{\tau}-\bar{\tau}')\langle \tensor{T}^{xy}(\bar{\tau},\mathbf{q})\tensor{T}^{xy}(\bar{\tau}',-\mathbf{q})\rangle $, the stress tensor-stress tensor response function. $\bar{\tau}$ denotes the imaginary time, $\theta(x)$ is the Heaviside step function, and $\tensor{T}^{xy}$ is the $xy$-component of the energy-momentum stress tensor $\tensor{T}$. Formally, $\tensor{T}=\frac{1}{m}\big(\nabla\psi^\dagger_{\uparrow}(\mathbf{x})\nabla\psi_{\uparrow}(\mathbf{x})+\nabla\psi^\dagger_{\downarrow}(\mathbf{x})
\nabla\psi_{\downarrow}(\mathbf{x})\big)+\tensor{I}\mathcal{L}$, where $\tensor{I}$ is the unit tensor and $\mathcal{L}$ is the Lagrangian density.
The momentum conservation law $\frac{\partial \mathbf{J}}{\partial t}+\frac{1}{m}\nabla\cdot\tensor{T}=0$ leads to the second expression of Eq.~\eqref{eta0} in terms of
the transverse current-current correlation function defined by $\chi_\textrm{T}=(\sum_{i=x}^z\chi^{ii}_{\textrm{JJ}}-\chi_{\textrm{L}})/2$. The longitudinal part is given by
$\chi_\textrm{L}=\hat{\mathbf{q}}\cdot\tensor{\chi}_{\textrm{JJ}}\cdot\hat{\mathbf{q}}$.
For BCS superfluids, the current-current response function $\tensor{\chi}_{\textrm{JJ}}$ can be evaluated from the CFOP linear response theory, and it has a tensor structure~\cite{OurPRD12,OurJLTP13} $\tensor{\chi}_{\textrm{JJ}}=\tensor{P}+\frac{n}{m}\tensor{I}+\tensor{C}$. Here $n$ is the particle density, $\tensor{P}$ is the paramagnetic response function, and
$\tensor{C}$ is associated with
the collective modes~\cite{KosztinPRB00,HaoPRL10} but is irrelevant to the shear viscosity~\cite{Haothesis} (see the Appendix for more details).
In Eq.~\eqref{eta0}, $\tau$ is the relaxation time measuring the broadening of the response functions. Moreover, $\eta$ is constrained by the sum rule~\cite{Kadanoff63}
$\lim_{\mathbf{q}\rightarrow\mathbf{0}}\int_{-\infty}^{+\infty}\frac{d\omega}{\pi}\Big(-\frac{\textrm{Im}\chi_{\textrm{T}}(\omega,\mathbf{q})}{\omega}\Big)=\frac{n_\textrm{n}}{m}$,
where $n_\textrm{n}$ is the normal fluid density.
Above the mean-field critical temperature $T^*$, $\Delta=0$ and Eq.~(\ref{eta0}) coincides with the expression of a normal Fermi gas~\cite{BruunPRA07}. In that case the shear viscosity reduces to
$\eta=\frac{1}{5}nv_F\tau m^\ast$, where $v_F$ is the Fermi velocity and $m^\ast$ is the effective mass (details can be found in the Appendix).
\section{Relations in unitary Fermi superfluids}\label{sec:relations}
\subsection{Relation from Leggett-BCS theory}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Figure1a.eps}
\caption{Plots of $\eta/\tau$ (black solid line), $\chi/\tau$ (green dot-dash line), $P$ (red solid line) and $\frac{2}{5}n_\textrm{s}\mu$ (blue dashed line) as functions of temperature (a) on the BCS side ($1/(k_F a)=-0.5$), (b) at unitary, and (c) on the BEC side ($1/(k_F a)=0.5$). The thick pink lines denote the difference, Diff$=\eta/\tau+\chi/\tau-(P-\frac{2}{5}n_\textrm{s}\mu)$.
}
\label{fig:BCS}
\end{figure}
Using the BCS-Leggett theory and the CFOP linear response theory, we found a relation connecting thermodynamic quantities and transport coefficients of a mean-field unitary Fermi superfluid:
\begin{eqnarray}\label{etaP}
\eta+\chi=(P-\frac{2}{5}\mu n_s)\tau.
\end{eqnarray}
The pressure $P$ and chemical potential $\mu$ are thermodynamic quantities. On the other hand, the shear viscosity $\eta$, superfluid density $n_\textrm{s}$, and the response function $\chi$ representing the Cooper-pair contribution to momentum transfer are transport coefficients which can be inferred from linear response theory (see the Appendix for the derivations). The two types of quantities are connected by the relaxation time $\tau$, which bears a similar structure as the Einstein relation.
A proof of the relation (\ref{etaP}) is summarized in the Appendix, and here we explain the physical meaning of the terms involved. The thermodynamic quantities such as $P$ and $\mu$ in the BCS-Legget theory has already been presented. The superfluid density can be obtained from the paramagnetic response function by \cite{Walecka,HaoNJP11}
$n_\textrm{s}=m\lim_{\omega\rightarrow0}\lim_{\mathbf{q}\rightarrow\mathbf{0}}\textrm{Re}[P^{xx}(\omega,\mathbf{q})]+n$. The response function
$\chi$ characterizes how momentum transfer can be conducted through the Cooper-pair channel in contrast to the density channel. As one will see, its derivation is almost identical to the shear viscosity except the anomalous density $\psi_{\downarrow}\psi_{\uparrow}$ replaces the normal density $\psi^{\dagger}_{\sigma}\psi_{\sigma}$ with $\sigma=\uparrow,\downarrow$. In conventional BCS theory, the anomalous Green's function also contributes to the linear response~\cite{Walecka}, and here we found another example.
Therefore, $\chi$ is identified as the anomalous shear viscosity from the Cooper-pair channel and can be obtained from the $\tensor{\Pi}$-$\tensor{\Pi}$ correlation function. Here
$\tensor{\Pi}(\mathbf{x})=\frac{1}{m}\big(\nabla\psi_{\downarrow}(\mathbf{x})\nabla\psi_{\uparrow}(\mathbf{x})+\nabla\psi^\dagger_{\uparrow}(\mathbf{x})\nabla\psi^\dagger_{\downarrow}(\mathbf{x})\big)$
is the anomalous counterpart of the stress tensor $\tensor{T}$, and the response function is
$\hat{\tensor{Q}}(\bar{\tau}-\bar{\tau}',\mathbf{q})=-i\theta(\bar{\tau}-\bar{\tau}')\langle[\tensor{\Pi}(\bar{\tau},\mathbf{q}),\tensor{\Pi}(\bar{\tau}',-\mathbf{q})]\rangle$.
The anomalous shear viscosity $\chi$ is defined via the $xy-xy$ component of the tensor response function $\hat{\tensor{Q}}$ similar to the definition of $\eta$. Explicitly, $\chi\equiv-\lim_{\omega\rightarrow0}\lim_{q\rightarrow0}\frac{1}{\omega}\textrm{Im}[Q^{xyxy}(\omega,\mathbf{q})]$.
The final expressions in the BCS-Leggett theory are
\begin{eqnarray}\label{ns}
n_{\textrm{s}}&=&\frac{2}{3}\sum_{\mathbf{k}}\frac{\Delta^2}{E^2_{\mathbf{k}}}\frac{k^2}{m}\Big[\frac{1-2
f(E_{\mathbf{k}})}{2E_{\mathbf{k}}}+\frac{\partial f(E_{\mathbf{k}})}{\partial E_{\mathbf{k}}}\Big], \nonumber \\
\chi&=&-\frac{2}{15}\sum_{\mathbf{k}}\frac{k^4}{m^2}\frac{\Delta^2}{E^2_\mathbf{k}}\frac{\partial f(E_\mathbf{k})}{\partial E_\mathbf{k}}\tau.
\end{eqnarray}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Graph1.eps}
\caption{$P-\frac{2}{5}n_\textrm{s}\mu$ at $T=0$ as a function of $1/(k_Fa)$. Inset (a) shows $P-\frac{2}{3}E$ vs. $1/(k_Fa)$ at zero temperature. Inset (b) shows $P-\frac{2}{5}n_s\mu$ vs. $T$ at unitarity. Here $P, E, n_\textrm{s}\mu$ are in units of $E_Fk_F^3$.
}
\label{diff}
\end{figure}
Above $T^*$, $n_\textrm{s}$ and $\chi$ both vanish due to the closing of the energy gap, and the relation reduces to the kinetic-theory result $\eta=P\tau$ in the normal phase. At zero temperature, $\eta$ and $\chi$ both vanish since the pure BCS ground state lacks dissipation associated with transport coefficients, and the relation reduces to
$P=\frac{2}{5}\mu n_{\textrm{s}}$, which is consistent with the experimental results of Ref.~\cite{Horikoshi17}. This zero-temperature relation remains valid when pairing fluctuations consistent with the BCS-Leggett ground state are present.
The quantities in the relation are visualized in Figure~\ref{fig:BCS}, where $\eta/\tau$, $\chi/\tau$, $P$, and $\frac{2}{5}n_\textrm{s}\mu$ as a function of temperature are shown for superfluids on the BCS side with $1/(k_Fa)=-0.5$, at unitaryity, and on the BEC side with $1/(k_Fa)=0.5$. Here $T^*/T_F=0.26, 0.50, 0.84$, respectively. The relation works well at unitarity but exhibits deviations in the other cases. However, from panel (a) this relation also works well at relatively high temperature ($0.6 T^*\lesssim T<T^*$) in the weak BCS regime. Although separate measurements of the normal and anomalous shear viscosity, $\eta$ and $\chi$, can be a challenge in experiments, they contribute to shear momentum transfer together and one may think of a composite shear viscosity $\tilde{\eta}\equiv\eta+\chi$ that accounts for the right hand side of the relation~\eqref{etaP}.
To better understand the deviation as the system moves away from the unitary point, we show in Figure~\ref{diff} the quantity $P-\frac{2}{5}n_\textrm{s}\mu$ at zero temperature as a function the interaction strength $1/(k_Fa)$ in the regime $-1\le1/(k_Fa)\le1$. At unitarity, the quantity vanishes as indicated by the relation \eqref{etaP}. We also show, in the insets, $P-\frac{2}{3}E$ as a function of $1/(k_Fa)$ at zero temperature and $P-\frac{2}{5}n_s\mu$ as a function of $T$ at $1/(k_Fa)=0$. The qualitative behavior of $P-\frac{2}{5}n_\textrm{s}\mu$ is similar to $P-\frac{2}{3}E$ at zero temperature, which implies that the deviation is associate with the scale-invariance which guarantees certain thermodynamic relations. Moreover, $P-\frac{2}{5}n_\textrm{s}\mu$ is close to $0$ already at low $T$, and this may ease its verification in experiments. Figures~\ref{fig:BCS} and \ref{diff} also suggest that even though the relation \eqref{etaP} is not exact on the BCS side, the deviation is relatively minor and one may continue using it there as an approximation.
\subsection{Pairing fluctuation effects}
\subsubsection{Beyond mean-field theory}
Due to the strongly attractive interactions in unitary Fermi gases, pairing fluctuation effects discerning the gap function and the order parameter should be considered ~\cite{Ourreview,OurAnnPhys,OurNSR10,OurJLTP13,ourlongpaper}. Moreover, the onset temperature of pairing $T^*$ and that of condensation (or superfluid transition temperature) $T_c$ are different, and from $t$-matrix calculations $T^*/T_F$ can be more than twice as large as $T_c/T_F$ \cite{OurAnnPhys,OurNSR10}. Here we follow a $t$-matrix theory consistent with the BCS-Leggett ground state and derive the thermodynamics first. The Green's function has the generic form $G(P)=[G_0^{-1}(P)-\Sigma(P)]^{-1}$, where $G_0$ is the noninteracting Green's function. The self energy comes from ladder-diagram corrections to the propagator and is given by $\Sigma(P)=\sum_{Q}t(Q)G_0(P-Q)$, where the $t$-matrix $t(Q)$ may be separated into the condensed ($Q=0$) and noncondensed ($Q\neq 0$) pair contributions. Thus, $t(Q)\approx t_\textrm{sc} + t_\textrm{pg}$ with $t_{\textrm{sc}}(Q)=-(\Delta_{\textrm{sc}}^2/T)\delta(Q)$ and $t_\textrm{pg}(Q)=[g^{-1}+X(Q)]^{-1}$ is constructed from $X(Q)=\sum_K G_0(Q-K)G(K)$ which consists of one bare and one full Green's functions.
Here $K=(i\omega_n,\mathbf{k})$ and $Q=(i\Omega_l,\mathbf{q})$ are fermionic and bosonic four-momenta, respectively. The total gap can also be decomposed into the order parameter (from the condensed pairs) $\Delta_\textrm{sc}$ and the pseudogap (from the noncondensed pairs) $\Delta_\textrm{pg}$ as $\Delta^{2}=\Delta_{\textrm{sc}}^{2}+\Delta_{\textrm{pg}}^{2}$, where the pseudogap is approximated by $\Delta_\textrm{pg}^2=-\sum_{Q}t_{\textrm{pg}}(Q)$, and the superfluid transition temperature is determined by where $\Delta_\textrm{sc}$ vanishes. The modified number and gap equations can be derived from $n=\sum_P G(P)$ and $\frac{1}{g}=\sum_PG(P)G_0(-P)$. The pairs have a finite lifetime, which contributes to the relaxation time $\tau$ in linear response.
The density of grand potential, $\Omega=-P$, can be derived from the same framework \cite{ourlongpaper}, and it leads to the aforementioned equations of states after minimization. There are also contributions from a fermionic part (including the condensed pairs and quasi-particles) and from a bosonic part (from the noncondensed pairs). Explicitly,
$P=-\Omega=-(\Omega_\textrm{f}+\Omega_\textrm{b})=P_\textrm{f}+P_\textrm{b}$,
where $\Omega_\textrm{f}=\Delta^2\chi_0+\sum_{\mathbf{k}}\big[(\xi_{\mathbf{k}}-E_{\mathbf{k}})-2T\ln(1+e^{-\frac{E_{\mathbf{k}}}{T}})\big]$ and $\Omega_\textrm{b}=T\sum_{\mathbf{q}}\ln(1-e^{-\frac{\Omega_\mathbf{q}}{T}})$.
Here $\chi_0=\frac{1}{g}-Z\mu_\textrm{pair}$, $\Omega_\mathbf{q}\approx\frac{q^2}{2M^*}-\mu_\textrm{pair}$, $Z=\frac{\partial X}{\partial \Omega}|_{\Omega=0,\mathbf{q}=0}$, $\mu_\textrm{pair}$ is the pair chemical potential, and $M^*=12(\frac{\partial^2\chi(Q)}{\partial k^2}|_{Q=0})^{-1}$ is the effective pair mass. The total energy can be decomposed in a similar fashion into
$E_\textrm{f}=\sum_{\mathbf{k}}(\xi_{\mathbf{k}}-E_{\mathbf{k}})+\Delta^2\chi_0+2\sum_{\mathbf{k}}E_{\mathbf{k}}f(E_{\mathbf{k}})+\mu n$ and
$E_\textrm{b}=\sum_\mathbf{q}(\Omega_\mathbf{q}+\mu_\textrm{pair})b(\Omega_\mathbf{q})$,
where $b(x)=[\exp(x/T)-1]^{-1}$ is the Bose distribution function. Similar to the mean-field theory results, for unitary Fermi gases one can show that $E_\textrm{f}=\frac{3}{2}P_\textrm{f}$ and $E_\textrm{b}=\frac{3}{2}P_\textrm{b}$, so the relation $E=\frac{3}{2}P$ still holds when pairing fluctuations are included.
The transport coefficients such as the shear viscosity can be evaluated from a modified CFOP theory with additional diagrams included to ensure the Ward identities~\cite{HaoPRL10,HaoPRL11}, which in turn guarantees charge conservation. Under a weak dissipation assumption~\cite{KosztinPRB00,HaoPRL10,Ourreview} the paramagnetic response function $\tensor{P}(\omega,\mathbf{q})$ can be evaluated and the shear viscosity can be derived accordingly (with the derivations summarized in the Appendix). An important observation is that since the pressure $P$ in Eq.~(\ref{etaP}) receives corrections from the noncondensed pairs (acting like composite bosons), it is natural to modify the shear viscosity to acquire similar (fermionic and bosonic) corrections, $\eta=\eta_\textrm{f}+\eta_\textrm{b}$. The former comes from a calculation similar to the BCS-Leggett theory, and the latter may be approximated by the shear viscosity of a free Bose gas:
\begin{eqnarray}\label{etapg}
\eta_\textrm{f}&=&\int_{0}^{+\infty}dk\frac{k^6}{15\pi^2m^2}\frac{E^2_{\mathbf{k}}-\Delta^2_\textrm{pg}}{E^2_{\mathbf{k}}}\frac{\xi^2_{\mathbf{k}}}{E^2_{\mathbf{k}}}\Big(-\frac{\partial f(E_{\mathbf{k}})}{\partial E_{\mathbf{k}}}\Big)\tau, \nonumber \\
\eta_\textrm{b}&=&-\frac{1}{30\pi^2M^{\ast2}}\int_0^\infty dkk^6\frac{\partial b(\Omega_\mathbf{k})}{\partial \Omega_\mathbf{k}}\tau.
\end{eqnarray}
Since the noncondensed pairs are in local equilibrium with the fermions, it may be reasonable to assume that the relaxation time for composite bosons is the same as that for fermions. Moreover, we found that $\eta_\textrm{b}=P_\textrm{b}\tau$, so in this approximation the noncondensed pairs satisfy the simple relation for normal scale-invariant systems.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Thermopga03.eps}
\caption{$\eta/\tau$ (black line), $\chi/\tau$ (green dot-dash line), pressure $P$ (red line), $\frac{2}{5}n\mu$ (blue dashed line) and the difference (thick pink line), and Diff=$(\eta+\chi)/\tau-(P-(2/5)n\mu)$ of a unitary Fermi superfluid as a function of temperature with pairing fluctuation effects included.
}
\label{fig:PG}
\end{figure}
The anomalous shear viscosity $\chi$ is due to momentum transfer through the correlations among Cooper pairs, and the phase rigidity of superfluidity is essential. Therefore, according to Eq.~\eqref{ns} its expression should be modified as
$\chi=-\frac{2}{15}\sum_{\mathbf{k}}\frac{k^4}{m^2}\frac{\Delta^2_\textrm{sc}}{E^2_\mathbf{k}}\frac{\partial f(E_\mathbf{k})}{\partial E_\mathbf{k}}\tau$. This approximation also avoids possible double-counting of the shear viscosity from noncondensed pairs (already included in $\eta_b$).
The terms on the right-hand side (RHS) of the relation (\ref{etaP}) should also receive bosonic corrections from the noncondensed pairs. In the pairing fluctuation theory consistent with the BCS-Leggett ground state~\cite{OurAnnPhys}, there is no noncondensed pairs at zero temperature, so the zero-temperature relation $P=(2/5)\mu n_s$ should remain the same.
To include the noncondensed-pair contributions at finite temperatures, we replace the superfluid density $n_\textrm{s}$ on the RHS of Eq.~\eqref{etaP} by the total number density $n$ since the two components (the superfluid and the noncondensed pairs) both contribute to thermodynamics, but $n_s=n$ at $T=0$ and this is consistent with the BCS-Legget ground state. Finally, we collect all the terms and modify the relation (\ref{etaP}) of a unitary Fermi superfluid as
\begin{eqnarray}\label{etaPpg}
\tilde{\eta}\equiv\eta+\chi\approx(P-\frac{2}{5}\mu n)\tau.
\end{eqnarray}
To verify the validity of this approximate relation, we perform numerical calculations to check each quantity. Figure~\ref{fig:PG} shows $\eta/\tau$, $\chi/\tau$, $P$, and $\frac{2}{5}n\mu$ as a function of temperature below the superfluid transition temperature $T_c$ for a unitary Fermi superfluid, with the convention of the labels the same as Figure~\ref{fig:BCS}. Here $T_c/T_F\approx 0.26$, and it is known that the $t$-matrix approximation overestimates the transition temperature~\cite{OurAnnPhys}, which should be around $0.16T_F$ in experiments~\cite{Nascimbene10}. The difference $(\eta+\chi)/\tau+\frac{2}{5}n\mu-P$ is also shown. For $T/T_c\le 0.5$, there is no observable difference, and the maximal ratio between the difference and the pressure $P$ in the regime $0.5\le T/T_c \le 1$ is less than $4.4\%$. Hence, the relation (\ref{etaPpg}) works reasonably for unitary Fermi superfluids and may serve as a condition for constraining the relevant thermodynamic and transport quantities.
\section{Conclusion}\label{sec:conclusion}
To summarize, we have presented a relation connecting thermodynamic quantities and transport coefficients of unitary Fermi superfluids. The relation involves the shear viscosity, pressure, superfluid density, particle density, anomalous shear viscosity characterizing momentum transfer via the Cooper pairs, chemical potential, and relaxation time. The relation is exact for the mean-field BCS Leggett theory, and an approximate relation is found when pairing fluctuations from noncondensed Cooper pairs are considered. The relations illustrate interesting properties of scale-invariant quantum fluids.
Our results of unitary Fermi superfluids also contrast different contributions from the condensate, fermionic quasiparticles, and noncondensed pairs and their interplays. The relations helps constrain elusive physical quantities such as the relaxation time if other quantities (the pressure, shear viscosity, etc.) can be measured experimentally. Moreover, the theoretical framework presented here may inspire searches for similar relations between thermodynamic quantities and transport coefficients in other many-body systems.
\textit{Acknowledgment:} We thank Qijin Chen for useful discussions. H. G. thanks the support from the National Natural Science Foundation of China (Grant No. 11674051).
|
2,877,628,090,849 | arxiv | \section*{Introduction}
Plain Kolmogorov complexity $\KS(x)$ of a binary string $x$ was defined in~\cite{kolm65} as the
minimal length of a program that computes $x$.
(See the preliminaries or~\cite{GacsNotes,LiVitanyi,shen00} for the details.)
It was clear from the beginning (see, e.g.,~\cite{ZvonkinLevin}) that complexity function
is not computable: no algorithm can compute $\KS(x)$ given $x$. In~\cite{gacs} a stronger
non-uniform version of this result was proven: for every $n$ there exists a string $x$ of length $n$
such that conditional complexity $\KS(\KS(x)|x)$, i.e., the minimal length of a program that maps
$x$ to $\KS(x)$, is at least $\log n - O(\log^{(2)} n)$. (If complexity function were computable, this
conditional complexity would be bounded.)
In Section~\ref{sec:gacs} we revisit this classical result and improve it a bit by removing the
$\log^{(2)} n$ term. No further improvement is possible because $\KS(n) \leq n + O(1)$, therefore
$\KS(\KS(n)|x) \leq \log n + O(1)$ for all~$x$. We use a game technique that was developed by Andrej
Muchnik (see~\cite{MuchnikGames,muchnik-game,ver-survey}) and turned out to be useful in many cases.
Recently Elena Kalinina (in her master thesis~\cite{kalinina}) used it to provide a proof of Gacs'
result. We use a more detailed analysis of essentially the same game to get a better bound.
For some $c$, a bit string $x$ is $\KS$-random if $n - \KS(x) \leq c$. Note that $n+O(1)$ is the smallest
upper bound for $\KS(x)$. A variant of plain complexity is prefix-free or self-delimiting
complexity,
which is defined as the shortest program that produces $x$ on a
Turing machine with binary input tape, i.e. without blanc or terminating symbol.
(See the preliminaries or~\cite{GacsNotes,LiVitanyi,shen00} for the details.)
The smallest upper bound for $\KP(x)$ for strings of length $n$ is
$n + \KP(n) + O(1)$. For some $c$, the string $x$ is defined to be
$\KP$-random if $n + \KP(n)- \KP(x) \leq c$.
Robert Solovay~\cite{solovay75} observed that $\KP$-random strings are also $\KS$-random strings
(for some $c' \leq O(c)$), but not vice versa.
Moreover, he showed that some $c$ and infinitely many $x$ satisfy $|x| - \KS(x) \leq c$ and
\[
|x| + \KP(|x|) - \KP(x) \geq \log^{(2)} |x| - c\log^{(3)} |x| \,.
\]
He also showed that for $\KS$-random $x$ the left-hand side of the equation
is upper-bounded by $\KP(\KP(n)|n) + O(1)$, which is bounded by $\log^{(2)} n + O(1)$.
\shortfull{Later Joseph Miller~\cite{miller-contrasting} and Alexander Shen~\cite{MuchnikGames}
generalized this, by showing that every co-enumerable set (i.e., the complement is enumerable)
containing strings of every length,
also contains infinitely many $x$ such that the above equation holds. (Note that the set of
$\KS$-random strings is co-enumerable but the set of $\KP$-random strings not.)}{}
In Section~\ref{sec:solovay} we provide a short proof for Solovay's result using the improved
version of Gacs' theorem. Then we generalize it by showing
that for some $c$ and every $n$ there are strings $x$ of length $n$ with
$n - \KS(x) \leq c$ and
\[
n + \KP(n) - \KP(x) \geq \KP(\KP(n)|n) - 3\KP(\;\KP(\KP(n)|n)\;|n) - c \,.
\]
This is very close to the upperbound $\KP(\KP(n)|n) - O(1)$, which was shown by
Solovay~\cite{solovay75}.
By the improved version of Gacs' result, we can choose $n$ such that $\KP(\KP(n)|n) = \log^{(2)} n + O(1)$.
For such $n$ we obtain Solovay's theorem with the $c\log^{(3)} |x|$ term replaced by a $O(1)$ constant.
\isfull{
In Section~\ref{sec:miller} we give a simple game-theoretic proof of Solovay's result which allows us to generalize it in two directions:
\begin{itemize}
\item Joseph Millers generalization \cite{miller-contrasting}: in every co-enumerable set (it is,
a set with computably enumerable complement) that contains a string of every length, there are
infinitely many strings $x$ satisfying the requirement of Solovay's theorem. This implies
Solovay's result as well as the following statement: for every $c$ the set of strings $x$ with
$c$-maximal prefix-free Kolmogorov complexity (it is, $\KP(x) \geq |x| + \KP(|x|) - c$) is not
co-enumerable.
\item \marginpar{TODO}|
Rather than looking to strings of
fixed length, we can look to effective finite set partitions of strings in some set, (it is, to disjoint
sets $S_n$ such that a program the finite set $S_1, S_2, \dots$). We show that if some
co-enumerable set intersects every $S_n$, then there are infinitely many $x$ such for $S_n \ni
x$: $C(x) = \log |S_n| \pm O(1)$ and
\[
K(x) \leq K(n) + \log |S_n| - \log^{(3)} |S_n| + O(\log^{(4)} |S_n|) \,.
\]
\end{itemize}
In the appendix plain and prefix-free randomness for more general discrete objects is defined
and related to typical models. We conclude that prefix-free randomness implies plain randomness,
and from the second generalization above, we conclude that this implication is not strict.
}
\shortfull{\textit{Preliminaries:} Let $U$ be a Turing machine. The {\em plain
(Kolmogorov) complexity} relative to $U$ is defined by
\[
\KS_U(x|y) = \min \left\{ |p|: U(p,y) = x \right\} \,.
\]
If the machine $U$ is prefix-free (i.e., for every $p,y$ such that $U(p,y)$ halts, there is no
prefix $q$ of $p$ such that $U(q,y)$ halts) then we write $\KP_U(x|y)$ rather than $\KS_U(x|y)$, and
refer to it as \textit{prefix-free (Kolmogorov) complexity} relative to $U$.
There exist plain and prefix-free Turing machines $U$ and $V$ for which $\KS_U(x|y)$
and $\KP_V(x|y)$ are minimal within an $O(1)$ constant.
We fix such machines and omit the indexes $U$,$V$.
If $y$ is the empty string we use the notation $\KS(x)$ and $\KP(x)$.
}{}
\section{Complexity of complexity can be high}\label{sec:gacs}
\begin{theorem}\label{th:main}
There exist some constant $c$ such that for every $n$ there exists a string $x$ of length $n$ such that $\KS(\KS(x)|x) \geq \log n - c$.
\end{theorem}
To prove this theorem, we first define some game and show a winning strategy for the game. (The connection between the game and the statement that we want to prove will be explained later.)
\subsection{The game}
Game $G_n$ has parameter $n$ and is played on a rectangular board divided into cells. The board has $2^n$ columns and $n$ rows numbered $0,1,\ldots,n-1$ (the bottom row has number $0$, the next one has number $1$ and so on, the top row has number $n-1$), see Fig.~\ref{fig:board}.
Initially the board is empty. Two players: White and Black, alternate their moves. At each move,
a player can pass or place a pawn (of his color) on the board. The pawn can not be moved or removed
afterwards. Also Black may blacken some cell instead. Let us agree that White starts the game
(though it does not matter).
The position of the game should satisfy some restrictions; the player who violates these restrictions, loses the game immediately. Formally the game is infinite, but since the number of (non-trivial) moves is a priori bounded, it can be considered as finite, and the winner is determined by the last (limit) position on the board.
\emph{Restrictions}: (1)~each player may put at most $2^i$ pawns in row $i$ (thus the total number of black and white pawns in a row can be at most $2^i + 2^i$); (2)~in each column Black may blacken at most half of the cells.
We say that a white pawn is \emph{dead} if either it is on a blackened cell or has a black pawn in the same column strictly below it.
\emph{Winning rule}: Black wins if he killed all white pawns, i.e., if each white pawn is dead in the final position.
\begin{figure}[h]
\begin{center}
\includegraphics{compcomp-1.mps}
\end{center}
\caption{Game board}\label{compcomp-1.mps}
\label{fig:board}
\end{figure}
For example, if the game ends in the position shown at Fig.~\ref{compcomp-1.mps}, the restrictions are
not violated (there are $3 \leq 2^2$ white pawns in row $2$ and $1\le 2^1$ white pawn in row $1$, as
well as $1 \leq 2^2$ black pawn in row $2$ and $1\le 2^0$ black pawn in row~$0$). Black loses
because the white pawn in the third column is not dead: it has no black pawn below and the cell is
not blackened. (There is also one living pawn in the fourth column.)
\subsection{How White can win}
The strategy is quite simple. White starts by placing a white pawn in an upper row of some column and waits until Black kills it, i.e., blackens the cell or places a black pawn below. In the first case White puts her pawn one row down and waits again. Since Black has no right to make all cells in a column black (at most half may be blackened), at some point he will be forced to place a black pawn below the white pawn in this column. After that White switches to some other column. (The ordering of columns is not important; we may assume that White moves from left to right.)
Note that when White switches to a next column, it may happen that there is a black pawn in this column or some cells are already blackened. If there is already a black pawn, White switches again to the next column; if some cell is blackened, White puts her pawn in the topmost white (non-blackened) cell.
This strategy allows White to win. Indeed, Black cannot place his pawns in all the columns due to the restrictions (the total number of his pawns is $\sum_{i=0}^{n-1} 2^i = 2^n -1$, which is less than the number of columns). White also cannot violate the restriction for the number of her pawns on some row $i$: all dead pawns have a black pawns strictly below them, so the number of them on row $i$ is $\sum_{j=0}^{i-1} 2^j = 2^i - 1$, hence White can put an additional pawn.
In fact we may even allow Black to blacken all the cells except one in each column, and White will still win, but this is not needed (and the $n/2$ restriction will be convenient later).
\subsection{Proof of Gacs' theorem}
Let us show that for each $n$ there exists a string $x$ of length $n$ such that $\KS(\KS(x|n)|x)\ge
\log n -O(1)$. Note that here $\KS(x|n)$ is used instead of $\KS(x)$; the difference between these
two numbers is $O(\log n)$ since $n$ can be described by $\log n$ bits, so the difference between
the complexities of these two numbers is $O(\log\log n)$.
Consider the following strategy for Black (assuming that the columns of the table are indexed by
strings of length $n$):
\begin{itemize}
\item Black blackens the cell in column $x$ and row $i$ as soon as he discovers that $\KS(i|x)<\log
n-1$. (The constant $1$ guarantees that less than half of the cells will be blackened.) Note that
Kolmogorov complexity is an upper semicomputable function, and Black approximates it from above,
so more and more cells are blackened.
\item Black puts a black pawn in a cell $(x,i)$ when he finds a program of length $i$ that produces
$x$ with input $n$ (this implies that $\KS(x|n) \leq i$). Note that there are at most $2^i$
programs of length $i$, so Black does not violate the restriction for the number of pawns on any
row $i$.
\end{itemize}
Let White play against this strategy (using the strategy described above). Since the strategy is
computable, the behavior of White is also computable. One can construct a decompressor $V$ for the
strings of length $n$ as follows: each time White puts a pawn in a cell $(x,i)$, a program of length
$i$ is assigned to $x$. By White's restriction, no more than $2^i$ programs need to be assigned. By
universality, a white pawn on cell $(x,i)$ implies that $\KS(x|n) \leq i + O(1)$. If White's pawn is
alive in column $x$, there is no black pawn below, so $\KS(x|n)\ge i$, and therefore
$\KS(x|n)=i+O(1)$. Moreover, for a winning pawn, the cell $(x,i)$ is not blackened, so $\KS(i|x)\ge
\log n -1$. Therefore, $\KS(\KS(x|n)|x)\ge \log n-O(1)$.
\textbf{Remark}: the construction also guarantees that $\KS(x|n)\ge n/2-O(1)$ for that $x$.
(Here the factor $1/2$ can be replaced by any $\alpha<1$ if we change the rules of the game accordingly.)
Indeed, according to white's strategy, he always plays in the highest non-black cell of some column,
and at most half of the cells in a column can be blackened, therefore no white pawns appear in the
lower half of the board.
\subsection{Modified game and the proof of Theorem~\protect{\ref{th:main}}}
Now we need to get rid of the condition $n$ and show that for every $n$ there is some $x$ such that $\KS(\KS(x)|x)\ge \log n - O(1)$. Imagine that White and Black play simultaneously all the games $G_n$. Black blackens the cell $(x,i)$ in game $G_{|x|}$ when he discovers that $\KS(i|x)<\log n-1$, as he did before, and puts a black pawn in a cell $(x,i)$ when he discovers an \emph{unconditional} program of length $i$ for $x$. If Black uses this strategy, he satisfies the stronger restriction: the total number of pawns in row $i$ \emph{on all boards} is bounded by $2^i$.
Assume that White uses the described strategy on each board. What can be said about the total number
of white pawns in row $i$? The dead pawns have black pawns strictly below them and hence the total
number of them does not exceed $2^i - 1$. On the other hand, there is at most one live white pawn on
each board. We know also that in $G_n$ white pawns never appear below row $n/2-1$, so the number of
live white pawns does not exceed $2i + O(1)$. Therefore we have $O(2^i)$ white pawns on the $i$-th row
in total.
For each $n$ there is a cell $(x,i)$ in $G_n$ where White wins in $G_n$. Hence, $\KS(x)<i+O(1)$
(because of property just mentioned and the computability of White's behavior), $\KS(x)\ge i$ and
$\KS(i|x)\ge \log n-1$ (by construction of Black's strategies and the winning condition).
Theorem~\ref{th:main} is proven.
\subsection{Version for prefix complexity}
\begin{theorem}
\label{th:prefix-max}
There exist some constant $c$ such that for every $n$ there exists a string $x$ of length $n$ such
that $\KS(\KP(x)|x) \geq \log n-c$ and $\KP(x) \geq n/2$. This also implies that $\KP(\KP(x)|x) \geq \log n-c$.
\end{theorem}
The proof of $\KS(\KP(x)|x) \geq \log n - c$ goes in the same way. Black places a pawn in cell
$(i,x)$ if some program of length $i$ for a prefix-free (unconditional) machine computes $x$ (and
hence $\KP(x) \leq i$); White uses the same strategy as described above. The sum of $2^{-i}$ for all
black pawns is less than $1$ (Kraft-inequality); some white pawns are dead, i.e., strictly above black ones, and for
each column the sum of $2^{-j}$ where $j$ is the row number, does not exceed $\sum_{j>i}^n 2^{-j} <
2^{-i}$. Hence the corresponding sum for all dead white pawns is less than $1$; for the rest the sum
is bounded by $\sum_n 2^{-n/2 + 1}$, so the total sum is bounded by a constant, and we conclude that for
$x$ in the winning column the row number is $\KP(x)+O(1)$, and this cell is not blackened.
\section{Strings with maximal plain and\\ non-maximal prefix-free complexity}
\label{sec:solovay}
In this section we compare two measures of non-randomness. Let $x$ be a string of length $n$; we
know that $\KS(x)\le n+O(1)$, and the difference $n-\KS(n)$ measures how ``nonrandom'' $x$ is. Let
us call it $\KS$-deficiency of $x$. On the other hand, $\KP(x)\le n+\KP(n)+O(1)$, so
$n+\KP(n)-\KP(x)$ also measures ``nonrandomness'' in some other way; we call this quantity
$\KP$-deficiency of~$x$.
The following proposition means that $\KP$-random strings (for which $\KP$-deficiency is small;
they are also called ``Chaitin random'') are always $\KS$-random ($\KS$-deficiency is small; such
strings are also called ``Kolmogorov random'').
\begin{proposition}[Solovay \cite{solovay75}]
$|x| + K(|x|) - K(x) \le c$ implies $|x| - C(x) \le O(c)$.
\label{prop:KrandImpliesCrand}
\end{proposition}
\begin{proof}
We use a result of Levin: for every string $u$
\begin{equation*}
\KP(u|\KS(u)) = \KS(u)+O(1),
\end{equation*}
and, on the other hand, for any positive or negative integer number $c$:
\begin{equation*}
\KP(u|i) = i+c,
\end{equation*}
implies $\KS(u) = i+O(c)$\footnote{Textbooks like \cite[Lemma 3.1.1]{LiVitanyi} mention
only the first statement. To show the second, note that the function $i \mapsto \KP(x|i)$ maps
numbers at distance $c$ to numbers at distance $O(\log c)$, hence, the fixed point $\KS(x)$ must be
unique within an $O(1)$ constant. Furthermore, for any $i$, the fixed point must be within distance
$O(\log |i-\KP(u|i)|)$ from $i$, hence $|\KS(u)-i| \leq O(\log |i - \KP(u|i)|) = O(\log c)$.
}.
Let $n = |x|$. Notice that
\[
n + K(n) \le K(x) - c = K(x,n) - O(c) \le K(x|n) + K(n) - O(c) \,.
\]
Hence, $K(x|n) \geq n - O(c)$, thus $K(x|n) = n + O(c)$ and thus: $C(x) = n + O(c)$.
\end{proof}
R.~Solovay showed that the reverse statement is not always true: a $\KS$-random string may be not
$\KP$-random. However, as the following result shows, the $\KP$-deficiency still can be bounded for
$\KS$-random strings:
\begin{proposition}[Solovay~\cite{solovay75}] \label{prop:solovayOptimal}
For any $x$ of length $n$ the inequality $\KS(x) \geq n-c$ implies:
$$
n + \KP(n) - \KP(x) \leq \KP(\KP(n) |n) + O(c)\,.
$$
\end{proposition}
Note that $\KP(\KP(n)|n)\le \log^{(2)} n+O(1).$
\begin{proof}
The proof uses another result of Levin~\cite{gacs,GacsNotes,LiVitanyi}: for all $u,v$ we have the additivity property
\begin{equation*}
\KP(u,v) = \KP(u) + \KP(v|u,\KP(u))+O(1) \,.
\end{equation*}
To prove Proposition~\ref{prop:solovayOptimal}, notice that $n = \KS(x) = \KP(x|\KS(x)) = \KP(x|n)$ with $O(c)$-precision. By additivity we have: $\KP(x) = \KP(n,x) = \KP(n) + \KP(x|n,\KP(n))$. Putting these observations together, we get (with $O(c)$-precision)
\begin{align}
n + \KP(n) - \KP(x) &= \KP(x|n) + \KP(n) - (\KP(n) + \KP(x|n,K(n)))=\nonumber \\
&= \KP(x|n) - \KP(x|n,\KP(n)) \,.\label{eq:gap}
\end{align}
Observe that $\KP(x|n) \leq \KP(x|n,\KP(n)) + \KP(\KP(n)|n)+O(1)$, hence the $\KP$-deficiency is
bounded by $\KP(\KP(n)|n)+O(c)$.
\end{proof}
The following theorem shows that for all $n$
the bound $\KP(\KP(n)|n)$ for $\KP$-deficiency for $\KS$-random strings
can almost be achieved. The error is at most $O(\log \KP(\KP(n)|n))$.
\begin{theorem} \label{th:solovay2allN}
For some $c$ and all $n$ there are strings $x$ of length $n$ such that $n - \KS(x) \leq c$, and
\[
n + \KP(n) - \KP(x) \geq \KP(\KP(n)|n) - 3\KP(\; \KP(\KP(n)|n)\;|n ) - c\,.
\]
\end{theorem}
By corollary, infinitely many $\KS$-random strings
have $\KP$-deficiency $\log^{(2)} |x| + O(1)$.
Indeed, for $n$ such that $\KP(\KP(n)|n) = \log^{(2)} n + O(1)$, we have $\KP(\; \KP(\KP(n)|n)\;|n ) \leq
O(1)$, and hence, a slightly stronger statement than proved by Solovay~\cite{solovay75} is obtained.
\begin{corollary}\label{cor:solovay2}
There exists a constant $c$ and infinitely many $x$ such that $|x| - \KS(x) \le c$ and
$|x| + \KP(|x|) - \KP(x) \ge \log^{(2)} |x|-c$.
\end{corollary}
Before proving Theorem \ref{th:solovay2allN}, we prove the corollary directly.
\begin{proof}
First we choose $n$, the length of string $x$. It is chosen in such a way that
$\KP(\KP(n)|n)=\log^{(2)} n +O(1)$ and $\KP(n)\ge (\log n)/2$ (Theorem~\ref{th:prefix-max}). (So the
bound of Proposition~\ref{prop:solovayOptimal} is not an obstacle.) We know already (see
equation~\ref{eq:gap}) that for a string $x$ with $\KS$-deficiency $c$ the value of
$\KP$-deficiency is $O(c)$-close to $\KP(x|n)-\KP(x|n,\KP(n))$. This means that adding $\KP(n)$ in
the condition should decrease the complexity, so let us include $\KP(n)$ in $x$ somehow. We also
have to guarantee maximal $\KS$-complexity of $x$. This motivates the following choice:
\begin{itemize}
\item choose $r$ of length $n - \log^{(2)} n$ such that $\KP(r|n,\KP(n)) \geq |r|$.
Note that this implies $\KP(r|n,\KP(n)) = |r|+O(1)$, since the length of $r$ is determined by the condition;
\item let $x = \langle K(n) \rangle r$, the concatenation of $K(n)$ (in binary) with $r$.
Note that $\langle K(n) \rangle$ has at most $\log^{(2)} n + O(1)$ bits for every $n$, and by choice of $n$
has at least $\log^{(2)} n - 1$ bits, hence $|x| = n + O(1)$.
\end{itemize}
As we have seen (looking at equation~\eqref{eq:gap}), it is enough to show that
$\KP(x| \KP(n),n) \le n - \log^{(2)} n$ and $\KP(x|n) \ge n$ (the latter equality implies $\KS(x)=n$); all the equalities here
and below are up to $O(1)$ additive term.
\begin{itemize}
\item
Knowing $n$, we can split $x$ in two parts $\langle \KP(n)\rangle$ and $r$.
Hence, $\KP(x|\KP(n),n) = \KP(\KP(n),r|n,\KP(n))$, and this equals $\KP(r|n,\KP(n))$, i.e.,
$n - \log^{(2)} n$ by choice of $r$.
\item
To compute $\KP(x|n)$, we use additivity:
\begin{equation*}
\KP(x | n) = \KP(\KP(n),r|n) = \KP(\KP(n)|n) + \KP(r | \KP(n), \KP(\KP(n)|n), n) \,.
\end{equation*}
By choice of $n$, we have $\KP(\KP(n)|n)=\log^{(2)} n$, and the last term simplifies to $\KP(r |
\KP(n), \log^{(2)} n, n)$, and this equals $\KP(r | \KP(n), n) = n - \log^{(2)} n$ by choice of~$r$.
Hence $\KP(x|n) = \log^{(2)} n + (n - \log^{(2)} n) = n$.
\end{itemize}
\end{proof}
\textbf{Remark 1:}
One can also ask how many strings exist that satisfy the conditions of Corollary \ref{cor:solovay2}.
By Proposition \ref{prop:solovayOptimal}, the length $n$ of such a string must satisfy
$\KP(\KP(n)|n) \geq \log^{(2)} n - O(1)$.
By Theorem~\ref{th:prefix-max}, there is at least one such an $n$ for every length
of $n$ in binary. Hence such $n$, can be found within exponential intervals.
\textbf{Remark 2:}
One can ask for these $n$, how many strings $x$ of length $n$ satisfy the conditions of
Corollary \ref{cor:solovay2}.
By a theorem of Chaitin \cite{LiVitanyi}, there are at least $O(2^{n-k})$ strings with
$\KP$-deficiency $k$, hence we can have at most $O(2^{n-\log^{(2)} n})$ such strings. It turns out
that indeed at least a fraction $1/O(1)$ of them satisfy the conditions of Corollary \ref{cor:solovay2}.
To show this, note that
in the proof Theorem \ref{th:solovay2allN}, every different $r$ of length $n-|q| = |n|-\log^{(2)} n + O(1)$
leads to the construction of a different $x$. For such $r$ we essentially need $\KP(r|n,\KP(n),q) \geq
|r|-O(1)$, and hence there are $O(2^{n - \log^{(2)} n})$ of them.
\noindent
\textit{Proof of Theorem \ref{th:solovay2allN}.}
In the proof above,
in order to obtain a large value $\KP(x|n)-\KP(x|n,\KP(n))$,
we incorporated $\KP(n)$ in a direct way (as $\langle \KP(n) \rangle$) in $x$.
To show that $C(x) = K(x|n)+O(1)$ is large we essentially used that the length of
$\langle \KP(n) \rangle$ equals $\KP(\KP(n)|n) + O(1)$. For general $n$,
this trick does not work anymore,
but we can use a shortest program for $\KP(n)$ given $n$ (on a plain machine).
For every $n$ we can construct $x$ as follows:
\begin{itemize}
\item
let $q$ be a shortest program that computes $\KP(n)$ from $n$ on a {\em plain} machine (if there are several shortest
programs, we choose the one with shortest running time).
Note that $|q| = \KS(\KP(n)|n) + O(1)= \KS(q|n) + O(1)$ (remind that by adding some fixed instructions, a
program can print itself, and that a shortest program is always incompressible, thus up to $O(1)$ constants:
$|q| \geq \KS(\KP(n)|n) \geq \KS(q|n) \geq |q|$), by Levin's result (conditional version), the last term also equals
$\KP(q|n,|q|)+O(1)$;
\item
let $r$ have length $n - |q|$, such that $\KP(r|n,\KP(n),q) \geq |r|$.
Note that this implies $\KP(r|n,\KP(n),q) = |r|+O(1)$, (since the length of $r$ is determined by the
condition).
\item
We define $x$ as the concatenation $qr$.
\end{itemize}
We show that $\KS(x) = n + O(1)$ and that the $\KP$-deficiency is at least $|q|-\KP(|q|\,|n) + O(1)$.
To show that this implies the theorem, we need that
\[
\KP(\KP(n)|n) - 3\KP(\; \KP(\KP(n)|n) \;|n) \leq \KS(\KP(n)|n) - \KP(\;\KS(\KP(n)|n)\;|n) + O(1)\,,
\]
which is for $a = \KP(n)$ the conditioned version of Lemma \ref{lem:plainPrefixHelp}:
\[
\KP(a|n) - 3\KP(\;\KP(a|n)\;|n) \leq \KS(a|n) - \KP(\;\KS(a|n)\;|n)+O(1) \,.
\]
Following the same structure as the proof above, it remains to show that
$\KP(x| \KP(n),n) \leq n - |q| + \KP(|q|\,|n)$ and $\KP(x|n) \ge n$ (the latter equality implies $\KS(x)=n$); all the equalities here
and below are up to $O(1)$ additive term.
\begin{itemize}
\item
Knowing $|q|$, we can split $x$ in two parts $q$ and $r$.
Hence, $\KP(x|\KP(n),n,|q|) = \KP(q,r|n,\KP(n),|q|)$.
Given $n, \KP(n), |q|$ we can search for a program of length $|q|$
that on input $n$ outputs $\KP(n)$; the one with shortest computation time is $q$.
Hence, $\KP(q,r|n,\KP(n),|q|) = \KP(r|n,\KP(n),|q|)$, i.e., $n - |q|$ by choice of $r$, and
therefore $\KP(x|\KP(n),n) \leq n - |q| + \KP(|q|\,|n)$.
\item
To compute $\KP(x|n)$, we use additivity:
\begin{equation*}
\KP(x | n) \ge \KP(x|n, |q|) = \KP(q,r|n, |q|) = \KP(q|n, |q|) + \KP(r | q, \KP(q|n,|q|), n) \,.
\end{equation*}
By choice of $q$ we have $\KS(q|n) = |q|$, and hence by Levin's result $\KP(q|n, |q|) = |q|$.
The last term is $\KP(r | q, |q|, n)$ which equals $\KP(r|q,n) = n - |q|$ by choice of~$r$.
Hence, $\KP(x|n) \ge |q| + (n - |q|) = n$. \qed
\end{itemize}
\begin{lemma}\label{lem:plainPrefixHelp}
$\KP(a) - 3\KP(\KP(a)) \leq \KS(a) - \KP(\KS(a))+O(1)$
\end{lemma}
\begin{proof}
Note that $\KP(a) - \KS(a) \leq \KP(\KS(a))$.
Indeed, any program for a plain machine can be
considered as a program for a prefix-free machine conditional to it's length.
Hence, we can transform a plain program $p$ to a prefix-free program by
adding a description of $|p|$ of length $\KP(|p|)$ to $p$.
Hence it remains to show
$2\KP(\KS(a)) \leq 3\KP(\KP(a))+O(1)$.
Solovay~\cite{solovay75} showed that
\[
\KP(a) - \KS(a) = \KP(\KP(a)) + O(\KP(\KP(\KP(a)))) \,,
\]
hence,
\[
|\KP(\KP(a)) - \KP(\KS(a))| \leq O(\log \KP(\KP(a))) \,.
\]
\end{proof}
\isfull{
\section{Generalizing Solovay's result using games}
\label{sec:miller}
Here we give a game-based proof of a generalization of Solovay's result due to
J.~Miller's~\cite{miller-contrasting}.
The original proof in~\cite{miller-contrasting} uses a different scheme that involves the Kleene fixed-point theorem.
\begin{theorem}
For any co-enumerable set $Q$ of strings (it is, its complement is enumerable), that for every $n$
contains at least one string of length $n$, there exist infinitely many $x$ in $Q$ such
that $\KP(x)< |x| + \KP(|x|)- \log^{(2)} |x| + O(\log^{(3)} |x|)$.
\label{th:solovay2Gen}
\end{theorem}
\noindent
Solovay's result follows
by choosing $Q$ to be the co-enumerable set of strings with $\KS(x) \geq |x| - O(1)$.
The proof below does not use Gacs' theorem, hence, by Lemma \ref{lem:solovayOptimal}, a weaker form becomes
a corollary of this theorem.
\begin{corollary}
For any $c$, the set of strings $x$ with $K(x) \geq K(|x|) + |x| - c$ is not co-enumerable.
\label{cor:KrandomNotCe}
\end{corollary}
\noindent
\textit{Proof of Theorem \ref{th:solovay2Gen}.}
Let us consider the following game specified by a natural number $C$ and a
finite family of disjoint finite sets $S_1,\dots,S_N$. During the game each element
$s\in S=\cup_{j=1}^N S_j$ is labeled by two non-negative rational numbers $A(s)$ and $B(s)$ called ``Alice
weight'' and ``Bob's weight''. Initially all weights are zeros. Alice and Bob make alternate moves.
On each move each player may increase her/his weight of several elements~$s\in S$.
Both players must obey the following total weight restrictions:
$$
\sum_{s\in S}A(s)\le1\quad\text{and}\quad \sum_{s\in S}B(s)\le1.
$$
In addition, Bob must be ``fair'': for every $j$ Bob's weights of all $s\in S_j$ must be equal. That
means that basically Bob assigns weights to $j\in\{1,\dots,N\}$ and Bob's weight $B(j)$ of $j$ is
then evenly distributed among all $s\in S_j$ so that $$ B(s)=B(j)/\#S_j $$ for all $s\in S_j$.
Alice need not be fair.
This extra requirement is somehow compensated by allowing Bob to ``disable'' certain $s\in S$. Once
an $s$ is disabled it cannot be ``enabled'' any more. Alice cannot disable or enable anything. For
every $j$ Bob is not allowed to disable \textsl{all} $s\in S_j$: every set $S_j$ should contain at
least one element that is enabled (=not disabled).
The game is infinite. Alice wins if at the end of the game (or, better to say, in the limit) there
exists an enabled $s\in S$ such that
$$
\frac{A(s)}{B(s)}\ge C.
$$
Now we have (as usual) to explain two things: why Alice has a (computable) winning strategy in the
game (with some assumptions on the parameters of the game) and why this implies Miller's theorem.
\begin{lemma}
Alice has a computable winning strategy if $N\ge2^{8C}$ and $\#S_j\ge 8C$ for all $j\le N$.
\label{lem:gameSolovayGen}
\end{lemma}
Let us show first why this statement implies the theorem. Let
$$
C=2^{c} \quad\text{and}\quad N= 2^{8C}=2^{2^{c+3}}
$$
Let us take the sets of all strings of length
$$\log 8C+1,\dots, \log 8C+N$$
as $S_1,\ldots,S_N$. [changed] If $S_j$ consists of less then $8C$ elements nothing needs to be
shown. Indeed, in this case we can produce the lexicographic first element $x$ in $S_j$
by a co-enumerable description of $S_j$ and the total number of elements in $S_j$,
because the co-enumerable description provides an initial estimate of $S_j$ and then
removes elements. Hence $K(x) \leq K(S_j) + O(1) \leq K(j) + O(1)$.
Otherwise, the conditions of the lemma are
satisfied and Alice has a computable winning strategy.
Consider the following Bob's strategy in this game: he enumerates the complement of $Q$ and disables
all its elements; in parallel, he approximates the prefix complexity from above and once he finds
out that $K(n)$ does not exceed some $l$, he increases the weights of all $2^n$ strings of
length~$n$ up to $2^{-l-n}$. Thus at the end of the game $B(x)=2^{-K(n)-n}$ for all $s\in S$ that
have length $n$ (i.e., for $s\in S_j$ where $j=n-\log 8C$).
Alice's limit weight function $x\mapsto A(x)$ is lower semi-computable given $c$, as both Alice's
and Bob's strategies are computable given $c$. Therefore (since prefix complexity is equal to the
logarithm of a priori probability) $$\KP(s|c)\le -\log A(s)+O(1)$$
for all $s\in S$. As Alice wins, there exists a string $s\in Q$ of some length $n\le N+\log 8C$ such
that $A(s)/B(s)\ge C$, i.e.,
$$
-\log A(s)\le -\log B(s)-c=\KP(n)+n-c.
$$
This implies that
$$
\KP(s|c)\le \KP(n)+n-c+O(1),
$$
and
$$
\KP(s)\le \KP(n)+n-c+2\log c+O(1).
$$
This is a bit weaker statement that we need: we wanted
$$K(s)< K(n)+n-c.$$
To fix this, apply this argument to $c'=c+3\log c$ in place of $c$. For all large enough $c$ we then
have $K(s)<K(n)+n-c$.
Note that this proof provides also some bound for $n$, which is $O(c^22^{2^c})$, hence the
theorem follows.
\qed
\smallskip
It remains to prove the Lemma by showing a winning strategy for Alice.
\smallskip
\textit{Proof of Lemma \ref{lem:gameSolovayGen}.}
The strategy is rather straightforward. The main idea is that playing
with one $S_i$, Alice can force Bob to spend twice more weight than she does. Then she switches to
the next $S_i$, and so on until Bob's weight is exhausted while she has solid reserves. To achieve her
goal on one set of $M$ elements, Alice assigns sequentially weights $1/2^M,
1/2^{M-1},\ldots,1/{2^1}$ and after each move waits until Bob increases his weight or disables the
corresponding element. Since he cannot disable all elements and is forced to use the same weights
for all elements while Alice puts more than half of the weight on the last element, Alice has factor
$M/2$ as a handicap, and we may assume that $M$ beats $C$-factor that Bob has in his favor.
Now the formal details. Assume first that $\#S_j=M=4C$ for all $j$ and $N=2^{M}$. (We will show
later how to adjust the proof to the case when $|S_j|\ge8C$ and $N\ge2^{8C}$.)
Alice picks an element $x_1\in S_1$ and assigns the weight $1/2^{M}$ to $x_1$. Bob (to avoid losing
the entire game) has either to assign a weight of more than $1/C2^{M}$ to all elements in $S_1$, or
to disable $x_1$. In the second case Alice picks another element $x_2\in S_1$ and assigns a (twice
bigger) weight of $2/2^{M}$ to it. Again Bob has a dilemma: either to increase the weight for all
elements of $S_1$ up to $2/C2^{M}$, or to disable $x_2$. In the second case Alice picks $x_3$,
assigns a weight of $4/2^{M}$ to it, and so on. (If this process continues long enough, the last
weight would be $2^{M-1}/2^M=1/2$.)
As Bob cannot disable all the elements of $S_1$, at some step $i$ the first case occurs, and Bob
assigns a weight greater than $2^i/C2^M$ to all the elements of $S_1$. Then Alice stops playing
with $S_1$. Note that the total Alice's weight of $S_1$ (let us call it $\beta$) is the sum of the
geometric sequence: $$
\beta=1/2^{M}+2/2^M+\dots +2^{i-1}/2^M<2^i/2^M\le1.
$$
Thus Alice obeys the rules. Note that total Bob's weight of $S_1$ is more than
$M2^{i-1}/C2^M=2^{i+1}/2^M$, which exceeds at least two times the total Alice's weight spent on
$S_1$. This implies, in particular, that Bob cannot beat Alice's weight for the last element if the
game comes to this stage (and Alice wins the game in this case.)
Then Alice proceeds to the second set $S_2$ and repeats the procedure. However this time she uses weights
$
\alpha/2^{M},2\alpha/2^M,\dots,
$
where $\alpha=1-\beta$ is the weight still available for Alice. Again she forces Bob to use twice
more weight than she does. Then Alice repeats the procedure for the third set $S_3$ with the
remaining weight etc.
Let $\beta_j$ is the total weight Alice spent on the sets $S_1,\dots,S_j$, and
$\alpha_j=1-\beta_j$ the weight remaining after the first $j$ iterations. By construction, Bob's
total weight spent on sets $S_1,\dots,S_j$ is greater than $2\beta_j$, so we have $2\beta_j<1$ and
hence $\alpha_j> 1/2$. Consequently, Alice's total weight of each $S_j$ is more than $1/2^{M+1}$.
Hence after at most $N=2^{M}$ iterations Alice wins.
If the size of $S_j$ are large but different, we need to make some modification. (We cannot use the
same approach starting with $1/2^M$ where $M$ is the size of the set: if Bob beats the first element
with factor $C$, he spends twice more weight than Alice but still a small amount, so we do not have
enough sets for a contradiction.)
However, the modification is easy. If the number of elements in $S_j$ is a multiple of $4C$ (which
is the case we use), we can split elements of $S_j$ into $4C$ groups of equal size, and treat all
members of each group $G$ as one element. This means that if the above algorithm asks to assign to
an ``element'' (group) $G$ a weight $w$, Alice distributes the weight $w$ uniformly among members of
$G$ and waits until either Bob disables all elements of the group or assigns $4C$-bigger weight to
all elements of $S_j$.
If $S_j$ is not a multiple of $4C$, the groups are not equal (the worst case is when some groups
have one element while other have two elements), so to compensate for this we heed to use $8C$
instead of $4C$.
Note that excess in the number of groups (when $N$ is bigger than required $8C$) does not matter at
all, we just ignore some groups
\qed
Note that instead of
classifying strings according to their length, we could split them (effectively) into arbitrary
finite sets $S_n$. Then for every
string $x\in S_n$ we have $\KP(x) \le |S_n|+K(n)+O(1)$ and for every co-enumerable set $Q$ that
intersects every $S_n$ there exists $n$ and $x\in S_n\cap Q$ such that $\KP(x)\le |S_n|+\KP(n)-c$
(for the same reasons).
\begin{theorem}
For every disjoint family of finite and uniformily computable sets $S_n$
(it is, a program enumerates $S_1, S_2, \dots$)
and for every co-enumerable set $Q$ interesecting every $S_n$,
there are infinitely many pairs $(x,n)$ with $x \in Q \cap S_n$ such that
$C(x) = \log |S_n| \pm O(1)$ and
\[
K(x) \leq K(|S_n|) + |S_n| - \log^{(3)} |S_n| + O(\log^{(4)} |S_n|) \,.
\]
\label{th:solovayGenGen}
\end{theorem}
}
\isfull{
\section*{Appendix: Randomness for discrete objects}
A string $x$ can be defined to be $c$-random if the Kolmogorov complexity of some type is
maximal within a constant $c$ among the strings of the same length.
Three types of complexities are often used: plain $C(x)$, monotone $Km(x)$,
and prefix-free $K(x)$. The corresponding maximal values of the complexities are within $O(1)$
constant: $|x|$, $|x|$, and $|x| + K(|x|)$.
Hence, a string is $c$-random for $C$, respectively, for $Km$, and for $K$ iff
$C(x) \geq |x|-c$, respectively, $Km(x) \geq |x| - c$, and $K(x) \geq |x| + K(|x|) - c$.
For these complexities, the amount of strings with complexity $i \pm O(1)$ increases exponentially fast with $i$;
hence most strings have complexity around the corresponding maximal value, and
only few strings are not $c$-random for large enough (but constant) $c$.
It is not so difficult to see (Lemma \ref{lem:finiteRandomnessEasyThings})
that randomness for $K$ implies randomness for $C$ implies randomness for $Km$,
and that the second implication is strict.
A discrete object (graph, subspace, \dots) can be defined to be $c$-random,
if the Kolmogorov complexity of some type of a description of the object
is not more than $c$ below the maximal Kolmogorov complexity
of an object in the set of objects with similar properties.
Note that we define Kolmogorov complexity of a discrete object as the minimal length of a program
that outputs a string that identifies the object.
Suppose the sets of objects of some type with similar properties define a collection of disjoint
finite sets $S_n$.
In many cases these sets are uniformly computable
(it is, a program enumerates $S_1, S_2, \dots$).
Natural examples for such sets $S_n$ are the strings of length $n$, or the set of graphs with $n$ vertices.
The maximal plain and prefix-free complexity of a member of $S_n$ is given by $\log |S_n|$ and
$K(n) + \log |S_n|$ by Lemma \ref{lem:discreteRandomMax}.
Note that a definition for monotone complexity in a natural way, depends on the type
of the objects (graphs, manifolds, \dots) or requires a more involved treatment.
For objects of these more general types, by Lemma \ref{lem:discreteRandomMax}
randomness for $K$ implies randomness for $C$.
However, by Theorem \ref{th:solovayGenGen}
this implication is not strict.
In \cite{algorithmicStatistics} an element $x$ is defined to be weakly typical for a
set $S_n$ if $K(x|S_n) = \log |S_n|$. It is defined to be strongly typical for $S_n$
if $K(x|S_n, K(S_n)) = \log |S_n|$. Notice that by Lemma \ref{lem:discreteRandomnessEquivalent}
strongly typical is equivalent with randomness for $K$,
and that if $\log |S_n| \geq O(n)$, then weakly typical
is equivalent with randomness for $C$. Notice that it also does not matter whether
typicality is defined using plain or prefix-free complexity.
Below the lemmas for this discussion are given.
\begin{lemma}
For some constant $c$ and for any constant $e$:
\begin{itemize}
\item if $x$ is $e$-random for $K$ then $x$ is $e+c$-random for $C$,
\item if $x$ is $e$-random for $C$ then $x$ is $e+c$-random for $Km$.
\end{itemize}
The second implication is strict: for any large enough constants $c$, $e$,
there is an $x$ that is $c$-random for $Km$, and not $c+e$-random for $C$.
\label{lem:finiteRandomnessEasyThings}
\end{lemma}
\begin{proof}
The first item is Lemma \ref{lem:KrandImpliesCrand}, and
the second item follows by definition and using $Km(x) \geq C(x) - O(1)$.
The third claim, follows by showing that for infinitely many $x$ one has
$Km(x) = |x| \pm O(1)$, while $C(x) \leq |x| - \log |x| + O(\log^{(2)} |x|)$.
To construct such $x$, let $y$ be arbitrarily large such that $Km(y) = |y| \pm O(1)$.
Let $x$ be the longest substring of $y$ such that $|x|$ in binary format equals $y_1 \dots y_k =
x_1 \dots x_k$ for some $k$. Notice that $k = \log |y| \pm 1$ and $k = \log |x|$.
$x$ can computed from $k$ and $x_{k+1} \dots x_{|x|}$, because the length of the second
item equals $|x|-k$ and from $|x|$ one can compute $x_1 \dots x_k$.
Hence $C(x) \leq |x| - k + O(\log k)$.
\end{proof}
\noindent
\begin{lemma}
Let $S_n$ be disjoint and uniformily computable family of sets.
The maximal value for $x \in S_n$ is $\log |S_n| + K(\log |S_n|) - O(1)$.
Obtaining this maximal value is equivalent with
$K(x|S_n, K(S_n)) = \log |S_n| \pm O(1)$. For such $x$ one also has $C(x) \geq \log |S_n|-O(1)$.
\label{lem:discreteRandomMax}
\end{lemma}
\begin{proof}
Indeed, from a description of the object, we can find it's category
hence by Equation \eqref{eq:additivity} we have $K(x) = K(x,S_n) = K(S_n) + K(x|S_n, K(S_n)$. The
second term can be at most $\log |S_n|$ and at least one string has this complexity.
Moreover, from $n$ we can compute $S_n$ and vice versa which implies $K(n) = K(S_n) \pm O(1)$.
Hence, maximal prefix-free complexity is equivalent with $K(x|S_n, K(S_n)) = \log |S_n| \pm O(1)$.
This also implies
$K(x|\log |S_n|) = \log S_n \pm O(1)$, which implies by Equation \eqref{eq:levinKC}
$C(x) = \log |S_n| \pm O(1)$.
\end{proof}
\begin{lemma}
Let $S_n$ be disjoint and uniformily computable family of sets such that
$\log |S_n| = O(n)$, the following statements are equivalent:
\begin{enumerate}
\item $C(x) = \log |S_n| \pm O(1)$, \label{it:plain}
\item $C(x|S_n) = \log |S_n| \pm O(1)$,\label{it:plaincond}
\item $K(x|S_n) = \log |S_n| \pm O(1)$.\label{it:prefixcond}
\end{enumerate}
\label{lem:discreteRandomnessEquivalent}
\end{lemma}
\noindent
\textit{proof.}
\\\ref{it:prefixcond} $\Rightarrow$ \ref{it:plain}: trivial.
\\\ref{it:plain} $\Rightarrow$ \ref{it:plaincond}:
note that for any $x$, we have that $C(x|C(x)) = C(x) \pm O(1)$.
Moreover, from $\log |S_n|$ one can compute $n$ and hence also $S_n$,
and vice versa. Hence $C(x|S_n) = C(x|\log |S_n|) = C(x|C(x)) = C(x)$,
within $O(1)$ error.
\\\ref{it:plaincond} $\Rightarrow$ \ref{it:prefixcond}:
follows from $C(x|C(x)) = K(x|C(x)) \pm O(1)$ for any $x$.
\qed
}
|
2,877,628,090,850 | arxiv | \section{\label{sec:level1} Introduction}
The Cosmological Constant (CC) problem \cite{Zeldovich:1967gd, Zel'dovich:1968zz, Wilczek:1983as, Weinberg:1988cp} is a shadow in modern Physics. Such a discrepancy between predictions and observations has been, in the history of Science, a sign of a paradigmatic change of our understanding of the Universe.
It has been tackled in the last decades in several different ways, all of them with a common negative outcome (see \cite{Burgess:2013ara} for a recent review). Nevertheless, given the relevance of the problem one should keep an open mind for new proposals like the recent one dubbed vacuum energy sequestering mechanism (VES) \cite{Kaloper:2013zca, Kaloper:2014dqa}.
Here we address the {\it new} version of the problem known as the coincidence problem. It consists on the coincidence between the epoch we live in and the beginning of an epoch where the Universe is dominated by what looks like a CC. Although the present data is consistent with a pure CC ($w=-1.019^{+0.075}_{-0.080} $ (95\% CL) \cite{Ade:2015xua}) there are several different ways to drive the present accelerated stage \cite{Copeland:2006wr}. One of the most interesting alternative ideas relies on a scalar (quintessence) field slowly-rolling its potential until the present time, when it starts dominating the energy content of the Universe driving a period of accelerated expansion \cite{Ratra:1987rm} (see also \cite{ Linder:2015zxa} for a more recent analysis).
In particular, a subclass of these quintessence models has received much attention due to its technical naturalness which protects a small mass for the field \cite{Frieman:1991tu, Hill:1988vm, Frieman:1995pm}, needed to explain the observations. That subclass consists of scenarios where the scalar field has a shift symmetry only broken by non-perturbative effects. Specific realizations include cosine but also some monomial potentials \cite{McAllister:2008hb, Kaloper:2008fb, Kaloper:2011jz} for the field. Here we will focus on the simple case of a linear potential.
The use of a linear potential to explain the current dark energy stage has been a recurrent idea in the literature \cite{Kallosh:2003bq, Avelino:2004hu, Avelino:2004vy, Linder:2007wa, Avelino:2014nqa, Barreira:2011qi}. Apart from the virtues mentioned above, the case of the linear potential is also appealing because its dynamics is quite insensitive to the initial conditions. A light field is essentially frozen in its potential, dynamically this stage is quickly achieved independently of the initial conditions, until it dominates the energy content of the Universe and drives a period of accelerated expansion. However, this period does not last forever. At some point in the future the slow-roll conditions are broken and the dark energy stage ends. At that time, the field quickly rolls down its potential to negative values such that the energy density of the Universe goes to zero, the Universe bounces and a collapsing stage starts dominated by the kinetic energy of the field \cite{Avelino:2004vy}.
In this work we perform a detailed analytical and numerical analysis of the dynamics. We discuss the coincidence problem in this context by establishing a relation between the total age of the Universe and the value of the equation of state $w$ today. A future observation of $w >-1$ would then indicate the time until the {\it doomsday}.
The paper is organized as follows. In section \ref{LinearPotential} we study the dynamics of a linear potential in an expanding Universe. We solve numerically and analytically the 3 relevant stages: matter domination, accelerated expansion and the collapse. Then, assuming that the linear potential drives the present stage of acceleration we derive relations between the equation of state and the time until the collapse. In section \ref{Conclusion} we conclude.
We assume a FRW metric in cosmic time ($t$) with a small positive curvature ($k$) which will not play any important role in the dynamics.
\section{Linear Potential and Dark Energy} \label{LinearPotential}
\subsection{Dynamics}
We consider a scalar field $\phi$ with a linear potential
\begin{equation} \label{4}
V(\phi) = m^3 \phi,
\end{equation}
where $m$ is the mass of the field. The equation of motion for a homogeneous scalar field in a linear potential is given by
\begin{equation} \label{5}
\ddot{\phi} + 3H \dot \phi +m^3=0,
\end{equation}
where $H=\dot a/a$ is the Hubble rate, $a$ is the scale factor and the dot means a derivative with respect to cosmic time.
\subsubsection{Radiation/Matter stage}
We start the analysis at a period where the Universe is dominated by some perfect fluid with equation of state $w$, hence, the scale factor evolves in time as $a=a_0 (t/t_0)^p$ where $p=2/(3(1+w))>1/3$. For such a background evolution the scalar field behaves as
\begin{equation} \label{6}
\phi(t)=C_1+ \frac{C_2 }{(1-3p) t^{3p-1}} - \frac{m^3 t^2}{2(3p+1)},
\end{equation}
where $C_1,C_2$ are two integration constants. For reasonable initial conditions one expects the second term in the right hand side to be diluted and so, at sufficient late times, we effectively have
\begin{equation} \label{7}
\phi = \phi_i - \frac{m^3 t^2}{2(3p+1)}.
\end{equation}
As one would expect, if the field is light, $H=p/t \gg m$, the second term on the rhs of Eq. (\ref{7}) is small and so the field remains effectively frozen in its potential. Even for masses close to the maximal value allowed by observations the value of $\phi$ today would only differ $\simeq 17\%$ from $\phi_i$.
\subsubsection{Accelerated stage}
While the Universe expands, $H$ decreases and we reach a time at which $\phi$ dominates the energy content of the Universe: $\rho_\text{total}= 3H_\text{DE}^2 M_p^2 \simeq \rho_\phi \simeq m^3 \phi_i$. If $\phi>0$ the field drives a period of accelerated expansion. Depending on the mass of the field, this period can be well described by the slow-roll approximation which requires the slow-roll parameters $\epsilon \equiv \dot \phi^2/(2 H^2 M_p^2) $ and $\delta = \ddot \phi/(H \dot \phi)$ to be $\ll1$ (in absolute value). The slow-roll condition for $\epsilon$ requires:
\begin{eqnarray} \label{8}
\left( \frac{ m^3 t_\text{DE}}{3p+1} \right)^2 \frac{1}{ 2 H_\text{DE}^2 M_p^2} \ll 1,
\end{eqnarray}
where $M_p$ is the reduced Planck mass. On the other hand, the condition for $\delta$ is
\begin{eqnarray}
(H_\text{DE} t_\text{DE})^{-1} \ll 1.
\end{eqnarray}
If these conditions are satisfied then the dynamics of $\phi$ is no longer given by Eq. (\ref{7}). Instead, using the slow-roll approximation to neglect the second derivative of $\phi$ in Eq. (\ref{5}) and assuming $\rho_\text{total}= m^3 \phi_i$, we have
\begin{equation} \label{9}
\phi (t)\simeq \left[ \phi_i^{3/2} - \frac{\sqrt{3m^3}}{2} (t-t_\text{DE}) M_p \right]^{2/3}.
\end{equation}
The accelerated period will end at a time $ t_s$ when the slow-roll conditions are violated $\epsilon (t_s) \simeq 1$. Using Eq. (\ref{9}) we can estimate the duration of the accelerated stage to be
\begin{eqnarray} \label{10}
t_s-t_\text{DE}= \frac{2}{\sqrt{3m^3} M_p} \left[ \phi_i^{3/2} - \left( \frac{M_p}{\sqrt{2}} \right)^{3/2} \right].
\end{eqnarray}
Note that this analysis is only valid if $\phi_i \gtrsim M_p /\sqrt{2}$. This can be seen as a consistency condition to have slow-roll in the first place.
For times close to the present time ($t_0$) then $H^{-1} \simeq t$ and the first slow-roll condition ($\epsilon$) requires $m^3 \ll H^2_\text{DE} M_p$. On the other hand the second slow-roll parameter ($\delta$) is never $\ll1$ around the present epoch. Therefore, the dynamics is only accurately described by slow-roll at a time $t_\text{DE} >t_0$. In particular, for the largest masses allowed by the observations, the slow-roll condition for $\delta$ is never strictly satisfied, even at late times. Nevertheless, we will assume the slow-roll to be is valid as long as $\epsilon, \delta <1$ (which is typically satisfied even before today). Naturally, this assumption will insert a mild error in the analytical results when comparing to the numerical results. In particular, the fact that the slow-roll conditions are not strictly satisfied today implies an error in estimating $\dot \phi (t_0)$ which, as we will explain in more detail later, affects the computation of the equation of state $w(t_0)$.
The initial condition for $\phi$ is not a free parameter if we want to explain the present period of acceleration. In that case\footnote{To be more precise $\phi_i m^3 + \dot \phi^2/2= \Omega_\text{DE} \rho_0$. However, in the cases allowed by observations this should give a small correction.}
\begin{equation} \label{8.2}
\phi_i m^3 = \Omega_\text{DE} \rho_0,
\end{equation}
where $\rho_0=3 H_0^2 M_p^2$ is the energy density today and $\Omega_\text{DE}\simeq 0.69 $ \cite{Ade:2015xua} is the dark energy density parameter today.
In Fig. (\ref{fig1}) we show the dynamics of the Universe described in the previous paragraphs containing only matter and a scalar field ($\phi$). We fixed $\phi_i=\phi(t_0)$ through Eq. (\ref{8.2}) and choose $m^3= 0.538\, H_0^2 M_p$ ($w+1=0.01$). The value of $\dot \phi$ is fixed by assuming that Eq. (\ref{7}) is satisfied well inside the matter dominated era.
We can verify that the analytical solution for $\phi$ in Eq. (\ref{9}) is accurate until the time the slow-roll conditions are violated. Moreover, for such a value of the mass Eq. (\ref{10}) predicts the duration of the dark energy stage to be $t_s-t_\text{DE} \simeq 11.5 \,H_0^{-1}$ which is in reasonable agreement with the numerical result of $12.9 \, t_0$. The difference corresponds to small deviations between $\phi_i$ and $\phi(t_\text{DE})$, and the fact that our conditions for slow-roll requirements are not exact. We verified that the error decreases for small masses while for the largest masses compatible with observations ($w+1\simeq0.1$), where the slow-roll is never strictly achieved, the error is $\simeq 50\%$.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.5]{PaperFig1.pdf}
\caption{\label{fig1} The total energy density of the Universe (black), the energy density in matter (blue) and the energy density in the field $\phi$ (red, dashed is analytical) from an early time until the collapse. We choose $m^3= 0.538 \,H_0^2 M_p$ ($w=0.01$), $ \Omega_\text{DE}=0.69$ and no curvature. The energy densities are normalized by $\rho_0$ and the times by $t_0$. In green and purple is plotted the slow-roll parameters $\epsilon$ and $\delta$, respectively.}
\par\end{centering}
\end{figure}
\subsubsection{ Bounce and collapse}
At the time the slow-roll conditions are violated the field starts rolling down the potential until it reaches a point where $\phi<0$ and $m^3 \phi + \dot \phi ^2 /2 =0$. Thus, the total energy of the Universe goes to zero ($H \rightarrow 0$) and a bounce occurs. The influence of other components like matter, radiation or curvature is, at this point, very much diluted and so it will no interfere in the dynamics.
During this particular time window, since the slow-roll condition is violated ($t_s$) until the bounce ($t_b$), it is difficult to have an analytical understanding of the dynamics. However, we can still get an analytical insight by noting that the kinetic $K(\phi)$ energy of $\phi$ is the dominant contribution for most of the time. Therefore, the equation of motion for $\phi$ will be approximately given by
\begin{eqnarray} \label{11}
\ddot{\phi} + \sqrt{ \frac{3}{2}} \frac{ \left| \dot \phi \right| \dot \phi}{ M_p} +m^3=0.
\end{eqnarray}
As $\dot \phi <0$ the solution for this equation, after matching with Eq. (\ref{9}) at $t_s$, yields \small
\begin{eqnarray} \label{11b}
\frac{\phi(t)}{M_p}&=& \frac{1}{\sqrt{2}}+ \frac{\log \left(\frac 1 2(3+\sqrt{3}) \right) }{\sqrt{6}} -\sqrt{\frac{2}{3}} b M_p\times \nonumber \\
&& \times \log \left[ \cosh \left[ b(t - t_s)+\text{arctanh} \left(3^{-1/4} \right) \right] \right]
\end{eqnarray} \normalsize
where $b = \left( 3/2 \right)^{1/4} \sqrt{m^3/M_p }$, while $\dot \phi$ is given by
\begin{eqnarray} \label{12b}
\dot \phi(t)= - \sqrt{\frac{2}{3}} b M_p \tanh \left[ b(t - t_s)+\text{arccoth} \left(3^{1/4} \right) \right].
\end{eqnarray}
The previous equation saturates before the bounce to $\dot \phi(t)\simeq- \sqrt{2/3} \, b M_p$. The bounce itself will happen when $m^3 \phi + \dot \phi ^2 /2 =0$. Using the saturated value for $\dot \phi$ we get $\phi (t_b) \simeq -M_p/\sqrt{6}$. Then, by inserting this result into Eq. (\ref{11b}) we find the time between $t_s$ and $t_b$ to be
\begin{equation} \label{12c}
t_b-t_s\simeq 1.34 \sqrt{\frac{M_p}{m^3}}.
\end{equation}
When $H=0$ the acceleration is negative so the Universe starts to collapse dominated again by the kinetic energy of $\phi$, in order to keep $\rho_\text{total}>0$, while the field continues to roll down the potential. Thus, for $t > t_b$ the equation of motion for $\phi$ becomes
\begin{equation} \label{11c}
\ddot{\phi} - \sqrt{ \frac{3}{2}} \frac{ \dot \phi |\dot \phi|}{ M_p} +m^3=0,
\end{equation}
which is similar to Eq. (\ref{11}) apart from the fact that $H<0$ during the collapse. As $|\dot \phi|= -\dot \phi$ the solution for $\dot \phi$ is now given by
\begin{equation} \label{12}
\dot \phi(t)= - \sqrt{\frac{2}{3}} b M_p \tan \left[ b(t - t_f)+\frac{\pi}{2} \right],
\end{equation}
where $t_f$ is an integration constant. Note that the solution is only valid for $0<b(t_f-t)<\pi/2$. When $t - t_f= 0$, $\dot \phi$ diverges. This divergence corresponds to the collapse of the Universe as we can verify by solving the Friedmann equation for the scale factor
\begin{equation} \label{13a}
a(t)= C_4 \left[ \cos \left(b (t - t_f)+\frac{\pi}{2} \right)\right]^{1/3},
\end{equation}
where $C_4$ is another integration constant.
When $t - t_f \rightarrow 0$ the scale factor tends to zero. Therefore, we can identify $t_f$ with the total age of the Universe. Close to $t_f$ the solutions can be expanded to leading order in $t_f-t$ giving
\begin{equation} \label{17}
a(t)= a_b \left( \frac{ t_f -t}{t_f-t_b} \right)^{1/3},
\end{equation}
where we have fixed $C_4 = a_b /(b (t_f-t_b))^{1/3}$, and
\begin{equation} \label{17b}
\dot \phi = -\sqrt{ \frac 2 3} \frac{M_p}{ t_f-t}.
\end{equation}
These results are in agreement with what we would expect from a Universe dominated by a kinetic stage ($w\simeq1$) \cite{Avelino:2004vy}.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.5]{PaperFig2.pdf}
\caption{\label{fig2}The total energy density of the Universe (black) and the kinetic $K(\phi)$ (red) and potential $V(\phi)$ (blue) energy of $\phi$ for $m^3= 0.538 \, H_0^2 M_p$ ($w=0.01$). The blue dashed line uses the analytical approximation for $\phi$ before the bounce Eq. (\ref{11b}) while the the red dashed line uses the analytical solution for $\dot \phi$ during the collapse Eq. (\ref{17b}). The energy densities are normalized by $\rho_0$.}
\par\end{centering}
\end{figure}
The final relevant quantity left to determine is the total age of the Universe $t_f$. In order to compute it we match Eqs. (\ref{12b}) and (\ref{17b}) at the bounce. Using the saturated value for $\dot \phi$ in Eq. (\ref{12b}) we simply get
\begin{equation} \label{19c}
t_f-t_b\simeq b^{-1}= \left(\sqrt{\frac{2}{3}} \frac{M_p}{m^3} \right)^{1/2}.
\end{equation}
In sum, both time windows, from the end of the dark energy stage to the bounce and from the bounce to the collapse are ${\cal O} (\sqrt{M_p/ m^{3}})$.
In Fig. (\ref{fig2}) we present the dynamics of the Universe, with the same conditions as in Fig. (\ref{fig1}), zoomed in the region from the end of the dark energy stage ($t_s$) to the complete collapse ($t_f$), where the energy density diverges. For the set of parameters chosen, the slow-roll conditions are violated at $t_s\simeq13.64\, t_0$, the bounce occurs at $t_b\simeq15.82\, t_0$ while the collapse happens at $t_f \simeq 17.07\, t_0$. In the region before the bounce we can verify that Eq. (\ref{11b}) is a reasonable approximation for $\phi$ by comparing the blue lines. Our expression is fully analytical and therefore uses the analytical approximations for $\phi$ and $\dot \phi$ computed at the end of slow-roll. That introduces a small error in this approximation. In Eq. (\ref{12c}) we estimated the time window from $t_s$ to $t_b$ as a function of the mass of the field. For the mass used here it yields $t_b-t_s= 1.91\, t_0$ which is close to the numerical value of $2.17 \, t_0$.
Regarding the collapsing stage, the total energy in the collapse is basically given by the kinetic energy of $\phi$, as expected. Furthermore, we can verify that the solution for $\dot \phi$ obtained in Eq. (\ref{17b}) is a good approximation during the collapse. Using Eq. (\ref{19c}) we estimate $t_f-t_b= 1.29\, t_0$ which is in very good agreement with the numerical value of $1.25 \, t_0$.
We verified for other values of the mass that our analytical estimations for $t_f-t_b$ is accurate with an error $< 3\%$ while the estimation of $t_b-t_s$ is also good with an error $< 15\%$. Moreover, we have also checked that the addition of curvature consistent with observations does not affect the dynamics.
\subsection{Predictions}
The previous analysis allowed us to derived relations between the mass of the field, the duration of the accelerated stage, the bounce and the collapse. Recall that we assumed $t_\text{DE}$ to be the time at which the slow-roll parameters are $<1$ and so $t_\text{DE} <t_0$. For the range of masses analyzed $t_\text{DE} \simeq 0.72 \,t_0$. Therefore, using Eqs. (\ref{10}), (\ref{12c}) and (\ref{19c}) we can write the total age of the Universe only as a function of the mass of the field as
\begin{equation} \label{20c}
t_f \simeq \frac{6 \Omega_\text{DE}^{3/2} H_0^{3} M_p^2}{m^6} + 1.56 \sqrt{\frac{M_p}{m^3}}+0.72 t_0.
\end{equation}
It is now interesting and useful to rewrite these relations in terms of the observed equation of state parameter $w$. Recall that the equation of state is measured by observations to be very close to a pure CC: $w=-1.019^{+0.075}_{-0.080}$ (95\% CL) \cite{Ade:2015xua}. In order to make predictions of what a given value of $w$ would tell us about the fate of the Universe we use the fact that during the dark energy stage
\begin{equation}
w=\frac{ K(\phi)-V(\phi)}{K(\phi) +V(\phi)} \simeq -1 + \frac{2 K(\phi)}{V(\phi)}.
\end{equation}
Therefore, using the slow-roll conditions, the equation of state today is given by
\begin{equation} \label{22c}
w+1 = \frac{\Omega_\text{DE} M_p^2}{3 \phi_i^2} = \frac{m^6 M^2_p}{3 \Omega_\text{DE} \rho^2_0},
\end{equation}
where we have used Eqs. (\ref{9}) and (\ref{8.2}). However, as we explained in the previous section, the present epoch is a transient regime where slow-roll is not a very accurate approximation. Both Eq. (\ref{7}) and Eq. (\ref{9}) overestimate $\dot \phi (t_0)$. A possible improvement would be to solve the equations assuming a Universe dominated by CC and matter until today and then solve the equation for the scalar field in such background. We take a more simplistic approach by noting that the previous expression overestimates the value of $w+1$ by a factor of $\gamma \simeq 1.5$. This is true for a wide range of $w+1$, which again suggests that an analytical improvement might be easily achieved. In what follows we take this factor $\gamma$ into consideration. Therefore, combining Eqs. (\ref{20c}) and (\ref{22c}) gives
\begin{equation} \label{w1}
t_f= \frac{\sqrt{\Omega_\text{DE}}}{H_0 (w+1) \gamma} \left[ \frac{2}{9 } + \frac{0.68 }{ \Omega_\text{DE}^{5/4}} \gamma^{3/4} (w+1)^{3/4} \right] +0.72 t_0.
\end{equation}
For $w+1 \ll 1$ the age of the Universe is inversely proportional to $w+1$. The present observational constraint of $w+1\lesssim {\cal O}(0.1)$ implies that the collapse of the Universe, an so the {\it doomsday}, will not happen in the next 56 billion years. The previous equation underestimates the age of the Universe for large masses with a maximal error of 16$\%$ which decreases for small masses, as explained in the previous section.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.5]{PaperFig5.pdf}
\caption{\label{fig5} The coincidence problem, defined as the ratio between the total age of the Universe over the time the Universe has been accelerating so far, as a function of the equation of state today. The blue line is the analytical result while the black is the numerical.}
\par\end{centering}
\end{figure}
We are now in position to properly analyze the coincidence problem in the context of the linear potential driving dark energy. The fact that the linear potential triggers a collapse of the Universe ameliorates per se the coincidence problem as the Universe is no longer eternal. We can quantify this improvement by defining the ratio between the time elapsed since the Universe started accelerating ($t_\text{acel}$) until today, with the total age of the Universe
\begin{equation}
f=\frac{t_0-t_\text{acel}}{t_f}.
\end{equation}
In the range of $w+1$ considered $t_0-t_\text{acel} \simeq 0.45 \,t_0$ although the precise value varies with the mass of the field. In Fig. (\ref{fig5}) we present this quantity evaluated both analytically, through Eq. (\ref{w1}), and numerically. The numerical results are in agreement with those ones obtained in \cite{Avelino:2004vy}. Our numerical expression is very accurate for small masses although it underestimates the age of the Universe for larger masses. In the context of the coincidence problem these results show that a valid solution requires a dynamical field with an equation of state not to far from the current bounds and, therefore, possibly observed with the forthcoming experiments. In particular, if $w+1$ is close to the maximal value allowed by observations the coincidence would only be at the level of $1/10$ while $w+1\simeq0.01$ would represent a $1/30$ level of coincidence.
\section{Conclusion} \label{Conclusion}
In this work we studied in detail the fate of the Universe assuming that the current accelerated stage is driven by a scalar field in a linear potential. The case of a linear potential is of great interest because of its predictive dynamics, quite insensitive to the initial conditions, but also because a small mass of the field can be made technically natural. Generically, a scalar field in a linear potential and in an expanding Universe will, at some point, dominate the energy density and drive a period of dark energy followed by a bounce and, finally, a crunch. We have derived analytical expressions for the scalar field and the scale factor during these epochs. This allowed us to estimate the time duration of the several stages as a function of the mass of the field. The analytical expressions are in good agreement with the numerical results, in particular, the collapsing time estimation in Eq. (\ref{19c}). The accuracy of our expressions increases for smaller masses while for masses close to the largest values allowed by observations there is a larger error because the slow-roll dynamics is no longer the best description for the scalar field. From the expressions derived in the three relevant stages we derived our main result, Eq. (\ref{w1}), where we predict the age of the universe for a given observed value of the equation of state $w$. Our expression is accurate for smaller masses although it underestimates the age of the Universe for the largest masses compatible with observations. The present observational constraints of $w+1 \lesssim 0.1$ enforces that the {\it doomsday} does not happen in the next $56$ billion years. Finally, we also compute how the linear potential addresses the coincidence problem by computing the ratio between the time the Universe was accelerating and its total age (see Fig. (\ref{fig5})). We find a $1/10$ level of coincidence for $w+1\simeq0.1$, which is very reasonable, and $1/30$ for $w+1\simeq0.01$, which although acceptable is border line to what we would consider a solution to the coincidence problem. Therefore, a solution to the coincidence problem prefers a more dynamical dark energy stage with a value of $w+1$ possibly observed by future experiments.
\bibliographystyle{apsrev4-1}
|
2,877,628,090,851 | arxiv | \section*{INTRODUCTION}
The denomination ``sigma term'' stands, in a generic way, for the
contribution of the quark masses $m_q$ to the mass $M_h$ of a hadronic
state $\vert h(p)>$.
According to the Feynman-Hellmann theorem~ \cite{FeynHell}, one has the exact
result (the notation does not explicitly take into account the spin degrees
of freedom)
\begin{equation}
\frac{\partial M_h^2}{\partial m_q}\,=\,<h(p)\vert ({\overline q}q)(0)
\vert h(p)>.
\end{equation}
In practice, and in the case of the light quark flavours $q=u,d,s$, one
tries to perform a chiral expansion of the matrix element of the scalar
density
appearing on the right-hand side of this formula. In the case of the pion,
for instance, one may use soft-pion techniques to obtain the well-known
result~ \cite{GOR} (here and in what follows, ${\cal O}(M^n)$ stands for
corrections of order $M^n$ modulo powers of $\ln M$)
\begin{equation}\label{gmor}
\frac{\partial M_\pi^2}{\partial m_q}\,=\,-\frac{<{\overline q}q>_0}{F_0^2}
+{\cal O}(m_u,m_d,m_s)\,,
q=u,d,\ {\mbox{and}}\ \frac{\partial M_\pi^2}{\partial m_s}\,=\,
0+{\cal O}(m_u,m_d,m_s),
\end{equation}
where $<{\overline q}q>_0$ denotes the single flavour light-quark condensate
in the $SU(3)_L\times SU(3)_R$ chiral limit, while $F_0$ stands for the
corresponding value of the pion decay constant $F_\pi=92.4$ MeV.
In the case of the nucleon, the sigma term is defined in an analogous way, as
the value at zero momentum transfer $\sigma\equiv\sigma (t=0)$ of the scalar
form factor of the nucleon ($t=(p'-p)^2$, ${\hat m}\equiv (m_u+m_d)/2$),
\begin{equation}\label{defsig}
{\overline {\mbox{u}}}_N(p'){\mbox{u}}_N(p)\sigma (t) \,=\,
\frac{1}{2M_N}<N(p')\vert {\hat m}({\overline u}u +
{\overline d}d)(0) \vert N(p)>,
\end{equation}
and contains, in principle, information on the quark mass dependence of the
nucleon mass $M_N$. Most theoretical evaluations of the nucleon sigma term
consider the isospin symmetric limit $m_u=m_d$, but this is not required by
the definition (\ref{defsig}).
Another quantity of particular interest in this context is the
relative amount of the nucleon mass contributed by the strange quarks of the
sea,
\begin{equation}
y\,\equiv\,2\frac{<N(p)\vert ({\overline s}s)(0)\vert N(p)>}
{<N(p)\vert ({\overline u}u + {\overline d}d)(0) \vert N(p)>}.
\end{equation}
Large-$N_c$ considerations (Zweig rule) would lead one to expect that
$y$ is small, not exceeding $\sim 30\%$. The ratio $y$ can be related,
{\it via} the sigma term and the strange to non-strange quark mass ratio,
to the nucleon matrix element of the $SU(3)_V$ breaking part of the strong
hamiltonian,
\begin{equation}
\sigma (1-y)\left(\frac{m_s}{\hat m}-1\right)\,=\,
\frac{1}{2M_N}<N(p')\vert (m_s-{\hat m})({\overline u}u +
{\overline d}d - 2{\overline s}s)(0) \vert N(p)>.
\end{equation}
For the standard scenario of a strong $<{\overline q}q>_0$ condensate,
$m_s/{\hat m}\sim 25$,
the evaluation of the product $\sigma (1-y)$ in the chiral expansion gives
$\sim 26$ MeV
at order ${\cal O}(m_q)$~ \cite{GLmasses}, $\sim 35\pm 5$ MeV at order
${\cal O}(m_q^{3/2})$~ \cite{GLmasses,gasser81}, and $\sim 36\pm 7$ MeV at
order ${\cal O}(m_q^2)$~ \cite{borasoy}.
\section*{THE NUCLEON SIGMA TERM AND $\pi N$ SCATTERING}
Although the nucleon sigma term is a well-defined QCD
observable, there is, unfortunately, no direct experimental access to it.
A link with the $\pi N$ cross section (for the notation, we refer the reader
to Refs.~ \cite{HoehlerLB,GLLS}) at the unphysical Cheng-Dashen point,
$\Sigma \equiv F_\pi^2{\overline D}^+(\nu=0,t=2M_\pi^2)$, is furnished by a
very old low-energy theorem~ \cite{cheng},
\begin{equation}
\Sigma = \sigma \big(1+{\cal O}(m_q^{1/2})\big).
\end{equation}
A more refined version of this statement~ \cite{brown} relates $\Sigma$ and
the form factor $\sigma (t)$ at $t=2M_\pi^2$,
\begin{equation}
\Sigma = \sigma(2M_\pi^2) + \Delta_R ,
\end{equation}
where $\Delta_R = {\cal O}(m_q^2)$. The size of the correction $\Delta_R$,
as estimated within the framework of Heavy Baryon Chiral Perturbation
Theory (HBChPT), is small~ \cite{bernard1}, $\Delta_R < 2$ MeV (an earlier
calculation to one-loop in the relativistic approach~ \cite{GSS} gave
$\Delta_R = 0.35$ MeV).
In order to obtain information on $\sigma$ itself, one thus needs to pin
down the difference $\Delta_\sigma\equiv\sigma(2M_\pi^2)-\sigma(0)$, and to
perform an extrapolation of the $\pi N$ scattering data from the physical
region $t\leq 0$ to the Cheng-Dashen point, using the existing experimental
information and dispersion relations.
The analysis of Refs.~ \cite{GLS1,GLS2}, using a
dispersive representation of the scalar form factor of the pion, gives the
result $\Delta_\sigma = 15.2\pm 0.4$ MeV. On the other hand, from the
subthreshold expansion
\begin{equation}\label{threxp}
{\overline D}^+(\nu=0,t) = d^+_{00} + td^+_{01} + \cdots
\end{equation}
one obtains $\Sigma = \Sigma_d + \Delta_D$, with $\Sigma_d =
F_\pi^2(d^+_{00} + 2M_\pi^2d^+_{01})$, and $\Delta_D$ is the remainder,
which contains the contributions from the higher order terms in the
expansion (\ref{threxp}). In Ref.~ \cite{GLS2}, the value
$\Delta_D = 11.9\pm 0.6$ MeV was obtained, so that the determination of
$\sigma$ boils down to the evaluation of the subthreshold parameters
$d^+_{00}$ and $d^+_{01}$.
Their values can in principle be obtained from experimental data on $\pi N$
scattering, using forward dispersion relations~ \cite{HoehlerLB,GLLS}
\begin{equation}
d^+_{00} = {\overline D}^+(0,0) = {\overline D}^+(M_\pi,0) + {\cal J}_D(0),
\ d^+_{11} = {\overline E}^+(0,0) = {\overline E}^+(M_\pi,0) + {\cal J}_E(0),
\end{equation}
where ${\cal J}_D(0)$ and ${\cal J}_E(0)$ stand for the corresponding
forward dispersive integrals, while the subtraction constants are expressed
in terms of the $\pi N$ coupling constant $g_{\pi N}$ and of the S- and
P-wave scattering lengths as follows:
\begin{equation}\label{subtract}
{\overline D}^+(M_\pi,0) = 4\pi(1+x)a^+_{0+} +
\frac{g_{\pi N}^2x^3}{M_\pi(4-x^2)},
\ {\overline E}^+(M_\pi,0) = 6\pi(1+x)a^+_{1+} -
\frac{g_{\pi N}^2x^2}{M_\pi(2-x)^2}.
\end{equation}
The dispersive integrals ${\cal J}_D(0)$ and ${\cal J}_E(0)$
are evaluated using $\pi N$ scattering data, which
exist only above a certain energy, and their extrapolation to the low-energy
region using dispersive methods. In the analysis of Ref.~ \cite{GLLS},
the two scatering lengths $a^+_{0+}$
and $a^+_{1+}$ are kept as free parameters of the extrapolation procedure.
In the Karlsruhe analysis, their values were obtained from the
iterative extrapolation procedure itself~ \cite{HoehlerLB}.
Using the partial waves of~ \cite{KochPiet,HoehlerLB}, the authors of
Ref.~ \cite{GLLS}
obtain the following simple representation of $d^+_{00}$ and $d^+_{01}$ (with
$a^+_{l+}$, $l=0,1$, in units of $M_\pi^{-1-2l}$),
\begin{eqnarray}\label{dparam}
d^+_{00} &=& -1.492+14.6(a^+_{0+}+0.010)-0.4(a^+_{1+}-0.133),
\nonumber\\
d^+_{01} &=& 1.138+0.003(a^+_{0+}+0.010)+20.8(a^+_{1+}-0.133).
\end{eqnarray}
This leads then to a value $\sigma\sim 45$ MeV, corresponding to $y\sim 0.2$~
\cite{GLS1}.
Further details of this analysis can be found in Refs.~ \cite{mikko1,mikko2}.
\section*{THEORETICAL ASPECTS}
In the framework of chiral perturbation theory, the sigma term has an
expansion of the form
\begin{equation}\label{sigexp}
\sigma \sim \sum_{n\ge 1}\sigma_n M_\pi^{n+1}.
\end{equation}
The first two terms of this expansion were computed in the framework of the
non-relativistic HBChPT in Ref.~ \cite{bernard2},
\begin{equation}
\sigma_1\,=\,-4c_1\,,\ \sigma_2\,=\,-\frac{9g_A^2}{64\pi F_\pi^2}.
\end{equation}
The determination of the low-energy constant $c_1$, which appears also in
the chiral expansion of the $\pi N$ scattering amplitude, is crucial for the
evaluation of $\sigma$. Earlier attempts, which extracted the value of $c_1$
from fits to the $\pi N$ amplitude extrapolated to the threshold
region using the phase-shifts of Refs.~ \cite{KochPiet,HoehlerLB},
obtained rather
large values, $\sigma\sim 59$ MeV~ \cite{mojzis}
($c_1=-0.94\pm 0.06$ GeV$^{-1}$), or even $\sigma\sim 70$ MeV~ \cite{fettes}
($c_1=-1.23\pm 0.16$ GeV$^{-1}$), as compared to the result of
Ref.~ \cite{GLS1}.
The threshold region in the case of elastic $\pi N$ might
however correspond to energies which are already too highy in order to make these
determinations of $c_1$ stable as far as higher order chiral corrections are
concerned. A new determination of $c_1$, obtained by matching the
${\cal O}(q^3)$ HBChPT expansion of the $\pi N$ amplitude {\it inside the
Mandelstam triangle} with the dispersive extrapolation of the data leads to a
smaller value~ \cite{buettiker1,buettiker2}, $c_1 = -0.81\pm 0.15$
GeV$^{-1}$, corresponding to $\sigma\sim 40 $ MeV. It remains however to be
checked that higher order corrections do not substancially modify this result.
Let us mention in this respect that the higher order contribution $\sigma_3$
(which contains a non-analytic ${\cal O}(M_\pi^4\ln M_\pi/M_N)$ piece)
in the expansion (\ref{sigexp}) has been computed in the context of the
manifestly Lorentz-invariant baryon chiral perturbation theory in
Ref.~ \cite{becher}, (see also~ \cite{leutwyler}). Once the expression of
the $\pi N$ amplitude is also known with the same accuracy~ \cite{leutwyler},
a much better control over the chiral perturbation evaluation of $\sigma$
should be reached.
Finally, let us also mention that the results quoted above were based on
the $\pi N$ phase-shifts obtained by the Karlsruhe group~ \cite{HoehlerLB}.
Using instead the SP99 phase-shifts of the VPI/GW group, the authors of
Ref.~ \cite{buettiker1} obtain a very different result, $c_1 \sim -3$
GeV$^{-1}$, which leads to $\sigma\sim 200$ MeV. Needless to say that the
consequences of this last result ($y\sim 0.8$) would be rather difficult to
accept.
\section*{EXPERIMENTAL DEVELOPMENTS}
We next turn to the discussion of several new experimental results which have
some bearing on the value of the nucleon sigma term. All numerical values
quoted below use $M_\pi = 139.57$ MeV and $F_\pi=92.4$ MeV.
Let us start with the influence of the scattering length $a^+_{0+}$ on the
value of the subthreshold parameter $d^+_{00}$, using Eq. (\ref{dparam}) and
$a^+_{1+} = 0.133 M_\pi^{-3}$.
The first line of Table 1 gives the result obtained
from the value of the phase-shift analysis of Ref.~ \cite{HoehlerLB}. In the
second line of Table 1, we show the value reported
at this conference~ \cite{leisi} and obtained from the data on pionic
hydrogen, $10^3M_\pi\times a^+_{0+} = 1.6\pm 1.3$.
The analysis of Loiseau {\it et al.}~ \cite{loiseau1}
consists in extracting the combinations of scattering lengths
$a_{\pi^-p}\pm a_{\pi^-n}$ from the value of pion deuteron scattering length
$a_{\pi^- d}$ obtained from the measurement of the strong interaction width
and lifetime of the 1S level of the pionic deuterium atom~ \cite{pid1,pid2}.
Assuming charge exchange symmetry ($a_{\pi+p} = a_{\pi^-n}$), they find
$10^3M_\pi\times a^+_{0+} = -2\pm 1$ (third line of Table 1).
Another determination of
$a^+_{0+}$ is also possible using the GMO sum rule (we use here the form
presented in~ \cite{loiseau1}, with the value of the total cross section
dispersive integral $J^-=-1.083(25)$, expressed in mb and $a_{\pi^- p}$,
$a^+_{0+}$ expressed in units of $M_\pi^{-1}$)
\begin{equation}\label{gmo}
g_{\pi N}^2/4\pi = -4.50\,J^- + 103.3\,a_{\pi^- p} -103.3\,a^+_{0+}.
\end{equation}
Using the value $a_{\pi^- p} = 0.0883\pm 0.0008$ obtained by~ \cite{loiseau1}
and the determination $g_{\pi N}= 13.51\pm 12$ from the Uppsala charge
exchange $np$ scattering data~ \cite{loiseau2}, one obtains
$a^+_{0+}=-0.005\pm 0.003$.
The resulting effect on $\Sigma_d$ is
shown on the fourth line of Table 1.
\begin{table}[htb]
\caption{$d^+_{00}$ for different values of the scattering length $a^+_{0+}$.}
\begin{center}
\begin{tabular}{c|c|c|c}
\hline\hline
& $a^+_{0+}\times 10^3M_\pi$ & $F_\pi^2d^+_{00}$ (MeV) & $\Delta\Sigma_d$ (MeV)
\\ \hline
KH \cite{HoehlerLB} & $ -9.7$ & $ -91.0$ & $0$
\\ \hline
$A_{\pi^-p}$\cite{leisi} & $+2\pm 1$ & $-80\pm 1$ & $+11$
\\ \hline
$A_{\pi^-d}$\cite{loiseau1} & $-2\pm 1$ & $-84\pm 1$ & $ +7$
\\ \hline
$g_{\pi N}$\cite{loiseau2}+GMO & $-5\pm 3$ & $-87\pm 3$ & $ +4$
\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
Several new determinations of the $\pi N$ coupling constant $g_{\pi N}$ have
also been reported at this meeting, with values which differ from the
``canonical'' value obtained long ago~ \cite{HoehlerLB}. Since most of these
recent determinations do not result from a complete partial-wave analysis of
$\pi - N$ scattering data, we can only compare the effect of variations in
the value of $g_{\pi N}$ on the subtraction terms (\ref{subtract}).
The results are shown in Tables 2 and 3, respectively. Again, we take the value of~ \cite{HoehlerLB} as reference point, and show the resulting changes
for the value $g_{\pi N}= 13.73\pm 0.07$ from the
latest VPI/GW analysis~ \cite{pavan2}. For comparison, we have also included
the determination of~
\cite{loiseau1}, using the published data on the $\pi^-d$ atom~
\cite{pid2} combined with the GMO sum rule~ (\ref{gmo}), as well as the value
determined from the Uppsala charge exchange $np$ scattering data~
\cite{loiseau2}. The repercussion on ${\overline D}^+(M_\pi,0)$ is
negligible in all cases shown in
Table 2, whereas in the case of ${\overline E}^+(M_\pi,0)$, the largest
effect comes from the rather low value of $g_{\pi N}$ obtained by the VPI/GW
analysis.
\begin{table}[htb]
\caption{The subtraction constant ${\overline D}^+(M_\pi,0)$ of
Eq. (\protect\ref{subtract}) for different values of the $\pi N$ coupling
constant, and for fixed value of the scattering length
$a^+_{0+}\times 10^3M_\pi = -9.7$.}
\begin{center}
\begin{tabular}{c|c|c|c}
\hline\hline
& $g^2_{\pi N}/4\pi$ & $F_\pi^2{\overline D}^+(M_\pi,0)$ (MeV) & $\Delta\Sigma_d$ (MeV)
\\ \hline
KH & $ 14.3\pm 0.2$ & $ 0.53$ & $0$
\\ \hline
VPI/GW\cite{pavan2} & $13.73\pm 0.07$ & $0.16$ & $-0.37$
\\ \hline
$A_{\pi^-d}$+GMO\cite{loiseau1} & $14.2\pm 0.2$ & $0.46$ & $-0.07$
\\ \hline
Uppsala\cite{loiseau2} & $14.52\pm 0.26$ & $0.67$ & $+0.14$
\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htb]
\caption{The subtraction constant ${\overline E}^+(M_\pi,0)$ of
Eq. (\protect\ref{subtract}) for different values of the $\pi N$ coupling
constant, and for fixed value of the scattering length
$a^+_{1+}\times 10^3M_\pi^3 = 133$.}
\begin{center}
\begin{tabular}{c|c|c|c}
\hline\hline
& $g^2_{\pi N}/4\pi$ & $F_\pi^2M_\pi^2{\overline E}^+(M_\pi,0)$ (MeV) &$\Delta\Sigma_d$ (MeV)
\\ \hline
KH & $ 14.3\pm 0.2$ & $ 105$ & $0$
\\ \hline
VPI/GW\cite{pavan2} & $13.73\pm 0.07$ & $108$ & $+6$
\\ \hline
$A_{\pi^-d}+GMO\cite{loiseau1}$ & $14.2\pm 0.2$ & $105$ & $+1$
\\ \hline
Uppsala\cite{loiseau2} & $14.52\pm 0.26$ & $104$ & $-2$
\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
Finally, we have summarized the various results in Table 4, where now the
complete results for the determination of the dispersive integrals
${\cal J}_D$ and ${\cal J}_E$ have beem included where possible, {\it i.e.}
in the case of the KH~ \cite{HoehlerLB,GLLS} and of the VPI/GW~
\cite{pavan1,pavan2} analyses (see also Table 1 in~ \cite{pavan1}).
The corresponding values of $\Sigma_d$ are
given in the last column of Table 4. The analysis of the VPI/GW group
increases the value of the sigma term by more than 25\%, as compared to the
value extracted from the KH phase-shift analysis. This would lead to a value
of $y\sim 0.5$, which is rather difficult to understand
theoretically. It should also be noticed that this large difference is due
for a large part to the value $d^+_{01}=(1.27\pm 0.03)M_\pi^{-3}$ (including
a shift in the value of the scattering length $a^+_{1+}$, which by itself
accounts for half of the difference between KH and VPI/GW in the $d^+_{01}$
contribution in Table 4) as quoted
by the VPI/GW group and obtained from fixed-$t$ dispersion relation. A
similar analysis, but based on so-called interior dispersion relation
(see for instance~ \cite{hoehler1} and references therein), yields
a much smaller value, $d^+_{01}=1.18M_\pi^{-3}$~ \cite{stahov}, which
{\it lowers} the VPI/GW value of $\Sigma_d$ in Table 4 by 10 MeV. It remains
therefore difficult to assess the size
of the error bars that should be assigned to the numbers given above. Also,
the VPI/GW phase-shifts have sometimes been criticized as far as the
implementation of theoretical constraints (analyticity properties) is
concerned (see for instance~ \cite{hoehler2}). Furthermore, the issue of
having a coherent $\pi N$ data base remains a crucial aspect of the problem.
The VPI/GW
partial wave analyses include data posterior to the analyses of the Karlsruhe
group, but which are not always mutually consistent (see {\it e.g.}~
\cite{mikko2} and references therein).
Hopefully, new experiments (see~ \cite{smith}),
will help in solving the existing discrepancies.
\begin{table}[ht]
\caption{Comparison of the values of the subthreshold parameters $d^+_{00}$ and
$d^+_{01}$ according to differences in the input discussed in the text.}
\begin{center}
\begin{tabular}{c|c|c|c|c}
\hline\hline
& $F_\pi^2d^+_{00}$ (MeV) &$2M_\pi^2F_\pi^2d^+_{01}$ (MeV) & $\Sigma_d$ (MeV)
\\ \hline
KH & -89.4 & 139.2 & 50
\\ \hline
VIP/GW\cite{pavan1} & -77.3 & 155.2 & 50 +12+16
\\ \hline
$A_{\pi^-d}+GMO$\cite{loiseau1} & -83 & $-$ & 50+6
\\ \hline
Uppsala\cite{loiseau2} & -86 & $-$ & 50+3.5
\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
Finally, it should be stressed that the above discussion is by no means
a substitute for a more elaborate analysis, along the lines of
Ref.~ \cite{GLLS},
for instance (see also~ \cite{hoehler1} and~ \cite{pavan1}).
Such a task would have been far beyond the competences of the present
author, at least within a reasonable amount of time and of work.
Nevertheless, very useful discussions with G. H\"ohler, M. Pavan,
M. Sainio and J. Stahov greatly improved the author's understanding of
this delicate subject. The author also thanks R. Badertscher and the
organizing committee for this very pleasant and lively meeting in Zuoz.
\bibliographystyle{unsrt}
|
2,877,628,090,852 | arxiv | \section{Charged hard-core interacting deterministic lattice gas}
We consider a deterministic lattice gas of charged particles with three local states $\mathcal{Q} = \{\emptyset, +, -\}$, with the corresponding charges $q \in \{0, +1, -1\}$ (which we identify with the states from $\mathcal{Q}$), moving on a light-cone lattice. The particles move to the right/left with velocities $\pm 1$ respectively. Each vertex of the lattice corresponds to a hard-core scattering of two particles according to a deterministic local rule
\begin{equation}
(\emptyset, \emptyset) \leftrightarrow (\emptyset, \emptyset), \qquad (\emptyset, q) \leftrightarrow (q,\emptyset), \qquad (q,q') \leftrightarrow (q,q'), \qquad \textrm{for}\quad q,q' \in \{+, -\}. \label{two_body_scattering}
\end{equation}
Time evolution of a particle with charge $q_\ell^t$ on a space-time lattice $(\ell, t) \in \mathbb{Z}^2$ is generated by staggered applications of the map elementary two-body map $\Phi: \mathcal Q \times \mathcal Q \rightarrow \mathcal Q \times \mathcal Q$ encoding the local update rule (\ref{two_body_scattering})
(see Fig.~\ref{figHPG1} for illustration)
\begin{equation}
(q_\ell^{t+1}, q_{\ell+1}^{t+1}) = \Phi(q_\ell^t, q_{\ell+1}^t), \quad \ell+t \equiv 1 \ (\textrm{mod}\ 2).
\end{equation}
\begin{figure}
\centering
\vspace{-1mm}
\includegraphics[width=\columnwidth]{process_schema_sm.pdf}
\vspace{-1mm}
\caption{Coordinate frame (time vertical, space horizontal) of a deterministic charged hardcore lattice gas (red: $+$ particles, blue: $-$ particles, while thin black lines indicate vacancies). Example of a pyramid section of a typical trajectory, for which initial data on a saw of $4t$ subsequent links uniquely determine the transport through the mid-point (dashed line) for
all times from $0$ to $2t$.}
\label{figHPG1}
\end{figure}
\section{Exactly solved full counting statistics}
\subsection{Time integrated current}
\noindent
We shall consider the total charge which is transferred between the left and right half of the system (through the origin $\ell=0$) in time $2t$ (after $t$ full time steps)
\begin{equation}
\mathfrak J(t) = \sum_{\ell > 0} q^{2t}_\ell - \sum_{\ell>0} q^0_\ell.
\label{eq:tranQ}
\end{equation}
This is exactly equal to the the time integrated current
\begin{equation}
\mathfrak J(t) = \sum_{t'=0}^{t-1} j_0^{2t'} = \sum_{t'=0}^{2t-1} (-1)^{t'+1} q_0^{t'}, \label{J_def}
\end{equation}
where $j$ is the local current that satisfies a pair (due to even-odd staggering) of continuity relations
\begin{equation}
q_{2\ell}^{2t+2} - q_{2\ell}^{2t} + j_{2\ell+1}^{2t+1} - j_{2\ell}^{2t} = 0, \qquad q_{2\ell+1}^{2t+2} - q_{2\ell+1}^{2t} + j_{2\ell+2}^{2t} - j_{2\ell+1}^{2t+1} = 0.
\end{equation}
A valid expression for the local current is a discrete forward difference, which was used in the second equality in (\ref{J_def})
\begin{equation}
j_\ell^t = q_\ell^{t+1} - q_\ell^t.
\end{equation}
\subsection{Exact moment generating function}
Our aim is to compute the moment generating function (MGF)
\begin{equation}
G(\lambda|t) \equiv \langle e^{\lambda \mathfrak J(t)} \rangle \equiv \sum_{\mathfrak{J}} \mathcal{P}(\mathfrak{J}|t) e^{\lambda \mathfrak{J}}, \label{g_def}
\end{equation}
where $\lambda \in \mathbb{C}$ is the counting field and $\langle \bullet \rangle$ denotes the average over an invariant separable measure of initial configurations $q_\ell \equiv q_\ell^0$
\begin{equation}
\mathbb{P}(\{q_{\ell}\})=\prod_{\ell}p(q_{\ell}), \qquad p(\pm)=\rho \frac{1 \pm b}{2}, \qquad p(\emptyset)=\ol{\rho}=1-\rho, \label{measure}
\end{equation}
with $0 \leq \rho \leq 1$ the density of particles, $-1 \leq b \leq 1$ the charge bias of the particles and $\overline \rho$ the density of vacancies. For later convenience we also introduce
\begin{equation}
\varDelta^2 = \rho \overline{\rho} \in [0,1/4].
\end{equation}
The idea is now to evaluate the average (\ref{g_def}) in a {\em nested} way - an outer average over all relevant charge-less (neutral) particle configurations $\{\Sigma\}$, $\Sigma\subset\mathbb Z$,
and an inner average, with a frozen particle configuration $\Sigma$, over all combinations of
charges $\bigl\{q_\ell \in\{+1,-1\}\bigr\}_{\ell \in \Sigma}$. Let us tag a particle starting from the initial configuration and follow its worldline. Note that distinct worldlines by definition
of the dynamics cannot cross. Let $\Lambda_\pm \subset \Sigma$ denote the coordinates of initial particles, which after
$t$ time-steps (i.e., at time $2t$) end on the opposite side of the lattice, i.e. passing from the interval $[-\infty, 0]$ to $[1, \infty]$ for
$\Lambda_+$ and the opposite for $\Lambda_-$. Note
that due to non-crossing of worldlines, at most one of the subsets $\Lambda_\pm$ can be non-empty (see Fig.~\ref{fig:worldlines} for an illustration).
Considering the defining expression for the transported charge (\ref{eq:tranQ}), an unbiased averaging over the measure (\ref{measure}) given a fixed occupancy configuration $\Sigma$ yields
\begin{equation}
G(\lambda|t;\Sigma) = \prod_{\ell \in \Lambda_+} \langle e^{\lambda q_\ell} \rangle \prod_{\ell \in \Lambda_-} \langle e^{-\lambda q_\ell} \rangle = \mu_+^{|\Lambda_+|} \mu_-^{|\Lambda_-|}, \label{sig_g}
\end{equation}
where
\begin{equation}
\mu_\pm = \cosh \lambda \pm b \sinh \lambda, \label{mu_def}
\end{equation}
and
$|\Lambda_\pm| \equiv \sum_{\ell \in \Lambda_\pm} 1$.
\begin{figure}
\centering
\vspace{-1mm}
\includegraphics[width=\columnwidth]{process_schema_sm_distinct.pdf}
\vspace{-1mm}
\caption{Distinct particle worldlines within a typical many-body trajectory, depicted with two different colors. Lines crossing the origin are thickened. In this specific example $\Lambda_+ = \{-9,-8,-5,-3,-1\}$, $\Lambda_-=\emptyset$,
i.e. there are five worldlines which in total duration $2t=26$ cross the boundary -- indicated by the dashed vertical line -- from left to right sublattice.
}
\label{fig:worldlines}
\end{figure}
Note that the only way
for a worldline to shift left/right is to “meet” a right/left moving vacant site. It follows that the total signed number of worldlines crossing the origin up to time $2t$ is equal to the number of right-moving vacancies to the left of the origin at $t=0$, minus the number of left-moving vacancies to the right of the origin again at $t=0$ within the causal cone
\begin{equation}
|\Lambda_+| - |\Lambda_-| = l-r, \quad l = \sum_{\tau=1}^{t} \delta_{s_{-2\tau+1}, 0}, \quad r = \sum_{\tau=1}^{t} \delta_{s_{2\tau}, 0}.
\end{equation}
Since at most one of the two sets $\Lambda_\pm$ is non-empty we have the identity
\begin{equation}
|\Lambda_+| + |\Lambda_-| = \big||\Lambda_+| - |\Lambda_-|\big|
\end{equation}
allowing us to express
\begin{equation}
|\Lambda_\pm| = \frac{|l-r| \pm (l-r)}{2}.
\end{equation}
By simply counting the number of configurations of the relevant sublattices of vacancies at $t=0$, weighting them with appropriate probabilities in terms of powers of $\rho$ and $\bar{\rho}$, and using the expression (\ref{sig_g}) for each fixed configuration, we obtain the exact moment generating function in the form of a double sum
\begin{gather}
G(\lambda|t) =\rho^{2t} \sum_{l=0}^{t}\sum_{r=0}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r} \mu_+^{|\Lambda_+|}\mu_-^{|\Lambda_-|}, \label{g_exact}
\end{gather}
where
\begin{equation}
\nu = \frac{\overline \rho}{\rho}.
\end{equation}
The moment generating function at the origin is normalized to unity
\begin{equation}
G(0|t) = 1, \label{g_norm}
\end{equation}
and is an even function of $\lambda$ as follows from microscopic reversibility of the equilibrium state (\ref{measure})
\begin{equation}
G(\lambda|t) = G(-\lambda|t). \label{even}
\end{equation}
When $|b|=1$ there is only one species of charged particles and the model reduces to that of free ballistically propagating particles, which we refer to as the \emph{free point} of the model.
At the free point the sum (\ref{g_exact}) can be computed exactly
\begin{equation}
G^{[1]}(\lambda|t) = \left[ 1+ 2\varDelta^2(\cosh \lambda -1) \right]^t.
\end{equation}
\subsection{Exact integrated current distribution}
The definition of the moment generating function (\ref{g_def}) can be inverted to obtain the probability distribution of the integrated current by introducing $\chi= e^\lambda$ and using the $Z$-transform (discrete Laplace transform)
\begin{equation}
\mathcal{P}(\mathfrak J|t) = \frac{1}{2\pi {\rm i}} \oint_{|\chi|=1} G(\lambda(\chi)|t) \chi^{-\mathfrak J} \frac{{\rm d} \chi}{\chi}. \label{Z_inversion}
\end{equation}
The normalization of the moment generating function (\ref{g_norm}) induces a normalization of the probability distribution
\begin{equation}
\sum_{\mathfrak J=-\infty}^{\infty} \mathcal P(\mathfrak J|t) = G(0|t) = 1.
\end{equation}
Expanding (for $|b|<1$) the hyperbolic functions in (\ref{g_exact}) with (\ref{mu_def}) using the binomial theorem
\begin{equation}
G(\lambda(\chi)|t) = \rho^{2t} \sum_{l,r= 0}^{t} \binom{t}{r} \binom{t}{l} \nu^{l+r} \sum_{i_\pm=0}^{|\Lambda_\pm|} \binom{|\Lambda_+|}{i_+} \binom{|\Lambda_-|}{i_-}
\left[ \frac{1+b}{2} \right]^{|\Lambda_-|-i_-+i_+ } \left[ \frac{1-b}{2} \right]^{|\Lambda_+| - i_+ + i_-} \chi^{2(i_++i_-) -|l-r|}, \label{G_binom}
\end{equation}
and computing the integral (\ref{Z_inversion}) by the residue theorem we find the exact current distribution
\begin{equation}
\mathcal P (\mathfrak J|t) = \rho^{2t}\sum_{l,r=0}^{t} \binom{t}{r} \binom{t}{l} \binom{|l-r|}{\frac{|l-r|+J}{2}} \nu^{l+r} \left[\frac{1-b^2}{4}\right]^{\frac{|l-r|}{2}}\left[\frac{1+b}{1-b}\right]^{ \frac{\mathfrak J\, \textrm{sgn}\, (l-r)}{2}}
\frac{1+(-1)^{\mathfrak J + |l-r|}}{2}, \label{P_exact}
\end{equation}
where we adopt the convention
\begin{equation}
\textrm{sgn}\, (x) =
\begin{cases}
\hfill +1 \hfill & \textrm{for } x >0,\\
\hfill \ \ 0 \hfill& \textrm{for } x = 0,\\
\hfill -1 \hfill & \textrm{for } x<0.
\end{cases}
\end{equation}
The free points $ b=\pm 1$ have to be considered separately and can be summed up using a hypergeometric function
\begin{equation}
\mathcal P^{[1]}(\mathfrak J|t) = \nu^{\mp \mathfrak J} \rho^{2t} \sum_{l=0}^t \binom{t}{l} \binom{t}{l \mp \mathfrak J} \nu^{2l} = \nu^{\mathfrak J} \rho^{2t} \binom{t}{ \mathfrak J} {}_2F_1\left(\mathfrak J -t, -t; \mathfrak J+1; \nu^{2}\right).
\end{equation}
\section{Asymptotic analysis}
\subsection{Preliminaries}
In this section we analyse the late-time asymptotics of the exact results (\ref{g_exact},\ref{P_exact}) and various aspects thereof. We emphasize that throughout this section we specialize our analysis to the case of a real counting field
\begin{equation}
\lambda \in \mathbb{R}.
\end{equation}
The free point of the model has already been considered, so we restrict ourselves to the interacting regime
\begin{equation}
-1 < b <1.
\end{equation}
Throughout the paper and supplemental material we use the following asymptotic notation
\begin{align}
f \stackrel{x}{\sim} g &\Leftrightarrow \lim_{x \to \infty} \frac{f(x)}{g(x)} = c \in \mathbb{C} \setminus \{0\},\\
f \stackrel{x}{\asymp} g &\Leftrightarrow \lim_{x \to \infty} \frac{f(x)}{g(x)} = 1,
\end{align}
where the variable above the asymptotic relation indicates which variables is sent to infinity. We most frequently send $t \to \infty$, in which case we suppress the variable above the asymptotic relation.\\\\
In what follows we frequently make use of Laplace's method \cite{miller2006applied} of localizing exponential integrals involving a large parameter over multi-dimensional domains spanned by coordinates $\vec{x} = (x_1, x_2, \ldots, x_n)$ as
\begin{equation}
\int {\rm d} \vec{x}\ g(\vec{x}) e^{t f(\vec{x})} \asymp \left(\frac{2\pi}{t}\right)^{n/2} \frac{g(\vec{x}_0)e^{tf(\vec{x}_0)}}{\sqrt{|\det[H[f](\vec x_0)]|}}, \qquad \nabla f(\vec x_0) = \vec 0, \label{Laplace}
\end{equation}
where $H[f]$ is the Hessian matrix of function $f$. We also make use of the Stirling formula
\begin{equation}
n! \stackrel{n}{\asymp} \sqrt{2\pi n} \left( \frac{n}{e} \right)^n, \label{Stirling}
\end{equation}
in particular to approximate binomials as
\begin{equation}
\binom{t}{l} \stackrel{t-l,t, l}{\asymp} \sqrt{\frac{t}{2\pi l(t-l)}}\left(\frac{l}{t}\right)^{-l} \left(1-\frac{l}{t}\right)^{l-t}, \label{binom_approx}
\end{equation}
where the order of limits, implied by the asymptotic notation, can be permuted.
\subsection{Moment generating function}
The dominant terms of the sum (\ref{g_exact}) are located away from the boundaries $l, r = 0$ and $l, r = t$.
It is then justified to approximate the binomials using (\ref{binom_approx}), convert the sums into integrals and introduce rescaled variables $x = l/t$, $y=r/t$
\begin{equation}
G(\lambda|t) \asymp \frac{t}{2\pi} \iint_{[0, 1]^2} \frac{e^{tf(x,y)}}{\sqrt{xy(1-x)(1-y)}}\, {\rm d} x {\rm d} y, \label{g_int}
\end{equation}
where
\begin{equation}
f(x, y) =\log \left( \rho^2 \frac{\nu^{x+y} \mu_+^{\frac{1}{2}(|x-y|+(x-y))} \mu_-^{\frac{1}{2}(|x-y|-(x-y))} }{x^x (1-x)^{1-x}y^y (1-y)^{1-y}}\right). \label{saddle_f}
\end{equation}
As $t \to \infty$ the integral localizes in the vicinity of the two extrema of the function in the exponent
\begin{gather}
\nabla f(x_\pm, y_\pm) = (0,0), \quad x_\pm= \frac{\nu}{\nu + \mu_\pm^{\mp1}}, \quad y_\pm = \frac{\nu}{\nu + \mu_\pm^{\pm 1}} \label{max1}.
\end{gather}
By inspecting the values at the these two points we observe that the global maximum is attained at
\begin{equation}
\max_{(x, y) \in [0,1]^2} f(x, y) = f(x_\gamma, y_\gamma), \quad \gamma = \textrm{sgn}\,(\lambda b), \label{max_func}
\end{equation}
so that generically there is only one global maximum. The generic picture breaks down when $\lambda = 0$ or $b=0$. For $\lambda = 0$ and $b \neq 0$ we have $x_+ = x_- = y_+ = y_-$ so that there is still a single global maximum on the diagonal of the unit square $[0,1]^2$. The case of $b = 0$, $\lambda \neq 0$ is more complicated since there the function $f$ attains its global maximum at two distinct points, symmetric with respect to the diagonal, $x=y$. This introduces a discontinuity in the function $G$ at $\lambda = 0$.
Evaluating the required objects at the maximum, namely
\begin{gather}
\det H(x_\gamma, y_\gamma) = \left((\mu_\gamma + \nu)(\mu_\gamma^{-1} + \nu)\nu^{-1}\right)^{2},\\
\frac{1}{\sqrt{x_\gamma(1-x_\gamma)y_\gamma(1-y_\gamma)}} = (\mu_\gamma + \nu)(\mu_\gamma^{-1} + \nu)\nu^{-1},
\end{gather}
and taking into account the localization of the integral (\ref{g_int}), we find
\begin{equation}
G(\lambda|t) \asymp e^{tf(x_\gamma, y_\gamma)} = D_b(\lambda) \left[\rho^2 (\nu + \mu_\gamma)(\nu + \mu_\gamma^{-1}) \right]^t, \label{g_res}
\end{equation}
where the factor $D$ accounts for the discontinuity at the origin when $b=0$
\begin{equation}
D_b(\lambda) =
\begin{cases}
2 & \textrm{for }\lambda = b = 0,\\
1 & \textrm{otherwise}.
\end{cases}
\end{equation}
\subsection{Scaled cumulant generating function}
The scaled cumulant generating function (SCGF) is defined as
\begin{equation}
F(\lambda) = \lim_{t \to \infty} \frac{1}{t} \log G(\lambda|t). \label{F_def}
\end{equation}
From the the result (\ref{g_res}) it immediately follows that
\begin{equation}
F(\lambda) = \log \left[1 + \varDelta^2 (\mu_{b} + \mu^{-1}_{b} - 2) \right] , \quad \mu_b= \cosh \lambda + |b|\sinh |\lambda|. \label{F_res}
\end{equation}
The SCGF has the following expansion around $\lambda = 0$
\begin{gather}
F^{[b]}(\lambda) = \lambda^2 \varDelta^2 b^2 + |\lambda|^3 \varDelta^2 |b|(1-b^2) + \lambda^4 \frac{\varDelta^2}{12}\left(3 - 14b^2 + 12b^4 - 6 \varDelta^2 b^4 \right) + \mathcal{O}(|\lambda|^5), \label{Fn0}\\
F^{[0]}(\lambda) = \lambda^4 \frac{\varDelta^2}{4} - \lambda^6 \frac{\varDelta^2}{12} + \lambda^8 \frac{\varDelta^2 (11 -10 \varDelta^2)}{320} + \mathcal{O}(\lambda^{10}), \label{F0}
\end{gather}
and around $|\lambda| \to \infty$
\begin{gather}
F(\lambda) = |\lambda| + \log \left(\frac{\varDelta^2(1+|b|)}{2} \right) + \frac{2(1-2\varDelta^2)}{\varDelta^2 (1+|b|)}e^{-|\lambda|}+ \mathcal{O}(e^{-2|\lambda|}). \label{inf_asymp}
\end{gather}
We note that while the asymptotic MGF (\ref{g_res}) for $b=0$ is discontinuous at the origin, the SCGF (\ref{F_res}) is in this case a real analytic function.
On the contrary the SCGF for $b\neq 0$ has a discontinuous third derivative at the origin
\begin{equation}
\frac{{\rm d}}{{\rm d} \lambda}\Big|_{\lambda = 0} F^{[b]}(\lambda) = 0, \qquad
\frac{{\rm d}^2}{{\rm d} \lambda^2}\Big|_{\lambda = 0} F^{[b]}(\lambda) = 2b^2 \varDelta^2, \qquad
\frac{{\rm d}^3}{{\rm d} \lambda^3}\Big|_{\lambda = 0^\pm} F^{[b]}(\lambda) = \pm 6b(1-b^2) \varDelta^2, \label{broken_der}
\end{equation}
where $0^\pm$ indicates from which side we approach the origin.
\subsection{Cumulants}
The cumulant generating function (CGF) is defined as the logarithm of the moment generating function (\ref{g_def})
\begin{equation}
\log G(\lambda|t). \label{CGF_def}
\end{equation}
We define the raw cumulants as derivatives of the cumulant generating function at the origin
\begin{equation}
c_n(t) \equiv \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big|_{\lambda = 0} \log G(\lambda|t). \label{cum_def}
\end{equation}
In equilibrium, the CGF is an even function [see Eq.~(\ref{even})], implying that all odd cumulants identically vanish
\begin{equation}
c_{2n-1}(t) = 0. \label{odd_zero}
\end{equation}
We note that the existence of a non-zero SCGF (\ref{F_def}) does not imply that at late times the cumulants all grow lineary with time since the limit $t \to \infty$ and the operation of taking derivatives with respect to $\lambda$ in general do not commute
\begin{equation}
\frac{{\rm d}^n}{{\rm d} \lambda^n} \Big|_{\lambda= 0 } \lim_{t \to \infty} \frac{1}{t}\log G(\lambda|t) \neq \lim_{t \to \infty} \frac{{\rm d}^n}{{\rm d} \lambda^n}\Big|_{\lambda= 0 } \frac{1}{t} \log G(\lambda |t). \label{non_commute}
\end{equation}
The discontinuity of the SCGF (\ref{broken_der}) already indicates that cumulants cannot be obtained by taking derivatives of the SCGF, but need to be computed directly from (\ref{cum_def}). The combinatorial sums inherent in the definition of the cumulants (\ref{cum_def}) can be resolved using the Fa\`{a} di Bruno's formula for higher derivatives of a composite function
\begin{equation}
\frac{{\rm d}^n}{{\rm d} x^n} f(g(x)) = \sum_{r=1}^n f^{(r)}(g(x)) B_{n, r}\left(g^{(1)}(x), g^{(2)}(x), \ldots, g^{(n-r+1)}(x) \right), \label{FdB}
\end{equation}
where $f^{(k)} \equiv \frac{{\rm d}^k f}{{\rm d} x^k}$ and $B_{n, r}$ are the incomplete Bell polynomials, defined by
\begin{equation}
\exp \left[{z \sum_{j=1}^\infty x_j \frac{t^j}{j!}}\right]=1+ \sum_{n=1}^\infty \frac{t^n}{n!}\sum_{k=1}^n z^k B_{n, k}(x_1, x_2, \ldots, x_{n-k+1}).
\end{equation}
\subsubsection{Localization of cumulants}
Consider first the computation of the $n$-th moment $G^{(n)}$ by taking derivatives of the MGF. Start by defining functions $d_n^k$ as
\begin{equation}
d_{n}^k(b) \equiv \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big|_{\lambda = 0} \left(\cosh \lambda + b \sinh \lambda \right)^{k} = \sum_{r=1}^n \frac{k!}{(k-r)!} B_{n, r}\left( b, 1,b, 1, \ldots \right) \label{Faa_cn}
\end{equation}
so that the $n$-th moment can be expressed as
\begin{equation}
G^{(n)}(t) \equiv \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big|_{\lambda = 0} G(\lambda|t) = \rho^{2t}\sum_{l=0}^{t}\sum_{r=0}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r} d_n^{|l-r|} (b\, \textrm{sgn}(l-r)). \label{Faa_gn}
\end{equation}
All odd functions $d_{2n+1}^k(b)$ contain only odd powers of $b$, while all even functions $d_{2n}^k(b)$ contain only even powers of $b$.
From the symmetric way in which $b$ enters $d_n^k$ in Eq.~(\ref{Faa_gn}) we see that all odd moments are zero
\begin{equation}
G^{(2n-1)}(t) = 0, \label{odd_moments}
\end{equation}
while the even terms do not depend on the sign of $b$.
To evaluate Eq.~(\ref{Faa_gn}) we consider sums of the form
\begin{equation}
S_n = \rho^{2t}\sum_{l=0}^{t}\sum_{r=0}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r} |l-r|^n. \label{S_def}
\end{equation}
Approximating the binomials with (\ref{binom_approx}), converting the sum into an integral and rescaling the integral (\ref{S_def}) as when computing the SCGF ($x=l/t$, $y=r/t$), we obtain
\begin{equation}
S_n \asymp \frac{t^{n+1}}{2\pi} \iint_{[0, 1]^2} h_n(x, y) e^{tf(x,y)}\, {\rm d} x {\rm d} y,
\end{equation}
where
\begin{equation}
f(x, y) =\log \left( \rho^2 \frac{\nu^{x+y}}{x^x (1-x)^{1-x}y^y (1-y)^{1-y}}\right), \quad h_n(x, y) = \frac{|x-y|^n}{\sqrt{xy(1-x)(1-y)}}.
\end{equation}
The gradient of $f$ vanishes at the critical point
\begin{equation}
\nabla f(x_c, y_c) = 0, \quad x_c = y_c = \frac{\nu}{1 + \nu} = 1 - \rho,
\end{equation}
with the Hessian and value
\begin{equation}
H(x_c, y_c) = - \mathds{1}/\varDelta, \quad \det H(x_c, y_c) = \varDelta^{-2}, \quad f(x_c, y_c) = 0.
\end{equation}
Note however, that $h_n(x_c, y_c) = 0$ for all $n > 0$ and simple localization (\ref{Laplace}) fails since there is a thin strip where $h$ is small, which splits the peak into two symmetric peaks. To capture this behavior we Taylor expand $f$ around the peak to quadratic order
\begin{equation}
f(x, y) \approx -\frac{(x - (1-\rho))^2 + (y-(1-\rho))^2}{2\varDelta^2},
\end{equation}
and introduce rotated coordinates
\begin{equation}
s_\pm = x \pm y, \label{rot_coord}
\end{equation}
in terms of which the critical point is at
\begin{equation}
s_-^c = 0, \quad s_+^c = 2(1-\rho).
\end{equation}
Given that the two peaks are symmetric around $s_-=0$ we compute twice the integral over the peak with $s_- > 0$
\begin{equation}
S_n \asymp \frac{t^{n+1}}{\pi} \iint_{s_- > 0} \frac{ s_-^n e^{-\frac{t}{4\varDelta^2}(s_-^2)} e^{-\frac{t}{4\varDelta^2}(s_+ - s_+^c)^2} }{\sqrt{(s_+^2 - s_-^2)(1 - \frac{s_+ +s_-}{2})(1 - \frac{s_+ -s_-}{2})}} {\rm d} s_- {\rm d} s_+,
\end{equation}
The expression under the square root is not singular at the peak and localizes to $\frac{1}{\varDelta^2}$
\begin{equation}
S_n \asymp \frac{t^{n+1}}{2\pi \varDelta^2} \int_{-\infty}^{\infty} {\rm d} s_+ e^{-\frac{t}{4\varDelta^2} (s_+ -s_+^c)^2} \int_{0}^\infty {\rm d} s_- s_-^n e^{-\frac{t}{4\varDelta^2} s_-^2}.
\end{equation}
Computing the Gaussian integrals we obtain the late-time behavior of the sum (\ref{S_def})
\begin{equation}
S_n \asymp t^{n/2} \frac{(2\varDelta)^{n} }{\sqrt{\pi}}\Gamma \left(\frac{n+1}{2}\right). \label{S_res}
\end{equation}
Having obtained the result (\ref{S_res}) we are in a position to extract the asymptotic behavior of the moments and cumulants.
Equation~(\ref{Faa_cn}) generates $d_n^k$ as a function of $k$. Expanding, we obtain a polynomial in $k$ of degree $n/2$ . Substituting the monomials as [cf. Eqs.~(\ref{Faa_gn},\ref{S_def})]
\begin{equation}
k^m \to S_m,
\end{equation}
we come to the expression for the even moments $G^{(2n)}$ in terms of $S_m$, for which we have the asymptotic result (\ref{S_res})
\begin{equation}
G^{(2n)}(t) \asymp \rho^{2t}\sum_{l=0}^{t}\sum_{r=0}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r} d_{2n}^{|l-r|} (b) \xrightarrow{k^m \to S_m} G^{(2n)}(S_1, S_2, \ldots). \label{Faa_final}
\end{equation}
To pass from the moments to the cumulants we again invoke (\ref{FdB}) while taking higher derivatives of the logarithm
\begin{equation}
c_n(t) \asymp \sum_{r=1}^n (-1)^{r-1} (r-1)! B_{n, r}\left(0, G^{(2)}(t), 0, G^{(4)}(t), \ldots, G^{(n-r+1)}(t) \right). \label{Faa_log}
\end{equation}
where we make use of the fact that all odd moments are zero (\ref{odd_moments}).
\subsubsection{Lowest cumulants}
Combining the results (\ref{S_res},\ref{Faa_final},\ref{Faa_log}) it is straightforward to extract the late-time behavior of cumulants order-by-order
\begin{align}
c_n^{[0]}(t) &= \sum_{l=0}^r c_{n|l}^{[0]} t^{(n-2l)/4} + \mathcal{O}(t^{(n-2(r+1))/4}), \label{asymp_series_b0}\\
c_n^{[b]}(t) &= \sum_{l=0}^r c_{n|l}^{[b]} t^{(n-l)/2} + \mathcal{O}(t^{(n-(r+1))/2}). \label{asymp_series_b1}
\end{align}
As an example we list the first few leading orders of the lowest cumulants. For $b=0$ we have
\begin{align}
c_{2|0}^{[0]} &= \frac{2\varDelta}{\pi^{1/2}}, \label{ref_c2}\\
c_{4|0}^{[0]} &= \frac{6\varDelta^2}{\pi} \left(\pi -2 \right), &
c_{4|1}^{[0]}&= - \frac{4\varDelta}{\pi^{1/2}}, \label{ref_c4}\\
c_{6|0}^{[0]}&= \frac{60 \varDelta^3}{\pi^{3/2}}(4 - \pi), &
c_{6|1}^{[0]} &= \frac{60 \varDelta^2}{\pi}(2-\pi), \label{ref_c6}\\
c_{8|0}^{[0]} &= \frac{3360 \varDelta^4}{\pi^2}(\pi - 3), &
c_{8|1}^{[0]}&= - \frac{1680 \varDelta^3}{\pi^{3/2}} \left(4 - \pi\right) , \label{ref_c8}\\
c_{10|0}^{[0]}&= \frac{7560\varDelta^5}{\pi^{5/2}}(3\pi^2 - 40 \pi + 96), &
c_{10|1}^{[0]} &= \frac{201600\varDelta^4}{\pi^2}\left(3 - \pi \right). \label{ref_c10}
\end{align}
while for $0<|b|<1$ we find
\begin{align}
c_{2|0}^{[b]} &= 2 \varDelta^2 b^2 &
c_{n>2|0}^{[b]} &= 0 \label{ref_cb0}\\
c_{2|1}^{[b]} &= \frac{2\varDelta}{\sqrt{\pi}}(1-b^2),\label{ref_cb2}\\
c_{4|1}^{[b]} &= \frac{24}{\sqrt{\pi}} \varDelta^3 b^2(1-b^2) &
c_{4|2}^{[b]} &= \frac{6\varDelta^2(1-b^2)}{\pi}\left[(1-b^2)(\pi-2) - \frac{8}{3}\pi b^2\right], \label{ref_cb4}\\
c_{6|1}^{[b]} &= -\frac{120}{\sqrt{\pi}} \varDelta^5 b^4(1-b^2) &
c_{6|2}^{[b]} &= \frac{360b^2 \varDelta^4}{\pi} \left(\pi - 2\right) (1-b^2)^2, \label{ref_cb6}\\
c_{8|1}^{[b]} &= \frac{1344}{\sqrt{\pi}} \varDelta^7 b^6(1-b^2) &
c_{8|2}^{[b]} &= - \frac{13440\varDelta^6 b^4}{\pi}\left(1-b^2\right)^2, \label{ref_cb8}\\
c_{10|1}^{[b]} &= -\frac{21600}{\sqrt{\pi}} \varDelta^9 b^8(1-b^2) &
c_{10|2}^{[b]} &= \frac{483840 \varDelta^8 b^6}{\pi}(1-b^2)^2. \label{ref_cb10}
\end{align}
In Figures~\ref{fig_cum1},\ref{fig_cum2} we show the comparison of asymptotics (\ref{ref_c2},\ref{ref_c4},\ref{ref_c6},\ref{ref_cb0},\ref{ref_cb2},\ref{ref_cb4},\ref{ref_cb6}) with direct numerical simulations for $b=0$ and $b=0.5$ respectively. In both cases we plot the ratio $\mathcal{R}_n$ of numerically estimated cumulants $\tilde c_n$ divided by the first respective non-zero asymptotic order (see Eqs.~(\ref{asymp_series_b0},\ref{asymp_series_b1}))
\begin{equation}
\mathcal{R}^{[0]}_n(t) \equiv
\frac{\tilde c_n(t)}{c_{n|0}^{[0]}}t^{-n/4},\\
\qquad
\mathcal{R}^{[b]}_n(t) \equiv
\begin{cases}
\frac{\tilde c_n(t)}{c_{n|0}^{[b]}}t^{-1}, & \textrm{for}\ n=2,\\
\frac{\tilde c_n(t)}{c_{n|1}^{[b]}}t^{-(n-1)/2}, & \textrm{for}\ n>2.
\end{cases}
\label{ratio_def}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[width=0.55\columnwidth]{SM_cum_b0-crop.pdf}
\caption{Ratio $\mathcal{R}^{[0]}_n$ (\ref{ratio_def}) (circles/stars/triangles) of numerically estimated cumulants $\tilde c_n(t)$ divided by leading order asymptotics $c_{n|0}^{[0]} t^{n/4}$ as a function of time $t$ for $\rho=0.5$ ($\varDelta=0.25$) and $b=0$. Dotted lines show the asymptotic results including an additional sub-leading order, $1 + t^{-1/2} c_{n|1}^{[0]}/{c_{n|0}^{[0]}} $. Number of samples for each time ${\rm N}_{\rm sample} = 10^{8}$.}
\label{fig_cum1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.55\columnwidth]{SM_cum_b1-crop.pdf}
\caption{Ratio $\mathcal{R}^{[b]}_n$ (\ref{ratio_def}) (circles/stars/triangles) of numerically estimated cumulants $\tilde c_n(t)$ divided by leading order asymptotics $c_{n|1}^{[0]} t^{n/2}$ (or $c_{n|0}^{[0]} t^{n/2}$ for $n=2$) as a function of time $t$ for $\rho=0.5$ ($\varDelta=0.25$) and $b=0.5$. Dotted lines show the asymptotic results including an additional sub-leading order, $1 + t^{-1/2} c_{n|2}^{[0]}/c_{n|1}^{[0]}$ (or $1 +t^{-1/2} c_{n|1}^{[0]}/c_{n|0}^{[0]} $ for $n=2$). Number of samples for each time ${\rm N}_{\rm sample} = 10^{8}$.}
\label{fig_cum2}
\end{figure}
The ratios converge towards unity for large times for both values of charge bias $b$, $ \lim_{t \to \infty} \mathcal{R}_n(t) \to 1$.
We observe that the cumulants scale with different powers of $t$ as
\begin{equation}
c_2^{[b]}(t) \sim t, \quad c^{[b]}_{2n>2}(t) \sim t^{n-1/2} \qquad {\rm and} \qquad c^{[0]}_{2n}(t) \sim t^{n/2}.
\label{cumulant_growth}
\end{equation}
\clearpage
\subsubsection{Generator of cumulant asymptotics for $b=0$}
\noindent To prove the assertion (\ref{cumulant_growth}) and obtain a convenient way of generating the leading coefficients of the expansion (\ref{asymp_series_b0}) we again make use of the Fa\`{a} di Bruno's formula (\ref{FdB}).
We start by considering the case of $b=0$ where the moments are to leading order in time equal to
\begin{equation}
G^{(2n)}(t) \asymp (2n-1)!! S_{n}(t) \asymp (2n-1)!! \frac{(2\varDelta)^n }{\sqrt{\pi}} \Gamma\left( \frac{n+1}{2}\right) t^{n/2}.
\end{equation}
From Eq.~(\ref{Faa_log}) the cumulants are expressed as
\begin{equation}
c_n(t) \asymp \sum_{r=1}^n (-1)^{r-1} (r-1)! B_{n, r}\left(0, 1!! \frac{2\varDelta\Gamma\left(1\right)}{\sqrt{\pi}}t^{1/2} , 0,3!! \frac{(2\varDelta)^2\Gamma\left(3/2\right)}{\sqrt{\pi}}t, \ldots \right). \label{Faa_b0}
\end{equation}
To resum the expression we can use (\ref{FdB}).
We construct a Taylor series from the prescribed derivatives and resum the series for an auxiliary function $h$
\begin{equation}
h(\lambda, t) \equiv \pi^{-1/2} \sum_{j=0}^\infty \frac{(2j-1)!!}{(2j)!} \Gamma\left(\frac{j+1}{2}\right) (2\varDelta)^j t^{j/2}\lambda^{2j} = e^{\frac{\lambda^4 \varDelta^2 t}{4}} \left(1 + \textrm{erf} \left[ \frac{\lambda^2 \varDelta \sqrt{t}}{2} \right] \right).
\end{equation}
Introducing a rescaled variable $\xi$
\begin{equation}
\xi^2 = \frac{\lambda^2 \varDelta\, t^{1/2}}{2},
\end{equation}
and noting that $h(0, t) = 1$ we can apply the Fa\`{a} di Bruno formula (\ref{FdB}) in reverse to resum the expression (\ref{Faa_b0})
\begin{equation}
c_n(t) \asymp \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big|_{\lambda = 0} \mathcal F^{[0]}_0(\xi(\lambda, t)) = \left( \frac{{\rm d} \xi}{{\rm d} \lambda} \right)^n \frac{{\rm d}^n}{{\rm d} \xi^n} \Big|_{\xi = 0} \mathcal F^{[0]}_0(\xi), \label{Faa_b0_0}
\end{equation}
where
\begin{equation}
\mathcal F^{[0]}_0(\xi) = \xi^4 + \log \left(1 + \textrm{erf}\ \xi^2 \right). \label{Faa_b0_1}
\end{equation}
The expression (\ref{Faa_b0_0}) recovers the leading asymptotics (\ref{ref_c2},\ref{ref_c4},\ref{ref_c6},\ref{ref_c8},\ref{ref_c10}).
\begin{equation}
c_{n|0}^{[0]} = \left(\frac{\varDelta}{2}\right)^{n/2} \frac{{\rm d}^n}{{\rm d} \xi^n} \Big|_{\xi = 0} \mathcal F^{[0]}_0(\xi). \label{leading_generator_b0}
\end{equation}
From (\ref{Faa_b0_0},\ref{Faa_b0_1},\ref{leading_generator_b0}) it follows immediately that $c_{2n}^{[0]}(t) \sim t^{n/2}$.
\subsubsection{Generator of cumulant asymptotics for $0<|b|<1$}
A similar procedure can be applied when $0< |b|<1$, with the proviso that the leading order contribution cancels (for $n>2$) and the next sub-leading contribution of the moments must be included
\begin{align}
G^{(2n)}(t) &\asymp b^{2n} S_{2n}(t) + n(2n-1) b^{2n-2}(1-b^2) S_{2n-1}(t),\\
& \asymp \frac{(2 b \varDelta t^{1/2})^{2n-1}}{\sqrt{\pi}} \left[ 2b \varDelta \Gamma (n+\tfrac{1}{2}) t^{1/2} + n(2n-1) (b^{-1}-b) \Gamma(n) \right].
\end{align}
We yet again construct the Taylor series, resum it and introduce a rescaled variable $\xi$
\begin{equation}
\xi = \lambda b \varDelta t^{1/2},
\end{equation}
yielding for $t\to \infty$
\begin{align}
h(\lambda, t) &\equiv \frac{1}{\sqrt{\pi}} \sum_{j=0}^\infty \frac{\left[ 2b \varDelta \Gamma (n+\tfrac{1}{2}) t^{1/2} + n(2n-1) (b^{-1}-b) \Gamma(n) \right]}{(2j)!} (2b \varDelta t^{1/2})^{2j-1} \lambda^{2j} \\
&= h_0(\xi) + t^{-1/2} \, h_1(\xi) + \mathcal{O}(t^{-1})
\end{align}
where
\begin{equation}
h_0(\xi) = e^{\xi^2}, \quad h_1(\xi) = \frac{1-b^2}{b^2 \varDelta} \left(\xi^3 e^{\xi^2} \textrm{erf}\ \xi + \frac{\xi^2}{\sqrt{\pi}} \right). \label{h_eqs}
\end{equation}
We now expand the logarithm for large $t$ with $\xi$ held fixed
\begin{equation}
c_n(t) = \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big |_{\lambda= 0} \log \left(h_0(\xi) + t^{-1/2}h_1(\xi) + \mathcal{O}(t^{-1}) \right) = \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big |_{\lambda= 0} \log\left( h_0(\xi) + t^{-1/2} \frac{h_1(\xi)}{h_0(\xi)} + \mathcal{O}(t^{-1}) \right).
\end{equation}
Using the expressions (\ref{h_eqs}) we come to
\begin{equation}
c_n(t) \asymp \frac{{\rm d}^n}{{\rm d} \lambda^n} \Big |_{\lambda= 0} \mathcal F^{[b]}(\xi(\lambda, t))=\left( \frac{{\rm d} \xi}{{\rm d} \lambda} \right)^n \frac{{\rm d}^n}{{\rm d} \xi^n} \Big |_{\xi= 0} \mathcal F^{[b]}(\xi), \label{Faa_b_0}
\end{equation}
where
\begin{equation}
\mathcal F^{[b]}(\xi) = \mathcal F_0^{[b]}(\xi) +\mathcal F_1^{[b]}(\xi) t^{-1/2} \label{Faa_b_1},
\end{equation}
with
\begin{equation}
\mathcal F^{[b]}_0(\xi) = \xi^2 , \qquad \mathcal F_1^{[b]}(\xi) = \frac{1-b^2}{b^2 \varDelta} \left[\xi^3 \textrm{erf}\ \xi + \frac{\xi^2e^{-\xi^2}}{\sqrt{\pi}} \right].
\end{equation}
The expression (\ref{Faa_b_0}) recovers the leading asymptotics (\ref{ref_cb0},\ref{ref_cb2},\ref{ref_cb4},\ref{ref_cb6},\ref{ref_cb8},\ref{ref_cb10})
\begin{equation}
c_{n|0}^{[b]} = \left(|b|\varDelta \right)^{n} \frac{{\rm d}^n}{{\rm d} \xi^n} \Big|_{\xi = 0}\mathcal F_0^{[0]}(\xi), \qquad c_{n|1}^{[b]} = \left(|b|\varDelta \right)^{n} \frac{{\rm d}^n}{{\rm d} \xi^n} \Big|_{\xi = 0} \mathcal F_1^{[0]}(\xi). \label{leading_generator_b1}
\end{equation}
From (\ref{Faa_b_0},\ref{Faa_b_1},\ref{leading_generator_b1}) it follows that $c_2^{[b]}(t) \sim t$, $c_{2n>2}^{[b]}(t) \sim t^{n-1/2}$.
\subsubsection{Generator of cumulant asymptotics within the Gaussian approximation}
A single generator of leading cumulant asymptotics, valid for all $|b|<1$, can alternatively be obtained by starting from the exact current distribution (\ref{P_exact}) and using the De Moivre-Laplace theorem to approximate the binomial
\begin{equation}
\binom{t}{l}p^l q^{t-l} \asymp \frac{1}{\sqrt{2\pi t pq}}\exp\left[ - \frac{(l - tp)^2}{2tpq} \right] \quad{\rm for}\ l - tp \sim t^{1/2} \ {\rm and }\ p+ q = 1. \label{Gaussian_approx}
\end{equation}
Note that the approximation (\ref{Gaussian_approx}) is less precise than the Stirling approximation (\ref{binom_approx}).
We start by splitting the probability distribution as
\begin{equation}
\mathcal{P}(\mathfrak{J}|t) = \sum_{l-r}\mathcal{P}_1(\mathfrak{J}|l-r) \mathcal{P}_2(l-r|t), \label{p_split}
\end{equation}
where
\begin{align}
\mathcal{P}_1(\mathfrak{J}|l-r) &\equiv \binom{|l-r|}{\frac{|l-r|+J}{2}} \nu^{l+r} \left[\frac{1-b^2}{4}\right]^{\frac{|l-r|}{2}}\left[\frac{1+b}{1-b}\right]^{ \frac{\mathfrak J\, \textrm{sgn}\, (l-r)}{2}} \frac{1+(-1)^{\mathfrak J + |l-r|}}{2}, \label{split_p1}\\
\mathcal{P}_2(l-r|t) &\equiv \rho^{2t} \sum_{l,r=0}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r}. \label{split_p2}
\end{align}
Using the Gaussian approximation (\ref{Gaussian_approx}), the two split distributions can be approximated as
\begin{align}
\mathcal{P}_1(\mathfrak{J}|l-r) & \stackrel{r-l}{\asymp} \frac{1}{\sqrt{2\pi(1-b^2)|l-r|}}\exp \left[-\frac{(\mathfrak{J}-(l-r)b)^2}{2(1-b^2)|l-r|} \right] \quad \textrm{for}\ \mathfrak{J} - (l-r)b \stackrel{|l-r|}{\sim} |l-r|^{1/2} ,\\
\mathcal{P}_2(l-r|t) &\asymp \frac{1}{\sqrt{4 \pi t \varDelta^2}} \exp\left[- \frac{(l-r)^2}{4 \varDelta^2 t} \right] \quad \textrm{for}\ l - t \rho,\, r - t \rho \sim t^{1/2}.
\end{align}
Using the definition of the MGF (\ref{g_def}), converting the sums over $\mathfrak{J}$ and $l-r$ into integrals, introducing
\begin{equation}
l-r = w \varDelta \sqrt{2t}, \qquad a_\pm \equiv \varDelta t^{1/2} \left(\frac{1-b^2}{2} \lambda^2 \pm b \lambda \right),
\end{equation}
and evaluating first the integral over $\mathfrak{J}$ we find
\begin{equation}
G(\lambda|t) \asymp \frac{1}{\sqrt{2\pi}} \sum_{\pm} \int_{0}^{\infty} {\rm d} w \exp \left[ - \frac{w^2}{2} + \sqrt{2} a_\pm w \right].
\end{equation}
Integrating over $w$ we arrive at the generator of cumulant asymptotics $\mathcal{F}$
\begin{equation}
G(\lambda|t) \asymp \frac{1}{2}\left[e^{a_+^2}\left(1 + \textrm{erf}\, a_+\right) +e^{a_-^2}\left(1 + \textrm{erf}\, a_-\right) \right] \equiv \exp\left[\mathcal{F}^{[b\geq0]}(\lambda) \right]. \label{Vincent_F}
\end{equation}
From Eq.~(\ref{Vincent_F}) we recover the leading-order cumulant asymptotics (\ref{ref_c2},\ref{ref_c4},\ref{ref_c6},\ref{ref_c8},\ref{ref_c10},\ref{ref_cb0},\ref{ref_cb2},\ref{ref_cb4},\ref{ref_cb6},\ref{ref_cb8},\ref{ref_cb10}) for all $|b|<1$
\begin{equation}
c_n^{[b \geq 0]}(t) \asymp \frac{{\rm d}^n}{{\rm d} \lambda^n}\Big|_{\lambda = 0} \mathcal{F}^{[b \geq 0]}(\lambda).
\end{equation}
\subsection{Probability distributions}
We now turn our attention to the asymptotics of the exact probability distribution of the integrated current (\ref{P_exact}). To this end we introduce the rescaled current at a scale $\zeta$, $ 0 \leq \zeta \leq 1$
\begin{equation}
\mathfrak{j}(t) \equiv t^{-\zeta}\mathfrak{J}(t),
\end{equation}
where we suppress the dependence of the scaled current on the scaling exponent $\zeta$ to streamline the notation.
Note that to preserve the normalization of the probability distribution of the rescaled current, $\int \mathcal{P}_\zeta(\mathfrak j|t) {\rm d} \mathfrak j = 1$, it is necessary to rescale the distribution as
\begin{equation}
\mathcal{P}_\zeta(\mathfrak j|t) \equiv t^{\zeta} \mathcal{P}(\mathfrak J=\mathfrak jt^{\zeta}|t) .
\end{equation}
The probability distribution of typical fluctuations on the scale $\zeta = 1/2z$ is independent of time. We accordingly define it as the limit
\begin{equation}
\mathcal{P}_{\textrm{typ}} (\mathfrak{j}) \equiv \lim_{t \to \infty} \mathcal{P}_{1/2z}(\mathfrak{j}|t).
\end{equation}
\subsubsection{Normal typical fluctuations for $0<|b|<1$}
For $0<|b|<1$ we have the dynamical exponent $z=1$.
The cumulants $\kappa_{n}(t)$ of the probability distribution $\mathcal{P}_{1/2}(\mathfrak j| t)$ are connected to the raw cumulants (\ref{cum_def}) by a rescaling
\begin{equation}
\kappa_n(t) = c_n(t) t^{-n/2}.
\end{equation}
The asymptotics of the cumulants (\ref{cumulant_growth}) for $0<|b| < 1$ implies that all but the second cumulant $\kappa_2(t)$ vanish as $t \to \infty$
\begin{equation}
\lim_{t \to \infty} \kappa_2(t) = 2\varDelta^2b^2, \quad \lim_{t \to \infty} \kappa_{n>2} (t) = 0.
\end{equation}
By the Marcinkiewicz theorem it follows that the probability of the rescaled current on the typical scale is a Gaussian with a variance fixed by the second cumulant
\begin{equation}
\mathcal P^{[b]}_{\rm typ}(\mathfrak j) = \frac{1}{2\sqrt{\pi} |b| \varDelta} \exp \left[{-\frac{\mathfrak j^2}{4 \varDelta^2 b^2}} \right]. \label{b_pos_dist3}
\end{equation}
\subsubsection{Asymptotic distributions for $b=0$}
We now turn our attention to the unbiased case $b=0$, where the asymptotics (\ref{cumulant_growth}) implies that all even cumulants $\kappa_n(t) = c_n(t) t^{-n/4}$ of the probability distribution $\mathcal{P}_{1/4}(\mathfrak{j}|t)$ are non-zero
\begin{equation}
\lim_{t \to \infty} \kappa_{n}(t) = \kappa_n, \quad \kappa_{2n} \neq 0.
\end{equation}
In what follows we consider the scales $ 0 \leq \zeta < 1$.
The dominant contribution to the sum (\ref{P_exact}) comes from two peaks, symmetric with respect to the diagonal $l=r$.
For $t \to \infty$ the dominant contribution to the sum will not be near the boundary so that we can again apply the Stirling approximation (\ref{binom_approx}) to the binomials in (\ref{P_exact}). Passing from a sum to an integral we must also account for the checkerboard pattern of the right-most sign factor, which we do by introducing an overall factor $\tfrac{1}{2}$. In terms of rescaled variables $x = l/t$, $y=r/t$ the distribution reads
\begin{equation}
t^{-\zeta}\mathcal P^{[0]}_\zeta(\mathfrak j|t) \asymp \frac{t^{1/2}}{\sqrt{2\pi}^3} \iint_{[0, 1]^2} \frac{e^{t(g-f) + \mathfrak jt^\zeta h}}{\sqrt{x(1-x)y(1-y)}} \sqrt{\frac{|x-y|}{(x-y)^2 -\mathfrak j^2/t^{2(1-\zeta)}}} \Theta \left(|x-y| \geq |\mathfrak j|/t^{1-\zeta} \right) {\rm d} x {\rm d} y,\\
\end{equation}
where $\Theta(x)$ is the Heaviside function and
\begin{align}
f &= \log \left( \frac{x^x(1-x)^{1-x} y^{y}(1-y)^{1-y}}{\rho^2 \nu^{x + y}} \right),\\
g &= |x-y|\log \left( \frac{|x-y|}{\sqrt{(x-y)^2 - j^2/t^{2(1-\zeta)}}} \ \right),\\
h &= \frac{1}{2}\log \left(\frac{|x-y| - \mathfrak j/t^{1-\zeta}}{|x-y| + \mathfrak j/t^{1-\zeta}} \right).
\end{align}
Given the geometry of the integral we again introduce rotated coordinates (\ref{rot_coord})
and anticipating localization of the integral, extend the integration across the entire plane
\begin{gather}
t^{-\zeta}\mathcal P_\zeta^{[0]}(\mathfrak j|t) \asymp \frac{t^{1/2}}{\sqrt{2\pi}^3} \iint_{\mathbb{R}^2} \frac{e^{ t(g(s_-)-f(s_+, s_-)) +\mathfrak jt^\zeta h(s_-)}}{\sqrt{(s_+^2 - s_-^2)((1-\frac{s_+ + s_-}{2})(1 - \frac{s_+-s_-}{2})}} \sqrt{\frac{|s_-|}{s_-^2 -\mathfrak j^2/t^{2(1-\zeta)}}} \Theta \left(|s_-| \geq |\mathfrak j|/t^{1-\zeta} \right) {\rm d} s_+ {\rm d} s_-.
\end{gather}
Taking into account the Heaviside functions and the two peaks we arrive at
\begin{gather}
t^{-\zeta }\mathcal P^{[0]}_\zeta(\mathfrak j |t) \asymp \frac{2 t^{1/2}}{\sqrt{2\pi}^3} \int_{\mathfrak j/t^{1-\zeta}}^\infty {\rm d} s_- \frac{\sqrt{s_-} e^{tg(s_-) +\mathfrak j t^\zeta h(s_-)} }{\sqrt{s_-^2 - \mathfrak j^2/t^{2(1-\zeta)}}} \int_{\mathbb{R}} {\rm d} s_+ \frac{e^{- tf(s_+, s_-)}}{\sqrt{(s_+^2 - s_-^2)((1-\frac{s_+ + s_-}{2})(1 - \frac{s_+-s_-}{2})}}.
\end{gather}
The inner integral can be evaluated by straightforward localization (\ref{Laplace}) around the critical point $s_c^+$
\begin{gather}
\partial_{s_+}\Big|_{s_+ = s_+^c} f(s_+, s_-) = 0, \quad s_+^c = \frac{2\nu^2 - \sqrt{4\nu^2 + (\nu^2-1)^2s_-^2}}{\nu^2 - 1},
\end{gather}
where the branch of the square root is taken so that $\lim_{\nu \to 1} s_+^c = 1$. This also ensures that for all $\nu \in \mathbb{R}_+$ we have $ 0<s_+^c <2$.
In the limit of small $s_-$ the critical value of $s_+^c$ becomes
\begin{equation}
s_+^c = \frac{2\nu}{\nu+1} - \frac{\nu^2 - 1}{4\nu} s_-^2 + \mathcal{O} \left( s_-^4 \right) = 2(1-\rho) - \frac{1-2\rho}{4 \rho(1-\rho)} s_-^2 + \mathcal{O} \left( s_-^4 \right).
\end{equation}
As $t \to \infty$ the maximum in $s_-$ approaches zero so we can expand the functions in the exponent
\begin{align}
g(s_-) \asymp \frac{\mathfrak j^2}{2s_-} t^{2(\alpha - 1)}, \quad
f(s_-) \asymp -\frac{\mathfrak j }{s_-}t^{\alpha-1},
\end{align}
and the integral becomes
\begin{gather}
t^{-\zeta} \mathcal P^{[0]}_\zeta(\mathfrak j |t) \asymp \frac{1}{ \sqrt{2} \pi \varDelta } \int_{\mathfrak jt^{\zeta-1}}^\infty {\rm d} s_- \, \frac{\sqrt{s_-}}{\sqrt{s_-^2 - \mathfrak j^2t^{2(\zeta-1)}}} e^{ - t \frac{s_-^2}{4\varDelta^2} - \frac{\mathfrak j^2}{2s_-}t^{2\zeta - 1}}. \label{eq10}
\end{gather}
To analyze the localization behavior in the exponential we rescale the integration variable
\begin{equation}
s_- = st^{-\omega}
\end{equation}
so that the expression in the exponential of Eq.~(\ref{eq10}) becomes
\begin{equation}
\frac{s^2}{4\varDelta^2}t^{1 - 2\omega} + \frac{\mathfrak j^2}{2s}t^{2 \zeta + \omega -1}.
\end{equation}
We equate the two algebraic exponents by setting
\begin{equation}
\omega = \frac{2}{3}(1 - \zeta) \quad \Rightarrow \quad t^{\frac{4\zeta - 1}{3}} \left(\frac{s^2}{4\varDelta^2} + \frac{\mathfrak j^2}{2s} \right),
\end{equation}
and, computing the localization, arrive at
\begin{equation}
\mathcal P^{[0]}_\zeta(\mathfrak j |t) \asymp \frac{t^{\frac{4\zeta - 1}{3}}}{\sqrt{2}\pi\varDelta} \int_{0}^\infty {\rm d} s \, \frac{1}{\sqrt{s}} \exp \left[{ - t^{\frac{4\zeta - 1}{3}} \left( \frac{s^2}{4\varDelta^2} + \frac{\mathfrak j^2}{2s}\right)} \right]. \label{b0_loc_eq}
\end{equation}
We now clearly distinguish between three regimes:
\begin{enumerate}
\item \emph{Small fluctuations} for $0 \leq \zeta < 1/4$. Here the exponential does not localize, but is instead close to unity on the entire domain, leading to an $\mathcal{O}(1)$ probability.\\
\item \emph{Typical fluctuations} for $\zeta = 1/4$, where we find a non-gaussian distribution of typical fluctuations
\begin{equation}
\mathcal P^{[0]}_{\textrm{typ}}(\mathfrak j) = \frac{1}{\sqrt{2}\pi\varDelta} \int_{0}^\infty {\rm d} s \, \frac{1}{\sqrt{s}} \exp \left[ - \frac{1}{2} \left( \frac{s^2}{2\varDelta^2} + \frac{\mathfrak j^2}{s}\right) \right].
\end{equation}
A more convenient integral representation is obtained by eliminating the square root factor by a change of variables $s \to u^2$ and extending the integral over the entire real line
\begin{equation}
\mathcal P^{[0]}_{\textrm{typ}}(\mathfrak j) = \frac{1}{\sqrt{2}\pi\varDelta} \int_{-\infty}^\infty {\rm d} u \, \exp \left[- \frac{u^4}{4\varDelta^2} -\frac{\mathfrak j^2}{2u^2} \right]. \label{b0_typical}
\end{equation}
\item \emph{Moderate fluctuations} for $1/4 < \zeta <1$, where the exponential in Eq.~(\ref{b0_loc_eq}) localizes around
\begin{equation}
s_c = \left(|\mathfrak j|\varDelta\right)^{\frac{2}{3}}
\end{equation}
and after some work gives the rate function in the absence of bias
\begin{equation}
\mathcal P^{[0]}_{\zeta}(\mathfrak j|t)\asymp t^{\frac{4\zeta - 1}{6}}\frac{\exp\left[-t^{\frac{4 \zeta - 1}{3}} \frac{3}{4} \left( \frac{\mathfrak j^2}{\varDelta}\right)^{2/3}\right] }{\sqrt{3 \pi} (|\mathfrak j| \varDelta)^{\frac{1}{3}}}. \label{rate_function1}
\end{equation}
\end{enumerate}
\subsection{Moderate deviations and the scaled cumulant generating function}
We demonstrate that while the SCGF (\ref{F_res}) does not generate (scaled) cumulants, we can use it to recover the rate function of moderate deviations for $1/2z<\zeta<1$.
For $t \to \infty$ we have from the definition of the SCGF (\ref{F_def})
\begin{equation}
e^{t F(\lambda)} \asymp G(\lambda| t) = \int {\rm d} \mathfrak j\, \mathcal{P}_\zeta(\mathfrak j|t) e^{\lambda \mathfrak j t^\zeta}. \label{F_P_rel}
\end{equation}
Making an exponential ansatz for the asymptotic probability distribution of the rescaled current
\begin{equation}
\mathcal{P}_\zeta(\mathfrak j|t) \asymp e^{-t^{\nu(\zeta)}I_\zeta(\mathfrak j)},\label{LD_ansatz}
\end{equation}
where $v(\zeta)$ is a `speed' associated to the scale $\zeta$,
we come to
\begin{equation}
e^{t F(\lambda)} \asymp \int {\rm d} \mathfrak j\, e^{\lambda \mathfrak j t^\zeta -t^{v(\zeta)}I_\zeta(\mathfrak j)}.
\end{equation}
To analyze the localization of this integral we introduce a dynamical rescaling of the counting field
\begin{equation}
\lambda = \eta t^{v(\zeta) - \zeta}, \label{dynamical_rescaling}
\end{equation}
in terms of which the exponential under the integrals is homogeneous in $t$
\begin{equation}
e^{t F(\eta, t)} \asymp \int {\rm d} \mathfrak j\, e^{t^{v(\zeta)}(\mathfrak j \eta - I_\zeta(\mathfrak j))}. \label{proto_moderate}
\end{equation}
The left hand side of (\ref{proto_moderate}) is to be understood as the limit $t\to \infty$ with $\eta$ held constant. In this limit, under the assumption $v(\zeta) < \zeta$, the expansions of the SCGF (\ref{Fn0},\ref{F0}) simplify since higher order terms in $\eta$ are suppressed
\begin{gather}
F^{[b]}(\eta, t) = F^{[b]}(\eta) t^{2 (v(\zeta) - \zeta)} + \mathcal{O}( t^{3(v(\zeta) - \zeta)}), \qquad F^{[b]}(\eta) \equiv b^2 \varDelta^2 \eta^2, \label{F_ser0}\\
F^{[0]}(\eta, t) = F^{[0]}(\eta) t^{4 (v(\zeta) - \zeta)} + \mathcal{O}( t^{6(v(\zeta) - \zeta)}), \qquad F^{[0]}(\eta) \equiv \frac{\varDelta^2 \eta^4}{4}. \label{F_ser1}
\end{gather}
\subsubsection{Moderate deviations}
For $v(\zeta) > 0$ the integral in (\ref{proto_moderate}) can be computed by localization (\ref{Laplace}) and yields
\begin{equation}
e^{t F(\eta, t)} \asymp t^{-v(\zeta)/2} \exp\left( t^{v(\zeta)} \max_{\mathfrak j} \{\mu \mathfrak j - I_\zeta(\mathfrak j) \}\right). \label{F_I_iden}
\end{equation}
Inverting (\ref{F_I_iden}) by Lagrange duality, we obtain an expression for the rate function at scale $\zeta$
\begin{equation}
I_\zeta(\mathfrak j) = \max_{\eta}\{\eta \mathfrak j - F(\eta) \}. \label{rescaled_Legendre}
\end{equation}
By matching the powers of $t$ in the exponents in Eq.~(\ref{F_I_iden}) we find the speeds $v(\zeta)$
\begin{equation}
v^{[b]}(\zeta) = 2 \zeta -1, \quad
v^{[0]}(\zeta) = \frac{4 \zeta -1}{3}. \label{speeds}
\end{equation}
We observe that the condition $v(\zeta) < \zeta$ is satisfied for $\zeta < 1$, while the `localization condition' $v(\zeta)>0$ is satisfied when $\zeta > 1/2z$, i.e. for
$\zeta > 1/2$ when $|b|>0$ and for $\zeta > 1/4$ when $b=0$. Provided these two conditions are satisfied Eq.~(\ref{rescaled_Legendre}) gives a ($\zeta$-independent) rate functions of moderate fluctuations for the range of scales
\begin{equation}
1/4<\zeta^{[0]}<1, \qquad 1/2<\zeta^{[b]}<1.
\end{equation}
Computing the Legendre transforms in (\ref{rescaled_Legendre}) and using (\ref{LD_ansatz}) we exactly recover the moderate deviations for $b=0$ (\ref{rate_function1}). For $0<|b|<1$ we extract the rate function of moderate deviations
\begin{equation}
\mathcal P^{[b]}_\zeta(\mathfrak j|t)\asymp \frac{t^\frac{2\zeta-1}{2}}{2\sqrt{\pi} |b| \varDelta} \exp\left[{-t^{2\zeta - 1}\frac{\mathfrak j^2}{4 \varDelta^2 b^2}}\right] . \label{b_pos_dist2}\\
\end{equation}
\subsubsection{Large deviations}
The same analysis also shows why the scale $\zeta=1$, i.e. large deviations, is distinguished. When $\zeta=1$ both scalings (\ref{speeds}) reduce to $v(1)=1$
implying that higher order terms in $\lambda$ of (\ref{F_ser0},\ref{F_ser1}) are not suppressed and the rate function is the Legendre transform of the full SCGF (\ref{F_res}), which we are however unable to compute in closed form.\\
\noindent Since all species move with unit velocity, the current distribution is supported on the interval $[-t, t]$, implying that the rate function on the scale $\zeta = 1$ is finite only on the interval $\mathcal{I} = [-1, 1]$. From the expansion of the SCGF around $\lambda \to \infty$ (\ref{inf_asymp}) we can extract the behavior of the rate function near the endpoints of the interval for $\delta \equiv 1- |\mathfrak j| $ as
\begin{equation}
I_{1}(\mathfrak{j}) = \log \frac{2}{(1+|b|) \varDelta^2} + \delta \log \delta + \mathcal{O}\left(\delta\right).
\end{equation}
\subsection{Large deviation rate function at the free point}
\noindent For $|b |= 1$ we can explicitly compute the Legendre transform of the SCGF to obtain the rate function
\begin{equation}
I^{[1]}_{1}(\mathfrak j) = \max_{\lambda}\{ \lambda \mathfrak j - F(\lambda) \}.
\end{equation}
The maximum is attained at
\begin{equation}
\lambda_c = \log \left[ \frac{\mathfrak j(1-2\varDelta^2 ) + \sqrt{\mathfrak j^2(1-4\varDelta^2)+4\varDelta^4}}{2\varDelta^2(1-\mathfrak j)} \right],
\end{equation}
where we find an even rate function
\begin{equation}
I^{[1]}_{1}(j) = \mathfrak j \log \left[ \frac{\mathfrak j(1-2\varDelta^2 ) + \sqrt{[\mathfrak j(1-2\varDelta^2)]^2+4\varDelta^4(1- \mathfrak j^2)}}{2\varDelta^2(1-\mathfrak j)} \right] - \log \left[ \frac{1-2\varDelta^2 + \sqrt{[\mathfrak j(1-2\varDelta^2)]^2+4\varDelta^4(1-\mathfrak j^2)}}{1-\mathfrak j^2} \right].
\end{equation}
\section{Numerical estimation of current fluctuations}
Efficient numerical estimations are facilitated by splitting of the exact current distribution as (\ref{p_split})
\begin{equation}
\mathcal{P}(\mathfrak{J}|t) = \sum_{l-r}\mathcal{P}_1(\mathfrak{J}|l-r) \mathcal{P}_2(l-r|t),
\end{equation}
where the distributions are given by (\ref{split_p1},\ref{split_p2}).
To independently sample random currents we follow a two step protocol. First we calculate the direction and amount of transfered charge at time $t$
\begin{equation}
d_t=l_t-r_t,
\end{equation}
by sampling $l,r$ from ($t$-dependent) binomial distributions
\begin{equation}
p(l) = \mathcal B\left(l|t, \rho\right), \qquad p(r) = \mathcal B\left(r|t, \rho\right),
\end{equation}
where $\mathcal B(k|n,\rho) \equiv \binom{n}{k} \rho^k (1-\rho)^{n-k}$.
Next we sample the number of $+$ particles out of $|d|$ particles $n^{+}_{|d|}$
from the binomial distribution
\begin{equation}
p(n^+) = \mathcal B\left(n^+ \big ||d|, (1+b)/2\right),
\end{equation}
leading to the total transfered current
\begin{equation}
\mathfrak{J}_t=2n^{+}_{|d_t|}\text{sgn}(d_t)-d_t.
\end{equation}
The estimated current distribution $\tilde{\mathcal{P}}$ is obtained by computing histograms
of ${\rm N}_{\rm sample}$ independently drawn currents
\begin{equation}
\tilde{\mathcal{P}}(\mathfrak{J}|t) = \frac{{\rm N}_\delta (\mathfrak{J}_t)}{{\rm N}_{\rm sample}},
\end{equation}
where ${\rm N}_\delta (\mathfrak{J})$ is the number of currents sampled in a window of width $\delta$ centered on $\mathfrak{J}$. The estimated moments of the distributions $\tilde G^{(n)}$ are computed as
\begin{equation}
\tilde G^{(n)}(t) = \sum_k (k \delta)^n \tilde{\mathcal{P}}(k \delta |t).
\end{equation}
Cumulants are estimated from current distribution's moments according to (\ref{Faa_log})
\begin{equation}
\tilde c_n(t) = \sum_{r=1}^n (-1)^{r-1} (r-1)! B_{n, r}\left(0, \tilde G^{(2)}(t), 0, \tilde G^{(4)}(t), \ldots, \tilde G^{(n-r+1)}(t) \right). \label{c_n_estimate}
\end{equation}
Monte Carlo errors
are estimated from the variance after spliting the total sample into
$10^{3}$ independent subsamples.
\section{Dynamical Lee--Yang theory in equilibrium}
\label{sec:LY}
We give a brief overview of Lee--Yang theory as applied to the MGF of time-integrated current (\ref{J_def}). We start with a general outline of the theory for equilibrium systems.
\subsection{Lee--Yang zeros}
Consider the definition of the moment generating function (\ref{g_def})
\begin{equation}
G(\lambda|t) = \sum_{\mathfrak J} \mathcal{P}(\mathfrak J|t) e^{\lambda \mathfrak{J(t)}}. \label{LY_Gdef}
\end{equation}
The probability distribution is by definition non-negative, $\mathcal{P}(\mathfrak{J}|t) \geq 0$, implying that the MGF cannot (for finite times) have zeros on the real axis $\lambda \in \mathbb{R}$. The main idea of the Lee--Yang theory is to instead consider Eq.~(\ref{LY_Gdef}) for complex values of the counting field
\begin{equation}
\lambda \in \mathbb{C},
\end{equation}
and study the properties of the moment generating function in the complex plane.
The probability distribution of the current is real implying an involutive symmetry of the MGF in the complex plane
\begin{equation}
G(\lambda|t) = \overline{G(\overline \lambda|t)}, \label{real_sym}
\end{equation}
where $\overline \bullet$ denotes complex conjugation. In equilibrium the condition of detailed balance, $\mathcal{P}(\mathfrak{J}|t) = \mathcal{P}(-\mathfrak{J}|t)$, implies an additional involutive symmetry of the MGF
\begin{equation}
G(\lambda|t) = G(-\lambda|t). \label{inversion_sym}
\end{equation}
Of particular interest are the (time-dependent) zeros $\lambda_j(t)$ (Lee--Yang zeros) of the MGF, since they are the singularities of the CGF (see Eq.~\ref{CGF_def})
\begin{equation}
G(\lambda_j|t) = 0. \label{LY_def}
\end{equation}
To avoid cluttering the notation we often suppress the time dependence of the zeros, $\lambda_j\equiv\lambda_j(t)$.
From the symmetries (\ref{real_sym},\ref{inversion_sym}) it follows that the zeros come in (possibly degenerate) quartets
\begin{equation}
(\lambda_j, \overline{\lambda}_j, -\lambda_j, -\overline \lambda_j). \label{LY_quartet}
\end{equation}
Given that the probability distribution of the current $\mathcal{P}(\mathfrak{J}|t)$ is supported on the interval $[-t, t]$ the MGF is a rational function in the variable $e^\lambda$. Consequently the MGF can be factorized as
\begin{equation}
G(\lambda |t) = g(\lambda |t)\prod_{j=1}^{2t} \left(1-\lambda/\lambda_j \right), \quad g(\lambda |t) = e^{-\lambda t} \prod_{j=1}^{2t} \frac{e^\lambda - e^{\lambda_j}}{1 - e^{\lambda_j}} \frac{1}{1 - \lambda/\lambda_j }, \label{G_factorized}
\end{equation}
where the normalization is fixed by the normalization of the MGF (see (\ref{g_norm})). By construction, the function $g(\lambda|t)$ does not have any zeros within the strip $ Z = \{x+ {\rm i} y\ |\ x \in \mathbb{R}, -\pi < y < \pi \}$ and is an analytic function near the origin. We refer to the zero of the MGF closest to the origin in the first quadrant as the \emph{leading} Lee--Yang zero $\lambda_1$
\begin{equation}
|\lambda_1| \leq |\lambda_j| \textrm{ for } 1< j \leq 2t, \quad 0 \leq \textrm{arg}\, \lambda_1 \leq \pi/2. \label{leading_LY}
\end{equation}
Noting that the modulus of the leading Lee--Yang zero is by definition equal to the convergence radius $r$ of the CGF we denote
\begin{equation}
\lambda_1(t) = r(t) e^{{\rm i}\varphi(t)}, \qquad r, \varphi \in \mathbb{R}.
\end{equation}
\subsection{Critical points}
As noted above, the MGF cannot vanish for $\lambda \in \mathbb{R}$ for finite times $t$, implying that the cumulant generating function (\ref{CGF_def})
is a real analytic function. From the factorization (\ref{G_factorized}) it can be expressed as
\begin{equation}
\log G(\lambda |t) = \log g(\lambda|t) + \sum_{j=1}^{2t} \log \left(1 - \lambda/\lambda_j \right). \label{F_factorized}
\end{equation}
On the other hand as $t \to \infty$ the Lee--Yang zeros can approach a \emph{critical} point\footnote{There can be more than one critical point. For simplicity we confine the discussion to only one.} $\lambda_c$ on the real axis
\begin{equation}
\lambda_j(t) \xrightarrow{t \to \infty} \lambda_c \in \mathbb{R},
\end{equation}
in the vicinity of which the CGF has a logarithmic singularity
\begin{equation}
\log G(\lambda|t) = \log (\lambda - \lambda_c) + \mathcal{O}(|\lambda-\lambda_c|^{-1}). \label{dpt_def}
\end{equation}
\subsection{Cumulants, Lee--Yang zeros and critical points}
We now establish a link between the higher cumulants and the dynamics of the leading Lee--Yang zero. We use this to demonstrate an intimate connection between critical points and diverging cumulants.
From the definition (\ref{cum_def}) and the factorization (\ref{G_factorized}) it follows that cumulants are related to Lee--Yang zeros as
\begin{equation}
c_n(t) = - (n-1)! \sum_{j=1}^{2t} \lambda_j^{-n}(t) + \frac{{\rm d}^n }{{\rm d} \lambda^n} \Big|_{\lambda=0} \log g(\lambda|t). \label{cum_split}
\end{equation}
We observe that the expression for the cumulants contains a term coming from Lee--Yang zeros and a derivative of a function analytic in the strip $Z$. From asymptotic analysis it follows by Darboux theorem \cite{Dingle} that for higher-order cumulants the expression (\ref{cum_split}) is dominated by the first term owing to universal oscillations of higher derivatives \cite{Berry}
\begin{equation}
c_{n}(t) \stackrel{n}{\asymp} - (n-1)! \sum_{j=1}^{2t} \lambda_j^{-n}(t).\label{cum_split}
\end{equation}
The expression (\ref{cum_split}) is in turn dominated by the leading Lee--Yang zero (\ref{leading_LY}). Assuming an absence of accidental symmetries, there is a quartet of zeros equidistant from the origin (\ref{LY_quartet}) that determines higher-order cumulants
\begin{equation}
c_{n}(t) \stackrel{n}{\asymp} -2(1 + (-1)^n)(n-1)! \frac{\cos(n \varphi(t))}{|r(t)|^{n}}. \label{LeeYang_cum}
\end{equation}
Note that $c_{2n-1}(t)=0$ already follows trivially from the symmetry (\ref{inversion_sym}). We now use the relation (\ref{LeeYang_cum}) to extract the dynamics of the leading Lee--Yang zero from the knowledge of cumulants. To this end, we define the ratios of (even\footnote{In equilibrium the ratios of subsequent cumulants are trivially either zero or divergent.}) cumulants
\begin{equation}
R_n^\pm(t) = \frac{c_{n\pm2}(t)}{c_{n}(t)}. \label{r_def}
\end{equation}
Using (\ref{LeeYang_cum}) we obtain the identities (for large $n$)
\begin{align}
\frac{R_n^+}{(n+1)n} r^4 + (n-1)(n-2)R_n^- &= 2 r^2 \cos (2 \varphi),\\
\frac{R_{n+2}^+}{(n+3)(n+2)} r^4 + (n+1)n R_{n+2}^- &= 2 r^2 \cos (2 \varphi),
\end{align}
which can be cast as a matrix system
\begin{equation}
\begin{bmatrix}
1 & - \frac{R_n^+}{(n+1)n}\\
1 & - \frac{R_{n+2}^+}{(n+3)(n+2)}
\end{bmatrix}
\begin{bmatrix}
z_1^2 + \overline z_1^2\\
|z_1|^4
\end{bmatrix} =
\begin{bmatrix}
(n-1)(n-2)R_n^-\\
(n+1)n R_{n+2}^-
\end{bmatrix} \label{matrix_system2}.
\end{equation}
The solution of (\ref{matrix_system2}) for $n \gg 1$ uniquely determines the location of the leading Lee--Yang zero $\lambda_1(t)$ in terms of higher order cumulants $c_n = c_n(t)$
\begin{equation}
r(t) \stackrel{n}{\asymp} n\sqrt[4]{\frac{c_n^2- c_{n+2}c_{n-2} }{c_{n+2}^2- c_{n+4}c_{n}}}, \qquad
\varphi(t) \stackrel{n}{\asymp} \frac{1}{2} \arccos \left(\frac{c_{n+2}c_n- c_{n+4}c_{n-2} }{2\sqrt{(c_{n+2}^2- c_{n+4}c_{n})(c_{n}^2- c_{n+2}c_{n-2})}} \right). \label{LeeYang_sol}
\end{equation}
Now consider a scenario when all cumulants at large times scale as $c_n(t) \sim t^{\gamma n+ \delta}$, $\gamma, \delta \in \mathbb{R}, \gamma>0$. From (\ref{LeeYang_sol}) it follows
\begin{equation}
r(t) \sim t^{-\gamma }, \label{LY_asymp}
\end{equation}
and the Lee--Yang zero approaches the origin as $t \to \infty$, implying a critical point (\ref{dpt_def}) with $\lambda_c = 0$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\columnwidth]{LY_zeros_0-crop.pdf}
\caption{Complex Lee-Yang zeros $\lambda_j$ (\ref{LY_def}) of the MGF of the charged hard-core gas (\ref{g_exact}) in the first quadrant $0 \leq \textrm{arg}\ \lambda_j \leq \pi/2$ at three different times $t$ for $b=0$ (+ symbols) or $b=0.5$ ($\times$ symbols) and $\rho=0.5$. The zeros for $b=0.5$ approach the origin faster, in agreement with (\ref{HPG_LY}). Circles show the leading Lee-Yang zeros $\lambda_1$ at different times, for both values of $b$.}
\label{fig_LY}
\end{figure}
\subsection{Leading Lee--Yang zero of the charged hard-core gas}
We are now in a position to apply Lee--Yang analysis to the charged hard-core gas. Taking the cumulant asymptotics (\ref{cumulant_growth}) for $n>2$ and using (\ref{LY_asymp})
we immediately obtain asymptotics of the modulus of the leading Lee--Yang zero
\begin{equation}
r^{[b]}(t) \sim t^{- 1/2}, \quad r^{[0]}(t) \sim t^{- 1/4}. \label{HPG_LY}
\end{equation}
The result (\ref{HPG_LY}) indicates a critical point at the origin, $\lambda_c=0$, in the charged hard point gas irrespectively of the bias $|b|<1$. For small times, the Lee-Yang zeros are shown in Figure~\ref{fig_LY}.
We emphasize that while the convergence radius of the CGF vanishes around the origin as $t \to \infty$ (for all $|b|<1$) this is not inconsistent with a finite radius of convergence of the SCGF (\ref{F_res}) for $b=0$ owing to the non-commutativity of limits (\ref{non_commute}).
\section{First order phase transition}
In this section we show that the model exhibits a line of first order phase transitions, by considering the bias $b$ as a control parameter.\\
Reparametrizing the pair of parameters $(\lambda, b)$ in terms of
\begin{align}
\beta(\lambda, b) &\equiv \log \sqrt{\mu_+ \mu_-} = \log \sqrt{\cosh^2 \lambda - b^2 \sinh^2 \lambda},\label{beta_param}\\
h(\lambda, b) &\equiv \log \sqrt{\mu_+/ \mu_-} = \log \sqrt{\frac{\cosh \lambda + b \sinh \lambda}{\cosh \lambda - b \sinh \lambda}}, \label{h_param}
\end{align}
while also introducing a magnetization $M = l-r$, the expression for the MGF (\ref{G_binom}) can be expressed as
\begin{equation}
G(\beta, h|t) = \sum_{M=-t}^{t} e^{\beta|M| + h M} P(M|t) , \label{G_thermo}
\end{equation}
where $P(M|t)$ is the probability distribution of $M$ at time $t$
\begin{equation}
P(M|t) = \rho^{2t}\sum_{\substack{l, r = 0,\\ l-r = M}}^{t} \binom{t}{l} \binom{t}{r} \nu^{l+r}. \label{Pm}
\end{equation}
The normalization (\ref{g_norm}), transcribed into the language of $(\beta, h)$, ensures that $P(M|t)$ is normalized to unity
\begin{equation}
\sum_{M=-t}^{t} P(M|t) = 1.
\end{equation}
The expressions (\ref{G_thermo},\ref{Pm}) can be formally identified with a thermodynamic partition sum of a Curie--Weiss type (i.e. mean-field) model of magnetism (see also \cite{Flindt_PRR}), where $t$ play the role of system size. Introducing rescaled magnetization as $m = M/t$ and sending $t \to \infty$ we can reuse the result (\ref{g_res}) to extract
\begin{equation}
G(\beta, h|t) \asymp e^{t f(x_\gamma, y_\gamma)},
\end{equation}
which gives the average magnetization and the average of it absolute values by taking logarithmic derivatives with respect to $h$ or $\beta$ respectively
\begin{align}
\langle m \rangle &\equiv t^{-1}\partial_h \log G(\beta, h|t),\\
\langle |m| \rangle &\equiv t^{-1}\partial_\beta \log G(\beta, h|t).
\end{align}
Using the explicit form of $f$ (\ref{saddle_f}) and (\ref{max1},\ref{max_func}), while noting that $\gamma = {\rm sgn}\, (\lambda b) = {\rm sgn}\, h$ , we find for $t \to \infty$\footnote{Note that the the limit $t \to \infty$ commutes with taking partial derivatives for $\beta \neq 0$, see Section~\ref{sec:LY}.}
\begin{align}
\langle m \rangle &= {\rm sgn}(h)\, \mathcal{M}, \label{avg_m}\\
\langle |m| \rangle &= \mathcal{M}.
\end{align}
where
\begin{equation}
\mathcal{M}(\beta, h) = \frac{\sinh(\beta + |h|)}{\cosh(\beta + |h|) + (\nu+\nu^{-1})/2}.
\end{equation}
Observing that $\mathcal{M}(\beta, 0) > 0$ for $\beta > 0$, while $\mathcal{M}(0, 0) =0$ it follows from (\ref{avg_m}) that the average magnetization jumps as $h$ is varied across the origin for $\beta > 0$, while it varies continuously for $\beta = 0$ (see Figure~\ref{fig_transition}). This indicates a line of first order phase transitions (in the parameter $h$) which terminates at a critical point $\beta_c = 0$.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{phase_transition-crop.pdf}
\caption{Average magnetization $\langle m \rangle$ (\ref{avg_m}) as a function of $h$ at four values of $\beta$. Circles indicate the left/right limits $\lim_{h \to 0^\pm} \langle m \rangle$. (inset) Jump in the average magnetization $\Delta \langle m \rangle \equiv \tfrac{1}{2} \left( \mathcal M|_{h=0^+} - \mathcal M |_{h=0^-} \right)$ as a function of $\beta$. Circle indicates the critical point $\beta_c = 0$. Parameters: $\nu = 1$}
\label{fig_transition}
\end{figure}
We note that the preceding discussion is exactly the behavior observed in localizing the integral (\ref{g_int}) (cf. discussion below Eq.~(\ref{max_func})), translated into a `thermodynamic' language using the parametrization (\ref{beta_param},\ref{h_param}). Moreover we observe that the appearance of the critical point at $\beta_c = 0$ is consistent with the Lee-Yang analysis in Section \ref{sec:LY}.
\end{document}
|
2,877,628,090,853 | arxiv | \section*{A. Generalization of the environmental dynamics to $N$ processes and $M$ jumps}
Let us consider a set of $N$ Langevin equations of the form
\begin{equation}
\label{eqn:langevin_equations}
\dv{x_\mu}{t} = - f_\mu(x_\mu) + \sqrt{2D_{i(t)}} \, \xi_\mu(t) \quad\quad \mu = 1 \dots, N
\end{equation}
where $\xi_\mu(t)$ are independent white noises such that $\ev{\xi_\mu(t)} = 0$ and $\ev{\xi_\mu(t_1)}\xi_\nu(t_2) = \delta_{\mu\nu} \delta(t_1 - t_2)$. As in the main text, all the variables $\vb{x}$ share the same diffusion coefficient $D_{i(t)}$, where $i(t)$ is a discrete stochastic process with $M$ states. The probability $\Pi_i(t) = \mathbb{P}[i(t) = i]$ is described by the master equation
\begin{equation}
\label{eqn:jump_process}
\partial_t \Pi_i(t) = \sum_{j = 1}^M \left[W(j \to i) \pi_j(t) - W(i \to j) \pi_i(t)\right]
\end{equation}
that is independent on all $x_\mu$.
We can write the corresponding Fokker-Planck equations as
\begin{align}
\label{eqn:fokker_planck}
\partial_t p_i(\vb{x}, t) = & \sum_{\mu = 1}^N\partial_\mu\left[f_\mu(x_\mu)p_i(\vb{x}, t)\right] + D_i\sum_{\mu= 1}^N \,\partial_\mu^2 \, p_i(\vb{x}, t) + \sum_{j = 1}^M \left[W(j \to i) p_j(\vb{x}, t) - W(i \to j) p_i(\vb{x}, t)\right]
\end{align}
where we used the shorthand notation $\partial_{x_\mu} := \partial_\mu$ and $\vb{x} = (x_1, \dots, x_N)$. These are $M$ equations for each of the discrete states of $D_i$. Similarly to the main text, the first row is associated to the continuous stochastic process described by the Langevin equations Eq. (\ref{eqn:langevin_equations} at a given diffusion coefficient, whereas the second row describes the jump process of Eq. (\ref{eqn:jump_process}) for the diffusion coefficient itself.
Since we have $N$ variables, we need to choose a suitable generalization of the mutual information. Such generalizations, however, are troublesome from an information-theoretic perspective \cite{ThomasCover2006}. Since the variables are not interacting but they only share the same environmental diffusion coefficient, we shall be interested in the factorizability of the joint probability distribution $p(\vb{x})$ with respect to its full factorization, that is
\begin{equation}
\label{eqn:multivariate_information}
I_N = \int \prod_{\mu = 1}^N dx_\mu p(x_1, \dots, x_N) \log\frac{p(x_1, \dots, x_N)}{\prod_{\mu = 1}^N p(x_\mu)} = \sum_{\mu = 1}^N H_\mu - H_{1, \dots, N}
\end{equation}
where $H_\mu$ and $H_{1, \dots, N}$ are the entropies of the corresponding probability distributions. This is nothing but the Kullback-Leibler divergence between the joint probability distribution and the product of the single-variable distributions. Thus, this quantity is always positive and for $N=2$ it gives exactly the mutual information.
\section*{B. Fast- and slow- jumps limit for $N$ processes and $M$ jumps}
We assume that the Langevin equations are associated with a timescale $\tau$ - e.g. the fastest timescale of $\vb{x}$ - whereas the jump process happens at a timescale $\tau_{\rm jumps}$ - e.g. $\tau_{\rm jumps} = \left(\sum_{i \ne j} W(i\to j)\right)^{-1}$. Hence we can write the rescaled equation
\begin{align}
\label{eqn:fokker_planck_rescaled}
\partial_t p_i(\vb{x}, t) = & \frac{1}{\tau} \sum_{\mu = 1}^N \biggl[\partial_\mu\left[\tilde{f}_\mu(x_\mu)p_i(\vb{x}, t)\right] + \tilde{D}_i \,\partial_\mu^2 \, p_i(\vb{x}, t) \biggl]+ \nonumber \\
& + \frac{1}{\tau_{\rm jumps}}\sum_{j = 1}^M \left[\tilde W(j \to i) p_j(\vb{x}, t) - \tilde W(i \to j) p_i(\vb{x}, t)\right]
\end{align}
where $\tilde f_\mu := \tau f_\mu$, $\tilde D_i := \tau D_i$ and $\tilde W(i \to j) := \tau_{\rm jumps} W(i \to j)$.
Let us now assume that the jump process is faster, i.e. $\tau_{\rm jumps}/\tau = \gamma \ll 1$. In this scenario, it makes sense to rescale the slow timescale $\tau$, namely $t \to t/\tau$ and to look for a solution of the form
\begin{equation}
\label{eqn:expansion_fast_jumps}
p_i(\vb{x}, t) = p_i^{(0)}(\vb{x}, t) + \gamma \, p_i^{(1)}(\vb{x}, t) + \mathcal{O}(\gamma^2).
\end{equation}
Thus, we end up with the Fokker-Planck equation
\begin{align}
\partial_t p_i^{(0)} = & \sum_{\mu = 1}^N \biggl[\partial_\mu\left[\tilde{f}_\mu(x_\mu)p_i^{(0)}\right] + \tilde{D}_i \,\partial_\mu^2 \, p_i^{(0)} \biggl]+ \nonumber\\
& + \sum_{j = 1}^M \left[\tilde W(j \to i) p_j^{(1)} - \tilde W(i \to j) p_i^{(1)}\right] + \nonumber\\
& + \frac{1}{\gamma}\sum_{j = 1}^M \left[\tilde W(j \to i) p_j^{(0)} - \tilde W(i \to j) p_i^{(0)}\right] + \mathcal{O}(\gamma).
\end{align}
At the leading order we simply have
\begin{equation*}
0 = \sum_{j = 1}^M \left[\tilde W(j \to i) p_j^{(0)}(\vb{x}, t) - \tilde W(i \to j) p_i^{(0)}(\vb{x}, t)\right]
\end{equation*}
which is the stationary condition of the jump process. Hence we can assume that the zero-th order solution can be factorized as $p_i^{(0)}(\vb{x}, t) = \pi_i P(\vb{x}, t)$, where
\begin{equation*}
0 = \sum_{j = 1}^M \left[\tilde W(j \to i) \pi_j^s - \tilde W(i \to j) \pi_i\right]
\end{equation*}
defines the dependence on the $i$-th index and we only need to find $P(\vb{x}, t)$.
At the order $\mathcal{O}(1)$ we can sum over $i$ to find
\begin{align*}
\partial_t P(\vb{x}, t) = & \sum_{\mu = 1}^N \biggl[\partial_\mu\left[\tilde{f}_\mu(x_\mu)P(\vb{x}, t)\right] + \left(\sum_i\pi_i\tilde{D}_i\right) \,\partial_\mu^2 \, P(\vb{x}, t) \biggl]
\end{align*}
which gives us the solution for $P(\vb{x}, t)$ as the solution for the Langevin equations in Eq. (\ref{eqn:langevin_equations}) with an effective diffusion coefficient $\sum_i\pi_i\tilde{D}_i$. This is nothing but a set of $N$ independent equations that can be solved separately. Thus we find that the solution in the limit of fast jumps is simply given by the factorization
\begin{align}
\label{eqn:solution_fast_jumps}
p(\vb{x}, t) = \sum_i p_i^{(0)}(\vb{x}, t) = \prod_{\mu = 1}^N g_\mu(x_\mu, t)
\end{align}
where $g_\mu(x_\mu, t)$ solves the one-dimensional equation
\begin{align*}
\partial_t g_\mu(x, t) = \partial_x[\tilde f_\mu(x)g_\mu(x, t)] + \left(\sum_i\pi_i\tilde{D}_i\right) \partial_x^2g_\mu(x, t).
\end{align*}
We immediately see that in this limit all the variables $x_\mu$ are independent and Eq. (\ref{eqn:multivariate_information}) is zero.
We are also interested in the opposite limit, where the Langevin equations in Eq. (\ref{eqn:langevin_equations}) relax faster to their own stationary state, that is in the limit $\tau/\tau_{\rm jumps} := \delta \ll 1$. As before, we rescale $t \to t/\tau_{\rm jumps}$ and we end up with the Fokker-Planck equation
\begin{align}
\partial_t p_i^{(0)} = & \frac{1}{\delta}\sum_{\mu = 1}^N \biggl[\partial_\mu\left[\tilde{f}_\mu(x_\mu)p_i^{(0)}\right] + \tilde{D}_i \,\partial_\mu^2 \, p_i^{(0)} \biggl]+ \nonumber\\
& + \sum_{\mu = 1}^N \biggl[\partial_\mu\left[\tilde{f}_\mu(x_\mu)p_i^{(1)}\right] + \tilde{D}_i \,\partial_\mu^2 \, p_i^{(1)} \biggl]+ \nonumber\\
& + \sum_{j = 1}^M \left[\tilde W(j \to i) p_j^{(0)} - \tilde W(i \to j) p_i^{(0)}\right] + \mathcal{O}(\delta).
\end{align}
Once more the leading order carries no temporal dependence so we assume that $p_i^{(0)}(\vb{x}, t) = \pi_i(t)P_i^s(\vb{x})$. Furthermore, we can write $P_i^s(\vb{x}) = \prod_\mu P_{\mu}^s(x_\mu, D_i)$
where $P_{\mu}^s(x, D_i)$ solves
\begin{align*}
0 = \partial_x\left[\tilde{f}_\mu(x)P_{\mu}^s(x, D_i)\right] + \tilde{D}_i \,\partial_x^2 \,P_{\mu}^s(x, D_i)
\end{align*}
which are nothing but the (independent) stationary solutions of each of the Langevin equations in Eq. (\ref{eqn:langevin_equations}) at constant diffusion coefficient.
The $\mathcal{O}(1)$ order, after an integration over $\vb{x}$, gives instead
\begin{align*}
\partial_t \pi_i(t) = \sum_{j = 1}^M \left[\tilde W(j \to i) \pi_j(t) - \tilde W(i \to j) \pi_i(t)\right]
\end{align*}
which leads to the overall solution
\begin{align}
\label{eqn:solution_slow_jumps}
p(\vb{x}, t) = \sum_{i = 1}^M \left[\pi_i(t) \prod_{\mu = 1}^N P_{\mu i}^s(x_\mu)\right]
\end{align}
where we denote $P_{\mu}^s(x_\mu, D_i)$ with $P_{\mu i}^s(x_\mu)$ for the sake of brevity. In this limit, the variables $x_\mu$ are not independent anymore and indeed their joint distribution is a mixture distribution.
\section*{C. Multivariate information for $N$ processes and $M$ jumps}
We now consider the limit of slow jumps, so that we end up with a probability distribution that is not trivially factorizable. Let us write the stationary limit of the one variable probability distributions as
\begin{equation}
\label{eqn:mixture_1D}
p_\mu(x) = \sum_{i=1}^M \pi_i P_{\mu i}^s(x)
\end{equation}
and the $N$ variables probability distribution as
\begin{equation}
\label{eqn:mixture_2D}
p_{1, \dots, N}(\vb{x}) = \sum_{i=1}^M \pi_i \prod_{\mu = 1}^N P_{\mu i}^s(x_\mu).
\end{equation}
In order to study the multivariate information in Eq. (\ref{eqn:multivariate_information}) we need to bound the entropies of these distributions, which do not admit a closed form. From \cite{Kolchinsky2017} we can write an upper and a lower bound starting from the estimator
\begin{equation}
\label{eqn:entropy_estimator}
\hat{H}_\mu = \sum_i \pi_i H(P_{\mu i}^s) - \sum_i \pi_i \log \left[\sum_{j}\pi_j e^{-d(P_{\mu i}^s || P_{\mu j}^s)}\right]
\end{equation}
where $d(P_{\mu i}^s || P_{\mu j}^s)$ is any distance function in the probability distributions space. We note that
\begin{align*}
H\left(\prod_{\mu = 1}^N P_{\mu i}^s\right) & = -\int dx_1 \dots dx_N \, \prod_{\mu = 1}^NP_{\mu i}^s(x_\mu) \log \left[\prod_{\mu = 1}^NP_{\mu i}^s(x_\mu)\right] = \sum_{\mu = 1}^N H(P_{\mu i}^s)
\end{align*}
so the first part of Eq. (\ref{eqn:entropy_estimator}) for the entropy of the joint probability distribution is exactly equal to the sum of the estimators of the entropy of the one variable distributions. Thus, in the corresponding estimator for the multivariate information we are left with
\begin{align}
\label{eqn:estimator_multivariate}
\hat{I}_{N,\rm{env}} = - \sum_i \pi_i \log \frac{\prod_{\mu = 1}^N\left(\sum_{j}\pi_j e^{-d_1(P_{\mu i}^s || P_{\mu j}^s)}\right)}{\sum_{j}\pi_j e^{-d_2(\prod_{\mu = 1}^N P_{\mu i}^s || \prod_{\mu = 1}^N P_{\mu j}^s )}}
\end{align}
where we denote as $d_1(\cdot ||\cdot)$ the distance function we choose for the one variable entropies and as $d_2(\cdot ||\cdot)$ the distance function we choose for the $N$ variables entropy.
Following \cite{Kolchinsky2017}, a lower bound for the entropy is achieved when we choose as a distance function the Chernoff-$\alpha$ divergence
\begin{equation*}
C_\alpha(p || q) = -\log \int dx \, p^\alpha(x) q^{1-\alpha}(x)
\end{equation*}
for any $\alpha \in [0,1]$, and an upper bound is instead achieved when we use a simple Kullback-Leibler divergence
\begin{equation*}
D_{KL}(p || q) = \int dx\, p(x) \log\frac{p(x)}{q(x)}.
\end{equation*}
Therefore, Eq. \ref{eqn:estimator_multivariate} is a lower bound if we choose the Chernoff-$\alpha$ divergence for the one variable entropy and the Kullback-Leibler divergence for the $N$ variables entropy, and it is an upper bound is we make the opposite choice.
Both this upper and lower bound saturate in two particular cases. The first is the one in which $C_\alpha(\cdot || \cdot)$ diverges for all $i \ne j$. In fact, the Jensen inequality implies that $C_\alpha(\cdot || \cdot) \le (1-\alpha) D_{KL} (\cdot || \cdot)$, hence if the Chernoff-$\alpha$ divergence diverges so does the Kullback-Leibler divergence. In this case, the estimator of the mutual information is exact and we find
\begin{align}
\label{eqn:residual_mutual}
I_{N,\rm{env}}^{\infty} = -\sum \pi_i \log \frac{(\pi_i)^N}{\pi_i} = (N-1) H_\text{jumps}.
\end{align}
Qualitatively, this means that the probability distribution $P_{\mu i}^s(x)$ is infinitely different from $P_{\mu j}^s(x)$, so the discrete jumps between the $D_i$ states generate an infinitely different dynamics in terms of its stationary states.
The second, albeit trivial, case is the one in which the distances between both $P_{\mu i}^s$ and $P_{\mu j}^s$ are zero. Once more, both the upper and the lower bounds given by Eq. (\ref{eqn:estimator_multivariate}) saturate and we find
\begin{align}
\label{eqn:residual_mutual_zero}
I_{N,\rm{env}}^0 = -\sum \pi_i \log 1 = 0
\end{align}
which amounts to the trivial statement that if the two mixtures of Eqs. (\ref{eqn:mixture_1D}-\ref{eqn:mixture_2D}) have the same components they are also factorizable.
These results have a nice intuitive explanation. In fact, as long as $D_i$ is fixed the processes described by Equation (\ref{eqn:langevin_equations}) are independent and thus they cannot share any information. The only moment in time in which they are effectively coupled is when a jump $D_i \to D_j$ happens, when they share the sudden change in the diffusion coefficient - from then on, as long as $D_j$ is fixed, they evolve independently once more. As these changes are instantaneous, the greatest amount of information the processes can share corresponds to the entropy of the jumps, which is achieved when the processes are infinitely distinguishable for different diffusion coefficients $D_i$. In terms of information theory, the entropy of the jumps corresponds to our ignorance of the system, that is, since the jumps are stochastic we do not know when they happen.
\section*{D. Bounds on the mutual information for two non-interacting Ornstein-Uhlenbeck processes}
We now focus on the model proposed in the main text, in the non-interacting case. The stationary solution in the limit of slow-jumps is the Gaussian mixture
\begin{align}
\label{eqn:joint_OU_process}
p_{12}(x_1, x_2) = \frac{1}{2\pi \tau}\left[\frac{\pi_-}{D_-} e^{-\frac{1}{2\tau D_-}\left(x_1^2 + x_2^2\right)} + \frac{\pi_+}{D_+} e^{-\frac{1}{2\tau D_+}\left(x_1^2 + x_2^2\right)}\right] = \pi_- \mathcal{N}(0, \Sigma_-) + \pi_+\mathcal{N}(0, \Sigma_+)
\end{align}
where $\pi_{+(-)}^s = w_{+(-)}/(w_+ + w_-)$ and $\Sigma_{-(+)} = D_{-(+)}\text{diag}\left(\tau, \tau\right)$. Similarly,
\begin{align}
\label{eqn:single_OU_process}
p_1(x_1) = \frac{1}{\sqrt{2\pi\tau}}\left[\frac{\pi_-}{\sqrt{D_-}} e^{-\frac{x_1^2}{2\tau D_-}} + \frac{\pi_+}{\sqrt{D_+}} e^{-\frac{x_1^2}{2\tau D_+}}\right] = \pi_- \mathcal{N}(0, \tau D_-) + \pi_+\mathcal{N}(0, \tau D_+).
\end{align}
We are now interested in both the Chernoff-$\alpha$ divergence and the Kullback-Leibler divergence between the components of these Gaussian mixtures. In particular, for the two one-dimensional components of Eq. (\ref{eqn:single_OU_process}) we have
\begin{equation*}
C_\alpha(\mathcal{N}(0, D_+) || \mathcal{N}(0, D_-)) = \frac{1}{2}\log\frac{(1-\alpha) + \alpha (D_-/D_+)}{(D_-/D_+)^\alpha}
\end{equation*}
which depends only on the ratio $D_-/D_+ := \varepsilon$. Since we are free to choose $\alpha$, we take $\partial_\alpha C_\alpha = 0$ so that the Chernoff divergence is minimum. We find
\begin{align*}
\alpha = \frac{1- \varepsilon - \varepsilon\log\varepsilon}{(\varepsilon-1)\log\varepsilon} \implies C(a || b) = \frac{1}{2}\left[-1 + \log \frac{(\varepsilon-1) \varepsilon^\frac{1}{\varepsilon -1}}{\log \varepsilon}\right]:= \frac{1}{2}\, z(\varepsilon).
\end{align*}
Similarly, for the components of Eq. \ref{eqn:joint_OU_process} we have
\begin{equation*}
C_\alpha(\mathcal{N}(0, \Sigma_+) || \mathcal{N}(0, \Sigma_-)) = \log\frac{(1-\alpha) + \alpha (D_+/D_-)}{(D_+/D_-)^\alpha}
\end{equation*}
and upon optimization over $\alpha$ we find the same result as before, up to a factor $1/2$, namely
\begin{equation*}
C(\mathcal{N}(0, \Sigma_+) || \mathcal{N}(0, \Sigma_-))) = -1 + \log \frac{(\varepsilon-1) \varepsilon^\frac{1}{\varepsilon -1}}{\log \varepsilon} = z(\varepsilon).
\end{equation*}
Notice that $z(\varepsilon) = z(1/\varepsilon)$ implies that the distance between the component with a diffusion coefficient $D_-$ and the one with a diffusion coefficient $D_+$ is the same as the reversed one. We also note that the function $z(\varepsilon)$ has the following properties:
\begin{gather*}
\lim_{\varepsilon \to 0} z(\varepsilon) = +\infty = \lim_{\varepsilon \to 0} z(1/\varepsilon) \\
\lim_{\varepsilon \to 1} z(\varepsilon) = 0
\end{gather*}
which means that if $D_- \ll D_+$ the Chernoff divergence between the components of the Gaussian mixtures diverge.
We also need to write down explicitly the Kullback-Leibler divergences between the mixture components, which are
\begin{align*}
D_{KL}(\mathcal{N}(0, D_+) || \mathcal{N}(0, D_-)) = \frac{1}{2}\left[\frac{1-\varepsilon}{\varepsilon}+\log\varepsilon\right] := \frac{1}{2} \, h(\varepsilon)
\end{align*}
and
\begin{align*}
D_{KL}(\mathcal{N}(0, \Sigma_+) || \mathcal{N}(0, \Sigma_-)) = \frac{1-\varepsilon}{\varepsilon}+\log\varepsilon = h(\varepsilon).
\end{align*}
These distances are not symmetric anymore, but the function $h(\varepsilon)$ is such that
\begin{gather*}
\lim_{\varepsilon \to 0} h(\varepsilon) = +\infty = \lim_{\varepsilon \to 0} h(1/\varepsilon) \\
\lim_{\varepsilon \to 1} h(\varepsilon) = 0
\end{gather*}
so the limit $D_- \ll D_+$ is, perhaps unsurprisingly, the limit in which the distances between the mixture components diverge and the mutual information is exactly equal to the jump entropy.
Overall, the bounds on the mutual information are given by
\begin{align}
\label{eqn:OU_bounds}
I_{\rm env}^{S,\rm{up}}\left(\frac{D_-}{D_+}, \frac{w_-}{w_+}\right) = -\pi_+\log\frac{\left[\pi_+ + \pi_- e^{-\frac{h(D_-/D_+)}{2}}\right]^2}{\pi_++\pi_- e^{-z(D_-/D_+)}} -\pi_-\log\frac{\left[\pi_+ e^{-\frac{h(D_+/D_-)}{2}} + \pi_-\right]^2}{\pi_+ e^{-z(D_+/D_-)} + \pi_-} \nonumber \\
I_{\rm env}^{S,\rm{low}}\left(\frac{D_-}{D_+}, \frac{w_-}{w_+}\right) = -\pi_+\log\frac{\left[\pi_+ + \pi_- e^{-\frac{z(D_-/D_+)}{2}}\right]^2}{\pi_++\pi_- e^{-h(D_-/D_+)}} -\pi_-\log\frac{\left[\pi_+ e^{-\frac{z(D_+/D_-)}{2}} + \pi_-\right]^2}{\pi_+ e^{-h(D_+/D_-)} + \pi_-}
\end{align}
and they only depend on the ratios $D_-/D_+$ and $w_+/w_-$.
\section*{E. Disentangling the environment and the internal interactions in the mutual information}
Let us now consider the corresponding generalization to $N$ variables and $M$ jumps of the interacting case studied in the main text,
\begin{align}
\dv{x_\mu}{t} = -\sum_{\nu}A_{\mu\nu} x_\nu + \sqrt{2 D_{i(t)}} \xi_\mu(t)
\end{align}
for $\mu = 1, \dots, N$ and $i = 1, \dots, M$. In the slow-jumps limit, the mixture components are multivariate Gaussian distributions with a covariance matrix $\bm{\Sigma}$ that solves the Lyapunov equation
\begin{equation}
\bm{A}\bm{\Sigma}_i + \bm\Sigma_i \bm{A}^T = 2D_i \mathbb{1}
\end{equation}
which we can rewrite as in the main text as
\begin{equation}
\label{eqn:lyapunov_equation_S}
\bm{A}\tilde{\bm\Sigma} + \tilde{\bm\Sigma} \bm{A}^T = 2\tau^{-1} \mathbb{1}
\end{equation}
where $\bm\Sigma_i = D_i \tau \tilde{\bm\Sigma}$. Thus, the covariance matrix receives separate contributions from the diffusion coefficient $D_i$ and the interactions $\bm A$.
In order to compute the bounds, we need the divergences
\begin{equation}
C_\alpha(\bm\Sigma_i || \bm \Sigma_j) = \frac{1}{2}\log\frac{\det \left[(1-\alpha)\bm\Sigma_i+ \alpha\bm\Sigma_j)\right]}{\det^{1-\alpha} \bm\Sigma_i \det^{\alpha} \bm\Sigma_j} = \frac{1}{2}\log\frac{\det \tau \tilde{\bm\Sigma}\left[(1-\alpha)D_i+ \alpha D_j\right]}{\det^{1-\alpha} \tau D_i\tilde{\bm\Sigma} \det^{\alpha} \tau D_j\tilde{\bm\Sigma}} = \frac{N}{2}\log \frac{\left[(1-\alpha)D_i+ \alpha D_j\right]}{D_i^{1-\alpha}D_j^\alpha}
\end{equation}
and
\begin{align}
D_{KL}(\bm\Sigma_i || \bm \Sigma_j) & = \frac{1}{2}\left[\log\frac{\det\bm\Sigma_j}{\det\bm\Sigma_i} + \Tr\bm\Sigma_j^{-1}\bm\Sigma_i - N\right] = \frac{1}{2}\left[\log\frac{\det\tau D_j\tilde{\bm\Sigma}}{\det\tau D_i\tilde{\bm\Sigma}} + \Tr \frac{1}{\tau D_j}\tilde{\bm\Sigma}^{-1}\tau D_i\tilde{\bm\Sigma} - N\right] = \nonumber \\
& = \frac{N}{2}\left[\log\frac{D_j}{D_i} + \frac{D_j}{D_i} - 1\right].
\end{align}
If we set $N=2$ we recover the two variables case considered in the main text, but in general these results hold for any $N$. As we can see, due to the factorization of the covariance matrix the bounds are the same as the ones of the non interacting case and they only depend on the ratios $D_i/D_j$.
Then, if we want to compute the multivariate information $I^{(N)}$ we need the entropies of the mixture components
\begin{equation}
H^{(i)}_{1, \dots, N} = \frac{1}{2}\left[N \log(2\pi e \tau D_i) + \log\det\tilde{\bm\Sigma} \right]
\end{equation}
and
\begin{equation}
H^{(i)}_\mu = \frac{1}{2}\left[\log(2\pi e \tau D_i) + \log\tilde{\bm\Sigma}_{\mu\mu} \right].
\end{equation}
Due to the interactions, it is not anymore the case in which $H^{(i)}_{1, \dots, N}$ is exactly equal to $\sum_{\mu} H^{(i)}_\mu$ and thus the bounds on the mutual information become
\begin{align}
\label{eqn:bound_interacting}
I^{S,\rm{up/low}}_N = \frac{1}{2} \log\left[\frac{\prod_\mu \tilde{\bm\Sigma}_{\mu\mu}}{\det \tilde{\bm\Sigma}}\right] + I_{N,\rm{env}}^{S,\rm{up/low}}\left(\left\{\frac{D_i}{D_j}\right\}, \left\{\pi_i\right\}\right)
\end{align}
so the contribution of the interactions is disentangled from the one of the switching environment. Hence, in the limit in which all the distances between the mixture components diverge, we are left with
\begin{equation}
I_N \to \frac{1}{2} \log\left[\frac{\prod_\mu \tilde{\bm\Sigma}_{\mu\mu}}{\det \tilde{\bm\Sigma}}\right] - (N-1) \sum_{i \in \pm} \pi_i \log\pi_i.
\end{equation}
Finally, in the case studied in the main text the interaction matrix is given by
\begin{equation}\renewcommand*{\arraystretch}{1.2}
\bm A = \begin{pmatrix}
\frac{1}{\tau} & -g_1 \\
-g_2 & \frac{1}{\tau}
\end{pmatrix}
\end{equation}
hence the solution to the Lyapunov equation is the covariance matrix
\begin{equation}\renewcommand*{\arraystretch}{1.2}
\tilde{\bm\Sigma} = \frac{1}{g_1 g_2 \tau^2-1}\begin{pmatrix}
\frac{1}{2}g_1 \tau^2(g_2-g_1) - 1 & -\frac{1}{2}\tau(g_1+g_2) \\
-\frac{1}{2}\tau(g_1+g_2) & \frac{1}{2}g_2 \tau^2(g_1-g_2) - 1
\end{pmatrix}.
\end{equation}
Therefore, the result presented in the main text is simply given by
\begin{equation}
\frac{1}{2} \log\left[\frac{\tilde{\bm\Sigma}_{11} \tilde{\bm\Sigma}_{22}}{\det \tilde{\bm\Sigma}}\right] = \frac{1}{2} \log\left[1 - \frac{4}{4 + \tau^2 (g_1 - g_2)^2}+ \frac{1}{1 - g_1 g_2 \tau^2} \right].
\end{equation}
Notice that in the fast-jumps limit, once we solve the Lyapunov equation Eq.~\eqref{eqn:lyapunov_equation_S}, the stationary probability distribution is the multivariate Gaussian distribution $\mathcal{N}(0, \ev{D}_\pi \tau \tilde{\bm \Sigma})$ that only depends on the single effective diffusion coefficient $\ev{D}_\pi$. In this limit we can compute the mutual information exactly
\begin{align}
I_\mathrm{fast} = \frac{1}{2} \log\left[1 - \frac{4}{4 + \tau^2 (g_1 - g_2)^2}+ \frac{1}{1 - g_1 g_2 \tau^2} \right] = I_{\rm int}(g_1, g_2)
\end{align}
thus in this limit the only - constant - contribution to the mutual information is the first term of Eq. (\ref{eqn:bound_interacting}).
\section*{F. Multivariate information in the fast-jumps limit with non-linear interactions}
In the presence of non linear interactions the Fokker-Planck equation of the system reads
\begin{align}
\label{eqn:fokker_planck_non_linear}
\partial_t p_i(\vb{x}, t) & = \mathcal{L}_{\rm FP}^{(i)}\,p_i(\vb x, t) + \sum_{j = 1}^M \left[W(j \to i) p_j(\vb{x}, t) - W(i \to j) p_i(\vb{x}, t)\right] \\
& = \sum_{\mu = 1}^N\partial_\mu\left[f_\mu(\vb x)p_i(\vb{x}, t)\right] + D_i\sum_{\mu= 1}^N \,\partial_\mu^2 \, p_i(\vb{x}, t) + \sum_{j = 1}^M \left[W(j \to i) p_j(\vb{x}, t) - W(i \to j) p_i(\vb{x}, t)\right]
\end{align}
where $\mathcal{L}_{\rm FP}^{(i)}$ is the Fokker-Planck operator and now $f_\mu(\vb x)$ contains the interaction terms between $x_\mu$ and all the remaining variables $x_{\nu \ne \mu}$. If we follow the same time-scale separation limits as in section B, the zero-th order stationary solution in the fast-jumps limit now solves the equation
\begin{align}
0 = \sum_{i} \mathcal{L}_{\rm FP}^{(i)}\,\left[\pi_i p(\vb x, t)\right] = \sum_{\mu = 1}^N\partial_\mu\left[f_\mu(\vb x)p(\vb{x}, t)\right] + \ev{D}_\pi\sum_{\mu= 1}^N \,\partial_\mu^2 \, p(\vb{x}, t)
\end{align}
where $\ev{D}_\pi = \sum_i \pi_i \tilde{D}_i$. Although this equation cannot be solved exactly, the solution is not factorizable, and thus the multivariate information is not zero unless $f_\mu(\vb x) = f_\mu(x_\mu)$ which corresponds to the non-interacting case.
Hence, a non-vanishing multivariate information is a distinctive signature of underlying interactions. Notably, the main difference with respect to the previous linearized case corresponds to the fact that in the linear case it is possible to show that the multivariate information depends only on the interaction matrix $\vb A$, whereas in the general non-linear case we cannot factor out the dependence on the environments through $\ev{D}_\pi$.
\end{document}
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